it fj. & I a;. ( I l: DATE DUE M NGV-W9Mr ^/^/gq interlibrary loan m \m 2 m^ PRINTED IN U S.A. .-y,^,^.«;.'.^.v-x.->«,,';^'v)!t.^^'^'i^!*?^;'< PRESENTED BY D. Van Nostrand Company, Publishers and Booksellers, NEW YORK. Cornell University Library TJ 265.Z62 y.1 Technical thermodynamics, 3 1924 003 989 559 Cornell University Library The original of this book is in the Cornell University Library. There are no known copyright restrictions in the United States on the use of the text. http://www.archive.org/details/cu31924003989559 TECHNICAL THERMODYNAMICS BY DR. GUSTAV ZEUNER FIRST ENGLISH EDITION FROM THE FIFTH COMPLETE AND EEVISED EDITION OF "GRUNDZtJGE DER MECHAKISCHEN WARMETHEORIE" Volume I FUNDAMENTAL LAWS OF THEEMODTNAMICS THEORY OF GASES AUTHORIZED TRANSLATION BY J. F. KLEIN, D.E. professor of Mechanical Engineering, Lehigh University NEW YORK D. VAN ISrOSTKAND COMPANY 23 MiTEEAT AND 27 Waeeen Steeets 1907 Copyright, 1906 BY D. VAN NOSTEAND COMPANY C«FVRIGn c 19U7, BY ». Van NosTRJk.NO Comfash;^ ROBERT DRUMMOND PRINTER, NEW YORK PREFACE. The first edition of the present book appeared in 1859 under the title "Grundzuge der mechanischen Warmetheorie " (Main Features of the Mechanical Theory of Heat). This small publication of only twelve printer's forms (signatures) had for its purpose a clear and connected presentation of the results of the scattered work of different authors which had appeared up to that time, and also of the outcome of my own investigations which dealt particularly with the behavior of vapors; another object was to direct the attention of mechanical engineers to the great importance which the new theory of heat might assume in the further development of the theory of machines. Under the same title there appeared in 1866 a revised and considerably enlarged second edition of the book; it was intended mainly for technical men, although the general theo- retical portion was given more fully than was necessary, as an introduction to the study of the here pertinent technical prob- lems. This edition was translated into French by Maurice A r n t h a 1 and Achille Cazin under, the title "Th^orie m^canique de la Chaleur," Paris, Gauthier-Villars, 1869; Hirn had already reproduced in translation the first edition of my book in a special section of his work, "Exposition analytique et exp^rimentale de la Th6orie m^canique-de la Chaleur," Paris, 1862. After the German edition was exhausted and for several years completely out of print, I resolved at the solicitation of the publishers to issue in 1877 a reprint of the second edition, because other obligations assumed by me rendered it impossible to prepare a revised, third edition; to this reprint I added, as IV PREFACE. an appendix, a paper which I published in the "Zeitschrift des Vereins deutscher Ingenieure," Vol. 11, 1866, under the title "Theorie der iiberhitzten Wasserdampfe "; a few omissions were made, but nothing else was changed in the arrangement and presentation of the whole material. When reworking the third edition of the "Grundziige der mechanischen Warme theorie " I had to completely give up the idea of simply revising the preceding editions and merely adding the necessary supplements. I then chose the title "Technical Thermodynamics" in order to bring out the fact that a wholly new treatise existed, and to emphasize strongly the new purpose which I had in view. In recent years there has appeared in the modern languages a whole series of works on Thermody- namics, some of them excellent and most of them written by physi- cists; the progress made by this comparatively young branch of Physics was certainly very considerable, but a keener glance at the results of these investigations also showed that scientific technology had reaped the main, and indeed we may say the greater, advantage from these results. This may be due in large measure to the fact that in the original development and further prosecution of Thermodynamics a prominent part was taken by engineers and the representatives of the engineering sciences — it is only necessary to mention the names ofCarnot, Clapeyron, Rankin e, Hirn, and G r a s h o f . To collect the new technical acquisitions in the Thermodynamic dominion seemed to me the most important task of this revision. There was also a special reason for distinguishing between the physical and technical treatment of Thermodynamics, and this was the need of the higher technical institutions of learning. The introduction into the curriculum of certain parts of technical mechanics and of the theory of machines demands at the technical high schools a different treatment for Thermodynamics than it can receive in the handbooks of physics and in the lectures on general physics. The arrangement and choice of material in the new book (accompanied by the use of the technical terminology and the peculiar graphical representation with which the young engineer must early become famihar) correspond in general to PREFACE. V the presentation which I have found suitable and fruitful, in my "Lectures on Thermodynamics as an Introduction to the Theory of Engines," during many years of activity as a teacher. The wealth of the worked-up material and the desire to make the book a handy one seem to require the division of the work into two volumes; the first volume covers the Fundamental Laws of Thermodynamics and the Theory of Gases, whereas the second volume treats of the theory of vapors. While the second volume in its special technical application naturally devotes itself largely to the theory of steam engines and of refrigerating machines, the first volume gives a thorough discussion of hot- and cold-air engines, and of internal-combustion motors the principal repre- sentative of which at the present day is the gas engine. In working up the fourth edition of the present book (1901) I did not expect to be again put in the position of undertaking another revision, for the book has been widely used during a long series of years and is particularly well known in technical circles. Nevertheless when the publisher after a comparatively short period again approached me with the request to prepare a new edition I could not decline his proposition; fo be sure I had to overcome many doubts and scruples, for now I hardly feel equal to the task. In the long series of years of its existence, since the introduc- tion of Thermodynamics into technical investigations, my book has pursued the aim of giving a bird's-eye view of all the work belonging to this field up to recent times; it therefore appears as a sort of review of the historical development of "Technical Thermodynamics," and to this is probably mainly due its favor- able reception in technical circles. I have clung to this point of view, it might possibly be called the standpoint of Regnault's experiments, in this new revision and therefore felt less obliged to lay stress on the recent observations — ^highly of course as I appreciate their value for current technical problems — because their theoretical discussion is amply provided for by the newer literary productions. Dr. Gustav Zeuner. Dresden, End of June 1905. VI PREFACE. By the same author have appeared : Abhandlungen aus der mathematischen Statistik. (Papers in Mathematical Statistics.) With 27 woodcuts and several tables. 8°. 1869. viii + 220 pages. Mk. 6. Das Locomotiven-Blasrohr. (The Locomotive Blast-pipe.) Ex- perimental and theoretical investigations on the production of draft by steam jets and on the suction action of liquid jets in general. With 25 woodcuts and 2 lithographed tables. 8°. 1863. viii+231 pages. Mk. 5. Die Schiebersteuerungen. (Valve Gears.) With special consider- ation of Locomotive Valve Gears. Sixth Edition. With 62 woodcuts printed in the text and 6 lithographed tables. 8°. xvi +259 pages. 1904. Paper, Mk. 8, bound, Mk. 9. Vorlesungen uber Theorie der Turbine. (Lectures on the Theory of Turbines.) With preparatory investigations in Technical Hydraulics. With 80 woodcuts printed in the text. 8' xii+372 pages. 1899. Paper, Mk. 10; bound, Mk. 12. o TRANSLATOR'S PREFACE. This treatise has maintained its leadership with the German technical public for forty years. It has molded their thought concerning engineering Thermodynamics and has given analytical expression to it. Dr. Z e u n e r was quick to see the necessity of taking this subject out of the department of Mathematical Physics, of casting it into simpler form, of bringing it within the range of engineers and technical students, and of limiting it to their interests. It will be found that his presentation combines German thoroughness with French elegance and simplicity. In the Fifth German Edition (just completed) the distinguished author has brought the whole subject well up to date. He has given an adequate and conservative presentation of the subject of variable specific heats in gases and vapors; has subjected the whole que tion of the characteristic equation of superheated steam to careful revision and tested it by Battelli's experiments; has established more practical and satisfactory standards of comparison in steam engineering, has given us the kernels of the subject of refrigerating machinery and has gone fully into problems that the rapidly growing sub ect of gas engineering now submits for solution to the technical world. The general plan of the work is to establish a few fundamental equations and deduce all results from these. The author has the engineer's ins inct to check everything by experiment, and he also has an engineer's appreciation of the limitations imposed by practical conditions. The great clearness and simplicity per- Viii TRANSLATOR'S PREFACE. vading the whole presentation is largely due to Dr. Zeuner's own active and fruitful participation in the discussion and ex- periments connected with the great questions of steam and gas engineering as they arose. This participation developed in him a sound sense of proportion which enables him to here maintain the proper relations between theory, experiment, and construction. It is no small achievement to have brought this advanced scientific subject within the reach and comprehension of that wide circle of efficient engineers who have only a moderate mathematical equipment. The author's constant reference to original sources, his frequent appeal to experiment, his many practical examples with real engineering data, and his convenient and far-reaching formulas and tables have always been highly appreciated by both the student and the practitioner of engineering. The indepen- dence of the several parts of the book, the simplicity and elegance of its mathematical presentation, its restriction to technical inter- ests, render this book particularly suitable for study in engineering schools. This English edition has had the benefit of the revision fur- nished by classroom work, for advance sheets of most of the treatise were supplied to, and studied with profit by, the Senior class in mechanical engineering at Lehigh University. J. F. K. Bethlehem, Pa., January, 1907. CONTENTS OF THE FIRST VOLUME. PAGE Introduction 1 FIRST SECTION. Fundamental Equations of Thermodynamics. § 1. Preliminary remarks 23 § 2. The inner work or energy of the body 25 § 3. The external work 28 § 4. Derivation of the first fundamental equation 32 § 5. Derivation of the second fundamental equation 33 § 6. Investigation of different pressure curves 37 § 7. Graphical determination of heat quantities 40 § 8. Relation between the functions S and P 43 § 9. Reversible cycles 46 §10. Carnot's cycle 51 § 11. Physical meaning of the function S 54 § 12. Transformation of the fundamental equations 60 § 13. Universal meaning of the C a r n o t function 63 § 14. The transformation of reversible pressure curves and of indicator diagrams 69 § 15. The n o n-reversible operations 74 § 16. Transformation of the non-reversible operation 78 § 17. Entropy or heat weight in the non-reversible operations 80 SECOND SECTION. The Theory of Gases. § 18. Preliminary remarks 89 § 19. Equation of condition for gases 93 § 20. Gas mixtures 107 ix CONTENTS OF THE FIEST VOLUME. PAGEI § 21. Specific heat of gases 116 § 22. First fundamental equation of Thermodynamics with respect to the behavior of gases 122 § 23. Second fundamental equation of Thermodynamics with respect to the behavior of gases. Determination of Carnot's function 128 § 24. The energy or inner work of gases 131 § 25. Heat equations and entropy of gases 133 § 26. The isothermal and isodynamic curves of gases 136 § 27. The adiabatic curve of gases 139 § 28. The general form of the equation of condition and the variability of the specific heat of gases 141 Experiments of Mallard and Le Chatelier 148 APPLICATIONS. PHYSICAL PART. I. Reversible Changes op State op a Gas. § 29. The polytropic curve of gases 151 § 30. Other properties of the polytropic curve 156 § 31. The construction of the polytropic curve and its transformation. 158 § 32. Pressure curve of gases, when the heat quantity is proportional to the change of pressure or to the change of volume 161 II. Non-reversible Changes op State op Gases. § 33. Expansion of gases under different conditions 163 § 34. Mixtures of different gases 171 § 35. Flow of a gas from one vessel to another for constant vessel- volumes 178 § 36. Flow of free atmospheric air into a vessel 182 § 37. Flow of atmospheric air from a vessel into the free atmosphere . 189 § 38. Flow of a gas from one vessel into another when heat is imparted and when the vessel-volumes vary 197 § 39. Applications and special problems 207 III. Flow and Epplux of Gases. § 40. Fundamental formulas for the flow of a fluid 225 § 41. On the flow of gases 231 § 42. Efflux of gases from simple orifices under constant pressure 233 § 43. Discussion of the formulas for efflux 240 CONTENTS OF THE FIRST VOLUME. xi PAGE § 44. Introduction of resistances into the efflux formulas for simple orifices. Concerning the efflux expoaent 245 § 45. Efflux under constant pressure with small differences of pressure. 254 § 46. Experiments on the efflux of air through simple orifices 257 § 47. New experiments on the efflux of air through well-rounded orifices.264 § 48. Experiments on the flow of air through long cylindrical pipes. Flow against great resistances 271 APPLICATIONS. TECHNICAL PART. I. Theory op Air Engines. § 49. Preliminary remarks 283 A. Hot-air Engines. (a) Closed Hot-air Engines. § 50. The C a r n o t cycle of a closed air engine 285 § 51. The cycle of a closed air engine between two pairs of polytropic curves 292 § 52. Cycle of a closed air engine between two adiabatics and two dif- ferent polytropic curves (L o r e n z cycle) 296 § 53. On the disposable work of heat engines 300 § 54. Cycle of an air engine between two isothermals and a pair of equal polytropic curves (R e i 1 1 i n g e r cycle) 309 § 55. Theory of the regenerator 314 § 56. Closed hot-air engine without regenerator. First arrange- ment (Rider system) 323 § 57. Closed hot-air engine without regenerator. Second arrange- ment (L e h m a n n system) '. 327 § 58. Theory of the engine arrangements just discussed 330 § 59. Applications and numerical examples 335 §60. Closed hot-air engines without regenerator. Third arrange- ment Lauberau-Schwartzkopff system 340 §61. Closed hot-air engine with regenerator. First arrange- ment (Rider system) 350 § 62. Closed hot-air engine with regenerator. Second arrange- ment 361 § 63. Hot-air engines with regenerator and continuous piston motion generated by cranks 366 § 64. Ericsson's closed hot-air engine 376 xii CONTENTS OF THE FIRST VOLUME. PAOB (6) Open Hot-air Engines. § 65. Engines with and without regenerator 382 B. Cold-air Engines. (a) Closed Cold-air Engines. § 66. Reversal of Ericsson's closed hot-air engine 384 § 67. Reversal of Rider's closed hot-air engine 393 (6) Open Cold-air Engines. § 68. Reversal of Ericsson's closed hot-air engine with omission of the heating apparatus 397 II. Theory of Internal-combustion Engines. § 69. Preliminary remarks 399 A. Otto's Qas Engine. § 70. Piston motion in the 4-cycle engine 402 § 71. Behavior of combustible gases during ignition 405 § 72. Behavior of (coal) illuminating gas during ignition 410 § 73. On the heating value of combustible gases 416 § 74. Thermodynamic equations for the process of combustion of a gas with invariable specific heats 423 § 75. The fundamental equations and the equations for process of combustion of gases with variable specific heat 428 B. Diesel's Heat Motor. § 76. Discussion of the motor and the new views concerning it 436 General Conclusions. § 77. On the "working value " of fuels 440 (a) External-combustion engines 444 (6) Internal-combustion engines 449 TECHNICAL THERMODYNAMICS. INTRODUCTION. When the second edition of this book appeared nearly forty years ago under its earliest title, "Grundziige der mechanischen Warmetheorie" (Main Features of the Mechanical Theory of Heat), Leipsic, 1866, it was prefaced by an Introduction the pur- pose of which was to present the general, physical, foundations of the Theory of Heat. This Introduction started with the Elastic-solid The- ory of Light for the purpose of showing that it is necessary to imagine the particles of the body in motion to explain the simplest heat phenomena, particularly heating by radiation. "The investigations of the properties of hght" — thus the Introduction began at that time — " have led to the generally prevailing view that the whole universe is filled with a fine and elastic substance, ether; a substance which permeates all bodies and whose particles are in vibrating, oscillating, motion. "Just as an oscillating body sets the surrounding air into oscillations which spread themselves wavelike in all directions and, reaching the ear, generate under certain circumstances the sen- sation of sound, so, according to the aforesaid undulatory theory, there proceed from an illuminating body oscillations of the ether which, entering the eye under certain conditions, produce the impression of light. " In the one case the particles of air oscillate, in the other case the particles of ether oscillate about their positions of equilibrium, and the propagation of the sound or of the light consists in start- i TECHNICAL THERMODYNAMICS. ing, from the center of the disturbance, oscillations in every direc- tion, so that constantly new particles are set into vibration by the action of the preceding ones. " The difference of tones in acoustic motion is due to the greater or less number of vibrations executed by the particles of air in a certain time, say in a second; the quicker the oscillations follow one another, the higher the tone; the velocity of propaga- tion is independent of the duration of an oscillation. "In like manner difference of color is explained by differ- ence in the number of ether vibrations in a certain time; here also the velocity, with which the waves of the differently colored rays of Hght are propagated in space, is the same for all rays. "The hypothesis that light is no substance but something analogous to sound, that it consists in the propagation of the oscillatory motion of the smallest parts of a peculiar medium (Undulatory Theory), was first suggested by Huyghens (1690), but not till the beginning of the century, till it was shown (by the work of Young and F r e s n e 1 , and by the mathematical investigations ofCauchy, Green, and others) that by means of this hypothesis there could be explained clearly and simply all the phenomena of light, diffraction, polarization, interference and double refraction, was the Undulatory Theory generally ac- cepted, and the Emission Theory set up by Newton rejected, the latter theory representing light as a substance whose particles were emitted with great velocity by illuminating bodies. "The general acceptance of the undulatory theory, for the explanation of the phenomena of light, has caused the imponder- ables to be generally banished from physics; the view spreads more and more that light and heat, electricity and magnetism, depend upon the motions of the smallest parts of one and the same substance, ether. " The acceptance of the undulatory theory for hght immediately ' led to the inference that radiant heat must be a motion of smallest particles. " It is an undoubted consequence of the beautiful experiments of M e 1 1 n i that radiant heat like light consists in the trans- verse oscillations of the ether permeating all bodies. INTRODUCTION. 3 " Heat rays are nothing but invisible rays of light (which con- sist of vibrations of the ether), whose time of oscillation is different from that of the visible rays. * " The eye can no longer detect such rays as light, just as the ear is incapable of hearing certain tones, in other words, incapable of recognizing vibrations of the air whose time of oscillation falls below a certain value. " If through a small aperture in a window-shutter of a dark room we allow rays of the sun to pass through a glass prism, there will appear, as is well known, on an opposite white screen a picture of colored stripes; they are the rainbow colors arranged in the order: red, orange, yellow, green, light blue, dark blue, and violet. The white sunlight thus shows itself to be composed of differ- ent colored rays which can be thus separated by virtue of their difference in refracting power; they cannot be decomposed any further, and when recombined will again produce white light. " Red is deflected the least and violet the most; the difference of deflection stands in a direct relation to the wave lengths of the different colored rays and to the number of vibrations which the ether particles make in a certain time. According to Fresnel, at the extreme red of the spectrum this nimiber of vibrations is 487 billions, and at the extreme violet the number is 764 billions. "The spread of the colors is different in the spectrum; it is greatest in the violet and least in the yellow; on the other hand, the strength of the light is least in the violet, increases up to the yellow, and then again diminishes toward the red. In the yellow, therefore, the light is of maximum strength. " Now a further investigation of the spectrum results in some- thing of special importance for our discussion: the area upon which the spectrum falls is warmed, but this warming has a distribution that is different from that of the strength of the light. "If we develop the spectrum by a prism of rock salt, because this most completely permits the heat rays to pass through, then the heating will be least at the violet end, will increase gradually toward the red end of the spectrum, but will not reach its maxi- 4 TECHNICAL THERMODYNAMICS. mum until it has passed into the dark space beyond the red, where it again diminishes. "The heating can be detected over two thirds of the visible spectrum and also beyond the red, that is, at the less refrangible end. The heat spectrum therefore falls partly on the optical spectrum and partly on the dark space beyond the red. " From this we must conclude that besides the visible rays, the rays of light proper, there are other rays which we do not see that make themselves known by heating. These rays, constituting the heat spectrum, are less refrangible, have greater wave length and a greater time of oscillation. The greatest heating therefore occurs with those rays in which the ether particles make less than 481 billions of vibrations per second. " For the sake of completeness we may add that investigation of the other end of the spectrum has led to no less interesting results. "Rays in which the number of ether vibrations exceed 764 billions per second also escape the eye, but they nevertheless do exist and express themselves principally by the chemical changes which they produce in certain bodies. " Photography depends on the ability of rays of light to decom- pose certain chemical combinations; a closer investigation of the spectrum shows that the red possesses this power to the least degree and the violet to the greatest degree, but that the greatest chemical action obtains with the invisible rays lying' beyond the violet, the so-called ultra-violet rays. "The presence of the ultra-violet rays manifests itself, more- over, by fluorescence. For example, certain colored plant-extracts glow when they are brought into the dark space beyond the violet, which is explained by the assumption that these bodies are capable of reducing the number of vibrations of the ultra-violet rays impinging upon them, and thus send forth visible rays. " On account of these properties of the spectrum near the vio- let end, this portion has been called the chemical or fluorescence spectrum. " From what has been said it appears that the visible spectrum is only a part of the larger one, and indeed the smaller part. " The visible series of colors has often been compared with a INTRODUCTION. 5 musical scale in which each color-tone can be determined on the basis of the known number of vibrations of the different colors; this acoustic division shows that the visible spectrum does not quite occupy an octave, but that, so far as we now know, the whole spectrum occupies four octaves. "The greater part of it falls on the heat spectrum, and the heat rays possess properties like those of the light rays; it has been shown that they, too, exhibit the phenomenon of refraction and of interference, which phenomena can only be explained by ether vibrations; the experiments on polarization and double refraction of the heat rays hkewise show that here, as with rays of light, we have to do with the trans- verse vibrations of the ether. " Consequently if we wish to embrace the ether vibration by one name, that name must be Heat. "A body which is capable of setting the surrounding ether into oscillation, and this property belongs to all bodies, sends out dif- ferent kinds of rays, and among these there is a certain kind which affects our nerves of feeling and exerts a warming effect in the ordinary sense. Another kind of these rays, for which the mmti- ber of vibrations falls between the just designated limits, can, in addition, produce in our eyes the sensation of light. Still other rays, and in general such as possess a still greater number of vibrations than the ' illuminating ' rays, announce their presence by their chemical actions and by fluorescence. "The results of the investigation of the spectrum which we have thus briefly emphasized, and according to which radiant heat doubtless consists in the propagation of oscillatory motion of the space-filling ether, lead at once to the very probable assumption that the heat contained in a body has its origin mainly in the oscillations of the smallest particles." Connected with this Theory of Light, so generally accepted forty years ago, and with the atomic view of the constitution of bodies, there were then separately and partially developed certain hypotheses concerning the internal motion of bodies which were reported in the 1866 Introduction as follows : 6 TECHNICAL THERMODYNAMICS. "That view of the constitution of bodies is the most wide-spread which considers the body as made up of invariable particles, 'atoms,' whose distances apart are relatively very great; it is the view that was first definitely expressed by Ampere, Pois- s n , and C a u c h y . " The atoms are so small that a quantity of matter imperceptible to our senses may contain an uncommonly large number of them, and there are as many kinds as there are chemically simple elements. "These atoms attract each other, and several of them, of the same or of different kinds, may combine to form a group, and such a group is called a molecule. " In this way chemically compound bodies arise. The different grouping of like atoms also explains why (according to C 1 a u - sius 1) .some bodies behave differently in a physical way, although chemically they may have the same composition. "The relatively large spaces between the molecules are filled with ether. The ether atoms, which are very small in comparison to their mutual distances, and in comparison to the atoms of the body, act so as to repel each other, and so as to be attracted by the atoms of the body. In consequence of the action of these forces the ether surrounds the molecules and atoms in atmospheric fashion. These atmospheres, whose density diminishes from within outward, constitute with their kernel an individual whole. An atom with its envelope of ether Redtenbacher called a 'Dynamide,' while, according to his view, a union of atoms (mole- cule) with a common ether atmosphere is called a compound dynamide .2 " If we imagine heat rays to fall on a body, then the oscillating particles of ether will transfer their motion in part to the particles of this body, or to ether particles contained within the body, or to both of them simultaneously, and strengthen the motion already existing within the body. " We then say that the body has been heated; it can, by contact ' C 1 a u s i u s , Uber die Natur des Ozons. Poggendorff's Annalen, Vol. 103, p. 644. 'Redtenbacher, "Dynamidensystem." " Grundziige einer mecha- nischen Physik." Mannheim, 1857. INTRODUCTION. 7 with our nerves of feeling, produce the same sensation as the heat rays themselves. " Conversely, motions within the interior of the body can be transferred to the surrounding ether; we can then say the body radiates heat. " Only in one point do the views diverge, namely, as to the question whether the so-called heat motion is caused by the motion of the atoms, i.e., of the material parts of the body, or by the motion of the ether particles collected in the body. "Rej^te nbache r (born July 25, 1809, died April 16, 1863) assumed that the heat motion consisted in the radial motion of the ether envelopes surrounding the atoms or molecules of the body; these envelopes contract and expand. When these envelopes are completely at rest the body is absolutely cold. If there is equiUbrium between the attraction and repulsion of the individual dynamides, the body has an independent volume of its own. On the other hand, if the repulsion is in excess, the collection of dy- namides must be enclosed by a solid envelope; this latter case corresponds to the gaseous condition; in the former case the body may exist in the solid or liquid state; in the fluid condition the dynamides are free to move relatively to one another without altering their relative distances and therefore without changing the total volume. " C 1 a u s i u s 1 (born Jan. 2, 1822, died Aug. 24, 1888) defended another view with far-reaching consequences. According to him the heat motion consisted in the motion of the molecules, i.e., of the material parts of the bodies, this motion differing in solid, liquid, and gaseous bodies. In the solid condition the molecules move about certain positions of equilibrium, they oscillate, and indeed under the influence of forces which they mutually exert on one another. Besides the rectilinear oscillatory motions of the molecules there can also arise rotary oscillations about the center of gravity, and there may occur motions of the constituents, i.e., of the atoms of the molecule. ' C 1 a u s i u s , Uber die Art der Bewegung, welche wir Warme nennen. Pog- gendorff's Annalen, Vol. 100, p. 353. — C 1 a u s i u s , fiber das Wesen der Warme, verglichen mit Licht und Schall. A popular address. Zurich, 1857, o TECHNICAL THERMODYNAMICS. "In the liquid condition there occurs an oscillating, rolling, and progressive motion; the kinetic energy of the motion, in com- parison with the mutual attraction of the molecules, is not great enough to completely separate them from one another; even without an external pressure they keep themselves within a cer- tain volume. " In the gaseous condition the molecules have passed wholly beyond the spheres of their mutual attraction, they move recti- linearly in accordance with the law of inertia, obey the laws of elastic impact when they collide, and they rotate; it is probable that in gases there may simultaneously occur oscillating motions of the constituents, of the atoms in the molecules. " That the principal properties of the gases can be explained by rectilinear progressive motions of the molecules was shown before ClausiusbyKronigi through simple calculations; indeed the idea underlying the latter's calculations and in part those of Clausius goes much further back. Fuller details about this matter may be found in the writings of Le Sage."^ The Introduction of 1866 thus reported on definitely formu- lated hypotheses concerning heat motion, but even then the scien- tific development was such as to raise doubts as to the reliability and value of such hypothetical details. At that time the only certain view seemed to be that heat was motion of the particles, but the kind of motion was even then doubtful and, for the pur- poses of the mechanical theory of heat, a matter of indifference. The Introduction of 1866 continues: "It would be of indis- putable importance for the further cultivation of the mechanical theory of heat if the problem in question could be decided, and if the kind of heat motion in bodies were known. "Grundzvige einer Theorie der Gase." Poggendorff's Annalen, Vol. 99, p. 315. ' Deux traitds de physique miScanique publics par Pierre Prevost. Geneva and Paris, 1818. The first part contains the work of Le Sage, in which the view expressed about the constitution of gases is quite similar to that recently adopted, in accordance with the laws of the mechanical theory of heat. Le Sage mentions a series of writers (page 126) who had already expressed similar views; this was done with particular definiteness by Daniel Bernoulli. INTRODUCTION. 9 "It is of great interest to compare the course of development] of our Mechanical Theory of Light with the development of the | Mechanical Theory of Heat or of Thermodynamics. ^ " The Theory of Light starts at once with a perfectly determined mode of motion of the ether; it is assumed that the particles of ether describe paths about their positions of equilibrium under the action of a force of attraction which is directly proportional to the distance of the particle from the position of equilibrium, and this simple hypothesis concerning the nature of Light motion led to discoveries which we count among the most splendid in science. "An essentially different path was pursued in the investiga- tions on the theory of Real after the view was given up that heat is a substance. Of the numerous works on this subject but few start with an assumption of a particular kind of heat motion, and these few investigations relate only to the behavior of gases. "In general the mathematical developments avoided making definite assumptions concerning the nature of the motion which we call heat, and this we will also do in the further course of the present treatise. " If we nevertheless here and there suggest the probable con- stitution of bodies and their condition of motion, it will only be for the purpose of rendering more easy the understanding and prosecution of the mathematical presentation. "We may therefore say that the Mechanical Theory of Heat rests upon simpler laws than the Theory of Light, for in the latter we not only assume that light is a motion of the smallest parts of a substance, but simultaneously adopt a particular hypothesis in the investigations concerning the nature of this motion. " In the newer theory of Heat the only assumption which is at first made is a very simple one, that the sensible heat of a body is a molecular motion. Heating or cooling a body consists in strengthening or weakening this motion. If the body is subjected to heat rays of a certain intensity, or if it is in contact with a body in which the motion is greater, then its own molecular mo- tion may be strengthened, it is sensibly heated; cooling consists in giving off motion to the surrounding ether or to other bodies. 10 TECHNICAL THERMODYNAMICS. , " The Kinetic Energy serves as a measure of the strength of the motion of a material particle of definite mass moving at a particular instant with a certain velocity ; this energy is the work which is necessary to bring the mass from a condition of rest to this velocity, or it is the work which is produced when the mass passes from a condition of velocity to one of rest, regardless of the manner in which the velocity changes are effected. " Now if we conceive of the body as a combination of material points (body and ether atoms) which act upon each other with certain forces, and which are engaged in any sort of oscillation and rotation, then there is stored at a certain instant in the whole system a definite amount of mechanical work, whatever the arrangement of the constituents and whatever the magnitude of the instantaneous velocity of every individual point may be. " First of all, the total kinetic energy is a quantity that can be stated definitely and is the half-sum of all the products of the masses of the various material points each multiplied by the square of the instantaneous velocity of the point in question; con- sequently a strengthening or weakening of the molecular motion in a body consists in an increase or decrease of the sum of the energies of all the separate oscillating motions. "If we (temporarily) designate this store of work in the body by the name 'vibration work,' then from all that has preceded we may at once conclude that the sensible heat in a body stands in a certain relation to this vibration work, and the thought nearest is that a sensible heating or cooling of a body consists precisely and only in the increase or decrease of the vibration work. But in sa)nng this we do not at all mean to state that a supply of heat to a body must always have as a consequence an increase of the vibration work; even without the ordinary experiences the contrary may be expected. " A system of connected material points (which act on each other by forces and are in oscillating motion) will experience changes in consequence of actions which it exerts on some other outside system, and this will be due not only to the change of vibration work, but in general also to the change in the rela- tive distances of the middle positions of the INTRODUCTION. 11 oscillating material points. Now whatever the forces may be which act or are overcome in, changes of this sort, these changes will always consume work or they will pro- duce work. The forces here mentioned are of two kinds; we have to distinguish between the forces (of attraction or repul- sion) with which the material parts of the body themselves act on one another and those external forces, acting from the outside on the system, which are due to the actions of a second system upon the first. These external forces usually make themselves felt as pressures against the total outer surface of the body or against a part of it. Now the whole work, which in such a case is consumed or produced in consequence of the change in arrangement of the smallest parts of the body or system of material points, will hereafter be briefly called disgregation work.^ " We can now say that every change of condition of a body is connected with an expenditure or production of vibration work or of disgregation work, or of both simultaneously. " If we consider more closely the changes which arise in a body that is receiving or rejecting heat, we will in general notice also changes of volume, usually accompanied by the overcoming of external forces (pressures). We conclude from this that the heat supplied to a body also, in general, effects a change in the arrangement of the constituents, and accordingly disgregation work is consumed or produced; without doubt in so doing the quality of heat stands in a certain relation to this work of dis- gregation. Since such a relation also exists with the simultane- ously occurring change of vibration work, the question becomes imminent as to what relations probably exist between the mag- nitudes mentioned. ' The word " disgregration" was first used by Clausius; he understood by this term the degree of division of a body, and when he indicated a change of arrangement of the smallest parts of a body, of the kind described above in ine text, he spoke of "changes of disgregation of the body." See "Ubdr die Anwendung des Satzes von der Aquivalenz der Verwandlungen auf die innere Arbeit." Poggendorff's Annalen, 1862, Vol. 116, p. 73. Also, Clausius, "Abhandlungen iiber die mechanische Warmetheorie. " Brunswick, 1864, Ab- handlung VI. 12 TECHNICAL THERMODYNAMICS. " Now in this respect the mechanical theory of heat starts with the following assumption : "The quantity of heat which is supplied to or withdrawn from a body is directly proportional to the sum of the simultaneously occurring changes in the vibration and disgregation work of the body. "It is this hypothesis which underhes all our future inves- tigations, and we must now more fully test its probability before we can utihze it as the basis of the mathematical de- velopment. " According to the assumption made we can directly meas- ure a certain quantity of heat by the work which it has performed, and therefore the above principle has also been simply enunciated in the words heat and work are equivalent. The absorption of heat on the part of a body appears to be synony- mous with an absorption of work. Heat absorption consists in an increase, and heat rejection in a decrease, of the vibration and disgregation work of the body. If this assumption is cor- rect, then we must immediately conclude that we can directly produce changes in a body by the performance of work (compres- sion and so on) similar to those occasioned by a supply of heat; conversely, through the performance of work by the body (by its expansion and the overcoming of an external pressure) changes take place in its interior similar to those observed when the body radiates heat or gives it off by conduction to other bodies. "We must furthermore conclude that the absorbed heat or work, which in general is expended to increase the vibration work (sensible heat) and to change the arrangement of the con- stituents, is solely expended in augmenting the vibration work (increasing the temperature) when the disgregation work is zero or infinitesimal. " We must also accept the possibility that under certain condi- tions the sensible heat may remain unchanged, and that then the whole heat supplied to a body is consumed in changing the arrangement of the constituents." INTRODUCTION, 13 Since these words were written, however, it is not only the kind of motion which constitutes heat thatis in doubt, but whether what we feel to be heat is at all due to the motions of par- ticles. The whole line of thought of the Introduction started with the Elastic-solid Theory of Light. But according to the electro- magnetic theory it is not at all necessary to transmit motions, changes of position, energy from one body to another, but only electrical and magnetic changes. And so to-day, more significantly even than forty years ago, it is true that the mechanical theory of heat can be founded on experience without entering into the question as to the nature of heat. Therefore what at the present time seems to be really con- vincing and fundamental for the Mechanical Theory is the pre- sentation, given in the Introduction of 1866 : It has long been known that a heating of bodies is always observed when there is friction between two bodies, when there is impact, and whenever work disappears during the interaction of two bodies or, as it is generally expressed, whenever losses of work take place; the accompanying heating may some- times be very considerable. Count E, u m f r d ^ was the first to appreciate more clearly this well-known phenomenon, and to draw the conclusion that heat could be generated by mechanical work; he also tried to determine the relation between the work consiuned and the heat quantity generated. He compared the work consumed during the boring of the tube of a cannon with the degree of heating of the tube, which he ascertained from the rise in temperature of the water surround- ing the tube. The experiment gave no decisive results because the losses of heat were not considered and because here evidently only a part of the work was expended in increasing the vibration work, the sensible heat, the rest being spent in disgregation work (separation of the metallic parts by drilling) which cannot be directly determined. ' Phil. Trans., 1798. "An experimental Inquiry concerning the Source of the Heat which is excited by Friction." 14 TECHNICAL THERMODYNAMICS Humphry Davy' adopted Rumford's view and sought to establish its correctness by other experiments. He showed that two pieces of ice rubbed against each other at 0° C. temperature, under the receiver of an air-pump, could be melted, and concluded that motion was the cause of the heat developed, which here showed itself by the melting of the ice. The great distinction of first expressing in definite fashion the principle of the equivalence of heat and work belongs to the Ger- man Dr. J. R. M a y e r of Heilbronn (born November 25, 1814, died March 20, 1878). His treatise "Bemerkungen iiber die Krafte der unbelebten Natur," ^ which appeared in 1842, contains upon a few pages the most interesting conclusions. Mayer expressed himself with great clearness on the ques- tion; he showed that one could heat water in a vessel by shaking (performance of work), i.e., could raise the temperature of the water; he concluded that by performing work under high pressure (compression) ice could be transformed into water, which was later on confirmed by Mousson's experiments; and he expressly says at the end of his treatise: "We must find out how high a particular weight must be raised above the surface of the earth in order that its falhng power may be equivalent to the heating of an equal weight of water from 0° to 1° C." The height in question he then gives as 365 meters. Mayer was therefore the first to definitely announce that the quantity of heat generated by the performance of work was proportional to the work consumed, and that the one could be directly measured by the other, while up to that time the idea had only been recognized and expressed that, in general, a certain relation between heat and work might exist. First of all, the proportionality mentioned had to be estab- lished by reliable experiments, before one could pass to the mathe- matical treatment of the problem with prospect of success. To measure quantities of heat the unit that must be taken is ' "Researches on Heat, Light, and Respiration," in Boddoe's West Country Contributions. Compare Joule, Phil. Trans, for 1842. * Annalen von Wohler und Liebig. May number, 1842. INTRODUCTION. 15 that quantity of heat which will raise the unit of weight of water (1 kg) from 0° to 1° C. [or 1 lb. of water from 32° to 33° F.]. Now if heat and work are really equivalent, we can measure quantities of heat in the same way as work, only we must know how large a quantity of heat (in units of heat, in calories) [British thermal units] corresponds to the unit of work (1 mkg) [1 foot-pound] or how much work corresponds to a unit of heat. This question was answered indubitably by the beautiful and manifold experiments of the EngUshman Joule (born December 24, 1818, died October 11, 1889). His experiments not only showed that heat could be generated by work, but also that the generated quantity of heat is always directly proportional to the work expended. The different experi- ments invariably led to almost the same relation, and indeed it appeared that the work of 424 mkg generated a quantity of heat exactly equivalent to the unit of heat, i.e., to a quantity of heat which could warm 1 kg of water from 0° to 1" C. [1 lb. of water from 32° to 33° F.]. The work value 424 mkg [772.83] is called "work equiva- lent of the unit of heat," or briefly "the me- chanical equivalent of heat"; conversely, the quantity of heat which corresponds to the unit of work, 1 mkg [1 ft.-lb.], is called the "Thermal equivalent of the unit of work." The value for the mechanical equivalent of heat found by Joule was confirmed later by other experiments. To Joule therefore undoubtedly belongs the merit of having first reUably determined this important quantity; it is just as certain, however, that Mayer was the first to direct attention to the existence of this important constant. If Mayer, as the quotation from his paper shows, gave too small a value to this constant (he determined the value from observation on the heat released during the compression of a gas), then it was due solely to the fact that he had to introduce into his computations some quantities which were not at that time determined with sufficient accuracy. The experiments for the determination of the value of the 16 TECHNICAL THERMODYNAMICS. mechanical equivalent of heat must be arranged and conducted with great care; it is a question of determining accurately the work which is completely expended in the generation of sensible heat and then to determine this quantity of heat itself. It is only possible to do both if we choose such bodies, for the absorption of the generated heat, that the experiment can leave no permanent change in thearrange- ment of their constituents. As was just mentioned, all the work must be utilized to increase the vibration work, i.e., the sensible heat; no part, or only an infinitesimal part, may here withdraw itself from direct observa- tion as disgregation work. Liquids , like water and mercury, are bodies which fulfill these conditions at low temperatures and small differences of tem- perature, and these were employed by J o u 1 e . To generate heat he, at the start, made use of the work which disappears with friction. In one series of experiments ^ a vessel filled with mercury con- tained two cast-iron disks, one of them stationary and the other pressed against the first by a lever and set into rotation by faUing weights. The magnitudes of the weights and the heights through which they fell gave, of course, after suitable corrections, the work which was expended in overcoming the friction. On the other hand the rise in temperature of the mercury and considera- tion of the heat lost by radiation enabled Joule to determine the quantity of heat generated by the friction. The experiments were conducted under different conditions, but all gave the same relation between the work expended and the heat generated. The mean valuQ of the mechanical equiva- lent of heat was thus found to be 425.18 mkg [775.01 ft-lb.]. ~ In another series of experiments friction was generated be- tween solid and liquid bodies. In a vessel there was placed water or mercury and a stirring apparatus which was set in rotation by falHng weights. In order to transfer the work more rapidly to the liquid and to convert it 'Joule, Phil. Transactions for the year 1850, p. 1. INTRODUCTION. 17 into heat, perforated partitions were placed in the vessel, the wings of the stirring-gear fitting these apertures quite closely. These experiments led to the same result as the preceding ones. Joule found 423.92 mkg [772.71] for brass in water, and 424.68 mkg [774.10] for iron in mercury, and concluded from all his experiments that the probable value was 423.55 mkg [772.04 ft-lb.], in place of which we will hereafter use 424 [772.86]. In a subsequent experiment Joule compressed atmospheric air in a pump and forced it into a spherical vessel till the pressure rose to 22 atmospheres, w hile the vessel was immersed in a water calorimeter. From the rise in temperature of the water in the calorimeter the heat generated was determined; the work of compression transformed into heat was the work consumed in running the force-pump. These experiments gave 437.77 mkg [797.96 ft-lb.] for the mechanical equivalent of heat; they are not as reliable as those already mentioned because the losses of work in the pump are difficult to ascertain, particularly the friction of the piston. After Joule, an ingenious experiment was conducted by H i r n 1 ; this experiment is of particular interest because H i r n transformed into heat the work which disappears during the impact of solid bodies, and because then some of the work ex- pended was changed into disgregation work which H i r n cleverly knew how to determine. Two heavy prismatic blocks, one of wood and the other of iron, were suspended pendulum fashion, so that their axes coincided when their ends touched. On the end of the wooden block, that toward the iron one, there was fastened an iron plate, against which a small lead cylinder could be placed. The iron block was now lifted like a pendulum to a certain height and allowed to fall freely, so that in its lowest position it struck the lead cylinder and knocked the backing of the latter, the wooden block, up to a certain height. In consequence of the impact the lead cylinder was com- pressed and heated. The heat generated was determined by filling, immediately after the impact, the hollow part of the lead ' H i rn , Th^orie m^canique de la chaleur, p. 58; 2d ed., Paris, IBS."). 18 TECHNICAL THERMODYNAMICS. cylinder with a measured amount of water of known temperature, noting its rise in temperature and applying suitable corrections. The energy contained in the iron block at the instant of impact was calculated directly from its weight and the height through which it fell. But this energy was by no means solely expended in the generation of heat; to determine the amount thus expended it is first necessary to ascertain how much was lost during impact, or rather how much energy remained in the blocks. For this purpose H i r n observed how high the iron block rebounded and how high the wooden block rose. From the weights and these hfts of the two blocks the work remaining in them was found, and then the work done on the lead cylinder and there trans- formed into heat could easily be ascertained. In this way H i r n found the mechanical equivalent of heat to be 425 mkg [774.68 ft-lb.]. Therefore by all these experiments it is shown that through mechanical work the same phenomena can be produced in a body as by the supply of heat; hence the only question still to be raised is, whether a body giving off work by overcoming an external pressure during expansion also experiences those changes which take place with the loss of heat of the body by radiation or con- duction, and whether the work performed by the body stands in the same relation to the diminution of the heat contained in the body. Although this question may be immediately answered in the affirmative from all that has preceded, nevertheless we call atten- tion to the fact that, in this particular also, observations exist. First of all, we may point • to the long-known fact that the temperature of a gas diminishes when it expands, doing work by the overcoming of pressure; heat therefore disappears, and the disappeared quantity of heat will bear to the work produced the definite ratio already given. Direct observations for this case do not exist, but the proof in question has been furnished for steam by H i r n .i H i r n observed the quantity of water which was converted in ' Hirn , "Recherches sur I'^qiiivalent m^canique de la chaleur, prfeent^ee k la soci^tS de physique de Berlin." Paris, 1868. INTRODUCTION. 19 a certain time into steam in the boiler of a large steam-engine and was then supplied to the steam-cylinder. As the pressure and temperature of this steam were both«observed, the quantity of heat which it contained was known from Regnault's ex- periments. Steam, after leaving the cylinder, was condensed, and from the quantity and temperature of the injected water and of the condensed steam the quantity of heat was computed which the steam still possessed when it entered the condenser. According to the mechanical theory of heat, and therefore according to the views developed above, the heat contained in the steam at its exit from the steam-cylinder must be "less " than the quantity of heat it contained at its entrance, and indeed the difference must correspond exactly to the work per- formed by the steam, which work was determined by experiments with the brake. Now H i r n not only observed such a difference, but C 1 a u s i u s ^ has also shown that the quantity of heat which disappeared in H i r n ' s steam-engine bears to the work performed the ratio demanded by theory. In the different experiments there was performed, for every unit of heat, 399 to 427 mkg of work, and the mean of all the values obtained amounted to 413 mkg [752.76]; there results, therefore, a value approximately like that given by Joule. The discrepancy is easily explained by the difficulties which must be overcome in experimenting on so grand a scale (the steam- engines developed over 100 horse-power) ; we must rather wonder at the cleverness with which the skillful experimenter conducted the experiments; the mechanical sciences are indebted to him also for the solution of other highly important problems. If, finally, we collect the results of the experimental investiga- tions adduced, we may well regard as proved the first law of Ther- modynamics, "that heat and work are equivalent," i.e., we are justified, in the following investigations, in starting with the assumption: "Work can be obtained from heat and, conversely, heat can be generated by work ; furthermore, that in so doing the consumed or generated unit of heat corresponds to a work ' Hirn . ibid., p. 134. (Report on the work in question to tlie physical society of Berlin by Prof. Dr. C 1 a u s i u s .) 20 TECHNICAL THERMODYNAMICS. of 424 mkg [772.83 ft-lb.], and, conversely, that a unit of work corresponds to tkt I y-,t> oo I of a unit of heat, understanding by a unit of eat the quantity of he hich is necessary to raise one kilogram [pound] of water from 0° to 1° C. [32° to 33° F.]." The more accurate determination of the mechanical equivalent of heat has repeatedly occupied experimenters during the last decades, and the reader is referred to the text-books on Physics for the methods of observation employed ; but for technical inves- tigations it is probably best for the present to stick to the mean value (424 mkg) [772.83 ft-lb.] given by Joule; following the practice of Clausius, this has been done throughout the pres- ent book. FIRST SECTION Fundamental Equations of Thermodynamics § 1. PRELIMINARY REMARKS. The changes of state of a body which are connected with an\ absorption of heat, and which we can certainly regard as a con- sequence of such an absorption, will now be subjected to an inves- tigation under the supposition that a condition of equilibrium exists, and that therefore neither the body as a whole nor its parts are undergoing visible changes of position connected with changes of velocity. ■-' Moreover, it is assumed that while there are changes in the state of aggregation, therejoja o.t simultane ously occur_ch e m i - cal changes; consequently, according to the prevailing the- ory", there doTior occur changes in the grouping of the atoms within their molecules". Under these limiting assumptions, to which we will almost immediately add another, the st ate of t he body is completely determined by its weight, its volume, and its pressure. Now, whether we are thinking of a body of finite extent or of l;he small- est part, we can refer, for the sake of simplicity, these determining magnitudes to the unit of weight of a body; for instance, only the volume referred to the unit of weight, or specific vol- ume, will here be considered. As specific pressure we designate the pressure in pounds acting upon the unit of area (kg per sq. meter) [lb. per sq. ft.], but here we introduce the additional limitation that the pressure is the same for all planes passing through the point in question; this is an assumption which at least always holds for liquid and gaseous bodies; furthermore, we assume that this pressure is the same at all parts of a body, and therefore is the 23 24 TECHNICAL THERMODYNAMICS. same at all parts of its surface; we now know fully the kind of body which will be the subject of the following investigation. A body of the described kind we will designate (in a wider sense than is customary) as homogeneous, or (following G r a s - h f ) as a body of uniform thermal condition. If we do not want to confine our attention to fluid bodies enclosed in a vessel, then insight into the following discussions may be rendered more easy by supposing the body investigated to be surrounded by an elastic envelope which, under the assumed equilibrium, exerts everywhere an equal pressure upon the surface elements in directions normal to these elements. Now if we supply from without the infinitesimal quantity of heat dO meas- ured in heat-units, the following changes will occur in the state of the body. First: There will occur an increase in the molecular motion; the smallest parts of the body oscillating about their positions of equilibrium experience changes of velocity and changes in their distances from the positions of equilibrium, in consequence of which the work stored in the sum total of all the molecular motions experiences a change. Let dW designate the incre- ment of energy of this molecular motion expressed in units of work (meter-kilograms) [foot-pounds]. Secondly : There will occur a change in the mid-positions of the smallest parts of the body, whose distances apart may be regarded as very great in comparison with the amplitude of the oscillations. But since these parts act upon one another with certain forces, a change of position will be connected with an expenditure of work, a change of energy of position, i.e., of potential energy. Let this quantity of work in the sense of expenditure be designated by dJ. Thirdly: There will occur a change in the total volume of the body. If we here consider expansion as taking place, then external pressure acting from the outside against the surface of the body will be overcome, the infinitesimal expansion corresponding to a quantity of work designated by dL. Now these three quantities of work just defined stand, accord- ing to the principles given in the Introduction, in a simple rela- THE INNER WORK OR ENERGY OF THE BODY. 25 tion to the quantity of heat dQ which is absorbed during the as- sumed change of state. If we designate by A (measured in heat- units) the quantity of heat which disappears or is generated when a unit of work (meter-kilogram) [foot-pound] is gained or lost, then in the case before us the work performed, dW +dJ +dL, cor- responds to a heat consumption of the amount A{dW +dJ +dL) and we get as the starting-point for all further investigation the equation dQ=A{dW+dJ+dL) (1) In this equation A means a constant value, according to the Introduction, and is to be taken as 1 : 424 [1 :772.83]; that in special cases some of the terms of the preceding equation may become zero or negative is self-evident. The positive and negative signs of dQ will respectively signify heat supply or heat withdrawal, while a positive sign for dL will indicate an increment of volume (expansion), and a minus sign will indicate a decrement of volume (compression). We must not overlook the fact that the foregoing equation is true only under the several limitations specified above. To be sure, the following investigation will show that in certain cases we can drop some of the specially made assumptions and extend and transform equation (1) under the assumptions that chemical changes simultaneously occur, but for the present we will disregard such cases. § 2. THE INNER WORK OR ENERGY OF THE BODY. As already mentioned, the changes of state of a body are com- pletely determined by the changes of volume and of pressure as soon as we assume the mass of the body to be of unit weight. If we designate "specific volume" by v and "specific pressure," in the sense indicated, by p, then we ought to try to transform equation (1) by expressing the individual terms of the right member by the magnitudes p and v. Such a transformation cannot be carried out because we have not succeeded (without hypothetical assumptions) in establishing a relation between the first and sec- 26 TECHNICAL THERMODYNAMICS. ond terms of the right-hand side of equation (1) and the two vari- ables p and V. Therefore the second term is combined witli the third or it is combined with the first term; both kinds of trans- formation were first given by CI a u s i u s . ^ If we make the sub- stitution dJ+dL=dH, then we get from equation (1): dQ=A{dW+dH) (2) We can therefore say that the heat dQ imparted to the bod}' is expended in two ways: the one part, AdW- is expended in in- creasing the energy of the molecular motion, in increasing the work which resides in the sum total of the smallest parts in consequence of their oscillations about positions of equilibrium. We assume that this part of the supplied heat expresses itself as an increase of the sensible heat (shown by thermometers). On the other hand the second term, AdH, of equation (2) corresponds to the work dH which is expended during the change of position of the smallest parts in overcoming not only the forces with which the parts act upon one another, but at the same time in changing the total volume by overcoming the external pressure acting on the surface of the body; according to Clausius this may be expressed by saying that the work dH or its part AdH of the supplied heat is expended in changing the disgregation. The second of the two mentioned transformations of equa- tion (1) is much the more fruitful one. If we place dW +dJ = dU, then we get dQ=AidU+dL), (3) and it is this expression which will serve as a starting-point for all the following investigations. The heat imparted to the body can be divided into two parts; the part AdU embraces the incre- ment dU of the kinetic and of the total potential energy, while the second part corresponds to the heat value AdL of the work which is expended in expanding the body against the external pressure acting against the surface, and is to be regarded as the, totheeye, visible effect of the absorption of heat. 1 Clausius, Abhandlungen, etc., 1864. See also his treatise "Die me- chanische Warmetheorie." Brunswick, 1876. THE INNER WORK OR ENERGY OF THE BODY. 27 The first-mentioned part dU includes, according to the fore- going, the total work which has been done within the body; therefore in what follows the value U will be called the inner work, the intrinsic energy or simply the energy of the body.^ Under the special assumptions made in establishing equation (1) the inner work U for a given body is completely determined by the pressure p and the volume v (of the unit of weight) and must therefore be capable of representation by a function of p and v. Accordingly if we put U=F{p,v), (4) differentiation gives aU = T~dp 4 -:—-dv, and if we respectively designate by X and Z the two partial differ- ential coefficients, that is, put Z=|^ and Z=f^, op dv' then we can write in simpler fashion dU=Xdp+Zdv; (5) but here there exists between the functions X and Z, according to the Principles of the Differential Calculus, the relation ^v dp' ^ ' a relation of which use will be made in the following investigation The finding of the form of the function given in equation (4) constitutes the main aim of all the following investigations. 'Kirchhoff calls the quantity U with a negative sign "acting function "; Clausius understands by inner work only that part of equation (1) which is designated by dj in this text. 28 TECHNICAL THERMODYNAMICS. § 3. THE EXTERNAL WORK. The second term dL of equation (3) gives the work which is expended during the infinitesimal expansion of the body in over- coming the external pressure; we call this value dL, but if we have to do with a finite expansion we call it L (following C 1 a u s i u s ), the "external work"; but here we at first hold fast to the assumption that the pressure p remains the same at all parts of the body during change of volume, and moreover is equal to the pres- sure exerted from the outside upon the surface of the body (say, for example, by the surrounding envelope). If we designate the pressure in the interior, and consequently the pressure exerted by the body from within against the external envelope, as "body tension," and, conversely, the pressure by the envelope against the body's surface as the external pressure, then we assume that during the change of state of the body there is constantly equilibrium between the body tension and the external pres- sure, and therefore the value p holds for the one pressure as well as for the other; if we suppose the body to experience an incre- ment of volume during heat absorption, that is, an extension or expansion, then the supposition made is valid only under the assumption that extension takes place slowly and uni- formly, and consequently that the surface elements advance outward slowly and uniformly in the direction of their normals; during decrement of volume or compression of the body the motion of the surface elements takes place in the reverse direc- tion, toward the inside. Under the circumstances mentioned the quantity of work dL can be easily expressed in terms of p, v. If we designate the surface elements by /i, /2, /s, etc., and the infinitesimal displacements of the body by si, S2, S3, etc., then the pressure from without against the individual elements of the area is /ip, /2P, /sP, etc., and since these pressures during the infinitesi- mal displacements can be regarded as constant, we will have for the separate quantities of work /ipsi, /2PS2, /3PS3, etc., and their sum represents the whole work dL expended in overcoming the THE EXTERNAL WORK. 29 external pressure; consequently we have the relation dL = p[/iSi+/2S2+/3S3-P- • •]• But now the expression in the bracket is simply the increment of the total volume v of the unit of weight of the body, and there- fore we have the relation dL = pdv, (7) and by substituting in equation (3) we get dQ = A(dU+pdv) (8) If we consider a finite change of state and designate the initial pressure and specific volume by pi and vi, then, according to equation (4), the value of the inner work at the beginning is Ui=F(pi, Vi), and at the end U2 = F{p2, V2), and we get from equation (8), with the help of (7), the whole heat supphed, Q = AiU2-Ui+L), (9) or Q=A[Fip2,V2)-F{pi,vi)+rpdv]. . . . (10) The first two terms of the right-hand side of this equation are completely determined by the initial and final state of the body, provided the form of the function U=F{p, v) is known; it is not so with the third term, which represents that part of the heat expended in external work. In carrying out the integration indi- cated in equation (10) it is still necessary to know the manner in which, during the whole course of the heat supply, the body ten- sion and the external pressure p (which we must repeatedly emphasize are to be here regarded as identical) vary with the volume, and in turn this variation depends on the manner in which the heat supply takes place in all parts of the process considered. If, for a particular case, the law is known, according to which the pressure p changes with the volume, then thejchanges^f_state can be very_simgly^hown to the eje in a graphical_wax; namely, if we lay off as abscissa The value of specific volume j; at any 30 TECHNICAL THERMODYNAMICS. Fig. 1. instant, and as ordinate the correspondin^pressure p (Fig^JJ, then the state of the body at the instant in question is completely determined by the location of the point U, and the curve U1UU2, which we will hereafter designate as pressure curve, shows also the way in which the pas- sage from the initial condition Ui into the final condition U2 takes place; the direction of this change can, moreover, be shown by an arrow a drawn in the direction of the curve U1U2', if this is connected with a supply of heat this will hereafter be shown by a second arrow, h, directed toward the curve JJi U2. If the figure is to indicate that heat withdrawal occurs, then we simply reverse the direction of the arrow in all the corresponding presentations; likewise the reversal of the arrow a will indicate the passage from the state U2 to the state Ui, that is, show a compression of the body. At the beginning and end of the process we always assume the condition of rest to obtain; now, if we are to start from the point Ui, it is only necessary to assume for the introduction of slow and uniform expansion that the external pressure diminishes a little for an instant. On the other hand, with the path reversed, during compression, we take U2 as a starting-point, and assume that its external pressure momen- tarily increases a little; during the whole path of the expan- sion U1U2 or of the compression U2U1 there will then be equi- librium between the body tension and the external pressure, and we can carry out the process either in going from Ui to U2 or in the reverse direction, U2 to Ui. During the reversal the arrows of the figure change their direction, and in equations (9) and (10) all terms change their signs; therefore if heat supply occurred during the forward motion, there will be heat withdrawal in all parts of the return motion, and if during the forward motion the external work L is performed, then during the return motion this work will have to be expended. Such a process is called a re- versible one, and this term expresses most briefly that during THE EXTERNAL WORK. 31 change of state of a body there is constantly equilibrium between the external pressure and the body tension. The external work dv of equations (9) and (10) is here represented simply by the area in Fig. 1 bounded by the pressure curve U1U2 and the two end ordinates. This quantity of work depends only upon the course of the pressure curve; but as an infinite number of pressure curves can be drawn through two given points Ui and U2, there will exist for each of these curves an external work L of different value, and consequently there will Ukewise vary the quantity of heat Q which, according to equations (9) and (10), is to be imparted to the body for the assumed passage. Only that part of the heat supply which is expended on inner work is completely determined solely by the initial and the final states. This important proposition, which was first definitely enun- ciated by C 1 a u s i u s, furnishes the important starting-point of the mechanical theory of heat; up to his time it was usually assumed in Physics, generally without further explanation, that the quan- tity of heat which was to be supplied to a body for certain changes of state was completely determined by its initial and its final states (pi, Vi) and (p2, V2), while we now know that there should also be given the path along which the passage occurs. Expressed mathematically equation (8), dQ = A{dU+pdv), is an incomplete differential; the integration can only be effected when we also know in what manner the heat supply varies and how the state of the body varies with the pressure p and the volume v. 32 TECHNICAL THERMODYNAMICS. § 4. DERIVATION OF THE FIRST FUNDAMENTAL EQUATION. If in the just-quoted equation we substitute for the change of inner work that given by equation (5), in which X and Z are those functions of p and v which represent the partial differential coeffi- cients of the inner work U with respect to p and v, that is, substitute dU = Xdp+Zdv, (11) we get dQ = A[Xdp + (Z+p)dv]. Because all the following deductions will be simplified thereby we here put Z+p^Y; (12) then we can write dQ = A{Xdp + Ydv) (13) If we differentiate equation (12) on both sides with respect to p, we have dZ ^ dY dp op and then utilizing equation (6), p. 27, dp dv ^ ' This equation may be designated as the first fundamental equation. It was first given by C 1 a u s i u s, though in different form, and is'simply an analytical expression for the fact that equa- tion (13) is an "incomplete differential," otherwise the right mem- ber of the first fundamental equation (I) would be equal to zero instead of being equal to unity. Since equation (13) followed from equation (8), and the latter was already recognized in the pre- ceding investigation as an incomplete differential, we see that what is given in equation (I) simply expresses in mathematical form what was before elucidated in words. DERIVATION OF THE SECOND FUNDAMENTAL EQUATION. 33 § 5. DERIVATION OF THE SECOND FUNDAMENTAL EQUATION. • The continuation of the general investigation demands the introduction of two more functions which, it will be seen later, assume a prominent significance in the theory of heat. Let S designate one of these functions, and let its relation to p and v be represented in general form by S=f(p,v) (14) Now if we divide both sides of the just given equation (13) by AS, we get dQ X^ Y, ~^=-^p + -dv (15) We can now so choose the form of function S that the right mem- ber of this equation will become a complete differential; that is to say, this expression will seem to have proceeded from the differ- entiation of another function : P = {p,v), (16) so that we have the relation §""'' (") or dQ = ASdP (18) But these assumptions completely fix mathematically the sig- nificance of the two functions S and P; the function S appears here as the integrating factor ^ of the right-hand side of equation ' Remark. — It ought to be emphasized that there are an infinite number of functions which, like S, possess the property of rendering expression (15) a complete differential. If we imagine both sides of equation (17) to be divided by f(P), any arbitrary function of P, we get dQ ^ _dP_ ASiiP) f{P) • Now since here the right member can be integrated, it follows that the left member can likewise be integrated, and therefore we can regard generally the 34 TECHNICAL THERMODYNAMICS. (13), and its connection with the functions A' and Y through equation (15), according to the Principles of the Calculus, is given by the formula dv\S/ dp\S/- If we carry out the indicated differentiation, we get dX ^dS_ dY dS ov ov op op' or also \3p dv' dv dv '3p dv' dp But since, according to the first fundamental equation (I), the expression in the parenthesis is equal to unity, we have ^'4'-^i m expression Sf(P) as the integrating factor. Here f(P), the arbitrary function of P, can be put equal to unity, as is done in the text. If in future applications another choice of this function presents advantages, then we can at any time make use of the foregoing proposition. Thus we can put /(P) equal to any constant fc, so that we can ascribe to the value kS all the properties which the future apphcations in the text may find for- the function S itself. It is just this last generalization of which profit- able use will be made hereafter. The step taken in the text, introduction of function S as integrating factor, was first taken in the second edition of the present book, 1866. R. R ii h 1 - mann notes in his article "Geschichte der mechanischen Warmetheorie," at the end of the second volume of his "Handbuch der mechanischen Warme- theorie," Brunswick, 1885, that before him the same course had been pursued by R e e c h (Journal de Math6matiques pures et appliqu6es von Liouville, 2d Series, Vol. 1, 1856, p. 58) ; but this remark is entirely incorrect. In the first edition of the present book (1860), I, of course, giving credit to the source in discussing the second fundamental law, pursued R e e c h ' s course, and that is also incidentally mentioned by C 1 a u s i u s ; the latter's remark was doubt- less sufficient reason for R.Riihlmann not to examine more closely R e e c h ' s work. R e e c h from the very start proceeds from the assumption that S is a function of the temperature, and at that time furnished nothing which was not already known through C 1 a u s i u s. It was not until the second edition that I pursued the above-mentioned course; this course at first avoids every assumption concerning the physical significance of said function, and herein lies the difference upon which, in my opinion, great weight should be laid. DERIVATION OF THE SECOND FUNDAMENTAL EQUATION. 35 and this may be put down as the second fundamental equation of Thermodynamics. It was first developed by C 1 a p e y r o n/ and later in another form by Clausius, alid both give from the start the particular physical significance to the function S to which the following investigation will likewise lead, but which cannot be obtained without setting up a new principle. It there- fore seems to me more suitable to base the general discussions for the present exclusively on the first fundamental law (the equiva- lence of heat and work) . Now if we find from equation (II) first Y and then X, and substitute the values in equation (13), at the same time making use of the relation dS=^dp + ^dv, (19) 'E. Clapeyron, Engineer of Mines (born Feb. 21, 1779, died Jan. 28, 1864). "M&noire sur la puissance motrice de la chaleurs," Journal de I'^cole polytech- nique. Twenty-third Number, Vol. XIV, Paris, 1834. It appeared as a German translation first in 1843 in Poggendorff's Annalen," Vol. 59, pp. 446 and 566. This masterly discussion byClapeyron, which is distinguished by great clear- ness and elegance in the mathematical presentation, is moreover the first mathematical work on Thermodynamics, and is based on the formerly very rare work of Sadi Carnot (born June 1, 1796; died Aug. 24, 1832), "Reflexions sur la puissance motrice du feu," Paris, 1824. Both works were for a long time unnoticed, or at least unused, and their high value was only recognized later through the discussions of Clausius, of which the first appeared in 1850. The work of Carnot was reprinted in the "Annales scientifiques de I'Ecole normale sup6rieure," 1872 (II. series, Vol. I) and more recently (1878) was pub- lished separately in Paris by Gauthier-Villars under the original title. This issue contains interesting biographical notices, extracts from the papers which Carnot left behind him, and was in 1878 presented to the Paris Academy of Sciences by his younger brother, Senator H. Carnot. The most remarkable statement in these papers is the following one, which was reproduced in fac- simile in the aforesaid issue: "D'aprds quelques id^es que je me suis form&s sur la thferie de la chaleur, la production d'une unite de puissance motrice n^cessite la destruction de 2.70 unites de chaleur." Here Carnot understands by unit of work that which is necessary to raise 1 cbm. of water to the height of 1 meter. His unit is there- fore 1000 mkg; the unit of heat would therefore correspond to 1000 „„. , 2;75=370mkg, and this is almost exactly the same value as was found by J. R. Mayer (1842) for the mechanical equivalent of heat, of course with the statement of the method by which it was obtained (Mayer gives 365 mkg) . 36 TECHNICAL THERMODYNAMICS. then we will get the second and third of the following three equa- tions : dQ^A[Xdp + Ydv], dQ = -^[XdS+Sdvl Vp [ (Ill) dQ = ^[YdS-Sdp]. dv For the purpose of ready inspection we will here reproduce equa- tions (8) and (18), dQ=A[dU+pdv] and dQ-^ASdP. These five equations are identical and differ from one another in having a different independent variable in each equation. The utilization of these formulas is only possible if the form of the functions X, Y, S, and P is known. But if we review the course of the present work, we recognize that it is only necessary to deter- mine o n e of the designated functions in order to determine the remainder from the given relations. For example, if the form of the function S=fip, v) is in any way known, then functions X and Y can be determined from the two fundamental equations (I) and (II). The substitution of Y in equation (12) then gives Z, and the integration of equation (11) thereupon brings out function U. Finally, the last function, P results when we integrate equation (15) and make use of equation (18). The line of thought here indicated is the one which we will now pursue, but the determination of the function S will demand a series of preliminary investigations whose results are separately of consequence in future applications, namely, in the solution of certain physical and technical problems. INVESTIGATION OF DIFFERENT PRESSURE CURVES. 37 § 6. INVESTIGATION OF DIFFERENT PRESSURE CURVES. If a body while absorbing heat passes from the initial con- dition a (Fig. 2), where its initial pressure is pi and initial volume Vi, to the final condition b (p2, ^2) along the pressure curve ab which represents a reversible path, then the equation of the pressure curve can be represented by R=4>(p,v), (20) «'i ^1 Pig. 2. provided the magnitude R can be regarded as a constant. The corre- sponding constant can be found if we substitute in this equation for p and v the initial values pi and Vi ; for another initial point, ai, equation (20) gives a pressure curve of the same sort, but for t h i s the constant has a different value, and indeed for the new curve we should put R' = {pi', vi), while for the first curve we should write R = (pi, Vi). Equation (20) therefore represents in general the curve equation of an infinite series of pressure curves of the same species which differ only by the different values of their constant R; and for a point a of the plane given by its coordinates pi and vi there is simultaneously determined through the corresponding constant the special pressure curve passing through a. Now if we luse equation (20) in formula (10), we can compute the quantity of heat Q which is to be imparted along the corresponding pressure curve during the passage from atob; similarly when Q is negative this quantity of heat is to be withdrawn, and we at once see that the sign of Q is the same when bodies starting from the initial condition a, ai, etc., describe separate pressure curves of the same species in the same direction. . Accordingly if heat supply was necessary along the path ab (Fig. 2), then such supply will also be necessary along the path aibi ; on the other hand, if we had transferred the body from the state b to the state 61 and by reversing brought it from &i to ai, then along this path biai a certain quantity of heat 38 TECHNICAL THERMODYNAMICS. would have to be withdrawn, provided heat s u p p ly had taken place before along the path ab, and conversely. If the two pressure curves of the same species ab and ai6i are infinitesimally near to one another, then their constants will differ by the infinitesimal R, which, according to equation (20), will be dR = —^dp+-^-dv. Of the numberless pressure curves of different species which can be considered there are especially three which play an impor- tant part in all future developments, and these will now receive fuller discussion. We discussed before, with some detail, the significance of the three functions: U=F(p,v), S=f(p,v), and P={-p,v). If the body in state a is given by the pressure pi and volume ui, then there exists for this point a definite value for each of the three functions, respectively: Ui=F{'pi,vi), Si=fipi,vi,) and Pi = ipi,vi). Accordingly in Fig. 3 we have, starting from point a, three curves aUi, aSi, and aPi, provided heat supply and pressure variation are so arranged that at one time U, another time S, and the third time P is maintained constant. The first of the three curves men- tioned is subject to the relation Fip,v) = Uu . . . (21) which expresses that during the change the inner work (energy) is kept constant. Therefore dU = 0, and the quantity of heat which is necessary along the path db to effect the change to the final condition (p2, ^2) can be found from equation (3), page 26, Q = AL, FiG. 3. INVESTIGATION OF DIFFERENT PRESSURE CURVES. 39 where L is understood to be the external work which, in Fig. 3, is given by the area ahv^vi. The corresponding curve may be called (according toCazin) the "isodj^namic curve," and for this we can say that during the expansion of the body, according to this curve, the whole heat supply is converted into outer work, and conversely that diu-ing compression the outer work expended has been converted into heat, and as such must be completely withdrawn. With the second curve aS\ (Fig. 3) it is assmned that the function S is to be maintained constant; the equation of the curve is therefore fip,v)=Si, :22) and this curve, for reasons which will be given later, will be called the isothermal curve. Finally, with the third curve aPi, the function P is to remain constant during the changes of state; the equation of the curve is therefore 'l>ip,-v)=Pi (23) Since P = Pi is a constant, there follows dP=0, and therefore from the equation ^accompanying set (III), namely, dQ = ASdP, we also get dQ = 0. It follows from this that equation (23) represents those changes of state of the body which occur if during the whole courseheat is neither supplied nor withdrawn. The corresponding curve we will in the fol- lowing call the adiabaticcurve, or briefly the adiabatic, as was done by R a n k i n e (born July 5, 1820, and died Dec. 24, 1872). If there is adiabatic expansion along the path ab' (Fig. 3) from the condition (pi, vi) to the final condition (p, v), and U2 is the inner work corresponding to the point b', then if we integrate equation (3), that is to say, dQ = AidU+dL), 40 TECHNICAL THERMODYNAMICS. we get, because dQ=0, and also Q=0, 0=AiU2-Ui+L), from which follows the external work which is represented by the area ab'vvi. In accordance with the foregoing discussion we will hereafter briefly speak of isodynamic, isothermal, or adiabatic expansion (or compression), according as one or the other of the three discussed changes of state; are in question. The last of the preceding equations therefore expresses that during adiabatic expansion the outer work requires an expen- diture of an equal amount of inner work, and conversely that dm-ing adiabatic compression the expended outer work finally completely reappears in the form of increased inner work. § 7. GRAPHICAL DETERMINATION OF HEAT QUANTITIES. To what has been said there may be added a few important remarks. If we imagine the body in state a to be given by pi and vi (Fig. 4), and supply it along the path ab with a quantity of heat Q till it has passed into the final state b, determined by p2 and V2, then its initial condition a will be given by the value ^^^1 = F{pi, vi), and its final condition b by the value U2=F{p2, V2), pro- vided the function U=F(p, v) is known. From equation (3), page 26, we can now find the quantity of heat Q which must be supplied along the arbitrarily assumed curve ah : Q=A{U2-Ui+L), (24) where L represents the external work, which is given by the" area ab V2V1. But to the two states a and b there correspond the perfectly definite values of the function P; that is, if we assume this func- GRAPHICAL DETERMINATION OF HEAT QUANTITIES. 41 tion as known, then Pi = {pi,vi) and P2 = {'p, V) = Pi and 0(p, v) = P2. Furthermore, let us consider the body to exist in any state Cq corresponding to the values po and vo; if the corresponding value of the inner work Uo=F{po, Vo) is computed, then we can draw through this point the isodynamic curve aoC^o, whose equation is Uo=F{p, v), and whose curve will cut the two adiabatics in the points c and d, having the pressure and volume Pi, Vi' and P2' i>2' respectively. Now if we imagine the body to be brought back into the initial state a and allow it to expand adiabatically from a to c, then because dQ=0 and Q=0, we get from integration of equation (3) = A{Uo-Ui+Li), the outer work Li being represented by the area a c Vi vi . On the other hand if we imagine the body to pass adiabatically from the state b to the state d, we have in Uke manner, for this change, 0=A{Uo-U2+L2), where the outer work L2 is determined by the area b d V2' V2. If we subtract the last two formulas from each other we get U2-Ui=L2-Lu and substitution in equation (24) then gives for the passage from o to & the quantity of heat : Q=A{L+L2-Li). V 42 TECHNICAL THERMODYNAMICS. If we further introduce for the separate quantities of work in the parenthesis the areas of Fig. 4 which represent them, we get the simple result that the hatched area abdv2' Vi c a of the figure represents the value L + L2—L1, or the value of the quan- tity of heat Q, measured in units of work, which is necessary for the change of state along the pressure curve db. ' We have here acquired a simple graphical procedure for ascertaining the quantity of heat Q for the change of state ah given by the diagram, provided that for the body in question we know the general course of the adiabatic and of the isodynamic curve. It is here worth noting that the position of the point ao, through which the isodynamic curve t/o is passed, can be chosen in a perfectly arbitrary manner; but the two adiabatics are fixed, one of them passes through the initial point a, and the other through the final point h. We now recognize clearly from the figure the influence exerted by the kind of passage along the curve ab both on the outer work L and upon the heat supphed; this was aheady fully described at the begimiing of the general investigation; every different pressure curve running from a to 6 leads to different values of L and Q. Ifjve let the point ao, through which the isodynamic curve C/o was passed, cdihcide~with~ the ihiliial point j JFigT^iyri heiLthis ■'Special--assuntpt-itjn'Tnn lead us to the ^aphical representation first~given by Ca z i n .1 , — On the other hand if we so choose ao that the intersections cd can coincide with the axis of abscissas OX, then we get the method given by Macquorne Rankine^; in this case the heat quantity Q expressed as work is represented simply by the area included between the pressure curve ab and the two adiabatics; to be sure, R a n k i n e here assumes that both adiabatics approach the axis of abscissas as asymptotes, which can be regarded as correct at least for gases and vapors. Hereafter, the more general representation of Fig. 4 will , i _^^_ ' Cazin, "Th^orie (51^mentaire des machines k, air chaud." Versailles, 1865. 'Macquorne Rankine, "A Manual of the Steam-engine and other Prime Movers." London and Glasgow, 18.5Q, RELATION BETWEEN THE FUNCTIONS S AND P. 48 be utilized in some discussions, but we may here and now remark that subsequent discussions will lead to a still simpler graphical method which is to be preferred in the 1;reatment of technical problems. In^ drawing the diagram of Fig. 4 a difficulty is en- countered; namely, it will be seen that with certain bodies (gases ^jHTvapors) the course of the adiabatic curve deviates buFTifEle ' from that oT T he isodynam ic curve ; ~ tEe intersections c"an5 ct of the~curves mentioned in Fig. 4 consequently occur at very acute angles, and therefore these points cannot be fixed with sufficient accuracy by the drawing of the curves. The same difficulty exists in the method given by Cazin mentioned above. The third method by Rankine, likewise indicated above, can be seen at once to be useless for the graphical solution of our problems because it demands the determination oFthe contents of an area enclosed by two curves which approach one another as asyinptatesT §8. RELATION BETWEEN THE FUNCTIONS S AND P. If a body receiving heat passes from the initial condition aiipi,vi) (Fig. 5), along any arbitrary but reversible path Uibi to the condition 6i(p2, 1^2), there will follow from AS "^^ by integration f^-P P J AS-^^~^'' where, according to equation (16), the value Pi corresponding to point tti is determined by the relation Pi = 4>ipi, vi); in like man- ner there corresponds to the point 61 the value P2 = 4>{p2, V2). Now if we pass through the points ai and 61 two adiabatic curves, then for the former, as was mentioned on page 39, the value of function P=cj){p, v) is the same for all points, namely =Pi, and likewise for the adiabatic passing through &i the corre- sponding value for P is constant for all points, and equal to ^(p, v) = P2. 44 TECHNICAL THERMODYNAMICS. Therefore if the body passes from another initial condition a^ or as in any reversible manner to the final state 62 or 63, and if these initial points 02 and 03 lie upon the adiabatic passing through Oi, and the terminal points 62 and 63 upon the adiabatic passing through hi, then for the second and third passages also there is valid the re- lation VfVg Fig. 5. / as" ■■P2-Pu From this follows the important proposition: "If a body in any (reversible) way passes from one point of a particular adiabatic to any other point of a secohd adiabatic, then the integral J AS' taken between the same limits, has always one and the same value." If, for simplicity, we designate the difference P2 — P1 by P, we have the magnitude ■/ dQ AS' (26) a constant quantity for every one of the infinite number of pas- sages possible. For the return path the proposition likewise holds; but then the sign of P changes and will be negative, provided it was positive upon the forward path, and conversely. We recognize from this that in Thermodynamics the adiabatic curves play a part similar to the altitudes, or differences of level in Mechanics. For reasons which will appear in the following discussions I have called the value P represented by equation (26) RELATION BETWEEN THE FUNCTIONS S AND P. 45 "heat weight";! but we will also hereafter use for it the term "entropy," one which is more widely used among physicists. In discussing, in Fig. 5, the passage ot a body from one. adia- batic to another, the pressure curves were assumed running in a perfectly arbitrary manner; we can now think of the transfer as taking place upon particular prescribed curves, and here that case is of special significance in which the transfer takes place along the isothermal curve, i. e., along the curve for which the function S = fip,v) is kept constant. If the body subjected to investigation exists in the state ai (Fig. 6), it will possess for this a particular value Si of the function 5, namely, Si=f(,pi, Vi); now, if we bring the body to the state bi and at the same time supply heat to the body along the isothermal curve of constant value S=Si, that is, along „ the curve having the equation ~ Fig. 6. Si-f(p,v), then equation (26) can be integrated, and, designating by Qi the necessary quantity of heat, we have Qi=APSu (27) If we again pass adiabatics through the point ai and bi, and im- agine this same body at one time to start from the point a2 and at another time from the point as, both of these points lying upon ' Clausius designates the integral I -w> ^iti> tlie addition of a constant, as the transformation value or as the entropy of a body in which S appears as the temperature function, and as such it will likev/ise appear in the subsequent developments of the present treatise. (Compare quotation on page 24.) — Rankine calls P=^(jp, v) the thermodynamic function. — C. Neumann designates P as the parameter of the caloric (adiabatic) curves. ("Vorlesungen iiber die mechanische Theorie der Warme.'' I.eipsic, 1S75.) 46 TECHNICAL THERMODYNAMICS. the first adiabatic, and to proceed along the corresponding iso- thermals to the second adiabatic, and if, furthermore, Q2 is the heat supplied for the path a2h2, and Qz that for the path a^hz — then, according to equation (27), we will have the relations Q2=APS2 and Qz = APS3. From this, and from equation (27), it follows that for all passages the heat weight is constant, namely, ^;Si ^^2 ASz ^'^^> The necessary quantities of heat are, therefore, proportional to the corresponding values of S. Use will be made hereafter of the foregoing relation (28). § 9. REVERSIBLE CYCLES. For the quantity of heat which must be imparted to a body undergoing changes of state along the reversible path, in order that the body may pass from a certain initial condition a into a given final condition h, we found from Fig. 4 and equation (24), page 38, Q=A{U2~Ui+L). But now if we suppose that the assumed pressure curve is a cl s e d curve (Fig. la), and that, consequently, the final state coincides with the initial state, the point h with the point a, then in the foregoing formula we have C/i = U2 and get the simple relation Q = AL (29) ~X Here L means the work gained, which is measured by the area enclosed by the pressure curve, and Q is the quantity of heat consumed in pro- ducing it. But doubtless this quantity of heat can be imparted, REVERSIBLE CYCLES. 47 during the whole process, only from without, that is, be furnished by other bodies, and therefore can not, either in whole or in part, be abstracted from the interior of the body with which the cycle was described, for, according to the hypothesis, the body has been led back to its initial condition: its total inner work is, consequently, the same at the end of the process as at the beginning. A process of the contemplated sort is called (according to Clausius) a cycle, and, moreover, since we are at first consid- ering only reversible changes of state, it is called a reversible cycle. The closer consideration of Fig. 7a shows, moreover, that the work gained can be considered as consisting of two parts : during the expansion of the body along the path aaia2, there was gained the work Li represented by the total hatched area; and when led back, or during compression along the path azaza, there was e x- pended the work L2, measured by the more closely hatched area. The total work L produced in the whole cycle, therefore, appears as the difference L1—L2. We see, moreover, that Fig. 7a represents nothing but what is known and familiar to the mechanical engineer as an indicator diagram. As the same process can be repeated as often as desired, we can easily refer the gained work L to the second of time and can express it in horse-powers, provided we know how many times the process takes place in a second. Steam engines, hot-air engines, gas (internal-combustion) engines, which I include under the general name of heat engines, do in fact produce diagrams or processes of the described kind; of course, in very special cases we must first investigate whether the process of the one or the other kind of engine really can be regarded as reversible, for only under this supposition do the above general propositions hold ; particularly should direct applica- tion to the occurrences in internal-combustion engines be preceded by a special investigation, because here the process is connected with a chemical change, a condition which has hitherto been expressly excluded from the investigation. In applying thermo- dynamics to technical problems we will come back to the foregoing. 48 TECHNICAL THERMODYNAMICS. Worthy of note is the reversal of the process indicated in Fig. la. If we reverse the direction of the arrow, then work L^ will be gained along the path 00302, and work Li must be expended upon the return path, 02010; consequently the work L=Li— L2 will become nega- tive, and so will the quantity of heat Q, according to equation (29) ; hence the carrying-out of the process here demands an expendi- ture of external work L, and the quantity of heat Q thus generated must be conducted away to the outside, to other bodies. This reversed process also plays an important technical part, in engines for the production of cold. We will now more fully discuss the question of heat consump- tion and of heat production for the cycle represented in Fig. 7o. In Fig. 76 the closed curve of the cycle considered in Fig. 7a is again represented; we may say, therefore, that we have before us for our investigation the same in- dicator diagram. Now if we draw two adiabatic curves Pi and P2 so that they Just touch the closed curve at Fig. 76. the points e and 63, and if, more- over, we draw anywhere under the diagram the isodynamic curve UqUo, cutting the adiabatics in points c and d, then, according to the discussion of Fig. 4, page 41, the whole hatched area of Fig. 76 represents in units of work the quantity of heat which must be supplied along the curve 66162; and the more closely hatched part measures in like manner the quantity of heat which must be withdrawn along the return path 62636. If we designate the first quantity of heat by Qi and the other quantity of heat by Q2, the two mentioned areas will be determined by 1 ^"'^ T- It is evident from the figure that the difference of these sections represents the work L which is given by the area enclosed by the cycle; therefore for this process we also have the relation REVERSIBLE CYCLES. 49 . Ql Q2 ^-T-lf' • (30) or AL=Qi-Q2, .' (31) and considering equation (29), Q=Qi-Q2 (32) We now recognize clearly whence comes the quantity of heat Q which, in this process, was converted into the work L. It is simply the excess of the imparted heat Qi over the abstracted heat ^2- If we reverse the process (Fig. 76), then along the path 66362 there will take place the supply of the quantity of heat Qz, and along the path 62616 the withdrawal of the heat Qi ; both quantities of heat, and therefore also the quantity of heat Q (equation 32), and likewise the work L (equation 29), change their signs: in this case the withdrawn quantity of heat is larger than the supplied quantity, and the excess corresponds exactly to the quantity of heat which has been generated in this process by the external work expended. It was shown above that, for any passage of a body from one adiabatic to another, the magnitude --/': ''-■AS (equation 26) is a constant; if the body in our cycle first traverses one of the branches 66162 or 66362, and then comes back upon the other, the magnitude of P has the same value for the forward motion as for the return motion; the only difference is a change of sign, and the sum of the two values is therefore zero. For a cycle, and, as we must expressly add, for the "reversible cycle," there accordingly holds the relation TS-" <33) / a proposition which is self-evident from our whole discussion. The function S was so chosen at the start that the expression under the integral sign of equation (33) represented a complete 50 TECHNICAL THERMODYNAMICS. differential. Now, if we effect this integration by a change of state in which the final condition coincides with the initial one, then the integral must be equal to zero, because the integration limits are alike. Returning once more to the consideration of Fig. 76, and imagin- ing the cycles to be conducted in the direction of the arrow, then the body, which we will call the "intermediate or mediating " body, receives the heat Qi along the path 66162. But the imparting of heat cannot be conceived without assuming at least another second body, Ki, which delivers said quantity of heat to the mediating body; we can also imagine a series of such bodies successively giving off heat. Furthermore, we must assume a third body K2 (or a commu- nity of such bodies) which withdraws along the path 62636 from the mediating body the quantity of heat Q2, that is, absorbs it. The cycle therefore includes the interaction of at least three bodies: one of them, Ki, furnishes the quantity of heat Qi, the other, K2, absorbs the quantity of heat Q2, and the third, the medi- ating body, describes the cycle; the latter body returns to the initial condition, while the other two must necessarily experience changes of state of a certain kind. It is only when the cycle is carried out first in one direction, and then along the same paths in the reverse direction, that all three bodies return to the initial condition. But then work is neither produced nor consumed. A glance at my graphical representation of the events in the reversible cycle of Fig. 7b shows that no such process is conceivable in which, with a gain or loss of work, there is not simultaneously connected a transfer of heat from a body Ki to another, K2, or conversely. 1 ' C 1 a u s i u s in his papers speaks of " transformations " of two kinds in the reversible cycle. By "transformation of the first kind " he understands a con- version of heat into work, or, conversely, effected by the mediating body. " Transformation of the second kind " includes that transformation of heat from a body Kj to another, K^ which accompanies such a cycle. Sadi Carnot has discussed the cycle, although only for a special case which will be more closely examined above in the text; he was led to his investigations as an engineer, by a close consideration of the occurrences in the steam engine. Later (1834) Clapeyron took up the question, and was the CARNOT'S CYCLE. 51 § 10. CARNOT'S CVPLE. The propositions just developed in general fashion for the reversi- ble cycle, under the assumption that the changes of state of the mediating body take place according to a closed curve (Fig. la and 76), also hold (as is directly shown by our "graphical representa- tion") for cases in which the work area L is bounded in parts by pressure curves of different species ; of the infinite number of cases of this sort, we will now pick out one cycle whose closer investigation leads to especially important results, namely, the one whose work area is bounded by two adiabatics and two isothermals; a cycle of this kind we may designate in advance as the Car not cycle. If in Fig. 8 we draw the two adiabatics Pi and P2 and then the two isothermals S\ and S2, we get a quadrilateral, 016162^2, of curves in place of the closed curve in Fig. 76. If the body starts from the initial condition corresponding to point Oi(pi, Vi), for which (S = jSi =/(pi, Vi), and expands from Oi to 61, with the corresponding supply of heat Qi, and then, shutting off the supply of heat, expands adiabatically from 61 to 62, then first to put Carnot's proposition in analytical form, and to also represent n graphical form the cycle with Carnot's assumptions, that is, the indicator diagram. But both writers, if I may use the phraseology of C 1 a u s i u s , had in view only the "transformation of the second kind," and ascribed the production of work in heat engines solely to the transfer of heat from the body Kx (according to the designation given above in the text) to the body K^, and assumed that here the quantities of heat were equal, that is, Qx = Q^. We must thank C 1 a u s i u s ' " Abhandlungen " (1850) for pointing out what was so important for the further development of thermodynamics, namely, that the above-mentioned equality could not exist; indeed, that, according to the theorem of the equivalence of work and heat, necessarily the relation Qx>Q2 must obtain, and that the difference between them corresponds exactly to the work produced in the cycle. It is what rarely occurs in science that just the most important parts of Clapeyron's analytical investigations were not touched by his false assump- tion; particularly that equation remained valid which was designated above in the text as the second fundamental equation of the mechanical theory of heat. It might therefore be designated as Clapeyron's equation if, corre- sponding to the historical development, the first fundamental equation were named after C 1 a u s i u s . 52 TECHNICAL THERMODYNAMICS. the body can be led back to its initial state by compression along the isothermal curve &2a2, and by compression along the other adiabatic from 02 to oi. In leading back along the path 6202, which occurs with 8 = 82 constant, the quantity of heat Q2 must be with- drawn. Now if we again draw an iso- dynamic curve Uo = F(p, ?;) = con- stant, which cuts the adiabatic in c _;5-and d, then, as was proved, the whole hatched area of Fig. 8 represents the heat Qi measured in units of work, and similarily the more closely hatched area represents the heat Q2 measured in the same units; the difference of the two is again the work L produced in the cycle : here equa- tion (30) and equation (31) likewise hold; moreover, according to equation (28), page 46, the heat weight, t he ent ropy^, is the same for each of the two isothermal passages, and, indeed, Fig. 8. Qi Q2 ASi AS2 (IV) Accordingly, Qi^APSi and Q2 = APS2, and, therefore, from equation (30), which we here rewrite, the work produced in this cycle becomes: Qi Q2 ^~ A~ A' L=P(Si-S2). (V) These equations are identical, and P is determined by equation (IV). Now the following is deserving of notice. Because Qi > Q2 (for the difference is the heat corresponding to work L), we also always have (Si > S2, according to equation (IV) . Now, although at this time nothing is known concerning the form of the equation (S = f(p, v), still we at least know that the value of the constant S in any single one of the infinite host of isothermal hues CARNOT'S CYCLE. 53 traversing the diagram is larger, the higher the curve lies in the diagram. For each point of the plane below the curve SiSi (Fig. 8) we have the corresponding value of S smaller; and for every point lying above it S is greater than the constant Si of the assumed isothermal. The following important remark is closely connected with the foregoing. If a body describes any cycle abed whatever, Fig. 9, and if we draw two adiabatics Pi and P2 and two isothermals SiSi and S2S2 which touch the curve, then for every point of the cycle the cor- responding value S is smaller than at the point of contact b, and greater than the value at the point of contact d. Now if the curve SiSi (Fig. 9) were given as a certain upper limit, and the curve S2S2 as a lower limit, that is, if it were impossible in the course of the cycle abed to pass beyond the upper and lower limits (Si and S2), then the area enclosed by the curve abed would always be smaller than the area aibib2a2. „ Fig. 9 From this follows a proposition of especial importance for the theory of engines: "Of all the cycles which can be described between two par- ticular adiabatics and two particular isothermals, the Car not Cycle will deliver the maximum of work L." ^ LefTIs how^return to the special consideration of the C a r n o t cycle with the help of Fig. 8. Along the first isothermal, along the path aibi, the mediating body receives the heat Qi, which we will suppose to come from a body Ki, which, during the course aibi, stands in a certain mutual relation to the mediating body, — for example, is in con- tact with it; but to this body Ki must evidently be ascribed a particular property, namely, that it will, as the case may be, successively impart heat to, or withdraw heat from, the mediating body in such measure that the function S will be maintained at the constant value »Si. The body Ki may therefore be designated as "body of the state Si"; for like reasons we must also imagine 54 TECHNICAL THERMODYNAMICS. a second "body of state Sz," which will affect the withdrawal of heat along the isothermal S2S2 on the path 62^2, or similarly affect the imparting of heat in the reversed path 0262. Now we will for the present dismiss the consideration whether bodies with the given properties really exist or not; the assumptions made do not alter the general character of the considerations; we can even say that it is undoubtedly correct that an exchange of heat would take place between the bodies Ki and K2 if they were brought into immediate contact; the only doubt remaining in the matter would be as to the direction in which the heat exchange would take place, — whether from Ki to K2 or in the opposite direction; at any rate, there would be no production of work in such a direct exchange of heat. We can now say briefly that in the Carnot cycle the heat weight P is transferred from the body Ki of the state Si to the body K2 of the state S2, and that thus the work L = P{Si —S2) is produced, and that this work, having a particular heat weight P, is a maximum as compared with all the other possible cycles which can be described between the limits Si and 82- During the reversal of the cycle an equal amount of work is consumed and the heat weight is inversely transferred from body K2 to body Ki. § 11. PHYSICAL SIGNIFICANCE OF THE FUNCTION S. Hitherto all propositions established in the foregoing have proceeded from the assumption of a single fundamental principle, namely, the equivalence of heat and work. But further progress now demands the setting up of another or second fundamental principle, in order to briefly determine the form and significance of the different functions thus far introduced. If we succeed in doing this for at least one of them , — for example, for the func- tion S=f{p, v), — then we know, from what has already been said, that we have the means given in the fundamental equations de- duced above of immediately expressing all the remaining functions. Before going into the further developments indicated, it will be useful to compare the propositions concerning the cycles, par- PHYSICAL SIGNIFICANCE OF THE FUNCTION S. 55 ticularly those concerning the C a r n o t cycle, with the analogous principles of Mechanics. Let us imagine three horizontal planes A, B, and C lying over one another. The uppermost plane, A (Fig. 10), lies at the height Hi, and the second, B, at the height H2 over the lowest plane, C; the lowest plane therefore lies at the zero point, f om which the heights are measured upward; furthermore, if we imagine there is at our disposal in the upper- most plane a body of the weight G, and that we can let it sink slowly and uniformly to the lowest plane, then there will be at our disposal in this body a work which may be designated by Wi, which is easily determined by the relation, G. ^ A 1 t B > '' Hi m \. r i Fig. 10. Wi^GEi. (34) The value TFi is what is designated in Mechanics as "energy of position," or as "potential energy due to the action of gravity." If the same weight G exists on the middle level B, then there will be produced, during its slow, uniform sinking to the lowest level, the work W2 = GE2; (35) finally, the work L, which is produced when the body sinks in like manner from A to B, is L=(?(Hi-^2). From equations (34) and (35) follows (36) (IVa) and substitution in equation (36), which we will write again, gives h^Wx-Wi 1 L=G{Hi-H2)-\ L (Va) If we imagine a body of weight G to be slid along in the plane A, then this corresponds to a "supply" of the work PFi, and 56 TECHNICAL THERMODYNAMICS. the body after it has descended to the level B, doing work in the man- ner indicated, must be there delivered, and this will correspond to a "withdrawal" of the work W2; the difference between the furnished work Wi and the led-away work W2 (which for our purposes can be considered as work withdrawn) is equal, accord- ing to equation (Va), to the work Lproduced. But the cycle is also reversible, and if the body is led to the lower level B and then lifted slowly and uniformly into the upper level A, then the furnished work W2 at B is smaller than the delivered Wia.t A; the work L has been expended in lifting and corresponds ex- actly to the difference Wi — W2. If we again imagine a cycle to be described in the first direction, then, under the assumptions made, the work L, produced in passing from A to B, is a maximum, because uniform sinking was as- sumed; for if the sinking were not uniform, that is, were accelerated, occurring with increasing velocity, then the work produced would be smaller, indeed this work might even be zero when the body of weight G simply falls from AtoB; in this case there dwells in the body at the lower level the whole initial work Wi, but the difference Wi — W2 exists in the form of living force, of kinetic energy. For the purpose of technical study, which, to be sure, is the main purpose of the present treatise, it is convenient to regard the body G as a liquid; if the level A is an infinitely broad and ex- tremely shallow reservoir, and likewise the level B,^ and if G is the weight of a quantity of water which sinks in a particular interval, ' for example in one second, then the process described in the fore- going will appear as the one carried out by the hydraulic motors, water wheels, turbines, and water-pressure engines; the reversal of the process would then correspond to hydraulic hoists, scoop wheels, centrifugal pumps, and cylindrical pumps. The choice of the lower level C is still perfectly arbitrary; in ' This assumption is always made in the investigation of hydraulic motors; if we proceeded more exactly and assumed finite values for the breadth and depth of the supply and discharge conduits, then we ought to take for the difference of heights H, — H^ the distance apart of the centers of gravity of the two conduit cross-sections; we should also assume that the water flows at all points with the same velocity and does the same in the discharge conduit. PHYSICAL SIGNIFICANCE OF THE FUNCTION S. 57 considering hydraulic motors we may imagine it to be at the level of the sea. , If we compare the formulas and principles in the foregoing with those which were given for the Carnot cycle (§ 10, p. 53), with the help of Fig. 8, then the complete analogy will appear; and where we there spoke of the heat quantities Qi and Q2, meas- ured in units of work, we here have the quantities of work TFi and W2; and where we there expressed the values of function S by (Si and *S2, we here have the heights Hi and H2. Equations (IVa) and (IV) are in structure and meaning like (Va) and (V). Function S can therefore be regarded as a linear quantity, the height of a level over a certain zero point, and the analogy between the two kinds of occurrences is just what incited us to designate the value P given by equation (26), page 44, as " h e a t weight," which designation can even be applied to the differential ^^=S' (^7) in this case the work produced in the Carnot cycle between the limits Si and S2 is dL = {Si-S2)dP (38) On the other hand, if we imagine the uniform sinking of a body of infinitesimal weight, dG, possessing at the height H the work dW=HdG, we have dG=^, (37a) and the work produced in falling from Hi to H2 is dL={Hi-H2)dG (38a) The last two equations and the cycle underlying them could easily be extended in such a way as to bring out the general propositions given for any cycle. But for the important questions before us it will suffice to keep in view the simple Carnot cycle. 58 TECHNICAL THERMODYNAMICS. Just as the consideration of the events based on Fig. 10 repro- duced occurrences in hydraulic motors and their inversions, the hydraulic hoists, so here the question arises whether a closer examination of the occurrences in h e a t engines and their inversions could not be utilized for the events in the C a r n o t cycle and lead to further information. If we think especially of the steam engine, then in fact a trans- fer of heat from the boiler to the condenser does accompany the productionofworkina steam cylinder; heat is imparted to the boiler, and cooling takes place in the cylinder, that is, heat is withdrawn. But here the heat supply takes place at high tem- perature, and the heat withdrawal at lower temperature. We can therefore say that in the steam engine we have before us a case of work production occurring simultaneously with a passage of heat of higher temperature to heat of lower temperature. The inverse process occurs in refrigerating engines, engines which serve for the production of cold; to run these engines work is ex- pended and the imparting of heat in the evaporator takes place at the lower temperature, the withdrawal of heat (in the con- denser) at the higher temperature. Here, therefore, we really have to do with a cycle described by engines which work in the one or the other direction according to the need of the moment, that is, we have to do with a cycle that is reversible. In the boiler, as in the condenser, the tem- peratures are maintained at constant height during the running. But now the foregoing considerations show that in the de- scribed cycle a new element comes into play which did not exist in our former discussions, or, to speak more exactly, existed there in disguised form, namely, temperature. Let us designate, according to Celsius, the temperature by t, whose zero is the freezing-point of water, and whose 100 is the boiling-point of water, both corresponding to the mean barometric height of 760 mm.merc ury; then it is a question whether all the discussed peculiarities of the C a r n o t cycle and of the properties belonging to the function S will really correspond with experience if we assume that in this cycle the expansion of the body takes place PHYSICAL SIGNIFICANCE OF THE FUNCTION S. 59 Fig. 11. along the path aifci (Fig. 11) at constant temperature ti, and that the compression takes place along the path 62^2 at the lower temperature ^2- In this case the body Ki of the state Si ap- pears to be one which maintains at constant temperature the mediating body during its expansion. This will occur when Ki itself is so large (strictly speaking, infinitely large) that its grant of heat does not cause any noticeable change in its own temperature. Likewise the body K2, which has to main- tain along the path 62^2 the mediating body at the constant temperature <2, must be as- sumed to be very great and to possess the temperature <2. Bodies of this sort do exist, and consequently the Car not cycle seems to be drawn within the realm of possibility, even if we disregard the experience of heat engines. Accompanying the production of work by means of the C a r - not cycle, there is a transfer of heat from the body Ki of the higher temperature ti to a "body K2 of the lower temperature <2, which is what actually happens in heat engines and was described when considering steam engines. Furthermore, experience shows that a direct transfer of heat from i?i to 1^2 without production of work is possible, a case which we just compared (in Fig. 10) with the falling of a weight from a higher to a lower level. We only need to bring the two bodies of different temperatures ti and (2 into contact with each other, in which case heat of its own accord flows from the warmer to the colder body, a procedure known under the name of heat conducti on. But never has there been observed during the direct contact of two bodies Ki and K2 a transfer of heat in the opposite direction that is, a flow from a body of lower temperature to one of higher temperature; such a transfer, according to the reversal of the Car not cycle (Fig. 11), is only possible with the simultaneous performance of external work, and without it could only be com- 60 TECHNICAL THERMODYNAMICS. pared with the phenomenon of a weight rising uniformly without expenditure of work, that is, alone and unaided, from a lower level to a higher one. From all the foregoing consideration of the function S=f{p, v) we conclude that the temperature is constant when the change of state of a body takes place with constant value of S. There- fore we must also regard S as a function of the tem- perature t alone. Accordingly we have the relation and an expansion or compression according to the curve )S = con- stant appears as a change of state at constant tempera- ture; for this reason this curve was designated from the beginning as the isothermal curve. § 12. TRANSFORMATION OF FUNDAMENTAL EQUATIONS. As S has been recognized as a function of the temperature t, we have the relation dS^^^dt (39) Now it has long been recognized in Physics that the tempera- ture i of a body is completely determined by the pressure p and the volume v. The analytical relation existing between t, p, and v for a particular body, and assumed to be derived from physical considerations, is called (according to Bauschinger) the "equa- tion of condition" of the body in question. This equation of condition appears to be different for different forms of the body, and is not known for all kinds of bodies; at the present time it is only known for gases, and, as will be discussed later, for vapors in certain conditions, but not for solid and liquid bodies. But the equation of condition when known invariably gives the relation d,t = -:^ap+i^av, TRANSFORMATION OF FUNDAMENTAL EQUATIONS. 61 and if we use this in equation (39) we get ,^ dS /dt ^ dt* \ , _^ '^^^dtid^'^P + Tv^')- ■ ■ ' ■ ' (40) If we compare this formula with the earlier one given under (19), page 35, there follows dp dt dp dv dt dv ' ' ' ' But now the second fundamental equation of the mechanical theory of heat (II, page 34) was found to be o Y^S dS op ov and if we here utilize equation (41) we get dS dp dv' As S is only a function of t, the expression in the left member is also only a function of t, and we can therefore put dt ^-^' (42) and this new temperature function T, whose special form is still to be determined, we will, till this form is established, call the Carnot Function. It stands in a simple relation to the func- tion designated by S, for from equation (42) follows dS_di and therefore logeS = J ~ + C, (44) 62 TECHNICAL THERMODYNAMICS. where C represents an integration constant. As soon, therefore, as the form of Carnot's temperature function is recognized, — and we will derive it below from physical principles, — so soon will equation (44) determine the magnitude *S as a function of the tem- perature t. For the purpose of the following investigations, and in order to easily refer to the principal results of all the foregoing investi- gations, we will here collect the principal formulas. The 'first fundamental equation (I), p. 32, is repeated unchanged in (la); on the other hand, the second fundamental equation (II), after utilizing the equation preceding equation (42), takes the form shown in (Ila) : 97 3Z 9p 'dv ' op ov (La) {Hay On the other hand, of equations (III), page 36, only the second and third change their form when equations (41) and (43) are utilized, a transformation which is easily followed, and we can therefore write dQ = A[Xdp + Ydv] = -^\X dt + T dv\ (Ilia) =-:^\Y dt-T dv\ dv = A[dU+pdv] = ASdP and from this grouping of five identical equations we will always pick out that one which most readily leads to the solution of the particular problem. " Clapeyron gave the second fundamental equation exactly in the form in which it is represented by (IIo) , only he used the letter C to designate the Carnot Function T. UNIVERSAL SIGNIFICANCE OF THE CARNOT FUNCTION. 63 § 13. UNIVERSAL SIGNIFICANCE. OF THE CARNOT FUNCTION. In the above investigation of cycles, particularly of the Car not cycle, only one body was mentioned, and its change of state considered. When the result was ascertained that the two functions S and T could be regarded as temperature functions of a particular kind, then evidently this result could for the time being hold only for the body in question describing the C a r n o t cycle. But it can now be shown, and this is the purpose of the fol- lowing discussion, that the form of this function must be the same for bodies of every kind, whether they exist in gaseous, liquid, or solid condition. If the proof is furnished for one of the two func- tions, then it will also hold for the other function on account of relation (44). If we assume two different bodies A and B, and describe with each a C a r n o t cycle between two isothermals corresponding to the same limiting temperatures h and <2, and indeed in such a way (as is always possible by a suitable choice of the corre- sponding adiabatics) that the work performed, or the work L expended during the reversal of the cycle, is the same, then we can utilize in both cases the bodies Ki and K2 for the supply and with- drawal of heat. A ^K A \KX^ Ki Fig. 12. If in Fig. 12a we describe the indicated cycle process in the direction of the arrow with the first body A, then the body Ki will furnish the quantity of heat Qi and the body K2 will absorb 64 TECHNICAL THERMODYNAMICS. the quantity of heat Q2, and the work L produced will bear, to the given quantities of heat, the relation AL=Qi-Q2. On the other hand, if we describe the same cycle in the same direction with the second body B (Fig. 12b), we may assume that the body Ki will supply to B the quantity of heat qi during iso- thermal expansion, and the body K2 will absorb during the iso- thermal compression the quantity of heat 52- As the work here must also be equal to L we have the formula AL = qi~q2. From this follows, by a combination of the two formulas : Qi-Q2 = qi-q2 (45) From the equating of the differences, however, it does not necessarily follow that the equations Qi=qi and ^2=92 must be true; rather the only permissible conclusion is that we have the relations qi=Qi+Qo, q2-Q2+Qo,\ ^^^^ where Qo represents a positive or negative quantity of heat. Let us assume that Qo is positive, and let us describe the cycle with the body A in the direction of the given arrow (Fig. 12a), then the body Ki will furnish the quantity of heat Qi ; now utihzing the same bodies Ki and K2 with the body B, let us conduct the cycle (Fig. 126) in the direction opposed to the arrows. Then the body Ki at the upper temperature h will absorb the quantity of heat qi=Qi+Qo, while it only gave out during the first cycle the quantity of heat Qi. Both mediating bodies -A and B are thus brought back to their initial condition; during the first cycle the work L was produced, while during the second cycle an equal quantity of work was consumed. The sole result after this double cycle is that the body K2 of the UNIVERSAL SIGNIFICANCE OF THE CARNOT FUNCTION. 65 lower temperature t2 possesses an amount of heat less by Qo than formerly, and the body Ki of the higher temperature an amount greater by Qo; it would therefore appear as if the quantity of heat Qo had simply of itself been transferred from the body K2 of lower temperature ^2 to a body Ki of the higher tempera- ture ti. The same result is produced when we take Qo in equations (46) as negative; in this case we would use Qi=qi+Qo, Q2=q2+Qo, and would get the same result as before, provided the second body B first described the cycle (Fig. 126) in the direction given by the arrow, and that then the cycle course were completed with the first body A in the direction opposed to the arrow. These considerations led Clausius to a simple proposition which has been universally accepted as the second fundamental law, namely: "Heat cannot of itself pass from a colder to a hotter body"; it has been placed alongside of the first law (that heat and work are equivalent) of the mechanical theory of heat. It follows that in the foregoing developments the introduced quantity of heat Qo can have neither a positive nor a negative value, but must be equal to zero. Equation (46) therefore becomes i'^^"\ (47) In the two cycles represented by Figs. 12a and 12& the body Ki therefore receives back the same quantity of heat Qi during the reversal of the cycle which it formerly supphed, and the body K2 gives out again the same quantity of heat Q2 which it previously received, no matter which body acted mediately in the first forward direct cycle or in the second return reversible cycle. The importance of this proposition justifies us in returning for a moment to the analogous case in Mechanics treated in § 11. 66 TECHNICAL THERMODYNAMICS. If we assume that a turbine consumes G kg [lb.] of water per second with a fall H^ —H2 (see Fig. 10, p. 55), then, neglecting the losses of work, its corresponding work will be G{Hi—H2). Now if this turbine drives a centrifugal pump, and we here also neglect the losses of work, then the same quantity of water G can be brought back to the original level Hi , and if the original quantity of work supplied at this level is Wi=GHi, then this also will be brought back again. Now if we try to conceive of a machinal arrangement which would bring a larger quantity of work TFi + Wo into the upper plane, then this would be equivalent to the assump- tion that the weight could be brought from a lower to a higher level without the expenditure of work. But since this weight Go could be al- lowed to sink while at work (for example, in a second turbine), this arrangement of the two turbines with a centrifugal pump would cdnstitute a case of perpetual motion, the impossi- bility of which was long ago demonstrated. But what is true of the hydraulic motors is also true of the heat engines; if we had two such engines describing the Carnot cycle, giA'en in Figs. 12a and 12b, both working between the same tem- perature limits, and the one engine running through the forward cycle and the other through the reverse cycle, and if, moreover, in the former the amount of work produced was numerically equal to the amount of work consumed in the latter, then it would be impossible for the quantity of heat supplied to the body Ki to be greater than the quantity of heat which it gave out during the first cycle; for if it were greater, and the excess were repre- sented by Qo, then this quantity of heat existing at the upper-tem- perature level could be utihzed and describe a third Carnot cycle by means of a third engine. In reahty Clausius's principle amounts to saying that in a mechanical sense no combination of heat engines is conceivable which will bring about perpetual motion. Now let us return to the discussion of the main question. Con- UNIVERSAL SIGNIFICANCE OF THE CARNOT FUNCTION. 67 sideration of the two cycles given in Figs. 12a and 12b has led to the result that, assuming the same temperature limits and equal quantities of work, the relations Qi=qi must be true, and it consequently follows that f = f (48) But still it does not follow that the function S= W{t), page 60, must have the same form for the two bodies A and B; to be sure, the two isothermal curves for the body A (Fig. 12a) certainly have a different course than for the body B when we consider the connection of S with p and v, and we may here say that this differ- ence really exists : because of the connection of the value S with the temperature a further discussion of the question is nec- essary. If for the body A we designate the temperature function by S, as hitherto, and on the other hand for the body B by s, and thus distinguish another form of the function, then, according to equa- tion (28), we have the relation, Qi Q2 _ Qi Si ASi AS2 Q2 S2 For the other cycle there is Ukewise, Asi As2 92 S2' According to equation (48) there immediately follows S2 S2' 68 TECHNICAL THERMODYNAMICS. If we subtract unity from both sides, we can also write S1-S2 S1-S2 S2 S2 As a consequence, if we suppose the two cycles in Fig. 12a and Fig. 12b to work between the limiting temperatures t2 = t and ti = t + dt, we have ds__dS s ~ S' and therefore, by integration, s = kS, (49) where the factor kis a, constant. This proves, first of all, that the temperature factor S for the one body A differs in value from that of the other body B only by this factor. But in § 5, p. 33, the fimction *S was introduced into equation (15) as an integrating factor, and, in the accompanying remark there made, it was em- phasized and proved that we could substitute for this function the value kS without altering at all any one of the following con- clusions; consequently the factor A; is perfectly arbitrary, and therefore may also be placed equal to unity. But independently of this we find, when we differentiate equation (49) with respect to temperature, ds dS If we divide equation (49) by this one and at the same time make use of equation (42), p. 61, we get dt dt and this magnitude, according to equation (42), was designated as the Car not temperature function T. TRANSFORMATION OF REVERSIBLE PRESSURE CURVES. 69 From this follows the important result, that the Carnot function must not only have the same form for all bodres of nature, but must also, at any particular temperature, have one and the same numerical value for all bodies without exception. § 14. TRANSFORMATION OF REVERSIBLE PRESSURE CURVES AND OF INDICATOR DIAGRAMS. If a body starting from the initial condition (pi, Vi), Fig. 13a, passes to the final condition (p2, V2), traversing the prescribed pressure curve acb while receiving the quantity of heat Q, there will correspond to the curve point c, having the coordinates p and V, a perfectly definite value of the function *S and of its tem- perature t, for we will continue to assume as known the two re- lations, S=f{p,v) and S = {t). Likewise the function P J as" ■4'{p,v) has for this point c a value which can be determined from this given equation. Fig. 13. If we now lay off from 0, in Fig. 13&, the value P as abscissa OP and the value S as its corresponding ordinate, we get point Ci, which we will call the transfer (copy) of the point c of the pressure curve. 70 TECHNICAL THERMODYNAMICS. In this way every point of the pressure curve can be repro- duced as soon as we know its course, and consequently we get in Fig. 136 a curve aiCibi which may be designated as the trans- formation of the pressure curve acb. If in a special case acb is the isothermal curve, for which we know, from what has preceded, that the temperature t and its corresponding value S=Si are constant, it will follow that the transformation of the isothermal will be a straight line 0162 lying parallel to the axis of abscissas OX. On the other hand, if the pressure curve acb is an adiabatic, then we know, from the discussion of Fig. 5, § 8, and from equa- tion (23), p. 39, that the function P = Pi is a constant; therefore the transformation cti&s of the adiabatic curve passing through the point a is a straight line lying parallel to the axis of ordinates. Now if we again return to the general case in Fig. 136, and allow the distance P to increase by the amount dP, then the area of the strip standing on dP will be S dP; consequently from the original relation dQ there follows also '"'-AS §-S,P. and therefore the area aibiP2Pi (hatched in the figure) lying under the transformed curve aiCibi represents nothing but 1 = /^''^' i.e., the quantity of heat Q, measured in units of work, which must be imparted to the body during its changes of state along the path acb; the area abv2Vi (Fig. 13a) under the latter curve measures the external work thus produced at the same time. TRANSFORMATION OF REVERSIBLE PRESSURE CURVES. 71 Besides the graphical determination of the heat quantity Q, given in connection with Fig. 4, § 7, p. 41, there follows from the above a second graphical method for the same purpose, and one free from the defects there emphasized. We must not overlook the fact that both methods are only true for reversible operations. If Fig. 13a refers to compression, the reversal of the operation before assumed, then the direction of the arrow in Fig. 136 will also be reversed, and the quantity of heat will be negative, that is, will be heat with- drawn, and the hatched area in Fig. 13a will represent the work consumed. It is self-evident that heat withdrawal maj exist during expansion, a case which would at once manifest itself in the transformation, by the distance OP2 being less than OPi . The propositions here developed, of course, also hold when the given pressure curve is a closed curve, therefore also when we are dealing with a reversible cycle. If Fig. 14a represents a certain indicator diagram, we know from earlier discussions that the area inclosed by the curve repre- sents the work produced in this cycle. The transformation (Fig. 146), constructed in the indicated manner, then leads directly to a closed curve, and if we draw the two end-ordinates, PiOi and P261, touching the closed curve at the points ai and 61, the M _ fbl iK Fig. 14. whole hatched area inclosed by the upper part aidibi of the curve and by the end-ordinates will represent the quantity of heat Qi (measured in units of work, that is, Q\:A) which must be sup- plied with this cycle ; the area inclosed by these end-ordinates and the lower part aiCift of the curve (area closely hatched in figure) will give in like manner the quantity of heat Q2 to be with- !'2 TECHNICAL THERMODYNAMICS. drawn, also measured in units of work. From equation (31), namely, . Q1-Q2 it follows that the widely hatched area of Fig. 14& Ukewise rep- resents the work L produced in this cycle; in other words, the transformation of an indicator diagram gives the (indicated) work L, but at the same time it furnishes the means of determining the supplied and withdrawn quantitiesof heat Qi and Q2; and this can be accomplished in a simple way by area determina- tions, which can be undertaken with the planimeter, as is very generally done at the present time in measuring indicator cards for the work L. Moreover, from the position of the two contact points Oi and 61 (Fig. 146), we can, by working backwards, determine the cor- responding points a and b (of the indicator diagram. Fig. 14a), between which the heat supply and the heat withdrawal can occur. Furthermore, if we draw in Fig. 14& two horizontal lines which touch the closed curve at the points di and ei, these points will correspond to the greatest and least values Si and S2 of the function S, and, working backward from this, we can likewise determine the highest and lowest temperatures h and ^2 occurring in this cycle; the points d and e in the indicator diagram, cor- responding to the points di and ei, can also be easily determined. If in the transformation several horizontal or vertical lines occur, then this is a sign that in the cycle given by the indicator diagram there respectively exist isothermal or adiabatic expan- sion or compression for the corresponding distances. The transformation of the C a r n o t cycle (Figs. 15a and 156) is very simple according to this method. Two isothermals, ab and cd, appear in the transformation as the horizontal Unes ai6i and Cidi, and two adiabatics, ac and bd, appear as vertical lines aiCi and bidi. The ordinate Piai=P26i represents Si, and hkewise PiCi=P2di represents the value S2, corresponding to the upper and lower limits of temperature h and tz; if in addition we desig- TRANSFORMATION OF REVERSIBLE PRESSURE CURVES. 73 nate the distance P^Pz by P, there follows, from the given signifi- cance of the individual areas, the supplied quantity of heat Qi and also the withdrawn heat Q2 by means of the formulas | = PSi and '^ = PS2, and the work produced is L = P{S^-S2), just as was explained some time ago. la) Fig. 15. We recognize from this that a large part, in fact the most im- portant part of thermodynamic problems, so far as they are tech- nical, are solved as soon as we know the form of the two functions S=f(v,v)=m and P = {p,v). Future investigations will show that these functions can be de- termined and fixed for the very bodies (gases and vapors) which are technically most important; and therefore the method devel- oped in the foregoing furnishes a way of solving graphically the problems of thermodynamics, and moreover provides us with an important and considerable extension in the utilization of indicator diagrams. But we must not overlook the fact that the propositions here developed are only true under the explicit assumption that the whole cycle is reversible. Our actually built heat engines, from which we can easily take indicator diagrams, by no means describe such reversible cycles. Later investigations will show that certain portions of the indicator curve are not rever- sible, so that the established propositions cannot be applied to 74 TECHNICAL THERMODYNAMICS. them without modification. For the present the given graphical method ^ must be regarded as merely a simple help in presenting to the eye a clear and general picture of propositions concerning the reversible cycle ; in attempting to employ it in technical inves- tigations, for example in the further utilization of indicator diagrams of the steam engine, air engine, etc., for the reasons given, we encounter difficulties which can only be overcome by a further investigation of non-reversible cycles. The aforesaid transformation of a closed curve, and therefore of an "indicator diagram," has recently and commonly been designated as "entropy diagram" and "heat diagram." i § 15. NON=REVERSIBLE OPERATIONS. In all the investigations hitherto made, it has been assumed that, during the change of a body from a certain initial condition (pi, vi) to the final condition (p2, V2), accompanied by a supply or withdrawal of heat, the external pressure is constantly main- tained equal to the body tension, and in such a case the opera- ' The first to follow out the thought of laying off the heat weight, the entropy P as abscissa and the corresponding value of S as ordinate, was Bel pal re (Bull, de I'acad. roy. de Belg., 1872, Vol. 34, p. 509); he represented especially the Carnot cycle, using for P and S the formulas which are still to be estab- lished in this treatise. Prof. Linde went into the question more thoroughly in his treatise, "Theorie der Kalteerzeugungmaschinen, graphische Darstellung der Leistungsverhaltnisse derselben" (Verhandlungen des Vereins zur Befor- derung des Gewerbefleisses, 1875, p. 365). Further use of the method was then made by Prof. Schroter in his work "Ueber die Anwendung von Regenera- torem bei Heissluf tmaschinen " (Zeitschrift des Vereins deutscher Ingenieure, 1883, Vol. 27, page 449); and recently Prof. Herrmann investigated the steam- engine cycle under the, to be sure, tacit assumption that this cycle is reversible in all its parts, which is by no means the case. See "Zur graphischen Behand- lung der mechanischen Warmetheorie" (Zeitschrift des Vereins deutscher In- genieure, 1884, Vol. 28). Moreover, J. W. Gibbs called attention to similar graphical representations ("On Graphical Methods in the Thermodynamics of Fluids," "On Representa- tion by Surfaces of the Thermodynamic Properties of Substances." Trans. Connecticut Academy, Vol. II, 1873). G i b b s ' s consideration of a curved surface possessing as 'coordinates the magnitudes P, U, and v proved very fruitful for general investigation. ("Thermodynamische Studien." Translated by Ost- vald, Leipsic, 1892.) NON-REVERSIBLE OPERATIONS. 75 Hl ' >i ■P^ h 4 k c 5 i ^^ ^^m 1?; Fig. It). tion can be reversed, i.e., it is called a reversible one. If we now drop the assumption of the equality o^ the two pressures the investigations take a wider range, but in so doing a series of propo- sitions, developed above, fail. There is now no difficulty in setting up corresponding equa- tions for the non-reversible operation. If the unit of weight of a body is given by the volume vi, and the pressure pi, say, represented by the point aj in Fig. 16, then for the condition of equilibrium and rest at first assumed we know the corresponding value Ui=F(pi, Vi), the inner work or energy. Now, if the external pressure is suddenly brought to the smaller value pi the body will expand, and, if the variabihty of the external pressure is given by the curve acb, then the area enclosed by this curve (hatched in figure) represents the external work L performed during the expansion from vi to V2. At the instant at which the attainment of the final volume V2 stops all further expansion, and also during the whole course of the transition, there is evidently within the body a disturbance of equilibrium which manifests itself, for example, in gases and vapors by visible stormy motion. But after stoppage of the expansion the condition of equilibrium and of rest will gradually be reestablished, and this will evidently be accompanied by a rise of pressure from 7)2' to p2- The final condition of equilibrium at which the external pressure is identical with the internal pressure P2 will, therefore, be determined by the point 61 (Fig. 16), and the corresponding inner work by the formula U2=F(p2, V2). We must at once understand that we cannot return to the initial condition by the path bca simply and solely by reducing the pressure from p2 to pz', and that therefore such a simple reversal of the operation is not possible here. If during the whole procedure the body receives from the outside a quantity of heat Q, then here also we have the equation Q=AiU2-Ui+L), (51) 76 TECHNICAL THERMODYNAMICS. but then the values Ui and U2 correspond to the condition of equiHbrium ai and bi. If the expansion had been stopped earUer at the volume v (Fig. 16), then during the transition to the con- dition of equilibrium the pressure would have risen from j/ to p, and the inner work U = F('p,v) would correspond to the point Clip, v). We see from this that there arises a second curve OiCi&i beside the curve acb of the external pressure. I will call the former curve the "curve of equilibrium" and the other the "curve of working pressure"; in the non-reversible operation the two curves are distinct, but are identical for the reversible procedure. If the operation is stopped at volume v, let Q' designate the quantity of heat supplied along the path ac, and L' the work rep- resented by the area Viacv; then Q'^AiU-Ui+U) (52) If the operation is stopped at the volume v+dv, Q' +dQ' = A{U -^-dU -Ui+U +dL', and we therefore get by subtraction dQ'.= A(dU+dL'), or, because dL' = p'dv, dQ' = A(dU + p'dv), (52a) or dQ' = A[dF{p,v)+p'dv] (526) Now in the last two equations we have already given the fundamental formulas for the non-reversible operation. The law of heat supply is determined as soon as the course of both pressure curves is given, and conversely one of the pressure curves can be determined when the law of the heat supply and the other pressure curve are given. Generally the question is to determine the t o t a 1 quantity of heat Q' necessary for a finite change; it is not then necessary to first determine the course of the pressure curve for equihbrium, provided we know pi, vi, and NON-REVERSIBLE OPERATIONS. 77 P2, V2, the pressure and volume for the condition of equiUbrium at the beginning and at the end. In this case we have, as in equa- tion (51), the heat quantity 1)2 Q'=A(U2-Ui+fp'dv), where naturally the course of the curve of working pressure must be known. Beside this procedure for the determination of the heat quan- tity Q' there is still another which was first given by Clausius,i and of which I myself have made considerable use ^ in handling technical problems. If the body is brought from the state ai to the state 6i (Fig. 16) by any reversible path aidibi instead of by the non-reversible path acb, and if in so doing the heat Qi is necessary and there is pro- duced an external work Li, represented by the area underneath the (dotted) curve aidibi, then there subsists the equation Qi=A(C/2-C7i+Li). Now if we consider a cycle composed in the first part of the non-reversible path acb, and in the second and returning part of any reversible path bidiai, then Qi is negative in the preceding equation, and its combination with equation (52) gives Q'-Qi=A(L'-Li), or Q'=Qi-A(Li-L'), and from this formula can also be computed the quantity of heat Q' corresponding to the non-reversible part of the cycle, pro- vided we know the work U corresponding to this part. The dif- ference L\—L' = L corresponds to the work expended with this •Clausius, "Abhandlungen."' Anwendung der mechan. Warmetheorie auf die Dampfmaschine. See also PoggendorfE's Annalen, 1856, Vol. 97. ^ " tJber die Wirkung des Drosselns," Civilingenieur, Vol. 21 , 1875. "Calori- metrisohe Untersuchung der Dampfmaschinen," Civilingenieur, Vol. 27 and 28, 1881 and 1882. 78 TECHNICAL THERMODYNAMICS. cycle, and the difference Qi—Q' is the heat thus consumed; a glance at the last equation shows that in a cycle with non- reversible parts the heat generated corresponds to the work ex- pentled, just as in a perfect reversible cycle. Therefore, if in any cycle ■S{Q) represents the algebraic sum of the heat quantities supplied and rejected, and i'(L) the algebraic sum of the various quantities of work (produced and consumed) then I{Q)=AI{L) (54) is a perfectly general equation, and is true even when the cycle contains some non-reversible parts. § 16. TRANSFORMATION OF THE NON=REVERSIBLE OPERATION. If a body passes from the condition of equilibrium ai (Fig. 17a) along a non-reversible path acb to the condition of equihbrium bi, then, according to earher propositions, we can easily transfer rs) Fig. 17. both points to Fig. 176. For point ai lay off the abscissa OPi equal to and as ordinate Piai lay off the value Si-f{pi,vi); correspondingly for the point bi take P2 = (f'{p2,V2) and S2=f(p2,V2). TRANSFORMATION OF THE NON-REVERSIBLE OPERATION. 79 For the non-reversible procedure let Q' again represent the heat imparted and L' the external work. , But we can also pass from oi to 61 in a reversible man- ner along the very same curves. First conceive the body to be brought from ai to a (Fig. 17a) under constant volume vi, then in a reversible manner along the path ach to h, and, finally, under the constant volume V2 belonging to h, brought to the state 61; the three pieces of curve appear transformed in Fig. 176, each point being computed for the mag- nitudes P = (f>ip, v) and S=f(p, v), which are laid off as abscissa and ordinate respectively. Now, according to the proposition developed above, the quantity of heat Qi to be withdrawn along the path aia is represented in units of work by the area Pi'aaiPi (Fig. 176). The areas Pi'ahPi and P^'hhiPi likewise give, in units of work, the quantities Q2 and Q3 (imparted) during the paths ach and 66i. The whole hatched area (Fig. 176) therefore represents the value Q2+Q3-Q1 but a part of the area is positive and the other part is negative; the planimetric determination of the area gives at once the dif- ference, and consequently the value of the preceding expression. Returning to the starting-point of the discussion, and assuming that the body passes along the non-reversible path from ax to 61, it can then be brought back to the initial condition along the reversible path 6i6caai. During the forward motion the quantity of heat Q' is supplied, and during the return motion the quantity of heat Q2+Q3— Qi is withdrawn; but as the same quantity of work U was produced during the forward motion as was expended during the return motion we have, as in equation (54), I{L)=Q, and we therefore get i'(Q)=0, or Q'-(Q2+Q3-Qi)=0, 80 TECHNICAL THERMODYNAMICS. that is, Q' Q2+Q3- Q1 ,„, A= A ' ^^^^ hence the hatched area in Fig. 176 represents (in. units of work) the quantity of heat Q' consumed during this non-reversible change from ai to bi. Returning to the initial condition by any reversible path bidiai (Fig. 17a), the transformation bidiUi (Fig. 176) encloses an area P1O161P2, which represents (in units of work) the quan- tity of heat withdrawn during this return. If we suppose this area (taken as negative) to be combined with the hatched por- tion there will remain the area enclosed by the curved quadri- lateral aib]ba, which represents, therefore, the heat generated during this non-reversible cycle; but as the amount of this area corresponds to the area Uibiba of Fig. 17a, representing the work expended, then from this also follows the proposition of C 1 a u - s i u s which was discussed when equation (54) was established. § 17. HEAT WEIGHT OR ENTROPY IN NON=REVER= SIBLE OPERATIONS. For the non-reversible operation (Fig. 16, p. 75) equation (52) gives the elementary quantity of heat to be supplied : dQ' = A{dU+p'dv), where U = F{p,v) belongs to the point Ci of the corresponding equilibrium-pressure curve. On the other hand, if the transition of the body takes place along the equilibrium-pressure curve aiCibi, that is, in reversible fashion, then at the point Ci there will be needed the quantity of heat dQ for the expansion dv, according to the formula dQ = A(dU + pdv). If we subtract these equations from each other there results dQ' = dQ-A(p-p')dv, HEAT WEIGHT IN NON-REVERSIBLE OPERATIONS. 81 and if we divide botii sides of this expression by AS, where S = fip, v) corresponding to the point Ci of the equilibrium-pressure curve, we get * dQ^_dQ {p-p')dv AS~AS S ^^^^ The first term of the right member is always a complete differen- tial whose integral is designated by 4>{p, v), and called the heat weight. But since the course of the two curves ach and aiCi&i and also the work of the equilibrium-pressure curve are to be regarded as given, then the second term of the right member of the preceding equation is also integrable. If we designate the latter integral for the passage from ai to 61 by iV, that is, if we put N-f^^^, (57) there follows 'dQ J AS J AS ^, 01;, using the designation on page 44, / i;=p.-p.-^. Now if we lead the body back to the initial condition along any path bidiGi (Fig. 16), we have P2 = P\, and therefore Jfs'"-' (5« consequently there follows for the non-reversible cycle: / 1=-". w in whatever manner the return motion may take place, because it is always reversible with com- 82 TECHNICAL THERMODYNAMICS. pression. On the other hand, equation (58) is valid for a cycle in which all parts are reversible. These latter propositions are likewise due to C 1 a u s i u s , only here a different path was pursued in the development, which, at the same time, through equation (57), brought out for N a par- ticular and clear meaning, at least for the non-reversible opera- tions. During the preceding investigations only the simplest and most obvious case was treated for the non-reversible operation and for the non-reversible cycle, but the propositions thus estab- lished also furnish the basis for future investigations of other non-reversible operations as they occur in the discussion of heat engines. The occurrences in a reversible cycle, and particularly in the C a r n o t cycle, were compared On page 55 with the slow and uniform lifting and lowering of a body under the action of gravity. The place of the body's weight G was taken by the heat weight or entropy rdQ J AS' and the place of the heights Hi and H2 of Fig. 10, page 55, was taken by Si and ^2- The analogy can now be easily transferred to the non- reversible operation. If at the upper level Hi the weight G is slowly slid along into position this corresponds to a s u p p 1 y of the work TFi = GHi . If the weight does not sink uniformly but in an accelerated fashion to the lower level H2, then the work there to be drawn away is W2 = G{H2+x), where x is the velocity head corresponding to the velocity attained at the lower level H2, and Gx is the kinetic energy existing there. The produced work L amounts to L = Wi — W2, and from the expressions given for Wi and W2 there follows Hi H2~ H2' HEAT WEIGHT IN NON-REVERSIBLE OPERATIONS. 83 01 using Wi = GHi, Hz Hi \ H2/' When the sinking takes place slowly and uniformly we have x = 0; we have then the reversible operation as in the Car not cycle. (See equation IVa, page 55.) In the non-reversible operation x>0; therefore, according to the preceding formula, the heat weight or the entropy at the lower level is greater than at the upper level, and becomes a maximum for x = Hi—H2 and consequently for W2 = Wi and L = 0; the case corresponds to the free falling of a weight, to the direct passing of heat from a body of higher to one of lower temperature, to heat conduction, and Q2 = Qi, if we express the quantity of work W2 and Wi in units of heat. Since a weight tends toward the lower level we can also say, according to the preceding, "entropy tends toward a maximum"; then, because this analogy, here developed for a particular case, can easily be extended to any cycle, we arrive at the proposition enunciated by Clausius for heat — "The entropy of the world tends toward a maximum," a proposition which, to be sure, seems very trivial, but, according to our analogy, can be compared with the statement that all the water existing on the earth tends toward the level of the sea. In the one case as in the other the proposi- tion thus generally stated possesses no special value as a basis for future investigation. For the reversible cycle our analogy is complete in every re- spect; for the non-reversible operation, on the other hand, as H e 1 m 1 properly emphasizes, it is only true when we regard the kinetic energy at the lower level as lost, but that is always the case in hydraulic motors.^ ' G. Helm, "Die Energetik," Leipsic, 1898, p. 259. ' The comparison of the cycle events in thermodynamics with a raising and lowering of weight was made in the eariier (1866) edition of the present book. Later Ma ch , at least for reversible operations, went more fully into the question (in a short note as early as 1871 and fully in 1892, Wiener-Sitzungsberichte, p. 1589. — "Die Principien der Warmelehre," Leipsic, 1896). 84 TECHNICAL THERMODYNAMICS. CONCLUSION. , Let us once more briefly return to the general consideration of the reversible cycle on page 47. ' There we can conceive the closed curve between the two limiting adiabatics Pi and P2 (Fig. 76) to be traversed by an infinite number of adiabatics lying close together J each elementary strip corresponds to an elementary C a r n 1 cycle hke the one represented for finite changes in Fig. 8, p. 52. But considering one such strip we have for every passage from one adiabatic to the next the constant magnitude ^^ AS' and from this follows, when we measure the quantities of heat dQ in units of work, and designate them by dW, so that dW = 1/A dQ, dW^S.dP (a) If at the upper limit S=Si, and at the lower 8=82, then the quantity of heat or work or energy supplied at the upper limit is dWi=SidP, and the amount withdrawn at the lower limit is dW 2 = 82 • dP; also the work produced by traversing the cycle is dL = iSi-S2)dP (^) In the heat cycle dP is the heat weight, or entropy, and 8 a function of temperature having the property 8i>82 when the temperature ti>t2. In our repeatedly discussed analogy, dP is the weight of the body and H the height of the plane above a given datum plane; in both cases dW is the disposable energy at the level 8. But the two formulas (a) and (JS) have a more general signifi- cance than appears from the foregoing. Thus for some time past attention has been called to the fact that when the electrical cur- rent moves in a conductor the term dL in equation (/?) represents the work necessary to overcome the resistance in the conductor (which work is converted into heat), but, on the other hand, that this differential of L represents the work of an electromotor when CONCLUSION. 85 dP means the quantity of electricity or strength of the current (in amperes), and 81-82 the difference of potential or electro- motive force (in volts) ; the quantity 8 represents nothing but the potential function, and dW its corresponding (electrical) energy, and it is these simple formulas which we employ in electrical engineering for the computation of great electrical plants and power transmissions and utilize in the most extended fashion in judging heat engines, particularly steam engines — and this occurs although the electrical and mechanical engineer, as well as the physicist, has no real conception of the nature of electricity or of heat; it completely suffices to assume that in both cases we are dealing with certain forms of energy. If in equation (a) we consider the magnitude 8 as a height and dP as a weight, and if we multiply the force by the specific weight 7-, then 8 appears as a pressure upon the unit of area, which may be designated by p; on the other hand dP divided by ;- represents the volume which may be designated by dv. Equation (a) then takes the form dW = pdv, (a') a relation which has already been considered by others from this same point of view, and in so doing W has been designated as volume-energy. If at the upper limit p = pi, and at the lower limit p = p2, then the supplied energy is dWi = pidv, the drawn-away energy dW2 ~ P2dv, and the corresponding work produced during the change dL = (pi-p2)dv, (^') which is entirely in accordance with formulas (a) and (/?). The question here indicated was first treated by Helm in the most general manner for the different forms of energy.^ The magni- tude 8 of the above equation (a) he designates by J and calls it intensity, and the magnitude P, which he designates by M, he 'George Helm, "Die Lehre von der Energie," Leipsic, 1887, as well as in the already mentioned treatise "Energetik." With respect to the propositions emphasized in the text see reference to the comparisons instituted in the former book, p. 62, and in the latter book, p. 291. 86 TECHNICAL THERMODYNAMICS. calls the quantity function and later on "Extensity" (" Energetik," p. 266), and then enunciates the proposition "That every form of energy has a tendency to pass from the position in which it exists at the higher intensity to a position of lower intensity. It is said to be released when it can obey this tendency." SECOND SECTION. Theory of Gases. § 18. PRELIMINARY REMARKS. During a comparatively recent period those kinds of air which were only known to exist in the air-hke condition were discussed as "gases," or also as "permanent gases." This was the case whenever they could not be brought into the liquid condition by any means, say by great compression and simul- taneous cooling; it was then naturally assumed that they were not originally produced by "evaporation" from liquid and solid bodies. But long ago and repeatedly the thought was expressed that it was highly probable that the so-called permanent gases did not exist, and it was simply the insufficiency of the means for producing unusually great reductions of temperature which rendered it apparently impossible to bring back such gases to the liquid conditior. This view has quite recently received full con- firmation by the experiments of Raoul Pictet, Cailletet, von Wroblewski, and Olszewski, who succeeded in liquefying even those gases which up to that time had usually been regarded as the principal representatives of the "permanent gases." Finally, great and wide-spread interest was created by the brilliant results of the experiments of L i n d e and D e w a r, particularly by those of L i n d e, on account of the cleverly conceived and constructed machine for liquefying atmospheric air on a large scale in a purely mechanical way. In doing this the first step was taken to utihze laboratory experiments in a large way for industrial purposes. Later in this book L i n d e ' s procedure will be fully explained. Under these circumstances it now seems to be time to desig- nate and treat all kinds of gases without exception as "vapors " ; 89 90 TECHNICAL THERMODYNAMICS. if in the whole arrangement of this treatise, however, we still maintain the difference between gases and vapors, we must justify this procedure in what follows. At present the remark may suffice that we conceive gases to be only a special condition of vapors and regard the adjective "permanent" as inadmissible. If we supply heat under invariable pressure to a solid body, say to a piece of ice, then at a particular tempera- ture this ice will gradually pass into the liquid condition (will melt), till it has all become water; with a further supply of heat, always under the same pressure and therefore accompanied by a corresponding change of volume, evaporation will begin, until finally the water has evaporated into steam, into the gaseous condition. If, while maintaining the aforesaid condition, the heat supply is still continued, then this steam (which we will designate as "superheated steam ") will show properties very much hke 'those which have hitherto been regarded as peculiar to the permanent gases. The conclusion seems obvious that gases are nothing but "highly superheated vapors," and that we may here conceive of a certain limiting condition in which complete agreement is real- ized and the well-known law ofMariotte and Gay-Lussac is followed, which law, moreover, will be discussed later. This limiting condition is designated as the one in which the ideally perfect gas exists, and is distinguished from that other limiting condition in which the steam is still in contact with the liquid (in this case the water) out of which it is formed. In this latter condition, the so-called "state of saturation," the steam exhibits the peculiar property of the temperature remaining con- stant so long as the pressure is constant. In spite of the heat supply and the corresponding expansion of the total mass this behavior continues as long as any liquid is on hand; not until the last element of the liquid has been evaporated will the further increase of temperature begin, and this will con- tinue until the ideal gaseous condition is reached. A similar behavior also accompanies the passage from the soUd to the liquid condition; now returning to the illustration assumed above, as long as ice is present with the water during the PRELIMINARY REMARKS. 91 melting period the temperature will remain constant while the pressure remains constant, provided the corresponding change of volume is possible. Not until the last element of ice has been converted into water will the continued supply of heat raise the temperature and raise it to that point at which evaporation begins ; and from this time on the circumstances shape themselves in the manner first described. The definition of a gas as highly superheated vapor, or as vapor which is far removed from its point of condensation, is by no means as definite as it seems at the first moment. If a vapor shut off from its hquid, and in the condition we call superheated, receives still more heat, then there occurs, at a par- ticular temperature, a new and significant change called "dis- sociation " which can be regarded as a chemical change of state, as decomposition, resolution, or breaking-up of the steam into its constituent parts; and these constituents can in turn be com- pound vapors or the elementary constituents of the decomposed vapor. Thus water-vapor decomposes into its constituents hydro- gen and oxygen; the formation of the so-called "water-gas " is effected by allowing the vapor of water to pass through glowing coal; this gas is already widely used for heating purposes, and its general use for purposes of illumination is probably only a question of time. Dissociation therefore seems to be a procedure analogous to fusion and evaporation. Return again to the example of ice and imagine it to be at 0° C; for the sake of simphcity consider it to be at atmospheric pressure during the various parts of its changes of state while heat is suppHed, then melting will occur at 0° C, and this temperature will be maintained till the fusion is completed; now the temperature will begin to rise up to 100° C, until the conamencement of the evaporation. This temperature in turn is maintained constant until the last element of water is evaporated, and now will begin the further rise of temperature until the commencement of dissociation. It has not yet been decided whether or not from here on to the end of dissociation the temperature remains constant, so long as the assumed atmospheric pressure is constant; fusion and evaporation are purely physical 92 TECHNICAL THERMODYNAMICS. procedures, and dissociation is at the same time a chemical pro- cess, and therefore it is quite possible that, according to the present prevailing views, the whole course of the dissociation takes place under increasing temperatures; doubtless the law of change of the temperature will be different from what it was before the beginning of the dissociation, and different from what it is after the end of the dissociation, even though we assume equal quantities of heat to be supplied in the cases compared. In hke manner the changes of volume during before and after dissociation are subject to laws which differ from those which obtain during, before and after either evaporation or fusion. There is still a complete lack of suitable experiments with respect to the process of dissociation; but this much may be expected, that in general during the dissociation (other things being equal) the increase of volume under constant pressure will take place more rapidly than before the beginning of the dissocia- tion. What has hitherto been said for this case of water in its various states of aggregation also holds for a whole series of other bodies; but bodies do occur in which certain intermediate stages of the changes of state considered, disappear. Thus some bodies pass directly from the solid to the vapor condition without any pre- liminary fusion, while certain other bodies do not even possess the intermediate stages of evaporation, but experience immediate dissociation. Concerning the temperature of dissociation, or, to speak more exactly, concerning the temperature at the beginning of dissocia- tion, there are at present very few observations; but we may say that it seems probable that certain gases dissociate, in whole or in part, at a comparatively low temperature, say at the atmos- pheric temperature; in the latter case they may be considered as consisting of a mixture of gases rather than of a gas of uniform chemical constitution throughout; but there are reasons for thinking that for those vapors which we will hereafter specially treat as gases a partial dissociation does not take place during the, to be sure, narrow range of temperatures assumed. EQUATION OF CONDITION FOR GASES. 93 On the basis of the foregoing remarks we will now briefly designate as g a s a superheated steam wl^ich before the beginning of dissociation obeys the law ofMariotte and G a y-L u s s a c, which law will be more fully discussed later on. At the same time the last modifying clause indicates that there may be vapors which at no point of their changes of state can be treated as gases, in the sense defined, unless it is after completing the dissociation, in which case vapor of the original chemical combination will not be before us. The proof that gases do exist which satisfy the given defini- tion is of the highest importance for thermodynamics; only under the assumption of the existence of such gases can we deduce the form and meaning of that temperature function which was designated above in § 12, p. 61, as the C a r n o t function. It was proved that this function was numerically the same for all bodies without any exception whatever. Nevertheless, no one has yet succeeded in deriving the form of this function from the physical behavior of solids, liquids, or vapors in general. Now in fact we do know some gases and mixtures of gases, to which latter atmospheric air belongs, which correspond to the given definition, and therefore it is justifiable in Thermo- dynamics to distinguish between gases and vapors, all the more as the equations of the mechanical theory of heat applied to gases have simpler and clearer forms than with vapors. § 19. EQUATION OF CONDITION FOR GASES. The equation of condition for a gas (§ 12, p. 62) expresses the relation between the specific volume, the pressure p, and the temperature t, and is given in Physics as the mathematical expression of the combined laws of Mariotte and G a y- L u s s a c. According to Mariotte or Boyle the pressure of a gas is inversely proportional to the volume when the gas expands or contracts under constant temperature, the product pv remaining constant under these conditions. 94 TECHNICAL THERMODYNAMICS But if the initial condition is given by pi and vi, then, accord- ing to M a r i o 1 1 e, we get the relation pv = pivi, (1) which represents an equilateral hyperbola when v is laid off as abscissa and p as ordinate; and as the temperature is here assumed to be constant, the foregoing expression at the same time represents the "isothermal curve " for gases (§ 6, p. 39) passing through the point (pi, vi). The other law, that of Gay-Lussac, says that the incre- ment of volume of a gas is proportional to the increment of tem- perature when the expansion of a gas takes place under constant pressure while the gas is being heated. Let a represent the expansion of a unit of volume of a gas when it is heated 1° C. [1° F.] under the assumption made, thenai [a(i— 32°)] is the increment of volume for heating from 0° C. [32° F.] to t°; let us designate by Vo the volume of the gas at 0° C. [32° F.] and its volume at t° by v, thus getting v==Voil+at) [v = vo[l+ ait -32°)]], and likewise for the volume Vi at temperature h Vi=Vo{l+ati) \vi = vo{l+a(ti-32°))^. Dividing one of these by the other we get the expression for Gay-Lussac's Law : V 1+at Vi 1 +ati rv l+a{t-32°) -l \.Vi'~l+aiti-32°)}' and this will be true for any pressure whatever provided it is constant. EQUATION OF CONDITION FOR GASES. 95 The value a is called the coefficient of expansion of the gas; if we divide the numerator and the denominator of the right-hand member by a and call a the reciprocal of a, then we get V a+t vi a+ti (2) wi~a + («i-32°)J" If the initial condition of the unit of weight of a gas is given by the magnitudes Vi-pih (Fig. 18), and if the gas expands under constant pres- sure pi and heat supply from Vi to Vm, then we have, according to equation (2), the relation a + t Vl vi~a a+ti a + (t- FiG. 18. •32°) + ik-S2°), Now if we allow the gas to continue its expansion under con- stant temperature t from Vm to v (which likewise requires suitable heat supply), then the pressure will fall from pi to p, and accord- ing to equation (1) we have the relation pv = piVm. If we eUminate from the last two expressions the intermediate volume Vm, we get pv PiVi a+t a+ti pv PlVi (3) .a + (<-32°) a + («i-32°). If a were experimentally determined and if for a particular gas the necessary temperature ti, the pressure pi, and the 96 TECHNICAL THERMODYNAMICS. volume Vi had been observed, we could compute the right mem- ber of the foregoing equation; if we designate the value of this member by B, then the equation can be written as follows : pv = B{a + t) (3a) [pv = B[a+{t-S2°)]], and this is the equation of condition for gases, which at the same time expresses in the simplest form the laws of Mariotte and Gay-Lussac. This form of the equa- tion of condition was used by Clapeyron. Now as regards the constants a and B of the equation of con- dition for gases we will first discuss the constant a. If we compute a from equation (3), we get ^_ pivit-pvti ^^^ pv-pivi PiVit — pvti pv-piVi +32 The experiments on this point have been conducted in two- fold fashion; either the pressure p was maintained constant and the volume observed at the temperatures t and ti, or the volume was kept constant and the pressures p and pi were determined for the corresponding temperatures. If we designate by ap the value of a derived from the experi- ments under constant pressure, and by a^ the value of a for constant volume (the subscript p or v indicating that in the experiments in question the designated magnitude is kept constant), then there follows from equation (4), respectively for p = pi and v = vi, .... (5) 4 321 ap = Vit- V- -Vti and a-v'- pit- P- -Vh -Pi ap- Vit- V- -Vi + 32 and a„ = pit- V- -ph -Pi EQUATION OF CONDITION FOR GASES. 97 If in both cases we imagine an infinitesimal change of tem- perature dt as occurring, then we get, as may easily be seen, fflp=J^-< and a, = pg--< (6) ap = v^^-it -32°) and a,=p^-(t- 32°)^. Under the supposition that the equation of condition is ex- actly correct both experimental methods should lead to one and the same value a for Up and a^, which is not the case; we must therefore conclude that the above equation of condition can only be regarded as an approximate form. Now there are, to be sure, gases in which the deviations are so slight that, without any hesi- tation whatever, one can regard equation (3) as rigidly exact; and it is just these gases which we will hereafter especially desig- nate as gases, in accordance with the remarks already made in § 18, p. 93. To these gases belong atmospheric air, which we will always put first on account of its great technical importance, although it is no simple gas but a mixture of two gases, Oxygen and Nitrogen. The mechanical mixture of such simple gases as Hydrogen, Oxygen, and Nitrogen, and also the chemically compound gases, nitric oxide and carbonic oxide, will be regarded as obejdng the law on account of their insignificant deviations from it. According to the experiments of R e g n a u 1 1 ,i for pressures deviating but httle from atmospheric pressure and with tem- perature hmits to 100° C. there were obtained, after a corre- 'V. Regnault," Relation des experiences pour determiner les prinoipales lois et les donn^es numeriques qui entrant dans le calcul des machines k vapeur.'' Paris. This classic work in three volumes, of which the first volume appeared in 1847, the second in 1862, and the third in 1870, contains the results of the num- berless wonderful experiments of Regnault. For the development of ther- modynamics, and particularly in its technical direction, the work of Regnault is of inestimable value — Henry Victor Regnau It, bom July 21, 1810, in Aix-la-Chapelle, died January 19, 1878, in Auteuil. This work of Regnault will, in the further course of this book, be so often referred to that we will designate it simply by Rel., the volume by Roman numerals, and the page by Arabic numerals. 98 TECHNICAL THERMODYNAMICS. spending recomputation of Regnault's^ data, for the mag- nitudes ttp and a» the following values : French Units. English Units. Atmospheric Air Hydrogen Nitrogen Carhonic Oxide . Carbonic Acid. . . Nitrous Oxide. . . 272.48 273. 1& 272 '52 269.54 268.89 272.85 272.70 272.63 272.70 271.15 272.03 490.46 491.67 490 ^54 485.17 484.00 491.13 490.86 490.73 490.86 488.07 489.65 (For Oxygen we would have to assume ai= 272.18 according to Jolly's data.) K = 489.92.] The numerical values show that with the exception of hydrogen the values of a^ are greater than the values of ap for all gases, but that for the earher tabulated gases the difference is insignificant. Only for carbonic acid and nitrous oxide are the differences consid- erable; but these are gases which are far nearer to their points of condensation, and can therefore not be designated as gases. In another experimental series Regnault^ observed the diminu- tion of volume for the following gases at different but constant pressures; the pressures measured in milli- meters of mercury gave the following values forop: Atmospheric Air Hydrogen Carbonic Acid. . . Pressure. Value of ap. [Value of ap.] r 760 272.431 490.37 2525 270.68 [ 487.22 2620 270.53 J 486.95 760 273 . 13 1 491.63 2545 273.10 ■ 491.58 760 269.55 485.19 1 2520 260.04/ 468.07 Here we again see the anomalous behavior of carbonic acid, and also see that the value of ap seems to be smaller at the higher pressures. Other investigations of Regnault related to atmospheric air and carbonic acid; the volume was kept constant ' Rel. I, 91. ' Rel. I, 115, 116, 117. EQUATION OF CONDITION FOR GASES. 99 and different initial pressures at 0° C. were employed, and then a corresponding final pressure was observed at 100° C. From the second of the equations (5) and from Ifegnault's data the values of a„ in the following tabulations have been derived : ATMOSPHERIC AIR. Pressure. At 0° C. At 100 C. French Value of English Value of [At 32° F.] [At 212° F.] «»• "v nun. mm. 109.72 149.31 274.11 493.40 174.36 237.17 273.87 492.97 266.06 395.07 273.65 492.57 374.67 510.35 273.32 491.98 375.23 510.97 273.43 492.17 760.00 — 272.85 491.13 1678.40 2286.09 272.03 489.65 1692.63 2306.23 271.96 489.53 2144.18 2924.04 271.05 487.89 3655.56 4992.09 269.61 485.30 CARBONIC ACID.= Pressure At 0° C. At 100° C. French Value of English Value of [At 32° F.] [At 212° F.] "v. "v. 785.47 1034.54 271.33 488.39 901.09 1230.37 270.68 487.20 1742.73 2387.72 266.50 479.70 3589.07 4759.03 259.08 466.34 We see from this experimental series that with the same limit- ing temperatures the- value a, diminishes as the initial pressure increases, and a similar result was evident in the foregoing tabula- tion of the values Up. The greater the pressure for the same tem- perature, the denser is the gas and the closer to one another will be the gas molecules. Now as the molecules act upon each other with certain forces, which doubtless grow very rapidly with diminishing distances, the thought lies nigh to ascribe to this circumstance the deviations exhibited by the gases from the be- havior assumed when estabUshing the above equation of condi- tion {3a). We further conclude that these deviations become the smaller the more rarefied the gas, and that the condition is finally attained in which the molecules have passed beyond the mutual 'Rel. I, 110. 'Rel. I, 112. 100 TECHNICAL THERMODYNAMICS. attraction of their spheres of action, or, to speak more exactly, in which these forces may be regarded as infinitesimal during the future changes of state of the gas; in this sense, we speak of an ideal gas, and for such a gas the differences, shown by experi- ments for the magnitudes ap and a^, disappear, and both values should approach for all gases that particular determinate value of a which has been introduced into the equation of condition. Disregarding for the present the values given for carbonic acid and nitrous oxide, and now directing our attention to the first two of the above given four tabulations of Regnault's experimental results, we see that for atmospheric air, hydrogen, and carbonic oxide the differences between Cp and a„ are very small; if furthermore we consider that hydrogen has, other things being equal, the slightest density among all the gases, and that for it the two values Cp and a^ differ least from one another and approximately show the value 273, then it seems thoroughly jus- tifiable that we should assume this value, generally, as the hmiting value. The third of the above tabulations shows that Regnault found for atmospheric air, under the least pressure occurring in his experiments, that the value of a^ turned out to be somewhat greater than 274, which has caused some authors to assume this value as the limiting value. But the circumstance that the first limiting value is generally accepted, and furthermore that the assumption of the one or the other limiting value causes only an insignificant difference in the computations (such slight differences can always be neglected in technical investigations, and can be disregarded in most cases in the more refined physical discussions), is the reason why, in all future investigations, the limiting value a = 273 [a = 491 .4] will be retained; this was done by C 1 a u s i u s in all his papers. If we substitute this hmiting value in the equation of condi- tion (3a), we get 'pv = B{21?, + t) [pi; = 5(459.4 + 0], EQUATION OF CONDITION FOR GASES. 101 and from this it is evident that a new zero-point of the thermo- metric scale has been obtained which iq a certain sense is pre- scribed by nature herself. As the freezing-point of water at atmospheric pressure serves as a starting-point for measuring the temperature t, according to Celsius, so that point of the thermometric scale lying 273° C. [491.4° F.] under the freezing- point, can in Hke manner serve as the starting-point for the meas- urement of temperature; we call this point the absolute zero, and the corresponding temperature a+t = 273+t [a+t =459.4 +i], the absolute temperature of the body in question. The equation of condition (3a) assumes that the volume v refers to the unit of weight of the gas in question; but if in the space V, G kilograms (G pounds) are enclosed, then V = Gv, and therefore if both members of. equation (3a) are multipUed by G we get Vp=GB{a + t) (7) This equation is the handiest one for computing the weight of a gas enclosed in a given space of V cbm. (F cu. ft.) at pressure p and temperature t, provided we know the constant B for the gas in question. If the capacity of the space V is just one cbm. (one cu. ft.), then G is the weight of a cubic unit of gas at the existing pressure p and its corresponding temperature t, and this weight we will, in what follows, always designate by ;-, and call it the specific weight of a gas. Equation (7) gives, therefore, j = B{a + t), (8) and from this follows, by combination with (3a), the following relation between the specific volume v and the specific weight y, vr = '^ (9) 102 TECHNICAL THERMODYNAMICS. It deserves to be emphasized that a special meaning can be said to underlie the two forms of the equations of condition (3a) and (8). The product pv represents the equation of work which is necessary to fill or empty the space v; this work is accordingly proportional to the absolute temperature a+t. On the other hand, in equation (8) p : ^ represents the height of a column of gas whose specific weight is ever3rwhere y and which by means of its weight exerts upon every unit of area of its base the pressure p. If we assume that equation (8) holds good for a particular gas, then for another gas, of a similar pressure p and the same tem- perature t, the specific weight ;- and the constant B will assume another value; if we designate these values respectively by jq and Bo, then we also have ^ = Bo(a+0, To and if we divide this equation by equation (8) there follows ^4" ao, This ratio, which in the future we will designate by £, is the weight of one gas relatively to the other. Hereafter the relative weight of gases will always be referred to hydrogen. It follows, therefore, from equation (10) that Bq = Bs, (11) from which it seems that the value Be is the same for all gases and that the equation of condition for gases, equation (3a), can also be written in the form Tpvs^Boia+t) (3c) For dry atmospheric air Regnault^ found the specific weight y, i.e., the weight of a cubic meter (cu. ft.) of air ' Rel. I, 157. EQUATION OF CONDITION FOR GASES. 103 measured in kilograms (pounds) at the temperature of 0° C, and at the average barometric pressure of J60 mm. of mercury, to be r = 1.293187 kg. [r= 0.0807288 lb.] in the latitude of Paris. Since, according to R e g n a u 1 t,i the weight of 1 cbm. of mercury amounts to 13596 kg. [1 cu. ft. weighs 848.747 lb.], then the barometric pressure of 760 mm. corre- sponds to a specific pressure of j3 = 0.760x13596 = 10332.96 kg. [2116.31 lb. per sq. ft.], or, accurately enough, p = 10333 kg. to the square meter, which value will be hereafter designated as the "mean atmospheric pressur e." Therefore with air ^ = 7990.34 r [^=26215.031, and from equation (8), for < = [32°] and a = 273 [491.4], we get 5 = 29.269 [B = 53.349]. On the basis of other experiments by Regnault^ the table on page 104 has been computed for those gases which more completely obey the given equation of condition. This table gives rise to some remarks. The product Bs, there- fore, of the values occurring in the last two columns, is the same for all the gases enumerated, and is 422.595 [770.259]; and more- over for the same pressure and volume the product By is also the same for all gases and amounts to 5;- = 37.850 [6/- = 4.3067], which value results directly from equation (8) for p = 10333 [p = 2116.31] and «=0 [< = 32°]. It is remarkable that the value of the first-mentioned product is almost equal to that of the mechanical equivalent of heat (424) [772.83]. That these two values are perfectly equal cannot be proved with our present knowledge of the internal constitution of ' Rel. I, 142, 144, 145. 104 TECHNICAL THERMODYNAMICS. Specific Weight. r Weight relatively to Hydrogen. Value of B. Atmospheric Air, Hydrogen Oxygen Nitrogen Nitric Oxide. . . . Carbonic Oxide. Atmospheric Air, Hydrogen Oxygen Nitrogen Nitric Oxide. . . . Carbonic Oxide. . kg. 1.293187 0.089566= fr^) 1 . 429786 1.256163 1.34284 1 . 25090 14.4384 1 15.9635 14.0250 14.9928 13.9662 29.269 422.691 = (BJ 26.472 30.131 28 . 186 30.258 Specific Weight. r Weight relatively to Hydrogen. Value of B. lb. 0.0807288 0.0055913= (ro) 0.0892562 0.0784175 0.083828 0.078089 14.4384 1. 15.9635 14.0250 14.9928 13.9662 53.349 770.259= (fio) 48.251 54.920 51.375 55.151 bodies ; nevertheless we can make use of this accidental coincidence, if we may so describe it, in certain approximate computations. As the thermal equivalent of the unit of work (A) is reciprocal of the value 424 [772.83], we would have for hydrogen ABo = l, and according to equation (11) for every other gas ABe = l; an equation of condition for gases could then be written, following equation (3a), p. 96, in the form Apv£ = a+t, (12) so that for all gases taken at the same temperature and at the same pressure the product ve would be the same, a result more- over which follows from the combination of equations (10) and (11); the product corresponds to specific volume of hydrogen, which becomes for atmospheric pressure 0° [32°] temperature Vo = - = 11.1649 cbm. To [.0=^ = 178.85]. Of special importance, however, is the following remark. If we consider the numerical values of the second column in the EQUATION OF CONDITION FOR GASES. 105 preceding table, leaving out atmospheric air, which is a mixture of gases, we see that these are almost exactly equal to the half values of the so-called molecular weights of the gases in question. If we designate for the different chemical elements Ei, E2, etc., the corresponding atomic weights by ei, 62, 63, etc., and if we assume that the molecule of any chemical combination con- sists of rii atoms of the first element Ei, of n2 atoms of the second element E2, etc., then the molecular weight of this body is m=niei+n2e2+ 11363 + . . . , or simply m = I{ne) (13) For the elements considered in the following investigations the atomic weights are as follows : Hydrogen (H). . . e= 1 Carbon (C) e = 12 Oxygen (0) 16 Chlorine (CI) 35 . 5 Nitrogen (N) 14 Sulphur (S) 32 We therefore have for Steam (H2O) m=2x 1 + 1X16 = 18 Ammonia Vapor (NH3) m = lXl4+3X 1 = 17 Sulphuric Acid (SO2) m = lX32+2xl6 = 64 Carbonic Acid (CO2) m = 1x12 + 2x16 =44 In the last tabulation; p. 104, all the gases enumerated are diatomic; if we designate the molecular weight of hydrogen by mo (mo = 2), then the gases there designated by £ appear to coincide almost exactly with the values m : mo of the following tabulation; it was Gay-Lussac who first called attention to this coincidence. The agreement also exists with a 1 1 other gases. Now if we utilize the value £=m:mo in equation (11), the constant B of the equation of condition of a gas becomes 5omo 845.182 B= = (14) mm ^ ' r^_^omo_ 1540.52 -1 L m m J' 106 TECHNICAL THERMODYNAMICS. provided we substitute the hydrogen values 5o= 422.591 L770.259]andTOo = 2. In place of the last tabulation we have therefore : Composi- tion. Molecular Weight. B French Value English Value „_Bomo Hydrogen Oxygen Nitrogen Nitric Oxide. . . Carbonic Oxide. Nb CO 2 = TOo 32 28 30 28 422.591 26.412 30.18.5 28.173 30.185 770.26 48.14 55.02 51.35 55.02 These values of B are in satisfactory agreement with the values given^'on page 104 and directly determined from experiments. The utihzation of relation (14) in the equation of condition gives Apmv=ABomo(a+t), (15) in which ^50^0 = 1-9933; this magnitude would be 2 if we were to assume Bo as identical with 424 [772.83], as recently suggested. But at any rate the foregoing equation shows that with the same pressure and the same temperature the product mv is the same for all gases. If we replace the specific volume v by the specific weight ;-, then from what has preceded we have for two different gases the relation mi m2 TT" "^) Now if we imagine that there exist in a space V at one time rii molecules of a gas of the molecular weight mi, and at another time in the same space ^2 molecules of the molecular weight m2, the ratio of the weights Gi and G2 of the two quantities of gas is Gi riimi G2 n2in2 On the other hand we also have G2 Vr, u GAS MIXTURES, 107 The combination of the two equations gives riimi _ n2m2 • and, comparing this with equation (16) we draw the conclusion that Wi=n2 must be true. It follows, therefore, that equal volumes of different gases at the same pres- sure and the same temperature possess the same number o f molecules. This is Avogadro's law, which is one of the most im- portant foundation stones of the Chemistry of to-day. § 20. GAS MIXTURES. The investigations concerning the behavior of mixtures, me- chanical minghng of various gases, are of importance for certain technical problems, and their closer examination is therefore justified. If in a space V there exist two gases, one weighing Gi kg. [lb.] and the other G2 kg. [lb.], and if the temperature of the two gases and that of their mixture is equal to t, and if moreover the constant B of the equation of condition is equal to Bi and to B2 of the constituent gases, and if finally the pressure of the one gas is pi and of the other pa, then we have, according to equation (7), the two relations Fpi=Gi5i(a+0 and Vp2=G2B2{a+t), . . (17) because, according to a well-known law, the one gas ejcpands in the space just as if the second were not present; adding these two equations we get Vp = iGiBi+G2B2Ka+t), (18) where p equals the sum of the pressures pi and p2, which sum can be a matter of direct observation. If V is the specific volume of the mixture, and G=Gi+G2 its 108 TECHNICAL THERMODYNAMICS. total weight, then we have also V = Gv, and from the foregoing formulas there follows for the mixture VV-[ g^^g^ ){a+t). If we designate the coefficient of (a + t) in the right member by Bm, then, just as for a simple gas, we have for the mixture pv = B^{a + t), for which the constant Bm is determined by BiGi +G2B2 We at once recognize that this law is true for more than two gases, and that in such a case we get for the mixture the general relation R ^(^^) no ^ KG) ^ -^ The pressures of the separate gases in the mixture are easily ob- tained if we divide equations (17) by equation (18); we then get respectively Pi GiBi p2 G2B2 p G1B1+G2B2 V G1B1+G2B2' • • ^^ which proposition can easily be transferred to the case in which more than two gases are present. If the constant Bm for a mixture of two gases has been deter- mined by observation, then the proportions of the mixture can easily be determined from equation (19). If the weight of the two gases isG=Gi +G2, we have G~ G' and we can directly determine from equation (19) 6ri Bm — B2 , G2 Bi — Bm G~Bi-B2 G~Bi-B2' • • • ^''^^ GAS MIXTURES. 109 If we utilize these two formulas in equation (20), we get the pres- sure pi of the one gas and p2 of the other gas from the equations Vx JBrn~B2)B, _ G,B, ] f {B^-B-2)Bm UB^ and i (22) P2 jBl — Bm)B2 G2B2 I p~{Bx~B2)Bra~GB„ J The most important case of gas mixture is presented by atmos- pheric air, which is composed of nitrogen and oxygen^; for air, therefore, the foregoing values can easily be computed. According to the tabulation on page 104 we have for nitrogen £1 = 30.131 [54.920], for oxygen 52 = 26.472 [48.251], and for atmospheric air 5,^=29.269 [53.349]. From equation (21) there- fore follows, for air, -^=0.7644 and ^^ = 0.2356. Accordingly atmospheric air is composed by weight of 76.44% nitrogen and 23.56% oxygen. Equations (22) on the other hand give ^=0.7869 and ^ = 0.2131. V V At the pressure p of atmospheric air, corresponding, for ex- ample, to an observed reading of the barometer, we have 78.69% 'Remark. — According to the recent investigations by Lord Rayleigh and WilliamRamsay atmospheric air does not consist solely of a mixture of oxygen and nitrogen, but also of other hitherto unknown gases, and each of them monatomic, which, as new chemical elements, have received the names Helium, Neon, Argon, Krypton, and Xenon; of these the first mentioned has been already subjected to a closer chemical and physical examination. This highly interesting discovery is one of the most important results which the production of liquid air has rendered possible. In the technical investigation of atmospheric air, only considered in the present treatise, we may neglect these tiny admixtures of other gases. 110 TECHNICAL THERMODYNAMICS. of the pressure due to nitrogen and 21.31% due to oxygen, pro- vided the proportions of the mixture of atmospheric air are invari- able and correspond exactly to the values just computed. Ac- cording to recent investigations the proportion of our atmospheric mixture is subject to variations, to be sure between narrow limits; consequently the just computed pressure ratios vary somewhat; but concerning this variation no conclusion can be drawn from the observation of the total pressure p, that is, of the variable reading of the barometer. Chemists usually compare gases by their volumes at the same pressure and the same temperature, and not by weights. In the present case, Gi kg. [lbs.] of the one gas at the pressure p and the temperature t has the volume Vi, and the G2 kg. [lbs.] of the other gas would occupy the volume V2 at this pressure p and temperature t, and for these gases the relations would subsist : Fip = (?iSi(a + and V2V = G2B2{a+t); and from the addition of the two equations, and because Fi + F2 = ^ represents the total volume, we would get yp = ((?i5i+(?252)(a+0, and according to equation (19) V-p^GBrrcia + t). If we use this as a divisor for the two preceding equations, we get the proportions of the mixture by volume: Fi_G^ . V2_G2B2 V~UB„, ^°^ V~GBZ ^^^^ But these expressions are identical with the pressure propor- tions of equation (22). From these computations it follows, for example, that atmos- pheric air consists by volume of 78.69% nitrogen and 21.31% oxygen. If we can conceive the mixture to consist of any number of GAS MIXTURES. Ill gases and designate by G the total weight 1(G), we get, according to equation (19a), GB,n = ^iGBy Moreover, if e^ designates the relative weight of t h e m i x - t u r e with respect to hydrogen, and if hkewise £i, S2, £3 • • • are the relative weights of the individual gases, then, according to equation (11), and by analogy ■Dm— 7 , £771 where 5o = 422.591 [770.259]; the foregoing formula gives, there- fore, G Gi G2 G3 — ~l~ ~r r • • • J «m Si ^2 ^3 from which ^-^ (24) ■-(!)' From equation (3c), p. 102, namely, pVmSm=Boia + t), we then get the specific volume Vm of the mixture for the given pressure p and the known temperature t. Of especial importance in technology are the gas mixtures which explode by ignition, either by means of gas flames or by the electric spark, and thus enter into chemical combination. Here we must distinguish whether the gas mixture, say of two gases, has been so chosen that there results from the two gases one single gas of definite chemical constitution, or whether there is an excess of one gas on hand and consequently the ignition results in a new gas mixture made up of the excess of the one gas and the chemical combination of the remainder with the other gas. 112 TECHNICAL THERMODYNAMICS. As we must enter more fully into this question later on in the trfeatise, we will only briefly elucidate the first case. Let us assume chemical combination of the elements . Ei E2E3 . . . of which the atomic weights are . . . Si 62 es . . . and furthermore let ni n2 na . . . represent the number of atoms existing respectively in the ele- mentary gases before the chemical combination is effected, then the molecular weights of the individual gases will be mj =niei, m2=n2e2, etc., and these values will enter equation (24) in place of the weights Gi, G2, etc. We therefore have I{G)=niei+n2e2 + . . . = I{ne) and 2:(?)-"f fG\ n-iei n2e2 £2 Now for the elements the relative weights s (with respect to hydrogen) are identical with the atomic weights, consequently the last formula gives simply s(!)-«. and hence we have from equation (24) the relative weight s^ of the mixture line) ''""i'(n) • But I{ne) signifies the molecular weight m of the chemical combination in question, and lin) is the number n of the atoms of which this consists; therefore the relative weight of the mixture is simply m ^'n=-, (25) GAS MIXTURES. 113 while the relative weight s of the corresponding chemical combination is in £ =— , (26 mo where mo = 2 is the molecular weight of hydrogen. If Vm is the specific volume of the mixture and v that of the chemical com- bination, then, according to equation (8c), p. 102, we have for the same pressure and temperature the relation and from this follows with the utilization of the preceding values, putting mo = 2, ^^^^^ (27) Vm e n From this we can draw the conclusion as to the change of volume when the mixture is converted by ignition, or by simply heating, into a chemical combination, provided that at the end the initial pressure and the initial temperature are again restored. Moreover, constant B of the equation of condition can be found from equation (11) : for the mixture Bm = n — , m for the chemical combination 5 = 2 — ; m of course it is here assumed that the chemical combination in a state of vapor obeys the equation of condition of gases. In dia- tomic gases, for which n = 2, no change of volume occurs, as equa- tion (27) shows. When a union of nitrogen and oxygen takes place in the proportions corresponding to the chemical compound nitric oxide (NO) we have the value of B the same for the mechani- cal mixture as for the chemical combination. By weight, moreover, the mixture is composed as follows : Gi ni G2 n2 G3 uz G=n''' G^m''' G=m'' ^^8) By volume, on the other hand, it is composed as per equa- tions (23), when we consider that, according to equation (11), 114 TECHNICAL THERMODYNAMICS. Bo = Bi£i, Bo = B2£2, etc., also Bo = Bmem, and we have besides ei = «i, 62 = £2, etc.; then Vi_rh_ YI-.V2. Yj.-V:3 (o(X\ V~m'"" V'tn'""' V'm'"" ■ * " ^^' These last expressions at the same time indicate, as compari- son of equation (23) with equations (22) shows, in what proportion the individual gases contribute in the way of pressure to the total pressure p of the mixture; we therefore also have Pl__rh . P2_W2 . PsTh .„„. — ^m ) ^m 1 ^wi" • • • loUl pm'pm'pm ' Thus we have, say, for a mixture of hydrogen and oxygen in the proportions necessary to form water, that is, for the so-called detonating gas, the relative weight of this gas £m = 6, according to equation (25), and that of the vapor of water (considered to be highly superheated) after explosion to be £ = 9, because, according to the chemical formula of water, H2O, the number of atoms n = 3 and the molecular weight of the water is m = niei +71262 = 2x1 + 1X16 = 18. The constant B of the equation of condition for detonating gas is Bo 422.591 ^^,^„ r770.259 Bm = - 6 6 .70.432 [IZI^^ 128.377]; and that for the vapor of water, of course considered in the state in which it can be regarded as a gas, is 5=f =46.954 [^-^^ = 85.584]. The ratio of the volumes of this steam and of the detonating gas is, according to equation (27), V__2 F."3' GAS MIXTURES. 115 therefore a marked reduction of volume is connected witii the explosion, provided the product is brought back to the same pressure and the same temperature. The mixture (detonating gas) and also this vapor of water are, according to equation (28), composed by weight as follows: Gi n, 2 1 G2 n2 1 ^,- 8. On the other hand the composition of the mixture (detonating gas) by volume, according to equation (29), is F^m'^^lS^ 3 , F2 n2 1 . 1 and T=m^"'=l8><^ = 3~- The same values exist for the pressure proportions. If p is the pressure of the mixture, then the pressure of the hydrogen is pi = §p, and that of the oxygen is p2 = Jp. In the same way we can find for ammonia, NH3, the number of atoms to be n=4, and the molecular weight m = 17. This com- pound is of technical importance because of its application in refrigerating machines. For the mixture of nitrogen and hydrogen in the corresponding proportions we have Sm = 17/4, and for the chemical compound in the form of ammonia we have £ = 17/2; the volumetric ratio between this steam and the mixture is V !_ The constant of the equation of condition for the mixture is Bm = 99.432 [181.24], and for the vapor of ammonia (highly super- heated) jB= 49.716 [90.618]. By weight the mixture consists of 14/17 nitrogen and 3/17 hydrogen; on the other hand, by volume it consists of 1/4 nitrogen and 3/4 hydrogen; the pressures of the two elementary gases in the mixture Ukewise possess the same ratio to each other. 116 TECHNICAL THERMODYNAMICS. It deserves to be emphasized that the just calculated values of B for the vapors of water and of ammonia have n o techni- cal value. These vapors, considered as subject to the equation of condition for gases, show too great deviations within the hmits for which they are employed. In the investigations concerning the behavior of vapors we will return to this question. §21. SPECIFIC HEAT OF GASES. The quantity of heat which is necessary to raise one k i 1 o- g r a m [1 lb.] of w a t e r from 0° C. [32° F.] to 1° C. [33° F.] is the unit of heat and constitutes the unit for heat measure- ment (it is known as the calorie in the French system of units, and as the British thermal unit in the English system); the quantity of heat dQ needed to heat water from 0° C. [32° F.] an amount dt is therefore dQ=dt. In order to heat a unit of weight of any other body from any initial temperature t an amount dt (and here we will consider solid or fluid bodies), the quantity of heat needed will be different and is simply written as dQ = cdt, (31) here assuming that for the body in question the factor c is deter- mined by experiment. The value c is. called the specific h e a t of this body, and therefore represents the ratio ^ dt' namely, the ratio of the quantity of heat which the body absorbs for a change of temperature dt compared with that needed by an equal mass of water for the same rise of temperature; in so doing, however, it is expressly understood that the water has the initial temperature 0° C. [32° F.], for which condition c = l, while the body in question has for the initial temperature any value of t (but assumed as known), because experiments have shown the specific heat c to vary with the temperature, that is, have shown c to be a function of t. This variability with the initial tempera- SPECIFIC HEAT OF GASES. 117 ture is also manifested by water. This is the reason that in the above comparison water was taken at thg initial temperature of 0° C. [32° F.], for thus only can a reliable basis for comparison be established. We have just compared the unit of weight of a body with the unit of weight of water, and therefore c is also called the specific heat for equal weight or weight capacity. We might also have taken a cubic unit, cubic meter [cu. ft.], of the body and compared it with a cubic unit of water at 0° C. [32° F.] with respect to its heat absorption for the incre- ment of temperature dt, and then we would be dealing with specific heat for equal volume, or volume capacity. But the one kind of specific heat can easily be determined from the other. Let Y be the specific weight of the body, i.e., the weight of a cu. meter of the body measured in kg. [the weight of a cu. ft. of the body measured in lbs.], and if jq is the specific weight of water (;-o = 1000) [7-0 = 62.425], then the body will need the quantity of heat Y cdt for the rise of temperature dt, and the water will need the quantity of heat Y„dt starting from 0° C. [32° F.]; the ratio w of the two is therefore ^ = 7^, (32) '0 and that is the specific heat for equal volumes; it is obtained by multiplying the specific heat c, for equal weight, by the quotient y : 7-0, i.e., by the relative weight of the body with respect to water. The foregoing presentation of the older works on Physics must, however, be extended according to the theory of Thermodynamics, for the quantity of heat needed by the body for the temperature rise dt is by no means solely dependent on this rise and on its initial temperature. The temperature is a function of the pres- sure and volume of the body, and as the body experiences a change of volume, while it is receiving heat under the external pressure p, 118 TECHNICAL THERMODYNAMICS. there mupt be considered here the external work performed by, or expended on, the body. In Section I, § 3, p. 27, it was dis- tinctly explained that the quantity of heat which can be imparted for a certain change of state depends upon the path along which the body is passed from the initial to the given final con- dition, and that there must be an infinite number of such passages, and equation (31) has therefore, in principle, no general validity, but presupposes in its application a perfectly definite law of change of the external pressure p with the volume v of the body. For gases this law will appear in subsequent developments; but for every other relation between p and v during heat supply equa- tion (31) will have no meaning. It is so with solid and liquid bodies, only with the difference that for such bodies the general law of the changes of state for which equation (31) is true is not even known, because the so-called equation of con- dition of solid and liquid bodies has not yet been found; this much can be regarded as certain, that equation (31) can only be used with such sohd and hquid bodies as experience during heating an imperceptible change of volume, which can be neglected in the computations, because in this case the outer work is of no account. With gases, heating is in general accompanied with considerable changes of volume, and therefore from the beginning the differ- ence could not be disregarded; consequently even before the question was clarified by Thermodynamics two kinds of heating of gases were distinguished from each other; the heating of the gas was considered as taking place under constant pressure or under constant volume; in the first case let the quantity of heat for the rise of temperature dt of the gas be dQp = Cpdt, (33) and, in the other case, let dQv = Cvdt, (34) where the subscript p or v indicates which magnitude is supposed to be constant during the heat supply considered. We call Cp the specific heat of the gas under constant pressure, and c» that under constant volume, and really we ought to add, in both cases, for equal SPECIFIC HEAT OF GASES. 119 weight, as a kg. [lb.] of the gas was compared with a kg. [lb.] of the water; but this supplementary remark is unnecessary, for in all Thermodynamic investigations the tinit weight of the bodies in question is assumed, and it is only in exceptional cases that the unit of volume is considered. Moreover the two cases of heating of the gas just mentioned will in the future investigations appear as only special cases of a general law; but for the questions to be treated here these two cases are of marked significance, because the values Cp and c„ for the different gases have been determined by experiment. The first reUable experiments with respect to the value Cp, the specific heat of gases under constant prsesure, we again owe to R e g n a u 1 1.^ The following tabulation gives the values of Cp and of the product yCp, where y is the specific weight, that is, the weight of a cubic unit of the gas (see p. 103) for those particu- lar gases which obey sufficiently closely the above equation of condition. If we divide the latter product yCp by the specific weight ?-Q = 1000 [^(, = 62.423], we get for the corresponding gas, as was already explained, the specific heat under constant pres- sure for equal volumes. Atmospheric Air Hydrogen Oxygen Nitrogen Nitric Oxide. . . . Carbonic Oxide. . Cp. 0.2375 3.4090 0.2175 0.2438 0.2317 0.2450 French rCp. 0.3071 0.3053 0.3110 0.3062 0.3114 0.3065 English rcp. 0.01917 0.01906 0.01941 0.01912 0.01942 0.01913 We see that the specific weight Cp is different for the different gases, but that hydrogen stands out in a remarkable fashion; its specific heat is, as other experiments also show, even greater than for any solid or liquid body. It is different with the values yCp of the last column, which are nearly equal for the given gases, as was early recognized by Delaroche and B e r a r d ; the smallest value appears with hydrogen, and as this gas, according to experience, stands n e a r - ' Rel. II, 303. 120 TECHNICAL THERMODYNAMICS. est to the perfect gas, we conclude, in accordance with the supposition under which the equation of condition was derived, that any individual gas deviates more from these suppositions the greater the deviation of its value yCp from that belonging to hydro- gen. But for the gases just adduced the deviations are so insig- nificant that for this reason also we can utilize the equation of condition with all of them. If we assume complete equality of the values ycp, then we may conclude that for equal volumes all gases require the same quantity of heat for the same rise of temperature under constant pressure, which proposition, according to A V o g a d r o's law, § 19, p. 107, can also be extended to the molecules themselves. But the investigations of Regnault concerning the spe- cific heat of gases under constant pressure have led to still other results which are important for the discussions which are to follow. First of all, his extended experiments show that for atmos- pheric air, hydrogen, and carbonic acid the specific heat is i n d e- pendent of the pressure, and that therefore the unit of weight of every one of these gases needs always the same quan- tity of heat for the 'rise of temperature dt under a constant but arbitrary pressure, whatever the volume may happen to be; this is true even of carbonic acid, which in other respects by no means obeys the equation of condition of gases. Furthermore it has been shown that for atmospheric air and hydrogen the specific heat Cj, is a constant quantity between wide tempera- ture limits ( -30° C. to 200° C.) [ -22° F. to 392° F.]. On the other hand the specific heat of carbonic acid grows not incon- siderably with rising temperatures. From this the conclusion has been drawn that for the gases mentioned in the above tabula- tions and which were formerly designated as the permanent gases, the specific heat under constant pressure within the ordinary pressure and temperature limits (which alone are to be considered for the present), may be regarded as independent of the pressure and temperature, and therefore considered as a constant value. But for high pressure and for high temperature, even for low temperatures at which SPECIFIC HEAT OF GASES. 121 the gas approaches the liquid condition, we must give up this assumption. , So far as the specific heat of gases under constant volume (c„) is concerned, no direct . determination has yet been successfully made, but by different experimental methods the ratio ^-y (35) of the specific heats Cp and c„ has been determined; the most important of these methods we will discuss more fully later on. According to the formula given by Laplace for the velocity of sound which contains this value k, there was determined for k on the average 1.403; for the velocity of sound is accurately known in air. D u I o n g found from experiments of the oscillations of gases in pipes, for atmospheric air k = 1.421, for carbonic oxide K= 1.428, and for carbonic acid 1.338, but then these experi- ments show certain imperfections. More exact experiments of a similar sort, using the procedure given by K u n d t , are due to W ii 1 1 n e r , who deserves the credit of taking account at the same time of the influence of the temperature. Among other results W ii 1 1 n e r found corre- sponding temperatures of 0° C. [32° F.] to 100° C. [212° F.] for atmospheric air « = 1.4053 and 1.4029 respectively, and also fof carbonic acid he found « = 1.4032 and 1.3946 respectively. According to other experimental methods, differing but little from each other in principle and which will be touched upon later in this treatise, there was found for atmospheric air by element 1.356, by Mass on 1.419, by Hirn 1.384, by Weisbach 1.402, by Cazin 1.410, and by Rontgen 1.405. Other careful and recent experiments by L u m m e r and Pringsheim have given for atmospheric air « = 1.4015, for hydrogen 1.4084, for oxygen 1.3962, and for carbonic acid 1.2961. As a mean value we will hereafter take « = 1.410, 122 TECHNICAL THERMODYNAMICS. and will assume that this magnitude, as well as Cp, is independent of temperature and pressure under ordinary conditions. Then, according to equation (35), the specific heat c„ under constant volume becomes constant. For atmospheric air, the technically most important gas, there is given Cp= 0.2375; consequently when « = 1.410 there follows c^ = 0.1684. §22. FIRST FUNDAMENTAL EQUATION OF THERMO= DYNAMICS WITH RESPECT TO GASES. The quantity of heat dQ which is to be imparted to a unit of weight of a body in order to increase its pressure, volume, and temperature by dp, dv, and dt respectively, was given by equa- tions (Ilia), § 12, p. 62: dQ- = A[Xdp + Ydv] ■ = = -^[Xdt + Tdv] dp = =^[Ydt-Tdp] di (36) In these three identical equations T means the C a r n o t function of temperature whose form for the present is still un- known; the two magnitudes X and Y are functions of p and v for the body in question, and it is these magnitudes which we must now ascertain for. gases. If for one case we assume the body to be heated under con- stant pressure and in another case under constant volume, then the third of these fundamental equations gives for the first case (for dp = 0) AY dQp- dv -dt, (37) FIRST FUNDAMENTAL EQUATION FOR GASES, 123 and the second equation, for dv = 0, gives AX dQ^ = -—dt. *. (38) dp A comparison of these two formulas with equations (33) and (34) now furnishes ^Y=cp-^ and AX = c,-^ (39) Both formulas are still valid for every body, and from what has preceded we can, especially for gases, consider the specific heats Cp and c„ to be constant, and, from the equation of condition pv = B{a+t) [pv = B(a + t-32°)], there follows, by differentiation, dt p . dt V Tv^B ^"^ d^=B' (40) and by substitution in equation (39) we get 7=^.p and X = ^.v (41) From this we see that for gases the function F is only depen- dent on p and the function X only on v, and furthermore that these functions are directly proportional to the corresponding values. Now the first fundamental equation of Thermodynamics, according to equation (la), § 12, p. 62, has the form dY dX_ dp dv The differentiation of equations (41) and the substitution of the two different coefficients in the preceding equation then give Cp~c^ = AB (42) 124 TECHNICAL THERMODYNAMICS. as the form into which the first fundamental equation passes when applied to gases. If we here use the symbols given in equa- tion (35), we get the following forms: K — 1 Cp-c^ = c^(K-l)=Cp =AB, .... (43) K and according to equations (41) the formulas for the two func- tions X and Y take the simpler form ^ = ~r -d Z=-^ (44) The important equation (42) was first given in this form by C 1 a u s i u s ; it contains four magnitudes, every one of which was determined by special experiments; the substitution in this equation of the experimentally determined quantities must give the corresponding connection. Thus if we substitute for atmos- pheric air the above-given values Cp = 0.2375, 5=29.269 [53.349], andA = l:424 [1:772.83], then the formula becomes c^ = 0.1685 and consequently « = 1.4095, or in round numbers « = 1.410, in agreement with the data ' given in the preceding paragraph. If we hold fast to Regnault's assumptions for B and Cp for air, and take «= 1.4015 according to Lummer and Pringsheim, then according to equation (43) the mechanical equivalent 1:A has the value 430.18 [784.09], which value agrees with that given by D i e t r i c i. From this we can conclude that a later determination of the mechanical equivalent of heat, completely free of all objections, will lead to a somewhat greater value than (424) [772.83] given by Joule, the value which we will still continue to assume for the present. Computing c in the same way from equation (42), the follow- ing table gives, in the first column, for the gases already enu- merated, the specific heat at constant volume ' If we pursue the line of thought of Robert Mayer (1842), who first computed the mechanical equivalent of heat, it will be found to correspond, when brought into mathematical form, to equation (42) of the text; although Mayer found 365 instead of 424, this must be ascribed to the fact that, follow- ing B 6 r a r d , he took the specific heat Cp or air as equal to 0.267, while the true value Cp = 0.2375 was not found until later byRegnault. FIRST FUNDAMENTAL EQUATION FOR GASES. 125 for equal weights; while in the second column are placed the products ^Cj, which are found with the help of the values y given in table on p. 104; dividing these products yc-c by 1x1000 [1x62.425] will give, for equal volumes of the different bodies, the specific heat at constant volume. French yc^. English tc Atmospheric Air, Hydrogen Oxygen Nitrogen Nitric Oxide. . . . Carbonic Oxide.. 0.1685 2.4123 0.1551 0.1727 0.1652 0.1736 0.2170 0.2161 0.2217 0.2169 0.2218 0.2172 0.01360 0.01349 0.01384 0.01354 0.01385 0.01356 We see from the values of the 2d [and 3d] columns that the specific heat c^, taken at equal volumes, is nearly the same for all gases; we conclude from this that all gases require the same quantity of heat for the same rise in temperature, pro- vided the unit of volume is used in the experiment ; this is also the case with the specific heat under constant pressure, and was discussed in connection with the table on p. 119. If we introduce into the following formulas the weight £ of the gas relatively to hydrogen, as was done in § 19 when discussing the equation of condition, there results a series of simple propositions which are worth emphasizing. Multiplying both members of equation (42) bye, we get £(cj, — cj =ABt. Becauseof the relation 5o = S£ of equation (11), where Bo = 422.591 ' [770.259] is the constant B for hydrogen belonging to the equation of condition (see Table p. 104), there follows £(Cp-C,) = ^5o, (45) according to which the left member of the equation is a constant quantity for all gases, a point to which Clapeyroni had already called attention. Moreover the value of AB^, as was incidentally noted with equation (12) in § 19, p. 104, is so nearly ' Poggendorif's Annalen, Vol. 59, p. 451 (1843). 126 TECHNICAL THERMODYNAMICS. equal to unity (or, more exactly, j4.So = 0.9967, for the values of the constants A and Bq used in this book) that in most computa- tions use can be made of the assumption ABq^I, but as there is so far no theoretical justification for this relation, no use will be made of it in the course of this book. Because in the gases quoted the relative weight c is identical with half the molecular weight m we can also write equation (45) in the form m(cp-cj= 24 Bo = 1.9934 (46) For a mechanical mixture of gases equation (24) gave ^m "), (55a) = -Po'+2 log. {Tv'-^), (556) = Po"+5log. /^i\ (55c) or c, or finally All the equations just developed are true for the unit of weight (1 kg.) [1 lb.] of gas; consequently if for such a unit there are given two of the three quantities p, v, and T, we can determine the third from equation (54) and can then determine easily the entropy P from one of the equations 55; on the other hand, the utilization of equation (53), for the determination of the quantity of heat Q for any finite change of state of the gas, demands a statement of the way in which the change has taken place. It will be the task of the following investigations to carry through the calculations for a series of the most important of these changes. But equation (536) calls for a particular remark ; if we replace in it, in accordance with the ordinary usage, the absolute tem- perature T by the temperature t, we get dQ = Cvdt + A'pdv. The heat to be imparted to the gas accordingly divides itself into two parts, of which the part Cydt consumed in the gas just corresponds to the quantity of heat which must be supplied to the gas if it is to experience this rise of temperature at constant 136 TECHNICAL THERMODYNAMICS. volume; therefore this part consumed in the interior of the gas must always be the same, whatever change the gas may experi- ence in the way of pressure and volume. The truth of this is so evident that we might have written the preceding equa- tion at once; the combination of the equation with the equa- tion of condition of gases (the law of Mariotte and G a y- L u s s a c) then easily and directly leads to the equations found above for gases. §26. THE ISOTHERMAL AND ISODYNAMIC CURVE FOR GASES. The "isothermal c u r v e " (§ 6, p. 39) gives the law according to which a body experiences reversible changes of state, during heat supply or heat withdrawal, while the temperature T is kept constant and the pressure p varies with the volume v. We get especially and immediately for gases from the equation of condition pv = BT, and for the equation of the curve sought Vv = 'P\Vi, when the initial condition is given by the quantities pi, vi, and T^ because T = Ti. The isothermal curve for gases is therefore an "equilateral hyperbola " whose constant piVi = BTi is determined by the initial condition, and indeed, if we here assume a unit of weight of a gas, it is completely determined by the initial temperature Ti alone. The "isodynamic curve" represents (§ 6, p. 37) the change of the pressure p with the volume v, when the energy or inner work U is kept constant during the change of state; the heat imparted or withdrawn is therefore completely transformed into outer work or generated by such work, respectively. But from equations (52) and (526), p. 132, and an assumption of a constant value of U, we see that then the product pv and the THE ISOTHERMAL AND ISODYNAMIC CURVE FOR GASES. 137 temperature T are each constant. In gases, therefore, the i s o- dynamic curve is identical with the isother- malcurve. * If we expand the unit of weight of gas isothermally or isody- namically from the volume (a) (h) 1 I V Fig. 19. S vi to the volume v (Fig. 19a), we can easily draw the transformation a'b' of the pressure curve ab, provided in Fig. 196 we lay off corre- sponding heat weights as abscissas and the tempera- tures Ti as ordinates. We compute according to equation (55a) the value Pi for the initial condition (pi, t;i) and the value P for the final condition (p, v); these in Fig. 196 make c c OPi=j\ogePiVi'' and OP = ~\ogepv', and draw the horizontal a'6' corresponding to the ordinate Ti. The area bounded by the pressure curve a6 represents the work performed during expansion, while the area P^a'b'P given in the transformation represents the quantity of heat Q, measured in units of work, which was supplied to the gas from the outside during the assumed change of state. The quantity of heat can be computed from equations (53d) and (53e), because T = Ti and dT = 0, by the integration Q = a«-l)7'i loge^ = c/—^T^ log,^\ (56a) or, considering the relations given in equation (54), it can be found from Q = ABT^ log -^^ABT^log,^, and considering the equation of condition: = ApiVi \0ge— = ApiVi loge — . (566) (56c) 138 TECHNICAL THERMODYNAMICS. Because, for an invariable value of V, the heat Q is supplied completely transformed into work, we have Q = AL, (57) and therefore we find the work performed or consumed during isothermal or isodynamic expansion or compression from the equations (566) or (56c) is L = BTi\og,^=BT^\og,'^, . . . .{58a) or L = pii;i log«^ = piViloge^ (586) From equation (57) it is evident that the transformation a'b' of the pressure curve here considered (Fig. 19) covers the same area L as the pressure curve itself, a proposition which is more- over true for the isothermal curve of every body. E X a m p 1 e. — Let us suppose one kilogram [one pound] of atmos- pheric air of the temperature ti=15° C. [59° F.] to expand isothermally from the initial pressure 2.5 atmospheres to 1 atmosphere, then pi=2.5 X 10333 and p = 1X10333 [p, =36.742 and p = 14.6967 lb. per sq. in.] and T, =288° C. [Tr =518.4° F.]; consequently it follows from the equation of condition because B =29.269 [B =53.349] for air, that the specific volume at the beginning and at the end is Vi =0.3263 cbm. [5.227 cu. ft.] and w=2.5vi =0.8158 cbm. [13.068 cu. ft.], and therefore according to equation (58) the work performed is L =7723.85 mkg. [25340.78 ft-lb.], and the quan- tity of heat imparted to maintain the temperature constant: Q=AL = 18.217 cal. [32.79 B.t.u.]. According to equation (55a) there results for the entropy of air at the beginning and at the end: P, =371.78 and P =398.60 [P, =902.27 and P =951.21], because the specific heat c» =0.1685 and « = 1.410; consequently we can find the work from the expression L = (P — Pi)Tu The formulas (586) given above for the external work L have long been extensively used in Physics and Mechanics and in the Theory of Engines, long before the rise of the Thermodynamics of the present day. The assumption that the expansion and com- THE ADIABATIC CURVE OF GASES. 139 pression took place according to the equilateral hyperbola was a natural one; but that such a change of state of the gas could only result from a suitable supply or withdrawal of heat was first made clear by investigations based on the laws of Thermodynamics; that is why this sort of change of state was assumed in cases in which it is not at all permissible, for example in the efflux prob- lem; it is remarkable that this occurs even now, and is an assump- tion which may still be found in some of the elementary manuals of Physics and of Mechanics. § 27. THE ADIABATIC CURVE OF GASES. If, during a reversible change of state of a gas, heat is neither imparted nor withdrawn either during the whole course or in any of its parts, then the pressure p of a gas will change with its volume according to a curve designated as the "adiabatic curve" (see § 6, p. 39). During the adiabatic change of state the temperature T also simultaneously changes with the pressure and volume according to a particular law; the changes in question follow directly from equations (55a) to (55c), p. 135, given for the entropy P of gases. Because dQ = 0, for all elements of the change of state under the assumption made, there follows, from equation (53/), dP = 0; consequently the entropy P is equally large for all points of the adiabatic; therefore from equation (55a) we have for the initial and final condition (Fig. 20) c c P = ^^0gePlVl' = -fl0gePV', .... (59) provided we make the arbitrary constant Po equal to zero ; hence the equation of the adiabatic curve can also be represented simply by pv''=piVi' (60) In the same way there can be determined from equations (556) and (55c) ""^ Tr\p) "' • • • (61) 140 TECHNICAL THERMODYNAMICS. from which we can find the temperature T for any point of the adiabatic corresponding to a given value of v or p. The transformation of the adiabatic curve ab (Fig. 20a) is given by the straight line a'V (Fig. 20&). The foregoing results (60) and (61) would also have been given by equations (53c) to (53e), if we had there made dQ = and integrated. The work L, which is performed or consumed during adiabatic expansion or compression respectively, can be computed from equation (536), p. 134, when we put dQ = Q, and integrate the equation : Apdv= —c^dT, from which follows L = |(7'i-7') (62a) If we use the relation (54) here, we can also write ^=;r^(pi'^i -?"')' (626) or by using formulas (61) In these two equations we can also replace the factor in front of the parenthesis by A K-Y Moreover equation (626) shows that for any adiabatic change of state the work is proportional to the difference of the two rect- angles fiVi and yv constructed from the initial and final coor- dinates. EQUATION OF CONDITION FOR GASES AND VAPORS. 141 E X a m p 1 e. — If a kilogram [pound] of atmospheric air expands adia- batically from the temperature ii = 15° C. [59° F.] and pressure of 2.5 atmos- pheres down to one atmosphere, then Ti=28^ C. [518.4° P.] and pi=2.5 X 10333 and jj = lX 10333 [p, =36.742 and p = 14.6967 lb. per sq. in.], and the initial volume t)i, as in the example of the preceding article, becomes «, =0.3263 cbm. [5.227 cu. ft.]. We now get, according to equation (61) for the final temperature T = 220.63 [397.13], or according to Celsius, < = -52.37° C. [«= -62.266° F.], and for the final volume « =0.6249 cbm. [10.0102 cu. ft.], and also the ratio V of expansion: — =1.915. Vi The work performed according to equation (62a) is L =4813.16 mkg. [15791.15 ft-lb.], and the corresponding quantity of heat which here dis- appears is AL = 11.352 cal. [20.434 B.t.u.]. Therefore there here occurs a considerable lowering of temperature. The example corresponds to con- ditions occurring in cold-air engines. After we have determined in tiie manner just shown the course of the adiabatic and of the isodynamic curve,, we can easily deter- mine for any change of state of the gas the quantity of heat Q by the method fully explained in connection with Fig. 4, § 7, p. 41, but we recognize now, what was already emphasized there, that the two intersections c and d are somewhat indeterminate when applied to gases, because in this case the curves in question cut one another at a very acute angle. The second of the above-given graphical methods, namely, the one found by the transformation of the pressure curve, is therefore to be preferred. § 28. THE GENERAL FORM OF THE EQUATION OF CON= DITION AND THE VARIABILITY OF THE SPECIFIC HEAT OF GASES. I The equation of condition for gases and vapors, and indeed for all bodies whatever, can always be written in the form pv=BT-R, . (64) where B is the constant corresponding to the body in question, and R is the function of two of the variables p, v, or T. At present nothing more is known of this function R than that 142 TECHNICAL THERMODYNAMICS. its influence is the more subordinate, the more the vapor approaches to the condition of a gas; if the latter obeys the law of Mariotte and Gay-Lussac, as was assumed in the above investiga- tions, then we must substitute i2 = 0. If we regard i2 as a function of p and v, and if we differentiate equation (64) partially with respect to p and v, and use for the absolute temperature the relation T = a+t = 273+t[4:59A+t], we get ^dt dR , ^ ^-v = P+^^' (65) ^dt dR , , ^9]^ = ^ + 3^ (66) Let us first compute the coefficients of expansion a„ and a^, which were already discussed on p. 97. We found for their recip- rocal values Oj, and a^, from equation (6) there given, and therefore there result for !r=a+i, and the preceding formulas (64), (65), and (66), 3fl B{a-a^) = R-p^, (67) 37? B{a-ap) = R-vir-- (68) According to these equations the two values Ov and ap and the two coefficients of expansion a^ and ap could be computed for the body in question if the function R in the equation of con- dition (64) were known. Now if heat supply under constant volume or constant pres- sure takes place, we have dv = or dp = respectively; from the second and third of equations (36), p. 122, there follows ,^ ^^ , dp dv EQUATION OF CONDITION FOR GASES AND VAPORS. 143 Let c„ and Cp be the specific heats of the body at constant volume and at constant pressure respectively, then dQv = Cydt and dQp = Cpdt, and by combining with the preceding expression there follows AX=c«Tr- and AY = Cp^r-. op ^av If we utilize these values in equations (la), (Ila), and (Ilia), p. 62, the first fundamental equation takes the form and the second fundamental equation (Cp — cj_ ^ —AT, dv dp (lib) and there further follow 3' , 3* J dQ = Cv-^dp + Cp-^dv 'dv From equation AT =c^t+ - dv dp =CpOi — a7~ ?* dv dQ=AdU + Apdv (1116) we can then compute the change of energy dU; for example, there follows from the second of equations (1116) AdU = c^dt + Al^^-p\d^- \3i / (IV6) 144 TECHNICAL THERMODYNAMICS. We will repeatedly emphasize the fact that these equations are perfectly valid and general for the equation of condition (64), that is, ioT pv+R = BT, the specific heats Cp and c^ being here regarded as functions of p and v. The formulas permit a number of transformations if we employ, 3< 'dt for the partial differential coefficients -g- and g-, equations (65) and (66), and as occasion may demand also utilize equations (67) and (68), in which a^ and ap are taken as functions of p and v; for example, we see that in place of equation (IV6) we can put AdU = c^dt + —^ — — — dv. The formulas become immediately available if we only know the law of change of two of the quantities R, Cp, and c^, regarded as functions of p and v. In subsequent investigations of vapors the different assumptions which have hitherto been made con- cerning the function R will be more fully discussed ; at t h i s place only one case will be subjected to special consideration; it will be assumed that we have before us a gas obeying the law of Mariotte and Gay-Lussac. Assume, therefore, the equation of condition, as in former investigations, to be pv = BT, consequently R = 0. Here equations (67) and (68) give at once a^ = ap = a = 273[4QlAl and from equations (65) and (66) follow Br^=p and B^r-=v. av '^ op Substitution in equation (116) then gives Cp~c^ = AB, (68o) and after a few simple transformations there follows from equa- tion (16) P3]^="9^ (69) EQUATION OF CONDITION FOR GASES AND VAPORS. 145 The next to the last equation shows that the difference of the specific heats is constant even if they dp vary with p and v; if we differentiate the equation with respect to v and substitute -~-^ = ^-^ in the last equation, there follows the relation dcp 9cp ^dp dv' which differential equation is only satisfied when Cp is regarded as a function of the product pv; but as this product is proportional to the absolute temperature T, there follows for R = 0, (1) That the specific heat Cp for constant pressure, if it is vari- able at all, can only be a function of the temperature t, and (2) That the specific heat c„ for constant volume, according to equation (68a), can differ from the value of Cp only by a con- stant. There exist valuable experiments by Eilhard Wiede- mann ^ on the variability of the specific heat Cp with the tem- perature. He found for atmospheric air Cp = 0.2389, for hydrogen Cp = 3.4100, for carbonic oxide Cp = 0.2425, values which differ very little from the results of Regnault. It was also found by E. Wiedemann that these values (within the ordinary pressure and temperature limits) were independent of the temperature. On the other hand he found an increase of the values Cp with the temperature, for the following gases : for<= 0° 100° 200° C. [32°] [212°] [392° F.] Carbonic acid, cp= 0.1952 0.2169 0.2387 Ammonia 0.5009 0.5317 0.5629 Ethylene 0.3364 0.4189 0.5015 Nitric oxide 0.1983 0.2212 0.2442 For that matter Regnault has already shown that for carbonic acid in particular there was an increase of specific heat with the temperature; he " found for 0°, 100°, and 200° [32°, 212°, ^E. Wiedemann, "Uber die sp^ifische Warms der Gase." Poggen- dorff's Annalen, vol. 157, p. 1. ' Rel. II, 130. 146 TECHNICAL THERMODYNAMICS. 392° F.], respectively, Cp = 0.1870, 0.2145, and 0.2396, values which differ little from those found byE. Wiedemann.. Later the question was again examined by Mallard and Le Chatelier,! and their work has created great interest among the engineers of Germany, because they touched questions which play an important part in judging gas- (explosive) engines and in the investigation of combustion phenomena in general ; the problem here is mainly the behavior of the products of combus- tion of exploding gases, namely, of carbonic acid and of vapor of water. Let m, as above, be the molecular weight of the gas or vapor, as the case may be, then we can write mc^=a+pt + rt^, (70) when c„ represents the specific heat at constant volume and a, /?, and y are experimental constants. Consequently the heat Q necessary for heating from 0° to t° is Q = J 'mc^t = at + -^+-^. The equation can also be written where we can consider the quantity enclosed in the bracket as the mean specific heat Cm between the temperatures 0° and t°; consequently we can write Accordingly we should get for equation (70) /? = 2/3' and r = 3r', provided /3' and / were known from experiments. ' "Recherches exp6rimentales et th^oriques sur la combustion des melanges gazeux explosifs," by Mallard and Le Chatelier, Ingfoieurs au corps de Mines. Annales des Mines, Vol. IV, 1883. EQUATION OF CONDITION FOR GASES AND VAPORS. 147 Mallard and Le Chatelier have recomputed their original observations, making use of the experimental results of S a r r a u and V i e i 1 1 e, and got for the mean specific heat Cm the following expressions : ^ for carbonic acid 0^=6.50+0.00387 i, i " steam =5.78+0.00286 i!, [ . (a) ■■' the simple gases (H2, O2, N2, CO) =4.76+0.00122 t. l From this follows, for molecular weight, the specific heat at constant volume : for carbonic acid mc^ = 6.50+ 0.00774 <, ■ " steam =5.78+0.00572 <, " the simple gases =4.76+0.00244 <. . m If we introduce the absolute temperature, putting t = T— 273 we get: for carbonic acid mc„ =4.387 +0.00774^,1 " steam =4.218 +0.00572 T,. (r) " the simple gases =4.094 +0.00244 T.J It follows from this, as is also pointed out by Le Chatelier, that at the absolute zero the real specific molecular heats would be nearly equal for the gases and vapors before us. As the relation mcp — mc^ = 1.9934, equation (46), p. 126, holds, we can compute the specific heat Cp at constant pressure from the foregoing formulas, but of course the values for car- bonic acid and the vapor of water must be regarded as approxi- mations only. If we substitute in the second group of the preceding expres- sions m= 44, we get for carbonic acid c„ =0.1477+0.000176 i, Cp=0.1930 +0.000176/; 'Wiedemann's Beiblatter zu den Annalen der Physik und Chemie 1890, XIV, p. 365. 148 TECHNICAL THERMODYNAMICS, for the vapor of water with m = 18 : c„ =0.3211 +0.000318 «, Cp = 0.4318+0.000318*; for hydrogen, because m = 2 : c„ =2.380+0.00122 « Cp = 3.377+ 0.00122 «. For other gases, oxygen, nitrogen, and carbonic oxide we would put respectively wi = 32, 28, and 28. These formulas give for carbonic acid and the vapor of water, values for Cp between the temperatures 0° and 200° [32° and 392° F.], which agree suffi- ciently well with the formerly given values of Regnault and E. Wiedemann; for hydrogen and the other simple gases the specific heats experience such a marked increase with the temperature that the author felt justified, when issuing the preceding edition of this book, in doubting the reliability of the experimental results of Mallard and Le Chatelier. In the meantime these investigations have been confirmed in a general way by the new experiments of A. L a n g e n i and have been utilized in the more recent technical investigations on internal-combustion motors by S t o d o I a, E. Meyer, and others, so that we may consider the doubt raised as removed although the problem cannot be regarded as solved in all its parts. At any rate there must be dropped from the theory of internal- combustion motors the former assumption of the constancy of the specific heats of the products of combustion, although we may still be far from a satisfactory theory of the occurrences in said engines. That the specific heats can be assumed to vary only with the tem- perature, in gases subject to the equation of condition pv = BT was discussed upon p. 145, but it is just the products of combus- tion, carbonic acid and steam, upon which everything here de- pends, that do not obey this law; here the specific heats must be regarded as also depending on the pressure. Considering this de- ' A. L a n g e n , Untersuchungen uber die Drucke, welche bei Explosionen von Wasserstoff und Kohlenoxyd in geschlossenen Gefassen auftreten. (Zeit- schrift des Vereins deutscher Ingenieure, 1903, p. 622.) EQUATION OF CONDITION FOR GASES AND VAPORS. 149 pendence on temperature and pressure at their higher values, there is at present complete uncertainty. To this must be added the circumstance that at high tempera- tures carbonic acid and steam experience "Dissociation," a de- composition into their constituents, a procedure concerning whose laws there likewise exists complete uncertainty. In the observations of L a n g e n, as well as in those of M a I- 1 a r d and Le Chatelier, connected with the cooHng curves of those experiments relating to temperatures above 1700° C. [3100° F.J after the combustion of carbonic acid, there occurred irregularities which can only be explained by the dissociatiop of carbonic acid. Therefore for combustion temperatures above 1700° C. [3100° F.] the above-given formulas for molecular heats cannot be employed. For gases (atmospheric air) there is on hand a special investi- gation by L i n d e .1 In a clever way L i n d e based and built his machine for the liquefaction of atmospheric air upon certain occurrences in an experimental series by W. Thomson and Joule, which have been known since 1862, and which previously had only occasionally given rise to theoretical considerations con- cerning the behavior of gases. On the basis of these experiments and by considering certain occurrences in his machine L i n d e (ibid.) obtained the following formula for the computation of the specific heats Cp of the air at constant pressure. Into the develop- ment of this formula we will go more fully later on, in the theory of vapors and when considering L i n d e ' s machine. But at this place we must emphasize what is of importance for the following investigations of air and gas engines. L i n d e ' s formula reads : _ r ^apl-i Here Cp„= 0.237, which value, according to Witkowski's experimental results, represents the specific heat Cp at very * L i n d e , "Uber die Veranderlichkeit der spezifischen Warme der Gase." Sitzungsberichte der mathem.-physik. Klasse der k. bayer. Akademie der Wissenschaften, Vol. XXVII, 1897, No. 3. 150 TECHNICAL THERMODYNAMICS. small values of the pressure p lying near to zero, and is inde- pendent of the temperature. We must write for the constant a when p is expressed in atmospheres (10333 kg. per qm.) [14.6967 lb. per sq. in.], a = 20570 [a =8163 for p in lb. per sq. in. and T in Fahr. degrees]. The following table is computed by the preceding equation and gives in somewhat more extended form L i n d e ' s table (ibid.) for the specific heat Cp of air. t=-ioo° -50° 0° + 100° C. t = [-\4S°] [-58°] f32°] [ + 212°F.] = 1 Atra. 2389 0.2.379 0.2375 0.2372 10 " 0.2579 0.2462 0.2419 0.2389 40 " 0.3650 0.2801 0.2583 0.2448 70 " 0.7856 0.3293 0.2779 0.2511 At the same temperature the values of Cp show an increase with the pressure, and this is more marked the lower the tempera- ture, that is, the nearer the air is to hquefaction. And for the same pressure the values diminish continuously with increasing temperature, and for all pressures would approach to the constant value 0.2370; the latter result contradicts the above-mentioned observations of Mallard and Le Chatelier and also those of L a n g e n, but of course it must here be remembered that L i n d e ' s formula for Cp is only to be regarded as decisive within narrow limits. In any case we can draw the conclusion that for temperatures from -50° to +100° [-58° to +212°] and over, and for pressures deviating but httle from one atmosphere, Regnault's con- stant average value 0.2375 is thoroughly reliable, so that in the theory of hot- and cold-air engines air can be assumed as a perfect gas. APPLICATIONS. PHYSICAL PART. I. Reversible Changes of State of a Qas. § 29. THE POLYTROPIC CURVE OF GASES. Of the numerous pressure curves, which a gas may follow when experiencing reversible changes of state, there is one case of special importance, because it includes numberless special cases and just the ones which are of physical and of technical import- ance. The problem indicated embraces the behavior of the gas under the assumption that "the quantity of heat" supplied to, or withdrawn from, the gas is directly proportional to the change of tem- perature." Suppose given the unit of weight of a gas, the pressure p, volume V, and temperature T; the pressm-e curve is to be deter- mined when the quantity of heat for an infinitesimal change of state is given by dQ = cdT, (1) in which c represents any positive or negative, whole or fractional, number. According to equation (54), p. 134, the equation of con- dition is written in the form If we differentiate it and substitute the value of dT in the foregoing equation (1), there follows, for the case assumed, A c dQ = — r -~(vdp + pdv) . 151 152 TECHNICAL THERMODYNAMICS. On the other hand we have generally, for every kind of change of state, according to equation (53c), p. 134, A dQ = r {vdp + Kpdv) . K J. By equating these two expressions and remembering that Cp = KCv, we get (c — c„)vdp + {c— Cp)pdv = 0. If we divide both sides by (c~Cy)pv and put -p c-c„ = n, (2) where n is a new constant quantity, completely determined by the value of c, by the specific heat for constant pressure {c^, and by that for constant volume (c„), we get dp dv „ p V and from this, by integration, Vv- = C, (3) where C represents a constant. If the initial condition is given by pi and Vi, we accordingly have pV^^PiVi" (3a) as the equation of the sought pressure curve, which we will here- after call the polytropic curve. If at the start there is given, instead of the factor c, the exponent n of the preceding formula, then we can compute backward from equation (2) and get n — K '^=;rri^'' (4) I call c the "specific heat of the gas for the pressure curve pv^^C." ^ ' I first called attention to this general case of the change of condition of a gas in the second edition of this book (1866). Since that time the correspond- ing formulas and propositions have been much used, particularly in technical investigations. THE POLYTROPIC CURVE OF GASES. 153 Because of the arbitrary choice of the exponent n there are therefore an infinite number of values o| the specific heat; but only on the assumption of this kind of pressure curve is the quantity of heat proportional to the change of temperature. If we write the equation of condition in the form pv PlVl its combination with equation (3) also gives (5) from which equations can be directly determined the temperature T for every pressure-curve point which is fixed by a correspond- ing value of the volume v or of the pressure p. li) "^ 4 1 \ T 1 1 1 ■5 P The quantity of heat which must be supplied for the change corresponding to the pressure curve ah (Fig. 21) can be found at once, from equation (I), to be Q=c(r-ro, (6) and the work performed is determined by the integration of equa- tion (536), p. 134: (T-Ti). Dividing this formula by equation (6) gives AL c — Cy (7) (7a) or 154 TECHNICAL THERMODYNAMICS. according to which the external work is hkewise proportional to the change of temperature, and therefore also directly propor- tional to the quantity of heat Q. If in equation (7) we replace the specific heat c by the value given in equation (4), we get, by considering equation (54), p. 134, c^JK-l) ,AB ''-''«-- n-1 "~n-r and from this also the external work : L = :^Z:^ipiVi-pv), (76) in which latter formulas the factor piDi can be replaced by BT^. Equation (76) seems to be particularly simple. The external work L, represented in Fig. 21a by the vertically hatched area, appears as proportional to the difference of the two rectangles formed by the pressure and volume at the beginning and at the end of the operation. If we determine in Fig. 21a by the planimeter the area L and the two rectangles, we can find from equation (76) the exponent n and then also the specific heat c, provided of course that the pressure curve is really subject to the law pu" = constant. In certain investigations into which we will enter later, the area piobp, bounded by the pressure curve a6 and distinguished in Fig. 21a by partially horizontal and partially close hatchings, plays an important part. If we designate this area by F we get, as the figure shows, F = piVi +L — pv, and therefore with the help of equation (76) -'^^^^(Pi^i-P^)' (8) THE POLYTROPIC CURVE OP GASES. 155 and by the combination of the two equations we get l^n, : (9) which formula will still more easily lead to the exponent n, with the help of the planimeter. The use of equation (6) in equation (7?)) leads to the formula Q=^^L, (10) or with the help of (9) to Q = ^JkL-F), (10a) K A. and with this and the planimeter we can directly determine the quantity of heat Q which can be imparted during the change of condition given by the drawing. The foregoing can even then be applied when the law of the change of state is not known, and when this change is not subject to the condition p?;" = constant. For if we suppose the pressure curve ab (Fig. 21a) to represent a part of the curve of an indicator diagram from a hot- or cold- air engine, and that part which corresponds to the expansion or compression of the air in the engine cylinder, then the curve can be conceived as decomposed into short pieces which approximately follow the foregoing law of change; as we do not here have to do with the unit of weight, let Vi and V2 represent the volume at the beginning and the end, and pi and 7)2 the corresponding pres- sure for any short piece of curve, both co-ordinates being deter- mined by measurement from the indicator diagram. Then we can put p2V2"' = piVi"' and from this derive logpi-lo gp2 log 72 - log y/ and then compute from equation (4) the specific heat c. If we perform this computation for the various parts of the curve, we get an insight as to whether, during the whole expansion or com- 156 TECHNICAL THERMODYNAMICS. pression, there has been a supply or withdrawal of heat and to what extent it has taken place; in general, different values will be obtained for n and c for the separate curve-intervals. This method of investigating the pressure curve has often been prac- tically carried out, but the solution with the help of the "trans- formation" of the pressure curve furnishes a still better insight into the problem. § 30. OTHER PROPERTIES OF THE POLYTROPIC CURVE. If we suppose different values given to the constant C of equa- tion (3), p. 152, we will get an infinite multitude of pressure curves having the same law (Fig. 22). The differentiation of the equation gives or n'pdv + vd'p=Q, dp p --f = n-. . av V B ■n 1 ; in both cases the curves pass through the origin of the coordinate axes (parabolas of higher order). For QT=0 there follows, from equation (12), n = oo, then the curve is transformed into a straight line parallel to the axis of ordinates OY. In this case equation (4) gives the specific heat c^c^; accord- ing to equation (6) there follows the suppHed heat Q = c^{T-Ti), and according to equation (7) the external work L=0; we here therefore strike a case, long ago treated in Physics, of heating the gas under constant volume. On the other hand, if the subtangent QT = , there follows n = from equation (12), and consequently from equation (4) c = Kc^ = Cp; the suppUed heat, according to equation (6), is Q = c^(T-T^), 158 TECHNICAL THERMODYNAMICS. and because here p = pi according to equation (3), the external work according to equation (76) becomes L = pi{v-vi). Here we have the long-known case of heating the gas under constant pressure. Furthermore if we substitute n = K in the given equations of the general case, then c=0, according to equation (4), and this assumption leads to the adiabatic curve; all the equations then pass into the forms which we find in § 27 for the adiabatic curve. Finally, if we assume n = l, equation (5) gives T = Ti, and hence pv = piVi, which result corresponds to the isothermal and isodynamic changes of state; equations {7b) and (10) then give the, to be sure, indeterminate expressions L=-^ and Q=-^, whose values, however, were found earlier by equations (566) and (58a), pp. 137 and 138. Consequently the polytropic curve embraces a 1 1 the special cases treated above. § 31. CONSTRUCTION OF THE POLYTROPIC CURVE AND ITS TRANSFORMATION. If through the point M (Fig. 24), whose co-ordinates OQ and QM are designated by vi and pi, there is to be passed a polytropic curve, for the law pi;"=pi?;i" = constant, we can derive a simple method of construction from the following considerations ^ : Draw the line OA, making any angle XOA=a with the axis OX, and hkewise draw the line OB, making the angle BOY=^ with the axis OY; we now pass vertically downward from the point M to the intersection S with the direction OA and then horizontally over to the intersection T with the axis OY; furthermore, pass ' According to B r a u e r : " Konstruktion gesetzmasiger Expansionskurven von der allgemeinen Form pri" = C." Zeitschrift des Vereins deutscher Ingenieure, 1885, vol. 29, p. 433. CONSTRUCTION OF THE POLYTROPIC CURVE. 159 through the points S and T the lines SR and TU, making an angle of 45° with the horizontal, and if we draw through R a vertical, and through U a horizontal, they will intersect in the point N, whose co-ordinates OR and RN = OV will be designated by v and p. According to the given construction QS=QR=v—Vi and VT = UV = pi —p, and consequently tan a V — Vi Vl and tan j3 = Pi-P p ' or v = vi{l+ta,na.) and pi = p(l + t3,n ^). If the point N is to be a point of the polytropic curve passing through M, there must hold the relation pv"'=piVi'^, or, utilizing the two preceding expressions, determining from them p and V and then substituting, we get the equation (l+tan /?) = (!+ tan a)". If the exponent n of the polytropic cruve is given and the angle « chosen arbitrarily, we can compute from this equation the angle /?, and in Fig. 24 make i:XOA=a and ^YOB=^. Now the rule for construction is simply the following, as may easily be seen. We start down vertically from the given point M to the point S, pass horizontally over to the point T and draw between the lines OX and OA, as well as be- tween OF and OB, the zigzag lines shown in the figure, in which a part of each broken line is inclined at an angle of 45° to the horizontal. The zigzag may also start from S and go to the left, and may start from T and be continued upward. If we now draw vertical Hues through all intersections with the axis OX and horizontal lines through all intersections with the axis OY, we will get a net of straight lines intersecting Fig. 24. 160 TECHNICAL THERMODYNAMICS. each other at right angles, and the corresponding intersections will lie on the curve sought. We see that in this way we not only- get the polytropic curve passing through the point M, but a whole host of curves of the same species. The smaller the angle a chosen for the construction the nearer to each other will be the sought points of the curves. Fig. 24 at once gives the course of the curve : In the net of rectangles we connect in each rectangle the lower left corner with the upper right corner by a straight line, and we get a series of curves corresponding to the preceding equa- tion. All of the curves pass through the origin (higher parab- olas), but are transformed into straight lines for n = l, forming a bundle of rays; the corresponding curves, for n=+l, then form a group of equilateral hyperbolas, for whose construction it is simply necessary to make the angles a and /? equal. We now attach to the given method of construction the proposi- tion concerning the "transformation " of the polytropic curve. If we substitute, in the formula for the computation of entropy ■/ AT' the quantity of heat dQ = cdT according to equation (1), we get, by integration, AP = APo+c\ogeT, (13) where Pq represents the arbitrary constant of integration. If, for the initial condition. Pi is the entropy and Ti the temperature, there follows also T ^(P-Pi)=clogeyr, (13a) and this expression is at the same time the equation for the trans- formation a'h' of the polytropic curve in question, provided P and Pi are laid off as abscissas and T and Ti as ordinates (Fig. 216, p. 153); this is an equation which can also be written in the exponential form A(P- Pi ) T = Tie <= , PRESSURE CURVE OF GASES. 161 where e = 2.71828 .... The curve is the known logarithmic Hne (Logistik). If we utiUze equation (5) in equation (13a) and replace c by expression (4), we have and ^(P-Pi) = (n-«)r„log,^ .... (136) ^(P-Pi) = (^)cJog,^ (13c) These formulas naturally embrace also all the special cases dis- cussed above. For heating under constant volume c = Cv and n = oo must be substituted, and for heating under constant pressure c = Cp and n=0. For adiabatic expansion c=0 and «. = «; consequently according to equation (13a) P = P\ and the transformation of the adiabatic is therefore a straight line lying parallel to the axis of ordinates, as was mentioned before. For isothermal and isodynamic expansion c = oo and n = 1 ; the transformation is therefore a horizontal line. If in any other case heat withdrawal is connected with an expansion, then P< Pi would result; in the trans- formation of the pressure curve ab (Fig. 216) the positions of a' and 6' would then be interchanged. § 32. PRESSURE CURVE OF GASES WHEN THE QUAN= TITY OF HEAT IS PROPORTIONAL TO THE CHANGE OF PRESSURE OR TO THE CHANGE OF VOLUME. Although the preceding investigations of the polytropic curve discussed the most important cases and included those treated in the books on Physics, it seems appropriate to examine those cases in which the heat supplied to the gas is n o t proportional to the change of temperature, at least to discuss the pressure curves mentioned in the heading of this article. 162 TECHNICAL THERMODYNAMICS. Let the heat to be imparted be given by the expression dQ= -[vodp + Kpodv], (14) K J. and here let vo and pobe arbitrary constants which, in accordance with the notation introduced, represent any value whatever of a volume and of a pressure respectively. If the values pi and vi correspond to the initial state of a gas, then integration of the preceding equation gives directly Q= ^K(p-pi)+«Po(«-?'i)] (14a) K J. ' The corresponding change of inner work follows from equation (52), p. 132, U-U^=--[VV-Viv^], (146) « — 1 and accordingly, with the help of the formula Q = A(C/-f7i+L), the external work becomes L= [vo{'p-'Pi)+i^'Po{'»-vi)+'piVi-'pv]. . . (14c) K J. For the special case that the quantity of heat Q is propor- tional to the change of volume v-vi we must place 110 = 0; on the other hand we would have to substitute po=0 if the heat to be imparted is assumed to be proportional to the change of pres- sure p — pi- If we combined the foregoing equation (14) with the always valid equation (53c), p. 134, there follows vdp + Kpdv = Vodp + Kpodv, and from this, by integration, we get the equation of the corre- sponding pressure curve: ip-po)(v-voy={pi-po)(vi-Vo)''. . . . (15) EXPANSION OF GASES. 163 The curve is therefore related to the adiabatic; the coordinate axes of the latter are only shifted parallel an amount equal to ^0 and po- * If for any value of v the pressure p has been computed from equation (15), the temperature T can be found from the equation of condition pv = BT for the point in question of the pressure curve, and then from one of the equations (55), p. 135, the cor- responding value of the entropy P. But in so doing the way is also indicated for "transforming " the pressure curve in hand. II. Non=reversible Changes of State of Gases. § 33. EXPANSION OF GASES UNDER DIFFERENT CIRCUMSTANCES. In the first section, § 15, the n o n-reversible process of any body was fully discussed; following those discussions, the proposi- tions established can now be easily applied to the behavior of gases. Suppose a unit of weight of gas to be enclosed by a cylinder and its piston, and let volume Vi, pressure pi, and temperature Ti correspond to the initial state repre- sented by the point ai (Fig. 25), and let this state be one of equilibrium and rest; then expansion of the gas will occur as soon as the pressure pi acting from without on the piston K suddenly drops to the smaller value pi, and the work L', which is thus given off to the outside by expansion to the volume V, is determined when (by the course of the pressure curve acb, the "working-pressure curve") the law is known according to which the counter-pressure p', exerted from the outside upon the piston, varies during expansion. Now if, after attaining the volume v, the piston (supposed to be without weight) is suddenly held fast, then the gas, which has got into a condition of stormy motion, will gradually pass into the state of rest, and, after the state or condition of equilibrium is Fig. 25. 164 TECHNICAL THERMODYNAMICS. established, the pressure of the gas will assume the value p, which corresponds to the point 61 of Fig. 25; corresponding to this point there is also a particular temperature T, which we will designate as the temperature of equilibrium, whose relation to p and v is given by the equation of condition of a gas. But similarly every intermediate point c of the working-pressure curve has a point at Ci, a certain pressure of equilibrium and a certain temperatiu-e of equilibrium, so that accompanying the working-pressure curve ach there is a second curve, aiCihi, which may be designated as the equilibrium-pressure curve (p. 76). If the temperature of equilib- rium is known at the beginning and at the end, then according to equation (52a), p. 132, the quantity of heat consumed by the inner work is A{U-U{) = c,(T-Tr), and consequently we have for gases the quantity of heat Q' which must be supplied from without for this n o n-reversible process, according to equation (52), p. 132: Q' = c,{T~T^)+AU, (1) or, passing to the differential, we have dQ'^c^T+Ap'dv, (la) and, taking account of equation (43), p. 124, and of the equation of condition, there follows finally dQ' = d{pv)+A'[/dv (16) « — 1 These equations are sufficient to solve problems of the pro- posed kind. If the course of both pressure curves is known, the quan- tity of heat Q' can be determined; on the other hand, if the law of heat supply is known and o n e of the two pressure curves, then the course of the other can easily be determined. For the special case p' = p both curves coincide and the process is transformed into a reversible one. We will apply the foregoing formulas to only one case, which, however, embraces an infinite number of cases, and just EXPANSION OF GASES. 165 those which have some technical interest. We assume that the heat imparted during the assumed n o n-reversible process i s directly proportional to thie change of the temperature of equilibrium; consequently dQ' is determined by the formula dQ'=cdT, (2) in which factor c is given as any constant quantity whatever, which may be designated as the specific heat for the proposed case; furthermore let the course of the working-pressure curve acb (Fig. 25) be givenby equation V'v''=^Pi'V, ........ (3) in which equation the exponent r likewise represents any constant quantity whatever, and possesses no relation to the first-mentioned quantity c. Now the integration of equation (2) gives us at once the quan- tity Q'=c{T-T{), (4) which is to be supplied during the assumed change, and by substitution in equation (1) there follows the external work U, AL' = {c-c,){T-T{) (5) If we here substitute the temperature values in accordance with the relations given in equation (54), p. 134, and for the sake of brevity write n = '-=^, (6) in which the new constant n is completely determined by c, then equation (5) also gives the work in the form L'= AviVi-vv) (5a) n — 1 ' 166 TECHNICAL THERMODYNAMICS. But., on the other hand, on account of the changes of external pressure assumed in equation (3), we get from equation {7b), p. 154, for the external work, also L'^~L{p,'v,-p'v), (56) r — L and therefore by equating the last two equations n — 1 pv = piVi+ Ap'v-pi'vi) .... (7) r — L for the sought equilibrium-pressure curve; by means of this there can be found the equilibrium-pressure curve p and the equilibrium- temperature T for every value of the volume v, because the initial values pi, vi, and pi' are supposed to be known, and p' is given by equation (3). According to the choice c or r, or of n and r, we have before us an infinite number of special cases; the process is transformed into a reversible one when the two curves of working-pressure and of equilibrium-pressure coincide; in this case p\=p\ and p' = p, and from equation (7) there follows r = n; then, if c is given, the constant r is no longer arbitrary, but bears a relation to c given by equation (6). We then have before us the poly tropic curve which was treated above. On the other hand, if, in the case under consideration, the two curves are not identical, then we are dealing with the n o n- reversible process, several special cases of which will be discussed. Case 1. The change takes place without heat being im- parted or withdrawn. Here c = 0, therefore according to equa- tion (6) n = K, and the equation of the equilibrium- pressure curve, which is here to be regarded as an adiabatic curve of the general kind, is written K — 1 pv^pivi+—^{p'v-pi'v{), .... (7a) in which formula we can substitute for p' its value derived from equation (3) EXPANSION OF GASES. 167 Case 2. If we assume r=0, then equation (3) gives p' = ??/, the expansion of the gas takes place while overcoming the con- stant external pressure, and the equilibrium-pressure curve result- ing from equation (7) is pv = piVi—{n — l)pi'{v — Vi) (76) The curve is therefore an hyperbola. If this is not accom- panied by a supply of heat, we must also make n = K, and then pv = piVi-iK-l)pi'{v-Vi) (7c) Finally if we assume the external pressure pi'=0, that is, assume the expansion to take place without performance of outer work, we shall have pv = pivi; consequently the equilibrium-pressure curve passes into an equi- lateral hyperbola; because of the equation of condition pv = BT and piVi =BTi the equilibrium-temperature appears as a constant quantity. The latter case, whose numerical results might have been found in a shorter way directly from the fundamental equation (1), arises when the space filled with gas is put in communication with a vacuum. The expansion of the gas here takes place with- out the performance of external work, and experiments by Joule have in fact shown that the gas, after the expansion and passage into the condition of equilibrium, possesses the same tempera- ture as at the beginning of the experiment. Let us once more return to the case given by equation (7c) for which there is expansion, under the constant external pressure pi, without supply of heat. As a limiting case we can assume the expansion to be continued so far that the final equilibrium- pressure p is equal to the external pressure pi'; with this case equation (7c) gives Vl Kp K from which the expansion ratio can be computed. 168 TECHNICAL THERMODYNAMICS. The ratio of the temperature values T and Ti can then be found from T 1 K-l-p and because n = « the external work becomes, according to equa- tion (5a), « — 1 A Example. If a kilogram [pound] of atmospheric air of the tem- perature t, =15 C. [59° F.] expands, overcoming an external constant pres- sure of one atmosphere, at the same time expanding from the initial pressure of 2.5 atmospheres down to the terminal pressure of one atmosphere, then the foregoing formulas give -=2.064 and ^=0.8255. Consequently T ==-237.44: or <= -35.56° C. [7=427.392° F. or <= -32.008° F.], and the external work L' =3612.2 mkg. [L' =11851 ft-lb.]. (Compare examples on pp. 138 and 141.) Returning to the general case, represented by Fig. 25, the remark may not now be superfluous that the variable external pressure p' does not measure at the same time the gas-pressure itself during the non-reversible change; on account of the existing stormy and irregular motion of the gas particles during the process neither the instantaneous pressure nor the corresponding tem- perature can be determined. The designations p and T used in the foregoing presentation refer expressly, we repeat, to the state of equilibrium and rest which ensues when the expansion is sud- denly stopped at the volume v; neither can the gradual passage, from the condition of stormy motion to the state of equilibrium at constant volume v, be followed more closely. As regards the "transformation " of the non-reversible process discussed here, we can only refer to the general presentation in § 16 of the first section, p. 78; there we have simply to replace EXPANSION OF GASES. 169 the function S by the absolute temperature T and employ for the function P=rfs = iV,v) for gases, equations (55), p. 135; but the propositions enunciated in § 17, concerning the entropy of the non-reversible process of the kind before us as applied to gases, well deserve closer investi- gation. According to equation (la), p. 164, the quantity of heat dQ' for the non-reversible process is given by the expression dQ'^c^T+Aip'dv, on the other hand, if the passage had taken place in reversible fashion along the equilibrium-pressure curve (Fig. 25), then the quantity of heat dQ, which is determined by dQ = c^dT+Afdv, would have been needed. The subtraction of the two equations gives dQ'=dQ-A{'p-p')dv, and, if we divide both sides by A T, we get dQ' _ dQ (p-p')dv AT" AT T The first term of the right-hand side is under all circumstances a complete differential, even when the change from the condition ai to the state 6i has taken place, not along the equilibrium- pressure curve aiCihi, but along another reversible path {a\d\hi,, dotted in Fig. 25). The integration of the first term on the right- hand side is the change of entropy for the reversible passage and is found from equation (55a), p. 134, P-P.=^ loge ^ (8) 170 TECHNICAL THERMODYNAMICS. The integral of the expression in the left member is, on the other hand, the change of entropy for the non-reversible process; we briefly designate this by P', and if we substitute, as in equation (57), p. 81, N--filzf± (9) there follows P' = P-Pi-N (10) Now since the non-reversible process of the assumed kind permits of the integration indicated in equation (9), as soon as the course of the working-pressure and the equilibrium-pressure curve is known, we get through equation (10) the change P' of the entropy for this non-reversible process. In illustration we use the case recently treated, a case shown to embrace an infinite number of special cases. Consequently, heat is to be again imparted according to the law dQ'=cdT, and the course of the working-pressure curve is given by pV=pi'?;i'', where c and r are two constant quantities independent of each other. Utilizing the equation of the equilibrium-pressure curve, r — L as given by equation (7), and using the relation c-c-o n = - c-c„ furnished by equation (6), we get, after some easily followed trans- formations, from equation (9) for the case before us : accordingly we can compute from equation (10), with the help of equation (8), the change P' of entropy. MIXTURES OF DIFFERENT GASES. 171 For exaxople, if heat exchange does not take place so that c =0 and n = k, we get • vv" and P'=0. On the other hand if ?-=0, then p' = pi', and if in addition v' = 'P\ =0, that is, if the expansion takes place without the performance of outer work, then pu = piUi for the equilibrium-pressure curve, and consequently, according to equation (11), AN=c^{k-\) log«— . Further pursuit of the non-reversible process before us is of purely theoretical interest only. § 34. MIXTURES OF DIFFERENT GASES. The heat equations (53), p. 134, were developed under the hypothesis that we were dealing with the unit of weight of gas, but if we now assume that there is included in a space of F cbm. [cu. ft.] G kg. [lb.] of gas of the pressure p, temperature T, and the specific volume v, we first have the relation F=(?j;, and in order to determine the quantity of heat dQ, which the weight G of the gas needs for an infinitesimal reversible change of state, we must employ the following identical equations which follow from equations (53), p. 134, when we multiply their right members by G : dQ=-^d(Vp)+ApdV (12a) =c^GdT+ApdV (126) =—-[Vdp + KpdV] (12c) K J. =c«(?[dr + («-l)rY]. . . . (12d) =c,G[dT-'^T^j] (12e) 172 TECHNICAL THERMODYNAMICS. Let us now suppose that a space V is filled with two different gases, say with Gi kg. [lb.] of the one (a) and G2 kg. [lb.] of the second (b). Let the specific heats at constant volume be respect- ively c/ and c„", at constant pressure Cp and Cp", and the cor- responding ratio of these two specific heats be «' and «"; further- more at the common temperature T of the mixture, p' will be the pressure of gas a, and p" that of gas b, so related that the sum p' +p represents the total pressure p. If we now supply the quantity of heat dQ to this mixture, there follows from equation (126) dQ = icJGidT+Ap'dV) + {c^"G2dT + Ap"dV), or, if we collect the corresponding terms and represent the total weight Gi +G2 of both quantities of gas by G, and remember that p = p' +p" , we will get dQ = G h'^' '^'^^^^^ \dT+ApdV. Comparison of this expression with equation (126) shows that so far as heat supply is concerned the mix- ture behaves exactly like a single gas, whose specific heat c„ at constant volume is determined by cJG,+c"G2 .,„. G1+G2 ^ ' In exactly the same way when equation (12d) is employed we find for the mixture the specific heat at constant pressure Cp'G, +Cp"G2 '"" Gi+G. ' ^^^' and for the ratio of the two specific heats of the mixture Cp_Cp'G\+Cp"G2 c„ c;Gi+c;'G2 "a J (15) MIXTURES OF DIFFERENT GASES. 173 and for the differences of these specific heats Op c„ p , w^ ^ • • • • V^oi Suppose the mixture to be composed of more than two gases/ then we can write for the mixture, in general fashion : ■■.-^'- (13«) "im <'="' c,-o.-^2£|=|ia (16a) From the relation Cp — c„=A5, derived from equation (42), p. 124, and when Bm is the constant for the equation of condition for the mixture, and similarly By, B2, etc., for the separate gas6s, there follows according to equation (16a) R _ ^iGB) " I{G) ' as was already expressed by equation (19a), p. 110. According to the equations there shown the pressures of the separate gases of the mixture are p ~I{GB)' p ~y{GB)' • • • ^^'' After these preparations the mixing of two separate kinds of gases can be closely followed; the operation is n o t a reversible one. Let us suppose two spaces Fi and V2 (Fig. 26) separated by ' Some writers designate the product Gcv and Gcp, of the weight of the gas by the corresponding specific heat, as "calorific capacity for con- stant volume and for constant pressure respectively." The equations of the text then enunciate that the calorific capacity of the mixture is equal to the sum of the calorific capacities of the separate parts before the mixture. 1 i " 1 (i±ViTk (ilPlT, l> 7 i ' 174 TECHNICAL THERMODYNAMICS. a partition ab; suppose that in one space there exists Gi kg. [lb.] of pressure pi and temperature Ti, and that there are in the other space 0-2 kg. [lb.] of another gas having the pressure p2 and the temperature Tz; if the partition ab is removed there is a mixture of the two gases, and the question now is as to the character of the mixture, its pressure p and temperature T. Let us here assume that during the process of mixing heat is neither p 26 supplied nor withdrawn, and that therefore before and after the mixture the total heat contents of the two gases is the same; but we must also assume, what is decisive for the following developments, that the partition ab is so removed that the two kinds of gases are perfectly free to diffuse and expand. Very different conditions would exist if the partition had an aperture through which the gas, in the space having the higher pressure, could flow into the other space, a case which will receive full consideration later on. According to equation (526), p. 133, the heat contents, or the gas heat in the unit of weight of a gas at temperature T, is when Jo is a constant corresponding to the particular gas in ques- tion. This constant quantity is, to be sure, not known for the different gases, either as regards absolute value or as regards its ratio to the value of any particular gas, say to hydrogen, but in a mechanical mixture, as is assumed in the case before us, this cir- cumstance causes no difficulty. If we designate the constant for the mixture by Jo, the one for the gas in the space Vi by Jo' and for the other space by Jo", and if c„, cj, and c/' mean the specific heats at constant volume, then before the mixture the heat content Jm of the two spaces is Jn. = (GJo' + cJG^Tr) + (G2J0" + c;'G27^2) ; on the other hand after the mixture the heat in the two spaces is J,n = GJo + cfiT, provided the total weight Gi +G2 is designated by G. MIXTURE OF DIFFERENT GASES. 175 By equating both expressions we get the total heat contents J of a unit of weight of the mixture : ♦ Gi +G2 Gi +G2 From this follows the constant Jo for the mixture, J GiJo +G2J0" /1CV •^0 G,+G2 ' ^^^^ and if we utilize equation (13) we get for the mixing temperature m_ Cv'GlTi +C"G2T2 ,^Q, c:g,+c:'G2 ^'^^ From the equation of condition for gases pv=BT, from Gv = V and the relation (54), p. 135, we have c^GT^^. K — 1 and from this and equation (19), also IP_ym,V2P2^ (On^ «-l~/-l+«"-l' ^"^^^ from which formula the pressure p of the mixture can be directly computed. It is worthy of note that in equation (19) \^e can write, in place of the absolute temperature, the temperature according to Celsius [Fahrenheit], that is, write "'-m^ <-> We moreover recognize that the preceding formulas can easily be written for the case in which we have to do with the mixture of more than two gases. If we assume that the two gases to be mixed are of the same 176 TECHNICAL THERMODYNAMICS. constitution, then c„' = cv" = c„, and k! = k!' = k, and then the mix- ture-'s temperature is rp G1T1+G2T2 /inj,\ ^= G^+G, ^^^^^ and the formula for computing the pressure of the mixture is Vp = Vipi + V2P2, (20a) two equations of which much use has been made, of the former in Physics and of the other in technical investigations. On the other hand if different gases are mixed, and if their temperature is the same before the mixture, ii = <2, then equation (19a) also gives t = ti; consequently after the mixture has been effected (after the condition of equilibrium is established) the temperature of the mixture is equal to that before the mixture. Finally, if the two gases before the mixture have different tem- peratures but equal pressures, so that Pi=p2, then equation (20) shows that the mixture's pressure p is only identical with the initial pressure when k' = k", a case which exists for the simple gases. As a special example we will consider the mixture of hydrogen and oxygen in the ratio necessary to form detonating gas. Therefore if in the one space Vi (Fig. 26) there exists Gi kg- [lb.] of hydrogen, and in the other (32 = 8 Gi kg. [lb.] of oxygen, and if we assume each to have the same pressure and the same temp*ature T, then, according to the preceding propositions, the temperature and the pressure of the mixture known as detonating gas will remain the same as before the mixture. Since c/ = 2.4123 for hydrogen and c„ = 0.1551 for oxygen (table, p. 126), there results for the mixture (detonating gas) the specific heat c^ according to equation (13) : c^ = 0.4059. If Jo is the constant in equation (526), p. 133, for the heat contents of hydrogen, and Jo" that for oxygen, which values, to MIXTURES OF DIFFERENT GASES. 177 be sure, are still unknown, then the corresponding constant for detonating gas is given by equation (18) : Jo = ^/o'+|jo", (21) and the heat contents of detonating gas at temperature T and pressure p is accordingly / = Jo+cX in which equation 'the foregoing values of Jq and c must be sub- stituted. Now if we suppose, in order to pursue the case somewhat farther, that the detonating gas explodes (say by an elec- tric spark) at constant volume, the vapor of water will be formed with heat development and a great rise of temperature. If from this steam at constant volume we withdraw the quantity of heat Q in order to reduce it to the initial temperature T of the detonating gas, then the heat contents of the steam at the tem- perature T will be given by when Cv" represents the specific heat of steam at constant volume, and Jo" the heat contents, according to equation (526), p. 133. From the preceding equation there can be determined for the unit of weight of detonating gas the quantity of heat dQ developed by the explosion: Q = (Jo-/o"')+c„-c/")7^ (15?) Q therefore appears as dependent upon the temperature T, at which the chemical combination of the two gases was permitted. a proposition which was also discovered by Kirchhoffi for the case before us. • Poggendorff's Annalen, Vol. 103, p. 205; compare also C. Neumann •'Vorlesungen iiber die mechanische Theorie der Warme," 1875, p. 175. 178 TECHNICAL THERMODYNAMICS. The equation of condition for detonating gas was given in § 20, p. 115, as pv = lBoT; the steam formed from the detonating- gas on the other hand had, for the same temperature T and the same volume, a pressure p' determined from the equation of condition p'v=^BoT (§ 20, p. 116); in the present case, therefore, the final pressure of the steam is p' = §p. The foregoing results of course only hold when it may be assumed that the equation of condition for steam is the same as for gases. §35. FLOW OF A GAS FROM ONE VESSEL INTO ANOTHER FOR CONSTANT VESSEL=VOLL]MES.i Let two vessels A and B (Fig. 27) be connected by a pipe pro- vided with a cock F, and let both vessels be filled with one and the same kind of gas, say atmospheric air. In vessel A, whose volume is Vi, there will be at the beginning (with closed valve) Gi kg. [lb.] of air having the pressure pi, specific volume vi, and tem- perature Ti ; the other vessel, B, which we will re- gard as the receiving vessel, has the volume V2, and contains G2 kg. [lb.] of air of pressure p2, volume V2, and temperature T2. We will suppose both vessels to be impenetrable to heat, conse- quently the occurrence to be considered must take place without heat being supplied to, or withdrawn from, the outside. Suppose the cock F to be open for a time, then after it is closed, provided pi > p2, there will remain behind in the space A a weight of air Gx, and therefore the quantity of air Gi — G^ will have passed into the receiver B. This vessel B now contains the weight of air Gy = Gi+G2—Gx; after the efflux let pressure, volume, and temperature of the gas in the discharging vessel A be p^:, v^, and Tx, in the receiving vessel B on the other hand be Py, Vy, and Ty. ' The problem here specified was first solved by Bauschinger in an excellent article entitled "Theorie des Ausstromens voUkommener Gase aus einem Gefas und ihres Einstromens in ein solches," published in S c h 1 6- m i 1 c h ' s Zeitschrift fiir Mathematik und Physik, Vol. VIII, p. 81. FLOW OF GAS FROM ONE VESSEL INTO ANOTHER. 179 If the quantity of gas Gx, remaining behind, is regarded as given, the problem will be to determine the just-mentioned six magnitudes; but because three of them are related to each other by PxVt = BTx and pyVy = BTy, only four equations must be set up for the solution of the problem, and let us assume that the unknown quantities to be determined are the pressures Px and py and the temperatures Tx and Ty. To be sure the determination of these values assumes that, after the closing of the cock, the condition of equilibrium has been restored in each of the two spaces, for during the flow and at the instant of closing there exists a disturbance of equilibrium : the gas will be in stormy, eddying, motion, and the disturbance will be greater the smaller the contents of the two vessels and the greater the aperture through which the flow takes place. The expansion which occurs in the discharging vessel and the expan- sion which occurs in the receiving vessel are neither of them reversible, but if we assume a still smaller aperture of flow and large contents for the vessels we can assume that the expansion and compression take place in reversible fashion. The gas flowing toward the orifice will move in a funnel-shaped space and will diffuse itself similarly in the receiving vessel; the contents of the two funnels, and therefore of the weight of the gas moving in them, can, under the assumption made, be regarded as very small in comparison with the remaining gas mass which is at rest in the two spaces, and can therefore be neglected. In this case the values px and py are the equilibrium pressures, and Tx and Ty the equilibrium temperatures, which already exist at the instant of closing the cock; under the assumption made it is not even neces- sary to think of the closing of the cock, and it may be assumed that the pressures px and py, experimentally found by observing a manometer, represent the equilibrium pressure in the two vessels during the flow. If the valve is kept open till the pressure in both vessels has become equal, that is, till the flow originating from the difference of pressure is ended, then the temperatures in the two vessels will 180 TECHNICAL THERMODYNAMICS. still be different, and not until the aperture of flow is kept open for quite a period will equalization of temperatures by diffu- sion obtain; when it occurs the conditions will be of the sort discussed in the preceding article. Under these limiting assumptions, part of which will be dropped in subsequent investigations, the occurrences can easily be fol- lowed by calculation in the following manner. (a) Occurrences in the Discharging Vessel. Let the hatched part of Fig. 27 (p. 177), enclosed by the dotted line, represent the space occupied, before the opening of the valve, by the quantity of gas discharged up to the moment of closing the cock, then the remaining gas quantity Gx originally possessed the volume Vx = GxVi, which is shown in Fig. 27 by that part of the contents of the vessel that is not hatched. This quan- tity of the gas now expands in reversible adiabatic fashion to the vessel volunie Vi=GxVx = GiVi. From this we can now find the specific volume Vx, of the gas remaining behind when the orifice is closed: v^Gx' ^'^^^ and hence according to equation (60), p. 140, can be found the pressure jpx from the equation and from equation (61), p. 140, the temperature Tx is given by If we provide the discharging vessel with a manometer we can easily observe the pressure p^ at the instant of closing the FLOW OF A GAS FROM ONE VESSEL INTO ANOTHER. 181 orifice or in the case of continuous flow, at any time intervals; according to equation (26) we can compute, for every observed quantity of air Gx remaining behind, the weight of air which flows out in the several intervals of time. Equations (23) and (25) then give the specific volume v^ and the temperature Tx of the quantity of air remaining behind in the discharging vessel at the end of each interval. (b) Occurrences in the Receiving Vessel. The total heat content of the gases in the two vessels taken together is, according to the presentation in the preceding articles, at the beginning of the experiment, {Gi+G2)Jo+c,iG,Ti+G2T2); on the other hand at the end of the flow, after the close of the orifice or at any instant during the flow, the total heat is (Gx+Gy)J0+C^(GxTx+GyTy). Both expressions are equal, for during the operation heat is neither supplied nor rejected, and because we also have Gy+Gx-Gi+G2, (26) we will get the equation GyTy+GxTx=GiTi+G2T2 (27) From this formula we can now compute the temperature Ty, for an already given quantity Gx, because Gy is known to us from equation (26) and T^ from equation (25). By substituting the value Tx we get, directly, GyTy=G2T2+G^T^^l-(pY~^ (27a) If we multiply both sides by the constant B of the equation of condition of the gas, there follows, on account of the relations V2=GyVy = G2V2 and Vi=GxVx = GiVi, 182 TECHNICAL THERMODYNAMICS, the pressure fy in the receiving vessel Vv- and, provided the pressure px has been computed or observed, we also have, from equation (24), ViVx + V2Vy = ViVi+V2V2 (28a) At the end of the flow Px = Py = P, therefore, Then with the help of this value we can easily compute the quan- tity of gas transferred, the temperature in the discharging vessel and in the receiving vessel, quantities which, to be sure, are only valid for one instant because diffusion begins immediately after the equalization of pressure. The equations of the preceding problem furnish a basis for the treatment of a series of important special cases, of which a few, possessing scientific interest and which have occasioned special experimenting, will be discussed. §36, FLOW OF FREE ATMOSPHERIC AIR INTO A VESSEL. Let us assume that the cubic capacity Vi of the discharging vessel A (Fig. 27, p. 178) is infinitely large, therefore Gi = oo. This case exists when free atmospheric aif flows into a vessel which contains rarefied air at the beginning ; pi is then the atmos- pheric pressure, and T], the temperature of the external air; the initial temperature in the receiving vessel, say in the receiver of an air-pump, may have any value T2 whatever. To solve the problem it is well to subject the corresponding formulas of the preceding articles to a transformation; let G desig- nate the weight of gas which, in a given time, has passed from one vessel into the other, then we must substitute in the formerly FLOW OF FREE ATMOSPHERIC AIR INTO A VESSEL. 183 given equations Gx=Gi-G and Gy = G2+G; hence, for the dis- charging vessel, we have, according to equations (24) and (25), Both formulas give for the present case, Gi = oo, Px = Pi and Tx = Ti, a result that is self-evident. On the other hand for the changes in the receiving vessel equation (27a) gives ((?2+G)r,=G27'2+G,ri[i-(i-^y]. If we develop the exponential term according to the binomial theorem, that is, put / G\ G K(K-l) (Gy there follows {G2+G)Ty=G2T2+>cGT, -1^^^+ _. . . , and from this we get for (ri = oo the temperature Ty in the vessel after flow of G kg. [lb.] of the external atmospheric air into the vessel : -'^ G2+G ^''^' Multiplying both sides of the equation by the constant B of the equation of condition, there follows, after some easily made transformations, the pressure py at the end of the influx Cp{K-l)GTi and also kGTi Py=P^ + ^-AVl ' ^^0) Py = P2 184 TECHNICAL THERMODYNAMICS. From equation (29), which can also be written in the form _ AkT,-T,)G Ty-T2+ G, + G ' ^2^"^ we see that a rise of temperature is connected with the influx into the chamber. A very remarkable result is reached, and Bauschinger (ibid.) has already called attention to it, if we assume that the space into which the outer atmospheric air flows is empty (is a vacuum) at the beginning. Equation (30) gives in this case, for P2 = and ^2=0, c,{k-\)GT , ^"^ IF^ — ' ^^^^^ and equation (29) furnishes Ty=KTi (296) From the last equation follows, that at the entrance of the atmospheric air into a vacuum, from the first instant of opening the orifice, the temperature there rises suddenly up to the value KTi, and then remains, constant as long as the influx into the vessel (assumed to be impenetrable to heat) lasts. If we replace the absolute temperature in equation (296) by the temperature according to Celsius (Fahrenheit), there follows /j, = 111.93 +Kh [ij, = 188.354 + «-i)ri ' (^1) or, if we put, in place of the weight G of the air which has entered, its volume V measured at the external pressure pi, taking account in so doing of the relation Vpi=GBTi, and determining the con- stant B from equation (54), p. 134, we get V J'^^P^-^'^ (31^) Kpi Therefore if we provide the receiver of an air-pump with a manometer and with an orifice through which atmospheric air can enter, and observe during influx the manometer reading py at stated intervals of time, then according to the preceding for- mulas we can compute the weight of air G or the volume of air V, measured at the external pressure pi, which passes through the orifice during the separate intervals. In this way a method may be established of determining experimentally the quantity of air flowing through a given orifice in a unit of time for a given differ- ence of pressure, and the values thus found can be compared with the theoretical results of investigations on the efflux of air through orifices. 186 TECHNICAL THERMODYNAMICS. This way was in fact pursued by de Saint-Venant and W a n t z e 1 1 (1839) ; they allowed atmospheric air to flow through different kinds of circular orifices, whose diameters were 2/3, 1, and 1.5 mm. [0.02824, 0.03937, and 0.05906 in.], into the receiver of an air-pump whose cubic capacity to be sure was only 0.0174 cbm. [0.615 cu. ft.], in which there was generally at the beginning of the various experimental series a pressure of 10 to 20 mm. (0.3937 to 0.7874 in.) of mercury. The investigators in one experimental method allowed the air to flow uninterruptedly till the pressure was equahzed, and observed at equal intervals of time (every 5 seconds) the increment of pressure; in the second method they allowed the air to flow in with interruptions, after each period of 5 seconds the orifice was closed for a few seconds with the finger and the pressure immediately noted. The work of de Saint-Venant and W a n t z e 1 remained almost unnoticed for thirty years; doubtless this was due to the circumstance that it was Poncelet who declared in his report to the Paris Academy that the experiments were not decisive because they were conducted on too small a scale (with too small orifices); one must agree with this judgment, but must add that neither could the results of the then given experi- ments be regarded as reliable. During the development of the above formulas it was assumed that the receiving as well as the discharging vessel was impermeable to heat, a condition that cannot be satisfied because during the influx into the receiver a marked rise of temperature occurs there, in consequence of which the air in the interior gives off heat through the walls of the vessel to the external atmosphere during the whole duration of the experiment; consequently in the first experimental method the increment of pressure Py-p2 takes place to a less degree than is assumed in equations (31) and (31a). In the second method, to be sure, it was assumed that on account of the short period of influx (5 seconds) the loss of heat by radiation might be neglected ; ' "M^moire et experiences sur I'^coulement de I'air, d^termind par des diffe- rences de pressions considerables," by Barr6 de Saint-Venant and Laurent Wantzel. (Presented to the Academy of Sciences. Feb. 25, 1839.) Journal de I'Ecole poly technique, XVI, 1839. FLOW OF FREE ATMOSPHERIC AIR INTO A VESSEL. 187 but the experimenters should have closed the orifice not only for a few seconds but till there had been complete equalization of temperature with the external atmosphete. The beginning of influx ought to be permitted only when the initial temperature T2 in the vessel is identical with the external temperature Ti, which can be recognized when the . manometer reading for the closed vessel keeps constant. Now for the beginning of the experiment we have the relation V2P2 = G2BTi, and for the end, after G kg. [lb.] air have entered, we have V2P = {G2+G)BTi, provided equalization of temperature has taken place and the manometer reading has become stationary at the pressure p; a combination of the two equations then gives for the quantity of air entered • G = G2^, (32) or, if we measure the quantity of air G in cubic meters (cu. ft.) at external pressure, V = V2^P:^ (32a) Pi There are therefore two ways of determining the quantity of air admitted, either with the help of equations (31) and (31a), by reading the pressure py in the vessel at the instant of closing, or with the help of the preceding equations (32) and (32a) by observ- ing the pressure p at the end after equalization of temperature; the latter procedure is evidently the more reliable one. De Saint-Venant and W a n t z e 1 of course did not know equations (31), which are derived from the laws of ther- modynamics, but calculated the air volume V according to equa- tion (32a), and in so doing committed the error of not waiting for the equalization of temperature by substituting py in the formula instead of p. In spite of these defects, however, their article is of high value, for the authors have developed laws concerning the efflux of gases to which men have recently returned and which will be dis- cussed more fully below. 188 TECHNICAL THERMODYNAMICS. The foregoing computations are of interest also for other reasons; the behavior of the two equations (31a) and (32a) gives « = ?^^^ (33) P-P2 ^ ' Therefore if we allow air to flow into a vessel in which there is rarefied atmospheric air of the pressure p2 and of the tempera- ture of the external atmosphere, and if we observe the pressure Pj, at the instant of closing the orifice, and also observe the pressure p after equalization of temperature, then the preceding equation furnishes us with the means of determining the important physical constant «. If the vessel is a vacuum at the beginning, that is if p2=0, we get still more simply «■ = — (33a) In this way, corresponding to equation (33), Clement and Desormes determine the value k. Of course to safely assume that there is no heat exchange of the gas with the walls of the vessel during the influx depends on the influx orifice being open only a very short time. However, this condition is hard to satisfy; the walls of the vessel have the external temperature Tx, the entering air has, according to equation (29a), a higher temperature after expansion (spreading), and because it comes in contact with the cooler walls during its stormy motion, the consequent lowering of temperature will cause the observed temperature py to be read too small and hence it will turn out smaller than was ex- pected, and this, in fact, is what occurred in the experiments of Clement and Desormes. By the method given, there- fore, a reliable determination of the important magnitude is not to be expected, and of this the author has convinced himself by recent experiments conducted on a large scale. FLOW OF AIR FROM VESSEL INTO FREE ATMOSPHERE. 189 § 37. flow of air from the vessel into the free atmospher'e. If air flows into the free atmosphere from a vessel impermeable to heat and possessing the volume Vi, in which there exists at the beginning the pressure pi, the temperature Ti, and Gi kg. (lb.) of air, then after a certain time the pressure will fall to px and the temperature to Tx- If at this instant the orifice is closed and if Gx is the weight of air remaining behind in the vessel, then there exist between these quantities the relations given by equations (24) and (25), p. 180. If G is the weight of air discharged, we must put Gx=Gi-G, and equation (24) gives for the assumptions there made (reversible adiabatic expansion of air in the discharg- ing vessel during efflux) : If we designate by p2 and T2 the pressure and temperature of the external atmosphere, which do not undergo any change during the experiment, and assume beforehand that the temperature of the air in the vessel is originally identical with that of the external atmosphere, so that Ti = T2, then, according to equation (25), T2 \pj -1 (35) Consequently if we observe with the help of a manometer the initial pressure pi and the terminal pressure px, at the instant of closing of the orifice, then by equation (34) the quantity of air can be computed which attainfe efflux, and by equation (35) the temperature T^ at the instant of closing. If we wish to measure the quantity of air discharged in cubic meters under the initial inner pressure pi, and under the initial temperature T2, and designate this volume by V, we have the relation Vpi=GBT2. 190 TECHNICAL THERMODYNAMICS. In like manner the weight G\ is determined from the relation Vi'pi=GiBT2; the use of these two formulas in equation (34) also gives '-="■[-(#] (^») If we drop the assumption that the vessel is impenetrable to heat, then from the moment of closing the orifice, and because Tx a careful calibration with water); the pressure of the air in the interior was observed by an open mercury manometer. The ratio of the cross-section of 194 TECHNICAL THERMODYNAMICS. the vessel and the cross-section of the manometer tube was exactly 200. Now when the vessel is filled with compressed air and, after equalization of temperature, the orifice of efHux opened and then closed again after a short period, there occur the pressure varia- tions represented schematically in Fig. 28. The abscissa OA represents the time which has elapsed from the opening of the orifice, and the ordinate AB represents the pressure of the air, meas- ured by the mercury column, for the as- sumed instant of time; at the beginning of the time, and therefore at the moment of opening the orifice, the pressure pi is meas- ured by OBo; at the instant of closing the orifice (at the time OAi, which in all my experiments on the average amounted to 10 seconds) a high pressure px is ob- served, and, as represented by the ordinate AiBi, this pressure can be read from the manometer with sufficient certainty; now here the observed phenomenon occurs, namely, that the pressure px jumps suddenly to the larger value py (from AiBi to ^1162), remains stationary for a moment, and gradually increases, somewhat like the course of the curve B2B3, at first rapidly and then more slowly, till finally, after a sufficiently long time {OA3), it remains stationary at the value ^=^4.353; the course of the pressures in the latter period corresponds to the gradual equalization of temperature. The observation of the pressure Py is tainted with the same uncertainty, but the jump Py — px in the following experiments amounted to 13 mm. [5.2 in.] of mercury column; the jump was greater the higher the initial pressure in the boiler; for initial pressures less than two atmospheres the jump phenomenon disappeared; Weisbach's experiments did not exceed this initial pressure, but a much less excess of pressure existed in the experiments of some of the other experi- menters, which perhaps explains why none of them mentioned the phenomenon. Only de Saint-Venant and W a n t z e 1 have observed a similar phenomenon in their experiments, dis- cussed above, and say expressly that the top of the mercury column FLOW or AIR FROM VESSEL INTO FREE ATMOSPHERE. 195 in the manometer made a jumping motion of about 1 to 3 mm. [0.4 to .12 in.] at the instant of closingjthe orifice. One is inclined to ascribe the phenomena exclusively to the inertia of the mercury column in the manometer tube ; during the efflux the surface of the mercury sinks in accelerated fashion, and the constantly diminishing mercury column assumes a certain "velocity so that its surface outruns its true position when the orifice closes and its immediate return to an instantaneous posi- tion of equilibrium seems explicable. But closer reflection shows that this procedure is not sufficient to explain the phenomenon; we should rather seek in it the proof that in reality the quantity of air remaining behind does not expand in reversible fashion during the efflux, and that the rapid rise of pressure is, in the main, to be ascribed to the passage of the stormy motion into a condition of equilibrium. From the great number ^ of my different observations, one will be picked out as an example. Example. In one experiment the pressures, measured in milli- meters [inches] of mercury, were as follows: The initial pressure pi =2809.5 [110.61], the pressure px at the instant of closing the orifice was equal to 2584.9 [101.77], at the end of the jump pj, =2597.4 [102.24], and after equal- ' A short account of these experiments with the description of the experi- mental apparatus and method can be found in Der Civilingenieur, 1874, Vol. 20, p. 1 : " Resultate experimenteller Untersuchungen iiber das Ausstromen der Luft bei starkem Uberdruck." The experiments of 1871 were conducted by me in Zurich with the apparatus which belonged to the machinery collection of the Confederation's Polytech- nikum, and which was constructed according to my directions. As I left Zurich a few months after the completion of the apparatus, I had to limit the experi- ments to the investigation of a question which will be discussed later in the text. The question touched upon above I could unfortunately not pursue more thoroughly in an experimental way. It would be of great value if the experiments with high pressure could be taken up again by others; the highest pressure inWeisbach's experiments only went to 2 atmospheres, in my experiments it went to 4 atmospheres. The greatest diflBculties are encountered by virtue of the fact that at high pressures the air penetrates the metallic pores; in my apparatus the air went through the inch walls of the metal dome. After much trying, and after the inner walls of the boiler were painted with red lead and all brass domes (mountings) were dipped into boiling linseed oil, we succeeded in making the apparatus air-tiq-ht and in avoiding losses of air. 196 TECHNICAL THERMODYNAMICS. ization of temperature p =2639.9 [103.93]; the temperature of the external air was «, =14.6° C. [58.28° F.]; the barometer reading was 722,1 mm. [28.43 in.], the air flowed for 9.9 seconds through a circular, well-rounded, orifice having a diameter of 5.73 mm. [0.2256 in.]. Boiler capacity 'V\ =0.81088 cbm. [28.637 cu. ft.]. Here we first compute the weight of air which the boiler contained at the beginning and at the end of the experi- ment according to the formulas V,p,=G,BTi and V.p^GxBT,, •where B =29.269 [53.349]; for the pressures we must substitute the specific values obtained by multiplying the pressure in millimeters of mercury by 10333 : 760 [2116.31 : 29.922]. In the present case there results: Gi =3.6796 and Gx =3.4575 kg. [Gi =8.1124 and Gx =7.6227 lb.]. The difference Gt—Gx is the quantity of air discharged in 9.9 seconds; these values are moreover independent of all occurrences which happen at the close of the orifice. According to equation (24), p. 180, if we replace the exponent by the letter r, and thus assume that the quantity of air Gi remain- ing behind expands during the period of efflux according to the law pt)' = con- stant, we can compute, with the help of the preceding experimental data, r = 1.338, which is essentially smaller than « = 1.410. The work L, which the air remaining behind performed on the discharged air, is, according to equation (76), p. 154, 1 — i when its initial volume at the pressure pi is designated by V. Now if py really corresponds to the equilibrium pressure, the change of inner work of the air remaining behind, according to equation (52), p. 132, is f;_[7,=-i-(y,p„-y», and then the quantity of heat Q, which the air remaining behind absorbs from the walls of the vessel during the period of efflux, can be computed from the formula Q = A{U-U +L, where the volume V of the preceding equations can be determined from t F'=y.H^. ^pi' FLOW OF A GAS FROM ONE VESSEL INTO ANOTHER. 197 The calculation can easily be pursued further even for this present example. However we will simply remark that the quantity of heat and the work just computed relate only to the air r e m a i^ i n g behind. But the quantity of air discharged has Ukewise absorbed heat and done work on the air preceding it. A further discussion of the problem would lead us too far for the purpose of the present book. § 38. FLOW OF A GAS FROM ONE VESSEL INTO ANOTHER WITH VARIABLE VESSEL=VOLLIME AND WHEN HEAT IS IMPARTED. In the general case, which was raade the basis of the investi- gations of the last article, the flow of the gas from one space to another, the capacity of both vessels was regarded as constant (§ 35, p. 178, Fig. 27), and at the same time the limiting assump- tion was made that no heat was imparted to either one vessel or the other during the efflux. If we drop the limitations mentioned, then we strike the problem which is of technical importance, because it arises in the fuller investigation of air engines.^ For the clarification of the problem, suppose the three pistons Ki, K, and K2 (Fig. 29) to form with the cylinder the spaces A and B which are filled with air. Let the middle piston K (called the transfer-piston in air engines) be regarded as impenetrable to heat, and let it contain perforations which are provided with valves. In a part of these holes the valves open in one di- rection, and in the other part in the opposite direction, so that all changes of pressure occurring in the two spaces are immediately equalized by the flow of a corresponding quantity of air, through the answering valve, out of one space into the other; then the pressure p, i n itself variable, nevertheless always possesses the same value in the two spaces. ' The problem was first treated by the author in Der Civilingenieur, 1883, Vol. 29, p. 557: "Uber die Wirkung des Verdrangers bei Heiss- und Kaltluft- maschinen." 198 TECHJNICAL THERMODYNAMICS. Let the space A have the variable volume Y x and, at the moment in question, let it enclose Gx kg. [lb.] of air of the tem- perature Tx\ in the other space B let there be at the same time the volume Y y, containing Gy kg. [lb.] of air of the temperature Ty, but, for the reasons mentioned, possessing the same pressure p as the space A. Now if we suppose the piston K to be stationary and, in any way whatever, heat to be supplied to, or withdrawn from, the two spaces A and B, the two pistons Kx and K^ simultaneously retreat- ing and performing work, then the problem arises of the state of the air in the one space, and in the other space for any piston position whatever; the question also is as to the quantities of work which have been produced or consumed, and finally as to the quantity of heat which has been absorbed or rejected by the one space and the other space. It is worthy of note that the same law of change of the spaces Y,x and V y can be brought about if we regard one of the two external pistons Ki and K2 (Fig. 29) as stationary and the other two pistons movable, which cases in fact occur in constructed air engines; the valves mentioned as closing the holes in the transfer piston do not exist in the actual constructions, but have been assumed here for the purpose of more easily following the theoretical developments. Let us first assume the valves in the holes to be closed, then the equation of condition of gases gives for the air in space A the relation VxP = BGxTx (38) and for the space B VyP^BGyTy (39) Let V designate the variable total volume, and let G be the total constant weight of the air in the two spaces, then we have V=Vx + Vy and G=Gx + Gy; FLOW OF A GAS FROM ONE VESSEL INTO ANOTHER. 199 from the addition of equations (38) and (39) follows Vp = B{G^T,+GyTy) (40) If under like circumstances the temperature in the two spaces is equally great and equal to T, the relation Vp = BGT holds, and the foregoing equation then becomes GT^GJ'^+GyTy, (41) from which formula the mean temperature T, i.e., the equalized temperature, can be computed ; the intro- duction of this magnitude renders some of the following investiga- tions easier. The combination of equation (38) and equation (39) moreover gives f 4:4: <«) because G=Gx+Gy, or, using the relation Vp=BGT, V Y^ Vy f = T^+T ^^^''^ Now supply to the closed space A the quantity of heat dQx and to the space B the heat dQy, and in so doing let the volume Vx increase by dV^ and Vy by dVy, and let the corresponding change of pressure be dp' in space A and dp" in space B, and let the corresponding changes of temperature be designated by dT^f and dTy'; then we can here use equations (12c) and (12e), p. 171. and obtain dQx= — 7{VJp'+pdV^) .... (43a) K A. and =CpGx(dTx'-'^Tx—\ . . . (436) dQy = — ::{Vydp" + KpdVy) .... (44a) ^^c.Gy^dTy'-'^Ty^y . . . (446) 200 TECHNICAL THERMODYNAMICS. After the two separate spaces have been supplied with the given quantities of heat imagine the valves in the passages (Fig. 29) to be opened until equalization of pressure has taken place; then, at the end, the pressure in both spaces will be p+dp. Furthermore, after equahzation of pressure, let Tx+dTx be the temperature in the space A, and Ty + dTy that in the space B. The introduced differentials can now be easily determined; first of all, as regards the relation between dp', dp", and dp we have in equation (20a), p. 176, when applied to the present case, simply to substitute p+dp in place of p, p+dp' in place of pi, and p + dp" in place of pzi also Vx+dVx in place of Vi, and Vy+dVy in place of V2, we then get , _ {V r+dV ,){p + dp') + {V y + dVy){p+dp") P'^'^P Vx + dVx + Vy + dVy Hence there follows, when the multiplication is effected and differentials of higher order are neglected in comparison with those of the lower order, VJp' + Vydp" dp = - Vx+Vy and because the denominator of this fraction simply represents the instantaneous total volume V we get Vdp = Vxdp' + Vydp" (45) We get besides, from differentiation of V = Vx + Vy, dV^dVx + dVy (45a) Now if we add the two equations (43a) and (44a) we get in the value dQx+dQy, which we will designate by dQ, the whole quantity of heat absorbed by the air in the two spaces.i4 and B, or dQ = —JVxdp' + Vydp" + Kp{dVx+dVy)]; K X or, utilizing equations (45) and (45a), dQ= — -AVdp + KpdV) (46) FLOW OF A GAS FROM ONE VESSEL INTO ANOTHER. 201 This equation gives for the present investigations an interesting and important result. The equation is identical with equation (12c), p. 171, and enunciates that the quantity of heat dQ, which has been absorbed by the whole quantity of air in- b o t h spaces and partly transformed into work, is determined exactly as if the air existed in only one space, V, under the pressure p. The same law also holds with respect to the external work pro- duced; the work in the space A is pdF^, and in the other space pdVy, the whole external work is therefore dL = pidV^+dVy)=pdV, and as if the air filled only one space V. We recognize, moreover, that the present propositions are valid for more than two spaces; no matter how the weight G of the air may be distributed in the several spaces, no matter what temperature may exist in each of the several spaces, the quantity of heat developed by the total mass of the air and the quantity of work which it produced or consumed can be always computed by equation (46), as if the air occupied only one space, pro- vided the spaces are so connected with each other that in all of them the same pressure p prevails. Equation (46) can easily be brought into the other forms given under (12), p. 171, if we introduce for T the mean or equalization temperature. We will present only one of these transformations. From the already used relation Vp=BGT there follows, by differentiation, Vdp+pdV = BGdT, and if we divide this expression by the preceding we get dp dV_dT p^ V ~T- Equation (46) can also be written as follows : 202 TECHNICAL THERMODYNAMICS. dV and from this results, if we eliminate -y- from the last two equa- tions and use the relation (54), p. 134, dQ = CpG[dT-^V^] (46a) Consequently what was said of equation (46) is true of this for- mula, only we have to remember in using it that T is the mean temperature determined according to equation (41). Finally as regards the main purpose of the problem, namely, the determination of the quantities of heat Q^: and Q„, which can be separately imparted to each of the two spaces, we must first know which is the discharging and which the receiving space during the equalization of pressure; if we assume in the following that the air flows from A to B, then during the equalization of pressure the quantity of air which originally amounted to Gx kg. [lb.] will at the end amount to Gr-dGx', this quantity of air remaining behind expands adia- batically, consequently for this procedure the equations (24) and (25), p. 180, are valid; then we must put p +dp in place of Pa; and p +dp' in place of pi, T^ + dT:, in place of T^ and T^ +dTJ in place of Ti, Gx - dGx in place of Gx and Gx in place of Gi, and therefore get p+dp /Gx — dGx^ and ( Gx-dGA ' p + dp' \ Gx ) Tx + dTx _( Gx~dGx \-'- Tx + dTx' \ Gx I ' If we multiply the left side of the formula above and below by ip-dp') and similarly the other by {T:,-dTx), and develop the expressions to t^e right according to the binomial thsorem, and cancel the differentials of higher order and of the higher powers, we get the following relations: dp' = dp + Kp-g- FLOW OF A GAS FROM ONE VESSEL INTO ANOTHER. 203 and With the help of equations (45) and (42) we could express in a similar manner the differentials dp" and dTy. Now if we substitute the preceding values of dp' and dTx' in equation (436) there results dQx=c,Gx(dT,-!^T,^y .... (47) or considering equation (38), p. 198, and equation (43), p. 124, dQ.=AV.p[^^^-^], . . . .(47a) and this formula likewise gives an interesting result which greatly facilitates the investigation of special cases; it enunciates that the quantity of heat dQx is determined as if the discharg- ing space were shut-off, and the quantity of air Gj. there existingwere constant, while the temperature rise dTx and the pressure rise dp need to proceed only so far as to correspond to the condition after the equal- ization of pressure. The determination of the other quantity of heat dQy, which is to be supplied to the receiving space, now easily follows from the relation dQ=dQx+dQy; we find, when we use equations (46a) and (47), dQy=cJGdT-GxdTx -^(GT -GxTx)^\ or if we remember that because G = Gx+Gy we have dGx = —dGy, and if in addition we employ the relation which comes from equa- tion (41), we get clQy=c^Gy[dTy-'^Ty'^'j+c^iTy-T.)dGy, . . (48) 204 TECHNICAL THERMODYNAMICS. or considering equation (39), p. 198, and equation (43), p. 124, K dTy dp' dQ „ = ^F,p[;^ y-"-^] + c,(r,-r,)d(?„. . . (48a) The first term of the right-hand side again gives the quantity of heat to be supplied to the s h u t - o f f receiving space if the temperature and pressure are to be brought directly to the equalization values; but the second term corresponds to the quantity of heat which can be supplied to the transferred quantity of air dGy, in order to heat it at the constant pressure p from Tx to Ty. Inversely, if the space B is the discharging space and A the receiving space, the second term of equation (48) will dis- appear, and should be added to the right-hand side of equation (47) in the form Cp{T^-Ty)dG. In most of the cases of the practical utilization of the preceding propositions it is guessed beforehand which of the two spaces A and B is the discharg- ing - and which the receiving- space. In a doubtful case a guide is secured by the following consideration. Division of the two equations (38) and (39) gives V T try— Wly ™ , consequently there follows, because the total weight of air in the two spaces G = Gi-VGy\^ constant, G G,= V T V T y z i y Now if A is the discharging space, the denominator on the right- hand side of this expression must increase, and therefore the condition « must be satisfied. The foregoing investigations form the basis for the solution of numerous problems of technical importance, but before discussing some of the most important we will give the graphical treatment FLOW OF A GAS FROM ONE VESSEL INTO ANOTHER. 205 77- ^ 17- -B» If V IJ ^ >^:\ 7 H^ p ^* i \ Fig. 30. of a part of the problems, and thus extraordinarily facilitate an insight into the following problems and into the occurrences which exist in hot-air and cold-air engines. In Fig. 30 suppose a cylinder, closed at one end, to have two movable pistons K and Ki, of which the first is the transfer-piston and is represented by a broken straight line. Let both pis- tons be supposed to be in motion and that the former at the beginning stands at K' and the other at K/; the space to the left of the trans- fer-piston, at the time t, is designated by F^, at the begin- ning by Vi, at the time t let the space between the transfer m piston and the outer piston, which latter we will call the power-piston, be designated by Vy and at the beginning of the time (say i=0) by V2. We recognize moreover that the difference from the arrangement in Fig. 29 consists in that here one of the outer pistons Ki and K^ is left out, and as a compensation the transfer-piston is here movable, while there it was regarded as stationary ; it amounts to the same thing in both cases, the arrangement in Fig. 29 is simply better suited for the development of the fundamental equations, while Fig. 30 is better suited for the explanation of the special investigations. In order to represent to the eye the law of change of the two spaces Vx and Vy, and hence the law of the piston motion, lay off OM in Fig. 30 from downward on the axis of OX for the elapsed time t, and draw through M a horizontal on which the distance MP is to represent the volume V^ and the distance PR the volume V^; the whole distance MR then repre- sents the total volume V = Vx + Vy of the air in the two spaces. At the beginning of the time (t=0), OPo similarly represents the volume Vi, and PqRo the volume V2; the sum of the two spaces Vi + Vz is designated by Vo and is given by the distance ORq. 206 TECHNICAL THERMODYNAMICS. From all this we see that when we think of continuous motion of both pistons the curves Pofl and RqR present to the eye the law according to which Vx, Vy, and V change with the time. I call this part of Fig. 30 designated by I the " piston. dia- gram " i; to each piston there corresponds a particular value of the pressure p of the air in the two spaces; if we again lay off the elapsed time' t, as O'M' from 0' along the vertical axis O'X', and make the horizontal distance M'h' equal to -p, and make, for the beginning of the time (i = 0), the distance O'a' equal to the initial pressure po, we will get, in the curve a'h' , the law according to which the pressure in the whole interior of the cylinder varies with the time; we designate the corresponding part II of Fig. 30 as the "piston force diagra m." Finally in the upper part of the figure, designated by III, there is drawn the ordinary "pres- sure- or indicator-diagra m" ; the curve ah (pressure curve) gives the variation of pressure which occurs when the total volume changes from Vo to F; the area enclosed by the curve ah gives the work which is performed during expansion. If, starting from a fixed point, we at one time lay off on a horizontal axis the variable volume MP = 'Vx as abscissa, and at another time the volume PR = Vy, and both times lay off the pressure p as ordinate, we shall get two more representations of the pressure curve, which have a practical significance for certain air engines because they can be obtained by the application of two indicators. In engines the pistons are as a rule moved by cranks which turn with nearly uniform rotation; in this case the time interval OM is proportional to the crank angle, and the curves in the piston diagram are sinusoids, provided we assume infinitely long ' The diagram presented, which I already gave in the secon'l edition of this boolj (1866, p. 200), has since been widely used, and from that time I have utilized this method of representation in my lectures on steam engines, and particularly for the graphical computation of engines with several cylinders, the compound engines. A full illustration of it, with practical examples, is given by Schroter in the Zeitschrift des Vereins deutscher Ingenieure. 18S4, Vol. 28, p. 191; he gave it, to be sure, without the addition of the piston force diagram, which latter renders good service in determining the turning force diagram and the degree of fluctuation in the motor of the engine. APPLICATIONS AND SPECIAL PROBLEMS. 207 connecting-rods, or curves resembling the sinusoids, if this condi- tion is not fulfilled. In engines moreover the two pistons return periodically after each revolution to the initial positions, and the curve ab in diagram III (Fig. 30) then forms a closed curve; the whole process is transformed into a cycle which for certain engines will be subjected later on to a closer examination. § 39. APPLICATIONS AND SPECIAL PROBLEMS. After the foregoing preliminaries we will now adduce a few examples which contain applications of the propositions given in § 38. Problem i. The power-piston Ki (Fig. 30) is held fast, on the other hand the transfer-piston K is shifted from right to left, so that space A experiences a decrease and space B an increase. Fig. 31 repre- sents the corresponding diagram; since the total volume V = Vj: + Vy = Vo is to be constant, the curve RqR for the power-piston is a straight line parallel to the ^. axis OX; the curve PqP gives the law according to which the shifting of the transfer-piston takes place with respect to the time, OPo gives the initial vol- ume Vi of the one space, and PoRq the initial volume V2 of second space; let Ti here be the temperature to the left of the transfer-piston and T2 that to the right, po the initial pressure in both spaces, time t the position of the transfer-piston is at P, Sd C -A- B Fig. 31. After the and now there should be calculated, as unknown quantities, some of the magnitudes T^i, Ty, p, Q^, and Qy corresponding to the values Vx and Vy = Vo-Vx- The problem presents many special cases of which a few will be treated. First of all there is no doubt that, under the assumptions made, the space A to the left of the transfer- piston is a discharging space and B a receiving 208 TECHNICAL THERMODYNAMICS. space, and that external work is neither produced nor con- sumed because it is assumed that the total volume V is constantly equal to Fo- Special case a. During the motion of the trans- fer-piston heat is n "either imparted to, nor withdrawn from, the space A or the space B. Therefore dQx=0 and (iQj,=0, and consequently also dQ = Because also dV=0 it follows from equations (46) and (46a) that dp=0 and dT=0; we therefore obtain the result that during shifting of the transfer-piston the pressure p and the mean tem- perature T remain unchanged in the two spaces, remain respect- ively at po and Tq. Of course in so doing (as it is now only a question of theoretical elucidation) we must imagine the transfer- piston to move slowly and must assume that the flow from one space into the other (through the transfer-piston) takes place without resistance. The substitution of dp=0 and of dQx=0 in equation (47) gives also dTx=0, from which is to be concluded that, during the whole occurrence, the temperature Tx remains constant and therefore Tx = 71. Matters are different in the receiving space B. Here the temperature Ty is variable and can be computed from equa- tion (42a), making use of the preceding notation, as follows: Tq T, As p was found to be constant, we have in the present case for the pressure curve a'h' (Fig. 31) a straight line parallel to the axis O'X'. Therefore, among all the quantities occurring in the calcula- tion, it is solely the temperature Ty which varies with F^. If we should now bring the transfer-piston back to the original position, then, for the return motion, the space B would be the discharging space and A the receiving space, the temperature Ty would remain constant and the temperature Ti would assume another value; we see at once without further cal- APPLICATIONS AND SPECIAL PROBLEMS. 209 culation that by the continual reciprocating motion of the transfer- piston a gradual equalization of temperature results till finally there exists in both spaces the mean temperature T^o- If we shift the transfer-piston immediately from position Pq (Fig. 31) to the left end Pi, then Vx=0, all air is pushed into the space Vy, and equation (49) gives Ty = To; the mean temperature is therefore immediately established, and further shifting of the transfer-piston will not cause any more changes in the pressure and temperature of the air in the two spaces. Special case b. The temperature of the air is to remain constant in the two spaces, while the power-piston is held fast and the trans- fer-piston moves. This assumption demands that heat be imparted and with- drawn, as the case may be, on both sides of the transfer-piston. Because here Tx = Ti and Ty = T2, we have from equation (42) V and, for the initial position of the transfer-piston, BG_V^ V2 po~T,'^T2' and hence, by division of the two equations, p V,T2 + V2T^ Po V.T2 + VyTi- (50) If we designate the constant ratio T2 :Tihy ^ and put V2 = Vo-Vi, also Vy = Vo — Vx, then from the foregoing formula we get p 7o + (^-l)F, po Vo + {X-l)V,' (50a) consequently the pressure p in both spaces changes with the shifting of the transfer-piston, and can be easily computed from every value of Vx- If the volume Vx diminishes, as is assumed in Fig. 31, an increase of pressure p occurs when A>1 or T2>Ti 210 TECHNICAL THERMODYNAMICS. or when, as is said in discussing hot-air engines, air is forced from the cold space A to the hot space B. On the other hand if, for the same motion of the transfer-piston, A were the hot space and B the cold space so that T2 Consequently Ty can be computed from every value of Vy and for the corresponding value of p. (^) Compression (Fig. 33, p. 213). Here the circumstances are very different from those of expan- sion. As the space B is now converted into the discharging space, we must omit the second term of the right member in equation (48), p. 203, and from this follows dQy^C,Gy(^dT-'^Ty^^. and from this in turn results, because dQy = 0, PA0^ <-) Consequently the temperature Ty in the space B is determined for every value of the pressure p. On the other hand, because A is now the receiving space, equation (47) can be written dQ:, = Cr,G,[dT,-~T,-^]+Cj,(T^-Ty)dG,. APPLICATIONS AND SPECIAL PROBLEMS. 219 But Tx = Tx and dTx = 0, and the equation of condition gives BGxTi = Vi'p and BTidGx = Vidp; consequently the equation for dQx reduces to the following: and if we here utilize equation (59), we get, by integration, the quantity of heat Qx which can be supplied to the constant space A : As regards the course of the compression curve ab (Fig. 33), here also the relation (68) holds, and if, for the sake of simplicity, we represent the ratio T2 : Ti of the initial temperatures by A and utilize the temperature Ty according to equation (69), we get for the equation of the curve Vy = [iXV, + V,)PJ>^XV,]{^J-^\ . . . (71) that is, a very complicated form. If we determine from this dV,, we finally get the work of compression L = j'pdVy, and, if we integrate between the corresponding limits, there finally follows For the case that the quantity of air existng in space B is forced completely into the space A we should have Vy = 0; equa- tion (71) then gives the pressure p at the end, by p W1 + V2 Po Ay, ' and then the foregoing equations will also permit the determina- tion of the corresponding values of Ty, L, and Qx. 220 TECHNICAL THERMODYNAMICS. Problem 3. In the foregoing investigations it was at one time assumed that only the transfer-piston moves, and at another time that only the power-piston moves. It is now assumed that both pistons are in motion, which leads us back to the general case represented by Fig. 30, p. 205, but for the purpose of the following investiga- tions it seems profitable to bring the equations there given into other forms. Using Fig. 30 as a basis and the given notation, we find from equation (42), when we designate the initial pressure in both spaces by -po, the initial temperatures by T^ and T2, and the initial volumes of the two spaces by Vi and V2, for the computa- tion of the pressure p, the formula Po \V.Ty+VyTjT,T2 ^'''^ and the external work L can be found by integration of the equa- tion dL = pd{V^+Vy) (74) Now if A is the discharging space and 5 the r e c e i v - i n g space, the heat Qx can be found from equations (47) and (38) to be dQ.= JL^AvJ^p-'-^^-2\ "-i LTj. K pj . . (75) and from equation (48) and equation (39), after suitable reduction, we get K— 1 1 y I. 1 y \ K I p ' y A These formulas assume that ^isa discharging space, which is the case when irM>' (77) as was proved on p. 204. If the inequality (77) is not fulfilled, the space B becomes the discharging space; we must then, in the two preceding formulas, everywhere interchange subscript x with subscript y. APPLICATIONS AND SPECIAL PROBLEMS. 221 Let US now discuss as a special case that one in which the temperature is constant in the two spaces, in which therefore Tx = Ti and Ty = T2, an assumption which has several times been the basis of investigations of hot-air engines. If we again designate the constant ratio T-z'-T^ by X, then the foregoing equations give the following expressions : 7o=AivrF,' (^3«^ dQ:,= - AV xd-p, (75a) " ^T7 /X-Kdp k-ldVy\ ,_„ , which presupposes that A is the receiving space; hence d (r)«> <"«' Special case a. During the motion of the two pistons the pressure remains constant in the two spaces. From equation (73a) and because p = po it follows that Vy = V2-\-XV,-Wx (78) and y„ V2+W '-X, from which d(0 = -(V2+AF:)^^ Therefore if A is the discharging space, according to equation (77a) dVj: must be negative, i.e., space V^ must decrease and hence, according to equation (78), Vy must increase; because V = Vx+Vy the latter equation gives also the total volume: F=F2+AFi-(A-l)F,. Accordingly this volume experiences a decrease (the power-piston like the transfer-piston [Fig. 30] moves from right 222 TECHNICAL THERMODYNAMICS. to left) when A<1 or r2Ti is assumed. The outer work is found from equation (74) L=(F-yo)po=(A-l)(7i-F,)po. Finally the quantity of heat which must be imparted to the dis- charging space A. is, with dp = 0, found from equation (75a) to be and the heat for the receiving space, according to equation (76o), is «(A-1) kU-i) Qy = J(^^^^^y- ^^yPo-^—^AiV,- F,)po, or also X{k-\Y Q,= —-,AL, which equation can easily be expressed in words. Special case b. The temperatures in both spaces are again constant, but both pistons move uniformly from left to right. Let C\ be the velocity of the transfer- piston and c that of the power-piston, then if F represents the cross-section of both pistons, we have, at the time t, V. = V, + Fcit and 3 Fig. 34 The combination of these equations furnishes c V=^Vo+Fct, and accordingly, by subtraction, Vy=V2+F{c~Cr)t. Vy y„--F. V. -+ C — Ci APPLICATIONS AND SPECIAL PROBLEMS 223 and accordingly i^y-iy'-i'^m- Now assuming that A is the discharging space, the right member must be positive according to (77a); this requires, since Vx was assumed to increase, the fulfillment of the condition ^^i <™) If we designate ci : c by m and eliminate the time from the three equations found for V^, V, and F„, there results V. = V, + m(V-Vo), Vy=V2+a-m)(V-Vo), so that both magnitudes are expressed in terms of the total volume V; the substitution in equation (73a) then gives the value of p expressed in terms of V, and consequently gives the course of the pressure curve db, whereupon, by integration, equation (74) easily determines the external work performed. The quantities of heat Qx and Qy can also be easily determined with the help of equations (75a) and (76a). There is no difficulty whatever in making the whole calculation, but we will here omit furnishing the results of the computation because very compli- cated expressions occur and the problem is of subordinate technical importance. Supplement. The foregoing problem was principally discussed in order to indicate in conclusion how the problem is to be handled (under the like assumption that the temperature is kept constant in both spaces) when the two pistons do not move uniformly, i.e., when the given piston-diagram has the character of that shown in Fig. 30, p. 205. If the two spaces Vx and Vy are given as functions of the time, so that y. = /'(«) and Vy = i"{t), 224 TECHNICAL THERMODYNAMICS, there follows F=/'(0+/"(0. Eliminating the time t from these equations, V x, and.Fj, as well, can be expressed by the total volume V. Substituting these quan- tities in equation (73a) will give the relation for the pressure curve ah, and then the heat quantities Qx and Qy can be computed from formulas (75a) and (76a) by integration between the corresponding limits. The velocities Ci and c are then variable and can be expressed by ci = -^- and c=-^, and the condition that A be the discharging space requires, ac- cording to equation (776), that during the course of the pressure curve (for which the integration of equations (75a) and (76a) is to be effected) the relation must be satisfied; the last two equations are to be interchanged from the moment in which We have to do with the last indicated case in certain, con- structed, air engines— a problem to which we will return. FUNDAMENTAL FORMULAS FOR THE FLOW OF A FLUID. 225 III. Flow and Efflux* of Gases. § 40. FUNDAMENTAL FORMULAS FOR THE FLOW OF A FLUID. In discussing the general problem of Hydrodynamics difficulties are met which up to the present time could only partially be overcome. To be sure they are not removed when the laws of Thermodynamics are employed in treating the question. Never- theless extensions result which have become irnportant for certain technical problems belonging to this field. Let us first assume that a fluid of the liquid or gaseous variety flows through a horizontal pipe (Fig. 35), possessing a variable cross-section, and flows without any action on the part of external forces; let us here assume a perma- nent or normal condition, and consequently that in a unit of time the same weight G flows through every cross-section. Let us assume a further r 1* limitation, namely, that through every - SZl^, S— - w element of the forward cross-section F the ^ J liquid flows with the same velocity w and ** that the direction of the velocities in all points of the cross-section are parallel, so that the volume of the fluid which passes the cross-section F in one second is repre- sented by Fw; similarly this volume is given by FiWi for the back section F^. li v is the specific volume and p the pressure in the forward cross-section, and if we designate these same quantities for the cross-section Fi by Vi and pi, then first of all we get for the normal condition the two reltaions Gvi = FiWi and Gv=Fw (1) Now let us follow the unit of weight of fluid on its path from Fi to F. Disregarding the present problem at first, let us suppose the unit of weight of fluid of volume v and pressure p to be enclosed in a vessel possessing no weight, and let U be the whole amount of the 226 TECHNICAL THERMODYNAMICS. inner work; now if this vessel is supposed to progress in space in a rectilinear and uniform fashion with a velocity w, then the value „- will be added to the store of work; let us designate this value by H, then if ^-2g' ^2) the term H will designate that part of the total energy which corresponds to the visible progressive motion, and the whole amount of work contained in the unit of weight will therefore be U+H. Returning to the present problem, the passage from the cross-section Fi to the cross-section F involves taking up an amount of work iU + H)-{Ui + Hr), where the difference H—Hi can be designated as the energy of flow. But on the way from Fi to F there will be also resist- ances to overcome which, in the main, consist in friction of the fluid along the walls of the vessel; let us designate by W the loss of work thus occasioned when referred to the unit of weight of the fluid, then the whole amount of work consumed on the way FiF, which we will temporarily designate by L, is L = {U+H)-{Ui + H^) + W, (3) but still another expression can be estabUshed for this quantity of work. During the onward motion of the fluid body FiF the back area, Fif travels the distance Widt, and the forward area the distance wdt. Now since Fipi is the pressure which the part of the fluid following cross-section Fi exerts on the fluid body FiF, the latter will receive at this back area from the flowing fluid the work FiWipidt, and the fluid body will exert at the forward area F the work Fwpdt on the fluid preceding it. From this follows that the fluid body FiF has taken up in the time dt the work expressed by FiWipidt— Fwpdt = {piVi — pv)Gdt, FUNDAMENTAL FORMULAS FOR THE FLOW OF A FLUID. 227 when account is taken of equation (1). But in the time dt the weight Gdt of the fluid has entered into the space FiF, and the same weight has passed out through the area F; as we re con- sidering the permanent or normal condition, the whole fluid body lying between Fi and F experiences no change in its state of motion, consequently the preceding expression can be regarded as the work which the weight Gdt of the fluid has taken u p during its motion from Fi to F. For the finite time t ^ this quantity of work amounts to {piVi-pv)Gt, and if we assume a unit of weight of fluid, that is, Gt = l, then the amount of work taken up by the unit of weight is (pivi-pv). Let us now further assume that during the motion from Fi to F the unit of weight of fluid has imparted to it from the outside the quantity of heat Q; but because on its way the above indicated work of resistance W is converted into heat, the result is, on the whole, as if the quantity of heat Q+AW had been supplied. This quantity of heat expressed in work (that is, divided by A) and combined with the already calculated quantity of work now gives, for the total quantity of work L imparted to the unit of weight of fluid on the way FiF, the expression L==piVi-pv+^+W (4) Equating this with expression (3) we get the first fundamental equation for the present problem in the form Q = A[pv-piVi+iU+H)-{Ui+Hi)l ... (5) ' In the following investigations the letter t has been used to designate the time that has elapsed, as is customary, although the same letter is also used in the present work to designate the temperature according to Celsius [Fah- renheit]. But confusion is here avoided by always expressing the tempera- ture, in the investigations of the present chapter, in absolute measure, by T. 228 TECHNICAL THERMODYNAMICS. or passing to the differential : dQ = A[dipv) + dU+dH] (5a) First of all it is remarkable that the work of resistance no longer occurs in this equation, and therefore the equation is valid no matter what resistances occur between Fi and F and what the character of these resistances may be. Gener- ally we have to do with the friction of the fluid along the walls of the vessel, but this work of resistance may also consist in the loss of work caused by the formation of eddies, such as would arise if there were sudden changes of cross-section between the areas Fi and F. Equation (5) is untouched by such assump- tions, provided the condition is satisfied that in the two limiting sections Fi and F parallelism is maintained in the directions of the velocities, provided, therefore, that in the forward section F the visible eddying motion of the fluid particles, which may have arisen between Fi and F, has disappeared. We must further remark that equation (5) remains unchanged in whatever manner and according to whatever law the total quantity of heat Q is imparted or withdrawn, as the case may be, from the unit of weight of fluid on its way from Fi to F. Only in one direction may the equation under certain circum- stances experience an extension; it was derived under the hypoth- esis that no external forces were exerted on the elements of the fluid; but under all circumstances, at least ^ -~ the force of gravity will act, and therefore equation (5) is only valid for a horizontal conduit as was assumed at the very start. If this latter condition is not satisfied, then in general it is necessary to take account of the action of the force of gravity. Let us Fig 36 ^ '"'assume for this purpose that the back sec- tion Fi (Fig. 36) lies at the distance hi, and the section F at the distance h, below the horizontal plane 00, then in consequence of the action of the force of gravity, the unit of weight of the fluid will take up the additional work h—hi, FUNDAMENTAL FORMULAS FOR THE FLOW OF A FLUID. 229 and this value should be added to the right member of equation (4) ; a combination with equation (3) then gives in place of equa- tion (5) the formula Q=A[pv-p,v, + {U+H)-iUi + H^)-{h-hi)l . . (I) or, passing to the differential, dQ = A[d{pv) + dU+dH-dh] (la) Finally we must emphasize, and this appears from the whole presentation, that, strictly speaking, the foregoing propositions apply only to the motion of the fluid in a conduit possessing an infinitesimal cross-section variable though it be; when applying the equation obtained to the case of finite cross- sections we enter the realm of approximate computations.^ In addition to the equations already developed, among which we will always choose the differential equation (la) for the sake of simplifjang the coming investigations, there is still another, second, relation between the introduced quantities, which is fur- nished by the fundamental equation (8), p. 29, of Thermody- namics; in the latter equation we must, besides the quantity of heat Q supphed from the outside, take account of the quantity of heat AW developed by the work of resistance W , consequently Q+AW = A{U-Ui) + Ajpdv, . . . . (II) 'The different forms in which equation (I) occurs in the text can easily be brought into the form in which it is to be applied when any external forces act on the fluid element, provided these forces can be regarded as derived from a force function V. Let x, y, and 2 be the coordinates of a fluid element referred to rectangular coordinates at the time t, and let X, Y , and Z be the three force components acting on the unit of weight of the fluid ; then, as is well known, dV = Xdx+Ydy + Zdz is the differential of the work of these forces. We then simply substitute dV for dh in equation (lo), so that we get dQ = A[d{pv) +dU + dH-dV], or integrated, Q = A[pv-p,v, + U-U,+H-H.-{V-V,)l where V—V, represents that part of the work which the unit of weight of fluid has received on the way F,F (Fig. 36) from the external forces. 230 TECHNICAL THERMODYNAMICS. or dQ + AdW = A{dU+pdv) (Ho) The two equations (I) and (II) now furnish the basis, for a full investigation of the flow of fluids under the prescribed limitations.^ The combination of equations (I) and (II) or (la) and (Ila) furthermore leads to the following expressions,^ whose use, in certain cases, simplifies the calculation: p H-H,^h~h,-W-fvdp (Ill) Pi dH = dh-dW-vdp (Ilia) In practical applications it is almost exclusively a question of determining the energy of flow H, and then, by means of equation (2), the velocity w in the cross-section is found by the formula w=\/2gH, and the weight G of the fluid passing in one second through the cross-section is determined according to equation (1) from G=^ (IV) V It is worthy of note that the integral p V — J vdp=piVi — pv+ J pdv (6) Pl Vi ' I first called attention to the connection of hydrodynamic equations with those of thermodynamics in my treatise, "Das Lokomotivenblasrohr. Experi- mentelleund theoretische Untersuchungen iiber die Zugerzeugung durch Dampf- strahlen und die saugende Wirkungder Fliissigkeitsstrahlen uberhaupt," Zurich, 1863. 1 there developed new formulas for the efflux of vapors and pointed out the connection which existed between the different formulas, which up to that time had been given for the efflux of gases. See also the author's articles " Ausfluss von Dampfen und bocherhitzten Fliissigkeiten aus Geiassmiindungen," Civilingenieur, Vol. 10, 1864, p. 87, and "Neue Darstellung der Vorgange beim Ausstromen der Gase und Dampfe aus Gefassmimdungen," Civilingenieur, Vol. 17. 1871. p 1. ' If any external forces act on the fluid element (provided of course they are derived from the force function), we must insert in equations (III) and (llla) in place of h and h, the force function V and V, respectively, as was already emphasized in the remark on the preceding page. THE FLOW OF GASES. 231 occurring in equation (III) can easily be represented graphically. If we lay off, as rectangular coordinates (Fig. 37), the pressure pi and the specific volume Vi of the fluid ^ in the back section Fi, and likewise the corresponding values p and v belonging to the forward section F, and if we as- sume that the law is known according to which the pressure p varies with the vol- ume while on the way FiF (Fig. 36), that is, assume the course of the curve ab as known, then the hatched area in Fig. 37 represents the value of the preceding integral and consequently determines the value -fvdp = H-Ht-(h-h) + W, (6a) in accordance with equation (III). The formulas developed still hold for other kinds of fluid; with liquid fluids in which, for widely separated pressure limits, the specific volume can be regarded as constant, so that we may write v = vi and dv=0, equation (III) gives with the help of equa- tion (2) the formula W^ — W]^ 29 = h-hi-W+Vi{pi-p), one long known in Hydraulics; but hitherto equation (I) or (II) has not been considered in such investigations. § 41. THE FLOW OF GASES. In applying the foregoing fundamental equations to the motion of gases, equation (III) remains unchanged in form. On the other hand the two equations (I) and (II) from which equation (III) was derived experience remarkable simplification. The change dU of energy is given, for gases, by the relation (51), p. 132, dU dipv) 232 TECHNICAL THERMODYNAMICS, substitution in equation (la) therefore gives AdH = dQ---^ipv) + Adh, a&) «— 1 and from equation (IIo) follows dQ+AdW = -^ivdp+Kpdv) (116) If we introduce the temperature T into equation (16) by utiliz- ing the equation of condition pv = BT, and also the relation «— 1 AT) Cj, = AB, K which was found upon p. 134, there will follow AdH = dQ+AdH-Cj,dT, (Ic) and similarly equation (116) can be brought into other forms if we consider the different equations (53), pp. 133 and 134. It is noticeable that the introduction of temperature T gives a very simple form of equation (Ic), but it deserves to be empha- sized that an experimental determination of the temperature of flowing gases, of flowing fluids generally, is not possible, at least with the help of thermometers, because friction and the impact of the fluid against the bulb influences the reading of the thermometer. We should note that in most writings on Physics and Mechanics, until very recently, equation (Ilia) given above has been exclu- sively employed in the solution of hydrodynamic problems, and, when integrating the last term vdp of the right member of the equation, different hypotheses were assumed with regard to the relation between the quantities p and v; but as the equation may be regarded as derived from equations (I) and (II), each of these hypotheses can always be regarded as a particularly definite assumption with respect to the quantity of heat dQ to be imparted to the fluid, and with respect to the law to which the resistances are subjected; not until there is a simultaneous consideration of EFFLUX OF GASES UNDER CONSTANT PRESSURE. 233 these circumstances does one gain complete insight into the occur- rences. It is judicious, when investigatiog certain cases, to choose as a starting-point the two fundamental formulas (la) and (Ila), and in gases to use the corresponding equations (I&) and (116). § 42. EFFLUX OF GASES UNDER CONSTANT PRESSURE THROUGH SIMPLE ORIFICES. Let us suppose a very large vessel filled with atmospheric air of pressure pi, temperature Ti, and specific volume Vi, but pro- vided with an orifice of efflux through which the air streams into a second space in which prevails the pressure p2, which is Ukewise kept at a constant height; the condition that the pressure pi in the discharging vessel is kept constant is satisfied if we suppose that simultaneously vessel and receiver are of great capacity, and that the orifice of efflux has a very small cross-section and that relatively the period of efflux is a small one. The ordinary case, which will at first be assumed in what follows, is the flow of air into the free atmosphere; in. the plane of the orifice (Fig. 38) whose cross-section amounts to F square meters [square feet] the pressure p prevails, and at first we will assume that it is different from ) /\ /Eiff^ the external pressure p2; let the temperature and specific volume in the plane of the orifice be T and v and outside of it be Ti and V2, then the temperature in this orifice will be identical with that in the discharging vessel. Finally, in order to ignore for the present the influence of the con- traction of the air-jet, suppose the orifice to be rounded on the inside. The problem which is now before us consists in determining the energy of flow H in the plane of the orifice and in determining the velocity of the efflux w and the discharge G per second measured in kilograms [pounds], for which latter purposes equa- tions (2) and (IV) are to be employed. In the large discharging vessel the air is assumed to be in a state of rest, and therefore Hi=0, because wi = 0; if the jet flows 234 TECHNICAL THERMODYNAMICS. out in a horizontal direction, and consequently h is constant, we must put dh = 0; finally, because in general the air, while flowing toward the orifice, neither receives nor rejects heat, we must also write dQ = 0, and we therefore get for the case before us, in place of equations (16) and (II), dH= —Mpv), (Ic) K— 1 dW = -{vdp+Kvdv), (lie) K— 1 and from these, by addition, dH+dW=-vdp, (III6) which also follows directly from equation (IIIo). Under the assumptions made we now get, from equation (Ic) by integration, H = ~!^{pivi-pv), (7) and consequently from equation (2) the velocity of efilux w = yj-i^{pivi-pv), (8) also according to equation (IV) the quantity of air measured in kilograms [pounds] flowing out in one second is G=F i^ (Pi^^i-H (9) But the last two equations are still indeterminate ; the quanti- ties pi and Di in the discharging vessel are given to be sure, and the pressure p in the plane of the orifice can also be assumed as known, say equal to the external pressure p2, but the specific volume V there is n o t known. Nevertheless we recognize that the relation between the four mentioned quantities is known as soon as the course of the curve ah (Fig. 37) is given, as soon, therefore, as we know how the EFFLUX OF GASES UNDER CONSTANT PRESSURE. 235 pressure of the air varies with the volume while the air in the discharging vessel flows from a state of ^ rest towards the orifice, a motion which doubtless, as indicated in Fig. 38, occurs in a funnel- shaped space; similarly the air outside of the orifice spreads itself, passing from the velocity w to a state of rest. In the latter case, after the spreading, we have in the receiv- ing vessel H=0, and because, in the discharging vessel itself, Hi=0 we have generally, from equation (I), Q = A[p2V2-Vi'Vi + U2-Ui-(h2-hi)], . . . (10) where the values possessing the subscript 2 relate to the state of the air after spreading, i.e., to the final condition of rest. If, as is assumed, heat supply from without does not occur, so that Q = 0, and if the jet of air flows out vertically upwards, then h2—hi represents the rise. For horizontal efflux there follows from the pre- ceding formula = p2V2-piVl+U2-Ul, (11) or divv) + dU = (11a) If, as is the case throughout this work, we consider air as a perfect gas, then for U we must employ equation (52), p. 132, and therefore follows P2V2 = PlVl, or, with the help of the equation of condition, T2 = Ti, i.e., the quantity of discharged air, after spreading outside, again possesses the same temperature as in the discharging vessel, what- ever resistances may have obtained during the efflux and whatever pressure p may prevail in the plane of the orifice, — so long as the condition is fulfilled that the air (during horizonta efflux) neither receives nor rejects heat during its flow toward the orifice. As the air, in accordance with our 236 TECHNICAL THERMODYNAMICS. assumption, already possesses the temperature Ti on the outside, the air thus added does not change this temperature. Now to be sure this proposition is only approximately true, as W. Thomson and Joule have already shown by experi- ment (in 1862).! In reality T2W, where (p was introduced as the velocity coefficient, as is customary in practical hydraulics. The effective energy of flow H^ in the plane of the orifice would consequently be and the work of resistance W, W = H-He-(l-^)H=(^j^-l^He==t:H,, . . (15) provided we introduce the notation C = ^-l (16), The value C is designated as "the coefficient of resist- a n c e," according to W e i s b a c h, to whom we owe the intro- duction of this factor into the formulas of practical hydraulics and at the same time its experimental determination for different kinds of resistances. Working backward from equation (16) we get the velocity coefficient '^=nIiT-c' ^'^^ and, as soon as it is known, equation (13) gives the effective efflux velocity We if one multiplies the radical on the right side by 4>. With orifices well rounded on the inside, and with short, cylin- drical adjutages, in each of which the jet leaves the orifice with its full cross-section, we get the true discharge G^ from Ge-. The beautiful experiments of Weisbach were carried out on a large scale and were in part re-calculated by G r a s h o f i; they were only intended to determine the coefficients 4> and // for a series of different forms of orifices. In general it turned out that these coefficients (at least within the limits employed by Weisbach) deviated but little from those observed during the efflux of water, likewise mostly found by Weisbach; more- over for one and the same orifice the velocity coefficient ^ and the coefficient of resistance (^ proved to be nearly invariable for various differences of pressure; these results had, indeed, already been accepted, on the basis of older experiments conducted by Schmidt (1820), Lagerhjelm (1822), Koch (1824), and d'Aubuisson (1828) ; the latter experiments, to be sure, were carried on throughout with very slight excess of pressure (fraction of one meter of water column) ; it was Weisbach who first went up to one atmosphere excess of pressure. Later on there will be an opportunity of coming back to Weisbach's experimental results (this experimental method has already been discussed on p. 191) ; here we need only remark that Weisbach omitted such a discussion of equations (1.3) and (14) as was under- taken by de Saint Venant and W a n t z e 1, and assumed from the start that in his experiments the pressure p in the plane of the orifice was identical with the pressure p2 of the external atmosphere. The assumption is correct for the pressures used by Weisbach; it will be shown that the assumption would not have been permissible in calculating the experiments if the pressure in the discharging vessel had been a little larger than it really was. ' G r a s h o f, "Theoretische Maschinenlehre," Vol. 1, Leipsic, 1875. 240 TECHNICAL THERMODYNAMICS. § 43. DISCUSSION OF THE EFFLUX FORMULAS. In the two equations (13) and (14) the state of the air (pi and Vi) in the discharging vessel is regarded as known; likewise the cross-section F of the orifice measured in square meters [square feet]. Then the efflux of velocity w, and the air weight G in kilo- grams [pounds] per second, can be computed, provided the pressure p, in the plane of the orifice, is given. Now it seems natural to assume that the pressure p2 (outside the discharging vessel) ex- tends up to the plane of the orifice, and that therefore p = p2 may be written. In this case equation (14) gives the quantity discharged as G = 0, as must be the case provided also the pressure in the dis- charging vessel and in the receiving vessel is equal, that is, p2 = Pi. On the other hand if we suppose the flow to take place into a vacuum, and if here also the pressure p2 = extends up to the plane of the orifice, then equation (14) would again give G = Ofor p=0, from which follows the absurd result that no flow takes place toward a vacuum. One must therefore draw the conclusion that the external pressure p2 will in general not be identical with the pressure p of the orifice, or only under certain circumstances. A closer examination of equation (14) shows that a certain value of p exists for which the discharge G is a maximum. The differentiation of the bracketed quantity under the radical gives, as is easy to see, a maximum for If pi is given, we can compute, by this formula, the corresponding pressure p in the plane of the orifice and then, with the help of equation (12), find the corresponding specific volume v there, from V /k+1\J— DISCUSSION OF THR EFFLUX FORMULAS. 241 and then also ^ = A. (20) PlVl K+1 At the same time the last formula, after substitution of the absolute temperatures T and Ti, gives Fr^i' «'> from which the temperature T in the plane of the orifice can be computed, when that of Ti in the discharging vessel is known. As « = 1.410 for atmospheric air, there follows from the preced- ing formulas -^ = 0.5266, (18a) -=1.5759, (19a) and T^= PL =0,8299 (20a) Ti pivi If we use relation (18) in equation (14), the maximum of dis- charge, which is designated by Gm, can be calculated : ^- = ^(41)-^^' • • • • (21) or with « = 1.410, G^ = 2.U99 fJ^, (21a) [(?,„ = 3.8942 ^\f^]. If we multiply both sides of this equation by i^i, then Gvi gives the maximum of air in cubic meters [cubic feet], measured under the internal pressure, and because piVi = BT there follows GmVi =UMlFV¥u lG,nVi = 15.703 FVTil 242 TECHNICAL THERMODYNAMICS. Now if we assume the temperature of the air in the discharging vessel to be 0° C. [32° F.], there will follow from this formula GmVl F = 192.18 [348.905]. Consequently when referred to a square meter [square foot] of orifice area, the maximum discharge per second, measured at the inner pressure, must be taken at 192.18 cbm. [348.905 cu. ft.], all this under the assumption which is here maintained throughout, that the air-jet leaves the orifice with full cross-section, and that therefore no contraction of the jet exists. In order to give a general view of the variation of the discharge G with the orifice pressure p, let us lay off in Fig. 39a the value x=— as abscissa, and as ordinate the discharge G calculated from Pi equation (14), p. 236, and thus get the curve oab; now if the orifice pressure p were really identical with the external pressure P2, then for the diminution of the pressure ratio P2'-Pi from the value 1 to Xm there would result, in accordance with equation (18), an in- crease of the discharge according to the portion of the curve ba and in entire accordance with expectation; the diminution of G would occur like that represented by the branch of the curve ao. Now because such a dimi- nution cannot really exist d e S a i n t V e n a n t and W a n t z e 1 first (ibid.) set up the hypothesis that for the interval in question the curve branch ao should be replaced by a horizontal line (ac) (Fig. 396); so long, therefore, as the pressure ratio p2'Pi lies between and the value Xm given by equation (18), so long the same quantity of air will flow out, namely, that given by equation (21) or (21a); in this case the orifice pressure p would be different from the external pressure V2, and indeed it would be greater and be determined by equation (18); the discharged air seems then to be independent of the external pressure within the limits indicated. Fig. 39. DISCUSSION OF THE EFFLUX FORMULAS. 243 The rule for computing should therefore be as follows : There is given the internal pressure pi, the external pressure p2, and the efflux cross-section F, and also the temperature Ti in the dis- charging vessel. First compute the orifice pressure p for the maximum discharge according to equation (18), namely, ^M^i)"^' ^''^ Now if P2>p, we get, from equation (14), for the discharge where Vi is to be computed from the relation piV\ = BTi. On the other hand, for P2p, then according to equation (13) the efflux velocity is w On the other hand if P2 integration will give pv^pivi"-, (30) and utilizing the equation of condition for gases there also follows r^/M«-^m^^=^, .... (31) Ti \v/ \pJ Pivi according to which the real value of the volume v and the tem- perature T can be calculated from the pressure p in the orifice. As the assumption of a constant value for ^ also makes n constant, it follows from equation (30) that the curve of expansion is the polytropic curve; for C = 0, i.e., neglecting the resistances, we have n = K, i.e., the curve passes, as already found, into the adiabatic line. I have called the magnitude n the "efflux EFFLUX FORMULAS WITH ORIFICE RESISTANCES. 247 exponent"!; it can easily be determined from equation (29) when ^ is known from experiment; conversely if n is given, we get C = ^^V (32) /c(n— 1) according to which n is always smaller than k. For the efflux of water, according to Weisbach, the coefficient of resistance i^ for a well-rounded orifice, and also for an orifice in a thin wall, is i^= 0.063, and for the cylindrical adjutage J =0.505. If we suppose these values to be also valid for the efflux of air, we shall get respectively n = 1.376 and n = 1.239. Now if we determine v from equation (30) and sub- stitute it in equations (8) and (9), p. 234, there follows accprding to which w and G can be calculated, when the pressure p in the orifice is known; moreover, for this case the temperature T in the orifice can be found from equation (31), and if we employ the equation of condition, in the sometimes used form ApiVi = Cp'^- — Ti, K we get from equation (33) the energy of flow H in the orifice (measured in units of heat) Aff = ^^ = cp(7'x-r), (33a) and the resistance, measured in like manner, AW=i:.Cp(T,-T). ' "Neue Darstellung der Vorgatige beim Ausstromen der Case und Dfimpfe aus Gefassmundungen," Civilingenieur, VoL 17, 1871, p. 71. 248 TECHNICAL THERMODYNAMICS. We must now decide the question as to what pressure p pre- vails in the plane of the orifice, when the pressure p2 is given in the receiving vessel. Here also it is shown by equation (34) that a certain value oi p:pi exists for which the discharge G is a maxi- mum, and indeed we get, from equation (34), by differentiation of the bracket under the radical, the corresponding ratio Xm and find it equal to ^n,=—=(-^)^' (35) Pi \n+l/ From the equation of the polytropic curve pv^=piVi^ we also find tH-y^ (36) If we substitute the value of Xm from equation (35) in equa- tion (33), we get, when there is a maximum discharge, for the velocity of efflux, -JMII)^ <^') and then, with the help of the relation GmV = Fw and equation (36), G.= (4-);r^.J^/?^l)^, .... (38) \n+l/ ^K-l\n+l/vi which two equations, when n = «, i.e., when resistances are neglected, pass over into equations (26) and (24), p. 243. If we should here also use the rule of de Saint Venant and Wantzel, as I did in earlier articles and former editions of this book, then the foregoing formulas would also be valid for efflux into a vacuum. Therefore if pi, Vi, n, and the pressure pa in the receiving vessel are given, we must carefully observe whether the ratio P2 : p\ is greater or smaller than the value of Xm given by equation (35) ; if it is g r e a t e r, w and G must be calculated from equa- tions (33) and (34) ; if it is smaller, equations (37) and (38) are to be employed. EFFLUX FORMULAS WITH ORIFICE RESISTANCES. 249 But with simple orifices, for which the efflux exponent n is always just a little smaller than «, equation (37) gives values for the efflux velocity in the orifice, so that it seems natural to assume that the two values are here also identical. New experiments, with great resistances in long pipes, which will be discussed below, confirm this assumption, therefore in sub- sequent investigations I proceed from the new hypothesis : "that the air flows into the vacuum with the acoustic velocity w = \ gK'pv, (39) corresponding to the state of the air in the orifice, n o matter what resistances exist during the flow toward the orifice," a proposition which may have a great range for certain technical investigations. If we return to the fundamental equation (8), p. 234, we get, with the help of equation (39), 2 P^ = -— -, Pii'i (40) Equation (39) gives no clue concerning the values p and v in the orifice; but if, for such simple orifices as those above, we pro- ceed on the assumption that the expansion of the air takes place poly tropically, we get, from the relation pi>" = pi^i", pv _/ p \2zJ PlVi \pj " ' and therefore, by combining with equation (40), also 250 TECHNICAL THERMODYNAMICS. These two relations, according to the new hypothesis, take the place of equations (35) and (36), and now, for the case that P2-Pi is smaller than the value Xm given by equation (41), there follows, from (33) and (34), w-J-~zPiVi, (43) 2l,^(^]n-^JWl^ (44) F V+1/ ^K+1 vi ^ ' If, for abbreviation, we use we get %^ = ^^ (45a) In accordance with the preceding formulas, the following table was calculated for a series of different values of the efflux ex- ponent n. n P. V p. In this case the external pressure p, extends up to the plane of the orifice, and therefore we must use equations (33) and (34) with P. =??=-. pi pi 3 ' Because r,=288° C. (518.4° F.) we get, from pm^BTi with 5=29.269 [5=53.349], for the specific volume of the air in the discharging vessel Vi =0.5438 cbm. [8.7068 cu. ft.]. Now from (^)hr ~ =0.895S there follows for the temperature T in the orifice, 257.8° or —15.2° C [464.04° or 4.64° F.]. The specific volume v in the orifice is computed from ^=(21)^^1.3430, Vl \p2 '' and from it we find w =0.7303 cbm. [i; = 11.693 cu. ft.]. 252 TECHNICAL THERMODYNAMICS. Equation (33) gives for the efflux velocity- Mi =244.03 m. [w =800.63 ft.], and from Gv = Fw there follows directly 1=1^=334.15 kg. [68.45 lb.], when the orifice of efflux F is measured in square meters [square feet]. The latter value can also be written 1 = 1.9793 The discharge in cubic meters per second, measured under the inner pressure, is Gv, =181.7 F [Gv, = 596.3 F]; on the other hand, measured in the plane of the orifice, Gu = 244.0 F [Gi; = 800.53 F]. After the jet has spread itself outside, the air again has the initial tempera- ture of 15° C. [59° F.] and the specific volume v, is found from the relation PiV2=piVi or ^2 = 0.8157 [13.067 cu. ft.]. Accordingly, the air discharged per second, measured under the outer pressure, is Gv, =272.6 F [Gi;^ =894.37 if]. The energy of flow in the plane of the orifice is ^ =3035.2 mkg. [9958.1 ft-lb.], and because the coefficient of resistance C, according to equation (32), is C =0.066, EFFLUX FORMULAS WITH ORIFICE RESISTANCES. 253 the loss of energy, in consequence of the resistances, is Z^ =200.2 mkg. [657.23 ft-lb.]. Example 2. Suppose that in a large vessel there is air of four atmos- pheres pressure, at a temperature of 15° C. [59° F.]. Then by the equa- tion of condition p, =4X10333, and Ti =273 + 15° [518.4], the specific vol- ume there is Vi =0.2039 cbm. [3.266 cu. ft.]. If this air flows, under constant pressure, through a well-rounded orifice, for which we again assume the efflux exponent n = 1.375, into the free atmosphere, so that pj corresponds to the pressure of o n e atmosphere, then again, as in the preceding ex- ample, we get from equation (41) ■^ =0.5047. Hence the pressure in the orifice is p =2.0188 atmospheres, and therefore P2!EM (47) As r is nearly identical with V\, and because the temperature in the plane of the orifice is nearly the same as that in the discharg- ing vessel, so that we can assume the relation p2V = BTi, we can approximately find the discharge: an equation which agrees in form with that given by Napier for the eflflux of steam under slight excess of pressure; we will return later to Napier's investigations. The assumption that the specific volume is constant really presupposes the with- drawal of heat, namely, from equation (lb), p. 232, there follows immediately, by integration, for v = vi, Q = AH-^v,{p,-p), K— 1 256 TECHNICAL THERMODYNAMICS. and if we determine the flow of energy from equation (46), and substitute it, there results the quantity of heat Q which, as the negative sign shows, must be withdrawn from the unit of weight of air during the flow toward the orifice : Q=-'j-^sVi{vi-V)- n{K—l) Alongside of equation (46a), which is valid for slight differences of pressure, there is also given in mechanical and physical writings one other efflux formula, which was first developed by N a v i e r, and is commonly said to hold also for the greater differences of pressure. If we assume that the temperature T of the gas in the orifice can be taken as identical with that (Ti) in the discharging vessel, there follows pv = piVi, and hence from equation (III6), when we again replace the work of resistance W by {^H, we get ^^Nrp^Pi'^i loge^, (48; and the discharge, disregarding contraction, V ^i+i:vi \pj " p' which equation can be brought, with the help of the relation pv = piVi = BTi, into other forms. Equation (48), as was men- tioned, is found in all the manuals on Physics, and yet it is derived from assumptions which, for several reasons, must be regarded as not at all permissible. If we assume first that the external pressure p2 extends to the plane of the orifice, and designate the ratio p:pi by x, then equa- EFFLUX OF AIR THROUGH SIMPLE ORIFICES. 257 tion (49) shows that for a particular inner pressure pi there is a value of X for which the discharge G becomes a maximum; this occurs with the maximum value of the expression and for acquires the value c2 log. - , 1 1 1 —=0.6055 or ^ = 1.687. Pi P2 Accordingly if the inner pressure amounts to more than, say, 1.65 of the external pressure, formula (49) will then become useless and we must have recourse to an hypothesis 3ike or similar to the one discussed above. Although the preceding proof was given by C o r i o 1 i s 1 (1838), nevertheless N a v i e r ' s formula was presented as suitable for the greater differences of pressure. We must also note that under the assumption pv = piVi equation (16) gives Q=AH; therefore, in order that the hypothesis may be realized, the unit of weight of gas must receive, whUe flowing toward the orifice, a quantity of heat corresponding to the full energy of flow in the plane of the orifice, a case which never occurs in ordinary efflux. § 46. EXPERIMENTS ON THE EFFLUX OF AIR THROUGH SIMPLE ORIFICES. The older experiments of Schmidt, Lagerhjelm, Koch, and d'Aubuisson (1820 to 1826), already men- tioned on p. 239), were all conducted with very small excess of pressure and in such a way that the air-filled discharging vessel was in communication with a vessel of water, from which the water flowed into the former vessel with a certain excess of pres- sure. The volume of the entering water gives the volume of the ■ Comptes rendus, 1838, VI, p. 239; also 1839, VIII, p. 295. 258 TECHNICAL THERMODYNAMICS. air (measured at the inner pressure) which at the same time is displaced and is then discharged through the orifice. As was- men- tioned, all these experiments were conducted with a very slight excess of pressure and also with very small orifice diameters, in fact, on a small scale ; i they furnish, as was shown later by B u f f,2 with the help of his own experiments, the proof that for the efflux of air there can be employed the same formulas and the same coefficients of efflux as for the efflux of water. But for greater excess of pressure the propositions in question are not directly available; these experiments, particularly, do not give any clue whatever as to the orifice pressure discussed above. The first experiments, on a larger scale and with a larger excess of pressure, are due to W e i s b a c h, as repeatedly mentioned above; very valuable as these experimental results are in general, nevertheless they are not suited to decide the question concerning the pressure in the orifice, because Weisbach's prps^ures in the discharging vessel were too small ; in only a few of his ex- periments did the initial pressure of the air in the boiler somewhat exceed two atmospheres, i.e., at most, it was double the external pressure. In the experiments of de Saint-Venant and W a n t z e 1 the ratio of the pressure in the discharging vessel to that in the receiving vessel was, to be sure, far greater than two; they allowed air to flow into the receiver of an air-pump through orifices of only 2/3, 1, and 1.5 mm. [0.026, 0.039, 0.059 in.], the receiver having a capacity of 0.0174 cbm. [0.6145 cu. ft.], in which the pressure at the beginning of the different series of experiments was generally from 10 to 20 mm. of mercury [0.4 to 0.8 in.]. They employed two methods. In the first they allowed the external air to flow in without interruption till the pressure in the receiver equaled the external pressure and ob- served at equal intervals of time (every 5 seconds) the increment of pressure; in the second method, which alone could have realized ' W e i s b a c h reports the older experiments on the efflux of air and, at the same time, gives full references to the literature, in "AUgemeine Maschinen- Encyklopadie," article "Ausfluss," p. 603. 'Buff, "Versuche fiber den Widerstand ausstromender Luft," Poggen- dorfi's Annalen, Vol. 40, 1837, p. 14. EFFLUX OF AIR THROUGH SIMPLE ORIFICES. 259 their aim, they allowed the air to flow in with interruptions; after every efflux, of 5 seconds' duration, the orifice was closed with the finger for a few seconds 'and the pressure quickly noted. The defective feature of this method is that the experi- menter did not keep the orifice closed long enough, and after clos- ing it, did not wait for the (diminishing) manometer reading of the receiver to become stationary. If we add to this the fact, criticized by P o n c e 1 e t, that the diameters of the orifices used and the capacity of the receiver were much too smalV then it seems justifiable to institute new experiments on a larger scale, with high boiler pressure and with more reliable experimental methods. Now I first made such experiments in Ziirich in 1S71 (see p. 193). The mechanical section of its Polytechnicum already pos- sessed at that time a series of experimental machines and appara- tus, and, in a modest way, followed the same aims now pursued in the mechanical engineering laboratories of technical high schools. I had especially constructed, for the contemplated experiments on the efflux of air, a wrought-iron, cyhndrical boiler 4.2 m. [13.776 ft.] long and 0.5 m. [1.64 ft.] in diameter, whose cubic capacity, carefully calibrated with water, gave 0.81088 cbm. [28.637 cu. ft.]. The boiler was tested up to ten atmospheres, and was provided with a pump by means of which external atmospheric air was compressed and forced into the boiler; the pump-piston was driven by a crank worked by hand, so that naturally the filling of the boiler with air, of the usual four atmospheres of pressure, took a long time. The boiler carried a dome ^ provided with large pipe connec- 'de Saint Venant and W a n t z e 1 replied to Poneelet's objections (Comptes rendus, 1845, XXI, p. 195) in the same volume, p. 366, whereupon Poncelet retorted somewhat violently, designating the experi- ments as "so to speak, microscopic," and characterized as unusual and odd the law of constancy of discharge with an external pressure which was smaller than about one-half the inner one. In the latter respect Poncelet did the experimenters mentioned decided injustice, as my experiments, given in the text, show. ' The boiler dome mentioned in the text is illustrated in Der Civilingenieur (1874), Vol. 20, Plate 1. 260 TECHNICAL THERMODYNAMICS. tions all of which were closed except the one used in the efflux experiments; this pipe was provided with a tight-fitting cock whose passageway had the same inner diameter as the pipe. In the plane of the open end of the latter, various orifices were inserted. A carefully graduated, open, manometer was con- nected with the interior of the boiler, and from it could be read off pressures up to four atmospheres (three atmospheres gauge pressure) . Experiments were conducted with three kinds of simple orifices well rounded on the inside and with short, cylindrical adjutages without interior rounding, and .with orifices in a thin wall. After the boiler was filled with air of about four atmospheres, and after equalization with the external atmosphere had taken place so that the manometer was stationary, the <30ck (with a quarter turn) was quickly opened and, after about ten seconds, was quickly closed and the manometer reading noted; in so doing the pressure pi was read before the opening, px immediately after the closing, py immediately afterward (after the jump — see remarks on p. 193), and, finally, p2 was read, namely, as soon as the ma^ nometer reading again showed itself stationary (because of the equalization of temperature) after the closing of the orifice; this equalization usually occurred after ten to fifteen minutes. Now the experiment was repeated with the pressure p2 as the new initial pressure pi, and this was continued till the boiler pressure had nearly fallen to the external atmospheric pressure. Thus there resulted for one filling of the boiler and for one and the same orifice a whole series of separate experiments, whose number was of course greater the smaller the diameter of the orifice used. From each individual experiment there was found, from the volume of the boiler, from the temperature, from the pressure pi at the beginning and from the pressure p2 at the end, after equal- ization of temperature, the weight of air in the boiler at the begin- ning and at the end, and then, from the observed time of efflux, the discharge per second was determined. The most exact possible determination of this period of efflux is of importance in this experimental method, efflux during short intervals. For this purpose a clock was used which kept fifths EFFLUX OF AIR THROUGH SIMPLE ORIFICES. 261 of a second and in which the moving index noted, on the dial immediately below the scale of seconds, ^a colored point, as soon as a button on the clock casing was pressed. The pressing of the button was effected by an electromagnet; to the cock of the efflux pipe there was fastened a metallic pointer which, at the middle position of the cock, dipped a little into the mercury that was in a vessel to one side of the cock; the dipping, during the turning of the cock, closed the metallic circuit and operated the magnet. In this way, at the instant of operating the cock and at the instant of closing the cock, there was put on the dial- plate of the clock a point whose position opposite the scale of seconds enabled one to read with great ease the duration. The experimental results also showed that this method of determining the period of efflux was completely accurate within 0.2 of a second; any uncertainties were neutralized by the great number of sep- arate experiments from which could be found, finally reUable, mean values of the quantities to be determined. The experiments themselves dissipated also my original doubt whether or no, with the short efflux period employed, the time which had elapsed till the efflux had attained its normal state might not be relatively too large; I convinced myself that with the size of the boiler and the orifice areas employed, this period must be almost infinitesimal, and inferred this from the peculiar tone of the violent noise which was coimected with the efflux of the compressed air; this tone was clap-like at the instant of opening the cock and was maintained at the same pitch during the short period of efflux; with a long period of efflux this tone changed slowly and continuously, and here, in a certain sense, one could follow with the ear a gradual diminution of pressure in the boiler. A preliminary report of these experiments was made public in 1874 in Der Civilingenieur,i after a lecture which I delivered in 1871 to the Saxon Society of Engineers in Leipsic. Later, with the same efflux apparatus, F 1 i e g n e r again took up the experi- ments 2 with short orifices and added to my experiments the ' "Resultate experimenteller Untersuchungen uber das Ausstromen der Luft bei starkem tjberdruck," Civilingenieur, 1874, Vol. 20, p. 1. 'Fliegner, Ergebnisse einiger Versuche uber das Ausstromen der atmos- pharischen Luft," Civilingenieur, 1874, Vol. 20, p. 13. 262 TECHNICAL THERMODYNAMICS. results of his own. Where I found for such orifices the efflux exponent n = 1 .403, Fliegner concludes that n = /c = 1 .410, so that here, in the main, the influence of the resistances could be neglected. The same conclusion is reached by Emil Herr- mann in a discussion of our experiments ("Compendium der mechanischen Warmetheorie," Berlin, 1879). By a further study of the problem Fliegner was led to new experiments which gave, for the kind of orifice mentioned, n = 1.37, after certain changes and improvements had been made in the experimental apparatus used.^ I reported, as completely as the space permitted, in an earlier edition of this book (Vol. 1, 1887), on the results of my numerous experiments made in 1871 with different simple orifices, and there- fore will not here return to them. The experiments were, to be sure, sufficiently reliable and realized what was their main pur- pose, namely, to thoroughly confirm the proposition of d e Saint- V e n a n t and W a n t z e 1 about the orifice pressure, whose correctness was doubted by Poncelet; but the experimental method showed certain defects which rendered impossible a per- fectly reliable determination of the resistances during efflux; on the whole, it does not seem suitable to compress the air in the discharging vessel by pumps and allow it to flow into the free atmosphere. With simple orifices, as was repeatedly emphasized above, the inner pressure p should be at least double the outside pressure p2', but we recognize from the formula given above that the controlling element is not the pressure difference Pi — P2, but the pressure ratio pi:p2; the latter ratio was at most — = 4 in the experiments, and while this lasted the pressure Pi diminished and p2 (the external atmospheric pressure) remained constant during the period of observation. Although the internal pressure pi could be taken higher, by using thicker boiler-plate, this would not remove another greater defect. To be sure one can determine the pressure and temperature shown in the discharging 'Fliegner, " Versuche iiber das Ausstromen der atmospharischen I.uft durch gut abgerundete Miindungen," Civilingenieur, 1877, Vol. 23, p. 443. EFFLUX OF AIR THROUGH SIMPLE ORIFICES. 263 vessel with great accuracy in the condition of equiUbrium, but cannot determine the degree of moistur^in the air, and, besides, the air was also rendered somewhat impure by the lubricants used for the pistons of the force-pump (i.e., by the oil vapors). I have cherished for years the desire to again take up the experiments in another way, moreover, theoretical studies of the motion of air in long pipes (with great resistances) showed that the experiments on efflux into the free atmosphere could only lead to the goal when very high pressures were employed in the discharging vessel, that is, with very large values of the ratio pi : p2. I therefore in 1897 conducted new experiments, by going back to the experimental method of de Saint-Venant and W a n t z e 1 (see p. 258 above), but carried out the experi- ments on a considerably larger scale with the means of recent times at my disposal. I allowed the outer atmospheric air to stream into a large boiler in which the air could be kept in a highly rarefied condition by an excellently made air pump, driven by a Schmidt water motor. The advantages of this experimental method are at once apparent : the external space, the free atmosphere, constitutes the discharging vessel in which the pressure remains absolutely con- stant during the flow and in which the state of the air with respect to moisture can be accurately determined. The pressure of the air in the receiving vessel (the boiler) can easily be brought to P2 = 25 mm. [1 in.] of mercury; with a total barometric pressure of pi = 750 mm. [30 in.] we should have therefore at the beginning of the experiment pi : p2 = 30, just as if air of 30 atmospheres pres- sure flowed into the free atmosphere, and then the difference of pressure would stiU be only pi — p2 = 715 mm. [29 in.], or about 0.95 of an atmosphere; experiments with long pipes are thus also rendered possible. 264 TECHNICAL THERMODYNAMICS. §47. NEW EXPERIMENTS ON THE EFFLUX OF AIR THROUGH WELL=ROUNDED ORIFICES. In the Mechanical Engineering Laboratory II (for prime movers) at the Royal Technical High School in Dresden there were set up, at my suggestion, two upright boilers each 1.3 m. [4.235 ft.] in diameter and 3.4 m. [11.155 ft.] in height (measured at the middle of the dished ends), and possessing a capacity of about 4.2 cbm.i [148.33 cu. ft.]. The boilers were tested up to 15 kg/qcm. [213.35 lb. per sq. in.] gauge pressure and were joined by a connecting pipe which was closed air-tight by a cock when only one boiler was employed. With the outfit it was possible to conduct the most varied pneumatic experiments; I will here briefly discuss the first experiments of this sort, which I made in April, 1897, with well-rounded orifices, but will first preface the discussion by some remarks on ordinary atmospheric air. Let Ti be the temperature and bi the barometric reading in millimeters [inches] of mercury. Then the corresponding specific pressure will be pi = 13.596 &i [pi = 70.73 6i]. Now air consists of a mixture of dry atmospheric air and of the vapor of water whose pressure is here always so small that we may assume that the vapor follows exactly the law of Mariotte and G a y - L u s s a c ; now in the equation of condition for gases, pv = BT, the constant for dry air is 5 = 29.269 [5 = 53.349] (see p. 104), and for steam it is 5 = 46.954 [85.584] (see p. 114); the ratio of the first value to the second, which we will designate by e, repre- sents the relative weight of steam to air, and is £ = 0.623. If we retain the letter B for the gas constant of dry air, then the 7? constant for steam is - . £ Now let p' be the pressure of the dry air in the mixture, f its specific weight (weight per cubic unit), and let p" be the pressure ' A cut of the boilers, taken from a photograph, is shown in the Zeitschrift fiir Architektur und Ingenieurwesen, 1898, Vol. 46, p. 549, and accompanies an article by Ernst Lewicki: "Das Laboratorium fiir Kraftmaschinen an der K. S. Technischen Hochschule zu Dresden." EFFLUX OF AIR THROUGH WELL-ROUNDED ORIFICES. 265 of the steam present (called vapor-pressure), and let Y' be its specific weight, then we have the relations t^^BT. and t^, = ^^, and the total air pressure pi is Pi = p'+p", and the specific weight 7-1 of the mixture n = /+/'• A combination of the equations gives 1 1 Now let us put ''' B'l\ Dml 1) ri where B^ represents the gas constant for the mixture, that is, for the (moist) air actually present, which is equal to 5™ = ^~-r, (51) 1-(1-.)^ Pi We see from these formulas that the specific weight of moist air is smaller, and the gas constant larger, than for dry air. As was mentioned, the pressure pi and the temperature T^ of the mixture are regarded as known; on the other hand the vapor-pressure p" is still to be determined. For this purpose Lambrecht's "dew-point mirror " is used, and by a gradual cooling of the air it is found at what temperature (dew-point) steam begins to con- dense on a smooth mirror surface. If Tq is the temperature in question we can find from the steam tables for saturated vapor the corresponding steam pressure, which we will designate by po. Then the relation holds r" ^ ' 266 TECHNICAL THERMODYNAMICS. which, combined with the above equation for p" : Y' , will enable us to compute the vapor-pressure, p"=^Vo, (52) and then will enable us to calculate the constant Bm according to equation (51). From the formula p,^,^ = Pl = 5^^l (53) ri there is finally determined the specific volume V\ and the specific weight y\ of the moist air. If we suppose the air to be completely saturated with vapor, and that po" is the vapor-pressure which corresponds to the tem- perature T\ (taken from the steam tables), and y^' the specific weight of the steam, then we have the relation ro" ^ ' and from the above equation for p" : y" there follows 7^'f'^- <^) where x now represents the "moisture" of the air in the form in which it is ordinarily presented in Meteorology. In the flow experiments themselves only one of the two boilers was employed, its capacity having been first determined by most careful calibration with water and fixed at F =4.2270 cbm. [149.28 cu. ft.]. The boiler was provided with a good, mercury, vacuum gauge whose reading in millimeters of mercury was subtracted from the barometer reading hi in order to get the pressure of the air in the boiler in millimeters of mercury. When the reading of the vacuum gauge was stationary it was taken as proof that the temperature of the air in the boiler agreed with the external temperature of the air. In the boiler shell there was an opening into which was inserted the orifice for the influx of air; ordinarily the orifice is closed on the EFFLUX OF AIR THROUGH WELL-ROUNDED ORIFICES. 267 outside by a plate (clack) which is made air-tight by being pressed with strong springs. By a cleverly designed closing mechanism the spring is released by a simple pull and the clack jumps back and can in a similar way be brought back to the orig- inal position by spring force. During the opening as well as the closing of the clack there is marked, by electrical means, on the dial plate of the clock mentioned (p. 261) exactly the instant of time, so that we can read the period during which the opening exists, and consequently the flow of air occurs, with an accuracy which is correct down to 0.1 to 0.2 of a second. The experiments were then conducted in the following maimer. After inserting the orifice and rendering it air-tight with re- spect to the outside, the boiler was pumped out till it was nearly in the condition of a vacuum, and after the pressure 62 became stationary it was read from the vacuum gauge. Then I allowed the air to flow for t seconds, and at the end observed the pressure ^3 after the equahzation of temperature. Thereupon the experi- ment was repeated with the pressure ps as initial pressure and the end pressure determined, etc. In this way there resulted a series of separate observations for each of which the value 63— &2 ■■ had to be computed. The determination of this quantity was the purpose of the observation. Let p be the specific pressure in the vessel, Ti the temperature there, and ;- the specific weight, V the boiler capacity, and Gjc the weight of air filling the boiler, then -Dm-' 1 If the composition of the air in the boiler is like that outside, we must put pi in place of p; on the other hand, if p2 is the pressure at the beginning and ps the pressure at the end of some one experi- ment, then the quantity of air which has entered in t seconds is 268 TECHNICAL THERMODYNAMICS. and consequently the weight of air G per second is (? = Vrihz—h2 61 t ' where ri refers to the outer air and h represents the pressures in millimeters [inches] of mercury. If F is the cross-section of the orifice in square meters [square feet] we get, for every separate experiment, the quantity of air per second in kilograms per sq. m. [lb. per sq. ft.], as follows: G_Vn h^-i2 Fbi (55) F Fbx t Of the different experiments conducted with well-rounded orifices we will here adduce a series for an orifice of 5.1 mm. [0.2008 in.] diameter. Well-rounded Orifices. Diameter d=5.1 mm. [0.20079 in.], cross-section i^ = 20.428 Xl0~^ sq. m. [0.03166 sq. in.], barometer reading 61 = 754 mm. [29.655 in.], temperature of air 7^1 = 273+ 15.8 = 288.8° [519.84°], dew point To = 273 +9.6 = 282 .6° [508.68°], and corresponding pres- sure of vapor 60 = 8.93 mm. [0.3516 in.]. No. Duration of Flow t" Equi- librium pressure b, 63 Values of 63 — 62 , No. Duration of Flow t" Equi- libnum pressure 62 63 • Values 01 First Part. Second Part. 1 46.8 27.8 2 48.5 60.1 3 64.1 93.9 4 67.4 138.3 5 66.9 184.9 6 65.2 231.3 7 67.0 276.2 8 65.3 322.4 9 66.2 367.6 413.3 0.6902 0.6969 0.6928 0.6914 0.6936 0.6887 0.6896 0.6922 0.6903 10 11 12 13 14 15 66.6 413.3 63.9 458.6 67.2 501.1 68.4 '544.1 110.8 585.1 126.6 644.2 697.6 0.6802 0.6651 0.6399 0.5994 0.5334 0.4218 Mean value ^ = 0.6917 [0.02723] EFFLUX OF AIR THROUGH WELL-ROUNDED ORIFICES. 269 For the given temperature of the air and of the dew point we have, according to equation (52), the vapor-pressure 6" = 9.12 mm. [0.3591 in.], and, according to equation (51), the gas constant for outer atmospheric air 5^ = 29.403 [53.593], therefore according to equation (53) for its specific weight and its specific volume we have n = 1.2072 kg., vi = 0.8283 cbm. [n =0.075366 lb., Vi = 13.269 cu. ft.]. For the temperature of air given we have the saturation pressure ?)o" = 13.37 mm. [0.5264 in.] of the vapor of water, and conse- quently, according to equation (54), the moisture of the air is x = 0.696. According to these investigations, the quantity of air passing through the orifice in a second must always be the same so long as the pressure in the boiler is about half of the external pressure, that is, smaller than 377 mm. [14.843 in.]. The value of in the first rows of the preceding table confirms this completely, for they seem to be constant and scarcely deviate from the mean value; according to equation (55), G is simply proportional to the value of (f). In the second part of the experiments in which the inner pressure was greater, the values of ^ diminish decidedly with the growing pressure. With the given values and the mean value 4>, equation (55) gives for the first part of the experiment 2 = 331.29^^^ = 229.1 kg. [46.923 lb.]. F t As we here calculate for the external atmospheric air J^= 111.23 [ = 12.577], we can also write J =2.0602x1^. 270 TECHNICAL THERMODYNAMICS. We compute 4> from and find it to be ^ = 2.0602 [3.7317], and then equations (45) and (45a) give, for the efflux exponent n, the value 71 = 1.375. For another series of experiments with an orifice diameter of d = 10.85 mm. [0.4272 in.] there was found ^ = 228.2, ^ = 2.0594, and n = 1.374 [1 = 46.63, ^ = 3.7302]; in the third series of experiments with an orifice diameter of d = 15.15 mm. [0.5965 in.] there was found ^ = 230.4, 9!' = 2.0785, and ?i=1.386 [^=47.18, ^=3.76481. According to equation (32) the three values of n correspond to the coefficient resistances i^ = 0.066, 0.068, and 0.044, which values lie within the limits also found for the efflux of water. The result therefore follows that in the efflux of air through simple orifices we can use the coefficient of resistance for water, not only for small differences of pressiu-e, but also with the largest efflux velocities, when it is a question of determining the quantities discharged. In the above-given efflux experiments there were observed for each separate experiment not only the initial and final pres- sures (in the state of equilibrium), but also the instantaneous pressure hx at the moment of closing the orifice, and the pressure hy after the close of the orifice (after the jump); the values were not tabulated because it has no significance for the problem before us. Thus, for the examples in experiment 1 of the above tabula- FLOW OF AIR THROUGH LONG CYLINDRICAL PIPES. 271 tion, there was found 62=27.8 mm. [1.112 in.], 6:^ = 61.4 [2.456], fc„ = 60.3 [2.412], 63 = 60.1 mm. [2.404 in,.] of mercury. The time needed for the final equahzation of pressure amounted to about ten minutes. As regards the jump of the pressure there appeared here also the phenomenon which had already been observed by de Saint Venant and Wantzel (see p. 194). During efflux into the boiler the temperature of the air there rises (on account of the compression of the air); consequently, during the equalization of pressure, heat is withdrawn through the boiler walls and given to the outside. The exchange of heat between the air and walls apears to be particularly vigorous im- mediately after the closing of the orifice, when the stormy air of the interior comes into contact with the walls. Very remarkable was the observation that the flow of air into the boiler took place with perfect quiet during the first part of the experiments, that is, so long as the pressure in the boiler was smaller than, say, half of the external pressure; but as soon as the pressure was greater a low rushing could be heard; the passage of the experiment from the first part into the second part (see preceding tabulation) could be determined by the ear. The efflux of air under high pressure into the free atmos- phere takes place, as is weU known, with a powerful, roaring, noise. § 48. EXPERIMENTS ON THE FLOW OF AIR THROUGH LONG CYLINDRICAL PIPES; MOTION AGAINST GREAT RESISTANCES. The investigation of the motion of gases in pipes is of tech- nical importance. In most cases we have to do with such slight differences of pressure that we can regard the specific volume of the gas at aU places of the pipe line as approximately invariable. Experience has shown that it is thoroughly reliable to deter- mine the velocity of the flowing quantity of gas or air exactly according to the formulas found for the motion of water in pipes, and to use then the same coefficient of resistance in the calculation. But hitherto there have been no experiments whatever which woiild enable one to draw a conclusion as to the behavior of air 272 TECHNICAL THERMODYNAMICS. when flowing through a long pipe toward a vacuum or to a highly rarefied space. Of interest here is the question as to the velocity and the condition of the air in the efflux opening, and of interest also is the law of the resistances which must be used in the cal- culation. It is at once evident that a rapid diminution of pressure and a marked expansion of the air, with rapid increase of velocity, will here take place on the way to the orifice, but it is also evident that the hypothesis employed with simple orifices, namely, that the air expands polytropically, is no longer permissible. Z' f-^£> Fig. 40. Now I have modified the experimental arrangement above described in such a way that atmospheric air could flow through pipes of different length into the air-boiler. In the following will now be described the results of a series of experiments in which the air went through a smooth pipe of d = 5.1 ram. diameter [0.201 in.] and of a length of about 4 m. [13.045 ft.]. The outer end A of the pipe (Fig. 40) was provided with a well-rounded influx orifice through which the outer air entered into the pipe. The other end B of the pipe projected for a short dis- tance into the interior of the air-boiler K. At the beginning of the experiment the cock C was completely opened, and it was closed at the end. At the places designated by 1, 2, and 3 there were vacuum gauges, glass tubes which at their upper ends opened into the interior of the pipe, and at their lower ends dipped into mercury. During the flow of the air the level of the mercury columns occupied the heights ^i, /12, and /13, and at a given signal these were simul- FLOW OF AIR THROUGH LONG CYLINDRICAL PIPES. 273 taneously read by three observers; these heights, subtracted from the barometer reading, gave the piezometer readings, the pressure of the air at the three places in millimeters of mercury, and in the following are designated by ai, 02, and as. The first part of the results of the observations is given in the following table, which after the above description needs no further explanation. Straight Line of Pipe. Diameter 5.1 mm. [0.2008 in.], cross-section i^ = 20.428 X 10" « qm. [0.03166 sq. in.], total height 1 = 3.976 m. [13.045 ft.], barom- eter reading 61 =750.3 mm. [29.54 in.], temperature of air, Ti =273 + 23.2 = 296.2° [533.16°], dew point To = 273 +17 = 290° [522°], and corresponding steam pressure is 60 = 14.42 mm. [0.5677 in.]. Number Duration Equilibrium Pressure of of Flow 62 xperiment t" 63 Value of 63 — &2 , 1 182.5 22.6 mm. 0.2110 2 187.0 61.1 0.2139 3 183.3 101.1 0.2128 4 181.8 182.0 140.1 178.1 0.2118 5 0.2099 6 184.5 216.8 255.0 0.2070 We see that in the first experiments the value of ^ is approxi- mately constant, but shows a tendency to decline from the fifth experiment on; the experiments were, to be sure, carried farther, but for the following purposes the given data will suflace. From the data just given for the short orifice we can compute the vapor pressure and find it to be &" = 14.73, the constant 5„ = 29.482 [53.639], also the specific weight yi and the specific volume Vi of the external atmospheric air, namely, ri = 1.1681kg. and ?;i =0.8561 cbm. [n =0.07292 lb. and vi = 13.714 cu. ft.]. 274 TECHNICAL THERMODYNAMICS. The cubic capacity V of the vessel is 7 = 4.2290 cbm.^ [149.35 sq. ft.], the mean value of 4> of the first four experiments is ^ = 0.2124, and we therefore get with the given values, from equation (55), ^ = 68.456 kg. [14.0211b.] for the weight of air per sq. meter [sq. ft.] which passes in one second through the efflux orifice. In the well-rounded orifice of the same cross-section this was found according to p. 269 to be 229.1 kg. [46,923 lb.]; we see from this how greatly the pipe resistances reduce the quantities flowing through. For the same series of experiments there were simultaneously read in each individual experiment the three vacuum gauges during the flow of the air; the first reading was taken 30 seconds after the opening of the cock, and also after 60, 120, and 150 seconds. The following tabulation gives the cor- responding piezometer readings only for the time corresponding to 30 and 150 seconds, and was determined from the readings of the vacuum gauge. Number Time after Piezometer Rea ding. of Opening the Cock. Experiment Qi a2 as 1 30" 725.8 545.3 192.7 150 725.8 545.3 192.7 2 30 725.8 545.3 192.7 150 725.8 545.3 192.7 3 30 725.9 545.8 194.3 150 725.9 546.2 198.3 4 30 726.0 546.6 202.8 150 726.0 548.3 215.8 5 30 726.3 549.4 224.3 150 726.7 552.4 242.8 6 30 726.9 554.4 251.8 150 727.4 558.9 273.1 ' During respective calibrations of the vessels of different temperatures there was found for the vessel capacity y = 4222.04 + 0.3 t cbdm for water at the temperature of t° C. FLOW OF AIR THROUGH LONG CYLINDRICAL PIPES. 275 We see that the pressures in the pipe maintain themselves at the same heights during the first two experiments; a marked rise of piezometer reading No. 3 only occurs with the third and fourth experiments. From experiment 5 the values Oi, 02, and as grow with the increasing boiler pressure. If at any place of the pipe p is the pressure, v the specific vol- ume, and w the velocity of the air, we shall have, according to equation (7), p. 234, the relation H=f^-^ip^vr-pv), (56) where H represents the energy of flow. But now the relation Gv=Fw holds, and as the same quantity G flows through all cross- sections, and because in a cylindrical pipe all cross-sections F are of the same size, we get also for the ratio G:F a, constant which we will designate by A, and there follows ^4-1 »" which value for the first four experiments in our case leaves A =68.456 [14.021]. The combination of the two preceding equa- tions gives -^ = — -ApiVi-pv) (58) Now since piVi is known for the exterior air and p is determined by the piezometer reading a by means of the equation p = 13 .596a [p = 70.73a], we can compute, from equation (58), for the bodies in question the specific volume v, then, according to equation (57), w=Xv, and next get H='U^ : 2g, or the energy of flow at this place. In this way there was found (Fig. 40) for the three piezometers respectively ai = 725.8, 02=545.3, a3 = 192.7mm. i/=0.8809, v" = 1.1635, v'" = 3.0819 cbm. «)i=60.30, W2 = 79.65, iy3 = 210.98m. Hi = 185.3, H2 = 323.4, H3 = 2268.7 mkg. 276 TECHNICAL THERMODYNAMICS. From this can be seen that particularly in the portion from 2 to 3 there is great expansion of the air and great increase of velocity. Now it is a question of taking into account the resistances to motion in the pipe. (k— 1)X^ If in equation (58) we temporarily make ■ =c, then we shall have ciP = piVi — pv, and from this vdv = PiJ^i^— cv dv, V or also, because the differentiation of equation (59) gives 2cv dv= —dipv), we get dv pdv = id(pv) + piVi— (60) But the work of resistance dW must be written, according to equation (lie), p. 234, as follows: dW = ^-^ + pdv, and with equation (60) we get or, if we utilize equations (56) and (57), dW^^-^'f-'AH. If in addition we introduce Ho as an auxiliary quantity with the significance ^o = ;^PiUi (61) FLOW OF AIR THROUGH LONG CYLINDRICAL PIPES. 277 there finally follows dW^!^(Ho^-nHJ (62) The equation can be integrated. In our case the work of resistance W on the way from the first to the second piezometer (Fig. 40) is found to be W'-'^(Ho log. ^- m-H^)y and the work W" on the way from the second to the third piezom- eter Tf" = "-^(ffolog.|-(H3-H.)). The auxiliary quantity Ho is found from equation (61) to be ffo = 5109.2 [Ho = 16,763ft-lb.], and according to the tabulation on p. 275 there follows W = 2312 mkg., W" = 6844 mkg. [W = 7585.4 f t-lb. W" = 22454 f t-lb.] referred to the unit of weight of air. The piezometers are equally far apart, and in our case (Fig. 40) ^2 = ^3 = 1.913 m. [6.2763 ft.]; therefore, in the second half of the way, the work of resistance is almost three times as great as in the first half. For the further investigation of the problem it is now neces- sary, in judging of the resistances, to start from a particular hypothesis. For the motion of water, in a cylindrical pipe of diameter dr and length I, we can calculate the loss of energy, in consequence of friction, by the formula ^'d72^=^^d; ' 278 TECHNICAL THERMODYNAMICS. where the velocity w of the water and the energy of flow H are the dame in all cross-sections, and where t^r represents the coeffi- cient of resistance, the so-called coefficient of friction of the pipe, which on the average is constant, is commonly taken (^,. = 0.025. With air, to be sure, under the assumption of small differences of pressure, the same hypothesis is generally employed; now in what follows it will be assumed that in the present case also l^r can be regarded, on the average, as a constant quantity. As H varies greatly, the foregoing formula is only valid for an elementary length dx, consequently we must put dW = t^r-^dx; introducing the auxiliary quantity a with the meaning 2k Cr i^idr""' (^^^ we get, from the preceding equation and with the help of equa- tion (62), , „ dH dH adx^Hcq^-^, adx=-dy-j^-\og^^\ (64) If we designate the quantity in the bracket by X, and in the present case introduce for the three piezometer readings the above values of Hi, H2, and H3, there will follow Zi = 24.249, Z2 = 13.040, ^3 = 1.440. Integration of equation (64) will respectively give for the pas- sage from the first to the second piezometer, and for the passage from the first to the third piezometer (Fig. 40), aZ2 = Xi-Z2 = 11.209 and a{l2 + l3)=Xi~X3 = 22.809. Measurement gives ^2 = ^3 = 1.913 m. [6.2763 ft.], hence we have, respectively, a = 5.862 [1.7867] and 5.965 [1.818], and because FLOW OF AIR THROUGH LONG CYLINDRICAL PIPES. 279 t4= 0.0051 m. [0.01673 ft.], equation (63) will give the coefficient of resistance l^r for the two pipe inteijvals, namely, Cr = 0.0256 and Cr= 0.0260 respectively, which are very nearly the same; the reliability of the hypothesis employed is therefore established and Cr is shown to be a nearly constant value, as with water. If we again start from the first piezometer, but go on to the orifice of efflux for which H represents the energy of flow, we get and because ^4 =0.074 [0.2428 ft.] (Fig. 40), we have Z=^-loge ^ = 24.249- 3.900a [Z= 24 .249 -12.795a], substituting here a = 5.965 [a = 1.818], there results Z= 0.986, a value differing but little from unity. The smallest changes of the separate quantities, lying wholly within the errors of observa- tion, lead to the value X=l, which in principle is the minimum value. With the help of equation (61) it therefore follows that The introduced auxiliary Ho means nothing but the energy of flow in the orifice. With the help of equation (56) then follows 2 V'»=-;^Vin, (65) and the efflux velocity consequently it is equal to the acoustic velocity corresponding to the state of the air in the plane of the orifice. 280 TECHNICAL THERMODYNAMICS. With it is also found ^=^^^-^1 <66) For the present case, calculation gives w^= 316.6 m. [1038.7 ft.]; furthermore from equation (57) the volume v in the orifice w becomes ?; = y = 4.6250 [74.088 cu. ft.], and equation (65) gives the pressure there as p = 1567 kg. [320.94 lb.] or & = 115.2 mm. [4.535 in.], and for the pressure ratio ^=0.1536,1 Pi which is a very different value from that given by the well-rounded orifice. Moreover the foregoing equations give directly the orifice pressure _ 1 2g^ri in which of course the value X=G:F must be given by experi- ment or must be specially calculated. If p2 is the pressure in the boiler, then the given propositions hold so long as p>p2- In this case X = l always, and if the distance of any cross-section of the pipe measured backward from the orifice is designated by I, and the energy of flow in this cross- section by Hx, we get, with the help of equation (63), Ho , Ho , 2k I H-r''^H-r'+^^^d.' (67) 'Remark. The value given for p:py is valid for a length of pipe of about 4 meters [13.12 ft.], with a diameter dr= 5.1 mm. [0.201 in.]. In another series of experiments, with a pipe half the length and the same diameter there is found — = 0.2090, and for the well-rounded orifice of the same diameter —=0.5047 P' . . ^' (see p. 251). Therefore if the air is to flow from a boiler into the free atmosphere, then in the three cases the boiler pressure must, respectively, be at least pi = 6.51, 4.78, or 1.98 atmospheres if the air is to flow out with the velocity of sound, as when passing into a vacuum. FLOW OF AIR THROUGH LONG CYLINDRICAL PIPES. 281 from which Hx can be found (by trial) and then the velocity Wx determined, in the cross-section under^ consideration. With the value A we can also find from Xvx=^Wx the specific volume Vx, and, with the help of equation (56), there can be found at this place the specific pressure p^, so that a picture can easily be made of the variations of 'px, Vx, and Wx from section to section. The proposition developed in the foregoing has also been con- firmed by experiments on a shorter pipe, so that we may now enunciate the following: 1. If air flows through a long pipe into a vacuum or into a highly rarefied space, the efflux velocity is identical with the acoustic velocity in the orifice, and can be computed according to equation (66) provided pressure, temperature, and moisture of the air in the discharging vessel are known. The efflux velocity is then a maximum and independent of the resist- ances of the pipe, the latter influencing only the quantity discharged, which becomes smaller as the resistances increase. 2. The resistances in a cylindrical pipe are to be estimated in the same way as with the motion of water, and the corresponding coefficient of resistance or of friction i^^ is also approximately the same ; the latter may, as with water, slightly change with the velocity of flo w,i which, however, should be settled by further experiments with air.^ 3. For the flow of air through pipes into rarefied spaces, or for the efflux of air with very great excess of pressure, there is still to be established the law according to which the air expands in the pipe; for this purpose the experiments discussed above will not, in general, suffice, and ought to be continued under varied conditions. On the other hand 4. It is thoroughly justifiable to draw from the above adduced experiments the conclusion that the hitherto customary method of treating, for very small differences of pressure, the flowing ' Zeuner, "Vorlesungen iiber Theorie der Turbinen," Leipsic, 1899, p. 50. 'Compare Weisbach, "Versuche uber Ausstromung der Luft unter hohem Druck," etc., Civilingenieur, Vol. 12, 1866, p. 85. 282 TECHNICAL THERMODYNAMICS. motion of air in pipes like the motion in water, is a sound method, provided we can regard the specific voliune of the air, during the flow, as approximately invariable. Conclusion. Recently the theory of the motion of gases has often been the subject of special investigations, but as they principally relate to the flow of steam the whole question will be more fully discussed in the second volume of this treatise; in this connection special reference may be made to the beautiful experimenta investigations of S t o d o 1 a. APPLICATIONS. TECHNICAL PART. I. On the Theory of Air Engines. § 49. PRELIMINARY REMARKS. We distinguish between h o t-air engines and c o 1 d-air engines; the purpose of the former is to transform, on a more or less large scale, heat into work, reUably and continuously, and to utDize this work for technical purposes, for the running of machines engaged in the performance of work. Hot-air engines are therefore prime movers; they are driven by a gas, the atmospheric air, which is the "motive power " in the mechanical, technical, sense, and the "medium or mediating body " in the thermodynamic sense; the quantity of air enclosed in the engine receives and rejects heat, and in conse- quence experiences periodically such changes of pressure and volume during expansion and compression that the corresponding pressure curve, or the several pressure curves occurring in the process, constitute an indicator diagram, and enclose an area whose amount gives directly a measure of the work performed during a full period; in a word, the engine describes a cycle. In the cycle of a hot-air engine the performance of work is accom- panied by the consumption of heat, as was fully explained in the first section. During expansion the air withdraws heat from a body of higher temperature, and during compression transfers it to a body of 283 ' 284 TECHNICAL THERMODYNAMICS. lower temperature; the difference of the heat received and the heat given off is thereby consumed in doing outer work. A cycle of the given kind can also be conceived as conducted in the reverse direction, and this case exists in the cold-air engine; here the air during expansion withdraws heat from a body of lower temperature and transfers it during compression to a body of higher temperature; in so doing the quantity of heat given off is greater than the quantity received, and the difference must be generated by work. The running of the cold-air engine requires mechanical work, and the machine therefore belongs to the class of operating machines (Arbeits- maschinen) ; its purpose is to withdraw heat, at low temperatures, from certain bodies, to produce cold; at times to cau^e liquids (water) to freeze and thus form i c e. (Ice machines.) In the one case, in which it is a question of generating work, as in the other case, in which it is a question of producing cold, it does not matter what the medium or mediating body is; if, therefore, in the engines discussed the air is replaced by a mix- ture of vapor and liquid, we have, in place of an air-engine, the steam-engine, and here we must in like manner distinguish the hot-vapor engine from the cold-vapor engine, according as the cycle is conducted in the one direction or the other. In all the engines mentioned, which are best designated by the general name "heat engines," we must, according to the con- struction, distinguish between the two different kinds, the closed and the open engines. In the closed heat engine the mediating body, air or gas, or a mixture of vapor and liquid of any sort, is confined in the engine, and at the end of the process is brought back to the initial condition in order that it may continually repeat the same pro- cess; in the open engine, on the other hand, the mediating body is discharged at the end of each process, i.e., expelled from the engine, and in its place there is taken in, from without, a new quantity of the same medium. Of course, in the latter case only such mediating bodies are employed as can be found everywhere, without running expendi- ture; consequently open-air engines are fed with atmospheric air THE CARNOT CYCLE OF A CLOSED AIR ENGINE. 285 in the condition (pressure and temperature) of the external atmos- phere, while the steani engines are fed yith water. In the engines here discussed, during the cycle described within the engine, no chemical changes take place; the open-air engines dismiss the atmospheric air with another temperature, in general a higher one, and, under certain condi- tions, with another pressm'e than it possessed when it was taken in; and in the open steam engine there exists only a change in the state of aggregation, the water that was taken in leaving the engine in the form of steam, or, to speak more correctly, as a mixture of vapor and liquid. But to the class of heat engines there also belong those engines which are fed by a mixture of bodies, ordinarily gases of various kinds which complete within the working cycle of the engine, and indeed in the interior of the engine itself, a chemical process — • combustion, explosion. They will be subjected to investigation later on under the name of internal-combustion engines. A. HOT=AIR ENGINES. (a) Closed Hot-air Engines. § 50. THE CARNOT CYCLE OF A CLOSED AIR ENGINE. . Let the piston K and its cylinder A (Fig. 41) enclose, at the left end of the latter, a space containing G kg. [lb.] of air; let the volume of the air be Vi, its pressure pi, and Ti the corresponding absolute temperature; if vi is the specific volume we have the relation Vi = Gvi. If we lay off on the axis OX the dis- tance Vi as abscissa, and pi as ordinate, then the point a corresponds to the accompanying temperature T', and de- termines completely the state of the air at the instant. 286 TECHNICAL THERMODYNAMICS. With this quantity of air the following cycle will be described, and in so doing we will for the present disregard the question whether it can be practically conducted in this manner. Let 1. the air expand isothermally, doing work by overcoming an external pressure which constantly corresponds to the pressure of the air at each successive instant (i.e., let it take place in a rever- sible fashion) ; in so doing the temperature Ti of the air remains constant, which presupposes a particular kind of heat supply. The quantity of heat Qi, which must be imparted from the outside to the air during its expansion from Vi to V3, and which is absorbed by the air at the constant temperature Ti, can be found according to the proposition given in § 26, pp. 137 and 138, from Qi^GABTi \og,^, (la) and the quantity of work Li thus produced by the expansion of the air is found from L, = GBTi\oge^, (16) where ps is the pressure corresponding to the point b and the volume V3. Now lay through the two points a and b (Fig. 41) the adiabatic curves be and ad and let 2. the air expand still further without receiving or rejecting heat, till it reaches the point c, corresponding to the volume V2, pressure p2, and temperature T2. The work Li thus produced is, according to equation (62a), p. 140, W = g'-1{T,-T2) (2) Now let 3. .the air be compressed along the isothermal curve cd at the constant temperature T3 till it reaches the point d which has V^, Pi, and T2 as volume, pressure, and temperature, and which lies THE CARNOT CYCLE OF A CLOSED AIR ENGINE. 287 on the adiabatic passing through a. In so doing a quantity of heat Q2 must be withdrawn from tjje air and a quantity of work L2 must be expended, which quantities can be found from the following formulas : Q2=(?ASr2loge^, (3a) L2=GBT2\oge'^ (36) Finally let 4. the air be brought back adiabatically along the path da to the initial state a; the work L2' which is necessary for this part of the compression can be found from L2' = G^-^{T,-T2) (4) During this cycle a quantity of heat Q1 — Q2 has disappeared and been transformed into work; the net work produced can be found from L=L\ + Li —L2—L2 , and by substituting in this the foregoing values we get L=(?5(rilog.g-r2log.g), . ... (5) the quantities of work Li' and L2' cancelling each other because of their equality. The two expressions given for the heat quantities Qi and Q2 of course lead to the relation AL=Qi-Q2 "(5a) Now, according to the propositions which were found for the adiabatic curve (§ 27, equation 61, p. 139), we have for the two passages be and ad, between the same temperature limits Ti and T2, the relations IlJP1.\-^ and ^^(P-A"^ T2 \pj ^"""^ T2 \P2) ' ■ ' (6) 288 TECHNICAL THERMODYNAMICS. and therefore between the four pressures there obtains the relation PlP2 = P3Pi, (7) in accordance with which, equation (5) gives L=GB{T,-T2) log,^, Ps and if, besides, we determine ps from the second one of equations (6), we get L=5(?(ri-7'2)logeg(^)^, .... (8) while with the help of relation (7) and equations (la) and (3a) we find the ratio of the two heat quantities Q2 and Qi to be and from this, with the help of equation (5a), we get. L=§^^{T^-T,), (10) where ATi AT2 ^^^' stands for the change of entropy for the two isothermal passages on the curves ab and dc that have taken place between the two adiabatic curves be and ad. The last two equations were already given in the first section, in the general discussion of the C a r n t cycle, and were there designated as (IV) and (V), p. 52, only there the function S was used in place of the absolute temperature T, and the identity of the two functions did not appear until the investigations in § 23, p. 130, were made. Moreover for the case before us the volume of the air can be found for each one of the four vertices of the curved quadrilateral THE CARNOT CYCLE OF A- CLOSED AIR ENGINE. 289 abed (i.e., for the indicator diagram, corresponding to the C a r n o t cycle) from the formulas , Vipi=GBTi; V2V2 = GBTr, l VsP3=^GBT^; V,p,=GBT,. \ ' ' • • (12) ^ Multiplying the two upper expressions and the two lower ones gives, with the consideration of equation (7), the additional relation ViV2 = VsV, (13) Finally from the second of equations (12) the value GB can be determined, and utilized in equation (8) ; we then get the work, which is designated in technical circles as the indicated work per period : i=7.p.^l„g.|(fi)^. (H) As V2 represents the greatest volume, assumed by the air weight G within the cycle, the preceding equation enables us to draw a conclusion as to the size of the power cylinder necessary for a determinate amount of work L; moreover pi is the greatest pressure and p2 the least pressure occurring in the cycle possessing the limiting temperatures Ti and T2- Now if a hot-air engine is to really describe a cycle like that of Fig. 41, and if not only to the left of the piston K, but also to the right of it, a quantity of G kg. [lb.] of air describes the given cycle, so that we have before us a double-acting engine, then there will be described, during every revolution, two cycles, during a complete double-stroke of the piston, when the oscillating motion is effected by a crank mechanism. Now if the engine makes u revolutions per minute, the work performed per second is 2Lu 60 ' and if we divide this value by 75 mkg. [550 ft-lb.] we get the 290 TECHNICAL THERMODYNAMICS. work produced in horse powers, which will be designated by N, and then Lu ^^30X75 (1^> L 30X550 J' If we here utilize equation (14) and let F represent the piston cross-section (Fig. 41) and let s represent the total piston stroke, then, disregarding the clearance space, we shall have V2 = Fs and accordingly This is the equation of work for this double-acting engine in the form in which it would be directly used in technical investiga- tions, provided the cycle treated here could be practically realized in the manner described. But of equal technical importance is equation (10), which gives the relation between the produced work L and the heat quantity Qi, which is absorbed during isothermal expansion ab of the air confined in the engine. If in equation (10) we reduce the heat quantity to the second, and designate it by Qi", then L will repre- sent the work produced in one second, expressed in meter kilo- grams [ft-lbs.], and if we designate this work expressed in horse- powers by N we shall get ^ = 75W.(^-^^^ (1^ b-5^^^^-''4 From equation (9) there will then follow the heat quantity Q2", which on the average must be given off to the outside, during every second, by the working air during its isothermal compression along the path cd (Fig. 41), namely, Q2"=^Qi" (18) THE CARNOT CYCLE OF A CLOSED AIR ENGINE. 291 Closely connected with the foregoing investigations is the ques- tion whether, and under what circumstances, it is possible to apply the ' C a r n t cycle to actually constructed engines. If first of all we consider only air-engines, we recognize at once that it is impossible to carry through, in one and the same cylin- der, the four separate parts of the cycle described in Fig. 41. Now as regards the first two parts of the cycle; during the first part, along the path ab, the air is to be heated at the constant upper temperature Ti, and during the second part along the path be it is to expand adiabatically, till the lower tempera- ture limit T2 is reached; even this requirement cannot be fulfilled because of the influence of the cylinder walls, for in the second part of the cycle these would take part in the adiabatic lowering of temperature, by imparting heat to the air; still less can the third and fourth parts of the cycle be carried on in the same space. These difficulties can, however, be overcome, at least in part, and are in fact approximately circumvented in the practical designs of the air-engine (as in' the steam-engine) by having the separate parts of the cycle take place, in at least two spaces, but mostly in three different spaces, by shifting the working quantity of air from one space to the other and by designing, for this purpose, engine arrangements which will be discussed later. Still another point is worthy of note. Equation (8) gives the work produced by the engine, but this formula has a positive value only when the value under the logarithmic sign is greater than unity, therefore the condition P2 \T2j must be satisfied; but pi represents the greatest, and p2 the least, pressure occurring in the cycle (Fig. 41),. consequently these limit- ing pressures must fulfill certain conditions, prescribed by the limiting temperatures. For example, if we assume 14.20. P2 292 TECHNICAL THERMODYNAMICS. The engine must therefore work between very wide pressure limits, which would separate still further if we wished to, and could, extend the temperature limits. The practical construction of the C a r n o t cycle in air- engines is therefore not feasible (in steam engines the conditions are far more favorable); nevertheless here, as later investigations will show, the application of the so-called regenerators offers a means of constituting the conditions more favorably. § 51. THE CLOSED AIR=ENGINE CYCLE BETWEEN TWO PAIRS OF POLYTROPIC CURVES. Let G kg. [lb.] of air be confined in the cylinder A (Fig. 42) to the left of the piston, and let the initial volume be F4, the pres- sure p4, and T2 the temperature; with this quantity of air the cycle dahc is to be described; here let da and he be two polytropic curves, subject to the law pv"» = constant, and v^ \ \ , -.' let ah and cd be like curves, following "^^ ^ ^ depends, as equation (29) shows, on the choice of the intermediate temperature T and be- comes greater, and therefore the cycle so much the more favorable, the larger T is assumed to be. The greatest permissible value is T=Ti; with this value we get Tz = T2 from equation (23) and the maximum Lm of work is then ■^m = ^^^^ (7^1 — 7^2) , the C a r n o t cycle stands forth and the two curves ah and cd are isothermals, as was to be expected. The foregoing formulas include almost all cases, even those which have hitherto been separately subjected to investigations by others. § 52. CLOSED AIR=ENQINE CYCLE BETWEEN TWO ADI= ABATICS AND TWO UNLIKE POLYTROPIC CURVES. (Lorenz Cycle.) Let Fig. 42 again be taken as a basis of the discussion and let the two curves ad and he of the curved quadrilateral be adiabatic curves; on the other hand let ah be a polytropic curve for which the specific heat is c' and let c" be a similar magnitude for the other polytropic. Accordingly the exponent n [from equation (19)] will be different for the two polytropic curves ah and cd. CYCLE BETWEEN ADIABATICS AND UNLIKE POLYTROPICS. 297 Let the highest and lowest temperatures occurring in the cycle again be Ti and T2, and the intermediate temperatures T and Ts. The quantity of heat Qi supplied along the path ab can then be found from Q,^c'G{T,~T), (30) when G is the weight of the mediating body, here the air in the working cylinder. Along the path cd the heat quantity Q2 is to be withdrawn and can be determined by the equation Q2 = c"G{Tz-T2), (31) and the work produced is found from AL=G[c'{Ti-T)-c"{Ts-T2)] (32) But the four temperature values are again related to each other in a particular way. For the passage, from the one adiabatic to the other along one or the other polytropic curve, the change of entropy is the same, and therefore we have the relation c' or c'log.^=c"loge^^ (33) To simplify let us put w=^, (34) then follows To /r,\"» (35) T2 [tJ ' from which the second intermediate temperature T3 can be cal- culated, when a choice has been made of the first one, T. 298 TECHNICAL THERMODYNAMICS. Combining equations (30) and (32) then gives the thermal efficiency : AL {T,-T,) or with the use of equation (35) 7^2 (rr-r-) The equation shows that here also there is a best intermediate temperature T for which In the perfect cycle the work Lm = 'Tjr {Ti — T2) is produced, and hence, for the present type of engine and with the help of equation (45a) and of the known relation we can calculate efficiency )j : c„(«-i)(ri-r2)iog.^^(J^)^ c{Ti - T2) + c,(«- 1)^1 log, g (^y^^ . . (48) THE AIR-ENGINE CYCLE AND rOLYTROPlC CURVES. 311 According to the assumed values of n and c there exists here an infinite number of cases; for the j)articular values n = K and c=0 we again turn to the C a r n o t cycle. If the engine is made practical, i.e., double-acting, with u revolutions per minute and developing a work of N horse powers, we get, by making use of equation (15), p. 290, in the preceding formula (47), the expression GJS(ri-r2)log,g(^^)^=30x75^, [=30X550^], (47a) from which can be calculated, for given temperature limits, the weight G of the air which must be confined in the engine, provided we also assume the pressure ratio pi:p2', this ratio, to be sure, is arbitrary but must not sink below a particular value. The fore- going formulas only lead to real values when the expression under the logarithmic sign is greater than unity, and therefore the relation !>©-' <-) P2 must be observed throughout. If of the four pressures only one is given, say pi, then p2 will be limited by the chosen value of the preceding pressure ratio, and the other values pa and pt will be determined from the already used relation frfAP)-' ™ The corresponding volumes then follow from the expression Vp = BGT, provided we substitute the corresponding values of the temperature and pressure belonging to the four vertices of the indicator dia- gram (Fig. 43a). We then get a measure of the size of the engine cylinder for the case in question, provided that the whole cycle really can be practically carried out in an engine with a single cylinder. 312 TECHNICAL THERMODYNAMICS. Moreover there will be opportunity below to further elucidate the preceding equations by numerical examples. The indicator diagram, assumed in Fig. 43a, p. 309, is trans- formed in Fig. 436; for this purpose (see Section I, p. 69) the change of entropy -/S is laid off as abscissa from and the temperature T as ordinate. The transformations of the two isothermal curves ob and cd are here given by the horizontals a'h' and c'd', while for the passage from any point of the polytropic curve to any arbitrary tempera- ture value T, lying between the chosen temperature limits, there is found the change of entropy (tC (because dQ=GcdT), where Po is an arbitrarily chosen constant. This equation represents the transformation d'a' of the polytropic curve da (Fig. 43a). (Compare p. 159.) Generally speaking, for the transformation of a point of an indicator diagram, equations (55), p. 135, will hold, provided the mediating body is a gas. If we here use the particular equation (55c), in the form AP=APo+Cp\ogeT-{cj,-c^)\ogep, . . (51) there will follow, for the change ab, the length of the portion a'b' in Fig. 436, a'6' = PiP3 = (c,-01og,Ji, and for the change dc we get the portion d'c' d'd = P4P2 = {cp- c^) logc 5-, P2 and then, because of the relation piP2 = P3P4, it results that a'6' = c'd' ; the transformations of the two poly- THE AIR-ENGINE CYCLE AND POLYTROPIC CURVES. 313 tropic curves d'a' and h'd are therefore hori- zontally equidistant, in coosequence of which the areas under the two curves, hatched in Fig. 43, are equally great, which is a proposition, moreover, that directly follows from the theorems on transformation, because the two hatched areas represent the heat quantities Qi and Q2 (Fig. 43, p. 292) measured in units of work, and because there was found above Qi'=Q2'=cG(ri-r2). In like manner the two rectangular areas a'b'PsPi and d'c'P2Pi represent the heat quantities Qi" and Q2", measured in units of work, and the curved quadrilateral a'b'c'd' represents the produced work L. If the curve da (Fig. 43a) is not polytropic, but any curve whatever, we still can easily construct for this case a second curve cb, under the assumption that the transformations of the two curves are horizontally equidistant, therefore likewise enclose equal areas. If, for example, T is the temperature and p the pressure for a point of the curve da, and if for the corresponding point of the curve cb there exists the same temperature T but the pressure p', then equation (51) will hold for the first point, and for the second AP' = APo + Cp loge T- (Cp-c^) log, p'; the difference of the two equations is ^P-AP' = (Cp-0 logX, and this value will be the same for all points when — = constant. P If we designate this constant by m, refer the formulas to the unit of weight of gas, and designate by v and 1/ the volumes which 314 TECHNICAL THERMODYNAMICS. belong to the corresponding points, we get, for the point of the second curve corresponding to a given point (pv), the relations p'=mv and v' = —, m with which the second curve can easily be constructed from the first. R a n k i n e 1 calls the two curves, connected in the manner indicated, isodiabatic; according to the preceding investigations the polytropic curves, possessing equal exponents, are at the same time always isodiabatic, i.e., their transformations are horizontally equidistant. These curves play an important r61e in subsequent investigations of the air engines. " § 55. THEORY OF THE REGENERATOR. Id the cycle represented in Fig. 43, p. 309, there can be com- puted from equation (45a) the heat quantity Qi, which must be supplied on the path dah to the air existing in the engine: Qi=g[c(2'j-7'2) + ^B2'i log,^(^^^;^J, and the heat quantity Q2, which is to be withdrawn along the path hcd, is found froin equation (46o) to be In the first equation, the first term Gc^T^ — Ti) represents the quantity of heat which must be supplied for the change da from one isothermal to the other, and the same term in the second equation, which is likewise GcCTi— Ta), represents the quantity of heat which must be withdrawn along the path he, i.e., on the return from the second isothermal to the first. ' R a n k i n e, "A Manual of the Steam Engine," 1859, p. 345. THEORY OF THE REGENERATOR. 315 Now if we could regularly store in a body this heat quan- tity during its return be, in order to u^tilize it again in the subsequent operation along the forward path da, then, accord- ing to both the preceding equations, there would result for the heat quantity Qi to be supplied from the outside, i.e., from the heating body, only the amount Qi^GABTi log, ^Jj^^^\ (52) and for the heat quantity Q2, to be given off to the cooling body cd, Q2=GABT2\os,^^(j^^^h (53) and from this follows the ratio of the two heat quantities, Qrr, ^^^ But for the cycle before us the work, according to equation (47), is L=GB(ri-r2)loge2i(^')^. . . . (55) If we divide this expression by equation (52) we get simply L=§^{T,-T,) (56) Equations (54) and (56) agree exactly with the formulas which have been foimd for the perfect engine describing a C a r n o t cycle. When therefore definite, limiting, temperatures are prescribed for the mediating body (medium) there will exist an infinite number of cycles through which the maxi- 316 TECHNICAL THERMODYNAMICS. mum of work can be produced, as in the C a r n o t cycle, provided the two adiabatics of the latter are replaced by any pair of polytropic curves whatever and the possibility of the assumed heat storage exists; yes, the polytropic pair of curves can be of any type whatever, if the two curves are only isodiabatic. (See closing remark in § 54.) But with this all difficulties would disappear which, according to the preceding presentation, p. 291, oppose the practical reahzation of the C a r n o t cycle in air engines; the partial removal of these difficulties is in fact ren- dered technically possible by the application of the regenerator .i Let AB (Fig. 44) represent the regenerator, i.e., a prismatic vessel which is filled uniformly with a dense weave of wire (which in hot-air engines is replaced by thin or perforated sheets of metal); in the main, it is filled with a body possessing a great surface which can easily absorb and reject heat, and which at the same time offers comparatively little resistance to the passage of the air; ' The regenerator was invented as early as 1816 by Dr. Robert S_Lir - / li n g7~arr(rit was not until later that James Stirling and Ericsson ' made use of it in their hot-air engines. The great importance which the regen- erator has attained in other parts of heat technology, through the brothers I W i 1 h e 1 m and Friedrich Siemens, is known to every engineer. \ The first theoretical treatment of the question as to the mode of action of '^he regenerator in hot-air engines is given by Macquorn R'aTn k i n e C^AManual of the Steam Engine," 1859, p. 344, etc., and still earlier in "Philo- sophical Transactions," 1854). Based on these presentations the question is again touched upon by Briot ("Th^orie m^canique de la chaleur," Paris, 1869, p. 84, in German by Heinrich Weber, "Lehrbuch der me- chanischen Warmethoorie," Leipsic, 1871, p. 88), also by de Saint Robert ("Principes de Thermodynamique," 2nd Ed., Turin and Florence, 1870, p. 287). In the second edition of the present book (1866) I did not know of R a n - k i n e ' s presentations, and I erroneously combated the advantage of the regenerator in the air engine because I had only investigated cycles between two adiabatics, for which, to be sure, the uselessness of the regenerator holds. But if, according to R e i 1 1 i n g e r, we replace the adiabatics by isothermals, the circumstances change. An enlarged theory of regenerator action is given by J. H i r s ch ("Thdorie des machines a^rothermiques," Annales des ponts et chauss^es, 1874, p. 409), and Schroter, making use of the propositions on the transformation of pressure curves, has discussed the question under con- sideration ("Uber die Anwendung von Regeneratoren bei Heisluftmaschinen," Zeitschrift des Vereins deutscher Ingenieure, 1883). The investigations in the present book embrace many additions and applications to specially practical THEORY OF THE REGENERATOR. 317 at the left end A there is a vessel K which is filled with air at the lower temperature limit T2, and at tlje right there is a vessel H which is filled with air at the higher temperature limit Ti ; sup- pose a certain quantity of air to have been pushed back and forth several times through the regenerator from the cold to the hot space; then, in the filling of the regenerator, in the wire mesh, there will exist a distribution of temperature whose law is represented by the curve CDE, provided we lay off T at any particular place P, i.e., at the cross-section F which is at the distance AF = OP=x from the left end, and lay off at P the ordinate PD = r = temperature, above the horizontal axis OX. In doing this let us assume that the temperature curve is nearly horizontal at its ends, and that there- fore the hnaiting temperatures T2 and Ti each extend for a certain distance into the regenerator. Now suppose that again a certain quantity of air has been driven from the cold space K to the hot space H through the regenerator, then the air will be brought from the temperature T2 to Ti, but simultaneously the temperature curve CDE will be changed and will now be represented by the curve C'D'E'; let us first deter- mine the connection between these two curves. Let ac (Fig. 45) represent the curve element corresponding to point D of the curve CDE in Fig. 44; the point P, whose abscissa is X, corresponds to the temperature T, and the point Q is at the distance dx from P and possesses the temperature T+dT. Let Go be the weight of the filling (wire mesh) of the regenerator per unit of length, Cq the specific heat of the filling, and G the weight of air which is forced through the regenerator per second; (we at first suppose a continuous flow of air to take place lasting for some time, and that the regenerator is considerably pro- longed to the right;) then the filling Godx, belonging to the length dx, must experience a diminution of temperature designated by c dx Q Fig. 45. 318 TECHNICAL THERMODYNAMICS. dT' and therefore gives off an amount of heat dQ which is deter- mined by the formula dQ = coGodxdT'. In the corresponding time dt there passes, through the cross-section F at the point P, the weight Gdt of air which is heated; let the corresponding rise of temperature be dT" and the specific heat of the air be c; then the quantity of heat which the air has absorbed along the path dx will be dQ = cGdt.dT". By equating the last two expressions we get coGo^dT' = cGdT". dt But the velocity w with which the air at the point P passes the cross-section F is dx ""^dl' hence the preceding formula gives coGowdT' = cGdT" (57) The two changes of temperature dT' and dT" are represented in Fig. 45 by the distances cd and hd, and here we also have dT = dT' + dT" (58) If through the point d the line da' is drawn, parallel to ac and continued until it cuts the horizontal ab, then a' will be that point of the curve C'D'E' (Fig. 44) which is assumed after an infinitesi- mal time dt by the point D possessing the temperature T. If we designate the horizontal displacement aa' (Fig. 45) by doc', then, from the similarity of the triangles aa'c' and abc, we have the relation dx dT THEORY OF THE REGENERATOR. 319 or dt dtdT ''dT"' Now the left member of this equation is nothing but the velocity of the horizontal displacement of the point a (Fig. 45) or of the point D of the temperature curve CDE in Fig. 44; if we designate it by u there follows dT and we get, if we eliminate dT" from the two equations (57) and (58), for this velocity Gc To be sure the velocity w, with which the air traverses the different cross-sections of the regenerator, is variable because changes of the volume of the air occur which are due to tempera- ture changes and eventually to the simultaneously occurring pressure changes; strictly speaking, therefore, the velocity u, with which the different points of the curve CDE (Fig. 44) progress, is different; but this difference is extraordinarily small, for in the first place the velocity w of the air in the regenerator is compara- tively large, and in the second place the fraction — ^, present (jC in equation (59), always has a very large value; hence in this equation we may unhesitatingly neglect the one (1) in the denomi- nator and then get the much simpler and sufficiently exact relation "=ll <«»> and from this derive the proposition that a 1 1 points of the temperature curve CDE move with the same velocity. Multiplying both sides of equation (60) by the differential dt of the time and integrating, there follows Lc 320 TECHNICAL THERMODYNAMICS. where L represents the quantity of air in kilograms [pounds] which has been driven through the regenerator in the given time, and s represents the displacement of the temperature curve thus occa- sioned; but the two temperature curves CDE and C'D'E' (Fig. 44), corresponding to the beginning and the end, are therefore horizontally equidistant in the direction of the axis of the regenerator. The area contained between the two curves will be designated by / and can be found from or Q=GoCof, (61) because Q = Lc(ri— 7*2) represents the quantity of heat which the air has absorbed in its passage through the regenerator. Equation (61) may be derived directly as follows. The initial temperature T in the cross-section F at the point Pis represented by the ordinate PD, and at the end it is T' and represented by the ordinate PD'; the part of the regenerator filling Godx, belonging to the length dx, has therefore experienced a temperature diminu- tion T—T' and has therefore given off a quantity of heat which is dQ=GoCodx{T-T'). Since (T—T')dx represents the strip of area lying between D and D', integration will give the whole area /, and the preceding formula will thus also lead to equation (61). If the air is forced through the regenerator in the reverse direction, a parallel shifting of the curve CDE takes place in the other direction ; consequently if the same quantity of air, after the change of direction, again passes through the regenerator, the curve will return to its original position. To be sure the preced- ing investigation gives no clue as to the course of the curve itself. The determination of the quantity of air contained in the regenerator is of importance; it varies, but can be graphically represented for a particular instant and for a particular position THEORY OF THE REGENERATOR. 321 of the temperature curve CDE. Let F represent the cross-section of the air-filled portion oi the free space of the regenerator at the distance x from the left end A (Fig. 44), then Fdx gives the volume of air for the portion dx of the length; if dG is the weight of the air in this space and v its specific volume there will exist the relation vdG=Fdx. Now multiplying both sides of this equation by the pressure p of the air, and making use of the equation of condition pv = BT, we get dG=^ — and this equation can be integrated when we know the course T = f{x) of the temperature curve CDE (Fig. 44) of the regenerator, for the pressure p is everywhere the same. In the lower part of the following figure the temperature curve CDE of Fig. 44 is repre- sented and repeated; if we designate the reciprocal value of T, i.e., l:T,hyy and lay off y in the upper part of Fig. 46 as an ordi- nate P'd, corresponding to the abscissa 0'P'=x, we get a new curve c c d e e , and integration of the preceding equation gives G, -^fydx, where Gr represents the weight of all the air in the regenerator. This integral is represented by the hatched area of Fig. 46; if we reduce this area to a rectangle having the base 0'X' = OZ=Z, the axial length of the regenerator, and if yo is the height of the rectangle, the preceding equation will give ^ _Flpyo (62) 322 TECHNICAL THERMODYNAMICS. or, if we designate the reciprocal value of yo by To and remember that Fl represents the whole space Vq filled with air, we shall get Gr=^^, (62a) where To is a sort of mean temperature in the regen- erator; it can also be found directly if we reduce the area CoCDEEo to a rectangle with the base OX = l; the hatched area, however, has the advantage, at once recognized, of showing the law according to which the weight of the air in the regenerator is distributed. Let us again suppose that L kg. [lb.] of air are forced from the cold space, through the regenerator, to the hot space, then the parallel displacement s of the curve CDE corresponds to an equally great displacement of the curve cde to c'd'e', and the area 5(2/2—2/1) contained between the two curves expresses the increment of the hatched area (Fig. 46); accordingly, after the passage of the air there will be a larger weight G/ of air in the regenerator than it originally contained; and consequently a smaller quantity of air will enter the hot space than was furnished by the cold space, — always assuming that the pressure p remains constant when the air is passing through. With the help of equation (62) this difference is found to be G/-Gr = ^s{y2-yi), where the displacement s is given by the equation (60a). The mean ordinate yo is increased to 2/0' after the passage of the air, and we have 2/o' = 2/o + y(2/2-2/i), from the reciprocal value of which we find the temperature To' after utilizing equation (60a) : „, r Lc To(T\-T2)l CLOSED HOT-AIR ENGINE WITHOUT REGENERATOR. 323 The second term in the bracket is a very small fraction with hot-air engines. We may therefore assume that the mean tem- perature in the regenerator only varies between narrow limits, and that the weight G of the air enclosed in the regenerator is principally dependent upon the pressure p and nearly proportional to it, according to equation 62a), in so far as the passage of the air through the regenerator is accompanied by any pressure change. § 56. CLOSED HOT=AIR ENGINE WITHOUT REGENERATOR. First Arrangement (Rider System). Following the above theoretical investigations of the different cycles of hot-air engines, we will now show how the practical realization of these cycles is possible. There exists in this direction a whole series of exceedingly ingenious suggestions and constructive efforts, for example by Stirling, by the clever Ericsson, by Laubereau- Schwartzkopf, Rider, Lehman n, and others; with the exception of perhaps Lehmann's engine arrangement, not one of the proposed constructions has been extensively used. The lack of success is probably partially due to the fact that in the recent suggestions the generator either does not exist at all, or is used in a very imperfect manner; but it may also be due to the fact that real theoretical investigations of the hot-air engines, with a basis of practically possible constructions, do not yet exist. The known investigations are based either on the practically im- possible assumption that all parts of the cycle are described in one and the same space, as was assumed in the above developments, or an effort has been made to derive the indicator diagrams for a certain class of engine and thus find the work of the engine; the equally important question as to the quantity of heat which must be supphed to, or withdrawn from, the working air in the con- structed engines, during the course of the cycle, has not been answered; the propositions which I have developed on pp. 197 to 324 TECHNICAL THERMODYNAMICS. 225, on the action of the transfer-piston, however, do furnish the means of pursuing the question further. Let us try to solve the problem of realizing the cycle, repre- sented in Fig. 47, which consists of two isothermals T2T2 and TiTi and of the two polytropic curves T2T1 and T1T2, and which has already been investigated above, by an engine which can be actually constructed. Fig. 48a gives a schematic representation of such an engine without a regenerator, an arrangement, moreover, which embraces several cases that have already been proposed and constructed. Two cylinders A and B of the same diameter are separated from each other by a perforated partition ah. Piston Ki moves in cylinder A, piston K2 in cylinder B; the former cylinder is open at the left, the other at the right, so that atmospheric air acts from the outside on both pistons. The working quantity of air is included between the two pis- tons Ki and K2. A jacket C surrounds cylinder A, and through the jacket the heating body (fire gases) flows, so that the air which is in this cylinder, or flows into it, is either kept at the upper temperature limit Tr or is at once brought to this temperature. Let the cylinder B, on the other hand, be provided with a jacket D through which the cooling body (water) flows, so that the air, existing in this cylinder, is kept at the lower temperature limit T2, and the air coming from the hot cylinder A is immediately cooled to the temperature T2. The two cylinder spaces will, in future, be distinguished as hot space and cold space. Fig. 486, given under Fig. 48a, represents the piston diagram; the vertical line OZ represents the time of a full period of the engine which is divided into four parts, M, N, P, and Z. Let us first suppose the pistons Ki and K2 to be at the right end of the stroke; at this instant all the air is in the cold space and possesses the volume V2, the corresponding pressure p2, and the temperature T2 (see Fig. 47). The working of the engine is as follows : In the first time interval OM, the piston Ki is held fast and the piston K2 is shoved from right to left till the air volume CLOSED HOT-AIR ENGINE WITHOUT REGENERATOR. 325 has diminished from V2 to V3; in so doing the compression takes place at constant temperature T2 an4 the pressure grows from i'2 toj93 (see Fig. 47). The assumed piston motion is given, in the piston diagram, by the lines 1"2" and 1'2', and in the adjacent piston force diagram (Fig. 48c) the increase of pressure from pz to ps is Pig. 47. Fig. 48d. shown by the portion 12 of the curve corresponding to the time interval OM. In the second time interval ikfiV, both pistons move from right to left; the path of the piston K2 in the cold space is repre- sented by the displacement curve 2"3", the air volume Vs is com- pletely crowded out of the cold space and pushed into the hot space, where the piston Ki finally hmits the volume Vi of the hot air; line 2'3' (Fig. 486) is the displacement curve of this piston in the course of the time MN, within which the pressure of the air grows from ps to pi according to curve 23 (Fig. 48c). In the third interval NP, piston i^2is held fast (displacement curve is 3"4") and piston Ki travels completely to the left end of 326 TECHNICAL THERMODYNAMICS. the stroke, as per displacement curve 3'4'; within this portion of time the air, which is now wholly collected in the hot space, ex- pands under the constant, highest temperature Ti from the volume Vi to V4, and in so doing its pressure diminishes from pi to p^ according to curve 34 (Fig. 48c). Finally in the fourth time interval PZ both pistons are brought back from their extreme position on the left to the right end of their stroke; in so doing the whole quantity of air is pushed out of the hot space back into the cold space, the volume Vi returns to V2, the pressure p^ to p2, and thus one period is completed. In the lower part of the drawing, namely in Fig. 48(i, there are exhibited the indicator curves of each of the two cylinders; it is easy to see how these two curves, as well as the piston-force diagram Fig. 48c, can be derived from the given cycle in Fig. 47. According to Fig. 486, the air volume mn = 7 corresponds to the instant of time R; if we lay off the distance OF in Fig. 47 for the volume V, we shall get in the ordinate VT the corre- sponding pressure p; this value is laid off at the time point R (Fig. 48c) as ordinate RS, and thus gets the corresponding point S on the curved line of the piston-force diagram. Finally if we go vertically down to the axis O'X' (Fig. 48d) from the two points m and n (Fig. 486), and lay off the correspond- ing pressure p, we get the two corresponding points ;S' and S" in the two indicator diagrams. In the engine period described the two indicator diagrams are generated in the direction of the two given arrows; the diagram of the hot space corresponds to positive work, i.e., to the work production of a single cycle, the diagram of the cold space corre- sponds to negative work, to work consumption. The difference of the two hatched areas gives the whole work delivered by this engine, and this is also measured by the hatched area of the fun- damental Fig. 47. It is worthy of note that the engine cycle just described em- braces an infinite number of special cases, for no definite assump- tion was made as to the twin polytropic curves T2T1 and T2T1 in Fig. 47. CLOSED HOT-AIR ENGINE WITHOUT REGENERATOR. 327 We easily see that a perfectly definite piston motion in the second and fourth time interval (MA'' and PZ in Fig. 486) must correspond to a particular value n of the exponent in the equation of the polytropic curve p?;'' = constant. For example, if the two displacement curves 2'3' and 2"3" are parallel straight lines, or even horizontally equidistant curves, and if the same is true of 4'1' and 4"1" (Fig. 486), then the air volume will remain constant both during the transfer from the cold to the hot space, and during the return motion; in this case the polytropic curves T2T1 and T2T1 are vertical straight lines in Fig. 47, and the cycle of the engine is one which has sometimes been called the Stirling cycle. In another special case the displacement of the air, from one space into the other and back again, which occurs in the second and fourth time intervals, can be so conducted that the pressure remains constant. In this case the two curves T^Ti become horizontal straight lines in Fig. 47, and the engine cycle is one which has been called the Ericsson cycle. § 57. CLOSED HOT=AIR ENGINE WITHOUT REGENERATOR. Second Arrangement (Lehmann System). The cycle of a feasible air engine just described can also be realized so far as the assumed cycle is concerned by another engine arrangement. In Fig. 49 let AB be a cylinder in which two pistons Ki and K2 move, of which the former possesses a great length in the horizontal direction, and does not hug the cylinder wall closely in order that the air may spread itself through the remaining annular space; it is called the transfer-piston, while the other piston, K2, is called the power-piston. Let a wall ab, impenetrable to heat, divide a cylinder exter- nally into two parts. To the left of the wall let the cylinder be surrounded by a jacket C in which the heating body circulates, and to the right of the wall let the cooling body flow through the jacket D, and let all this take place in such a way that when the 328 TECHNICAL THERMODYNAMICS. air is to the left of the transfer-piston Ki it will be maintained at the upper temperature limit Ti, while the air to the right, i.e., between the transfer-piston Ki and the power-piston K2, possesses the lower temperature T2. The cycle of Fig. 47 is again represented in the upper part of Fig. 49, and is to be executed by the engine under consideration. Let the whole time of a period be given by the vertical OZ; at the beginning both pistons assume the positions in which they ^ \, " \ p. / i" V / z w ±' Fig. 49. are drawn in cylinder AB. The working of the piston must take place as follows : Time interval OM. The transfer-piston K-^ is held fast and the piston K2 is shoved from right to left, so that the air in the cold space is compressed at constant temperature T^ from the volume Y2 to Vz, and the pressure grows from pressure p2 to ps. The displacement curves of both pistons are given by the lines 1'2' and 1"2". Time interval MN. Both pistons are pushed to the right till they nearly touch each other, the displacement curves in the piston diagram being represented by the lines 2'3' and 2"3"; in CLOSED HOT-AIR ENGINE WITHOUT REGENERATOR. 329 SO doing the total quantity of air is pushed from the cold space into the hot space, which possesses the, volume Vi to the left of transfer-piston Ki, and its temperature is raised to T^; the pres- sure has grown from ps to pi. Time interval NP. Both pistons move, nearly touching each other during the whole way, i.e., move together from left to right, the two displacement curves 3'4' and 3"4" being parallel. The air, which is all in the hot space, expands therefore at constant temperature Ti from the volume Vi to the volume Vt, and the pressure sinks from pi to pi. Finally in the last Time interval PZ both pistons move from right to left back to their original positions, whereby the whole quantity of air is driven from the hot space back into the cold space. It is moreover evident that in the present case, as in the preceding one, the length of the separate time intervals is of no further consequence. If, for the engine arrangement shown in Fig. 49, we suppose the pipe of an indicator to communicate with the space in which the working air exists, and suppose the indicator drum to be driven by the power-piston K2, we shall get directly the indicator diagram (Fig. 49), i.e., the cycle prescribed for the engine. If the indicator drum is connected with the transfer-piston Ki, there will be obtained an indicator diagram of the form found in Fig. 48d for the hot space. The cycle under contemplation also en- closes an infinite number of special cases, according to the manner in which the piston is moved in the second and fourth time inter- vals {MN and PZ, Fig. 49). The engine arrangement discussed corresponds to the general case of Lehmann's hot-air engine, while that given in Fig. 48 represents Rider's engine working without a regenerator. We see from what has been said that these two kinds of engines do not differ in principle, but only in constructive particulars. In the actually constructed engine of Rider, Lehman n, and others, the curved lines 1'2'3'4'1' and 1"2"3"4"1" of the piston diagrams (Figs. 48 and 49) are continuously running curves and, indeed, approximately sinusoids, for the pistons are driven by crank mechanisms, a matter to which we will return later. 330 TECHNICAL THERMODYNAMICS. § 58. THEORY OF THE ENGINE ARRANGEMENTS JUST DISCUSSED. The following calculations are simultaneously true for the arrangements given in Fig. 48 and Fig. 49; but the first one will be made the basis of the discussions. As regards the work L produced by the engine in one period, if we suppose (Fig. 47, p. 325) the cycle to consist of two isothermals and a pair of polytropic curves, then we will have at once equation (47) developed in § 54, p. 310, namely, L=G5(7'i-r2)log,g(^')^i,. . . . (63) where G means the weight of the air enclosed in the engine; at the same time n is the exponent belonging to the members of the assumed pair of polytropic curves T2T1. However, as regards the quantities of heat Qi and Q2 which must be absorbed or rejected respectively during a period, the two equations (45a) and (46a), p. 310, no longer hold here. We must instead make use of the propositions which were developed in § 38, p. 197, for the action of the transfer-piston. Here each of the separate time intervals of Fig. 48 must be specially investigated. In the first interval OM, air is compressed at constant temperature T2 from F2 to Vz, and the pressure grows from p2 to ps; the quantity of heat which must, in so doing, be here withdrawn from the air in the cold space, which is designated by Q2, is found from equation (566) upon p. 137, Q2' = 450^2 log, ^. P2 (64) In the third interval all the air in the hot space expands at constant temperature Ti from the volume Fi to the volume F4, and the pressure sinks from pi to ^4; the quantity of heat Q/ THEORY OF THE ENGINE ARRANGEMENTS JUST DISCUSSED. 331 which must hereby be supplied to the hot space, according to the just cited formula, is • (65) We must now specially investigate the occurrences in the second interval MN in which all the air is pushed out of the cold space into the hot space; at the beginning the volume is V3 and the pressure ps; at the instant of time R (Fig. 486) the volume Fig. 47. Fig. 48d. is V and the pressure p, and, in accordance with the hypothesis, the relation p7» = p3T^3" (66) must obtain, where we will for the present avoid making any definite assumptions as to the constant exponent n. The differen- tiation of this equation gives Vdp+npdV=0. (67) 332 TECHNICAL THERMODYNAMICS. Now we get for the total heat quantity Q which must be sup- plied to both spaces, to the hot as well as the cold space, by integration of equation (46), p. 200, which, after the utilization of the preceding equauuu yoi), oecomes dQ = A'^pdV, and, if besides we determine p from equation (66), we get the value or, if we substitute the corresponding pressure limit, according to equation (66) and, finally, if we consider the laws concerning the polytropic curve in § 29, p. 153, equation (5), Q-^AG ^Jl~^^_^^ B{T^-T^). . . . (686) But this heat quantity Q is made up of two parts, namely, the one, Q2" , which must be supplied to the cold space, and the one, Qi", which is supplied to the hot space. The first quantity of heat is found by integration of equation (75a), p. 221, Q2"=-fAVJp, (69) where 7x is the instantaneous volume of the air in the dis- charging space, here the cold space. At the same instant R of time {V— V^) is the volume of the hot space, and therefore, according to equation (42), p. 199, we have the relation V T2' Ti THEORY OF THE ENGINE ARRANGEMENTS JUST DISCUSSED. 333 If we determine Vx from this and substitute its value, as well as the value of V proceeding from equq^tion (66), in equation (69), then, by integration and some easily followed transformations, we get for the heat quantity Q2", which must be supplied to the air when it is forced out of the cold space : «-"=-« [S-^.'*£]- • • • ™ On the other hand the quantity of heat Qi" which must be im- parted to the hot space at the same time is Qi"=Q~Q2", and, combining preceding equation with equation (68&), Finally, in order to determine Qi'" and Q2", which in the fourth time interval PZ must be imparted to the two spaces, when the air is pushed back from the hot space to the cold space, it is only necessary in the two preceding equations to interchange the temperature values and to substitute the pressures pa and Pi for the pressures pi and ps ; we then get Now we finally get the total heat quantity Qi which must be imparted by the heating body to the hot space during one engine period, from the relation Qi=Qi'+Qr+Qr, and if we here use the preceding formulas and, at the same time, utilize the relations (Fig. 47) ^ = ^ and 4B=Cp— , (74) 334 TECHNICAL THERMODYNAMICS. also substitute Pi P2 P2\T we get Pl=p3^Pi/^,^„-^,^ (75) Qr=Glcj,{Ti-T2) + ABTi\oge~(py-'\.. • (76) Furthermore the quantity of heat Q2, which during a whole period must be withdrawn from the cold space, is found from /// Q2 = Q2' — Q2" — Qi and from this, with the utilization of the preceding formulas and the several relations last given, Q2 = G[c,(ri-r2) + ^B7'2log,g(^j^i]. . . (77) The two preceding equations (76) and (77) now conclude the theory of the two engine arrangements given in §§ 56 and 57, pp. 325 and 328, and are distinguished by their great simplicity in spite of the very complicated occurrences involved in the action of the transfer-piston. The difference Q1 — Q2 leads again, as it ought, to equation (63), which gives the work L produced by this engine; but it is interesting that the two expressions, found for Qi and Q2, for the actually feasible hot-air engines, completely agree in structure with the formulas which were found for the purely theoretical cycle in § 54, p. 310 (equations 45a and 46a). In place of the specific heat n — K there corresponding to the polytropic curve employed, we have here in the feasible engine simply the specific heat Cp of the air under constant pressure; in other respects, however, according to the assumed value of n, the foregoing propositions hold for an infinite number of cycles, and it is only a question as to the kind of motion of the two pistons Ki and K2 (Fig. 48& and 49) in the second and fourth time intervals {MN and PZ). The discussed feasible engines are single-acting; if the engine in question APPLICATIONS AND NUMERICAL EXAMPLES. 335 has u periods per minute, then its work N in horse-powers (indi- cated) according to equation (63) is N = In- BGu 60X75 BGu 60X550 (78) As the expression under the logarithmic sign must be positive in the preceding equations, the condition must be satisfied (see p. 311). § 59. APPLICATIONS AND NUMERICAL EXAMPLES. If we assume that the cycle at the base of the preceding engine arrangements consists of two isothermals and two horizontal straight lines, as indicated in Fig. 50, then we must substitute n=0 in the given equations (this cycle may be called the Ericsson cycle, although no Ericsson engine ever really described this cycle exactly) ; let the ratio of the two limiting temperatures Ti : T2 be des- gnated by A, then equation (78) gives N BGT2 u 60X75^^-1) ^«g»g (78a) [ N BGT2 u 60X550 (A-l)log< Pi P2 ]■ On the other hand equations (76) and (77) give the quantities of heat which must be respectively supplied and withdrawn during every engine period: Qi = 45Gr2[^^^^+Alog«g], . . . (76a) Q2=A5Gr{^+loge|]. .... (77a) 336 TECHNICAL THERMODYNAMICS. If we consider the L e h m a n n engine arrangement given in Fig. 49, p. 328, and assume that the corresponding piston motion in the second and fourth time intervals is arranged in accordance with the here basal indicator diagram (Fig. 50), and if F desig- nates the cross-section of the two pistons, si the whole stroke of the transfer-piston Ki, and S2 the stroke of the power-piston K-i, then (Fig. 49) ^4 = ^-51 and V,-V^ = Fs2.' According to the equation of condition of gases and the assump- tions of Fig. 50 we have V4P2-BGT1 and V3Pi = BGT2, therefore Fs,V2-BGT, (79) Fs2P2 = Bg(t,-^T2^, (80) from which, by division, follows ^=1-^^ (81) From this, according to preceding equations, N FS1P2 A-1 pi ^ = 60X75 -rl°g«^' ^78^) [550] e,=^Fs,p2[^^ + log«g] (766) Q.=^F.4,^+ilogeg] (776) As regards the value of the ratio X, i.e., the ratio of the tem- peratures Ti and T2, actual observation of constructed L e h - m ann engines shows A to lie between 1.5 to 3; these engines, to be sure, only incompletely describe the here assumed cycle. Example. In order to elucidate the preceding formulas by numer- ical example let ^ =2 and pi =2p2, then from equation (786) there follows i^s,pj = 12980- u Ni [Fsip, =95217-]. APPLICATIONS AND NUMERICAL EXAMPLES. 337 If the engine makes m = 100 revolutions (per minute), and if the lower pressure p, amounts to just one atmosphere (p = 10333 kg. [2116.3 lb.]), we get for iV =2 horse powers (indicated) * Fs, =0.025 [Fsi =0.8998 cu. ft.]. If we make the stroke of the transfer-piston si =0.250 m. [0.821 ft.], the necessary piston cross-section becomes f =0.100 qm [F = 1.097 sq.ft.], and accordingly the diameter is d =0.357 m. [d = 1.1819ft.]. On the other hand the stroke of the power-piston, according to equation <81), can be taken s, =0.187 m. [0.614 ft.]. Under the assumptions made, equation (766) gives the heat quantity that must here be supplied per minute to the hot space : Q, = 147.72 cal. [Qi =594.35 B.t.u.], and the heat quantity which must be withdrawn from the cold space in the meantime is Q, = 126.50 cal. [Qj =508.97 B.t.u.]. If the cooling is effected by cold water which, in flowing through the cylinder jacket D (Fig. 49, p. 328), is heated, say, through 10° C. [18° F.], then the quantity of cooling water necessary per minute is 12.65 kg. (28.28 lb.].' The computed heat quantity Qi would give, if the cycle were conducted in a perfect engine describing the C a r n o t cycle, an amount which ex- pressed in horse powers is ^^ 60X75Ar.^''' ^="""60X754 A ' [60X550Ari] [60X550A1 or, after substituting the corresponding values, iVm= 6.959 horse powers, [iVm= 6.86 horse powers,] ' For comparison, with observations on a really constructed L e h m a n n engine, those experimental results will serve which were published by E c k - erth in the "Technischen Blattern " (Prague, 1869, Vol. I, p. 104). The general agreement of my theoretical results with the dimensions and experimen- tal results there given seems to be a very satisfactory one. 338 TECHNICAL THERMODYNAMICS. and accordingly the efficiency of this engine, in the formerly given sense, is N V^TT =0.287. The two equations (76&) and (776) for this efficiency lead to a formula from which we see that it approaches unity more nearly, the larger A and the pressure ratio pi:p2 are taken. In the pre- ceding example the efficiency is very small because the pressure ratio is comparatively small and because the difference of the limiting temperatures is assumed small. It is moreover worth noting that the Ericsson cycle, assumed in the preceding, gives formulas for the work and Jfor the heat quantities Qi and Q2, which are identical with those which result from formulas (47), (45a), and (46a), p. 310, under the sup- position that c = Cp and n=0 for the theoretical cycle assumed in Fig. 43. The identity of these equations with the preceding equations (76), (77), and (78) disappears when the L e h m a n n engine describes a cycle which is different from the Ericsson cycle. When we consider the Stirling cycle, that is, when we consider the transfer-piston Ki (Fig. 49, p. 328) moved in such a way that the displacement of the air takes place at constant volume, and accordingly the contemplated cycle is represented by Fig. 51, then we must substitute n = 00 in equations (76), (77), and (78). We will here employ the same method of transformation previously used with the Stirling cycle, and again repre- sent the stroke of the transfer-piston by Si and the stroke of the power-piston by S2. Here Fig. 51. I5=i and 71 = 73 and 73 = ^4=^81, i^S2 = 74-72=72-7i (see Fig. 49) ; we also have the relations V2V2-BGT2 and Vipi = BGTi. APPLICATIONS AND NUMERICAL EXAMPLES. 339 From the combination of these equations we get s, ViT, ^^2) As can easily be seen we furthermore get [60X550] Qi=Ai^sxpi[^ + Alog.g^], . . . (76c) Q^=^F^^p{~r+^'^'fM' • • • • (^^^^ where both quantities of heat apply to one period. Example. In order to add here also a numerical example, we will assume a Lehmann engine which describes the assumed Stirling cycle; here again let the temperature ratio ^=2; on the other hand let pi =4p2, that is, if the lower pressure pj is again one atmosphere, P2 = 10333 kg. [2116.31 lb.], a compression to four atmospheres will take place. If this engine also is to have an indicated performance of JV=2 horse powers with u = 100 periods per minute, we calculate, from equation (78c), i^s, =0.0125 cbm. [0.4414 cu. ft.]. for the space displaced by the transfer-piston per stroke; according to equation (82) for the power-piston the space in question becomes ' ii'sj =0.00625 [0.2207 cu. ft.]. The same values as in the preceding example are obtained for the heat quantities Qi and Q2 and for Nm. and ij as well (in consequence of the assump- tions here specially made for ^, pi, and ps); this is in accord with the equa- tions (76c) and (77c). If in these two preceding examples we had substituted the engine arrangement prescribed by Fig. 48 instead of the Leh- mann arrangement (Fig. 49), then the volume swept through per stroke by the piston Ki in the hot space would be Vi=FiSi 340 TECHNICAL THERMODYNAMICS. and the corresponding volume for the piston in the cold space V2=F2S2, in case the piston cross-sections i^i and F2 were different. According to the propositions concerning the polytropic curve (see Fig. 47, p. 325), V2 \TiJ ' hence follows F2S2P2 = BGT2 and F2S2 \tJ • If the engine has the Ericsson cycle, then n=0, and there- fore the ratio of the two cylinder volumes is FiSi_Ti F2S2 T2 When the Stirling cycle is assumed n= 00, and therefore FiSi^F2S2 must be assumed. In the actually constructed Rider engines, which to be sure are provided with a regenerator, the cylinders are in fact of nearly equal capacity. § 60. CLOSED HOT=AIR ENGINE WITHOUT REGENERATOR. Third Arrangement (Laubereau-Schwartz- kopff System). Cylinder AA (see schematic Fig. 52) is closed at both ends and is divided into two parts by the partition cd; the left half is sur- rounded by a jacket C through which the heating body flows, the right half is surrounded by the jacket D in which the cooling body (cold water) circulates. In the cylinder there is a transfer-piston Ki, whose length is half that of the cylinder; if the piston is at the right CLOSED HOT-AIR ENGINE WITHOUT REGENERATOR. 341 end of its stroke (as in the figure), the v/hole enclosed air is to the left of the transfer-piston and then, has the highest tempera- ture Ti] if the transfer-piston Ki is pushed to the left end, all the air is forced to the right through the annular space between the transfer-piston and the cylinder surface, where it is now brought down to the lower temperature limit T2. By suitable heat supply and heat withdrawal on the part of the jacket fluids, the enclosed air, to the left of the transfer-piston, is kept at the temperature Ti, Fig. 52. and to the right of it, at the temperature T2, wherever the transfer- piston may stand. To the left of cylinder AA there is a second cyhnder B, the power-cylinder, in which moves the power-piston K2 ; this cylinder is open on the left so that the atmospheric pressure rests on the left side of the piston K2; on the other hand, the space to the right of piston K2 is connected with the cylinder AA by the pipe ab; the pipe opens into this cylinder at the middle of its length. The working of this engine (the normal cycle of the Laubereau-Schwartzkopff engine) takes place as follows, provided we take into consideration the lower part of Fig. 52, the piston diagram, and the upper left part of Fig. 52, the corresponding indicator diagram ; the time of a period is given by the vertical distance OZ, and is divided into four parts. 342 TECHNICAL THERMODYNAMICS. In the beginning both pistons stand at the right end of their stroke, as shown in the figure; all the air weight G enclosed in the engine (disregarding the clearance space) is here in the hot space, possessing the volume Vi and the highest pressure pi. In the first time interval OM, the transfer-piston Ki is held fast and the piston K2 is pushed from right to left ; in so doing the air expands from the volume Vi of the hot space to the volume Vi + V-z, and indeed the expansion then takes place in the hot space, receiving heat at the constant temperature Ti, but in the power-cylinder expansion takes place without the absorption or rejection of heat. Here we have before us the special case c of problem 2, treated in § 39, p. 217, and accordingly there follows from equation (65), p. 217, the equation of the pressure curve P1PP4 (Fig. 52) : H^v)' <^> and therefore the terminal pressure p^ of the expansion in the power-cylinder is ^"^{-£h)' <«) The work Li herewith transmitted to the power-piston is (85) or, utilizing equation (84), Li= SKViiVi-Vii-y^Vi] (85a) /C 1 The final temperature Ti in the power-cylinder is found from the relation BT^ BTi BTi' Ti_ F2P4 CLOSED HOT-AIR ENGINE WITHOUT REGENERATOR. 343 Finally the heat quantity Qi, to be supplied to the hot space in the first time interval, is found from equation (64), p. 217, Qi' = 4Fi(pi-p4) (87) Occurrences in the second time interval MN (Fig. 52) : The power-piston K2 is held fast at the left end of the stroke, and the transfer-piston Ki is pushed from right to left; the total volume yi + F2 of the enclosed air herewith remains constant and all the air is forced out of the hot space ; in so doing work is neither pro- duced nor consumed, but the pressure sinks from pt to p2 and the temperature of the air in the power-cylinder from Ti to T3. With this from which is found Tz= J:\^ (88) %2 tJ During this motion of the transfer-piston, the air is distributed through three spaces; the total volume Fi4-F2 is invariable, and consequently the heat quantity dQ which, altogether, at the instant of time R (Fig. 52), is suppUed to the three spaces, is given by equation (46), p. 200: K—l or if we integrate, and consider that, during the whole time inter- val MN, heat is neither supplied to, nor withdrawn from, the power-cylinder, and also assume that Qi" and Q2' are the quan- tities of heat which are respectively supplied to the hot and the cold space, we get Qi"+Q2'= ^A(V, + V2)(p2-Pi). ..... (89) 344 TECHNICAL THERMODYNAMICS. Accordingly this determines the sum of the two quantities of heat ; but as regards the determination of the separate heat quantities Qi" and Q2 we here encounter a problem demand- ing an essential extension of the theory of the action of the transfer- piston, developed above in § 38. Considering the subordinate im- portance of this special problem we will only make an approxi- mate determination of these two heat quantities, after we have presented the investigation of the whole cycle. In the third time interval NP (Fig. 52), the transfer-piston Ki is held fast at the left end of its stroke, and the power-piston K2 is pushed from left to right, and thus the air contained in the power-cylinder B is brought entirely into the cold space. In so doing the pressure grows according to the curve P2P3, and a certain amount of compression work L2 is consumed. Let us designate the variable values of pressure and tempera- ture by p and T and consider that here the cylinder B is the dis- charging space, and that it receives no heat while the air is being forced out, so that we must substitute dQx = 0, and T in place of Tx, in equation (47), p. 203, thus getting by integration Tz U2/ (90) \P2/ and r:=(g) ■' <") from which can be determined how the temperature varies with the pressure, and what pressure ps exists at the end. If V is the volume of the air in the power-cyhnder at the instant when the pressure p prevails there, we have the relation Combination with the preceding two equations then gives the equation of the compression curve P2P3 (Fig. 52) CLOSED HOT-AIR ENGINE WITHOUT REGENERATOR 345 and then follows, as is easily seen, the compression work V, L2 = fpdV Now follows the heat quantity Q2", which in this time interval is supplied to the cold space, because n o heat is imparted to the power-cylinder during compression, namely, according to equation (46), p. 200, dQ = -A. {Vdp+ KpdV) =.-A^d(Fp) - A-pdV, whose integration gives Q2" = ^[FiP3-(Fi + 72)p2J-4L2 (94) «— 1 Finally in the fourth time interval PZ (Fig. 52) the power- piston K2 is held fast, and the transfer-piston Ki is pushed from left to right. The pressure herewith grows from ps to pi and the relation obtains, according to which T P3=yPi (95) The total volume is constant, the external work is zero, and the quantity of heat which is to be supplied to both spaces follows from equation (46), p. 200 •. Qi"' + Q2"' = ^(p-P3) (96) 346 TECHNICAL THERMODYNAMICS. Here the cold space is the discharging space and therefore, from the propositions on p. 210, the heat which must be im- parted to it is - „, AVi. . AVipi pi Q2"'=.pi(pi-P3) — JZi^^Se—, • • • (97) if we again designate the temperature ratio 7'i:7'2 by A; the heat quantity Qi", which must herewith be supplied to the hot space, can then be found from the combination of equations (96) and (97). Only the question raised with equation (89), p. 343, is still undecided. As the hot space is the discharging space in the second time interval MN (Fig. 52), we can express the heat quantity Qi" approximately by equation (52a), p. 210, namely, dQi" = AVdp, where V is the volume of the hot space at the instant; during the time interval MN the temperature in the power-cylinder sinks from Ti to T3; if we assume for this a certain constant mean temperature To there will obtain for the instant of time R j^n y2pVv. (yx-V)p Vxp^ D^- T^^ Ti T2 Ti • If we substitute in this formula 7fr=^ and ^ = J 2 Jo we derive (A-i)y = Ao72+AFi-^. Substitution in the preceding differential formula and the subsequent integration then give an approximate expression for CLOSED HOT-AIR ENGINE WITHOUT REGENERATOR 347 the heat quantity Qi", which must be suppUed to the hot space during the second time interval : • ^„ ^(Ao72+AFi) , AVvpx, Pi ,_-, Qi = xzi — (P*-P2)— liT^^g^^" • (98) and thus there will also be determined, according to equation (89), the heat quantity Q2' which must be supplied to the cold space in the assumed time interval. All the separate formulas are now to be grouped in the follow- ing manner. First there results the work L which is produced by this engine per period, L = Li — L2, where equations (85a) and (93) come into use; furthermore, the heat quantity Qi, which is to be imparted per period to the hot space, is Qi=Qi'+Qi"+Qi"', and the heat quantity to the cold space The latter value naturally turns out to be negative, and by change of sign gives the quantity of heat which the cooling body must withdraw from the cold space per period. The separate terms of the right sides of the equations involved are given above and substitution then gives the relation AL = Q1 — Q2, as it ought; but the substitution leads to very compli- cated expressions which we will therefore avoid reproducing here; what has been given enables one to follow all the separate phases of the normal cycle of the Laubereau-Schwartzkopff engine. For the sake of completeness I wished to insert the investigation of this engine, although it shows an evident defect, namely, that the hot air during its expansion in the power-cylinder acts on the power-piston, while usually it is a rule (followed also in the Rider and L e h m a n n systems) that the power-piston is exposed to the pressure of the cold air. 348 TECHNICAL THERMODYNAMICS. The engine, in the form appUed by Schwartzkopff, has been beautifully thought out and was subjected by Tresca^ to thorough tests; moreover there have subsequently arisen propositions concerning the construction of hot-air engines which in principle return to the arrangement of Laubereau- Schwartzkopff ; this also may justify us in here investigat- ing this arrangement more fully. Numerical Example. For an engine of the character under consideration the following ratios are given : Then by Fig. (52) are found from equations (84) and (95): ^=2.1224 and ^ = 2. p. P2 Furthermore, according to equations (86) and (88): ^=0.9044, -^=0.4000. The expansion- and compression -work, according to equation (85o)and (93): Li =2.8946 y,p, ; L, = 1 .4457 7jp„ and therefore the engine work per period is L =L,-L, = 1.4489 7jPj. The separate heat quantities moreover are: g/ =2.3470^FjPj, equation (87), Qi" =0.7269 AFjPj, equation (98), Qi'" = 7.0631 AV,j),, equations (96) and (97), 4' = -6.8864 4 F,Pj, equations (89) and (98), Q/' 0.8360 AV^Tp^, equation (94), 4"'= -0.9657 AVjp^, equation (97). ' T r e 8 c a, "Annales du Conservatoire des arts et metiers," IV p. 113- D e 1 a b a r, Dingler's Polytechn. Journal, 1864, Vol. 172, p. 81. CLOSED HOT-AIR ENGINE WITHOUT REGENERATOR. 349. Accordingly the first three values give for the total heat Qi, which must be supplied per period to the hot space, Q, =10.1370 AF^Pj. The last three values give the heat quantity Q^ which must be withdrawn from the cold space per period : Q^=8.6881 AV,p,. At the end of the expansion in the power-cylinder there prevails there a temperature Tj which lies even below the temperature T^ in the cold space, and indeed here T'i=0.8 T^; in this cylinder therefore a very con- siderable lowering of temperature occurs during expansion. Let us further assume that the engine works between the pressures of 2 and 0.5 atmospheres, consequently p =0.5X10333 kg. [1008.16 lb.], and the diameter of the power-cylinder d=0.5 m. [1.64 ft.], and the stroke s=0.40 m. [1.312 ft.], then calculation gives AV,p, =0.9570. From this follows the indicated work per period of the engine L=588mkg. [L=4253ft-lb.], and the corresponding heat quantities Qi= 9.701 and Qj= 8.314 Cal. [Q, =38.497 and Q^ =32.993 B.t.u. If the engine makes m=36 revolutions (periods) per minute, then the (indicated) work in horse-powers is N=7rz — rr =4.70 horse power 60X75 Lu \N = z ;: =4.64 horse power . L 60X550 ^ J If the cooling water is heated through 15° C. [27° F.], then the water per minute necessary is 20 kg. [44 lb.] or per hour 1200 1. [about 300 gallons]. Now, to be sure, the above-mentioned experiments by T r e s c a on an engine of the dimensions here assumed gave results which 350 TECHNICAL THERMODYNAMICS differed considerably from the preceding, computed, ones; the indicated work was only 0.4 of that here calculated, and the effec- tive performance measured at the brake amounted to only about 0.8 horse-power ; the indicator diagrams also have a very different form from that indicated in Fig. 52. To be sure the greatest pressure was 1.45 and the least 0.85 atmospheres, instead of 2 and 0.5 atmospheres as was here assumed. The deviation is due to the fact that the cycle of the actually constructed engine is far from approaching the normal cycle. We may well say that, on the whole, the separation of the power- cylinder from the heating and cooling space is not to be recom- mended. Accordingly the Rider and L e h m a n n arrange- ment is to be preferred, and for the additional reason that here the regenerator can be employed without difficulty. § 61. CLOSED HOT=AIR ENGINE WITH REGENERATOR. First Arrangement (Rider System). The schematic representation of the engine here given in Fig. 53a is only distinguished from that in Fig. 48 by regenerator R being inserted between the two cylinders A and B; the interior of the regenerator is in free communication with these two spaces A and B. The piston diagram, Fig. 536, the piston-force diagram, Fig. 53d, and the indicator diagram. Fig. 53c, do not differ in general character from the assumed diagrams in Fig. 48. Let the cycle which the engine before us is to execute be pre- scribed by the diagram. Fig. 53e, and let it again take place between the two isothermals T^Tl and T2T2 and the two polytropic curves T2T1 and again T2T1, which obey the law p7" = constant, where V represents at any instant the combined volumes of the air in the hot space and the cold space (with the exclusion of the volume Fo of the air in the regenerator). CLOSED HOT-AIR ENGINE V/ITH REGENERATOR. 351 For the four varieties of the cycle we therefore have the rela- tions , P3^3" = PiT^i" and p4V4'* = p2F2", from which we can also derive Fig. 53a. Fig. 53e. Now, as regards the course of the two isothermals, we get for the point 1 of the first isothermal T2T2 the equation or BGT2=(l+Yf^y2P2, when To represents the mean temperature in the regenerator (in the sense indicated above) at this instant. For any other point of the same isothermal for which V is the instantane- or 352 TECHNICAL THERMODYNAMICS. ous volume, and p the pressure, and for which T is a corresponding mean temperature in the regenerator, we find I now set up the hypothesis that during the isothermal changes the mean temperature in the regenerator varies in such a way that the product TV remains invariable, so that we can write TV^ToVi. If, for the sake of simplicity, we make in what follows ?^="^' (100) then the last of the preceding equations gives BG7'2 = (l + m)7p. In the same way, for any point of the second isothermal TiTi we get the relation BGTi = {\ + Xm)V'p when the ratio Ti:T2 of the limiting temperatures is again repre- sented by X. For the four vertices of the cycle assumed in Fig. 53e there follow the relations BGT2={l + m)V2P2-{l + m)Vz'pz, J ABGr2 = (l + Am)7ipi = (l + Am)F4P4.i' * * ^^^^^ From this results F3P3 = F2P2 and ViVi^ViVu • • • (102) and in combination with equation (99) Pl = Pi and Jj = Zf. P3 P2 Fs F2 (103) CLOSED HOT-AIR ENGINE WITH REGENERATOR. 353 If we regard the limiting pressures pi and p2 as given, likewise X and the value m according to equation (100), we shall get "-^r^ "«' and when the exponent n of the polytropic curve is given, we can also determine, according to equations (99) and (102), the values V3 and Vi, Pa and pt, as follows : Vi r 1 + mA -|^ , Pi rA(l + m)l^L- ,,„, , V2 LA(l + m)J p2 L 1 + mA J ' Here Vt is the volume of the hot cyUnder; V3 and ps can be found from equations (103). The application of the formulas developed and also of the following propositions of course assumes the reUability of the hypothesis just set up, concerning the variability of the tempera- ture in the regenerator; but the assumption made appears to me thoroughly reliable; the variability always falls on the side ex- pected and is always confined between comparatively narrow limits, because it always happens that only a part of the air enclosed in the engine fills the regenerator. At any rate we must take refuge in an hypothesis, for an exact theory of the regenerator, closely fitting the facts, is hardly possible. At the first glance it seems more natural to regard the mean temperature in the regen- erator, during the whole course of the cycle, as constant, and this assumption has in fact been made by Gustav Schmidt,^ Slaby,2and Schottler^; but if we test this hypothesis by the propositions developed above on pp. 197 to 225, concerning the action of the transfer-piston, we get the absurd result that the regenerator, during a period of the engine, delivers more heat to the air flowing through it than it has absorbed in the same time 'Gustav Schmidt, "Theorie der geschlossenen kalorischen Maschinen " Civilingenieur, Vol. 8, 1862, p. 285. ^ S 1 a b y, "Beitrage zur Theorie der geschlossenen Luftmaschinen," Ver- handlungen des Vereins zur Beforderung des Gewerbefleisses, Berlin, 1878. 'Schottler, "Uber die Heisluftmaschine von Rider," Zeitschrift des Vereins deutscher Ingenieure, 1881, Vol. 25, p. 633. 354 TECHNICAL THERMODYNAMICS. in the corresponding part of a period, i.e., has stored up for sub- sequent utilization. The authors mentioned could not recognize the unreliability of their hypothesis, for they confined themselves to the determination of the engine work and to the investigation of the indicator diagrams; they left out of consideration just the most important part of the problem, namely, the determination of the heat quantities which must be supplied to, and withdrawn from, the engine per period; for up to that time the action of the transfer-piston had not been closely examined. With the help of the presentations in § 38, p. 197, we can approach the investigation of the cycle described by the engine arrangement assumed in Fig. 53, p. 351; in so doing, however, it will be assumed that the air coming from the regenerator always attains there that tem- perature which prevails in the space into which the air enters. (a) Occurrences in the first time interval (/), Fig. 53. The piston Ki is held fast and the piston K2 is pushed from right to left; at the instant in which the volume of the cold space has decreased from V2 to V and for which the pressure is p, there exists the relation BGT2 = {l+m)Vp, which was given on p. 352. The work L2' during the whole time interval I is therefore found by integrating the equation dL2' = pdV, with the help of the preceding formula, L/ = -^Mog.^ (105) 1 + m ° p2 The heat quantity Q2' herewith supplied to the cold space B can be found by integrating the formula: dQ2' = -AVdp=-Ad{Vp) + ApdV, because the space B is here the discharging space and a part of the air is forced into the regenerator (see equation (47a), p. 203), in which also Tj: = T 2 is constant; hence we get Q2' = AL2' + AV2P2-AVsP3 (106) CLOSED HOT-AIR ENGINE WITH REGENERATOR. 355 (6) Occurrences in the second time interval (//). The piston X^2 goes to the very end of the left stroke and piston Ki sweeps through the volume Vi in the hot space ; in so doifig the air is forced through the regenerator from the cold to the hot space. At a certain moment when the volume of air in the hot space is Vx, and in the cold space Vy, and the mean temperature in the regenerator is T for the instant, the pressure p can be found from the relation If we make Vx+Vy = V and corresponding to our hypothesis VT = V2To, there will follow, with the use of equation (100), _A(l+m) A BGT2 and from this, provided we differentiate and multiply by p, A—\ p /—I But in general we have for the polytropic curve (p. 152) and because there follows pdV= ^MVp), it X pdVy = pdV-pdVx and Integration of the first equation gives the work L/, and in- tegration of the other equation the work L2", which is respect- ively produced in the hot space and in the cold space : 353 TECHNICAL THERMODYNAMICS. u = _ ji-ser, log. a_ _i|H^^(y„._ ^.,3), . (107) «'.+ji-B(?I'.log,2l+jji±^,(y.p.-F,p.) . (108) Here the hot space is the discharging space, for which equation (46a), p. 202, will give the heat dQ' to be imparted, and we must there substitute Tx~Ty = T\ because the air coming from the regenerator already possesses the temperature Ti ; therefore dQ'^-AV:dv=-Ad{VxP)+ATpdVx. . . (109a) The heat quantity dQ" for the discharging space is, according to equation (47a), p. 203, dQ" = -AVydp=-Ad{Vyp) + ApdVy. . . (110a) The integration of the first equation gives the quantity of heat Qi' for the hot space, and the second equation the heat quantity Q2" which must be supplied to the cold space : Qi' =AW-AVipi (109) and Q2"=AL2"+AV3ps (110) (c) Occurrences in the third time interval {III), Fig. 53, p. 351. Here the piston K2 stands at the left end of its stroke, piston Ki goes completely to the left end of the stroke. If p is the pressure and V the variable volume of the hot space, then according to p. 343 there will obtain the relation XBGT2 = 0- + Xm)Vp. The produced work Li" is found by integrating the equation for pdV, and the heat quantity Qi" by integrating the expression -AVdp=-Ad{Vp) + ApdV. CLOSED HOT-AIR ENGINE WITH REGENERATOR. 357 Utilizing the preceding relation there follows ^^"=rr^^^^^i°^»g (Ill) and Qi"=^Li"+^(7ipi-74P4). . . . (112) (d) Occurrences in the fourth time interval {IV), Fig. 53. Both pistons go back from the left to the right end of the stroke; the quantities of work can evidently be determined by the integration of the two equations (107a) and (108o). We get for the work Li" in the hot space W" = - j^.BGT. log. g- ^X^^^^ iV^P.- v. v.), (113) and for the work LJ" in the cold space L/"= + J^5Gr. log. g+ (,_\+;:ii) (73P3-7.p.), (114) and finally the heat quantities Qi" and Q2" are determined by the integration of equations (109a) and (110a) : Q,"'=AW" + AV^n (115) and Q^"' = AL2"'-AV2V2 (116) With the help of the preceding formulas, which embrace an infinite number of special cases according to the values assumed for n, the main questions can now be answered. The whole work Li, which can be produced in the hot cylinder during one period, is found from the expression Li=Li +Li +Li , and the work L2, which is consumed in the cold cylinder, is given by L2 — L2 +i2 +■£'2 • 358 TECHNICAL THERMODYNAMICS. If in the last equation but one we utilize equations (107), (111), and (113), and utilize in the last equation formulas (105), (108), and (114), and consider besides the relation (101) to (104), pp. 352 and 353, we obtain the remarkably simple formulas and I'2 = F2P2l0ge-^. If in addition we employ equation (104) and consider the relation P3Pi=PiP2 from equation (103), we also have Xjl + m) pi L2 = 72p2l0ge^ (118) The difference of the two values gives the work L of the engine per period ^=rT4^^p^i«g^g' (119) in which, with the help of equation (104a), p^ is determined: and the auxiliary quantity m is to be chosen in accordance with equation (100) . Now, as regards the important question of the heat quantities, we can find the quantity of heat Qi which is to be imparted to the hot space per period from the expression Qi=Qi'+Qi"+Qi"', and the quantity of heat Q2, which is to be withdrawn from the cold space, is given by CLOSED HOT-AIR ENGINE WITH REGENERATOR. 359 if we substitute in the first expression equations (109), (112), and (115), and in the last expression use equations (106), (110), and (116), we get the surprisingly simple propositions Qi = ALi and Q2 = AL 2, where Li and L2 are determined by equations (117) and (118). From the foregoing follow the ratios Li X{l+m) Li X{l+m) L2 l + Xm and L2 1+Am' L X-\ ' L~ \-\' • ^^^^) Qi A(l+m)' ^^^^> also AL l-\ Q^ = W+mj (124) If the same heat quantity Qi were utilized in a theoretically perfect engine, the work L^ would have to be estimated from equation AL^_ Ti-T2 _X-l accordingly, by dividing the last two formulas there follows the efficiency for the here investigated hot-air engine, '=£ = 1?^ (125) The quantity m, according to equation (100), is directly pro- portional to the air capacity Fo of the regenerator; therefore if the metallic filling, or the wire mesh, is so dense in the regenerator that its air capacity Fo can be neglected as too small, then, in all the preceding formulas, we must substitute m = 0; but then we get exactly the same formulas as were found before when considering the purely theoretical cycle on p. 288; the hot- air engine, with a regenerator of the assumed arrangement, there- 360 TECHNICAL THERMODYXAMICS. fore approaches more closely the perfect engine in its action, the smaller the volume Fo of the air confined in the regenerator; we see from this that this air volume, as was to be expected, has exactly the action of a hurtful space (clearance), and that the whole advantage of the regenera- tor disappears for a particular size of this space. Therefore these practical rules result : 1. Hot-air engines are always to be provided with a regen- erator ; 2. the free (air) space Vo of the regenerator is to be kept as small as possible, relatively to the capacity of the cylinder. Since the assumed regenerator action demands a certain, and not too small, volume of metallic filling, we must be sure that its intermediate spaces are kept as small as possible. Definite ex- periences on this point are not recorded, for just those systems of hot-air engines which are in vogue possess no regenerator, except the Rider engine. I only know of observations on the regen- erator of hot-air engines, from some remarks of Combes (Moni- teur industriel, 1853, No. 1788), that experiments in Havre showed that the resistance offered by the wire mesh in the regenerator to the passage of the air was extraordinarily small, and that the action of the regenerator was almost perfect. The cold air passing through the regenerator was brought almost completely to the upper temperature, and the air on its return gave its heat back completely to the regenerator, the engine in so doing making 50 periods per minute. The theoretical investigations presented above speak for the view that success may be expected in the construction of hot-air engines with the regenerator. It will therefore be appropriate to illustrate more fully the propositions estabhshed by numerical examples. CLOSED HOT-AIR ENGINES WITH REGENERATOR. 361 § 62. CLOSED HOT=AIR ENGINES WITH REGENERATOR. Second Arrangement. Before we enter into the special investigation just mentioned, let us discuss an engine arrangement describing exactly the same cycle as the one before subjected to discussion in Fig. 53, p. 351. Fig. 54 is a schematic representation of the engine in question, which we recognize as identical with the L e h m a n n engine, Fig. 49, p. 328, only it has this modification that the transfer- piston Ki is hollow, and its cavity is filled with wire mesh, thus constituting the regenerator R, through which the air flows when the transfer-piston is shifted, because the interior of the regen- erator is in free communication with both ends of the cylinder. Concerning the working cycle of the engine and the piston motion, there holds here what was already said in § 57 when con- sidering Fig. 49, p. 328. The idea of putting the regenerator into the interior of the transfer-piston is due to Dr. Robert Stir- ling; but the engine proposed by him (1832) had a special power- cylinder Uke the Laubereau-Schwartzkopff (Fig. 52, p. 341); the latter is therefore identical in principle with the Stirling engine, only with the difference that the regenerator is left u t.i The engine sketched in Fig. 54 has not been built so far as I know; if the introduction of the regenerator is an improvement, as was proved above to be really the case to the highest degree, then an attempt at practical construction is to be recommended. Let us now enter upon the special investigation promised; the results of the calculations will simultaneously hold for both the engine arrangements Fig. 53 and Fig. 54. Special case 1. A hot-air engine with a regenerator, with the arrangement shown in Fig. 53 or Fig. 54, is supposed to describe the Ericsson cycle, and therefore the cycle given in Fig. 50, p. 335, is taken as a basis. Here the curve portions 1-4 and ' Compare Civilingenieur, Vol. I, 1854, p. 92, Table 11, Fig. 1. 362 TECHNICAL THERMODYNAMICS. 2-3 are horizontal straight lines in the indicator diagrams, Figs. 53 and 54; therefore we must put P3 = Pi, Pi=P2, and 71 = 0. Concerning the limiting values of the temperatures and of the pressures, we assume the following : A=^ = 2 and ^ = 2.5. Tz P2 Fig. 54. Furthermore let the regenerator volume Fo = 0.7 V2 and the mean temperature To of the air in the regenerator at the initial position, i.e., when all the air is crowded into the cold space and regenerator, has the value To=VTiT2 = T2\^ X=1AUT2; then, according to equation (100), the auxiliary quantity m becomes VoT; m = V2T0 = 0.495. CLOSED HOT-AIR ENGINES WITH REGENERATOR. 363 Now according to equation (119) we have for the work L of the engine per period, * L = 0.4600 F2P2. If we assume the Rider arrangement, Fig. 53, then, accord- ing to equations (122), the indicated work in the hot and cold cylinders respectively will be Li = 1 .3754 72^2 ; L2 = 0.9154 72P2. If we substitute the lower pressure p2 = 10333 kg. [2116.3 lb.], i.e., one atmosphere, there follows L=4753 V2 [L=973.5 Fg], and the heat quantities per period : Qi = 33.518 72 and ^2 = 22 .301 72 [Qi = 3.7664 72 and ^2 = 2.5067 72]. The efficiency of this engine, according to equation (125), is )? = 0.669. Moreover we find from equations (104) and (102) 7i = 0.6010 72, 73 = 0.4000 72, and 74=1.5025 72. If both pistons have the same cross-section F, if Si is the stroke of the piston in the hot space, and S2 for the Rider arrange- ment, Fig. 53, the stroke of the piston in the cold space, then because V^^Fsi and 72=i^S2 we have from the preceding S2 = 0.665 si. In the L e h m a n n arrangement (Fig. 54) the volume de- scribed by the piston in the cold space (power-piston) is Fs2 = Vi— 73 = 1.1025 72, from which we get S2 = 0.734 si. If this engine had no regenerator, it would, according to equa- tions (65) and (76), pp. 331 and 334, for an equal value of A and 364 TECHNICAL THERMODYNAMICS. pi :p2, only possess an efficiency ij =0.347, so that here the advan- tage of the regenerator is measurable and stands forth distinctly. Special case 2. A hot-air engine with a regenerator of the arrangement Figs. 53 and 54, describes a Stirling cycle and therefore the cycle given in Fig. 51, p. 338, is taken as a base; here we make ^3 = ^1, ^4=^2, and n= 00 , and from equation (102) we get the intermediate pressures P3 = y^P2 and P4=y-Pl, and Vi again follows from equation (104). If we assume, as in the preceding case, A =2, ^^ = 2.5, and m= 0.495, for the present cycle, we again have 7i = 0.6010 72, and in like manner the work quantities Li, L2, and L, the heat quantities Qi and Q2, as well as the efficiency, are of the same magnitude. The intermediate pressures are found to be P3 = 1.664p2 and p4 = 1.502p2. If we assume the same cross-section F for both pistons, and let Si be the stroke of the piston in the hot space, and S2 that of the piston in the cold space, then with the Rider arrangement we have S2 = Si, because V^ and V2 are equal, and represent the spaces swept through by the pistons per period. On the other hand, if the L e h m a n n arrangement (Fig. 54) is taken as a base, then the space which is swept through by the power-piston per stroke is Fs2 = Vi-V: 3, CLOSED HOT-AIR ENGINES WITH REGENERATOR. 365 from which follows S2 = 0.299 si. ^ The practical construction of the Rider engine in fact shows V4 = V2 and S2 = si approximately; it would also describe the Stirling cycle if its piston motion were that assumed in Fig. 53. If for such an engine the diameter of each of the two pistons is d=250 mm. [10 in.] and the stroke of each of the two pistons is si = S2 = 300 mm. [12 in.], and if moreover the limiting pressures are p2 = 10333 kg. [2116.3 lb.] and pi = 2.5 p2 = 25833 kg. [5290 lb.], and if the engine makes per minute w = 140 periods (revolu- tions), then, if N represents the indicated work in horse powers, we get the relation 60X75 N==Lu [60X550 N=Lu], and therefore if in the preceding example we utilize L=4753 Vz [L = 973.5 Vz] and substitute 72 = — S2= 0.014726 cbm. [72 = 0.52006 cu. ft.] we get for the work of the engine in horse powers iV = 2.177 [AT = 2.148]. The heat quantities per period follow from the calculated results of the preceding example : Qi = 0.4936 Cal. and ^2=0.3285 Cal. [Qi = 1.9588 B.t.u. and ^2 = 1-3036 B.t.u.]. The quantity of heat which must therefore be withdrawn per hour from the cold space is given by 60 Q2U, and consequently the quantity of cooling water needed hourly, if it is heated through 10° C. [18° F.], is given by 276 kg. [608.35 lb.]. The found values are of course theoretical ones, because they were found by neglecting the clearance spaces and all other losses of work, and by assuming complete action of the regenerator. 366 TECHNICAL THERMODYNAMICS. § 63. HOT=AIR ENGINE WITH A REGENERATOR, AND WITH CONTINUOUS PISTON MOTION GENERATED BY CRANKS. The kind of piston-motion assumed for the investigations con- nected with Figs. 48 to .54 rendered possible very different cycles for one and the same engine arrangement, according to the choice of the poly tropic curve in question; but the assumed piston motion is not employed in the actually constructed engines; instead of a jerky, discontinuous, motion there are employed motions which are attainable by cranks, occasionally modified by intermediate leverage. We will therefore more fully discuss the hot-air engines, and particularly the Rider and L e h m a n n engines, under the hypothesis that they are provided with the regenerator, and that each of the two pistons is moved by a crank, but, for the sake of simplicity, it is assumed that very long connecting-rods are used. First the Riderengine. Here the two lines which prescribe the law of piston-motion in the piston diagram, Fig. 53, p. 351, are represented by sinusoids. Let Vi be the volume swept through by the piston in the hot space, V2 that for the piston in the cold space, and Vq the air volume in the regenerator. Lay off (Fig. 55) the values Vi, Vo, and V2 on a straight line; bisect the distances Vi and V2 at the points Oi and O2 respectively; with these points as centers describe semicircles with the radii OiRi = ^Vi, and 02/22 = iT^2, then these radii can be regarded as the corresponding crank radii. If the piston in the hot space stands at the right end of the stroke in the beginning, then its crank OiRi will be at the right dead-center; but at this instant the crank O2R2 of the piston in the cold space may already be at a given angular distance S (in the figure ^=90° is assumed) from its own right dead point; in short, let d be the angle of advance of the crank 02^22 relatively to the crank OiRi, the direction of motion of the two cranks being determined by the given arrows. In place of the time for a full period, and for a part of it, we can substitute on the vertical HOT-AIR ENGINE WITH A REGENERATOR. 367 OZ the angle of turning (crank angle) o), through which both cranks have moved from their initial position; for a full turn we must substitute 2/t = OZ for w. ^ ^ .y^ ^„„^^— -^ Y ■» « ^ -» Fig. 55. There is now no difficulty in determining, for a turning angle w of the cranks, the corresponding volume V^ of the air in the hot space, and the instantaneous volume Vy of the air in the cold space; we get at once V, = ^Vi{l-cos (143) V-\.—V2 cos 8' ^ ' ' Compare referer.cs on p. 345. HOT-AIR ENGINE WITH A REGENERATOR 375 and consequently from equation (140) 7i + Fa 1 • V,= Vo+-^^-^Vcos{do- as abscissa OM, and lay off from M, to the right, the ordinates V^, Vx+ Vq and V^+Vg+Vy, where F^ and Vy are to be computed from equations (126) and (127), p. 367, then we shall 376 TECHNICAL THERMODYNAMICS. get three curves; the first two curves I and II give the travel of the transfer-piston (piston diagram) per revolution, and curve III gives the travel diagram for the power-piston (see Fig. 54, p. 362). If the pressure curve DD is likewise drawn as a function of w, we find, as is at once evident from the indications of Fig. 56, the several points of the indicator diagram, which is identical with diagram III of Fig. 55, and which in the L e h m a n n engine is directly obtained by means of an indicator. If Vi, V2, and d are given for the Rider engine, then V and ^0 are readily found from equations (142) and (143) for the L e h m a n n engine, which is to describe the same cycle. For example if Fi = Fa and § = 90°, then in the L e h m a n n engine the piston displacement V of the power engine must be y = yjV2 and the crank of this piston must follow the crank of the transfer-piston at the angle ^o=45°. On the other hand if, in the L e h m a n n engine (with a regenerator), Vi, V, and ^0 are given, then V2 and d are calculated by equations (141); we find V2 = V 7i2 +72-771 cos and we get the volume of the air at the several, main, points of the cycle (Fig. 59a) from the equations Ti Ts T2 ~ T ^*^ The work Li, consumed per revolution in cylinder C, which here acts as compression cylinder, is found in the same way as that taken when deriving equations (148) and (149) , p. 379; it is L^=j^BG{T,-T^) (5) On the other hand the work L2 produced per revolution in cylinder D, which is here the expansion cylinder, is L2 = ^BG{T2-T) (6) The difference of these two quantities of work gives the d r i v - ingwork Lof the engine per double stroke, i.e., the indicated work taken theoretically, provided we use relation 54, p. 134, AL = c^[iTs-T^)-iT2-T)], or, eliminating T3, with the help of equation (3), L=''^{T2-T){T^-T) (7) This equation is identical with equation (150), p. 380, if we there assume adiabatic expansion and compression in the hot-air engine, i.e., assume n = «, only the work quantity L appears with the opposite sign, as it ought, for in the hot-air engine the work L is produced, while in the cold-air engine it is consumed as driving work. 388 TECHNICAL THERMODYNAMICS. If we divide equation (7) by equation (1) we get the simple relation AL_T,-T Qr~^r~ ^^^ As the cooling water has the atmospheric temperature, the temperature T2 need only be taken a little greater; the preceding equation therefore teaches that with a particular value of Qi the work L will be larger the lower we go with the tempera- ture T; as this value represents the lowest temperature occurring in the cycle, the rule ought to be followed not to place the lower limit of temperature any further down than is absolutely neces- sary for the cooling process contemplated. In the designing of refrigerating machines we regard as given the heat quantity which must be continuously withdrawn from the brine in a unit of time. If we assume the second as the unit, and designate the heat quantity in question by Qa, then if we sub- stitute Qs for Qi, equation (8) will give the theoretical (indicated) driving work L, per second, L,=^%{T2-T), (9) or, if we express the work in N horse powers, and assume the engine as double-acting and making u revolutions per minute, we get ^ = 75^(7'2-r) (10) If we determine G from the first of equations (4), namely, from ABG=^, .und substitute it in equation (7) we get ^^7^1 ff[ ^'P'' REVERSAL OF THE ERICSSON CLOSED HOT-AIR ENGINE. 389 and therefore follows, because N L = 30X75- • u N L = 30X550- u ]• for the computation of the volume Vi of the compression cylinder C, the formula K— 1 TTi N [f,p,=^ From equations (4) we then get volume Vz of the expansion cylinder ¥2=^/1 (12) The ratio of compression in cylinder C and the ratio of expan- sion in cylinder D follow from the same equations : I'h^ '-> Numerical Example. In a closed cold-air engine of the pre- sented sort the brine in casing B (Fig. 59, p. 385) is to be permanently kept at -10° to -15° C. [ + 14° to +5° F.] for the purpose of ice production. We assume that the air is suclced in by the compression cylinder at the temperature Ti=258° (-15° C.) [+5° F.] from the heating apparatus A, further that the air leaves the coohng apparatus at the temperature T^ = 293° (+20°C.) [ + 68° F.] to enter the expansion cylinder; let the pressure in the latter p2=3pi, hence three times pi in the heating apparatus; then from equation (3) we get T T ±?=4^' = 1.3764. Ti T From this follows the temperature at the end of the compression in the compression cylinder T, =355.1° (+82.1°C.) [179.78° F.], and the tempera- ture at the end of the expansion in the expansion cylinder is T =212.9 ( -60.1° C.)[ -76.18° F.]. 390 TECHNICAL THERMODYNAMICS. From equation (10) we have N =2.1279 Q, [iV =0.52866 Q,], and from equation (11) N yip,=9944— u ry,pi=72923— ]. In order to produce ice of 0° C. [32° F.] from 1 kg. [lb.] of water at 0° C. [32° F.], it is necessary to withdraw from the water 79 calories [142.2 B.t.u.] of heat; if the water has a temperature of t° C. [t° F.] and if there is to be produced from it ice at —1^° C. [to° F.], then there must be withdrawn from the water the heat quantity 79 + i + ci(i [142.8 + < + c(<^2 + l) (l+mA)Fi + A(l+m)F2 {4> + lf sin 5; . . (15) the heat quantity Q2, withdrawn from the brine per revolution, is, according to equation (132), p. 369, Q2 = ^-j^AL, (16) and the quantity of heat which is to be absorbed in the same time, from the coohng water, follows from equation (131), p. 369: Qx = ^^AL (17) If in equation (16) the heat quantity Q2 is referred to the second REVERSAL OF THE RIDER CLOSED HOT-AIR ENGINE. 395 and designated by Q,, we get the driving work of the machine in horse powers : , JV = i^,A- (18) L l+mA 550^ J while in equation (15), when the engine makes u revolutions per minute, we must substitute L = 60X75- (19) rL = 60X550-.] Numerical Example. If we assume the ratio ^ of the two temperatures T\ and T^ to be A = 1.25, which gives for a temperature 7^=238° (-35° C.) [-31° F.] in cylinder B, to a temperature Ti =297.5 ( + 24.5° C.) [-76.1° F.] in cylinder A (Fig. 60), and if we estimate the value m, discussed on p. 368 and belonging to the regenerator, to be m =0.5, then the preceding equation (18) gives the driving work: N =0.870 Q, [iV = 0.216 QJ. If we choose the angle of advance ^ =90°, make the two cylinders equally large, i.e., take ^1 = ^2, and assume that the greatest pressure pi occurring in the cycle bears to the least, Pz, the relation pi =2.5 p,, then combination with equations (15) and (19) gives 72^2=24143- rF2P2 = 177050-1' 1- u -■ because here (p. 372) 0=,^= 1.5811. N/P2 If the machine is to withdraw hourly from the brine 25000 calories [99208 B.t.u.] of heat, then, according to the preceding formula, the theoreti- cal driving work for A'^ is A?^ = 6.04 horse powers [5.953]. If the lower value of the pressure is o n e atmosphere, then p2 = 10333 kg. [2116.3 lb.], and if the engine makes u =50 revolutions per minute, then 396 TECHNICAL THERMODYNAMICS. according to the last of the preceding formulas the cubic capacity of the two cylinders must be 7, = v^ =0.2822 cbm. [9.961 cu. ft.]. If we divide equation (17) by equation (16) we get ^■=^fi±^-) =1.154. If we refer the heat quantities to the second, then, for the present case, the heat quantity which the cooling water must withdraw from the heat per second is 8.0140 cal. [31.8 B.t.u.] or 28850 cal. [114500 B. t.u.] per hour. The driving work of this engine seems considerably smaller than that found for the machine of the preceding example on p. 389; this is due to the use of the regenerator and to the circumstance that the lower temperature limit is not taken so low; of course the cylinder dimensions are much larger, but the engine is single- acting and there is no special heating or cooling apparatus. Although some uncertainty still attaches to the assumptions underlying the calculations, which can only be removed by experi- ments on, and experience with, actually constructed engines, nevertheless it does' seem possible that the construction of closed cold-air engines (refrigerating machines) of the kind described, will be attended with success. Engineers ought therefore to attack the problem; the cold-producing engines, of course in the form of cold-vapor engines, have been extensively used and not simply for purely technical purposes, but have also been proposed for the cooling of dwellings in hot countries. The cooling of air in dwellings, by a system of pipes in which the cold fluid circulates, is accompanied by a condensation on the pipes of the moisture of the atmospheric air, and experience shows, and L i n d e first called attention to it, that this condensation con- tains an appreciable amount of absorbed carbonic acid. There- fore the cooling of the air is accompanied by a purification of the air which ought to be of great importance hygienically for home life in tropical countries. REVERSAL OF THE ERICSSON CLOSED HOT-AIR ENGINE. 397 (6) Open Cold-air Engine. § 68. REVERSAL OF THE ERICSSON CLOSED HOT=AIR ENGINE WITH THE HEATING APPARATUS LEFT OUT. In the schematic representation of the closed hot-air engine (Fig. 57, p. 377) suppose the heating apparatus AA and its jacket to be left out, and reverse the cycle there discussed, then we get the open cold-air engine as it is often built and as it is schematic- ally represented in the following figure (61). Fig. 61. C is again the compression cylinder which, through the pipes riiiia, alternately sucks in the external atmospheric air, com- presses it, and forces it through the cooling apparatus EE to the expansion cylinder D; there the cooled air expands from the upper pressure limit p2, again to the atmospheric pressure pi, and is pushed out of the cylinder under the constant pressm-e pi through the pipes mim2; the two pipes mim2 unite in forming a conduit through which the cooled air is conducted to the space in which cooling is to be effected. The cooling of the air, which has been heated in the machine by compression, is effected in the cooling apparatus EE by cold water, which is supplied at a and discharged through the pipe h. Of the two indicator diagrams I and II the former belongs to the expansion cylinder and the latter to the compression cylinder. The theory of this engine is contained simply and completely ' in the formulas which have been developed for the closed engine on pp. 386 to 390. 398 TECHNICAL THERMODYNAMICS. The whole difference is that the values of the pressure pi and the temperature Ti in the open engine correspond to those of the outer atmosphere; it therefore seems unnecessary to further explain this engine cycle by numerical calculations. In the practical constructions, however, in spite of the great simplicity of these engines a pecuUar obstacle is encountered, which is the reason why this machine, well known in principle, was introduced so late and with such difficulty.! As the external atmospheric air always contains more or less vapor, or water, snow is formed in the expansion cyhnder, and because new quantities of air are continuously driven through the machine it is formed in such quantities that in a short time the air channels are stopped up with snow; further, when high pressure is used, i.e., great compression in the compression cylinder, the marked rise in tem- perature connected with it acts injuriously upon the packing. The removal of the latter evil has been attempted by injecting cold water into the compression cylinder, but this increased the moist- ure of the air in the expansion cylinder and increased the forma- tion of snow. Although the stoppage of the conduits could be removed by special designing, say by inserting special snow cham- bers, nothing would be gained by it, for the performance of the machine is extraordinarily reduced by the snow formation itself. On board ships and in great slaughter-houses the open air engines of Bell-Coleman have been employed for the cool- ing of meat. In these engines water is injected into the com- pression cylinder and from this the air leaves the cooling space, where cooling-off results from more water jets (not from surface cooling) ; from this space the air enters the upper part of a tower- like vessel and deposits the water part of its mixture; from here the air passes through a pipe, which is led through the space that is to be cooled by the machine, and then goes to the expansion cylinder. The whole arrangement is based on the correct idea of supplying the expansion cylinder with air, already cooled as much as possible, for the moisture capacity of air is smaller, the lower its temperature. The whole manipulation occurs at the expense ' The first open cold-air engine was built by an American, G o r r i e. PRELIMINARY REMARKS. 399 of the engine performance, and it is therefore very questionable whether the suppression of snow precipitation is not too dearly bought, particularly as it is not even completely removed in this way. In the cold-vapor engines all these difficulties disappear,- unfortunately there is still a complete lack of experimental results for comparison. Of course the experiments should not be con- fined simply to the cooling action of the two engine types, but should help to decide the question as to the amount of driving work needed by the two kinds of engines for an equal production of cold. One can hardly doubt that the comparison will be un- favorable to the open cold-air engine; but the circumstances are doubtless different in the closed cold-air engines, for in them the loss of air by leakage can be covered by the introduction of new air and this can be artificially dried beforehand. Kirk suggested, for this purpose, a vessel filled with a solution of chloride of calcium. II. Theory of Internal-combustion Engines. § 69. PRELIMINARY REMARKS. By an internal-combustion engine is always understood one in which the process of combustion, with the accompanying heat development and the production of a gas mixture capable of doing work, takes place in the interior of the engine. In the air engine considered above (as in steam engines) the fuel and the air necessary for its combustion were separated from the mediating body in the working cyfinder ; consequently the real working cycle was distinguished from the combustion process in the heating plant and was treated separately in the theoretical elucidations, as was particularly emphasized in § 53, p. 300. In the internal-combustion engines the fuel, the air necessary for combustion, and the products of combustion constitute the working body which here takes the place of the mediating body 400 TECHNICAL THERMODYNAMICS. of the air engine considered above and, like it, serves to produce work by suitable changes of pressure and volume. Here the important difference exists that in the course of the working cycle chemical changes of the working body occur ; these were hitherto specially excluded in all the developments of the present treatise, and particularly so in the derivation of the general laws of thermodynamics of the first section. Therefore, as far as is possible at the present time, account will be taken of the influence of the chemical occurrences in the internal-combus- tion engines. The view has always been held, although not always definitely expressed, that the process of converting heat into work would be more perfect in the internal-combustion engines; from this thought, in the course of time, there has sprung a whole series of proposals for experimental constructions of such engines, in which solid, liquid, and gaseous fuels and explosives should, and did, find application. The success of all these efforts of recent times must be characterized as perfectly extraordinary, although a whole series of questions still remains, which can only be answered experimentally. Among engines which at the present time seem to be most perfectly designed, we must emphasize the "gas engine" or "gas m o t o r " in the form presented by Otto. In these engines illuminating gas mixed with atmos- pheric air and a part of the gaseous products of combustion left behind from the preceding period, are ignited in the working • cylinder, after being subjected to a preliminary compression. At the present time engines are extensively used which work with the so-called producer gas which is generated by passing air and steam through the glowing layers of coal in a fur- nace. The name Dowson-gas comes from the name of the Eng- lishman D w s n who invented the process ; the working part of this gas consists principally of carbonic oxide gas, while in the engine run with illuminating gas different hydrocarbons con- stitute the explosive constituent. The motors run with illuminating gas can of course only be driven in connection with a gas plant, while the motors run by PRELIMINARY REMARKS. 401 producer gas generate their own gas supply; consequently these motors are in general only available for comparatively small powers. Very considerable powers are developed by engines which utilize blast-furnace gas, and in recent times their details have been greatly perfected; these gases likewise mainly contain car- bonic oxide. In all the engines mentioned at the end of the explosive-like combustion the cylinder contains the gaseous products of com- bustion in a condition of high pressure and high temperature, which are then allowed to expand so far that at the end of the piston stroke the pressure has come down nearly to atmospheric pressure; then, during the return of the piston in the working cylinder, a part of the cylinder contents are retained and the rest are thrust into the open air. The engine is therefore an open one, for during each period a new charge of illuminating gas and atmos- pheric air must be taken in. In addition to gas engines, motors have been extensively used and have led to ingenious constructions in which liquid fuels were employed, for instance benzine, petroleum and alcohol. For the production of the fuel mixture of the charge for the working cylinder, the atmospheric air is driven through liquid benzine and thus saturated with benzine vapors; in petroleum engines, on the other hand, the petroleum is sprayed, evaporated by contact with hot walls, and then mixed with air. The further procedure, ignition and work, resembles the occurrences in the gas engine. It is not in keeping with the plan of the present book to go more fully into the construction and peculiarities of the aforesaid engine types; for the thermodynamic side of the question it completely suffices to discuss the principal representative of this type, the Otto gas engine, and take up first, theoretically, the process of combustion of a mixture of illuminating gas and air. As an appendix the starting points and successes of the new- est efforts of the Diesel Motor will be discussed. 402 TECHNICAL THERMODYNAMICS. A. OTTO'S GAS ENGINE. § 70. PISTON MOTION IN THE FOUR=CYCLE ENGINE. In Fig. 62 are shown the cylinder and the piston diagram of an Otto engine; Ki and K2 give the piston position at the two ends of the stroke; to the left of the extreme piston position Ki there is a dead space which may be called the compression space, and its volume be desig- nated by Vi. In the piston diagram the vertical distance OZ represents the time for two revo- lutions of the crank, or, if uniform rotation is as- sumed, it represents the path then traversed by the crank-pin. The whole cycle, taking the figure as a basis, takes place to the left of the piston; to the right of it, because here the cylinder is open, the atmospheric pressure prevails, whose action is here left out of account, because the work expend- ed in overcoming it, during the forward motion, is regularly pro- duced on the return motion. At the beginning of the period the piston stands at 1, and the compression space is filled with gaseous products of combustion of approximately atmospheric pressure, but of comparatively high temperature, i.e., with gases which have been left over PISTON MOTION IN THE FOUR-CYCLE ENGINE. 403 in the cylinder from the preceding period. The vertical stretch OZ is here divided into four parts; in the first portion OM of the time, or during the first half revolution df the crank, the piston goes from left to right; at the time OR = t, the piston has traveled the distance AB and the volume of the gas in the cylinder at this instant is measured by the distance RB; in the second por- tion MN of the time the piston goes back, and in the third and fourth portions of time the piston motion repeats itself. The working cycle is therefore distributed over four portions of time, and hence the engine has been called half-acting, or is said to be of the four-cycle type. Fig. 626 represents the cor- responding piston-force diagram, the ordinate Oo repre- sents the atmospheric pressure, hence the vertical fine oo repre- sents the so-called atmosphere and OZ the line of zero pressures. In the first stage (I) the piston sucks in atmospheric air and illuminating gas, and thus, to a certain degree, there occurs a mixture with the products of combustion initially pres- ent in the space Vi; the pressure in the cylinder is represented by the ordinates of the curve portion 1-2 of Fig. 626; the pres- sure is here, and nearly to the end of the piston stroke, smaller than the atmospheric pressure. At point 2, shut-off takes place, and then during the return stroke of the piston the gas quantity is compressed to the volume Vi and in so doing the pressure rises in accordance with curve 2-3. At the point 3 (i.e., at the dead-point) ignition now takes place, the pressure increases rapidly, almost suddenly, according to curve 3-4 (Fig. 626) ; as in so doing the piston is already upon the return path, this portion of the curve, in reality, rises in a somewhat sloping fashion; during the rest of the third phase (III) expansion takes place, and shortly before the end of the piston stroke the exhaust port opens so that the last portion 4r-5 of the curve represents expansion with exhaust; finally in the fourth phase the quantity of gas developed by the com- bustion and corresponding to the piston displacement V2 is pushed into the open air; the pressure, which is represented by the portion 5-6 of the curve, is nearly constant and somewhat greater than the atmospheric pressure. 404 TECHNICAL THERMODYNAMICS. Tl^e real indicator diagram in Fig. 62c was derived from the piston-force diagram, Fig. 626; but as regards the connection between these two diagrams what is necessary has been said dur- ing the discussion of hot-air engines; besides, it is so simple that no further explanations are here required. The indicator diagram in Fig. 62c (the sketch corresponds in general to the practically observed diagram) is enclosed in a loop- like curve; the upper part of the area corresponds to positive, produced work, the lower part, on the other hand, to negative; the latter area nearly disappears in the actually obtained dia- grams, for here the curved portions 5-6 and 1-2 nearly coincide with the atmospheric line o-o. It is worth remarking that according to what has preceded Otto's engine is a compression pump in the first and second phases, it sucks in gas and air and compresses the mixture; it is only in the third and fourth phases that it is really a driv- ing engine and conducts the real working process. The two processes can be separated; the suction and compression can be undertaken in a special cylinder, a compression pump, and dur- ing the compression the gas mixtures can be forced into the second cylinder (the power-cylinder), in which explosion, expansion, and discharge into the open air occur. This arrangement under- lies the K 6 r t i n g gas engine; here two indicator diagrams appear which, superimposed, likewise produce the diagram rep- resented in Fig. 62c. The examination of physical occurrences, however, now de- mands the study of the real process of combustion in the power- cylinder and of the experimental data on hand concerning it. BEHAVIOR OF COMBUSTIBLE GASES DURING IGNITION. 405 § 71. THE BEHAVIOR OF COMBUSTIBLE GASES DURING IGNITION. In order to embrace all cases which arise, let us suppose a combustible gas to consist of ni atoms of carbon (C), n-z atoms of hydrogen (H), ns atoms of oxygen (0), and m atoms of nitro- gen (N), then the chemical notation for this gas is {C„,HrJ)„,NJ (1) If we designate in their order the atomic weights of the several elements C, H, 0, and N by Ci, e-^, e^, and e^, then the molecular weight m of the contemplated gas is m = niei+ 71262 +n3e3+ 71464 =i'(ne); .... (2) here we may substitute (see p. 105) 6 =12, 62 = 1, 63 = 16, 64 = 14. The constant B of the equation of condition of this gas, namely, of the equation pv=BT, can be found from relation (14), p. 105: Bm = Bomo, where the factors in the right member refer to hydrogen; now since mo =2 and because we can use with sufficient accuracy 5o=424 for all the following investigations (see p. 106), we can determine B from the equation ABm=2, (3) and then, for a given pressure p and given temperature T, immediately find the specific volume v of this gas from the equation of condition. Now let this gas be mixed with another gas which consists of a mixture of oxygen and nitrogen in the proportion of a kg. [lb.] of oxygen and /? kg. [lb.] of nitrogen in the unit of weight; let q kg. [lb.] of this gas be employed with one kg. [lb.] of the given gas, which can be designated as combustible gas. 406 TECHNICAL THERMODYNAMICS. If we assume m kg. [lb.] of the last-mentioned gas, then the total mixture consists of m kg. [lb.] combustible gas, qma kg. [lb.] of oxygen, and gm/? kg. [lb.] of nitrogen. The constant B of the equation of condition for the combus- tible gas follows from equation (3), the constants for oxygen and nitrogen are to be taken from the table on p. 104; hence the constant Bm, which corresponds to the total mixture, is found from equation(19a), p. 108: -^ + (26.472a +30.131/3)0 '>•-- iT. <^> r ;^+ (48.251 a + 54.920/3)g"|. lBm = — ^^ J Moreover, if for the combustible gas the specific heat at constant pressure is Cp, and c^ that for constant volume, and if c/ and cj designate similar values for the whole mixture, we get from equations (13o) and (14a), p. 173, and ,_ Cp + (0.2175a +0.2438/?)? ._. ''^"'' Fl ^^ ,_ c, + (0.1551a +0.1727^)g .„. by utilizing the values given on pp. 119 and 125. If the mix- ture supplied to the combustible gas is ordinary atmospheric air, then we may substitute in the preceding formulas, according to p. 126, a =0.2356 and /?= 0.7644; on the other hand if pure oxygen instead of air is added we must substitute a = 1 and /? = 0. This gas mixture is now to be ignited; if the combustion is a perfect one the carbon burns to carbonic acid, the hydrogen burns to water, and the result is a new gas mixture consisting of car- bonic acid, steam, oxygen, and nitrogen, provided the tempera- tiu:e after combustion is so high that the resulting water is pres- ent as vapor. Since one atom of carbon takes up two atoms of BEHAVIOR OF COMBUSTIBLE GASES DURING IGNITION. 407 oxygen when changing to carbonic acid (CO2), i.e., when ei kg. [lb.] carbon requires 2e3 kg. [lb.] oxygen, then on the whole the rii atoms of carbon in the gas con^dered require 2nie3 kg. [lb.] of oxygen, and the weight Gi of carbonic acid, resulting from combustion, amounts to (?i=ni(e,+2e3)kg. [lb.] (7a) On the other hand when water (H2O) is formed there will be a half atom of oxygen for every atom of hydrogen, or for 62 kg. [lb.] there will be needed OS.ea of oxygen ; as the contemplated gas possesses 112 atoms its combustion will require 0.571263 kg. [lb.] of oxygen; hence the weight G2 of the resulting water amounts to G2 = 712(62 +0.663) kg. [lb.] (76) Further if, in the contemplated gas, there are present, according to equation (2), 71363 kg. [lb.] of oxygen, the added mixture con- tains qma kg. [lb.] of oxygen. The weight G3 of the oxygen left over is therefore G3 =71363 +g7na -271163 -0.571263 kg. [lb.]. . . (7c) Finally the weight G4 of the nitrogen, at the end of combustion, is Gi={niei + qmp)kg.l\h.], (7d) because the nitrogen effects no changes. The addition of the four preceding weights, when we consider equation (2), again leads, as it ought, to the value miq + 1), i.e., to the weight of the whole mixture before combustion. Now taking in order the four gases, carbonic acid, steam, oxygen, and nitrogen, we have for the corresponding constants B of the equation of condition : 19.204, 46.954, 26.472, and 30.131 [35.003, 85.583, 48.251, and 54.920], furthermore the specific heats at constant pressure (cp) are ■ 0.2169, 0.4805, 0.2175, and 0.2438, 408 TECHNICAL THERMODYNAMICS. and the specific heats at constant volume (c„) 0.1718, 0.3695, 0.1551, and 0.1727; then we can calculate for the mixture resulting from the combustion from the formulas given on pp. 108 and 172: I{G) ' " I{G) ' " I{G) • ' • ^^^ while the similar values before ignition were calculated above and designated by Bm, cj , and cj . If the pressure, specific volume, and temperature before ignition are p', xf , and T' , and after it p, v, and T, then the rela- tions pv = BT and p'v' = B„T', hold. If we suppose the pressure and temperature after combustion to be brought back to the values which they possessed before ignition, then the last two expressions give ?=#. <») and from this we can conclude whether a change of volume is connected with combustion. To be sure the given calculations lay no claim to great accu- racy because just for carbonic acid and for steam the above-given specific heats are uncertain, particularly for carbonic acid, for which these magnitudes vary with the temperature; neither are the two vapors subject to the equation of condition for gases, and here it is especially the vapor of water for which it is only permissible at low pressures and high temperatures to assume that it follows the law of Mariotte and Gay-Lussac. The combustible gases have usually atmospheric temperature before ignition; then if we lead the gas mixture, after com- bustion, back to the initial temperature, the water will no longer be present in the form of vapor but will appear almost completely BEHAVIOR OF COMBUSTIBLE GASES DURING IGNITION. 409 condensed; as the water volume is almost infinitesimal in com- parison with the other gases it would have been more correct in determining B from equation (8) to have substituted ^2 = 0, for the weight of the steam present after combustion, in place of the weight given by equation (76). Equation (7c) gives rise to a special remark bearing on the weight of oxygen left over after combustion and enables us to calculate it. If we substitute G3 = we get here, from the rela- tion qma = {2ni+0.5n2— 113)63, (10) the minimum weight q of the mixture composed of oxygen and nitrogen which must be supplied for the combustion of 1 kg- [lb.] of gas in order to effect perfect combustion. If this mixture consists of atmospheric air, we must substitute, as was mentioned above, a =0.2356, while we must assume a = l when pure oxygen is used. Let us assume as a special case that the principal constituent, namely marsh gas (light hydrocarbon), is to be burned with a supply of atmospheric air. For this gas we have the chemical formula Clij, therefore, according to equation (1), ni = l, n2 = 4, 713 = 0, n4=0, and hence, according to equation (2), the molecular weight is wi = lxl2 + 4xl = 16, and according to equation (3) the constant B of the equation of condition is 5 = 53.000 [5=96.604]. From equation (10) then follows, because eg — 16, qa=4:. For complete combustion, therefore, at least 4 kg. [lb.] of pure oxygen must be supplied for 1 kg. [lb.] of marsh gas; the mini- mum of the atmospheric air necessary is, on the other hand (because a =0.2356), g = 16.978kg. [lb.]. Let us now assume that in reality 1 kg. [lb.] of this gas is, before ignition, mixed with 5 = 24 kg. [lb.] of atmospheric air; 410 TECHNICAL THERMODYNAMICS. as we must assume for marsh gas Cp = 0.5929 and c„= 0.4680, there is found from equations (4), (5), and (6), p. 406, for the total mixture before ignition, 5^ = 30.218 [B,„ = 55.079], c/ =0.2518, c/ =0.1872, ^ = 1.345. If we have m = 16 kg. [lb.] of marsh gas before ignition, the total weight of the mixture will be m(9 + l)=400kg. [lb.]. If we substitute the given values in equations (7a) to (7d) we get, after combustion: Gi=44 kg. [lb.] of carbonic acid; (?2 = 36 kg. [lb.] of water, ^3=26.47 kg. [lb.] of oxygen, and ^4 = 293.53 kg. [lb.] of nitrogen. From equations (8) therefore follows, for the whole mixture resulting from the combustion, 5 = 30.201 [5 = 55.048], Cp = 0.2604, c„ = 0.1891, ^ = 1.377. The values differ but little from those which belong to the mix- ture before combustion. In this comparison the water present is regarded 'as a vapor; now if in consequence of subsequent cooling of the products of combustion it had become hquid, we should get 5 = 25.975 [47.345]; in the first case, for the same pressure and the same temperature, the volumes before and after com- bustion, v' and V, will, according to equation (9), be almost exactly equal ; in the second case there will be a contraction and we shall have i; = 0.859 1)'. § 72. BEHAVIOR OF COAL (ILLUMINATING) GAS DURING IGNITION. Illuminating gas is composed of a series of separate gases, and indeed the proportions of the mixture are very variable; for the purpose of general investigation it will be best to assume a certain mixture of average proportions; i n order to facilitate compari- sons, let us assume in the following that illuminating gas has a composition like the gas employed by G r a s h o f .i ' Grashof , "Theorctischc Liaschinenlehre.'' Leipsic, 1875, Vol. l,p. C07. BEHAVIOR OF COAL GAS DURING IGNITION. 411 In order to facilitate calculation of this sort we will first give, in the following Table a, the gas constants belonging to those gases contained in the mixture of the there assumed illuminating gas. Table a. Chemical Formula. Molecular Weight. 3. Coastant. B Specific Volume. 5. 6. Specific Heat. Marah gas. . . . Ethylene Butylene Hydrogen. . . . Carbonic oxide Nitrogen CH, 16 52.824 1.3956 0.5929 C.H, 28 30.185 0.7975 0.4040 '^b"- 56 15.092 0.3987 0.4040 2 422.591 11.1649 3.4090 CO 28 30.185 0.7975 0.2450 N^ 28 30.185 0.7975 0.2438 0.46S0 0.3326 0.3326 2.4123 0.1736 0.1727 [Tabl e a.] 1. 2. 3. 4. 5. 6. Chemical Formula. Molecular Weight. Constant. Specific Volume. Specific Heat. m B V ■j) Ct, Marsh gas. . . . Ethylene Butylene Hydrogen. . . . Carbonic oxide Nitrogen CH, 16 96.283 22.356 0.5929 C.H, 28 55.018 12.775 0.4040 C4H, 56 ■27.508 6.387 0.4040 H,' 2 770.259 178.85 3.4090 CO 28 55.018 12.775 0.2450 N, 28 55.018 12.775 0.2438 0.4680 0.3326 0.3326 2.4123 0.1736 0.1727 The constants B of the equation of condition were calculated from formula (14), p. 105, namely, B 845.182 m r^_ 1540.52 1 L m J The specific volume of the several gases is calculated from the equation of condition fv^BT for 0° C. [32° F.] or r = 273° 412 TECHNICAL THERMODYNAMICS. [!r=491.4°], and the mean atmospheric pressure p = 10333 kg. [2116.3 lb.], with the help of the preceding formula for B, gives 22.3297 , v= cbm, m r 357.70 v = cu, L m . ft. .1 The following Table b gives in columns 1, 2, and 3 the con- stitution of the assumed coal (illuminating) gas, according to the weight and volume of the several gases. Table b. Kind of Gaa. 1. G kg. 2. G„ cbm. 3. V cbm. 4. Oxygen. 8 kg. 5. Carbonic acid. k kg. Marsh gas. . . . Ethylene Butylene Hydrogen. . . . Carbonic oxide Nitrogen 0.54 0.7536 0.469 4.000 2.750 0.10 0.0797 0.050 3.428 3.143 0.08 0.0319 0.020 3.428 3.143 0.05 0.5582 0.347 8.000 — 0.15 0.1196 0.074 0.571 1.571 0.08 0.0638 0.040 — — 1kg. 1.6068 1 cbm. Water. kg. 2.250 1.286 1.286 9.000 [Table b.] Kind of Gas. 1. G lb. 2. cu. ft. 3. V cu. ft. 4. Oxygen. 8 lb. 5. Carbonic acid. k lb. Water. w lb. Marsh gas. . . . Ethylene Butylene Hydrogen. . . . Carbonic oxide Nitrogen 0.54 12.072 0.469 4.000 2.750 0.10 1.277 0.050 3.428 3.143 0.08 0.511 0.020 3.428 3.143 0.05 8.942 0.347 8.000 — 0.15 1.916 0.074 0.571 1.571 0.08 1.022 0.040 — — lib. 25.740 1 cu. ft. 2.250 1.286 1.286 9.000 .The summation of column 2 gives the volume of 1 kg. [lb.] of the contemplated illuminating gas, i.e., gives its specific volume BEHAVIOR OF COAL GAS DURING IGNITION. 413 !; = 1.6068 [25.740]; its reciprocal value r=0.5797 [r=0.03885] gives the specific weight of the gas. ^ Column 3 gives the several gas quantities in cubic meters [cu. ft.], contained in 1 cbm. [cu. ft.] of illuminating gas. The values are obtained by dividing those of column 2 by 1.6068 [25.740]. In column 4 is given the quantity of oxygen expressed in kilo- grams [pounds] and designated by s, which is necessary for the combustion of o n e kilogram [pound] of the gas in question. It has been calculated, according to the presentation of the preceding article, from the formula _ 2nie3 +0.571263 -n^ e3 32ni+8n2-16n3 ,^,, s = . . . (11) m m Column 5, designated by k, gives the weight of the carbonic acid, and column 6, designated by w, gives the weight of water, developed by the combustion of 1 kg. [lb.] of the gas considered. The first value is found from the formula ^^n,{e,+2e3) _^^^n^ m m and the second value from ^^n,{e,+Q.be3) ^^n^ m m Let us assume that a certain illuminating gas weighing one kilogram [pound] is composed of Gi kg. [lb.] of marsh gas, G2 kg. [lb.] of ethylene, etc., also let the values in columns 3 to 6 be desig- nated by the letters of the headings and by the subscripts 1, 2, etc.; then the constant B of the equation of condition of this illuminating gas can be computed with the help of the formula B,= I{GB), (14) where Bg is employed to avoid confusion. Similarly the amount of oxygen s, necessary for combustion of 1 kg. [lb.], and the car- bonic acid and water resulting from the combustion can be found, respectively, from s = I{Gs), k = I{Gk), and w = I{Gw) . . . (15) 414 TECHNICAL THERMODYNAMICS. and finally the specific heat of this illuminating gas for constant pressure and constant volume respectively is c^ = I{Gcj,) and c„ = 2'((?cj (16) For butylene there are no observations on its specific heat, and therefore the ethylene values were substituted for it. For our coal (illuminating) gas (Table b, p. 412) we get, with the help of the preceding formulas and the numerical values, for the constant of the equation of condition S^ = 60.823 [5^ = 110.86]. Further we find for the oxygen requisite for 1 kg. [lb.] of illu- minating gas the quantity s = 3.262 kg. [lb.], for the produced car- bonic acid k = 2.286 kg. [lb.], and for the quantity of water w = 1.896. Finally there is found Cp=0.6196, c^ = 0.4730, «=^ =1.310. Now if the gas considered is mixed with q kg. [lb.] of air, and V is the volume of this air, then its equation of condition can be written where, for air, we have 5j = 29.269 [53.349] (see p. 104); on the other hand, for the unit of weight of gas we have Consequently for equal pressure and equal temperature we have the ratio of the air volume to the volume of the illuminating gas under consideration : -=0.481g (17) V Since 1 kg. [lb.] of air contains a =0.2356 kg. [lb.] of oxygen, we determine the minimum quantity of air necessary for combus- tion from q a=s, or g = 13.85 kg., from which the minimum of the air volume, V = 6.66 v, results. BEHAVIOR OF COAL GAS DURING IGNITION. 415 If the contemplated gas is mixed with q kg. [lb.] atmospheric air, where 5 > 13.85, then we find fjpr the mixture before ignition D 60.823 + 29.269 V^ 110.86 + 53.349 gl g+l L g+1 J also ,0.6196 +0 .2375 g , ,0.4730+0.1685? C'p z 3.11a. Cy , g+l g+1 On the other hand, after combustion we have _, 48.984+29.269 g r„ 89.284+53.345 g" 9 + 1 also r., 89.284+53.345 gl r^ ^n — I 0.7169+0.2375 g 0.6011+0.1685 g Cp- ^^1 and c,- ^^ , because here we have on hand 2.286 kg. [lb.] of carbonic acid, 1.896 kg. [lb.] of water, (0.2356g - 3.262) kg. [lb.] of oxygen, and (0.7644 g+ 0.08) kg. [lb.] of nitrogen, and because the constants, given on p. 406, are used. For example there follows from this, for the minimum quantity of air g = 13.85 kg. [lb.] before combustion, B„ = 31.394 [B,„ = 57.222], c/=0.2632, c/=0.1890, ^ = 1.392; after combustion, 5=30.597 [5 = 55.769], Cp=0.2698, c„=0.1976, ^ = 1.365. These last results of calculation lead to propositions which greatly facilitate and simplify the computations connected with gas engines. Although the combustion of our illuminating gas presupposes the minimum quantity of air, nevertheless the physi- cal constants before and after ignition are of nearly 416 TECHNICAL THERMODYNAMICS. eq-ual magnitude; with a greater quantity of air the equahty is still more apparent. If we consider that some of the values introduced into the calculation are not yet certainly established experimentally, we may for the present, in technical calculations, assume : 1. that combustion of illuminating gas in a gas engine, after a preliminary mixing with atmospheric air, in- volves no change of volume provided the mixture, after com- bustion, is brought back to the initial pressure and initial tempera- ture, and that the water generated is present in the gaseous state; 2. that the specific heat both for constant pressure and con- stant volume may be taken as equal before and after combustion, and its determination can be obtained from the pro- portions of the mixture before ignition ; finally 3. that the values thus calculated are also valid in the case in which the illuminating gas is not only mixed with atmospheric air, but also with gases left Over from a preceding combustion. These propositions, however, only hold under the express assumptions maintained for the present that the specific heats Cp and c„ of the gases are independent of pressure and temperature. § 73. THE HEATING VALUE OF COMBUSTIBLE GASES. For the purpose of technical investigations of the present kind it is permitted to utilize the experimental results of F a v r e and Silbermann on the heating value of combustible gases, although the extensive and later investigations of Berthelot and J. Thomsen have far surpassed those older ones; for the gases here considered, however, the discrepancies are but slight. F a V r e and Silbermann observed for these gases the heat quantity W released when one kilogram [pound] of the gas was burned in the calorimeter. The experimental values of W are given in column 2 of the following table : THE HEATING VALUE OF COMBUSTIBLE GASES. Table. 417 1. 2. • 3. 4. 5. W H H, H.J.') cal. cal. caL cal. Marsh gas, CH4 Ethylene, C2H4.... Butylene, C4H8. . . . Hydrogen, H2 Carbonic oxide, CO 13063 11710 2930 11858 11090 3235 ? 10840 3162 34462 29060 3630 2403 2400 2100 2342 2504 2447 3229 1528 [Table.] 1. 2. 3. 4, 5. W H H> H, B.t.u. B.t.u. B.t.u. B.t.u. Marsh gas, CH4. . . . Ethylene, C2H4. ... Butylene, C4H8. . . . Hydrogen, H2 Carbonic oxide, CO 23513 21078 5274 21344 19962 5823 9 19512 5692 62032 52308 6534 4325 4320 3780 4216 4507 4405 5812 2750 It is to be noted that the experimental values given by F a v r e and Silbermann do not represent the real heating values. Since the gases were burned in the calorimeter at constant atmos- pheric pressure, and then cooled to the initial (atmospheric) temperature, the steam generated was, in so doing, almost com- pletely condensed ; the calorimeter thus withdrew from it a certain heat quantity which is contained in the value W and should be subtracted from it, in order to determine the real heating power H. Now, as later investigations will show, the quantity of heat which must be withdrawn from one kilogram [pound] of satu. rated steam at atmospheric temperature, in order to convert it under constant pressure into water, amounts to about 600 cal. [1080 B.t.u.] in round numbers; therefore if the combustion of ' The values given in the table for H, differ but slightly from the values which Berthelot found (Ann. de chim. et de phys., 4th Series, Vol. 22, p. 130). 418 TECHNICAL THERMODYNAMICS. 1 kg. [lb.] of gas generates w kg. [lb.] of steam, the heat quantity under consideration is 600 w [1080 w] and there consequently follows H = W-600 w [^ = PF-1080 w]. For the gases given in the preceding table the corresponding values of w are given in the table on p. 412; thus the heating values H given in column 3 can be calculated and the results agree with Grashof's data; the value given for butylene gas was estimated by G r a s h o f . If we suppose the combustion to be effected by pure oxygen and if we divide the heating power H by the weight s of the oxygen assigned to 1 kg. [lb.] of gas (see table, p. 412) we get the values Hi given in column 4 of the preceding table. (With carbonic oxide gas the divisor was 2x0.571.) These values give the heat quantities per kg. [lb.] of oxygen. Welter concluded from the older experiments of Lavoisier and R u m f o r d that equal heat quantities must correspond to equal quantities of oxygen; the values of Hi ought therefore to be of nearly the same magnitude; we see, however, that Wel- ter's proposition is not confirmed and this has been pointed out before. On the other hand if the heating power is referred to 1 kg. [lb.] of mixture of gas and oxygen we get the values H2 of column 5; they were found from the division of H by (k+w) or by 1+s, the latter values being taken from the table on p. 412. It follows therefore that H "^-TTS (i«> We see that the values H2 also follow no simple law. Let us consider a mixture of combustible gases (illuminating gas) and designate the gas-weights contained in a unit of weight of mixture, as before, by Gi, G2, etc., and their heating power H (taken from column 3 of the preceding table) by Hi, H2, Hz, etc., then the heating power of this mixture is determined by H = I{GH) (19) THE HEATING VALUE OF COMBUSTIBLE GASES. 419 For example, in the illuminating gas, assumed on p. 412, we have H = 10113 cal. [18203*B.t.u.]. According to the investigations in § 24, p. 132, the heat con- tents J of the unit of weight of a gas, or of its inner work U, expressed in units of heat, is given by where Jo is a constant belonging to the gas in question; in what follows let J be simply called the "gas heat" and Jo the "con- stant of the gas heat." This constant can be found for the unit of weight of a mixture of different gases from equation (18), p. 175,, Jo = I(GJo), (20) where Gi, G2, G3, . . . represent the weights of the several gases, and Jo', Jo", Jo'", . . . their constants. Now for a mixture of one kilogram [pound] of combustible gases let Ji be the constant of the gas heat, ci the specific heat at constant volume for this mixture, Bi the constant of the equa- tion of condition; moreover let these same magnitudes for the mixture after combustion be designated by J2, C2, and B2 respectively; let pressure, volume, and temperature before ignition be pi, Vi, and Ti, after combustion p2, V2, and ^2; then the process of combustion can be represented, in a general fashion, by the curve ACB, Fig. 63, which I have called the "combustion curve." The points A and B represent the gas condition before and after combustion, and if, during combustion, pressure and vol- ume have changed according to the course of the arbitrary curve AB, then in so doing a work L is produced, which is represented by the area V1ABV2. If the combustion j¥ v» takes place without heat supply from, and -p^^ ^3 without heat withdrawal by, the outside, then evidently the gas heat at the end is equal to that at the 420 TECHNICAL THERMODYNAMICS. beginning, diminished by the heat quantity converted into work, and therefore J2+C2T2=Ji +CiTi -AL, from which follows Ji-J2 = C2T2-CiTi+AL, (21) and at the same time the relations p^Vi=BiTi and p2V2 = B2T2 .... (22) hold. Let the mixture of gases be led back by the same, but reversed, path BA to the initial values of pi and ^i. The heat quantity Q, which must be withdrawn in so doing, is found from Q=C2(T2-T)+AL, (23) where T is the temperature of the mixture at the point A, after the return. Combination of equation piVi = B2T with the first of equa- tions (22) gives r=|-Vi (24) Utilization of this expression and elimination of AL from the twoi equations (21) and (23) give J.-J2=Q+'^^^P^T, (25) In F a V r e and Silbermann's experiments the here- described procedure actually occurred, only the combustion and return took place at constant pressure pi and so the curve AB was a horizontal straight line. The value here designated by Q is identical with the value H2, column 5, for the gases adduced in the table on p. 417, and for these gases the difference Ji —J2 can be computed, Ti being assumed as the mean atmospheric temperature, because from the known proportions of the mixture before and after com- bustion, the values Ci and C2 and also Bi and B2 can be deter- mined. Under the hypothesis that the gas considered burns with pure oxygen, and assuming the data in the table on p. 412, and considering the data already given, we get the values given in the following tabulation. THE HEATING VALUE OF COMBUSTIBLE GASES Tabulation. 421 B, J I — J2 Marsh gas. . . . Ethylene Butylene Hydrogen. . . . Carbonic oxide 0.2177 0.2607 31.74 31.74 0.1952 0.2292 27.31 27.31 0.1952 0.2292 39.90 27.27 0.4059 0.3695 70.42 46.95 0.1669 0.1718 28.80 • 19.20 2355 2514 2449 3272 1551 [Tabulation.] Ji — Jz Marsh gas. . . . Ethylene Butylene Hydrogen. . . . Carbonic oxide 0.2177 0.2607 57.720 57.720 0.1952 0.2292 49.778 49.7.78 0.1952 0.2292 72.726 49.705 0.4059 0.3695 128.355 85.576 0.1669 0.1718 52.494 34.996 4239 4525 4408 5890 2792 The value! Ji — J2 of the last column are computed from equation (25) for the mean temperature ti=20° C. [68° F.], i.e., Ti = 293°C. [527.4 F.] and differ but httle from the values H2, p. 417. In the difference (J,— J 2), we repeat, the first term Ji repre- sents the constant of the gas heat for the mixture of the com- bustible gas with pure oxygen, and J2 refers to the mix- ture after combustion, consisting of carbonic acid and water. Now if we represent this constant for the gas in question by Jo, and its value for oxygen, carbonic acid, and water respectively Jj, Jk, and J„, then, according to equation (19), the following relations : {l + s)Ji=Jo+sJ„ (l+s)J2 = kJk + wJn hold, when, as was assumed in calculating the table on p. 412, s kg. [lb.] of oxygen are furnished for 1 kg. [lb.] of gas, and when the result is k kg. [lb.] of carbonic acid and w kg. [lb.] of water. The difference then gives for the gas considered^ Jo = {l + s) {Ji-J2)+kJk+wJy,-sJ,. . . , (27) If, therefore, the values Jk, Jw, and J. were known, the con- stant Jo could be computed for all gases of the composition (26) 422 TECHNICAL THERMODYNAMICS. (Cnjinflm)- As regards this problem only one experiment exists for its solution, which is by Nordenskioldi; from the experi- mental results of F a v r e and Silbermann on the heats of combustion of organic fluids of the preceding composition there were found the following values for this constant Jo : for Oxygen : J, = 7317 cal. [13170.6 B.t.u.], Carbonic acid: Jfc = 43I2 cal. [7761.6 B.t.u.], Vapor of water: J„ = 7275 cal. [13095 B t u.], of course under the hypothetical proviso that the heats of com- bustion of these fluids are inversely proportional to their specific weights. With the help of these values and the corresponding values s, k, and w in Table b, p. 412, we can compute from equations (26) the following tabulation. Combustible Gas. Gas Heat Constants. J2 J, Jo Marsh gas 5645 5173 5173 7275 4312 [Table.: 8000 7687 7622 10547 5863- 10732 Ethvlene 8955 Butylene 8667 Hvdrogen 36387 Carbonic oxide 5033 Combustible Gas. Gas Heat Constants. J2 J, J. Marsh gas 10161 9311 9311 13095 7762 7200 13837 13720 18985 10553 19318 Ethylene 16119 Butvlene 15601 Hvdroeen 65497 Carbonic oxide 9059 ' Poggendorft's Annalen, Vol. 109, 1860, p. 184.— W ii 1 1 n e r, Lehrbuch der Experimentalphysik, Vol. 3. THERMODYNAMIC EQUATIONS FOR COMBUSTION. 423 The contents J 2 are computed from formula (26) : {X + s)J2 = kJi, + wSy„ while Ji results from the relation where the corresponding values J1-J2 are taken from the table on p. 421. The values Jo are computed according to the first of equa- tions (26): Jo = 0-+s)Ji-sJs. Moreover we may especially emphasize that J2 is the constant of the gas heat after the combustion of the gas with pure oxygen, while Ji is the constant of the mixture of the combustible gas with the necessary oxygen before combustion. Jo is the constant for the combustible gas considered. The value Jo could therefore also be calculated for every mix- ture of the gases here given. An exact determination of the quantities considered would certainly be of importance for the further development of thermo- dynamics; for the present the values given by Nordenskiold and the values just derived from them must be regarded as hypo- thetical. § 74. THERMODYNAMIC EQUATIONS FOR THE PROCESS OF COMBUSTION OF A GAS WITH INVARIABLE SPECIFIC HEATS. If the unit of weight of a combustible gas is mixed with s kg. [lb.] of oxygen, i.e., with just enough of it to effect perfect com- bustion, and if the products consist of k kg. [lb.] of carbonic acid and w kg. [lb.] of water, then, at a particular instant, say at the point C of the combustion curve AB (Fig. 63, p. 419), in which X kg. [lb.] of gas have just been burned, the total weight 1 + s* = {k+w) kg. [lb.] of the gas mixture will consist of (1— x) kg. [lb.] of gas, (l-x)s kg. [lb.] of oxygen, kz kg. [lb.] of carbonic acid, 424 TECHNICAL THERMODYNAMICS. and wx kg. [lb.] of steam; the specific heat Cx at constant volume for this mixture can be calculated in the manner given, also the constant Bx of the equation of condition, and finally from this equation itself the temperature can be computed for every given value of the pressure p and volume v. The constant Jx for the gas heat of the mixture can be deter- mined, provided Jo is the constant for the combustible gas itself, from the relation (1 +s)Jx = (1 -x)Jo + {l -x)sJs-\-kxJ k+WXJw, or (1 -\-s)Jx=Jo + sJa + {hJk + Wjy)x — (Jo + SJ,)X, or, taking into consideration equations (26), Jx=Ji-{Ji-J2)x (28) This formula also holds when there is mixed with the com- bustible gas, at the beginning, not only oxygen, but also an indif- ferent (neutral) gas; enough oxygen must be on hand, however, to burn the whole gas; any excess of oxygen would have to be counted with the indifferent gas; if i kg. [lb.] of such a gas were on hand arid J,- its constant, we should write in place of equations (26) (1 + s + i) Ji = Jo + s J, + Hi, 1 (l + s + i)J2=kJk+wJw+'i^i- J For the considered point C of the combustion curve AB (Fig. 63, p. 419) there can be found, in the same way, the specific heat Cx at constant volume, also the constant Bx of the equation of condition : Cx = Ci-{Ci-C2)x, (30) Bx=B,-iBi-B2)x, (31) and for the equation of condition of the mixture at this instant pv = [B,-(Bi-B2)x]T (32) for the corresponding temperature T. THERMODYNAMIC EQUATIONS FOR COMBUSTION. 425 Taking a unit of weight of a gas mixture, the heat content at the point C of the curve AB is given by J^ + CxT; on the other hand, for the beginning, it is given by Ji+CiTi; the difference gives the increment of the heat content when equations (28) and (30) are used: Ci{T-Ti)-[Ji-J2 + (c,-C2)T]x. The bracketed quantity designated by p is a simple function of the temperature because the constant quantities Ji, J^, Ci, and C2 are known for the case in hand, i.e., substitute P=Jx-J2 + {ci-C2)T, (33) and we shall get for the total increase c,{T-T{)-px. Now if during combustion, i.e., during the corresponding chemical change, there is also supplied to the mass, from without, the heat quantity Q along the path AC of the curve AB, and if in so doing the external work AL is produced, then the relation Q = cx{T-T{)-px+AL obtains, or, passing to the differential, dQ=CidT-d{px)>rA'pdv, (34) where the function p is given by equation (33). The preceding equation is the result of an attempt to extend the fundamental equation of thermodynamics for gases to the case in which there exist simultaneously chemical equa- tions of condition. If combustion did not take place during heat supply, then x would be and constant; then we should again obtain the equation as it was presented on p. 134. In its structure, moreover, the preceding formula is identical with the one which subsequent investigations will furnish for the heat quan- tity that is necessary for the evaporation of liquids. The preceding investigations exist in the earlier edition of this book (1887, Vol. I, p. 413). Recently S t o d o 1 a in the article 426 TECHNICAL THERMODYNAMICS. "Die Kreisprozesse in der Gasmaschine," Zeitschrift des Vereins deutscher Ingenieure (Vol. 42, 1898, pp. 1045 and 1086), has given for the process of combustion equations which agree in structure with the formulas given above in the text, and for gases in particular are identical with them. More recently still Hans Lorenz in "Technische Warmelehre," 1904, p. 392, has derived the same equations for gases in a somewhat different form, but extends the investigations, like S t o d o 1 a, to the case of variable specific heats in accordance with the experimental results of Mallard and LeChatelier, a supposition which we will Ukewise make later on. Example. Hydrogen detonating gas, a mixture of hydrogen and oxygen in the proportion in which the two gases unite to form water, is to be burned at constant volume. Let p, and T^ be the pressure and temperature of the mixture at the beginning, and let p^ ^-nd T.^ be the pressure and temperature at the end of combustion, in the state of vapor; then, because v^ =v^, we find from equations (22) when perfect combustion obtains, i.e., x = \, while at the beginning x=Q. From the integration of equation (34) there follows here, because dv=0, and with equation (33) Q+J,-J,=c,T,-c,T„ (/?) where Q represents the quantity of heat which must be supplied from with- out during the process of combustion. During the explosion heat is neither supplied nor withdrawn, i.e., let Q=0. If the initial temperature of the mixture is «, =20° [68° F.] and T, =293° [527.4° F.], and if we utilize the constants B,, B^, etc., given for hydrogen in the tabulation on p. 421, there will result from equations (a) and (^) r,=9177°, <2=8904°, and ^'=20.88 [r, = 16518.6, 4 = 16059.2]. Bunsen found, by experiments, that the pressure ratio —=9.6 obtained; here therefore are conditions which still require explanation. The most natural and probably the most correct idea is, that dissocia- THERMODYNAMIC EQUATIONS FOR COMBUSTION. 427 tion (see p. 91) must here play an important part; ' for does not the calcu- lation in the preceding example give the enormously high temperature of combustion i^ =8904° C. [16060° F.]? • As a special case we will discuss gas combustion at constant temperature. If the constant value of the combustion temperature is desig- nated by T, then the value of p, from equation (33), namely, p=J,-J2 + ici-C2)T,. .' (35) is constant, and because dT=0, equation (34) gives dQ= —pdx+Apdv. But the differentiation of equation (32) gives d{pv) = -{Bi-B2)Tdx. If we determine dx from this, we get dQ = .^^_^^^y rf(pt') +Apdv. It for simplification we substitute ^^^^ AiB,-B2)T ^ (36) P there follows, after simple reduction, dQ=AELdlogeipv"') (37) m — 1 Therefore according to the method of heat supply the i s o t h e r- mal curve of combustion will have a different course. If, during combustion, heat is neither supplied nor withdrawn, 'Mallard and Le Chatelier found that a noticeable dissociation did not occur as long as the temperature of carbonic acid was below 1800° (3240° F.) and as long as the temperature of steam was below 2000° [3600° F.]. Annales des Mines, Vol. IV, 1883, p. 456. 428 TECHNICAL THERMODYNAMICS. then dQ=0, and from equation (37) follows for the course of the combustion curve pv'"=pivi"', (38) where m is to be determined with the help of equations (35) and (36). For example, we find for detonating gas, which is to burn at the constant temperature t = 500° C. [932° F.] or 7' = 773° [1391.4°] for the exponent m, m = 1.13, from the values given for hydrogen in the tabulation on p. 421. On the other hand for gases or gas mixtures, in which the gas constants are taken equally large before and after combustion, i.e., for which we substitute Bi=B2, we find m = l; the isother- mal curve of combustion is then the isothermal of ordinary gases, i.e., an equilateral hyperbola. § 75. THE FUNDAMENTAL EQUATIONS AND THE EQUA= TIONS FOR THE COMBUSTION PROCESS OF OASES WITH VARIABLE SPECIFIC HEAT. If we continue to hold fast to the assumption that gases are subject to the ordinary equation of condition pv=BT, we find, " as was shown on p. 145, that the specific heat at constant pres- sure can only be a function of the temperature, but that its differ- ence from the specific heat at constant volume again appears as a constant and that the equation Cp Cj) ^ xl. Ij is here also valid. Now if we substitute Cj=Co+T and Cp=koCo+T, .... (39) where t represents any temperature function and the quantities Co and ka are regarded as constants still to be determined, then EQUATIONS FOR COMBUSTION PROCESS OF GASES. 429 in place of equation (51a), p. 132, there may be written for the variation of energy of a gas for any change of state whatsoever dJ-=AdU = {co+z)dT, (40) and for the supplied heat quantity dQ there is found dQ = {co+z)dT+Apdv, (41) which equation, like the one (53), p. 133, can be written in different forms. It results that at constant temperature or constant energy the isothermal curve, and the isodynamic, can be represented by an hyperbola. The circumstances are different, however, with the adiabatic curve, and as this is of importance right here, because of the course of expansion in gas engines, it will be examined more closely. In so doing the propo- sitions of Mallard and LeChatelier will at once be applied. Accordingly the temperature function z is represented by aoT, where ao is a constant, valid for the gas considered. (See equations (;-) upon p. 147.) We therefore write in place of equation (40) AdU=dJ = {co+aoT)dT (42) and dQ = {co + aoT)dT+Apdv (43) The difference of equations (39) gives Cp-c^=(ko-l)co=AB (44) If we utilize this relation and divide both sides of equation (43) by AT, then there is found for the differential of entropy P, AdP=Cod{log.pv^)+^^, Co Kq — i- and from this, by integration, AP = cologeipv''^) + -^^+APo. . . . (45) 430 TECHNICAL THERMODYNAMICS. But for the adiabatic curve the entropy is the same for all points, ■ and we therefore get for this curve the equation Co loge ipv'"') +— ^^ = Constant. Co n^O ~ 1 . . (46) Compare this with equation (59), p. 139, with respect to the ordinary case of invariable specific heat at constant volume. Closer investigation shows that according to this equation (46) the adiabatic approaches the axis of abscissas less rapidly than in the case treated on p. 139; it approaches more to the iso- thermal curve, which is, in general, confirmed by the indicator diagrams of gas engines. Since equation (39) represents the specific heat at constant volume as c^ = co+aoT the constants Cq and ao are found from equations (r),.p. 147, when we divide their two members by the molecular weight m (accord- ing to data on p. 147) to be as follows : «o (C.) [«o (F.)] Hydrogen, H2 Oxygen, O2 Nitrogen, N2 Carbonic oxide, CO. Steam, H2O Carbonic acid, CO2 . 2.047 0.128 0.146 0.146 0.234 0.100 0.00122 0.000076 0.000087 0.000087 0.000318 0.000176 0.000678 0.0000422 0.0000483 0.0000483 0.000177 0.0000978 To these preliminary investigations we now add the investiga- tions concerning the combustible process of gases with variable specific heat. Let the unit of weight of a combustible gas be mixed with s kg. [lb.] of oxygen and i kg. [lb.] of an indifferent (neutral) gas. The total weight G of the gas mixture is therefore G = l + s+i. (47) EQUATIONS FOR COMBUSTION PROCESS OF GASES. 431 Now let the constant B of the equation of condition pv = BT for the combustible gas be Bq, for oxygen be Bs, for the indifferent gas be Bf, in like manner we have for carbonic acid Bk and for the vapor of water B^, provided, after complete combustion, k kg. [lb.] of carbonic acid and w kg. [lb.] of steam are on hand. Now suppose that at the point C of the combustion curve, Fig. 63, p. 419, just x kg. [lb.] of combustible gas have been burnt; let V be the total volume of the gas mixture and let the partial pressures be respectively represented by po, Pi, Ps, Pk, and p„, and their sum by p, and then the following relations obtain : poV = (l-x)BoT, PiV=iBiT, p,V = {l-x)sB,T, pjJ=xkBkT, pj^=xwBjr, and from them follows, by addition, Y2 = {Bo+iBi+sB,)-x{Bo+sB,-kBk-wBJi. . . (48) If we introduce the specific volume v, we have 7=Gv = (l + s+i)u, and equation (48) gives for the beginning of the combustion, i.e., for a;=0, the constant Bi of the equation of condition, Bo+iBj+sB, ^'= 1 + s+i ' ^^^^ and likewise Ba for the end of the combustion, i.e., for x=l, _ kBk+wB^+iB i ^ 1+s+i ^ ' The difference gives (l+s+i) {Bi-B2)=Bo-^sB,-kBk-wBy„ 432 TECHNICAL THERMODYNAMICS. hence there follows, according to equation (48), the simple relation ^ = 5i-(5i-52)x, (51) in which Bi and B2 are given by equations (49) and (50). Equa- tion (51) is identical with equation (32), p. 424. For a single gas we get, according to equation (42), p. 429, for the variation of energy measured in units of heat, AdU=dJ = {co + aoT)dT, and from this, by integration, J=Jo+CoT + ^aoT2, (52) where Jo is again the "constant of the gas hea t." Let the total weight G of the V mixture be G = l + s+i and again designate the constants of the preceding equation of the several gases of this mixture by subscripts i, s, k, and w respect- ively; then the heat constant GJ for the point C of the com- bustible curve, Fig. 63, p. 419, is GJ = {\-x){jQ + coT + \aoT^)+i{Ji+CiT + \aiT^) + 0—x)s{J,+c,T + \a,T^) + xk{Jk + CkT + \akT^) From this, by addition and transformation of the terms, GJ = {Jo +iJi+sJ,) + (Co + iCi + scs) T + (iao + ¥oi.i + istts) T^ - (Jo + sJg - kJk — wJJ)x i -(co + sCs-kck-wcJxT L (53) — {^ao + isas-ikak-^wa^)xT^. J Introducing the following notation for simplification : Jo+sJ,+ iJi=GJi, (54) iJi+kJk + wJ^ = GJ2, (55) EQUATIONS FOR COMBUSTION PROCESS OF GASES. 433 Co+ sc,+ iCi=Gci, (53) ici+ kck+ wCy,=Sc2, (57) ao+ sa,+ iai = Ghi, (58) iai+kaic+way,=Gh2, (59), we finally get, in place of equation (53), We designate the bracketed expression by p, that is, if we make [J,-J2 + {Cl-C2)T + ^{hi-h2)T^]=p, . . . (60) there follows J=Ji+ciTi + ihiT2-xp (61) for any corresponding stage of combustion and, from this, by differentiation, AdU=dJ=CidT+hiTdT-d{xp) (62) The last equations refer to the unit of weight of the whole gas mixture. If combustion takes place under heat supply, we get the fun- damental formula dQ=CidT+hiTdT-dixp)+Apdv, . . . (63) which, with the help of the equation of condition and equation (51), permits different transformations. Example. Hydrogen detonating gas, a mixture of hydrogen and oxygen in the ratio in which the two gases unite to form water, is to be burnt at constant volume. Let Ti be the initial and T^ the final temperature; then for complete combustion a; = 1 and we get from equation (63) Q=c.(T.-T,)+ih.{T^-T.')-p„ (63a) in which, according to equation (60), we must substitute P,=J,-J.+ {c:-c,)T,+i{h,-h,)T,' .... (60o) 434 TECHNICAL THERMODYNAMICS. Now here s =8, w = 9, and therefore, using the values of J on p. 422, Ji = 10547, .72=7275; hence J, -J, =3272 [Ji = 18985, ^2 = 13095; hence Ji -Jj = 5890], and according to the tabulation on p. 433, Ci =0.341, C2 =0.234, c.-Cj =0.107; Ai =0.000203 and A^ =0.000318 [fc =0.000113 and ^2=0.000177]; and ft =3272 +0.107 Tj -0.0000575 r^^ [P2=5889. +0.107 72-0.000032 T/]. If the combustion takes place without heat supply, then we must sub- stitute Q =0 in equation (63a) : 3372.63=0.234 Tj + 0.0001586 T^ [6070.73=0.234 7^2+0.00000881 T^^\ from which can be determined the temperature at the end of the combus- tion: T2 — \ oQoo [ or, according to Celsius, is = j „„„„ I ^2 = I 7Q794 J or, according to Fahrenheit, %=\ g2'>0 J J ' The constants for the equation of condition before and after combustion are: B, =70.42 and B^ =46.95 (Table, p. 421) [B, = 128.35 and 52=85.576]. From piV=BiTi and p2V=B2T2 follows, according to the found temperature values, the pressure ratio P?_ [12.301 pi 1 8.95/- From what has been said other examples of the burning of simple combustible gases might easily be adduced; but the cal- culations are extraordinarily complicated for compound gases, for example with illuminating gas. If we compare the results of the calculation of the preceding example with the results of the example on p. 426, in which the invariability of the specific heats was assumed, then such con- siderable differences manifest themselves that we must probably EQUATIONS FOR COMBUSTION PROCESS OF GASES 435 forbear for the present pursuing the calculations in the indicated way. In the theory of the combustion process and in the treatment of the experimental results of Mallard and LeChatelier and of those of L a n g e n it was expressly assumed that the combustible gases, as well as the products of combustion, were subject to the equation of condition pv = BT of gases and that therefore the specific heat could be regarded as only a func- tion of the temperature. This assumption is certainly not valid for the products of combustion, carbonic acid and the vapor of water, for with these the specific heat must be regarded as a function of the temperature and of the pres- sure. The most recent investigations are directed to this very problem; so long as this is not satisfactorily solved there exists a very unfortunate gap which renders every further attempt to pursue these questions theoretically very doubtful of success. Nevertheless progress in gas-engine construction has been perfectly extraordinary in the last few years and is due to the common efforts of many which have called forth numberless separate observations and practical experiences. Systematic experiments on gas engines, which will embrace all the separate problems, are of pressing need; the mechanical en- gineering laboratories of the technical high schools, which now exist everywhere, are in some places directing their attention to the experimental investigation of the gas engine, in accordance with the demands of the present day, and will probably soon furnish the necessary basis for a more rigorous, theoretical, treatment of the processes occurring in the engines. The first complete and carefully conducted experiments on gas engines are due to S 1 a b y (Verhandlungen des Vereins zur Beforderung des Gewerbfleisses) ; in latter years the distin- guished experimental investigations of Eugen Meyer have excited well-deserved attention among engineers. The reports of his experiments are found scattered over the last ten years in the "Zeitschrift des Vereins deutscher Ingenieure," and include different combustion motors of recent date. 436 TECHNICAL THERMODYNAMICS. B. DIESEL'S HEAT MOTOR. § 76. DISCUSSION OF THE MOTOR AND OF THE NEW VIEWS UNDERLYING IT. In 1893 R. Diesel published an article "Theorie und Konstrucktion eines rationellen Warmemotors," Berlin, 1893, in which he, starting with special thermodynamic studies, developed views concerning the construction and mode of working of com- bustion motors which deserved the highest consideration. The article found on one side complete assent, on the other objections were encountered with respect to the possibility and suitability of practical construction, and with respect to certain portions of the theoretical presentation. In the meantime, with the cooperation of prominent men and firms and with rare persistence and great sacrifices, different con- structions were subjected to careful experimental tests, which indubitably showed that an important invention was under con- sideration. It was to be foreseen that all the proposals contained in Diesel's article could not be completely realized; Diesel's merit is not diminished thereby, nor ought we to underestimate beforehand the constructive possibilities of our highly developed engineering resources to attain the goal set by Diesel.^ Fig. 62, p. 402, which refers to a four-cycle Otto gas engine, will be taken as the basis of the fuller discussion of the new motor; but the dead, or compressing, space 71, indicated in Fig. 62a, must be considered much smaller relatively to the space V2 swept through by the piston; we will likewise assume that work is performed in the four-cycle process. ' A lecture by Diesel gives an insight into the gradual development of the motor: "Diesel's rationeller Warmemotor," Zeitschrift des Vereins deutsoher Ingenieure, 1897, Vol. 41, pp. 785 and 817; also Schroter's report on his excellent experiments with a motor (ibid., p. 845). Further: Diesel's "Mitteilungen iiber den Dieselschen Warmemotor " in Zeitschrift, 1899, pp. 36 and 128. DISCUSSION OF THE DIESEL MOTOR. 437 The working of the motor is now as follows : (a) During the first forward stroke of the piston only at- mospheric air is sucked in, and during the adiabatic return it is compressed in the actual constructions to from 30 to 50 atmos- pheres. In the compression space there is now pure atmospheric air of high pressure and of high temperature, which has been produced exclusively in a mechanical way by adiabatic com- pression, as was assumed in discussing the C a r n o t cycle. Into this space Diesel introduced the fuel under great ex- cess of pressure and, from the first, thought of utilizing a solid fuel (coal dust), liquid fuel (petroleum, naphtha, alcohol), or gaseous fuel (compressed illuminating gas) . (6) The motor does not, hke the gas engine, possess a special ignition device, therefore does not have hot tubes, or flames or apparatus for electrical ignition, for the temperature reached by the air during compression is sufficient to effect the ignition of the fuel. The temperature of ignition has a par- ticular value for each fuel under given conditions, and is lower, the higher the pressure at which ignition is introduced; petroleum behaves particularly well in this respect; in fuels which ignite with difficulty the ignition is promoted by a slight admixture of petroleum. (c) Combustion begins immediately after ignition; the tem- perature of combustion is a consequence of the chemi- cal occurrences connected with combustion and depends on the quantity of air present, which Diesel uses in great excess in opposition to the hitherto customary view that in ordinary combustion the air supply should, comparatively speak- ing, be but a little more than is just needed for complete com- bustion. (d) The blowing of the fuel into the compression space filled with dense air occurs during the first part of the return stroke of the piston and occurs gradually, being regulated by the admis- sion nozzle, and thus a certain control is obtained over the course of the curve of combustion to which the expansion curve (adiabatic) then attaches itself. The starting of the engine is effected by the introduction of 438 TECHNICAL THERMODYNAMICS. compressed air from a storage reservoir, which is kept filled with compressed air by the motor itself, when running. We must add that originally the engine worked without a cooling jacket, but that it was added later on with advantage; furthermore, that with a variable load the performance was regu- lated by varying the supply, i.e., the period of admission of the fuel. With the first engine built in the engine works at Augsburg, which was specially designed for experimental purposes by Diesel, the very careful and extensive experiments, already mentioned on p. 436, were conducted by Schroter, a fluid fuel, petroleum, being used; for fuller details we must refer to Schroter's report; on account of the importance of the question, we will here adduce at least the principal results. The petroleum employed had a heating power H which was on the average /? = 10206 cal. [18371 B.t.u.], i.e., this amount of heat was released in the calorimeter when one kilogram [pound] of petroleum was burned completely. This heat quantity is really developed in the cylinder of the engine and can therefore be re- garded as the heat quantity available for the performance of work. According to all the general investigations made above it is of course theoretically impossible to produce all the work corresponding to this heat; but if we compare the work actually produced with that just indicated, we get a measure for the excel- lence of the engine's performance. In so doing there can be taken from the indicator diagrams the indicated work in horse powers (iVj), or the effective work {Ne) can be obtained with the help of the brake dynamometer and be made the basis of further discussion. Let Gh be the weight of the petroleum in kilograms [pounds] which is consumed per hour by the engine, and let H represent, as before, the heating power, then, according to the usual con- ception, there will correspond to the available heat quantity a work A^'o in horse powers ^<' = 7-5^0^^^ = l^-027^^ [772 8S 1 ^'' = 5W360()^^''-7-17H DISCUSSION OF THE DIESEL MOTOR. 439 for the engine investigated; the indicated "efficiency " is then 7]i=Ni:No . and the effective efficiency is The following table gives the most important of Schroter's experimental results (arranged somewhat differently) ; two sets of experiments (I and II) refer to full load and the other two (III and IV) to half load. The first three rows give Schroter's observations, the other three rows the results of calculations according to preceding formulas. Full Load. Half Load. I. II. III. IV. 4.92 26.56 19.87 78.85 0.337 0.252 4.24 23.60 17.82 67.95 0.347 0.262 2.66 16.57 9.58 42.63 0.389 0.225 2.72 kg. (hourly) 16 . 52 borse powers 9.84 43.59 0.379 0.226 The efficiencies here attain values which have hitherto never been observed in any other heat motor, and especially deserving of attention are the slight differences they exhibit at different loads. The consumption of petroleum in kilograms [pounds] per hour, per brake horse power at full load, is in the mean Gi,:N e = 0.2i2 [0.541], and at half load is 0.276 kg. [0.617 lb.], which are far more favorable values than have been reached by ordinary petroleum motors. More favorable results still than those found by Schroter are said to have been attained in experiments with engines built later. Diesel's motor has probably met the high expectations of the inventor so far as the use of Uquid fuels is concerned; as regards the employment of highly compressed il- luminating gas and the use of solid fuel in dust form, in place of the Uquid fuels, no report has thus far been made concerning the 440 TECHNICAL THERMODYNAMICS. results of the experiments, but here also success should not be lacking. In the meantime the construction of the ordinary gas motor has progressed, and its improvement is largely to be sought in the use of a high preliminary compression; it has the appearance of having here reached about the limit of improvement of the working process, while with the Diesel motor there seems to be a .wider margin in this direction. Diesel's theoretical preliminary investigations started with the idea of making the cycle of his engine approach as nearly as possible that of the C a r n o t cycle; Diesel had to give up from the beginning the idea of bringing about a perfect agree- ment, for, as was already emphasized, this is impossible; on the other hand he sought to follow as closely as possible certain parts of the C a r n t cycle. The agreement exists with respect to the expansion and compression curves, but the former is not continued down to the external back pressure, but is broken off earlier, as is moreover the custom with other heat engines, because while the loss of work thus incurred is small, the cylinder dimensions are con- siderably decreased. The lower curve, isothermal compression at the constant lowest temperature (path cd, Fig. 41, p. 285), as in gas engines, is replaced by the initially slightly rising part of the compression curve. As was said before, the compression curve is broken off when the ignition temperature is reached; here begins the introduction of the liquid or gaseous fuel and the regulation of the combustion, in such a way that, so far as possible, it may take place at constant temperature. In this part of the cycle the main difference between the Diesel motor and the gas engine manifests itself to the advantage of the former. General Conclusions. § 77. ON THE " WORKING VALUE " OF FUELS. When considering the closed air engine in § 53, under the head "Disposable Work of Heat Engines " (p. 300), mention was made of the "working value " Lq of the heat in the heating body and fuller elucidation was reserved. There the efficiency rj = L:Lm, where L is the work actually produced in the engine and L^, repre- ON THE "WORKING VALUE" OF FUELS. 441 sents the disposable work, the latter, calculated under the hypoth- esis that the engine describes the p^fect cycle (according to C a r n t ), is determined by the formula ALm = p^iT,-T2), (1) but in so doing it is expressly assumed that the working or m e- d i a t i n g body is supplied with the heat quantity Qi from the outside, the heat coming from a special heating plant and passing through walls before entering the medium, and that Ti and T2 represent respectively the highest and lowest temper- atures occurring in the cycle. Engines of this sort we will here designate as "external-combustion engines"; steam and hot-air engines are of this sort. In applying the preceding formula the analogy of the heat engine to the hydraulic motor was repeatedly mentioned and made manifest; when the disposable work L^ is to be determined, the place of the weight or of the "water supply" is taken by the heat weight or the change of entropy QiiATi and the place of the "fall" is taken by the temperature difference Ti — r2. In this view the occurrences in the heating plant proper are ignored and so is the heat quantity which is released at the same time by the combustion of the fuel on the grate. If B is the fuel in kilo- grams [pounds] per second and H the heating power, then the heat released is BH; it has a very different and indeed a larger value than Qi in the last formula, provided that Qi and hence Lm are likewise reduced to the second; neither do the temper- atures Ti and T2 stand in any direct relation to the combustion temperature in the furnace and to the temperature with which the products of combustion leave the heating surface. The cost of running a heat engine depends upon the fuel con- sumed and therefore one is accustomed, in comparing the excel- lence of different engines, to immediately base the comparison on the fuel which they consume. If L is again the work per second performed by the engine, and here we may think of either the indicated work L^ or of the effective work L^, then the heat value is AL, and its comparison 442 TECHNICAL THERMODYNAMICS. with the heat quantity BH, simultaneously generated on the grate, will furnish the measure. If we substitute ^"'^BH' ^^^ we can (according to G r a s h o f ) designate jj„ as the eco- nomic efficiency of the engine. If the work of the engine is given in horse powers and is desig- nated by N, and if B^ represents the fuel consumed per hour, then L = 75N [L=550iV] and 5^ = 36005; hence we also have jj„ = 75 X3600|^r550x 3600-^1. . . . (2a) Here we can, at pleasure, choose for N the indicated work iV,- or the effective work Ne- If in equation (2) we substitute for L the value Lm, according to equation (1), and replace the value Tjy, by %, then for exter- nal-combustion engines '^^-bF (3) can be regarded as the "economic efficiency of the heating plant." The value Bk,:l>i in equation (2a) means the fuel per hour per horse power; with indicated work we estimate, for the best steam engines of the present day, Bh:Ni = l kg. [2.2 lb.] of coal (Steinkohle), and asff = 7500 cal. [13500 B.t.u.] may be assumed for this coal, it follows from equation (2a) that the economic in- dicated efficiency is ij„== 0.085, and referred to the effective power {Ne:Ni = 0.85 is assumed) we have ij„= 0.072; therefore the heat utilization in the best steam engines seems to be imperfect to the highest degree; it appears absolutely as a fright- ful destruction of the precious and constantly diminishing fuel coal (Steinkohle), furnished us by nature. In this way the best engines have been judged since the time of Redtenbacher. Although no objection can be urged against the preceding propo- sitions from the purely practical stand- ON THE "WORKING VALUE" OF FUELS. 443 point, SO far as it involves a comparison of different engine types with the help of the economic efficiency ij^, based on equa- tion (1), nevertheless it is thoroughly inadmissible to employ )j„ offhand in the manner just given, for estimating the excellence of a particular engine, or for judging our best steam engines. In so doing it is tacitly assumed that it is possible to completely transform into mechanical work the heat of combustion H be- longing to 1 kg. [lb.] of fuel. If we designate the corresponding work by Lq, then this assumption is that ALo=H; (4) now if we could conduct the perfect cycle (C a r n o t), with the fire gases, and in equation (1) replace Q by H, then we should get the preceding relation by supposing ^2=0, as has already been emphasized. The expansion in the C a r n o t cycle would there- fore have to be prolonged till the lowest limit of temperature, the absolute zero, was reached, which is impossible; it resembles charging an hydraulic plant in the interior of the country with a fall that is measured from the level of the head-race to the level of the sea. Therefore if, for a particular type of engine, it is desired to express the efficiency in terms of work which is actually available, we must determine a quantity Lo, already designated as "working value of the fuel,' but which cannot at all be determined according to equation (4) but must be calculated in a special way. The working value Lq depends on the process of combustion and on the temperature limits between which this process can, in principle, be con- , ducted. The process will be different in the various engine types ; it is only necessary to think of the ordinary steam-boiler plant or of the combustion in the gas engine, or in the Diesel motor; in each discussion that combustion process is taken as a base, which actually exists with the engine considered, and in so doing we at the same time consider this process the most profitable for the case under consideration. Of course this does not exclude the possibility of considering some other unknown process of combustion (following some other combustion curve) which will lead to other results. 444 TECHNICAL THERMODYNAMICS. (a) External-combustion Engines. Let us next consider the ordinary steam-boiler plant in which solid fuels are burned under constant atmospheric pressure po. ' Let the atmospheric air enter the furnace with the specific volume vo = OAo and the atmospheric temperature To, and let combustion begin at the point a and stop at the point c. The curve of com- bustion is therefore represented by the "^ horizontal straight hne ac. 4j Now from the point c let the process with the fire gases be continued in the following manner. Let the products of combustion first expand adiabatically according to the curve cd till it reaches, at the point d, the lower temperature limit To and then compress these products at the constant lowest temperature To (isother- mally with a corresponding withdrawal of heat) till atmospheric pressure po is again reached at the point e; in this condition the mass of gas is technically worthless and is now pushed into the open air under the constant pressure po which corresponds to an expenditure of work represented by the rect- angular area standing over OA3 (Fig. 64). It is evident that we have here a closed working cycle before us; the indicator-diagram area, hatched in Fig. 64, gives the work produced in this cycle, and this now represents the working value of the fuel. If the work area ecd is first reduced to a kilogram [pound] of the fire gases (products of combustion), and if we designate by Vi and V the specific volumes belonging to the points e and c, and the corresponding temperature values by To and T, then the work produced along the path ec, expressed in units of heat, is ALi^Apo{v-vo)=ABiT-To); (5) here it is always permissible to assume that the gas constants {B, Cp, and c„) are the same for air and for the fire gases; T is the temperature of combustion. ON THE "WORKING VALUE" OF FUELS. 445 The work L2, performed during adiabatic expansion along the path cd, is found (equation 62a), p. 140^ from equation AL2=c,iT-To) (6) Finally the work L3 expended along the path de is found from isothermal compression, i.e., ALs^ABTolog,^ (equation 58a), p. 138, when the pressure at the point d is desig- nated by p2- For adiabatic expansion the equation P2 \To/ (equation 61, p. 139) holds, and because K— 1 Cp—c^ AB' it follows that T ALz^CpTo log, Y' (7) The work area Lo, hatched in Fig. 64, is determined by Lo = Li+L2—L3, and therefore with the utilization of the preced- ing formulas, ALo = Cp[r-To-Tolog,jr], ' reduced to the unit of weight of the products of combustion. Let G be the weight of the products of combustion per kilo- gram [pound] of fuel, then the working value Lq of the fuel is ALo^CpG^T-To-TologejT^. o . . . (8) The weight of air necessary for combustion here amounts to (G-l)kg.[lb.]. 446 TECHNICAL THERMODYNAMICS' If, for example, in a particular boiler plant we assume the temperature of the atmospheric air to be io=20° C. [68° F.] and the temperature of combustion in the furnace to het = 1200° C. [2192° F.], and accordingly To =293° [527.4°] and 7 = 1473° [2651.4°], which to be sure already presupposes a certain imper- fection of combustion, then for the mean value Cp=0,24, equation (8) gives ALo = 169.64 (? [ALo = 305.35 G]. If we suppose coal (Steinkohle) firing and estimate the quan- tity of air, per kilogram, of coal to be 16 kg. [lb.], and therefore G = 17 kg. [lb.], then for the present case the working value of the coal, measured in units of heat, is ALo = 2884 cal. [ALo = 5191 B.t.u.], while its heating value, based on the combustion in the calorimeter is taken up at the mean value H = 7500 cal. [13,500 B.t.u.]. The heat Qi here released upon the grate is Qi =CpG{T-To) =4814 cal. [8665.2 B.t.u.]. Hence follows from equation (8) ALo=Qi[l-^^logey (9) and with the assumed temperature values ALo = 0.599 Qi =2884 cal. [5191 B.t.u.] as before. If we now substitute the value ALo for H, in equation {2a), assume the indicated performance Ni for N, and designate the efficiency by tj, we have N- jj=0.221^\ ■DA If we again estimate for the best steam engine 1 kg. [2.2 lb.] of coal (Steinkohle) per indicated horse power, we get )j = 0.221, ON THE "WORKING VALUE" OF FUELS. 447 which is nearly three times greater than was found on p. 442, even though this value seems still unfaMorable enough. In order to get a greater insight into the conditions, let us emphasize the following. In the here assumed engines (steam or hot-air engines) the fire gases, while flowing along the so-called heating surface, deliver a part of their heat to the mediating body confined in the interior of the engine. Let T' be the temperature with which the fire gases leave, and flow away from, the heating surface, then the heat quantity Q, per kg. [lb.] of fuel, which is placed at the disposal of the engine itself, is : Q=cp(?(r-r); and the work L^, which is here produced when the engine describes a C a rn t cycle between the limiting temperatures Ti and T2, is or, utilizing the value of Q, L^ = ^^iT-T')iTi-T2), . . . (10) which magnitude can be presented as the disposable work of the prime mover. Now the ratio Lm'-Lo represents the important magnitude ija, which can be regarded as the efficiency of the heating plant, and in combination with equation (8) it can be determined from (T-T'){T^-T2) ^h = T^^T-To-TologeY^ (11) In order to again insert a numerical example let the tempera- tures be i = 1200° C, io=20° C. [2192° F. and 68° F.], and let the temperature at which the fire gases leave the heating surface be i' = 300° C. [572° F.] and let the Carnot cycle be described between the temperatures ii = 180° C. [356° F.] and <2 = 20° C. [68° F.]. Equation (11) then gives for the efficiency of the heating plant )jA = 0.450. 448 TECHNICAL THERMODYNAMICS. From the smallness of this value it undoubtedly follows that the efficiency, found above for the best steam engines, is not to be charged against the steam engine proper, but is due to the kind of heating and to the steam production in the boiler. That is why in all my earlier publications on steam engines I have rigorously distinguished the efficiency of the engine from the efficiency of the whole plant (inclusive of the heating), as I did above in air engines. Equation (11) gives rise moreover to an additional remark. The temperature t' with which the fire gases leave the heating surface is always greater than the upper temperature limit t\ of the cycle described by the engine, greater, for example, than the steam temperature in the steam boiler. If we substitute i' = ii+r and again t2 = to, equation (11) gives Vh= — p 7rr=i (11a) Tr^T-To-Tolog^Y^ For a particular value of the combustion temperature T, for the atmospheric temperature to, and for the excess r, this equation gives a most profitable value for the upper temperature limit Ti or ii of the C a r n o t cycle, and thus furnishes a maximum for the efficiency of the heating plant. We get, as is easily seen, Ti=VTo{T-z). For T = 100° C. [180° F.], for the above-used values i = 1200° C, [2192° F.] and to=20° C. [68° F.] we get Ti^Q33° [1139°], or ii = 360° C. [680° F.], and 1?^ =0.562 for a maximum. The temperature value ti here found may occur in hot-air engines, but in the earlier discussion of these engines it was stated that this upper temperature limit ought to be chosen as high as possible; now the limitation presents itself, that with these engines a certain upper value should not be exceeded; for while this excess increases the efficiency of the engine, it dimin- ishes that of the heating plant. The foregoing considerations confirm the view, expressed ON THE "WORKING VALUE" OF FUELS. 449 long ago, that external-combustion engines, hence such as possess special heating plants, utiUze imperfectly in a mechanical way, the heat of the fuel, which fact unfortunately applies to steam engines in which really extraordinary results have been accom- plished with the prime mover proper. It seems that no essential betterment of the heating plant itself is to be expected, and that the effort to produce greater and more perfect combustion en- gines are thoroughly justifiable.^ (&) Internal-combustion Engines. Let the investigation be again confined to a special case and let the "working value of illuminating gas" be determined for the Otto gas motor. At the beginning of the cycle let there be in the cylinder the volume Vq of a mix- ture of illuminating gas and atmospheric air, and let it include a remnant of the products of combustion (Fig. 65a); the pres- sure po and the temperature To correspond to the values of the external atmosphere. First let the mixture be compressed adiabatically along the path TqTi till the pressure has risen to pi and the temperature to Ti. Let combustion begin at the point Ti and let it be concluded at the temperature T2; let the products of combustion expand adiabatically from T2 to To till the lower temperature limit To is again attained; let the corresponding pressure 72^0 be desig- nated by p2- Finally let the mass be compressed isothermally till the initial point po is again reached, which will be the case ' The investigation given above, with the derivation of the fundamental equation (11), is already found in the edition of the year 1887. Later L o r e n z in the Zeitschrift deutscher Ingenieure, 1894, p. 1450, and in 1895, p. 1239, under the title "Die Beurteilung der Dampfkessel," developed the same formulas in a somewhat different way, assuming a cycle presented above as the L o r e n z cycle. 450 TECHNICAL THERMODYNAMICS. when we make what was proved above to be a permissible as- sumption that the gas constants B, Cp, and c» may be taken as equally large for the mixture and for the products of combustion. The area, hatched in Fig. 65a, now represents the working value Lo of G kg. [lb.] of the mixture, when G represents the weight of the mixture consumed per period. Since, in accordance with the hypothesis, the mass has returned to the initial state we can trans- form the indicator diagram by means of the known propositions. Fig. 656 shows the corresponding entropy diagram. Its line T1T2 represents the transformation of the curve of combustion, while the area P1T1T2P2 lying under it is proportional to the heat Q released during combustion, and at the same is proportional to the produced work L; the two values Q and L should be reduced to the unit of weight of mixture. The hatched area in the entropy diagram gives the working value ALo measured in units of heat; therefore we at once have ALo = G[Q-ATo(P2-Pi)] (12) Now, in order to include an infinite number of special cases and all technically important ones, we will assume a curve of com- bustion to be a poly tropic curve having the course pv" = constant, of which the corresponding specific heat c is given by (p. 152) n — K dT then dQ=cdT and AdP^c-^, and there follows at once from equa- tion 12 ALo=cG^T2-Ti-To log.p'j, .... (13) because Q = ciT2-T,) (14) The work L produced during combustion is given by equation (7) (p. 153): AL = {c-c,){T2-Ti) (15) ON THE "WORKING VALUE" OF FUELS. 451 If the compression relation is pi : po, we can get the temperature at the end of compression from To ^=(r ™ If we refer the weight G of the mixture to one kilogram (pound) of illuminating gas, equation (13) will at once give the working value Lq per kilogram (pound) of gas. If there were no compression, then we should have pi=po, and should have to make Ti = To. In equations (14) and (15) the values Q and L refer to 1 kg. [lb.] of mixture; with G kg. [lb.] of mixture we must therefore write Q=cGiT2-Ti), (14a) where Q refers to G kg. [lb.] of illuminating gas. Unfortunately the experiments known at the present time do not render it possible to utilize the preceding formulas with any degree of certainty. If we assume, as seems permissible for many cases, combustion at constant volume, we must substitute specific volume c„ at constant volume in place of c. The uncer- tainty indicated Ues in our ignorance of the combustion temperature T2, and hence, according to equation (14a), the heat quantity Q released during combustion cannot be determined; the basis for these problems will only be furnished by experiments which may be expected in the future. In order, however, to get an insight with the help of an ex- ample let us assume c = c^; then follows ALo = c,GfT2-Ti-To\oge'^] (13a) If it is assumed that the combustion was perfect, we must put in place of Q in equation (14a) the heating power H of illuminating gas, where G is the weight of the mixture for 1 kg. [lb.] of illuminating gas; then H = cMT2~Ti) (Ub) 452 TECHNICAL THERMODYNAMICS. Here the influence of the coohng of the cylinder walls should cer- tainly find expression, for this action prescribes a smaller value for the combustion temperature T2. If there is preliminary compression, Ti can be computed from equation (16), and by the known value of G the temperature T2 can be found, and finally the working value Lq, according to equa- tion (13a). Example. For the illuminating gas of average composition (p. 412), there is found /f = 10113 cal. [18203 B.t.u.]. For this illuminating gas there was found B = 60.823 [B = 110.86], and for a mixture of gas of the proportions 19 kg. 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