r REESE LIBRARY ov THI-: UNIVERSITY OF CALIFORNIA , i8(>u a .-J mv*ds. Hence we have mgdz + mo^pdp = % md (v*) + mg |- + \ ^mv*ds, . . . (7) But the last term cannot be integrated unless v be a known function of s, and since this is not known, we make v = v 2 , the terminal velocity. The coefficient /< 2 is determined independently of the length, and includes the value ^ 9 when s is the length of a bucket. Integrating between initial and terminal limits gives The fall ^ is so small in practice compared with the next term of the equation, that it may, and will, be omitted, giving {1 + ^> = rf +(,,> _,fl_ ty SLZ, . . . (9) which gives v* At exit the pressure will be (10) The velocity of exit, relative to the earth, will be (11) The work done upon the wheel will be the initial (poten- tial) energy of the water less the energy in the water as it quits the wheel, still further diminished by the energy due to frictional losses ; or ..... (13) Three tubes having small orifices, A, B, C, on the same arc, ro- tate about a common axis, 0, one discharges to the left, another radially, and the third to the right ; they receive water at the ends near to the axis, with so small a velocity as to be negligible ; required the velocity of exit. Equation (9) gives for all three cases v^ = &? r* for the velocity relative to the tubes; and equation (11) gives for 1st, F~ 2 2 GO r^ " 2d, 7, = V2.a>r " 3d, F a = 0. HYDKA.ULIC MOTORS. 111 .s 9.2 OD S. PH r y cC cQ M -2 o .T! 5 41 0. .g $ o . -S ^ " |a s 02 '5^ PH cur + S!^ 02 i s ' c? I + 'S + &. ! s " I I 1 8 HYDRAULIC MOTORS. = L [ when cos . . . (15) -Jf= - 1 + 4- Pi\ra 2 cos ex sin sin (a + ri) sin 2 (a + ^i) ' 2 \ (15a) For maximum efficiency make 6?^ -r- doo = in (15) and solve for GO, calling this particular value GO', then 'I?'-* ?"' (16) Ta which value substituted in equation (15) will give the maxi- mum efficiency. Then equations (5), (6), become -r-r V= sin sin (a -f y sin a sin (a + ; Also from (9), (10), (4), (17), (18), GOT } , (17) (18) But sin 2 or sin 2 (a + y\} sin 2 o: sinVi + 2 sin (a + y) cos a sin ^ t 2 sin (a 4- ;KJ) cos a sin ^, = 1- 2 cos a sin y , sin (a + ^,) ' -which substituted above will give equation (14). Remark. Substituting eq. (16) in (15) gives, by reduction, E -= v. Ji-r^rW- VJf'-^eosVJ (168) , 7^* tfA<3 terminal angle of the guide, a, the initial and ter- minal angles of the bucket y 1 and y n respectively, the ratio of the radii r 1 ?\ and the frictional resistances constant y then for all such wheels M and N will be constant and the efficiency will be constant y the velocity of the initial rim, QD' / the velocity through the gate V, the initial velocity in the bucket, will each and all vary as VH. HYDRAULIC MOTORS. ,,'- 2 rf-K . , ^ . (19) sin - The normal sections of the buckets will be The depths of those sections will be 77 Z. If ~\7~ 1 2 /O1 \ J. sin ' ^ ~ i sin ^ t ' ^ 2 " <^ 2 sin x a ' DISCUSSION. 4. Three simple systems are recognized. r x < r 2 , called outward flow. r x > r 2 , called inward flow. r = r 2 , called parallel flow. The first and second may be combined with the third, mak- ing a ra{#ed system. The third, in theory, is really an inward or outward flow, with an indefinitely narrow crown, although the analysis applies to a parallel flow wheel, in which the width is indefinitely small, and depth small compared with the total head. 5. Value of / the quitting angle. Equations (14) and (16a) show that the efficiency is increased as cos y 2 is increased, or as y 2 decreases, and is greatest for y 2 =& Hence, theoretically, the terminal element of the bucket should be tangent to the quitting rim for best efficiency. This, how- ever, for the discharge of a finite quantity of water, would re- quire an infinite depth of bucket, as shown by the third of equa- tions (21). In practice, therefore, this angle must have a finite 10 HYDRAULIC MOTOES. value. The larger the diameter of the terminal rim the smaller may be this angle for a given depth of wheel and given quantity of water discharged. Theoretical considerations then would require, for best efficiency, a very large diameter for the quit- ting rim, and a very small angle, y 2 , between the terminal ele- ment of the bucket and the rim ; but commercial considera- tions require some sacrifice of best efficiency to cost, so that a smaller diameter and larger angle of discharge is made. If wheels are of the same diameter and depth, the inward flow wheel requires a larger quitting angle for the same volume of water than the outward flow, since the discharge rim will be smaller in the former than in the latter wheel, and the velocity v 2 , eq. (19), will also be less. In practice y 2 is from 10 to 20. 6. Relation between y 2 and GO'. Equation (16) when put under the form NT J N' 2 COS 2 y * (22) shows that <*>' increases as y 2 decreases, and is largest for y 2 = ; that is, in a wheel in which all the elements except y 2 are fixed, the velocity of the wheel for best effect must increase as the quitting angle of the bucket decreases. If the terminal element be radial, then y 2 = 90, and equa- tion (22) appears to give GO' = ; but the discussion really fails for this case. See article 89. 7. Values of a -f y^ If a + Kl = JL80, and a and y l both finite, then will If and N in (15a) both be infinite ; but equation (5) gives that is, the wheel will have no motion, and no work will be FOURNEYRON TRIPLE WHEEL. * V ~ OF THF '^iy UNIVEBSITY HYDRAULIC MOTORS. 11 done. If a + y l = 180, then the terminal element of the guide and the initial element of the bucket have a common tangent, in which case the stream can flow smoothly from the former into the latter only when the wheel is at rest. (See Fig. 5.) If a 4- ;/j exceed 180, GO' would be negative, and it would be necessary to rotate the wheel backwards in order that the water should flow smoothly from the guide into the bucket. It follows, then, that a -t- y^ must be less than 180, but the best relation cannot be determined by analysis ; however, since the water should be deflected from its course as much as possible from its entering to its leaving the wheel, the angle a for this reason should be as small as practicable. 8. Values of a. If a 0, equation (14) will reduce to + r 2 cos y* V ZgH + (rf - 2n 2 - w?) a . . (24) which is independent of y } ; hence, for this limiting case, the efficiency will be independent of the initial angle of the bucket. This is because the water enters the wheel tangentially and therefore has no radial component that would give an initial velocity in the bucket ; and equation (18) shows that the ini- tial velocity v l would be zero, while (17) shows that the velocity of the initial rim must equal that of the water flowing from the guides, or V GO'TI. For the limiting, or critical case, a 0, y 2 = 0, Ui 0, A/ 2 = 0, the velocity producing maximum efficiency will be, from equa- tion ^16), ....... (25) 12 HYDRAULIC MOTORS. or the velocity of the initial rim, if the wheel be frictionless, will be that due to half the head in the supply chamber. If r} = 2n 2 , then ...... (26) or the velocity of the terminal rim will equal that due to the head. Substituting in (19) the values a = 0, y 2 = 0, /^ = 0, ^2 0, r} = Zr*, and it will reduce to ....... (27) as it should. The following table gives the values of quantities for the three classes of wheels : TABLE I. 72 = =0, =0. VELOCITI r OF Velocity of Exit VELOCITY IN BUCKET. Velocity DIMENSIONS OF WHEEL. Inner Rim. Outer Rim. from Guide V. Initial t'j. Terminal r a . of Exit. w. CL>>! GO'TZ f i=V$ r * r } =r z VffS yfi* V*ffH VffH~ VffH V&2 0.00 0.00 ^/Zgll VffZ 0.00 0.00 1.000 1.000 Ti=lAr 2 GO'TZ 0.714%/^ GO'TI V~gH Vffff 0.00 .nt^H 0.00 1.000 In the first case the inner rim is the initial one, in the third case the outer rim is initial, it being an inward flow wheel. Since, in this case, the velocity of admission to the wheel in reference to the earth is that due to half the head in the supply chamber, and the velocity of exit is zero, it follows that the energy due to the velocity is all imparted to the RADIALLY INFLOW TURBINE. HYDRAULIC MOTORS. wheel ; and the energy due to the remaining half of the head is imparted to the wheel by pressure in the wheel. If the velocity of entrance to the wheel be that due to the head, or F 2 = %gH then will no energy be imparted to the wheel on account of pressure exerted by any part of the head 77, but if V- < 2^/77, then will some of the work be done by this press- ure, 10 being zero. For the cases in Table I., the energy im- parted to the ivheel ivill be due one-half to velocity and one-half to pressure / or in symbols, TF. +^WH= Wff, . . (28) or, the entire potential energy of the water will be expended in work upon the wheel. Whenever F 2 < 2(/77, the pressure at entrance must exceed the external pressure at exit, and F- H - _?2, ........ (29) H then will be the part of the head producing pressure in the wheel. In practice, a cannot be zero and is made from 20 to 30. When other elements of the wheel are fixed, the value of a may be determined so as to secure a certain amount of initial pressure in the wheel, as will be shown hereafter. The value ?\ = 1.4r 2 makes the width of the crown for internal flow about the same as for i\ *Jfy\ for outward flow, being approximately 0.3 of the external radius. 9. Values of /./, and /< 2 . The frictional resistances depend not only upon the con- struction of the wheel as to smoothness of the surfaces, sharp- 14 HYDRAULIC MOTORS. ness of the angles, regularity of the curved parts, but also upon the manner it is run ; for if run too fast, the initial ele- ments of the wheel will cut across the stream of water, pro- ducing eddies and preventing the buckets from being filled, and if run too slow, eddies and whirls may be produced and thus the effective sections be reduced. These values cannot be definitely assigned beforehand, but Weisbach gives for good conditions, ^ = ^ = 0.05 to 0.10.. . . . . ' . . (30) They are not necessarily equal, and //, may be from 0.05 to 0.075, and // 2 from 0.06 to 0.10, or values near these. 10. Values of y^ It has already been shown that /, must be less than 180 (x. If y, = 90, equation (14) shows that the efficiency of the frictionless wheel will be independent of a. The effect of different values for y l is best observed from numerical results as shown in the following table : TABLE II. Let a = 25, = 12, INITIAL ANGLE. yi (i) TI = r^t. r, = 1.4r 2 . 'r a . (2) y. (3) CO'TV (4) w'r 2 . (5) E. (6) "ffir 60 1.322V<7# .812 ,934y7# .780^/^ff .911 1.092-y/^ 90 1.226 " .827 .866 " .689 " .908 .964 " 120 1.078 " .838 .762 " .576 " .898 .806 " 150 .518 " .744 .366 " .271 " .752 .379 " The values o?V a in columns (2) and (5) are velocities for the terminal rim, which in column (2) are for the exterior rim, but PARALLEL FLOW TURBINE, GIRARD TYPE. HYDRAULIC MOTORS. 15 for column (5) it is the interior rim, while column (7) is for the exterior rim. Columns (2) and (7) show that the velocity of the outer rim is less, for maximum effect, for the inflow than for the out- flow, for the same size wheel. Column (3) shows that the efficiency, E, decreases as the initial angle of the bucket, ;/ increases up to 120. This maximum will be for this wheel with this amount of friction. Column (6) shows that for the inflow wheel the efficiency continually decreases as ;/, increases. If the head and quan- tity of water discharged be constant, the work would be pro- portional to the efficiency ; for, from equation (14), U=dQHE ....... (31) The effect of y l on the velocities is shown in Table III. TABLE III. Let a = 25, = U Z = 0.10, INI- r ,w: ,, = >.4,, TIAL ANGLE. F i K* ft, x ft 2 x F Vi & A'x ft, X * 2 X VgH Vffff VgH VgH VgH 4/<7// 4/^ VgH V^ VgH 4/<7# VgH 60 .820 .396 1.447 1.2192.525 .691 .959 .463 .761 1.043 2.160 1.314 90 .955 .403 1.378 1.0472.481 .725 1.063 .449 .676 .940 2.227 1.479 120 1.150 .560 1.153 .8691.785 .874 1.217 .593 .605 .821 1.686 1.653 150 2.1001.775 .621 .476 .563 1.610 2.060 1.741 .296 .485 .574 3.378 For commercial considerations it may be necessary to sac- rifice some efficiency to save on first cost, and to avoid making the wheel unwieldy. From equation (4) it appears that the pressure in the wheel at entrance, p l9 diminishes as the velocity of admission, V, in- 16 HYDRAULIC MOTORS. creases, and. according to equation (5), V depends upon y\ when a is fixed. Since the crowns are not fitted air tight nor water tight it is desirable that p l should exceed the pressure of the atmosphere when the wheel runs in free air, or the press- ure p 2 + p a when submerged, to prevent air or water from flow- ing in at the edge of the crown. It will be shown hereafter, in discussing the pressures in the wheel, that we should have - tan y l > tan 2^, (32) or, 180 - Yi> 2 of, or, ;/! < 180 - 2. If a = 30, then y l < 120. To be on the safe side, the angle y l may be 20 or 30 degrees less than this limit, giving y 1 = 180 - 2 - 25 (say) = 155 - 2. Then if a = 30, y l = 95. Some designers make this angle 90, others more, and still others less than that amount. Weis- bach suggests that it be less so that the bucket will be shorter and friction less. This reasoning appears to be correct for the inflow wheel, for the size and conditions shown in Table II., but not for the outflow wheel. In the Tremont turbines, de- scribed in the Lowell Hydraulic Experiments, this angle is 90, the angle ^, 20, and y 2 , 10. Fourneyron made y t 90, and n from 30 to 33. In Table III. it appears that for y l = 150, V 2.1 VgH, which exceeds VSgfB $ that is, the velocity of exit from the supply chamber exceeds that due. to the head, hence the pressure at entrance into the wheel must be less than that of the atmo- sphere. For zero pressure for the frictionless wheel, the above condition gives = 180 - 2o PARALLEL FLOW. JONVAL WITH DRAFT TUBE. HYDRAULIC MOTORS. 17 which for a 25, gives j\ = 130 : , and for f/ t == 150, the press- ure would be negative, and for 120 J it would be positive. It appears that for the wheel with friction, considered in the table, that this pressure is also positive for y l 120, and negative for 150. 11. Form of Bucket. The form of the bucket does not enter the analysis, and therefore its proper form cannot be determined analytically. Only the initial and terminal directions enter directly, and from these and the volume of the water flowing through the wheel, the area of the normal sections may be found from equations (20). . But well-known physical facts determine that the changes of curvature and section must be gradual, and the general form regular, so that eddies and whirls shall not be formed. For the same reason the wheel must be run with the correct velocity to secure the best effect ; for otherwise the effective angles a and YI may be changed to values which cannot be determined beforehand, in which case the wheel cannot be correctly ana- FIG. 6. lyzed. In practice the buckets are made of two or three arcs of circles mutually tangential at their points of meeting. Also, if the normal sections, K, k lt &, of the buckets as constructed do not agree with those given by computation, the stream will, if possible, adjust itself to true conditions by the formation of 2 18 HYDRAULIC MOTORS. eddies. If the terminal sections at the guides, or the initial section of the bucket, be too small, the action may be changed from a pressure wheel to one of free deviation. So long as the pressure in the wheel exceeds the external pressure, the pre- ceding analysis is applicable for the wheel running for best effect, observing that the sections K, k jt . &,, are not those of the wheel, but those which are. computed from the velocities F, v l9 v v 12. Value of ; or direction of the quitting water. From Fig. 1 it may be found that and ', cos B = v. 2 cos y-2 - r 2 sin 6 = v. 2 sin y z ; . cot cot y.> sin (33) (34) (35) These formulas are for the velocity giving maximum effi- ciency. If the speed be assumed, GO in place of oo' becoming known, v- 2 is given by equation (19). It is apparent for such a case that 6 may have a large range of values from 6 = y 2i when the wheel is at rest, to 6 exceeding 90 for high velocities. The following table gives some results : TABLE IV. a. = 25, = 12, =// 2 = 0.10. TI = T* 4/t. r,=1.4r, Yi 0) 6 CO 6 60 .314 V^H 72 14' .160 VjH 102 43' 90 .310 " 66 59' .143 " 101 17' 120 .241 " 60 24' .126 " 82 52' 150 .157 " 55 26 .043 " 74 51' INFLOW PARALLEL CROWNS. HYDRAULIC MOTORS. 19 According to this table the water is thrown backward, or in the direction opposite to the motion of the wheel for the outward flow wheel, and for the inflow it is thrown forward for Yi less than 90, and backward for YI greater than 120. In the Tremont turbine a device was used for determining the direction of the water leaving the wheel, and for the best effi- ciency, 79J per cent,, the angle 6 was about 120. Lowell Hydraulic Experiments, p. 33. The angle thus observed had a large range of values ranging from 50 to 140 for efficiencies only two or three per cent, less than 79^ per cent. 13. Of the vahte of GO. So far as analysis indicates, the wheel may run at any speed ; but in order that the stream shall flow smoothly from the supply chamber into the bucket thus practically maintaining the angles a and Y\. the relations in equations (5) and (6) must be maintained, or = ^n^v | .I.... (36) sin Y,. and this requires that the velocity V shall be properly regu- lated, which can be done by regulating the head hi or the press- ure pi or both hi and p { , as shown by equation (4). This however is not practical. In practice, the speed is regulated, and when the condition for maximum efficiency is established, the velocities V and Vi are found from equations (17) and (18). Since ;/ 2 , in practice, is small we have, for best effect, v 2 = c> 2 , approximately, .... (37) and, adopting this value, a more simple expression may be found for the velocity of the wheel. For equation (19) gives V?. r* GO' . /^s = .(Approx.)(38) A cos asm y l fr 20 HYDRAULIC MOTORS. If Ml = ^ = 0.10, r 2 * ?', = 1.40, = 25, yi = 90, the velo- city of the initial rim for outward flow will be VgH &}'?, =. y n Q9Q Vi + 0.159 ~ The velocity due to the head would be 0.659 ..... . (39) hence, the velocity of the initial rim should be about - 928 1.414 of the velocity due to the head. For an inflow wheel in which r? = 2r 2 2 , and the other dimen- sions, as given above, this becomes of the velocity due to the head. The highest efficiency of the Tremont turbine, found experi- mentally, was 0.79375, and the corresponding velocity, 0.62645 of the velocity due to the head, and for all velocities above and below this value the efficiency was less. Experiment showed that the velocity might be considerably larger or smaller than this amount without diminishing the efficiency very much. In the Tremont turbine it was found that if the velocity of the initial (or interior) rim was not less than 44 nor more than 75 per cent, of that due to the fall, the efficiency was 75 per cent, or more. Exp., p. 44. This wheel was allowed to run freely without any brake except its own friction, and the velocity of the initial rim was observed to be 1.335 V2gII, half of which is 0.6675 VfyH, ...... (41) THE THE " HERCULES." MIXED FLOW, INWARD AND DOWNWARD. HYDRAULIC MOTORS. 21 " which is not far from the velocity giving maximum effect ; that is to say, when the gate is fully raised the coefficient of effect is a maximum when the wheel is moving with about half its maximum velocity." Exp., p. 37. M. Poncelet computed the theoretical useful effect of a certain turbine of which M. Morin had determined the value by experiment. The following are the results (Comptes Rendus, 1838, Juillet) : TABLE V. Velocity of initial rim or 7'ito' Number of turns of the wheel per Ratio of useful to theoretical effect. Means of values by experiment. ^2gH~ minute. 0.0 0.00 0.000 0.2 33.80 0.<>64 0.4 47.87 0.773 0.700 0.6 58.61 0.807 0.705 0.7 62.81 0.810 0.700 0.8 67.67 0.806 0.675 1.0 75.76 0.786 0.610 1.2 82.88 0.753 0.490 1.4 89.52 0.712 0.360 1.6 95.70 0.664 0.280 1.8 101.51 0.612 0.203 2.0 107.00 0.546 0.050 3.72 145.00 0.000 Poncelet states that he took no account of passive resist- ances, and hence his results should be larger than those of experiment as they are ; but here both theory and experiment give the maximum efficiency for a velocity of about 0.6 that due to the head, and the efficiency is but little less for velocities perceptibly greater and less than that for the best effect. For velocities considerably greater and less, theoretical results are much larger than those found by experiment, for reasons already given, chief of which is the fact that eddies are induced, and the effective angles of the mechanism changed to unknown values. 22 HYDRAULIC MOTORS. 14. Pressure in the wheel. Dropping the subscript 2 from v, r, p, in equation (9), the resulting value of p will give the pressure per unit at any point of the bucket providing that /^ 2 be considered constant. Chang- ing r to p, equation (9) thus gives i jf - (42) To solve this requires a knowledge of the transverse sec- tions of the stream, for the velocity v will be inversely as the cross section. From equations (20) and (6) ft, ft, sin. From (4) and (5), . 1 + u, sin 2 ! a ^ . 2^r sin 2 (a + These reduce equation (42) to (44) - i [( 1 + ^sinV, + [fc 4- - The back or concave side of the bucket will be subjected to a pressure which may be considered in two parts : one due to the deflection of the stream passing through it, the other to a pressure which is the same as that against the crown, and is uniform throughout the cross section of the bucket, due to the pressure of a part (or all) of the head in the supply chamber. It is the latter pressure which is given by the value of p in equation (45). The construction of the wheel being known, the pressure p may be found at any point of the wheel for any assumed practical velocity ; although, for reasons previously SEGMENTAL FEED. RADIALLY INWARD FLOW. TANGENTIAL WHEEL. HYDKAULIC MOTORS. 23 given, it will be of practical value only when running near the velocity for maximum efficiency. " There are two cases : 1. That in which the discharge is into free air ; 2. That in which the wheel is submerged. * In the first case if the pressure is uniform, the case is called that of 'free deviation' in which the entire pressure upon the forward side of the bucket is due to the deviation of the water from a right line, and will be considered further on. If equation (45) shows a continually decreasing pressure from the initial element to that of exit, or if the minimum pressure exceeds p a , the preceding analysis is applicable. But if it shows a point of minimum pressure less than p a , it will be in a condition of unstable equilibrium, in which the slightest inequality would cause air to rush in and restore the pressure to that of the atmosphere ; so that the pressure in the wheel and the flow would be changed. The point of mini- mum pressure may be found by plotting results found from equation (45), substituting values* for p taken from measure- ments of the wheel, and ~k from computation. From the entrance of the wheel up to the point of minimum pressure the preceding analysis applies ; and the remainder of the wheel must be analyzed for ' free deviation ' and the two results added. In the second case the equations will apply, since air can- not enter, provided that p does not become negative, to realize which requires a tensile stress of the water. This is impos- sible and eddies would be formed ; and the effect of these on the velocity and pressure cannot be computed. Such a case cannot be analyzed." 15. To find the pressure at the entrance to the bucket when running at best effect. In (45) let p r,, k = ki and p p\. To simplify still more, let the wheel be frictionless, or ^ = H 2 = 0, and find from equation (38) , 9 sin (a + i/ 2 cos of sm , . (46) 24 HYDRAULIC MOTORS. also hi = H + A-2, and (45) becomes Pl = 611+ 6h* + p a - d * - 2 cos a sin ( + xO If the wheel is not submerged h. 2 = 0, and let the pressure PI equal that of the atmosphere, or p a , then - sinri 2 cos (v sin (a + If the wheel be submerged, let p { 180 -- y lt . . . . . . (50) the pressure within will be less than that without ; but if 2 < 180 -- xi, ..- .- .'. (51) the pressure within will exceed that without a condition which is considered desirable. If frictional resistances be considered the value of r^ri from equation (38) will be less than that given by (46), and hence the last term of equation (47) will be less unless a be greater than the value given by equation (51) ; hence with frictional resistances the terminal angle of the guide blade may exceed somewhat 90 - ^y l ; therefore, if the value of a be found for a frictionless wheel it will be a safe value when there is friction. If y = 90 and a = 90 - xi = 45 > tnen (47) g ives 8h,+p a , . . (52) SCOTTISH on WHITELAW. BARKER MILL. WHEELS WITHOUT GUIDES. |P i V^ or TH* UNIVERSITY HYDRAULIC MOTORS. 25 as it should. If y l = 90 and a 30, then or, p, > 6k, +p a + 0.33tf//. . . .... . (53) The angle a should not be so small or y\ so small as to pro- duce excessive pressure at the entrance to the wheel. Example. Find the pressure per square inch at the entrance to the wheel when the head is 10 feet, the terminal angle of the guide is 30, the initial angle of the bucket y l = 90 ; the wheel being one foot under the water in the tail race. 16. Number of buckets. The analysis given above is true for a wheel with a single bucket, provided the supply is constantly open to the bucket and closed by the remainder of the wheel. But for practical considerations the wheel should be full of buckets, although the number cannot be determined by analysis. Successful wheels have been made in which the distance between the buckets was as small as 0.75 of an inch, and others as much as 2.75 inches. Lowell Hyd. Exp., p. 47. Turbines at the Centen- nial Exposition had buckets from 4V inches to 9 inches from centre to centre. 17. Ratio of radii. Theory does not limit the dimensions of the wheel. In practice, for outward flow, r 2 -f- ^ is from 1.25 to 1.50 { ,~. for inward flow, r 2 -f- r l is from 0.66 to 0.80 ) It appears from Table II. that the inflow wheel has a higher efficiency than the outward flow wheel (columns 6 and 3), and these wheels have about the same outside and inside diameters. The inflow wheel also runs somewhat slower for 26 HYDRAULIC MOTORS. best effect. The centrifugal force in the outward flow wheel tends to force the water outward faster than it would other- wise flow; while in the inward flow wheel it has the contrary effect, acting as it does in opposition to the velocity in the buckets. It also appears that the efficiency of the outward flow wheel increases slightly as the width of the crown is less, and the velocity for maximum efficiency is slower ; while for the inflow wheel the efficiency slightly increases for increased width of crown and the velocity of the outer rim at the same time also increases. Let r, = nr t , Yl = 90, y, = 20, /i, = ^ = 0, a = 80 ; then for n = 0, 0.5, 0.8, 1.4, we have GJ'T, = 0, 0.761^//, 0.972^ #", 1. coV 1 = 1.3910//, 1.52%/#, 1.215*7 //, 0. E = 0.6594, 0.8050, 0.9070, 0.9784. 18. Efficiency, E. The method of determining the theoretical value of E has already been given ; but to determine the actual value, resort must be had to experiments. These have been made in large numbers and the results published. By assuming the mini- mum values of the several losses, a maximum limit to the effi- ciency may be fixed. Thus, if the actual velocity be 0.97 of the theoretical, the energy lost will be (1-0.97 2 ) or 6 per cent. Friction along the buckets and bends . . . 5 " " Energy lost by impact, say 2 " " Energy lost in the escaping water .... 3 " " Total .... 16 " " Leaving 84 " " available for work, This discards the friction of the mechan- ism and frictional losses along the guides, and if 2 per cent. be allowed for the latter, there will be left 82 per cent. It WHEEI JET DIRECT- ACTING. WHECL IS SHOWN MOUNTE1D ON TEMPORARY TRLSTLES JET REACTION JET WHEELS. HYDBAULIC MOTORS. 27 seems hardly possible for the effective efficiency to exceed 82 per cent., and all claims of 90 or more per cent, for these motors should be at once discarded as being too improbable for serious consideration. A turbine yielding from 75 to 80 per cent, is extremely good. The celebrated Tremont turbine gave 79 f per cent. Lowell Exp., p. 33. Experiments with higher efficiencies have been reported. A Jonval turbine (parallel flow) was reported as yielding 0.75 to 0.90, but Morin suggested corrections reducing it to 0.63 to 0.71. (Weisbach, Mech. of jEng., vol. ii., p. 501.) Weisbach gives the results of many experiments, in which the efficiency ranged from 50 to 84 per cent. See pages 470, 500-507. See also Jour. Frank. Inst., 1843, for efficiencies from 64 to 75 per cent. Numerous experiments give E =0.60 to 0.65. The efficiency, considering only the energy imparted to the wheel, will exceed by several per cent, the efficiency of the wheel, for the latter will include the friction of the support, and leakage at the joint between the sluice and wheel, which are not included in the former ; also as a plant the resistances and losses in the supply chamber are to be still further deducted. 19. The Crowns. The crowns may be plane annular discs, or conical, or curved. If the partitions forming the buckets be so thin that they may be discarded, the law of radial flow will be determined by the form of the crowns. If the crowns be plane, the radial flow (or radial component) will diminish as the distance from the axis increases the buckets being full for the annular space will be greater. 20. Designing. The dimensions of a wheel must be determined for a definite velocity. Thus far it has been assumed that the angles ar, y }> etc., are given, and the normal sections of the stream thus deduced. We will now assume that all the dimensions of the buckets are known, and the angle a and the section JTare 28 HYDRAULIC MOTORS. to be determined. The velocities v l and v 9 must now be found independently of a. From Fig. 1 we have F 2 = v* -\- G^V!* -- 2v l Gor l cos y\ (55) which combined with equations (4r), (9), (10), vfa vjc as in (20), and If = /^ A 2 , will give A\ (1 + Mi) kji\(&)i\ cos Y\ + /*,) V + J*M ' L (i + k)X' 2 + M a J . - ' (56) - (56a) where ^ _ (1 -h /^i) ^i^'i eos 7i, Equations (11), (12), (13), (55), (56), and (57), after making a = I give = 1+A + /V' I = ^ a cos 7, 4- //^j cos y. } - aA) ooV^HB + O'oi] (58) Remark. Since jn l is a comparatively small fraction and k l exceeds & 2 certain terms may be omitted giving for v^ the approximate but very nearly exact value, . , cog I 2 and corresponding reductions in A, B, J. HYDRAULIC MOTORS. 29 Let D = gll (1 4 aB), ^ = a U" + * S) - I A + * rf + i ^S G I aA. Then (59) For a maximum rZ^E 7 -f- r/cj = (9, This value of co is the one to be used in the other equations. The form of equation (60) is the same as that of equation (16). Substituting in (59), observing that aB = 1, and hence I) = 0, we find j ^- 3 (61) (600) To find the terminable angle a of the guide blade that will enable the stream to flow smoothly, subject to the preceding conditions. Fig. 1 gives T r cos a GOJ\ t\ cos y^ which, combined with equation (55), gives cos a = *"'' " * COS ^ . (62) Vv? -f- c^X 2 2^,0?^ cos y l Eliminating v l by means of equations (57) and (56) gives cos a in terms of the six constants r jy r^ y l y Jc l and # 2 , which are fixed and known from the dimensions of the wheel, and of the velocity GO of the wheel. Since the wheel may run at dif- 30 HYDRAULIC MOTORS. ferent velocities the angle must vary, and this will be done in practice by the piling of the water in the passages. Each turbine, however, should be designed to run at the speed giv- ing maximum efficiency, and its angles and dimensions should satisfy equations (60) and (62). From equation (9), + 6Jh . (63) in which, if v 2 and v\ be substituted from above, p l becomes known. Similarly, V from equation (55) becomes known, and finally, from (60), aor l v l cos cos a = J (64) 2L Path of the Water. Let aA be the position of the bucket when the water enters at b. The bucket being drawn in position to a scale, divide it into any number of parts equal or unequal aa,, a^, etc., and find the time re- quired for it to go from a to Ox. The distance being small, assume that the velocity is uniform from a to a lt and equal to -y,, which will be given by equation (20), O FIG. 7. or better, (64) THE COLLI NS WHEEL THE KNIGHT WHEEL HYDRAULIC MOTORS. 31 Then will the time t be t = ^ (65) V During this time the rim has gone from a to b a distance ab=?\cot (66) If the bucket bB be drawn through b, and the arc a t bi through a l5 their intersection h will be the position of the particle at the end of the time t. In a similar manner, the successive points Z> 2 , b a , etc., may be found, through which a continuous curve may be drawn representing the path of the stream. The line tangent to the termination of the bucket, will indicate the direction of the water at entrance of the wheel, and if the water drives the wheel, the path should be entirely outside this line and convex toward it. 22. Design a guide blade, outflow turbine. Assume the effective fall, H ft. Assume the required horse-power, . . . HP = Assume the exit angle, . . . ... . y- 2 = Assume the entrance angle of bucket, . . y\ = 90. Fix the exit angle of the guide, Eqs. (32), (51), a = 30. ? Assume efficiency, . . . . . . . . . E= 0.65, or 0.70; and after the wheel is fully designed, re- compute this value and if necessary correct the dimensions. Required quantity of water per second without loss, . ; . Q U 6H. Required quantity, . . .. . . . Q -4- E ' = Assume ^ = 0.10, /* 2 = 0.075. Velocity of the initial rim, Eq. (40), approx., ovr\ = (The corrected, final value will be found by Eq. (16) or (60). 32 HYDKAULIC MOTORS. Let r. 2 = 1.3r b then velocity of outer rim, oor 2 ~ The velocity into the bucket, Eq. (18), . 1% Initial section of buckets, Eq. (20), . . . ki = The inner circumference will be 2^. Let the walls of the buckets be T V of the circumference, then the effective open- ings will be f| of the circumference, or i-^nr^ Assume a depth, y, between the crowns. Try y Jr x . Then will the initial cross section of all the buckets be \\nrf ; hence, xt 77 "^ 2 &i 5 " f\ = If the radius is not what is desired, it may be changed to some other value and the depth y computed. Then, ... r 2 = The cross section at outer rim will be, if the crowns are planes, ^7rr 2 y sin y 2 k 2 = The number of buckets will be assumed . = Having determined these elements, the final velocity v 2 in reference to the bucket may be computed by equation (67), G0i\ from (62), V from (55), and a from (64). If the turbine revolves in air, at least half the depth of the wheel is to be deducted from the head H. If the circular opening between the wheel and gate be ^ of an inch, or y|^- of a foot, and the coefficient of discharge be 0.7, the discharge will be 0.7 x 2w, x i = q, . . . (67) PI being determined from equation (63). The loss of work will be 62.2g x // or 623g(H-$y) ..... (68) The work lost by friction, if the radius of the axle be r s , the TWIN TURBINES ON A HORIZONTAL Axis, OF THE UNIVERSITY HYDRAULIC MOTORS. 33 weight of the loaded wheel, FF 3 , and coefficient of friction //, will be nearly o g fo TF 3 r 3 cj per sec (69) The work done by the water must be the effective U plus the work due to the losses. The work done by the water pass- ing through the wheel must be U plus that given by equation (69). Call this C7j. Compute the work done by the water, and let it be U% ; then will the required depth be The efficiency may now be recomputed. The following tables, YI. and YIL, though for wheels of special dimensions, give some general results as shown in the following "Conclusions." The sections 1', and &, are assumed to be those of the wheel, and further it is assumed that the wheel passages (or channels) are filled with flowing water, and hence without eddies. SZ 1-1 If. V 00= " 2, 8 d d d ,-1* oooo S fe I > ^ > O 05 05 l' Esr l S S So S co S ddoo i-5 d i-^d PIS H, <5> fe ' e i-Jr-c i-id I-HO od lffMl*U ?2b5 S S ^ S o es CJD o i- i- >S o'd do dd dd 1-! d ~ d !-' d d '= ^ 1 i i i I I g s s 1-1 r-J O O S 58 -i s I" fc . 15 000 q q q d o i-J j co i i dd dd i-; d r-id - S?|fe! Ifeil^ HirMS 1 53 53 fe<|fe| l fe H Sliliyei dd dd o'd od ll^l W W V ns ss ^s & o TT ec -t es * 5 < d d do' d d do dd do" dd do m% *& m m w v5 v > v > 1-1 O 00 00 00 1111 ddoo HYDRAULIC MOTORS. 35 Conclusions. From an examination of Tables II., III., YL, Vli., the following conclusions are drawn : 1. The maximum theoretical efficiency of the inflow wheel is perceptibly larger than that of the outflow, the width of crowns and the initial and terminal angles of the buckets being the same. One reason for this is due to the flow through the wheel being opposed by the centrifugal action, but more particularly to the smaller velocity of discharge from the inflow wheel. 2. Columns (10) in Tables YI. and VII. show that for the wheels here considered the loss of energy due to the quitting velocity is from 2.2, 5.1 per cent, from the outflow, and from 0.9 to 1 per cent for the inflow. 3. The same tables show that in column (2) the efficiency is almost constant for the varying conditions here considered, while for the outflow there is considerable variation. 4. One of the most interesting and profitable studies to the theorist and practitioner is the effect upon the efficiency due to properly proportioning the terminal angle, ex, of the guide blade. It will be observed that all the efficiencies in Tables YI. and VII. exceed the corresponding ones in Table II. except the first in column (3) of Table II. In Table II. the terminal angle, a, is constantly 25, while in Tables VI. and VII. it is less than that value, and in the highest efficiencies very much less. 5. It appears from these tables that the terminal angle, <*, has frequently been made too large for best efficiency. 6. That the terminal angle, a, of the guide should be compara- tively small for best effect ; for the inflow less than 10, and that theoretically, when the angle is about 7, the efficiency is some 10 per cent, greater than when it is 25 in the wheels here considered. 7. Tables II. and VI. indicate that the initial angle of the bucket should exceed 90 for best effect for outflow wheels. 8. Tables II. and VII. show that the initial angle should be less than 90 for best effect on inflow wheels, but that from 60 to 120 the efficiency varies scarcely 1 per cent. 36 HYDRAULIC MOTORS. 9. The most marked effect in properly proportioning the ter- minal angle, , of the guide is shown when the initial angle of the bucket is 1 50. In this case the efficiency for the outflow when a is 25 is 0.7-14, Table II., but when a is 13, as in Table YL, it becomes 0.921. For the inflow, in the former case, it is 0.752, but when the angle is 3, as in Table YIL, it becomes 0.918. 10. Since the wheels here considered have the same width of crowns and the same terminal angle of the bucket, the depths of the wheels will be proportional to 2 for discharging equal vol- umes of water. Tables III., YL, YII. show that the section & a in- creases as the initial angle of the buckets increases, and that it must be greater for the inflow than for the outflow ; hence the depth of the wheel must be greater for the inflow for delivering the same volume of water. 11. But the same volume of water delivered by the inflow does more work than that of the outflow ; the depths should be as &,, divided by the efficiency. Thus in Tables YL and YIL, for y = 90, and for the same heads, II, the relative depths should be for equal works (0.759 ^- 0.828) -j- (150 -j- 0.920) = 1.67. 12. In the outflow wheel, column (9), Table YL, shows that for the outflow for best effect the direction of the quitting water in reference to the earth should be nearly radial (from 76 to 97), but for the inflow wheel the water is thrown forward in quitting (column [9] Table YIL). This alone shows that the velocity of the rim should somewhat exceed the relative final velocity back- ward in the bucket, as shown in columns (1) and (5). 13. In these tables I have given all the velocities in terms of V 2 g h, and the coefficients of this expression will be the part of the head which would produce that velocity if the water issued freely. In Tables YL and YIL there is only one case, column (5) of the former table, where the coefficient exceeds unity, and the excess is so small it may be discarded ; and it may be said that in a properly proportioned turbine with the conditions here CASCADE WHEEL. HYDRAULIC MOTORS. 37 given, none of the velocities will equal that due to the head in the supply chamber when running at best effect. 14. The inflow turbine presents the best conditions for con- struction for producing a given effect, the only apparent disad- vantage being an increased first cost due to an increased depth, or an increased diameter for producing a given amount of work. The larger efficiency should, however, more than neutralize the increased first cost. 15. Column 7 and equation (29) show that the pressure at the initial rim decreases as the initial angle y l increases. 16. Tables VI. and VII. are for parallel crowns. Examples of buckets of variable depths will be given later, and are illustrated in Figs. 19 and 24. SPECIAL WHEELS. 23. Fourneyron Turbine. All wheels having guide blades are of the Fourneyron type, although the wheels made by him were outward flow. The preceding analysis is a general solution of this turbine. 24. Francis and Thomson's vortex viheels are inward flow wheels with guide blades. The preceding analysis is also applicable to these wheels. FIG. 8. 25. The Jonval Turbine is a parallel flow wheel with guide blades, to which the preceding analysis is applicable by making r t r 2 . (For the details of these and many other forms, see Hydraulic Motors by Weisbach.) 26. Rankine Wheel. This is a wheel of the Fourneyron type, but Rankine having made certain modifications in its 38 HYDRAULIC MOTORS. assumed construction it is indicated by his own name. (Fig. 8.) It is an outflow wheel and the crowns are so made that the radial velocity of the water in passing through the wheel will be uniform. If x be the abscissa from the axis of the wheel to any point of the crown, and y the distance between the crowns at that point, v r the radial component of the velocity, then y 27raj - v r = Q, or, yx = a constant, .... (71) which is the equation to an hyperbola referred to its asymp- totes. This determines the form of the crowns. If the wheel were inward flow, the depth would be greatest at the inner rim. In this wheel the initial element of the bucket is radial, or y l = 90 ; and Rankine assumed that the velocity for best effect must be such that the water will quit the wheel radially, or 90. These conditions given, from Fig. 1 and equations (5), (6), (34), for & frictionless wheel, G^T! = V cos a. . . ..... (72) 2 r { into equal parts by concentric arcs, and the space aB into the same number of equal parts, and through the points of division a, &, c, d, trace the buckets ; then will aD, drawn through the proper intersections of the arcs and buckets, be the required path. VERTICAL SECTION OF A PROPOSED DOUBLE FOUBNEYRON WHEEL WITH TRIPLE CHAMBERS FOR GREAT FALLS. TRIPLE DOUBLE WHEEL. HYDRAULIC MOTORS. 41 9. Analyze a Kankine turbine, having given : H = 12 feet, X 2 -- 15, >'i = 2 feet, ?' 2 2 = 2rv Depth of outer rim, 6 inches. Find Eadius of outer rim, ...... r 2 = Angular velocity, ....... &> = Velocity of initial rim, .... ^co = Velocity of outer rim, ..... r 9 a> = Angle of guide plates, ...... a = Velocity from the supply chamber, . V = Initial velocity in bucket, . . . . Vi = Terminal velocity in bucket, . . . . v 2 = Velocity of quitting water, . . . . w = Depth of inner rim, . - . . , . . . y, = The horse-power, ...... HP = The efficiency, ......... E = If the partitions for the buckets occupy ^ of the wheel, and the losses due to frictional resistances in the wheel and friction of the wheel be 20 per cent., what will be The horse-power, ...... HP = The efficiency, ........ E = Find the pressure at the inner rim, . . . pi = Find the path of the water. 29. Velocity of a particle along a tube rotating about an axis perpendicular to its plane. This problem has already been solved in establishing the general equations of turbines, and the following is given to present it from another point of view. If the particle at A, whose mass is m, be confined while the tube rotates about 0, Fig. 11, with the angular velocity GO, the centrifugal force would be ....... (84) If 6 be the angle between the radius vector prolonged and the normal upon the tangent, the component in the direction of the tangent to the tube will be 42 HYDRAULIC MOTORS. moo 2 p sin 6, and when the particle is free to move, this component will be effective for producing motion, and if the pressures at the opposite ends of the element are not equal, but differ by an amount dp, we have the equation cPs -T = moo 2 p sin dp. (85) D PIG. 12. But cfesin 6 = dp, which combined with the preceding equation gives dsd*s 1 dp , ,, 2 = co 2 pdp ; - a dp. . . dt 2 m sin 6 r (86) dp_ sin d But -^-7, = dSj and if 6 be the weight of unity of volume, . u then m = -ds, and the last term becomes ^ dp. The integral J/ will be . (87) HYDRAULIC MOTORS. 13 If the friction be ^ 2 v 2 2 , the equation becomes . (88) which is the same as equation (9). If p 2 Ph an d fa = 0> then v} = V + *9 2 (r a * - rf) ...... (89) This gives the velocity relative to the tube whether it revolves to the right or left, and whatever be its curvature. If it revolves to the left, the resultant velocity will be AD, Fig. 12 ; if to the right, it will be A C. If y 2 be measured from the arc backward of the motion, or y% = BAF for rotation to the left, and y 2 = EAB for motion to the right ; then AD 2 = w 2 - = v} + Go-rf 2v 2 . oor 2 cos y z . . . (90) - = tv 2 = v 2 + co 2 r 2 2 4- 2v 2 . oor 2 cos y 2 . . . (91) In the latter case the quitting velocity will exceed the ter- minal velocity in the tube, and therefore increased velocity will have been imparted to the water a condition requiring that energy be imparted to the wheel from an external source. Im the former case the wheel is a motor, in the latter it is a re- ceiver or transmitter of power ; in the former the water drives: the wheel, in the latter the wheel drives the water and virtually becomes a centrifugal pump. If the water issues tangentially to the path described by the orifice, then y 2 0, and F 2 = v 2 T Gar* . ...... (92) the upper sign belonging to the motor, and the lower to the pump. Exercise. If ^ = 1 ft. r 2 = 5 ft. /v 2 = 0.1, v l = 5 ft. per sec- ond, and the bucket rotates about a vertical axis 30 times per 44 HYDRAULIC MOTORS. minute, and discharges the water directly backward, making Yi = 0, required the terminal velocity along the tube and the velocity of discharge relatively to the earth. 30. Wheel of Free Deviation. In this wheel the water in the buckets has a free surface, or, in other words, is subjected only to the pressure of the atmosphere. For this case Pz = Pi = p a 5 *i = H, and h- 2 = 0, and equation (4) gives (1 + ,1,) F 2 - 20/7, (93) which will be the velocity of discharge from the supply chamber into the wheel ; it is the velocity due to the head in the supply chamber when frictional resistance is included. The triangle ABC, Fig. 1, gives V? = F 2 + a? r f - 2 Ffi>n cos a, ... (94) which substituted in equation (88) gives (1 + /< 2 ) v 2 2 = F 2 + GJ> r} - 2 F. cor, cos a, . (95) and this in equation (90) gives F 2 , and equation (12) will give the required work. An exact general solution involves a solu- tion of the general equation of the fourth degree. See article 89. 'The following is an approximate solution. If y a be small, and the wheel be run for best effect, that is, so as to make the velocity F 2 very small, and considering y 2 = 0, equation (92) makes V 2 = cor 2 nearly. Using this value as if it were the exact one, also neglecting friction, (95) gives 2 V. vr } cos a = F 2 = or 2^! cos = HYDRAULIC MOTORS. which gives the proper velocity of the initial rim ; and for the terminal rim cos Number of revolutions per minute (97) (98) To find the velocity at any point of the bucket relative to the bucket, drop the subscript 2 from equation (95) giving (1 + // 2 ) v 2 = - 2 cos a. . (99) FIG. 14. FIG. 13. From Fig. 1 or equations (4) and (5) find sin y l = sm a. In the Motionless wheel, the work done will be and the efficiency will be U (100) (101) (102) 31. To find the form of the free surface, let the bucket be very narrow, so that a normal to one of the curves will be approx- imately normal to the other. Divide one side of the bucket 46 HYDRAULIC MOTORS. into any convenient number of parts, as c, ce, etc., and erect normals to the arc, as ah, cd, etc. Lay off these arcs on a right line. Compute the velocity at any point, as d 9 Fig. 13, by for- mula (99). Let x be the required depth at d, then because the velocity into the section equals q, the volume passing through one of the buckets per second, we have x.dc.v = q; dc.v (103) and similarly for all other sections. If only relative heights are to be found, the quantity q need not be found, for if y be the height at b, Fig. 14, then y.ba. Vl = q\ ba.v l ' and by assuming any arbitrary value for y the relative value of x becomes known. Similarly, the relative heights at all other sections may be found. 32. To find the path of the fluid in reference to the earth, pro- ceed as in Article 21 of the discussion of the general case. 33. Exercise. Design a 30 horse-power inflow turbine of free deviation, given an effective head of 16 feet. Assume the depth of gate opening to be 4 inches (J foot), and after the computation has been completed if it does not give 30 horse-power the depth may be changed by proportion. Let the radius of the outer or initial rim be 1 ft. ; of the inner rim, f of a foot ; terminal angle of the bucket, y 2 = 15 ; termi- nal angle of the guide, a = 30, ^ = 0.10 /* 2 . Then, velocity of exit from supply chamber, Eq. (93), .... ...... V= Velocity of outer rim, Eq. (96), . . . . <&ri = Velocity of inner rim, Eq. (97), .. HYDRAULIC MOTORS. 47 Number of turns per minute, = Initial angle of bucket, Eq. (100), . . . y l = Initial velocity in bucket, Eq. (94), ... v l = Terminal velocity in bucket, Eq. (95), . . v 2 = Velocity of exit, Eq. (90), w = Direction of outflow, Eq. (35), . .... 6 = Coefficient of discharge 0.60, volume of water, Q = Weight of water (S = 62.4), . . . ' . , dQ = Work per second, Eq. (101), Z7 = Horse-power, HP = Efficiency, . . E If 90 per cent, of U is effective work, and if this does not give 30 horse-power, then the depth of the wheel should be 30 4 inches. Find the profile of the stream in the buckets. 34. The following is taken from the report of the Commis- sioners of the Centennial Exposition, 1876, on Turbines, Group XX. The tests were for two minutes each. The revolutions and horse-powers here given are those corresponding to the best efficiencies : Diameter of wheel. Inches. Head in sup- ply chamber. Feet. Revolutions per minute. Horse- power. Efficiency, per cent. No. of Buckets. Kind of wheel. 30 31 255 95 85.0 10 Inflow. 24 31 302 67 77.0 14 Parallel. 24 30* 310 64 74.5 13 27 30 291 76.8 80.3 16 30 30 257 74 75.5 18 25 31 288 46 82.0 12 Parallel. 30 29.2 258 80.5 78.7 13 In and down 25 30 279 62.5 83.7 15 In and down. 27 30.4 246 53.2 73.6 14 Parallel. 36 29.6. 197 66.2 83.8 26 Parallel. HYDRAULIC MOTORS. These tests were by no means exhaustive. It is not known that they were run for best effect. The distance from centre to centre of buckets varied from 4.3 inches to 9.5, and at these extreme values the efficiencies were about the same. The number of gate openings was less than the number of buckets. TUKBINES WITHOUT GUIDES. 35. Barker's Mill. As ordinarily constructed, this motor has two hollow arms connected with a central supply chamber, with orifices near their outer ends and on oppo- site sides of the arms. There are no guide plates The supply chamber rotates with the arms. The arms may be cylindrical, con- ical, or other convenient shape. Since the water issues perpendicularly to the arms y- 2 = ; and since the initial elements of the arms are radial, yi = 90, and as the water must flow radially into the arms, a 90. The inner radius is necessarily small and may be considered zero. Hence, making y , = Q, ri = 90, a = 90, n = 0, FIG. 15. equation (14) gives U = w& = Equation (19) gives . .- (105) HYDRAULIC MOTORS. 49 hence, the efficiency reduces to, for the frictionless wheel, E = 2 (107) V 2 4- oor. z This has no algebraic maximum, but approaches unity as the velocity increases indefinitely. Practically it has been found that the best effect is produced when the velocity of the orifices is about that due to the head, or . . . . (108) for which value the efficiency will be, if yw 2 = 0.10 =0.70. . . . . (109) If & 2 be the area of the effective section of the orifice, then . . . , .<.'.." (110) The pressure on the back side the arms opposite the orifices will be . , .... (Ill) / Of this pressure there will be required P 2 = M.Gor 2 =^.oor 2 , ..... (112) to impart to the water the rotary velocity oor 2 which it has when it reaches the orifice. The effective pressure will be PI P 2 , and the work done per second will be this pressure into the distance it traverses per second, or U=oor,[P l -P 2 ], which reduces to the value found from equations (105) and (106). 36. Exercise. Let the supply chamber be square, and from two of its opposite sides let pyramidal arms project. Let 50 HYDRAULIC MOTORS. H 10 feet, orifices each 2 square inches, vertical section of arms through the orifice each 4 square inches, section of the arms where they join the supply chamber each 8 square inches, horizontal section of the supply chamber 36 square inches, r 2 = 36 inches, velocity of the orifice oor 2 = V^glf, coefficient of discharge 0.64, and // 2 = 0.10. Required : Velocity of discharge relative to the orifice, v- 2 = Velocity of discharge relative to the earth, V t Velocity at entrance to the arms, ... Vi = Velocity in the supply chamber, . . . The volume of water discharged, ... Q = The weight of water discharged, . . . 6Q = The work per second, 17= The horse-power, ... . . ... HP = The efficiency, E ' = The pressure on arm opposite orifice at A per square inch, p l = The pressure at base of the arms at C, . p = The equation to the path of the fluid. 37. . Scottish and Whitelaw Turbines. These wheels have no guide plates, and differ from Barker's mill chiefly in having curved arms. The analysis is precisely the same as for the Barker mill. The only practical difference consists in pro- viding a curved path for the water, instead of compelling the water to seek its path, forming eddies, etc. 38. Jet Propeller. We first show how this problem may be solved by the preceding equations, and afterwards make an independent solution. Let a narrow vessel, Fig. 16, be carried by an arm E about a shaft BA. Let water, by any suitable device, be dropped into the vessel, the horizontal velocity of the water being the same as that of the vessel. At F, the lower end of this chamber, let there be an orifice from which water may issue horizontally. The water may then be con- HYDRAULIC MOTORS. 51 sidered as entering the vessel or bucket without velocity, and passing downward finally curve towards, and issue from, the orifice. It thus becomes a parallel flow wheel without guides, and we have, for tlie frictionless wheel, - r 2 , Yl = 90, 7 / 2 =5 0, #J = Pi = Pa, = // 2 - 0, in equation (8) ; hence, the velocity of exit relative to the ori- fice will be v} = 20*2 (113) FIG. 17. FIG. 16. where % equals the head in the supply chamber. Under these conditions the velocity of discharge will be independent of the velocity of rotation, if the rotation be uniform. Equation (11) gives for the velocity of discharge relative to the earth T7" / 1 -i A\ K 2 = v 2 cor 2 (114) Equations (19) and (14) give = - V 2 9 (115) 52 HYDRAULIC MOTORS. which acts radially inward and its components are moo cos y -si along the vane, moo cot Y cos y -j~ normal to the vane. r/ v (No. 7.) By moving from a' to d\ v will be increased by an amount dv . -j- dp, dp in the time dt, and the reaction will be dv dp m -j- . -j,, dp dt* which will be outward along the vane, and the reaction will be directly backward along the vane, and hence is dv dp , ,, m -j-. -j- along the vane, normal to the vane. (No. 8.) In passing from a' to d, v is changed in direction by two amounts : the angle y changes an amount HYDEAULIC MOTORS. 61 This is negative, for a differential is the limiting value of the second state minus the first, and the first is here larger. But this is not the total change, since y "is measured from a radius making an angle dp cot y ~^~' with Oa' as in No. 6 ; hence the total change will be the sum of these, and the rate of change will be the sum divided by dt, which result, multiplied by the momentum mv 9 will give the reaction, which will be normal and in the direction l>'n or along the vane, fcot y dp dy dp~\ , , mv - . -T, -T- -T. i normal to the vane. L p dt dp dtj This completes the reactions. Next consider the pressure in the wheel. The intensity of the pressure on the two sides ab and cd differs by an amount The area of the face is dc x x = xpdO sin y, and the force due to the difference of pressures will be xpdO sin y ~- dp. If dp is positive, which will be the case when the pressure on dc exceeds that on ab, the force acts backwards, and the preceding expression will be minus along the vane. In regard to the pressure normal to the vane, if a uniform pressure p existed from one end of the vane V IF to the other, the resultant effect would be zero, since the pressure in one direction on V W would equal the opposite pressure on XT. If, however, in passing from d to a, the pressure increases by an amount dp, 62 HYDRAULIC MOTORS. since Va is longer than Xb, the pressure on Va will exceed that on Xb by an amount dp . x x ah = dp.x.pdO cos y = xp cos ydO -f-dp. Collecting these several reactions, we have NORMAL TO THE VANE. (1.) (2.) + moo 2 p cos y. (4.) moot. ,. .dp (5.) - (6.) moo cot y cos y -jr. (7.) \~coiy dp dy dp~] (8.) + mv\ - . -JT . JT \. L P dt dp dt_\ (9.) xp cos y J- dpdO. ALONG THE VANE. + ma> 2 p sin y. 0. dp + mcocosy-j-. dp mcocos y-Ji m dp dv dt ' dp' 0. ap , , xp sin y -j- dpdO The sum of the quantities in the second column, neglecting friction, will be zero ; hence motfp sin y m -rr . -, xp sin y -j- dpdO = 0. . (123) Substituting dp . , 7/3 , m r. = v sin y, and xpdudp = ~^ and dividing by m sin y, we have co 2 pdp - dp = vdv. (124) HYDRAULIC MOTORS. 63 Integrating, r ~| limit r -I limit UcV--P =U^ 2 .... (125) L- $J limit L_ -I limit The sum of the quantities in the first column gives the pressure normal to the vane, which, multiplied by p sin y, gives the moment. This done, we have fP r\ . dy ojy GO cos y 2 ) pv sin y -j \v / dp = mv sin p - cos y dp 4- v cos y p ^- -r~ vo dp fi O Putting mv sin y = o dpdft, where Q is the quantity of water flowing through the wheel per second, and integrating in refer- ence to 6 between and 2?r, we have dM<=M \o*P (^ cos y- 2] -pv*m y^+v^y-p^ ^1 dp L \v dp vd dj g L \v dp vd dp Multiplying (124) by P cos y, v we have > 2 p 2 7 p cos v dio 7 dv 7 cos yelp L - L -/ dp = pcos y . dp, v v6 dp dp which substituted above gives d J/ t _ ^~ dp the integral of which is J/; = tf Q [ G?p 2 + p^ cos y] -r- g ^ C os r ] !i S Jt : . . . (127) <7 64: HYDRAULIC MOTORS. But GO p v cos y is the circumferential velocity in space of the water at any point, and S Q p [GO p v cos y] is the moment of the momentum ; hence, integrating between limits for inner and outer rims, the moment exerted l>y the water on the wheel equals the difference in its moment of momentum on entering and leaving wheel. Let the values of the variables at the entrance of the wheel be p n Xi, ^,,^i, and at exit p n y v,, p. 2 . Equations (125) and (127) become M = " r, 1 ; .-. ?7 = il' (v, - r, co) r, A/- - j \ 2 + tan* a as in equation (76). 42. Again, if the crowns are parallel discs and the initial element of the bucket is radial, and if the water quits the wheel radially, and if the velocity of whirl equals the velocity of the initial rim, we have U= Mr: ?, (136) as in equation (134). But y^ will not be the same as in (135). To find it we have, neglecting the thickness of the walls of the buckets, 2 7t >\ v l = 2 7t i\ sin y^ - v^ V^ V sin a r t GO = V cos a F" 2 t\ oj tan / 2 v, = i\ & ; /\ a tan OL /. tan y, =: - - ; (137) V r * ~ /J i 4 i an * a .\U= d QII-^M V; ?\* ta?i* a l M r? v* - -- - 7 -. (138) /* r* tan 3 a Equations (136) and (138) give 3 4 r* tan 9 a (139) HYDRAULIC MOTORS. 67 Practical and Experimental Data and Results. A STUDY OF THE TEEMOXT TURBINE. (Revision of a paper by the Author in Vol. XVI. of Trans, of Am. Soc. of Mech. Engineers.) 44. The Tremont turbine furnishes an excellent example for testing the theoretical formulas for the proportions and deliver- ance of turbines. The one here analyzed was made by J. B. Francis, engineer, after the general pattern of the celebrated U. A. Boyden, Esq., turbines, which yielded, from careful ex- periments, an admitted efficiency of 88 per cent. The dimen- sions of a Tremont turbine and the careful and somewhat ex- haustive efficiency tests are fully set forth by Francis, in his work entitled Lowell Hydraulic Experiments. (Fig. 19.) The workmanship on these wheels was of high grade. The crowns, which were of cast iron, were accurately turned in a lathe, and the partitions (or walls) of the buckets were of Russia sheet-iron plates, ^ 9 of an inch thick. These plates fitted into grooves carefully cut in the crowns, on which were tongues pro- jecting through mortices in the crowns, and the former headed down, thus securing the crowns to each other without bolts or rods, and forming smooth, unobstructed passages for the flow of water. In the Boyden wheel a "diffuser" was added, which consisted of two crown -like pieces outside the wheel, but not rotating with it, the space between which was diverging, produc- ing a diminution of velocity of the escaping water, causing more of its energy to be imparted to the wheel. This device, it is said, added some 2 or 3 per cent, to the efficiency of the wheel. The Tremont turbine was not provided with this device. (Fig. 30.) We have selected for analysis the experiment by Francis which yielded the highest efficiency ; for our turbine formulas are strictly applicable only when the wheel is so proportioned and run as to give a maximum efficiency. 68 HYDRAULIC MOTORS. auBA am ^0. pajBOipm SB paqAv aqj Sui -ABai iajBAY aqi jo uoijoajtQ; C050 OSt-OGO t-OGOO t- ^COO roo CO puooas J2 Ja'l laaj ui paqAv aqj uo Sin ^ -JOB i[t:j aqj oj anp XjjoopA cox-^co^co^rcooci ^5 tc S co co t^ t- <--~ t- x X' X' X' X X X X X X GO X X X X X GO x' CQ O"> OJ Ci C3 C"< (?>? d O1 C "popuadxa ja.vvod am t 1C C5 ^ IC-^fC^O'^t 1 OO -^fCOO ^HO'?- TtHco^oco xco oicSco t^ CO t^ CO CO O t- X X X S5 CI X- GO puooas ,iad looj auo S pasuu aiodnpaioAW npnuod ui 9 aajBAV am jo aaAvod ^^f.-c--rtioxo>^j.-rcixo'*cocs l> CO CO O> X O '-fi CO CO L~ i.-O T- CO X CO CO 1-1 x co LT 07 x j.t- cocoaoec5^ "^ CO O? I-H O O OS '-' C5 X Cl GO GO t^ aqi -jaaj in ^COOOOCOOOCOXClClCOOOi-'O 1O O O O T-H LO CI X X CI O C5 O CO T- "t CO 03 oi oj o? o? 03 oi 07 oi oi oi oi oi oi oi o? o> puooas aad jooj auo pasiBa siodnp 1-1 -JIOAB apunod ui a^Baq aqj jo uoijouj aqj JO 'j CO CO IO CD SS Ci C5 X X X C- C7 "^ CO XX puooas .iad aqj jo suoi}ii[OAaa jo jaqumx; ,o OOXXCOCi^t ( ^CtXXOX"cOXrf I ^J -4 ,-i T_; r-i o o o o ~ o" o T-* o' o" o saqoui ni aiB SnilBinSaa aqi' ^H -juauiuadxa am HYDRAULIC MOTORS. 69 O O7 l> O7 r- O O7 GO O O7 OS O CO CO TH 07 O O 07 CO CO xH o TH TH t~ ^eswcftGOJGOTHOoiTHt-^THOicttt-ocososocfcsoo&o ^r O O O IQ r^ ^ T< 30 O 6 CO T* OS 00 CO CO C* 00 D O > O6 -3 SO CO SO N t> CO TH CS CO i CS O Cl TH O OO CO OO TH L- c- tO TH CS O CS O7 O? Xi O CS GO CO TT CO TH CO CO CO to tO O 00 t- O SO tO 1C O7 30 30 GO t- t- TJH CO O to OS ?C *$ CO 09 70 TH :OOSO7O7X>C-t>OSTH OOX>THO3O3COTtltOO5 OS T Q T-I cs CS -H JO TH -i 07 CO ^ TH f ^iS O7 TH to O7 * f- CO 00 ^4OQOiH O OS CO CO 00 O TH O7 O TT O* t- O TH IO C^ CO O O CS CO O CO CO TH O O O? GO O CO I- CS O? 00 T-I TH CO O O T-I O7 CO 1C O O CO OJ -: CS GO CO IO X> Oi CO OO O Tf O CO O >-H t- T~i CO O O CO ^f 1 CQ t-- L- CO O O Tt< 00 O OS c; ^p O O T-I t- T CO C< QO T-H O O> " - OS OO t- A o i -- 1 O CT O OS CO O O T-I GO CO O OO ^ O C* OT - ^ t- r-i O C5 !?07DClCJO>?OiOO^r-iOi>i-i^^OOOTHCl^t-t-C30'-(Oi.':"^-O?GOCiOCO 00 C? O) 70 O CO 00 -X) QO O .00 GO OO OS O5 OS O T-H O i T C^J C-7 CO i>> ; 30 CO CO "^_ O -^ OS O O GO ;r 06 a5 -06 od ad a5 TD ad 06 ad 06 06 a5 od a> cs' os' ci ri cs' os os 06 a5 cs os" os" cs' as' as' o o OlC^Ol^l 0> ^< 07 O? O? 07 Oi 07 07 07 O< O7 O7 O7 O< O7 O7 O7 O7 O7 O7 O7 O7 O7 O7 O7 O7 CO CO O O7 O GO t^ TP O T-t OS t^ O * rf O C- O7 OS 00 ~X> *# t- O O O7 O7 <^ O O O T-H O OS I- rt< O O -^ t- rH CO O7 CO O Cl T-H O CO O -H ^r 1 X) 7D CO O O7 -rf O -f DC5oiot^t^ co ^ -^ ^ T-I 07 O 03 ^H 30 CO O 10 l> O Z> 07 iO CO O i-- O_ IT OS ^ T-H as O 00 O O TH CO O O 'JO O O t^ O O "* OS* O7 CO O t^ Tjn O7" O7 O 3 O7 ^' OS* ^ 00 C^ O O O CO OS GO Tf' O7' t^ O YJ CO O CO T-I CO TH O O 30 rf O i-T OS C5 O OO O O7 T- 3D t- J> T-t O7 ^ CO T 70 O O7 O O7 CO O T-H O C5 ~H t- I- ' CO OS C5 L*>TH:o:O:OL-~THOCSCSGOOTHTHOO7OlOTHC5t~O7 lOGOCSTHCOOGOCOOCSCO ^ 2 j , /3 jo CO TH OS O TH 07 t- TH O CO t- GO -H cs CS TH TH rH O CO ' 10 10 OS T-, CS CO -H os as 10 to o* O7 o to co to o as o as as TH o to to -^ TH co TH o TH to 07 c~ i to 07 o t- T-, O t-- i - O O O TH TH CO t- O O CO OS GO to OS to O7 IO O tO IO CO tO TH O GO 00 CO O7_ to TH jjj ^5 ^j5 TO" to' O7* GO* GO* GO CO' t*-' J>" io" CO' -H'O OS* T-H GO* TH TH GO' O O l^ O O O TH t> O GO" GO* l> CO iOCOCOCOCOCOCOCOTfCOCOCOCOCOCOTHO>THTHT-iOOOCOCOaOt~t-OOCOCOCO t J> OS C- O *** O iO O C O os as t'* ^ as as as t~~ as as as O7' O7 O7 O7 O7 O7 O7 O7 O7~ Oi O7 O7~ O7 O7 CO CO O"> CO CO CO CO CO CO CO O7 O7 CO CO CO C CO 07 OS T-I X) T^ 07 ^ O iO 07 T-I t- ^ T-I 07_ T-I r-t so TJ ^ O TH OS _ GO J> O o os' O7 o o co" o o 07' c^ o T-' 10 DgOG>O^cit-ec$t^*OP *o o> o x> e> o OasOTt*oio07rHT-iOSOOX> iO^O7t-OOOO 10000 OS07TH TH-HH OO7GO^fOOO7a5'-^{>C~- O --H {>- C O7 O >? T I O7OS-HH OS-HH30 ooDt-ciHJo O7THOS osco30x>:o:r>o:o rHr-HH >GOX>COit'aOC/OGOGOt~ CC65dOO^t l fr-O i> to co o cs c- o oo i- 07 as o n< GO 70 HYDRAULIC MOTORS. The coefficients /*, and /* 2 are uncertain quantities. Weisbach considers /<, = 0.10, and /* 2 from .05 to 0.10. We will assume ju l 0.10, and for the assumed Boyden wheel yw 2 = 0.05. Our analysis will show that /^ Q = 0.15 nearly, for the Francis wheel. For the wheel we now analyze we have (see Lowell Hyd. ^ = 3.375 feet ; r, 4.146 feet. a - - 21 } as measured from Plate III. of Experiments* y^ = 10 > (We have used ;/ a = 13 and 15 for reasons Yi = 90 ) which appear later.) Q = 138.2 cubic feet. // = 12.9 feet. A, = 15.11 feet. A, = 2.2 feet. U' = 111,218.1, total power of the water in foot-pounds per second. E 0.79375 efficiency. H.P. 161, about, as measured by the brake. n 0.85106, number of revolutions per second. y^ = 0.9368 feet, depth between the inner edges of the crowns. y 2 0.9314 feet, depth between the outer edges of the crowns. JV 44, number of buckets. g = 32.16. With this data we solve as follows : I2ev. per sec. -- . (1^0) 2 71 * Rodmer in his work gives a = 28, y\ = 90, y 2 = 22, but these do not agree with the plates given by Francis, neither do they produce results agree- ing with Experiments. Rodmer considers these as mean angles of the stream. FIG. 19. HYDRAULIC MOTORS. TABLE IX. 71 a = 21 V 2 = 13 j*a = 0.20 a= 21* Y 2 = 13 im 2 = 0.10 a= 21" V2 = 13 M 2 = 0.05 Measured Values. w Eq (16) 5 369 5 476 5 514 Rev per second. 855 872 879 351 Emu P er cent. Eq. (16a) vt feet. Eq. (18) Vi feet Eq (19) 79.308 6.968 23 115 84.84 7.102 23.989 87.76 7.157 24 525 79.375 ^2 fc\ T\ /rn\ - = T = Eq. (20) Vi KI 26 feet. Eq. (21) yi feet Eq (21) 3.316 0.957 1.103 3.37 0.939 1.063 3.426 0.932 1 039 0.9368 0.9314 It will be observed that for /* 2 = 0.05 the efficiency is over 87 per cent., and if a diffuser would add 2 per cent., we would have an efficiency exceeding that admitted for the Boyden wheel. But it seems improbable that the prejudicial resistance can be so low ; and it is well to observe that for a = 20 degrees, ftj = 0.10 = jw 2 , and y^ 10 degrees, a theoretical efficiency nearer 90 per cent, will be found. It will be seen that for // 2 = 0.20, some of the numbers given }n the first column of the preceding table agree well with actual values. Thus the revolutions are : Computed . .... . . , . 51.3 per minute. Measured -, . . 51.1 per minute. Also the efficiency is : Computed .. 79.308 per cent. From measurements . . , * 79.375 per cent. The initial depth between the crowns is : Computed .... .... 0.957 feet. Measured 0.9368 feet. Difference .0202 feet. 72 HYDRAULIC MOTORS. The outer depths between the crowns are : Computed 1.103 feet. Measured . . .9314 feet. Difference ; . . . .1716 feet. or . . ... .. . . . 2.059 inches. Not only do the computed depths exceed the measured ones, but our computed depth is less for the inner rim than for the outer, while the measured values are the reverse. This cannot be dismissed with the remark that the wheel was improperly proportioned; for the 138.2 cubic feet per second passed the weir ; therefore this requires further investigation. 45. The depths between the crowns were determined as follows : Dropping the subscripts in the notation and placing p for ?\ or ?' 2 , we have y (2 7t p Nt -4- sin y) v sin y = Q (1^1) in which 2 n p = whole circumference at distance p from centre, JV7 = thickness of N bucket plates. ^Yt -r- sin y thickness measured on an arc. y (2 n p Nt -r- sin y) = free ring section for passage of water. v sin y = radial component of the velocity in the bucket, and the product gives the quantity of flow in feet per second, or Q. Hence, for initial and terminable values, we have, since Q is constant, y ._^ 8 ,r>nr.-37)_*-( 8 x8 - M -ir) y. v~ (2 7t i\ sin y. JV?) / 44 X 9 *' 1 * ^ * * * / At I VI Tf 'i OT 11 /* * If the thickness of the bucket plates be neglected, we have simply (y l being 90 degrees), ?/., ?*, HYDRAULIC MOTORS. 73 which for = 13 degrees gives 3.38 6.968 23.115 X 0.2+5 -- or the depths would be equal ; and that the outer depth shall be less than the inner, y^ must exceed 13 degrees. We tried 15 degrees and made other computations, as follows : TABLE X. = 138.2 a = 21 72 = 15 /it a = 0.20 = K35.5 a = 21 Y 2 = 15' M 2 = 0.15 q = 138.2 a = 23 y 2 = 14- M 2 = 0.20 Q = 138.2 a = 23 V 2 = 15 M 2 = 0.20 Measured . (t) . . . 5 339 5 407 5 364 5 346 Rev. per second 853 867 854 851 851 E max . per cent Vi feet 78.33 6 954 80.9 7 017 78.90 7 696 78.10 7 67 79.375 V? leet 23.048 23 573 23 15 23 19 y\ feet 9439 9317 862 870 9368 y 2 feet ... 09346 9104 1 00 945 9314 In the first column the buckets are still too deep and the efficiency too small. Since the gate does not fit water-tight, a part of the 138.2 cubic feet which passed over the weir may have escaped at the gate, and hence the entire quantity may not have done work on the wheel. If there be 4- of an inch clearance at o the gate (this assumption is gratuitous, but will serve for illustra- tion) there will be an annular opening of 2 X 3-J- X 3.38 X -J- X r V = 0.222 square feet. It will be found hereafter that the in- ternal pressure at the gate is 2648 pounds per square foot, and the atmosphere and 2.2 feet head in the wheel pit gives an out- side pressure of 2254 pounds, leaving 394 pounds inside effective pressure at that point, which would produce a velocity = |/2 g X | 20 feet nearly ; 62.2 hence the volume of water which would escape under these con- 74 HYDRAULIC MOTORS. ditions would be 0.62 X 0.222 X 20 = 2.75 cubic feet nearly, allowing 0.62 for the coefficient of discharge. This would leave 135.5 cubic feet of water to do work on the wheel. The leakage produces a slight discontinuity. The computed (hydraulic) efficiency ought to equal the brake efficiency plus that of the shaft friction plus that lost by leakage ; so another computation was made, using Q = 135.5, a = 21, y, = 15, ^ = 0.10, ^ = 0.15, the results of which are in the second column of Table X., and are very good. The revolutions are one more per minute than those observed ; the efficiency, 1.525 per cent, greater than that observed, which difference is probably about right ; the inner depth, 2/j, is almost exact, but y^ is perceptably less than that measured. A reduction of 2 per cent, of the speed to reduce it to that observed would reduce the efficiencies but slightly, as shown by the experiments on page 68. Thus, in experiment 32 the speed is about 2 per cent, less than in No. 30, which gave the maximum, but the efficiency is reduced from 0.79375 only to 0.79294. In No. 48 the speed is nearly 6 per cent, less, but the efficiency is reduced about 0.7 per cent., or to 0.789. It will be observed that the wheel delivers nearly the maximum work when the speed is within 5 or 6 per cent, above or below that which gives the maximum efficiency. There is some uncertainty about the correct measure of the section of discharge. In equation (141) gives the thick- sin 7 ness of a bucket- wall measured on the arc whose radius is p, on the hypothesis that the arc is a straight line, which is sufficiently accurate when the wall is very thin as in this case. Also v sin y is assumed to be constant for all points in a ring section which is accurate for a fillet, and sufficiently so for a very narrow stream. But these conditions are changed when applied to finite streams, as in actual wheels. This will be more clearly seen if equation HYDRAULIC MOTORS. 75 (141) is so transformed as to apply to a single bucket, for which it becomes at the terminal end ft in which /2 = qf, Fig. 20, the distance measured on the arc covered by one bucket and partition-wall. t = he, the thickness of the partition-wall. Francis assumed that the correct section of discharge was not the arc ad, but was the least section, and experiments and com- putations combined confirm, at least approximately, the correct- ness of this assumption. Hence with a as a centre find by trial an arc which will be tangent at h draw ah, it will be normal to dh, and will be the base of the least section of fiow. Through the middle point of ah at cj pass the middle arc gt of the bucket. Then considering g as the terminus of the middle fillet, find 7, r, and v for that point, and call that velocity the mean, and let ^ m be the mean velocity ; then v m ah . y . N = Q. In this case y at g will exceed y. 2 at , r at g will be less than r 9 at a and v m will be less than /y 2 . We have not determined these quantities at g, for our knowledge of what takes place in the section ah is riot definite, for it is a state bordering on discontinuity, since the stream follows the bucket to that section, and from that flies off into space in the direction nahm more or less radial. We have not positive means of determining whether this view is correct, for we do not know if the bucket was filled with a live stream from end to end ; but we assume that it was, and are seeking conditions which will be consistent with this assumption. We have already seen that by increasing y t from 13, the vane angle, to 15 gives results which agree closely with our assumption, and we anticipate that where the terminal angle y t is small and the buckets numerous a similar result will follow in other cases. 76 HYDRAULIC MOTORS. Prolong ah to , being less than the least base, there is a partial compensation in the product b v 2 , so that y 2 = Q -T- 2? b v^ approximates toward the correct value. According to our analysis, y, v, (of sin 13 - t) = J. gave too large a value for y 2 ; then we found by trial that y 2 v, (of sin 15 - t) = j gave a value for y 2 a little too small. Had the mean velocity, v w ' been found, it would have been necessary to increase y still more? that we may have the equality y 2 v m (af sin y t} = j^. But in this case y 2 would be a mean depth along ah. If the bucket be divided into very small imaginary ones, say n such, we have FIG. 20. HYDRAULIC MOTORS. 77 ad . O y* v* sm r 2 = -4^ n nN but this is equivalent to assuming that the velocity is uniform throughout the cross-section and equal to v a , which is true only for very narrow streams. Cancelling n will reduce the equation to the same as (141). From the results in Tables IX. and X., we conclude that the prejudicial resistance ^ is less than 0.15 and exceeds 0.10 in this wheel. The terminal angle of the guide a perhaps ought to be increased in the analysis, for the same reason as that applied to Yv but not to the same amount, for the pressure at the gate will cause the water to follow the back of the guide more nearly to its' end. We try another computation, making Q = 136, a = 22, y, = 15, ^ = /i, = 0.13, with the following results : Hev. = 0.862 ; E = 0.821 ; y l = 0.884, y, = 1.038, which re- sults are not as good as those previously found. Since this value of y 9 exceeds y l9 y^ ought to exceed 15. Making y, = 17, //, = 0.10 = /i,, and other quantities as before, we find Rev. = 880; E = 0.8361 ; y l = 0.872, y, = 0.869, where the ratio of y 1 -f- y a is good, but the other values are too large. In regard to the value of or, measurements on Francis plates (Lowell Hydraulic Experiments] show that a, as it is measured, will not exceed 22, and yet Rodmer gives the mean angle a as 28. We solve the problem with Rodmer's values y 9 = 22, a = 28, assuming /*, = 0.10, ^ = 0.05 ; also /*, = 0.10 = /; with the following results : 78 HYDRAULIC MOTORS. TABLE XI. r t = 3.38 ft., r, = 4.0 ft., y l = 90, a = 28, y, = 22, 11 12.9 ft., Q = 136 cu. ft. Mi = 0.10 M 2 = 0.05 Mi = 0.10 M 2 = 0.10 Rev per second. 869 8636 -^max P er cent 81 96 79 30 i feet per second 9 822 9 755 #2 feet per second 23 525 23 059 V 20 9012 20 770 Vi Ea (141) . . 668 673 Vo Eq. (141). . 650 663 7/,/W 1 027 1 015 The brake efficiency of this w r heel being 0. 79375, its hydraulic efficiency will probably be 81 or 82 per cent., so that the last hypothesis above is not admissible. Rodmer's values for a and y t with low resistances give a satisfactory result for the effi- ciency, and about 1 revolution more per minute than the meas- ured one ; but they do not produce the proper values for the depths y l and ?/ 2 . The total least section of the buckets is &' 3 = 7.687 sq. ft., and if the velocity be v^ 23.5, the depth would be 136 7 f - 23.5 X 7.687 sq. ft. ~ so that the proper depth cannot be found with this data, and it would require a coefficient of contraction of 0.76 to give 0.93 +. The total least section of the guides, as measured, was K' = 6.5371 sq. ft. ; the total initial arc of the buckets was k\ = 19.380 sq. ft. To show the effect of a large terminal angle of the bucket, a computation was made with the data HYDRAULIC MOTORS. 79 r, = 3.38, ^=4.00, a = 22, y,= 90V, = /< a = 0.10, and y n _ = 28, and the result gave E rfMX . = 0.738, Rev. = 0.850, an efficiency nearly 6 per cent, less than the observed, at about the same speed. "With the same data, except j.i l = 0.20, // 2 0.05, it was found : EM*. = 0.780, Rev. = 0.820, so that with less revolutions the efficiency approximated to the actual. As seen, the method of " trial vane angles" may be made to fit the conditions of the wheel when the results are known ; but as a method of design it is not satisfactory, if not worthless. The " least section" as used by Francis is better, but the condi- tions of that cannot be realized, for the stream cannot change suddenly from the tangent al>' to a . AVe now suggest the fol- lowing, which, though not perfect, offers the most promising condition for a practical solution. Draw the tangent a //, Fig. 200, and through it,, the middle of the arc a d, erect a perpendicu- lar^ r to a T)'\ it will be found that in this case p is the point of tangency to p d of a line parallel to a I)'. A particle at a would, if free, pass off in the direction a 7/ tangent to the bucket, and would trace that tangent on a plane rotating with the crowns. Similarly, a particle at Ji would, if free, go off in a tangent at li ; but this tangent is not parallel to a V . Similarly at all points in a h the particles would, if free, move in tangential directions to their paths ; but those tangents all have varying directions. The particles, however, are not free. Those moving along the back of the bucket tend continually to leave it, and, with other particles in a h, force the stream toward a I', and the stream may leave the back of the bucket before reaching p, while the mass of water in the vicinity of a opposes such deflection, the result being a contraction of the stream. If the bucket be full at exit, the 80 HYDRAULIC MOTORS. coefficient of contraction may be found. The middle line of the bucket will pass through u, and at u the terminal angle y 9 will be the same as at a (10). With the data, r t = 3.375, >- 2 4.146 (which call 4.1, since the buckets do not extend to the outer rim) ; a = 21, Yl = 90, y, = 10 ; /i, = 0.10, // 2 = 0.20, Q = 136, we recompute, finding Rev. per sec. 0.867. -Z^max. per cent, 81.18. v l9 feet 7.072. tf a , feet 22,57. y 1? feet 0.93. ?/ 2 by equation (141) would exceed 1.3 ; but we will find it as follows : Measure^* r on the drawing, finding p r = 0.166, and assume that the value of v t 22.57 is the computed mean velocity of the section, and assume (for trial) a coefficient of contraction of 0.88 ; then v, X 0.88 (44 X 0.166) y, = 136 ; . . y 2 0.9304 feet, = 0.93 to the nearest hundredth, which is not. only sufficiently exact, but, in our ignorance of the depth of the stream, may be actually too large ; hence the co- efficient 0.88 is sufficiently small. If the coefficient of discharge be 0.90 the computed depth will be 0.9166 feet, which is about J- of an inch less than the measured depth between the crowns, and this may possibly be greater than the depth of the live stream at that point. HYDRAULIC MOTORS. 81 The exact character of the stream in the buckets is unknown, especially at the terminus, so that perhaps the computed depth ought not to agree exactly with that of the wheel, but it is very certain that the initial end of the bucket was full, with gate fully open, for with the least closing of the gate there was a diminu- tion in the volume discharged. Many other computations have been made, but they add little if anything to our knowledge. As a result of this study it is inferred that for an outflow turbine similar to that of the Francis, the efliciency may be determined quite accurately by equation (16#), using the dimensions of the wheel, including the guide and bucket angles ; and the initial depth between crowns to deliver a given amount of work may be determined by the second of equations (20) ; but the terminal depth is more accurately deter- mined by means of a measurement across the stream, as shown above. It will be seen hereafter that the same general principles apply to other turbines. No motor is designed which secures exactly a required result. There are minor elements which cannot be figured out exactly. After all the labor and study which has been given to the steam engine, it is necessary to " test " one to determine exactly what it will do, and it is the same with other motors. The perform- ance of a w r ell-proportioned turbine may, apparently, be deter- mined as accurately by theory as that of any other motor. 46. In this wheel the inside of both the crowns is convex in- ward, as shown in the figure, the curvature being such as to re- duce the depth between the crowns f of an inch at 5 inches from the inner rim. This reduction is about y 1 ^ of the initial depth. No reason is assigned for this form. If the object of this curvature was to conform to the form of the vena con- tracta, it appears to be too small. In any case, its effect will be to increase the velocity, and hence reduce the pressure in the wheel at that point ; but if the internal pressure exceeds the ex- 82 HYDRAULIC MOTORS. ternal pressure (the external being that of an atmosphere 2116 pounds per foot, and the weight of 2.2 head of water in the wheel pit about 138 pounds, or a total of 2254 pounds) it seems to be unnecessary. The internal pressure was computed from equation (45), page 22, which reduces to P = Pa + # h, 4 ) sin . Yi +; sin 2 (a + Yl ) with the following results for mid-section TABLE XII. ,)-- f] sin (143) If Then Pressure Ibs. per Square Foot. Pressure Ibs. per Square Inch. r-* *i = 1.205 2,637 18.31 y = 0.9 #! ^ = 1.339 2,630 18.26 y - 0.8 y! T = L51 2,610 18.12 y = 0.7 #1 ^ = 1.720 A/* 2,581 17.92 y = 0.6 i/i ^ = 2.01 /c 2,540 17.64 y = 0.5. y, ^1 = 2.41 2,380 16.52 In all cases where k is involved it is assumed that the stream completely fills the successive sections, otherwise k ought to be the actual section of the flowing stream : But it is more important, in this case, to find the pressure at other points in the wheel. We have found the pressure, p^ at the entrance into the bucket where p = i\, and at the width of HYDRAULIC MOTORS. 83 the crown from the inner rim, at -J, f- , and J, giving pounds per square foot as follows : Pi P\ P\ Pi Pi P* 2648 2670 2667 2610 2420 2254 ug = 0.986, ^ = 1.20, -r- 1 = 1.67, p = 2.2, ^ = 3.3. The pressure p^ is at the gate, where the velocity is Y = 7.02 4- sin 21 = 19.6 feet. At J the width of the crown the velocity will be v = 7.02 >< 0.986 =.- 6.29 feet, and the pressure is greater than at the gate. Since the terminal velocity is about 3.3 times the initial, the velocity ought to in- crease continually from the entrance into the bucket, to the exit, and therefore the normal sections should continually de- crease, which, as is seen, may not be the case with flat crowns and partitions of uniform thickness. Thus, as shown just above, the, initial section \ is 0.986 of the section at one-fourth* the width of the rim from the initial element ; hence the normal sections increase at first, and then decrease. To secure a continual dimi- nution of sections, the depths of the buckets may at first diminish and later increase by curving the crowns, as in Fig. 21, the least depth being near the inner rim ; or still more accurately as shown in Fig. 54. If the wheel be cast the diminution of section may be secured by increasing the thickness of the w T alls, as in Fig. 22, the crowns being plane. If the terminal angle be less than 10 degrees, the outer depth (in this wheel) should be greater than the inner, and the crowns should flare outward, as in Fig. 23. The wheel may be so submerged as to produce any desired terminal pressure. If the Tremont wheel were sub- merged 394 -^ 64.2 = 6.1 feet more (p. 73), or 8.3 feet below the surface of the water in the wheel pit, then p^ p^ or the pres- sure at the ends of the bucket would be equal. 84 HYDRAULIC MOTORS. 47. The direction of the water as it leaves the outer rim is given by equation (35), cot 6 = cot y -- . ^ sin y z where B is the angle measured from the rear arc. This gives, in this case, 8 = 91 33', or the water will be thrown forward of the radius prolonged 1 degree 33 minutes ; hence the direction should be nearly radial. Francis found, by a movable vane, about 34 degrees ; but it is difficult to account for so large an angle, and one is inclined to think that there was a defect in the measurement. The measured angles appear to be very erratic. It is asserted by some writers that the quitting direction should be radial for best effect, but theory and experiment unite in showing that the assumption is not true, except in special cases. According to the latter part of Article 45 the particles of a finite stream at quitting will not pass off in parallel lines, and may account, at least in part, for the discrepancy between the computed and measured angles. 48. The radial component of the velocity at quitting will be v, sin y, = 5.990, or, say, 6 feet per second. The velocity of the quitting water will be F~ 2 in equation (34), which is F 2 sin B v<> sin y 9 ; . . F 2 = 5.99, which is nearly the s'ame as the radial component, as it ought to be, since the direction is nearly radial call it 6 feet. The initial velocity v- = 7.1, or, say, 7 feet, is radial in this case, so that the radial component of the velocity diminishes from about 7 feet to about 6 feet. For intermediate points it will depend upon the ring sections. HYDKAULIC MOTORS. 85 49. To find the diameter of the shaft. The shaft will be subjected to a twisting stress. Let P be a twisting force, and a its arm in feet ; n, the number of revolutions of the wheel per minute; II. P. the number of horse-powers delivered, then P. 2 nan 33,000 = JB.P. .. n 16 where d is in inches, J the modulus of rupture to torsion = of 50,000, say. _ . . d = 100 H ' P ' nearly. (144) If H.P. = 161, n = 52, then d 6f inches, nearly. In the Tremont wheel, the diameter of the shaft was 7 inches from the wheel to the upper bearing, and larger in the hub. This gave a large margin for safety, the factor being more than 16. 50. To find the path of the water in reference to the earth. Divide the space between the outer and inner rims into any number of parts, by concentric circles. Suppose there are 8 equal parts ? r t = 4.2 r, = 3.38 /< 8)0.82 0.1025 feet between consecutive rings. Let /> p 2 , etc., be the radii of the successive annuli, then P 2 p l = p, r, = 0.1025 feet. The initial velocity being radial and v 1 = 7.0 feet, the time required to go from the inner circle to the next will be, consider- ing the velocity uniform over this space, 86 HYDRAULIC MOTORS. but in this time the point a l of the bucket. Fig. 24, will have gone forward a distance #, d^ = p l &>T l = 0.27 feet. The radial velocity at the second arc will be --^ -5- Bin r , JA ! > L j/ i _ ! ; i rf ( } - ^ 5 sin r y Pl y in which ^ and y must be measured from the elevation in Fig. 19. It will be a little more accurate to take v, midway between r l and p,, and y midway between p, and p 2 ; but as the working of the wheel and the efficiency do not depend upon the path of the water in reference to the earth, and only gives some information in regard to the course of the water, the method given is con- sidered sufficiently accurate. Then T . . P 2 " Pi 2 ? v r , and , <7, = A (T, + T,). and so on through the wheel. It will be seen from Fig. 24 that the terminal direction is nearly radial. A still more simple, but approximate, method, is to consider the diminution of the radial velocity as uniform. The initial being 7 feet and the terminal 6 feet (say), the diminution of velocity from one annulus to the succeeding one will be of (T - 6) = 0.125 feet, Then ^=0.1025-7-7; T a =0.1025+-6.875 ; r s =0.1025-f-6.750, etc. (The Tremont turbine run continuously from 1849 to 1892, when it was removed to give place for another wheel of larger power.) 51. The Bucket. In the study of the Tremont turbine, some- thing was said, in Article 45, in regard to the form of the bucket, but it is such an important element that more may properly be said. The parallel flow wheel offers the simplest solution. Let the hori- HYDRAULIC MOTORS. 87 zontal lines through a and erect perpendicular b e, and with o as a centre on ~b e prolonged and o e as radius describe the arc e I' I i, and similarly for the other buckets. Then from I draw Ip perpendicular to o j as a partial back vane, and similarly for the others. The width of such a bucket is variable, increasing from Ij to I' m', thence decreasing to f and again increasing to the end, which feature is objection- able. If the initial angle, y^ is to be 90, b e must be prolonged to meet * tangent at i, where the point i may be found by trial. The buckets may be made of uniform width as follows : Lay off the terminal angles, y^ at a, , e, Fig. 29, as before. At 0, a small arbitrary distance from J, erect the perpendicular e o, and describe e f. Similarly with o' as centre describe h i. From f erect/" I perpendicular to o' h, and with o' as centre describe p I to the intersection with o' i, and from p draw p I parallel to c i. The thicker part of the partition wall may be cored as indicated in the figure. Since the Boyden wheel with Russia sheet-iron plates for buckets, and the Swain and Hercules tur- bines, hereafter described, with cast-iron buckets, give high efficiencies and are durable, it is hardly necessary to give further discussion on forms. Guides JSn chcls FIG. 26. HYDRAULIC MOTORS. 89 To find the depth of the buckets, when of uniform breadth, we have in all cases of continuity, lev k, v l = & 2 -y a , and since the breadth, i p, Fig. 29, in this case is uniform, y v = y, V* = y, ^> ( 146 ) and if -y increase by equal increments, y will decrease correspond- ingly, and the equation will be an equilateral hyperbola, in which the axis of v is the developed arc of the bucket. As shown in Tables VI., VII., IX., X., v 2 exceeds v t \ hence the terminal depth should be less than the initial for this case. This form of bucket has not, so far as known to the writer, been used, but it seems to have commendable features. 52. The Hoyden d iff user shown in Fig. 30 consists of a coni- cal ly diverging stationary piece, A, outside the buckets. Its office is to produce a diminishing velocity of discharge, and thus impart more energy of the water to the wheel. 53. Turbine at Boott Cotton Mills, Lowell, Mass. This is an inward flow or vortex wheel, Fig. 31. It was made from the design of Mr. Francis in 1849, and tested by him in 1851, and was to develop 230 horse-power with 19 foot head. The regulating gate was a cylinder of cast iron placed between the guides and wheel, and was made water-tight by means of leather packing. The water was conducted to the wheel by a wrought-iron riveted pipe, about 130 feet long, 8 feet in dia- meter, plates f " thick. DIMENSIONS OF THE BOOTT TURBINE. Outside diameter, . . . ,. , . . 2/>, = 9.338 ft. Inside " ; .' . . . . . 2/-, = 7.987 " Least depth of guide passages, ... Y 0.999 " Outer " " buckets, . . . . . y t = 1.000 " Inside " " buckets, y, = 1.230 " Number of buckets N = 40 " " guides, 40 90 HYDRAULIC MOTORS. Thickness of wheel vanes, .... Jin.: " " guide u .... T V n - : Terminal angle of guides, a \ " " " stream, mean, a' . " " buckets, . . . . ft Mean angle of outflow from buckets, y\ Initial " " buckets, . t , . . a Measured area of guide passages, . . K' u " " outflow from buckets, k\ Calculated least area of buckets, . , & If contraction be 0.9 effective area, . K" ii it .. ( |<( U U 7. " 0.0208 ft 0.0156 " 8 12 10 C 15 62 5.904 sq. ft. 6.8092 " " 4.343 5.3136 sq. ft. 6.1283 " " Ratio of depths, & = 0.82. y TABLE XIII. ABSTRACT OF EXPERIMENTS ON THE BOOTT CENTRE-VENT WATER- WHEEL. | 3 ' g 1* 3 a 1 111 1^ u 1| Sco> h * 2 oS g J C G a jN C-^ ir,^, ^^3 ?-fes 'QtH C p. A O 02 *3 >^ Kr & *o "^ fe r* ^ ife >>5 3 - s Terminal vane angle, . .... a Mean exit angle, . ... . . . a' Initial angle of bucket, . . . , . y l Terminal angle of bucket, . . . . y 2 Mean angle of flow, . . . . , . . ;/ 2 Initial depth of wheel, . . ... y 1 Terminal depth of wheel, . . . . y 2 dumber of buckets, J\ T 34. " guides, 54. Terminal least section of guides . . K' 6.814 sq. ft. " " " buckets . . k\ = 5.660 " " TABLE XV. ABSTRACT OF A TABLE OF FRANCIS EXPERIMENTS WITH A BOYDEN WHEEL (OF FOURNEYRON TYPE). 90.00 " 24 29 90 26 29 8.64 in. 9.125 in. z > ^ t- --^ orf fc; "c g fe 2 ^ I'gg^. g ^ 2 Is .^ ^-g^ ?"i oS j|N f ^ h ss g ^' S S 1 ^ "S3 tB2 K S K s $ 1.000 57.00 16'. 61 145.35 216.79 79.17 0.560 1.000 59.33 16.57 146.18 216.55 78.82 0.584 1.000 63.50 16.60 147.10 222.04 80.17 0.624 ..000 66.50 16.62 148.32 220.28 78.79 0.653 0.833 55.67 16.66 142.04 203.19 73.71 0.546 58.75 16.63 142.82 207.23 76.93 0.577 0.773 56.00 16.74 136.12 188.94 73.11 0.548 0.662 55.75 16.80 128.86 169.28 72.42 0.545 0.442 50.63 17.10 105.65 113.36 55.32 0.461 FIG. 31. HYDRAULIC MOTORS. 93 The hydraulic efficiency would probably be about 82 per B are guide vanes, directrices, or distributors. The cross-section of Girard's turbines are bell-mouthed ; but as first made by Jonval, the sides were parallel, as shown in Fig. 36. The Jonval turbine may be placed at any point between the level of the water up stream and that in the tail race, provided there be a closed tube, called a " suction tube," through which the water passes before being discharged, as in Fig. 36. 59. The " Collins" Turbine. The Collins turbine is the only parallel flow wheel of American make for which we have the results of a test. It was tested in 1SS3. DIMENSIONS. = iTi, r Mean diameter, ... , '". ' S . 2>' m = 4.170 ft. Terminal depth of guides, ..... .. Y =0.836 " Number of buckets, ...... N = 24 " " guides, ...... N, = 30 Measured terminal arc of guides, . K' = 2.912 sq. ft. " " " " buckets, . k\ = 2.882 " FIG. 32, FIG. 33. HYDRAULIC MOTORS. 95 TABLE XVI. ABSTRACT OF TEST OF A 60-iNCH COLLINS TURBINE SUBMERGED IN THE WHEEL PIT (JONVAL TYPE). , *fe * .< 5 fl "rt a; O/O jf* ^ C Ofl | |1^* ft fc w M E- 3 HI* lip |5 t- ^ 1!" P S O EH B2 K A 1.000 16.57 64.35 96.41 79.85 85.25 0.573 1.000 16.55 64.40 97.03 80.40 89.17 0.600 1.000 16.56 64.40 96.65 80.03 93.00 0.625 0.748 .79 58.50 86.69 77.49 81.50 0.544 0.600 .86 53.61 69.53 67.93 77.90 0.650 0.503 17.18 45.14 56.22 64.02 69.67 0.460 0.303 17.41 34.25 34.72 51.42 71.00 0.218 0.161 17.84 24.80 19.12 38.16 53.00 0.984 The same turbine was tested with a suction tube, with the following results : TABLE XVII. SIXTY-INCH COLLINS TURBINE WITH A SUCTION TUBE. i - IJl. fc ft .2 I ^5 H ^^ 1 j3 ^^ ^ *5 "^ a* l ^ j|! r |S 1* V O Mi P5 1.000 16.56 64.88 102.18 84.01 0.604 1.000 16.55 64.99 102.70 84.34 0.646 1.000 16.56 64.88 101.25 83.25 0.709 0.548 17.01 68.87 64.83 65.97 0.457 The same turbine was also tested with a suction tube, in which was a cone with its base placed just under the centre of the wheel, so as to act as a diffuser, with the following results : 96 HYDRAULIC MOTORS. TABLE XVIII. SIXTY-INCH COLLINS TURBINE WITH SUCTION TUBE AND INVERTED CONE. fe . gf i . 1 1!^ o 6 1 8, ^ K 2 < f 3- f ll w 3 v w CJ PH tf n W 1.000 16.56 65.18 101.46 83.05 0.578 1.000 16.57 65.30 102.62 83.76 0.606 1.000 16.58 65.37 101.73 82.92 0.712 0.548 16.96 52.09 69.66 69.66 0.492 The efficiency was increased with the suction tube, but not with inverted cone. 60. A 96-inch " Collins " (parallel now) turbine was tested at the Holyoke testing station about 1883, which gave the highest efficiency of any recorded experiments with a parallel flow wheel so far as we know. NINETY-SIX INCH COLLINS TURBINE (JONVAL TYPE). Gate opening, . ..... . . fully open. Total head, . . . -. , -. . ' . H 16.59 ft. Volume of water per second, ... Q = 113.46 cu. ft. Number of revolutions per minute, . ^=63.38 Brake horse-power, . ..... . HP = 131.49 Efficiency of wheel, per cent., . . E = 85. 06 Hydraulic efficiency , possibly, . . 87 per cent. 61. Segmented Feed.- A turbine works with better effect when the buckets are full ; and when the wheel is too large to secure, at all times, full buckets with variable supply, means have been devised to shut off a part of them, leaving others fully open. Fig. 37 shows such a device, where the upper part of the guide forms the gate by sliding downward to close the passage of the guides, or distributors. Those at the right marked B are fully open, while those marked A at the left are shown partly closed. They may be so constructed that two or three gates may be closed at a, FIG. 34. FIG. 35. FIG. 36. HYDRAULIC MOTORS. 97 time, while the others remain open, and so be adapted to great range of the supply of water. Segmental feed does not produce quite as efficient a wheel as full feed, but, as will be seen, the difference is not as great as might be anticipated, considering that a portion of the water will have but little effect during the tilling and emptying of the buckets. 62. HaenePs Turbine. Haenel, manager of machine works at Magdeburg, constructed several parallel floAv wheels, and made an extensive set of experiments which are very instructive. Ad- mission to the guide wheel was regulated by a pair of rubber strips, supported by iron stays, and rolled upon two conical roll- ers, the object being to admit water to as few passages as desired, or to open all 32 of them at the same time, Fig. 38. A A are the buckets, B B the guides. C C conical rollers, E E the rubber flap. The wheel passages were designed to be .of equal sections normal to the flow of the water, so that the velocity through the wheel would be uniform when submerged. - This was accomplished, in part, by the use of back vanes. The ventilating pipes connected with the vane passages were sometimes open and sometimes closed, but there was no appreciable difference in the efficiency due to the difference of the two conditions. The turbine tested was one of eight, all alike, having the fol- lowing dimensions. The dimensions in feet are approximate, as the original were in Prussian feet, and for our purpose it is not considered necessary to be particular about the fractions of an inch in English units : 1 Prussian foot = 1.02972 English feet. 1 square Prussian foot = 1.06032 English square feet. 1 cubic Prussian foot = 1.09183 English cubic feet. DIMENSIONS OF HAENEL 's TURBINE. Outer diameter of wheel and guide at inflow, 5 ft. 9J in. Inner " " " 4 " 6 Mean " " " d = 5 " 1 " 98 HYDRAULIC MOTORS. Outer diameter of wheel at outflow, . . . Inner " " " .... Initial depth (width) of wheel buckets, . . y l = Terminal " " . . y a = Depth of wheel, / Angle of outflow from guides, a = Initial angle of buckets, y l = Terminal angle of buckets on concave side, . y^ = u a a u convex Measured area between guides, . .... . K = Effective area 0.9 of measured, . . ... . . K" = " " of buckets at outflow, . . . & a " = TABLE XIX. RESULTS OF TESTS OF HAENEL'S TURBINE. 6 ft. 5 in. 3 " 10 (< 1 " 3< L a 2 " 7' it, 1 " \ L a 22 30' 45 26 20' 23 00' 3.17 sq. ft. 3.12 " u 6.38 " a co h 3 - 1 .d _-J a u o <5 O c ^ ^ ^ V^ > 32 C "~ o K Z * "32 s ,^ t " ;^ v-i **" SH C ^ ^ H HH ^ir ^ ~ "S c 3 s - C ^ ^2nK S-S g JE""! C a NUMI GATE 01 P 1 lit! *o "~^ ll |^ 2 S| II | pq 4 6.65 0.00 5.80 2250.8 1457.7 29.0 0.6476 0.7101 8 6.58 0.11 12.98 5052.5 3340.6 27.5 0.6612 0.6876 12 6.48 0.21 19.80 7540.9 5351.2 36.5 0.7096 0.1331 16 6.44 0.21 26.10 10217.0 6369.9 35.0 0.6816 0.6982 24 6.18 0.39 47.61 15844.0 10556.0 31.5 0.6662 0.6759 32 6.26 0.29 53.66 19765.0 1214.8 72.5 0.0615 0.0703 32 5.90 0.54 62.20 21579.0 14703.0 39.0 0.6813 0.6901 Another test. 32 4.07 2.03 48.93 12060. 8042.5 32.0 0.6669 Still another tf st, 32 5.12 1.48 45.66 14156 9676.2 33.0 0.6836 0.6949 FIG. 37. FIG. 38. HYDRAULIC MOTORS. 99 According to these results, the prejudicial resistances were 6J per cent, with 4 buckets working, and was about 1 per cent, with all the gates open. One test gave only j of one per cent, prej- udicial resistance. 63. Tangential wheels are radial with segmental feed. They are more especially adapted to high heads with a limited supply of water. They are made larger in diameter than pressure wheels of the same capacity, and hence, when the speed of the periphery is the same, the revolutions will be less in the same time. They may be inflow, as in Fig. 39, or outflow, as in Fig. 40. 64. Industries gives an interesting account of the turbines used in the steel-manufacturing establishment at Terni, Italy. They are segmental feed, radially outward flow, of free deviation, Fig. 40. TABLE XX. SOME DIMENSIONS OF TURBINES AT TERNI. HEAD. H. Horse-power. H.P. Volume of water. Q- Rev. per minute. n. Inner diameter. V Ft. Cu. ft. per sec. Ft. In. 595.5 1000 19.77 210 7 10.5 595.5 800 15.89 200 8 2.4 595.5 500 9.89 240 6 5.9 595.5 330 7.06 200 7 10.5 595.5 150 3.00 250 6 4.7 595.5 50 0.99 850 1 10.25 595.5 50 0.99 850 1 10.25 595.5 40 0.85 450 3 6.1 595.5 40 0.85 450 3 6.1 595 5 30 0.60 600 2 7.5 595.5 20 0.42 450 3 6.1 . 65. Dimensions of a segmental feed, tangential turbine, radially outward flow of free deviation, with horizontal axis, Fig. 40. 100 HYDRAULIC MOTORS. 2/' 2 = 8 " 11.1 " T = 4.32 in. y> = 15.75 " N = 110 t 0.2 in. & ft. Inner diameter, . ' 2/ 1 , = 7 ft. 10 J in. Outer u Width of guide passages, . . " buckets, initial, . " " terminal, . Number of wheel vanes, . . Thickness of steel vanes, Length of supply pipe, . . . (3481 feet of cast iron, remainder wrought iron.) Thickness of cast-iron pipe, . . 0.71 in. Diameter " . . 21.66 " " of wrought-iron pipe, . 18.9 " Thickness " ... 0.20 "to 0.47 in. Designed for, 400 H.P. With head of, H 570.8 ft. And water supply of, . . . . Q 8.5 cu. ft. per sec. TABLE XXI. TEST OP THE PRECEDING WHEEL AT IMMEKSTADT. (A r r THE TIME OP THE TEST 8.5 Cu. FT. WERE NOT AVAILABLE.) I 1 h IV ti , BJj || < f 1 3 0* l!i 1 Is $ ii 6 1 O s-, 2 w 10 211 570.84 2.111 136.8 81.5 0.595 15 211 570.84 3.350 217.1 144.3 0.665 20 214 570.84 4.375 283.4 196.8 0.694 25 216 570.84 5 187 336.0 253.9 0.755 31 210 570.84 6.840 440.9 336.8 0.764 The power absorbed by the friction of the shaft was 5 H.P. with 10-inch opening of gate and 4.5 with 25-inch opening, or about 3 per cent, of the total power in the former case and FIG 39. FIG. 40. HYDRAULIC MOTORS. 101 1 per cent, in the latter; or about 5J per cent, of the brake power to less than 1^ per cent. Turbines acting under a fall of about 650 feet are described by Knoke. They are radially outflow and of cast iron. They re- ceive water through two guides at opposite extremities of the diameter (four guide passages in all). DIMENSIONS OF Two OF THEM. Inner diameter, ..... . . 11.81 in. 11.81 in. Outer . , . . . . . 16.41 14.84 " No. of vanes, ........ 45 cast iron 84 wrt. iron. No. rev., . . . 583. 928 Yol. water per min., . ... . . . 4.52 cu. ft. 5.085 cu. ft. Head, feet, 129. 131 Power of the water, d Q II, ft. Ibs., 36440. 53806 Brake power, ........ 16498. 28809 Efficiency, ......;.. 45.27. 53.5 MIXED FLOW. 66. Risdon Wheel. Mixed flow wheels, so far as known, are inward and downward, of which the " Risdon," Fig. 41, is a type, giving also a view of the cross-section of the wheel jusfc under the upper crown, in which B D are guides and A buckets. The illustration shows the construction so clearly that a detailed description seems unnecessary. This gave the highest efficiency of any wheel tested at the Centennial Exposition, 1876, it being reported as 87 per cent. Those tests lasted but a few minutes, and may have been fortunate in showing a higher efficiency than they would maintain for a long period ; but this wheel has main- tained a reputation for high efficiency, and in some more recent tests has been reported as giving even 90 per cent. But all such high figures should be received with caution, and writers dis- card the last figure as being too improbable to be true. 102 HYDRAULIC MOTORS. 67. The Humphrey Wheel. This is an inward and downward flow wheel, Fig. 42. The wheel A A has 13 buckets, a a extend- ing below the casing. C C is a regulator, also containing 12 guides, the whole moving on spherical rollers, and operated by a hand wheel, G, through shafts and bevel gearing F. By this means water is admitted to or cut off from the wheel. The wheel is enclosed in metal having curved surfaces, into which the water was conducted by a riveted pipe. The lower end of the shaft had a pivoted step, while the upper end was supported by collar bearings. Fig. 43 shows a plan of the wheel casing, con- duit, and brake used in the test. The diameter of the brake pulley was 5.44 feet; width, 2.7 feet; arm of brake lever, 15.9 feet. The wheel was a design of the " Humphrey" Machine Co., of Xew Hampshire, and was designed to deliver 270 horse- power with a fall of 13 feet. It was tested by Mr. James B. Francis about 1880. The head was determined by the difference in heights of the water in two tubes, one entering the supply shaft above the wheel and the other in the tail race ; and the quantity of water by means of weirs and hook gauges. There being no contraction at the ends of the weir, the quantity was .computed from the formula Q = 3.33 Z 7/f, where L = length of weir in feet = 11.92 for one weir, and 10.98 for the other. H = head of water above sill in feet, measured some dis- tance above weir. DIMENSIONS. Outer diameter, 2/' 2 = 8.1 feet; inner, i\ = 2.0; a = 15, y i = 80, Y not given, but the result of the test indicates that it was small, probably about 12, Y = 2 feet. Fia 41. UNIVERSITY HYDRAULIC MOTORS. 103 TABLE XXII. TEST OF HUMPHREY TURBINE. Brief extract of results. OPENING OF REGULATOR. W Volume Water per second. late ?o^ |H,o Brake Power, U. Efficiency. E. fff'fe r l s 3 Per cent. 101.00 88.50 68.21 52.90 40.66 27.94 19.15 ft. 12.478 12.839 13.103 13.456 13.688 14.069 13.998 cu. ft. 207.82 181.59 166.80 131.52 110.70 79.26 59.02 Ft. Ibs. 161744 145421 136326 110387 94520 69558 51534 Ft. Ibs. 132475 111527 99084 67363 53065 18517 8264 Per cent . 81.9 76.69 72.68 61.02 56.14 26.62 16.04 0.7301 0.7831 0.687 0.7508 0.6702 0.8073 0.7225 The velocity about 3.458 feet in the supply pipe was neg- lected in determining the head, for which, if a correction be ap- plied, it will reduce the brake efficiency to about 80.59, and hence the hydraulic efficiency would be about 82 per cent. 68. The Swain Turbine. The Swain turbine is a mixed now wheel, inward and downward, of which a sectional view is shown in Fig. 45, and an external view with vertical shaft in Fig. 44. TF, Fig. 45, is the wheel, A the guides which are secured to the gate 6r, and are raised and lowered with it, and pass into and out of the chamber E. By this arrangement the inner ends of the guides are brought nearer the buckets than when the gate is between the bucket and guides, the space in this wheel being If inches. The gate is opened by being lowered, so that the water first enters just under the crown, and passing inward and downward is discharged nearly radially, while that which enters lower is discharged nearly axially. The guides and upper ends of the buckets are shown in Fig. 46. There are three heavy cast-iron guides, one of which is shown in Fig. 46, through which pass the lifting rods, as shown at e. The other 21 guides were 0.23 inch thick of bronze, sharpened at the ends to 0.04 of an inch thick, nearly 19 inches long. It had 25 buckets of bronze, pressed into shape and cast 104: HYDRAULIC MOTORS. into a crown above and band below, and were 23.35 inches deep. The gate G was made of two cylinders M and JV, joined by a disc, Q. At the lower end of M is a narrow flange, to which was attached a leather packing to prevent the escape of water. The leakage was so small as to be unimportant. The wheel rested on an oak pivot, S, conical at its ends, free to turn or rest, and sup- plied with water through the pipe f. It was so arranged be- tween the connecting piece a and crown T r , that the step S could be replaced by another, and the adjustment of the height of the wheel made by means of the screws #, t. Opposite the thick vanes are stationary supports, of which one is shown at 0, Fig. 45, resting on the cast-iron base C, and support the chamber E and cover of the wheel L. The openings in the base C for the passage of water into the wheel pit flow outward, as shown in the right-hand part of the figure. An elaborate report of the test of this turbine designed to replace the centre-vent turbine established in 1849 at the Boott Cotton Mills, previously described is given by Francis and published in the Journal of the Franklin Institute for April, 1875, to which we are, through the courtesy of the Swain Manu- facturing Co., indebted for the facts and tables herein given. Table XXIII. contains only the best result for each gate opening up to "full gate." Mr. Francis states that Table XXIV. is made from a curve of the results plotted on section paper. It will be seen that the highest efficiency was obtained with the gate closed, 1.08 inches. This indicates that there may have been interferences of the stream of water in the Bucket at full gate opening. It is also worthy of special note that the effi- ciencies are well maintained down to the smallest opening of 2 inches. The hydraulic efficiency for the best test was about 85 or 86 per cent. DIMENSIONS. Outer diameter, . . . . . . 2/\ = Oft. Least inner diameter, . . ... 2/' 2 = 2f ft. FIG. 42. FIG. 48. HYDRAULIC MOTORS. Depth of guide passages, . . . " buckets at inflow, . . Mean terminal angle of guides, . u initial angle of buckets, . . " terminal angle of buckets, estimated, . . . Number of buckets, . .... a guides, . . . . , Thickness of guides, . . . -. . Measured area between guides, Least measured area of the buckets, . ... . . . . /,-/ = 9.558 " " Where the water leaves the wheel axially (outer diameter), y <2 = 26, and where it leaves it radially, ;/ 2 = 22, so it is esti- mated that the effective angle is about 25. TABLE XXIII. ABSTRACT OF THE RESULTS OF EXPERIMENTS WITH " SWAIN" TURBINE WHEN Rux FOR BEST EFFECT. T = 13.08 in. y* = 13.285 in. a = 25 Yi = 90 7, = 25 ^ = 25 t i = 0.23 in. to t = 0.04 in. at end. K' = 9.880 sq. -ft. OPENING OF GATE. IN. No. of rev. per minute. n. Head. Feet. H Vcl Water passing weir per second. Cu. ft. Q. Energy of the fall Ft. Ibs. & QH. Brake Power Ft. Ibs. U. Efficiency per cent. E. Ratio. ?*2 W 1 v *~g~H 2 60 5 14 2S3 51 177 45518 21572.5 47.39 0.6274 3 60.3 13 977 68 009 59189 34806.5 58.81 0.6623 4 60.0 13.699 83.894 71562 47672.5 66.02 0.6478 5 60 12 13.482 97 736 82048 59179.7 72.13 0.6537 6 60 21 13 281 110 092 91044 6.2263.5 76.08 0.6689 7 66.33 13 102 119.268 97309 76932.3 79.06 0.7184 8 60.40 12 968 130.253 105184 84794 3 80.61 0.6979 9 60 5 12.836 138.839 110956 91961.2 82.88 0.7098 10 60.43 12 680 144.774 134202 95583.9 83 63 0.7082 11 60 56 12.640 151.501 139226 99206.1 83 26 0.7227 12 67 6 12 581 156.703 122158 102435.2 83.85 0.7490 13.08 108 ..36 13.099 120 392 98204 20237.7 20 61 1.1738 Full Gate. 97 8 12 880 137.871 110580 53189.4 49.91 1.0683 87.7 12.603 149.133 117041 80804.4 69.04 0.9683 78.0 12.480 158.877 123471 99109.8 80.27 0.8660 72.7 12.432 161 225 124814 103522.0 82.94 0.8083 69.1 12.172 162.538 125223 104652 9 83.57 0.7701 1 68.0 12.408 163.572 126:387 105401.9 83 40 0.7588 ' 66.5 12 361 163.607 125986 104857.4 83.23 7412 IOC HYDRAULIC MOTORS. 1 V , 3 s d d o o d d Q- d d d o d 1 _>> 1 R 1 dddddddddddd J > I- Sg^Tcj^psf-iMo^oS- V d d do '000000000 1 i ? -L ~ 5 - - g dddddddddddd 5 Ss s ^l 11 d do'ddoddddddd 1 ? ^ g, c S || d 1 1 1 I 1 1 g g S 1 S o'ddoo'ddddddd 1 1 ^ S llSlil8Snll 1 a oooooooooooo 1 - 1 ?i siiiiiigrEiii > 3 - i oooooooooooo ^ - o C V r J3 e illliSSISill > i -S - oooooooooooo H M c c a = It 8 d do'do'dddddddd S PQ ^ " 1 5 S > s 3 ^ 8 d ssli!!i!!is! do'dddod'ddo'dd S 1 1 o 1 I'- ll e d dddddddddddd OS 3 *$ !l d GO CO X 00 X OD i. t~ i- S O "^f dododo'ddodoo' o g 1 1 Si 5 d S I 1 1 B i s I ?i i 1 1 dddddddddddd tc 1 ? -g rj ^ 1 o' 1 1 1 1 1 g g g g 1 SI dddddddddddd 1 1 o * B sa 2 * j 2 d dddddddddddd 1 ^ 1 xScsSfeSSS^SSi- 1 1 P d dddddddddddd w 3 d g||ggg[:g5ii^ dddddddddddd I 1 S o E s ' s 8 i 6 i S 5 1 S S do'odddoo'odd'd o 5 S gggggggg g g g 3 00000000 0000 1 1 If 8 * if S8888S888SS8 The Swain Turbine FIG.' 44 FIG. 44a. HYDRAULIC MOTORS. 107 69. The Hercules. The Hercules turbine is also a mixed flow wheel, inward and downward. The entrance surface A A, Fig. 47, is conical, and divided horizontally by several partitions, so as to be equivalent to several wheels superimposed, but cast and working as one. The partitions are for the purpose of securing a good effi- ciency with partial gate opening ; indeed, the best efficiency was for partial opening, from which it is inferred that at full opening the streams within the wheel interfered with each other, and so prevented their producing the best effect. The passages are so curved that the inner filaments passed downward and issued nearly radially, while the outer ones escaped at more nearly axially. A cylindrical gate surrounded the wheel and was opened by being raised, similar to the Fourneyron style, or as was done in the Francis wheels, Figs. 19 and 31. The guides were outside the gate and stationary, as in Fig. 31, and divided by hori- zontal partitions. In 1883 a "Hercules" was tested at the Holyoke testing station. The wheel was of the following dimensions : DIMENSIONS OF THE " HERCULES" TURBINE. Mean external radius, %r m = 36 in. Terminal angle of centre line guides, a = 14f Initial angle of buckets, . . . . y\ 98 Terminal angle of buckets (say) . . / 2 14 Measured least section of guides, . K' 4.752 sq. ft. " " terminal section of buckets, . . .... A- a ' = 7.925 sq. ft. Number of buckets, N = 24 " " guides, N, = 17 The first test showed such remarkable results in regard to effi- ciency, that it was followed by two other tests, all of which seemed to confirm each other. The following are from the third test: 108 HYDRAULIC MOTOKS. TABLE XXV. ABSTRACT OP (THIRD) TEST OF A GO-INCH HERCULES TURBINE. 6 CO a || I Id 1 o| t* u o 'o^ v g; I* h" r* 1 O d g "o O. s ~ fc * ^ M 1.000 Still 16.82 129.10 16.94 89.66 145.59 84.55 0.614 135.33 16.95 89.00 146.02 85.38 0.644 140.62 16.96 88.33 145.72 85.80 0.669 144.80 16.98 87.79 143.88 85.14 0.688 0.806 123.00 16.88 79.38 129.71 85.39 0.586 130.25 16.88 78.63 131.01 87.07 0.621 137.00 16.92 78.00 130.28 87.07 0.652 143.00 16.94 77.38 129.01 86.82 0.680 0.647 126.50 17.09 67.25 111.04 85.22 0.599 134.67 17.15 66.52 111.65 86.33 0.637 140.40 17.16 65.72 109.55 85.69 0.664 0.489 115.60 17.25 55.54 85.27 78.50 0.545 123.00 17.22 54.80 85.48 79.90 0.581 130.00 17.28 54.07 84.79 80.05 0.613 0.379 112.75 .17.61 45.20 65.29 72.36 0.526 122.67 17.65 44.50 65.06 73.06 0.572 129.40 17.66 43.76 63.89 72.93 0.603 The hydraulic efficiency was probably about 87 or 88 per cent. Analysis of the " Hercules" Wheel. Making r l = r^ = r m = 18", a = Uf, Y, = 98, y, = 14, ^ = 0.10, // 3 = 0.05, gives E = O.8998 = 9O per cent, nearly. With the same data except /*, = ^ = 0.10 gives E O.88O3 = 88 percent., which exceeds the brake efficiency only one per cent., and indi- cates that the resistances are low and about 0.10 for //, and /v FIG. 46 HYDRAULIC MOTORS. 109 70. The Victor. The " Victor" turbine followed the Her- cules in its introduction to the public. It is a mixed-flow tur- bine, the water entering radially inward at the circumference, thence discharging downward and outward. The whole body of the wheel, except that for the shaft and step, is occupied by the buckets, which are deep axially, thus giving great capacity for its size. The water is regulated by two styles of gates : one is called " the register gate ;" the other, " the cylindrical gate." The former opens the passages for the water by turning about the axis of the wheel, thus opening the passages their whole length and making the opening wider and wider as it is turned more and more, and at the same time gives direction to the w r ater ; the latter is a cylinder moving axially> and opens the passages their full width as the gate is raised. The latter is preferable when the water supply is variable and not sufficient to fill the passages at full gate, or when the work is variable ; for a better efficiency is obtained with it at " partial gate." A view of the latter is shown in Fig. 48, for which we are indebted to the courtesy of the manufacturers, Stillwell-Bierce and Smith- Yaile Co., Day- ton, Ohio. 110 HYDRAULIC MOTORS. TABLE XXVI. RESULTS OF TESTS OF THE "VICTOR" TURBINE AT THE HOLYOKE TESTING STATION, WITH THE "CYLINDER GATE." SIZE OF WHEEL AND GATE OPENING. Head in Feet. H. Revolutions of Wheel per minute. It, Cubic Feet Water per minute. 60 Q. Horse-power Developed by Wheel. H.P. Percentage Tseful Effect. E. 30-inch Full Gate. . . 17.51 168 4440 119.56 81.35 4 17.82 163 3892 104.93 80.03 4 17.95 163 3392 88.24 76.66 f " 18.10 155 2893 70.97 71.28 J- " 18.20 159 2265 51.42 63.46 36-inch Full Gate 16.78 135 6106 158.18 81.80 1 " 17.14 135 5422 141.58 80.71 17.35 140 4708 118.22 76.68 4c 4 " 17.05 129 3982 91.62 71.50 4 " 17.48 134 3202 66.87 63 30 39-inch Full Gate 14.66 116 6873 152.66 80.37 1 " . ... 14.53 118 5920 129.41 79.80 1 " 16.84 125 5517 135.56 77.40 i |' 17.06 123 4695 108.22 71.67 17.39 124 3856 81.00 64.07 48-inch Full Gate 13.23 91 10072 201.71 80.11 i \ 14.36 89 9042 192.41 78.42 14.75 89 7869 165.23 75.34 I " ."'.! 14.87 85 6744 132.76 70.06 i " 15.28 87 5526 100.66 63.09 NOTE. In the above calculations "fractional gate" means "fractional water, 11 regardless of the position of the gate of the turbine ; that is. " three-quarters gate' 1 means that the gate i closed to a point where the discharge of water is only three-fourths of what it is at full gate tinder the same head. This is nearly but not perfectly exact, THE VICTOR TURBINE* 11 CYLINDER GATE" FIG. 46. HYDRAULIC MOTORS. Ill TABLE XXVII. RESULTS OF TESTS OF THE " VICTOR " TURBINE AT THE HOLYOKE TESTING STATION, WITH THE " REGISTER GATE." SIZE OP WHEEL. FULL GATE. Head in Feet. H. Revolutions per minute. n. Horse- power. H.P. Cubic Feet of Water per minute. 60 q. Percentage Useful Effect. E. 15-inch ] 18.06 368 30.17 990.19 .8932 17} inch ) 18.08 18.02 355 280 30. 12 35.51 996 . 83 1164.60 .8849 .8960 20-inch j 17.96 18.22 292 286 36.35 48.75 1197 1660.17 .8950 .8532 25-inch ] 18.23 17.79 275 205.5 48 . 75 67.72 1660.17 2362.72 .8528 .8530 30-inch ] 7.96 11.65 209 144.5 68.62 52.54 2356 . 54 2751.87 .8584 .8676 35-inch ... -j .DO 17.31 147.5 151.7 ol . yo 135.68 27o5 . 09 4895 . 8564 .8489 40-inch ] i .29 16.49 160 130 133.19 148.93 4806 5789 fJQ1 R .8497 .8253 44-inch lO.4< 15.50 109.25 14o.oo 155.78 Oolo 6643 .OZ4L .8003 48-inch 15.51 102 179.29 7456 .8202 It will be seen that the highest efficiencies, over 89 per cent., border very closely upon the figure which many writers reject without discussion. The engineer who made the tests has re- marked, in effect, that the smallest wheels were too small for determining the efficiency with great accuracy. Even if these be excluded, the efficiencies are high. 71. Pelton Wheel. The Peltoii wheel, or, as it was formerly called, the " hurdy-gurdy" wheel, has been popular, especially in California, for utilizing the power of high waterfalls. It con- 112 HYDRAULIC MOTORS. sists of a series of curved double buckets, as shown in Figs. 48 and 49, attached to the circumference of the wheel. A is the nozzle, B the valve case, in Fig. 49# . The water impinges against the buckets at the partition and flows outward in opposite direc- tions, discharging at the outside planes of the wheel. It is then an impulse wheel, or wheel of free deviation, with axial flow and segmental feed, and may be analyzed according to Article 30. For the limiting values of the angles, we have a = 0, Yl = 180, y, = ; r, = r, ; ^ = ^ = ; for which we find GO i\ = velocity of circumference = -J 4/2 g H, equation (36), which is half the velocity due to the head. The velocity of discharge will be, equation (34), F~ 2 = 0, and the efficiency will be unity. But these are impossible conditions in practice. The angle EEC must be less than 90, so that the water will escape before it is struck by the succeeding bucket. Draw B D parallel to the line of the jet (or in the plane of the wheel), to represent the velocity of the wheel, and B C tangent to the bucket at its terminus, to represent the velocity of the water as it leaves the bucket relative to the bucket, and complete the parallelogram ; then will EB represent the velocity of exit from the bucket relative to the earth. Considering the angle at A as zer\>, F, the velocity of the rim of the wheel B D C E, and V the velocity of the jet, then will the relative velocity of the water along the concave surface of the bucket be V V, = B C = ED = ^ V^JI V,. If the velocity of the wheel be such that E B is perpendicular to the jet (or to the plane of the wheel), then HURDY-GURDY WHEEL. FIG. 48. FIG. 49. HYDRAULIC MOTORS. 113 V, = & C cos E C B = ( V FO cos y f COS - cos y t y _ g _ gin _ SIR K 2 y- 1 + <^ os 72 Thus, if ;i = 0.98, EC B = 20, then F= 0.98 ^2, V l = 0.4T6 F, F a = 0.18 V, E = 0.928, or nearly 93 per cent. If the velocity of the wheel be half that of the jet, then EC= V, = j- F; BC = V - F, = i F; .'.B C = EC', .-. F, = ^^ = 2 sin J- (7; and 7 and ^ the same as that given above. When the angle is small say less than 25, "2 sin -J C = sin (7 with sufficient accuracy for this case, and the results will be very nearly the same as those given above, so that for the conditions assumed above, the theoretical efficiency will be about 93 per cent, for this case. The efficiency will be nearly the same for a speed several per cent, above or below that for the maximum. If the angle C 30, and the half angle at A also 30, and the speed be one-half the component of the velocity of the jet, and p = 0,96; then F = 0.96 4/2^7?; F, = i F cos 30 = 0.433 F, B C = F--F, = 0.433 F, F 2 = 2 B C sin 15 = 0.224 F; . . E = 0.92 (1 - 0.224*) = 0.874. 114: HYDRAULIC MOTORS. In these computations no allowance lias been made for the fric- tion in the feed pipe, nor for imperfect working at the junction of the buckets, nor for loss due (if any) to imperfect discharge. These all operate to reduce the efficiency. In a paper read by Mr. Hamilton Smith, Jr., before the Ameri- can Society of Civil Engineers, February 6, 1884, a test is reported in which the efficiency was given as 87.3 per cent, with a speed of 0.51 t/2 y //, under a head of 386 feet, wheel 6 feet in diam- eter. He states that the wheel carries over a large amount of water. This efficiency is remarkable, but if it be admitted that there were incidental errors in the test, it still shows that the efficiency was high, and Mr. Hamilton infers that it was certainly as high as 85 per cent. A small " hurdy-gurdy" was tested at Stevens' Institute by some students in 1891, for which the efficiency was found to be less than 70 per cent. The wheel is especially commendable for great heads : it is simple in construction, durable, efficient, easily managed, and easily repaired. It was invented by a village carpenter, who, after reading Francis' Lowell Hydraulic Experiments, and resid- ing near a fall of water, made a Prony friction brake and a weir, and by continuous experiments arrived at the particular form and setting of buckets which he adopted as the best. The juncture A in the actual bucket is back of the line, joining the extremities through B. 72. Poncelet Wheel. The Poncelet wheel is the invention of him whose name it bears. In accordance with hydraulic prin- ciples, the inventor changed the plane float wheel, which had an efficiency of about 16 per cent.> to one with curved buckets, Fig. 50, raising the efficiency to 60 or 70 per cent. The water enters the buckets tangentially, moving up the concave side with a dimin- ishing velocity since it works against gravity, until it ceases to as- cend, when it descends and leaves the bucket with a backward Fw.60. HYDRAULIC MOTORS. 115 velocity in reference to the bucket, its velocity in reference to the earth depending upon the velocity of the wheel. The energy lost in its work against gravity is restored by gravity during descent, neglecting friction in the buckets, so that it is only necessary to consider the energy of the jet at entrance and quitting. It is virtually a turbine of " free deviation," with segmental feed. Considering the limiting case, in which the terminus of the bucket is tangent to the outer circumference, and that the water enters tangentially, then a 0, y l 180, y, = 0, /*, = /* r, = r then Equation (93), V = V% g H, (94), v l = V -- w r, (97), G* r = J V^~H = i F, " (101), U=$M F 2 , (102), E = 1. But these conditions cannot be realized in practice. The ter- minal angle of the bucket has an angle of 15 or more with the circumference, or y l 165. The angle of the guide, or chute, will be 20 or 25 or even more, and y^ = 15 or more, and yu 1 = 0.10 or more. These would give an efficiency of less than 90 per cent. There may be a further loss by water escaping be- tween the wheel and apron below the wheel, and there may be an imperfect action of a finite stream ; since the bucket may strike the stream, and the stream strike the crown, thus reducing the theoretical efficiency. Actual tests give E = 0.55 to 0.70, when v = 0.55 F about. 116 HYDRAULIC MOTORS. 73. In the preceding tests we have given the highest effi- ciencies, but it should not be inferred that these high figures are attained in the majority of cases. In Europe, Rittenger tested eight turbines, parallel flow, mean radii from 6 to 11 inches, having various vane angles, under heads varying from about 6 to 18 feet, for which he found efficiencies varying from 0.63 to 0.71. It may be observed that tests of American wheels in America or we might say in Massachusetts have given higher efficiencies than tests in Europe, and the question has been raised, if the difference may not be due to the different methods of measuring the quantity of water ; but the difference is not accounted for in this way. There appears to be essential differences in the wheels in favor of the American types. In designing, it is not advisable to use the highest attainable figures for the efficiency, for when tested the wheels are supposed to be in their best condition, well lubricated and bearing perfect, and after long service, some parts may be so worn as to make the wheel less efficient ; hence 70 per cent., or certainly not to exceed 75 per cent., should be assumed. In ordinary practice, with variations of speed above or below that which would pro- duce a maximum, with wheels not constructed with great care, and with a lack of proper attendance they may fall below 60 per cent. HYDRAULIC MOTORS. 117 CLASS-ROOM EXERCISE. (The following exercise was conducted by the author with a class.) Design a turbine to utilize the power of a stream having an available fall of 16 feet. 74. To find the power of the fall. Let be the weight of a cubic foot of water, Q, the volume of water falling per second, H, the height of the fall ; then the energy of the fall per second will be SQH, and the horse-power, 77 P _ * Q H 550 ' The weight of a cubic foot of water depends upon its tem- perature, latitude and elevation of the place ; but in the use of water wheels the temperature will vary so little from 60 or 70, and the latitude and elevation will have so small an effect, we will use tf = 62.4.* * If T be temperature, degrees Fah. , " I, the latitude of the place, " e, the elevation above sea level, " 6 = 62,375, the weight of a cubic foot of water at its maximum den- sity, 39.1, at the level of the sea at latitude 45, then 6 = 6 (1 - 0.0002) (T- 39.1) (1 - 0.000256 cos 2l)(l ?^ If r be the radius of the earth at the place, r fl " '" " " 45, then r To (i + 0.00164 cos 2 I), r = 20,892,200 feet. 118 HYDRAULIC MOTORS. In case of accurate tests, the proper allowance for these changes may be made. The head over the crest of the weir will vary from zero at the crest to some finite value depending upon the depth of water over the crest. To deduce a formula for this case, first assume that the water flows through an orifice B D, Fig. 51, in which C D is the breadth, B C the depth, and A the free surface. Let E be a rectangular element = d x d y, in which y is vertical and x horizontal, and y the depth of E below the free surface ; then d Q = ft V% g y , d y d x. Let I* be a coefficient of discharge, A 2 , the depth C A, A,, BA, I, the breadth C D ; then Q = v VZ g \'y dydx = ^ V^gl V - * O) If the upper surface of the orifice be free in other words, if the orifice be a weir, as in Fig. 52 it has been found by experi- ment that Aj may be taken as zero when 7? 2 = B D, measured from the level A B of the water several feet above the weir down to the crest of the weir D. The coefficient ^ varies be- tween 0.60 and 0.64, depending upon the depth C D over the weir and B F the breadth ; and if one were gauging a stream where great accuracy was essential, it would be necessary to re- sort to tables to determine the exact coefficient to be used (D'Aubisson, Hydraulics, or Weisbach, Hydraulics), or resort to Francis' formula as given in Lowell Hydraulic Experiments, which is Q = 3.33 (I 0.1 n h) h%, FIG. 51- B 1 1 // l&&m*tt ^^:^M^m F 3T FIGK 52. HYDRAULIC MOTORS. 119 in which n is the number of contractions (two, if both ends are contracted ; one, if only one end ; and zero is neither end). If n = 0, Q = 3.33 I $, to which the preceding formula will reduce if k 1 = and fj. = 0.622. Streams vary continually in the quantity of water discharged, so we will use the smaller coefficient, or /* = 0.60, and our equation becomes Q = 3.21 I A* (J) Let a weir be constructed of boards having bevelled edges the sharp edges being placed up stream and by measurements sup- pose it be found that I = 6 feet, /< = 1.83 feet. The height of the surface may be accurately found by a " hook-gauge,'' a device invented by Mr. Francis, in which a hook is submerged and then gradually raised until the point is just visible in the surface. In this way the height of the surface .may be found with much accuracy to a small fraction of an inch. We have Q = 3.21 X 6 X (1.83) * = 48 cubic feet per second, nearly, and sufficiently accurate for our present pur- pose. The theoretical power of the fall will be H. P. = 4S X 62 ' 4 X 16 = 87.13 550 horse-power. If the turbine has 75 per cent, hydraulic effi- ciency, it will have 65.35 horse-power ; and if the frictional re- sistances be 3 per cent, of this power, it will have at the brake 63 horse-power. From this we can judge whether the stream will supply the required power. 120 HYDRAULIC MOTORS. 75. To determine the diameter of the ivheel. We choose for this exercise a radially outflow wheel. The greater the diameter of the wheel, the less will be its depth in order to discharge the given amount of water. There is no recognized rule for deter- mining the diameter. The depth between the crowns of the Tre- inont wheel was about \ the outside diameter. If proportioned for space or to economize material, the depth would be more nearly equal to the radius. We will try a depth equal to J the radius, r v Table VI., column 10, second case, gives F 2 2 = 0.04 X 2 g H. .'. F 2 = 0.2 1/2 g H. The Tremont wheel gave F 2 = 0.24 VY^H = V^YH, nearly. Assuming that the free opening at the outer rim (the circum- ference less the space occupied by the walls of the buckets) is f of the outer circumference, and that the radial velocity at quit- ting is 0.2 1/2 g H, and depth J r we have | . 2 n r, . J r, . 0.2 1/2 g H = 48 ; . . /* a = 2.6 feet, nearly = 31.2 inches. The object of this investigation is not to fix exactly the pro- portions of the wheel, but to determine approximately such pro- portions as may be previously desired. We may now assume, arbitrarily, the outer radius, and if not satisfied with the final re- sult, recompute with another assumed radiiis. We observe that the smaller the diameter, the greater will be the number of revolutions per minute, and the wheel may be proportioned to give the required revolutions ; but the analysis HYDRAULIC MOTORS. 121 is complex, as shown by equations (15#) and (16), page 8. By the aid of Table VI., a first approximation may be made ; thus, if n be the number of revolutions per minute, then ,/ {JeJ n . 2 n n 60 "30' and from Table VI. r t = 0.8T5 Thus, if II = 16 feet, n = 100 per minute ; then / = 2.67 feet = 32.04 inches ; but the coefficient 0.875 depends upon values we have not yet fixed. In several cases given in the preceding pages, the velocity of the initial rim to the velocity due to the head is between 0.62 and 0.70. We now assume a value near those found above, but otherwise arbitrarily, /> 2 = 30 inches. The width of the crown will now depend upon the inner radius, and no rule exists for determining this. In the Trernont wheel ?'/?, = 1.23. Rankine gives r^ i\ |/2 = 1.41 r r These give, for our problem, 1\ = 24.4 inches and 21.27 inches respec- tively ; and for width of crown 5.6 and 8.73 inches respectively. The analysis on page 26 shows that, theoretically, the crown should be comparatively narrow for the outflow wheel, but from physical considerations the crown should be sufficiently wide to secure the full effect of the stream ; but if too wide, friction and difficulty in proper feeding might be prejudicial. For the pur- pose of study, a wide crown will show more clearly all the pecu- liarities of the buckets than a very narrow one, especially in the graphical construction. Then assume r^ = 20 inches, giving 10 inches for the width of crown. (The width of crown in the Tremont was nearly 9J inches.) 122 HYDRAULIC MOTORS. 76. Initial angle of the bucket. For reasons given previously, we make Xl = 90. 77. Terminal angle of bucket. The discussion in Article 5, page 9, shows that y 9 should be small for high efficiency. The smaller it is the greater must be the outer depth, and if the wheel is to be cast, the angle should not be too small. In the Tremont wheel, in which the walls of the buckets were of Russia sheet iron and fitted into place by special tools, this angle was 10 de- grees. Other wheels mentioned in this work are made with larger angles. We will assume 78. Terminal angle of the guide vanes, directrices, or dis- tributors. The proper value of a is discussed on page 16. and that the internal pressure at the gate should exceed that of an atmosphere it should be less than 45 for y l = 90. The smaller it is the higher the efficiency, although the gain in efficiency is small per degree of decrease of the angle. It may be seen in Table X., that for an increase of a from 21 to 23, the loss of efficiency was 0.23 of one per cent., or not far from y 1 ^ of one per cent, per degree for those values. We will assume the fairly practical v? ] ue, a = 30. 79. Buckets. There is no recognized rule for determining the number of buckets or their form. Francis' rule for the number is ^=3 (D +2), where D is the outside diameter in feet ; but he did not follow it in the construction of his wheel. He made N = 44, while the rule would give 30. In our example it gives 13, and 12 or 14 would be used ; or if in the proportion of the diameters to the Tremont, about 26. The wheels tested at the Centennial Expo- HYDRAULIC MOTORS. 123 sition.liad less buckets, page 41, than given by this formula. Cir- cumstances pertaining to the ease and certainty of construction, or of obstacles entering the wheel, may have controlling influ- ences. A case is related of a wheel choked and stopped by eels, but it is rare for a wheel to act as an eel-trap ; but if there is ^danger of obstructions, it would be wise to make the buckets larger. We will try ^"=18, and 16 guides. The thickness of the partitions will depend upon the mode of manufacture. If of sheet steel, they may be -J inch thick ; if of bronze, less than J of an inch ; but if of cast iron, they should be thicker to secure sound castings, and some allowance may be made for imperfect setting of the cores. So many cores will be necessary that dry sand moulds should be used, in which case sound castings may be secured if f inch thick and possibly if less ; but to provide for contingencies, we will design them T 7 g inch thick and bevel the initial ends to a sharp edge. With this data we make the first calculation. The data are r t = 20", x, = 90 > Ai = - 10 > H = 10 ft - n = 30", y., = 14, yu 2 = 0.20, Q = 48 cu. ft. per sec. t = T y, a = 30, lf=lS buckets. RESULTS. M 2 = 0. 608 A 7 * 0.052 Ang. vel. GO' = 10. 92 E = 0.737 Rev. = 1.738 per sec. K = 0.127 sq. ft. 104. 28 per min. k, = 0.254 a a F = 21. 016 ft. & 2 ~ 0.089 u a v l 10. 501 a 2/i =B 0.466 ft. v 9 = 29.824 a 2/2 1=. 0.511 " a, = 0. 545 a e = 77 10' 0, = 0. 722 a H P = 64.31, hydraulic F,= f-r i. 403 u horse -power of the wheel. 80. Form of the bucket. If the bucket is to be described with several arcs of circles, proceed as in Fig. 19. Let F, Fig. 19 12-t HYDRAULIC MOTORS. or 53 be the terminus of a bucket, draw the radius F 0, and lay off F c, making an angle of 14 with F 0. Choose a con- venient point ) A B A u' Similarly at (7, v > Vi + $v 9 = 14.36. v" = | i\ + | v, - 18.23. TABLE XXVIII. By Measurement on the Drawing . From Computation as above. From Equation (141). 7i = 90, >'i 20 inches ! = 10.50 feet y\ 0.466 feet r' = 63, P 23 v' = 14.37 y 1 0.332 y" = 44, P" 25.6 v" = 18.23 y" rr 0.304 r'" = 30, P" 27.6 v"' 22.10 y'" 0.331 y"" = 20, P"" 29.0 v"" - 25.96 v"" 0.408 Y* = 14, r t 30.0 v 2 = 29.82 y* 0.511 The values of y,, y' ', y' 1 ', etc., will be the depths of the buckets at A,B, C, etc. Since these will be the depths at all points in the circumference of the arcs passing through those points respec- tively, draw a radius GH and prolong the arcs to an intersection with this radius. On a line A u\ Fig. 54, equal to the width of the crowns, lay off the divisions of the radius, and through those divisions erect ordinates y l = 0.-466, y 1 0.332, y" 0.304, etc. r and trace a smooth curve through their extremities these will be the form of the crowns for an indefinitely narrow bucket. But for a bucket of finite widtli as we have in practice, the- normal widths should be measured. These widths will be strictly a curve passing through the points A,B, (7, etc., cutting normally the traces of an indefinite number of buckets between GF and a" d, as shown by the dotted line through D, but it will be sufficiently exact to consider the lines as straight. "With dividers- find by trial the shortest distance B V and B l>", and similarly for all the points C, D, etc. We find HYDRAULIC MOTORS. TABLE XXIX. 127 Distances. Velocities as found above. Hence. a' A a" - 0.556 = 10.50 y t = 0.466 V Bb" = 0.572 v' = 14.37 y' = 0.331 c C c" = 0.492 v" = 18.23 y" = 0.304 d D d" = 0.396 v'" = 22.10 y'" = 0.311 e' Ee" = 0.328 v"" = 25 96 y"" = 0.320 pr = 0.284 V ', C c', etc., and with the ex- cess B & 2 = V B y B ~b', an arc is described with B as a centre, and similarly at 6 Y , D, E, etc., and a curve a" & 2 c 2 d* traced tangent to the arcs. This forms, substantially, a back vane, the form of which will depend upon the law governing the velocity of the water along the bucket. 128 HYDRAULIC MOTORS. Fig. 55# shows the form of back vane when the velocity in the bucket is uniform from a" to a?, and equal to 10.50 feet ; and from x to exit increasing to 29.82 feet. 82. The pressure at the entrance into the wheel will be given by equation (143), making p . i\, k = #,, then Pi p A _|_ , = 13 IT g = 32.16 ft., 2 r = 38 in. outside diameter of the tubular part plihe shaft. 134 HYDRAULIC MOTORS. t = f in. thickness of tube of shaft. d = 11 in. outside diameter of solid part of shaft. "We find from equations (15) to (22) ; and the equation preced- ing (144) : M * = 0.478 N* = - 0.156 tf = 642.99. . . GO' = 25.357 angular velocity per second ; . . N = 242.2 rev. per minute ; E = 0.8108, or 81 per cent, efficiency; v 1 = 22.98 ft. velocity of entrance into bucket ; v^ 82.91 " terminal velocity in the bucket ; V = 55.27 " velocity of quitting the guide; T^ 19.13 quitting velocity in reference to the earth ; 8 = 85 35'; H.P. 5,500 horse-power ; Q = 433.27 cu. ft. per second ; y, = 1.55 ft. y, = 1.52 oo r r l = 66.56 ft. velocity of inner rim ; to' 7\ = 79.24 " " outer " J 867 Ibs. stress on the tubular part of the shaft. Several of these results differ perceptibly from those given in Fig. 56. Assuming 250 revolutions per minute, we have 250 X 2 n , -, GO = CL = 26.16 ft. per second. 60 GO r, = 68.30 ft. velocity of initial rim. GO 7\ = 81.81 " " " terminal rim. And if y, = 180 - 110 40' = 69 20' and a = 19 00 r , the triangle of velocities gives V = 64.31 ft. the velocity of quitting the guides. v l 22.5 ft., nearly, the initial velocity in the bucket. HYDRAULIC MOTORS. 135 And if the ratio of the initial normal section of the bucket to that of the terminal be as 4.275 to 1.25 3.42, then ^ 2 = 3.42 X 22J = 76.95 ft. terminal velocity in the bucket, V^ = 19 ft. actual velocity of quitting, and 8 = 113, direction of F 2 . These figures agree so nearly, almost exactly, with those given in Fig. 56, that we assume that they were obtained in this man- ner. If the wheel makes 250 revolutions per minute when pro- ducing its highest efficiency, under a head of 136 feet, and the other data be as given or assumed, the solution is correct ; but otherwise, it is only an approximation, more .or less rough. As stated above, the data in Fig. 56 gives about 232 revolutions per minute for best effect, and our computation with y^ 51 gives 242 ; so that if it had 75 per cent, efficiency and gave 5,000 horse -power at 250 revolutions, it ought to have a higher effi- ciency and greater capacity at a slower speed. If 430 cubic feet were discharged at a velocity of 76.95 feet, through 32 buckets each 1J inches wide, the depth should be A of 1.25 X 32 X 76.95 The actual depth of the six chambers is 1.81 feet ; hence the capacity of the wheel should exceed 5,000 horse-power in the ratio 1.81/1.68, giving 5320 horse-power. This assumes that the buckets are properly made. It appears that the cross-section a d, Fig. 56, is slightly less than that at f e, whereas the former ought to be perceptibly larger, since the velocity of the water increases as it goes outward, so that if the section at a d is filled, that at/ e will not be full, and the wheel will be a " pressure" wheel from the initial element to a d, and one of " free deviation" from a d to exit. This being the case, 136 HYDKAULIC MOTORS. as determined from the plan of the wheel, the correct depth would be found by using the velocity at a, which will be some- what less than at e, and also the width at , which is also some- what less than at e / and both these elements conspire to make the depth y^ greater than 1.68 as computed. We would modify the design of the wheel by terminating the long arc at, or near,/*, and using a shorter radius to, or near, c. Figs. 54 and 54& are suggestive of a better form than that given in this wheel. But retaining the width e f and increasing it at a d ought to in- crease the capacity of wheel somewhat without decreasing the efficiency. We now return to our computation. If the results obtained from equations (15) to (21) do not agree with those found in the wheel, then equations (56) to (62) must be used, since the sections of the buckets are fixed. First, the cross- sections of the initial and terminal sections of the buckets must be inversely as the velocities vjv, = 82.91/22.98 = 3.679. To determine the ratio of the sections of the stream we assume that the initial section of the stream is the same as that of the bucket, but, as has been shown above, the terminal section of the stream is less than that of the bucket ; and to find what the former is or what the section of the bucket ought to be re- quires a knowledge of the capacity of the wheel. This requires an extended analysis, not here given, according to which and to other information, it is for 138 foot head about 5,500 horse- powers ; and we wdll assume this value for the present computa- tion, and test its correctness by the results which follow : The volume of water flowing through the wheel, in produc- ing 5,500 horse-power, will be ft. lb. per sec. H.P. per H.P. Q = 5 ' 500 X 55 100 = 433 cu. ft. per sec. 81 X 138 X 62.4 per ft. \vt. cent. cu. ft. - PLAN OF WHEEL OF THE NIAGARA POWER CO. / Estimated 5000 Fra. 56. HYDRAULIC MOTORS. 137 The terminal velocity being 83 ft. to the nearest entire foot, as found above, the aggregate depth of the six chambers being 1.81 feet, the aggregate width of the 32 buckets should be 433 + (83 X 1.81) = 2.882 ft., and the width of each would be 12 X 2.882 -7- 32 = 1.081 in., instead of 1J in. as marked on the plan. The thickness of the initial end of the partitions between the buckets as measured on the drawing is -J- of an inch each, and for the 32 buckets the aggregate thickness will be 32 X -- ft. LA /\ O If, therefore, the buckets were indefinitely narrow, the aggregate thickness of all the partitions being the same, the aggregate nor- mal widths would be 2 n r, sin 51 32 . . _J - = 12.27 ft, 12x5 and the ratio of the sections in this wheel being as their widths, we have , . 12.27 _ . , 95 ' - But for buckets of finite width, the ratio is found more accu- rately by tracing a curved line cutting normally the traces of the buckets, if there were an indefinitely large number. This proc- ess gives a ratio of about 4 or a little more. We find that the results do not differ largely for the ratios 4 and 4.25, except for the value of v^ the initial velocity in the bucket ; so we take the ratio 4 as representing more accurately the actual wheel, and as being sufficiently accurate for our purpose, and this dif- fers so much from the inverse ratio of the velocities (3.679) as to make a computation with equations (56) to (62) desirable. With the data 138 HYDRAULIC MOTORS. K = kj\ = 4.00, ^ = 0.10, r, = 2.675, ^ = 0.10, Yl = 51, r, = 3.125, J7 = 138 ft. r , = 13 IT, we find c = 1/JT= 0.25, a = 1.11203, = 3.092, J. = 0.503, B = 0.897, J* = 4.221. <7 2 = - 3.9678 F* = 1.602, = 2.531. . . ^ max = 0.834, a? = 26.45, n = 252.5, F = 54.54, v, = 85.29 ft., v l = 21.32 " F 2 = 19.62 " a/ r, = 69.43 " GO' r, = 82.65 (9 = 88 53 r , a = 14 10 r , H. P. = 5,500, Q = 433 cu. ft. per sec., Least breadth of bucket = 1.04 in. No allowance is here made for leakage through the T Vm. clearance of the wheel. There are three such clearances for the escape of water, two at the lower part and one at the upper part of the wheel. From equation (44) it is found that the pressure at entrance into the wheel is p l = 7,468 pounds per sq. foot ; hence, if the coefficient of discharge be 0.80, the volume of dis- charge will be FIG. 57. HYDRAULIC MOTORS 139 / - y ^g p -i = 15.4 cu. ft. per sec. Hence, the quantity of water passing into the penstock, when running with full gate and at best effect, should be about Q = 433 + 16 = 449 cu. ft. per sec. This computed leakage will be TO - - 0344 ' ' or 3.44 per cent, of the water delivered to the penstock ; or 96.56 per cent, of the water delivered to the penstock passes through the wheel ; hence the efficiency of the wheel system referred to the water consumed will be E '= 83.4 X .9656 = 80.53 per cent. The terminal angle of the guide (19) is somewhat larger than given by theory. We make the following abstract of a statement in regard to a test furnished by Dr. Sellers. At the time of the test the total head from the surface of the water above the penstock to the centre of the wheel was II 135.113 ft., and the water delivered to the penstock per minute was Q = 26867 cu. ft. per min. = 447.8 cu. ft. per sec., and the theoretical horse-power of the water, HP 447.8 X 135.113 X 62.3 = ^^ 550 There was an electrical output of 5335 horse-powers ; hence the actual efficiency of the wheel and dynamo combined was 77V = 140 HYDKAULIC MOTORS. or, 77.85 per cent. ; and if the dynamo yielded 97 per cent, as guaranteed by the makers, then the efficiency of the wheel sys- tem, including friction and leakage, would be E = TT - 85 = 80.26 per cent, 0.97 and the power delivered at the upper end of the shaft would be H. P. = ^ = 5,500. The head during this test was less than that assumed in the computation, but if 1.4 ft. for the head due to the velocity in the penstock, be added, the effective head will be 136.5 ft., which is only 1.5 ft. less than the effective head assumed. This difference will not affect the efficiency, but would affect the com- parative speed. The speed was not measured, but was regulated for about 245 to 250 revolutions per minute, and the experi- mental efficiency and power involve an assumption ; and the theo- retical computation is founded on the supposition that the wheel is a pressure wheel throughout ; so that it cannot be said whether a more exact analysis would agree more nearly with a test ex- periment, if the data were precisely the same and the quantities directly measured. As they stand, the two results theory and experiment agree remarkably well. The indication is the re- sistances are less than those assumed, the leakage greater than that computed, and the hydraulic efficiency greater than 85 per cent, of the power of the water passing through the wheel. The volume of water may now be recomputed, and will be O = 5,500 X 550 _ V " 62.4 X 138 X 0.8053 which is 3 cu. ft. more than that before found, which differ- ence is chiefly due to the difference in the efficiency used FIG, 58. HYDRAULIC MOTORS. 141 in the computation. Using this result, the terminal normal width of the bucket will be : Width = _ x 12 _ = 1.04 in., 1.81 X 85.29 X 32 which is 0.04 of an inch less than that found by the former computation, which is due to the larger terminal velocity now found. This again emphasizes the remark previously made in regard to increasing the capacity of the wheel. According to our computation, the velocity of the water in the penstock will be The velocity as it enters the case will be ..... 12.2 " " " in the case just before entering the dis- tributers will be ........ 28.5 " " " entering the distributers will be . , . . 30.4 " " " quitting " ... - 55.3 " " " entering the wheel relative to the bucket will be ....... ...... 21.3 " " " quitting the bucket will be . .. . . . 85.3 " " " u " wheel in reference to the earth will be ......... 19.6 " The main part of the shaft is a tube of steel rolled and without longitudinal riveted seam, 38 in. outside diameter and j in. thick. There are two solid parts joining the tubular parts, as shown in Fig. 59, which form journals for the support of the shaft and wheel, and are 11 in. in diameter, one of which is shown in Fig. 58. The moment of stress is given in Article 49, and is 12Pa = 63 ' 000 H ' P ' inch pounds. n For the resistance, let r be the mean radius of the tubular shaft in inches ; 142 HYDRAULIC MOTORS. t, the thickness of the tube ; t/, the modulus of torsional shear ; then for a thin tube 2 7t r . t will be the sectional area of the tube, 2 n T t . J will be the resisting force of the tube, and 2 TT r t J.r will be the moment of resistance ; , . 7 63,000/7: P. . . 2 TT r t J = - - r _ 31,500 II. P. * which in this case becomes, J= . 81600 X5,500_. = 867poimd , 3.U X ISf X f X 252 The torsional stress on the solid part will be given by the equa- tion 63,000 H.P. = j x jjp n in which R is the external radius of the solid part and is 5 in. ; T , 126,000 H.P. , . J = 8 - - - = 5,260 pounds. it n . 11 s The resistance to shear of steel or iron in large masses is not well known. If homogeneous, theory indicates that it will be of the tenacity of the material, and experiments indicate that the shearing resistance is nearly the same as that of the tenacity. The tenacity of mild steel is 65,000 pounds and upward per square inch ; hence its shearing strength ought to be 50,000 pounds at least, according to which the solid part will be strained HYDRAULIC MOTORS. 143 to about -j-^ of its ultimate strength when running steadily and delivering 5500 horse-powers, which is no more than ought to be allowed for safety, considering that in starting and stopping and for variations of loads, the stress may be considerably in- creased. The stress on the tubular part is small compared with that on the solid part less than ^ as great. If the shaft be a uniform tube f in. thick, 19 in. radius, 140 ft. long, and if the modulus of elasticity to shear be 10,000,000 pounds, then will the amount of torsion, when running steady at 252 revolutions, delivering 5500 horse-powers, be . 63,000 X 5,500 X 140 x 12 = Q " 10,000,00,0 X 2 n r t . r* . n which is the arc for radius unity. The number of degrees will be 9 3.14 X 180 = 4 13'. Fig. 59 shows the penstock, shaft, and relative position of the wheel. They are supported by heavy cast-iron beams resting on the solid rock. Pressure due to deflecting a stream. 87. The pressure resulting from the deflection of stream of water may be determined as follows : Let a particle whose mass is in enter a stationary tube with a velocity, -y, and follow the tube to its exit. Let the tube be frictionless, then will the veloc- ity be v in reference to the tube from entrance to exit. Let the tube be the arc of a circle with as the centre. The centrif- ugal force will be radially outward and equal to v 2 m ; r which will be the pressure against the outside of the tube, and may be represented in magnitude and direction by the line A B 14-i HYDRAULIC MOTORS. on the radius A prolonged. The centripetal force will be of equal magnitude, and will be the reaction of the tube acting upon the particle toward the centre 0. Assume that the particle fills the tube for a distance d s ; then if k be the cross-section of the tube ; d the weight of unity of volume ; 6 the angle D A, Fig. 60 ; r the radius A ;

-: - r?) ; (160) equation (11) becomes and (12) and (13) give fi_r r * a,* + Fcos a - r, GO] (162) 9 .-.E= -J* = -Tr'C-n' ^+ Fcos ' r* <\ d (J H g 1 which is a maximum for r, GO = \ V T cos a, (164) ** in which 1\ GO will be the velocity of the outer rim = V" (say) and if the buckets are very narrow, or if they move in a right 148 HYDRAULIC MOTORS. line, as in the preceding article, or practically in a right line, we may consider r, = r, ; then F" = iFcosar, (165) or the velocity of the bucket will be one half the velocity of the component of the velocity of the water as it issues from the supply chamber in the direction of motion of the wheel. If a 0, the velocity of the bucket should be one half that of the jet ; and for this case we have v> = 4 F. (167) (1 + /<) < = i F 2 . (168) F 2 2 - 4 F ;;: ,, (169) ?7 = LJ? .j V = %8 Q H. (170) ^ = i- (171) Equations (162) and (163) show that the work and efficiency are independent of the frictional resistances of the water, but (166) and (168) show that F and v. t are diminished by such re- sistances. If y z be finite and the wheel one of " free deviation," we have from equations (158), (159), (11), (12), and (13), after making dlf+ dtv&ty = VI + ^ ( Vr, cos a - 2 r? GO) + 2 F 2 3 Vr, cos of . GO 4-4 r 2 GO* r * C S y * V^T - 2 Vr cos a co 4-7^7 ' This produces a complete equation of the fourth degree, and the complete solution will be more lengthy than for the pressure wheel, as given in equations (16) and (60). FIG. 60. FIG. 61. HYDRAULIC MOTORS. 149 90. -Cascade Wheel. The James Leffel & Co. have designed two wheels to be established at Ward, Col., to work under a head of 730 feet. One is to be 38 inches diameter, to be driven by one nozzle, producing about 25 horse-power at about 552 revolu- tions per minute ; the other, 50 inches diameter, fed by a nozzle a little larger than 1-J- inches, will have a capacity of about 140 horse-power. Some wheels of this design are now working under high heads. Fig. 62 shows one of these wheels with pulley. In actual running the wheel is inclosed in a case, which was removed in order to show the construction of the wheel. OF THB UNIVERSITY INDEX. PAGE Angular velocity 10, 19, 29 Barker's mill 48, 64 Boott turbine 89 Boyden diffuser 89 Bryden turbine 92 Bucket, depthof 89 " formof 17,86,123 " initial angle of 10, 14, 122, 133 " terminal angle of 122 Buckets, number of 25, 122 Centrifugal force 5, 41 Class-room exercise 117 Coefficient of effect 21 Collins turbine 94 Comparison of inward and outward flow wheels 35 " various makes of turbines .. 130 Crown, width of 121 Crowns 27 " depth between 72 " formof 125 parallel 66 Designing 3,27,117 Diameter of wheel 120 Diffuser, Bryden 89 Direction of quitting water 9, 18, 84 Efficiency 26 " effect of v on 15 " equation for 7, 28, 29 " maximum 8 " for inflow 35 " u " outflow 35 " wheel of free deviation 45 Energy imparted to wheel 13 " lost by impact 26 " " in escaping water 26 " loss of due to quitting velocity 35 Escaping water, volume of 73 Exercises 25, 31, 41, 43, 46, 49, 70, 90, 117 Fall, power of 117 PAGE Form of bucket 17,86,123 " " crowns 125 Fourneyron turbine 37, 93 Francis and Thomson's vortex wheel . 37 Free deviation 18, ^3, 44, 147 1 ' surface, form of . 45 Friction along buckets 26 " head lost by 4 Frictional resistance 5, 13 Frictionless wheel, work done in 39 " efficiency of 39 " path of water for 40 General solution of pressure turbine. . . 3 Guides, terminal angle of 10, 11, 35, 122 Haenel turbine 97 Head due to pressure and velocity 4 " lost by friction 4 " total 4 " virtual 3 Hercules turbine. , 107 Humphrey wheel 102 Hurdy-gurdy wheel 114 Inflow wheel, efficiency of 15, 26 " " maximum efficiency of 35 Initial angle of bucket 10, 14, 122, 133 " velocity 4 Inward flow 9 " ratio of radii for 25 Jet propeller 50 Jonval turbine . . . . 37 Leakage 137 Maximum efficiency 7 Moment of stress 85, 141 Momentum, moment of 56 Niagara wheel . . 130 Notation 1 Number of buckets 25 Outflow wheel efficiency compared with in- flow 35 Outward flow 9 INDEX. PAGE Outward flow efficiency of . . . 26 " " ratio of radii for 25 Parallel crowns 66 flow turbine. ./. 9,93 Path of water 30, 85, 128 Pelton wheel 111 Poncelot wheel J14 Pressure at entrance to bucket 23 " " exit from bucket 6 " due to deflecting stream 143 " in wheel 22, 82 " theoretical at entrance to bucket. .. 4 " turbines, general solution of 3 Quitting angle, value of 9, 18, 84 Radii, ratio of 25 Rankine wheel 37, 65 Relation between y 2 and w 1 10 Revolutions for best efficiency 47 Risdon wheel 101 Scottish and Whitelaw turbine 50 Segmeutal feed 95, 96 Shaft, diameter of 85 Stress, moment of 85, 141 Submerged wheels .23, 24 Swain turbines 103 Tables, effect of different values of y 1 14 " " Y! on velocities 15 ' for revolutions and H.P. for best efficiency 47 Tangential wheels . . 99 Terminal angle, effect of large 78 " " of bucket 122 " "guide 10,11 " velocity 4 Tests, Boott turbine 90 " Bryden turbine 92 " Collins turbine 94 " Haenel turbine 97 " Swain turbine 105 " Tremont turbine 68, 69 '-' Victor turbine ... UO Total head 4 Tremont turbine 67 Turbine, Boott 89 Boyden 92 PAGK Turbine, Collins 94 " Fourneyron. . . . 37, 93 cut 4 triple, cut 10 " Guideless, cut 24 " Haenel 97 " Hercules 107 ' ' Inflow, cut 12 " Jet, cut 26 " Jonval 37, 93 " mixed flow, cut 20 " outflow, cut .... 2 " parallel flow 93 " " " cut 14. 16 " Scottish and Whitelaw 50 " Swain 103 " Tangential, cut 22 " Tremont G7 Victor 109 Values of a , 11 " + 7i 10 " e is " MI and M 13,84, 70, 71,78, 138 " ! .\ '?; '. . .18mo, 1 50 " Permanent Fortifications. 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