is moving in the same direction as the jet is Pu: (G/g)<-1('v-u)'u
foot-pounds per second. There issue from the jet wv cub. ft.
per second, and the energy of this quantity before impact is
(G/2g)wU3. The efhciency of the jet is therefore n=2(v-u)'u/93.
The value of u which makes thisa maximum is found by differentiating
and equating the differential coefficient to zero:-
dv;/du=2(v2-4vu+3§ ¢**)/'v'*=o;
u =v or ga.
The former gives a minimum, the latter a maximum efficiency.
Putting u = 51' in the expression above,
11 max. = 58,
(3) I f, instead of one plane moving before the jet, a series of planes
are introduced at short intervals at the same point, the quantity of
water impinging on the series will be mv instead of w(°v-u), and the
whole pressure=(G/g)a>v(v-u). The work done is (G/g)wvu(v-u).
The efficiency 17 = (G/g)w°vu(v-u)+(G/2g)w11° =2u(v-u)/vz. This becomes
a maximum for dn/du=2(v-2u) =0, or u=&v, and the 71=%.
This result is often used as an approximate expression for the velocity
of greatest efficiency when a jet of water strikes the floats of a water
wheel. The work wasted in this case is half the whole energy of the
jet when the floats run at the best speed.
§ 156. (4) Case nfa Jet impinging on a Concave Cug Vane. velocity
of water v, velocity of vane in the same direction u (g. 155), weight
impinging per second = Gw(v A u).
If the cup is hemispherical, the water leaves the cup in a
direction parallel to the jet. Its relative velocity is 'U-u when approaching
the cup, and
-(v-u) when leaving it.

Hence its absolute ve ocity

5* when leaving the cup is M, V, E u-(11-u)=2u-v. The { ; i ~:g v Www change of momentum er

- '"""' “ ” ' second = (G/g)w(v-u) fi:-

M/ (2u~'v)} = 2(G/g)w(v-u)'. AQ(Comparing this with case 2, //' it is seen that the pressure on a hemispherical cu is 2u-v double that on a flat plane. The work done on the cup=2(G/g)w (1/-'u)'u foot pounds per second. The efficiency of the jet is greatest when v=3u; in that case the efficiency = If a series of cup vanes are introduced in front of the jet, so that the quantity of water acted upon is tw instead of a»(v-u), then the whole pressure on the chain of cups is ((i/g)w'U{U-(Zu-'U)}='2(G/g)w1/(U-M). In this case the efficiency is greatest when v=2u, and the maximum efficiency is unity, or all the energy of the water is expended on the cu s. lg 157. (5) Case of a Flat Vane oblique to the Jet (fig.156).-This case presents some difficulty. The water spreading on the plane in all Fig. 155. / D

/ Vr .' "'. )7'?)»*-> /f " '- i?;f:~~-»< -L —; ' ~' V, " B is gs.; ¢ A . .

- W ¢ gp—" .f 'Ji

// /7 -"" C FIG. 156. directions from the point of impact, different particles leave the plane with different absolute velocities. Let AB=v=velocity of water, AC=u=velocity of plane. Then, completing the parallelogram, AD represents in magnitude and direction the relative velocity of water and plane. Draw AE normal to the plane and DE parallel to the plane. Then the relative velocity AD may be regarded as consisting of two components, one AE normal, the other DE parallel to the plane. On the assumption that friction is insensible, DE is unaffected by impact, but AE is destroyed. Hence AE represents the entire change of velocity due to impact and the direction of that change. The pressure on the plane is in the direction AE, and its amount is = mass of water impinging per second X AE. Let DAE =0, and let AD =-v, . Then AE='v, cos 0; DE=v, - sin 0. If Q is the volume of water impinging on the plane per second, the change of momentum is (G/g)Qv, cos 0. Let AC=u=velocity of the plane, and let AC make the angle CAE=5 with the normal to the plane. The velocity of the plane in the direction AE= u cos 5. The work of the jet on the plane= (G/g)Q'v, cos 0 u cos 5. The same problem may be thus treated algebraically (fig. 157). Let BAF =o., and CAF =5. The velocity 11 of the water may be decomposed into AF=v cos a. normal to the plane, and FB ='v sin u. parallel to the plane. Similarly the velocity of the plane =u =AC = BD can be decomposed into BG =FE =u cos 5 normal to the plane, and DG ==1¢ sin 5 parallel to the plane. As friction is neglected, the velocity of the water parallel to the plane is unaffected by the impact, but its component v cos a normal to the plane becomes after impact the same as that of change of velocity during change of momentum per 87 the plane, that is, u cos 5. Hence the impact=AE='v cos a.-u cos 5. The second, and consequently the normal (D fi " Y/ / (y " / Jil fl pressure on the plane is N = in the direction in which the FIG. 157. (G/g) Q (v cos a-ucos 5). The pressure plane is moving is P =N cos 5 = (G/g)Q (v cos o.-u cos 5) cos 5, and the work done on the plane is Pu(G/g)Q(v cos a.-u cos 5) u before, since AE ='v, - cos 9 cos 5, which is the same expression as ='v cos a-u cos 5. In one second the plane moves so that the point A (fig. 158) comes to C, or from the position shown in full lines to the Position shown in dotted ines. If the plane remained stationary, a length AB =1/ of the jet would impinge on, ,H the plane, but, since the plane A “ I/2/' B moves in the same direction /jf as the °et, only the length 5 HB =AE-AH impinged on U the plane. But AH =AC cos 5/ cos o. = u cos 5/ cos a, and therefore N HB='u-u cos 5/ cos a. Let wTsectional area of ljet; . vo ume impinging on ane per second=Q=w(v-up cos FIG' 158 5/cos a) =w(v cos a-u cos 5)/ cos a. Inserting this in the formulae above, we get N=Q l(v cos a-u cos 5)2- (I) g cos a P=Q (v cos a-ucos 5)2; (2) g cos a Pu=§ -wugikvcos G.-1uC0S5)2. (3) COS U. 3 . . .. . Three cases may be distinguished:- (a) The plane is at rest. work done on the plane and Then u=o, N = (G/g)w1/2 cos a; and the the efficiency of the -jet are zero. (b) The lane moves arallel to the jet. Then 5=a., and Pu-P P (G/g) wu cos 2a('v - u)2, which When u=§ v then Pu max.= =11=2, ' cos 2a. is a maximum when u=§ ~v. 547 (G/g)<»113 cos za, and the efficiency (c) The plane moves perpendicularly to the jet. Then 5=9o°-a; G cos 5=sin a; and Pu=-wu Z mum when u=§ v cos a. S111 U. COS 0. (v cos a-u sin a)2. This is a maxi-When u=§ v cos a, the maximum work and the efficiency are the same as in the last case. § 158. Best Form of Vane normally or obliquely on a after impact, and the work c ally lost, from the impossibili to receive Water.-When water impinges plane, it is scattered in all directions arried away by the water is then generty of dealing afterwards with streams of water deviated in so many directions. By suitably forming the vane, B » »' . "~ li, “" 5 /f ... ... -. - ..- Cz( T }T'-, f/V ss / f ' D

however, the water may be FIG. 159. entirely deviated in one direction, ana the loss of energy from agitation of the water is entirely avoided. Let AB (fig. 159) be a vane, on which a jet of water impinges at

the point A and in the direction AC. Take AC=v=velocity of