Document OEXmB8ZneNd8jbVRrQN4bRQQw

84 CHAPTER 4 1950 Guide --/!??["-'] -'/sksFDH If this is computed and the figures are plotted, the curved line (partly solid and partly broken) of Fig. 6 is found. The maximum value of -- may be computed by differentiating to with respect to p2 and equatinPgi the result to zero. This operation produces the formula: For air, with k = 1.40, -- = 0.53. Pi Actually, the broken part of the curve is not attained for the flow in the nozzle. If the ratio of p2 to pi is decreased from unity, the mass Fig. 6. Relation of Flow of Gasto Pressure Drop in a Converging Tube 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 p2/pi is decreased further, the discharge is constant, as indicated by the horizontal line. The value of p2 at the maximum point is called the critical pressure, or pc, and it is seen that p,, is approximately 53 per cent of pi when air is flowing. To find the velocity at the critical pressure, it is assumed that the upstream velocity Vi is so small as to be negligible. Using the subscript c to indicate conditions at the critical point, from Equation 31 p. k l*-l (42) or (43) Fluid Flow 85 Substituting the critical pressure ratio from Equation 41 it follows that M = 1 (44) or that the velocity at the throat is equal to the local sonic velocity aryIntodesvtealortpiwngiththethweoirnkcinogmepqreusastiibolnes ffoorrmoriofifcethseanfldonwozEzoleusa, tiitnins cVuistomrl SuSse'''>tir.".n' d ,. .maillyqfi""ua>",.S ulaTU-'.fJKSl so that f k--1 1+-- 1 - M?/M} -- Vl - (A,M,) Vi - P (46) where P -- Dt/Di. The quantity l/\/l -- P* is the velocity of approach factor as generally used, with p being the ratio of the throat or orifice di ameter to the pipe diameter. Since Ap/pa is small 2k k-1 2Ap Vi (47) and the mass flow is to (48) The volume flow is then 0. = A, Vl-P* \/2ffAp/p = At Vl -P` V2ghi .(49) Actual Flow Through Orifices and Nozzles .. The actual rate of flow through an orifice, nozzle, or Venturi tube is rarely equal to the theoretical, and generally the actual rate is less than the theoretical. In the case of the nozzle and Venturi tube, this is due to losses from wall roughness, fluid friction, and turbulence during the ex pansion in the section following the throat. While wall roughness is not a factor in a sharp-edged orifice, fluid friction and turbulence are im portant, as is the fact that the discharge contracts to a degree variable with the ratio of outlet to inlet pressure after leaving the orifice, so that the limiting area is somewhat less than the opening in the orifice plate. Accordingly, Equation 49 must be modified by a correction factor, C. Usually, the velocity of approach factor is included with this correction factor, and, if K = C ----Vl-P (50) Q. = KAt y/2gh, (51) Multiplying by 3600 to convert from cubic feet per-second to cubic feet