Cornell University Library 
 TJ 170.B841 
 
 Water-wheels, or hydraulic motors, 
 
 3 1924 004 406 561 
 
Cornell University 
 Library 
 
 The original of this book is in 
 the Cornell University Library. 
 
 There are no known copyright restrictions in 
 the United States on the use of the text. 
 
 http://www.archive.org/details/cu31924004406561 
 
¥ATEE-¥HEELS 
 
 HYDRAULIC MOTORS. 
 
 TRANSLATED FROM THE 
 
 "COUES DE MEGA ISTI QUE APPLIQUEE." 
 
 Profetaeur de Mdcanique d rjtcole des Fonts et Chausees. 
 
 F.^A. MAHAN, 
 
 LIEUTENANT U. S. CORPS OF ENGINEERS. 
 
 -,'-' /REVISED BS 
 
 D/' H/MAHAN", LL.D., 
 
 PHOFKSSOR OF 01V1L ENGINEERING, AC., UNITED STATES MILITARY ACADEMY. 
 
 NEW EDITION, 
 WITH THE FRENCH MEASURES TRANSLATED INTO ENGLISH. 
 
 NEW YORK: 
 JOHN WILEY & SONS, 
 
 15 ASTOR PLACE. 
 1876. 
 
Entered according' to Act of Congress, in the year 1869, by 
 
 F. A. MAHAN, 
 
 In the Clerk's Office of the District Court of the United States for the Southern District of 
 
 New York. 
 
 John F. Trow & Son, Printers, 
 205-213 East 12TH St., New York., 
 
PKEFACE. 
 
 The eminent position of M. Bresse in the scientific 
 world, and in the French Corps of Civil Engineers, is 
 my best apology for attempting to supply a want, felt 
 by the students of civil engineering in our country, of 
 some standard work on Hydraulic Motors, by furnish- 
 ing a translation of the chapter on this subject con- 
 tained in the second volume of M. Bresse's lectures 
 on Applied Mechanics, delivered to the pupils of the 
 School of Civil Engineers (JZcole des Fonts et Ohaus- 
 sees) at Paris. 
 
 In making the translation, I have retained the French 
 units of weights and measures in the numerical exam- 
 ples given, as the majority of our students are conver- 
 sant with them. 
 
 F. A. MABAN. 
 
 Willett's Point, N. T., July, 1869. 
 
 November, 1875. 
 
 In the present edition the French units of weights 
 and measures have been translated into English. 
 
CONTENTS 
 
 § I.— PRELIMINARY IDEAS ON HYDRAULIC MOTORS. 
 
 * ART. PAGK. 
 
 Definitions ; theorem of the transmission of work in machines 1 9 
 
 Analogous ideas applied to a waterfall 2 11 
 
 General observations on the means of securing a good effective 
 
 delivery for a waterfall driving a hydraulic motor 3 14 
 
 § II,— WATER WHEELS WITH A HORIZONTAL AXLE. 
 
 Undershot wheel with plane buckets or floats moving in a confined 
 
 race 4 17 
 
 Wheels arranged according to Poncelet's method 5 28 
 
 Paddle wheels in an unconfined current 6 34 
 
 Bbeast Wheels. 
 
 Wheels set in a. circular race, called breast wheels 7 36 
 
 Example of calculations for a rapidly moving breast wheel 8 48 
 
 Overshot Wheels. 
 
 Wheels with buckets, or overshoti wheels 9 51 
 
 § III.— WATER WHEELS WITH VERTICAL AXLES. 
 
 Old-fashioned spoon or tub wheels 10 60 
 
 Turbines. 
 
 Of turbines 11 69 
 
 Fourneyron's turbine 12 70 
 
 Fontaine's turbine 13 73 
 
 Kcecklin's turbine 14 76 
 
 Theory of the three preceding turbines . . . . 15 78 
 
 Eemarks on the angles 0, y, 0, and on the dimensions b, V, r, r', 
 
 h, K 16 90 
 
-VU1 CONTENTS. 
 
 ART. PAGE. 
 
 Examples of the calculations to be made in constructing a turbine. . 17 93 
 
 Of the means of regulating the expenditure of water in turbines ... 1 8 98 
 
 Hydropneumatic turbine of Girard and Callon 19 103 
 
 Some practical views on the subject of turbines 20 105 
 
 Reaction wheels 21 107 
 
 § IV.— OF A FEW MACHINES FOR RAISING WATER. 
 
 Pumps 22 111 
 
 Spiral noria 23 122 
 
 Lifting turbines ; centrifugal pump 24 127 
 
 Authorities on Water Wheels „. 134 
 
 APPENDIX. 
 
 Comparative table of French and United States measures 135 
 
 Note A, Article 1 135 
 
 Note B, Article 2 136 
 
 Note C, Article 4 137 
 
 Note D, Article 9 140 
 
 Note.E, Boyden turbines, from Lowell Hydraulic Experiments, by 
 
 James B. Francis, Esq .41 
 
HYDRAULIC MOTORS, 
 
 AND SOME 
 
 MACHINES FOR RAISING WATER. 
 
 § |. — Preliminary Ideas on Hydraulic Motors. 
 
 1. Definitions ,' theorem of the transmission of worlc in ma- 
 chines. — The term machine is applied to any body or collection 
 of bodies intended to receive at some of their points certain 
 forces, and to exert, at other points, forces which generally 
 differ from the first in their intensity, direction, and the velocity 
 of their points of application. 
 
 The dynamic effect of a machine is the total work, generally- 
 negative or resisting, which it receives from external bodies 
 subject to its action. It happens that the dynamic effect is 
 sometimes positive work : for example, when we let down a 
 load by a rope passed over a pulley, the weight of the load pro- 
 duces a motive work on the system, of the rope and pulley. 
 
 Let us suppose, to make this clear, that the dynamic effect is 
 a resisting work. Independently of this, the machine is affected 
 by some others which are employed to overcome friction, the 
 resistance of the air, &c. The resistances which give rise to 
 these negative works have received the name of secondary 
 
10 GENERAL THEOREM. 
 
 resistances; whilst the dynamic effect is due to what are called 
 principal resistances. 
 
 The general theorem of mechanics, in virtue of which a 
 relation is established between the increase of living force of 
 a material system and the work of the forces, is applicable 
 to a machine as to every assemblage of bodies. To express 
 it analytically, let us suppose the dynamic effect to be taken 
 negative, and let us call 
 
 T m , the sum of the work of the motive forces which have 
 acted on the machine during a certain interval of time ; 
 
 T e , the dynamic effect during the same time ; 
 
 T /5 the corresponding value of the work of the secondary 
 resistances ; 
 
 v and v m the velocities of a material point, whose mass is m, 
 making a part of the machine, at the beginning and end 
 of the time in question ; 
 
 H and IT , the corresponding distances of the centre of grav- 
 ity of the mechanism below a horizontal plane ; 
 
 2, a sum extended to all the masses m. 
 
 From the theorem above mentioned there obtains 
 i*m v'-i^m «„■ = T m - T e - T f + (H — TL a )2mg. 
 
 In a machine moving regularly, each of the velocities v in- 
 creases from zero, the value corresponding to a state of rest, to 
 a certain maximum which it never exceeds ; the first member 
 of the equation has, then, necessarily, a superior limit, below 
 which it will be found, or which it will, at the farthest, reach, 
 whatever be the interval of time to which the initial and final 
 values v and v are referred. The same holds with the term 
 (H — 1J ) 2 m g, when the machine moves without changing 
 its place, as its centre periodically occupies the same posilions. 
 On the contrary, the terms T m , T e , T r increase indefinitely with 
 the time, if the motion of the machine is prolonged, because 
 
GENEKAL THEOREM. 11 
 
 new quantities of work are being continually added to those 
 already accumulated. These terms will at length greatly 
 surpass the others, so the equation, therefore, should, after an 
 unlimited interval of time, reduce to 
 
 i-m, = # 1-c 4- J-/- 
 
 This is what would really take place, without supposing the 
 time unlimited, if the beginning and end of this interval cor- 
 responded to a state of rest of the machine, and if, at the same 
 time, II = H„. We can then say that, as a general rule, the 
 motive work is equal to the resisting work; but as this last 
 includes, besides the dynamic effect for which the machine is 
 established, the work of the secondary resistances, we see that 
 the action of the motive power is not all usefully employed, 
 since a portion goes to overcoming the work of Ty. 
 
 T 
 It is evident that the ratio -frf- measures the proportional 
 
 -*-m 
 
 T T 
 
 loss; the ratio jyr, or 1 — ~j-r, gives, on the contrary, a clear 
 
 idea of the portion of the effective work. This last ratio is 
 what is called the delivery of the machine; it is evidently 
 always less than unity, since in the best-arranged machines H f 
 still preserves a certain value. The skill of the constructor is 
 shown in bringing this as near unity as possible. 
 
 2. Analogous ideas applied to a waterfall. — A waterfall 
 may be considered as a current of water flowing through two 
 sections C B,E F (Fig- 1), at no great distance apart, but 
 with a noticeable difference of level, H, between the surface 
 slope at C and E, and which may be assumed to yield a 
 constant volume of water for each unit of time. The mate- 
 rial liquid system thus comprised between the sections C B, 
 E F, at each instant may be regarded as a machine that is con- 
 tinually renewed, the molecules which flow out through E F 
 
12 
 
 GENERAL THEOREM. 
 
 being replaced by those which enter through B. The motive 
 work in this case will be that of the weight combined with that 
 of the pressures on the external boundaries of the system ; the 
 dynamic effect will be the work of the resistances against the 
 fall of water caused by any apparatus whatever, a water-wheel 
 
 Fig. 1. 
 
 for example, exposed to its action. In its turn the water-wheel, 
 considered as a machine, will receive from the fall a motive 
 work sensibly equal to the dynamic effect that we have just 
 spoken of,* and will change only a portion of it into useful 
 work, which will be its dynamic effect proper. But in what is 
 to follow we shall limit ourselves to studying the dynamic effect 
 of the fall, and not that of the wheel. 
 
 Although the motion of the liquid cannot always be strictly 
 the same, because the wheel does not always maintain exactly 
 the same position, still it can be so regarded without material 
 error; for after an interval of time d, generally very short, 
 occupied by a float or paddle in taking the place of the one that 
 preceded it, everything returns to the same condition as at the 
 beginning of the interval. Supposing the motion of the wheel 
 regular and the paddles to be uniformly distributed, there is 
 such a frequent periodicity in the state of the system that it 
 
 * We say sensibly, for the equality between the mutual actions of the water 
 and wheel do not involve that of corresponding work. This equality is only 
 strict in supposing the friction of the liquid against the solid sides of the 
 wheel zero, which friction is in reality very slight. 
 
GENERAL THEOREM. 13 
 
 almost amounts to a permanency. We will now apply Ber- 
 noulli's theorem to any molecule whatever, having a mass m, 
 which, departing from the section C B with the velocity Y , 
 reaches E F with a velocity Y. The entire head is H, if we 
 allow the parallelism of the threads in the extreme sections, for 
 the pressure then varies according to the hydrostatic law in 'each 
 of the two surfaces C B and E F, so that the points C and E 
 can be considered as piezometric levels for the initial and final 
 positions of the mass m. Let — t e and — ty be the respective 
 work referred to the unit of mass, and considered as resistances 
 which m has encountered in its course between C B and E F, 
 in consequence of the action of the wheel and of viscosity. 
 Then from the general theorem of living forces we have the 
 equation — 
 
 H - - (t e +tf)~ ^r-~ = ; 
 9 2 g ' 
 
 whence, 
 
 Y 2 Y^ 
 
 Consequently we see that the weight m g of each molecule 
 which passes from C B to E F gives rise to a dynamic effect 
 m t e , the value of which is expressed in the second member of 
 the equation. The sum of the weights of the molecules m in- 
 cluded in the entire weight P, which the current expends in a 
 second, will produce a dynamic effect equal to a sum 2 of analo- 
 gous expressions extended to all these masses ; considering Y 
 and Y as constant velocities in the respective sections C B and 
 E F, this summation will give — 
 
 Moreover, in each second a new weight P is supplied by the 
 current ; there is then produced a new dynamic effect 2 m t e , 
 which thus represents the mean dynamic effect in each second. 
 
 m t, = m g ( H + <fZ ~ ^) mt f 
 
14 GENERAL THEOREM. 
 
 "V" 2 v a 
 The quantity P f H + — — — - — J reduces to P H, in the 
 
 case in which Y and Y are sensibly equal to zero, which hap- 
 pens in measuring the difference of level between the basins 
 from which the water starts and that into which it flows, when 
 the water is nearly at a stand-still ; we then call the product 
 
 P H the effective delivery of the fall. The ratio -~p~rr i s the 
 
 productive force ; 2 m t f \s the work lost. Dividing the last 
 equation by P or 2 m g, and supposing Y = 0, Y — 0, there 
 obtains, 
 
 L^=H-l£Jr .... (2 ) 
 
 g 2 ?u g 2 m x ' 
 
 H in this expression may be regarded as the total head of water; 
 
 e the productive head, that is, the height which, multi- 
 
 g 2 m J ' ' ° ' 
 
 plied by the weight P expended, would give the dynamic effect 
 
 per second ; the head lost, to be subtracted from the 
 
 1 ' g 2 m ' 
 
 total head when the productive head is required. We see that 
 
 — - — r is the mean loss of head experienced by the molecules in 
 g 2 m r j 
 
 their passage from C B to E F ; for this expression represents 
 exactly the mean work of viscosity on a molecule, referred to 
 the unit of mass and divided by g. 
 
 3. General remarks on the means of securing a satisfactory 
 delivery from a head of water which moves an hydraulic 
 motor. — In order to obtain a good satisfactory delivery, we 
 must seek to diminish as much as possible the term 2 m t f , or 
 
 the mean loss of head f - A few of the causes that pro- 
 
 g 2 m r 
 
 duce this loss will now be pointed out, and the manner in 
 
 which they may be diminished. 
 
GENERAL THEOREM. 15 
 
 Firstly, if the water enters the wheel, and can in consequence 
 act on it, it is because it possesses a certain relative velocity w ; 
 now, in the majority of cases this relative velocity gives rise to 
 a violent agitation of the liquid and to vibratory motions, from 
 
 which a loss of head is experienced equal to g— , like to what 
 
 has been observed to obtain in the collision of solid bodies; 
 
 hence jr- would be a portion of the head lost.* It is, then, 
 
 generally a matter of importance to make the water enter with 
 as small a relative velocity as possible. However, that is not 
 rfeeessary when w has its direction tangent to the sides that it 
 comes in contact with, and when the particular arrangement 
 of the apparatus allows the water to continue its relative mo- 
 tion in the wheel without there being any shock of the threads 
 on the solid sides or on the liquid already introduced, since 
 then we have no longer to fear the violent disturbance that we 
 have just spoken of. 
 
 When the water on leaving the wheel is received into a race 
 of invariable level, in which it loses its absolute velocity of 
 
 * Let us suppose that the water that has once entered the wheel passes at 
 once to relative rest : the destruction of the velocity w being then attributa- 
 ble only to the resisting work of the molecular actions, this work for a fluid 
 molecule having the mass m would be i m w s , a quantity which, when referred 
 
 to the unit of mass and divided by g, would give the loss of head — , which 
 
 « 3 
 
 is the same for all the molecules. But the supposition of the instant produc- 
 tion of relative rest is not, strictly speaking, true ; weight, for example, can 
 combine sometimes with molecular actions to bring about this result at the 
 
 end of a sensible time. Consequently the expression c r— must be considered 
 
 less as the exact value of the quantity whose value we here wish to find, than 
 as a superior limit which we approach more or less nearly, according to the 
 particular case considered. 
 
16 GENERAL THEOREM. 
 
 departure v', we readily see that this velocity v' must be re- 
 duced as much as possible. In fact, the water that leaves the 
 
 wheel carries with it, in each second, a living force, —^ — > 
 ' - a ' 2 g 
 
 which might have been taken up by a resisting work of the 
 hydraulic motor, and have thus increased by its amount the 
 dynamic effect T e ; whereas this living force will only serve to 
 produce a disturbance and eddies in the interior race, and will 
 enter the term 2 m t f . There are cases, however, in which we 
 are obliged to give v' a value more or less considerable; we 
 shall see presently, by a few examples, how it is sometimes 
 possible to diminish this unfavorable condition. 
 
 Ordinarily, the considerations above mentioned are under- 
 stood when we say that the water must enter without shock, 
 and leave without velocity. We can also add, that it is well 
 not to deliver it too rapidly through channels of too little 
 breadth, as this would involve losses of head to be included in 
 
 2 m t f 
 
 the expression -. 
 
 c g 2 m 
 
 We shall now proceed to examine the most widely known 
 hydraulic motors, keeping principally in view the best means 
 of making use of the head in each case, and showing the man- 
 ner of calculating the dynamic effect that can be realized with 
 the means adopted. 
 
 Water-wheels are divided into two great classes — those hav- 
 ing a vertical and those having a horizontal axis. The varieties 
 included in these two classes will form the subject of the fol- 
 lowing paragraphs. 
 
UNDERSHOT WHEELS. 
 
 § ||. "Water- "Wheels with a Horizontal Axis. 
 
 4. Undershot wheel with plane buckets or floats moving in a 
 confined race. — These wheels are ordinarily constructed of wood. 
 Upon a polygonal arbor A (Fig. 2) a socket C, of cast iron, 
 
 Fig. 2. 
 
 is fastened by means of wooden wedges h. Arms D are set 
 in grooves cast in the socket, and are fastened to it by bolts ; 
 these arms serve to support a ring E E, the segments of which 
 are fastened to each other and to the arms by iron bands. In the 
 ring are set the projecting pieces F F . . . . , of wood, placed at 
 
18 UNDERSHOT WHEELS. 
 
 equal distances apart, and intended to support the floats GG. . , 
 which are boards varying from ^ to l^ ia in thickness, situated 
 in planes passing through the axis of the wheel' and occupying 
 its entire breadth. A single set of the foregoing parts would 
 not be sufficient to give a good support to the floats. In 
 wheels of little breadth in the direction of the axis two parallel 
 sets will suffice ; if the wheels are broad, three or more may be 
 requisite. 
 
 The number of arms increases with the diameter of the 
 wheel. In the more ordinary kinds, of 10 to 16.50 feet in 
 diameter, each socket carries six arms. The floats may be 
 about l ft 2 ln to l n 4™ apart, and have a little greater depth in 
 the direction of the radius, say 2 ft to 2 ft 4" 1 - 
 
 From this brief description of the wheel, let us now see how 
 we can calculate the work which it receives from the head of 
 water. The water flows in a very nearly horizontal current 
 through a race B G II F (Fig. 3), of nearly the same breadth 
 
 as the wheel, a portion G H of the bottom being hollowed 
 out, in a direction perpendicular to the axis, to a cylindrical 
 shape, and allowing but a slight play to the floats. The liquid 
 molecules have, when passing C B, a velocity v, but shortlv 
 after they are confined in the intervals limited by two consecu- 
 tive floats and the race. They entered these spaces with a 
 mean relative velocity equal to the difference between the hori- 
 zontal velocity v, and the velocity v'%$ the middle of the im- 
 mersed portion of the floats, the direction of which last velocity 
 is also very nearly horizontal. There result from this relative 
 
TOTDEKSHOT WHEELS. V.) 
 
 velocity a shock and disturbance which gradually subside, 
 while the floats are traversing over the circular portion of the 
 canal ; so that if this circular portion be sufficiently long, and 
 if there be not too much play between the floats and the canal, 
 the water that leaves the wheel will have a velocity sensibly 
 equal to v'. The action brought to bear by the wheel on the 
 water is the cause of the change in this velocity from v to v', 
 which gives us the means, as we shall presently see, of calculat- 
 ing the total intensity of this action. 
 
 For this purpose let us apply to the liquid system included 
 between the cross sections C B, E F, in which the threads are 
 supposed parallel, the theorem of quantities of motion projected 
 on a horizontal axis. Represent by 
 
 i the constant breadth of the wheel and canal ; 
 
 h, hi the depths C B, E F, of the extreme sections which are 
 supposed to be rectangular ; 
 
 F the total force exerted by the wheel on the water, or in- 
 versely, in a horizontal direction ; 
 
 P the expenditure of the current, expressed in pounds, per 
 second ; 
 
 II the weight of a cubic foot of the water ; 
 
 6 the short interval of time during which C B E F passes to 
 C B' E' F'. 
 
 The liquid system C B E F, here under consideration, is 
 analogous to the one treated in Note A (see Appendix), in 
 which a change in the surface level takes place ; and the man- 
 ner of determining the gain in the quantity of motion during 
 a short time 6, and calculating the corresponding impulses, 
 during the same time, produced by gravity and the pressures 
 on the exterior surface of the liquid system, are alike in both 
 cases. 
 
 Employing the foregoing notation, we obtain 
 
20 TXNDEKSHOT WHEELS. 
 
 1st. For the mean gain in the projected quantity of motion, 
 
 — (V - v) ; 
 9 K ' 
 
 2d. For the impulses of the weight and pressures together, 
 also taken in horizontal projection, \ XI h 6 (A 3 — A' 3 ) ; to these 
 impulses is to be added that produced by F, or — F 6, to have 
 the sum of the projected impulses. 
 
 We will then have 
 
 whence we obtain 
 
 F = - - v') - i n I (A' 3 -A 2 ). 
 
 The forces of which F is the resultant in horizontal projec- 
 tion are exerted in a contrary direction by the water on the 
 wheel, at points whose vertical motion is nearly null, and 
 whose horizontal velocity is approximately v'. The wheel will 
 then receive from these forces, in each unit of time, a work 
 F v', which represents the dynamic effect T e to within a slight 
 error. So that 
 
 T e = Ft/ = -v'{v - v') - 1 n b v'{h n - A 2 ). 
 
 P = Ill>hv = llbh'v / ; 
 
 Moreover we have 
 
 whence 
 T. = ?»'(»-»')-Jp(i'-|) = j »'(»-.') 
 
 or finally, observing that p = — 
 
UNDEKSHOT WHEELS. 21 
 
 In order that this formula should be tolerably exact, the depths 
 h and hf mast be quite small, without which the floats would 
 make an appreciable angle with the vertical at the moment 
 they leave the water ; the velocity of the points at which are 
 applied the forces, whose resultant is F, could no longer be con- 
 sidered horizontal, as heretofore, and a resistance due to the 
 emersion of the floats would be produced, on account of the 
 liquid uselessly raised by them. The water must also be con- 
 fined a sufficiently long time to assume the velocity v'. 
 
 "We can consider in formula (1), v and h as fixed data, and 
 seek the most suitable value of the velocity v' of the floats, to 
 
 v' 
 make T e a maximum. For this purpose, if we place — = x, 
 
 T e = AP r — , formula (1) becomes 
 
 A = 8.(l-")-^(5--), 
 
 and we must choose x so as to make A a maximum. This 
 
 value will be obtained by taking the value which reduces the 
 
 d A 
 first differential co-efficient -j — ■ to zero, which gives the equa- 
 
 tion 
 
 »--&(?+ 0=* 
 
 1 
 
 The value of x deduced depends on — — ; we find approxi- 
 mately 
 
 For 9 — = 0.00 x = 0.500 A = 0.500 
 v 
 
 0.05 0.553 0.431 
 
 0.10 0.595 0.373 
 
 Generally we neglect the term T ( x\ ; then we have 
 
22 
 
 UNDERSHOT "WHEELS. 
 
 X 
 
 0.50, A = 0.50 ; and recollecting that the maximum value 
 
 of A is verv much changed, even for small values of — j-. J3e- 
 
 sides, experiment does not show that there is any use in taking 
 
 x greater than 0.50, as we have just found ; it would rather 
 
 show a ratio of v' to v of about 0.4, which obtains probably 
 
 because of the motion of the wheel being too swift to allow the 
 
 liquid to pass completely from the velocity v to v', during the 
 
 time that it remains between the floats : a portion of the water 
 
 passes without producing its entire dynamic effect, and the 
 
 formula (1) ceases to conform to fact. 
 
 Smeaton, an English engineer, made some experiments, in 
 
 1759, on a small wheel, 2 feet in diameter, having plane floats. 
 
 This wheel was enclosed in a race having a flat bottom, which 
 
 is a defect, because the intervals included between the floats 
 
 and the race are never completely closed. The weight P 
 
 varied from l lb .9 to 6 lb .25. In each experiment the work 
 
 transmitted to the wheel was determined by the raising of a 
 
 weight attached to a rope which was wound around the axle. 
 
 v' 
 The most suitable value of as = — was thus found to vary be- 
 
 v J 
 
 tween 0.34 and 0.52, the mean being 0.43.. The number A, 
 comprised between 0.29 and 0.35, had for a mean value about £. 
 
 "We can then take definitively the ratio — = 0.4, as found by 
 
 v 
 
 experiment. The expression for A becomes then 
 A = 0.48 - 2.1 g -4, 
 
 ■which, for "— = = 0.05 and —- — 0.10, gives the numbers 
 
 A = 0.375 and A = 0.27, 
 very nearly those found by Smeaton. 
 
TJNDEBSHOT WHEELS. 23 
 
 A few remarks remain to be made, to which it would be well 
 to pay attention in practice. 
 
 1st. It is well, as far as possible, to have the depth h say 
 from 6 to 8 inches. Too small a thickness of the stratum of 
 water whicli impinges on the wheel would give a relatively 
 appreciable importance to the unavoidable play between the 
 wheel and the race, a play which results in pure loss of the 
 motive water. Too great a thickness has also its inconve- 
 niences : for from the relation -=- = — „ it follows that for — = 
 
 h v v 
 
 0.4 and h = 8 inches, we will then have 
 
 8 
 A' = -^— = 20 inches, 
 0.4 
 
 the floats will then be immersed 20 inches, adopting the thick- 
 ness of 8 m , and were they more so, they would meet with 
 considerable resistance in leaving the water, as we have already 
 said. It is necessary then that h be neither too large nor 
 too small : the limits of 6 in to 8 1 " 1 are recommended by M. Be- 
 langer. 
 
 2d. We should avoid as much as possible losses of head 
 during the passage of the water from the basin up .stream to 
 its arrival at the section C B near the wheel, losses whicli 
 result in a diminution of effective delivery (No. 2). In order 
 to give it the shape of a thin layer, from 6" 1 to 8" 1 in thickness, 
 the water is made to flow under a gate through a rectangular 
 orifice : care should be taken to avoid abrupt changes of direc- 
 tion between the sides of this orifice and the interior of the 
 dam, in order to avoid a contraction followed by a sudden 
 change of direction of the threads, as in cylindrical orifices. 
 The gate should be inclined (as in Fig. 4), in order to leave 
 but a small distance between the orifice and the wheel, whicli 
 will diminish the loss of head produced by the friction of the 
 
24 
 
 UNDEBSHOT WHEELS. 
 
 water on the portion of the race M C, through which the water 
 reaches the wheel. 
 
 3d. The water leaves the wheel at E F with a velocity v' = 
 
 0.4 v, in the shape of a horizontal band of parallel threads. 
 
 If, in order to flow into the tail race through a section G K, 
 
 where its velocity will be sensibly zero, it had to undergo no 
 
 loss of head, there would be between E F and G K a negative 
 
 v" v 2 
 
 head, the absolute value of which would be r— , or 0.16 « — ; that 
 
 2g' 2g' 
 
 is, the point G would be at a height 0.16 „— above E, since the 
 
 piezometric levels at EF and G H may be confounded with 
 those of the points E and G. It is not possible so to arrange 
 every part that all loss of head shall be suppressed between 
 E F and G ~K ; but these losses are much diminished by means 
 of a plan first recommended by M. Belanger. The bottom of 
 the race, beyond the circular portion A D, presents a slight 
 slope, for a distance D I = 3 or 6 feet ; thence it connects 
 with the tail race by a line I K, having an inclination from 
 1 inch to 1.25 inches for each foot in length; the side-walls are 
 prolonged for the same distance, either keeping their planes 
 parallel, or very gradually spread iug outwards, but never ex- 
 
UNDEESHOT WHEELS. 25 
 
 ceeding 3 ,or 4 degrees. The point D is placed at a height 
 h' + -. 0.16 -J— (*), or 2.5 h + 0.11 ^~ below the level of the 
 
 water in the tail race. From this the following effects take 
 
 place : the water overcomes the difference of level between E 
 
 v* 
 and G, or the height 0.11 - — , in virtue of its velocity 0.4 v, 
 
 ° 2 g ' 
 
 either by a surface counter-slope, or by a sudden change of 
 
 level with a counter-slope, so that the head lost reduces to 
 
 v* 
 0.05 5—. 
 
 In many wheels this precaution has been neglected, and the 
 level of the tail race has been placed at the same height as the 
 point E, and sometimes even below it. We will now show 
 that a loss of effective delivery is thus produced. For this pur- 
 pose let us see what must, with the above-described arrange- 
 ment, be the position of the race relative to the pond, and the 
 expression for the effective delivery. In view of simplifying 
 this investigation, we will admit that the lines MA, D F, 
 slightly inclined, are a portion of the same horizontal. In the 
 hypothesis of no loss of head up to B C, the velocity v would 
 be due to the height z of the level N in the pond, above the 
 highest thread of the fluid vein thrown on the wheel ; but, on 
 
 s 
 
 * The co-efficient q is simply assumed: in replacing it by unity the 
 
 water would no longer be able to attain the level of the point G-, since this 
 
 would require a loss of head which would be null in the interval between 
 
 E F and G K ; consequently the water in the lower race would probably 
 
 drown the floats and impede their motion. It is for the purpose of avoiding 
 
 such an inconvenience that the number in question is taken less than unity ; 
 
 1 v* tf 
 
 the value assumed gives a disposable head expressed by g g— , or 0.05g— , to 
 
 counterbalance the loss of head of the liquid molecules after escaping from the 
 wheel, and to secure for the wheel their free discharge. 
 
26 UNDERSHOT WHEELS. 
 
 account of the losses of head, we give z a co-efficient of reduc- 
 tion, which we will take (for want of precise data) equal to 
 0.95 ; that, is, we will write 
 
 whence 
 
 if 
 =f- = 0.95 z. 
 
 ~ 0.95 2 g 
 The distance from the bottom M A to the level N will then be 
 
 1 v* 
 h + — ; this can also be expressed in another way, thus : 
 
 H + 2.5 h + ry 0.16 --, 
 
 by calling H the total head, or the difference in height between 
 G- and K". Hence 
 
 1 i)' 2 v' 
 A + 0l52-, = H + 2 - 5A + 3 - 16 275 
 
 and consequently 
 
 fg = hm ( H + 1 4 
 a relation which gives v, for any given head, when the value of 
 h has been determined. From this we can deduce z and h + z, 
 which is sufficient to determine the position of the race. The 
 dynamic effect T e , from what we have just seen, will be 
 
 T, «' , V'' 
 
 A P „ — . or, making; — = 0.4, 
 2 g' v 
 
 substituting for — — its value, this relation becomes 
 
 2 g 
 
 T e = P [0.48 x 1.057 (H + | h \ - 2.1 - h ] 
 
 = P (0.507 H - 0.289 h). 
 
UNDERSHOT WHEELS. 27 
 
 T 
 
 The effective delivery p-£x will then be expressed in round 
 
 numbers by 
 
 m 7 
 
 " -0.50 — 0.3 tj; 
 
 PH " v "■" H ! 
 
 for h = 8^ and H included between 3 feet and 6 feet, it would 
 vary from 0.44 to 0.47. 
 
 Now let us suppose that, without changing the head H, we 
 wish to place the level of the tail race below E, or, at most, on 
 the same level ; it is plain that we will have to raise the bottom 
 of the race. Then v will diminish, and h will have to increase 
 in order that the expenditure P may remain the same; for 
 these two reasons T e will diminish, as the above formula (2) 
 shows. For example, supposing that the tail race is at the 
 
 height of E, the equation that determines - — will become 
 
 A + i<5 = H + 2 - 5A ' 
 whence we derive successively, regard being had to equa- 
 tion (2), 
 
 V* nn ./rr . 3 
 
 2 
 3 
 
 2, ™»( H + 5 A ) 
 
 T c = P [0.48 x 0.95 (H + ^h) - 1.05 hj 
 = P (0.456 H- 0.375 h), 
 and in round numbers 
 
 j -0.45- 0.375 - h 
 
 PH ~~.~ H 
 
 The data h = 8" 1 and H = 3 feet would give an effective de- 
 livery of 0.37 instead of 0.44; with H = 6 feet, we would ob- 
 tain ah effective delivery of 0.41, whilst we had found 0.47. 
 
28 "UNDERSHOT "WHEELS. 
 
 There would be a much more marked falling off, if we supposed 
 the bottom of the tail race on the same level as the bottom of 
 the portion preceding it, as was the manner of constructing the 
 race formerly. 
 
 The arrangement of the channel through which the water 
 flows off, which we have mentioned as by M. Belanger, can* be 
 advantageously employed in all systems of hydraulic wheels 
 from which the water flows with a sensible velocity, in the 
 form of a horizontal current, with parallel threads. The ris- 
 ing of the surface of this current taking place beyond the 
 wheel, this latter will experience the same action from the 
 water, if everything is similarly arranged from the head race 
 to the outlet of the wheel. With the canal in question, the 
 level rises, instead of remaining the same or falling ; we 
 then obtain the same action on the wheel with a less head, 
 and consequently we can have a greater effect the head remain- 
 ing the same. 
 
 However, we see by formula (3) that the effective delivery 
 of these wheels never reaches 0.50, in spite of all possible pre- 
 cautions ; this system is, then, not comparatively as good as 
 those which we are now going to take up. 
 
 5. Wheels arranged according to Poncelefs method. — The 
 principal cause of loss of work in the undershot wheel with 
 plane floats is the sudden change from the velocity v to v', 
 twice and a 'half less, which necessarily produces in the liquid 
 a violent disturbance. From this disturbance proceed great 
 inner distortions and a negative work produced by viscosity, 
 all of which diminishes the dynamic effect. The water also 
 possesses a great velocity of exit, which is at best only partially 
 turned to account. General Poncelet proposed to avoid these 
 inconveniences, continuing, however, to preserve to the wheel 
 its special character, which is rapid motion ; that is, 'he has 
 
TTNDEKSHOT WHEELS. 29 
 
 endeavored to fulfil for the undershot wheel the two general 
 conditions of a good hydraulic motor, viz. : the entrance of the 
 water without shock, and its exit without velocity. To this 
 end he has contrived the following arrangements : 
 
 The bottom of the head race, which is sensibly horizontal, is 
 joined without break to the flume, the profile of which is com- 
 posed of a right line of T \-, followed by a curve. 
 
 The right line forms a slope near the wheel, and its prolon- 
 gation would be tangent to the outer circumference of this lat- 
 ter ; it ends' at the point at which the water begins to enter 
 the wheel. The curved portion is composed of a special curve, 
 to the shape of which we will return presently, and which 
 stops at the point at which it meets the exterior circumference. 
 Finally, the floats are set in a cylindrical portion of the race, 
 having a development a little greater than the interval beween 
 two consecutive floats, and terminated by an abrupt depression ; 
 this depression has its summit at the mean level of the water 
 down-stream ; its object is to facilitate the discharge of the 
 water from the wheel. 
 
 The water enters the race under a sluice inclined at an 
 angle of from 30 to 45 degrees with the vertical ; the sides of 
 the orifice are rounded off, so as to avoid the loss of head analo- 
 gous to that in cylindrical orifices. 
 
 The floats are set between two rings or shrouds, which pre- 
 vent the water from escaping at the sides ; the interior space 
 between the rings is somewhat greater than the breadth of the 
 orifice opened by the sluice. The floats are curved ; they in- 
 tersect the outer circumference of the crown at an angle of 
 about 30 degrees, and are normal to the inner circumference ; 
 beyond this, their curvature is a matter of indifference. There 
 are ordinarily 36 for wheels of from 10 to 13 feet in diameter, 
 and 48 for those from 20 to 23 feet. 
 
30 TTNDEKSHOT "WHEELS. 
 
 The exact theory of this wheel it is almost impossible to 
 explain in the present state of science. It is simplified, first, 
 by considering the mass of water that enters between two con- 
 secutive floats as a simple material point, which, during its 
 relative motion in the wheel, would experience no friction. 
 "We suppose, moreover, that the absolute velocity of this point 
 is in the direction of the horizontal that touches the wheel at its 
 lowest point, and that the floats are so put on as to be tangent to 
 the exterior circumference. Now, let v be the absolute velocity 
 of the water on striking the wheel, and u the velocity of the 
 wheel on this circumference ; the water possesses relatively to 
 the wheel, at the moment of entering the floats, a horizontal velo- 
 city v — u, in virtue of which it takes up a motion towards the 
 interior of the basin formed by two consecutive floats If we 
 liken, during this very short relative motion, the motion of the 
 floats to a uniform motion of translation along the horizontal, 
 the apparent forces will reduce to zero, so that the small liquid 
 mass that we have spoken of will ascend along the floats to a 
 
 ( v 
 
 height — ^ -, on account of its initial relative velocity, which 
 
 is gradually destroyed by the action of gravity. Then this 
 mass descends, and again takes up the same relative velocity, 
 v — u, when on the point of leaving the float ; but this relative 
 velocity will be in a direction contrary to the preceding, and, 
 consequently, also in a direction contrary to the velocity u of 
 the floats. The absolute velocity of the water on leaving will 
 then be equal to the difference between u and v — u, or 2w — v ; 
 
 we see that it will be zero if we have u = - v, that is, provided 
 
 that the velocity at the outer circumference of the wheel is half 
 that of the water in the supply channel. 
 
UNDERSHOT WHEELS. 31 
 
 "We could thus realize the two principal conditions for a 
 good wheel. But, as M. Poncelet has observed, such favorable 
 circumstances are found by no means in practice. 
 
 The water cannot enter the wheel tangent to its circumfer- 
 ence, as we have here supposed. In fact, let us call ds the 
 length of an element of this circumference immersed in the 
 current, h the breadth of the wheel, /3 the angle formed by ds 
 and the relative velocity, w, of the water referred to the wheel ; 
 there will enter during a unit of time, through the surface ids, 
 a prismatic volume of liquid having for a right section bds. sin ,<S, 
 and a length w ; in. other words, a volume bds.w sin /?, a 
 quantity that reduces to zero, for /3=0. Hence w must inter- 
 sect the circumference at a certain angle which cannot be zero ; 
 besides, we must make it as small as possible, in order that, at 
 the point of exit, the relative velocity of the liquid and the 
 velocity of the floats may be sensibly opposite, and give a resul- 
 tant zero ; on the other hand, it must not be so small as to 
 make the entrance of the water difficult or impossible. It is 
 in order to reconcile these two contradictory conditions that 
 the angle /3 has been fixed at 30 degrees, which is also that made 
 by the floats with the exterior circumference, since the threads 
 must enter in the direction tangent to the floats. But then 
 the absolute velocity of the water is no longer zero on leaving 
 the wheel, for its two components are no longer following the 
 same right line, but make an angle of 180°— -30° or 150°. 
 !N~ow, supposing v — u — u, the resultant v' would have, for a 
 value 2u cos 75°, or v cos 75°, or finally, 0.259^; this resultant 
 lies furthermore in the direction of the bisecting line of the angle 
 between u and v — u ; that is to say, it is almost vertical, and 
 consequently it is impossible to turn it to account by means of 
 a counter-slope ; its effect is 'destroyed in producing a useless 
 disturbance in the lower portion of the canal, whence there 
 
32 UNDERSHOT WHEELS. 
 
 v" 
 results a negative work equal to P — , P being the expenditure 
 
 9 v" 
 
 of the head. It is then . a loss of head expressed by — - , or 
 
 0.067-^. 
 
 As a cause of loss we may still mention the friction of the 
 water against the race and against the floats. Another very 
 serious objection to the above-mentioned theory is that the 
 liquid molecules do not move as though they were entirely 
 
 isolated ; when one of them, having reached the height ^— - — -, 
 
 • ° 2 g ' 
 
 between the floats, is in its descent, another has just entered, 
 
 and it is not clear that no sensible disturbance in the motion 
 
 of the molecules follows. 
 
 On account of all these reasons, experiment indicates only 
 
 an effective mean work of 0.60 for these wheels. Nevertheless, 
 
 the improvement is very great on the old undershot wheels 
 
 with plane floats, whose effective work was scarcely 0.25 or 
 
 0.30, and could never reach the limit 0.50. As to the most 
 
 favorable ratio between the velocities w and v, experiment 
 
 gives it 0.55 instead of -. 
 
 It has been observed that the straight portion of the race was 
 followed by a curve; the following condition determines its 
 form. The relative velocity of the water v — u at its entrance, 
 equal to that which exists at its exit, is also the same as the cir- 
 cumference velocity u; then the direction of the absolute velo- 
 city, resulting from these two velocities, is on the line bisecting 
 the angle formed by these two, and since the angle between 
 the tangent to the exterior circumference and the relative 
 velocity is 30 degrees, that between the same tangent and the 
 absolute velocity will be 15 degrees. All the threads, then, 
 
TTNDEKSHOT WHEELS. 33 
 
 must make an angle of 15 degrees with the circumference. 
 
 To deduce from this the shape of the bottom, let us assume 
 . that every normal to the bottom is at the same time normal to 
 
 all the threads that it intersects. Let A then (Fig. 5) be the 
 
 point of entrance of a thread, the absolute velocity A Y of which 
 
 makes an angle of 15 
 
 degrees with the tan- /'"" "\ 
 
 gent A U ; the perpen- B /( J ° } 
 
 dicular B A B' to A Y \ j\ \J. ./ 
 
 is a normal to the \ / i / 
 
 bottom of the race. 
 
 At the same time, if 
 
 we draw the radius 
 
 A (), the angle A B' 
 
 will be equal to IT AY, as their sides are perpendicular; hence 
 
 O A B' = 15 degrees. The perpendicular O B' let fall from the 
 centre to the prolongation of B A, has .then a constant value 
 equal to A sin 15°. Consequently, all the normals to the 
 bottom of the race are tangent to the same circle, having a 
 radius O B' ; the curve C B D is then the involute of which this 
 circle is the evolute. It is terminated .at one end by the cir- 
 cumference OA, at the point D, and at the other at a point C, 
 such that the normal E, taken as far as the circumference 
 O A, may have a length equal to the thickness of the fluid 
 stratum. This thickness, moreover, varies with the head ; it 
 must be, according to experiment, from 8 inches to 1 foot for 
 heads below 5 feet, and may be diminished to about 4 inches 
 for heads of more than 6.5 feet. 
 
 (v — 16) 1 $ 
 
 The water rises in the float to a height — = , 
 
 2 g 4 2^' 
 
 which differs little from — H, calling H the height of the head. 
 
34 UNDEKSHOT WHEELS. 
 
 The distance between the two circumferences that limit the 
 
 float should be at least - H ; in order more certainly to avoid 
 
 the possibilitj' of the water still possessing any relative velocity 
 on reaching the extremity of the floats, which would give rise 
 to a spirting of the water to the interior of the wheel, M. Pon- 
 
 celet has advised increasing this advance to - H. 
 
 6. Paddle wheels in an unoonfined current. — These wheels 
 are placed in a current whose section has a breadth much 
 greater than that of the wheel ; frequently they are supported 
 by boats, and are called hanging wheels. It being impossible 
 to calculate theoretically the dynamic effect of the current on 
 these wheels, we will be satisfied with the following general 
 ideas. 
 
 The horizontal force F which the wheel exerts on the liquid 
 being constant, its impulse in the unit of time lias numerically 
 the same value as its intensity. If, then, owing to this impulse, 
 a mass m of water passes, in a second, from the velocity v of 
 the current to the velocity u of the wheel, we shall have 
 
 F.= m (v — u), 
 and consequently the work produced on the wheel in the same 
 time, sensibly equal to the dynamic effect of the current, will 
 be 
 
 E u = m u (v — u). 
 
 Moreover, General Poncelet supposes that the mass m must 
 be proportional to v, and, furthermore, it is quite natural to 
 admit that it is proportional to the area S of the immersed 
 portion of the floats. He then places, calling B a constant 
 co-efficient and n the weight of a cubic foot of water, 
 
 m = BS» — 
 
UNDERSHOT WHEELS. 35 
 
 whence we have 
 
 9 
 
 This formula has been found quite true by experiment, in 
 taking B = 0.8. 
 
 When v is given, the maximum state of F u corresponds to 
 u = v — u, since the sum of these two factors is constant : from 
 
 this we deduce u = - v, as we found for the two wheels pre- 
 
 £1 
 
 viously discussed. Experiment indicates the ratio - = 0.4 as 
 
 v 
 
 being the most suitable; this ratio only changes very slightly 
 
 the theoretical maximum, the value of which is 
 
 v* 
 
 2 9 
 
 The depth of the floats must be from -to t the length of 
 
 the radius. Flanges placed on the edge, on the side which re- 
 ceives the shock of the water, will increase the mutual action. 
 The diameter is ordinarily from 13 to 16.5 feet, and the floats 
 are 12 in number. 
 3 
 
BREAST WHEELS. 
 
 7. Wheels set in a circular race, called breast wheels. — These 
 wheels resemble very closely, in their construction, the under- 
 shot wheels noticed in No. 4 ; but one essential difference, which 
 can produce a great change in their effective delivery, con- 
 sists in the method of introducing the water; this no longer 
 enters the wheel at its lower portion, but at a slight depth only 
 below the axle — that is to say, on the side. The questions now 
 to be successively examined will be, 1st, the most suitable ve- 
 locity of the wheel; 2d, the method of introducing the water; 
 3d, the situation of the race as regards the upper end and 
 tail race ; 4th, the manner of determining the dynamic effect. 
 Finally, several practical ideas will be put forth on the subject 
 of this wheel. 
 
 (a) To determine the velocity of the wheel. — Let Q be the 
 volume that the head expends in a second ; 
 
 b the breadth of the wheel, which is the same as that of the 
 flume ; 
 
 h the depth of the immersed portion of the floats directly 
 beneath the axle; 
 
 e the thickness of the floats ; 
 
 C their distance apart, measured between their axes along 
 the exterior circumference ; 
 
 R the radius of this circumference ; 
 
 u the velocity of any one of its points. 
 
BKEAST WHEELS. 37 
 
 The water comprised between any two consecutive floats, 
 directly beneath the axle, will have for its depth A, for its mean 
 
 breadth C (l — -p= j — c, and b for its thickness parallel to the 
 
 axis of the wheel ; its volume is then hbC (l — — ^ — — ), and, 
 
 supposing that it expends during one second a number -^ of 
 iheie volumes, we shall have 
 
 A c 
 
 In practice — = and -y are always small fractions, the sum of 
 
 which rarely exceeds 0.10 ; we would then have, except a 
 slight error, 
 
 Q = 0.9 A b u. 
 
 Q being given, we can satisfy this relation by assuming u and 
 h arbitrarily, and then determining b. In this case, we would 
 
 take A 'from 6 ta to 10* u = 4 ft .25, and finally, b — —— ; the 
 
 y th ti> 
 
 expenditure — |- per foot in breadth, expressed by 0.9 h u, would 
 
 then vary from 14.1 to 21.5 gallons. 
 
 The values just given for A and u may be satisfactorily ac- 
 counted for as follows : 
 
 The calculation of the losses of head sustained by the water, 
 in its passage from the head race to the tail race, shows, as we 
 shall see further on, that all these losses increase with the velo- 
 city of the wheel. "We should naturally be inclined to make 
 this velocity very small, in order to increase the effective deliv- 
 ery. But several conditions show that it is not well to allow 
 
38 BKEAST WHEELS. 
 
 the wheel to move too slowly. In fact, we see that if u were 
 
 very small, the product lib = — — would be large, and one of 
 
 y to 
 
 the two dimensions b or h would have to be quite large. Now 
 a great breadth b would give us a' heavy wheel, expensive to 
 put up, losing a great deal of work by the friction of the axle ; 
 a great value of h would produce inconveniences already men- 
 tioned in speaking of the undershot wheel (Ho. 4). Finally, it is 
 well to have the wheel perform, up to a certain point, the func- 
 tions of a fly-wheel for the machinery that it sets in motion, 
 which is another reason for allowing it to retain a certain 
 velocity. The value u = 4*25 has been given by experiment. 
 As regards the depth h, it is well, on account of the unavoida- 
 ble play between the wheel and the race, that it should not 
 descend much below 6 in , in order not to lose too great a propor- 
 tion of water. 
 
 But it sometimes happens that, in taking u = 4*25, and h 
 within the limits above mentioned, we arrive at a great value 
 for b, or a valne that goes beyond a fixed limit, either on ac- 
 count of local circumstances, or through economy in construc- 
 tion. "We are then obliged to increase h or u. It is not well 
 to have h greater than l ft .50 or 1*66, and when ' this limit is 
 reached, we must then begin to increase u. 
 
 For example: let Q = 20 ou -*. Taking u = 4*25, h = 8 m , 
 
 we would have b = — -. — = 7*92, a value that in general ia 
 9 hu 
 
 quite admissible. But if, by reason of particular circum- 
 stances, we could not exceed a breadth of 2*64, the third or" 
 7*92, we should first increase h, bringing this up to 1*66 ; we 
 
 would then find u = n T 7 = 5.05. Or else, as the velocity 5.05 
 95 h ' J 
 
BEEAST WHEELS. 
 
 39 
 
 is not yet very great, we would be content to take h = l ft .3, 
 which would give u = 6 £t .40. 
 
 (h.) Method of introducing the water. — As has already been 
 said (No. 3), in order to diminish as much as possible the loss 
 of work produced by the introduction of the water on the 
 wheel, we must so arrange matters as to have a small relative 
 velocity of the water at its point of entrance, or, when that 
 cannot be obtained, the relative velocity must be tangent to 
 the first element of the floats, and the water must move to the 
 interior of the wheel without shock, merely gliding along the 
 solid sides. 
 
 If the wheel be moving slowly — that is, if the velocity at the 
 circumference be about 4 ft .25 per second, we should let the 
 water in by the means indicated in the following figure. 
 
 Fis. 6. 
 
 The flume A B, constructed of masonry, is prolonged by a 
 piece of cast iron B C, called a swarfs neck, or guide bucket ; 
 A B C is the arc of a circle nearly coincident with the exterior 
 circumference of the wheel, leaving the slightest amount of 
 
40 BKEAST WHEELS. 
 
 play. A gate D, furnished at its upper extremity with a small, 
 rounded metallic appendage E, can slide along the guide bucket 
 while resting on it ; a system of two racks connected by cog- 
 wheels allows this sluice to be raised to any convenient point. 
 The metallic appendage E forms the sole of a weir over which 
 the water flows to get to the wheel ; this sole must be from 
 8 111 to 10 to .5 below the level of the pond. 
 
 The weir being thus very near the wheel, the absolute 
 velocity of the water at its entrance, due only to the slight fall 
 which takes place in the surface of the pond, will consequently 
 be small; and as the wheel itself moves slowly, the relative 
 velocity will be moderate, and the disturbance produced hardly 
 perceptible. We see that, in order to attain this end, we should 
 reduce the head over the sole of the weir, for in the contrary 
 case the velocity of flow would have a greater or less value ; we 
 should not, moreover, carry the reduction too far, in order that 
 the loss of water between the wheel and flume may not be too 
 great. The limits 8" 1 to 10^.5 fulfil this double condition quite 
 well; they correspond very nearly to the limits 6 m and 10 m 
 above mentioned for the depth h of the current, jnst beneath 
 the axle; for a weir without lateral contraction can yield about 
 13.7 gallons per foot of breadth, with a head of 8™ above the 
 sole, and 21.3 gallons with a head of 10 to .5. 
 
 The rounded metallic appendage E has for its object to 
 diminish the contraction of the sheet of water flowing over the 
 weir, and thus for the same head to yield more water: the head 
 and velocity of flow are then less for a given yield, which 
 diminishes the disturbance of the liquid on entering. The 
 piece E may also serve to direct the vein so as not to intersect 
 the exterior circumference of the wheel at too great an angle; 
 this is as it should be, for, all other things being equal, the 
 relative velocity of the water increases with the angle in ques- 
 
BREAST WHEELS. 41 
 
 tion, as the following considerations show, although they more 
 particularly apply to a different case. 
 
 Let us now pass to the case of rapidly moving wheels. The 
 velocity u of the wheel Ht its circumference has been fixed as 
 has been shown ; but we still, in order to lessen the loss due to 
 the introduction of the water, dispose of the absolute velocity 
 v of this latter, in intensity and direction. Let, at the point 
 of entrance, M (Fig. 7), M U be the velocity 
 u of the wheel, M Y the absolute velocity v 
 of the water, y the angle formed by these two 
 right lines. The line M "W, equal and paral- 
 lel to U Y, will be the velocity w of the water 
 relatively to the wheel. The first point is to 
 make w as small as possible. JSTow we should 
 F 7 have w = o, if y were zero and v equal to u ; 
 
 but it is not possible for the water to enter 
 without any relative velocity, and at an angle zero. One 
 thread for which 7 may have the value 0, would be tangent to 
 the exterior circumference of the wheel ; but it is evident that 
 the other threads, placed in juxtaposition parallel to this, could 
 not fulfil the same condition, unless the total thickness of these 
 threads taken together were itself zero, which cannot be admit- 
 ted in the case of a finite expenditure. We have seen that in 
 Poncelet's wheel the angle 7 was taken equal to 15 degrees ; in 
 the breast wheel it is generally taken to be 30 degrees, in order 
 to somewhat facilitate the introduction of the water. Then, 
 since U is a point determined as well as the direction MY, the 
 smallest possible value of U Y = w would be the perpendiuular 
 
 let fall from U on MY, orw sin 7, or finally ~ u if 7 be taken 
 
 equal to 30°; v would have for a corresponding value u cos 7, 
 or 0.866 v, 7 being 30°. 
 
42 BREAST WHEELS. 
 
 Still these values are not those taken for v and w. It is well, 
 in fact, to have the first element of the floats in the direction 
 of w, in order to diminish the disturbance of the water within 
 the wheel. IS r ow, if this first element were perpendicular to 
 M V, it would make an angle of 120° with M U — that is, with 
 the circumference of the wheel, and consequently an angle of 
 30° with the radius through M. When this radius, by the act 
 of revolving, reaches the vertical, immediately below the axle, 
 the first element of the float will still be inclined at an angle 
 of 30° with the vertical, and it would be still more inclined at 
 the point of emersion, which would obstruct the motion of the 
 wheel. On this account it is better that the relative velocity 
 M¥ should pass through the axis of rotation and be perpen- 
 dicular to Mil; the parallelogram of velocities is then repre- 
 sented by the rectangle MDY'f', in which M U = u, 
 
 M W ' = w, M V ' = v. The angle 7 still preserving its value 
 of 30°, we have these relations : 
 
 v=z — ^775 = L155 M > 
 cos 30° 
 
 w — u tang 30° = 0.577 u. 
 The intensity of v and its angle with u being henceforth de- 
 termined, it remains to be seen how in practice these two con- 
 ditions can be realized. To obtain them, we will first deter- 
 
 v' 
 mine the height - — ; we will increase it a little, say by one- 
 tenth, in order to compensate approximately for the losses of 
 head sustained by the liquid between the head race and the 
 
 point of entrance, and 1.1 - — will be the depth of the point of 
 
 2 9 
 entrance below the level of the head race. The direction of 
 the velocity v will be obtained by drawing through the point 
 of entrance, which we have just, found (since it is on the ex- 
 
BREAST WHEELS. 43 
 
 tenor circumference of the wheel, and an a known horizontal 
 line), aline making an angle of 30° with the said circumfer- 
 ence. The threads should then be obliged to take this direc- 
 tion by means of a canal of from l ft .66 to 2 ft .00 in length, the 
 last element of which should be tangent to v, and into which 
 the water would flow, either by passing under a sluice, or by 
 flowing out without any obstacle, the bottom of this canal 
 being then the sole of the weir. 
 
 (o.) Position of the flume as regards the head race and tail 
 race. — We have only to repeat here what has already been said 
 in discussing the undershot wheel with plane floats. When 
 the water possesses, on leaving the wheel, a sensible velocity, 
 it is well to place the level of the portion between the floats, 
 which is directly under the axle, below the level of the lower 
 
 2 u s 
 section, by a quantity equal to - — , u being the velocity of 
 
 the wheel at the circumference. The velocity u, having a 
 value previously determined, as well as the height h of the 
 water between the floats, the situation of the bottom of the 
 flume is then also fixed, at least that part below the axle. On 
 the up-stream side this bottom has a circular profile, with a 
 radius sensibly eqiial to that of the wheel ; it should be joined 
 with the tail race in accordance with M. Belanger's rules, of 
 which we have previously spoken (No. 4). 
 
 When the wheel is moving slowly, u being about 4 ft .25, 
 
 9, 
 
 u" 
 
 gives a depth a little below 2^.5. However great the 
 
 head may be, the loss or gain of 2 in .5 is of but small import- 
 ance, and we would then be able, by economy, to do away with 
 the portion of the race beyond the wheel, or at least to do 
 without a masonry-lined canal with a regular section. The 
 level of the water between the floats would then coincide with 
 
44: BREAST WHEELS. 
 
 that of the tail race. But in the case of large values of u, es- 
 pecially if, at the same time, we have only a slight head, the 
 
 2 v? 
 height - - — may afford a sensible gain that we should not 
 & 3 2 g ■' to 
 
 neglect. 
 
 id.) To calculate the dynamic effect of a head of water 
 
 which sets a breast wheel in motion. — Calling P the weight of the 
 
 water expended in a second, H the head measured between the 
 
 two portions of the canal supposed to be in a state of rest, we 
 
 know (No. 2) that the dynamic effect T e of the head, during 
 
 each second, has for its value the product of the weight P by 
 
 the height H, diminished by the mean loss of head that the 
 
 liquid molecules sustain by reason of friction in their passage 
 
 from one portion of the canal to the other. Moreover, this loss 
 
 of head is composed of several portions, which we shall now 
 
 proceed to analyze, using the notation adopted (No. 7, a, 
 
 1st. Loss of head "between the head race and the point at 
 which the water enters the wheel. — This is reduced by avoiding, 
 by means of rounded outlines, contractions followed by a sud- 
 den expansion, and by placing the point of entrance as near 
 the head race as possible. Nevertheless there is always a cer- 
 tain loss : this we may assume at first sight as comprised 
 
 v' v* 
 
 between 0.05—- and 0.1—- . If there were a canal between 
 2 9 2 g 
 
 the wheel and the head race, we should have to take the fric- 
 tion in this canal into consideration, by a calculation analogous 
 to that which will be presently given for the circular flume. 
 
 2d. Loss of head due to the introduction of the water. — After 
 what has been said (in No. 3), this loss may be valued at 
 
 w' 1 
 
 — -, or at — - (v* + u' — 2 u v cos y). In the case of slowly 
 
BREAST WHEELS. 45 
 
 moving wheels it is of slight importance, like all the other 
 losses that we consider ; in one moving quickly, if 7 = 30°, 
 
 v = , we shall have 
 
 cos 7 
 
 2 2 12 
 
 w u , „ \ u 
 
 ■^ — = ~ — tan 7 = „ n . 
 2g 2g 3 2g 
 
 "When the precaution is taken to introduce the water with a 
 relative velocity tangent to the floats, and a surface is presented 
 along which it can ascend in virtue of this relative velocity, it 
 is probable that the action of gravity assists in overcoming this 
 velocity, and thus so much less effect will be consumed in a 
 violent disturbance of the water. This consideration justifies 
 the use of polygonal floats, such as are shown in (Fig. 6), the 
 arrangement of which was contrived by M. Belanger in 1819. 
 They are composed of three planes making angles of 45 degrees 
 with each other, of which the furthest from the centre is in the 
 direction of a radius, the nearest touches the circumference of 
 the ring, and the third connects the two others. The planes 
 that are fixed to the crown have vacant spaces left between 
 them to facilitate the disengagement of the air. The point of 
 entrance of the water must, moreover, be lower than the centre 
 of the wheel, in order that the water introduced may be 
 received upon an ascending inclined plane. 
 
 Data are wanting to estimate the effect produced by the use 
 of this kind of float. 
 
 3d. Loss of head produced oy the friction of the water 
 against the circular flume. — We know that the friction of a 
 current on its bed, per square foot, is expressed by 0.4 U a , U 
 being the mean velocity. Moreover, calling V and W" the 
 velocities at the surface and at the bottom, it has been found 
 that these quantities are connected by the approximate rela- 
 tions : 
 
46 BKEAST WHEELS. 
 
 whence 
 
 U = ^(V + W), U = 0.80 V; 
 
 u = fw. 
 
 The friction per square foot of bed would then be 
 
 0.4. ^W 2 or 0.71 W\ 
 
 Now, if L be the length of the circular race, L (b 4- 2 h) will 
 be the surface wet, and the entire friction is expressed by 
 0.71 L (b + 2 h) u 2 , since the velocity at the bottom is the same as 
 the velocity at the circumference of the wheel. The points of 
 application of all the forces that compose this friction moving 
 with the velocity u, their negative work in the unit of time 
 will be 0.71 L (b + 2 A) u a ; we shall obtain the corresponding 
 loss of head by dividing by P, or by 62.3 bhu, which gives 
 
 0.0114 M» + 8*)*, or else 0.733 M» + »*)*. 
 b h o h 2 g 
 
 4th. Imss of head from the point just beneath the axle to the 
 ' tail race. — If the level of the water between the floats, beneath 
 the axle, exceeds by a height *i the level of the portion lower 
 down, the veloeit}' of the water will increase on account of this 
 difference of level. This velocity, sensibly equal to u on leav- 
 ing the wheel, will become Vw 1 + 2 g *i, for the point at which 
 the water begins to enter the lower portion. As we have seen 
 
 (No. 3), a loss of head *i 4- ■= — corresponds to this velocity. 
 
 r 
 "When we adopt the arrangement of the tail race recom- 
 mended by M. Belanger (No. 4), t\ becomes negative and equal 
 
 to - - — ; the loss then reduces to . 
 
 3 2^' 3 2^ 
 
 (e) Some practical data. — With regard to the nnmber of 
 
BREAST WHEELS. 47 
 
 arms, the same rule here applies as to undershot wheels 
 (No. 4). 
 
 The number of floats is a multiple of the number of arms ; 
 their distance apart may be from once and a third to once and 
 a half the head above the top of the weir, in the case of slowly 
 moving wheels. In wheels of rapid motion, it will always be 
 necessary to take this distance apart a little greater at the por- 
 tion of the circumference intercepted by the stratum of water 
 that falls upon the wheel. It is not well to place the floats too 
 far apart, because some of the threads might fall from quite a 
 considerable height before reaching them ; it is also bad to 
 place them too close together, because the water could with 
 difficulty enter the wheel, and a portion would be thrown oif. 
 
 The depth of the floats in the direction of the radius is but 
 little greater than 2 ft .30. In its normal condition, the interior 
 capacity formed by any two consecutive floats should be very 
 nearls' double the volume of water that they contain : we might 
 then take the depth in question equal to 2 h, whenever h is not 
 greater than l ft .15. 
 
 The diameter of the wheel should be at least 11 ft .5 ; it is sel- 
 dom greater than 20 to 23 feet. The axis of the arbor is placed 
 a little above the level of the head race. 
 
 Breast wheels are suitable for falls of from 3.25 to 6.5 feet, or 
 even 8 ft .25. Beyond these limits they can still be frequently 
 used to advantage. 
 
 "When a breast wheel of slow motion is well constructed, the 
 dynamic effect of the bead may approximate quite near to what 
 is due to the entire head. General Morin, in some experiments 
 on these wheels, found an effective delivery of 0.93. But, 
 since the measure of the expenditure of water is always some- 
 what involved in uncertainty, and consequently the total effect 
 
48 BEEAST WHEELS. 
 
 due to a given head can be but imperfectly ascertained, it will 
 be prudent not to count beforehand on an effective delivery, in 
 practice, greater than 0.80. 
 
 8. Example of calculations for a rapidly moving breast 
 wheel. — The wheel in question was experimented on by Gene- 
 ral Morin; it belonged to the foundry at Toulouse. 
 
 The water left the head race under a gate that was raised 
 6™ above the sole, which was 4 ft .66 below the above-mentioned 
 level. The orifice being prolonged by a very nearly horizontal 
 race, the velocity v with which the water reaches the wheel 
 will be due to the head at the upper portion of the orifice, 
 except a co-efficient of correction very nearly 1, which we will 
 value at 0.95; hence 
 
 v = 0.95 V 2g (4 ft .66 - 6*5) = 15 ft .54 
 
 The velocity at the circumference of the wheel was u ■=. 10 feet, 
 and the angle made by u and v, at the point of introduction, 
 was valued at 30°. It follows directly from this, that the loss 
 of head sustained in bringing the water from the head race 
 
 to the wheel should be — of the head (-i ft .66 - ft .5) or 5 in ; we 
 
 can also calculate that produced by the introduction of the 
 water, which was expressed by (No. 7, d) 
 
 — - = — (v 2 + u 2 - 2 u v cos 30° = l».ll. 
 2g 2g\ I 
 
 The level of the water between the floats, just below the 
 axle, being at the level in the down-stream portion of the canal, 
 
 we must again count upon a loss equal to — (No. 7, d), or 
 
 2 g ' 
 
 l ft .57. 
 
 ' Finally, there is a loss in the circular flume. The depth of 
 
 the water beneath the axle and the breadth of the wheel being 
 
BKEAST WHEELS. 49 
 
 valued respectively at 8™ and 5* and the length of the flume 
 
 being 8 ft .20, we find for this loss .733 8 ' 2 ° X 6-32 . —, or 5*28. 
 ° ' 0.66 x 5 2g 
 
 All these losses together make up a head of ft .42 + 1*11 
 
 + l ft .57 + ft .44, or of 3 ft .54. The head being 5 ft .64. we see 
 
 that the productive force as calculated would be only 
 
 — — £-o7- — ) tnat i fl » °-37 ; M. Morin found experimentally 0.41, 
 a number which corresponds to an available head of 
 
 5 ft .64 x 0.41 = 2 ft .21, 
 instead of 2 ft .16, which the preceding calculation gives. This 
 difference, otherwise hardly noticeable, of ft .05, belongs proba- 
 bly to a somewhat greater value of the head lost by the in 
 traduction of the water ; in fact we have seen that, in certain 
 cases, a portion of the relative velocity could be annulled by 
 the action of gravity, which would diminish by so much the 
 disturbance within the floats, and would give rise to a smaller 
 loss of head. 
 
 To increase the effective delivery of this wheel without 
 changing its velocity, the following arrangements might have 
 
 2 1$ 
 been made: First, to place the flume at - — -, or l ft .03 lower 
 
 than its actual position, taking care to arrange the race as 
 described by M. Belanger (No. 4) — that is, without any sudden 
 variation in section, and with a bottom having a moderate 
 slope, to its junction with the tail race; then to raise the point 
 of entrance of the water so as to reduce the velocity v to 
 
 or = ll ft .54. The loss from the wheel to the 
 
 cos 30° 0.866 
 
 1 u* 
 tail race would then have been reduced to - — - or to ft .52, 
 
 dig 
 
 instead of l ft .57 ; the loss for the entrance of the water would 
 
50 BREAST WHEELS. 
 
 also be reduced to the same value tt .52 instead of l ft .03, which 
 would procure a total benefit of l ft .55. The other losses remain- 
 ing sensibly the same, the available head would be 2 tt .16 + l ft .5o 
 = 3 ft .71, and the effective delivery would be raised to about 
 
 — = 0.66. 
 
 5.64 
 
OVERSHOT WHEELS. 
 
 9. Wheels with hxtckets, or over-shot wheels. — These wheels 
 are not, like the preceding, set in a flume. The water is let in 
 at the upper portion ; it enters the buckets, which are, as it were, 
 basins formed by two consecntive floats, terminated at the sides 
 by the annular rings, and closed at the bottom or sole by a con- 
 tinuous cylindrical surface concentric with the wheel. The 
 questions that the organization of this kind of motor present 
 are as follows : 
 
 {a) Introduction of water into the wheel. — Two arrange- 
 ments are employed which are represented hereafter (Figs. 8, 
 
 9). In (Fig. 8) the top D of the wheel is placed a little below 
 4 
 
52 OVEESHOT WHEELS. 
 
 the level N JST of the pond, at ft .60 to ft .75 lower than that 
 level; the water is led to a point 0, situated about l ft .50 up- 
 stream, and is delivered directly above the axle, by means of a 
 canal A B, or pen trough made of planks, terminated by a very 
 thin metallic plate B C, which, being prolonged, would be very 
 nearly tangent at D to the exterior circumference. The lateral 
 boundaries of the canal ABC are prolonged about 3 feet beyond 
 the point C, to prevent the water from falling outside of the 
 wheel. The water passes over the distance CD in virtue of its 
 acquired velocit}', and enters the wheel nearly at the top. As 
 quite a narrow opening only is left between the soles of the 
 buckets, the water that flows in the canal ABC is given the 
 form of a thin stratum, by making it pass under a sluice placed 
 near A, and which is only raised about ft .20 or ft .33. This 
 sluice presents an orifice with rounded edges, so as to avoid the 
 eddies consequent to the exit of the liquid threads. As has 
 been shown, there is little difference in height between the 
 point at which the water enters and the level of the head race ; 
 consequently the water enters the wheel with a slight absolute 
 velocity, and if the wheel turn slowly, as it should do, to attain 
 a good effective delivery, the relative velocity will itself be 
 slight, as well as the loss of work that it involves. 
 
 The arrangement in (Fig. 9), which has been frequently em- 
 ployed, does not appear to be so good ; but we are sometimes 
 obliged to make use of it if the pond level be very variable. 
 This portion is terminated near the wheel by a wooden shutter 
 AB, with openings C C, having vertical faces like those of a 
 window-blind ; a movable gate allows of covering as many 
 of these openings as may be requisite, so as to expend only 
 the disposable volume of water. The inconvenience of this 
 method is, that the water falls through sufficient height into 
 the buckets to give it a considerable increase of velocity ; the 
 
OVERSHOT WHEELS. 
 
 53 
 
 disturbance of the water in the wheel thus becomes much 
 greater. It tends, moreover, for the same head, to increase the 
 diameter of the wheel, which makes it more heavy and expen- 
 
 se. 9. 
 
 sive. Besides, the point at which the buckets take a sufficient 
 inclination to begin to discharge the water in them is situated 
 at a greater height above the lowest point of the wheel, because 
 this height is proportional to the diameter ; there is thus, then, 
 a greater loss of head, seeing that the work of the weight of the 
 molecules that have left the buckets, whilst they are falling 
 into the race below, is evidently lost to the wheel. 
 
 (b) Shape of the surface of the water in the buckets ; velo- 
 city of the wheel. — It can be shown that a heavy homogeneous 
 liquid cannot be in equilibrio relatively to a system that turns 
 uniformly about a horizontal axis. If, however, we admit that 
 the relative equilibrium of the water can exist approximately in 
 the buckets, which may arise when the disturbance due to the 
 entrance of the liquid has nearly ceased, we can determine the 
 shape assumed by the free surface as follows : 
 
54 
 
 OVERSHOT WHEEL8. 
 
 Let M (Fig. 10) be a liquid molecule, having a mass m, situ- 
 ated at the distance O M = r from the axis of rotation O ; it is 
 in eqnilibrio relatively with a system which turns around this 
 axis with an angular velocity ". This equilibrium exists under 
 the action : 1st, of the weight mg, which acts vertically along 
 the line M G ; 2d, of the centrifugal force m <•>" r, along the 
 prolongation M C of M, an apparent force to be introduced, 
 as regards solely a relative equilibrium ; 3d, of the pressures 
 produced by the surrounding molecules. We know from the 
 principles of hydrostatics that the resultant of the two "first 
 forces is normal to the surface level (or of equal pressure) which 
 
 passes through M. If, then, M 
 be found at the free surface, as 
 the pressure there is entirely 
 the atmospheric pressure, the 
 resultant in question will be 
 normal to this surface. Let us 
 take MG = mg,MG = mur, 
 the diagonal M B of the paral- 
 lelogram M G B C represents the 
 resultant of mgemd of mu'r, 
 and consequently it is normal to 
 the free surface. Moreover, pro- 
 longing the vertical O A until it intersects this normal, we 
 obtain from the property of similar triangles, 
 
 Fig. 10. 
 
 OA 
 
 MG 
 
 OM 
 GB' 
 
 whence 
 
 OA = 
 
 MClx 01 __ mgr __ g 
 
 G B m c a r «' ' 
 
 hence the distance O A is constant, which shows that in a plane 
 section perpendicular to the axis all the normals to the free 
 
OVERSHOT WHEELS. 55 
 
 surface meet at the same point. The profile of the free sur- 
 face, if there be relative equilibrium, is then necessarily a circle 
 described from the point A as a centre. 
 
 This result proves that, in effect, the relative equilibrium is, 
 strictly speaking, impossible ; for, in proportion -as the bucket 
 leaves its place, the point A not changing position, the free 
 surface would have an increasing radius, which is incompatible 
 with the hypothesis of relative equilibrium, since in this case 
 the form of the free surface ought not to change. The form 
 that has been determined is that which the water endeavors to 
 assume without being able to preserve it. 
 
 To finish determining the circle which limits the water in a 
 given bucket, a circle of which we as yet know only the centre 
 A, we must take into consideration the quantity of water that 
 the bucket is to hold. To this end, let N be the number of 
 buckets filled, h the breadth of the wheel parallel to the axis, 
 Q the volume expended by the pond per second. Each bucket 
 
 occupies on the circumference an angle ^- (expressed in terms 
 of an arc of a circle having a radius 1), and as the wheel turns 
 with an angular velocity u, — — will represent the number of 
 buckets filled in a unit of time. Each bucket then contains a 
 volume — ~, so that the area occupied by the water in the 
 
 cross section of the bucket has for its value -^ — *=-. The arc 
 
 D M E will then be determined, since we know its centre and 
 the surface DMEF1 which it must- intercept in the given 
 profile of the bucket. 
 
 In a certain position I' E' F' of the bucket, the free surface, 
 determined in the way just mentioned, just touches the edge of 
 
56 0VEKSH0T WHEELS. 
 
 the exterior side E' ; this position may be found by trial. As 
 soon as the bucket passes it, the water begins to run out ; for 
 every position below this, it is clear that the free surface will 
 have for a profile a circle having A for a centre and just touch- 
 ing the outer edge of the bucket, and which will allow the 
 volume of water remaining in the bucket to be determined. 
 When the circle in question passes entirely below the profile of 
 the bucket, the discharge will be complete. 
 
 In practice, if the question relates to wheels possessing only 
 
 a slow angular velocity, -^- will be so great that the circles de- 
 
 scribed from A as a centre, to limit the surface of the water in 
 the buckets, may be assumed as horizontal lines. Example : 
 The wheel being 12 feet in diameter, and having a velocity of 
 
 3 feet at the circumference, then o> = ~ and —„ = 128*68, or 
 
 2 or 
 
 the distance of the centre A above the axis. 
 
 "When an overshot wheel turns rapidly, the distance -^ may 
 
 become so small that the free surface may present a noticeable 
 concavity below the horizontal ; thus the more the angular 
 velocity increases the less water the bucket can hold in a given 
 position, which is easily seen, since the centrifugal force becomes 
 greater and greater, and this force tends to throw the water out 
 of the bucket. This is an inconvenience attendant upon wheels 
 that turn rapidly; they lose a great deal, of water by spilling, 
 and consequently yield a smaller effective delivery. 
 
 The losses of head produced by the introduction of the water 
 into the buckets, and by the velocity of the water when it 
 leaves the wheel, also increase with the angular velocity. We 
 will then be led, in order to economize the motive force as 
 much as possible, to make the wheel turn very slowly. But 
 
OVERSHOT WHEELS. 57 
 
 we have already seen, in speaking of breast wheels, that it is 
 not well to make a water wheel move very slowly, because, in 
 order to use up an appreciable volume of the water, it would 
 bo necessary to establish a machine of immense size. A velo- 
 city of from 3 ft to 4 ft .5 at the circumference gives good 
 results. 
 
 (c) Breadth of the wheel ; depth of the buckets in the direc- 
 tion of the radius. — We have said above that, in a well- 
 arranged wheel, the water leaves the up-stream portion of the 
 troughs by passing under a gate raised from fl .20 to ft .33 
 above the sole ; which is itself from ft .66 to ft .82 below the 
 level of this portion. If we call 
 
 Q the expenditure per second ; 
 
 o the breadth of the wheel, and of the orifice under the 
 sluice ; 
 
 x the height to which the sluice gate is raised ; 
 
 h the depth of the sole below the level of the water in the 
 up-stream section ; 
 
 v the velocity with which the water leaves the sluice ; 
 the velocity v will be due very nearly to the head h — x; and 
 as the adjustments are so arranged as to have but little contrac- 
 tion, we can place 
 
 Q = 0.95 bx V 2 g (A - x), 
 
 the co-efficient 0.95 being intended to account, at a rough esti- 
 mate, for the loss of head that water always undergoes in any 
 movement whatever, and for the contraction that would yet 
 partially exist. By making, in this expression, h = ft .66, x — 
 
 ft -20, we deduce $ = l ouffc .03 ; in like manner, for h = 0*82, 
 o 
 
 x — ft .33, we find j = V^^.TZ ; that is, with the sluice ar- 
 ranged as we have said, we can expend from 7.67 to 13.1 
 
58 OVERSHOT WHEELS. 
 
 gallons per foot of breadth of the wheel. It would be easy to 
 expend less than 7.67 gallons, by diminishing h and x a little; 
 we can, when necessary, expend more than 13.1 gallons by in- 
 verse means. But experience shows that, to be in the best con- 
 dition, the expenditure should be but little more than 8 gallons 
 per foot of breadth; for otherwise we might be led either to 
 make deep buckets, or to cause the wheel to turn rapidly, which 
 would tend to increase the velocity of the water when it enters 
 or leaves the wheel, and consequently to diminish the effective 
 delivery. 
 
 To show the relation that exists between the depth p of the 
 
 buckets in the direction of the radius, and the expense -y- per 
 
 foot in breadth, let us preserve the notation already employed 
 in the present number, and furthermore let us call 
 
 R the radius of the wheel ; 
 
 ■u its velocity at the circumference ; 
 
 2 it R ' . 
 
 C = — =r= — the distance of the buckets apart ; c their thick- 
 ness. 
 
 The volume of a bucket will be equal to the product of its 
 three mean dimensions, viz. : its length i. its depth p, and its 
 
 breadth C (1 — -~^ j — c ; this volume has then for its value, 
 
 p b C ( 1 — ^-= — - ). Moreover it would be well, to retard 
 
 the discharge from the bucket, not to have it more than one- 
 third full ; the volume of the water it contains would then be 
 
 -p i C (l — nTT - p)> aD dj s 18 we nave previously seen, by 
 — ^ T -, there obtains 
 
0VEKSH0T WHEELS. 59 
 
 3^ V 2E c; «N ' 
 
 whence, because C = — =— _ and u = — : 
 JST E ' 
 
 As the factor in the parenthesis in the second member differs 
 but little from 1, we may simply place 
 
 Q 1 
 
 This equation shows that when -£ is large, one of the factors p 
 
 or u must be so too. For example, if -£ = 0.100 of a quart, and 
 
 u — 3 feet, we find p = 1 foot. It is desirable that p should 
 not much exceed 1 foot. However, if we had an ample supply 
 of water to expend, we might either go beyond this limit, or 
 use a faster wheel, or finally fill the buckets more than one- 
 third. 
 
 The expenditure per foot of breadth having been fixed, 
 from what precedes, as much as possible below 25 gallons per 
 second, the breadth of the wheel results naturally frpm the 
 total volume of water to be expended. It is seldom that 
 wheels having a greater breadth than 16.5 feet are con- 
 structed. 
 
 (d) Geometrical outline of the buckets. — The distance of the 
 buckets apart is a little greater than their depth ; generally, 
 this last dimension is from 10™ to 11™, and the other about 
 from 12 m to 14™. Their number must be a multiple of the 
 number of arms for facilitating the connections, unless the 
 crown and arms are composed of a single piece. 
 
 As to their profile, the annexed outline is frequently made 
 
CO OVERSHOT WHEELS. 
 
 use of (Fig. 11). After dividing the exterior circumference 
 O A into portions A A', A' A", all equal to the distance of 
 the buckets apart, we take A D = p, the depth of the buckets 
 in the direction of the radius, and describe the circumference 
 O D ; a third circumference is then drawn, B, at equal dis- 
 tances from the first two. The radii O A, O A', A" . . . 
 
 A' A 
 
 V 
 
 
 Fig. 11. 
 
 being then drawn through the points of division, A B', A' B", 
 . . . will be joined, and we shall then have the profiles A B' D', 
 A' B" D", . . . which, excepting the thickness, will be those 
 of the buckets. 
 
 Skilful constructors think that, instead of the lines such as 
 A B', A' B", ... we might employ the lines a B', a' B", . . 
 which produce a certain degree of mutual covering between 
 the buckets ; in like manner for the lines B D, B' D', B" D", 
 . . . , the inclined right lines B d, B' d\ B" d", . . . , have 
 been sometimes substituted.- These two changes have the one 
 end, that of increasing the depth of the buckets in the direction 
 parallel to the circumference, and consequently to retard the 
 emptying. They are inconvenient, because they make the 
 construction more difficult ; besides, this overlapping A a must 
 
OVEESHOT WHEELS. 61 
 
 not be carried to excess, otherwise the remaining free space 
 between the point B and the side a B' would perhaps be too 
 much diminished. This minimum distance should be a few 
 centimetres greater than the height to which the sluice gate is 
 raised, in order that the water may enter well into the wheel, 
 and not be thrown to the outside. 
 
 When the buckets are made of sheet-iron, the broken pro- 
 files just mentioned are replaced by curved profiles, which 
 should differ as little as may be from them. 
 
 Wooden buckets are generally from half an inch to an inch 
 in thickness; the sheet-iron ones are only from three-sixteenths' 
 to quarter of an inch, which increases slightly their capacity, 
 all other things being equal. They are limited at the sides by 
 the annular rings, which are fastened to the axle by arms, 
 which increase in number with the diameter of the wheel. 
 They present a continuous bottom or sole throughout the 
 entire circumference D D' D". . . . ; in very large wheels this 
 bottom must be sustained by supports at one or two points 
 placed between the e'xterior crowns. We might also use in this 
 case one or two intermediate crowns. 
 
 (e) To calculate the dynamic effect of a head that causes an 
 overshot wheel to turn. — The two main causes which give rise 
 to the losses of head to be subtracted from the entire head, to 
 obtain the head that is turned to account, are the relative velo- 
 city when the water enters the wheel, and that which it pos- 
 sesses at the moment that it falls to the level of the tail race. 
 
 It is almost impossible to obtain an accurate value of the 
 first. During the time that a bucket is being filled, the point 
 of entrance of the molecules, which come in successively, is 
 changing in a continuous manner. The first impinge against 
 the solid sides ; those that come after, against those that are 
 already in; and thence result phenomena very difficult to 
 
62 OVERSHOT WHEELS. 
 
 analyze. The study is greatly simplified by admitting, as we 
 did in (No. 3), that the height — - , due to the relative velocity 
 
 2 y 
 
 w of the water at its point of entrance, represents the loss of 
 head in question. Besides, if we call v the absolute velocity 
 of the water, u the velocity of the wheel, y the angle formed 
 by the two velocities ; as w is the third side of the triangle 
 formed by v and u, we shall have 
 
 w 1 = if 2 + v* — 2 u v cos y. 
 
 In reality, the impinging of the water on the wheel takes place 
 at different points along the depth of the bucket. Recollecting 
 now that the radius of the wheel is great compared with the 
 thickness of the shrouding of the buckets, this will not mate- 
 rially affect u; but to determine v and y, it would be well per- 
 haps to suppose the point of entrance, not at the exterior cir- 
 cumference, but at the middle of the depth of the buckets. 
 
 Let us pass to the second loss. Let a molecule of the mass 
 m leave the wheel at a height z above the tail race. . This 
 molecule, having only an insensible relative motion in the 
 bucket, possesses, at the moment that it leaves it, the velocity 
 u of the wheel, and at the moment it reaches the level of the 
 tail race it has a velocity v' equal to V u* + 2 g z. Then it 
 gradually loses all its velocity while moving in this portion, 
 without its piezometric level changing (for we suppose the free 
 surface horizontal in this portion) ; it undergoes then a loss of 
 head equal to 
 
 ~^ + 2 9 z) = f g + z. 
 
 For all the molecules composing the weight P expended in a 
 second, there will be a mean loss expressed by -=5- 2 m g 
 
OVERSHOT WHEELS. 63 
 
 ( h z\ or else by - — + =r- 2 m a z, the sum 2 including all 
 
 \2 g r J 2 g P v ' b 
 
 the molecules. This will be the second height to be subtracted 
 from the total height of the head ; it is composed of two term.;, 
 of which the first is at once given, and it only remains to be 
 
 seen how we can calculate the term ^2 mg s, which expresses 
 
 the special effect of the emptying of the buckets. 
 
 The quantity -=- 2 m ■ g z is nothing more than the mean 
 
 height comprised between the point of exit of a molecule and 
 the level of the tail race ; as the circumstances of all the buck- 
 ets are exactly the same, it is evidently sufficient to seek this 
 mean for the molecules contained in one bucket. To this end, 
 we will first determine, as stated above (5), the positions of the 
 bucket at which the emptying begins and ends, and, for a cer- 
 tain number of intermediate positions, we will ascertain the 
 amount of water that remains in the bucket. Let then 
 
 c be the height of the outer edge of the bucket above the 
 level of the tail race when the emptying begins, and let the 
 bucket, still full, hold the volume of water q a ; 
 
 c' the analogous height when the emptying has just ended ; 
 
 y the distance that this same edge has descended whilst the 
 volume of water q„ was being reduced to q. 
 
 During an infinitely small displacement of the wheel, to 
 which the descent d y corresponds, an infinitely small volume 
 — dqis emptied out, which falls into the tail race from a 
 height c — y; the mean height of the outflow will then be 
 
 2_fi* t c — y)d q. JSTow integrating by parts there obtains 
 
 J"(e-y)dq = q(c-y)+fqdy, 
 
64 0VEKSH0T WHEELS. 
 
 and, observing that y = c — c' and y = o correspond to the 
 limit q = o and q = q , 
 
 f g * (c-y)dq = q c +f°_ c ,9.dy = V° G -f° ' ° <? d V > 
 
 the mean sought is then — 2mgs = c- / qdy. 
 
 r q J o 
 
 qdy will be 
 
 effected by Simpson's method, for want of a strict analysis, since 
 we have the means of determining the value of q correspond- 
 ing to a given, value of y. Were we satisfied with a greater or 
 less approximation, but generally one sufficient, we could, 
 
 under the sign/", replace the variable ^by the mean - q a of its 
 
 extreme values ; we should then find, 
 
 -^s mg z = e- -(c - c') = -(c + c'). 
 
 To the losses already determined we must still add one for 
 bringing the water from the upper portion of the canal to the 
 wheel. As we have seen in examining other wheels, it will be 
 very slight if a good arrangement be adopted ; its value would 
 
 v* 
 then be 0.1 - — . 
 
 2 9 
 (f) Practical suggestions. — The liquid molecules taking, 
 during their fall which follows the discharge, a velocity sensi- 
 bly vertical, there are scarcely any means for turning this velo- 
 city to account by a counter-slope, and consequently the down- 
 stream level should just graze the bottom of the wheel. For 
 wheels that turn rapidly, it would perhaps be advantageous to 
 set them in a mill race, which would only allow the water to 
 escape at the lower portion, and with a velocity nearly horizon- 
 tal; we should take care to furnish each bucket with a valve, 
 
OVERSHOT WHEELS. 65^ 
 
 placed near the bottom, and opening from without inwards, to 
 allow the air to enter when the water runs out. It would then 
 be possible to diminish greatly the loss of head occasioned by 
 the velocity that the water possesses on leaving the wheel ; but, 
 on the other hand, we would increase, in no small degree, the 
 expense of erecting the wheel, and very likely also that of 
 keeping it in repair. 
 
 Overshot wheels answer very well for heads from 13 to 20 
 feet; less than 10 feet the breast-wheel is to be preferred. 
 Besides, as their diameter is nearly equal to the height of the 
 
 ■ 
 
 water fall, their employment would become practically impos- 
 sible for very high falls. 
 
 Experience shows that with an overshot wheel, well set up 
 and moving slowly, the productive force of the head of water 
 may rise as high as 0.80, and sometimes even more. But, in 
 wheels that turn rapidly, it sometimes falls as low as 0.40. 
 
TUB WHEELS. 
 
 §111. Water "Wheels with Vertical Axles. 
 
 10. Old-fashioned spoon or tub wheels. — The paddles of 
 spoon-wheels are of slightly concave form in the direction of 
 their length and breadth. They are arranged around a vertical 
 axle, and receive in an almost horizontal direction the shock of 
 a fluid vein which, leaving a reservoir with a great head, is led 
 near the wheel by a wooden trough. To obtain a greater 
 action, the water is made to strike against the concave side of 
 the paddles. 
 
 Let us call 
 
 v the absolute velocity, supposed to be horizontal and per- 
 pendicular to the paddle struck, of the vein which strikes the 
 wheel ; 
 
 w the section of this vein ; 
 
 u the velocity of the paddles at the point at which they re-, 
 ceive the shock ; 
 
 n weight of the cubic foot of water. 
 
 The relative velocity of the Avater and paddles will be hori- 
 zontal and equal to v — u ; then, if this phenomenon be assimi- 
 lated to that of a fluid vein impinging against a plane, the 
 
 force exerted on the wheel will be n u i L • the work 
 
 y 
 
 which this force performs on the wheel in the unit of time will 
 
 , . . u (v — uY . ., , , 
 
 be expressed by n u — i y -, a quantity sensibly equal to the 
 
TUB WHEELS. 6 i 
 
 dynamic effect of the head (No. 2). This expression varies 
 
 with u, and reaches its maximum for u = -v; this maximum 
 
 o 
 
 4 v 3 
 is n« — — , or, seeing that n w v gives the weight P expended 
 
 8 v* . v* 
 
 per second, — P— - . The height - — can only be a fraction 
 
 27 2 g ft 2 g J 
 
 of the head H ; hence, the dynamic effect is found below 
 
 8 8 
 
 — - P H, aDd the effective delivery below — -, or about 0.30. 
 
 This number, moreover, cannot be considered as strictly cor- 
 rect, because the imperfectness of the theories relating to the 
 resistance of fluids has caused us to give a more or less approxi- 
 mate value for the reciprocal action of the water and the wheel ; 
 however, experience confirms the result of calculation, at least 
 inasmuch as it indicates for this class of motors an effective 
 delivery that is always very small, varying from 0.16 to 0.33. 
 
 The tub-wheel does not give a much better result. A verti- 
 cal-cylindrical well made of masonry receives the water from 
 the head race through a conducting channel A (Fig. 12), whose 
 sides incline towards 
 each other with an in- 
 clination of about - as 
 5 
 
 they recede from the. 
 well ; that is to say, 
 that the sides make an 
 angle of 11 or 12 de- 
 grees with each other. 
 One side is also tangent to the circumference of the well. The 
 wheel, whose axis coincides with that of the well, consists of a 
 certain number of paddles regularly distributed around a ver- 
 
 PlG. 12. 
 
68 TUB WHEELS. 
 
 tical axle. The horizontal section of the paddles presents a 
 slightly curved form, having its concavity towards the side 
 from which the action of the water comes ; cut by a cylinder 
 concentric with the well, they would give inclined lines more 
 or less like arcs of helices. The working of the machine is 
 easily understood : the water comes through the trough A with 
 considerable velocity, endeavors to circulate all around the 
 well, and, meeting the paddles in its road, obliges them to turn, 
 as well as the axle that supports them. At the same time, the 
 water obeys 'the law of gravity, passes through the wheel by 
 means of the free space between the paddles, and falls into the 
 tail race, which ought to be a little lower. We see that the 
 water must undergo a good deal of disturbance in entering the 
 wheel, and, moreover, that it acts upon the latter for too short 
 a time to entirely lose its relative velocity. Also the effective 
 delivery, sometimes very slight and about 0.15, never exceeds 
 0.40. 
 
 We will pass over these primitive machines in order to study 
 others more perfect. 
 
TURBINES. 
 
 11. Of turbines. — The principle of reaction wheels, such as 
 are ordinarily mentioned in Treatises of Physics, has long been 
 known ; but it appears that it was only towards the beginning 
 of the last century that the idea was entertained of making use 
 of and applying it to the construction of water wheels with a 
 certain power. Segner, a professor at Gottingen, and more 
 recently Euler, in 1752, made them the object of their 
 researches. 
 
 In 1754 Euler constructed another machine, still founded on 
 the principle of reaction wheels, but differing from them in 
 several important arrangements ; this machine offers the most 
 striking resemblance to a powerful wheel now in use, called in 
 the arts Fontaine's turbine, from the name of the skilful con- 
 structor, who has set up a great many within late years, besides 
 extending and completing, in the details of their application, 
 the idea first advanced by Euler. It appears that this kind of 
 wheel was not much used until towards 1824, the period at 
 which the question was again studied by M. Burdin, engineer 
 in chief of mines, who constructed a similar machine which he 
 called a reaction turbine. In the years following, M. Fourney- 
 ron, inspired by the ideas of M. Burdin, established some tur- 
 bines in which he introduced marked improvements. Since 
 that time turbines have greatly spread and multiplied ; and 
 there exists a great number of models which differ more or less 
 from each other. 
 
70 THEBINKS. 
 
 Without undertaking to follow up more completely the his- 
 tory of the changes successively undergone by this kind of 
 wheel, we shall briefly describe the three principal classes into 
 which the turbines now in use may be divided ; then we will 
 ■give a general theory for them, and finally we will mention 
 some details to which a particular interest is attached. 
 
 12. Fourneyrmi's turbine. — The essential parts of this tur- 
 bine are represented in (Fig. 13). 
 
 The water from the head race A descends into the tail race 
 B by following a tube, with a circular horizontal section, of 
 which D is the upper opening. This tube, which is perma- 
 nently fixed, rests on supports of timber or masonry; it is pro- 
 longed by another circular cylinder of cast iron E G I F, mov- 
 able vertically, which can be lowered more or less, by means to 
 be presently explained. The bottom K K' K" L" L' L of the 
 tube is joined to a hollow cylinder or pipe abed, supported at 
 its upper end ; this pipe is moreover intended to keep the ver- 
 tical shaft e/from contact with the water, a motion of rotation 
 being given to the shaft by the flow of the water due to its 
 head. In fact, we see that if the water reached the axle, it 
 would be necessary, in order to avoid leakage, to make this 
 latter pass through tightly closed packings, which would occa- 
 sion friction, independently of that caused by the contact of 
 the fluid. The arbor, as well as the pipe, are, moreover, con- 
 centric with the tub. 
 
 Between the bottom G I of the cylindrical sluice E G I F and 
 the annular plate K L, there is an opening G K, 1 L, entirely 
 around the perimeter of the bottom, through which the water 
 can flow. But as it is important, as we shall see, that the 
 threads should not flow in any direction whatever, they are 
 guided in their exit by a certain number of cylindrical par- 
 titions, with vertical generatrices, which are supported by the 
 
TURBINES. 
 
 71 
 
72 TURBINES. 
 
 plate K L, and of which the arrangement is sufficiently well 
 indicated in the horizontal section ; amongst these directing 
 partitions or guides, some, such as g A, are joined to the sides 
 K' K", L' L" ; others, such as i k, are shorter, in order to avoid 
 too great a proximity in the extremities of the partitions 
 towards the axle. 
 
 With regard to the opening SI, IL, the turbine proper 
 is included between the two annular plates or crowns 
 S R M jST, UTPQ; these plates are connected together by 
 means of the floats of -the turbine, which are cj'lindrical 
 surfaces with vertical generatrices, giving in horizontal sec- 
 tion a series of curves, such as lm,j>q, &c. ; the lower plate 
 is further connected with the arbor by a surface of revolu- 
 tion T a. j3 P, bolted to it, so as to form a perfectly solid whole. 
 The axle rests on a pivot at its loweY end ; a lever y <5, 
 moved by a rod 5 s, which ends at a point easily reached, 
 allows this pivot to be raised a very little, when the wear 
 and tear of the rubbing surfaces has produced a slight settling 
 of the axle. 
 
 To see how the action of the water sets the machine in 
 motion, let us suppose first that the arbor is made fast : then 
 the liquid threads, leaving the well -through the directing par- 
 titions, will strike against the concavity of the floats ; they will 
 thus exert a greater pressure on the concave than on the con- 
 vex portion of a channel such as Impq, formed of two con- 
 secutive floats, first by virtue of the shock, and secondly the 
 curved path that they are obliged to describe. Hence there 
 would result a series of forces whose moments, relative to the 
 axis, would all tend to turn the system of the floats in the 
 direction of the arrow-head indicated in the horizontal section ; 
 thus there will be produced effectively a rotation in the direc- 
 tion indicated, if the axle be allowed to turn, even in opposing 
 
TUEBINEB. 73 
 
 it by a resistance of which the moment should be inferior to 
 the entire moment of the motive forces. 
 
 To diminish as much as possible the loss of head experienced 
 by the liquid molecules during their passage from the head 
 race to the wheel, we should take care : 1st, to give the opening 
 C D a sufficiently great diameter and to round off its edges ; 
 2d, to furnish the circular sluice with wooden appendages G G', 
 1 1', placed at the lower portion, and having their edges rounded 
 off, as shown in the figure. We shall thus sufficiently avoid 
 the whirls and eddies caused by the successive contractions of 
 the threads of water. The wooden packing, moreover, is not 
 continuous ; it is composed of a series of pieces, each occupying 
 the free space between two consecutive partitions, so that the 
 sluice can be lowered without any hindrance as far as the bot- 
 tom K K' L I/. 
 
 We shall see, in considering the general theory of turbines, 
 how the other conditions essential to a good hydraulic motor 
 are fulfilled. 
 
 13. Fontaine's turbine. — Fig. 14 shows a general section of 
 this machine by a vertical plane. A pillar or vertical metallic 
 support A B is set as firmly as possible into the masonry form- 
 ing the bottom of the tail race ; this supports at its upper end 
 A a hollow cast-iron axle GDEF, which surrounds it; this 
 axle is prolonged above by a solid one, upon which is the mech- 
 anism for transmitting the motion. A screw and nut C allows 
 the position of the axle to be regulated in a vertical direction. 
 Nearly at the level of the water in the tail race (or, if desirable, 
 below it) is placed the turbine II I K L M N P, permanently 
 fastened to the bottom of the hollow axle ; it is comprised 
 between two surfaces of revolution concentric with the vertical 
 axis of the system ; these surfaces having H K and 1 L for 
 meridian lines ; in the intermediate space are placed the floats, 
 
74 
 
 TUBB1KES. 
 
TURBINES. 75 
 
 which receive the action of the water, and at the same time 
 strengthen the two surfaces. The water comes from the head 
 race a to the floats by flowing through a series of distributing 
 canals, of which the quadrilaterals Q R H I, STMN represent 
 the sections. These channels are distributed continuously over 
 an annular space, directly over the floats; they are limited 
 laterally by the surfaces QHTN, B.ISM; the space between 
 these surfaces, moreover, remains free, except the volume occu- 
 pied by the directing partitions, which divide it into a certain 
 number of inclined channels, in which the liquid threads move 
 with a determinate figure and direction. 
 
 To give a clear idea of the shape of the directing partitions 
 and floats, let us suppose a section made by a cylinder or a 
 cone, concentric with the axis of the system, passing through 
 the middle of the spaces Q R, H I, K L, and this section deve- 
 loped on a plane. The developed section of the directing sec- 
 tions will give a series of curves such as c d, ef, . . . comprised 
 in a. straight or curved row ; in like manner, for the floats of 
 the turbine, we shall obtain the curves dg,fh, . . . also com- 
 prised in another row. These curves having been drawn con- 
 formably to rules which we shall consider further on, let us 
 suppose reconstructed the cylinder or cone that had been devel- 
 oped, aitd let us conceive the warped surfaces generated by a 
 right line moving along the axis and on each of the curves in 
 question successively ; we shall in this way have determined 
 the surfaces of the partitions and floats. 
 
 It is deemed unnecessary to describe the arrangement for the 
 water from the head race to flow into the tail race in no other 
 way than through the channels formed by the directing parti- 
 tions ; in this respect the figure gives sufficiently clear indica- 
 tions. 
 
 "We can, as in the case of Fourneyron's turbine (No. 12), ac- 
 
76 
 
 TTJEBINES. 
 
 count for the direction in which the machine should turn on 
 account of the action of the water ; and which is that of the 
 arrow-head drawn below the development of the floats. 
 
 14. Kacldin's turbine. — Koecldin's turbine, of which the 
 entire arrangement was first imagined by a mechanic named 
 Jonval, does not differ essentially from Fontaine's turbine, 
 either in the arrangement of the floats and directing partitions, 
 or the mode of action of the water. The most noticeable dif- 
 
 R 
 
 A 
 
 ference consists in this, that the turbine is above the level of 
 the water in the tail race, as shown in (Fig. 15), a vertical sec- 
 tion of the apparatus. The directing partitions, set in an 
 
TURBINES. 77 
 
 annular space, of which the trapezoids QEHI, STMK, 
 indicate sections, are fastened to a kind of cast-iron socket, 
 which embraces the axle A B, without, however, forming part 
 of it, or pressing it hard ; they form a set of inclined channels, 
 through which the water from the head race flows and reaches 
 the floats of the turbine, placed immediately below, as in Fon- 
 taine's turbine. These floats are fastened to another socket, 
 which is bolted to the axle; they occupy the annular space 
 HIK^MNOP. 
 
 The inclined channels included between two consecutive 
 floats or partitions are limited on the outside by a fixed tub of 
 cast-iron, resting on the edges of a well of masonry ; it forms a 
 surface of revolution about the axis A B, having QHKD for 
 a meridian section. At the bottom of this tub are found a cer- 
 tain number of arms which support a centre piece, on which is 
 placed the pivot of the revolving shaft. 
 
 The water having left the turbine, by the apertures K L, 
 O P, flows into the tail race by descending through the mason- 
 ry well, and then passing into an opening which we may con- 
 tract, or, if need be, close at will, by means of a sluice Y. 
 
 The situation of the turbine above the level of the water in 
 the tail race allows it to be easily emptied, and herein lies its 
 principal advantage ; for this purpose we have merely to leave 
 the sluice of the tail race open, and to prevent the water reach- 
 ing the distributing channels QEHI, STMK We can then 
 visit the machine and make the necessary repairs. Besides, 
 the height included between the horizontal plane K L OP and 
 the level of the water in the tail race should not be considered 
 as a loss of head, because it belongs to a diminution of pressure 
 on the water which leaves the turbine, and we shall see by the 
 general theory, now to be investigated, that this causes an 
 exact compensation. 
 
78 TURBINES. 
 
 15. Theory of the three preceding turbines. — A complete 
 theory of the turbines that we have just summarily described 
 ought to include first the solution of the following general 
 problem. Having given all the dimensions of a turbine, its 
 position as regards the head and tail races, and finally its angu- 
 lar velocity, to determine the volume of water that it expends, 
 and the dynamic effect of the head, we should then seek the 
 conditions necessary to make the effective delivery for a given 
 head and expenditure of water a maximum. 
 
 But we shall not treat the question in such general terms. 
 In order to simplify the researches 'with which we are to be 
 employed, we shall allow in what follows that all the dimen- 
 sions have been chosen and the arrangements made, so that the 
 turbine may fulfil in the best possible way the conditions for a 
 good hydraulic motor. Then, from the pond to the exit from 
 the directing partitions, care will have been taken to avoid 
 contractions and sudden changes in direction of the threads ; 
 to have smooth and rounded surfaces in contact with the water 
 that flows through, in order that, in this first portion of its 
 passage, it may meet with no sensible loss of head. At the 
 point of entrance into the turbine, the water possesses a certain 
 relative velocity ; matters will be so arranged that this velocity 
 shall be directed tangentially to the first elements of the floats, 
 in order that no shock or violent disturbance may follow. This 
 is a condition that it is possible to fulfil by choosing a suitable 
 velocity for the wheel, as well as proper directions for the floats 
 and partitions where they join. Finally, as the water leaves 
 the turbine in every direction about a circumference, and as it 
 is hardly possible to prevent the absolute velocity which it 
 then possesses being used up as a dead loss in producing eddies 
 in the tail race, we shall suppose that we have taken care to 
 make this velocity small. All this combination of circum- 
 
TURBINES. 79 
 
 stances will considerably simplify our calculations, by allowing 
 us to neglect in them, without a very sensible error, the differ- 
 ent losses of head experienced by the water up to the point of 
 its exit from the turbine, a loss of which the analytical expres- 
 sion, more or less complicated, would overload our formulas 
 and make them much less manageable. Only, it should be 
 understood that our results will be exclusively applicable to the 
 case in which the machine works according to the conditions 
 for a maximum effective delivery. 
 
 This granted, let us call 
 
 v the absolute velocity of the water when it leaves the 
 directing partitions and enters the turbine ; 
 
 u its impulsive velocity and w its relative velocity, at the 
 same point, with respect to the turbine taken for a system of 
 comparison ; 
 
 v\ u' and w' the three analogous velocities for the point at 
 which the water leaves the wheel ; 
 
 p and jp' corresponding pressures at these two points; 
 jp a the atmospheric pressure ; 
 
 r and r' the distances of the same two points from the axis 
 of rotation of the system* ; 
 
 H the height of the head, measured between the level of the 
 
 (*) These two distances are often equal in the Fontaine and Koecklin tur- 
 bines ; but they differ in a marked degree in Fourneyron' s turbine. More- 
 over, it should be observed that, in the Fontaine and Kceeklin turbines', the 
 outlets of the water, from the guide curves, have a certain dimension perpen- 
 dicular Db the axis of rotation ; the lengths r and »•', be it well understood, 
 should be referred to mean points of these outlets. Thus in Fig. 15, for 
 example, r should be the mean of UM and UN"; r', in the same way, should 
 
 be equal to = . MH and P should also be so taken as to be small 
 
 relatively to r and r ', in order that the consideration of a single thread of 
 water only may not cause an appreciable error. 
 
80 TURBINES. 
 
 water in the pond and tail race, supposed to be sensibly at 
 rest; 
 
 h the depth, positive or negative, of the point of entrance of 
 the water into the interior of the turbine, below the level of 
 the tail race ; 
 
 h' the height the water descends during its motion in the 
 interior of the turbine, a quantity equal to zero when M. Four- 
 neyron's arrangements are adopted ; 
 
 n weight of the cubic foot of water. 
 
 Now, all the intervals between two consecutive directing 
 partitions being considered as a first system of curved channels, 
 and all the intervals between two consecutive floats as a second 
 system. For the first system of these channels, represent by, 
 
 /3 the acute angle nnder which they cut the plane of the 
 orifices that terminate them at the distance r from the axis; 
 this angle /3 is also that at which the circumference 2 * r is cut 
 by the partitions ; 
 
 i the depth or breadth of the orifices of which we have just been 
 speaking, measured perpendicularly to the circumference 2 * r. 
 
 For the second system : 
 
 6 the angle made by the plane of the orifices of entrance 
 with the direction of the floats, or, what amounts to the same 
 thing, the angle of these floats with the circumference 2 * r, 
 to which we will give, moreover, a direction opposed to the 
 velocity u, the direction of the float being taken in that of the 
 relative motion of the water ; 
 
 y the acute angle at which the channels cut their orifices of 
 exit, or, in other words, the angle of the floats with the cir- 
 cumference 2 * r' about which the above-mentioned orifices are 
 distributed ; 
 
 V the depth or breadth of the orifices of exit, measured per- 
 pendicularly to the circumference 2 * r'. 
 
TTJKBINES. 81 
 
 The question now is to establish the relations between all 
 these quantities, under the supposition that the conditions of 
 the maximum effective delivery are satisfied. For that pur- 
 pose we shall first follow the motion of a molecule of water 
 along its path between the two races, and write out the equa- 
 tions furnished by Bernouilli's theorem. 
 
 Between a point of departure taken in the pond, where there 
 is no sensible velocity, and the point of exit at the extremity 
 of the directing partitions, there is a head expressed by H + /; 
 
 T) — 7} 
 
 H — — ; the velocity being v, we have then, under the 
 
 supposition of a loss of head that can be disregarded between 
 the two points in question, 
 
 v> = 2g(R. + h+^J > -) (1) 
 
 The water then moves along the floats of the tnrbine with a 
 velocity at first equal to w and finally to w' : in this second 
 pei'iod, if we call to the angular velocity of the machine, we 
 know, applying Bernouilli's theorem to the relative motion, 
 
 there must be added to the real head h' + — — — -, a fictitious 
 
 head - , or — : still, neglecting the losses 
 
 2 g 2 g ' ' ° ° 
 
 of head, which is approximately admissible when the water 
 enters with a relative velocity tangent to the floats, we shall 
 then have to place 
 
 w n - w* = 2 g (h' +"^f^) + u" - u* . . . . (2) 
 
 When the turbine is immersed and h is positive, the point of 
 exit of the water is found at a depth h + h' below the level of 
 the tail race ; besides, as, for the maximum effective delivery, 
 the water must go out with a slight absolute velocity, we can 
 
82 TURBINES. 
 
 without material error admit that the pressure varies, in the 
 tail race, according to the law of hydrostatics, which gives 
 
 %* + A + h' = £- . . . . (3) 
 
 n n ■ ; 
 
 This relation is also true in Kcecklin's turbine, although 
 h + h! becomes negative, provided that the well placed below 
 the turbine and the orifice by which it communicates with the 
 tail race are sufficiently large ; for then the water will take up 
 in it but a slight velocity and can there be considered as in 
 equilibrio ; the pressure p' would then be less than p a by a 
 quantity represented by the depth — (h + A'), as equation (3) 
 shows. We could also preserve it, if the lower plane of the 
 turbine, constructed according to one of the first two systems, 
 were on the same level as the water in the tail race : we would 
 then, in fact, have^ a =jp' and h + h' = o, the quantities h and 
 h! being equal with contrary signs, or else both equal to zero, 
 according to whether we are considering Fontaine's or Four- 
 neyron's turbine. 
 
 The incompressibility of water will furnish us with the 
 fourth equation, showing that the volume of water that has 
 flowed between the directing partitions is equal to that which 
 leaves the turbine. The distributing orifices, left free between 
 the partitions, occupy a total development 2 « r (excepting the 
 slight thickness of the partitions) and a breadth b, from whence 
 there results a surface 2 * b r y as they are cut by liquid threads 
 moving with the velocity v, at an angle p, we have for the first 
 expression of the volume Q expended in a unit of time 
 
 Q = 2 * b r sin /3. v. 
 In like manner, the orifices of exit at the extremity of the 
 floats have a total development 2«r\ a breadth b', a surface 
 2 ir V r', and they are cut by the threads moving with a rela- 
 tive velocity w' at an angle y' ; then 
 
 Q = 2 * V r' sin y. iv'. 
 
TUKBINES. 
 
 83 
 
 Strictly speaking, on account of the thickness of the floats 
 or partitions, these two expressions for the value of Q ought to 
 
 undergo a slight relative reduction of— or — ; but in all cases, 
 
 the reduction being the same for both, we shall have by the 
 
 equality of the values of 
 
 Q 
 
 2 v 
 b r v sin /3 = V r' w' sin y (4) 
 
 The three following relations are in a certain degree geo- 
 metrical. 
 
 Let us represent (Fig. 16) a float B and a directing par- 
 tition A B ; a liquid mole- 
 cule having followed the 
 path A B arrives at B with 
 an absolute velocity v, and a 
 velocity w relatively to the 
 turbine which itself, at the 
 point B, possesses the velo- fig. i6. 
 
 city u. This last being what 
 
 is called the propelling velocity, we know that v is the diagonal 
 of the parallelogram constructed on u and w ; and as the angle 
 between v and u is exactly g, the triangle B IT V will give 
 
 UV =BU + BV-2BU. BVcos/3, 
 
 that is, 
 
 w* = u* + v* — 2 u v cos /3 (5) 
 
 In like manner the liquid molecule, after having traversed, 
 
 relatively to the turbine, the path B C, arrives at C with the 
 
 velocity w', which, taken as a component with the propelling 
 
 velocity u', gives the absolute velocity v' ; then the angle y 
 
 being the supplement of that made by u' and w\ we shall have 
 
 v" = u n + w" - 2 v! w' cosy (6) 
 
 6 
 
84 TURBINES. 
 
 On the other hand, the velocities u and u' belong to two points 
 of the turbine situated respectively at the distances r and r' of 
 
 the axis ot rotation, we have then — = — ; , or 
 
 r v 
 
 u' r = ur' (7) 
 
 There still remains to express two conditions necessary for 
 
 obtaining the best effective delivery. It is necessary first, at 
 
 the point B, that io be directed tangentially to. the floats B C 
 
 without which there would be a sudden change of relative 
 
 velocity, whence would result disturbance and a loss of head 
 
 that we have not considered. Now the angle between w and u 
 
 is the supplement of 6, hence the triangle BUY gives 
 
 BIT _ sin B Y U _ sin (B TJ Y + Y B U ) 
 BY ~~ sin B U Y ~~ sin BUY ' 
 
 or 
 
 u_ _ sin (6 + g) 
 
 v sin'0 { ) 
 
 It is then necessary that the absolute velocity v' possessed by 
 
 the water on leaving the turbine should be very slight, since 
 
 v" 
 
 - — enters in the loss of head (No. 2) : this condition is suf- 
 
 2 g . 
 
 ■ ficiently -satisfied by taking the angle y small, and placing 
 u' = w' ; .... (9) 
 for then the parallelogram C U' Y' W is changed into a lozenge, 
 very obtuse at one angle and very acute at the other, and the 
 diagonal joining the obtuse vertices is short ; .in other words, 
 the velocities u' and w' are equal, and almost directly opposed, 
 which makes their resultant very small. 
 
 We have thus obtained, in all, nine equations between six- 
 teen variable quantities in a turbine, namely — 
 
 six velocities u, v, w, u', v\ w\ 
 
 two pressures _p, j>', 
 
TURBINES. 85 
 
 two ratios — r , ^-r, 
 r o 
 
 three altitudes H, h, h', 
 
 three angles p, y, 6. 
 
 These equations will serve us for solving two distinct problems, 
 
 which may be thus stated : 1st, having given a turbine and all 
 
 t h 
 its dimensions (that is to say, the eight quantities £, y, I, — „ ^ 
 
 H, h, h'), to show the conditions these dimensions must satisfy, 
 in order that the turbine may work with the maximum effective 
 delivery — that is to say, so that the nine above equations may 
 •obtain ; and, under the supposition that these conditions are 
 fulfilled, to show the most suitable velocity of the turbine, as 
 well as the expenditure of water' corresponding to this velocity, 
 its effective delivery, and its dynamic effect ; 2d, having given 
 the expenditure and the height of a head, to establish under 
 this head a turbine with the best conditions. 
 
 The first question involves eight unknown quantities, which 
 are u, v, w, u', v', w',p,p' / the elimination of these unknown 
 quantities between the nine equations will then give an equa- 
 tion of condition to be satisfied by the dimensions of the ma- 
 chine — an equation to which we shall have to add two others 
 in order to show that the pressures p and^>' are essentially posi- 
 tive. The following calculation has for its object to bring out 
 these three conditions, and, at the same time, to give the value 
 of the unknown quantities. 
 
 Adding equations (1), (2), and (5), member to member, there 
 obtains 
 
 10' 
 
 = 2ff(B. + h + ti +^ L ~~) + u" — 2uv cos /3, 
 
 or, considering (3) and. (9), 
 
 u v cos /3 = g H . . . . (10) 
 
86 TUBBINES. 
 
 The combination of equations (4) and (9) readily .give 
 
 b v r sin /a = V u' r' sin y ; . . . . (11) 
 multiplying equations (7), (10), and (11), member by member, 
 there obtains 
 
 v' b r* sin /s cos /a = y H. 5' r' 3 sin 7 ; 
 
 whence one of the unknown quantities 
 
 + = 9 kV£^JEJL- ...(12) 
 * t»r sin (3 cos /3 
 
 J "TT2 
 
 "We have besides, from equation (10), u' = -j j— ; whence 
 
 , „ 5 r s tan /3 n „. 
 
 y V r n sin 7 
 and from equation (7) 
 
 »» = jH*^- (14) 
 
 y J' sin 7 v y 
 
 To obtain v', we will first make w' — u' in equation (6), which 
 
 will give 
 
 v"' = 2 u" (1 — cos 7), 
 
 and consequently from the value of u' (14), 
 
 v" = 2 o H A ^ (1 - C0S7) . . . . (15). 
 y 5' Bin 7 v ' v 
 
 Knowing zs' we obtain w', and, if 10 were required, we could 
 easily obtain it by substituting in equation (5) the values of v 
 and u. Thus, all the velocities may be considered as known ; 
 we could deduce from them the angular velocity w with which 
 the turbine must move, when it is working with the conditions 
 of the maximum effective delivery ; we would then have prac- 
 tically 
 
 _u u' 
 
 The corresponding expense Q has for its value 2 * b' w' r' sin 7, 
 or, substituting in place of w' the value of its equal u' from 
 equation (14), 
 
TUKBOTES. 87 
 
 Q = 2 * r' Vbb' V g H tan /3 sin y, . . . . (16) 
 a formula whose second member we should probably have to 
 multiply by a number less than unity, in order to take iuto 
 consideration the space occupied by the floats, and also to com- 
 pensate for the influence of the losses of head neglected in the 
 calculation. 
 
 Let us now seek the three equations of condition to be satis- 
 fied by the dimensions of the turbine. First, by dividing equa- 
 tions (13) and (12), member by member, and extracting the 
 square root of the quotient, we will find 
 
 u b r* sin p 
 
 v V T n sin y' 
 
 and by reason of eq. (8) 
 
 sin (fl + /3) _ b r* sin /3 
 
 sin fl ~b'r"sm 7 ' * ' ' ' X } 
 this is the condition obtained by. eliminating eight unknown 
 quantities between nine equations. There remains yet to ex- 
 press that^? > 0,j>' > 0. As to this last condition, we see from 
 (3) that it is itself satisfied for Fourneyron's and Fontaine's 
 turbines, by supposing that they are on a level with the Water 
 in the tail race or below it, as we have admitted in the preced- 
 ing calculations ; because then h + h' is positive, and we have. 
 Jp' > Pa- I n Koecklin's turbine, on the contrary, the bottom 
 of the turbine is in reality above the level of the water in the 
 
 p' 
 tail race, by a positive height expressed by — (h + h') ; — has 
 
 for its value ^, 33 ft .90 less this height ; then it is absolutely 
 
 necessary to have 
 
 - (h + h') < 33 ft .90, 
 and perhaps even, on account of neglected losses of head, it 
 would be well to place 
 
88 TURBINES. 
 
 - (A + A') < 20 feet, (18) 
 
 in order to. make perfectly sure of the continuity of the liquid 
 column in the cylindrical well, above which the turbine is 
 found. As to the pressure^, it will be found from eq. (1) after 
 substituting in it for i? its value in eq. (12), that is 
 
 £ = i + u i«-b^ _^ilL2_ ( i9) 
 
 n n 2 I r* sin p cos /3 v ' 
 
 The second member of this equation should, of course, be 
 greater than zero ; but we may assign it a higher limit. In 
 fact, if we examine the arrangement of the different systems of 
 turbines, we see that there is always an indirect communication 
 between the distributing orifices, situated at the end of the 
 directing partitions, either with the tail race, or with the exter- 
 nal air. This communication is effected by the play necessarily 
 left between the turbine proper and the distributing* orifices. 
 When it takes place with the tail race, p cannot differ much 
 from the hydrostatic pressure^ + n h, which would take place 
 in a piezometric column communicating with this race, and at 
 the height of the point of entrance of the water above the 
 turbine ; otherwise there would be, on account of the play that 
 we have just spoken of, either a sudden gushing out, or suction 
 of the water, which would produce a disturbance in the motion. 
 When it is with the atmosphere,^? must, for a similar reason, 
 be equal to^> a . It is then prudent, in the first case, to impose 
 the condition that the two terms in which the factor H appears, 
 eq. (19), should nearly cancel each other, or, designating by k 
 a number that differs little from unity, to make 
 
 o r sin p cos p ' 
 
 h is moreover rigorously subjected to the condition that 
 
 A + & + H (1 - h) 
 
TURBINES. 89 
 
 should be positive. And in like manner, for the second case, 
 we should establish the condition 
 
 H (1 -%- r 4 — ™L2—) + h = h" .... (20 bis) 
 y br* 2 sin /S cos /?/ x ' 
 
 h" being a very small height. 
 
 Again it might be proposed, for a turbine known to be work- 
 ing with the maximum effective delivery, to seek this delivery 
 as well as the dynamic effect. As we are supposing that all 
 losses of head, other than that due to the velocity of exit v', 
 may be disregarded, the head that is turned to account will be 
 
 and consequently the productive force p will be expressed by 
 
 TT _ V 
 
 2 I — 1 - V "* 
 
 or replacing i>" by its value 
 
 . b tan j3 . . , . 
 
 f* = 1 — 77 ~ C 1 _ cos 7) ( 21 ) 
 
 4 Bin 7 
 
 The dynamic effect T e would be obtained by finding the pro- 
 duct (a n Q II of the effective delivery by the absolute power 
 of the head ; then We would have, from eqs. (16) and (21) 
 
 T e = n II V'g H. 2 * r' V b V. V tan /s sin 7 J 
 
 j - i tarns , ) f ■ • • • (22) 
 
 X 1 t; — (.1 — cos 7) f I 
 
 I 6 sin y ) J 
 
 Thus have we now solved the first of the two general prob- 
 lems proposed. "When we take up the second, which consists 
 in setting up a turbine for a given head, Q and H become the 
 known quantities, and we have, between the nine quantities 
 j8, 7, d, r, r', b, b', h, h', which define the unknown dimensions, 
 only equations (16), (17), (20), or (20 bis), to which must be 
 
90 TCEBINES. 
 
 added (if a Koecklin's turbine be in question) the inequality 
 (18) ; still this inequality leaves a certain margin; and it is the 
 same with equations (20) and (20 bis), because th% quantities k 
 and h" have not a definite value. It appears, then, that the 
 problem is very indeterminate, and that we may assume almost 
 all the above-mentioned dimensions arbitrarily ; however, the 
 following remarks impose restrictions that it will be well to 
 keep in mind. 
 
 16. Remarks on t7ie angles /3, y, 0, and on the dimensions b, 
 b', r, r', h, h'. — If we only considered the expression for the 
 effective delivery eq. (21), we should be tempted to make one 
 of the angles /3 or y equal to zero ; the theoretical effective 
 delivery would then become practically equal to unity. But 
 we see that the expenditure Q would reduce to zero, as well as 
 the dynamic effect T e : hence the value zero is not admissible 
 for either of these angles. 
 
 Making y very small, the channels formed by two consecu- 
 tive float's would be very much narrowed at the point of exit 
 of the water; the water would flow with difficulty through 
 these narrow passages, and there would be danger of its not 
 following exactly the sides of the floats, which would occasion 
 eddies and losses of head. On the other hand, a large value 
 for y would diminish, very likely, the effective delivery too 
 much. Between these two points to be avoided, experience 
 gives a value of 20 or 30 degrees as affording satisfactory 
 results. 
 
 As to the angle £, besides the reason already given, there is 
 still another for not making it zero : this second reason is that 
 from equation (19) p would be negative for /3 = o and /3 = 90°. 
 We must not then approach too closely to zero or to 90° ; the 
 limits from 30 to 50 degrees have been advised by some 
 experts ; but there is none that is absolute. 
 
TURBINES. 9] 
 
 Let us suppose that we are about to apply equation (20) ; 
 multiplying it, member by member, by equation (17) we find 
 h sin (6 + /3) 1 
 
 sin 8 2 cos ' 
 
 whence 
 
 1 _ 1 _ 2 cos /3 sin (8 + j8) — sin 4 
 & sin d 
 
 or, by developing the sin (8 + /3), 
 
 2 cos /s sin (d + /s) — sin 4 = sin 8 (2 eos s /S — 1) + 2 sin /3 cos j8 cos d 
 = sin 8 cos 2 jS + cos d sin 2 £ 
 
 = sin (2 /3 + d) ; 
 we can then write 
 
 . ' ip (V + '> *-! (23) 
 
 sin d A y 
 
 We have previously seen that & should be a number very near 
 unity ; it follows that sin (2 jS + d) should be small, and conse- 
 quently that 2 /S + d should differ but little from 180 degrees. 
 If, for example, we took /3 about 45 degrees, d would be about 
 a right angle. Besides, it is not well to have 8 greater than 
 90° ; for, if we refer to Fig. 16, we see that if 8 be obtuse the 
 floats should have a form like B' C, presenting a considerable 
 curvature; and experience shows that in a very much curved 
 channel the water meets with a greater loss of head, all other 
 things being equal : the liquid molecules then tend to separate 
 from the convex portion, which gives ripe to an eddy. We see 
 also in Fig. 16 that, in taking 8 very acute, the side V U of the 
 triangle B TJ V — that is, the relative velocity at the entrance, 
 would tend to become more or less great, which would be hurt- 
 ful, since the friction of the water on the floats would be in- 
 creased. Hence 8 should be an acute angle, but at the same 
 time almost a right angle : we might make it vary, for example, 
 between 80 and 90 degrees. 
 
 If it be equation (20 his) and not equation (20) that we have 
 
92 TUKBUfES. 
 
 to apply, the same reasons obtain for taking 6 acute and nearly 
 90 degrees ; but the sum 2 /J + 6 need no longer differ much 
 from 180 degrees. 
 
 After having fixed the values of /a, y, and 6, we will find from 
 
 b r* . . i 
 
 equation (17) the ratio p— «, whence we can obtain either — or 
 
 — , the other having been assumed. 
 
 It is in favor of the effective delivery to have -p less than 
 
 unity, as formula (21) shows ; the difference b' — b, however, 
 must not be too great, and must be proportional to the length 
 of the floats, in order that the channels between two con- 
 secutive floats may not be emptied too rapidly, because this 
 emptying would give rise to a loss of head. We can impose 
 
 the condition that V — b should be less than — the length of 
 
 10 & 
 
 the floats. 
 
 r' 
 As we have already said, the ratio — is often taken equal to 
 
 unity in Fontaine's and Koscklin's turbines, but it is neces- 
 sarily greater than unity in Fourneyron's turbine. If it be 
 taken different from unity, we must not, except for- particular 
 reasons, increase the difference r' — r, or r — r\ for we should 
 thus lengthen the floats and increase friction. In Fourneyron's 
 
 turbine — varies ordinarily between 1.25 and 1.50. 
 
 The height h', from which the water descends into the inte- 
 rior of the turbine, is always zero in Fourneyron's turbines ; in 
 the other two systems it is so taken that the floats may be suf- 
 ficiently, but not too long, regard being had to the difference 
 V — b. As to the height h, if it be a question of a turbine of 
 
TTJ&BINES. 93 
 
 Kcecklin's, it is fixed according to local circumstances, the in- 
 equality expressed in (18) being considered; if it be of one of 
 Fourneyron's or Fontaine's, we so arrange matters that its 
 lower plane may be on the same level as the waters in the tail 
 race when at their minimum depth. 
 
 Finally, M. Fourneyron recommends giving the circular sec- 
 tion of the tub, in which the directing partitions of his turbines 
 are placed, a surface at least equal to four times the right sec- 
 tion of the distributing orifices, in order that the fluid threads 
 may easily pass from the vertical to the horizontal direction, 
 which they must have at their point of exit. With the nota- 
 tion employed in (No. 15), we can write 
 
 * r 1 > 4. 2 * r b sin /3, 
 or else 
 
 t > 8 I sin /J, . . . . (24) 
 
 the sign > not being exclusive of equality. 
 
 Let us now show, by two examples, how we shall be enabled, 
 
 * 
 by means of these considerations, to determine the dimensions 
 
 of a turbine to be established. 
 
 17. Examples of the calculations to he made for constructing 
 a turbine. — Let it first be determined to establish a Fourney- 
 ron turbine with the following data : 
 
 Height of fall, H = 20 feet. 
 
 Volume expended per second, Q = 54 cubic feet. 
 
 The absolute power of the head is 3364 lbs .2 x 20 ft = 67284 
 foot pounds per second, or 122 horse power. 
 
 Since the angle 7 is not fixed theoretically, we will take it 
 (No. 16) equal to 25 degrees ; we will al&, make k — 1 (*), 
 which secures thatjp shall be positive (No. 15); finally, we will 
 take — 90°. Equation (23) then gives 
 sin (2 § + 6 ) = 0, 
 ■whence 
 
94 TUBBLSTES. 
 
 2,3+0 = 180° and/? = 45°. 
 As we have satisfied equation (23), which results from the 
 
 elimination of -, — •„- - — - between formulae (17) and (20), it is 
 
 sufficient to preserve one of these last ; we deduce from both 
 
 7 2 
 
 1L- = s i n 25° = 0.4226 («). 
 
 b r' 4 
 
 The condition of expending 54™- ft is expressed by formula (16). 
 
 which here becomes 
 
 54 cu - ft = 2nr' VbVVsOg. 0.4226, 
 
 or else, by reducing 
 
 r' Vbb' = 0.5211 {t'j. 
 
 We have still to express the inequality (24), which gives 
 
 r > 8 b sin 45° or r > 5.662 b ; 
 
 we will take 
 
 r = 6 b («"). 
 
 We have thus only three equations involving b, V, r, r'; but 
 
 on account of their particular form we may still deduce the 
 
 values of b and r. Extracting the square root of equation (a) 
 
 and multiplying it member by member by («'), we make r' V b' 
 
 disappear and find 
 
 b r = 0.031107, 
 
 a relation which, combined with r = 6 b, gives 
 
 r = l ft .425, b = ft .234. 
 
 This being done, the system of the three equations («), («'), (a") 
 
 would no longer give anything but r' V b' ; to avoid any inde- 
 
 termination, we will take b' arbitrarily and deduce r\ except 
 
 that the conditions mentioned in (No. 16), and not thus far 
 
 expressed, must subsequently be verified. If we take, for 
 
 example, b' = ft .3, equation («') will become 
 
 r' V0:3 x 0.234 - 0.5211, 
 
TURBINES. 95 
 
 whence we obtain 
 
 r' = l ft .966. 
 
 These values of V and r' may be retained, because - = 1.37, 
 
 and the difference V — i = 0^.7 is only - of r' — r, a quan- 
 
 tity which, on account of the obliquity of the floats to the 
 exterior circumference, should be but little greater than two- 
 thirds of the length of these last ; the discharge of water will 
 not then be too rapid. 
 
 The height h now alone remains to be determined: if the 
 level of the tail race were constant, we should make A = o ; 
 however, we would have to consider what is said (No. 16) on 
 this subject. 
 
 The« theoretical effective delivery will be obtained from 
 formula (21) ; we find 
 
 , 0.072 . ,„ 1- cos 25° „ QOQ 
 
 " = l ~ O090- tan ^° • -1EW- = °- 823 - 
 
 In practice, we only rely upon a net effective delivery of from 
 0.70 to 0.75 at most ; this it is well to do, on account of all the 
 losses of head that we have neglected, and also because it is 
 very difficult to make a machine move exactly with the velo- 
 city and expenditure of water corresponding to the maximum 
 effective delivery. 
 
 Finally, to obtain the velocity with which the turbine should 
 revolve, formula (14) should be applied, and we would deduce 
 
 therefrom u' = 34 ft .63, since the angular velocity i» = -— = 
 
 17.61, and finally the number of revolutions per minute N = 
 
 3 -l» = 168.1. 
 
 n 
 
 Again, let, it be proposed as a second example to set up one 
 
06 TURBINES. 
 
 of Fourneyron's turbines, with a head of 6 ft .5, expending 
 19 cu - ft .2 of water per second, which corresponds to an absolute 
 work of 8660 foot pounds, or about a 16 horse-power. "We 
 will suppose that the water in the tail race only rises to the 
 level of the lower plane of the turbine, so that the interval or 
 play between the turbine and the directing partitions may 
 communicate directly with the atmosphere, and that the pres- 
 sure p is sensibly equal to the atmospheric pressure. "We will 
 then assume equation (20 bis), making in it h" = o, in other 
 words we will place 
 
 b r s 2 sin /3 cos p 
 
 Following the ordinary rules, we will make r=r' j besides, it 
 
 may be remarked that, on account of the position assigned to 
 
 the plane of the tail race, h is equal to h' with a contrary sign. 
 
 The above equation can then be written 
 
 H V siny \ h , = (f {S) 
 
 v b 2 sin /3 cos jS / w 
 
 As 6 can only differ slightly from 90 degrees, we will give it 
 
 this value ; the equation of condition (17) then takes the form 
 
 b r* tan /3 1 
 
 b' r'* sin y 
 and, because r = r' 
 
 V sin y = b tan /3 (8 1 ) 
 
 Introducing V sin y in the place of b tan /3 in the expression 
 (16) of the expenditure, it becomes 
 
 Q = 2*/J' sin y Vl/TK. (8") 
 
 The equations (S, 5', 8") are those pertaining to the problem. 
 They contain six unknown quantities, viz. : /3, y, b, b', r', h' / 
 we consequently see that they are indeterminate, and that we 
 can assume three of the unknown quantities or three new 
 equations. The angle y not being determinable by theory, we 
 
TURBINES. 97 
 
 will take it at first equal to 30 degrees (No. 16) ; equation (<5") 
 will become, by substituting numbers for letters, 
 
 b'r' = ._°^L = 0.04312, 
 it V 2g 
 
 a relation which is satisfied by the values 
 
 r' = l ft .97, V = 2 in .81. 
 
 "We thus see that the ratio - is only 0.12 ; consequently the 
 
 inequality in the velocities of the liquid threads in the orifice 
 of exit -will not be too noticeable. Now as there remain three 
 unknown quantities, V , ti', p, connected only by the two equa- 
 tions (8), (5'), we will assume h' = ft .50 ; then eliminating 
 
 =r between (8) and (8'), we will obtain 
 
 1- 1 = £. = _L 
 
 2 cos 2 (3 H 40' 
 whence 
 
 2 cos 2 /?= :=,2cosV-l = cos2/? * 
 oi 37 
 
 and consequently 
 
 /? = 42° 40 very nearly. 
 
 Knowing /?, we obtain from equation (#') 
 
 I = l in .52. 
 
 The difference V — b = l 1 "^ is perhaps too great' relatively 
 
 to the height ft .5 of the turbine, because the floats having a 
 
 development of from ft .66 to ft .82 at the most, their spread 
 
 would reach the amount of - very nearly. We then try an- 
 other value of h' ; for example, let 
 
 h' = 1 foot. 
 Proceeding as before, we will have successively 
 
 90 30 
 
 2 cos 2 ft = ~, cos 2 /? = ~, § = 40°, I = l ta .67. 
 
98 TUKBINES. 
 
 The difference V — b would then be l in .14; but as the floats 
 would be nearly l ft .33 in length (on account of their inclination 
 to the lower plane of the turbine), this number appears per- 
 fectly admissible. We should then obtain the results 
 Y - 30°, p = 40°, 6 = 90°, r = r' = l ft .97. 
 b = l in .67, V = 2 in .81, - h = h! = ft .98. 
 The effective delivery (i would be given by equation (21), 
 which, considering equation (8'), becomes 
 
 ft = cos y - cos 30° = 0.866. 
 In like manner equation (14) would be simplified and give 
 
 u'* = gll, 
 whence 
 
 u' = 14 ft .46, 
 from this we deduce finally the angular velocity to be given to 
 
 the machine 
 
 u' 
 
 O) 
 
 = h = ?- 34 , 
 
 r 
 
 and the number of revolutions per minute 
 N = ^ =7 0.0. 
 
 n 
 
 In general, as we have seen, the problem which consists in 
 fixing the dimensions of a turbine for which we have the ex- 
 penditure and the head is indeterminate ; we take advantage 
 of this to assume in part the unknown dimensions, and by the 
 method of trial, if that be necessary, endeavor to satisfy the 
 different conditions to be fulfilled, but which the equations do 
 not express. 
 
 18. Of the means of regulating the expenditure of water in 
 turbines. — A turbine constructed with assigned dimensions, in 
 order to move with the maximum effective delivery, should ex- 
 pend a perfectly determinate volume of water, so long at least 
 as the head remains constant. However, in practice, we are 
 
TURBINES. 99 
 
 obliged to regulate the expenditure by the volume furnished 
 by the pond ; for if we expended more we would be exposed to 
 the lack of water, after some time, and we would then be 
 forced to suspend the operation of the machine. Consequently 
 we so calculate the dimensions as to expend suitably the great- 
 est volume of water at our disposal, and we make proper dis- 
 positions to expend less when the supply diminishes. For this 
 purpose several methods have been employed. 
 
 In Fourneyron's turbine, the movable tub E G F I (Fig. 13) 
 allows this end to be attained: it is "merely necessary to lower 
 it more or less, to contract the openings G K, I L, or to close 
 them entirely. The vertical motion of translation of this 
 tub is obtained by means of three vertical rods, as rs, tu, 
 which are attached to it at three points, which form the vertices 
 of an equilateral horizontal triangle. These rods are termina- 
 ted at their upper ends by screws, and enter into nuts which are 
 forced by their construction to turn in their places. The three 
 nuts, moreover, are furnished with three cogged wheels, all 
 exactly alike, which are geared on the same wheel, which is 
 loose on the axle of the turbine. By turning one of the nuts 
 by means of a wrench, the other two are turned exactly the 
 same amount, and the tub is raised or lowered by the three 
 rods at once. 
 
 There is one great inconvenience attendant upon the partial 
 obstruction of the openings G K, I L ; it is that the fluid veins 
 which issue through these openings immediately enter canals 
 of greater section, in which they flow necessarily through a full 
 pipe, since the turbine is below the tail race. There is a sud- 
 den change of section thus produced, and consequently a great- 
 er or less loss of head. This influence is sometimes so great 
 that General Morin has mentioned, in different experiments on 
 a turbine, a diminution in the effective delivery from. 0.719 to 
 
100 TURBINES. 
 
 0.24, when the free opening under the wheel descended from 
 its greatest elevation to about £ of this height. The inconve- 
 nience is so much the greater as the diminution in the effective 
 delivery corresponds to that of the volume of water expended, 
 which tends to make the dynamic effect of the machine very 
 irregular. To remedy this, M. Fourneyron has proposed to 
 subdivide the height of the turbine into several stages, by means 
 of two or three annular horizontal plates, like the plates 
 SEIN,UTPQ, the distance between which is divided into 
 three or four equal parts. Supposing, for example, that there 
 are three stages, we see that there will be no sudden change of 
 section when the cylindrical sluice is raised ^, f , or the whole 
 height of the turbine ; in all cases the phenomenon of sudden 
 expansion will only affect a portion of the liquid vein. But, 
 on the other hand, the construction of the machinery is compli- 
 cated, and the friction of the water against the solid walls is 
 increased. M. Fourneyron has again proposed using only the 
 two plates SEMN, UTPQ; the lower one would carry the 
 floats only, and the upper one be pierced with grooves which 
 would allow it to settle freely between the floats, under the 
 action of its weight alone. This upper plate would bear on a 
 flange placed at the bottom of the cylindrical sluice on the out- 
 side. When the sluice is lowered, the plate S K M 1ST is low- 
 ered with it, and the height of the turbine would be always 
 equal to the height to which the sluice is raised. The movable 
 iplate carried around by the turbine would, while turning, rub 
 :against the flange which serves it as a support ; but, the pres- 
 sure between the two being slight, this would not produce an 
 increase of resistance worthy of mention. 
 
 M. Fontaine, to regulate the expenditure of his turbines, 
 uses a series of valves similar to that represented in Fig. 17. 
 A B is a directing partition, B C a float of the turbine, D a 
 
TURBINES. 
 
 101 
 
 .'I 
 
 valve winch can be sunk to a great or less depth into the space 
 between A B and the next directing partition to the left. In 
 this way we can narrow as much as we choose the free pas- 
 sage into this interval, and as the same effect is produced on all 
 in the same way, it is plain that we have the means of reducing 
 the volume of water expended as much as circumstances may 
 require. The vertical motion of translation is given simultane- 
 ously to all the rods E F by means similar to those used by 
 M. Fourneyron : these rods are all 
 connected with a metallic ring, 
 at three points of which are at- 
 tached vertical screws, furnished 
 with nuts which can be turned 
 only in their beds. These three 
 have each a cogged wheel secure- 
 ly attached, the wheels being all 
 of the same size, surrounded by 
 an endless chain, which causes 
 them to turn equally and at the 
 same time. It is then necessary to turn one of the three 
 wheels by means of a winch and pinion, in order that the whole 
 system of valves may take up a vertical motion. The partial 
 closing of the in-leading canals here, as in Fourneyron's tur- 
 bine, is not without inconveniences, for the sudden change of 
 section in these canals still gives rise to a loss of head ; how- 
 ever, this loss is found to be diminished to a considerable 
 extent. 
 
 For the valves M. Koecklin has substituted clack-valves re- 
 volving on a hinge, so as to fit exactly over the entrance to the 
 distributing canals, when the closing is complete. The ar- 
 rangement of the Koecklin turbine also allows the expenditure 
 of water to be regulated by making use of the sluice V (Fig. 
 
 —- 
 
 
 1 
 v'B 
 
 
 
 \ "-***.. 
 
 
 
 
 
 ,..--' 
 
 
 
 cj^' 
 
 
 
 Fig. 17. 
 
102 TURBINES. 
 
 15), which can close the communication between the well 
 placed below the turbine and the tail race. But experience 
 shows that a greater portion of the head is lost in this way than 
 by using the valves. 
 
 The inconveniences attendant upon the partial closing of the 
 distributing channels are so great, when considered from the 
 point of view of economy of the motive power, that all possible 
 means for remedying them have been sought for. We have 
 already mentioned two devised by M. Fourneyron. M. Charles 
 Oallon, civil engineer, and a constructor of reputation, has pro- 
 posed another way, which consists in making all the partial 
 sluices, which close the channels in question, independent of 
 each other ; to diminish the expenditure, a certain number of 
 these sluices can be closed completely, leaving the remainder 
 entirely opened. But as the channels formed by the floats of 
 the turbine pass alternately before the opened and closed ori- 
 fices, there is still a cause of unsteadiness and trouble in the 
 motion. 
 
 M. Callon's idea has been reproduced under another form by 
 M. Fontaine. The orifices of entrance of the distributing chan- 
 nels occupy a horizontal surface comprised between two circles 
 concentric with the axis of the turbine, M. Fontaine arranges 
 two rollers, of the shape of a truncated cone, which can roll 
 over this annular surface. The two rollers are mounted on the 
 same horizontal spindle which forms a collar surrounding the 
 axle of rotation. When, they move in one direction, each one 
 of them unrolls a band of leather, which lias one end fastened 
 to the roller and the other to the plane of the orifices of en- 
 trance : some of the orifices are thus entirely closed while the 
 others remain wide open. When the truncated cones move in 
 the contrary direction, they roll up the two leathern bands and 
 uncover the orifices. M. Fontaine has also imitated the tur- 
 
TURBINES. 103 
 
 bines of several stories of M. Fonrneyron, in proposing to di- 
 vide the turbines into several zones by means of surfaces bi 
 revolution about the axis of the system ; each of these zones 
 could be separately obstructed, 
 
 19. Uydrojmeumal/io turbine of Girard and Gallon. — The 
 problem of regulating the expenditure, without too great loss, 
 appears to have been solved in the best manner' in a kind of 
 turbine called by its inventors, MM. Girard and Gallon, the 
 hydropneumatic turbine* Their system consists essentially in 
 surrounding Fourneyron's turbine with a sheet-iron bell, the 
 lower plane of which is nearly at the height of the points at 
 which the water leaves the floats. In this bell, by means of a 
 small pump set in motion by the machine itself, the 1 air is com- 
 pressed, which gradually forces the water entirely out of the 
 bell ; then, if we suppose the cylindrical sluice to be partially 
 raised, the liquid vein which escapes below has a depth less 
 than the distance between the two plates of the turbine ; but 
 no sudden change in the section of liquid results from this, be- 
 cause the turbine moves in compressed air, and it is not cover- 
 ed by the water in the tail race. The water flowing into the 
 turbine has a depth which, at the beginning of the floats, is 
 equal to the height to which the sluice is raised ; the upper 
 plate is no longer wet, and as it only serves to hold the floats, 
 it can be hollowed out, in order to make sure of a free circula- 
 tion of air above the liquid vein. Thus the principal catise of 
 the loss of head, due to the partial raising of the sluice', is 
 found to be suppressed, and we should succeed in Obtaining a 
 very slightly varying effective delivery. 
 
 The calculations to be employed for the hydropneumatic tur- 
 bine may be regarded as a particular case of those in l$o. 15. 
 Preserving the same notation, we must consider V as an un- 
 known quantity, and at the same time suppose 
 
104 TURBINES. 
 
 in fact, h represents the depth to which the turbine is immersed 
 below the level of the tail race, and p a + n h is properly the 
 pressure of the air in the bell. Thus, from this value of p equa- 
 tions (19) and (20) show first that k = 1, and consequently (No. 
 16) that we have 2 p + 6 = 180°, the only condition to be satis- 
 fied by the dimensions of the machine. It gives p + & = 180° 
 — p ; equation (17) then becomes 
 
 V r n sin j = i r* sin A = i r* sin 2 p ; 
 whence we can deduce V in terms of the height i to which the 
 sluice is raised, for a turbine moving with the maximum effec- 
 tive delivery. In virtue of the preceding relation, equations 
 (13), (16), and (21) take the form 
 
 tan p g H 
 
 u* = gK 
 
 sin 7 2 c.)S" /3 
 
 Q = 2 * h r sin p V 2 g H, 
 
 these formulas, which can very readily be demonstrated direct- 
 ly, give : 1st, the angular velocity - of the turbine, which an- 
 swers to the maximum effective delivery ; 2d, its expenditure 
 Q, and its effective delivery ^ under the same condition. If we 
 had to set up a turbine for a given head and expenditure, the 
 equation of which Q is the first member, together with the in- 
 equality (24) (No. 16), would give us the means for finding r 
 and b sin p, and consequently h, after p has been chosen. As 
 to 7, it should always be taken between 20 and 30 degrees ; & 
 should be equal to 180° — 2/3; finally, r' should be taken as 
 small as possible (on account of the expression for (a), but not, 
 however, so as to run any risk of making the floats too short. 
 
 The method of MM. Girard and Gallon could be equally well 
 adapted to Fontaine's turbine. 
 
TURBINES. 105 
 
 20. Some practical views on the subject of turbines. — The 
 directing partitions and floats are generally made of sheet iron ; 
 they are fastened to the surfaces that are to support them either 
 by angle irons, or by setting them in cast-iron grooves on these 
 surfaces. They should be sufficient in number to give to the 
 velocity of the water their own direction. The distance of any 
 two consecutive floats or partitions apart should not be, at any 
 (joint, more than 2 ln .34 to 3 in .12, measured along the normal to 
 the surfaces, and generally it is made less. However, it must 
 not be made too small, for then the friction of the water against 
 the solid sides would be too great. 
 
 As the floats in Fourneyron's turbine are placed further from 
 the axis than the partitions, that is to say distributed over a 
 greater circumference, their number is from one-third to one- 
 half greater than that of the partitions, in order to have every- 
 where a suitable distance apart. 
 
 Excepting the condition of cutting the planes of the orifices 
 under a determinate angle, the curvature of the floats and par- 
 titions is almost a matter of indifference. However, as too 
 great a curvature, or a sudden change in the curve, may pre- 
 vent the threads from following the sides, and thus produce 
 losses of head, we must avoid these two faults. It would be 
 well to have the radius of curvature at least three or four times 
 the distance apart measured along the normal. 
 
 Turbines can be used for all heads and every expenditure. 
 For example, some are mentioned the head being only from m .30 
 to m .40, whilst, in the Black Forest, there is a turbine set up by 
 M. Fourneyron which has a fall of 108 metres. The expendi- 
 ture may be great even with quite small dimensions. In one 
 Of the examples given in No. 17, we have seen that a turbine 
 of m .60 external radius and m .09 in height, expended without 
 loss 54 cubic feet, or 400 gallons per second. Some turbines 
 
106 TURBINES. 
 
 are constructed wh/>se expenditure reaches as high as 141.2 
 cubic feet, or 1060 gallons, per second, and, if need be, more 
 could be expended. 
 
 Under ordinary circumstances, turbines move with sufficient 
 rapidity, and thus allow the gearing for transmission to be 
 economized. 
 
 For each turbine organized with fixed conditions, the theory 
 of No. 15 gives a certain velocity to be imparted to it, in order 
 to obtain the maximum useful effect. But if, in reality, a velo- 
 city different from this be given to it, and deviate from 25 per 
 cent, more or less, experience shows that the effective delivery 
 does not change much, a very important property for many 
 manufacturing purposes, in which, in spite of the variations 
 that take place in the expenditure of the head of water, it is 
 important to have the machine always move with a very nearly 
 constant velocity. 
 
 These motors have then a very decided advantage over 
 wheels with a horizontal axis. Unfortunately their propor- 
 tional effective delivery is not always constant, even approxi- 
 mately, for heads with a very variable expenditure ; their con- 
 struction and repairs can only be intrusted to very skilful me- 
 chanics, and consequently are quite expensive ; while, on the 
 other hand, overshot and breast wheels can be very economi- 
 cally constructed, and at the same time give an effective delivery 
 at least equal to, if not greater than, that of turbines^ These 
 wheels will frequently, on this account, be preferred when the 
 expenditure and head of water are favorable to their con- 
 struction. 
 
REACTION WHEELS. 
 
 21. Reaction wheels. — Let us conceive of one of Fourney- 
 ron's turbines deprived of its directing partitions, care being 
 taken to prolong the floats to a slight distance from the axis of 
 rotation ; let us suppose that the water comes to the floats 
 through a pipe concentric with the axle, having for a radius 
 precisely the free distance that we have just mentioned. Fur- 
 thermore, the expenditure of water in this pipe will be consid- 
 ered as very little, in order that the absolute velocity of the 
 liquid in it may not be sensible. We shall thus have the idea 
 of reaction wheels. 
 
 To give their theory, we will consider the point of entrance 
 of water into the wheel as being on the axis of rotation itself; 
 at this point the velocity of the wheel being nothing, as well 
 as the absolute velocity of the water, it will be the same for 
 the relative velocity of this last. In the calculations of ISTo. 
 15, we will have to make v — 0, u = 0, w = y equations (1), 
 (2), and (3) then become 
 
 = H + h + 2s~iP. 
 
 w" = 2g{h' + ^-^A + v/ 
 
 n 
 
 whence we find 
 
 P± + h + h > = Z. 
 
 a n 
 
 w"' = 2 g H + tt" 
 
108 EEACTION WHEELS. 
 
 .Neglecting friction, the only loss of head is as yet the height 
 
 v" ... 
 
 — - due to the absolute velocity of exit ; we calculate its value 
 
 by means of equation (6), which, combined with the preceding, 
 will give 
 
 v" = 2 u 1 ' + 2 g H — 2 u' cos y V 2 g H + u". 
 The effective delivery would then be 
 
 v 
 R ~2~a 1 v n u« «' 
 
 *"■ = Tt — * — n tt — ff + cos 7 —, ■ 
 
 H 2^H ff B. ' 4/^H 
 
 "^ 
 
 , . u' 
 or, placing — = x 
 
 VgH. 
 fx- = — a? + x cos yV 2 + x'. 
 "We can consider p as a function of a:, and seek its maximum 
 when x varies. To this end we will take off the radical sign 
 by writing 
 
 (^ + xj = » a cos 2 y (2 + a; 2 ), 
 or by reducing 
 
 a? 4 sin 2 y — 2 a; 2 (cos 2 7 — /x) + j*, 2 = 0. 
 If we deduced from this ar" in terms of p, the roots would 
 become, from their nature, real; consequently, we have the 
 condition 
 
 (cos 2 y — (x) 2 — |A a sin 2 7 > 0, 
 
 or developing and reducing 
 
 cos 4 y — 2 j/. cos 2 7 + (a 2 cos 2 7 > 0. 
 If we suppress the positive factor cos 2 7, we find 
 
 cos 2 7 — 2(* .+ fA 2 > 0, 
 or 
 
 (l-M-) 2 -sin 2 7>0, 
 
BEACTION WHEELS. 109 
 
 and, observing that 1 — f* is necessarily positive, as well as 
 sin r , 
 
 1 — (i > sin 7, 
 (* < 1 — sin y. 
 The limit of the effective delivery that we can reach is then 
 
 (a, = 1 — sin y. 
 The corresponding value for x is easily obtained by the relation 
 between x and p. ; if we take the equation deprived of radicals, 
 and make in it x = x 1 and n = 1 — sin y, it becomes 
 
 a;, 4 sin 2 y — 2 x' sin y (1 — sin y) + (1 — sin 7)* = 0, 
 or, more simply, by extracting the square root 
 
 x* sin 7 — (1 — sin 7) =: 0, 
 whence 
 
 xA 
 
 sin 7 
 
 Bin 7 
 
 If we supposed 7 = 0, we would find f*, = 1 ; but #„ and 
 consequently u', would become infinite. Strictly speaking, the 
 value 7 = might be realized ; it would only be necessary 
 that the channels that make 
 nn flip, whfifil shnnlrl he ar- ^srq _^\ 
 
110 BEACTION WHEELS. 
 
 not become infinite, nor \t., consequently, reach unity : we only 
 see that it would be necessary to make the wheel turn very 
 rapidly. Moreover, it is difficult to expend a large volume of 
 water without giving a considerable diameter to the central 
 pipe A 0, which would make the hypothesis, that u, v, and w 
 were equal to zero, inadmissible ; there are equally great diffi- 
 culties in regulating the expense according to the required cir- 
 cumstances. It is undoubtedly on these accounts that this kind 
 of wheel is but little used. 
 
 Preserving the arrangement of the floats of Fourneyron's 
 turbine, which gives a series of contiguous channels, we can no 
 longer make y = 0, and then the limit of the theoretical effec- 
 tive delivery decreases quite rapidly as y increases ; so that, for 
 7 = 15°, 1 — sin y would no longer be greater than 0.741. On 
 the other hand, as we offer the water a wider outlet, we should 
 perhaps lose less in friction, and the theoretical effective deliv- 
 ery would differ less from the real. 
 
PUMPS. 
 
 § IV. Of a few Machines foe Eaising Water. 
 
 22. Pumps. — The arrangement and shape of the parts of 
 pumps are of infinite variety, according to the notions of the 
 constructor. A special treatise would be necessary merely to 
 describe the principal kinds. We will suppose, then, that the 
 reader has seen a summary description of these machines, and 
 confine ourselves to general ideas. 
 
 (a.) Effort necessary to make the piston move. — Two cases 
 must be distinguished : pumps of single stroke, and pumps of 
 double stroke. In the former case, the piston only draws up 
 the water into the pump, or else only drives out the water pre- 
 viously drawn up, by forcing it up through the delivery pipe, 
 when it moves in a determined direction ; in the second, these 
 effects are produced simultaneously, whatever be the direction 
 of the stroke. Let. us first take the single stroke ; let 
 
 h be its height ; 
 
 o the section of the piston ; 
 
 p a the atmospheric pressure ; 
 
 n the weight of a cubic metre of water. 
 The side of the piston in contact with the column of water drawn 
 up would support, supposing that it remained in equilibria, a 
 pressure equal to n (p a — n h), whilst the other side, generally in 
 direct communication with the atmosphere, would sustain a 
 pressure in a contrary direction equal to p a Q ; the difference 
 
112 PUMPS. 
 
 n. ah would be the resultant pressure on the piston. If, on 
 the contrary, there be but a single stroke forcing to a height h', 
 we will find, in like manner, that the piston supports, exclu- 
 sively of its motion, a resultant pressure n a h'. Finally, if 
 the pump were one of double stroke, these two resultants would 
 be superposed, and the value for the total pressure would be 
 n.n (h + h') or nflH, H being the height included between 
 the level of the basin from which the water is drawn, and that 
 of the basin receiving it. In a certain class of single-stroke 
 pumps, the same superposition of resultant pressure on the two 
 faces of the piston takes place during the motion in one direc- 
 tion, and these resultants are in equilibrio when motion occurs 
 in the opposite direction : this is the case in the kind of pump 
 called the lifting-pump. If the piston have no horizontal 
 motion, we must, besides, consider its weight as well as that of 
 its rod, the component of which, parallel to the axis of the 
 pump, would be added to or subtracted from the preceding 
 expressions, according to circumstances. The friction of the 
 piston against the barrel of the pump must also be added, as 
 well as that of the rod against the packing-box, if there be any, 
 through which "it passes. 
 
 But these expressions only give the value of the force capable 
 of maintaining the piston, as well as the water drawn or forced 
 up, in equilibrio in a given position. When motion takes 
 place, the effort brought to bear on the piston may be very dif- 
 ferent from this force. In the first place, the water does not 
 begin to move in the pipes, and does not pass through the nar- 
 row openings of the valves without experiencing losses of head 
 which are to be added to the heights h and h'. For ex- 
 ample, in the case of the sucking pump, if there be a loss of 
 head £ in the length of the column drawn up, the pressure 
 a (p a — n h) will be reduced to a [p a — n (A + £) ), and the 
 
PUMPS. 113 
 
 resultant Tin h would become n n (A + £). In like manner, 
 if we consider a single forcing stroke, and that there is in the 
 entire column forced up a l.oss of head %', the piezometric level 
 in this c'olumn, at the point at which it touches the piston, 
 would be increased by £', and the resultant pressure would be 
 n n (h' + £'). If the pump be one of double stroke, the ex- 
 pression nnH should in like manner be replaced by n n (II 
 + I + D- Besides the heights £ and £', others must be added, 
 if the mass of water set in motion is not uniformly displaced. 
 Let P be the weight of the piston and its rod,y its acceleration, 
 P' the weight of the water set in motion and which fills the 
 pipes through which it is drawn up or forced out : the weight 
 P' being supposed to move with a mean acceleration f, we see 
 
 that an additional force ~(Pj + P' j') would be necessary to 
 
 overcome the inertia of the water and piston. This additional 
 force, at one time a pressure, at another a resistance, may pro- 
 duce considerable variations in the total force to be applied to 
 the piston, which is always inconvenient : since we must, in 
 the first place, determine the dimensions of the pieces, not ac- 
 cording to the mean, but according to the maximum effort sus- 
 tained, which generally produces a clumsy and expensive 
 machine; in the second, it is seldom that great variations in 
 resistance do not give rise indirectly to some loss of motive 
 
 Pi 
 power. "We diminish the value of the term — - by means of 
 
 weights acting as a counterpoise to the piston, if P is large ; f 
 
 P' j' 
 and consequently — — are also diminished, in case it is found 
 
 to be worth . the trouble, by means of an air-chamber, placed 
 at the entrance- to the ascent pipe, which makes the motion of 
 a great part of the weight of the column forced up uniform, 
 
114 
 
 PUMPS. 
 
 Fia. 19. 
 
 and consequently suppresses or diminishes to a very great 
 extent the corresponding force of inertia. 
 
 Another means of obtaining approximate uniformity of mo- 
 tion in the ascent pipe consists in making it answer for the 
 delivery of several pumps working together, in such a way 
 that their total delivery in a series of equal 
 times shall vary but little: it is accomplished 
 in this way. Let O (Fig. 19) be the axis of 
 rotation of an arbor that receives from a 
 motor a motion rendered regular by a fly- 
 wheel, and consequently nearly uniform. 
 This arbor carries two arms, OB, B', mak- 
 ing a right angle with each other; to each 
 of them is attached a connecting rod, fasten- 
 ed at its other end to a piston running be- 
 tween guides, and which belongs to a double-stroke pump. As 
 an example, we will suppose the arbor O horizontal, the piston 
 rods vertical and their prolongations intersecting the axis of 
 rotation ; the connecting rods will generally be of the same 
 length, about five or six times as long as the arm O B. It fol- 
 lows from this that the obliquity of the connecting rods with 
 the vertical being always small, the velocities v and v' of the 
 pistons are practically those of the projections of B and B' on 
 the vertical B B, ; calling w the angular velocity of the arbor, 
 b the length OT>,x the angle formed by B with the vertical 
 
 B B„ we will then have 
 
 (if 
 + x ) = w b cos x. 
 
 Let fi again be the common cross-section of the two pistons, 
 and 6 a very short interval of time; exclusively of the losses 
 by the play of the two mechanisms, the volume of water fur- 
 nished during the time 6 to a common ascent pipe, by the two 
 
PUMPS. 115 
 
 pumps together, will be the arithmetical sum of the volumes 
 generated by the two pistons, viz., n (v + v') 6 or n u h d (sin x 
 + cos x), a formula in which the sine and cosine should have 
 their absolute values given, since it is a. question of an arith- 
 metical sum, and therefore the velocities are essentially positive. 
 Consequently it is sufficient, in order to obtain the maximum, 
 the minimum, and the mean of the variable quantity sin x + 
 
 it 
 cos x, to suppose x included between o and -. Now we find be- 
 
 tween these limits 
 
 Two minima equal to 1 for x = o and x = » 
 
 One maximum equal to 1.414, for x = j 5 
 
 The mean value equal to J * (™ "+*»«)*? = * = 1.272. 
 
 P 
 
 J rfa; 
 
 There would thus be between the minimum and the mean a 
 
 272 
 relative difference of ^-^--, or about 0.214 ; whilst, with a sin- 
 
 gle pump, the elementary delivery, proportional to sin x, would 
 
 2 
 vary between and 1, and would have .- or 0.637 for a mean 
 
 value, which would produce a much greater relative difference 
 
 between the minimum and the mean. 
 
 We obtain a still more satisfactory result when we use three 
 
 arms making angles of 120° with eacli other. The elementary 
 
 delivery of the three pumps together is then proportional to 
 
 • / , 5J*\\ • / , 4*\ 
 sin x + sin ( x + — J + sin (x + — J, 
 
 each sine to be always taken as positive, whatever may be x. 
 8 ' " 
 
116 POMPS. 
 
 We readily see, moreover, that the arithmetical sum of the 
 
 three sines will not change by increasing the arc by 60 degrees, 
 
 tff 
 so that it is sufficient to make x vary between and -. "Within 
 
 J 3 
 
 these limits, the first two sines are positive and the third nega- 
 tive ; hence, the sum of the absolute values will be expressed 
 
 by 
 
 sin x + sin (w + — ) — sin (x +-^\ 
 
 2* 4* 
 or else, by developing and observing that the arcs — and — 
 
 together make an entire circumference, 
 or finally 
 
 sin x + 2 cos x sin --- 
 
 sin x + V 3 cos x. 
 
 The minima of this quantity correspond to x = and x = Q , 
 
 o 
 
 and have for their value V 3 or 1.732 ; the maximum, corre- 
 sponding to x = -, is - + - V 3. V 3, that is 2 ; the mean 
 
 f 3 v~ 
 
 ' — — '- becomes- or 1.910. The relative 
 
 or TT 
 
 J „ dx 
 
 difference between the minimum and the mean is consequently 
 
 \ 910 j Y32 
 
 lowered to - — ^-j-r- — - or to about 0.093. 
 1.910 
 
 There is also a great de"al of regularity in the elementary 
 delivery of the three pumps united as above, when they are 
 supposed to be of a single stroke. Let us admit, for example, 
 
PUMPS. 117 
 
 that each piston only forces up water when its crank B de- 
 scends from B to B s ; the sum of the elementary deliveries will 
 still be proportional to the expression 
 
 sin x + sinf x + — J + sin [x + — J ; 
 
 hut the pumps being only of single stroke, instead of changing 
 the sign of the negative sines, we must suppress them altogether. 
 
 This granted, let us first increase x from to 5 : x and x 4- 
 
 o 
 
 — will be less than the semi-circumference, and x + — will be 
 o 6 
 
 included between <s and 2 it. "We shall then only have to keep 
 
 x + ^- J, which can 
 be put under the form 
 
 2 sin fa? -^ cos -, or sin fx + j, 
 
 since cos 5 = -; this sum, equal to sin ^, or 0.866 for a; = 0, 
 
 becomes a maximum and equal to 1 for x = ^, then it decreases 
 
 to 0.866 when x passes from ^ to -. In the second place, if we 
 
 take the values of x between - and — , the sines of x + — and 
 
 6 6 6 
 
 x -1 are both negative, so that the sin a? alone can be pre- 
 served, which has still a minimum value 0.866, corresponding 
 to two limits, and 1 for a maximum equidistant from these 
 limits. Moreover, it is useless, to consider the values of x 
 
 greater than — , for a rotation of 120 degrees not producing 
 o 
 
 any change of figure in the combination of the mechanism, we 
 
118 PUMPS. 
 
 would again find the same sines. "We see, then, that the ele- 
 mentary delivery of the three pumps working together varies 
 as certain numbers which are always comprised between 0.866 
 and 1, and consequently that it is sufficiently regular : the 
 minimum and maximum are respectively half of what they 
 were in the case of three pumps of double stroke. 
 
 We have heretofore supposed that the two cranks at right 
 angles to each other, or the three cranks following each other 
 at distances of 120 degrees, are fixed to the same arbor ; it is 
 plain that they can be attached to different arbors, provided 
 that they all have the same angular velocity, the cranks being 
 of equal length ; or, more generally, provided that the centre 
 of the joint of each of them with the corresponding connecting 
 rod, in the three systems, has the same velocity of rotation 
 about its arbor. 
 
 It is always well, as has been already said, to avoid great 
 variations in the force that is to be transmitted to the piston of 
 a pump ; this becomes almost indispensable when it is moved 
 by means of horses. An essential condition for employing the 
 work of horses to advantage is that their pace and the exertion 
 they have to make shall be as steady as possible ; it would be 
 difficult to accomplish this with one single stroke pump forcinc 
 up a long column of water, the piston of which should receive 
 its motion from the arbor of the gearing-wheel by a system of 
 a crank and connecting rod, and we could only succeed by em- 
 ploying fly-wheels, more or less clumsy. It would generally 
 be better to regulate the resistance by means now to be men- 
 tioned. 
 
 (b) Work to be transmitted to the piston. — If we knew exact- 
 ly, in each position of the piston, the force to be applied to give 
 it motion, it would be easy to deduce the amount of work to 
 be transmitted to it. But this force cannot be exactly deter- 
 
PUMPS. 119 
 
 mined; thus, the- determination of friction of the piston against 
 the pump barrel, or against the packing that it passes through 
 (if it be a plunger), is necessarily uncertain, because it depends 
 on the skill of the constructor ; we can say the same for the 
 losses of head encountered in the sucking and forcing pipes, on 
 account of the defect of permanence and uniformity in the 
 motion. However, when the forcing pipe is very long, we 
 have seen that it is well so to arrange matters that the motion 
 in it shall be uniform, and then we can calculate with sufficient 
 exactness the entire head £" between the two extremities of 
 this pipe. This granted, let us admit first, that we are consid- 
 ering a double-stroke pump : the resultant pressure exerted on 
 the piston being expressed by n ft (H + £ + £'), the force to 
 be transmitted to it will be represented by n ft (H + £") + F. 
 We will include in the additive term F the friction against the 
 barrel of the pump and the packing, the excess of £ + ?' over 
 £", and lastly inertia. The total work of this force, in a linear 
 distance I, divided into elements d x, will be 
 
 nnZ(H + n +f l Fdx. 
 
 
 
 Moreover, ft I represents very nearly the volume of water raised 
 by one stroke of the piston ; if then we wish the work expend- 
 ed in raising each cubic foot of water to the height H, we will 
 have to calculate the quantity 
 
 (^ + K")+-^ T f l Fdx. 
 
 In practice, on account of the difficulty of determining exactly 
 the integral/ Fdx, we simply multiply the term n (H + I") 
 
 D 
 
 by a co-efficient n such as 1.10 or 1.15 or 1.20, according to 
 the greater or less perfectness of the machine. 
 
 If the pump were one of single stroke, we would obtain the 
 
120 pumps. 
 
 same expression for the work, by adding together two consecu- 
 tive strokes of the piston. 
 
 When we wish to determine the amount of horse-power to 
 be transmitted to the piston, we must also know the mean velo- 
 city u of the piston. We easily deduce the mean amount 
 pumped up per second 0. u, if the pump is one of double stroke, 
 
 or — — for a single-stroke pump ; we multiply this amount by 
 2i 
 
 n n (H + £") ; dividing finally by 550, we have the horse-power 
 
 sought. 
 
 (c) Mean velocity of the piston ; delivery of pumps.— The 
 mean velocity of the piston ought not to be very small, because, 
 in order to pump up any considerable quantity of water, we 
 would have to make the body of the pump and the lifting pipe 
 very large, which would increase the cost of construction. But 
 too great a velocity possesses also great inconveniences: firstly, 
 we increase the losses of head in a very rapid proportion ; then 
 it may happen that the water furnished by the sucking pipe 
 may not come up sufficiently fast to follow the piston, and that 
 the pump barrel may not be filled at each stroke, which would 
 cause a loss in the delivery, and a shock on the return of the 
 p'iston in an opposite direction. On account of the difficulty 
 of determining exactly the velocity of the water sucked up, we 
 ordinarily adopt a mean velocity of the piston, about ft .66 per 
 second ; we rarely go so high as 1 foot. It is clear that the 
 limit may be increased in proportion as the piston moves to a 
 less height above the basin from which the water is drawn, and 
 the more care that has been taken to avoid losses of head in 
 the sucking pipe. 
 
 The piston having a stroke the length of which is I and the 
 cross section R, describes, during one of the periods employed 
 in raising water to the upper basin, a volume ii I; this volume 
 
PUMPS. 121 
 
 would also be that of the water raised during the same time, 
 if there were no leakage around the valves between the piston 
 and the hollow cylinder in which it moves. On this account, 
 the volume raised varies from 0.75 n I to n I ; the co-efficient 
 by which the volume described by the piston is to be affected 
 varies with the care displayed in the construction and in keep- 
 ing the pump in good order ; under ordinary circumstances, we 
 may take it from 0.90 to 0.92. 
 
SPIRAL NORIA. 
 
 23. Spiral Noria. — This wheel consists essentially of a hori- 
 zontal arbor (Fig. 20), to which are fastened a certain num- 
 
 Fis. 20. 
 
 ber of cylindrical surfaces having their generatrices parallel to 
 the axis; the right sections of these cylinders are involutes of 
 a circle. The space between two consecutive cylinders thus 
 forms a canal with a constant breadth, as well in the direction 
 of the normal as perpendicularly to the plane of the figure. 
 One of the canals, for example, will have its outer opening at 
 A B and the other at I G. The entire system turns around 
 the axis O, in the direction of the arrow-head ; the centre O is 
 above the basin whence the water is drawn, and the level of 
 this basin covers the lower part of the wheel more or less. 
 During the time that the opening A B is wholly or partially 
 
SPIRAL NOEIA. 123 
 
 below the surface level of the basin, a certain quantity of water 
 enters the canal A B I Gr, by the effect of the rotation ; the ro- 
 tation continuing. A B rises until finally it comes directly over 
 
 1 G ; then the water taken in flows into I G- through holes left 
 open all around the arbor, and falls into a canal which leads it 
 off to the basin that is to receive it. 
 
 We will propose two questions : 1st, a given spiral noria 
 turns with a known angular velocity, and occupies a given po- 
 sition with respect to the lower basin ; what will be the amount 
 raised per second ? 2d, what will be the work that the motor 
 will have to transmit to it ? 
 
 Let us call S the section A B projected on the plane passing 
 through the axis O and the centre of A B ; !N the number of 
 revolutions of the wheel per minute ; n the number of invo- 
 lutes ; H the height O C of the point O above the level of the 
 water to be raised; r', r" the distances of the points A and B 
 from the axis of rotation. The point A will describe under- 
 neath the water an arc DAD', of which we will designate the 
 angle at the centre by 2 a. ; in like manner the point B will de- 
 scribe the arc E B E', corresponding to the angle at the centre 
 
 2 /3. First we shall have 
 
 TT TT 
 
 COSa= — C0S/3= — , 
 
 arc D A D' = 2 r' a. = 2 r' cos ~ - r , 
 
 r 
 
 TI 
 
 arc E B E' = 2 r" H = 2r" cos ^; 
 the arc described below the water by the centre of A B differ- 
 ing little fr 
 pressed by 
 
 ing little from the mean - (D A D' + E B E') will then be ex- 
 
 _1 H , „ ~'R T 
 r cos -7 + r cos -— = li. 
 r r 
 
124 SPIEAL NORIA. 
 
 Now, the volume that entered at the opening A B is E D E' D', 
 or the prodnet of this arc by the perpendicular section S ; then, 
 since there are n canals that raise the same volume in each 
 revolution of the wheel, the volume raised will be, per revolu- 
 tion, ?iLS; finally, the number of revolutions per second being 
 
 N 
 
 — , the amount Q raised by the wheel in the same time will be 
 
 60 J 
 
 expressed by — N n L S, that is, we ought to have 
 
 Q = lNnS(/cos "** + r"cos - 1 -?). 
 
 But this calculation supposes that there enters, during each 
 element of time d t through the opening A B, a volume of 
 water equal to that generated by A B in the same time ; in 
 this way the contraction which the liquid may experience on 
 entering is not considered, nor is the motion communicated to 
 the surrounding water, which, up to a certain point, may give 
 way before the surface A B, instead of crossing it. For these 
 reasons it would be well in practice to admit a certain reduc- 
 tion in the value of Q above given ; we could effect it, for ex- 
 ample, by'a co-efficient which we will value, at a rough esti- 
 mate, at 0.80, for want of exact experiments on this subject. 
 
 Here is an example for calculating Q. Let N = 12, n = 4, 
 r' = 2 m .50, r" = 3 m .O, H = 2 m .O, S = O 89 - m .17. We shall have 
 
 5 = 0.8000, cos !? = 0.410 - ; 
 r' ' r' 2 ' 
 
 5 = 0.6667, cos ~*-5 = 0.535 — : 
 r" r" 2 
 
 r' cos -^ = r" cos *5 = J (1-025 + 1.605) = 4.131 ; 
 
 whence we deduce 
 
 Q = mo .562, 
 
SPIEAL NOKIA. 125 
 
 a number which would be reduced to mo .45 about, by multi- 
 plying by 0.80. 
 
 As to the motive work to be expended in raising a certain 
 weight P of water, it is composed : 1st, of the work P II des- 
 tined to overcome that of the weight ; 2d, of the work of fric- 
 tion on the trnnions and shoulders of the' arbor O, which can 
 be determined by means of known formulae ; 3d, of the work 
 necessary to overcome the friction of the water against the 
 solid walls with which it is in contact; this work being very 
 slight, if the involutes form tolerably large channels ; 4th, the 
 work necessary to give to the water the absolute velocity with 
 which it leaves the wheel. This last work will also be very 
 slight, if we take care to make the wheel turn slowly ; for the 
 lowest point of any involute whatever being always on the 
 vertical through 1, we see that the water that has already en- 
 tered the interior of the canal A B I G, and that which will 
 still enter in the course of the same revolution, will only be 
 completely emptied out after an entire revolution, reckoning 
 from the position indicated by the figure. The water rises 
 then with little absolute velocity into the machine, and conse- 
 quently a small portion of the motive work is employed in 
 giving to it an unproductive living force. But it must not be 
 forgotten that this supposes slowness of revolution around the 
 axis O. 
 
 To sum up, we will calculate the first two portions of the. 
 motive work, which are the most important, and in order to 
 account approximately for the other two, we will multiply the 
 sum of the calculated portions by a co-efficient a little greater 
 than unity. 
 
 The first idea of the noria is very old, since Yitruvius speaks 
 of a similar machine ; it was Lafaye who, in 1717, proposed 
 giving it the form we have described above. This machine 
 
126 SPIRAL NOEIA. 
 
 seems susceptible of a very good delivery, and is well adapted 
 to raising large volumes of water; but the height to which the 
 water is raised, always less than the radius of the wheel, is 
 necessarily very limited ; besides, this wheel is heavy, and on 
 this account hard to transport. 
 
CENTRIFUGAL PUMP. 
 
 24. Lifting turbines ; centrifugal pump. — The greater part 
 of the machines which ar.e used to turn the motive power of a 
 head of water to account, can, with a few changes, be converted 
 into machines for raising water, and the reverse. Thus, for 
 example, if a breast-wheel, set in a water-course, receives a 
 motion about its horizontal axis, by the action of any motor, so 
 that the floats may ascend the circular flume, these floats will 
 carry up with them the water from the tail race and throw it 
 into the head race: we would then obtain, in principle, the 
 lifting wheel. In like manner, let us take one of Fourneyron'a 
 turbines, and make the intervals between the directing parti- 
 tions communicate directly with the tail race, and let the outer 
 orifices of the turbine open into a compartment from which the 
 ascent pipe leads ; when a motion of rotation is impressed on 
 the wheel, the water contained in the floats will be urged to- 
 ward the exterior by the centrifugal force, and will reach the 
 enclosed comjjartment with an excess of pressure which will 
 cause it to ascend the pipe to a certain height, the greater as 
 the rotation becomes more rapid. If the pipe is not too high, 
 a delivery of water will take place at its end ; and this, more- 
 over, will be continuous, the water thrown out by the centrifu- 
 gal force being incessantly replaced by that from the tail race, 
 which tends to fill up the empty space between the partitions. 
 
 The theory of such a turbine, which we might call a lifting 
 turbine, resembles very closely that- of (X<>. 15). But as the 
 
128 
 
 CENTRIFUGAL PTJMP. 
 
 machine there discussed has not as yet been set up or experi- 
 mented upon, it need no longer be dwelt upon. We will pass 
 to the study of a pump called the centrifugal pump, which 
 belongs to the same class of machines, but which bears, how- 
 ever, a greater resemblance to reaction wheels. 
 
 A wheel composed of a series of cylindrical floats, such as 
 B (Fig. 21), assembled between two annular plates, is caused 
 
 Fig. 21. 
 
 to turn around a horizontal axis projected at A. The water, 
 from the basin to be emptied, comes freely within the circle 
 A B, which limits the floats on the inside, either because the 
 centre A is a little below the level 1ST N of this bay, or by 
 means of suction pipes. The motion of rotation impressed r>n 
 this wheel drives the water from the canals B C, B' C, into the 
 annular space D, where it acquires a pressure sufficient to drive it 
 up the pipe E, the only means of escape open to it, and by which 
 it reaches the upper basin. The angular velocity of the arbor 
 A being known, as well as all the dimensions of the machine, 
 and its position relatively to the basins of departure and arrival, 
 
CENTRIFUGAL PUMP. 129 
 
 we can find the amount of water pumped up per second, the 
 motive work that it requires, and its delivery. 
 
 To show this, let us call 
 
 H the difference of level between the two basins ; 
 
 h the depth of the centre A below tHe level of the lower; 
 
 r the exterior radius A C of the wheel ; 
 
 i the distance apart of the two annular plates, which confine 
 the floats ; 
 
 w the angular velocity of the arbor A ; 
 
 v the absolute velocity of the water when it leaves the floats ; 
 
 u the velocity w r at the outer circumference of the wheel ; 
 
 w the relative velocity of the water at the same point ; 
 
 y the acute angle formed by the velocities w and u — that is, 
 the angle at which the floats out the outer circumference ; 
 
 p the pressure of the water at its point of entrance into the 
 interval between the floats ; 
 
 p' its pressure at the point of exit ; 
 
 p a the atmospheric pressure ; 
 
 n the weight of the cubic foot of water. 
 
 We will begin by simplifying the question a little by means 
 of a few hypotheses. First, we will neglect the absolute velo- 
 city of the water in the ascent pipe and in the conduit which 
 conveys it to the floats, which may be allowed if the cross sec- 
 tions of these conduits are sufficiently large relatively to the 
 volume pumped out. The radius A B, however, should still 
 be sufficiently small so that the velocity of rotation of the point 
 B may be neglected ; in other words, we will consider the in- 
 troduction of the water into the wheel as taking place along 
 the axis, without any velocity occasioned by the motion, and 
 consequently without any relative velocity. Secondly, we will 
 corjduct our argument as though the water were displaced hori- 
 zontally in its passage across the wheel ; the height of this last 
 
130 CENTRIFUGAL PUMP. 
 
 will be supposed slight relatively to H, so that it can be left 
 out of account. Besides, nothing in practice would prevent 
 our assuming the arbor A as vertical ; but this would be a 
 matter of very little importance in the result. 
 
 This granted, the pressure varying according to the hydro- 
 static law from the lower basin to the point of entrance, and 
 from the point of exit to the other basin, we will have 
 p=p a + nh, 
 
 whence, by subtraction, 
 
 F—T — 
 
 H. 
 
 n 
 
 Now, if we apply Bernoulli's theorem to the relative motion 
 of a molecule following the curve B C, the fictitious gain of 
 
 2 2 2 
 
 head will be expressed by -r — or — — , and we shall find 
 ■' 2 g 2 g 
 
 v? _ p ~~p' u 2 
 
 2 g " n 2 g 1 
 
 or else 
 
 v? = - 2 g H + u\ . . . (1) 
 an equation giving w since u is known. This first result gives 
 the means of calculating the amount Q pumped up in each 
 second. In fact, the water leaving the floats cuts a cylindrical 
 surface 2 * i r at an angle y and with the relative velocity w ; 
 then the total orifice of exit, measured perpendicularly to w, is 
 2 * i r sin y, and consequently 
 
 Q = 2 * b r w sin y. . . . (2) 
 The motive work consumed per second in making the wheel 
 turn includes first the work nQH; then the water reaching 
 the annular space D with a velocity v, this is lost in useless 
 
 disturbance ; whence there results a molecular work n Q V 
 
 2 9' 
 
CENTRIFUGAL PUMP. 131 
 
 Thus then, throwing out of account the other frictions, the 
 
 work expended per second will be n Q (H + —-\ ; and as the 
 
 2 gJ 
 
 useful work is only n Q H, the effective delivery f* will have for 
 
 its value 
 
 " = — —^=— ^r- ( 3 ) 
 
 2 g 2^H 
 
 There remains to determine v ; now v is the resultant of w and 
 u, hence we have 
 
 v* = u* + w* + 2 uw cos 7, 
 or from eq. (1) 
 
 v° = — 2 ^ H + 2 u* — 2u cos 7 V^~2g H + i? (4) 
 
 Equations (1), (2), (3), and (4) give the means of solving with- 
 out difficulty the questions proposed. 
 
 Let us again see by what means we could obtain the greatest 
 possible result of the motive power. Expression (3) for the 
 effective delivery becomes, substituting for v its value, and 
 u 
 
 making . 
 
 V'gR 
 
 x — x cos 7 \/ x * _ 2 
 we shall then have the maximum of i^, considered as a function 
 of x, in seeking the minimum of the denominator, or, what 
 
 amounts to the same, the minimum of -. We shall conduct 
 
 this research as in (No. 21) ; we will write 
 
 1 . 
 
 x 1 — - = x cos 7 V x* — 2 
 V- 
 
 or, by making the radical disappear and transposing, 
 
 x l sin" 7 — 2 as" f cos 2 7 J -I — ; = 0. 
 
 9 
 
132 CENTRIFUGAL PUMP. 
 
 Now (a can only receive values that, substituted in this biquad- 
 ratic equation, will give ar" real and positive ; hence we have 
 
 ( cos 2 y) 3 sin" 7 > 0, 
 
 or successively 
 
 1 2 
 — ^ cos" y cos" v + cos 4 y > 0, 
 
 4-- + cos' 7 >0, 
 
 (- - 1 y > sin 2 y. 
 
 As sin y and 1 are positive quantities, we can extract the 
 
 square root of both members and place 
 
 1 , ^ • 1^ -, 
 
 — 1 > sin r, or - > 1 + sm y • 
 
 f* ¥■ 
 
 the minimum of - has then for its value 1 + sin y, and the 
 
 limit of the effective delivery (a, will be ; — . The corre- 
 
 J ' 1 + sin 7 
 
 sponding value s», of x is obtained from the above biquadratic 
 
 equation, which gives 
 
 1 
 
 - — cos y . . 
 
 „ _(*, ' _ 1 f sin y - cos y _ 1 + sm y 
 
 sin 2 y sin 2 7 sin 7 
 
 Thus the most favorable velocity u for the effective delivery is 
 
 .obtained from this equation 
 
 a -□- 1 + sin 7 
 
 u = a H — '- ; 
 
 sin 7 
 
 -whence we get the angular velocity u = -, and the number of 
 
 r 
 
 revolutions per minute N = . The effective delivery being 
 
CENTRIFUGAL PUMP. 133 
 
 then r — we may be tempted, in order to increase it, to 
 
 1 + sin/ ' 
 
 make y very small ; but we see that the velocities u and w 
 would become very great, and we should thus lose a great deal 
 in the friction of the water against the floats. Besides, we 
 have, from equations (1) and (2), 
 
 Q = 2 * h r sin y Y u' — 2gK = 2*hr Y gH. sin y. Y as' — 2~; 
 the amount Q', which corresponds to the maximum effective 
 delivery, will then be 
 
 Q' = 2 * h r YjR. sin 7 .i/l±™±Z— 2 
 
 v sin y 
 
 = 2*br Y gK Y sin y (1 — sin y). 
 This amount reduces to zero at the same time as y ; when 
 7 alone varies, Q' becomes a maximum for sin y = 1 — sin y, or 
 
 sin 7 = - or y = 30° ; the effective delivery is then - 
 
 1 + 2 
 
 2 
 that is -. The value y = o is consequently inadmissible as re- 
 
 O 
 
 ducing the amount to zero ; but from this point of view it is 
 not well to go beyond y = 30\ On the other hand, this last 
 value does not give a very high theoretical effective delivery; 
 perhaps the best thing to do in practice would be to take y be- 
 tween 15 and 20 degrees. For y = 15°, for example, the effec- 
 tive delivery increases to ^-^rr^ = 0.794, and the product 
 
 J , 1 + 0.2588 ' ^ 
 
 Y sin 7 (1 — sin y) is decreased to 0.438, whereas it is. 0.50 for 
 y = 30° ; it is a diminution that could be compensated for by 
 a slight increase of r or i. 
 
 Instead of arranging the wheel as represented in Fig. 21, we 
 might adopt, two separate canals, as in Fig. 19. In this case 
 the expression for the amount would change, and the angle y 
 
134 CENTRIFUGAL PUMP. 
 
 might become zero; but, whereas u cannot increase to infinity, 
 
 the value — would no longer be admissible for — — , and 
 
 sin 7 & g R 
 
 we should have to depart more or less from the limit of the 
 
 effective delivery. Furthermore, for an equal expenditure, we 
 
 should probably lose more in friction. 
 
 AUTHORITIES OF WATER WHEELS. 
 
 Experiences sur les Roues Hydrauliques a auhes planes, et sur les Roues 
 Hydrauliques d augets, by Morin. 
 
 Experiments made by the Committee of the Franklin Institute on Water 
 Wheels, in 1829-30. See Journal of the Franklin Institute, 3d, Series, Vol. 
 I, pp. 149, 154, &c, and Vol. II., p. 2. 
 
 Experiments on Water Wheels, by Elvvood Morris. See Jour. Frank. 
 Ins., 3d Series, Vol. IV., p. 222. 
 
 Memoire sur les Roues Hydrauliques a auhes courhes, rnues par dessous, by 
 Ponoelet. 
 
 Experiences sur les Roues Hydrauliques a axe vertical, appelces Turbines, by 
 Morin. 
 
 Experiments on the Turbines of Fourneyron. by Elwood Morris. See 
 Jour. Frank. Ins., Dec. 1843, and Jour. Frank. Ins., 3d Series, Vol. IV., p. 
 303. 
 
 Lowell Hydraulic Experiments, 2d Ed., 1868, by Jas. B. Francis. 
 
APPENDIX. 
 
 Comparative Table of French and United States Measure*. 
 
 Pounds avoirdupois in a kilogramme 2.2 
 
 Inch in a millimetre 0.039 
 
 Inch in a centimetre 0.393 
 
 Inch in a decimetre 3.937 
 
 Feet in a metre 3.280 
 
 Yard in a metre. . 1.093 
 
 Square feet in a square metre 10.7643 
 
 Cubic inch in a cubic centimetre 0.061 
 
 Cubic feet in a cubic metre 35.316 
 
 Quart in a litre 1.0567 
 
 Note. — A cubic metre of distilled water weighs one thousand 
 kilogrammes. 
 
 The litre contains one cubic decimetre of distilled water, and 
 weighs one kilogramme. 
 
 The horse-power of the French is 75 kilogram metres, equiva- 
 lent to 542J foot-pounds per second ; the English horse-power 
 being 550 foot-pounds per second. The value of g is 9.81 
 metres; and that of the height of a column of water equivalent 
 to the atmospheric pressure is taken at 10.33 metres. 
 
 Note A. Art. 1. 
 The formula given in this article is to be found, with the 
 exception of variations in the notation, in all works of applied 
 
136 APPENDIX. 
 
 mechanics in which the subject of the theory of machines is 
 discussed (see, for example, Moseley's Engineering and Archi- 
 tecture, Am. Ed., p. 146). The only term in it which is not 
 generally found in other works is the one (H — H„) ^mg, which 
 expresses the work expended in overcoming the weight of any 
 part of the machine, when its centre of gravity is raised from 
 one level to another, represented by the vertical height (H — 
 H ), and the corresponding work by (H — H ) 2mg ; as, for 
 example,, in the case of a wheel revolving on a horizontal axle, 
 the axis of which does not coincide with its centre of gravity ; 
 or in that of a revolving crank ; in both of which cases the 
 work expended will be equal to the product of the weight 
 2 mg raised, and the vertical height (II — H„) passed over by 
 its centre of gravity. But, in all like cases, as in the descent of 
 the centre of gravity from its highest to. its lowest position, the 
 same amount of work will be restored by the action of gravity, 
 the total work expended will be zero for each revolution, and 
 the term (H — H ) 2 rn g will disappear from the formula. 
 
 Note B. Art. 2. 
 
 1 U ! — IP 
 
 The equation H (t 6 + tj) °- = o is the modified 
 
 g K 2g 
 
 form of what is known as Bernouilli's theorem as applied to 
 
 the case treated of in Art. 2. 
 
 This theorem, applied to the phenomena of the flow of a 
 
 heavy homogeneous fluid, may be generally thus stated : The 
 
 increase of height due to the velocity is equal to the difference 
 
 between the effective head and the loss of head. 
 
 In this case H is the effective head ; (t e + t f ) is the loss 
 
 9 
 of head from the dynamical effect t e imparted to the wheel by 
 
APPENDIX. 137 
 
 the action of the water, and the work t f due to the various sec- 
 IP — TJ a 
 ondary resistances : and the term - the height due to 
 
 the velocity. 
 
 Multiplying each member of the above equation by g, we 
 obtain 
 
 gK-t e -t f =\w-\w a , 
 
 or, in other words, the modified expression of the general for- 
 mula Note A, as applied to this case. 
 
 See Bresse. Mecanique Ajypliquee. Vol. 2, Nos. 12 and 
 15, pp. 23 and 30. 
 
 Note 0. Art. 4. 
 In the equation 
 
 9 A 
 
 which expresses the force applied horizontally at the centre of 
 
 the submerged portion of the bucket, the second term of the 
 
 second member - n I (h™ — A 2 ), represents the diminution of 
 
 the force imparted to the wheel by the current, arising from 
 the increase of depth of the water as it leaves the wheel, or by 
 the back water ; or, in other words, the difference of level be- 
 tween the point C, before the depth of the current is affected 
 by the action of the wheel, and the point E, where the depth 
 of the current has increased from the back water. This differ- 
 ence of level receives the name of a surface fall. 
 
 The relations existing between the two terms of the second 
 member of the equation, leaving out of consideration the action 
 of the wheel, may be established in the following manner. 
 
 Considering the portion, of the current comprised between the 
 
138 APPENDIX. 
 
 two sections C B, E F (Fig. 3), at a short distance apart, be- 
 tween wliich the surface fall takes place, we can apply to the 
 liquid system CBEF, comprised between these sections, the 
 theorem of the quantities of motion projected on the axis of 
 the current, which, in the present case, may be regarded as 
 horizontal. Now, during a very short interval of time 9, the 
 system CBEF will have changed its position to C B' E' F', 
 and, in virtue of the supposed permanency of the motion, each 
 
 EE 
 
 c r ■ /rr - / ■'"' 7 ' 1 "- \~^\^" — 
 
 
 
 — ; 
 
 
 
 =4=p^--^^=— — / / — — H^ — \ — 
 
 -f 
 
 B V ' y "" 'cT'^^»W^^ H F F' 
 Fig. 3. 
 
 point of the intermediate portion C'B'EF will have equal 
 masses moving with the same velocity at the beginning and 
 ending of the time 6 ; the variation in the projected quantity of 
 motion of the system CBEF, during the time 8, will therefore 
 be equal to the quantity of motion of the portion included be- 
 tween the final sections E F, E' F', and that comprised between 
 the initial sections CB, C B'. 
 
 To find these quantities of motion. Represent by u' a super- 
 ficial element of the section E F, and by v' the velocity of the 
 fluid thread wliich flows through it ; v' 6 will then be the length 
 of this thread for the time 6, between the sections EF and 
 E' F' ; and w' v' 6 will be the volume of the thread which has 
 w' for its base and v' 6 for its length. Representing by n the 
 
 weight of a cubic metre of the liquid, — w' v' 6 will be the cor- 
 responding mass of this volume, and — w' v" its quantity of 
 motion ; and, designating by s the sum of all the elements u', 
 
APPENDIX. 139 
 
 n 
 
 — 2 «' v" 6 will be the quantity of motion of the portion 
 
 E F E' F' of the liquid comprised between the two final sec- 
 tions. In like manner, v being the velocity with which each 
 thread flows through an element u of the section B 0, the quan- 
 tity of motion of the portion of liquid between the sections BC 
 
 and B' C will be expressed by — 2 u v* 6. The increase, there- 
 fore, in the quantity of motion during the time i will be ex- 
 pressed by 
 
 H. (s u' v " 6 — 2 u v* 6). 
 
 9 
 But as v and v' may be assumed as sensibly equal to the mean 
 velocities of the current in sections. E F, CB, then Zu'v' and 
 2 u v will be the volumes corresponding to these velocities; and 
 
 — s u' v' and — 2 u v the corresponding masses. But since. 
 9 9 
 
 from the permanency of the motion u' v' = " v, the expression 
 for the increase of the quantity of motion, for the time 8 will 
 therefore take the form 
 
 g («'-*)> 
 
 in which P represents the weight of the water expended in each 
 
 P . 
 
 second, and — its corresponding mass. 
 
 The expression here found, is equal to the sum of the impul- 
 sions of the forces exterior to the liquid system considered 
 during the time 6, also projected on the horizontal axis of the 
 current. From the form given to the section of the race, which 
 is rectangular, the direction of the axis of the current, which is 
 assumed as horizontal, between the extreme sections, and the 
 short distance between these sections, the only impulsions of the 
 pressures upon the liquid system are those on the sections E F 
 
14:0 . APPENDIX. 
 
 and CB. Representing, then, by b the breadth of the sections, 
 by h' and h their respective depths, their respective areas will 
 be expressed by b h! and b h ; the .pressures on these areas will 
 
 be n b hi x - h' and n b h x - A ; and for the respective pro- 
 
 2 2 
 
 jected sums of the impulsions of. these pressures, during the 
 
 time 6, we shall have -Jib 6 h" and - n J A', since from the 
 2 2 
 
 circumstances of the motion the pressures follow the hydro- 
 static law. The impulsion in the direction of the motion will 
 
 therefore be expressed by - n b 6 (A 2 — A' 2 ), from which we ob- 
 tain 
 
 — t(v'-v) = -n bo(k 2 - A' 2 ). 
 
 g K ' 2 v ; 
 
 to express the relation in question. 
 
 See Bresse. Mecanique Ajppliquee. Vol. 2, No. 83, p. 245 
 
 Note J). Art. 9. 
 
 The term Ofl — ■— J — c, which expresses the arc inter- 
 
 cepted between two buckets, taken at the middle point of their 
 depth, is obtained as follows : 
 
 R being the exterior radius of the wheel, corresponding to 
 the arc C, the radius of the arc at the middle point of the 
 
 bucket will be R — - p ; calling x the arc corresponding to 
 
 this radius, we have 
 
 R:R--V-:C:,. ;. x = 0(l-J£; 
 
 and for the arc intercepted between the two buckets at their 
 middle point the expression above. 
 
APPENDIX. 141 
 
 Note E. 
 
 By the courteous permission of James B. Francis, Esq., granted through 
 Gen. John C. Palfrey, the following extracts were taken from the val- 
 uable work of Mr. Francis, under the title of " Lowell Hydraulic Experi- 
 ments." 
 
 A vast amount of ingenuity has been expended by intel- 
 ligent millwrights on turbines ; and it was said, several years 
 since, that not less than three hundred patents relating to them 
 had been granted by the United States Government. They 
 continue, perhaps, as much as ever to be the subject of almost 
 innumerable modifications. Within a few years there has been 
 a manifest improvement in them, and there are now several 
 varieties in use, in which the wheels themselves are of simple 
 forms, and of single pieces of cast iron, giving a useful effect 
 approaching sixty per cent, of the power expended. 
 
 In the journal of the Franklin Institute, Mr. Morris also 
 published an account of a series of experiments, by himself, on 
 two turbines constructed from his own designs, and then ope- 
 rating in the neighborhood of Philadelphia. 
 
 The experiments on one of these wheels indicate a useful 
 effect of seventy-five per cent, of the power expended, a result 
 as good as that claimed for the practical effect of the best over- 
 shot wheels, which had heretofore in this country been con- 
 sidered unapproachable in their economical use of water. 
 
 In the year 1844, Uriah A. Boyden, Esq., an eminent 
 hydraulic engineer of Massachusetts, designed a turbine of 
 about seventy-five horse power, for the picking-house of the 
 Appleton Company's cotton-mills, at Lowell, in Massachusetts, 
 in which wheel Mr. Boyden introduced several improvements 
 of great value. 
 
 The performance of the Appleton Company's turbine was 
 
142 APPENDIX. 
 
 carefully ascertained by Mr. Boy den, and its effective power, 
 exclusive of that required to carry the wheel itself, a pair of 
 bevel gears, and the horizontal shaft carrying the friction-pulley 
 of a Prony dynamometer, was found to be seventy-eight per 
 cent, of the power expended. 
 
 In the year 18A6, Mr. Boyden superintended the construc- 
 tion of three turbines, of about one hundred and ninety horse- 
 power each, for the same company. By the terms of the con- 
 tract, Mr. Boyden's compensation depended on the performance 
 of the turbines ; and it was stipulated that two of them should 
 be tested. In accordance with the contract, two of the turbines 
 were tested, a very perfect apparatus being designed by Mr. 
 Boyden for the purpose, consisting essentially of a Prony dyna- 
 mometer to measure the useful effects, and a weir to gauge the 
 quantity of water expended. 
 
 The observations were put into the hands of the author for 
 computation, who found that the mean maximum effective 
 power for the two turbines tested was eighty-eight per cent, of 
 the power of the water expended. 
 
 According to the terms of the contract, this made the com- 
 pensation for engineering services, and patent rights for these 
 three wheels, amount to fifty-two hundred dollars, which sum 
 was paid by the Appleton Company without objection. 
 
 These turbines have now been in operation about eight 
 years, and their performance has been, in every respect, entirely 
 satisfactory. The iron work for these wheels was constructed 
 by Messrs. Gay & Silver, at their machine-shop at North 
 Chelmsford, near Lowell; the workmanship was of the finest 
 description, and of a delicacy and accuracy altogether unpre- 
 cedented in constructions of this class. 
 
 These wheels, of course, contained Mr. Boyden's latest im- 
 provements, and it was evidently for his pecuniary interest that 
 
APPENDIX. 143 
 
 the wheels should be as perfect as possible, without much regard 
 to cost. The principal points in which one of them differs 
 from the constructions of Fourneyron are as follows : — 
 
 The wooden flume conducting the water immediately to the 
 turbine is in the form of an inverted truncated cone, the water 
 bevng introduced into the upper part of the cone, on one side of the 
 axis of the cone {which coincides with the axis of the turbine), in 
 such a manner that the water, as it descends in the cone, has a 
 gradually increasing velocity and a spiral motion ; the horizontal 
 component of the spiral motion being in the direction of the 
 motion of the wheel. This horizontal motion is derived from 
 the necessary velocity with which the water enters the trun- 
 cated cone; and the arrangement is such that, if perfectly pro- 
 portioned, there would be no loss of power between the nearly 
 still water in the principal penstock and the guides or leading 
 curves near the wheel, except from the friction of the water 
 against the walls of the passages. It is not to be supposed that 
 the construction is so perfect as to avoid all loss, except from 
 friction ; but there is, without doubt, a distinct advantage in 
 this arrangement over that which had been usually adopted, 
 and where no attempt had been made to avoid sudden changes 
 of direction and velocity. 
 
 The guides, or leading curves (Figs. A, B), are not perpen- 
 dicular, but a little inclined backwards from the motion of the 
 wheel, so that the water, descending with a spiral motion, meets 
 only the edges of the guides. This leaning of the guides has 
 also another valuable effect : when the regulating gate is raised 
 only a small part of the height of the wheel, the guides do not 
 completely fulfil their office of directing the water, the water 
 entering the wheel more nearly in the direction of the radius 
 than when the gate is fully raised ; by leaning the guides it will 
 be seen the ends of the guides near the wheel are inclined, the 
 
Hi APPENDIX. 
 
 bottom part standing farther forward, and operating more 
 efficiently in directing the water when the gate is partially 
 raised, than if the guides were perpendicular. 
 
 In Fourneyron's constructions a garniture is attached to 
 the regulating gate, and moves with it, for the purpose of di- 
 minishing the contraction. This, considered apart from the 
 mechanical difficulties, is probably the best arrangement; to 
 be perfect, however, theoretically, this garniture should be of 
 different forms for different heights of gate ; but this is evi- 
 dently impracticable. 
 
 In the Appleton turbine the garniture is attached to the 
 guides, the gate (at least the lower part of it) being a simple thin 
 cylinder. By this arrangement the gate meets with much 
 less obstruction to its motion than in the old arrangement, un- 
 less the parts are so loosely fitted as to be objectionable ; and it 
 is believed that the coefficient of effect, for a partial gate, is 
 proportionally as good as under the old arrangement. 
 
 On the outside of the wheel is fitted an apparatus, named by 
 Mr. Boyden the Diffuser. The object of this extremely inter- 
 esting invention is to render useful a part of the power other- 
 wise entirely lost, in consequence of the water leaving the wheel 
 with a considerable velocity. It consists, essentially, of two 
 stationary rings or discs, placed concentrically with the wheel, 
 having an interior diameter a very little larger than the exte- 
 rior diameter of the wheel-; and an exterior diameter equal to 
 about twice that of the wheel ; the height between the discs at 
 their interior circumference is a very little greater than that of 
 the orifices in the exterior circumference of the wheel, and at 
 the exterior circumference of the discs the height between them 
 is about twice as great as at the interior circumference ; the form 
 of the surfaces connecting the interior and exterior circumfer- 
 ences of the discs is gently rounded, the first elements of the 
 
APPENDIX. 145 
 
 curves near the interior circumferences being nearly horizon- 
 tal. There is consequently included between the two surfaces 
 an aperture gradually enlarging from the exterior circumference 
 of the wheel to the exterior circumference of the diffnser. When 
 the regulating gate is raised to its full height, the section through 
 which the water passes will be increased, by insensible degrees, 
 in the proportion of one to four, and if the velocity is uniform 
 in all parts of the diffuser at the same distance from the wheel, 
 the velocity of the water will be diminished in the same pro- 
 portion ; or its velocity on leaving the diffuser will be one-fourth 
 of that at its entrance. By the doctrine of living forces, the 
 power of the water in passing through the diffuser must, there- 
 fore, be diminished to one-sixteenth of the power at its entrance. 
 It is essential to the proper action of the diffuser that it should 
 be entirely under water, and the power rendered useful by it 
 is expended in diminishing the pressure against the water issu- 
 ing from the exterior orifices of the wheel ; and the effect pro- 
 duced is the same as if the available form under which the 
 turbine is acting is increased a certain amount. It appears 
 probable that a diffuser of different proportions from those above 
 indicated would operate with some advantage without being 
 submerged. It is nearly always inconvenient to place the 
 wheel entirely below low-water mark ; up to this time, however, 
 all that have been fitted up with a diffnser have been so placed ; 
 and indeed, to obtain the full effect of a fall of water, it appears 
 essential, even when a diffuser is not used, that the wheel should 
 be placed below the lowest level to which the water falls in the 
 wheel-pit, when the wheel is in operation. 
 
 The action of the diffuser depends upon similar principles 
 to that of diverging conical tubes, which, when of certain pro- 
 portions, it is well known, increase the discharge; the author 
 has not met with any experiments on tubes of this form dis- 
 
140 APPENDIX. 
 
 charging under water although there is good reason to believe 
 that tubes of greater length and divergency would operate more 
 effectively under water than when discharging freely in the 
 air, and that results might be obtained that are now deemed 
 impossible by most engineers. 
 
 Experiments on the same turbine, with and without a dif- 
 fusa*, show again in the coefficient of effect, due to the latter, 
 of about three per cent. By the principles of living forces, and 
 assuming that the motion of the water is free from irregularity, 
 the gain should be about five per cent. The difference is due, 
 in part at least, to the unstable equilibrium of water flowing 
 through expanding apertures ; this must interfere with the uni- 
 formity of the velocities of the fluid streams, at equal distances 
 from the wheel. 
 
 Suspending the wheel on the top of ths vertical shaft (Fig. 
 A) , instead of running it on a step at the bottom. This had been 
 previously attempted, but not with such success as to warrant 
 its general adoption. It has been accomplished with complete 
 success by Mr. Boyden, whose mode is to cut the upper part of 
 the shaft into a series of necks, and to rest the projecting parts 
 upon corresponding parts of a box. A proper fit is secured by 
 lining the box, which is of cast-iron, with Babbitt metal — a soft 
 metallic composition consisting, principally, of tin ; the cast- 
 iron box is made with suitable projections and recesses, to sup- 
 port and retain the soft metal, which is melted and poured into 
 it, the shaft being at the same time in its proper position in the 
 box. It will readily be seen that a great amount of bearing- 
 surface can be easily obtained by this mode, and also, what is 
 of equal importance, it may be near the axis; the lining metal, 
 being soft, yields a little if any part of the bearing should re- 
 ceive a great excess of weight. The cast-iron box is suspended 
 on gimbals, similar to those usually adopted for mariners' 1 com- 
 
APPENDIX. ]4'J 
 
 passes and chronometers, which arrangement permits the box 
 to oscillate freely in all directions, horizontally, and prevents,' 
 in a great measure, all danger of breaking the shaft at the 
 necks, in consequence of imperfections in the workmanship or 
 in the adjustments. Several years' experience has shown that 
 this arrangement, carefully constructed, is all that can be de- 
 sired ; and that a bearing thus constructed is as durable, and 
 can be as readily oiled and taken care of, as any of the ordinary 
 bearings in a manufactory. 
 
 The buckets are secured to the crowns of the wheel in a 
 novel and much more perfect manner than had been previously 
 used ; the crowns are first turned to the required form, and 
 made smooth ; by ingenious machinery designed for the pur- 
 pose, grooves are cut with great accnracy in the crowns, of the 
 exact curvature of the buckets ; mortices are cut through the 
 crowns in several places in each groove ; the buckets, or floats, 
 are made with corresponding tenons, which project through the 
 crowns, and are riveted on the bottom of the lower crown, and 
 on the top of the upper crown ; this construction gives the re- 
 quisite strength and firmness, with buckets of much thinner 
 iron than was necessary under any of the old arrangements ; it 
 also leaves the passages through the wheel entirely free from 
 injurious obstructions. 
 
 In the year 1849, the manufacturing companies at Lowell 
 
 purchased of Mr. Boyden the right to use all his improvements 
 
 relating to turbines and other hydraulic motors. Since that 
 
 time it has devolved upon the author, as the chief engineer of 
 
 these companies, to design and superintend the construction of 
 
 such turbines as might be wanted for their manufactories, and 
 
 to aid him in this important undertaking, Mr. Boyden has 
 
 communicated to him copies of many of his designs for 
 
 turbines, together with the results of experiments upon a por- 
 10 
 
148 APPENDIX. 
 
 tionofthem; he has communicated, however, but little theo- 
 retical information, and the author has been guided principally 
 by a comparison of the most successful designs, and such light 
 as he could obtain from writers on this most intricate subject. 
 
 Summary description of one of the turbines at the Tremont 
 Mills, Lowell. Figs. A, B, C. 
 
 Fig. A is a vertical section of the turbine through the axis of the wheel 
 shaft ; Fig. B is a portion of the plan, on an enlarged scale, showing the 
 disposition of the leading curves and buckets and diffuser; Fig. C is a cross 
 section of the wheel and diffuser on an enlarged scale, and the more adja- 
 cent parts. The letters on the corresponding parts of the figures are the 
 same. 
 
 The water is conveyed to the wheel of the turbine, from the 
 forebay by a supply pipe, the greater portion of which, from 
 the forebay downwards, is of wrought iron, and of gradually 
 diminishing diameter towards the lower portion I, termed the 
 curbs, which is of cast iron. The curbs are supported on col- 
 umns, which rest on cast-iron supports firmly imbedded in the 
 wheel-pit. 
 
 The Disc K, K', K", to which the guides for the water, or 
 the leading curves, thirty -three in number, are attached, is sus- 
 pended from the upper end of the cast-iron curb, by means of 
 the disc-pipes M M. 
 
 The leading curves are of Russian iron, one-tenth of an inch 
 in thickness. The upper corners of these, near the wheel, are 
 connected by what is termed the garniture L, L', L", intended 
 to diminish the contraction of the fluid vein when the regulat- 
 ing gate is fully raised. 
 
 The disc-pipe is very securely fastened, to Sustain the pressure 
 of the water on the disc. The escape of water, between the 
 upper curb and the upper flange of the disc-pipe, is prevented 
 by a band of leather on the outside, enclosed within an iron 
 
APPENDIX. 149 
 
 ring. This pipe is so fastened as to prevent its rotating in a 
 d irection opposite to that in which the water flows out. 
 
 Ths regulating gate is a cast-iron cylinder, R, enclosing the 
 disc and curves, and which, raised or lowered by- suitable 
 machinery, regulates the amount of water let on the wheel B 
 B' B", exterior to it. 
 
 The wheel consists of a central plate of cast-iron and two 
 crowns, C,C, C, C", of the same material to which the buckets 
 are attached. These pieces are all accurately turned, and pol- 
 ished, to offer the least obstruction in revolving rapidly in the 
 water. 
 
 The huckets, made of Russian iron, are forty-four in number, 
 and each ^ of an inch thick. They are firmly fastened to the 
 crowns. 
 
 The vertical shaft D, from which motion is communicated 
 to the machinery by suitable gearing, is of wrought-iron. In- 
 stead of resting on a gudgeon, or step at bottom, it is suspended 
 from a suspension box, E', by which the collars at the top are en- 
 closed. These collars are of steel, and are fastened to the upper 
 portion of the shaft, which last can be detached from the lower 
 portion. 
 
 The suspension hox is lined with Babbitt metal, a soft compo- 
 sition consisting mostly of tin, and capable of sustaining a 
 pressure of from 50 lbs. to 100 lbs. per square inch, without 
 sensible diminution of durability. The box consists of two 
 parts, for the convenience of fastening it on, or the reverse. 
 The box rests upon the gimbal G, which is so arranged that the 
 suspension box, the shaft, and the wheel can be lowered or 
 raised, and the suspension box be allowed to oscillate laterally, 
 so as to avoid subjecting it to any lateral strain. 
 
 The lower end of the shaft has a cast-steel pin, 0, fixed to it. 
 This is retained in its place by the step, which is made of three 
 
150 APPENDIX. 
 
 parts, and lined with case-hardened iron. The step can he ad 
 justed by horizontal screws, by a small lateral motion given by 
 them to it. 
 
 Rules for iwoTportioning turbines. In making the designs 
 for the Tremont and other turbines, the author has been guided 
 by the following rules, which he has been led to, by a compari- 
 son of several turbines designed by Mr. Boyden, which have 
 been carefully tested, and found to operate well. 
 
 Rule 1st. The sum of the shortest distances between the 
 buckets should be equal to the diameter of the wheel. 
 
 Rule 2d. The height of the orifices of the circumference of 
 the wheel should be equal to one-tenth of the diameter of the 
 wheel. 
 
 Rule 3d. The width of the crowns should be four times the 
 shortest distance between the buckets. 
 
 Rule 4th. The sum of the shortest distances between the 
 curved guides, taken near the wheel, should be equal to the in- 
 terior diameter of the wheel. 
 
 The turbines, from a comparison of which the above rules 
 were derived, varied in diameter from twenty-eight inches to 
 nearly one hundred inches, and operated on falls from thirty 
 feet to thirteen feet. The author believes that they may be 
 safely followed for all falls between five feet and forty feet, and 
 for all diameters not less than two feet ; and, with judicious 
 arrangements in other respects, and careful workmanship, a 
 useful effect of seventy-five per cent, of the power expended 
 may be relied upon. For falls greater than forty feet, the sec- 
 ond rule should be modified, by making the height of the 
 orifices smaller in proportion to the diameter of the wheel. 
 
 Taking the foregoing rules as a basis, we may, by aid of the 
 experiments on the Tremont turbine, establish the following 
 formulas. Let 
 
APPENDIX. 151 
 
 D = the diameter of the wheel at the outer extremities of 
 the buckets. 
 
 d = the diameter at their inner extremities. 
 
 H = the height of the orifices of discharge, at the outer ex- 
 tremities of the buckets. 
 
 W = width of crowns of the buckets. 
 
 ]V= the number of buckets. 
 
 n = the number of guides. 
 
 P = the horse-power of the turbine, of 550 fl - lb3, per second. 
 
 h = the fall acting on the wheel. 
 
 Q = the quantity of water expended by the turbine, in cubic 
 feet per second. 
 
 V = the velocity due the fall acting on the wheel. 
 
 V = the velocity of the water passing the narrowest sections 
 of the wheel. 
 
 v = the velocity of the interior circumference of the wheel ; 
 all velocities being in feet per second. 
 
 C= the coefficient of V, or the ratio of the real velocity 
 of the water passing the narrowest sections of the wheel, to the 
 theoretical velocity due the fall acting on the wheel. 
 
 The unit of length is the English foot. 
 
 It is assumed that the useful effect is seventy-five per cent, 
 of the total power of the water expended. 
 
 According to Rule 1st, we have the sum of the widths of the 
 orifices of discharge, equal to I). Then the sum of the areas 
 of all the orifices of discharge is equal to D II. 
 
 By the fundamental law of hydraulics, we have 
 F= VTgh. 
 Therefore V = C VY^h. 
 
 For the quantity of water expended we have 
 Q= IIDV = II D CVYJk. 
 
 IP_ 
 10 
 From the extremely interesting and accurate experiments of 
 
 H=~ () D,Q=^Y.V2gxCxVh. 
 
152 APPENDIX. 
 
 Mr. Francis on the expenditure of water by one of the Tremont 
 wheels, recorded in his work, the following data are obtained 
 from it : — 
 
 For the sum of the widths of the orifices of discharge, 
 
 44 x 0.18757 = 8.25308 feet. 
 Q = 138.1892 cubic feet per second; 
 A = 12.903 feet; 
 VJg = 8.0202 feet. 
 
 Substituting these numerical results in the preceding value 
 of Q, there obtains 
 
 138.1892 = 7.68692 x 8.0202 x O x VlMM, 
 hence 
 
 {7=0.624. ' 
 By Rule 2d we have 
 
 //= 0.10 J), hence HI) = 0.10 B\ 
 hence Q = H D V = 0.10 D" G VTg~h. • 
 
 Calling the weight of a cubic foot of water 62.33 lbs., we 
 have 
 
 p = 0.75 x 62.33 ,j A = Q 085 Q h 
 550 v v ' 
 
 or, substituting for Q the value just found, 
 P = 0.0425 IP h V~h, 
 hence 
 
 D 
 
 = 4.85 J 
 
 H 
 
 The number of buckets is to a certain extent arbitrary, and 
 would usually be determined by practical considerations. Some 
 of the ideas to be kept in mind are the following: 
 
 The pressure on each bucket is less, as the number is greater ; 
 the greater number will therefore permit of the use of thinner 
 iron, which is important in order to obtain the best results. 
 The width of the crowns will be less for a greater number of 
 
APPENDIX. 153 
 
 buckets. A narrow crown appears to be favorable to the useful 
 effect, when the gate is only partially raised. As the spaces 
 between the buckets must be proportionally narrower for a 
 larger number of buckets, the liability to become choked up, 
 either with anchor ice or other substances, is increased. The 
 amount of power lost by the friction of the water against the 
 surfaces of the buckets will not be materially changed, as the 
 total amount of rubbing surface on the buckets will be nearly 
 constant for the same diameter ; there will be a little less on 
 the crown, for the larger number. The cost of the wheel will 
 probably increase with the number of buckets. The thickness 
 and quality of the iron, or other metal intended to be used for 
 the buckets, will ' sometimes be an element. In some water 
 wrought iron is rapidly corroded. 
 
 The author is of opinion that a general rule cannot be given 
 for the number of buckets ; among the numerous turbines work- 
 ing rapidly in Lowell, there are examples in which the shortest 
 distance between the buckets is as small as 0.75 of an inch, and 
 in others as large as 2.75 inches. 
 
 As a guide in practice, to be controlled by particular circum- 
 stances, the following is proposed, to be limited to diameters 
 of not less than two feet : — 
 
 iT=3(Z> + 10). 
 Taking the nearest whole number for the value of If, 
 
 The Tremont turbine is 8£ in diameter, and, according to the 
 proposed rule, should have fifty-five buckets instead of forty- 
 four. With fifty-five buckets, the crowns should have a width 
 of 7.2 inches instead of 9 inches. With the narrower width, it 
 is probable that the useful effect, in proportion to the power 
 expended, would have been a little greater when the gate was 
 partially raised. 
 
 By the 3d rule, we have for the width of the crowns, 
 
154 APPENDIX. 
 
 N ' 
 
 « 
 
 and for the interior diameter of the wheel, 
 
 8Z> 
 
 d = D 
 
 N " 
 
 By the 4th rule, d is also equal to the sum of the shortest 
 distances between the guides, where the water leaves them. 
 
 The number n of the guides is, to a certain extent, arbitrary. 
 The practice at Lowell has been, usually, to have from a half to 
 three-fourths of the number of the buckets ; exactly half would 
 probably be objectionable, as it would tend to produce pulsa- 
 tions or vibrations. 
 
 The proper velocity to be given to the wheel is an impor- 
 tant consideration. Experiment 30 (the one above used for 
 data) on the Tremont turbine gives- the maximum coefficient 
 of effect of that wheel ; in that experiment, the velocity of the 
 interior circumference of the wheel is 0.62645 of the velocity 
 due to the fall acting on the wheel. By reference to other ex- 
 periments, with the gate fully raised, it will be seen, however, 
 that the coefficient of effect varies only about two per cent, 
 from the maximum, for any velocity of the interior circumfer- 
 ence, between fifty per cent, and seventy per cent, of that due 
 to the fall acting upon the wheel. By reference to the experi- 
 ments in which the gate is only partially raised, it will be seen 
 that the maximum corresponds to slower velocities ; and as tur- 
 bines, to admit being regulated in velocity for variable work, 
 must, almost necessarily, be used with a gate not fully raised, 
 it would appear proper to give them a velocity such that they 
 will give a good effect under these circumstances. 
 
 "With this view, the following is extracted from the experi- 
 ments in Table II. : — 
 
APPENDIX. 
 
 15.3 
 
 Number of the ex- 
 periment. 
 
 Height of the regulat- 
 ing gate in inches. 
 
 Ratio of the velocity of the in- 
 terior circumference of the 
 wheel, to the velocity due the 
 fall acting upon the wheel, cor- 
 responding to the maximum 
 coefficient of effect. 
 
 30 
 62 
 73 
 
 84 
 
 11.49 
 
 8.55 
 5.65 
 
 2.875 
 
 / 
 
 0.62645 
 0.56541 
 0.56205 
 0.48390 
 
 By this table it would appear that, as turbines are generally 
 used, a velocity of the interior circumference of the wheel, of 
 about fifty-six per cent, of that due to the fall acting upon the 
 wheel, would be most suitable. By reference to the diagram 
 at Plate VI,* it will be seen that at this velocity, when the 
 gate is fully raised, the coefficient of effect will be within less 
 than one per cent, of the maximum. 
 
 Other considerations, however, must usually be taken into ac- 
 count in determining the velocity ; the most frequent is the vari- 
 ation of the fall under which the wheel is intended to operate. 
 If, for instance, it were required to establish a turbine of a given 
 power on a fall liable to be diminished to one-half by back- 
 water, and that the turbine should be of a capacity to give the 
 requisite power at all times, in this case the dimensions of the 
 turbine must be determined for the smallest fall ; but if it has 
 assigned to it a velocity, to give the maximum effect at the 
 smallest fall, it will evidently move too slow for the greatest 
 fall, and this is the more objectionable, as, usually, when the 
 fall is greatest the quantity of water is the least, and it is of 
 the most importance to obtain a good effect. It would then be 
 * " Lowell Hydraulic Experiments." 
 
156 APPENDIX. 
 
 usually the best arrangement to give the wheel a velocity cor 
 responding to the maximum coefficient of effect, when the fall 
 is greatest. To assign this velocity, we must find the propor- 
 tional height of the gate when the fall is greatest ; this may be 
 determined approximately by aid of the experiments on the 
 Tremont turbine. 
 
 We have seen that P= 0.085 Qh. 
 
 Now, if h is increased to 2 h, the velocity, and consequently 
 the quantity, of water discharged will be increased in the pro- 
 portion of V h to Vsh; that is to say, the quantity for the fall- 
 2A will be \/% Q. 
 
 Calling P' the total power of the turbine on the double fall, 
 we have 
 
 P'=0.085 4/2£2A, 
 
 or, 
 
 P' = 0.085x 2.8284 Qh. 
 
 Thus, the total power of the turbine is increased 2.8284 times, 
 by doubling the fall ; on the double fall, therefore, in order to 
 preserve the effective power uniform, the regulating gate must 
 be shut down to a point that will give only -g-.-ji-g-f part of the 
 total power of the turbine. 
 
 In Experiment 15, the fall acting upon the wheel was 12.888 
 feet, and the total useful effect of the turbine was 85625.3 
 lbs. raised one foot per second ; -s-.-gVsr P art 0I " ^' s ^ s 30273.4 
 lbs. ; consequently the same opening of gate that would give 
 this last power on a fall of 12.888 feet, would give a power of 
 85625.3 lbs. raised one foot per second, on a fall of 2 x 12.888 
 feet=25.776 feet. To find this opening of gate, we must have 
 recourse to some of the other experiments. 
 
 In Experiment 73, the fall was 13.310 feet, the height of the 
 gate 5.65 inches, and the useful effect 58830.1 lbs. In Ex- 
 periment 83 the fall was 13.435 feet, the height of the gate 
 
APPENDIX. 157 
 
 2.875 inches, and the useful effect 27310.9 lbs. Reducing 
 both these useful effects to what they would have been if the 
 fall was 12.888 feet, 
 
 the useful effect in experiment 73, 58830.l(^-— *)*=56054.5; 
 
 83, 27310.9f 1|4W=25660J 
 \13.435/ 
 
 By a comparison of the useful effects with the corresponding 
 .heights of gate, we find, by simple proportion of the differences, 
 that a useful effect of 30273.4 lbs. raised one foot high per 
 second, would be given when the height of the regulating gate 
 was 3.296 inches. 
 
 By another mode : — 
 As 25660.1 : 2.875 :: 30273.4 : 2.875 xffrfflft =3.392 in., a little 
 consideration will show that the first mode must give top little, 
 and the second too much ; taking a mean of the two results, we 
 have for the height of the gate, giving ^-'wr^s °? ^ ie total power 
 of the turbine, 3.344 inches. Referring to Table II., we see 
 that, with this height of gate, in order to obtain the best coeffi- 
 cient of useful effect, the velocity of the interior circumference 
 of the wheel should be about one-half of that due to the fall 
 acting upon the wheel ; and by comparison of Experiments 74 
 and 84, it will be seen that, with this height of gate and with 
 this velocity, the coefficient of useful effect must be near 0.50. 
 
 This example shows, in a strong light, the well-known defect 
 of the turbine, viz., giving a diminished coefficient of useful 
 effect at times when it is important to obtain the best results. 
 One remedy for this defect would be, to have a spare turbine, 
 to be used when the fall is greatly diminished ; this arrangement 
 would permit the principal turbine to be made nearly of the di- 
 mensions required for the greatest fall. As at other heights of 
 the water economy of water is usually of less importance, 
 
158 
 
 APPENDIX. 
 
 the spare turbine might generally be of a cheaper construc- 
 tion. 
 
 To lay out the curve of the buckets, the author makes use of 
 the following method : — 
 
 Referring to Fig. D, the number of buckets, 1ST, having been 
 
 nD 
 determined by the preceding rules, set off the arc GI=--=. 
 
 Let u = G H = I P', the shortest distance between the buckets ; 
 
 t = the thickness of the metal forming the buckets. 
 
 Make the arc GK = 5u. Draw the radius K, intersecting 
 the interior circumference of the wheel at L; the point L will 
 be the inner extremity of the bucket. Draw the directrix L M 
 tangent to the inner circumference of the wheel. Draw the 
 arc N, with the radius w + t, from I as a centre; the other 
 directrix, G P, must be found by trial, the required conditions 
 being, that,- when the line M L is revolved round to the position 
 G T, the point M being constantly on the directrix G P, and 
 another point at the distance M 6 = E S, from the extremity 
 of the line describing the bucket, being constantly on the di- 
 rectrix M L, the curve described shall just touch the arc N O. 
 A convenient line for a first approximation may be drawn by 
 making the angle O GP = 11°. After determining the direc- 
 trix according to the preceding method, if the angle GP 
 should be greater than 12°, or less than 10°, the length of the 
 arc G K should be changed to bring the angle within these 
 limits. 
 
 The curve G S S' S" L, described as above, is nearly the quar- 
 ter of an ellipse, and would be precisely so if the angle G M L 
 was a right angle ; tbe curve may be readily described, me- 
 chanically, with an apparatus similar to the elliptic trammel; 
 there is, however, no difficulty in drawing it by a series of 
 points, as is sufficiently obvious. 
 
APPENDIX. 159 
 
 The trace adopted by the author for the corresponding guides 
 is as follows : — 
 
 The number n having been determined, divide the circle in 
 ■which the extremities of the guides are found into n equal 
 parts T¥,fX, etc. 
 
 Put w' for the width between two adjoining guides, 
 
 and if for the thickness of the metal forming the guides. 
 
 d 
 We have by Eule 4, w' = — 
 
 With W as a centre, and the radius u' + t' , draw the arc 
 TZ; and witli X as a centre, and the radius 2(w' + t'), draw 
 the arc A' B'. Through Y draw the portion of a circle, V C, 
 touching the arcs T Z and A' B'; this will be the curve for the 
 essential portion of the guide. The remainder of the guide, 
 C D', should be drawn tangent to the curve C T; a convenient 
 radius is one that would cause the curve C D', if continued to 
 pass through the centre 0. This part of the guide might be 
 dispensed with, except that it affords great support to the part 
 C'T, and thus permits the use of much thinner iron thau would 
 be necessary if the guide terminated at C, or near it. 
 
 Collecting together the foregoing formulas for proportioning 
 turbines, which, it is understood, are to be limited to falls not 
 exceeding forty feet, and to diameters not less than two feet, 
 we have for the horse power, 
 
 P = 0.0425Z> 2 A Vh ; 
 
 for the diameter, 
 
 for the quantity of water discharged per second, 
 = O.5Z>V^/ 
 
160 APPENDIX. 
 
 for the velocity of the interior circumference of the wheel, when 
 the fall is not very variable, 
 
 v = 0.56 |/2^A, 
 or, v = 4.491 4/ h; 
 
 for the height of the orifices of discharge, 
 
 H= 0.10 D; 
 for the number of buckets, 
 
 l¥= 3 (D + 10) ; 
 for the shortest distance between two adjacent buckets, 
 
 D 
 U =2T> 
 for the width of the crown occupied by the buckets, 
 
 ]V - JV > 
 for the interior diameter of the wheel, 
 
 for the number of guides, 
 
 n = 0.50 JV to 0.75 N; 
 for the shortest distance between two adjacent guides, 
 
 w , = £ 
 
 n 
 Table has been computed by these formulas. 
 For falls greater than forty feet, the height of the orifices in 
 the circumference of the wheel should be diminished. The 
 foregoing formulas may, however, still be made use of. Thus : 
 supposing, for a high fall, it is determined to make the orifices 
 three-fourths of that given by the formula ; divide the given 
 power, or quantity of water to be used, by 0.75, and use the 
 quotient in place of the true power or quantity, in determining 
 the dimensions of the turbine. JSTo modifications of the dimen- 
 sions will be necessary, except that -^ of the diameter of the 
 
APPENDIX. 161 
 
 turbine should be diminished to -fo of the diameter, to give the 
 height of the orifices in the circumference. 
 
 It is plain, from the method by which the preceding formulas 
 have been obtained, that they cannot be considered as estab- 
 lished, but should only be taken as guides in practical applica- 
 tions, until some more satisfactory are proposed, or the intrica- 
 cies of the turbine have been more fully unravelled. The 
 turbine lias been an object of deep interest to many learned 
 mathematicians, but up to this time the results of their investi- 
 gations, so far as they have been published, have afforded but 
 little aid to hydraulic engineers. 
 
 Diffuser. — As previously stated, the principles involved in 
 the flow of water through a diverging tube find a useful appli- 
 cation in Mr. Boyden's diffuser. This invention, applied to a 
 turbine water wheel 104.25 inches in diameter, and about seven 
 hundred horse-power, is represented on Fig. B at X' and on 
 Fig. C at X". 
 
 The diffuser is supported on iron pillars from below. The 
 wheel is placed sufficiently low to permit the diffuser to be 
 submerged at all times when the wheel is in operation, that 
 being essential to the most advantageous operation of the 
 diffuser. 
 
 When the speed gate is fully raised, the wheel moves with 
 the velocity which gives its greatest coefficient of useful effect. 
 On leaving the wheel it necessarily has considerable velocity, 
 which would involve a corresponding loss of power, except for 
 the effect of the diffuser,- which utilizes a portion of it. When 
 operating under a fall of thirty-three feet, and the speed gate 
 is raised to its full height, this wheel discharges about 219 
 cubic feet per second. The area of the annular space where 
 the water enters the diffuser, is 0.802 x 8.792 * = 22.152 square 
 feet ; and if the stream passes through this section radially, its 
 
162 APrENDIX. 
 
 219 
 
 mean velocity must be = 9.886 feet per second, which 
 
 J 22.152 l 
 
 is due to a head of 1.519 feet. The area of the annular space 
 
 where the water leaves the diffuser is 1.5 x 15.333 * = 72.255 
 
 219 
 
 square feet, and the mean velocity —— = 3.031 feet per 
 
 72.255 
 
 second, which is due to a head of 0.143 feet. According to this 
 
 the saving of head clue to the diffuser is 1.519 — 0.143 = 1.376 
 
 1 376 
 feet, being - — ■- 1 or about 4-| per cent, of the head available 
 
 without the diffuser, which is equivalent to a gain in the coeffi- 
 cient of useful effect to the same extent. Experiments on the 
 same turbine, with and without the diffuser, have shown a 
 gain due to the latter of about three per cent, in the coefficient 
 of useful effect. The diffuser adds to the coefficient of useful 
 effect by increasing the velocity of the water passing through 
 the wheel, and it must of course increase the quantity of" water 
 discharged in the same proportion. If it increases the availa- 
 ble head three per cent., the velocity, which varies as the 
 square root of the head, must be increased in the same propor- 
 tion. The power of the wheel, which varies as the product of 
 the head into the quantity of water discharged, must be 
 increased about 4.5 per cent. 
 
APPENDIX. 163 
 
 EXPLANATION OF FIGURES. 
 
 Fig. A. Section through the axis of the Turbine without the Diffuser. 
 
 I Cast-iron Curbs through which the water passes from the wrought-iron 
 supply-pipe to the Disc. 
 
 K Cast-iron Disc on which the Guide Curves are fastened. 
 
 |_ Garniture fitted to lower end of Lower Curb. 
 
 |V| Disc-pipe suspending the Disc from Upper Curb. 
 
 N Columns of cast-iron supporting Curbs. 
 
 R Regulating Gate of cast-iron. 
 
 S Brackets for raising and lowering the Gate. 
 
 B Wheel. 
 
 Q f C Crowns of the Wheel between which the curved buckets are in- 
 serted. 
 
 D Main Shaft of the Wheel. 
 
 E' Suspension Box, lined with babbit metal, from which the Wheel hangs 
 by the cast-steel Collars around the upper end of the Shaft. 
 
 F Upper portion of Shaft fastened to lower portion, with bearings p of 
 cast-iron lined with babbit metal. 
 
 G Gimbal on which the Suspension Box E' rests. 
 
 H Support of the Gimbal. 
 
 O Step to receive cast-steel Pin on lower end of Shaft. 
 
 Fig. A. Plan of Disc, K " i Garniture, L', Wheel, Q', and Diffuser, M' N'. 
 
 Fig. B. Section of Wheel, Q", Garniture, |_ ", Regulating Gate, R", and 
 upper and lower Crowns of the Diffuser. . 
 
 Fig. D. Diagram for laying out the curves of the Buckets and Guide 
 Curves. 
 
164: 
 
 APPENDIX. 
 
APPENDIX. 
 
 165