wKMMMMKamm MMRNwH Hflilif MfMMRPM 1 IMHHHM|HHHM iHHIIiilHnliiiiiiinMiinl ■■innMHi LIBRARY OF CONGRESS. 717 " g 7 — Ofpit Oop^rig^t fa Shelf/ UNITED STATES OF AMERICA. THEORY OF TURBINES. DE YOLSON WOOD, Professor of Mechanical Engineering, Stevens Institute of Technology. J. WILEY & SONS, NEW YORK. 18 9 5. Copyright, 1893, by DE VOLSOST WOOD. / Remark. About forty-live years ago M. Poncelet made a solution of the Fourneyron turbine which, for its thoroughness and the direct- Dess of its analysis, has become classical (Comptes jRendus, 1838). But that writer neglected the frictional (and other) resistances within the wheel, and assumed that the buckets, or passages in the wheel, were constantly full. The former is an important element in the theory, and its consideration makes the analysis but little more complicated. AVeisbach, in his Hydraulic Motors, gives a solution in which frictional resistances are involved, and the sections of the stream at the outlet of the supply chamber, the entrance into the wheel, and all the sections of the buckets are determined when the wheel runs for best efficiency. The formulas, however, are so complex that but little practical knowledge can be gained from •their general discussion. I have, therefore, assumed that the wheels here discussed have about the proportions made for com- mercial purposes, and deduced certain numerical results which are entered in tables ; and a simple examination of these furnishes certain desirable information. The driving power here considered is that of an incompressible fluid, which in practice will be water. The steam turbine or those driven by a compressible fluid are not in practice con- structed like water turbines, and no theory for such is here at- tempted. HYDRAULIC MOTORS. 1. The motors here analyzed may be called "reaction wheels," or "pressure turbines." Some writers call those wheels " pressure turbines " in which the water has a "free surface," and these by others are called " turbines of free devi- ation. (Weisbach, Hydraulics, etc., p. 421.) The former term is somewhat ambiguous when applied to wheels in which the water in the buckets has a free surface, and, therefore, the term " free deviation " will be applied to such. There will first be given a general solution of the " pressure turbine," and the other turbines will be considered as special cases of the more general one. 2. Notation. Let Q be the volume of water passing through the wheel per second, 6, the weight of unity of volume of the water, or 62 J pounds per cubic foot ; then 6Q=^Y will be the weight of water passing through the wheel per second, li x be the head in the supjDly chamber above the entrance to the buckets, lu, the head in the tail race above the exit from the bucket, 2,, the fall in passing through the buckets, H—h { ^ r z l — Jt 2 , the effective head, U, the useful work done by the water upon the wheel, R, the work lost by frictional resistances, whirls, etc., Mi, the coefficient of resistance along the guides, M2, the coefficient of resistance along the buckets, HYDRAULIC MOTORS. r lt the radius of the initial rim, r 2 , the radius of the terminal rim, p, the radius to any point of the bucket, n=7\-^-r 2 , the ratio of the initial to that of the terminal radius, V, the velocity of the water issuing from supply chamber, Vi, the initial velocity of the water in the bucket in reference to the bucket, ) _ vj - v x 2 p-2-Pi v 2 2 , . Zl + ' Yg 2^" + ~^ +/ * 2 %" * (8) The fall z y is so small in practice compared with the next term of the equation, that it may, and will, be omitted, giving (9) (1 + M 2 ) v} = v? + erf (r 2 2 — r, 2 ) - _O a P2~Pl • which gives v 2 . At exit the pressure will be P-2=Pa + $K • (10) The velocity of exit, relative to the earth, will be w 2 = 1) 2 2 + G0> V — 2v 2 .^^ 2 cos^ 2 (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 U=dQH-6Q~-B (12) /? = ^-£ + MG |° (13) HYDRAULIC MOTORS. © >. c3 2 c3 o c3 © M CO S3 P oa Sh CD © -*3 © £ 4- O P O p3 — © > r* ^ > f-4 C '* H — ^ DQ og P .2 "-3 'p ■ — c3 ri > •i-l © p p P "2 © rH p fcD — © rP ^ — "© pJW ^3 £ - b P s > O cc CD © rH © -C •rH A P £ o ^ * X ^ q X -1 ^N ?s 1 ^ CO O © /- -\ + X n >S + © CO + + > + > I X. T. o U £ 3 C3 L > 5 s £ U c^ + V 3 CM > + a, I S 84,1 J a u 4- a s CO | 1 tq S? Css 1 o? 1 II eq ^o> 55J r^ II a v i' o s £ I 1 "3 c b l^ V rd r. O bD o c £ 3 3 -t-= OJ +3 02 .O 3 m V 6 HYDRAULIC MOTORS. = L [— JfV 2 oti + cos ;/ 2 V2gH + iWgca 3 ] r 2 &?, when Z= l (15) gllVl + av 2 ■M 2 = Vl~+7, become -,-t sin v\ V = -t— ; " r^/ 1 ], sin (or 4- i/ T ) sin a v i = zr. sin (a + ;/ 2 ) Also from (9), (10), (4), (17), (18 GOT (17) (18) But siircr — ^in 2 ^) Sid- a — sin'-/, sin 2 (a + Y\) — sin 2 a — sin-^i -+- 2 sin (a + y ^ cos a sin y 2 sin (a + y^) cos a sin jk, — ~~ Am a + y ^ sin (a — y ^) + 2 sin(a + ^])cos a siny 1 2 cori a sin v , sin (a + Y\) ' which substituted above will give equation (14)= HYDRAULIC MOTORS. V 2 v& a /o tt . / 2 IcosTTsin Ki 2 sin 2 ;/! 2 \ « m y y \- sm (« +y,) sin 8 (a + n) V v ; The normal sections of the buckets will be K=$; h = Q-; h=^\ *=2. . . (20) I ^ y 2 v The depths of those sections will be Y= J£— 9 y x = ft , ^ = A ' 2 , . (21) i sib « * a x sin yi a 2 siny 2 DISCUSSION. 4. Three simple systems are recognized. r x < r 2 , called outivard flow. it i > T 2i called inward flow. r t = r 2 , called parallel floiv. The first and second may be combined with the third, mak- ing a mixed 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 y-i, the quitting angle. Equations (14) and (15) show that the efficiency is increased as cos y 2 is increased, or as y 2 decreases, and is greatest for ;/ 2 =0. 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 MOTORS. 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 now, since the discharge rim will be smaller in the former tha'n 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 betiveen y 2 and go'. Equation (16) when put under the form , VgH A / 1 m = ~kV ,/ T^^T. - 1 - (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) gives go' = ; that is, for minimum efficiency the wheel must be at rest, and no work will be done. 7. Values of a + y v If a + y t = 180°, and a and y l both finite, then will M and N in (15a) both be infinite ; but equation (5) gives sin 180° V A /00x go — — : ■ — = ; (23) sm y x 7\ J that is, the wheel will have no motion, and no work will be HYDRAULIC MOTORS. 1 1 done. If a -f- y t = 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 + y l 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 + y x 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 at = 0, equation i!4) will reduce to E = (/HVl + M-2 VI + /< 2 (- r} + r, 2 ) go ■ 2 cos y 2 V 'IgH + (r 2 2 — 2r L 2 — Min 2 ) go 2 (24) which is independent of y 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 x would be zero, while (17) shows that the velocity of the initial rim must equal that of the water flowing from the guides, or For the limiting, or critical case, a — 0, y 2 = 0, u x = 0, m 2 = 0, the velocity producing maximum efficiency will be, from equa- tion (16), r^' = Vg~H~ (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£ = 2?y, then r 2 GD' = V2gH, (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, u x = 0, u. 2 = 0, r} = 2?*! 2 , and it will reduce to v 2 = V2gH, (27) as it should. The following table gives the values of quantities for the three classes of wheels :* TABLE I. a = 0, y-i =0, u l = 0. JU-2 = 0. Velocity of Velocity of Exit from Guide V. Velocity Initial in Bucket. Terminal v 9 . Velocity of Exit. w. Efficiency, E. Dimensions of Wheel. Inner Rim. Outer Rim. co >, go r 2 r l = ^/ir, y/gfi ^/2g~H Vg~s 0.00 y/tgii 0.00 1.000 r ] =r. 2 go r. 2 Vgs y/W 0.00 \/gM 0.00 1.000 r { =1.4r 2 0.7lWgH \ ffH y/g~S 0.00 lU^/jH 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 HYDRAULIC MOTORS. 13 wheel ; and the energy clue 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 2 ill then will no energy be imparted to the wheel on account of pressure exerted by any part of the head //, but if V- < 2 ///, then will some of the work be done by this press- ure, w being zero. For the cases in Table I., the energy im- parted to the wheel will be due one-half to velocity and one-half to pressure : or in symbols, .w ~>~ . gH + ±WH= Wff 9 . . (28) or, the entire potential energy of the water will be exjoended in work upon the wheel. Whenever V' < ZgH, the pressure at entrance must exceed the external pressure at exit, and H (29) then will be the part of the head producing pressure in the wheel. In practice, ol 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 )\ — 1Ai\ makes the width of the crown for internal flow about the same as for r x — ^/lr 2 for outward flow, being approximately 0.3 of the external radius. 9. Values of )* x and pt .' The frictional resistances depend not only upon the con- struction of the wheel as to smoothness of the surfaces, sharp- 14 HYDRAULIC MOTOES. 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, IA X = /.i 2 = 0.05 to 0.10 (30) They are not necessarily equal, and /*, may be from 0.05 to 0.075, and m 2 from 0.06 to 0.10, or values near these. 10. Values of y x . It has already been shown that y 1 must be less than 180° — a. If y x — 90°, equation (14) shows that the efficiency of the Motionless wheel will be independent of a. The effect of different values for y x is best observed from numerical results as shown in the following table : TABLE II. Let a = 25°, y 2 = 12°, 0.10. Initial Angle. Yi- (1) r } - /•„ >/} / i = \Ar !• (2) E. (3) (4) 1;.YU,1C 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 loss, for maximum effect, for the inflow than for the out- How, for the same size wheel. Column (3) shows that the efficiency, £] decreases as the initial angle of the bucket, y v 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 y i 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\ The effect of U= SQHE ....... (31) on the velocities is shown in Table III. TABLE III. Let a = 25 c r-> 12\ iio =0.10, Ini- r x .-= r^/h /' 1= 1.4r 2 . tial Angle. V VgH .396 «2 K x fc, x VgH Vf/H 1.219 2.525 * 2 * VgH .691 F VgH .959 VgH .463 V2 VgH .761 A'x VgH *, X VgH 2.100 A- 2 x VgH .820 VgH 1.447 VgH 60 3 1.043 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 .869 1.785 .874 1.217 .593 .605 .821 1.686 1.653 150° 2.100 1 . 775 .621 .476 .568 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 u diminishes as the velocity of admission, T , in- 16 HYDRAULIC MOTORS. creases, and. according to equation (5), V depends upon y t when a is fixed. Since the crowns are not fitted air tight nor water tight it is desirable that j^ should exceed the pressure of the atmosphere when the wheel runs in free air, or the press- ure p> + Pa when submerged, to prevent air or water from flow- ins 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^, . or, 180 -ri>**, or, ;/, < 180 c - 2« If - a = 30, then n < 120°, (32) To be on the safe side, the angle y l may be 20 or 30 degrees less than this limit, giving n = 180° - 2a - 25 (say). = 155 - 2a. Then if a = 30°, y x = 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 wj.eel. In the Tremont turbines, de- scribed in the Lowell Hydraulic Experiments, this angle is 90°, the angle a, 20", and y%, 10°. Fourneyron made y 1 = 90°, and a from 30° to 33 . In Table III. it appears that for y x = 150 = , V = 2.1 Vgft, which exceeds V \gH; that is, the velocity of exit from the supply chamber exceeds that due to the head, which condi- tion must result from a negative pressure at entrance into the wheel. For zero pressure for the frictionless wheel, the above condition gives y x = 180° - 2 a, HYDRAULIC MOTORS. which for <> = 2d , gives )\ = 130 , and for v x = 150", the press- ure would be negative, and for 120 it would be positive. It appears that for the wheel with friction, considered in the table, that this pressure is also positive for y i = 1*20', 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 y x may be changed to values which cannot be determined beforehand, in which case the wheel cannot be correctly ana- Fig. 5. 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, \, h, 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 18 HYDRAULIC MO TOES. eddies. If the teiminal 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 l9 &,, are not those of the wheel, but those which are computed from the velocities V, v„ i\. 12. Value of ; or direction of the quitting water. From Fig. 1 it may be found that and w cos = v 2 cos y to sin 6 = v 2 sin y .'. cot 6 = cot y 2 - got* go r 9 v 2 sin y 2 (33) (34) (35) These formulas are for the velocity giving maximum effi- ciency. If the speed be assumed, go in place of go' 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 2} when the wheel is at rest, to 6 exceeding 90° for high velocities. The following table gives some results : TABLE IV. ar=25 C r- 2 = i2 c /J, = jli 2 = 0.10. ?' 1 = r* Vh r,= 1.4 r a Vi CO e CO 60° .314 V^H 72° 14' .160 VgR 102° 43' 90° .310 " 66° 59' .143 " 101° 17' 120° .241 " 60° 24' .126 " 82° 52' 150° .157 " 55° 26' .043 " 74° 51' 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 How wheel, and for the inflow it is thrown forward for j'i less than 90 , and backward for y l 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, 79} 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 5(V to 140° for efficiencies only two or three per cent, less than 79] per cent. 13. Of the value of oo. 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 Y — the relations in equations (5) and (6) must be maintained, or sin a jr oa . v. = — I , (oo) sin y x 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 li x and p u 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 v Y are found from equations (17) and (18). Since y 2i in practice, is small we have, for best effect, v 2 = Go'r 2i approximately, .... (37) and, adopting this value, a more simple expression may be found for the velocity of the wheel. For equation (19) gives v 2 = r 2 oo' : . (Approx.) (38) 1/ cos asm y, /rA 2 ., sin 2 y. /r,\ 2 . sin(a + v 1 )\r , 2 / 2 'sin'- (a + y)\r 20 HYDRAULIC MOTORS. If fj x = f < 2 = 0.10, r, -f-n == 1.40, a = 25°, y Y = 90°, the velo- city of the initial rim for outward flow will be " ri = whm = °- 929 v ^ The velocity due to the head would be v h = V2gff= 1.414 VgS; hence, the velocity of the initial rim should be about °- 928 ,g = 0.659 (39) 1.414 VgR 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 0.954 1.414 = 0.689 (40) 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 V2gH, (41) HYDRAULIC MOTORS. 21 "which is riot far from the velocity giving maximum effect; that is fco say, when the gate isfuUy 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 o l initial rim or Number of turns of the wheel per minute. Ratio of useful to Means of values by /•jto' theoretical effect. experiment. J&JH 0.0 0.00 0.000 0/2 33.80 0.664 0.4 47.87 0.773 "oiioo" 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 jj 2 be considered constant. Chang- ing r to p, equation (9) thus gives V>-(1 + V*)V 2 + Go'ip 2 -^ 2 ) 1 P = 2g + ^' * W v^v^ h ' sin ( ..CO n From (4) and (5), Pi = #h + p a - d 1 + Mi sin 2 y 1 sin 2 (a + n) These reduce equation (42) to 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) h_h sin a _ caVV . (44) - 1 [(1 + ^AV. + Wd + *) " Wy] ,^^ 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 BYDItAULIC 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 ^ 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 h 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 — k { and p — p x . To simplify still more, let the wheel be frictionless, or Mi = fA 2 — 0, and find from equation (38) 2 , 2 sin (a + v\) TT ,.„. r?oo - = ^ =— ^ all, (46) cos a sin y x 24 HYDRAULIC MOTORS. also hi =■ H + h 2 , and (45) becomes Pi = dH+dh,+p a -^ dB ^p , ,5: . (47) 1 x 2 cos or sin (or + Ki) v 7 If the wheel is not submerged lu = 0, and let the pressure Pi equal that of the atmosphere, or p a , then 0=1-3 Sl nr , 1 + , («) 2 cos ex sin {a + Yi) If the wheel be submerged, let pi = Sh 2 + p a , and the equa- tion reduces to that of the preceding. Equation (48) gives tan 2^ = — tan y l9 or, 2a = 180° -y x \ (49) for which value the pressure at the entrance to the wheel will equal that just outside. If, 2a > 180° - yi, (50) the pressure within will be less than that without ; but if 2a < 180 = - yi, (51) the pressure within will exceed that without — a condition which is considered desirable. If frictional resistances be considered the value of rico 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\ '■> therefore, if the value of ex be found for a frictionless wheel it will be a safe value when there is friction. If y = 90° and a = 90° - iYi = 45°, then (47) gives p l = dR + dhv +p a - 6H=z dhz + p a , . . (52i HYDRAULIC MOTORS. 25 as it should.- [f y i = 90° and a = 30°, then or, ^> , -fO.SSrt//". Tlie angle l C ° S y ^> V ' l0D = ~[- S 2 go* + Tv 2 gj] (58) in which 8 2 = r 2 2 -r t 2 (58a) fa T — r 2 cos y 2 — 77 n cos y\ (586) fC-2 HYDRAULIC MOTORS, 29 Substituting v 2 from equation (56) gives E= \ [(- S 2 + TP)g£ + T VZgHQc*? + <7^ = -U[- F 2 ^ + T V2gHQ<*> i fi 2 G**-\ . . . (59) in which F 2 = ^p _ ^2 # ( 59a) For is'a maximum, make -=— = 0, giving » ^ 2 V2glTQTTlh 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 a 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), 2g^ = (l + M 2)v\-v 2 a^ir 1 + 2? (63) in which, if v 2 and V\ be substituted from above, p x becomes known. Similarly, V from equation (55) becomes known, and finally, from (60), oor x — Vi cos y x cos a = — ^ r -\ (64) 21. Path of the Water, — Let a A be the position of the bucket when the water enters at b. Q »> The bucket being drawn in position to a scale, divide it into any number of parts — equal or unequal — aa lf a x a 2> etc., and find the time re- quired for it to go from a to aj. The distance being small, assume that the velocity is uniform from a to a lt and equal to v, which will be given by equation (56) by dropping all the subscripts 2 Fig. 7. and changing r to p, thus, _ (1 + fii) h x koori cos y x (1 + J*i) W + fix W t* gH+ !« + &9 2 (r 2 2 - 7^) (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 2 = BAF for rotation to the left, and y 2 = EAB for motion to the right ; then AD 1 = id 2 = v-2 + oo'W — 2v 2 . oor 2 cos y 2 . . . (90) A C 2 = iir - v} + oo 2 r 2 2 + 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. In 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 w — v 2 T oor 2 , (92) the upper sign belonging to the motor, and the lower to the pump. Exercise. — If r x — 1 ft. r 2 — 5 ft jj 2 = 0.1, v x — 5 ft. per sec- ond, and the bucket rotates about a vertical axis 30 times per 40 HYDRAULIC MOTORS. minute, and discharges the water directly backward, making y 2 = 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 Pi — Pi = p a ) h — H, and k 2 = 0, and equation (4) gives (1 + yuO F 2 = 2(//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 2 = V 2 + oo 2 r 2 - 2 Ywr, cos a, ... (94) which substituted in equation (88) gives (1 + ;/ 2 ) v 2 = V 2 + go 2 n 2 - 2 V. av, cos a, . (95) and this in equation (90) gives w 2 , and equation (12) will give the required work. Equation (14) will give the velocity for best effect. But this involves a long analysis, and the follow- ing approximate solution is sufficiently accurate. If y 2 be small, and the wheel be run for best effect, that is, so as to make the velocity to very small, and considering w = 0, equation (92) makes v 2 = GDV 2 nearly. Using this value as if it were the exact one, also neglecting friction, (95) gives 2 V. gdT] cos a — V 2 — 2gH t or 2 awi cos (x = \/2gJI; ,\ gdt 1 = — '^— ■ (approx.) .... (96) COS a ^ r A ' HYDRAULIC MOTORS. 41 which gives the proper velocity of the initial rim ; and for the terminal rim cor, _ Vijjr//. cos a r x Number of revolutions per minute .Y=30". (97) (98) To find the velocity at any point of the bucket relative to the bucket, drop the subscript 2 from equation (95) giving ■ (99) (1 -f yw 2 ) v- — 2gII + go 2 r 2 2 — 2 V^gH. oor l cos A Fig. 14. Fig. 13. From Fig. 1 or equations (4) and (5) find V2gR . sin i/i = — sin a. In the frictionless wheel, the work done will be U = i3f(V 2 -w 2 ) 9 . . and the efliciency will be U E = 8QH (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 42 HYDRAULIC MOTORS. into any convenient number of parts, as ac, ce, etc., and erect normals to the arc, as a ] >, ed, etc. Lay off these arcs on a right line. Compute the velocity at any point, as d, 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 ; 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.la.Vy = q; ba.v l , . - a!= ^r^ •••••• um) 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 (^ 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, | of a foot ; terminal angle of the bucket, y 2 = 15° ; termi- nal angle of the guide, a = 30°, j.i v = 0.10 — yu 2 . Then, velocity of exit from supply chamber, Eq. (93), V= Velocity of outer rim, Eq. (96), .... oo^ = Velocity of inner rim, Eq. (97), .... oor 2 = HYDRAULIC MOTORS. 43 Number of turns per minute, = Initial angle of bucket, Eq. (100), . . . y x = Initial velocity in bucket, Eq. (94), ... v x = Terminal velocity in bucket, Eq. (95), . . v 2 == Velocity of exit, Eq. (90), w = Direction of outflow, Eq. (35), .... = Coefficient of discharge 0.60, volume of water, Q = Weight of water (S = 62.4), 6Q = Work per second, Eq. (101), U = 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 d — TTTT77 ^ i ncnes - 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. 44 HYDRAULIC MOTOES. 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. TUEBIKES 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, y± = 90°, and as the water must flow radially into the arms, ex — 90°. The inner radius is necessarily small and may be considered zero. Hence, making Y2 = 0, ri = 90°, a = 90°, n = 0, in equation (14) gives _ U _ GOTz *-*QH. gH Equation (19) gives aor 2 + \/ML + afr? 1 + \x 2 . (105) (106) HYDRAULIC MOTORS. 45 hence, the efficiency reduces to, for the frictionless wheel, E=^- (107) V 2 + &>T 2 v 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 gdt 2 - V2gH; (108) for which value the efficiency will be, if yu 2 = 0.10 E= 2 [~- 1 + V n] = 070 (109) If k 2 be the area of the effective section of the orifice, then Q = k 2 v 2 (110) The pressure on the back side the arms opposite the orifices and useful in driving the mill, will be P 1 = Mv 2 = ^-v 2 (Ill) Of this pressure there will be required P 2 = M.Gor 2 = 6 Q.Gor 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 P x - P 2i and the work done per second will be this pressure into the distance it traverses per second, or U=cor 2 [P l -P 2 l 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 46 HYDRAULIC MOTORS. 11=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 = \ / "2gH i coefficient of discharge 0.64, and jj 2 = 0.10. Be quired : Velocity of discharge relative to the orifice, v 2 = Velocity of discharge relative to the earth, w = Velocity at entrance to the arms, ... v x = Velocity in the supply chamber, ... = The volume of water discharged, ... Q = The weight of water discharged, . . . d Q = The work per second, . U= The horse-power, HP = The efficiency, . E = The pressure on arm opposite orifice at A per square inch, pi = The pressure at base of the arms at C, . p = The equation to the path of the fluid. 37. Scottish and Whitelaiv 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's 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. 47 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 the frictionless wheel, >'i = r,, Y\ = 90 °> n = °> Mi = fh = 0, II — 0, Pz = Pi = Pa, along the vane, — mvGj normal to the vane. (No. 5.) In passing from a' to d' ; at d' the circular velocity will be greater than at a' by the amount at dp, and the acceleration will be dp requiring a force moo tt tangentially to the wheel in the direc- tion of motion, the reaction of which will be dp but backwards, and its components will be moo -jr cos y along the vane, — moo — sin y normal to the vane. dt (No. 6.) In passing from a to d', oop will be changed in direction by the angle between Tea and k"d', or , n ,, _a'g _dp cot y p p and the rate of angular change will be cot y dp ~~/T 'dt' 56 HYDRAULIC MOTORS. and the momentum being moop, the reaction will be dp moo cot y -T7, which acts radially inward and its components are — moo cos y -T. along the vane, — moo cot y cos y -j- normal to the vane. dt (No. 7.) By moving from a to d\ v will be increased by an amount dv , a P df> > in the time dt, and the reaction will be dv do dp' dtf which will be outward along the vane, and the reaction will be directly backward along the vane, and hence is dv dp , ,, — m -=- . ^r; alone the vane, dp dt 5 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 )=-p P . HYDRAULIC MOTORS. 57 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, will give the reaction, which will be normal and in the direction b'ri or mv along the vane, "cot y dp dy dp~\ , , , , . -T- — ~ . -r, , normal to the vane. . 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 de x x = xpdB sin y, and the force due to the difference of pressures will be xpdO sin v ~- dp. ' 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 TT to the other, the resultant effect would be zero, since the pressure in one direction on V W would equal the opposite pressure on XY. If, however, in passing from d to #, the pressure increases by an amount — dp, 58 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. pdd cos y — — xp cos ydO -±- dp. Collecting these several reactions, we have NORMAL TO THE VANE. (1.) (2.) + mou 2 u cos y. (4.) — juojv. (5.) — moo sin y dp TV (6.) — moo cot y cos y dp dt' (7.) fe% N fcot y dp dy dpi (8.) + mv . -jr- 4- . 4r I (9.) — #p cos y j- dpdd. ALONG THE VANE. -f moo-p sin y. 0. dp + racy COS y-rr. dp — mc^cos y-5T. W2 ^ ' dp* 0. xp sin y -y- dpdd The sum of the quantities in the second column, neglecting friction, will be zero ; hence dp dv dp j Jn /ACkri moo-p sin y — m -^- ■ -* xp sin y -~- dpdu = 0. . (123) Substituting -j = v sin y, and xpdOdp = -^ and dividing by m sin y, we have oo 2 pdp — —dp = vdv. (124) HYDRAULIC MOTORS. 59 Integrating, r n limit T- -| limit IWp'-P =\iv>\ .... (125) L- °-l limit L. J 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 dr2T = mv sin p ooy (— go cos y — 2 J — pv sin y cos y dp + v cos y — p sr- -/- dy Putting mv sin y = —^-dpdft, where Q is the quantity of water flowing through the wheel per second, and integrating in refer- ence to 6 between and 27T, we have dM=dQ\ copl -go cos y—2j — pvsmy-J- + vco&y — p ^ y ~ Multiplying (124) by dp P — cos y, we have &rp 2 7 p cos y dp 7 dv , cos yap r — -f dp = p cos y ^-dp v r vd dp r dp which substituted above gives a\3/=d() —2Gopdp + pcosy-j-dp + vcosydp—pvsmy-f-dp (126) the integral of which is M = SQ [— Gjp 2 + pv cos y] = -tfgp[«p-t>eoBrti88:. • • (127) 60 HYDRAULIC MOTORS. But go p — v cos y is the circumferential velocity in space of the water at any point, and $ Q p [go p — v cos 7] is the moment of the momentum; hence, integrating between limits for inner and outer rims, the moment exerted by 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„ y„ v^jpu and at exit p 2 , y„ v„ p. 2 . Equations (125) and (127) become i co> ( P : - p:) ._ £_p& = 1 « _ <). . . (128) M = S Q [go {p* - p 2 2 ) - Pi Vi cos y x + p 2 v 2 cos yj: (129) U = Mao = 6 Q go [go (p, 2 — p 2 2 ) — p l v 1 cos y 1 + p 2 v 2 cos y 2J (130) which is essentially the same as equation (58). It, however, in- volves the velocity of entrance, v iy and of exit, v 2 . The former may be found by equation (6), when a is known, or assumed when a is to be determined, and v 2 may be found by (19) or (56). This principle does not appear to be of great value in the solution of the general problem, but may be of much service in certain special cases. Thus, in the Barker Mill, page 44, the moment of the momentum of the water entering the wheel will be zero, but of exit will be M V, where V is the velocity of exit, relative to the earth, perpendicular to the arm, and the moment will be M Vr 2 ; .-. U = M V • r 2 go = Ps, (131) where P is the pressure on the arm opposite the orifice, and s the space passed over by the orifice in a second of time. HYDRAULIC MOTORS. 61 But M V = >\ : _ r , co : 6 Q 6 \ y 8 (1 + M) v, = 2 g H + or r 2 2 ; # A' v. .-. U = LJ (r, - /<, c») /•„ <* (132) 9 6Q / t /ZgH+a?r? as in equation (105). 40. Again, if the water quits the wheel radially, then the moment of the momentum of the quitting water will be zero, and U = 31 V )\ cos a ■ go. But V cos a = Y t , the tangential component of the velocity, or velocity of whirl ; .-. U = 31 r t i\ go. (133) 41. In the frictionless Rankine wheel the velocity of whirl equals the velocity of the initial rim of the wheel. .-. V t = r x co; .-. U = Mr? oo\ (134) The work will also equal the potential energy of the water, W H = C) Q H, less the kinetic energy of the quitting water, \ 31 w* (less the energy lost in resistances, ju v*, which in this case we neglect) ; .-. U = 6 Q H - \3Iw\ and since the water is assumed to quit radially w = i\ go ■ tan y 2 = r x co tan a. t (135) 62 HYDRAULIC MOTORS. The three preceding equations give 2 -f- tan 2 a r, 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? oo% (136) as in equation (134). But y 2 will not be the same as in (135). To find it we have, neglecting the thickness of the walls of the buckets, 2 n r 1 v l — 2 n r 2 sin y 2 • v 2 v 1 — Y siii a r x go = V cos a w = r 2 go tan y 2 v 2 = r % go ; r? tan a .-. tan y 2 = ; (137) \/r? — r? tan 2 a ,:U= 8 Q H-\Mw 2 r? tan 2 a = 8QH-\Mr!oS --—• (138) r? — r? tan 2 a Equations (136) and (138) give r x go = it/ *9H I / r? r? tan 2 a y 2 + ^ r* — r* tan 2 a (139) HYDRAULIC MOTOR8. 63 43. Conclusions.- From an examination of Tables II., III., VI., VIL, 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 VI. 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. 1. 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, a, of the guide blade. Jt will be observed that all the efficiencies in Tables VI. 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 ('•distantly 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, or, 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. 64 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 150°. In this case the efficiency for the outflow when a is 25° is 0.744, Table II., but when a is 13J°, as in Table VI., 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 \ for discharging equal vol- umes of water. Tables III., VI., VII. show that the section ~k % 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 A' 2 , 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 -f- 0.828) -=- (150 -^ 0.920) == 1.67. 12. In the outflow wheel, column (9), Table VI., 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 (4) and (5). 13. In these tables I have given all the velocities in terms of V 2 g A, 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 HYDRAULIC MOTORS. 65 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 rirst cost. Column (3) shows that the efficiency, E, increases as the initial angle of the bucket, y iy increases, up to 120°. The maxi- mum will be for about 120° with this amount of friction. MECHANICS— MACHINERY. Text-Books and Practical Works. A TEXT-BOOK OF ELEMENTARY MECHANICS FOR THE USE OF COLLEGES AND SCHOOLS. By E. S. Dana, Assistant Professor of Natural Philosophy Yale College. Twelfth edition 12mo, cloth, $1 50 " All Students and Mechanics will find the above a most admirable work. ' '— In dustrial World. PRINCIPLES OF ELEMENTARY MECHANICS. By Prof. De Volson Wood. Fully illustrated. Ninth edi- tion 12ino, cloth, 1 25 This work is designed to give more attention to the fundamental principles of mechanics. Analysis is subordinated, and whatia used is of a very ele- mentary character. No Calculus is used nor auy analysis of a high character, and yet many problems which are generally considered quite difficult are here solved in a very simple manner. The principles of Energy, which holds an important place in modern physics, is explained, and several problems solved by its use. Every chapter contains numerous problems and examples, the former of which are fully solved ; but the latter, which are numerical, are unsolved, and are intended to familiarize the student with the principles, and test his ability to apply the subject practically. At the close of each chapter is a list of Exercises. These consist of questions of a general character, requiring no analysis in order to answer them, but simply a good knowledge of the subject. The mechanics of fluids forms an important part of the work. Supplement and Key to ditto 1 25 THE ELEMENTS OF ANALYTICAL MECHANICS. With numerous examples and illustrations. For use in Scientific Schools and Colleges. By Prof. De Volson Wood. Sixth edition, revised and enlarged, comprising Mechanics of Solids and Mechanics of Fluids, of which Mechanics of Fluids is entirely new. About 500 pages. Seventh edition. 8vo, cloth, 3 00 The Calculus is freely used in this work. One of the chief objects sought is to teach the students how to use analytical methods. It contains many problems completely solved, and many others which are left as exercises for the student. The last chapter shows how to reduce all the equations of mechanics from the principle oi d'Alembert. STRENGTH OF MATERIALS AND THEORY OF STRUCTURES. By Henry T. Bovey, Dean of School of Applied Science, McGill University, Montreal, Canada 8vo, cloth, 7 50 ELEMENTS OF ANALYTICAL MECHANICS. By Col. Peter S. Michie, of U. S. Military Academy. Fourth edition 8vo, cloth , 4 00 " A revised edition, as taught to the Cadets of U. S. Military Academy, West Point." MECHANICS— MACHINERY. A TEXT-BOOK ON THE MECHANICS OF MA- TERIALS, And on Beams, Columns, and Shafts. By Prof. M. Merriman. Fourth edition, revised and enlarged , 8vo, cloth, $3 50 " We cannot commend this bock too highly." — American, Engineer. "The well-earned reputation of the Author renders any comment on the quality of the work superfluous."— Fan, Nostrand's Magazine. MECHANICS OP ENGINEERING. Comprising Statics and Dynamics of Solids, the Mechanics of the Materials of Construction or Strength and Elasticity of Beams, Columns, Shafts, Arches, etc., and the Principles of Hydraulics and Pneumatics with Applications. For the use of Technical Schools. By P*of. Irving P. Church, C.E., Cornell University .8vo, cloth, 6 00 "The work is very abundantly illustrated, and the information is given in a style which cannot fail to make the student thoroughly master of the subject. Prof. Church may certainly be congratulated upon compressing a vast amount of instruction into a very small space without in any degree interfering with the necessary minuteness of detail or clearness of descrip- tion."— London Industrial Review. MECHANICAL PRINCIPLES OF ENGINEERING AND ARCHITECTURE. By Henry Mosely, M.A., F.R.S. From last London edition, with considerable additions by Prof. D. H. Mahan, LL.D , of the U. S. Military Academy. 700 pages. With numerous cuts 8vo, cloth, 5 00 MECHANICS OF ENGINEERING AND MACHINERY. By Dr. Julius Weisbach. Designed as a Text-book for Tech- nical Schools and Colleges, and for the use of Engineers, Draughtsmen, etc. Second edition, thoroughly revised and greatly enlarged, by Gustav Herrmann, Prof, at the Royal Polytechnic School, Aachen, Germany. Translated by J. F. Klein, D.E., Prot. of Mechanical Engineering, Lehigh Uni- versity, Pa. With numerous fine illustrations. Second edition. 1 vol., 8vo, cloth, 5 00 " Weisbach is a standard iu all matters of Engineering and Mechanics, and his teachings are accepted as correct.' ' — Mechanical Engineer. MECHANICS OF THE MACHINERY OF TRANS- MISSION. Being Vol. III., Part I., Section II. of Mechanics of Engineering and Machinery. By Dr. Julius Weisbach. Edited by Prof. Gustav Herrmann and translated by Prof. J. F. Klein, Lehigh University, Bethlehem, Pa 8vo, cloth, 5 00 NOTES AND EXAMPLES IN MECHANICS. With an Appendix on the Graphical Statics of Mechanism. By Prof. I. P. Church, Cornell University. 135 pages, with blank pages for problems 8vo, cloth, 2 00 RfECHANTCS MACHINERY. APPLIED MECHANICS AND RESISTANCE OF MATERIALS. By Prof. Gr. Lanza. Showing Strains on Beams as determined by the Testing Machines of Watertown Arsenal and at the Massachusetts Institute of Technology. Practical and Theo- retical. Designed for Engineers, Architects, and Students. With hundreds of illustrations. Sixth edition, revised. 1 vol., 8vo, cloth, $7 50 " The whole work is a valuable contribution to the subject of which it treats, and we can cordially recommend it." — London Builder. WEISBACH'S MECHANICS. HYDRAULICS AND HYDRAULIC MOTORS. With numerous practical examples for the calculation and construction of Water- wheels, including Breast, Undershot, Back-pitch, Overshot Wheels, etc., as well as a special discus- sion of the various forms of Turbines. Translated from the fourth edition of Weisbach's Mechanics, by A. Jay Du Bois. Profusely illustrated. Second edition 8vo, cloth, 5 00 WEISBACH'S MECHANICS. THEORY OF STEAM- ENGINE. Translated from the fourth edition of Weisbach's Mechanics by A. Jay Du Bois. Containing notes giving practical examples of Stationary, Marine, and Locomotive Engines, showing Amer- ican practice, by R. H. Buel. Numerous illustrations. Second edition 8vo, cloth, 5 00 MECHANICS OF THE GIRDER. A Treatise on Bridges and Roofs, in which the necessary and sufficient weight of the structure is calculated, not assumed, and the number of Panels and height of Girder that render the Bridge weight least for a given Span, Live Load, and Wind Pressure are determined. By John D. Crebore, C.E. Illus- trated by over 100 engravings, with tables, etc 8vo, cloth, 5 00 "The work bears internal evidences of patient industry and scholarly ability— is a valuable contribution to science and to the literature of Bridge building."— W. H. Seakles, C.E. KINEMATICS; OR, PRACTICAL MECHANISM. A Treatise on the Transmission and Modification of Motion and the Construction of Mechanical Movements. For the use of Draughtsmen, Machinists, and Students of Mechanical En- gineering, in which the laws governing the motions and various parts of Mechanics, as affected by their forms and modes of connection are deduced by simple geometrical reasoning, and their application is illustrated by accurately constructed diagrams of the different mechanical combinations discussed. By Prof. Chas. W. MacCord. Fourth edition 8vo, cloth, 5 00 "The work can be confidently recommended to Draughtsmen, and all who have occasion to design machinery, as well as to every earnest student of Mechanics, young or old."— American Machinist. MECHANICS— MACHINERY. TREATISE ON FRICTION AND LOST WORK IN MACHINERY AND MILL WORK. Containing an explanation of the Theory of Friction, and an account of the various Lubricants in general use, with a record of various experimenters to deduce the laws of Friction and Lubricated Surfaces, etc. By Prof. Robt. H. Thurston. Copiously illustrated. Fourth edition 8vo, cloth, $3 00 "It is not too high praise to say that the present treatise is exhaustive and a complete review of the whole subject.'"— American Engineer. CAR LUBRICATION. Treating of Theoretical Relations, Coefficient of Friction, Bearing Metals, Methods of Lubrication, Journal-box Con- struction, Heated Journals, and the Cost of Lubrication. By W. E. Hall ; 12mo, cloth, 1 00 " A very useful book on a subject upon which literature is very scarce. While the author gives full credit to Prof. Thurston and to Mr. Woodbury for their researches in this direction, he puts the various theories and re- sults of experiment in a very practical shape and shorn of all but the plainest mathematical dress. The volume is evidently the work of a practical investigator, and is correspondingly valuable." — Engineering News. A HISTORY OP THE PLANING MILL. "With Practical Suggestions for the Construction, Care, and Management of Wood- working Machinery. By C. R. Tompkins, M.E 12mo, cloth, 1 50 ■' Each of these chapters is as full of meat as an egg : they give the results of long experience and intelligent observation, and no proprietor of woodworking machinery and employer of labor can afford to be without a copy, nor should any young mechanic, ambitious to excel in his calling, fail to send for it."— The Lumberman, Chicago, June 15, 1889. DYNAMOMETERS, AND THE MEASUREMENT OF POWER. By J. J. Flather, Prof, of Mechanical Engineering in Purdue University, Lafayette, Ind 12mo, cloth, 2 00 A Treatise on the Construction and Application of Dynamometers. Com- prising Determination of Driving Power, Friction Brakes, Absorption and Transmission Dynamometers, Power to Drive Lathes, Measurement of Water-Power. ELEMENTS OP MACHINE CONSTRUCTION AND DRAWING; Or, Machine-Drawing, with some elements of descriptive and rational Kinematics. A Text-book for Schools of Civil and Mechanical Engineering, and for the use of Mechanical Estab- lishments, Artisans, and Inventors. Containing the principles of Gearing, Screw Propellers, Valve Motions, and Governors, and many standard and novel examples, mostly from present American practice. By Prof. S. Edward Warren. Seventh edition 2 vols., 8vo, text, and small 4to plates, cloth, 7 50 EXTRACTS FROM CHORDAL'S LETTERS. Comprising the choicest selections from the Series of Articles which have been appearing for the past two years in the columns of the American Machinist. With over 50 illustrations. 12mo, cloth, 2 00 "The author discusses shop work and shop management with more practical shrewdness, and in a manner that Mechanics, Artisans, and wide awake working men, generally, cannot help but en joy." — Scientific American. MECHANICS—MACHINERY. THE LATHE AND ITS USES ; Or, Instruction in the Art of Turning Wood and Metal. Including a description of the most modern appliances for the ornamentation of plane and curved surfaces, with a de- scription also of an entirely novel form of Lathe for Eccentric and Rose Engine Turning, a Lathe and TurniDg Machine com- bined, and other valuable matter relating to the Art. 1 vol., 8vo, copiously illustrated. Sixth edition, with additional chapters and Index 8vo, cloth, $6 00 "The most complete work on the subject ever published."— American Artisan. "Here is an invaluable book to the practical workman and amateur."— London Weekly Times. A TREATISE ON TOOTHED GEARING. Containing complete instructions of Designing, Drawing, and Constructing Spur Wheels, Bevel Wheels, Lantern Gear, Screw Gear, Worms, etc., and the proper formation of Tooth Profiles. For the use of Machinists, Pattern Makers, Draughts- men, Designers, Scientific Schools, etc. With many plates. By J. Howard Cromwell. Fourth edition 12mo, cloth, 1 50 "Mr. Cromwell has accomplished good work in bringing together in this volume a great deal of information only to be found by searching many works, and by adding the results of his own experience in the field of Mechanical Engineering." — American Machinist. A TREATISE ON BELTS AND PULLEYS. Embracing full explanations of Fundamental Principles ; proper Disposition of Pulleys ; Rules for determining widths of leather and vulcanized rubber belts, and belts running over covered pulleys ; Strength and Proportions of Pulleys, Drums, etc. Together with the principles and necessary rules for Rope Gearing and transmission of power by means of Metallic Cables. By J. Howard Cromwell, Ph. B., author of a Treatise on Toothed Gearino- 12mo, cloth, 1 50 "This is a very complete and comprehensive treatise, and is worthy of the attention of all Mechanics who have anything to do with the manage- ment of belts and pulleys, etc."— National Car Builder. SAW FILING. The Art of Saw Filing Scientifically Treated and Explained on Philosophical Principles. With explicit directions for putting in order all kinds of Saws, from a Jeweler's Saw to a Steam Saw-mill. Illustrated by 44 engravings. By H. W. Holly. Fifth edition 18mo, cloth, 75 SAW FILING. A Practical Treatise on Filing, Gumming, and Swageing Saws. By Robert H. Grimshaw Fully illustrated 1 vol., 16mo, 1 00 MACHINERY PATTERN MAKING. A Discussion of Methods, including Marking and Recording Patterns, Printing Press, Slice Valve and Corliss Cylinders ; How to Cast Journal-boxes on Frames, Differential Pulleys, Fly-wheels, Engine Frames, Spur, Bevel and Worm Gears, Key Heads for Motion Rods, Elbows and Tee Pipes, Sweeping Straight and Conical Grooved Winding Drums, Large Sheaves with Wrought and Cast-iron Arms, 128 full-size Profiles of MECHANICS— MACHINERY. Gear Teeth of different pitches for Gears of 14 to 800 Teeth, with a Table showing at a glance the required diameter of Gear for a given number of teeth and pitch, Double Beat, Governor, and Plug Valves, Screw Propellor, a chapter on items for Pattern Makers, besides a number of valuable end useful Tables, etc., etc. 417 illustrations By P S. Dingey, Foreman Pattern Maker and Draughtsman. . . .12mo, cloth, $2 00 " A neat little work that should be not only in the hands of every pattern maker, but, read by every foundry foreman and proprietor of foundries doing machinery work."— Machinery Moulder's Journal. THE BOSTON MACHINIST. Being a complete School for the Apprentice as well as the advanced Machinist, showing how to make and use every tool in every branch of the business ; with a Treatise on Screws and Gear-cutting. By "Walter Fitzgerald. Third ed'n. 18mo,cloth, 75 STEAM HEATING FOR BUILDINGS. Or, Hints to Steam Fitters. Being a description of Steam Heating Apparatus for Warming and Ventilating Private Houses and Large Buildings, with Remarks on Steam, Water, and Air in their Relations to Heating. To which are added useful miscellaneous tables. By Wm. J. Baldwin. Thirteenth edition. With many illustrative plates 12mo, cloth, 2 50 "Mr. Baldwin has supplied a want long felt for a practical work on Heating and Heating Apparatus. 1 '— Sanitary Engineer. THE COST OF MANUFACTURES— AND THE AD- MINISTRATION OF WORKSHOPS, PUBLIC AND PRIVATE. A System of Mechanical Book-keeping, based on the Card- Catalogue method, dispensing with skilled clerical labor and the use of books, by which the cost of manufactures may be promptly determined, either in gross or in any detail, as to component parts and operations thereon. Comprising a simple method of recording all dealings with materials which relate to its procurement, expenditure, or possession. Applied, with numerous practical illustrations, to the trust, accounta- bility for public property, and funds required of the U. S. Ordnance Department, with a review of its present practice. ByCapt. Henry Metcalfe, U. S. Ordnance Department. Illustra- ted with tables, forms of cards, etc., etc. Second ed'n. 8vo,cloth, 5 00 " I feel sure that by the use of your methods I can determine a cost I hare never been able to arrive at." — Ewart Manufacturing Co., Chicago. " We tind that it enables us to keep a more accurate record of each piece of work. We can locate the responsibility for any delay or omission."— Rathbone & Co., Stove Works, Albany. WRINKLES AND RECIPES. Compiled from the SCIENTIFIC AMERICAN. A collection of practical suggestions, processes, and directions, for the MECHANIC, ENGINEER, FARMER, and Housekeeper. With a COLOR TEMPERING SCALE, and numerous wood engravings. By Park Benjamin. Revised by Profs. Thurston and Van der Weyde, and Engineers Buel and Rose. Fifth revised edition 12mo, cloth, 2 00 "Hundreds of Trade Secrets and Mechanical Shop Wrinkles.' 1 JOHN WILEY & SONS, 53 E. Tenth St., New York. ■H ■111 HUH SffiSHlfi UHmMiil^ffi^^^m nnHnH liIlHHnRK