Document rxv5oN8Mkg5Z9VXMaRYVkqG1v

76 CHAPTER 4 1953 Guide solid and partly broken) of Fig. 6 is found. The maximum value of -- Pi may be computed by differentiating to with respect to p and equating the result to zero. This operation produces the formula: For air, with k = 1.40, -- = 0.63. Pi Actually, the broken part of the curve is not attained for the flow in the nozzle. If the ratio of pi to pi is decreased from unity, the'mass rate of discharge, as well as the volume, increases from zero to a maximum, as shown by the solid section of the curve in Fig. 6; thereafter, as pi/p is decreased further, the discharge is constant, as indicated by the horizontal line. The value of pj at the maximum point is called the critical pressure, or p., and it is seen that p,, is approximately 53 percent of pi when air is flowing. -Fluid Flow 77 In developing the working equations for orifices and nozzles, it is custom ary to start with the'incompressible form of the flow Equation 38. In this case both ;M, and M, are smail quantities, in - pi, and .(pi -- pj)/,pi = Ap/pj is small. Retaining only first order terms, it follows from Equation 35 that Mi = A,i Mi = A, (44) so that k -1 1 + ---- Mi1 1 - AfiVAf,' y/l - (4VA0* Vl - 0` . (45) where 0 = D2/Dt. The quantity l/Vl - P is the velocity of approach Fiq. 6. Relation or Flow or Gas to Pbessube Drop in a Converging Tube To find the velocity at the critical pressure, it is assumed that the upstream velocity Fi is so small as to be negligible. Using the subscript c to indicate conditions at-the critical point, from Equation 29 CiP-] < Substituting the critical pressure ratio from Equation 40 it follows that M, = 1 (43) or that the velocity at the throat is equal to the local sonic velocity. 4*I0J I04 2 3 I05 2 3. pViOl Fig. 7. Dimensions and Flow Coefficient for Standard Sharp Edged Orifice (Coefficient shown as a function of Reynolds Number and Ratio, A/A.) Note: From Reference 4. Used by permission. factor as generally used, with being the ratio of the throat or orifice di ameter to the pipe diameter. . Since Ap/p2 is small 2k k-1 (46) and the mass flow is w V2ffpAp Vi-01 The volume flow is then (47) Q = A, Vi -p '2g&p/p = At Vl ~ P y/tyht (48)