512 HYDROMECHANICS [HYDRAULICS. face in the direction and with the velocity eb. As this relative velocity is unaltered by contact with the surface, take cd eb, then cd is the relative motion of the water with respect to the surface ate. Take df equal and parallel to ae. Then/c (obtained by compound- in 01 the relative motion of water to surface and common velocity of water and surface) is the absolute velocity and direction of the water leaving the surface. Take ag equal and parallel to/c. Then, since ab is the initial and ag the final velocity and direction of motion, gb is the total change of motion of the water. The resultant pressure on the plane is in the direction gb. Join eg. In the tri angle gae, ae is equal and parallel to (//, and ag to fc. Hence eg is equal and parallel to cd. But cd = cb = relative motion of water and surface. Hence the change of motion of the water is represented in magnitude and direction by the third side of an isosceles tri angle, of which the other sides are equal to the relative velocity of the water and surface, and parallel to the initial and final direc tions of relative motion. SPECIAL CASES. 141. (1) A Jet impinges on a plane surface at rest, in a direction normal to the plane (fig. 155). Let a jet whose section is u impinge with a velocity v on a plane surface at rest, in a direction normal to the plane. The particles approach the plane, are gradually deviated, and finally flow away parallel to the plane, having then no velocity in the original direction of the jet. The quantity of water impinging per second is cav. The pressure on the plane, which is equal to the change of momentum per second, is wv" 1 . (2) If (lie plane is moving in the direc tion of the jet with the velocity u, the quantity impinging per second is ea(v^fu). The momentum of this quantity before p impact is ca(v^fu) v. After impact, the water still possesses the velocity u in the direction of the jet; andthemomen- .. turn, in that direction, of so much water p as impinges in one second, after impact, is * u(v : fu)u. The pres sure on the plane, which is the change of momentum per second, is the difference of these quantities or <a(v^pu^. This differs from g the expression obtained in the previous case, in that the relative velocity of the water and plane v^u is substituted for v. / T~ 2 The expression may be written P = 2xGxco( ), where the last term is the volume of a prism of water whose section is the area of the jet and whose length is the head due to the relative velocity. The pressure on the plane is twice the weight of that prism of water. The work done on the plane in this case is Pu &> (v^fu} 2 9 foot-pounds per second. There issue from the jet cav cubic feet per second, and the energy of this quantity before impact is &rv 3 . The efficiency of the jet is therefore ?j = 2 Jftt. The f -n ^, a maximum lor -~ 2(v du 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. 142. (4) Case of a Concave Cup Vane, velocity of water v, velocity of vane in the same direction u (fig. 156). If the cup is hemispherical, the water leaves the cup in a direction value of u which makes this a maximum is found by differentiating and equating the differential coefficient to zero :
- L_ 2
du . . u = v or Jr. The former gives a minimum, the latter a maximum efficiency. Putting u = ^ v in the expression above, i} max. = -j7 (3) If, 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 uv instead of ca(v - u), and the C G whole pressure = ur(v - u). The work done is uvu(v - u). The efficiency ij = um(v - n) -^^-tav^=1u( ) This becomes This result is often used parallel to the jet. Its relative velocity is v- u when approaching the cup, and -(v- u) when leaving it. Hence its absolute velocity when leaving the cup is u - (v - u) = 2u - v. The change of momen- C" 1 ci /^i turn per second = u(v - u}v u(v - u}(1u - v) = 2 u (v - Comparing this with case 2, it is seen that the pres sure on a hemispherical cup is double that on a flat plane. The work done on P the cup = 2 ta (v- iifu foot-pounds per second. The efficiency of the jet is greatest when v = 2u ; in that case the efficiency 2u-v 2 17 * n 1 K ff If a series of cup vanes are introduced in front of the jet, so that the quantity of water acted upon is <av instead of w(v - u}, then the whole pressure on the chain C P C of cups is cov 2 <av(1u-v) = 2 avlv-u}. In this case the 9 9 ff efficiency is greatest when v = 2u, and the maximum efficiency is unity, or all the energy of the water is expended on the cups. 143. (5) Case of a Flat Fane oblique to the Jet (fig. 157). This case presents some difficulty. The water spreading on the plane in all directions from the point of impact, different particles leave the plane with different absolute velocities. Let AB = f= velocity of water, AC = M= 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 D Fig. 157. the plane. Then the relative velocity AD may be regarded as con sisting 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 = e, and let AD = v r . Then AE = v r cos 6 ; DE = ?v sin 0. If Q is the volume of water impinging on the plane per second, C the change of momentum is Qiv cos 0. Let AC = u = velocity g of the plane, and let AC make the angle CAE = 8 with the normal to the plane. The velocity of the plane in the direction AE = u cos 8. The work of the jet on the plane = Qv r cos u cos 8. The same problem may be thus treated algebraically (fig. 158). Let i BAF = o, and CAF = 8. The velocity v of the water may be decom- D Fig 158. posed into AF = v cos o normal to the plane, and FB = r sin a parallel to the plane. Similarly the velocity of the plane = u = AC = 151) can be decomposed into BG = FE = it cos 8 normal to the plane, and DG = u sin 8 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 the plane, that is, u cos 8. Hence the change of velocity during impact = AE=y cos a-u cos 8. The change of momentum per second, and consequently the normal pressure on G
the plane is N = Q (v cos a-u cos 8). The pressure in the
