514 HYDROMECHANICS [HYDRAULICS. face of the same projected area, -which deviates the water through the same angle. Case 2. Cylindrical Surface moving in the Direction AC with Velo- --- u . The relative velocity =v-u. The final velocity BF 163) is found by combining the relative velocity BD=v- Fig. 163. tangential to the surface with the velocity BE = M of the surface. . Gt (v- w) 2 . The intensity of normal pressure, as in the last case, is The resultant normal pressure K = 2 U (v- u) z siiiy This resultant pressure may be resolved into two components P and L, one parallel and the other perpendicular to the direction of the vane s motion. The former is an effort doing work on the vane. The latter is a lateral force which does no work. P = R sin 4- = M0> - ) a (1 - cos 0) ;
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L = R cos -|- = - bt(v-u)* sin <j>. G The work done by the jet on the vane is Ptt = Uu(v - w) 2 (l - cos </>), which is a maximum when u = %o. This result can also be ob- tfli4d by considering that the work done on the plane must be equal to the energy lost by the water, when friction is neglected. If = 180, cos0=-l, 1 -cos = 2; then P = 2 bt(v - u) 2 , the same result as for a concave cup. 147. Position which a Movable Plane takes in Flowing Water. When a rectangular plane, movable about an axis parallel to one of its sides, is placed in an indefinite current of fluid, it takes a position such that the resultant of the normal pressures on the two sides of the axis passes through the axis. If, therefore, planes pivoted so that the ratio -^ (fig. 164) is varied are placed in water, and the angle they Fig. 164. make with the direction of the stream is observed, the position of the resultant of the pressures on the plane is determined for different angular positions. Experi ments of this kind have been made by Herr Hagen. Some of his results are given in the following table : Larger Plane. Smaller Plane. 4L-1-0 </>=... = 90 b 0-9 75 724 8 60 57 07 48 43 6 25 29 0-5 13 13 0-4 8 6| 0-3 6| 0-2 4 148. Effect of Friction during Impulse. Thus far the effect of the friction between the water and the surface which deviates it has been neglected. Nothing precise is known of its mode of action, and tli", following investigation is in part conjectural (Rankine, Steam Engine, p. 171, 146). Let it be assumed that the friction causes a loss of energy per second proportional to the height due to the velocity of the wate. relatively to the surface ; that is, the head due to the relative o > velocity being , the loss of head due to friction will be/ ; the loss of
whole energy due to the relative head being GQ^- energy due to friction will be GQ/ . Cylindrical Surface with Water deviated wholly in one Direction, Friction taken into account. In Case 2, discussed in 146, the velocity of the water relatively to the surface is v-u. The quan tity of "water impinging per second is bt(v-u). The loss of head due to friction is/ v ~ u > . The loss of energy due to friction is <f ( v u > . The energy exerted on the surface, after deducting 2</ the loss due to friction, is Pu = ~ bt(v - u)*u( - cos 0) - Gbtf Q-^L u(l - cos <) -/ jr The efficiency when friction is taken into account becomes p (v-u) 2w(l - cos 0) -f(v-ii) [ and this becomes a maximum if 1-COS0+/ ~2-2cos0+/ being greater than the speed when friction is neglected in the ratio 2(1 -cos <)+/: 3(1 -cos <+/). Suppose that the speed of greatest efficiency u has been found by experiment ; (2u-v)(l -cos 0) ^ V-U 149. Direct Action distinguished from Reaction (Eankine, Steam Engine, 147). The pressure which a jet exerts on a vane can be distinguished into two parts, viz : (1) The pressure arising from changing the direct component of the velocity of the water into the velocity of the vane. In fig. 154, 140, ab cos bac is the direct component of the water s velocity, or component in the direction of motion of vane. This is changed into the velocity ae of the vane. The pressure due to direct impulse is then .p rc ab cos bae - ae PI _GQ For a flat vane moving normally, this direct action is the only action producing pressure on the vane. (2) The term reaction is applied to the additional action due to the direction and velocity with which the "water glances off the vane. It is this which is diminished by the friction between the water and the vane. In Case 2, 146, the direct pressure is (v - u}* That due to reaction 9 S~ g > S 4>- If 0<^90, the direct component of the water s motion is not wholly converted into the velocity of the vane, and the whole pressure due to direct impulse is not obtained. If (/>90, cos <p is negative and an additional pressure due to reaction is obtained. 150. Reaction of a Jet issuing from a Vessel. Suppose a vessel filled with water (fig. 165), having an orifice of area o>, from which water issues horizontally with a velo city v = /2gh. The volume dis- chcarged per second, neglecting con traction, = a>v . The momentum gene rated per second in a horizontal direc-
tion = uv 2 ; and this is equal to the g force producing the change of mo mentum. Hence the horizontal force or reac tion R, acting on the side of the vessel, opposite to the orifice, and equal and Fig. 165. opposite to the force producing the momentum, is E= G - wt; 2 = 2Gft>/i ; this is the weight of a column of water the section of which is the area of the orifice, and the height is twice the head. If the vessel moves in a direction opposite to that of the jet, with the velocity u, the absolute velocity of the water leaving the vessel
is v - u. The momentum generated per second is uv(v - u) = K-
