Document vBXNXXkJdj7dnR50xGOMq3LER

CHAPTER 4 solid and partly broken) of Fig. 6 is found. The maximum value of ^ may be computed by differentiating w with respect to p2 and equating -y the result to zero. This operation produces the formula: SR V1! pi V* +1/ f(40) For air, with k = 1.40, Pi = 0.53. Pi V? IActually, the broken part of the curve is not attained for the flow in the nozzle. If the ratio of p2 to p2 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 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 percent of pi when air is flowing. & Fig. Relation of Flow of Gas to Pressure Drop in a Converging Tube To find the velocity at the critical pressure, it is assumed that the up- tstream velocity Vx is so small as to be negligible. Using the subscript I* c to indicate conditions at the critical point, from Equation 29 (41) I' or Me = % (42): Substituting the critical pressure ratio from Equation 40 it follows that || . Mc = 1 (43) f or that the velocity at the throat is equal to the local sonic velocity. M 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 case1 both Mi and M2 are'.small quantities, pi =. '--/pi)/,P.ir = Ap/p2 is small. Retaining only fimt order terms, it. follows frofiijEqua- tion 35 that . i- - - : ... Mi M Mi= A, (44) so that 1 +. --k --- 1 MY i - ,=/*, -- Vi (Ai/AlY 1 (45) where |3 = D2/Di. The quantity 1/ vV -- |34 is the velocity of approach' t-JHN <t>) Jr* Wo- O d o $0.02 Dj 1W - SO.IOD, 1- <a03 Df- $0.03 Dt 0.80 * 0.76 $0.72 . u w 0.68 ' oO 0.64 _A A| 0.70 5-10*2 pVlPl OAO 030 0O..2fO0 5 10* .0.05 Fig. 7. Dimensions and Flow Coefficient for Standard Square-Edged Orifice with Corner Taps or Annular-Slits (Coefficient shown as a function of Reynolds Number and Ratio, A/A.) Note: From Reference 4. Used by permission. pfactor as generally used, with being the ratio of the throat or orifice di ameter to the pipe diameter. Since Ap/p2 is small and the mass flow is fc2_k 1LVV J~ Pi (46) The volume flow is. then Vl-F \/2gpAp (47) Q~At v'2^P'/p = 2Bh' (48) ill