HYDRAULICS.] HYDROMECHANICS 513 direction in which the plane is moving is P = N cos 8 = _ Q(v cos o- u cos 8) cos 8, and the work done on the plane is /- P(( = Q(y COS O - U COS 8) u cos 8, which is the same expression as before, since AE=r r cos = v cos a - u cos 8. In one second the piano moves so that the point A (fig. 159) comes to C, or from the position shown in full lines to .the posi tion shown in dotted lines. If the plane remained sta tionary, a length AB=y /[/ of the jet would impinge on the plane, but, since Fig. 159. the plane moves in the same direction as the jet, only the length HB = AB- AH impinges on the plane. Tint A TT f co a o cos 8 i ,1 , c , up cos S let ff cos a cos a cos a positioi w = sectional area of jet ; volume impinging on plane per second A and / cos 8
v cos a - u cos 8 T , . , , . height V cos a / cos o formulae above, we go t then de vr ^i == G (v cos a- u cos 8) 2 .... (1); float ha U A 2 and p G a cos 8 (2V a relati ff cos a of grav G cos 8 therefo la . (3). bucket g cos a Three cases may be distinguished :-
and the (it) The plane is at rest. Then = 0, N = <av- cos work done on the plane and the efficiency of the jet are zero. (ft) The plane moves parallel to the jet. Then 8 = 0, and G Pit wu cos -a (r - u) - , which is a maximum when u ^i /-i When K = ^ then Tu max. = >, o>y 3 cos 2 a, and the efficiency =TJ = * cos 2 a. (c) The plane moves perpendicularly to the jet. Then 8 = 90 - a ; cos 8 = sin a ; and Pi<= - uii : (v cos A- u sin a) 2 . This is a <J cos a maximum when u = ^r cos o. When ?< = ^.> cos o, the maxiimim work and the efficiency are the same as in the last case. 144. Best Form of Vane to receive Water. When water impinges normally or obliquely on a plane, it is scattered in all directions after impact, and the work carried away by the water is then generally lost, from the impossibility of dealing afterwards with streams of water deviated in so many directions. By suitably forming the vane, however, the water may be entirely deviated in one direction, and the loss of energy from* agitation of the water is entirely avoided. Let AB (fig. 160) bo a vane., on which a jet of water impinges at the point A and in the direction AC. Take AC = i> = velocity of water, ami let AD represent in magnitude and direction the velo city of the vane. Completing the parallelogram, DC or AE repre- up the vane in the direction AB. This is sometimes expressed by saying that the vane receives the water without shock. 145. Floats of Poncelct Water Wheels. Let AC (fig. 161) repre sent the direction of a thin horizontal stream of water having the velocity v. Let AB be a curved float moving horizontally with velo city u. The relative motion of water and float is then initially horizontal, and equal to v u. In order that the float may receive the water without shock, it is necessary and sufficient that the lip of the float at A should be tangential to the direction AC of relative motion. At the end of u seconds the float moving with the velocity u comes to the position AiB l5 and during this time a particle of water received at A and gliding up the float with the relative velocity v - u, attains a 1= <j . At E the water comes to relative rest. It then descends along the float, and when after "^~ " seconds the g float has come to A 2 B. 2 the water will again have reached the lip at A 2 and will quit it tangentially, that is, in the direction CA 2 , with a relative velocity - (v -u)= - V 2</DE acquired under the influence The absolute velocity of the water leaving the float i:- - (v - u) = 2u - v. If M = v, the water will drop off the bucket deprived of all energy of motion. The whole of the work of the jet must therefore have been expended in driving the float. The water will have been received without shock and discharged without velocity. This is the principle of the Poncelet wheel, but in that case the floats move over an arc of a large circle ; the stream of water has considerable thickness (about 8 inches) ; in order to I get the water into and out of the wheel, it is then necessary that the lip of the float should make a small angle (about 15) with the direction of its motion. The water quits the wheel with a little of its energy of motion remaining. 146. Pressure on a Curved Surface when the Water is deviated wholly in one Direction. When a jet of water impinges on a curved surface in such a direction that it is received without shock, the pressure on the surface is due to its gradiral deviation from its first direction. On any portion of the area the pressure is equal and opposite to the force required to cause the deviation of so much water as rests on that surface. In common language, it is equal to the centrifugal force of that quantity of water. Case 1. Surface Cylindrical and Stationary. Let AB (fig. 162) be the surface, having its axis at and its radius =r. Let the water impinge at A tangentially, and quit the surface tnngentialJy at B. Since the surface is at rest, v is both the absolute velo city of the water and the velocity relatively to the surface, and this remains unchanged during con tact with the surface, because the deviating force is at each point perpendicular to the direction of motion. The water is deviated through an angle BCD - AOB = <p. Each particle of water of weight p exerts radially a centrifugal D force ^ rg Let the thickness of Fig. 162. Fiy. 100. sents the direction in which the water is moving relatively to the vane. If the lip of the vane at A is tangential to AE, the water will not have its direction suddenly changed when it impinges on the vane, and will therefore have no tendency to spread laterally. On the contrary it will be so gradually deviated that it vililgUde the stream =t feet, Then the weight of water resting on unit of surface =Gt Ib ; and the normal pressure per unit of Mirfiice = M = . . The resultant of the radial pressures uniformly dis- <7 f ti-ilmtod from A to B will be a force acting in the direction OC bisecting AOB, and its magnitude will equal that of n force of intensity =n, acting on the projection of AB on a plane per pendicular to the direction OC. The length <>f the chord AB = 2/- sin ~- ; let 6 = breadth of the surface perpendicular to the i plane of the figure. The resultant pressure on surface -K-2r6Bin^x^.^-2^W^8ir f - , g r ;/ 1 whiK is independent of the radius of curvature. It may be in- fen-od that the resultant pressure is the same for any curved sur-
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