eee 27 F726 L 
 
 AN INQUIRY 
 ye FC) VA MORE PERFECT FORM 
 
 OF 
 
 WATER-WHEEL 
 
 BY 
 
 J. P. FRIZELL 
 | i. 
 
 BOSTON 
 60 CONGRESS STREET 
 1897 
 
tga. 
 
 
 
pe? oo Se arc 
 
 ‘Boker, ; s 
 
 ‘ Tro - 
 
 
 
 An Inquiry ase toma 
 
 More Perfect Form of Water-Wheel 
 
 A WATER-WHEEL is a wheel for applying the power of water to 
 some useful purpose. 
 
 Water-power implies two things: a flow of water and a fall. 
 We will express the former generally by the symbol g in cubic feet 
 per second, the latter by h in feet. 
 
 Work is mechanical effect. It implies resistance overcome, and is 
 conveniently represented by the raising of a weight. <A pressure of 
 1 pound overcome through a distance of 1 foot is equivalent to the 
 raising of a pound one foot and is called a foot-pound. 550 foot- 
 pounds per second is a horsepower. ‘The weight of a cubic foot of 
 water at ordinary temperatures is very nearly 622 Ibs. It varies 
 slightly with the temperature, and will be here represented by the 
 letter w. In computations relative to water-power, w is usually 
 taken at 62.5 pounds. Another symbol of frequent use in hydraulie 
 computations is the velocity imparted per second by gravity, desig- 
 nated by the letter g. This varies slightly with the latitude and 
 altitude of the place. 
 
 Energy is capacity to perform work. <A quantity g of water in a 
 mill-pond, at a height # above the tail-water, has the energy wqh 
 foot-pounds, and it could exert that amount of energy if we neglect 
 the losses incident to the application. A quantity ¢ of water at the 
 level of the tail-water, supposing it enclosed in a penstock communi- 
 eating with the head-water, could exert upon a piston of 1 square 
 foot cross-section the pressure wh pounds, and could move the piston 
 q feet, representing an amount of energy of wqh foot-pounds. En- 
 
 ergy in these forms is called potential energy, being the energy of 
 position or the energy of pressure. When water, in the last named | 
 3 
 
4 A More Perfect Form of Water - Wheel. 
 
 case, issues from an orifice at or below the level of the tail-water, it 
 
 9 
 
 Oe y 
 takes a velocity, v = 2gh, and its energy is wq ino wah. This 
 
 form of energy is called kinetic energy, or the energy of velocity. 
 
 Two remarks of importance may be here made as bearing on what 
 follows : — 
 
 1. A head of water can exert a pressure or generate a velocity, 
 but it cannot increase one of these effects without diminishing the 
 other. When it has its full effect of pressure it can produce no veloc- 
 ity. When it has its full effect of velocity it can exert no pressure. 
 This, of course, implies that the velocity is directly communicated 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 by the pressure. Water under pressure may be put in motion by an 
 extraneous cause without any effect upon the pressure. 
 
 2. The energy with which water leaves a hydraulic motor is a de- 
 duction from the efficiency of the motor. 
 
 The efficiency of a wheel is its energy as compared with the energy 
 of the water acting on it. 
 
 A wheel which could raise the water that acts on it to the full 
 height of the fall would have perfect efficiency in the organs by 
 which it is acted on as well as in the organs by which it acts. We 
 know this to be a mechanical impossibility, by reason of the natural 
 resistances to motion. We should regard it as a very perfect appa- 
 ratus that could perform this duty with an efficiency of 70 per cent. 
 Nature, however, accomplishes the same result with an efficiency of 
 more than 99 per cent. Consider the system of pipes represented in 
 Fig. 1. Suppose the pipes a and b to be 20 feet in length. Water 
 
AW More Perfect Form of Water - Wheel. 5 
 
 goes down @ and rises through b. The descent of the water in a 
 may be regarded as the motive power; its elevation through 0 as the 
 effect produced. A difference of one inch on opposite sides of the 
 partition ¢ will cause a perceptible current through the pipes. This 
 machine acts with an efficiency of 99.6 per cent. 
 Water acts to turn a wheel in three ways : — 
 1. By weight. 
 2. By impulse. 
 3. By reaction. 
 The first is exemplified in the now nearly obsolete forms of over- 
 shot and breast wheels, which it is not our present purpose to discuss. 
 The action of water by impulse depends upon certain well-known 
 mechanical principles. Force is required to impart velocity to water, 
 Cc 
 
 Fig. 2 
 
 and the velocity imparted is a correct measure of the force employed 
 in imparting it. When water is in motion, force is required to change 
 the direction or velocity of its motion, and the change of motion is a 
 correct measure of the force. Suppose, for instance, a jet of water, 
 moving from D toward B, in the line DAB. At A it encounters a 
 smooth vane which so deflects it that it reaches C instead of B, at 
 the end of one second, A B representing the velocity of the jet, BC 
 is the change of motion occasioned by the vane. ‘The effect of the 
 vane is to impart to the water a velocity BC, and the pressure on 
 the vane is in a direction parallel to CB. To find the pressure of 
 the water on the vane, which is equal and opposite to that of the 
 vane upon the water, we reason thus : — 
 
 Gravity acting freely for one second would impart to the water a 
 velocity of g feet per second. The pressure of the vane acting for 
 one second imparts to the water a velocity BC per second. There- 
 fore, the pressure on the vane is to the weight of water flowing in 
 one second as BC is tog. If a@ represents the cross-section of the 
 
 ; BC 
 stream, the weight of water is-wav, and the pressure is wav 
 
 
 
6 A More Perfect Form of Water - Wheel. 
 
 pounds. This is the pressure in the direction BC. To find the 
 normal pressure we should use, instead of BC, its projection on a 
 line perpendicular to the vane. 
 
 The impulse of water upon a stationary vane is attended with no 
 material loss of energy. The water glides along the vane and glances 
 off at the extremity, in a direction tangent to the latter, with sub- | 
 stantially undiminished velocity. When the vane moves under the 
 action of the water, a portion of the energy is imparted to the vane, 
 and the energy of the stream is correspondingly diminished. It is 
 manifest that if the energy of the stream is wholly imparted to the 
 vane, it must leave the latter with no absolute velocity. That is, its 
 velocity must be equal and opposite to that of the vane at the point 
 of exit. 
 
 
 
 Fig. 3 
 
 Let us now consider the action of a jet on a flat vane perpendicu- 
 lar to its direction and moving in the same direction as the jet. In 
 Fig. 3, let BC=v represent the original direction and velocity of the 
 jet, B D = u the velocity of the vane. Were the vane not present, a 
 particle of water at B would have reached C at the end of one sec- 
 ond. By the action of the vane, the particle finds itself at the point 
 A, at the end of one second, ) A being =v—wu. AC is the change 
 of motion due to the action of the vane. ‘The pressure on the vane, 
 
 hee aA CE Sen ; 
 in a direction parallel to 4 C, is wav ——. The change of motion 
 4 
 
 normal to the vane is DC. ‘The normal pressure on the vane is 
 
 
 
 D Di a 
 P=wav —=wav (1) 
 y g 
 The energy imparted to the vane is 
 UAV — Ww) 
 Pas PAO (2) 
 
 Y 
 
A More Perfect Form of Water - Wheel. 7 
 
 This expression has its maximum value when wu = $v, and becomes 
 in that case 
 aes 
 Pu=fF wav ——4wavh (3) 
 29 
 h being the head to which the velocity is due. In other words, the 
 maximum energy that can be imparted to a flat vane normal to the 
 stream is one-half that of the stream, or the maximum efficiency is 
 50 per cent, and the best velocity for such a vane is one-half that 
 due the head. 
 There are two cases in which the energy becomes 0, viz. : 
 1. When w is 0, i.e. when the vane does not move. In that case, 
 
 Eq. 1 spit eae (4) 
 
 That is, the pressure on the vane is twice that of the head to which 
 the velocity is due. 
 
 
 
 Pg. +4} 
 
 2. When u = v, that is, when the velocity of the vane is equal to 
 that of the stream. 
 
 A cup-shaped vane, Fig. 4, reverses the direction of the water’s 
 motion; so that if such a vane be moving in the direction of the 
 stream with a velocity u, the change of motion will be 2 (v— w). 
 
 2(v—u 
 Paw OV geen ls (5) 
 y) 
 And the energy exerted on the vane is 
 Dab Ul 
 Pu=wav ace a Pe (6) 
 g 
 
 As before, the expression has its maximum value when v = 2 u, 
 in which case 
 
 9 
 
 PUES wav y=" avh. (7) 
 Or the total energy of the stream is imparted to the vane, and the 
 efficiency is 100 per cent. We have proceeded, however, upon as- 
 sumptions which cannot be perfectly realized. In any practical appli- 
 cation the direction of the water’s. motion cannot be exactly reversed, 
 the vanes and their attachments cannot move without friction, the 
 
8 A More Perfect Form of Water - Wheel. 
 
 water cannot approach and leave the vanes without velocity and con- 
 sequent loss of head. The practical interpretation of this result is, 
 that the arrangement described is consistent with the highest effi- 
 
 ciency. Equation 5 shows that when the vane does not, move, the 
 pressure 
 
 . qs 2 
 
 Gas ea eae 9° (8) 
 
 cy 
 
 That is, the pressure on the vane is four times that of the head to 
 which the velocity is due. As in the former case, the energy be- 
 comes 0 when vu = 0, and when wv = v. 
 
 To trace the application of the above principles to different forms 
 of vanes, and to vanes which do not move in the same line as the 
 water, would be foreign to our present purpose, which is merely to 
 put the reader in a position to understand what follows. We are, 
 nevertheless, even now, in a position to notice two points of impor- 
 tance which are commonly lost sight of in the design of water- 
 wheels : — 
 
 1. The purpose of the vanes or floats in an impulse wheel is to 
 effect the greatest possible change in the motion of the water. Their 
 length need be no greater than is necessary to accomplish that 
 change. 
 
 2. It is a condition of the highest efficiency that the water should 
 leave the vane with a velocity equal and opposite to that of the vane 
 at the point of exit. Where the edge of the vane is radial to the 
 wheel, different parts of it move with different velocities, and the ful- 
 fillment of this condition is impossible. 
 
 Reaction is the pressure exerted on the walls of a pipe or vessel 
 from which water is discharged. Strictly speaking, the discharge of 
 water from an orifice does not create pressure within the pipe or 
 vessel from which it issues. It destroys the equilibrium of pressures 
 previously existing. Suppose the pipe, Fig. 5, filled with water 
 under pressure, and free to revolve around the center c. When the 
 orifice o is closed, there is no tendency in the pipe to revolve. The 
 water presses equally upon every part of the interior, and the force 
 tending to turn it toward the right is exactly balanced by that tend- 
 ing to turn it toward the left. When the orifice 0 is opened, the 
 conditions are changed. ‘There is now a small space relieved of 
 pressure on one side of the pipe, while the pressure acts in full force 
 on the other side. The pipe will revolve around the center ¢ in a 
 ‘direction opposite that of the stream. 
 
A More Perfect Form of Water - Wheel. 9 
 
 In the arrangement indicated by Fig. 6, water issues from the 
 orifice o and impinges upon a flat vane or plate P, which is in a line 
 with the center c. The pipe in this case would have no tendency to 
 revolve, as is evident from this consideration. In the arrangement 
 of Fig. 5, the energy of the stream diminishes as the pipe revolves, 
 u being the velocity of the orifice, and v that of the stream, the 
 velocity relative to a fixed point will be v — uw, and the energy of the 
 
 2 
 Ou oi 
 stream = Saat) wav. In the case of Fig. 6, the energy of the 
 ‘ 4 g ry 
 
 re 
 
 
 
 : Coa e we : 
 water will be wav —,-—. That is to say, the energy of the stream 
 * rp cc 
 
 we 
 
 must increase if the system revolves; so that, instead of developing 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 Fig. 6 Fig. 5 
 
 energy, a constant expenditure of energy would be required to keep 
 it in motion. We do not here consider the centrifugal force devel- 
 oped in the water when the pipe revolves... These considerations 
 enable us to estimate the force of reaction. Since the system has no 
 tendency to revolve, the reaction on the pipe must be exactly equal 
 to the pressure on the vane. ‘This we have already found equal to 
 2wah. Let Fig. 7 represent the rim of a wheel containing the 
 orifices 00, so disposed as to discharge water in a direction, as 
 nearly as may be, tangential to the wheel. We assume for our 
 present purpose that the direction is absolutely tangential. Let a, 
 as before, represent the cross-section of the stream. The water is 
 supposed to be at a greater pressure within the wheel than without, 
 the difference of pressure being represented by the head h. 
 
 The best velocity of the circumference is that with which the water 
 
10 A More Perfect Form of Water - Wheel. 
 
 issues, being the velocity due the head h. In this case, the absolute 
 tangential velocity of the water leaving the wheel is o. 
 
 The energy imparted to the wheel is 2 wa uh = twice the energy of 
 the water under the given head. 
 
 This does not imply that the wheel is capable of yielding an 
 efficiency of 200 per cent. In order that the water may issue from 
 the orifices while the wheel is in motion, it must receive a tangential 
 velocity equal to that of the wheel. To impart this velocity requires 
 the energy wavh. The wheel then, under the conditions supposed, 
 is capable of exciting the energ 
 
 2wavh—wavh=wavh 
 
 
 
 Fig 8 
 
 
 
 and from this must be deducted the several losses incident to motion, 
 together with that due to the deviation of the issuing stream from the 
 direction of a tangent. 
 
 In the above illustration of the principle of reaction no account is 
 taken of the effect of the centrifugal force developed by the whirling 
 motion of the water. The introduction of this element makes the 
 problem more complex. The full effect of centrifugal force appears 
 in the arrangement of Fig. 5,.in which a pipe filled with water, under 
 pressure, revolves around a center ¢, discharging from an orifice 0, 
 the entire mass of water being in rotation with uniform angular 
 
A More Perfect Form of Water - Wheel. 11 
 
 velocity. This principle is demonstrated later on: — If we give the 
 orifice o a velocity, v = u = V2gh, we develop at any point in the 
 whirling mass of water a centrifugal force equal to the force repre- 
 sented by the head due the velocity at that point. The pressure 
 acting on the orifice under this condition will be 2wah, and the 
 velocity of the discharge will increase to v= Y2q K 2h=1.4143 
 V2gh. If we increase the velocity wu of the orifice so as to make 
 u = 1.4143 29h, we develop a still greater centrifugal force, and so 
 on. So that the condition of maximum efficiency v = — wu is impos- 
 sible. 
 
 If, for example, we attempt to determine the value of uw on the 
 assumption that it is equal to the velocity with which the water issues 
 from the orifice 0, we should have the equation 
 
 he 
 waaf2g (ht —) 
 29 
 
 or u?= 2gh + u?, which is only possible when h =o. 
 In the arrangement of Fig. 7 the entire mass of water is not set in 
 motion with uniform angular velocity from the center outwards, and 
 
 the pressure head developed by centrifugal force is less than that due 
 u? 
 
 the velocity of rotation. Suppose it to be represented by coon 
 Y 
 
 2 
 
 
 
 being less than unity. 
 In this case 
 
 = 42 eet 
 Uu= g (h 29 
 
 when u?=2gh+ mu? and 
 
 
 
 The fulfillment of the condition » = — uw would be possible in such 
 a wheel, but w would be so great that the losses from friction and 
 resistance to motion would exceed the waste of energy incident to a 
 lower velocity. 
 
 A wheel of the form indicated at Fig. 8 would operate wholly by 
 reaction. The water within the wheel would have a whirling motion, 
 the tangential component of which is equal to the velocity of the 
 inner circumference. We know this because otherwise the water 
 could not enter the wheel. This whirling motion is not the direct 
 effect of the head. It is imparted to the water mechanically by the 
 
12 A More Perfect Form of Water - Wheel. 
 
 wheel. The pressure of the water does not diminish till the latter has 
 entered the buckets. The energy of reaction exerted at the orifices of 
 discharge exceeds the total energy due the head. ‘This, as we have 
 already seen, is partly absorbed in imparting the whirling motion to 
 the water before it enters the wheel. 
 
 Impulse and Reaction Wheels. —- Most wheels act partly by im- 
 pulse and partly by reaction. In a wheel acting purely by impulse 
 the water issues from orifices with the full velocity due the head and 
 impinges upon vanes moving, preferably, with half that velocity. In 
 a purely reaction wheel the work of the water is finished when it 
 issues from the orifices of discharge. A tangential velocity equal to 
 that of the influx orifice is imparted to the water, not directly by the 
 head, but indirectly through the action of the wheel. Neglecting 
 
 friction and losses incident to the movement of the water, the energy 
 
 “imparted to the wheel by the water is twice that due the head and 
 quantity, but of this, one-half or more is useless energy, being that 
 expended in imparting the necessary tangential movement to the 
 water. 
 
 A wheel which has found many useful applications in cases of high 
 head and small flow is the Pelton or hurdy-gurdy wheel. A jet of 
 water acts upon cup-shaped vanes (Fig. 4) disposed around the 
 circumference of the wheel. ‘This is an impulse wheel pure and _ 
 simple. The Barker Mill (Fig. 5) and the forms of Figs. 7 and 8 
 are reaction wheels pure and simple, observing that in 5 and 7 the 
 orifices of influx and discharge are the same. 
 
 All wheels with guides may be regarded as acting partly by impulse 
 
 and partly by reaction. ‘The tangential velocity of influx is imparted 
 directly by the head. The velocity of the efflux orifices is less than 
 that due the head and greater than half the same. At full discharge 
 they have more the character of reaction wheels. At diminished 
 discharge they become impulse wheels. 
 + Impulse and reaction wheels have this in common: Their effi- 
 ciency depends upon the change which they effect in the direction of 
 the water’s motion, perfect efficiency implying exact reversal. Perfect 
 efficiency also requires the water to leave the wheel with no tangen- 
 tial velocity, a condition inconsistent with radial orifices of dis- 
 charge. 
 
 Centrifugal Force. — We shall have occasion in what follows to 
 refer to certain propositions relative to the action of centrifugal force 
 ‘in the proposed form of wheel. I give in this place the demonstra- 
 
AW More Perfect Form of Water - Wheel. 13 
 
 tion of these propositions so far as required for our purpose. It is 
 impossible to set forth and demonstrate these propositions without 
 the aid of mathematical symbols and operations which are only 
 intelligible to those who have studied that subject. Those who have 
 not must ask the advice of those who have, or else take the results 
 upon trust. 
 
 A cylindrical vessel, open at the top and partly filled with water, 
 revolves around its vertical axis. What form will the surface as- 
 sume ? 
 
 Let Fig. 9 represent the vertical section of such a vessel. 
 
 The ordinates of any point o, in the curve, are x= AD and 
 y= Do; ow is the angular velocity. The surface must take such a 
 form that the resultant of the forces acting at o will be normal to the 
 surface. 
 
 
 
 Fig..9 
 
 The forces acting upon any given small mass, whose weight is w, 
 
 . TO 
 at o, are: vertical w, horizontal — wy. If we take y to represent 
 J 
 
 the centrifugal force, i.e. the horizontal force, then 2 will represent 
 W 
 the vertical force, and the forces acting at o may be represented by 
 a) O,and =f = Fo=cD. The resultant of these two forces will 
 W 
 
 be represented by co, which must be normal to the surface. Now 
 
 a = cD is independent of the position of o. In other words, the 
 W@W 
 
 subnormal is constant, which is a characteristic of the parabola. 
 Any vertical section of the surface through the axis, therefore, is a 
 parabola. 
 
 The equation of the parabola referred to its axis and vertex is 
 ordinarily written, 
 Thee ted Ga (9) 
 2 P being the value of « when 7 = y. 
 
14 A More Perfect Form of Water - Wheel. 
 
 
 
 
 
 Stee Bee ; . Cy tla 
 Differentiating (9) 2y¥dy=2 Pda, whence = =-—. The sub- 
 wv y 
 bd d Y ry. q e 
 normal is 7¥ “=P, Therefore P—-—,, and the equation of the 
 7 w~ 
 curve may be put in the form 
 ses 10) 
 a 
 24 ( 
 
 wy? is the velocity of rotation at 0; therefore, at any point in the 
 vessel, the height of the surface above its lowest point is the height 
 due the velocity of rotation. 
 
 The quantity of water in the cylinder remaining constant, and the 
 cylinder being supposed of indefinite height, the rotation will not 
 alter the aggregate pressure on the bottom, which is the weight of 
 the liquid. ‘The rotation will only alter the distribution of the 
 pressure. 
 
 The question often arises, What is the pressure on the ends of a 
 filled cylinder in rotation? Referring to Fig. 9, suppose, at the 
 lowest point of the curved surface, a horizontal diaphragm ab to be 
 inserted, forming a closed cylindrical vessel abcd. ‘The pressure 
 on both sides of the diaphragm will be equal, that is, the centrifugal 
 force developed in the mass below the diaphragm will be exactly 
 equal to the weight of water above it. The water will be the con- 
 tents of the cylinder /mba, less the contents of the paraboloid 
 1Am. The volume of a paraboloid is one-half that of the cireum- 
 scribing cylinder. The pressure on « b, therefore, is $7 wX a A®?Xal. 
 
 Putting + for the radius of the cylinder, the pressure on ab is 
 w- y2 
 
 4warr? oe being the weight of a cylinder of water of radius 7, 
 
 2 2y 
 
 and height equal to one-half the head due the velocity of the exte- 
 rior circumference. It must be observed that the effect of atmos- 
 pheric pressure is not here considered. 
 
 A vessel filled to the height H, the radius being 7, is set in motion 
 with the angular velocity ». ‘To find the highest and lowest points 
 of the water. 
 
 The total volume of the water is 77? 77; ditto of the paraboloid, 
 
 
 
 a = aa Late eethe height of the lowest point in the surface of 
 : | 
 w- y? 
 
 
 
 water above the bottom of the vessel. Then 77? (a+ 4 aa 
 
 = 71? H, whence 
 
A More Perfect Form of Water - Wheel. 15 
 
 
 
 (11) 
 
 2 2 
 Ww”) 
 
 yy 
 
 il 
 e 
 
 The highest point of the surface will be at a height above the 
 
 
 
 
 
 
 
 2 mr2 
 lowest point, or at a height # + 3 : above the bottom. Its height 
 above the bottom will be 
 w- yr? 
 = Tye. (12) 
 
 ae 
 
 Suppose the vessel abcd revolving around a horizontal axis, and 
 it is required to find the pressure on the ends. We must remember 
 that the centrifugal force developed in any particle of water is in no 
 way affected by the direction of the axis of rotation, and will be the 
 same as before. To this must be added the pressure due to the 
 weight of the water in the cylinder. The surfaces on which the 
 water acts are each r7?. The average static head, in the horizon- 
 tal position of the axis, is7. This part of the pressure will be w77°, 
 and the total pressure | 
 
 Mar? (es th) (13) 
 
 h being the head due the velocity at the exterior circumference. ‘The 
 same result is obtained in a more direct manner, as follows. Put R 
 for the exterior radius of the cylinder, and 7 for the distance, from 
 the axis, of any point under consideration. ‘The centrifugal force, 
 
 at any point, is 7° 7. This is the force with which a cubic foot of 
 
 water pulls away from the center. The actual mass of water to 
 which this applies is wh2a7rdvr, 6 being the axial length of the 
 warrbar w 
 
 cylinder. The force per square foot is — ———_— w?r =— o’ rdr. 
 Gem U0 g 
 
 This.is the pressure which the particles at a distance + from the axis 
 of rotation exert on all the surface outside of them. The end sur- 
 face on which this pressure takes effect is (??— 7°). The total 
 pressure therefore due to centrifugal force is 
 
 rR 
 a of 1) (liao. Or 7, 
 O 
 
 
 
 where A is the area of the end of the cylinder. 
 
16 A More Perfect Form of Water - Wheel. 
 
 Adding the pressure due to the weight of the water, we have for 
 the total end pressure 
 om Fe (hi hy, (14) 
 
 the same result as before, putting / in the place of r. 
 
 We are now prepared to consider the special question which will 
 arise later, viz.: A cylindrical vessel in rotation, immersed in water, 
 discharges water through a small orifice in its exterior circumference. 
 What will be the pressure on the ends, from without inward? 
 
 First, suppose the axis of rotation vertical, and let d be the depth 
 of water on the upper end of the cylinder. Let P be the pressure of 
 the atmosphere in pounds per square foot. Then the total downward 
 pressure on the upper end of the cylinder is tr? (P+wd). The up- 
 ward pressure per square foot at the outer circumference is P + dw, 
 
 2.9 
 
 “a 4s 
 
 @ . 
 and at the center P+ dw— ay w. ‘The pressure varies from the 
 
 « 
 
 center outward, according to the parabolic law. ‘The aggregate up- 
 , h 
 ward pressure is, therefore, 77? (P+ wd — Ws ), h being the head 
 
 due the velocity of the exterior circle, and the effective pressure act- 
 
 ing downward is 
 = tar wits (15) 
 
 The effective upward pressure on the lower end will be the same, 
 being increased by the pressure due to the height of the vessel, and 
 diminished by the weight of water in the vessel. 
 
 Next, suppose the axis of rotation to be horizontal, and let d rep- 
 resent the depth of the axis below the surface of the water. The 
 depth of the orifice, while in rotation, will vary from d—r tod+7, 
 its average depth being d. 
 
 The pressure per square foot directed inward will be P + wd, the 
 ageregate pressure being r7?(P+ wd). The pressure directed 
 outward will be P+wd pounds per square foot at the circum- 
 
 2 pad 
 
 W@W 
 ference, and P+ wd — at the center, the aggregate outward 
 
 pel 
 pressure being rr?>(P+wd—gwh). As before, the effective 
 pressure tending to force the cylinder heads inward is $77? wh= 
 
 
 
 h : ; 
 mrws; that is to say, the pressure represented by one-half the 
 
 in 
 
 head due the velocity of the orifice. It appears that the orifice 
 would play an important part in the pressure on such a revolving 
 
A More Perfect Form of Water - Wheel. 17 
 
 h 
 cylinder. If open, the pressure represented by . would act inward, 
 
 if closed, outward. It is manifest that opening a larger orifice at 
 the center of the cylinder would prevent the lowering of the pressure 
 within the same, and have the same effect as closing the outer 
 orifice, and the closing of the central orifice would have the same 
 effect as opening the outer. 
 
 This result would not hold if the velocity were so great as to make 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 Pio 10 
 
 h greater than the height due the atmospheric pressure increased by 
 d, which would imply a vacuum at the center of the cylinder. 
 
 It is not the purpose of this little essay to enter into a general de- 
 scription of existing turbines. Some of their most prominent features, 
 however, must be referred to in order to understand the advantages, 
 if any, of the proposed form. 
 
 Floats. — Figures 10, 11 and 12 may be taken as types of the 
 existing form of floats. In Fig. 10 the letter f below the band 
 denotes the same float as f above. ‘The water enters the wheel in a 
 horizontal direction above the band and leaves it in an inclined 
 
18 A More Perfect Form of Water - Wheel. 
 
 direction below. The orifices of discharge extend from the circum- 
 ference nearly to the center. The same characteristics appear in 
 ll and 12. ab, Fig. 11, is the length of the orifice of discharge, 
 in a radial direction. The length of the passage traversed by the 
 water appears clearly in 12. ‘These floats are at variance with both 
 conditions of efficiency referred to on page 8. ‘They present a 
 longer passage to the water than is necessary for the required change 
 of direction. The orifices of discharge are radial, and, in Fig. 10, the 
 
 
 
 inner end of the orifice moves with a velocity less than half that of 
 the outer end; whereas, the velocity imparted to the water by the 
 head acting on the wheel is substantially the same at the inner end 
 as at the outer. The highest efficiency cannot be obtained from a 
 wheel with floats of this form. The extent to which this form of | 
 float affects the efficiency at full gate is not wholly a matter of 
 conjecture or theory. The Boyden-Fourneyron wheel (Figs. 13, 14 
 and 15), on the most rigorous trial, gave an efficiency of 88 per cent. 
 The Swain wheel (Figs. 11, 12 and 16) has shown a maximum of 
 
A More Perfect Form of Water- W heel. 19 
 
 85. This falling off of 3 points must be charged to the form of the 
 floats. Wheel No. 10 is still more faulty in this respect. At a 
 competition test at Holyoke several years ago, where a large number 
 of wheels were entered for trial, this wheel gave, as a maximum, not 
 quite 83 per cent. 
 
 One maker of wheels —no doubt a very excellent mechanic and 
 skillful man of affairs —says that the great power of his wheels 
 ‘* consists in the use of long buckets gradually leaning forward with- 
 out any short or abrupt bend to prevent the natural flow or passage 
 of water, and in turning the upper and outer half of the buckets 
 forward in the direction the wheel runs, thus compelling the water 
 that comes through the chutes to gather in largest bulk on the outer 
 
 
 
 parts of the buckets and therefore exert more pressure because 
 
 2 
 
 pushing chiefly where there is most leverage.’’ This quotation shows 
 how loose and vague are the ideas of the action of water entertained 
 by makers of wheels. 
 
 Guides. — Practically inseparable from every existing form of 
 water-wheel are the guides for giving a suitable direction to the 
 water entering the wheel. figures 13 and 14 show these parts as 
 applied to the Boyden-Fourneyron Turbine, in which the water flows 
 
 from within outward. They consist, here, in a series of thin steel 
 
 
 
 blades inserted in the disc and forming a series of narrow and deep 
 channels. In addition to their function of guiding the water, they are 
 attached, at their upper outer corners, to the supply pipe, and serve 
 the purpose of firmly uniting this part with the disc, a purpose which 
 
PAY AA More Perfect Form of Water - Wheel. 
 
 it would. be impossible to secure in any other way without introducing 
 parts that would interfere with the action of the water. It must be 
 observed that the dise is an entirely indispensable part of the 
 apparatus, serving to relieve the wheel and its bearings of the 
 pressure of the water. In almost every form of center-vent wheel 
 the guides sustain a plate through which the shaft passes in a stuffing- 
 box and which serves, to relieve the wheel of pressure. ‘The point to 
 
 
 
 Fig. 13 
 
 which I wish to direct attention now, is this: The necessity of these 
 parts to the structural solidity and strength of the wheel and case has 
 diverted the attention of wheel-makers from the question: Are the 
 guides necessary to the mechanical action of the wheel? 
 
 In a wheel whose design and construction is otherwise consistent 
 with the highest efficiency, it is possible to so adjust the angle of the 
 guides, the opening of the guides and the discharge orifices of the 
 wheel as to secure, at full gate, a high degree of efficiency. Nearly 
 the highest that any wheel admits of. ‘This adjustment requires a 
 
A More Perfect Form of Water - Wheel. 21 
 
 degree of knowledge of hydraulics that few men possess. But how- 
 ever complete this adjustment may be at full gate, it begins to 
 be deranged as soon as the gate begins to close and, generally, when 
 the discharge is diminished one-half, is wholly destroyed, and the 
 efficiency much reduced. Wheels of the form shown in Figs. 13, 14 and 
 15 have shown, at full gate, the highest efficiency ever attained by : 
 turbine. But observe how the conditions of efficiency are changed 
 when the gate has descended to the position shown in Fig. 15. In 
 this case, the water does not enter the wheel in the direction that the 
 
 
 
 
 
 
 
 
 
 
 
 
 N : 
 
 PATON NT NG Row \ 
 ITE x ! Wy ‘hh 
 CLIN) | Ny AHN | Natt. 
 \\|\\ NI LIN iN 
 N NY IN ING 
 
 NS AS ESQ ‘i 
 BS y | NS Na 
 
 
 
 SANSass > 
 
 
 
 
 ~ == 
 
 
 
 
 
 GAGBtAUE Ts 
 
 
 
 
 
 
 
 y 
 Y 
 Y 
 y 
 Y 
 
 
 
 
 
 it 
 
 aT 
 
 [ No ng 
 
 TN 
 
 SFISIGNS WA 
 
 4 
 1 
 
 
 
 Fig. 14 
 
 ceuides would lead it. The opening formed by the gate and two 
 consecutive guides is an orifice which the water approaches from 
 various directions and from which it issues in a direction at right 
 angles to the plane of the orifice, or radial to the wheel, at least 
 tending to do so, and becoming so when the gate is nearly closed. 
 The closing of the gate alters the direction of the water entering the 
 wheel and so impairs its efficiency. But this is not the whole effect, 
 nor the worst. The stream entering the bucket doesnot fill and pass 
 smoothly through it as it does at full gate, but wastes its energy in 
 commotions and eddies. It tends to fill the bucket at its exit, and so 
 is discharged with a much lower velocity than it possessed at its 
 entrance. . 
 
bo 
 bo 
 
 AA More Perfect Form of Water - Wheel. 
 
 Here another remark might be offered as to the length of the 
 passages through the wheel. ‘These, as before observed, need be no 
 longer than will suffice to impart the necessary change of direction to 
 the water. At part gate the portion of the passage not filled by the 
 stream is filled with dead water. ‘The shorter the passage is, the less 
 the stream mingles with the dead water, and the less the diminution 
 of velocity sustained by it during its passage. ‘The longer the 
 passages are, the more the stream expands in its passage, and the 
 lower (up to the point where the stream completely fills the passage) 
 will be the velocity of discharge. 
 
 
 
 
 
 
 
 
 Yl hyp aps 
 
 YW 
 
 
 
 
 \y 
 
 
 
 This is the inherent and unavoidable defect of the turbine: That 
 it cannot use a diminished quantity of water with the same efficiency 
 as the full quantity that it is fitted to discharge. A wheel which 
 shows an efficiency of 88 per cent at full discharge generally shows 
 not above 65 at half discharge, and below half it is so low that 
 makers do not usually care to publish it. 
 
 Nevertheless it is exceedingly desirable that water-wheels should 
 be able to use a small quantity of water as efficiently as a large 
 quantity. Streams vary constantly in flow. During several months 
 in every year the flow of the stream is greatly below the capacity of 
 the wheels. Steam is employed to make good the deficiency, and 
 part gate is the normal condition of the wheels. It is extremely 
 unfortunate that, at the period when water is the scarcest, it should 
 
 
 
 
A More Perfect Form of Water - Wheel. 23 
 
 have to be used at a greatly diminished efficiency. The ability to 
 work economically at $, + or even ¢ of full power is just as desirable 
 in a water-wheel as in a steam-engine. 
 
 Innumerable have been the devices for alleviating this defect. 
 I would by no means attempt to describe or even name them. I will 
 mention several of the most prominent. 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 TL 
 APS p 
 
 
 
 
 
 
 
 
 
 
 
 
 OT 
 
 sveeeut 
 
 ZZ 
 
 
 
 
 \ ae 
 
 i 
 
 aanuuue 
 SD 
 QAAAAAN 
 S 
 
 
 
 
 
 
 
 
 a= 
 
 Seecece 
 Sespan 
 
 = 
 se 
 
 apse/ 
 See! % ay 
 3 nF Se y 
 aoa 4 R Re aK 
 ark SK 
 N N N N 
 N N N N 
 N N N N 
 N N N N 
 N N N N 
 N N Ny 
 N N N NN 
 N N N N 
 N N N N 
 N N N N 
 N N N N 
 N Ns Go SS \ \ 
 N N S SS N 
 
 Fig. 16 
 
 1. In the wheel shown in Figures 13, 14 and 15, the guides have 
 been given a sharp backward inclination, so as to stand at an angle 
 of 45 degrees or less with the horizontal. In this arrangement a 
 particle of water at the top of the guides, near the outer end, 
 approaches the opening when the gate is low, not in a vertical, but in 
 an inclined, direction, with a large component of tangential velocity. 
 In a low position of the gate, however, the water does not reach the 
 outlet of the guides with a suflicient velocity to overcome its tendency 
 to issue in a radial direction. Moreover, this disposition does not 
 affect the most serious source of loss, which is the tendency of the 
 
24 A More Perfect. Form of Water - Wheel. 
 
 water to fill the orifices of discharge from the wheel, and issue there- 
 from with a greatly diminished velocity. 
 
 2. In the wheel just referred to, several horizontal partitions have 
 been introduced separating the wheel into a number of compartments, 
 or ‘* stories,” to prevent the stream from expanding during its pas- 
 sage. This arrangement appears well calculated to diminish one 
 source of loss incident to this form of wheel, but the results were not 
 encouraging enough to lead to the extended adoption of this method. 
 The tendency of the water to escape from the guides in a radial 
 direction leads to losses of head which cannot be obviated by these 
 means. In this condition of the wheel the guides cease to guide. 
 Were the guides removed and the water allowed to enter the wheel 
 entirely free from their control, there is reason to suppose that it 
 would take a direction more conducive to efficiency. 
 
 3. In the Swain wheel (Figures 11, 12 and 16) the guides are 
 attached to a broad flange in the gate G, which closes by rising. As 
 the gate rises, the guides enter an annular chamber above, which 
 diminishes the free height of the guide passages in the same degree 
 that it diminishes the opening of the gate. This appears to largely 
 obviate the loss arising from a change of direction in the water 
 entering the wheel at part gate, and greatly improves its action. 
 Tests of this wheel have shown an efficiency of 85 at full discharge, 
 60 to 70 at half, and some 45 at one-fourth of the full discharge. 
 These results exceed any existing wheels on part gate, and are not 
 inferior on full gate to any but that shown in Figures 13, 14 and 15, 
 which has shown an authentic result of 88 per cent, and for which 
 higher results have been claimed. The record of the Swain wheel, 
 however, shows that there is still much to be desired in point of 
 efficiency at part gate. It shows that this arrangement of gate does 
 not obviate the second source of loss, viz. the expansion of the 
 stream in traversing the buckets. 
 
 4. Fig. 17 represents another attempt to use large and small 
 quantities of water with good efticiency in the same wheel. This is 
 nothing less than a double wheel, with two sets of floats, two sets of © 
 cuides, and two gates, all on the same shaft. The outer wheel w w 
 is governed by the cylindrical gate GG, which opens downward. 
 The floats ff and guides yg belong to this wheel. ‘The inner wheel 
 w' w' is controlled by the gate G’ G’, opening upwards, and has the 
 cuides g/g’ and floats f’ f’. In times of abundant water both gates 
 are open. In lower stages the inner gate is closed, leaving the outer 
 
A More Perfect Form of Water - Wheel. 
 
 be 
 Or 
 
 one open. In still lower, the outer is closed and the inner is used. 
 Or, these several dispositions are made not in accordance with the 
 stage of the stream, but with the requirements of the mill. 
 
 This arrangement has the advantage that it fixes three different 
 quantities of water which can be used with good effect, instead of one 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 | a 
 |S 
 Penstock || mill 
 N | i 
 N roll 
 N ol) 
 a lis id 
 N | 3 ' 
 N t ins 
 N TR | ye tr N 
 N Y till g N 
 N Y's a Z N 
 N YY ‘\3 if las N 
 N y vie i! A) Sy N 
 N y \ Vill Yj Ss N 
 N Y} \ (iM a N 
 N Y \ > - Ws N 
 . N---- 4H Ah S 
 WAI N | j SN \ Ns 7 
 VSS) N N 
 S ZZ, N RSSSSSSSJ | Seeerecemnnees N iP N 
 UN N \ N VY N 
 \ N N : \ N YW. A 
 V GH, Sf | Ne Myc . 
 eee NS YA A 
 NY 
 ; N Nh Z S 
 SSG) Hh AA || "om Ao iN 
 \ Fr E YZ) We F Y fF iN 
 Ni A \s EY Ns JI 7] FY Z j 
 Ni Z A. AC WA * > y t 
 Yi Z 
 
 
 
 Draft-tube 
 
 SJfJu 
 eae 
 
 (ZIZILLLL LLL LLL LLL LAY 
 A 
 43 
 
 SS 
 OS 
 
 ZZZZZZLE 
 
 Fig 17 
 
 as in the ordinary case. Any quantity between the capacity of both 
 wheels and the capacity of the small wheel can be used with little 
 loss. Below the latter quantity the defect appears in full force. The 
 form of bucket adopted in this wheel, although of good efficiency at 
 full gate, is one of the worst forms extant at part gate. However 
 small the quantity of water admitted to the wheel, it is certain 
 to issue in a uniform stream from the orifices of discharge with 
 
26 A More Perfect Form of Water - Wheel. 
 
 greatly diminished velocity. This wheel is also liable to losses from 
 which ordinary types are exempt. A turbine, in order to give its 
 best effect, must run with a velocity which bears a certain ratio to 
 that due the head. The connections must be such as to admit of this 
 velocity when the shafting and machinery run at their normal speed. 
 Two wheels of different diameter on the same shaft, cannot both run 
 at this velocity. Moreover the wheel out of action is kept in rotation 
 as a useless drag upon the other. 
 
 
 
 Fig. 18 
 
 5. In the forms of wheel shown in Figures 18 and 19, the guide 
 serves the double purpose of guide and gate. It controls the width 
 of the guide passages by turning upon a hinge or joint under the action 
 of the regulator. In 18, the outer end of the guide is fixed; the 
 inner end is susceptible of an outward movement, limited by the 
 adjoining guide. In 19, the inner end is jointed, the outer end 
 rotates. The passages are closed when each guide touches the 
 adjoining guide. 
 
 The same remark may be made of this arrangement as of the 
 Swain wheel. It obviates the loss incident to a change of direction 
 and velocity of the eiftering water. But it in no way affects the 
 second source of loss, viz. that arising from the expansion of the 
 
~l 
 
 A More Perfect Form of Water - Wheel. 2 
 
 stream in the wheel passages. To avoid this latter loss would require 
 the wheel passages to be contracted in the same proportion as the 
 guide passages, when the discharge is reduced. 
 
 From this brief survey of the present state of the art of wheel- 
 building we are, I think, justified in asserting that no existing form 
 of wheel is free from grave, inherent and unavoidable defects, defects 
 which are material at full discharge, and become more and more 
 marked as the discharge diminishes. No existing form is consistent 
 
 
 
 in design with the highest degree of efficiency or with the well estab- 
 lished principles of hydraulics. 
 
 They have resulted from tentative methods and from partial and 
 incomplete knowledge, not from a thorough and comprehensive 
 study of the whole subject. 
 
 It appears to me, also, that the only hope of developing a perfect 
 water-wheel lies in a radical departure from existing forms, every one 
 of which is intrinsically defective. 
 
 The whole subject of improvement turns upon this question: Are 
 the guides necessary and indispensable to the efficient action of the 
 wheel? Iam convinced that they are not. It appears to me that in 
 a wheel surrounded by a free space sufficient to allow the water to 
 
28 A More Perfect Form of Water - Wheel. 
 
 move without constraint, it would naturally take the direction and 
 velocity most conducive to the efficient action of the wheel. ‘This, in 
 a center-vent wheel, means a velocity, the tangential component of 
 which is equal to the velocity of the exterior circumference of the 
 wheel. What reason have we for supposing that the water will take 
 that velocity’?* We have the very simple and satisfactory reason 
 that the water cannot otherwise enter the wheel. ‘This answer, how- 
 ever, carries another question with it, viz.: Though there is no doubt 
 that the water would take the required velocity, would not that move; 
 meut be attended with serious loss of energy? I think not, for these 
 reasons : — 
 
 K 
 
 
 
 Fig. 2O 
 
 1. When an orifice is opened for the escape of water under press- 
 ure, the water will approach the orifice. If the orifice recedes, the 
 water will follow and overtake it. 
 
 2. A movement of rotation, under the conditions supposed, is in 
 accordance with the natural tendency of the water. Water in a cir- 
 cular vessel, discharging through a central orifice, spontaneously 
 takes a movement of rotation.+ 
 
 * This would be true for an orifice opening normally, as 7, Fig. 20, or backwards, as 
 atk. It would not be true for an orifice opening toward the direction of rotation, as @. 
 In that case, tle water would be “ scooped” into the wheel without taking the full ve- 
 locity of the wheel. 
 
 + When a heavy particle (i.e. a particle having weight) moves freely under the action 
 
 of a force directed toward a fixed point, the line joining it with the fixed point describes 
 equal areas in equal times. This general proposition appears from Fig. 21. Suppose a 
 
A More Perfect Form of Water - Wheel. 29 
 
 3. It is a universal principle of nature that every movement is 
 performed with the minimum expenditure of energy. Water, in the 
 case supposed, will enter the orifices of the wheel with the least loss 
 of energy possible under the existing conditions. Now we know 
 that if the water reaches the orifice with the rotatory velocity of the 
 wheel, the loss of energy will be slight; and since we know that ex- 
 isting conditions admit of this velocity, and that the actual loss will 
 be as small as is consistent with existing conditions, we are entitled 
 to assume that the actual loss will be slight. 
 
 4. If we assume that the water enters the wheel with a less ve- 
 locity than that of the floats, and suddenly acquires the motion of 
 the latter, this sudden accession of velocity does not necessarily im- 
 ply any loss of energy. Loss of energy may occur in several ways. 
 
 
 
 D Cc B A 
 
 Fig. ra | 
 
 heavy particle to move with uniform velocity.in the line A D,— A B, BC, CD, being 
 the equal distances moved in successive elements of time. Let O be any point whatever. 
 The triangles A OL, BOC, COD, are the areas described in equal times with reference 
 to 0. These triangles are all equal to each other, having equal bases and the common 
 height A H = the perpendicular distance of O from A D. 
 
 Now introduce the supposition that, at @, the particle is acted on by a force directed 
 toward QO, which would cause it to move a distance ( / in the element of time. At D 
 draw DG, parallel to CO, and on it lay off DE=CF. Draw CE, then C EO is the 
 area described in the element of time under consideration. Join E O, and draw OG, 
 perpendicular to DG. The triangle C DZ = O D £, both having the base D /, and the 
 common altitude 0G. 
 
 The triangle CH O=CDO—(CDE—DKE)+O0DE—DKE=CDO. There- 
 fore, the areas described in successive elements of time are equal; and in order to make 
 these areas equal, the velocity must increase as the particle approaches the fixed point. 
 This is the law discovered by Kepler, and it governs the motions of the planets. 
 
 A particle of water in the wheel-case is under conditions very similar to those of a 
 planet in free space. It enters the case with a certain velocity. As soon as it enters, it 
 is acted on by a force directed toward a fixed point, i.e. the center of the wheel. It 
 does not move straight toward the wheel, but circles around it with an increasing 
 
 velocity. 
 
30 A More Perfect Form of Water- Wheel. 
 
 One of these is the communication to the water of motions other 
 than those necessary for its entrance into the wheel, i.e. useless 
 movements. Commotions in water commonly arise from irregulari- 
 ties in the channels, or in the circumstances of movement. No such 
 irregularity exists here. Loss of energy occurs when water passes 
 orifices with contraction. No appreciable loss can occur here from 
 that cause. ‘The contraction, if any, is only that due to the radial 
 component of the velocity, which is not over one-fifth of the tangen- 
 tial component. Each orifice of entrance has the contraction sup- 
 pressed on one side, and it is rounded on two others. On the fourth 
 side, contraction can only exist in virtue of the excess of the wheel’s 
 velocity over the tangential component of the water’s velocity. Nei- 
 ther contraction nor commotion therefore can exist as a source of 
 serious loss. 
 
 The most natural conception of the phenomenon is this: The water 
 which has passed the tips of the floats is in motion with the full ve- 
 locity of the wheel, the next outside film a little slower, the next 
 slower still, etc., the acceleration being communicated by fluid fric- 
 tion. The loss of energy consists in the fact that, of two consecu- 
 tive films of water, the one nearest the wheel moves a little faster 
 than the one more remote, so that the energy expended is not fully 
 represented by the velocity acquired. ‘The energy represented by 
 the velocity imparted to the film of water is not lost. 
 
 The loss, however, from fluid friction in this wheel is no greater 
 than in any wheel of equal surface (speaking now of the general 
 surface of the wheel, not that containing the orifices) and equal 
 velocity. In any wheel, the film in contact with the wheel moves 
 with the velocity of the wheel, the adjoining film a little slower, the 
 next slower still, etc., precisely as in this. 
 
 I have, therefore, become convinced that the guides, although in 
 existing forms of wheel, necessary for constructive reasons, are in 
 no sense essential to the efficient action of the wheel. That they are 
 attempts to force upon the water a direction and velocity which it 
 would take spontaneously if relieved of constraint. That, ordina- 
 rily, they but imperfectly fulfill their purpose at full gate, and are a 
 prolific source of waste at part gate. The first step in the improve- 
 ment of the water-wheel, therefore, is, as it appears to me, to dis- 
 pense with the guides, and to adopt a form of wheel that does not 
 require them for constructive reasons. 
 
 The second essential to the perfect action of the water is a gate 
 
9 
 
 A More Perfect Form of Water - Wheel. 31 
 
 
 
 
 » 
 i y 
 wrevesssrnanpwaswawae sede PLD PPP PET I PPP PLD PI PIS a 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 ee, 
 
 
 
 
 
 
 
 
 
 
 
 
 ee Bor? rot | -__ N 
 Ze az" 
 
 
 
 COLI LILA “a 
 t) 
 \) 
 \ 
 rest 
 Sacks 
 ” 
 2 4 
 all! 
 | iD RF oo 
 
 nn 
 LY 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 RY sS 
 
 nl a rn 
 
 ! AS ina al 
 
 is a \ | 
 
 ; SEED | See | SRE 
 
 f Ke tN oN 
 
 : aa ee : 
 N Nissi} N 
 N : 
 
 
 
 a ZZ 
 
 
 
 
 
 
 
 
 
 tas 
 "ee, 
 Pe, 35 
 5 
 > 
 ea. 
 ‘a. 
 ooo 
 > 
 
 
 
 
 
 
 
 
 
 
 
 \ 
 \' 
 \\ 
 \ 
 \\ 
 \ 
 ‘ x 
 % 33 
 g y 
 ree 3 Ka 
 N N N 
 N N 1 
 : N N 
 : N f . 
 ; ) ontal sh aft N N 
 & y 
 : N Vertical section. Scales! N 
 : N i N 
 \ i : 
 | | 
 : Draft- tube H H Draft tube . 
 : i N N 
 N N N N 
 N \ . N L 
 : N Fig. Pal ae : \ 
 N N 
 : . | 
 N 
 
 e 
 
 that shall, in closing, contract not only the influx of the wheel, but 
 
 the entire passages through it. 
 
 The attainment of these conditions necessitates an abrupt and 
 
 radical departure from all existing forms of water-wheel. 
 
 The figures which follow illustrate my idea of a form of water- 
 It must 
 
 wheel calculated to obviate all the above described defects. 
 
fect Form of Water - Wheel. 
 
 ev 
 
 A More Per 
 
 SSSR AS AAAS anennnnnnnn ny //74 
 
 COSSSSUS SSSI SG 
 
 
 
 
 
 aioe 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 v2 
 
 
 
 
 
 shaft 
 
 on horizontal 
 
 No 2 
 
 Wheel 
 Sectional plan- half, looking up, and half. looking down. 
 
 Fig. £3 
 
A More Perfect Form of Water - Wheel. 33 
 
 be observed that these are, in no sense, working drawings. They 
 are intended to exhibit the principles on which such a wheel could be 
 constructed, and are sufficiently detailed to show that the construc- 
 tion of such a wheel involves no mechanical impossibility, and is 
 entirely within the resources of modern engineering. 
 
 Fig. 25 shows the form of the floats and float passages, Figs. 22 
 and 23 the form of the gate. The gate g is a conoidal dise with a 
 central hub, which slides on the shaft. A heavy hub, secured to the 
 shaft, carries the flat dise or discs dd, to which the floats f are se- 
 
 
 
 Wheel Na2 .on horizontal 
 shaft 
 
 Elevation, 
 Fig. a4 
 
 cured. ‘The outer part of the gate is provided with openings through 
 which the floats pass. ‘These openings admit of being packed as in- 
 dicated at Fig. 28. Stout rims are attached to the outer ends of the 
 floats in the manner to be described later. ‘The water approaches 
 the wheel from the exterior with a revolving motion. Its rotatory 
 velocity at its entrance to the wheel is equal to the velocity of the 
 exterior circumference. If this was not so, the water would not 
 enter the weeel. It is discharged in the reverse direction with a 
 velocity nearly equal to that due the head. Its entrance to the mov- 
 ing orifices of the wheel implies no greater shock or loss of head 
 than would be involved in its entrance to rounded stationary orifices. 
 It passes the wheel under conditions consistent with maximum efli- 
 ciency. These conditions are in no way changed by the opening or 
 closing of the gate. ‘The energy of the water in this wheel is devel- 
 oped wholly by reaction. It is, as shown at page 10, nearly double 
 
34 A More Perfect Form of Water - Wheel. 
 
 fe 
 y 
 y 
 ; 
 
 
 
 
 
 on y y of fig 22 
 
 Fig. 25 
 
 Section 
 
m of Water - Wheel. 
 
 A More Perfect For 
 
 LA oe: 
 
 
 
 is 
 
 Fig. &6 
 
36 
 
 A More Perfect Form of Water - Wheel. 
 
 Pn ss 
 eT 
 ~ 
 
 ~ 
 
 
 
 ei 
 
 
 
 y fO X-X UO UO!LI—aS 
 
 
 
 gee 
 =- 
 = 
 - - 
 - — 
 
 a 
 - 
 
 but half of this energy | 
 
 2 the whirling movement to the 
 
 ) 
 
 that due to the head and quantity of water ; 
 
 is absorbed in the work of impartin 
 
 water before or during its entrance to the wheel. 
 
A More Perfect Form of Water - Wheel. 37 
 
 We will now describe the construction of the wheel and its ad- 
 juncts. Figs. 22 to 28 relate to a wheel on a horizontal shaft. 
 
 The Gate is in two parts: — The first part is a dise of conoidal 
 shape which slides horizontally upon the shaft; the second is a cylin- 
 der with a broad internal flange, as shown at Fig. 28. Both parts are 
 pierced with openings for the passage of the floats. At the junction 
 of the two parts is a thick sheet of fibrous packing, pierced in like 
 manner. ‘This being compressed by the screws which fasten the two 
 parts together packs the floats. The cylindrical part of the gate has 
 the packing ring 7, so that, in a single wheel, the space between the 
 gate and disc, in a double wheel the space between the two gates, 
 becomes a water-tight compartment. It would be a very simple 
 matter, also, to introduce packing at the hub where it slides on the 
 shaft. 
 
 The Discs. — The wheel under consideration is a double wheel. 
 The central hub which is fastened to the shaft carries two flat circular 
 discs. These have openings, 0, which allow the water to pass freely. 
 The rim of the dise is thickened and notched like a ratchet wheel, 
 as shown at Fig. 27. 
 
 The Floats. —'These are continuous through both wheels. The 
 part traversed by the gate is of the form shown in section, Fig. 25. 
 The part resting on the dises, and lying between them, has the sec- 
 tion shown by the cross natching in Fig. 27, the blade of the float 
 coming to a close shoulder against the disc, and preventing any ten- 
 dency to endlong displacement. After being put in place and tem- 
 porarily fastened, the wheel is put in a lathe, and the central part of 
 the floats turned off smoothly, leaving the diameter of that part of 
 the wheel a little greater than that of the outer ends of the floats. 
 Then a heavy band, B, isshrunk on. Fig. 28 shows this band extend- 
 ing oyer the whole space covered by the discs, but two narrower 
 bands would do as well. The float may have a thickness of 14 inches 
 
 
 
 near its outer edge, forming a very strong stiff bar. The outer end 
 of the float is turned down to a cylindrical stud, 14 inches diameter, 
 and 14 inches long, and threaded. Some 4 of the floats in a wheel 
 are bored longitudinally with holes, 0, Figures 25, 26 and 28. These 
 holes reach from the outer end to the disc, and are there met by holes 
 cut from the outside of the float. These holes connect the interior 
 of the wheel with the low pressure compartment of the case. The 
 band, B, is thicker at one disc than the other to allow the cylindrical 
 parts of the gates to telescope when open. A groove is turned in 
 
38 A More Perfect Form of Water - Wheel. 
 
 the band, at each disc, for the packing ring 7. After shrinking on 
 and finishing the bands, the cylindrical part of the gate is put on, the 
 flange straddling the floats. Then the sheet of fibrous packing is 
 slipped on. Then the discoidal part of the gate is adjusted, and the 
 screws turned up which connect the two parts and compress the 
 
 = N 
 SS 
 LikiiiiiiiiiiiDieliisidiiViidildllldlde 
 
 < \ : : By , 
 
 yy 
 ei 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 \ | 
 Wiss 
 
 SSN 
 
 FY 
 Se 
 
 Section on k-kK of fig 27 
 
 Fig. 28 
 
 packing. Then the rim, which is bored to receive the threaded studs 
 on the outer ends of the floats, is put in place and solidly fastened 
 by countersunk nuts. ; 
 
 The Shaft has an internal bore-hole communicating with the inte- 
 rior of the wheel. At the end of the shaft this bore-hole communicates 
 with a pipe by a stuffing-box. 
 
A More Perfect Form of Wuter - Wheel. 39 
 
 
 
 The Case. — The wheel being set above the level of the canal of 
 discharge, the case has high pressure and low pressure compartments. 
 The case is shown in Figures 22, 23 and 24. It is a short cylinder of 
 cast iron with flanges for the attachment of the penstock and draft 
 tubes. The interior compartment is in communication, by means of 
 the penstock, with the upper level of the mill site. Being exposed to 
 a bursting pressure, it is formed by two conical diaphragms joining 
 the large cylinder. The junction is marked by two broad ribs run- 
 ning around the latter and widening into flanges for the attachment 
 of the penstock and draft tubes. An annular portion of these dia- 
 phragms, next the wheel, is made detachable for convenience of fin- 
 ishing. Ribs also run around the ends of the exterior cylinder, and 
 expand into flanges for the attachment of the draft tubes. The pres- 
 sure on the ends of the cylinder is inward, and these ends are of dish 
 shape, bending inward. ‘This form not only gives great strength to 
 resist the pressure, but shortens the unsupported part of the shaft. 
 The flattened portion of the penstock, at its junction with the case, 
 can be strengthened by external ribs, if required. ‘The draft tubes 
 being rectangular in section are divided by partitions into several dis- 
 tinct passages, as shown in Fig. 24, for greater strength. As the 
 rim w, of the wheel, Fig. 26, must revolve without touching the case 
 c, an annular space must exist, allowing for wear and imperfection 
 of workmanship, through which the escape of water would be objec- 
 tionable. To alleviate this difficulty, the small ring, 7, is confined 
 to a seat turned on the case so as to be capable of slight lateral 
 movement. This ring can fit the rim of the wheel much closer than 
 the fixed case, while yielding to any slight displacement of the wheel, 
 from wear or other cause. 
 
 The Bearing, }, of the shaft, is an undivided cylinder. It has a rib 
 or flange around the middle by which it is riveted to a circular plate 
 of wrought iron. This latter is fastened, at its outer circumference, 
 to the end plate of the case, covering a circular opening in the same, 
 some 30 or 36 inches in diameter. The bearing has an oil-cup and 
 a packing gland not shown. This bearing, I presume, will be the 
 subject of some criticism, but I think it a correct design. Bearings 
 are usually made in two halves, for convenience ih setting up machin- 
 ery rather than any inherent advantage in that method. The ring of 
 boiler plate surrounding the bearing gives it a slight but sufficient 
 degree of flexibility, which is favorable to uniform wear. The press- 
 ure being from without inward favors the admission of oil. The 
 

 
 40 A More Perfect Form of Water - Wheel. 
 
 admission of water to a bearing which is well lubricated is no disad- 
 vantage, as is shown in the Westinghouse engine, where the bearings ~ 
 run constantly in water covered with oil. 
 
 Regulation. — Fig. 29 is a schematic sketch of the proposed regu- 
 lator, and Fig. 380, a section of the valve for controlling the move- 
 ment of water through the central bore-hole in the shaft. The plug 
 of the valve carries an arm resting on the spindle of the regulator, 
 
 
 
 Lib 
 
 med LDL if 4 
 Sa 
 
 Je 
 
 Sketch of governor and 
 section of valve to regulate 
 Speed of wheel—— 
 
 and weighted as indicated, so that it follows the movement of the 
 spindle, under the action of the revolving balls, rising when the 
 speed diminishes, and falling when it increases. An increase of 
 speed opens the passage leading from the upper level to the wheel; 
 a decrease closes it. Now, in the normal running of the wheel, 
 water is escaping from the closed chamber through the small orifices 
 o, in the floats, and entering through the valve. When the valve opens 
 wider, an increased quantity of water enters the chamber, raises the 
 pressure therein and moves the gate to close. When the speed sud- 
 
 
 
 
A More Perfect Form of Water - Wheel. 41 
 
 denly starts forward, the valve opens wide and closes the gate rapidly. 
 When the valve closes to less than the normal opening, water escapes 
 faster than it enters, the pressure in the chamber falls below that in 
 the low pressure compartment, and the gate opens. The rapidity with 
 which the gate will open when the valve is entirely closed will depend 
 on the size and number of the openings 0, that is, upon the quantity 
 of water constantly wasted. ‘The wheel under consideration has an 
 external diameter of 5 feet and is supposed to draw some 200 eubic 
 feet of water per second, 12,000 per minute, under a head of 20 feet. 
 The cross-section of the closed chambers may be taken at 20 square 
 feet. To move the gate at the rate of 12 inches in a minute would 
 involve a loss of 20 cubic feet per minute, which in comparison with 
 12,000 is not worth considering. A movement at the rate of 12 
 inches per minute would suffice for any ordinary use of a water- 
 wheel. A closing movement of any desired rapidity can be attained 
 by suitable proportions of valves and passages without waste of 
 water. To start the wheel from rest, the weighted arm is thrown 
 into the position ‘* to start,” Fig. 29. Then the interior of the wheel 
 is in communication with the lower level, and water from the upper 
 level rushes through the nozzle into the pipe leading to the lower 
 level, forming a jet pump which draws the water or air from the in- 
 terior of the wheel, and opens the gate. ‘To stop, when running, the 
 weighted lever is thrown into the position marked ‘‘ to stop.” Then 
 the valve is free from control of the regulator, and the interior of the 
 wheel is in full communication with the upper level; the gate closes. 
 On a low head it might be doubtful whether the wheel could be 
 started with certainty by this method. In that case it might be ad- 
 visable to temporarily connect the pipe ‘‘to upper level,” with a 
 municipal water-main, or with the fire-tank of the mill, which would, 
 if required, make a complete vacuum in the wheel chamber. In start- 
 ing the wheel after the water has been shut out of the penstock, the 
 draft tubes and wheel chamber would be filled with air. The latter 
 would remain filled with air after the starting of the wheel, and, as it 
 filled with water, the air would collect at the center and obstruct the 
 regulation of the wheel. In this case the gate would be opened far 
 enough to admit a large volume of water and expel the air from the 
 draft tubes, raising the water in the latter above the wheel. ‘Then 
 admit water from the upper level, close the gate and expel the air, 
 which escapes through the highest orifices 0. Then the gate can be 
 opened without admitting air. Of course the air can be drawn out 
 
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 ct Form of Water - Wheel. 
 
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 42 
 
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A More Perfect Form of Water - Wheel. 43 
 
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 by the jet pump, but this would involve opening the gate to its full 
 width, which might not be desirable. 
 
 Equations 11 to 15 were deduced with reference to the regulation 
 of the wheel by the action of centrifugal force. We will inquire 
 what amount of force we have at disposal for the movement of the 
 gate. ‘The wheel under discussion is 5 feet in diameter on a head of 
 20 feet. This wheel takes a higher velocity than one of the common 
 form, in which the head is partly expended in imparting the velocity 
 through the guide passages. The exterior circumference would move 
 
 
 
 Wheel on vertical shaft-— Plan 
 
 Fig. 32. 
 
 _ with very near the velocity due the head. When the influx of water 
 to the wheel chamber is shut off, the internal pressure at the cireum- 
 ference is equal to the external pressure at the center. The internal 
 pressure at the center is less than this by the head due the velocity, 
 which we may call the centrifugal head. This head, in the present 
 case, would not be less than 16 feet. The force tending to open the 
 gate is equal to 8 feet depth of water acting on a surface of some 20 
 square feet — over 10,000 pounds. The force to be overcome is the 
 friction due to the weight and packing — nothing approaching the 
 above. Consider the most unfavorable case that could occur, 12 
 feet may be taken as the lowest head that a horizontal wheel would 
 be used on, and 36’/’ diameter would be about as small a wheel as 
 would be used on such a head. The centrifugal head would not be 
 less than 10 feet. The pressure tending to open the gate is, roughly, 
 
fect Form of Wuter- Wheel. 
 
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 44 
 
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A More Perfect Form of Water - Wheel. 45 
 
 yx 38x 3 XK 7854 XX 62.5 = 2209 lbs. For a head less than 12 
 feet we should employ a wheel on a yertical shaft. 6 feet may be 
 regarded as the lowest head worth developing by water-wheels. On 
 such a head there is seldom occasion to use a small wheel. » A 5-ft. 
 wheel on a 6-ft. head would expose to pressure something over 
 20 square feet for moving the gate. The centrifugal head would 
 be about five feet. The pressure would amount to about 20 K 25 x 
 62.5= 3125 lbs. Of course we may conceive of a head so low and 
 a wheel so small that this mode of regulation would be inapplicable. 
 To meet such causes I am prepared to say that a regulator may 
 
 
 
 Single wheel on vertical shaft 
 Plan. 
 
 Fig. 34 
 
 be devised capable of creating a total or partial vacuum within the 
 wheel chamber. 
 
 There is never any question as to the amount of force available for 
 closing the gate. With full communication between the wheel 
 chamber and the upper level, the water enters the former much 
 faster than it can be discharged. The centrifugal head acts in con- 
 cert with the static head to close the gate. 
 
 In every application of this method, the water issuing from the 
 floats changes its direction by a right angle in leaving the wheel. 
 This change of direction causes a pressure on the gate tending 
 to open it. The water should not leave the wheel with a velocity of 
 more than 6 or 7 feet per second. With 7 feet at full gate, the 
 pressure would amount to a head of some 15 inches, diminishing to 
 
46 AW More Perfect Form of Water - Wheel. 
 
 nothing as the gate closes. No reliance can be placed on this force 
 as aiding the opening of the gate. ; | 
 
 The foregoing description contemplates a wheel on a horizontal 
 shaft. Figures 31-2-3-4 show that the proposed construction is just 
 as applicable to a wheel on a vertical shaft; 31-2 show a double 
 wheel; 33-4 a single wheel. Neither of these presents any peculiar 
 difficulty, or requires any special description. For a small quantity 
 of water on a low head, a single wheel would be preferable to a 
 double one of smaller diameter. In such case it would probably be 
 better to let the wheel discharge upward instead of downward as 
 indicated, in order that the weight of the gate might assist in open- 
 ing. 
 
 Wheels often run under conditions to which the foregoing mode of 
 regulation is inapplicable, as, for instance, in connection with a 
 steam-engine, which controls the speed, and meets all variations in 
 the demand for power. In this case, the wheel, when under no 
 limitation as to quantity of water, runs at full gate, the regulator 
 disconnected, and the valve set to keep the gate open. Sometimes, 
 however, the wheel is under limitations as to the quantity of water. 
 Natural limitations resulting from diminished flow of the stream. 
 Legal limitation of leases and grants. In such case the gate cannot 
 be set at any unalterable opening. It can only be left in control of 
 the valve. It is obvious that the latter must be controlled by condi- 
 tions other than the speed of the wheel. 
 
 It would be easy to point out modes of regulating the discharge, 
 without reference to the velocity, to conform to the varying flow of a 
 stream, to a uniform draft of water or to a uniform output of power. 
 But such inquiries would carry us far beyond the contemplated limits 
 of this paper. 
 

 

 
 
 
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