Class / xJ / fi & < S Book HC 5 Copyright^ COPYRIGHT DEPOSIT: HEAT AND THERMODYNAMICS Published by the Mc G raw - Hill Book Comp any NewYork 6uccessons to the J&ookDepartmenis of the McGraw Publishing Company Hill Publishing" Company Publishers of ]B>qoLs for* Electrical World The Engineering" and Mining Journal Engineering Record American Machinist Electric Railway Journal Coal Age Metallurgical and Chemical Engineering Power \V)A*\t>At\f:t\t)£\i\t\t\*\to£\At\toS\AMI\At\*\S\fi\to£iMe\£\A*\t\totototof\*\AtotoAA HEAT AND THERMODYNAMICS BY F. M. HAETMAO n ment of Physics, and Electrical and Cooper Union Day and Night Schools t) In charge of the Department of Physics, and Electrical and Mechanical Engineering. McGRAW-HILL BOOK COMPANY 239 WEST 39TH STREET, NEW YORK 6 BOUVERIE STREET, LONDON, E.C. 1911 \9 *>V Copyright, 1911 BY McGRAW-HILL BOOK COMPANY !*<&> ©CU305235 no. r PREFACE With so many good works in existence, both on Heat and Thermodynamics, it may perhaps appear presumptuous to publish the following text. The author, however, has long felt the need of a text, in teaching the subject of thermodynamics, which properly covers, without introducing too much material, the fundamental principles of heat measurements. To expect an average student to cull from his text book on physics, or some treatise on heat, no matter how well the subject may have been taught, an introduction to thermodynamics is, in general, expecting somewhat more of him than he can accomplish. But it has been found, by experience, that a short course on the fundamental principles of heat, given as an introduction to the subject of thermodynamics, greatly reduces the difficulties, experienced by most students, in pursuing this subject. Since it is almost impossible for a student to understand a complex piece of apparatus, unless he can actually examine it, long and tedious descriptions have been purposely avoided. Likewise, for the reason that photographs are seldom, if ever, of any value, all pictorial illustrations are diagrammatic. It is, of course, impossible to teach the subject of thermo- dynamics without the application of differential and integral calculus; but the aim has been throughout to keep within the bounds of elementary mathematics. However, a fair knowledge of the calculus, on the part of the reader, has been assumed. Very few teachers, if any, can present an unbiassed view of a speculative theory; furthermore, before a student has thor- oughly mastered the groundwork of any subject, he is not in a position to properly discriminate between the various arguments that may be advanced, either for or against a speculative theory. vi PKEFACE It must also be remembered that the average student looks upon his instructor as an infallible authority; and that he accepts a theory on the mere say so of his instructor, no matter how flimsy the arguments upon which it may be based. How fre- quently one meets those who are in a condition so deplorable that they can talk very glibly about electrons, ionization, etc., and are driven helplessly into a corner by one or two well directed questions. Whether there is or is not such a thing as an atom has nothing to do with the law of definite propor- tion. Facts will always remain and theories change to fit them. It is for these various reasons that speculative discussions, such as that of the kinetic theory of gases, have been avoided, and that very hypothetical medium — the ether — has found no place in this text. It cannot be too strongly emphasized that before we teach metaphysics to a student we must first give him a thor- ough training in mathematics and physics. It is the author's opinion that the best that can be done in any technical course is to thoroughly teach the fundamental principles underlying the subject, and that it is impossible to give a training which makes the student a practical engineer. This part must be learned in practice, and the engineer must keep up to date and in proper touch with his profession by reading the current engineering literature, and by studying individual problems as they arise. It must not be understood that the author expects this text to supersede such admirable works as Peabody's treatise on " The Thermodynamics of the Steam Engine," Zeuner's " Tech- nische Thermodynamic " etc.; but rather as a proper prepara- tion for the reading of such works. Finally, the author cannot express his feelings too strongly in regard to the pleasure he experienced, as a student, while reading TyndalPs " Heat a Mode of Motion," and Ewing's " The Steam Engine and other Heat Engines." Thanks are hereby expressed to Mr. Albert Goertz for the care with which he read the manuscript. F. M. H. Cooper Union, July, 1911. TABLE OF CONTENTS HEAT CHAPTER PAGE I. Temperature and Thermal Units 1 II. Calorimetry 11 III. Production of and Effects of Heat 21 IV. Expansion of Solids and Liquids 34 V. Fundamental Equations of Gases 41 VI. Elasticities and Thermal Capacities of Gases 68 VII. Propagation of Heat 80 THERMODYNAMICS VIII. Fundamental Principles ....-! 93 IX. Steam and Steam Engines 116 X. Entropy 128 XI. Applications of Temperature-Entropy Diagrams 137 XII. Elementary Steam and Engine Tests 151 XIII. Compound Engines 180 XIV. Internal Combustion Engines and Fuels 192 XV. Ideal Coefficient of Conversion and Elementary Tests . . . 210 XVI. Compressed Air and Compressors 230 XVII. Refrigeration 265 XVIII. Steam Turbines. 289 vii HEAT CHAPTER I TEMPERATURE AND THERMAL UNITS 1. The fundamental conception of hotness or coldness is one of bodily sensation. That is, an object is said to be hot or cold depending upon whether it gives us one sensation or another when we are near it or in contact with it; and the more intense the sensation, the hotter or colder the object is said to be. Exper- ience, however, teaches us that the estimates so formed are not accurate; since the intensity of the sensation experienced, in any given case, depends not only upon the condition of the body under consideration, but very largely upon our experience imme- diately preceding. Observation shows that, in general, as bodies are heated or cooled they change in volume; and in most cases, other things being equal, bodies increase in volume when heated and decrease in volume when cooled. Observation further shows that there is a continual exchange of heat among bodies; i.e., if in any system of bodies some are gaining heat, others are losing heat. Or, in other words, there is a continual tendency toward equilibrium. 2. Temperature. Assume now, that we are dealing with two bodies, A and B, and that it is desired to determine which is the hotter. To do this, some test specimen, upon which 2 HEAT previous observation has shown a continuous expansion with continued application of heat, may be used in the following manner: The test specimen is put into contact with A, and after a suitable interval of time its length, say, is accurately measured; it is then put into contact with B, and its length is again measured. If the length is now greater than it was before, i.e., if the test specimen expanded when put into contact with B, after having been in contact with A, B is hotter than A; for, by previous observation it was found that the test specimen expanded con- tinually as it became hotter. The body B, therefore, was capable of imparting more heat to the test specimen than the body A could impart to it, and B is said to be at a higher temperature than A . The difference of temperature, then, between two bodies may be measured by the amount of change in dimensions which a test specimen undergoes when, after having been in contact with one of the bodies, it is brought into contact with the other body. Such a test specimen is called a thermometer; and, for accurate measurements, the nature of the thermometer must be such that no appreciable change is brought about in the body whose temperature is sought. If the two bodies, A and B, of the previous discussion, are now brought into contact, and after a suitable interval of time the test specimen is put into contact with A and then with B, there will be no change in its dimensions; i.e., the two bodies are in thermal equilibrium, or, in other words, at the same temperature. But, the body B will have lost heat and the body A will have gained heat; hence, when a body has the capability of imparting heat to another body, it is said to be at a higher temperature. Solids, liquids, or gases may be employed in the construction of thermometers; but it is important that the substance used does not change its state during the change of temperature. For, the rate at which a body changes in volume, with respect to change in temperature, depends upon its physical state; i.e., though a substance may exist in the three different states, its rate TEMPERATURE AND THERMAL UNITS 3 of expansion will, in general, be entirely different in the various states; being usually the greatest for gases and the least for solids. The rate of expansion, in general, changes abruptly in passing from the solid to the liquid, and from the liquid to the gaseous state. The three substances most generally employed in the con- struction of thermometers are: Mercury, alcohol, and dry air; the most convenient and most commonly used being mercury. There are two standard temperatures, arbitrarily chosen, upon which all thermometric scales are based; the one is that of melting ice, and the other that of the vapor of boiling water under a pres- sure of one standard atmosphere. The pressure of one atmosphere being taken equal to that of a column of mercury, at the tempera- ture of melting ice, whose height is 76 cm., at 45° latitude and sea level, where the acceleration of gravity is 980.60 cm. per sec. per sec; or, in c.g.s. units, a pressure of 1.01325 X10 6 dynes per square centimeter. Thermometric Scales and Thermometers 3. There are three thermometric scales in use: The centigrade scale, the zero of which is the melting-point of ice, and the point corresponding to the temperature of the vapor from boiling water under standard conditions, called the boiling-point, is marked 100. Hence there are 100 units, called degrees, for the interval between the melting-point and the boiling-point. The Fahrenheit scale is marked 32 for the melting-point and 212 for the boiling-point; hence, there is an interval of 180 degrees between the two fixed points. The Reaumur scale is marked for the melting-point and 80 for the boiling-point; hence, there is an interval of 80 degrees between the two fixed points. Fig. 1 is a diagrammatic representation of the relation of the three scales. From the foregoing it is obvious that the value of a degree on the Fahrenheit scale is 5/9 of a degree on the centi- HEAT 100 i 212 Ujl h Ql Ul I U| < 0| u. o ! 32 80 grade scale, and the value of a degree on the Reaumur scale is 5/4 of a degree on the centigrade scale. Since both the centi- grade and Fahrenheit scales are in common use, it is convenient to have a simple method of conversion from one scale to the other. Let it be desired to convert the temperature on the centigrade scale to the Fahrenheit scale. Since one degree centigrade equals 9/5 degree Fahrenheit, it follows that an interval of 6°C. is equal to an 8 Fig. 1. interval of — 6°F. ; but, since the melting-point on the Fahrenheit scale is marked 32, we must add 32 to give the Fahrenheit reading. Therefore, to convert a temperature on the centigrade scale to the Fahrenheit scale, we must multiply the reading by 9/5 and add 32. And similarly, to convert Fahrenheit to centigrade we must subtract 32 from the reading and multiply by 5/9. A method of conversion, which is simpler, is to solve for that temperature for which both scales read the same. This temperature is obviously below zero, and is therefore negative. 9 Let be that temperature on the centigrade scale; then, —0+32 5 is the reading on the Fahrenheit scale, are by condition equal. Hence, But these two readings from which = |0+32; 0=-4O. Therefore, to convert from one scale to the other, the most con- venient way is to add 40 to the reading, multiply this by the con- TEMPERATURE AND THERMAL UNITS 5 version factor and deduct 40. To convert from centigrade to Fahrenheit, add 40 to the reading, multiply by 9/5 and deduct 40. To convert from Fahrenheit to centigrade, add 40, multiply by 5/9 and deduct 40. 4. Mercurial Thermometer. The mercurial thermometer con- sists of a capillary tube of, as nearly as obtainable, uniform bore, on one end of which is blown a bulb. The bulb is filled with mercury and heated so as to drive out all the air. When this has been satisfactorily performed, so that nothing but mercury remains in the bulb and tube, the tube is hermetically sealed. The bulb and tube are then immersed in ice from which the water is allowed to drain away, and the point to which the mercury falls marked on the stem. Next, the thermometer is immersed in saturated steam, under standard pressure, and the point to which the mercury rises marked on the stem. It is, however, necessary to allow considerable time to elapse between the seal- ing of the tube and the determination of the fixed points; since glass, after having been heated, does not, upon being cooled, immediately return to its original volume. If the bulb contracts after the fixed points have been placed on the stem, the ther- mometer will read too high. Joule found that the bulb of a certain thermometer, upon which he had taken observations for twenty years, was still changing slightly at the end of that time. After the fixed points are determined, a thread of mercury is detached, and, by means of it, the tube calibrated to the desired scale. In this way the units on the scale represent equal volumes, and not necessarily equal lengths. But in good thermometers the tube is of so nearly uniform bore that the lengths of a degree do not differ by any considerable amount over different parts of the scale. Since glass changes in volume when its temperature changes, it is obvious that the indications of the mercurial thermometer are proportional to the relative changes between the mercury and glass, and do not necessarily indicate the true changes in 6 HEAT temperature; i.e., the indicated temperatures between the two fixed points depend upon the substances used in construction. It is readily seen that two mercurial thermometers, if constructed of different qualities of glass, which have not the same rates of expansion with respect to mercury throughout the entire scale, will differ slightly in their readings for some parts of the scale, even though they read alike for the fixed points. 5. Alcohol Thermometer. The alcohol thermometer is con- structed in a manner similar to the mercurial thermometer. Its chief advantage lies in the fact that it may be used for temperatures below the melting-point of mercury, which is — 39°C. However, on account of its low boiling-point, which is 78.2°C, alcohol cannot be employed for high temperatures; on the other hand, the boiling-point of mercury is 357°C. The discussion of the air thermometer will be deferred until after the discussion of the laws of gases. 6. Thermo Couple. When the junction of two dissimilar metals is heated, an e.m.f. is developed; and since this e.m.f. is a function of the temperature, such a combination, called a thermoelectric couple, or simply thermo couple, may be employed to indicate temperatures. The thermo couple is, in many cases, where a bulb thermometer cannot be employed, a very conven- ient device for measuring changes of temperature; and it is par- ticularly valuable in enabling us to estimate changes of temper- ature above the boiling-point of mercury. By employing proper metals, such as platinum and iridium, very large ranges of tem- perature can be measured; the melting-point of iridium being about 2500°C. and that of platinum about 1775°C. 7. Resistance Thermometer. The fact that the ohmic resist- ance of a metal is a function of its temperature makes it possible to estimate changes in temperatures, by noting the changes in resistance of a particular conductor employed for the purpose. From the foregoing, it is obvious that if any body or combina- tion of bodies manifest some change, which is a function of the TEMPERATURE AND THERMAL UNITS 7 temperature and readily measurable, such body or combination of bodies may be employed to indicate temperatures. But it is to be carefully noted that the device employed must be such that the temperature of the body, upon which the measurements are made, is not appreciably altered by the test body; and that, in any case, the changes produced in the test body are peculiar to it, and not necessarily proportional to the changes that would be produced in some other instrument of a different type. There- fore, temperatures must always be referred to some scale chosen as a standard. This will be dealt with more fully in the discussion of thermodynamics. Heat as a Measurable Quantity 8. If a quantity of water at a temperature ti, be mixed with an equal quantity of water at a temperature T2, the resulting temperature of the mixture is very nearly the mean between the two initial temperatures. If it requires a certain quantity of heat, q, to raise n grams of water through a given temperature interval, then it obviously requires a quantity of heat, mq, to raise mn grams of water through the same temperature interval. If the water be cooled through the same temperature interval, then there is imparted to the surrounding bodies a quantity of heat which is equal to that absorbed by the water while being raised through that temperature interval. If the quantities of heat, required to raise a given mass of water through equal temperature intervals throughout the chosen thermometric scale, were all equal, then the resulting temperature obtained when mixing equal masses of water would be the exact mean between the two initial temperatures. This is shown by experiment to be very nearly, but not quite true. Hence, there is not strict proportionality, in the case of water, between change of temperature, according to our thermometric scale, and change of heat. 8 HEAT If equal masses of water and some other substance, say, mercury, be mixed, the resulting temperature will differ considerably from the mean between the two initial temperatures. Experiment shows that if mercury at a temperature ti, be mixed with water at a temperature T2, the mass of the mercury must be 29.85 times the mass of the water so as to give a resulting temperature equal T1+T2 to -3-. 9. Thermal Capacity. The thermal capacity of a body is numerically equal to the ratio of change in heat to the corresponding change in temperature produced by it. The preceding paragraph states that water has, mass for mass, a greater thermal capacity than mercury in the ratio of 29.85 : 1; or the thermal capacities of equal masses of mercury and water are to each other as 0.0335 : 1. If copper be compared with water the ratio is found to be as 0.0933 : 1. In general, the ratio is less than unity, has different values for different substances, and varies somewhat with change of temperature. One notable exception is hydrogen gas, where the ratio is found to be, at constant pressure, as 3.409 : 1; and at constant volume, as 2.42 : 1. To compare different quantities of heat, it is necessary to choose some substance as a standard. On account of convenience, water has been so chosen. 10. The Calorie. The quantity of heat required to raise the temperature of 1 kilogram of water through 1 degree centi- grade, is called a calorie, and is the unit adopted for heat measure- ments. But, since this quantity varies slightly for different temperatures it becomes necessary, in making exact measure- ments, to specify some particular quantity. There are three different definitions for the calorie: (1) The quantity of heat required to raise the temperature of 1 kilogram of water from 0°C. to 1°C, called the zero calorie. (2) One hundredth part of the heat required to raise the tern- TEMPERATURE AND THERMAL UNITS 9 perature of 1 kilogram of water from 0°C. to 100°C, called the mean calorie. (3) The quantity of heat required to raise 1 kilogram of water from 15°C. to 16°C, called the common calorie. Since it is impossible to realize accurately the first or second of these, on account of the difficulty experienced in working with water at 0°C, the last, or common calorie, is the one most gen- erally used. Furthermore, due to the fact that a great many heat measurements are made in the range between 15°C. and 25° C, no large corrections for change in thermal capacity, due to change in temperature, when the common calorie is employed, are necessitated; hence, this unit is more convenient than the others. Since, in ordinary heat measurements, masses are generally specified in grams, a secondary unit, called gram calorie, having the gram instead of the kilogram for the unit mass, is usually found more convenient than the calorie. In what follows, unless otherwise specified, by calorie is to be understood the quantity of heat required to raise the temperature of 1 kilogram of water from 15°C. to 16°C, and by gram calorie, one thousandth part of the calorie. 11. British Thermal Unit. The thermal unit most commonly employed in engineering practice, in England and America, is the British Thermal Unit or B.T.U.; it is the quantity of heat required, at ordinary temperatures, to raise the temperature of 1 pound of water through 1 degree Fahrenheit. 12. Thermal Capacity per Unit Mass and Specific Heat. The ratio of the quantity of heat required to raise the temperature of a given mass of a substance through a given temperature iriterval, to the quantity of heat required to raise an equal mass of water through the same temperature interval, is called the specific heat of the substance. And, since the unit of heat — the gram calorie — is the quantity of heat required to raise the temperature of 1 gram of water through 1 degree centigrade, it follows 10 HEAT that the quantity of heat, measured in gram calories, required to raise the temperature of 1 gram of a substance 1 degree centigrade, is numerically equal to the specific heat of the sub- stance; or, in other words, the specific heat of a substance is numerically equal to its thermal capacity per unit mass. 13. Water Equivalent. By the water equivalent of a body is understood the mass of water which has a thermal capacity equal to that of the given body, and is numerically equal to the mass of the body multiplied by its thermal capacity per unit mass. 14. Method of Mixtures. Assume a mass of water mi, at a temperature ti, to be mixed with a mass wi2, of some other sub- stance, at a higher temperature T2, yielding for the mixture a resulting temperature of 6. Then, if no heat is lost to or gained from the surroundings, during the operation, and the thermal capacities of the water and substance are sensibly constant for the temperature ranges experienced, it follows that, since the heat gained by the water is equal to that lost by the substance, we must have mi(0— Ti)=ra 2 c(T 2 — 0); (1) where c is the thermal capacity per unit mass of the substance, amd m2C is its water equivalent. To take a numerical example, assume 268 grams of water at a temperature 10°C. to be mixed with 1000 grams of mercury at a temperature 100°C, giving a temperature of 20°C. for the mix- ture. Substituting in equation (1), we have 268(20 - 10) = 1000c(100 - 20) ; from which, the thermal capacity of mercury, per unit mass, is • 268(20-10 ) . C ~ 1000(100-20) - UM66b - CHAPTER II CALORIMETRY 15. As thermometry has for its object measurement of tem- peratures, so has calorimetry for its object the measurement of quantities of heat. In equation (1), Art. 14, it was shown what must be the relation between the masses involved, the changes in temperature, and the thermal capacity per unit mass of a substance, when two substances are mixed and assume a common temperature. The actual determination of thermal capacities is, however, not so simple. Since, in general, the vessel in which the mixing takes place suffers a change in temperature, its thermal capacity must be taken into consideration. Furthermore, there is usually an exchange of heat between the vessel, in which the experiment is performed, and the surrounding medium during the progress of the experiment. The vessel, specially designed, in which the mixing takes place, is called a calorimeter. It therefore follows, from what has just been said, that in making heat measurements it is necessary to know not only the thermal capacity of the calo- rimeter, but also, the rate at which, for a given difference of tem- perature, exchange of heat takes place between the calorimeter and the surrounding medium. 16. Law of Cooling. The rate at which a body loses heat to surrounding bodies is independent of its thermal capacity, and depends upon the nature and area of its exposed surface, and the difference in temperature between it and the surroundings. For small differences of temperature, i.e., up to a difference of about 11 12 HEAT 15°C, from ordinary room temperature, the rate of cooling is very nearly proportional to the difference of temperature. This is known as Newton's Law of Cooling. The foregoing then states that the rate at which a body loses heat at any instant is a func- tion of its surface, and of the difference of temperature between it and surrounding bodies. Newton's law may be stated as follows: §--'*■> 0) where Q is quantity of heat, t time, K some constant, depending upon the surface of the body, and t the difference of temperature. If m is the mass of the body, and c the thermal capacity per unit mass, then equation (1) may be written 4r- & - • • w If c is constant, then equation (2) becomes dt ~™ ; where k= — . mc Separati ing the variables, we find — = — kdt, from which log f r -kt; and T=fcis" te . (3) To determine the constant of integration A>i, assume that we begin to reckon time when t=ti; i.e., t=ti when £ = 0. Making this substitution, in equation (3), we find &i=ti; hence, finally T=Tl£-^ (4) CALORIMETRY 13 If m, c, and K are known, thus fixing the value of k, then, by means of equation (4), the difference of temperature, t, between the body under consideration and the surroundings at any time, t, can be predicted, provided the initial difference of temperature be known, and the surroundings remain constant in temperature. It is found, by experiment, that bodies having highly polished surfaces, other things being equal, lose heat less rapidly than do those having rough dark surfaces; hence, calorimeters should have their exposed surfaces highly polished. Furthermore, the thermal capacities of calorimeters should be small in comparison with those of the bodies contained in them, and upon which measurements are being made. Also, while the experiment is under progress, the calorimeter should be protected from draughts of air. We are not as yet in a position, nor is it essential, to enu- merate all precautions that must be taken to give results of abso- lute precision. 17. Thermal Capacity of a Calorimeter. Since the calo- rimeter in which the bodies, upon which measurements are to be made, are contained, always suffers a change in temperature, it is necessary to know its thermal capacity. But, since the thermal capacity of the calorimeter, in general, is small in comparison with that of the bodies upon which measurements are being made, it follows that a small error in the determination of the thermal capacity of the calorimeter will not seriously affect the results obtained for these bodies. The thermal capacity of a calorimeter may be obtained, though not with absolute precision, in the following manner: The calorimeter containing water has its temperature noted, the water being first stirred to insure uniform temperature, then immediately a quantity of water at some other temperature is poured into the calorimeter, the contents stirred and the resulting temperature is noted. If C is the thermal capacity of the calo- rimeter and stirrer, wii the mass of water originally in it, ti the temperature of calorimeter and contents before mixing, and 6 14 HEAT the common temperature after mixing, then on the assumption that ti is higher than 0, the loss in heat, suffered by the calorimeter and water originally in it, is (mi+C)(T!-0). The gain in heat, by the water poured into the calorimeter, is m 2 (e-T 2 ); where ra 2 is the mass, and t 2 the temperature of the water poured into the calorimeter. But, if there are no other heat exchanges, the loss on the one side must be equal to the gain on the other, hence from which (mi+C)(Ti-6)=ra 2 (0-T 2 ); c= m 2 (0^T 2 )_ Tl-6 • There will always be an exchange of heat between the calo- rimeter and surroundings; but this can be reduced to a small quantity by choosing the masses of water such that the result- ing temperature of the calorimeter is as much below the room temperature as was its initial temperature above the room tem- perature. When it is possible, large differences of tempera- ture, between the calorimeter and room, should be avoided. 18. Cooling Constant of a Calorimeter. When it is impossible to have the initial and final temperatures differ by equal amounts from the room temperature — one, of course, being above and the other below — then, to obtain accurate results, correction must be made, as the case may be, for loss or gain in heat. To do this, the calorimeter is filled with water, at about 15°C. or 16°C. above room temperature, to the same height as it will be when the experiment proper is performed. The temperature is then CALORIMETRY 15 noted at short intervals of time, the water being continuously stirred to insure a uniform temperature throughout at any instant. If the room temperature has remained constant during the progress of the experiment, then, obviously, equation (4), Art. 16, applies and the constant k of this equation is determined. In general, however, better results are obtained by plotting the observations; using times as abscissas and differences in temper- atures, between the calorimeter and the room, as ordinates, and passing a smooth curve through the points so found. The slope of the tangent then, to the curve at any point, is the rate of change of temperature at that point; and this slope, divided by the dif- ference of temperature, or in other words, by the ordinate of the point, is, according to Newton's law of cooling, a constant for any point on the curve. Drawing a number of tangents and divid- ing the slope of each by its ordinate, will give quotients nearly equal; and the mean of these quotients will be a fair value for the rate of change of temperature for unit difference of temper- ature. The rate so obtained multiplied by the thermal capacity of the calorimeter and contents is numerically equal to the quan- tity of heat lost, by the calorimeter, per unit time per unit differ- ence of temperature. 19. Determination of Thermal Capacities. Thermal capacities may be determined in various ways; the simplest, though not necessarily the most accurate, and not applicable in all cases, is the method of mixtures. The substance, whose thermal capacity is sought, is heated to a temperature ti, which is noted. It is then immediately transferred to a calorimeter of thermal capacity C, containing a mass of water m 2 , at a temperature T2. If the result- ing temperature is 6, and no heat has been lost to or gained from the surroundings during the operation, then the heat lost by the one side must be equal to the heat gained by the other; hence we have mic(Ti-0) = (m 2 +C)(0-T 2 ); . . . . . (6) 16 HEAT where mi is the mass of the substance, and c its thermal capacity per unit mass. From equation (6), we find c = (m 2 +C)(e-T 2 ) ...... (7) wi(ti-g) ' v ' which determines the thermal capacity per unit mass of the substance. If the resulting temperature differs materially from the room temperature, then corrections will have to be made, from the curve of cooling of the calorimeter. 20. Method of Cooling. If heat is generated or absorbed when two substances are mixed, the thermal capacity of a sub- stance cannot be determined by the method of mixtures. As an example, if sulphuric acid is mixed with water, considerable heat is evolved; hence, recourse must be had to some method, other than the method of mixtures, in determining the thermal capacity of sulphuric acid. This may conveniently be done by what is known as the method of cooling. Assume that we have a calorimeter of known thermal capacity, C. Let the calorimeter be filled to a definite height with water, and the time noted which is required for the calorimeter and contents to cool, from a temperature ti to a temperature t 2 , when exposed to a definite and constant room temperature. Next, the calorimeter is filled to the same height with the liquid, whose thermal capacity is sought, and the time which is required to cool from ti to t 2 , when the calorimeter and contents are sub- jected to precisely the same conditions as when filled with water, is again noted. Then, since the average difference of temperature between calorimeter and contents and the surroundings is the same in both cases, it follows that the thermal capacities, in the two cases, are to each other directly as the times required in cooling through the same temperature intervals. Therefore, if h is the time required for the calorimeter and water to cool from ti to T2, and t2 is the time required for the calorimeter and the sub- CALORIMETRY 17 stance whose thermal capacity is sought, to cool through the same temperature interval, it follows that C+mi C+m 2 c , Q . -ir = -~h"' (8) where wi is the mass of water, W2 the mass of the liquid, and c its thermal capacity per unit mass. From equation (8), since all quantities excepting c are known, the thermal capacity per unit mass is determined. It is, however, not necessary to know the thermal capacity of the calorimeter. For, if the calorimeter be first cooled through the temperature interval, ti— T2, when empty, then when filled with water, and again when filled with the liquid whose thermal capacity is sought, and the times t, h, and fe are noted, we shall have C C+mi C+7112C t h t 2 (9) where t is the time required when empty, t\ when filled with water, and h when filled with the liquid whose thermal capacity is sought, and the other symbols having the same significance as before. Eliminating C, from equation (9), we find c=^=| (10) A convenient form of apparatus, for the method of cooling, is an alcohol thermometer with its bulb greatly enlarged and having the form of a hollow cylinder. The substance is then placed directly inside of the bulb and the whole thermometer, while cooling, exposed to a constant temperature; this may readily be brought about by placing the thermometer inside of a vessel which is surrounded by melting ice. 21. Mechanical Equivalent of Heat. From mechanics we have the following statement: " The change in kinetic energy 18 HEAT that a body undergoes in passing over a given path is equal to the work done in traversing that path." The foregoing statement is very simple, and readily comprehended when considered in the purely mechanical sense. That is, it is merely stated that when- ever a given amount of kinetic energy, has been destroyed or developed in a system, an equivalent amount of work has been done by the system or, as the case may be, on the system. The statement, however, does not concern itself with the transforma- tion of one form of energy into another. The most general experience, common to all, is that of the destruction of energy, in the form of mechanical motion, by fric- tion or impact and the simultaneous evolution of heat. However, it was not until 1842 that a clearly formulated statement was made, by J. R. Mayer, to the effect, that when heat is converted into work, or vice versa, the ratio of the numbers representing the two quantities involved is constant. Unfortunately, the figures upon which Mayer based his calculations were in error, and consequently the value obtained for the mechanical equivalent was also in error. Shortly subsequent to Mayer's enunciation, Joule began his series of experiments to determine the mechanical equivalent of heat by direct measurement. Joule's method was essentially as follows: A vessel having fixed vanes, was filled with water, and a paddle-wheel was caused, by means of falling weights, to rotate in the water. The fixed vanes prevented the water from assuming a rotary motion. The heat developed manifested itself by a rise in temperature, and the work done was measured by the weights and distance fallen. A series of experiments was then made, using mercury instead of water. Another series of experiments was made by causing one iron plate to rotate with friction over another iron plate under water. It is, of course, understood that experiments like these are attended by great difficulties, and various precautions must be taken and corrections made which cannot here be enumerated. CALORIMETRY 19 Still, Joule obtained fairly consistent results; and, the figures he finally published were not greatly in error. The results were expressed in meter-kilograms of work per calorie; i.e., Joule found when using water that it required 423.9 m.kgs. of work to develop one calorie, 424.7 m.kgs. of work per calorie when using mercury, and for the experiment with iron, the number was found to be 425.2. It must be remembered that, if comparisons are to be made between results obtained by experiments, which have been performed in different localities, corrections will have to be made for variations in the value of g. Rowland varied Joule's method by using a motor drive instead of falling weights. The vessel was suspended and the torque required to prevent rotation measured. This enabled a much more rapid expenditure of energy, and a consequent rapid rise in temperature; thus making the correction due to cooling much smaller. Rowland's experiment covered the range from 5°C. to 36 °C. Since the thermal capacity of water varies for different temperatures, a variation was found for the mechanical equivalent of heat. Rowland found, at Baltimore, where g = 980.0, for the mechanical equivalent of heat, of the common calorie, 427.3 m.kgs. This may be taken as being substantially correct. Anthony modified Rowland's method by having a continuous flow of water through the calorimeter, the temperature of the inflowing water was constant, and its rate of flow was so regulated that the vessel was always at room temperature. The mass of water flowing for a given time was determined by weighing, the number of rotations made by the paddle was recorded on a speed counter, and the torque was noted, together with the tem- perature of the inflowing and outflowing water. By this method, gince the vessel is always at room temperature, no corrections for cooling are required; and furthermore, since the vessel suffers no change in temperature, its thermal capacity need not be known. The values obtained by this method were in practical concordance with those found by Rowland. 20 HEAT In the English system of units, the value for the mechanical equivalent of heat, usually employed, is 778 ft. -lbs. per B.T.U. Various experiments have been performed, by sending an elec- tric current through a conductor wound upon an insulating support and submerged in water. By noting the current, the applied e.m.f., and the time, during which the current has been flowing, the energy input is readily computed. Results obtained by this method agree almost precisely with those obtained by the methods previously described. A safe value to use, which if in error is only slightly so, and which is correct for all practical purposes, is 4.195 X10 7 ergs per common gram calorie. It will now readily be seen that, since we can measure electrical quantities with great precision, the best method for obtaining a definite quantity of heat is by sending a steady current for a given time through a given resistance; it being remembered that there are alloys whose resistances are practically independent of temperature. CHAPTER III PRODUCTION OF AND EFFECTS OF HEAT 22. Since, in a great many cases, those changes which evolve heat during their progress, require the application of heat to bring about a change in the reverse order, it is inadvisable to consider the production of heat and the effects of heat independently. Whenever a change of a chemical nature takes place there is either an absorption or a liberation of heat. In general, heat is absorbed when a compound is split up into the elements com- posing it; and heat is liberated when the elements recombine to form the compound. Those compounds which evolve heat, during their formation, are called exothermic compounds; and those rare compounds which absorb heat, during their formation, are called endothermic compounds. It will now be instructive to consider some particular substance and the various changes which take place with the continuous application of heat. Suppose that we are dealing with a definite mass of ice, under a given pressure, whose temperature is below that of its melting-point for the applied pressure. (The melting- point of ice changes slightly with change of pressure; i.e., the melting-point is lowered about 0.0075°C. for each increment in pressure equal to one atmosphere.) A definite amount of heat then, must be applied to raise the temperature of the ice to the melting-point. If heat be then further applied the temperature will no longer change; but, a change of physical state takes place together with a continuous absorption of heat until all the ice is melted. 21 22 HEAT 23. Heat of Fusion. The quantity of heat required to convert unit mass of a solid into the liquid state without change of tem- perature is called the heat of fusion of the substance. In the case of ice, the heat of fusion is approximately 80 gram calories per gram. If now, after all the ice has been converted into water, heat be continuously applied, the temperature will rise progress- ively until, if the liquid be under a pressure equal to one stand- ard atmosphere, the temperature of 100 °C. is reached. The temperature will then cease to rise, provided the pressure be maintained constant, and a change of physical state, namely, vaporization at constant temperature, takes place progressively with the continuous application of heat until all the water is evaporated. The heat of fusion of ice may be determined as follows: Let L, in, and ti, respectively, represent the. heat of fusion, the mass, and the initial temperature of the ice; and M, and 12, respectively, represent the water equivalent of calorimeter and contents, and the initial temperature of calorimeter. Let the ice now be sub- merged in the water in the calorimeter until it is all melted, and the calorimeter and total contents assume a common temper- ature 6. The total heat, then, consumed by the ice in having its temperature raised from ti to 0, being converted into water at this temperature, and in raising the temperature of the liquid from to 6, is racTi+ra(L+0); where c is the thermal capacity per unit mass of ice. But this quantity of heat must be equal to hence, From which M(t 2 -0); racTi+ra(L+6) =M(t 2 -8), L = -(T 2 -6)-(cTi+e). m PRODUCTION OF AND EFFECTS OF HEAT 23 24. Heat of Vaporization. The quantity of heat required to convert unit mass of a liquid into a vapor without change of temperature is called the heat of vaporization of the substance. In the case of water, at 100°C, the heat of vaporization is approx- imately 537 gram calories per gram. The heat of vaporization for most substances becomes less as the temperature rises. The heat of vaporization of water may be determined by the method of mixtures as follows : Let superheated steam, at a tem- perature ti, be passed into a calorimeter containing water at a temperature T2. This is continued until a convenient rise of temperature is obtained in the calorimeter, and a mass of steam m has been condensed. The quantity of heat, given up by the steam, is , rac(Ti— T)+mr-+m(T — 0); where t is the temperature at which the condensation takes place, r the heat of vaporization, c the thermal capacity per unit mass for superheated steam, and.0 the resulting temperature. But this quantity of heat must be equal to M(6-T 2 ); where M is the thermal capacity of the calorimeter and water initially contained in it. Hence, rar+ra{c(Ti-T) + (T-6)S=Af(e-T2); from which r=^(8-T 2 )-{c(T 1 -T)+(T-6)|. fit To obtain accurate results by means of calorimetric methods, as previously stated, it is always necessary to take certain pre- cautions and apply proper corrections. 25. Sublimation. Under certain conditions a substance may pass directly from the solid to the gaseous state without passing 24 HEAT through the liquid state. Such a change is called sublimation. Substances such as camphor and iodine, when gently heated, pass readily from the solid to the gaseous state without melting. Ice, under normal pressure, also sublimes at temperatures lower than the melting-point. 26. Superheating. If now, after all of the liquid has been converted into vapor, heat be further applied, the temperature will rise progressively with continued application of heat, and the vapor will become superheated. All of the foregoing may, instructively, be represented diagram- matically; bearing in mind that the thermal capacities per unit mass of water for the three states are: Solid 0.504, liquid 1, and gaseous 0.481. Fig. 2. Let, as in Fig. 2, the temperatures be taken as ordinates and quantities of heat as abscissas. Then, if we assume some arbitrary zero, such as 0, for the initial condition of the ice, we have for the application of the quantity of heat Oe, the increment of tempera- ture ea, to the point of fusion. The application of the quantity of heat ef then brings about the conversion from the solid to the liquid state at constant temperature. The further application of the quantity of heat fg brings about the elevation of temper- ature, from the temperature of fusion, to that of vaporization. The application of the quantity of heat gh brings about complete PKODUCTION OF AND EFFECTS OF HEAT 25 vaporziation at constant temperature. The further application of heat brings about superheating, as shown by the line di. In general, the physical history of substances with continued application of heat will be similar to the process just discussed; but, the ratios of the quantities involved will be entirely different for each substance. Assume now, that the process takes place in the reverse order, step by step; i.e., the steam is first cooled to the point of conden- sation, then condensation takes place at constant temperature, and so on, step by step, until the initial condition is reached. Then during each change, heat is liberated precisely equal in amount to that which was absorbed when the change was taking place in the opposite direction. 27. Reversible Processes. The changes just described and depicted in Fig. 2 are, however, not the only changes involved. Assume the applied pressure to be maintained constant throughout the entire change, the volume then of the substance will be chang- ing continually; and, in general, will be increasing with the tem- perature. When the volume is increasing, work is being done by the substance in overcoming the applied pressure. When the volume is decreasing, work is being done on the substance by the applied pressure. From the initial condition up to 4°C, this being the temperature corresponding to the maximum density of water, work is being done on the substance. For all tem- peratures higher than this, the substance expands continuously with the continued application of heat; and work is being done by the substance in overcoming the applied pressure. When the process takes place in the reverse order, then, wherever heat was absorbed and work done by the substance during the direct process, heat will be liberated by, and work will be done on the substance during the reverse process. And if these quantities be mutually equal, and no permanent changes have been made to take place in the surrounding bodies, by this cycle of operations, either process is said to be a reversible process. That is, when a system 26 HEAT undergoes a change, or a series of changes, the process is said to be reversible if, after it has taken place, a second process can be made to take place, in a manner, such that when the system is again in its initial condition, there remain, due to these various changes, no changes outside of the system. A little consideration will show that vaporization at constant pressure, and hence at constant temperature, provided we could have perfect insulation and no. friction, would be a reversible process. For, under the assumed conditions, the amount of work done by the vapor, during its formation, in overcoming the external pressure, is precisely equal to the amount of work done on the vapor during its condensation. Furthermore, the quantity of heat absorbed, during vaporization, from a reservoir of heat at constant temperature, is precisely equal in amount to the quantity of heat rejected, to the reservoir, during condensation. Hence, since all quantities involved balance each other, and no changes have been brought about in the surroundings, the process is reversible. The cooling and heating of a substance at constant pressure, together with its consequent changes in volume, can be made a reversible process only by the aid of a perfect regenerator. The following discussion will make this clear. Assume that a body at a temperature t», which is also the temperature of the first reservoir, cools to a temperature ti, by being put, successively, into contact with n reservoirs, perfectly insulated from each other, and each reservoir differing in temperature from the one adjacent to it by an amount equal to (x n — Ti)/(n— 1). The* body then, in cooling, gives up to each reservoir, excepting the first, a definite quantity of heat, and has done upon it, by the constant external pressure, a definite amount of work. Let the process now take place in the reverse order, i.e., the body at a temperature ti is put into contact with the reservoir at a temper- ature T2; a definite quantity of heat will be absorbed, which will be precisely equal to that rejected when put into contact with PRODUCTION OF AND EFFECTS OF HEAT 27 the reservoir at a temperature ti, after having been in contact with the reservoir at a temperature 12. Let this be continued until the temperature T n is again reached. Now, the work done by the external pressure on the body, while cooling from the tem- perature Tn to the temperature ti, is precisely equal in amount to the work done by the body, in overcoming the external pressure, while being heated from the temperature ti to the temperature Tn. This process, however, is not perfectly reversible; since the reservoir at the temperature i n has given up heat and received none, and the reservoir at the temperature ti has received heat and given up none. In the limit, however, as the fraction (: n - Ti)/(n— 1), approaches zero for its value, the process becomes perfectly reversible. But this implies a perfect regenerator; i.e., a series of reservoirs which are perfectly insulated from each other, and still have a continuous variation in temperature throughout the series; but this is practically impossible. Hence, it is obvious that, since in the case of vaporization, we must assume no radiation and no friction to make the process reversible, and in the case of cooling and heating of a body, we must assume a perfect regenerator and no friction and radiation to make the process reversible, the process as described and represented diagrammatically in Fig. 2 is reversible only in an ideal sense; i.e., an ideally reversible process. 28. Irreversible Processes. If a constant e.m.f. be applied to the terminals of a homogeneous conductor, a current will flow which is directly proportional to the applied e.m.f., and inversely to the resistance of the conductor. After a time the conductor will reach a constant temperature; i.e., the rate at which the energy is being converted into heat by the conductor, due to its resistance, will be equal to the rate at which heat, expressed in the same units, is given to the surroundings by the conductor. This, however, does not mean that there is thermal equilibrium. In this case thermal equilibrium can only be brought about by dis- connecting the applied e.m.f., and consequently, discontinuing the 28 HEAT dissipation of energy. This being done, the conductor will finally assume the temperature of the surroundings, and thermal equilib- rium will have been established. This process, viz, the con- version of energy, in the form of an electric current, into energy, in the form of heat, differs essentially in one particular feature from the process discussed in Art. 27. The former process, which we termed an ideally reversible process, can be made to take place, barring various losses, in the reverse order. The latter process, however, cannot be made to take place in the reverse order; i.e., it is absolutely impossible to cause a current to flow in a homogeneous conductor by applying heat to it. Such a process is called an irreversible process. Another example of an irreversible process is that of the con- version of energy, in the form of mechanical motion, into heat by friction. For it is impossible to restore a system of bodies to their initial positions by the application of a quantity of heat to the surfaces, equal to that which was evolved, due to friction, during their displacements. The same is true for the case of impact. When impact takes place between two or more bodies, a certain amount of kinetic energy is always converted into heat. But it is absolutely impossible, by the direct application of heat, to restore the kinetic energy which was destroyed during impact. Also, when thermal equilibrium is established by mixing sub- stances, initially at different temperatures, the process is abso- lutely irreversible. 29. Dissociation. Under Art. 26 we discussed the physical history of water with the continued application of heat up to the point of its superheated vapor. If heat be still further applied to the superheated vapor its temperature will continue to rise, up to some very high temperature, when complete dissociation takes place; i.e., the vapor splits up into its two constituent elements, viz, hydrogen and oxygen. Such a change is called a chemical change. If, while the pressure is maintained constant, heat be further applied, the gaseous mixture will rise in temper- PRODUCTION OF AND EFFECTS OF HEAT 29 ature and increase in volume progressively with continued appli- cation of heat. If now, the process be reversed, i.e., the mixture be cooled, the temperature and volume will diminish until the temperature of dissociation is reached. When this point is reached the two gases recombine to form steam, and precisely the same amount of heat is evolved as was absorbed to bring about decom- position. The quantities of heat, however, which are involved in chemical decomposition and recomposition are very large in comparision with those quantities involved during changes in temperature and changes of physical state. Indeed, our greatest source of supply of energy, in the form of heat, is that due to chemical combination; viz, the combination of carbon, in the form of coal, with oxygen. It is true that the amount of dissociation is a function of the temperature; i.e., even water at ordinary temperatures has a small percentage of dissociation. This, however, does not invalidate the statement that, the energy absorbed during dis- sociation is equal to that liberated upon recombination. To give an illustration of the quantities of heat evolved during chemical combination, we may take as examples the combination of hydrogen and oxygen to form steam, and the combination of carbon and oxygen to form carbon-dioxide. In the former case, 1 gram of hydrogen combining with oxygen to form steam (H2O), about 34,000 gram calories are evolved, or expressed in mechanical units, 1.43 X10 12 ergs. In the latter case, i.e., when 1 gram of carbon combines with oxygen *to form carbon-dioxide (CO2), about 8000 gram calories are evolved, or, expressed in mechanical units, 3.36 X10 11 ergs. 30. Electrolysis. Dissociation may also, in general, be brought about by electrolysis. That is, if an electric current be passed through a chemical compound, in the form of a solution, the com- pound will be split up into its constituents. Suppose that we are dealing with a solution of copper sulphate (CUSO4), and that the two electrodes are absolutely inert as regards 30 HEAT chemical reactions. Then if an electric current be passed through the solution, copper will be deposited on the negative electrode (cathode), and the radical SO4 will be liberated at the positive electrode (anode). The SO4 thus liberated will combine with hydrogen of the solvent to form sulphuric acid (H2SO4) ; and at the same time oxygen will be liberated. The equation, repre- senting this reaction, is CuS0 4 +H 2 = H 2 S04+Cu+0. If an electric current be passed through water, then the water will be split up into its two elements, hydrogen and oxygen; hydrogen being given off at the cathode and oxygen at the anode. In the case of dissociation by heat, a definite quantity of heat disappears for a given amount of dissociation; and an evolution of an equal quantity of heat upon recombination. But, since a given quantity of heat represents a definite amount of energy, it follows that dissociation involves storing of energy. Likewise, when dissociation is brought about by electrolysis a definite amount of energy is consumed for a given amount of dissociation, which must necessarily be equal to that consumed when the same amount of dissociation is brought about by the application of heat; since the energy stored is the same in amount for both cases. It is true that a solution becomes heated when conveying a current; but, this has nothing to do with the dissociation. The development of heat being merely due to the resistance of the solution, the same as when any other conductor is conveying a current. It is immaterial, so far as the foregoing argument is concerned, whether we consider the solution initially partly ionized and the current merely a icarrier of the ions, or that the current actually splits up the compound. 31. Faraday's Discoveries. Faraday showed experimentally that the amount of dissociation is directly proportional to the PKODUCTION OF AND EFFECTS OF HEAT 31 time and the intensity of the current; and furthermore, that the amount of chemical action is the same for all parts of the circuit. The latter part may perhaps be best illustrated as follows : Assume that there are two voltameters connected in series; the first containing a solution of copper-sulphate and the second water. Then upon passing a steady current through the circuit a definite amount of copper will be deposited on the cathode of the first voltameter, for a given interval of time, and a definite amount of hydrogen liberated at the cathode of the second voltameter, during the same interval of time; and these two quantities will be in the same ratio as their chemical combining numbers. That is, for every gram of hydrogen set free at the cathode of the second voltameter, 31.59 grams of copper will be deposited on the cathode of the first voltameter; where, if hydrogen be taken as unity, 31.59 is the chemical equivalent of copper in copper-sul- phate. During the same time that the 31.59 grams of copper are being deposited on the cathode of the first voltameter, 1 gram of hydrogen must be liberated to combine with the sulphion (SO4), set free to form H2SO4. 32. Counter Electromotive Force. Since the amount of dis- sociation, other things being equal, varies directly as the cur- rent, and the amount of energy stored during dissociation depends upon the compound dissociated, it follows that every compound offers a definite counter e.m.f. to dissociation. And any applied e.m.f. less than this cannot bring about dissociation. To make this clear, the e.m.f. necessary to dissociate water will here be cal- culated. The amount of hydrogen set free, per coulomb of elec- tricity conveyed, is 0.000010357 grams. The number 0.000010357 is called the electro chemical equivalent of hydrogen. Let, in the c.g.s. system of units, z be the electro chemical equivalent of hydrogen, I the current, and t the time, then the mass of hydrogen liberated, during the time t, is m = Izt . . (1) 32 HEAT Let h be the heat, expressed in mechanical units, required to dissociate 1 gram of hydrogen, then mh = IEt; (2) where E is the applied e.m.f. Substituting in equation (2), the value of m as given in equation (1), we find Izth = IEt; from which zh = E. .. . . (3) If now, in equation (3), we substitute for z and h their values, remembering that for 1 gram of hydrogen, combining with oxygen to form steam, h = l.43Xl0 12 ergs; and that z, in the c.g.s. system of units, equals 0.00010357 grams per unit quantity of electricity, we find or, E = 1.43 X10 12 X 0.00010357 c.g.s. units e.m.f., w 1.43X10 12 X0.00010357 , ._ ., E = -^ = 1.48 volts. 10 8 It must, however, be emphasized that, in general, the e.m.f. for most cells is a function of the temperature; and, therefore, to calculate the e.m.f. this function must be known. We are not prepared, here, to take up this matter. The counter e.m.f. is readily determined by experiment. 33. Junction of Dissimilar Metals. If the junction of two dissimilar metals be heated an e.m.f. is developed which is a func- tion of the temperature, and the metals which form the junction. To take a concrete example, assume a junction of antimony and bismuth. Such a junction, if heat be applied to it, develops an e.m.f. which tends to send a current from bismuth to antimony; and if a current be sent from bismuth to antimony, by the applica- PRODUCTION OF AND EFFECTS OF HEAT 33 tion of an external e.m.f., there will be a tendency to reduce the temperature of the junction. On the other hand, if a current be sent through the junction from' antimony to bismuth, heat will be developed. 34. As a resume, we may then state that the most general effects of heat are: To change the volumes of bodies; to bring about physical changes of state; in general, to promote chemical dissociation; to develop an e.m.f. at the junction of dissimilar metals. For the production of heat we may state the following examples : The production of heat by the mechanical compression of bodies which expand upon the application of heat; when the physical state of a body changes in the reverse order from the change when heat is applied; in general, by chemical combination; at the junc- tion of two dissimilar metals when a current is passed in a direc- tion opposite to that of the developed e.m.f. when heat is applied. Also, the production of heat when an electric current is conveyed by a homogeneous conductor; and, in general, when friction is being overcome, and mechanical motion destroyed by impact. 35. The Principle of Energy. The various relations discussed in this chapter may now be summed up and stated quantitatively in a very simple manner. This generalization, known as the principle of energy, or conservation of energy, is one of the most extensive of generalizations, and may be stated in substance as follows: If in any system, from which no energy escapes and into which no energy enters, account be taken of all forms of energy, then no matter what transformations take place within the system, the sum total is a constant quantity. So far as experience goes, the foregoing statement is consistent with all phenomena; and hence, in all subsequent demonstrations its truth will be assumed. CHAPTER IV EXPANSION OF SOLIDS AND LIQUIDS 36. Linear Expansion. It has been previously stated that, in general, bodies expand when the temperature is augmented. It is found by experiment that a body, such as a metal rod, increases in length by approximately equal amounts, between 0°C. and 100°C, for equal increments of temperature. But, even though there is an approximate proportionality between change in length and change in temperature for moderate ranges, such as just specified, it must not be inferred that this is generally true for large ranges or high temperatures. If a body of unit length expand in length by an amount a for unit increment of temperature, then a body of length I will expand in length by an amount la for an increment of 1 degree; and for an increment of t°, between 0°C. and 100°C, the body will expand in length, approximately, by an amount Ion. Hence, if the length of the body at 0° be denoted by Zo, and at t° by l T , we have £ z = £o+ZoaT; from which Z t = Zo(1 + «t) (1) The quantity a is called the coefficient of linear expansion, and may be defined as the ratio of the change in length, per unit change in temperature, to the length at zero. The quantity in the parenthesis, viz, 1 + or, is called the factor of linear expansion. 34 EXPANSION OF SOLIDS AND LIQUIDS 35 37. Voluminal Expansion. Homogeneous isotropic bodies will change in amount by like fractional parts of their original dimensions in all directions when the temperature changes. Assume that we are dealing with a rectangular parallelopiped whose three edges at zero temperature are ao, bo, and Co, its volume then, at zero, is vo = aob co (2) If the temperature, now, be changed to t°, the three edges become: ao(l + 1.678X10 -5 Steel (annealed) 1.095XKT 5 Zinc 2.918X10" 5 Brass . .0.187X10" 4 Glass 0.083X10 -4 Invar (steel containing 36% nickel) 0.087 X10~ 5 It is only necessary to substitute the values of a, as given in the foregoing table, in equation (5) and it becomes evident that equation (6) is approximately true. Hence, we may say that, the coefficient of voluminal expansion is practically equal to three times the coefficient of linear expansion. 38. Non-Isotropic Bodies. There are certain bodies which have different physical properties in different directions. Such bodies are termed non-isotropic. A notable example is that of Iceland spar, in which it is found that the coefficient of linear expansion in one direction is 2.63 X10~ 5 ; whereas, in a direction normal to this it is found to be only 0.544 X 10 ~ 6 . It is also interesting to note that Iceland spar manifests different optical properties in different directions. 39. Expansion of Liquids. Liquids, in general, change more rapidly in volume than do solids for equal changes in temperature. However, since, in the case of liquids, the term linear expansion is meaningless, we deal only with voluminal expansion. The determination of the coefficient of linear expansion is quite simple; hence, the description of the methods employed in its determination was omitted. The determination of the coef- ficient of voluminal expansion of a liquid is attended by various difficulties; and, since the discussion of the principles involved will prove instructive, a few methods will be described. EXPANSION OF SOLIDS AND LIQUIDS 37 Assume that we have a glass flask terminating in a tube of capillary bore, and that the mass of the flask when empty is known. The flask is then filled, to a definite mark on the capillary tube, with the liquid at a temperature ti, whose coefficient of voluminal expansion is sought. The mass of the vessel and contents is now determined. The difference between this mass and the mass of the flask gives us the mass of the liquid at the temperature ti. If Di is the density of the liquid at the temperature ti, and V the volume of the liquid in the flask at the same temperature, then D^f; (7) where M i is the mass of the liquid in the flask at the temperature ti. Let, now, M 2 be the mass of the liquid in the flask, when its temperature is T2, the flask being filled to precisely the same mark on the tube as when the temperature was ti. The volume of the flask will now be r = F{l + WT 2 -Ti)|; where @ is the coefficient of voluminal expansion of the glass of which the flask is composed. @ may be computed from the coef- ficient of linear expansion, or determined directly by experiment, as will be shown subsequently. The density of the liquid at T2, will now be M 2= M 2 m U2 V Fil + P(T2-Tl)} K) Dividing equation (7) by equation (8), we find g-gu+tt"-o)»- •••••• w 38 HEAT If the coefficient of voluminal expansion of the liquid, between 0° and the other two temperatures under consideration, is approx- imately constant, we have ft-.^. ... (10) where Do is the density of the liquid at 0° and a the coefficient of voluminal expansion. Similarly, we find D 2 =j^; (11) l+aT2 from which, by dividing equation (10) by equation (11), we find D\ 1 + 0CT2 Z>2 1 + GCTl (12) Finally, by equating the right-hand members of equations (12) and (9), we find 1 + «T2 Mi 1 + «T1 M 2 1 + &(t 2 -ti)}. .... (13) In equation (13), all quantities excepting a are known; hence, its value is determinate. Another method is as follows: A solid, whose coefficient of voluminal expansion is accurately known, and which does not react chemically with the liquid whose coefficient of voluminal expansion is sought, is weighed in the liquid, first at temperature ti, and second at temperature T2. The weight of the solid being known, its loss of weight for the two temperatures is known; and, since the volume of the solid for any temperature may, from its known coefficient of voluminal expansion, be computed, the densities of the liquid for the two temperatures ti and T2 are readily found, and from these, as previously shown, the coefficient of voluminal expansion. EXPANSION OF SOLIDS AND LIQUIDS 39 I" w tt 40. Direct Measurement of Coefficient of Voluminal Expan- sion. The method about to be described, and the one by which Regnault determined the coefficient of expansion of mercury, depends upon the principle that when communicating columns of liquids are in equilibrium their heights are inversely as their densities. In Fig. 3, AB and CD are two vertical iron tubes, cross connected by the horizontal tube BD. The two horizontal tubes, AE and CF, terminate in the vertical tubes EG and FI, which are connected by the inverted glass U-tube GJL The tube containing the stop cock s, is connected to the receiver of a compression pump; and after the apparatus has been filled with mercury, air is forced in from the compressor until the mercury in the tubes EG and FI is at a convenient height. The stop cock s is then closed. If, now, the temperature, and consequently the density, of the mercury in the tubes AB and CD be the same, the columns in EG and FI will be at the same level. On the other hand, if the temperatures of the two columns AB and CD be not the same, then the columns in EG and FI will not be at the same level. For, since there is free communication between B and D, the pressure must be the same at these two points. Further- more, the pressure of the air inside the U-tube being everywhere the same, and, since this pressure plus the pressure, due to the column in FI, balances the pressure due to the column DC, and the same pressure plus the pressure, due to the column in EG, balances that of the column BA, it follows that if the pressures, due to the two columns AB and CD, are not the same, the two columns in EG and FI cannot be at the same level. i__. E Fig. 3. 40 HEAT Suppose, now, that the column in EG, FI, and CD be main- tained at 0°C. and the column AB at t°C, then the column in FI must exceed the column in EG, in height, by an amount /i,such that (H-h)(l + aT)=H; (14) where a is the coefficient of volummal expansion of mercury. From equation (14) we find (15) i(H-h) 41. If, now, it be desired to determine the coefficient of expan- sion of some other liquid, it becomes only necessary to take a flask and determine its mass when empty, then the mass of flask and contents when filled to a given mark at various temperatures, first with the liquid whose coefficient is sought, and then when filled with mercury to the same mark for the various temperatures. From the mass of mercury required to fill the flask at various temperatures, and from the known density of mercury for these temperatures, the volume of the flask is readily computed. From the known volume of the flask and the mass of liquid required to fill it at the various temperatures, the densities corresponding to those temperatures are found. CHAPTER V 'FUNDAMENTAL. EQUATIONS OF GASES 42. Isothermal Equation. Experiment shows that, between certain limits, for the so-called permanent gases, such as hydrogen, oxygen, nitrogen, etc., the product of pressure and volume is a constant, for constant temperature. Expressed symbolically pv = k; ........ (1) where p is the applied pressure per unit area, * v the corresponding volume of the gas, and k some constant whose value depends upon the units chosen. Equation (1) is usually designated as Boyle's Law. But, in any case, the equation which expresses the relation between the pressure and volume of a gas, at constant temperature, is its isothermal equation. 43. Gay-Lussac's Law. As a further result of experiment it is found that all gases which obey Boyle's Law have the same constant temperature coefficient; i.e.,' all gases, under constant pressure, expand by the same fractional part of their volumes at zero temperature, for equal increments of temperature. This is known as Gay-Lussac's Law. If we denote by a the increment in volume, for a unit volume of a gas, under a constant pressure po, when its temperature changes from zero to unity, then the volume of a gas at t degrees is v x = v +voon; (2) *In all subsequent equations, unless otherwise stated, p will be used to denote pressure per unit area. 41 42 HEAT where vo is the volume of the gas at zero temperature, and v x the volume,, under the same pressure, at t degrees. Writing equation (2) in another form, we have z; t =^o(1 + gct) . (3) If, after the temperature t, and the corresponding volume v z , under the constant pressure po, has been attained, the pressure is augmented, the temperature being maintained constant, until the gas assumes its original volume vo, we must, from Boyle's law, have the following relation: p x = p (l + on) (4) As a matter of fact the experiment is most conveniently per- formed, by varying the pressure, so as to maintain the volume constant, as the temperature is varied. Equation (4) may there- fore be considered as being the expression of experimental results. Multiplying both sides of equation (4), by vo, we obtain p T vo = p vo(l-\-(xx) (5) If, now, while the temperature is maintained constant, the pressure be varied, the volume will vary according to Boyle's Law; i.e., pv = p x vo = p vo(l+cn); (6) where p is any pressure and v the corresponding volume at the temperature t. Since Gay-Lussac' s Law also holds for temperatures below zero, a, of course, being a decrement, we have pv = poV (l — oct) (7) Equation (7) reduces to zero when o(l + (k) ', hence p t «o(l+(fc)=;po«D(l+aT). Fig. 4. Now, p T = po(l + 2''t); where a' is the apparent coefficient of expansion of the gas. Making this substitution, for p T , and eliminating, we find (l+«'T)(l+PT) = l+arc; from which T=- «- («'+£) «'(* 46 HEAT When extreme precision is sought, corrections must also be made for changes in volume of the bulb, due to changes in pressure. Gas thermometers are used only for purposes of standardization. 46. Expansion without Doing External Work. Experiment has shown that there is no energy consumed in the simple expan- sion of a gas; i.e., when a gas expands in such a manner that no external pressure is overcome, and hence, no external work is being done, no energy is consumed by it. This experiment was first performed by Gay-Lussac (who apparently did not realize its full significance) in the following manner: Two vessels, one of which was exhausted and the other filled with air under a pres- sure, were placed in a calorimeter and surrounded by water. When the stop cock in the tube, connecting the two vessels, was opened, the air from the one vessel expanded into the other, bringing about an equalization of pressures. During this process the gas increased in volume without doing any work external to the system. That is, work was done by the gas under a high pressure, in the one vessel, in expanding against the increasing pressure of the gas in the other vessel. But, since the temperature of the system after the completion of the process was found to be precisely equal to that of the system before the process ^began, it follows that the total energy of the system is unchanged by the expansion. If energy were required to bring about an increase in volume, the temperature of the system at the end of the process would necessarily be less than at the beginning of the process. Or, to express the result in another way, since the temperature of the system is unchanged by the change in volume, the work done by the gas in the vessel, initially under the higher pressure, is pre- cisely equal in amount to the work done on the gas in the other vessel. The results just deduced may be embodied in a simple statement; i.e., the intrinsic energy of a perfect gas is a function only of the temperature. Or, to put it still in another way, the heat of disgregation of a perfect gas, when no external work is being done, is zero. FUNDAMENTAL EQUATIONS OF GASES 47 The foregoing experiment was subsequently repeated, in the most careful manner, by Joule and found to be approximately, though not rigidly, true. 47. Thermal Capacities of Gases. If a gas, under a pressure p, expand by an amount in volume dv, the external work done will be numerically equal to pdv; where p is the pressure per unit area. This is shown as follows: From the definition of work, we have dw = Fds; (10) where F is the applied force and ds the displacement. But, since F, the applied force, is numerically equal to the product of p, the pressure per unit area, and the area A, we have dw = pAds (11) But Ads = dv; hence, by substituting in equation (11), we have dw = pdv (12) Since, according to the experiments of Gay-Lussac and Joule, when the temperature of a gas is augmented, that part of the heat which is required to elevate the temperature of the gas is practically the same for equal ranges of temperature, no matter whether the volume be varied or maintained constant, it follows that the quan- tity of heat required to bring about a given elevation of temper- ature, when the pressure is maintained constant, is greater than the quantity of heat required to bring about an equal elevation of temperature, when the volume is maintained constant. For, in the former case, heat is required, not only to elevate the temper- ature of the gas, but also to do the external work due to the expan- sion of the gas; whereas, in the latter case, the heat consumed is only that required to elevate the temperature of the gas. 48 HEAT The ratio of change in heat to the corresponding change in temperature, in a unit mass of gas, when the volume is maintained constant, is a measure of the thermal capacity per unit mass at constant volume, and is denoted by Cv Hence, for a unit mass of gas, we have In a like manner ST 7 (13) = C P ; (14) where C p represents, for a gas, the thermal capacity per unit mass, under a constant pressure. The ratio C p /C v = n is practically a constant for all the per- manent gases; and, in the case of air, is approximately 1.405. The quantity (Cp—C v ), is evidently a measure of the external work done by the unit mass of gas in expanding against the pressure p, while it is being heated through a range of 1 degree. Assume that we have a given mass of gas m, whose volume is v at 0°C, confined in a cylinder by a piston of area A, under a pressure p, per unit area. If the piston be perfectly free to move, and heat be applied bringing about an elevation of temperature t, the pressure being maintained constant during the process, then the volume will be increased by an amount von. The dis- tance through which the piston moves during the expansion is von /A; and the external work done, which is numerically equal to the product of force and displacement, is W = pA-~- = pvai: (15) The heat consumed in doing the external work, expressed in mechanical units, is H = Jm(C p -C v )'z; ........ (16) FUNDAMENTAL EQUATIONS OF GASES 49 where J is the mechanical equivalent of heat. But, under the assumed conditions, H and W are numerically equal; hence, from equations (15) and (16), we have Jm(C p — C P )T = pyax; from which j- Wl (m Equation (17) * enables us to compute the mechanical equiv- alent of heat from the known constants of a gas. . The constants of dry air are as follows: C p = 0.2375, C„ = 0.1690, a = 0.003665, and the mass of 1 c.c. of air at 0°C, under a pressure of 1.01325 X10 6 dynes per sq.cm., is 0.001293 grams. Substituting these values in equation (17), and assuming 1 c.c. for the initial volume, we find T 1.0132 X10 6 X 0.003665 . 1ft o vin7 , • J = 0.001293(0.2375-0.169 0) =4 - 193X1 ° ergS per gram Calone ' The value of J thus obtained differs by a small percentage from that obtained by the direct conversion of mechanical work into heat. We have here then a complete verification of the numer- ical relation between heat and work. That is, in the one case, mechanical work is directly converted into energy, in the form of heat, and the numerical ratio of the two quantities involved is determined. In the other case, heat is converted into work, and the numerical relation between the two quantities is again determined; and the difference, between the two values so deter- mined, is well within the limits of observational error. * It was by this method that J. R. Mayer first computed the mechanical equivalent of heat. Various writers have attempted to take some of the credit from Mayer, by asserting that it was not then known that no energy- is required for the simple expansion of a gas. But Gay-Lussac had per- formed this experiment, and anyone reading Mayer's original papers will see that he was aware of this and interpreted the experiment properly. 50 HEAT 48. Adiabatic Equation. If a gas is compressed, work is being done on it and heat is necessarily developed; and since the pres- sure of a gas, other things being equal, rises with the temperature, it follows that, unless the heat developed by the compression is abstracted from the gas as rapidly as it is developed, the pressure must rise more rapidly, with respect to the amount of compres- sion, than it would during isothermal compression. There are certain processes where compression and dilatation take place so rapidly that there are practically no exchanges of heat between the various parts of the system. Changes, during which no heat enters or escapes, are called adiabatic changes. A good example of adiabatic changes are the compressions and rarefactions which take place in a medium when a sound wave exists in it ; the time of compression, or rarefaction, being so small that practically no heat is transferred from particle to particle. In general, if we are dealing with a unit mass of a perfect gas, we may write dQ = C v dT+pdv; (18) where all quantities, of course, are expressed in the same units. dQ, expressed in mechanical units, is the quantity of heat absorbed by the gas, or else abstracted from it, C v dT is the quantity of heat involved in bringing about the change of temperature dT, and pdv represents work done either by the gas or on the gas. As a matter of illustration, assume work is being done on the gas in a manner such that its temperature rises and that there is also heat given to the surroundings. If we consider work done by the gas and heat absorbed by the gas positive, then, in equation (18), if applied to this case, both dQ and pdv become negative; on the other hand, C v dT remains positive. If, now, during the change in volume, no heat enters or escapes, the process will be adiabatic; and equation (18) becomes C v dT+pdv = (19) FUNDAMENTAL EQUATIONS OF GASES 51 In equation (19), if dv is positive, dT must be negative; i.e., if external work is done by the gas it is done at the expense of the intrinsic energy of the gas, and the temperature must fall. Like- wise, if work is done on the gas, since according to our assumption no heat escapes, its temperature must rise. To solve equation (19) we will substitute for dT its value, as found from equation (9), Art. 43; i.e., by differentiating pv = RT, we find dT = pd»+vdp (20) Now, R can be expressed in terms of the two thermal capacities. To find this relation, assume that we are dealing with a unit mass of the gas and allow it to expand under a constant pressure p, while its temperature is increased by unity. If v\ is the initial and V2 the final volume, the external work done is p(V2 — Vi)=Cp — Cv. It being understood that all quantities are measured in mechan- ical units. But, from equation (9), it follows directly, that p(v 2 -v 1 )=R(T 2 -T 1 ); and, since by the conditions we have T 2 — T\ equal to unity, it follows that it = O p Op. Substituting this value of R in equation (20), and the value of dT so obtained in equation (19), we find ^pdv+vdp Cv C -C +P dv = > from which /Cp\ dv dp C> v + J = °- ....... TO 52 HEAT But, as previously explained, C v /C v is practically a constant. Representing this constant by n and substituting in equation (21), we have do .dp n — h— = 0; v p from which, by integration, \ogv n -\-\ogp = ki; where k\ is the constant of integration. Or, expressed in another form, log po n = ki; from which po n = k; (22) where k is a constant depending upon the units chosen. Equation (22) gives the relation of pressure and volume for a gas, during adiabatic changes, and is known as the adidbatic equation. General Equations of Gases 49. The change of heat involved when a gas suffers a change is a function of the temperature, pressure, and volume; i.e., Q=f(T,p,v). But, since any two of these quantities may vary independently of the third, we may write Q-f(T,p), Q=f'(T,v), and Q=f"(p,v). By partial differentiation of these functions , we obtain and dQ-C . '■->■(%) (fWS>° FUNDAMENTAL EQUATIONS OF GASES 55 Substituting the value of dp as given in equation (33), in equation (29), we have CpdT+m [ (£)*+(!£) W\ =C„dT+ldv; from which hence, by equating like coefficients, we find C p -C,= -m(||) o (34) By equating the right-hand members of equations (26) and (28) , we find C p dT+mdp=jdp+odv (35) From the fundamental statement T=f(v, p), we have dT=( and substituting this value of dT, in equation (35), we obtain -) dv+(^)dp; | /8T\ Cp J \to) p dv+ X^) v dv I +mdp =J d V+odv, hence, by equating like coefficients, we find = ^(£)/ ■•-.... 06) From equations (27) and (28), we have CvdT+ldv =jdp + odv ; 56 HEAT and, by substituting for dT its value, we find from which *-o. <*> We then have the following values : : (sr), l\ ^m) Co L>V) l[h £)r m > »Mg|) =-(.G,-C.), and 1 if > - ^ - From the characteristic equation pv = RT, we find hence Again hence 'iv\ R lJLrr _m l -pK^P ^vj» • • • lv\ RT IvJt p 2 '' T m= (C p — C V )o o e p o . . (38) (39) FUNDAMENTAL EQUATIONS OF GASES 57 Also (— \ = -• hence j=^fi, (40) Finally \lv] v R' and o = \Cv (41) Substituting in equations (26), (27), and (28) the values of I, m, j, and o as just determined, we find dQ = C p dT--(C P -C v )dp, (42) dQ = C v dT+^(C p -C v )dv, (43) and dQ = ^C v dp+^C p dv (44) These equations, viz, (42), (43), and (44), may be put into different forms; since, from the characteristic equation pv = RT, we have T/p = v/R, and p/R = T/v. From the assumption, then, that in the fundamental statement Q=f(T,p,v), 58 HEAT any two of the variables may vary independently, while the third is maintained constant, we have obtained three distinct equa- tions, viz, equations (42), (43), and (44). It is of further interest to note that if in any one of these three equations the right-hand member is equated to zero, the adiabatic equation is obtained. Assuming that the change in equation (44) is adiabatic, then dQ = 0, and we have C v vdp = —C p pdv; from which and from which C p dv _ _dp fdv (dp n\ogv= — logp+fci. Or, expressed in another form log pv n = k\ ; and, finally pv n = h (45) This may also be obtained from the other equations, as may be readily shown. From pv = RT, we have dT= pdv±vd2 R Substituting this value of dT in equation (43) and equating to zero, we find C v (pdv-\-vdp) = —p(C P — Cv)dv; from which C v vdp= —C v pdv; FUNDAMENTAL EQUATIONS OF GASES 59 which is identical with the result obtained from equation (44) under the same assumption. Again, substituting in equation (42), for dT its value and equating to zero, we find from which c vMp^ {Cp _ Ct)dp . C p pdv = —Cvvdp; which is again identical with that previously obtained under the same assumption. If it be desired to find the temperature of a gas, corresponding to a given pressure and volume, during an adiabatic change, in terms of the initial temperature and pressure and the given pres- sure, or in terms of the initial temperature and volume and the given volume, we proceed as follows: Let T\, pi, and v\ be, re- spectively, the initial temperature, pressure, and volume, p and v, respectively, the pressure and volume, for which the corre- sponding temperature, T, is sought. Then, since the two points are on the same adiabatic, we have pivi n = pv n ; (46) and from the characteristic equation, we have piv^RTu (47) and pv = RT (48) Substituting in equation (46) , the values of v\ and v, as found from equations (47) and (48), we find 60 HEAT In a similar manner, by substituting in equation (46), the values of pi and p, as found from equations (47) and (48), we obtain T=Ti (j) n ~- (50) Vapors 50. Vaporization. The gaseous states of bodies, which under ordinary conditions of temperature and pressure are either liquids or solids, are called vapors; and the process by which the vapor is formed is called vaporization. In general, vaporization takes place in two distinct ways. In the one process, called evaporation, vapor is continually formed at the exposed surfaces of liquids; and in the other process, called ebullition, bubbles of vapor are formed in the body of the liquid or at the heated surfaces. 51. Evaporation. If a liquid be enclosed in a space, only part of which is occupied by the liquid, then vapor immediately forms and occupies the space above the liquid. This continues until the vapor has reached a certain density which depends upon the temperature, and is greater as the temperature is higher, but is always the same for the same temperature. In other words, for any given temperature there is a maximum density and hence, a maximum pressure, which the vapor is capable of exerting. When this state is reached the vapor is said to be saturated. That is, for the given temperature the space contains the maximum possible amount of vapor. If, after this state has been reached, the temperature be maintained constant and an attempt be made to increase the pressure by the application of an external force the result will be, not an increment in pressure, but a diminution in vdlume, at constant pressure, and a corre- sponding amount of condensation. In other words, the pressure for a saturated vapor at constant temperature is constant. Or, FUNDAMENTAL EQUATIONS OF GASES 61 to put it in still another way, the temperature of a saturated vapor is uniquely defined by its pressure. 52. Addition of Vapor Pressures. The rate of evaporation depends, of course, upon the rate at which heat is being supplied; but, as has just been stated, the final pressure reached depends merely upon the temperature. Furthermore, evaporation takes place more rapidly in a vacuum than in a space occupied by the vapor of some other substance ; however, the final pressure reached by the vapor will be almost, though not quite, as high, when the space is partially occupied by some other gas or vapor, than it would be were the space originally a vacuum, provided always, that the temperature be the same and that there be no chemical action between the vapors. This statement was first made by Dalton, viz, when evaporation takes place in a space filled by another gas, which has no action on the vapor, the final pressure reached by the mixture is equal to the sum of the pressures of its constituents. Careful experiment shows Dalton's statement to be approximately, though not rigidly true. 53. Ebullition. As has been previously stated, when heat is applied to a liquid, the temperature rises progressively with continued application of heat until a certain point, which depends upon the pressure, is reached, when the temperature remains constant. This is the boiling-point for the given pressure; and is that temperature for which the pressure of the vapor is equal to the superimposed pressure. Since the pressure at any point in the liquid, is equal to the pressure at the surface plus the pres- sure due to the liquid, from the surface to the point under con- sideration, it follows that, the temperature varies for different depths below the surface of the liquid. Hence, the temperature of the boiling liquid is not a constant throughout; but increases slightly with the depth. When equilibrium has been attained, i.e., the temperature becomes constant, then all the energy that is supplied, in the form of heat, is consumed in converting the liquid into a vapor. This 62 HEAT energy consists of two parts, viz, one part being that energy which is required to overcome the inherent forces, that is, to separate the particles so as to form vapor, and the other part to overcome the external pressure during the augmentation of volume. The former is called the heat of disgregation and the latter the heat of expansion. Pressure, however, is not the only factor that fixes the boiling- point of a liquid. As examples, the following may be cited: The nature of the material of the containing vessel has some influ- ence. If the liquid be first carefully freed from the imprisoned air, the temperature may be raised considerably above the tem- perature at which ebullition ordinarily takes place. Impurities in the liquid influence the boiling-point. And finally, salts dis- solved in ajiquid always raise the boiling-point. As an example, the boiling-point of a saturated solution of water with common salt is about 109°C. But the temperature of the saturated vapor of a liquid is always the same for the same pressure, no matter what the temperature of the liquid. It is for this reason that the temperature of steam, rather than that of water, under a pressure of one standard atmosphere has been chosen as the boiling-point. 54. Critical Temperature. When a liquid is heated in a closed vessel the vapor accumulates above the liquid and augments the pressure. Up to a certain point, differing for different liquids, there is a sharp definition between the liquid and vapor; but, for every liquid there is reached, finally, a temperature when this definition ceases, and the liquid disappears and is completely converted into vapor, even though the volume occupied by the vapor is but little greater than that occupied by the liquid. The temperature at which this takes place is called the critical tem- perature for the substance. And, it appears that, for temperatures, higher than this, no matter what the applied pressure, the substance can exist only in the gaseous state. The following table gives a few substances together with their approximate critical temperatures, and the corresponding pressures: FUNDAMENTAL ^EQUATIONS OF GASES 63 Substance. Temperature in Degrees, C. Pressure in Atmospheres. Carbon-dioxide Sulphur-dioxide 31 156 194 365 -118 -146 -234 77 79 36 195 50 33 20 Ether Water Oxvsen Nitrogen Hydrogen 55. It is interesting to note that there apparently is a relation between heat of disgregation of a substance and its critical tem- perature. Let a be the specific volume of the liquid; i.e., the volume occupied by unit mass of the liquid, and s the specific volume of the dry saturated vapor, then the increment in volume, when a unit mass of a liquid is converted into vapor, is /j. = s- K is the ratio of the quantity of heat, passing any section, to the product of the area of the section, the time, and the temperature slope at that section. This ratio is called the coefficient of con- ductivity of the substance; and, of course, differs for different substances. From equation (6) it follows that the coefficient of conductivity K, of a substance, is numerically equal to the quantity of heat which flows across a section of unit area, in unit time, when the tempera- ture slope is unity. In the c.g.s. system, and using the centi- grade scale, the coefficient of conductivity of a substance is numer- 86 HEAT ically equal to the quantity of heat, measured in gram calories, which flows across a section 1 sq.cm. in area, in 1 second, when the temperature slope at the section is 1°C. per centimeter. 78. Flow of Heat along a Bar. If a bar be maintained at a constant temperature at one end, and the remainder of the bar be exposed to a space of lower temperature, which is also main- tained constant, the fall of temperature along the bar will not be the same as that of the prism previously discussed. For, since the bar is at a higher temperature than the enclosure, it will con- tinually give up heat to the surroundings by radiation and con- vection. Eventually, heat will be supplied to every portion of the bar, by conduction, as rapidly as it is dissipated by radiation and convection. That is, the temperatures along the bar will finally assume steady values. But, as previously stated, the tem- perature slope along the bar will not be constant. For since, when a steady condition has been assumed by the bar, the quantity of heat which passes any section, for a given interval of time, is neces- sarily equal to the quantity of heat which is dissipated from the bar beyond that section, for the same interval of time, it follows that the quantity of heat which passes a section of the bar becomes less as the distance from the end, which is maintained at a con- stant temperature by the application of heat, increases. Hence since, other things being equal, the quantity of heat, which passes any section of the bar, is directly proportional to the temperature slope at that section, it follows that the temperature slope decreases with increase of distance from the heated end. 79. Determination of Coefficient of Conductivity. Since, it is impossible to realize in practice those ideal conditions which were assumed in the discussion of the flow of heat between two parallel walls having areas of indefinite extent, recourse must be had to other methods. A bar maintained at a constant tem- perature at one end, and having the remainder exposed to a space of constant temperature, furnishes a convenient means for deter- mining the coefficient of conductivity. PROPAGATION OF HEAT 87 To do this, we proceed as follows : After the bar has assumed a constant condition throughout, its temperature is ascertained at a number of definite points along it; this is most conveniently done by means of a thermo couple, which is calibrated by com- paring with a standard thermometer. The results are then plotted, differences of temperature between the bar and its enclosure as ordinates and distances along the bar as abscissas. The curve passed through the points so found, shows the difference of tem- perature between the bar and the enclosure, throughout the length of the bar; and the slope of the tangent, drawn to any point of this curve is numerically equal to the temperature slope at that section. This gives us r for equation (6) ; and A of this equation, viz, the area of the section, is determined directly from the dimen- sions of the bar. It now remains to determine Q/t, i.e., the quan- tity of heat which passes a section per unit time. To do this, a second experiment is necessary. The bar is now heated until its temperature is uniform throughout and slightly higher than the highest temperature on the curve for the rate of fall of temperature along the bar. The bar is then placed in the enclosure, under precisely the same conditions as obtained when the curve for the rate of fall of temperature was determined, and its temperature noted at definite intervals of time. From the data so obtained, a second curve is plotted, differences of temperature between the bar and the enclosure as ordinates and times as abscissas. The curve so obtained is the curve of cooling; and the slope of the tangent, drawn to any point of this curve, is numerically equal to the rate of change of temperature of the bar, with respect to time, for the particular difference of temperature between the bar and its enclosure at that time. Let it now be desired to determine the quantity of heat, which passes in a unit of time, some particular section of the bar, repre- sented by the point a, on the curve A, of Fig. 8. Curve A is the curve representing the temperatures along the bar, and curve B, the curve of cooling. If now, that part of the bar to the right 88 HEAT of a be divided into elements, such as ab, so short, that without appreciable error, the fall of temperature along the element may be considered constant, then the temperature of the element may be taken as the mean of the two temperatures at the points a and b. If this mean temperature be then projected across to the curve of cooling B, and at the point c, so found, a tangent be drawn, then the slope of this tangent is numerically equal to the rate of change of temperature with respect to time, for a difference of temperature equal in amount to the difference between that of the mean temperature of the element ab and its enclosure. If Fig. 8. now, we take the product of the thermal capacity of the element ab, and the rate of change of temperature just found, we obtain q/t, the quantity of heat lost, per unit of time, by the element ab at the instant when its temperature is defined by the point c. But, since the temperatures of the various parts of the element ab are constant, the mean temperature is a constant, and differs con- tinually from the temperature of the enclosure by an amount pre- cisely equal to the difference of temperature as found from the curve of cooling for the instant when the temperature is repre- sented by the point c. Therefore, the element ab is continuously losing heat, at a constant rate, equal in amount to the quantity just found from the curve of cooling for the temperature represented PROPAGATION OF HEAT 89 by the point c. In a similar manner, the quantities of heat, escap- ing per unit of time, from the various elements to the right of the element ab, are found. Taking the sum of the quantities of heat so found, for all the elements to the right of ab, the quantity of heat Q/t, of equation (6), which passes the section a, in a unit of time, is found. From which, by substitution, K is found. Experiment shows, that, in general, the conductivity of solids decreases slightly with increase of temperature. 80. Conductivity in Non-isotropic Substances. If there be a source of heat at a point in an isotropic substance, i.e., a substance having like physical properties in all directions, then other things being equal, heat will be propagated with equal speeds in all directions; and the temperatures at equal distances in all directions from the source of heat, at any instant, will be found the same. Or, in other words, the source of heat will be the center of spherical isothermal surfaces. On the other hand, substances which are non-isotropic do not conduct heat with equal speeds in all directions. As an example, the conductivity of Ice- land spar is greatest in the direction of the axis of symmetry, and equal in all] directions at right angles to this axis. It will be remembered that the coefficient of expansion for Iceland spar is also greatest in the direction of the axis of symmetry, and equal in all directions at right angles to this axis. 81. Non-homogeneous Solids. Tyndall found, by experiment- ing with cubes of wood, that the speed of propagation of heat is greatest, in the direction of the fibers; i.e., parallel to the length of the tree, and least, parallel to the annual layers. And in a direction normal to both the fibers and the annual layers, i.e., radial to a section of a tree, a value was found for the conductivity slightly greater than that parallel to the annual layers; but, con- siderably less than that parallel to the fibers. Wood, however, on the whole is a very poor conductor in comparison with metals. It will be of interest here to note that the speed of propagation of sound through wood is different for the three directions; i.e., 90 HEAT the speed of propagation is greatest, parallel to the fibers, least, parallel to the annual layers, and radial to a section of the tree, it is somewhat greater than it is parallel to the annual layers, but considerably less than that parallel to the fibers. 82. Conductivity of Liquids. The determination of the co- efficient of conductivity of a liquid is attended by difficulties which are not experienced when dealing with solids. For, in the case of liquids, if we wish to determine the true conductivity, convection currents must be avoided. It is, therefore, necessary to heat the column of liquid from the top. It is impossible here, to consider all the necessary precautions which must be taken to insure accurate results. The principle involved, however, is precisely the same as for solids. That is, to determine accurately the temperature slope along the column and the quantity of heat passing a given section for a definite interval of time. 83. Conductivity of Gases. The determination of the coef- ficient of conductivity of a gas is still more difficult than is the determination of the coefficient of conductivity of a liquid. For, in a case of a gas, not only must convection currents be eliminated, but radiation must also be taken into account. This makes it extremely difficult to obtain even fairly accurate results. As a matter of interest, the following coefficients of conductivity for a few substances are given. They are all expressed in the c.g.s. system with the gram calorie as the unit quantity of heat. That is, the numbers in the table represent in each case, the quantity of heat, in gram calories, which passes a section 1 sq.cm. in area, in 1 second, when the temperature slope is 1°C. per cm. Silver 1.01 Glass 0.002 Copper 0.891 Firebrick 0.0017 Aluminum 0.344 Cork 0.0007 Zinc 0.265 Paraffine 0.0002 Iron 0.167 Water 0.0014 Mercury 0.0152 Ether 0.0003 Ice 0.0057 Hydrogen 0.0004 Granite 0.005 Air 0.000056 PROPAGATION OF HEAT 91 The student must always remember that the results given in the tables for the coefficients of expansion and conductivity must be taken as being only approximate. For, the physical properties of a substance depend very largely upon its chemical purity; and, furthermore, the properties any substance may manifest, will depend very largely upon its physical history and composition. This is especially true for alloys, such as brass, organic growths, such as cork, and complex compositions and mixtures, such as glass. It is interesting to note that, for metals, the order is the same for electrical conductivity as it is for thermal conductivity; i.e., good conductors of heat are also good conductors of electricity, and vice versa. However, there is not, as was at one time sup- posed, strict proportionality. THERMODYNAMICS CHAPTER VIII FUNDAMENTAL PRINCIPLES 84. First Principle of Thermodynamics. The first principle of thermodynamics is merely the application of the principle of energy to the special case of mechanical work and heat; and may be stated as follows : When heat is converted into work, or work into heat, the ratio of the numbers representing the two quantities involved is a constant. The foregoing statement is, of course, the result of direct experiment. 85. Second Principle of Thermodynamics. The second prin- ciple of thermodynamics is stated variously by different authors. Indeed, in some cases, the statement is preceded by discussions which involve almost the whole theory of heat. For our purposes, however, the statement first enunciated by Clausius will suffice. This statement is essentially as follows: Heat cannot pass from a body of lower temperature to one of higher temperature without the aid of some external agent. This statement, though not the result of direct experiment, is in conformity with our common experience. As an example, we know from experience that heat passes by conduction and radiation from regions of higher tem- perature to regions of lower temperature. To illustrate further, assume that we are dealing with two bodies A and B, and that the temperature of the former is lower than that of the latter; then 93 94 THERMODYNAMICS heat may be made to pass from A to B, by applying heat to A, until its temperature is the same as that of B, bringing the two bodies into contact, and by the further application of heat to A, heat will pass from it to B. Heat may also be made to pass from A to B, if work be first done on the former, such as compressing it, until its temperature is the same as that of B; then by bringing the two bodies in o contact and developing, by a further expendi- ture of work, more heat in A, heat will pass from it to B. But, until there is a tendency to raise the temperature of A above that of B, no heat will pass from the former to the latter. Assume, now, a third body, C, under compression and at the same temperature as A. By bringing the two bodies A and C into contact, and allow- ing C to expand against the external pressure, thus performing work, its temperature will fall and a certain quantity of heat will flow from A into C. The body C may now be removed from A, and compressed adiabatically until its temperature is equal to that of B } and then by bringing C into contact with B, and by a further expenditure of work on C, heat will flow from it to B. At the end of this process, C may be removed from B and allowed to expand adiabatically, and, if the various ranges have been properly chosen, it will at the end of this cycle of operations be in precisely the same condition as it was at the beginning. But, A now contains less heat, and B contains more heat than it did when the process began; and since C is in the same condition as it was at the beginning, heat has been transferred from a body of lower temperature to one of higher temperature by the aid of an external agent. 86. Heat Motors. Heat Motors, or Heat Engines, are devices by means of which energy in the form of heat, is converted into energy, in the form of mechanical motion. All heat motors consist of three parts; viz, a source of heat, a working substance, and a refrigerator. Furthermore, all actual heat motors act periodically; i.e., operate on cycles; and for each cycle a certain quantity of heat is abstracted from the source by FUNDAMENTAL PRINCIPLES 95 the working substance, part of which is converted into work, and the remainder, neglecting radiation, etc., is rejected to the refrigerator. 87. Simple Thermodynamic Engine. Before making a general demonstration it will be instructive to give, as an illustration, a simple concrete, though by no means economical, method for the conversion of heat into mechanical work. Assume that the source of heat is a reservoir of boiling water at 100°C, the refrigerator melting ice at 0°C, and the working substance, as depicted in Fig. 9, a metallic rod ab. One end of Fig. 9. the rod rests against the support SS f , and the other end against one of the teeth of the disk, supported on an axis through 0. A pawl P, pivoted on the support SS', also engages one of the teeth of the disk. Connected to this disk is a drum, having wound over it a cord, supporting a resistance R. Suppose now, that the length of the rod has been so chosen with respect to its coefficient of linear expansion, that when its temperature is raised from 0°C. to 100° C, it expands by an amount such that the disk turns through an angle equal to that subtended by a tooth. The resistance R, will suffer a certain displacement and the pawl will engage the 96 THERMODYNAMICS next tooth. If the rod be now surrounded by a bath of melting ice, it will contract and engage the next tooth; the pawl, in the meantime, holding the disk in position. This process may be repeated indefinitely, until the resistance has been displaced through any desired distance. The cycle is then as follows: The rod at the temperature of melting ice is put into position, and sur- rounded by a jbath of boiling water at a temperature of 100 °C. In consequence of this elevation of temperature, the rod expands and turns the disk through a certain angle and in this manner does work in overcoming the resistance R. The heat taken from the source consists of two parts : One part being consumed in elevating the temperature of the rod, and is numerically equal to the prod- uct of the mass of the rod, its thermal capacity per unit mass, and the elevation of temperature. The other part consists of the heat equivalent of the work done in displacing the resistance R. The rod, now being disconnected and surrounded by melting ice, gives up to the refrigerator, in cooling from 100°C. to 0°C, an amount of heat precisely equal to that absorbed in being heated, without any external work being done, from 0°C. to 100° C. Therefore, the difference between the heat taken from the source and that given to the refrigerator is equivalent to the work done in displacing the resistance R. Since this completes a cycle it may be repeated indefinitely without any change in the relation of the quantities involved. If, now, we represent by Qi, the quantity of heat absorbed from the source, during a cycle, and by Q2, the heat rejected to the refrig- erator, then the external work done is W = J(Qi-Q 2 ); (1) where W is the external work done, and J the mechanical equiv- alent of heat. Since for every cycle the quantity of heat Qi has forever disappeared from the source, and only the part Q1 — Q2 has been converted into work, it follows that, with the contrivance FUNDAMENTAL PRINCIPLES 97 just described, it is impossible to convert all the heat, taken from the source, into external work. 88. Heat of Expansion. As explained in Arts. 46 and 47, when a gas is heated and expands against an external pressure, the heat required is practically equal to that required to elevate the temperature of the gas, plus the heat equivalent of the exter- nal work done; i.e., the external work done, expressed in heat units, is practically equal to the heat of expansion. This, however, is by no means the case when a metal rod is heated and expands against an external pressure; for, in this case, the heat of expansion consists of two parts, viz, the heat equivalent of the external work done, and the heat required to expand the rod against its own inherent forces. The former may be called the external heat of expansion, and the latter the internal heat of expansion. In the present state of our knowledge we are unable to assign the proper relative values for the heat consumed in elevating the temperature of a substance and the internal heat of expansion; but, for most substances, the latter is a relatively large quantity. Since, now, in the cycle discussed in Art. 87, the internal heat of expansion is not recovered as work, but is rejected to the refriger- ator, it follows that such a contrivance cannot use heat eco- nomically. 89. Carnot's Cycle. The first scientific discussion of a peri- odically acting thermodynamic engine is that due to Sadi Carnot, published in 1824. In this discussion, ideal conditions are assumed ; i.e., it is assumed that there are no losses due to radiation and friction. In other words, it was Carnot's object to show that under certain given conditions, assuming ideal processes, a definite fractional part of the heat taken from the source, by a periodically acting engine, is converted into work; and that, for the given conditions, this is the maximal amount of work that may be realized. The following demonstration will make this clear. Assume that we are dealing with any working substance whatso- ever, confined in such a manner that it may be put into contact 98 THERMODYNAMICS with either the source or the refrigerator, whose temperatures remain constant throughout the process, and that the conduction between the working substance and both the source and refrig- erator is perfect. Furthermore, for ideal conditions, it must be possible to insulate the working substance such that adiabatic processes may take place. Let now, in Fig. 10, the point A, on the p-v (pressure-volume) diagram, represent the pressure and volume of the working sub- stance at that part of its cycle when it is removed from the refrig- erator. Pressures are represented by ordinates and volumes by abscissas. The working substance is now insulated and com- pressed adiabatically, in consequence of which its temperature rises. This compression is continued until the temperature of the working substance is equal to that of the source, and its con- dition, as regards pressure and volume, is represented by the point B. The work done on the substance during this compres- sion is represented by the area under the curve; i.e., the area GABK. The working substance is now put into contact with the source and allowed to expand isothermally , by any desired amount, and its condition, as regards pressure and volume, at the end of this expansion, is represented by the point C. During this FUNDAMENTAL PRINCIPLES 99 isothermal expansion a certain quantity of heat Qi, has been abstracted from the source, and work has been done by the working substance, represented by the area BCFK. The working sub- stance is now again insulated and allowed to expand adiabatically, in consequence of which its temperature falls. This expansion is continued until the temperature of the working substance has fallen to that of the refrigerator, and the work done by it, during this expansion, is represented by the area CDEF. The working substance is now put into contact with the refrigerator and com- pressed isothermally until its condition, as regards pressure and volume, is again represented by the point A. During this isother- mal compression, work was done on the working substance repre- sented by the area DEGA ; and, a quantity of heat Q2, was rejected to the refrigerator. Since now, the working substance, as regards pressure, volume, and temperature, is in precisely the same condition as it was at the beginning of the cycle, its intrinsic energy is also the same; it therefore follows, from the first principle of thermodynamics, that the difference between the heat abstracted from the source and that rejected to the refrigerator, expressed in mechanical units, is equal to the net work done. By an inspection of Fig. 10 it is obvious that the net work done is represented by the area ABCD; and, from equation (1), we have Wi=J(Qi-Q 2 ); where Wi is the net work done. But the heat, expressed in mechanical units, abstracted from the source is W 2 =JQi; therefore, the ideal coefficient of conversion^ the maximal fractional part of the heat, abstracted from the source, which in an ideal process can be converted into work, is „ J(Qi-Q2) Q1-Q2 m 100 THERMODYNAMICS The result, just found, has been deduced without making any assumption in regard to the nature of the working substance; it is therefore perfectly general. Suppose that the working substance suffers a physical change of state during the cycle; the foregoing demonstration still holds. For, since the working substance is in precisely the same condition as regards pressure, volume, and temperature at the end of the cycle as it was at the beginning, it follows that whatever physical changes of state have taken place during any part of the cycle, changes of a like kind must have taken place in the reverse order during some other part of the cycle; and hence, are balanced. Therefore, the difference between the heat taken from the source and that rejected to the refrigerator, expressed in mechanical units, is equal to the external work done. 90. Since the relation, expressed in equation (2), was deduced without considering the properties of the working substance, it must be independent of those properties. There being, however, no other quantities involved in the right-hand member of this equation, excepting quantities of heat, and since these do not depend upon the properties of the working substance, they must be functions of the two temperatures. That is, the quantity of heat taken from the source must be some function of the tempera- ture of the source, and the quantity of heat rejected to the refrigerator must be some function of the temperature of the refrigerator. Just what values are to be assigned to these func- tions must be determined for some specific case, which is con- sistent with the demonstration. 91. Carnot's Cycle a Reversible Process. The ideal cycle just described is a reversible process. For, if the working substance, at that part of its cycle when its condition, as regards pres- sure and volume, is represented by the point A, Fig. 10, and its temperature is the same as that of the refrigerator, is put into contact with the refrigerator and allowed to expand isothermally to the point D, it will abstract from the refrigerator, a quantity FUNDAMENTAL PEINCIPLES 101 of heat Q2, and do an amount of external work, represented by the area ADEG. The working substance is then insulated and com- pressed adiabatically , in consequence of which its temperature will rise; let this be continued until its temperature is the same as that of the source, and its condition, as regards pressure and vol- ume, is represented by the point C, and an amount of work, represented by the area FEDC, has been done on the working substance. The working substance is now put into contact with the source, and compressed isothermally until its condition, as regards pressure and volume, is represented by the point B, a quantity of heat Qi being rejected to the source, and an amount of work, represented by the area FCBK, has been done on the work- ing substance. The working substance is now insulated and allowed to expand adiabatically until its temperature has fallen to that of the refrigerator; its pressure and volume being the same as at the beginning, and the external work, represented by the area BAGK, having been done by it. Taking the sum, we find that the work done by the working substance is represented by the area KBADE; and the work done on the working substance is repre- sented by the area BKEDC. Finally, the net work done on the working substance is represented by the area ADCB. But, during this process, the quantity of heat Q2 has been taken from the refrigerator, and the quantity of heat Qi has been transferred to the source. Since now, the working substance is in precisely the same condition as regards temperature, pressure, and volume, as it was initially, it follows that the difference between the heat rejected to the source and that taken from the refrigerator, expressed in mechanical units, is equal to the net work done on the working substance ; i.e., W = J(Qi-Q 2 ). 92. It will now be shown that, for a given source and refrig- erator, an engine operating on the Carnot cycle, that is, a reversible M)2 THERMODYNAMICS engine, converts into work as large a fractional part of the heat taken from the source as is possible under the assumed conditions. To do this, we assume that we have two engines A and B, operating between the same source and refrigerator, the former acting direct and driving the latter, which is running reversed. Let Ha and H a " be, respectively, the heat taken from the source and that rejected to the refrigerator by the engine A during a given interval of time; and likewise, let H b ' and H b " be, respectively, the heat transferred to the source and that abstracted from the refrigerator by the engine B, during the same interval of time. Assume, now, that the engine A, which is non-reversible, can convert a larger fractional part of the heat taken from the source into work than could the engine B if it were running direct. We then have TJ ' rr // tj t tj // tig —tig tit —lib , Q s H a ' > H»' {6) Also, the work done by the engine A must be equal to the work done on the engine B, since the former is driving the latter; hence, we have W = J(Hg'-H a ")=J(Hb'-H b ") (4) From equation (4) it follows that the numerators of the inequality, expressed by statement (3), are equal; hence Hg V Fig. 11. work has been done on the piston by the gas, measured by the area under the curve BC. The pressure is now 7^3 and the volume v%. (3) The cylinder is now put into contact with the refrigerator and the gas is compressed isothermally , until its pressure is p4, and volume v±, as represented by the point D. During this com- pression a quantity of heat Q2 is developed, and work is done by the piston on the gas, measured by the area under the curve DC. It being assumed that the temperature of the refrigerator, during the absorption of the heat Q2, remains constant; this may be brought about by abstracting heat from it at the proper rate. (4) The cylinder is now removed from the refrigerator, is perfectly insulated, and the gas is compressed adiabatically until 106 THERMODYNAMICS its temperature is T\, that of the source, and its pressure and vol- ume are, respectively, pi and vi. During this compression work was done, by the piston on the gas, measured by the area under the curve AD. The gas being now in precisely the same condition as it was initially, its intrinsic energy must also be the same. Since, now, A and B are on the same isotherm, and likewise, C and D are on the same isotherm, we have, from the character- istic equation, piVi = p 2 V2 = RT 1} (5) and P3V3 = P4V4 = RT2 (6) Also, since B and C are on the same adiabatic, and likewise, A and D are on the same adiabatic, we have p 2 V2 n = P3V 3 n , (7) and PlVi n =P±V4: n (8) From equation (5) we find and From equation (6) we find RTi , . RTi /inN p2= ^- (10) v *=~w (11) and p* = -^~ w Substituting the value of P2 as given in equation (10), and that of pz as given in equation (11), in equation (7) we find V2 V3 FUNDAMENTAL PEINCIPLES 107 from which 1 V3 /TAn-l n \T 2/ < 13 > Again, substituting in equation (8) the values of pi and p4,as given by equations (9) and (12), we find RT 1 n RT 2 n Vl n = V4 n . Vl V± from which vr\T 2 ) w From equations (13) and (14) it follows that Vz _V4 V 2 Vl V3 = _V2 V4 Vl from which (15) Equation (15) shows that the volume at C must be to the volume at D, as the volume at B is to the volume at A, so that when the gas is compressed adiabatically from D, it will come to the point A. Since we are dealing with a perfect gas, its intrinsic energy is a function of the temperature only, and is, therefore, independent of pressure and volume. Therefore, the work done by the gas in going along the adiabatic from B to C, is exactly equal to the work done on the gas in going along the adiabatic from D to A. Hence to obtain the net work done during the cycle, and the quan- tities of heat involved, it is only necessary to consider the two isothermal processes; viz, the heat abstracted from the source, and the external work done, by the gas in going from A to B, along the isotherm Ti, and the heat rejected to the refrigerator, 108 THERMODYNAMICS and work done, on the gas, in going along the isotherm T 2 from CtoD. Since the temperature, during the isothermal expansion, from A to B is constant, it follows that the heat abstracted from the source is directly proportional to the external work done. Hence we have Qi=A I pdv: J n where A is the heat equivalent of a unit of work. But p, for any part of this process is equal to RTi/v; hence "2 dv r v2 d\ Q 1 =ART 1 - Jh v =ART 1 \og V ^ (16) By similar reasoning we find, that the heat rejected to the refrigerator, during the isothermal compression, in going from C to D, is fn Q 2 = A pdv. But, for any part of this process p is equal to RT2/V; hence * V3 dv Q 2 = ART 2 r v3 dv Jv 4 V =ART 2 \og V ^- (17) V4 The net work done, measured in heat units is, by condition, proportional to the difference between the heat taken from the source and that rejected to the refrigerator; hence AW = Qi-Q 2 = AR(T 1 log V ^-T 2 log^); \ V\ V4J FUNDAMENTAL PRINCIPLES 109 and the ideal coefficient of conversion, since Qi has forever dis- appeared from the source, is Qi-Qj W^log^-^log^) \ Vi ° v 4 / Ql AR Tl log** Vi from which, since by equation (15), 02/01 = 03/04, we find Qi-Q2 _ T 1 -T 2 . Y1 ~ Qi -Ti (18) Equation (18) shows that, for a perfect gas operating on a Carnot cycle, the ideal coefficient of conversion is the ratio of the difference in temperature of source and refrigerator, to the tem- perature of the source, as measured on the ideal gas thermometer. Equation (18) is usually written in the following form: 1=-^; • (19) where S is the temperature of the source and R that of the refrig- erator, both being measured by means of the ideal gas thermometer. 95. A little consideration will show that the foregoing discussion, and result obtained, is perfectly consistent in every way with that of the Carnot cycle using any working substance. We are there- fore justified (Art. 90) in assuming that even under ideal conditions the maximum quantity of work that can be realized from an engine working between a given source and refrigerator and ab- sorbing the quantity of heat H from the source, is W=Jh(^) (20) Writing equation (19) in another form, we have 1 R 110 THERMODYNAMICS from which it is obvious that, for y] to approach unity, R must either approach zero, or S must approach infinity. Experience, however, shows that it is not economical to attempt to maintain the refrigerator at a temperature lower than that of the surround- ings. Also, as the temperature of the source is increased, a point is soon reached for which radiation and pressures become excess- ive, and lubrication becomes difficult. It therefore follows, with conditions such as obtain on the earth's surface, that even a perfect engine can convert only a small fractional part of the heat, taken from a source, into work. 96. Reversible Engine and Refrigeration. An engine operating in a reverse order, i.e., one that is taking heat from a body of lower temperature, transferring heat to a body of higher temper- ature, and absorbing external work, constitutes a refrigerating machine. Let, for any given time, Hi be the quantity of heat transferred to a body of higher temperature, and H2 the quantity of heat abstracted from a body of lower temperature, by a per- fectly reversible engine; i.e., a perfect refrigerating machine. We will then have the following relation : H1—H2 _ S—R . . #1 " s {2l) From equation (20) we have, for the amount of work that must be done, to transfer the quantity of heat Hi, to the body of higher temperature, W = JHi^^ (22) In general, however, in the case of refrigerating machines, we are concerned principally with the work that must be done to bring about a certain absorption from the body of lower temperature; i.e., the amount of refrigeration. It is therefore advisable to deduce an expression for the amount of work that must be done in terms of H 2 , the quantity of heat taken from the body of lower FUNDAMENTAL PRINCIPLES 111 temperature, instead of the quantity of heat Hi, rejected to the body of higher temperature. From equation (21) we find H2 = R Hx S' from which #i=# 2 § (23) Substituting in equation (22) the value of Hi, as given by equation (23), we find W = JH 2 ^j^; ...... (24) which gives the desired relation. 97. In Art. 95, it was stated that it is not economical to attempt to maintain the temperature of the refrigerator lower than that of the surrounding medium. We are now prepared to demon- strate this mathematically. Let Hi be the heat taken from the source, at a temperature S, and let Ri be the temperature of the surroundings. If then the temperature of the refrigerator be also Ri, the work that, under perfect conditions, may be realized is Wi=JHi^f±. ..... (25) Assume now, that the refrigerator, by means of a reversible engine, is maintained at some temperature R2, lower than Ri. The work that can now be realized, by means of a perfect engine, is % W 2 = JhJ^^ (26) Subtracting equation (25) from equation (26), member by mem- ber, we obtain, due to lowering the temperature of the refriger- ator, for the gain in work, W 2 -W 1 =J^(R 1 -R 2 ) (27) 112 THERMODYNAMICS To maintain the temperature R 2 we must, by means of a revers- ible engine, abstract heat from the refrigerator at the same rate that the direct engine is rejecting heat to it, and transfer heat to the surroundings. The heat rejected by the direct engine is H 2 = H 1 -AW 2 =H 1 -H 1 ^~^ = H 1 ^. . . (28) The work that must be expended in transferring this quantity of heat from the body of temperature R 2 , to the surroundings at a temperature Ri, is = J I ^(R 1 -R 2 ) (29) By comparing equations (29) and (27), it is obvious that, even under ideal conditions, the amount of work that must be done by the reversible engine, to maintain the temperature of the refrigerator, below that of the surroundings, is equal to the gain in work by the direct engine, due to the lower temperature of the refrigerator. It therefore follows that, even without consid- ering losses, there can be nothing gained by attempting to have the temperature of the refrigerator lower than that of the earth's sur- face. As a matter of fact, if the temperature of the refrigerator is lower than that of the surroundings, heat will continually pass from the surroundings to the refrigerator, and the reversible engine must do an amount of work greater than that given by equation (29). Furthermore, due to imperfections of the engines, the gain in work realized by the direct engine will be less than that specified by equation (27), and the work that must be done on the reversible engine will be greater than that specified by equation (29) ; hence, there is a decided loss when the refrigerator is main- tained at a temperature lower than that of the surrounding media. FUNDAMENTAL PRINCIPLES 113 98. Thermodynamic Scale of Temperatures. The thermo- dynamic scale of temperatures, which was first proposed by Lord Kelvin, will be made clear by the following considerations. Assume a series of n perfect heat engines arranged in such a manner that the refrigerator of the first engine is the source of the second engine, the refrigerator of the second engine is the source of the third engine, etc., and furthermore, that the heat rejected by any engine is absorbed by the engine next lower in the scale. To show that, if the difference in temperature between source and refrig- erator for the various engines is the same, they are all doing the same amount of work. Let, as in Fig. 12, the two adiabatics, AB and CD, be cut by the isotherms T 1} T 2 , T 3 , etc., such that the temperature intervals are all equal, and each equal to t; i.e., T 1 -T 2 = T2-Tz = T n -T n + 1 = 'z. The ideal coefficients of conversion for the various engines, begin- ning with the first, then are TV zy 5y T (30) 114 THERMODYNAMICS If H is the quantity of heat absorbed by the first engine from its source, during a given interval of time, then b-b—tT - h T! is the heat rejected to its refrigerator, and absorbed by the second engine, during the same interval of time. In a similar manner, the quantity of heat supplied to the third engine is rp rp rp rp rp jjl2 rri2 v l2— i3 = TTJ3 " m ■" rp S^ rp •" rp • The quantities of heat supplied to the various engines, beginning with the first, then are Uy U Y'y U T~> ' ' ' n ~f — » n jT' • • \ 6i -J Since, now, the work done by any engine of the series is equal to the product of its ideal coefficient of conversion and quantity of heat, expressed in mechanical units, absorbed by it, it follows from expressions (30) and (31), that all the engines are doing the same amount of work; i.e., is the work done by each engine of the series. The results just deduced, being independent of the properties of any substance, a thermodynamic scale of temperature may be established in the following manner: Assume a series of n heat engines, working between a given source of temperature T\, and refrigerator of temperature T n + i, in such a manner that the n engines are all doing the same amount of work, and each engine is absorbing the heat rejected by the engine next higher on the scale. If we then designate the difference of temperature between FUNDAMENTAL PRINCIPLES 115 the source and refrigerator of any one of these ideal engines, as a unit of temperature, we will have a scale of temperatures inde- pendent of any substance, and depending only upon the perform- ance of a perfect engine. But, from the discussion just given, we found that by assuming the temperature intervals, as measured on the ideal gas thermometer, equal, the engines were all doing the same amount of work; hence, the thermodynamic scale is identical with that of an ideal gas thermometer; and differs but slightly, for temperatures not exceeding 500 °C, from those as found by means of the ordinary gas thermometer. CHAPTER IX STEAM AND STEAM ENGINES 99. The proper design of a heat engine presupposes, on the part of the designer, a knowledge of the construction of mechan- ical contrivances; i.e., how to construct a machine which shall withstand the stresses imposed upon it in the performance of its duties, with the lowest cost. The expression, lowest cost, must not be interpreted as meaning lowest first cost; but it must be understood to mean that the interest^on the capital invested, for both machinery and ground rent, plus depreciation, plus cost of power lost, must be a minimum. This part of the subject comes under the heading of machine design; and, properly speak- ing, has nothing to do, except in so far as fuel economy is affected by the design, with the subject of thermodynamics. But, a thorough knowledge of the characteristics of the working substance and the changes it undergoes, during its various stages, is fully as impor- tant, if not more so, in the designing of an engine, as is a knowledge of machine design. It is for this reason, since steam is so widely used as a working substance, that so much research work has been done, to accurately determine its characteristics. 100. Steam Operating on Carnot's Cycle. Assume that we are dealing with a unit mass of water, at a temperature I2, which corresponds to that of the refrigerator, and let its condition, as regards pressure and volume, be represented by the point D of Fig. 13. The water is compressed adiabatically until its tem- perature is T\, that of the source, and its condition, as regards pressure and volume, is represented by the point A. If the water 116 STEAM AND STEAM ENGINES 117 is now placed into contact with the source, and the pressure is maintained constant, vaporization will take place. Assume this to be continued until all the water has been converted into satu- rated steam, whose condition, as regards pressure and volume, is represented by the point B. The steam is now allowed to expand adiabatically until its temperature has fallen to T2, that of the refrigerator; its pressure and volume being now represented by the point C. During this adiabatic expansion a certain amount of condensation, which will be discussed later, has taken place. The mixture of steam and water is now put into contact with the \ .Tt B To D L V Fig. 13. refrigerator and compressed isothermally until complete conden- sation has taken place, and its condition, as regards pressure and volume, is again represented by the point D. Since, now, the condition of the working substance, as regards temperature, pressure, and volume, is precisely the same as it was initially, its intrinsic energy is also the same. Therefore, the net work done during the cycle is measured by the area DABC. Further- more, since the process is ideally reversible, the ideal coefficient of conversion is T l -T 2 f\ = Tx the same as previously deduced for any working substance. 118 THERMODYNAMICS 101. Relation of Temperature and Density of Saturated Steam. It is frequently of prime importance to know the density of satu- rated steam for a given temperature; and it being difficult to determine this relation by direct experiment, it will be shown how it is found from the relation of pressure and temperature of a saturated vapor, this being easily determined by direct experiment. To show how to determine the relation of temperature and density of the saturated vapor of a substance, it will be assumed that we are dealing with a unit mass operating on a Carnot cycle, as just described, and an indefinitely small difference of temperature, AT, between source and refrigerator. This is represented dia- grammatically in Fig. 14, where T-\-AT is the temperature of the T+ AT ^ V Fig. 14. source, and T the temperature of the refrigerator. This being a reversible process, the ideal coefficient of conversion is Y) = AT T+AT> which, in the limit, becomes dT T) = If the quantity of heat taken from the source, in going from A to B, is Q, then the work done is W = JQ dT (1) STEAM AND STEAM ENGINES 119 250 225 200 175 • 150 5l25 Q. 100 75 50 25 ■ / / / RELATION OF 1 TEMPERATURE AND PRESSURE / / / SATURATED STEAM ' / / / 1 I / I / / 1 / / 1 ' 1 ) 1 j / / / / / / - / / ^ / 100 200 300 TEMP. IN DEGREES FAHR. 400 1 20 THERMODYNAMICS The work done during the cycle may also be expressed in terms of the initial and final volumes and the change in pressure dp, corresponding to the change in temperature dT. That is, if a is the volume of unit mass of the liquid, and s the volume of unit mass of saturated vapor, then the work done, during the cycle, is W = (s-a)dp (2) Now the right-hand members of equations (1) and (2) must be equal; since they are expressions for the same amount of work, hence dT JQ-y = (s-°)dp; from which and JQ K dT •«+f *§ « The quantity Q, in equation (3), represents the quantity of heat required to convert unit mass of the liquid into a saturated vapor at the temperature T, and may be replaced by r, the heat of vaporization; hence, equation (3) becomes S = °+T X Jp (4) In equation (4), a, the volume of unit mass of the liquid, for the temperature T, is readily found by experiment; and likewise r, the heat of vaporization. dT I dp is found from the curve giving the relation of temperature and pressure of the saturated vapor. Hence, since J, the mechanical equivalent of heat, is known, s is determinate ; and the reciprocal of this gives the density of the saturated vapor. STEAM AND STEAM ENGINES 121 As a matter of interest, the curve showing the relation of temperature and pressure, for saturated steam, is given on page 119. 102. Perfect Steam Engine and Boiler. In the previous dis- cussions it has been assumed that all of the heat is taken in at the highest temperature. This, however, is by no means the case, even under perfect conditions, with a steam engine and boiler. For the present, we will confine ourselves to the operation of a reciprocating engine, which has supplied to it saturated steam from a boiler. The reciprocating engine consists essentially of the following parts: A source of heat, the boiler, where steam is generated under a constant pressure, and hence, at a constant temperature, a cylinder and piston, and a refrigerator, or condenser, at constant temperature, by means of which the steam, after expanding and doing work against the piston, is converted into water and returned to the boiler. The cycle of operations is as follows: The piston P is at the position as represented in the diagram, Fig-. 15, and the condition of the steam, as regards pres- sure and volume, is represented by the point A, the point of admission. That is, at this point, the valve in the pipe connect- ing the boiler with the cylinder is opened, and steam is freely admitted. The piston advances to the point B, while vapori- zation takes place at the temperature T\. To simplify matters, we will assume that we are dealing with unit mass of water and that complete evaporation has taken place when the volume is represented by the point B. The quantity of heat then, taken from the boiler, is n, the heat of vaporization at the temperature T\. The line of admission, A B, is a straight line and parallel to the axis of volumes, since vaporization has taken place at constant temperature; and hence, at constant pressure. The external work done, during this advance of the piston, is measured by the area ABFE. The point B is the point of cut-off; i.e., the admission- valve is closed, and the steam is allowed to expand adiabatically until its temperature has fallen to T 2 , that of the condenser. 122 THERMODYNAMICS In the meantime, the external work, measured by the area BCGF } has been done. The exhaust-valve now opens, and the steam remaining in the cylinder, is compressed isothermally, in contact with the condenser, until complete condensation has taken place, and the work represented by the area DCGE, has been done by the piston. The condensed steam, at the temperature T2, is returned to the boiler and heated from the temperature I2 to that of T\, thus completing the cycle. The net work done during A T i B d !k t 2 ^^ ^ E If _[g v i 1 2 ^ i Fig. 15. this cycle is evidently measured by the area ABCD, and is neces- sarily less, as will now be shown, due to not taking in all the heat at the maximum temperature, than that which could be realized by a Carnot cycle. The maximum amount of work that could be realized from an engine taking in an elementary quantity of heat dQ, at the temperature T, between the temperatures T2 and T\, operating on a Carnot cycle, and rejecting heat to the condenser at the tem- perature T2, is T-T 2 dW = JdQ (5) STEAM AND STEAM ENGINES 123 But, dQ is equal to cdT; where c is the thermal capacity and dT the change in temperature. And, since we are dealing with unit mass of water, dQ is practically equal to dT; since for water c is almost constant and equal to unity. Therefore, equation (5) may be written dW=J^^dT; ....... (6) from which we obtain, for the total work that could be realized, under ideal conditions, from the heat required to elevate the tem- perature of unit mass of water from T 2 to T\, T t rp__rp J1 T Tr-Tz-TzlogpJ. The work that, under ideal conditions, could be realized from the heat taken from the source during vaporization, since this is absorbed at constant temperature, is TF " =/n ^TT ?; ••••••• (8) where n is the heat of vaporization at the temperature T\. Adding equations (7) and (8), we obtain for the total work that may be realized, for the given conditions, Wi = W' + W"=j(T 1 -T 2 -T 2 log^+r 1 ^^y . (9) Since the total heat, abstracted from the source, expressed in mechanical units, is J(T 1 -T 2 +r 1 ), 124 THERMODYNAMICS the maximum work which would have been realized, had the operation been on a Carnot cycle, is W 2 = J(T 1 -T 2 +r 1 ) J ^^ (10) ^ 1 Dividing equation (9) by equation (10), we find W 2 T 1 +r 1 -T 2 { } If now, in equation (11), yr^Jrlog ^~>T 2 , (12) then W1/W2 is less than unity. To prove that the expression, given by the inequality (12), holds, we assume that T 2 , the tem- perature of the condenser, is fixed, and that T\ is a variable, which may be represented by T; remembering that T is always greater than T 2 , and that both are positive. Expression (12) may then be written TTo T ^jr-\og-^==kT 2 ; (13) where k is a proportionality factor. From this, we find log^^(l-f) (14) Substituting, in equation (14), for T/T 2 , a new variable, x,we have log x = k( 1 and, differentiating with respect to x, we obtain k =*=Y 2 , (15) STEAM AND STEAM ENGINES 125 from which, if T/T2 equals unity, k equals unity; and, if T/T2 becomes greater than unity, k must be greater than unity or equation (15) cannot hold. But this means that the left-hand member of equation (13) must be greater than T2', and hence W1/W2 is less than unity. It therefore follows that a steam engine, which rejects condensed steam to a boiler cannot, even under perfect conditions, convert into work as large a fractional part of the heat taken from the source as can an engine operating on a Carnot cycle, between the same limits of temperature. To illustrate the foregoing, we will deal with a concrete case; i.e., assume the temperature of the entering steam, and of the con- denser, respectively, 356°F. and 140°F. This gives: ri = 816,* 72 = 600, and ri = 865. Substituting these values, in equation (11), we find ciaiqa* 816X60 0, 816 Wl 816 + 865 -8i6^600 log 6 00 mn W 2 = 816+865-600 = 9L0 per Cent ' giving a loss of about 9 per cent due to not taking in all the heat at the maximum temperature. 103. Unresisted Adiabatic Expansion of Steam. If dry saturated steam is allowed to expand adiabatically, from a chamber of given pressure to one of lower pressure, without doing work, the steam becomes superheated. This is due to the fact that, when the steam enters the chamber of lower pressure, eddy cur- rents are developed; and as they subside, the kinetic energy, possessed by them, is converted into heat. Since the process is adiabatic, and no external work is done, the total heat content, i.e., the total quantity of heat contained by the steam, will be the same at the end of the process as it was at the beginning. But since, the total heat of steam decreases as the pressure is * According to recent experiments, the zero for the thermodynamic scale is 491.65°F. below the melting-point of ice; but, in general, 492 is sufficiently accurate. 126 THERMODYNAMICS decreased, and the final pressure is lower than the initial pressure, the steam must become superheated. If the steam is not initially dry, then it will become drier by unresisted adiabatic expansion. Assume that we are dealing with a unit mass of a mixture of steam and water under a pressure pi, for which the heat of the water and the heat of vaporization are, respectively, hi and T\. The total heat of the mixture, then is # = /*i+2iri; (16) where q\ is the dryness; i.e., the fractional part of the liquid which is present as steam. After expansion, since the total heat content remains the same, we have H = h 2 -\-q 2 r 2 ; (17) where h 2 , q 2 , and r 2 are, respectively, the heat of the liquid, the dryness, and the heat of vaporization for the final pressure p 2 . Equating the right-hand members of equations (16) and (17), we obtain hi+qm = h 2 -\-q 2 r 2 . . ... . . (18) By priming is meant the percentage of moisture present; and if this is low, the steam may become superheated by the unresisted adiabatic expansion, and q 2 , in equation (18), becomes unity. It will be shown later how, under certain conditions, advantage may be taken of this, and the initial priming determined exper- imentally. 104. Resisted Adiabatic Expansion. If steam, the initial priming of which is low, expands adiabatically in such a manner that external work is done, it will become wetter. If in equation (9), Art. 102, it is assumed that complete evaporation has not taken place before the adiabatic expansion begins, then the work, expressed in heat units, yielded per cycle, is W=T 1 -T 2 -T 2 \ogp+qir 1 ?^^; . . (19) STEAM AND STEAM ENGINES 127 where qi is the dryness. The total heat absorbed, in elevating the temperature of the water from T2 to T\, and evaporating it to the dryness qi, is Hi=Ti-T 2 +qiri (20) And since, under the assumed conditions, the difference between the heat abstracted from the source and that converted into work, must be equal to H r , the heat rejected to the condenser, we find, by subtracting equation (19) from equation (20), ff'=girig+!T 2 logg (21) But, the heat rejected to the condenser, after adiaoatic expansion to the temperature T2, must be equal to the heat liberated during condensation, i.e., H' = q 2 r 2 ; (22) where qi is the dryness and ? 2 the heat of vaporization corre- sponding to the temperature TV Equating the right-hand mem- bers of equations (21) and (22), we obtain from which T2 , m 1 ^1 q 2 r 2 = qin ^r + T 2 log ^- J g2= ^fe +l0g rJ (23) Equation (23) enables us to compute the dryness, during resisted adiabatic expansion, provided the initial dryness be known. In the next chapter, the relation expressed in equation (23) will be deduced by a much simpler and shorter method. CHAPTER X ENTROPY 105. It is obvious that a substance, in going from one isotherm to another, always suffers the same definite change in temperature; and furthermore, that this change in temperature is independent of changes in pressure and volume. That is, the change in tem- perature in going from one isotherm to another is independent of the path pursued during the change. A good analogue of this is the change in potential a body undergoes in going from a sur- face of potential Vi, to a surface of potential V2, the change in potential, V2 — V\, being independent of the path pursued in bringing about the change. It will now be shown that, in going from one curve to another, both curves representing reversible adiabatic processes, there is some definite constant change. Equation (19) of Art. 48 specifies for reversible adiabatic processes C v dT+pdv = 0; ....... (1) from which, by substituting for p its value as obtained from the characteristic equation, pv = RT, and separating the variables, we find C.f+1%-0 (2) By integrating equation (2), between limits, we obtain C.log^+JKlogf = 0; (3) 128 ENTEOPY 129 where T\ and v\ are, respectively, the temperature and volume before the change, and T and v, respectively, the temperature and volume after the change. Since equation (3) is equal to zero, no matter what the limits of integration, it follows that there is something which does not change during a reversible adiabatic process. Integrating equation (2) for the primitive, we find CAogT+R\ogv = k', (4) where k is a constant of integration. Equation (4) shows that the fundamental differential equation for a perfect gas, yields upon integration for a reversible adiabatic process a constant. But since T and v, in equation (4), may have any values whatsoever, pro- vided, always, they are so related that the process is adiabatic, it follows that, no matter what the range, there is some function which remainsconstant; which conclusion is the same as that drawn from equation (3). Hence, since there is some function which remains constant during a reversible adiabatic change, there must be some definite constant change in going from one curve, repre- senting a reversible adiabatic process, to another curve, represent- ing a reversible adiabatic process. In equation (4), the constant k evidently represents some particular condition for the gas, which remains constant, during an adiabatic process; and its value depends upon the unit of measure and zero chosen. The condition of a gas, as expressed by equation (4), was called by Clausius the entropy of the gas; and, as just stated, the numerical value of the entropy depends upon the units chosen and the arbitrary zero from which it is measured. Equation (4) may now be stated as follows: The entropy of a substance during a reversible adiabatic change remains constant. 106. Change of Entropy. If the left-hand member of equation (1) is not equal to zero, i.e., heat is either added or abstracted 130 THERMODYNAMICS while the gas changes in volume and temperature, the process is no longer adiabatic, and the equation becomes dQ = C v dT+pdv; from which, by substituting for p its value as obtained from the characteristic equation, we have dQ = C v dT+RT— (5) Dividing equation (5) by T, we obtain dQ dT dv If we represent the entropy of the gas by ( 8 ) i.e., for a reversible process, the change in entropy is numerically equal to the ratio of the change in heat to the temperature at which the change takes place; the temperature being measured on the thermodynamic scale. The foregoing may be illustrated by equation (18) of Art. 94, which states that for a Carnot cycle operating on a perfect gas, Qi-Q 2 = T 1 -T 2 Qi T x ' from which *"■* » ENTROPY 131 Equation (9) shows that the change in entropy in going from the adiabatic AD, (Fig. 16), to the adiabatic BC, is the same whether the change takes place along the isotherm AB or DC; since the ratio of change in heat to the temperature at which the change takes place is the same in both cases. It is obvious that the same ratio holds for any other isotherm cutting the two adiabatics AD and BC. It is, however, not necessary that the change take place along an isotherm. For, assume as depicted in Fig. 16, the irregular path ef to be cut by the two adiabatics aa! and W , which differ by an indefinitely small interval. The change in v Fig. 16. entropy, in going from g to h, along the irregular path ef, may be resolved into the two component changes; i.e., the change in entropy, in going along the isotherm gi, which is dy = dQ/T; where dQ is the change in heat and T the temperature at which the change takes place. The other component ih, being adiabatic, involves no change in entropy; hence, the change in entropy, in going from g to h, is dQ T' dy But since, as has just been shown, the change in entropy, in going from one adiabatic to another is the same for all isotherms, it follows, since gi is an isotherm, that the change in entropy in 132 THERMODYNAMICS going from g to h is equal to the change in entropy in going from a to b along the isotherm Ti, and also to the change in entropy in going from a' to b' along the isotherm TV Similarly, it can be shown that the change in entropy in going along the irregular path ef, between any two adiabatics, is equal to the change in entropy in going between the same two adiabatics along either the isotherm T\ or TV It therefore follows that the change in entropy in going from the adiabatic AD, to the adiabatic BC, is always the same and is independent of the path by means of which the change is brought about. The foregoing demonstrations establish the fact that we are justified in making the assumption that there is a constant definite change in going from one reversible adiabatic to another; and, this being the case, it follows that during a reversible adiabatic process, some function, which has been termed entropy, must remain constant. It also follows that, since for a Carnot cycle Tx T 2 ' the source suffers a diminution of entropy during a cycle, which is precisely equal in amount to the entropy gained by the refrig- erator. 107. Universal Increment of Entropy. The conduction of heat, such as discussed in Art. 77, is an irreversible process; and if, during a given interval of time, a quantity of heat Q is abstracted from a source at the temperature Ti, then, if steady conditions have been assumed by the prism, an equal quantity of heat will be rejected during the same interval of time, to some receiver at a lower temperature, say Ti. The loss in entropy, of the source, is then Q. ENTROPY 133 and the gain in entropy of the receiver is . - Q 92 - ¥ 2 - The gain in entropy of the system, since that of the prism is unchanged, is „ n _Q Q Q(T 1 -T 2 ) . n-n- ¥2 - ¥i - TiTi (10) Since the right-hand member of equation (10) is positive, it fol- lows that the entropy of the system, due to conduction, has increased; and further, since the work which would have been realized on a Carnot cycle, for the quantity of heat Q, oper- ating between the same temperature limits, is Q Tl ~ T2 we see that the work, expressed in heat units, which has been irrevocably lost, due to the quantity of heat Q being transferred by conduction from the temperature T\ to T2, is numerically equal to the product of change in entropy and temperature of the receiver. We will now consider this in a wider sense. Assume, first, an engine working direct, which is thermodynamically perfect; i.e., one which maintains, during its operation, the sum of the entropies of source and refrigerator constant. If now, the mechan- ism upon which the engine does work is perfect and capable at any time of restoring all the energy imparted to it, then the process is perfectly reversible. This, however, is never the case; since all processes are attended by friction and a consequent develop- ment of heat, which is imparted, by conduction and radiation, to the surrounding bodies, there is necessarily an increment in entropy. To put it still more broadly, since heat can be only partially converted into work, and all energy, by friction, ohmic 134 THERMODYNAMICS resistance, hysteresis, impact, etc., is finally degenerated into heat, it would appear that the entropy of the Universe, such as we know it, is tending toward a maximum. And the most gener- alized definition we can give, is :* The change in entropy that a system undergoes during a given irreversible process is a measure of the irreversibility of the process. This is indicated, in a limited way, by equation (10). 108. The concept of entropy has been here introduced, not on account of its great scientific value, in the domain of theo- retical physics, but rather because so many of the discussions of practical thermodynamics are simplified so largely by its use. For our purposes, the two most important statements are: The change in entropy during a reversible process, is numerically equal to the ratio of change in heat to the temperature at which the change takes place; and reversible adiabatic processes are also isoentropic. 109. Temperature Entropy Diagrams. From the equation -J: dQ T' where 9 is the change in entropy, it follows, immediately, that dQ = Tdr, and, for a reversible isothermal process, we have Q=rjj^=ru 2 - 91 ) (ii) Applying equation (11) to a Carnot cycle, we have for the heat abstracted from the source, during isothermal expansion, Qi = ri(q>2- (15) and, the condition of the working substance, as regards temperature and entropy, is represented by the point B. Since, during this change, the temperature is constant, the line representing the change in entropy is a straight line parallel to the

as before, and the Carnot cycle on the T- 9 diagram is represented by a rectangle; the heat abstracted from the source, during a cycle, being measured by the area FABE=T 1 (y 2 -n), and the heat rejected to the refrigerator is measured by the area FDCE=T 2 (n-n)- The difference between these two areas is a measure of the heat converted into work; i.e., the area ABCD = (T 1 -T 2 )(n~n) is a measure of the external work done. CHAPTER XI APPLICATIONS OF TEMPERATURE-ENTROPY DIAGRAMS 110. In Art. 102, it was shown analytically that, even under perfect conditions, a steam engine and boiler cannot convert into work as large a fractional part of the heat taken from a source as can an engine operating on a Carnot cycle. We will now show this by means of the T-

Fig. 21. But, as has been previously shown, a relation must subsist, such that mn q= mp (id hence, by combining equations (10) and (11), we find mn = mp—. s (12) Finding a number of points in this manner, the curve BnF is determined; and the loss of work, due to using the steam non expansively, is obviously measured by the area BFE. 115. Loss of Work Due to Incomplete Expansion. If there is a partial adiabatic expansion before exhaust or condensation, 146 THEKMODYNAMICS then the T-y diagram takes the form as depicted in Fig. 22. DA is the T- 9 curve for the heating of unit mass of water, AB the curve for complete evaporation, at the temperature Ti, BC the saturation curve, BG the curve- for adiabatic expansion, to the temperature V ', GnF the curve of condensation at constant volume, and FD the curve of condensation at the temperature T2. To determine the curve of condensation at constant volume, we must find a point n for the temperature T, such that mn q= — ; mp (13) A T, B „/ T' G V / / T 2 _T_ / / / / a. \P D F b E C Fig. 22. where q is the dryness corresponding to that temperature. Now, the volume occupied at the point G, corresponding to the tempera- ture T', is where q' is the dryness at the temperature T', and s' the volume of unit mass of saturated vapor at T' . Also, for constant volume ! qs = q's'\ (14) where q is the dryness at the temperature T, and s the correspond- ing volume for unit mass of saturated vapor. From equation (14), we have ff-fff (15) APPLICATIONS OF TEMPEEATUEE-ENTEOPY DIAGEAMS 147 Finally, combining equations (13) and (15), we obtain s mn = q f -mp (16) o Finding a number of points in this manner, the curve GnF is determined. If the steam is initially not dry, the curve GnF is determined in precisely the same manner; but, the curve BG is shifted toward the left by a fractional part of the length AB, depending upon the amount of initial priming. The work lost, due to incomplete expansion, is measured by the area GFE; and, by an inspection of the figure, it becomes obvious that the loss of work decreases very rapidly as the expan- sion is increased. As an example, were the expansion continued up to the point a, the loss of work, due to incomplete expansion, would be measured by the small area abE. The greater the amount of expansion, after cut-off, the longer, necessarily, the stroke of the piston; but, the longer the stroke, other things being equal, the higher the first cost of the engine, and the greater the loss of work due to friction. Hence, there must be a point beyond which it is uneconomical to carry the expansion. Besides increasing the friction, there are still other losses introduced, by carrying the expansion too far; these will be considered later. Just how far to carry the expansion so as to give the best economy is a problem far too complex to be solved theo- retically. At best, theory can only serve as a guide, and the most economical expansion must be determined experimentally. For a simple engine, the point of cut-off may vary from about one-third to one-sixth of the total stroke; depending upon whether the engine is running non-condensing or condensing. But, it must always be remembered that the ratio of cut-off to length of stroke depends upon various conditions, which will be better understood after we have dealt with the actual behavior of the steam in passing through the cylinder. 148 THERMODYNAMICS 116. Gain of Work Due to Superheating. If the steam, after being completely evaporated, be superheated, the ideal coefficient of conversion is increased. But, this must not be under- stood to mean the same proportional gain in work; for, lubrica- tion and packing become more difficult as the temperature is increased; and when the temperature becomes very high, radiation becomes excessive. Let, in Fig. 23, DA be the T-

y/. If the steam be superheated to a temperature, such that the adiabatic EF passes through the point C, then the steam will be just saturated after it has been expanded to the temperature 7 7 2. To find the amount of superheating that will bring about this condition, it is only necessary to equate entropies, for the points E and C. The change in entropy, in going from D to E, is and the change in entropy, in going from D to C, is * =T- 2 ' But, in order that the adiabatic EF pass through the point C, •p' must equal x<' III III ' 1 ! i i i pi qi It jb 9 Fig. 30. to the cylinder walls is to that taken from them as the area pnBq is to the area pnut. As previously explained, the pressure in the cylinder during admission, due to wire drawing, is less than the pressure in the supply pipe; and as stated in Art 125, superheating may occur. In general, however, on account of initial priming, even if there were no condensation during admission, due to the incoming steam coming into contact with the cylinder walls of lower temperature, there would still be present a certain amount of moisture. Most authors assume, in discussing the exchange of heat between the steam and cylinder walls, that the steam is 170 THERMODYNAMICS dry at cut-off. This assumption is neither justifiable nor nec- essary. No prediction can be made unless the dryness of the supplied steam is known. If, however, the dryness of the steam in the supply pipe is known, together with its pressure and the pressure in the cylinder, during admission, the dryness of the steam in the cylinder, during admission, had there been no con- densation, is readily computed. Let this hypothetical dryness be represented on the T- 9 diagram (Fig. 30) by Aa Since, however, the actual dryness at cut-off, as found from the indicator diagram, is AB q = AC it follows that an amount of condensation, represented by the change in entropy Ba, has taken place during admission. Hence, the heat given up to the cylinder walls by the steam, during admis- sion, is measured by the area Babq. Heat is also given to the cylinder walls during compression; this, however, is not entirely lost. Since, due to this, the tem- perature of the walls is raised, and the condensation during admission, is partially reduced. 132. Steam Jackets. The fluctuations in temperature of the cylinder walls, as described in Art. 129, are the more pronounced the lower the speed of the engine. In other words, the higher the speed of the engine, the smaller the interval of time during which exchanges can take place between the cylinder walls and the steam, and as the speed becomes very high the exchange becomes very small. There is, however, another element to be considered, viz, the cooling of the cylinder, due to the fact that it is always at a higher temperature than the surroundings. This loss of heat must continually be made up by the incoming ELEMENTARY STEAM AND ENGINE TESTS 171 steam; and hence, increases the condensation. This loss of heat is partially prevented by having the cylinder jacketed by some non-conducting material. In some cases, a steam-jacket is used, which is maintained full of live steam, • taken directly from the supply pipe; and therefore, the pressure of the steam, in the jacket is usually slightly higher than the pressure of the steam, during admission, in the cylinder. There is, therefore, less condensation, during admission, than there would be were the steam jacket absent; and reevaporation begins earlier. On the other hand, the jacket increases the area of the exposed surface; hence, a greater loss of heat, due to radiation. If complete reevaporation takes place before the exhaust-valve opens, the steam during the exhaust-stroke is dry, and very little heat is absorbed by it from the steam in the jacket. The question then is, whether the thermodynamic gain, obtained by applying the heat at a higher temperature, to bring about reevaporation at the earlier part of the stroke, is greater than the energy lost, in the jacket steam, to bring about this reevaporation, plus the greater radiation and heat imparted to the exhaust steam. This question can be answered only by experiment. Experiments performed, on slow and moderate-speed engines, appear to indicate a decided gain in economy, by using a steam-jacket. In a great many cases, however, such discrepant results have been obtained, that it is extremely difficult to say under just what conditions steam jackets are beneficial. 133. Brake Power. The output of an engine of low power, is most conveniently measured by a friction brake, which is a device by means of which the power, developed by the engine, is absorbed in overcoming the friction applied to the surface of its fly-wheel; the force required to prevent rotation of the brake, being measured by a balance. The most common form assumed by the friction brake is depicted in Fig. 31. It consists of a number of wooden blocks fastened by means of bolts, to steel bands, wrapping, approx- 172 THERMODYNAMICS imately, two-thirds of the circumference of the fly-wheel. The wing-nut w on the bolt b makes it possible to vary the pressure to any desired value. The tie-rod t, going from the lower part of the bolt b to the lever, is merely to give rigidity to the brake. The rim of the fly-wheel is provided with flanges, so that water may be contained in it, to absorb the heat developed by the work done, in overcoming the friction. Assume, now, that the fly-wheel is rotating in the direction as indicated by the arrow. Then, due to friction, the brake will tend to rotate in the same direction; and to prevent this, a certain force is applied to the lever, at the point p. This force Fig. 31. is most conveniently measured by a balance; which may be either a spring balance or a beam balance. Let the fly-wheel be making N r.p.m. (rotations per minute), the net weight registered by the balance, to prevent rotation, be W lbs., and d be the horizontal distance between the center of the shaft and point of contact p. Then, since power is numerically equal to the product of angular velocity and torque, we have, employing the minute as the unit of time, P = 2%NWd ft.-lbs. per min. ; and since one horse-power is the equivalent of doing work at ELEMENTARY STEAM AND ENGINE TESTS 173 the rate of 33,000 ft.-lbs. per minute, we have, for the brake horse-power, BJLR = ^ooo a4) In the case of very small units, the torque is frequently meas- ured by wrapping a canvas belt around the pulley, and applying tensions to its two free ends. The tensions are then varied, until the machine is loaded to the desired amount, and measured. The torque is then found by taking the product of the difference between the two tensions and the radius of pulley plus one-half the thickness of the belt. In this case, the heat developed by the work done, in overcoming the friction, is also absorbed by water contained in the pulley. When testing high-power machines, it is neither convenient nor desirable to make friction tests. One method used is that of connecting the engine under test to an electric generator, whose efficiency is known, and by means of an ammeter and voltmeter, or else by a wattmeter, determining its output. From the efficiency of the generator and the power delivered by it, the power delivered to it, by the engine, is readily found. Another method for determining the power delivered by an engine, is to make the shaft, through which the power is being transmitted, take the place of a transmission dynamometer. This is accomplished by determining the amount of twist, which a definite length of the shaft experiences, when transmitting the given power. Then, from the length and diameter of shaft, its modulus of rigidity, and the angle of torsion, the torque is readily found. 134. Indicated Power. The power expended on the piston of an engine, by the working substance, as found by means of the indicator diagram, is called the indicated power. During admis- sion and expansion, work is being done by the working substance on the piston; and during exhaust and compression, work is being done by the piston, on the working substance. Hence, the net work done by the working substance, during a cycle, is measured 174 THEEMODYNAMICS by the area enclosed by the indicator diagram. If then, the area of the indicator diagram be determined and divided by the length of the stroke, reduced to the proper scale, the average ordinate is found. The average ordinate, so found, multiplied by the scale of the spring, used in taking the diagram, gives the mean effective pressure. The area of the diagram is most conveniently found by means of a planimeter. There are certain types of planimeters, which are specially designed for determining the mean effective pressure from an indicator diagram. This type of planimeter is very convenient, inasmuch as it is only necessary to set it to the length of the diagram, employing a scale corresponding to the scale of the spring, used in taking the indicator diagram, and following the outline of the diagram with the tracing point of the instru- ment. The mean effective pressure is then given directly by the reading on the scale. The mean effective pressure is the average pressure on the piston, during admission and expansion, minus the average pressure during exhaust and compression; hence, it is the effective pressure, due to which external work is obtained. If the indica- tor spring has been calibrated to lbs. per square inch, then the mean effective pressure is also given in lbs. per square inch; and the total effective pressure on the piston is numerically equal to the product of the mean effective pressure and the area, expressed in square inches, of the piston. If we represent by P, the mean effective pressure, in lbs. per square inch, by A the area of the piston, in square inches, by L the length of the stroke in feet, and by N the number of cycles per minute, then the net work done on the piston, per minute, is W = PALN ft. -lbs.; and the indicated horse-power is ELEMENTARY STEAM AND ENGINE TESTS 175 135. Mechanical Efficiency. The indicated power of an engine is always greater than the power delivered by the engine, by an amount which is equal to the power consumed in overcoming the engine friction. The ratio of the brake horse-power, to the indi- cated horse-power gives the mechanical efficiency; i.e., ^ m = I H P ^^ 136. Thermal Efficiency. The thermal efficiency of an engine is given by the ratio of the power delivered by the engine to the power due to the heat taken from the source. As an example, assume a steam engine to be taking M pounds of steam per minute from a boiler, the total heat of which, per pound, is H. Let the heat of the water in the condenser be h, which we will assume is returned to the boiler without losses. Then the heat, expressed in mechan- ical units, which is taken per minute from the boiler, is JM{H-h) ft. -lbs.; and if W represents the number of ft .-lbs. of work delivered per minute by the engine, then the thermal efficiency is W Eh = JM{H-h) (17) We will now illustrate equation (17) by a numerical example. Assume an engine making 300 r.p.m., doing work against a friction brake whose arm is 5 ft., and which requires a force of 135 lbs., applied at its end, to prevent rotation. If the engine is consuming 16 pounds of saturated steam per minute, under a pressure of 80 lbs., and returns the water without losses directly to the boiler, from the condenser, where the pressure is 2 lbs., what is the thermal efficiency? Substituting, in equation (17), we find „ 2xX300Xl35X 5 n . ^ = 778Xl6(1182-94.2) =9 - 4 per Cent; 176 THERMODYNAMICS where 1182 * is the total heat of steam under a pressure Of 80 lbs., and 94.2 the heat of the liquid, corresponding to the temper- ature of the steam, under 2 lbs. pressure. 137. Commercial Efficiency. The commercial efficiency of an engine is given by the ratio of the power delivered by the engine to the power which a perfect heat engine, working between the same temperature limits, would deliver. Let the symbols have the same significance as in Art. 136, then the work, per minute, which a perfect heat engine would deliver, is JM(H-h)^K ft.-lbs.; and the commercial efficiency is W Ec = „ ff (18) o Substituting in equation (18), the numerical data given as an illustration in the preceding article, we find „ 2^X300X135X5 qo E c = t^w = 39.0 per cent. 778X16(1182-94.2)^1 This is the proper method of comparison; i.e., comparing the actual performance of the engine with an ideally perfect engine, operating between the same temperature limits. When an engine exhausts to the atmosphere there is, of course, no heat returned to the boiler by means of the condensed steam, and the heat h, in equations (17) and (18), is lost. It is, however, not proper to charge this entire loss of heat against the engine; since, by proper arrangements part of the heat at least, contained by the liquid, can be returned to the boiler. There are other methods for rating the performance of engines, which are in certain cases, very convenient. One is, specifying * Taken from Peabody's Steam Tables. ELEMENTAEY STEAM AND ENGINE TESTS 177 the number of pounds of steam per B.H.P. hour, consumed by the engine. Another is, specifying the number of B.T.U. per B.H.P. hour, or the number of B.T.U. per K.W. hour of energy delivered to the bus-bar. The latter is especially expressive; giving, as it does, the rating of the power plant as a whole. 138. Rankine's Cycle. Another important comparison may be made by the aid of Rankine's cycle, the indicator diagram of which is shown in Fig. 32. This indicator diagram is based on the assumption that the cylinder of the steam engine has no clearance and is perfectly insulated. AB represents the admis- sion at constant pressure pi, BC represents the adiabatic expan- sion to the pressure p2, and CD represents the exhaust, at constant pressure p2. Assume now, that we are dealing with a unit mass of liquid, whose specific volume is a, and that the dryness, during admission, is qi. If the specific volume of the steam, at the pressure pi, is sit "then the volume of the mixture, at the point of cut-off, is vi = qisi + (l- qi)a = qi(si — a) + a = qi^i+a; . (19) where \n is the increment in volume due to complete evaporation at the pressure pi. Since the pressure, during admission, is con- stant, the work done by the steam, on the piston, is piVi = pi(qi[Li + Qi MC V (T2-T!) from which ^-JWY ••••••• (4) Since both DE and BC represent adiabatic changes for the same changes in volume, we have, from equation (50), Art. 49, ' V2 y-i Tz T± T3-T4 >\ n - 1 = T 1= T4 L = 7 T 2 Tx vil T 2 T x T 2 -T^ where v\ is the volume of the mixture before compression, and v 2 the volume after compression. Hence equation (4) becomes ,-i-^r'. in, -©■• IDEAL COEFFICIENT OF CONVERSION AND TESTS 213 Equation (5) shows that the ideal coefficient of conversion is a function of the ratio of the volume before compression to the volume after compression; and increases with the amount of precompression. 162. Theoretical Temperatures. The temperature which would obtain upon complete combustion, if there were no losses, is readily computed for any given case. That is, the theoretical rise in temperature, viz, T2 — T1, is numerically equal to the ratio of the heat of combustion to the thermal capacity of the products of combustion. However, it is found to be necessary, in order to have proper lubrication between the piston and cylinder walls, so as to prevent deterioration of material, to abstract heat from the cylinder walls, either by water jacketing, or else by air cooling. The former, that is water cooling, is brought about by having water, at a comparatively low temperature, circulate in a jacket surrounding the cylinder; and the latter, viz, air cooling, is brought about by increasing the surface of the exposed part of the cylinder by means of ribs, and having a stream of air playing over it continuously, by means of an air blower of some kind, or else, as is the case in some automobile engines, the circulation of air is brought about by the motion of the car. In any case, the heat abstracted, due to either water or air cooling, limits the rise in temperature. Therefore, the temperature found in the cylinder of an internal combustion engine, is always less than that pre- dicted from the heat of combustion and the thermal capacity of the products of combustion. Frequently, the actual temperature is found to be only 50 per cent of the theoretical temperature. 163. Standard Diagram. In deducing the expression for the ideal coefficient of conversion for the internal combustion engine, Art. 161, certain assumptions, in regard to thermal capacities and volumes before and after combustion, were made, which are only approximations. But the errors involved in these assump- tions are very small in comparison with the difference between the actual and theoretical temperatures obtaining in the cylinder. 214 THERMODYNAMICS However, the diagram described in Art. 161, and the results deduced therefrom, though differing materially from what can be realized in practice, are very convenient as a basis for com- paring the performances of internal combustion engines. Elementary Engine Tests 164. Brake Power and Indicated Power. The power delivered by an internal combustion engine is determined in precisely the same manner as is that of a steam engine. This has been fully described in Art. 133. Furthermore, the indicated power of an internal combustion engine is also found in the same manner as is that of a steam engine, as described in Art. 134. But it must be emphasized that N, in equation (15) of Art. 134, represents not the number of revolutions per minute of the fly-wheel, but the number of cycles per minute in the cylinder under test. The ratio of Brake Power to Indicated Power is, of course, in the case of an internal combustion engine, as well as in the case of a steam engine, a measure of the mechanical efficiency. It is found, however, that the mechanical efficiency of an internal combustion engine, other things being equal, is always less than the mechanical efficiency of a steam engine. This is principally due to the fact that, owing to the high temperatures existing in the cylinders of internal combustion engines, the lubrication is not as good as that obtained in steam cylinders. 165. Thermal Efficiency of Internal Combustion Engine. The thermal efficiency of an internal combustion engine is, of course, the ratio of the power delivered by the engine, to the power due to the fuel consumed. In making a test, the engine is loaded by means of a brake, or some other contrivance, to the desired amount. Then, in the case of a gaseous fuel, the volume of gas consumed is measured by means of a meter. Simultaneously with this, as described in Art. 158, the calorific value of the gas is determined. The best results are obtained if continuous tests IDEAL COEFFICIENT OF CONVERSION AND TESTS 215 are made for the calorific value of the fuel; that is, if the supply to the fuel calorimeter is tapped directly onto the main, supplying fuel to the engine, and samples of the fuel are tested, for calorific values, throughout the entire run. The ratio, then, of the work done by the engine, during the test, to the work, expressed in the same units, due to the fuel consumed, which is equal to the product of the volume of gas consumed during the run and the mean calorific value of the gas, as found by means of the gas calorimeter, is a measure of the thermal efficiency. Or, if a liquid fuel be used, the work due to the fuel consumed, is found from the product of the mass of liquid consumed, during the run, and the mean calorific value per unit mass. The calorific value, per unit mass of the liquid, is determined as described in Art. 159. 166. Actual Indicator Diagram of Internal Combustion Engine. By means of the indicator diagram, taken from an internal com- bustion engine, the behavior of the working substance may be conveniently studied. The actual indicator diagram differs, of course, from the ideal indicator diagram, as depicted in Fig. 40. Whereas, in the ideal indicator diagram, the line representing the aspirating stroke, is parallel to the axis of zero pressure, in the actual indicator diagram the line representing the aspirating stroke approaches the axis of zero pressure, as represented by AB in Fig. 41. This is due to the throttling effect of the inlet- valve, on account of which, the pressure in the cylinder decreases as the piston advances. In Fig. 41, 01 and OH are, respectively, the axes of zero pressure and of zero volume, and A A' is the atmos- pheric line. The curve AB, as just stated, represents the aspira- ting stroke; and shows the pressure in the cylinder at the end of this stroke, less than the atmospheric pressure, by an amount A'B. The compression of the mixture, which is approximately adiabatic, is represented by the curve BC. The combustion of the mixture, and consequent rise of pressure in the cylinder, is represented by the curve CD, which is, if the ignition has been properly timed, practically parallel to the axis OH. DE is the 216 THEEMODYNAMICS curve representing the expansion of the products of combustion. At E the exhaust-valve begins to open, and the pressure decreases rapidly to the end of the stroke F. The expulsion stroke then begins, and the pressure continues to decrease rapidly up to the point G. At this point, the exhaust-valve is fully open and the pressure decreases gradually as represented by the curve GA, to the end of the stroke, where the pressure is practically atmos- pheric, and the cycle has been completed. The pressure in the cylinder, during expulsion, is higher than that of the atmosphere due to the resistance offered by the exhaust-valve, to the outflow of the products of combustion. That part of the diagram, which represents the effects due to throttling, has been purposely exaggerated. The curve DE is usually, more or less, wavy; this may be due to various causes. If the vibrations appear to be regular and of decreasing amplitude, they are principally due to the inertia of moving parts of the indicator. On the other hand, if the pressure is apparently constant for a time, then suddenly decreases, etc., the waves are due to friction between the piston and cylinder of the indicator. This trouble is readily removed by proper cleaning and lubrication of the piston and cylinder. Furthermore, waves may be established in the mixture in a manner similar to that described in Art. 69; i.e., as the piston begins to compress the IDEAL COEFFICIENT OF CONVERSION AND TESTS 217 mixture a wave of compression travels through the medium to the other end of the cylinder, where it is reflected, with change of sign. This reflected wave then travels toward the piston; and when it meets the piston reflection again takes place, etc. In this manner, inequalities in pressure may be established, which under certain conditions may persist throughout the compression and expansion strokes. However, in general, these inequalities will not be manifested to any marked degree on the indicator diagram, since the inertia, of the moving parts of the indicator, will tend to suppress them. By an inspection of Fig. 41, it is obvious that the work done on the piston during the aspirating stroke is measured by the area JABK; and, likewise, the work done by the piston during the com- pression stroke is measured by the area KBCJ. During combus- tion, since there is no displacement of the piston, the work done is zero. During expansion the work done on the piston is measured by the area JDEFK. And, during expulsion, the work done by the piston is measured by the area KFGAJ. By taking the algebraic sum, we find that the net work done by the working sub- stance, during the cycle, is measured by the difference between the areas CDEFGi and AiB. Hence, if the mean effective pressure is determined by means of a planimeter, the tracing point of the planimeter, in tracing the area AiB must travel in a sense opposite to that pursued in tracing the area CDEFGi. That is, if i be the starting point, then, to find the difference between the two areas, the tracing point of the planimeter must follow, in order, the path i, C, D, E, F, G } i, A, B, i. The area AiB represents the work lost, due to valve throttling, and is, in well designed engines, small in comparison with the area CDEFGi. If the power lost, due to valve throttling, is large in comparison with the total indicated power, the valves must be readjusted. In general, the spring which gives good results for measuring the indicated power, has a modulus so high that the 218 THERMODYNAMICS part of the diagram, representing the power lost, due to valve throttling, is too small to be accurately measured. But, by using a stop, so as not to injure the spring, a much lower scale spring may be employed. In this manner the power lost, due to throt- tling, and also the amount of precompression may be accurately determined. 167. Efficiency and Precompression. In Art. 161, equation (5), it was shown from theoretical considerations that, other things being equal, the thermal efficiency increases with the amount of precompression. This is found to be so in practice. There are, however, limits, beyond which the precompression may not be carried, due to the severe strains to which the engine is sub- jected during the explosion of the mixture. Tests made, in the Cooper Union Laboratories, on a Fair- banks 8 H.P. gas engine, gave the following results: R.P.M. B.H.P. Efficiencies. vjvz Thermal, Per Cent. Mechanical, Per Cent. 4.72 4.96 5.09 5.31 368 395 414 478 6.54 7.60 7.92 9.14 13.9 15.8 16.1 19.8 74.0 72.0 68.0 67.0 The value given for the B.H.P. is, in each case, the maximum load the engine would carry for the given precompression. On attempting to carry the precompression higher than that given by i>i/fl2 = 5.31, it was found that the vibrations set up in the engine were so violent that satisfactory operation could not be obtained. From the table it is seen that the thermal efficiency increases rapidly with increased precompression. The mechanical efficiency, however, is considerably reduced. The fuel used during these tests was illuminating gas having a calorific value of about 590 B.T.U. per cubic foot. The amount of precompression which, in any case, gives the best results depends, of course, upon the quality of the fuel used. IDEAL COEFFICIENT OF CONVEBSION AND TESTS 219 It must, however, be emphasized that in any case, without considering the severe strains to which the engine is subjected, the amount of allowable precompression depends upon the tem- perature of ignition for the fuel used. For, if the temperature of the mixture due to the heat developed during the compression, becomes higher than that of ignition, premature explosions will occur, and the engine will not operate successfully. 168. T-9 Diagrams and Internal Combustion Engines. The T-

»> Since now the volume, at the end of the adiabatic compression, is i_ _i_ Vi = Pa n V a pi », the work done, during expulsion, is n-l 1 2zi /©A n W3 = PiV 1 =p a n V a pi n =PaVal—J . • (11) Taking the algebraic sum of the right-hand members of equa- tions (8), (10), and (11), we find, for the net work done by the compressor, n— 1 n— 1 which reduces to n-Y [\p a J j It was stated, in Art. 176, that the cycle having isothermal compression may be taken as a standard cycle. The ratio of the work done, during a cycle, when the compression is iso- thermal, to that when the compression is adiabatic, may be termed the theoretical efficiency of compression. Dividing equation 238 THERMODYNAMICS (6) by equation (12), we find, for the theoretical efficiency of compression, log Y] = n-l n J Ip n-l] \p a/ (13) As a matter of convenience a table is here given, which was obtained by computing the theoretical efficiencies, by means of equation (13), on the assumption that n has the value of 1.4. Pi Pa Per Cent. El Pa Per Cent. 1.5 94.3 6 76.6 2 90.4 7 74.8 3 85.1 8 73.2 4 81.5 9 71.9 5 78.8 10 70.7 It will be noted that, when the ratio pi/p a = 3, the theoretical efficiency of compression is approximately 85 per cent, and when pi/p a = / l, it is approximately 81.5 per cent. Hence, when the ratio of pi to p a is greater than 3 or 4, the losses, from a ther- modynamic standpoint, become excessive. Therefore, if it be desired to operate economically, it becomes necessary to limit the ratio of final to initial pressure to a value between 3 and 4. Equation (12) may be transformed so as to express the work done by the compressor, in terms of the initial and final tem- peratures. Since n-l Ta \Pa) where T a is the initial and T\ the final temperature, and since p a Va = BTa, COMPRESSED AIR AND COMPRESSORS 239 we find, by substituting in equation (12), w =^i RT ^¥ a - 1 ) (14) And finally, by substituting, in equation (14), for n and R, respectively, the equivalent values, C P /C V , and J(Cp—C v ), we find W = JC P (T 1 -Ta) (15) That is, equation (15) shows that the net work consumed by an air compressor, per cycle, per pound of air, when the com- pression is adiabatic, is precisely equal in amount to the heat, expressed in mechanical units, consumed in elevating the temper- ature of the air, at constant pressure, from that before compres- sion to that after the compression is completed. Equation (15) may be deduced in an entirely different, though very simple, manner. The work done by the piston, during admission, is W 1 =-p a v a =-RT a =-J(C p -C v )T a . ... (a) And, during adiabatic compression, since the intrinsic energy of the air is a function of its temperature only, the work done by the piston is W 2 = C V (T 1 -T a ) .... (6) The work done by the piston, during exhaust, is Ws = Vl v 1 =RT 1 =J(C p -C v )T 1 ; ( c ) taking the sum of Wi, W2, and W3, as given by equations (a), (6), and (c), we find W = JC P (T 1 -T a ). 178. Multi-stage Compression. Various methods, in which an attempt is made, to bring about compression, approaching 240 THEKMODYNAMICS an isothermal process, have been tried; but, it appears impossible to reduce the exponent, n, to a value approaching unity. The various methods used are: Water-jacketing, playing a jet of water into the cylinder while compression is taking place, and spraying, by means of an atomized jet of cold water, the air while it is being compressed. Water-jacketing appears, so far as the results of investigations show, to give very little, if any gain in economy. That is, when recourse is had to jacket cooling, the heat developed by compression is absorbed so slowly that the com- pression is practically adiabatic. When the cooling is attempted by means of a jet of water, played into the cylinder, the exponent, n, may be reduced to a value of about 1.35; and when the cooling is brought about by an atomized spray, the value of n may be reduced to about 1.25. Hence, at best, the compression is far from approaching an isothermal process; and, for efficient operation, when the ratio of the final pressure to the initial pressure is greater than 4, recourse must be had to multi-stage compression. We will consider, first, a two-stage compressor. That is, during the aspirating stroke, a certain quantity of air, at a pressure p a , flows into the cylinder, which on the return stroke is compressed to some pressure p±; the relation of pressure and volume being given by the equation pv n = k ; where the exponent n, depending upon the method of cooling applied, may have a value ranging from about 1.25 to 1.4. At the end of this compression stroke, when the pressure pi has been attained, the air is expelled into a receiver under a con- stant pressure p\. The receiver has a jacket through which water is circulated in a manner such that the heat developed, during compression, is removed; and the product of pressure and volume, after cooling in the receiver, is equal to the product of pressure and volume at the instant the compression began. COMPRESSED AIR AND COMPRESSORS 241 In other words, the condition of the air, as regards pressure and volume, in the receiver, is the same as though the compression had been isothermal; i.e., PlV%= PaVa. Let, in Fig. 47, the line AB represent the aspirating stroke, for the low-pressure cylinder, and the pressure and volume, corresponding to the point B, be given, respectively, by p a and v a . The compression then takes place approximately adiabat- err-- f Fig. 47. ically, as represented by the curve BE. At the point E the exhaust-valve opens and expulsion takes place, to the receiver, at a constant pressure pu The volume of the receiver being large in comparison with that of the first cylinder, the pressure in it is sensibly constant; and the volume of air, in cooling from T\, the temperature at the end of the compression BE, to T a , the atmospheric temperature, shrinks from the volume, represented by FE, to that represented by FG, such that Pivi=p a v a ; and the point G is on the isotherm through the point B. For the same quantity of air, then, the line FG represents the aspirating 242 THERMODYNAMICS stroke for the second, or high-pressure, cylinder. The curve GJ, represents the compression, which is practically adiabatic, to the pressure p2, existing in the reservoir in which the air is stored. Finally, the expulsion stroke to the reservoir is represented by the line JD. The various quantities of work involved during the cycle are as follows: During the aspirating stroke, for the low-pressure cylinder, the work done on the piston is measured by the area ABKO; and during compression, to the pressure pi, the work done by the piston is measured by the area KBEL; and during expulsion, to the receiver, the work done by the piston is measured by the area LEFO. Hence, the net work done by the piston, in the low-pressure cylinder is measured by the area ABEF. In a similar manner, we find that the net work done, by the piston in the high-pressure cylinder, is measured by the area FGJD. The total net work done, therefore, by the two-stage compressor, during the cycle, is measured by the area ABEGJD. Had the compression taken place in a single-stage compressor, between the same limits of pressure, the net work done, by the piston, would be measured by the area ABEND; where the curve BEN repre- sents an adiabatic through the point B. Hence, the saving in work, neglecting losses, by employing a two-stage compressor, is measured by the area ENJG. And the work done by the piston of this two-stage compressor, in excess of that which would have been done had the compression been isothermal, is measured by the sum of the areas GBE and CGJ, where the curve BGC repre- sents an isotherm. By equation (12), we have for the work done in the low-pres- sure cylinder, per cycle, when 1 pound of air is taken in at a pres- sure p a , is compressed adiabatically to a pressure p\, and expelled at this pressure to a receiver, "'^•-{(gf - 1 ! 2* (20) 244 THEEMODYNAMICS Substituting the value of pi as given by equation (20), in equations (16) and (18), we find and, *-£H (£»*-' » .-V>.((e)"*"'-i n-r [\Pa . . . (21) (22) From equations (21) and (22) it is seen that, if the work done, during a cycle, by a two-stage compressor, is to be a minimum, it must be equally divided between the two cylinders. Taking the sum of the right-hand members of equations (21) and (22), we find the net work done, when employing the most efficient compression possible, by a two-stage compressor, in taking air under a pressure p a and expelling to a receiver, under a pressure V2, is n-l W = ~iPaV a { (-) ^ - 1 1 ft.-lbs. per pound. . (23) If the compression is brought about by three stages, the final pressure being p%, and the pressures of the intermediate receivers, respectively, p\ and p2, then, on the assumption that, in the two intermediate receivers, the temperature is reduced to that of the atmosphere, the work done in the first, second, and third cylinders is given, respectively, by and Wi = n—1 PaVa W2 = -- 1 PaV, n — 1 n-l n Pa n- l ^ n Pi. n-l (24) (25) (26) COMPRESSED AIR AND COMPRESSORS 245 Taking the sum of Wi, W2, and W3 we find, for a cycle, the net work done by the three-stage compressor, is The right-hand member of equation (27) is a minimum when the expression included in the brace is a minimum. Differentiating this expression, first, assuming pi variable, and p a , p2, and pz constant, equating to zero, and solving for pi, we find Pi = Vp^P2 (28) Equation (28) gives the relation of p\ to p a and p2 such that the process in going from p a to P2 shall involve a minimum amount of work. Differentiating again, this time, however, assuming pi and p3 constant, and P2 variable, in order to obtain the relation 2?2 must bear to pi and p% such that a minimum amount of work is involved, while the process takes place from pi to p%, we find V2 = ^pm (29) By elimination we find, from equations (28) and (29), vi=^vln, (30) and V2 = Now, the effective displacement is given by FB=AB-(EF-EA); (41) hence by substituting, in equation (41), the value of EF, as given by equation (40), and the value of EA, as given by equation (38), we find The effective piston displacement not being equal to the actual piston displacement, does not affect the expression deduced for the work done on an air compressor; for, the air remaining in the cylinder, at the end of the expulsion stroke, does an amount of work * K usually has a value of about 50. COMPRESSED AIR AND COMPRESSORS 249 on the piston, in expanding, which is practically equal to that which was done on it while being compressed. The effect of the clearance, then, is merely to reduce the capacity of the cylinder. 180. Throttling and Other Imperfections. The capacity of a cylinder of an air compressor is very frequently more seriously affected by other causes than it is by clearance. In the first place there is always, due to valve friction, a certain amount of throt- tling, which causes the pressure in the cylinder, during the aspira- ting stroke, to be less than atmospheric. Further, due to imper- fect valve action, i.e., the valves not opening or closing at the proper time, the capacity is reduced. And, finally, the temperature of the cylinder walls is usually higher than that of the incoming air, which again tends to reduce the capacity of the cylinder. These combined causes may reduce the apparent capacity, depending upon the speed of the machine, from 5 to 20 per cent. 181. Adiabatic Expansion in Motor. The cycle of an air motor is practically the reverse of that of an air compressor. The admission-valve opens and air from the mains, under a practically constant pressure, forces the piston forward to the point of cut- off, and the work done on the piston, per pound of air is Wi=jpivi) (43) where pi is the pressure in the main, and vi the volume of one pound of air at cut-off. The expansion is then practically adia- batic, and the work done in expanding from the pressure pi, to p a , that of the atmosphere, is n-l (44) w f Pa a l ~ C Va ~-j VVJi W2= I pdv= p\ n vi I p n dp = I - — - Jvx n Jpi n ~ l !_i* Pi The exhaust-valve then opens, the air is expelled under a pressure p a , and the work done, by the air, is 1_ n-l W3=-PaVa=—pi n Vip a « . . . ' . . . (45) 250 THERMODYNAMICS Taking the sum of the right-hand members of equations (43) (44), and (45), we find, for the net work done on the motor per pound of air, n-l w=- p - lV -\< > n — 1 r + * n-l n 1 (Pa n-r \pi 1_ n-l Wi — pinViPa n ft.-lbs. (46) Since the temperature in the mains is practically atmospheric, pivi is the product of pressure and volume, for one pound of air under ordinary conditions, and may be replaced by the constant 27,800. Hence, equation (46) becomes n-l IF = 27,800— — \ n—1 (47) The indicator diagram for the preceding discussion is shown in Fig. 49, in which AB represents the admission, BC the expansion, and CD the expulsion. The work done on the piston during admission and expansion, is measured by the area ABCGF; and the work done by the piston, during exhaust, is measured by the area CGFD. Hence, the net work done, by the air, is measured by the area ABCD. COMPRESSED AIR AND COMPRESSORS 251 Equation (46) may be put into another form, by substituting n-l for pivi, its value RT\, and for (p a /pi) n , its value T a /T\. Making these substitutions, we find from which, for adiabatic processes, since in that case n = C P /C V , we obtain And, since we have finally R = J(Cp — Cv), W = JC P (T 1 -T a ) (48) It must be noted that, in equation (48), T\ is the temperature in the mains, which is practically that of the atmosphere, and T a , the temperature of the air after expanding adiabatically from the pressure pi, that existing in the mains, to p a , that of the atmos- phere. 182. Reheating. When air at atmospheric temperature, and under a high pressure pi, expands to atmospheric pressure p a , the corresponding temperature, T a , will be very low. As an exam- ple, if air under a pressure of five atmospheres, and at atmospheric temperature Ti, expands adiabatically to a pressure of one atmos- phere, its temperature becomes approximately, n-l 2_ T a = Ti(^\ n =522^|V =330=-130°F.; n having been assumed to have the value 1.4. Temperatures as low as this, due to the fact that the moisture present in the air freezes, makes lubrication difficult and clogs the valves, are undesirable at the exhaust of an air motor. To prevent this, 252 THEKMODYNAMICS recourse must be had to reheating; i.e., the air from the mains is passed through a heater before being admitted to the motor. In being heated at constant pressure, the volume of the air is increased, and the ratio of the two volumes is given by T/Ti; where T is the temperature of the air after heating. Hence, equation (46) , giving the work done, per pound of air, on the air motor, becomes =^x'- n -^i{i-(|) n \ Tin-V If, now, T a r is the temperature at the end of the adiabatic expansion, to the pressure p a , and since n = C P /C V , pivi =RT\= J(C P — C v ) Ti, n-l and (pa/pi) n = Ta'/Ti, we find, by substituting in equation (49), Wl= ¥ 1 x c^/ iCp ~ Cv)Tl { 1 ~^)' ' ' (50) And further, since the ratio of final to initial pressure is the same whether there be reheating or not, the ratio of final to initial temperature must also be the same for both cases; hence, T rp t rp±J* . J-a — -L m ) where T a is the final temperature when there is no reheating. Substituting this value of T a ' in equation (50), and simplifying, we find Wi-JcSiTi-T.) (51) J- 1 Equation (51) is the expression, in terms of the three temper- atures, with reheating, for the work done per pound of air, on the air motor. Equation (48) is the expression for the work done per pound of air without reheating. Taking the difference between equations (51) and (48), we find, due to reheating, for the gain in work W' = JC v ~(T l -T a )-JCATi-T a )=JC v (T 1 -T a ) T -~±. (52) COMPRESSED AIR AND COMPRESSORS 253 The heat consumed, expressed in mechanical units, in raising the temperature of 1 pound of air at constant pressure, from T\ to T, is and the work which could be realized from this quantity of heat, by means of a Carnot cycle, is W" = JC V {T-T 1 ) T ^-. ..... (53) Taking the ratio of W, as given by equation (52), to W", as given by equation (53), we find W'_T Ti-Tg ■\ty" Ti T—Ti ^ ' In equation (54), the first factor, viz, T/Ti, is always greater than unity, and for any given case, T\ — T a is a constant. Hence the ratio, W ' /W" ', is greater than unity until the air is reheated to a temperature such that ^ = ^A (55) And, for reheating to a temperature higher than this, the ratio becomes less than unity. Solving equation (55), for T, we find T 2 T = ~r (56) Hence, for reheating to temperatures lower than that given by equation (56), there is a thermodynamic gain; i.e., the gain in work, due to the heat applied in reheating the air, is greater than that which could be realized if an equal quantity of heat were utilized on a Carnot cycle for the same limits of temperature. To illustrate, we will assume a particular case and solve for T. Let pi, the pressure in the mains, be six atmospheres, T\ be 522, and p a , the final pressure, be one atmosphere; then M ter= 522 (^ f 254 THERMODYNAMICS From equation (56), we find _ T x * (522)2 5- = 871 + . 522( 6 - Giving a temperature for reheating, above that existing in the mains, of practically 349° F., which is higher than ever employed in practice. The heat consumed in reheating is applied to much better advantage than in the case of a steam engine and boiler. Further- more, since the fuel used in reheaters may be of a much lower grade than that ordinarily employed for heat motors, there is a saving in cost of fuel. 183. Loss of Head in Transmission Pipes. When a liquid flows in a pipe, there is always, due to friction between the liquid and the surfaces with which it comes into contact, a resistance to be overcome; and on account of this, there is a loss in pressure. That is, when friction is taken into account, Bernouilli's Theorem, which states that, for the steady flow of a liquid in parallel stream lines without friction, the pressure head plus the velocity head plus the static head is a constant for any section under consideration, no longer applies. The energy consumed in overcoming fric- tion, is manifested by eddy currents. These eddy currents in turn subside; and the energy, possessed by them, is converted into heat, which in turn is lost by being dissipated to the surroundings. The loss of energy thus experienced by a given mass of the liquid is usually expressed by a loss of head. That is, the loss of head, experienced by a unit mass of the liquid, is numerically equal to the vertical height through which it would have to fall to do an amount of work equal to that consumed in overcoming the friction. From a great number of experiments upon the flow of liquids in pipes, the following facts have been adduced : The loss of head is very nearly proportional to the square of the speed of flow, COMPRESSED AIR AND COMPRESSORS 255 varies directly as the length of the pipe, as the wetted perimeter, and inversely as the cross-sectional area of the stream. These relations may be stated symbolically as follows : H =f s if' < 57 > where / is an experimental constant depending upon the nature of the liquid and inner surface of pipe, and H, s, L, P, and A are, respectively, the loss in head, speed of flow, length of pipe, wetted perimeter, and area of stream. For any particular cross-section the ratio of A to P is a constant; which is termed the hydraulic radius, and may be replaced by the symbol K. Hence, equation (57) may be written H =fSk ^ Since the temperature of air, flowing in a pipe of any consider- able length, is sensibly constant, the product of pressure and volume is also practically constant; and hence, as the pressure falls the speed of flow must increase. Therefore, since equation (58) assumes a constant speed, it is not directly applicable to the flow of air, or any other gas. In the limit, however, we have dH=f~dL. ....... (59) And, since the loss of head is numerically equal to the work done by a unit mass of the substance, we have dH = pdv; (60) where p is the pressure, and dv the change in volume, per unit mass, for the section under consideration. From equations (59) and (60), we find pdv=f£ K dL (61) 256 THERMODYNAMICS There is, of course, due to change in speed, also a change in kinetic energy; but, in general, this is so small in comparison with the total loss of head that it may be neglected. Substituting, in equation (61), for dv its value as obtained from the equation pv = RT, we find f dp= - f m dL • • (62) Under steady flow the mass of air passing any section, for a given interval of time, is a constant throughout the entire length of pipe. Hence, we have, for the speed of flow, s= ~a=^pa> (63) where M is the mass passing any section per unit time, v the volume per unit mass, and A the cross-sectional area of the pipe. Sub- stituting the value of s, as given by equation (63), in equation (62), we find , M 2 RT „ pdp= ~ f 2gKA^ dL; from which )j dv= - f w^) dL; (64) where pi and p2 are, respectively, the initial and final pressures, and L the length of the pipe. Finally, integrating, as indicated in equation (64), we find v*-*-*^ <«> From equation (63) we have Vl sM 2 '' (66) COMPKESSED AIR AND COMPRESSORS 257 where si is the initial speed. Dividing equation (65) by equation (66), member by member, we find Pi 2 ~P2 2 =f sJL pi 2 ] gKRT { °° Solving equation (67), respectively, for p2, si, and/, we find »-»(?-{gm?: (68) Sl and K^^)' « '^>m <™ By means of equations (65), (66), (68), and (69), the necessary calculations, for any given case, may be made; and, by means of equation (70), the coefficient / may be found for a given set of observations. The ratio A/P is, for cylindrical pipes, a function of the diam- eter only; i.e., xP 2 /4 ^D %D 4* We may substitute, then, in equation (68), the following con- stants: = 32.2, K=D/4:, and # = 53.3, and find f Sl 2 L V /i M 2 L \i *-* 1 - Mx .wxfr) ■"(-«»)■ <7 " In a similar manner, the various equations may be simplified. Equation (71) is, perhaps, best illustrated by assuming a concrete case, and solving for the terminal pressure. As an example, let it be required to find the final pressure, for the case when the initial pressure is six atmospheres, the temperature 258 THERMODYNAMICS 62 °F., the quantity of air required 1200 cu.ft. per minute, the length of pipe 5 miles, and the diameter of the pipe is 1 ft. First of all, from equation (71), it is obvious that the pressure may be specified in any units whatsoever. From a series of observations made by Riedler and Gutter- muth upon the compressed air system of Paris, extending over a distance of about 10 miles, the diameter of the cast-iron pipe being very nearly 1 ft. (exactly 300 mm.), Professor Unwin deduced for the coefficient /, in this particular case, the value of 0.0029.* It must be remembered that this is not the coefficient for a straight piece of cast-iron piping; including as it does, bends and joints, and also a small amount of leakage. Though transmissions to such distances are unusual, the value just quoted for the coefficient is probably a good average value to use for a practical case for the same diameter of piping. That is, in any practical case, for piping of an equal diameter, we should probably find the average losses per given length, approximately the same. From the conditions we have pi = 88.2 lbs. per square inch, %D 2 T = 522, L = 26,400 ft., D = l ft., and §i = (1200/60)/ = 25.5 ft. per second. For /, we will use the value 0.0029. Substituting these values, in equation (71), we find QQ o/i 0-0029X25.5 X26,400 \l _ _ ., P2 = 88.2^1 4 29x52 2 ) =77.8 lbs. per sq. in. Thus giving a loss in pressure of about 11.8 per cent. It must, however, not be understood from this, that the percentage loss of power in transmission is also 11.8 per cent. The efficiency of transmission is found by taking the ratio of the work which the air motor can do in expanding adiabatically from the pressure P2 to that of the atmosphere, to that which would have been obtained had adiabatic expansion taken place before transmission. * "On the Development and Transmission of Power," by W. C. Unwin. COMPRESSED AIR AND COMPRESSORS 259 That is, the efficiency of transmission is , .-M'-feTl -(ST _ ,-v4-feH Htr Substituting in equation (72), for p a , pi, and p2, respectively, 14.7, 88.2, and 77.8, we find, for the efficiency of transmission, 2_ 1 /14.7 = — \UA^_ u.7\t "°..2 where n is assumed equal to 1.4. It is thus seen that, though the loss in pressure is about 11.8 per cent, the loss in power, due to this loss in pressure, is only about 5.5 per cent. The efficiency of transmission may also be defined, depending upon the point of view, as the ratio of the work that could be realized, before transmission, by allowing the air to expand isothermally, to that which would be realized by means of isother- mal expansion after transmission. In any case, for pressures such as are ordinarily employed, the value found, for the efficiency of transmission, by this comparison will not differ materially from that found by means of equation (72) . If we make the computa- tion for this particular case, we find, by assuming isothermal processes, , 77.8 l0g lA7 *= -88 is the work actually required, then the commercial efficiency is W 5 W 5 X Ti UU; The heat which must be carried away by the circulating water, in the cooler, per pound of air, is Hi = C v (T 2 -T a ) (11) The cycle of an air refrigerating machine may be conveniently represented by means of the T-y diagram. By equation (12), Art. 169, the change in entropy, when both the pressure and volume vary, is n -n = CAog^+C P \og v ^ (12) Pi Vi In the cycle just discussed it was assumed that the pressure, during the absorption and rejection of heat, remains constant. Hence, equation (12) becomes ?2-9i = C,logg = C,log^; .... (13) and the heating and cooling curves, on the T-

B c^/ T a T L A D To^^ E F Fig. 51. to Ti, is represented by the curve DA ; and the cycle is completed. The heat abstracted from the refrigerator is measured by the area FADE, the heat rejected to the cooler is measured by the area FBCE, and the work done, on the compressor, is measured by the area A BCD. Finally, the ideal coefficient of performance is given by _ Area FADE V Area ABCD' Since the rejection of heat to the cooler, and the abstraction of heat from the cold storage room, both take place at constant REFRIGERATION 271 pressure, equation (8) may be deduced in a very simple manner. If the heat abstracted from the cold storage room, for a given interval of time, is H 2 = C P (T 1 -T ), then the heat rejected to the cooler for the same interval of time, is Hi = Cp(T2 — To) Therefore, the ideal coefficient of performance is H 2 C^Tx-To) T x T} = Hx-H 2 C^-Taj-CviTt-To) T 2 -7Y which is the same as previously found. It must be emphasized that the equations deduced, in this article, do not represent conditions as found in actual practice. For the pressure, in the cooling pipes, of any actual refrigerating machine will vary considerably throughout the cycle. Hence, the actual coefficient of performance, even when all other losses are neglected, will be less than that indicated by equation (8) . Due to the fact that air has a low thermal capacity, air refrig- erating machines are necessarily bulky, and therefore, commer- cially uneconomical. However, there are certain places, as for example on board of ships, where it is inadvisable to use machines employing a volatile liquid. For, in the first place, there are possi- ble dangers from injurious escaping gases. But, even if the escap- ing gas is not injurious, there is always the possibility of a large leak, and consequently a total loss of the working substance, which cannot be replaced until the end of the trip. This, however, means a complete disablement of the plant. Hence, air machines are used only as a matter of expedience and not economy, in place of refrigerating machines employing a volatile liquid as a working substance. 190. Compression Machines Using Volatile Liquids. Com- pression refrigerating machines, using a volatile liquid for the 272 THERMODYNAMICS working substance, consist essentially of the parts as represented diagrammatically, in Fig. 52. A is the compression cylinder where the vapor is compressed, and then expelled into coils immersed in water in B; B being the condenser, or cooler. If the vapor is just saturated as it leaves the refrigerating coils, superheating may take place, during compression; this however is usually very small in comparison with the heat of condensation. Due to the high pressure in the condenser, and the low temper- ature, maintained by the circulating water, the vapor condenses, gives up the superheat and heat of condensation, which is carried away by the water, and the liquid flows into C, the storage tank. Fig. 52. In the tank C, the liquid is under a pressure corresponding to that of its vapor, for the existing temperature; the temperature of the liquid in the storage tank, usually does not differ materially from that of the surroundings. As an example, if the liquid employed be ammonia and the temperature in the tank is 75°F., then the pressure of the vapor is approximately 140 lbs. per square inch. Due to this high pressure, under which the liquid is in C, it flows, through the expansion valve D, into the coils in the refrig- erator E. The pressure in the coils, due to the aspirating action of the compressor, is low. By regulating the expansion valve, or the speed of the compressor, or both, the pressure in the refriger- ator coils may be varied at pleasure. Since, when the liquid passes through the expansion valve, the process is adiabatic, and no work REFRIGERATION 273 is being done, the total heat content remains the same. There- fore, for thermal equilibrium to obtain, when the pressure falls fcom.pi, that existing in the storage tank, to p 2 , that existing in the refrigerating coils, there must take place a certain amount of evaporation, such that h=h 2 +qr 2 ; (14) where hi and h 2 , respectively, are the heats of the liquid corre- sponding to the pressures pi and p 2 , q the amount of dryness, and r 2 the heat of vaporization at the pressure p 2 . From equation (14), we find hi — h 2 /1C * q= ~^ ; (15) and the remainder of the liquid can, then, if completely vapor- ized, take from the surrounding medium the quantity of heat H 2 = (l-q)r 2 (16) In order that heat may flow from the medium in E, into the coils it is, of course, necessary that the temperature of the medium be higher than that of the liquid, in the coils, corresponding to the pressure p 2 . If the difference of temperature is sufficient, the liquid will be completely vaporized; and the quantity of heat, as expressed by equation (16), will be removed from the refrigerator. If the difference of temperature be greater than this, the vapor becomes superheated ; and the quantity of heat removed from the refrigerator will be greater than that indicated by equation (16). The ideal p-v diagram, of the cycle just discussed, is represented in Fig. 53. The point A represents the condition, as regards pressure and volume, of the vapor at the beginning of the aspira- ting stroke, and the point B represents the condition at the end of the aspirating stroke; the line AB, therefore, represents the volume, due to complete vaporization under constant pressure. The curve BC represents the compression, which is nearly adia- 274 THERMODYNAMICS batic, CD represents the expulsion, and also the condensation, under constant pressure, in the condenser, and DE the change in pressure, and consequent change in volume, due to partial evaporation in passing through the expansion-valve. Therefore, the net work done, during the cycle, • is measured by the area A BCD. Fina ly the ideal coefficient of performance is given by the ratio of the work equivalent of the heat removed from the refrigerator to the work equivalent of the area A BCD. The refrigerating coils, in which the vaporization takes place, may be placed directly in a cold storage room, in the form of pipes, or else placed in a tank containing a solution of some salt, called brine. The freezing point for the brine must, of course, be lower than the temperature in the coils. The brine may then be em- ployed, by circulating it through pipes, to bring about refrigeration in some place remote from the plant, or else, to produce ice, by abstracting heat from water, contained in tanks, immersed in the brine. The cycle of a compressor refrigerating plant, using a volatile liquid as a working substance, is most instructively represented by the T— 9 diagram. However, before plotting the T— 9 dia- gram, it will be necessary to deduce an expression for the change REFRIGERATION 275 in entropy, for the substance, when passing through the expan- sion valve. To do this, let T\ be the temperature of the liquid in the storage tank, then by assuming some arbitrary temperature, say To, from which the entropy is measured, the entropy of a unit mass of the liquid, before passing through the expansion valve, is C Tx dT . Ti n = c\ -Y=c\og^; (17) where c is the thermal capacity, of the liquid, per unit mass. Assume some temperature T, in the refrigerating coils; T being, of course, less than T\. The entropy, then of a unit mass of liquid and vapor, measured from the same zero, is 'T C T dT ar * 2 = C ) T -T+T Tar. - clog^+l^; .... (18) where q is the amount of dryness, and r the heat of vaporization corresponding to the temperature T. Taking the difference between equations (18) and (17), we find, for the change in entropy in passing through the expansion valve, T Ti or 9 =

T, that as the temperature increases, the entropy decreases, and vice versa. Hence, the 276 THERMODYNAMICS entropy of the substance is increased by passing through the expansion valve. This is necessarily so, since the process is irre- versible. The T-

i)(l+cose). . . (12) If M is the mass of water, per unit of time, impinging on the cup, then, the force moving the cup is numerically equal to the product of change in velocity and mass ; hence we find for the force acting, F = M(V 1 -v 1 )(l+cosQ) (13) Finally, since power is numerically equal to product of force and speed, we have, by multiplying both sides of equation (13) by vi, for the power developed by the cup, P 2 = Ft;i=Jlft;i(7i-!;i)(l+cose). .... (14) It is obvious, from Fig. 59, that the motion of the water cannot be completely reversed; i.e., the direction of motion of the water leaving the cup must be inclined to the direction of motion of the cup. For, otherwise, the stream at exit will interfere with the forward motion of the following cup. By assuming, in equation (14), the power and the velocity of the cup variable, and the other quantities constant, we find, by differentiating for a maximum, ^- = M(Fi-2t;)(l+cos 0) = O; from which That is, for maximum power, the velocity of the cup must be one- half the velocity of the stream. STEAM TURBINES 295 Substituting in equation (14), this value for vi, we find, for the maximum power developed by the cup, P2 = M^(Vi-^)(l+cos 6)=^p- 2 (l+cos 0). . (15) Since the energy of the impinging jet is the efficiency of the cup becomes, P 2 _Fi 2 (l+cos 0)_l+cos 6 • *=K~~ 27? - — 2 * ' * ' (16) We may deduce the expression for the efficiency of the cup by considering the kinetic energy at entrance and exit. By an inspec- tion of Fig. 59, it is obvious that the absolute velocity at exit is the vector sum of (Fi — vi) and v\. Designating the absolute velocity at exit by V2, we find V 2 z = (V 1 -v 1 ) 2 +v 1 2 -2(V 1 -v 1 )vi cos 6 = Fi 2 -2(Fiz;i-z;i 2 )(l+cos 0). And the efficiency becomes, since it is given by the ratio of the kinetic energy absorbed by the cup, to the kinetic energy of the stream, 2(Fiz;i-z;i 2 )(l+cos0) ri = yj . Assuming Vi and constant, and the velocity of the cup and the efficiency variable, the expression for the efficiency becomes, f] 2(Fi?;-z; 2 )(l+cos0) Vi and solving this for a maximum, we find dri 2(Fi-2^)(l+cos0) dv Vi 2 from which Vi = 0; 296 THERMODYNAMICS Substituting this value of v, in the equation for efficiency, we find for the maximum efficiency l+cos0 which is identical with equation (16). Equation (16) shows that if the angle 6 equals zero, so that the motion of the water is completely reversed, the efficiency equals unity. But, this also means, and necessarily so, since for max- imum efficiency wi = Fi/2, that the water is brought to rest; and the kinetic energy possessed by it, before impact, is completely absorbed by the cup. However, as previously stated, it is Fig. 60. impossible to have the angle equal to zero, on account of inter- ference with the following cup. Assume the angle to vary between the extreme limits of 0° and 90°. Then, as just stated, for the angle = 0, the efficiency is unity; and for the angle = 90°, the efficiency becomes 1/2. When equals 90°, the result is equiv- alent to normal impact without shock, as represented in Fig. 60. If Vi is the absolute velocity before impact, then, since for maxi- mum efficiency the velocity of the cup equals Fi/2, the absolute velocity of the stream after impact, is equal to 7i/V2. There- STEAM TURBINES 297 fore, since the efficiency is equal to the ratio of ihe kinetic energy absorbed by the cup to the kinetic energy of the stream, before impact, we find v?-vm _l (17) Hence, theoretically, the efficiency of an impulse turbine may vary between 50 and 100 per cent; depending upon the value of the angle 0. In practice it is attempted to have the angle just sufficiently large so that the stream, leaving the cup, does not interfere with the following cup. In properly designed impulse wheels the actual efficiency may be, considering all losses, as high as 90 per cent. The results deduced clearly indicate, that in any case, it is essential, if a high efficiency is to be realized, to reduce the absolute velocity of the impinging stream, in going through the turbine, as nearly as possible, to zero. And this, in the case of steam turbines, is just as necessary a prerequisite for high efficiencies, as it is in the case of water turbines. The question, why is it possible, in the case of hydraulic motors, to convert so large a fractional part of the theoretical energy, due to the difference in topographical level, into actual work, and in the case of heat motors, so small a fractional part of the energy, due to the difference in " temperature level," naturally suggests itself. The answer is obvious. Every heat motor must act periodically. Even though the identical working substance is not used in the succeeding cycle, the result is just the same. For, the condition of the working substance, for the best results, must be at the beginning of each cycle the same as it was at the end of the preceding cycle. This is equivalent to cyclic operation. In the case of the hydraulic motor, however, the cycle is only partially completed. That is, the water, after having performed work, in falling through a certain height, is restored to its original condition by the action of the sun, which completes the cycle automatically. In other words, the water 298 THERMODYNAMICS at the height Hi, of the headrace, falls to the height H2, of the tailrace, and performs, theoretically, the amount of work w(H 1 -H 2 ); where Hi is the height of the headrace, H2 the height of the tail- race, and w the weight of water. But, to complete the cycle, the water must again be raised from the level H2 to H\. This is done at the expense of the radiant heat from the sun, by means of which the water from streams, lakes, the oceans, etc., is evapo- rated and carried, by means of convection currents, to higher elevations, where condensation takes place, and the difference in elevation, H1—H2, is again established. 197. Flow of Fluids in Pipes of Varying Section (De Laval Nozzle). The flow of a gas or vapor, under steady conditions, Fig. 61. through a pipe of varying cross-section, is very similar to the flow of a liquid under similar conditions ; but, the flow of a gas differs materially from that of a non-compressible liquid in two respects. That is, in general, the weight of the gas, or statical head, is negligibly small in comparison with the pressure head and velocity head; but, on the other hand, account must be taken of the expansion. Consider a pipe CC , such as is represented in Fig. 61, and assume a steady flow, i.e., the mass of gas entering the section at C, for a given interval of time, is equal to the mass leaving the section at C", during the same interval of time. And further, that the gas flows, without friction, in straight stream lines. Let m be the mass of gas that enters the channel at C, per unit STEAM TURBINES 299 time, with a speed s a , and pressure p a , then an equal mass will leave the channel at C, during the same interval of time, with some speed s b , and pressure p b . Assume the pipe CC, to be divided into an indefinitely large number of sections, such that the thick- ness of each section is indefinitely small. Let the pressures on the left-hand side of the various sections, be, respectively, Pa, P2, P3, • • • Pn-1, Pn', and on the right-hand side of the sections, V2, P3, m • • • Pn, p&. Representing the respective cross-sectional areas by ■A-a, A.2, -A-3, . . . A. n , A b , and the corresponding thicknesses of the elements by ds a , ds2, dss, . . . ds n , then the work done by the positive pressures, during the time that the displacement ds a takes place at C, and the displacement ds n takes place at C ', is W ' = PaA a ds a -\-p2A 2 dS2-\- • • • +Pn-lA n - 1 dSn-l+PnAndSn- ■ (18) Likewise, the work done by the back pressures is W"= —p2 A 2 ds2 — p3A3ds3— ••• —pnAndsn — pbAbdsb. . . (19) Taking the sum, of equations (18) and (19), we find, for the net work done, due to change in pressure, W^W'+W'^PaAadsa-ptA-tdsi,. . . . (20) Changing, in equation (20), the subscripts a and & to 1 and 2, we have Wi=piAidsi — p 2 A 2 ds 2 . ' (21) 300 THEKMODYNAMICS If dsi is the distance passed through, at C, in the time dt, and ds2 the distance passed through at C in the same interval of time, then ds\=s\dt, and dS2 = S2dt', where si and S2 are, respectively, the initial and final speeds. Substituting, in equation (21), these values of dsi and ds2, we obtain Wi = (piA 1 s 1 -p 2 A 2 s 2 )dt (22) Since, m is the mass of gas, flowing per unit time, we have, for steady flow, _ Aisi A 2 s 2 m— = ; Vi V2 where v\ and V2 are, respectively, the volumes per unit mass of the gas corresponding to the pressures pi and P2. Substituting in equation (22), we obtain Wi — ijpivi^^v^mdt . (23) The change in kinetic energy is and expressed in engineer's units, this becomes W 2 = Sj ~^mdt . (24) The work due to expansion is, Ws = (mdt) ( 2 pdv; ...... (25) and for the assumed conditions, Tfi + T^2+Tf 3 = 0. STEAM TUKBINES. 301 Hence, Sl 2 _ g 2 rv 2 (piVi — v 2 V2)mdt-i s — mdl-\-{mdt) I pdv = 0; Zg Jv l from which s 2 2 — s i 2 C v * — o - = pivi-p2^2+ I pdv (26) But, ri>2 rv\ p\Vi—p2V2 J r I pdv= I vdp; J v i J P 2 2g--k ^ ■■■■■■ W hence, S2 2_. Ql 2 rVi If, now, the process be such that the equation pv n = k i_ _i_ holds, then v = k n p n ; and by substitution, equation (27) becomes s 2 2 — s i 2 — C Vx -— n — n ~ 1 VlzA ___ = fc»J p ndp = ~_ I jkn(p 1 n -p 2 n ). . (28) 1_ Substituting for & ra its value, we find s 2 2 -si 2 n L »=1 «=* w f /p 2 \^1 ™ ___ = __ pi ^ 1 (p 1 B _ p2 n )= ___ Wl | 1 _^j , j (2g) Or, since L L L fcn z=p 1 n Z ; 1= p 2 n Z ; 2j 1_ equation (28) may be simplified by dividing by k n , and multiply- i_ ing the first term in the parenthesis by pi n vi, and the second term i_ by p2 n V2. Performing this operation we find S2 2 — si 2 n , /o ^ x -^— = —^ViVi-p2V2). ..... (30) 302 THERMODYNAMICS If the pipe is curved, then the pressure on the convex surface is less than on the concave surface. However, unless the change in direction is considerable, the difference in pressure is very small. The foregoing conclusions are the result of a modification of of Bernouilli's theorem; i.e., applying the theorem to a compress- ible fluid of negligible weight. If the initial speed is negligible, as is the case when discharge takes place from a comparatively large vessel, then si, inequation (29), may be omitted, and we have n-l 52 _ n PlVl 1- (31) 71-1 1 Solving for S2, we find Now, the mass of fluid conveyed, per unit time, through any section is As m = — ; where A is the area of the section, s the speed, and v the volume per unit mass. But for steady flow this is a constant throughout the pipe. Hence we have As A2S2 V2 (33) where A 2 is the area corresponding to the pressure P2, and V2 the corresponding volume per unit mass. By combining equa- tions (32) and (33) we find A2S2 (V2 m = V2 V2 [n — 1 n-\ pm l (34) Assuming again, the flow to be such that the equation pv =pivi (35) STEAM TURBINES 303 holds, we find Substituting this value of V2 in equation (34), we find -4ss(© ! -(gfif • • • <=>«» Since, as previously stated, m is constant, equation (34) may be written where A is any section, p the corresponding pressure, and v the corresponding volume per unit mass. Since As/v is a constant, s/t; must be a maximum, when A is a minimum. But, s/v becomes a maximum, and hence, A a minimum, when the value of the expression included in the bracket, of equation (37), becomes a maximum. Hence, we may write L =Kx= (W IP and from which Pi/ \Pi 2-n 1 dp n L n n + i > P\n pi n ™fahl) ^ The value of p, as given by equation (38), is that value which makes s/v a maximum, and, therefore, A a minimum. Substitut- ing this value of p, for p2, in equation (32), and reducing, we find i^ / 2gn \ 2 •■^"i (39) 304 THEKMODYNAMICS Finally, by means of equations (35), (38), and (39), we find i As A/2gn \2 v v \n-\-r n+l vi\n+l, • (40) Equating the right-hand members of equations (36) and (40), we find A 2 A 2 - n-1/ 2 \n-i - n+l\n+l/ 2 » + l /pgV _ (v2\ n W W i 2 ; . . . . . . (41) which gives the ratio of the area, for the pressure p2, to minimum area. Solving equation (38), on the assumption that the fluid is saturated steam, for which n equals 1.135, we find p = 0.577pi. By means of equation (32), the final speed may be determined, when the initial and final pressures are known; and with the aid of equation (37) any one of the three quantities, viz, m, A, and p, may be found, if two of them are given. Since, in deducing the foregoing equations, we equated work, expressed by the product of pressure and volume, against energy, expressed by the product of mass and square of the speed, we must in substituting numerical values, in these equations, use the same system of units. As an example, if in equation (39), s is to be given in feet per second, p\ must be given in lbs. per square foot, and v\ in cubic feet. Reducing equation (39), so that p\ is expressed in lbs. per square inch, and substituting for g and n the proper values, we find j_ /2X32.2X1.135 v/1 AA \2 __ ^ s=( 2^35 XUApiVi) =70.2(pit; 1 )l. STEAM TURBINES 305 For steam under a pressure of 100 lbs. per square inch, the volume per pound is approximately 4.43 cu.ft.; hence, by substitution, we find s = 70.2(443) * = 1478 ft. per sec. If the pressure be 200 lbs. per square inch, for which the volume, per pound, is approximately 2.29 cu.ft., we find s = 70.2(458)* = 1503 ft. per sec. These two computations show that the variation in speed is small when compared with the variation in pressure. Reducing equation (40) in a similar manner, we find A f 2X32.2X1.135/ 2 XoIm Pl | 2" m = l44{ 2JL35 Ul35 ) X 144 ^} i_ = 0.30A f — J pounds per sec. ; where A is now in square inches. 198. Two Principal Types of Turbines. From the previous discussions on the flow of steam through pipes it is obvious that the speed of flow, for any considerable difference in pressure, is very high; and that if any single-stage turbine, i.e., a turbine consisting of a set of nozzles and only one rotating part, were to utilize practically all the kinetic energy of the steam, due to its speed at nozzle exit, the speed of the turbine would have to be abnormally high. As an example, some of the De Laval tur- bines, which were single stage, had speeds as high as 40,000 r.p.m. Though the efficiencies of the De Laval turbines, from the stand- point of steam consumption, were not exceptionally low, the enormously high speeds were a serious disadvantage. For, in no other mechanical contrivance, not even dynamo electric machines, which are operated at relatively high speeds in com- parison with other machines, are such high speeds ever approached. 306 THERMODYNAMICS Therefore, to utilize the power developed by a single-stage turbine it is necessary to employ a reduction gear. But, a reduc- tion gear means an additional first cost, and a lowering of mechan- ical efficiency. Furthermore, proper lubrication becomes exceed- ingly difficult when machines are operated under speeds such as are attained by single-stage turbines. The difficulties, however, stated in the preceding paragraph, were overcome by the introduction of multi-stage turbines* That is, by allowing the steam to act successively upon the rotors of a multi-stage turbine, its speed is gradually reduced, and the speed of the turbine need not be abnormally high. The turbine must of course be so designed that the steam expands, and the temperature is reduced continuously to the lowest possible value at exit. That is, the kinetic energy of the steam at exit must be, as nearly as possible, equal to zero. There is then the choice of the following types of multi-stage turbines: Combined impulse and reaction, and impulse. 199. The Parsons Turbine. The Parsons turbine, at the pres- ent time, represents one of the commercially successful types of turbines; and may be considered a combined impulse and reaction turbine of the parallel-flow type. That is, the steam passes through the first set of guide blades approximately parallel to the shaft of the turbine, and has given to it the proper direction so that it may enter into the channels of the first rotor without shock. At exit from the first rotor, the steam enters a second set of guide blades, where it is again directed so as to properly enter the chan- nels of the second rotor, etc. In this way the steam reacts, expands, and falls in pressure continuously as it travels, from * There appears to be considerable confusion in regard to the meaning of the word " stage." In some cases authors designate a turbine, as an n-stage turbine when there are n rotors, which is consistent with the nomenclature employed in the case of hydraulic turbines. In other cases, however, namely the Curtis turbine, by number of stages is meant the number of sets of expanding nozzles. STEAM TURBINES 307 entrance to exit, through the turbine. Since the steam is con- tinually expanding, the length of the blades and spacing, for both guides and rotors, must be increased so that the ratio of steam speed and blade speed, upon which the efficiency of the turbine depends, is maintained constant. 200. The Curtis Turbine. The Curtis turbine is of the impulse type. The steam expands in a set of nozzles, where the pressure head is converted into velocity head, and then impinges on the curved blades of a rotor. Part of the kinetic energy of the steam is absorbed by the first rotor; the steam then at reduced speed passes through a set of guide blades where it is directed so as to properly enter the channels of a second rotor, where the speed is still further reduced, etc., until, finally, the speed is very low. The steam is then expanded through a second set of nozzles, and again passes through a series of rotors and guides, precisely as in the first stage. This is continued until the pressure of the steam has been reduced to the desired exhaust pressure. The number of stages, other things being equal, depends, of course, upon the range in pressure. Due to the fact that the speed of the steam is reduced in each rotor, the passages traversed by the steam must be continuously enlarged. This is brought about by reducing the curvature of the blades as well as lengthening them.* 201. Comparison of Parsons and Curtis Turbines. Since the speed of the steam entering a Parsons turbine is moderately low, and for high efficiency its absolute velocity at exit must approach zero in value, it follows that the relative velocity must be high. That is, the relative velocity at exit being, approximately, the vector difference between the absolute velocity at entrance and the velocity of the wheel, it follows that, since the velocity of the steam at entrance is low, the relative velocity at exit will be high, and therefore, the velocity of the wheel must be high * For a comprehensive discussion on the design and testing of turbines see "The Marine Steam Turbine" by J. W. Sothern. 308 THERMODYNAMICS in order that the absolute velocity at exit may be low. On the other hand, in the case of an impulse turbine, the velocity of the blade, for the best efficiency, is approximately one-half that of the entering steam. Hence it is obvious that, other things being equal, the Parsons turbine is inherently a higher speed prime mover than is the Curtis turbine. 202. Turbines and Reciprocating Engines. No matter how operated the steam turbine is inherently a high-speed prime mover; and since power is proportional to the product of torque and angu- lar velocity, it follows that for equal output, the steam turbine, with its high rotative speed, will be of smaller dimensions than a reciprocating engine. Furthermore, where rotative motion is required, which is usually the case, the turbine needs no connect- ing rod and crank, as does the reciprocating engine. Again, where electric generators are direct connected, as in power plants, high speeds, up to a certain point, are desirable. Since, the power developed is equal to the product of e.m.f. and current, high-speed generators, for equal output, will have a lower first cost and occupy less floor space than low-speed generators. Finally, the turbine has the further mechanical advantage of having a uniform turn- ing moment. On the other hand there are certain cases where low speeds are essential either to successful operation or economy; under such conditions the reciprocting engine is superior. As an illustration of this we may consider present conditions in marine engineering. As previously stated, the turbine, using high pres- sure steam is, for high efficiencies, inherently a high-speed prime mover; on the other hand the propeller of a ship, is, for high efficiencies, inherently a low-speed mechanism. On passenger liners, the increased rates, which passengers are ready to pay for a reduction of time in transit, more than pay for the increased cost of operation. However, on freight steamers, such is by no means the case ; and it appears that for such steamers the recipro- cating engine combined with a low-pressure turbine, as regards economy, is at least equal if not superior to the turbine. STEAM TURBINES 309 Thermodynamically, the steam turbine is far superior to the reciprocating engine. For, in the turbine there is no alternate heating and cooling of the surfaces with which the steam comes into intimate contact. In other words, in the case of a turbine, very shortly after starting, steady conditions will prevail ; and the incoming steam, therefore, does not come into contact with sur- faces which have been previously chilled by the low-temperature exhaust steam. That is, the steam changes gradually in pressure and temperature from admission to exhaust. And this means that -G v Fig. 62. there is no condensation excepting that due to radiation. Hence, in a turbine, condensation is largely eliminated in comparison with a reciprocating engine; and herein lies one of the great factors that makes the turbine thermodynamically superior to the reciprocating engine. Another important factor is the fact that in a turbine a good vacuum is utilized to much better advan- tage. This is illustrated by Fig. 62. Let, in the figure, ABODE be the indicator diagram of a reciprocating engine operating between the pressures as indicated by the points A and E. Then the net work done by the engine is measured by the area ABODE', and the net work realized by means of a turbine, for the same limits in pressure, is measured by the area ABCFE. If, now, the back pressure be reduced, from that as represented by the line 310 THERMODYNAMICS ED, to that represented by the line HI, the net gain in work, by means of a reciprocating engine, is measured by the area EDIH, and that, in the case of a turbine, by the area EFGH. That is, due to mechanical considerations, the length of stroke of the reciprocating engine is fixed; and hence, full expansion cannot be realized. But, in the turbine full expansion is realized and the toe of the indicator diagram is utilized in doing useful work. This gain in work, in the case of low-pressure turbines, is quite appreciable. 203. Turbine Tests. It has been found impossible, up to the present time, to devise any method by means of which to determine the indicated power of a turbine, in the same manner that the indicated power of a reciprocating engine is determined. There is, however, no difficulty experienced in determining the output, or brake power. The output is determined in precisely the same manner as described in Art. 133; and the thermal effi- ciency is determined as described in Art. 136. If it be desired to determine the commercial efficiency, the method of procedure is precisely the same as that described in Art. 137. The com- parison frequently made is that between the actual output of the turbine, and that which would have been obtained on a Rankine cycle, as discussed in Art. 138. 204. Reciprocating Engines and Low-pressure Turbines. In 1906, H. G. Stott presented a paper* before the American Institute of Electrical Engineers on " Power Plant Economics," which gave a complete analysis of the various losses, from the coal bunkers to the bus-bars, of the Interborough Power Plant, located at Fifty-ninth Street and Eleventh Avenue, New York City. The prime movers employed at that time were of the Manhattan type compound Corliss engines; two engines being connected to one generator of 7500 K.W. maximum capacity. The following quotation is an extract from Mr. Stott's paper, * "Power Plant Economics," Transactions of the A.I.E.E., Vol. XXV. STEAM TUEBINES 311 which is one of the most complete and instructive analyses that has ever been made of a power plant: " Three years ago the steam-power plant for the generation of electricity had apparently settled down to an almost uniform arrangement of standard apparatus in which one power plant differed from another only in details of construction of engines, generators, and auxiliaries. As only about twenty years had then elapsed since the first central station was put in operation on a commercial basis, this uniformity of design seemed to indicate that in the near future it would only be necessary to purchase a standard set of power-plant drawings, and make the necessary changes in size of units in order to have a station of the best type known to the art. " The internal combustion or gas engine had from time to time been brought forward as a candidate for the position of prime mover, with every prospect of improved economy in fuel consumption; but with the exception of a few special instances it was not looked upon with favor, as shown by the almost universal use of the steam engine. " After a long period of development a new factor in power- plant design; namely, the steam turbine, was placed on the market in commercial sizes. It is safe to say that during the last three years no other piece of apparatus has had so stimulating an effect upon the power plant. Its effect upon the entire plant has been most beneficial,- for it has revived the apparently moribund superheater. This has now been so developed and improved that superheat of 200° or 300° fahr. can be safely and economically obtained. With the development of the superheater further study of the problem of combustion has improved the efficiency of the furnace; and this most important subject is apparently susceptible of still further development. " One other important result of the steam-turbine develop- ment has been the development of condensing apparatus to such a point of efficiency that a vacuum within one inch of the simul- 312 THEEMODYNAMICS taneous barometer reading can now be maintained without difficulty. " Another change in the power plant has been the reversion to high-speed generators, resulting in decreased cost of the gen- erator and its foundations, as well as saving in floor space. " Last but not least the steam turbine has put the recipro- cating engine and the gas engine on the defensive and has actually been unkind enough to throw out hints in regard to the applica- tion of Dr. Osier's proposed methods to the treatment of older apparatus. " The reciprocating engine and internal combustion engine have not been slow in accepting this challenge; they have responded by showing so improved an economy (especially in the gas engine) that the situation has become most interesting to the power-plant designer. It is safe to say that the develop- ments of the next ten years will show very marked improvement in power plant efficiency. " In regard to this development the author wishes to direct attention to the basic fact that in power plants one should not look merely for increased efficiency in the prime mover, but should also investigate and analyze the entire plant from the coal to the bus-bars: first, in regard to efficiency; secondly, in regard to the effect of load-factor upon investment; and thirdly, the effect of the first and second upon the total cost of producing the kilowatt- hour, which is the ultimate test of the skill of the designer and operator. " Efficiency. " In Table 1 will be found a complete analysis of the losses found in a year's operation of what is probably one of the most efficient plants in existence to-day and, therefore, typical of the present state of the art. STEAM TURBINES 313 " Table No. 1 ANALYSIS OF THE AVERAGE LOSSES IN THE CONVERSION OF ONE POUND OF COAL INTO ELECTRICITY. B.T.U. Per Cent. B.T.U. Per Cent. 1. B.T.U. per pound of coal supplied 2. Loss in ashes 14150 441 960 100.0 3.1 6.8 340 3212 1131 28 223 203 152 51 31 111 36 28 8524 29 2 4 3. Loss to stack 22 7 4. Loss in boiler radiation and leakage . . 5. Returned by feed-water heater 6. Returned by economizer 8.0 7 Loss in pipe radiation 2 8. Delivered to circulator 1 6 9. Delivered to feed-pump 1 4 10. Loss in leakage and high-pressure drips 11. Delivered to small auxiliaries 12. Heating 1.1 0.4 2 13. Loss in engine friction 8 14. Electrical losses 3 15. Engine radiation losses 2 16. Rejected to condenser .... 60 1 17. To house auxiliaries 2 15551 14099 109.9 99.6 14099 99.6 Delivered to bus-bar 1452 10.3 " Discussion of Data in Table 1 " Item 1. B.t.u. per Pound of Coal Supplied. The thermal value of the coal used is evidently of prime importance, as it affects the cost efficiency of the entire plant. The method of purchasing coal used in the plant from which this heat balance is derived is that of paying for B.t.u. only, with suitable restrictions on the maximum permissible amount of volatile matter, ash, and sulphur. " A small sample of coal is automatically taken from each filling of the weighing hoppers, so that the final sample represents a true average of a boat-load of coal. This final average sample is then pulverized and tested for heat value in a bomb calorim- eter, after which a proximate analysis is made of another por- 314 THERMODYNAMICS tion of the sample. This method of purchasing coal has been in use for two years, with highly satisfactory results. " Item 2. Loss in Ashes. It is doubtful whether a further saving in this item can be made, as the extra care and labor necessary to accomplish any improvement would in all probability offset the saving in coal. " Item 3. Loss to Stack. This is one of the most vulnerable points to attack, as the loss of 22.7 per cent, is very large. Recent investigations show that promising results may be obtained by the use of more scientific methods in the boiler room. In prac- tically all cases it will be found that this loss is due almost entirely to admitting too much air to the combustion chamber, result- ing in cooling of the furnace. This result is usually produced by " holes " in the fire; these " holes " may be due to several causes, but usually are due to carelessness on the part of the fireman. " Fortunately, a very valuable piece of apparatus has been placed upon the market in the shape of a CO2 recording instru- ment. The results of a series of tests made with this instrument are shown in Figs. 63 to 66. " Fig. 63 shows the average condition of a furnace using small sizes of anthracite, with forced draught. The conditions are such that approximately 40 per cent, of the thermal value is being lost. " Fig. 64 shows what improvement may easily be obtained by watching the CO2 record, and indicates a saving of about 19 per cent, over the previous case. " In the combustion of the small sizes of anthracite it is neces- sary to use a draught of not less than 1.5 in. of water; this breaks the crust of the fire in the thin spots, allowing the air to come through in such volumes that an enormous amount of heat is wasted in raising the temperature of the surplus air and at the same time causing inefficient combustion in the entire furnace. " Fig. 65 shows a record taken from a stoker boiler whilst the recorder was covered up to prevent the fireman from seeing the record. STEAM TUEBINES 315 •^ - < - e- p- p° u_ =» - PER CENT. C0 2 RECORD DEC. 18-19, 1905, BOILER NO. 29. AVERAGE PER CENT.C0 2 = 4.7 HAND-FIRED, NO. 2 BUCK, UNDER FORCED DRAUGHT. — < r ' 7 -"•" J r , — J * ~<\ ^ *> ^ ^ V ^ s •s! _ »>. — *s ed C^? - « J- ' fe 5. - < >" e ' Ml CO ■«* m CO 33VlN30a3d 316 THERMODYNAMICS s i — »» < II . ol Ol- h2 zlu UJQ. oo 8*2 Ouig oo< cr ce , e? IXl 1— ZCN< Ul .£ oo Q w ceui o£ CO O Ol o q; T ~il Si ?< .1 o LU Q ?H *=L P=* - o» QO t „ - CO >o ^ >__ CO c>4 a - 13 TIME Fig. 6 7-- ^ > - T-t r s " OS " vz br CO z ~tr 1 " l -~« > CO lO ■* *-X CO CXI rH. a O z - - " _ ' - C - < >» >> \ s _ - 1, «? - K» ~ r~ V. -2 ! 2-. ,> - / =-£ - " . / s ^ T - li > 1 — - - Si " 5 CO 30VlN30y3d 318 THERMODYNAMICS " Fig. 66 shows a record taken from the same stoker boiler with the fireman watching the CO2 indications, resulting in a sav- ing of over 12 per cent. Later records show that even better results than an average of 11.4 per cent, of CO2 can be obtained. " Fig. 67 shows the calculated losses in fuel corresponding to various percentages of CO2 for three different temperatures of flue gases. " From a consideration of the above tests it seems reasonable to assume that the 22.7 per cent, loss to stack can, by scientific methods in the fireroom, be reduced to about 12.7 per cent, and possibly to 10 per cent. " Before the installation of the CO2 recorder a long series of evaporative tests was made to determine the most economical draught to carry when a high-grade semi-bituminous coal was burning on the automatic stokers. The results shown in Fig. 68 were so remarkable that they were repeated under different conditions in order to confirm them. Since the installation of the CO2 recorder, however, the explanation is apparent; as the draught giving maximum evaporation per pound of combustible corresponds to the point of maximum CO2, illustrating the inher- ent difficulty of maintaining efficient conditions in the combus- tion chamber with high draught. This is well illustrated by Fig. 69, showing the draught, per cent, of rating, and percentage of C0 2 . " Item 4- The loss in boiler radiation and leakage, amounting to 8 per cent., is largely due to the inefficient boiler setting of brick which, besides permitting radiation, admits a large amount of air by infiltration. This infiltration will increase with the draught, thus tending to exaggerate the maximum and minimum points on Fig. 68. The remedy for this radiation and infiltration loss is evidently to use new methods of boiler setting, such as an iron plate air-tight case enclosing a carbonate of magnesia lining outside the brickwork. STEAM TURBINES 319 3 - rH £ r-t cs V. — =• 00 t-» J o PER CENT C0 2 RECORD DEC. 22-23, 1905-BOILER NO. 31 AVERAGE PER CENT C0 2 =11.4 STOKER OPERATED tH CO N g «j >•"- 13 s, i—l I *> ' =* " T— 1 ^ J** OT - CO S - CO lO ^n - CO s — •" — ■*, * ' - ^H F T-t CD uj CO 2 • ft 33V±N30U3d 320 THERMODYNAMICS CURVES SHOWING THE RELATIVE FUEL LOSS TO PER CENT. C02 , BY VOLUME, IN FLUE GAS. DIFFERENCE BETWEEN TEMPERATURES OF ENTERING AIR AND EXHAUST GAS: CURVE "A"= 400° FAHR. ,, "B"= 500 FAHR. "C"= 600° FAHR. 1 ,1 1 1 1 1 II l / 1 ' / / / / / / f // / / / / / / / / / / / / / / / / / / / u ffl- < Z UJ o a. CO CO O _l 3IAin"10A -JLN30 UQd -00 STEAM TURBINES 321 " Mr. W. H. Patchell,* of London, who recently visited us, has introduced very large boilers, assembling two in one setting; each boiler has a normal evaporation of 33,000 lb. per hour and in this way has cut down to a minimum the radiating surface per square foot of heating surface. He has also introduced the iron case with magnesia lining, and with good results. " The question of boiler leakage is one in which the choice of the lesser of two evils is necessary; for in the tubular or cylindri- < o o > cc a Q 2 O °-10. cc 9.5 ? q a. o a. < > UJ z / s CURVES SHOWING RELATIVE ECONOMIES ATDIFFERENT DRAUGHT IN / \ ' \ THE COMBUSTION OFSEMI'BITUMINO COAL UNDER A BABCOCK AND WILCC IS / \ Y / \ BOILER, UPON A RONEY STOKER. 1 \ 1 *v pv A n vIC — s s / / / / / 1 1 0.1 rooq 600, 500 iOO 0.3 0.4 0.5 0.6 0.7 draught: inches water. Fig. 68. 1.0 cal boiler the leakage will undoubtedly be less than in the water- tube type, owing to the smaller number of joints in the water space. But these two advantages are offset by the increased difficulty of construction, and the danger of using large boilers of the tubular type, especially with high-pressure steam. "It is now generally admitted that there can be no more difference in the efficiency of different types of boilers under * See paper read December 7, 1905, before the Institution of Electrical Engineers, by W. H. Patchell. 322 THERMODYNAMICS similar conditions than there can be in electric heaters, press agents to the contrary notwithstanding. " Item 5. Returned by Feed-water Heater. The importance of getting the feed water to the maximum temperature obtainable is generally recognized, and would seem to indicate that all auxili- L 2 3 4 HOUR - 23-HOUR RUN, JAN. 1-2-06 5 6 7 8 9 10 11 12 13 U 15 16 17 18 19 20 21 22 23 20 UJ GRAPHICAL LOG OF AVERAGE READINGS TAKEN ON BOILER NO. 31, JAN. 1, 2, 1906. 2 3 15 o 810 \ \ / / ul UJ \ / \ v OL o a 100 UJ r£ O-50 CC uj aJ d 25 o n CO 0.25 ,_0.20 -?0 15 S\ \ -■^ \ / < 01 \ / cc Q 0.05 \ / \ Fig. aries should be steam driven so that their exhaust may be utilized in the feed-water heater; in this way the auxiliaries may operate at about 80 per cent, thermal efficiency. " Item 6. Owing to the difficulty of pumping water at tem- peratures above 150 degrees fahr., when under pressure, it becomes necessary to install economizers for the purpose of increasing the feed- water temperature to 200 or 250 degrees fahr. STEAM TUEBINES 323 As this increase of temperature is obtained from the waste gases at no expense for fuel, it only becomes necessary to consider the load-factor, as will be shown later, in order to decide whether economizers should be installed or not. In practically all cases where the load factor exceeds 25 per cent, the investment will be justified. ' " In deciding upon the size of economizer to be installed it is important to consider first, the influence of the economizer upon the available draught due to the obstruction it offers and also due to the reduced stack temperature; the second important consider- ation is to equate the interest and depreciation charges against the saving in fuel, and so determine the amount of investment justified in each particular case. " Item 7. Loss in Pipe Radiation. By the use of two-layer pipe covering, each layer being approximately 1.5-in. thick, and sections put on in such manner that all joints are broken, the radiation losses have become practically negligible. " Items 8 and 9. Heat Delivered to Circulating and Boiler-Feed Pumps. As these auxiliaries may be either electrically driven or steam driven it is interesting to note that the thermal efficiency of the electrically-driven pumps would be equal to the thermal efficiency of the plant, multiplied by both the efficiency of con- version from the alternating to direct current and by the motor efficiency. In this case, there would be a net thermal efficiency of 10.3X0.93X0.90 = 8.63 per cent., whereas the thermal efficiency of the steam-driven auxiliary discharging its exhaust into a feed- water heater at atmospheric pressure would be approximately 87 per cent. " Item 10. Loss in Leakage and High-Pressure Drips. The loss in leakage should be infinitesimal, and the high-pressure drips can be returned to the boilers, so that practically all the loss under this heading is recoverable. " Items 11, 12, and 17 are probably unavoidable and of so small a magnitude as not to merit much consideration. 324 THERMODYNAMICS " Item 13. Loss in Engine Friction. Recent tests of a 7500- h.p. reciprocating engine show a mechanical efficiency of 93.65 per cent, at full load, or an engine friction of 6.35 per cent. As this forms only 0.8 per cent, of the total thermal losses it is relatively unimportant. "Attention is called to the method of lubricating all the principal bearings by what is known as the flushing system, whereby a large quantity of oil is put through all the bearings by gravity feed from elevated oil reservoirs common to all the units; after passing through the bearings the oil is returned by gravity to oil filters in the basement and then pumped up to the reservoir tanks again. About 200 gallons per hour are put through each engine, and of this quantity only about 0.5 per cent, is lost. This method of oiling undoubtedly contributes to the general results. " Item 14- As large electrical generators can now be obtained which give from 98 to 98.5 per cent, efficiency, it would seem as if the limit in design had been reached and that hereafter the problem of design is to be merely one of altering dimensions to suit varying sizes and speeds. While this is true as far as the efficiency is concerned, other problems are continually arising, such as the design of generators for an overload capacity of 100 per cent, to meet the demand for apparatus capable of taking care of great overloads economically for short periods, corresponding to peak loads of a railroad or lighting plant. " Item 15. Engine Radiation Losses. This source of loss has evidently been reduced to a negligible quantity by the use of improved material and methods of heat insulation. " Item 16. Rejected to Condenser, 60.1 per cent. This imme- diately introduces the thermodynamics of the steam engine, a subject so broad that it will be impossible to do more than touch upon some of the most important points in considering steam- engine efficiency. " The efficiency * of any heat engine can be expressed by the * Defined as ideal coefficient of conversion, Art. 89. — Author. STEAM TUEBINES 325 rp m ratio of E = — ^ where T\ is the absolute temperature of ± 1 the steam entering the engine and T2 the absolute temperature of the steam leaving the engine. Thus in the engine whose steam-consumption curve is given in Fig. 70, if the initial pressure is 175 lb. gauge and the vacuum at the low-pressure exhaust nozzle is 28 in., then the maximum thermal efficiency is — === — = 33 per cent. This would be true for any form of engine or turbine working between the same temperature limits. I 1 1 1 A. ECONOMY CURVE FOR 7 500 H.P. ENGINE LOAD EQUALLY DIVIDED BETWEEN CYLINDERS. B. ECONOMY CURVE FOR 7 500 H.P. ENGINE LOAD UNEQUALLY DIVIDED BETWEEN CYLINDERS. J / 20 / / cc zz t- $v> O 1£ or Ul ^v 2 El8 p ^ \ S s / cc \ \ > Y y \ s ^ ^~o £« \ A u \ ' S 17 4 000 5 000 6 000 load: kilowatt- houp ( switchboard reading) Fig. 70. 7 000 " In Fig. 70, however, it is seen that the point of maximum economy shows a steam consumption of approximately 17 lb. per kilowatt-hour, which is equivalent to 20,349 B.t.u. per hour. One kilowatt-hour is equal to 3412 B.t.u. per hour, so that the actual efficiency of the steam engine and generator . 3412 is 20349 16.7 per cent. As the generator efficiency at this 326 THEKMODYNAMICS load is approximately 98 per cent, the net engine thermody- namic efficiency * is tt-^= 17 per cent. U.yo " The difference between the theoretical efficiency and the actual is then 33 — 17 = 16 per cent., of which 0.8 per cent, has already been accounted for in engine friction, so that the balance of 15.2 per cent, is due to cylinder condensation, incomplete expansion, and radiation. " As the engine friction in a two-bearing engine with high- pressure poppet valves and low-pressure Corliss valves has by care- ful design been reduced to less than 0.8 per cent, gain cannot be expected here, so attention must be centered on the loss due to cylinder condensation, etc., amounting to 15.2 per cent., in order to effect any improvement. " Superheated steam is the only remedy at hand and with it we can probably effect an improvement of 5 or 6 per cent, by using such a degree of superheat in the boilers that dry steam will be had at the point of cut-off in the low-pressure cylinder. " Any greater amount of superheat than this will merely result in loss to the condenser; for it should be remembered that the cylinder losses increase with the difference in temperature between the steam and exhaust portions of the cycle; in other words, the greater the thermal range of temperature the greater the condensation loss. This would seem to point to the use of more cylinders; but this involves additional first cost and fric- tion as well as more space and higher maintenance charges. " Fig. 71 shows what may be gained by reducing the temper- ature at the end of the cycle by means of increased vacuum, but in the case in point the maximum vacuum obtainable in practice was used so that no additional economy can be expected in this way. * Defined as thermal efficiency under Art. 136. — Author. STEAM TURBINES 327 " Summary of Analysis of Heat Balance. " The present type of power plant using reciprocating engines can be improved in efficiency as follows : Reduction of stack losses 12% Reduction in boiler radiation and leakage. . . . 5% Reduction in engine losses by the use of superheat 6% 28 25 *, s V s \ " \ \ v s s s \ \ S s \ S s k N | 17 18 19 water rate-pounds per kilowatt- hour-load 4 000 kilowatts Fig. 71. resulting in a net increase of thermal efficiency of the entire plant of 4.14 per cent., and bringing up the total thermal efficiency from 10.3 per cent, to 14.44 per cent." It was subsequently found that the capacity of the plant was inadequate to generate sufficient power to take care of the increas- ing demand, due to the increment in traffic; and it was finally decided to install low-pressure turbines to operate on the exhaust steam of the compound reciprocating engines. 328 THERMODYNAMICS The following quotation * indicates the all-around gain by this combination. " During the year 1908 it became apparent that owing to the ever-increasing traffic in the New York subway, it would be necessary to have additional power available for the winter of 1909-1910. " 2. The power plant of the Interborough Rapid Transit Com- pany, which supplies the subway, is located on the block bounded by 58th and_59th Streets, and by 11th and 12th Avenues, ad- jacent to the North River; it contains nine 7500-kw. (maximum rating) engine units, besides three 1250-kw. 60-cycle turbine units which are used exclusively for lighting and signal purposes. " 3. The 7500-kw. units consist of Manhattan-type com- pound Corliss engines, having two 42-in. horizontal high-pressure cylinders and two 86-in. vertical low-pressure cylinders. Each horizontal high-pressure cylinder and vertical low-pressure cylinder has its connecting rod attached to the same crank, so that the unit becomes a four-cylinder 60-in. stroke compound engine with an overhanging crank on each side of a 7500-kw. maximum rating 11,000-volt, three-phase, 25-cycle generator. The generator revolving field is built up of riveted steel plates of sufficient weight to act as a flywheel for the two engines con- nected to it. This arrangement gives a very compact two-bear- ing unit. The valve gear on the high-pressure cylinders is of the poppet type, and on the low-pressure of the Corliss double-ported type. " 4. The condensing apparatus consists of barometric con- densers, arranged so as to be directly attached to the low-pressure exhaust nozzles, with the usual compound displacement circu- lating pump and simple dry- vacuum pump. " 5. These engine and generator units are in general probably the most satisfactory large units ever built, as five years' experience * "Tests of a 15,000-KW. Steam-Engine-Turbine Unit," by H. G. Stott and R. J. S. Pigott. Transactions of the A.S.M.E., Vol. XXXII. STEAM TURBINES 329 with them has proved; their normal economic rating is 5000 kw., but they operate equally well (water rate excepted) on 8000 kw. continuously. " 6. In considering the problem of how to get an additional supply of power, every available source was considered, but by a process of elimination only two distinct plans were left in the field. " 7. The electric transmission of power from a hydraulic plant was first considered, but owing to the high cost of a double transmission line from the nearest available water power, and the impossibility of gettng reliable service (that is, service having a maximum total interruption of not more than ten minutes per annum) from such a line, further consideration of this plan was abandoned. " 8. The gas engine, while offering the highest thermo-dynamic efficiency, at the same time required an investment of at least 35 per cent more than ordinary steam-turbine plant with a prob- able maintenance and operation account of from four to ten times that of the steam turbine. " 9. The reciprocating-engine unit of the same type as those already installed, was rejected in spite of its most satisfactory performance, on account of the high first cost and small range of economical operation. Reference to Fig. 72 will show that the economic limits of operation are between 3300 kw. and 6300 kw.; beyond these limits the water rate rises so rapidly as to make operation undesirable under this condition, except for a short period during peak loads. "10. The choice was thus narrowed down to either the high- pressure steam turbine or the low-pressure steam turbine. There was sufficient space in the present building to accommodate three 7500-kw. units of the high-pressure type, or a low-pressure unit of the same size on each of the nine engines, so that the questions of real estate and building were eliminated from the problem. $m THERMODYNAMICS "11. The 'first cost of a low-pressure turbine unit is slightly/ lower than that of a high-pressure unit, due to the omission of the high pressure stages and the hydraulic governing apparatus, but the cost.x>f ^he condensing^apparatiis would be. the. same in both cases. .The foundations and the.steam piping in both cases 23 30 22 o25 < o CD wlio 18 16 VARIATION OF WATER-RATE & BEST RECEIVER PRESSURE WITH LpAD . ORIGINAL ENGINE A:- WATER RATE B:- REC. PR. V - -- " - - - - ^y^Q> - " "'" " -' ■* ~ ^^ - - -v.". ._ :.. - rf B ■-' -•" s j - _ _ '- " - - L A \-£ ^""^C > J - -- - -'- "2000 3000 4000 " "5000 ENGINE. LOAD K.W. 6000" 700U j .Eig. 72. .:. -.,;. ;._' . .-1 .;; .,. „\_ :• . ..:„.■_•..:. j. - _c n . ; m ::■. j .vjaj_q would not differ greatly.. The economic^ results, so far as the fisst cost is concerned, would then be approximately the same, if we consider the general case only; but in this particular instance the installation of high-pressure turbines would have meant a much greater investment for foundations, flooring, switchboard STEAM TURBINES 331 apparatus, steam piping and water tunnels, amounting to an addition of not less than 25 per cent to the first cost. " 12. The general case of displacing reciprocating engines and installing steam-turbine units in their place was also con- sidered. The best type of high-pressure turbine plant has a thermal efficiency approximately 10 per cent better than the best reciprocating-engine plant, but the items of labor for operation and for maintenance, together with the saving of about 85 per cent of the water for boiler-feed purposes and the 10 per cent of coal, reduce the relative operating and maintenance charges for the steam-turbine plant to 80 per cent, as compared to 100 per cent for the reciprocating-engine plant. " 13. Assuming that the reciprocating engine plant is a first- class one and has been well maintained, about 20 per cent of its original cost (for engines, generators and condensers) may be real- ized on the old plant and so credited to the cost of the high-pres- sure turbine plant. But on the other hand, if the high-pressure turbine installation is to receive credit for the second-hand value of the engines, it must also have a debit charge for 100 per cent of the original reciprocating-engine plant which it displaced. The relative investments, therefore, upon this basis would be approximately equal for the high-pressure or the low-pressure turbine; but 80 per cent of the cost of the original engine plant would have to be charged against the high-pressure turbine plant, as against an actual increase in value (to the owner) of the engine by reason of its improved thermal efficiency, due to the addition of the low-pressure turbine. " 14. The preliminary calculations, based upon the manu- facturers' guarantees for the low-pressure and high-pressure turbines, showed that the combined engine-turbine unit would give at least 8 per cent better efficiency than the high-pressure turbine unit, so that it was finally decided to place an order for one 7500-kw. (maximum rating) unit, as by this means we would not only get an increase of 100 per cent in capacity, but 332 THEKMODYNAMIOS at the same time give the engines a new lease of life by bring- ing them up to a thermal efficiency higher than that attained by any other type of steam plant. " 15. The turbine installed is of the vertical three-stage impulse type having six fixed nozzles and six which can be operated by hand, so as to control the back pressure on the engine, or the division of load between engine and turbine. An emergency overspeed governor, which trips a 40-in. butterfly valve on the steam pipe connecting the separator and the turbine and at the same time the 8-in. vacuum breaker on the condenser, is the only form of governor used. The footstep bearing, carrying the weight of the turbine and generator rotors, is of the usual design supplied with oil under a pressure of 600 lb. per sq. in. with the usual double system of supply and accumulator to regulate the pressure and speed of the oil pumps. " 16. The condenser contains approximately 25,000 sq.ft. of cooling surface arranged in the double two-pass system of water circulation with a 30-in. centrifugal circulating pump having a maximum capacity of 30,000 gal. per hr. The dry vacuum pump is of the single-stage type, 12-in. and 29-in. by 24-in., fitted with Corliss valves on the air cylinder. The whole condensing plant is capable of maintaining a vacuum within 1.1 in. of the barometer when condensing 150,000 lb. of steam per hr. when supplied with circulating water at 70 deg. fahr. " 17. The electric generator is of the three-phase induction type, star-wound for 11,000 volts, 25 cycles and a speed of 750 r.p.m. The rotor is of the squirrel-cage type with bar winding connecting into common bus-bar straps at each end. This type of generator was chosen as being specially suited to the conditions obtaining in the plant. "18. With nine units operating in multiple, each one capable of giving out 15,000 kw. for a short time, operating in multiple with another plant of the same size, it is evident that it is quite possible to concentrate 270,000 kw. on a short circuit. If we STEAM TURBINES 333 proceed to add to this, synchronous turbine units of 7500-kw. capacity, which, owing to their inherently better regulation and enormous stored energy, are capable of giving out at least six times their maximum rated capacity, the situation might soon become dangerous to operate, as it would be impossible to design switching apparatus which could successfully handle this amount of energy. The induction generator, on the other hand, is entirely dependent upon the synchronous apparatus for its excitation, and in case of a short circuit on the bus-bars would automatically lose its excitation by the fall in potential on the synchronous apparatus. "19. The absence of fields leads to the simplest possible switching apparatus, as the induction generator leads are tied in solidly through knife switches, whichare never opened, to the main generator leads. The switchboard operator has no control whatever over the induction generator, and only knows it is present by the increased output on the engine generator instru- ments. " 20. The method of starting is simplicity itself — the exciting current is put on the engine generator before starting the engine, and then the engine is started, brought up to speed and synchron- ized in exactly the same way as before. While starting in this way, the induction generator acts as a motor until sufficient steam passes through the engine to carry the turbine above synchronism, when it immediately becomes a generator and picks up the load. Three of these 7500-kw. low-pressure turbine units have been installed and tests run on Nos. 1 and 2. No. 3, having been just started, has not yet been tested. "21. Instead of inserting in this paper the enormous accumu- lation of data incident to these tests, we have divided the paper into two parts in the hope that it would thus be more accessible for reference, the first part giving the reasons for adopting this particular type of apparatus, with a brief description of the plant and a summary of the results obtained, and the second part con- 334 THERMODYNAMICS taining all the principal data acquired during the tests, with sufficient explanation to make their meaning clear without refer- ence to the text." " 24. The net results obtained by the installation of low- pressure turbine units may be summarized as follows : "a. An increase of 100 per cent in maximum capacity of plant. " b. An increase of 146 per cent in economic capacity of plant. " c. A saving of approximately 85 per cent of the condensed steam for return to the boilers. " d. An average improvement in economy of 13 per cent over the best high-pressure turbine results. " e. An average improvement in economy of 25 per cent (between the limits of 7000 kw. and 15,000 kw.) over the results obtained by the engine units alone. "/. An average unit thermal efficiency between the limits of 6500 kw. and 15,500 kw. of 20.6 per cent." 205. Summary. The two preceding quotations are self-explan- atory; hence no comment is necessary. But, before concluding, it must be remarked that the internal combustion engine and the steam turbine are still in the experimental stage; and that it is impossible to predict what the final adjustment will be. It is true that the internal combustion engine has a higher thermal efficiency than has any other heat motor. But, due to complexity of construction, the internal combustion engine has a higher first cost; and furthermore, its regulation is inherently inferior to a reciprocating engine or turbine. Due to this, in spite of the fact that the reciprocating engine has a lower thermal efficiency, it still holds its place, on account of its simplicity and high over-load capacity; the latter being especially important in most power plants where it is necessary to take care of large " peak loads." It must be further remarked, that the installation of every power plant is finally affected by the economy of transmission. STEAM TUEBINES 335 Whether power can be developed economically at any locality depends upon whether the cost of power for the particular locality is greater or less if developed at this particular point, or developed at some other point and transmitted to the point under considera- tion. This, of course, depends largely upon the economy of transmission. At the present time electrical engineers are giving consider- able attention to the subject of high-tension transmission. And if it develop that methods can be devised by means of which corona losses can be eliminated, or partially avoided, for potential differences far in excess of those employed at present, the subject of power plant economics will require revision. For, if corona losses can be avoided, the cost of power for any particular locality will be materially changed. And hence, the cost for the produc- tion of power will, likewise, be changed. To illustrate concretely: Assume that it becomes possible to transmit with a potential difference of 300 kilo-volts instead of 125 or 150 kilo-volts. Under these conditions the economy of transmission is considerably increased; and the distances to which coal can be transported, to compete with the increased efficiency of transmission, is considerably reduced. This, however, is not the only governing factor. Ground rent also influences the choice. That is, when the saving in transmission and the saving in ground rent, by locating the plant at the coal fields, is balanced against the hauling of the coal, and the ground rent for a large city, it may develop that it is more economical to locate the power plant where the coal is mined. A similar argument, of course, applies to water-power plants. That is, the initial cost of a water-power plant is high, and therefore the distance, for a given potential difference, over which power can be profitably transmitted is limited; and, of course, the lower the cost of transmission, the greater the area over which profitable transmission may take place. Hence, as the potential difference, which may be employed in transmission, is increased, the smaller, 336 THERMODYNAMICS relatively, due to high ground rent, becomes the economy of a localized plant. Therefore, if it develop, that potential differences, far in excess of those employed at the present time, may be used, power plants in large cities, where ground rent is high, will dis- appear; and the future power plant will be located at the point where the raw material, for the development of power, is found. INDEX PAGE Absolute scale 43 Absolute zero 43 Adiabatic changes, change of dryness with 139 change of temperature with 59 Adiabatic equation for gases 50 Air, and adiabatic compression in compressor 235 and air compressors 230 composite diagram of compression, transmission, and expansion .... 260 compression and expansion 231 isothermal 233 constants of 230 loss of head in transmission pipes 254 Air compressor 230 throttling and other imperfections 249 Air motor, adiabatic expansion in 249 Air refrigerating machines 266 ideal coefficient of performance of 267 T-§ diagram of 270 Air transmission system, theoretical efficiency of 262 Ammonia, coefficient of absorption 280 heat of dilution 281 Ammonia refrigerating machines, absorption 277 compression 271 Analysis of power plant losses 310 Anode 30 Boyle's law 41 departure from 43 Brake power 171 Brayton cycle 195 British thermal unit 9 Calorie - 8 common 9 gram 9 mean 8 zero 8 337 338 INDEX PAGE Calorimeter 11 condensing 154 cooling constant of 14 Joly's differential steam 70 Joly's steam 69 separating 156 thermal capacity of 13 throttling 151 Calorimetry 11 Carnot's cycle 97 a reversible process 100 coefficient of conversion 99, 109 steam operating on 116 with a perfect gas 104 Cathode 30 Centigrade scale 3 Characteristic equation , 43 Chemical reactions of fuels 198 Clearance 156 equivalent length of 157 of air compressor 247 Commercial efficiency of steam engine 176 Compound engines, cross 189 tandem 183, 186, 188 Compressed air 230 Compressed air system, composite diagram of 260 theoretical efficiency of 262 Compression, adiabatic 235 isothermal 233 multi-stage 239 theoretical efficiency of 237 Condenser 121 Condensing calorimeter 154 Conduction 83 flow of heat along a bar 86 Conductivity, coefficient of 85 determination of coefficient of 86 in non-isotropic substances 89 non-homogeneous solids 89 of gases 90 of liquids 90 table of coefficients 90 Convection 83 Cooling, constant of calorimeter 14 Newton's law of 11 Stefan's law of 82 INDEX 339 PAGE Critical temperatures, table of 63 Cross-compound engine 189 Curtis turbine 307 Curves, theoretical and actual 163 Cushion steam 157 Cycle 76 Brayton 195 Carnot's 97 with a perfect gas as a working substance 104 Diesel 196 four phase 193 of perfect steam engine and boiler represented by means of T-§ diagram 137 Rankine's 177 reversible, as a standard 103 two phase 194 Cylinder feed 157 De Laval nozzle 298 Density and temperature of saturated steam 118 Dew point 66 Diesel cycle 196 Disgregation, heat of 62 Dissociation 28 Ebullition 61 Economics of power plants 310 and transmission 334 Efficiency and precompression 218 commercial 176 mechanical 175 of power plants and transmission 334 thermal 175 Elastic medium, propagation of wave motion 74 speed of 77 Elasticities, determination of ratio from speed of propagation of wave motion 77 isothermal and adiabatic and ratio of two thermal capacities . 72 Electro-chemical equivalent 31 Electrolysis 29 counter e.m.f . of 31 Faraday's statement 30 Emissivity 81 Endothermic 21 Energy, principle of 33 340 INDEX PAGE Engine and boiler, perfect, with steam 121 studied with aid of the T — diagram 137 clearance 156 internal combustion 192 actual indicator diagram of 215 actual p-v and T — c|> diagrams of 222 double-acting cylinders 227 multi-cylinder 226 standard diagram 213 thermal efficiency of 214 T-§ diagram of 219 reversible 101 and refrigeration 110 as a standard 103 ideal coefficient of conversion 99, 109 simple thermodynamic 95 steam, condensing 150 double-acting 150 Engines, compound 180 cross 189 double expansion 181 tandem, with large receiver 183 tandem, with small receiver 188 tandem, without receiver 186 triple expansion 190 heat 94 reciprocating and turbines 308 reciprocating, combined with low-pressure turbines 310, 328 Entropy 128 and temperature diagrams 134 change of 129 during reversible and irreversible processes 134 universal increment of 132 Evaporation 60 Exchanges, Prevost's theory of 80 Exothermic 21 Expansion, adiabatic, in motor 249 change of dryness during 167 double, in steam engine 181 external heat of 97 heat of 62 ■ internal heat of .... r 97 isothermal 233 triple, in steam engine 190 without doing external work 46 INDEX 341 PAGE Expansion, linear 34 coefficient of .• 34 factor of 34 table of coefficients of 36 voluminal 35 determination of coefficient of, for liquids 36 direct measurement of 39 Fahrenheit scale 3 Four-phase cycle 193 Frequency 76 Friction brake 171 Fuel, calorific value of 203 determination of 204 Fuels, and chemical reactions 198 and fuel tests 198 liquid 207 table of calorific values of 208 Fusion, heat of 22 Gas, characteristic equation of 43 city 203 perfect or ideal .' . . 44 thermometer 44 Gases, adiabatic equation 50 change of temperature with adiabatic changes 59 conductivity of 90 elasticities of 72 expansion of, without doing external work 46 Gay-Lussac's law 41 general equations of 52 isothermal equation 41 table of constants 208 thermal capacities of 47 at constant pressure 48, 68 at constant volume 48, 69 determination of 68, 69, 70, 74 Gasoline 200 Gram calorie 9 Head, loss of, in transmission pipes 254 Heat, as a measurable quantity 7 content 125 effects of 33 engines or motors 94 flow along a bar 86 342 INDEX PAGE Heat, mechanical equivalent of 17 from constants of air 48 of dilution 281 of disgregation 62 of fusion 22 of vaporization 23 for water 65 production of 33 total, of steam 65 unit quantity 8 Heating machine 287 Humidity, absolute 67 relative 66 Hydraulic-radius 255 Hygrometry 66 Impact on curved surfaces 289 Indicated power 173 Indicator 159 Indicator diagram 161 and valve adjustment 161 comparison of theoretical and actual curves 163 ideal of internal combustion engine 210 standard of internal combustion engine 213 Internal combustion engine 192 actual indicator diagram of 215 actual p-v and T — <£> diagrams 222 and double-acting cylinders 227 efficiency and precompression 218 multi-cylinder 226 standard diagram of 213 thermal efficiency of .• 214 Irreversible processes 27 Isothermal equation 41 Isotropic 35 Kerosene 201 Mean effective pressure 174 Mechanical efficiency of steam engine 175 Mechanical equivalent of heat 17 from constants of air 48 Mixtures, method of 10 Motor, adiabatic expansion of air in 249 reheating air before expanding in 251 Motors, heat 94 INDEX 343 PAGE Multi-cylinder internal combustion engines 226 Multi-stage air compressors 239 Non-isotropic 36 Parsons turbine 306 Pelton cup 293 Power, brake 171, 214 indicated 173, 214 Power plant economics 310 analysis of losses 310 efficiency and transmission 334 Precompression and efficiency 218 Pressure and temperature of saturated steam 119 Pressure, mean effective 174 Priming 126 Principle of energy 33 Radiation 80 Dulong and Petit's formula 82 Prevost's theory of exchanges 80 Stefan's formula 82 Rankine's cycle 177 Reaumur scale 3 Reciprocating engines and turbines 308 engines exhausting to low-pressure turbines 310 Refrigerating machine 110 absorption 277 air 266 ideal coefficient of performance of 267 T-cJ> diagram of 270 commercial 265 commercial efficiency of 269 comparison of air and ammonia 284 compression, using volatile liquids 271 T-4> diagram of 276 Kelvin heating 287 Refrigeration 265 and reversible engine 110 Refrigerator 94, 266 Resisted adiabatic expansion of steam 126 Reversible process, ideally 27 Reversible processes 25, 104 Separating calorimeter ' 156 Specific heat 9 344 INDEX PAGE Steam, behavior throughout the cycle 166 change of dryness during adiabatic expansion 126, 139 by means of T — diagram 139 cushion 157 cylinder feed 157 double expansion engine 181 dryness during expansion in cylinder 167 engines, reciprocating and turbines 308 engines, reciprocating with low-pressure turbines 328 exchange of heat with cylinder walls 168 jackets 170 operating on Carnot's cycle 116 relation of temperature and entropy 139, 141 resisted adiabatic expansion 126 saturated, density and temperature of 118 relation or pressure and temperature 119 total heat of 65 triple expansion engine 190 turbine 289 unresisted adiabatic expansion 125 wire drawing V 158 with perfect engine and boiler 121 work and incomplete expansion 145 work and superheating 148 work without expansion 144 zero curve 142 Zeuner's equation for adiabatic changes 151 Stefan's formula 82 Sublimation 23 Superheated vapors 66 Superheating 24 Tandem compound engines 183, 186, 188 Temperature 1 critical 62 table of 63 difference of . : 2 Temperature and density of saturated steam 118 Temperature and pressure of saturated steam 119 Temperature-entropy diagrams 134, 219 dryness by means of 139 of air refrigerating machines 270 Temperatures, theoretical in cylinder of internal combustion engine 213 thermodynamic scale of 113 Thermal capacities, determination of by method of cooling 16 INDEX 345 PAGE Thermal capacities, determination of by method of mixtures 15 Thermal capacities of gases 47 determination of, at constant pressure 68 at constant volume 69 ratio of, by method of Clement and De- sormes 70 ratio of, from speed of propagation of wave motion 77 Thermal capacity 8 by method of mixtures 10 per unit mass and specific heat 9 Thermal efficiency of internal combustion engine 214 of steam engine 175 Thermal equilibrium 2 Thermo couple 6, 32 Thermodynamic drop 158 scale of temperatures 113 Thermodynamics, first principle of 93 second principle of . 93 Thermometer 2 alcohol 6 gas 44 mercurial 5 resistance 6 Thermometric scales 3 Throttling and other imperfections of air compressor 249 calorimeter 151 Transmission of air and loss of head in pipes 254 of power and economy of plants 334 Turbine blades, impact on 289 Turbines, and reciprocating engines 308 comparison of Parsons and Curtis 307 Curtis .307 low pressure on exhaust of reciprocating engine 310, 328 Parsons 306 steam 289 tests 310 two principal types of 305 Two phase cycle 194 Unresisted adiabatic expansion of steam 125 Valve adjustment and indicator diagram 161 Vaporization 60 heat of 23 heat of, for water 65 346 INDEX PAGE Vapor pressures, addition of 61 Vapors 60 saturated 60 superheated 24, 66 Water equivalent 10 Water, relation of temperature and entropy 137, 141 Wave length 76 Wave motion, propagation of 74 speed of propagation 77 Wire drawing 158 Work, gain of, due to superheating steam 148 loss of, due to incomplete expansion of steam 145 loss of, due to using steam non-expansively 144 \j &V 1 One copy del. to Cat. 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