HYDRAULICS,] account. Suppose, as before, that AB (fig. 77) comes to A B in a short time t. Let p v o>j, v v G l be the pressure, sectional area of stream, velocity, and weight A A B B , L_f of cubic foot at A, and p. 2 , w v r 2 , G 2 the same quan tities at B. Then, from the steadiness of motion, the weight of fluid passing A in any given time must be equal to the weight passing B : Fig. 77. Let ~i, z 2 tc the heights of the sections A and B above any given datum. Then the work of gravity on the mass AB in t seconds is where W is the weight of gas passing A or B per second. As in the case of an incompressible fluid, the work of the pressures on the ends of the mass AB is Pi u i v i^ ~Pa u ? v yt > The work done by expansion of W< pounds of fluid between A and B is W</ *pdv . The change of kinetic energy as before is (r. 2 2 - v^}t . Hence, equating work to change of kinetic energy, g- - Now the work of expansion per pound of fluid has already been given. If the temperature is constant, we get (eq. In-, 59) G- I ~!+p- + ^- = Z2+-p + Uj 6<J ljr. 2 But at constant temperature (2); or, neglecting the difference of level, Uj 2 Vj 2 p 1 p l 2g G 1 e p. 2 Similarly, if the expansion is adiabatic (eq. 2a, 59), 2 +^i + !lL= 2 . +^ + ^!_^i * 1 Gj 2g 2 G. 2 2g G 1 y - 1 -1 |] -a. .(*.). or neglecting the difference of level ^"27^ It will be seen hereafter that there is a limit in the ratio ?l beyond P-2 which these expressions cease to be true. 61. Discharge, of Air from an Orifice. The form of the equation of work for a steady stream of compressible fluid is the expansion being adiabatic, because in the flow of the streams of air through an orifice no sensible amount of heat can be communi cated from outside. Suppose the air flows from a vessel, where the pressure is p 1 and the velocity sensibly zero, through an orifice, into a space where the pressure is p. 2 . Let v. 2 be the velocity of the jet at a point where the convergence of the streams has ceased, so that the pressure in the jet is also p. 2 . As air is light, the work of gravity will be small compared with that of the pressures and expansion, so that z v z 2 may be neglected. Putting these values in the equation above ,Y-1 P-L-Pl + VjL.-.Pl. G! G 2 ^2;7 G x 1- -l_L G,-y- = M _y l_(P* G x ( y-1 y-lp 1 (P,
P
But Pi G., y^l Gz7-l < - pj IvLjy p_i_p_* 2g y-l (i); 481 an equation commonly ascribed to "Weisbach (Civilingenieur, 1856), though it appears to have been given earlier by St Venant and Wantzel. It has already ( 9, eq. 4a) been seen that where for air Po = 2116-8, G _ G T -08075 and 492 6 . 2</ G T y - or, inserting numerical values, (2); which gives the velocity of discharge v 2 in terms of the pressure and absolute temperature, p 1} TI , in the vessel from which the air flows, and the pressure p. 2 in the vessel into which it flows. Proceeding now as for liquids, and putting o> for the area of the orifice and c for the coefficient of discharge, the volume of air dis charged per second at the pressure p. 2 and temperature r 2 is - n- If the volume discharged is measured at the pressure p l and absolute temperature TJ in the vessel from which the air flows, let Q! be that volume ; then Let The weight of air at pressure p l and temperature TJ is G! = =^5 pounds per cubic foot. OO ^Tj Hence the weight of air discharged is (4). (5). "Weisbach has found the following values of the coefficient of dis charge c : Conoidal mouthpieces of the form of the ] c = contracted vein with effective pressures > 97 to 99 of 23 to I l atmosphere ) Circular sharp-edged orifices 563 ,,0788 Short cylindrical mouthpieces - 81 ,, 84 The same rounded at the inner end 92 ,, 93 Conical converging mouthpieces 90 ,, 99 62. Limit to the Application of the above Formula;. In the formulae above it is assumed that the fluid issuing from the orifice expands from the pressure p 1 to the pressure jt? 2 , while passing from the vessel to the section of the jet considered in estimating the area o>. Hence p. 2 is strictly the pressure in the jet at the plane of the external orifice in the case of mouthpieces, or at the plane of the contracted section in the case of simple orifices. Till recently it was tacitly assumed that this pressure p^ was identical with the- general pressure external to the orifice. Mr E. D. Napier first dis covered that, when the ratio ^ exceeded a value which does not ^i greatly differ from 5, this was no longer true. In that case the expansion of the fluid clown to the external pressure is not com pleted at the time it readies the plane of the contracted section, and the pressure there is greater than the general external pres sure ; or, what amounts to the same thing, the section of the jet where the expansion is completed is a section which is greater than the area c c o> of the contracted section of the jot, and may be greater than the area o> of the orifice. Mr Napier made experiments with 5, the formula? above steam which showed that, so long as ^* Pi
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