HYDRAULICS.] HYDROMECHANICS 4G3 Co"ffidoit of Discharge. In applying the filaments have a common velocity the general formula Q, = uv to a stream, it is assumed that the i normal to the section co. But if the jet contracts, it is at the con tracted section of the jet that the direction of motion is normal to a transverse section of the jet. Hence the actual discharge when contraction occurs is Qa = C T V x c c d> = c c c v u/2gh , or simply, if c = c c c c , where c is called the coefficient of discharge. Thus for a sharp- edged plane orifice c = 97 x 64 = 62. 17. Experimental determin ation of c v , c c , and c. -The coefficient of contraction c c is directly determined by measuring the dimensions of the jet. For this purpose fixed screws of fine pitch (fig. 21) are convenient. These are set to touch the jet, and then the distance between them can be measured at leisure. The coefficient of velocity is determined directly by measuring the parabolic path of a horizontal jet. Let OX, OY (fig. 22) be horizontal and vertical axes, the origin being at the orifice. Let h be the head, and x, y the coordinates of a point A on the parabolic path of the jet. If v a Fig. 22. is the velocity at the orifice, and t the time in which a particle moves from to A, then x=v a t ; y=%gt*. Eliminating t, Then If the jet is not initially horizontal, let OB (fig. 23) be any hori zontal datum line, Mid let the vertical distances OC, AD, BE be Fig. 23. measured, the point A being taken conveniently midway between and B. Then j/! = OC-AD, and 7/ 2 = OC-BE. Let o be the inclination of the jet atC to the horizontal, so that v a cos a is its horizontal and v a sin o its vertical velocity at that point. If t is the time in which a particle moves from C to D, then = v a cos a t . 2 Eliminating t, Similarly, Hence /i = IT tan a ~ r^ 14 " tan2 a) / 2 = x tan a - --^ (1 + tan 2 a) . A V = Ac + ("4yi V M(*Ji where for h is to be put the depth of C below the free water sur face. The coefficient of discharge is determined independently, by measuring the discharge in a gauging tank for a given time. Then, if Q is the measured volume discharged in one second,
18. Coefficients for Bellmontlis and Bcllmouthcd Orifices. If an orifice is furnished with a mouthpiece exactly of the form of the contracted vein, then the whole of the contraction occurs within the mouthpiece, and if the area of the orifice is measured at the smaller end, c c must be put = l. It is often desirable to bellmouth the ends of pipes, to avoid the loss of head which occurs if this is not done ; and such a bellmouth may also have the form of the con tracted jet. Fig. 24 shows the proportions of such a bellmouth or bellmouthed orifice, which approximates to the form of the con tracted jet sufficiently for any practical purpose. For such an orifice Weisbach has found the following values of the coefficients with different heads. Head over orifice, in fcet=/i 6G 1-64 11-48 5577 337-93 Coefficient of velocity = c v . . Coefficient of resistance = c , 959 087 967 069 975 052 994 012 994 012 As there is no contraction after the jet issues from the orifice, = 1 , c = c, ; and therefore l+c r 19. Coefficients for Sharp-edged or virtually Sharp-edged Orifices. The coefficient of velocity for sharp-edged orifices of diilVrent areas and under different heads is not very accurately determined. Its mean value is about 96. The coefficient of contraction is also dependent on circumstances the relative influence of which is not so perfectly known as is de sirable. Its mean value for well-placed orifices in a plane surface is 64. For conditions similar in other respects, the contraction is less (that is, the area of the stream is greater) the smaller the orifice and the less the head. If the surface surrounding the orifice is not plane, the coefficient is greater for a surface convex to the interior of the reservoir and less for a concave surface. The thickening of the edges of the orifice modifies the contraction in a slight degree, and if a border or rim is placed round part of the edge of the ori fice, and projects inwards or outwards, the coefficient is very consider ably altered, and the contraction is then termed incomplete. If the orifice is placed in a contracted part of the vessel so that the water approaches the orifice with considerable velocity, the coefficient is
increased, and the contraction is said to be imperfect.
