F • CONVECTIVE HEAT TRANSFER IN GASES 



for 0.7 ^ Prt ^ 1, where the term f (1 — Pr^ is a numerical approxi- 

 mation. 



Collecting Eq. 11-10, 11-13, 11-14, and 11-16 into Eq. 11-5 gives 

 finally, for the recovery factor: 



= Pr 



^ + 1 



+ (In 6) In [l + §(^ - l)] - (In 6) In [l +\{^ - l)j} 



(11-17) 

 This formula should be valid for normal gases. 



In order to utilize Eq. 11-17 to calculate the recovery factor, one re- 

 quires numerical values for the molecular and turbulent Prandtl num- 

 bers. While data on the molecular Prandtl number are readily available 

 in the NBS-NACA Tables of Thermal Properties of Gases, yet only a 

 few direct measurements [27,28] of the turbulent Prandtl number have 

 been made, and the accuracy of such measurements is apparently not suf- 

 ficient to establish a definitive value for the turbulent Prandtl number. 



Since it is much simpler to take temperature recovery measurements 

 than direct turbulent Prandtl number measurements, Eq. 11-17 might 

 well be used to determine indirectly the turbulent Prandtl number from 

 measured turbulent recovery factors. However, it is first necessary to 

 state that, while the above-derived expression for recovery factor is 

 essentially an incompressible flow formula, the measurements of recovery 

 factors are obtained only with supersonic flows on insulated surfaces 

 [20,29,30]. Certainly, it is only for supersonic flow that the recovery factor 

 has value. Fortunately, it appears from measurements with air that the 

 turbulent recovery factor is fairly independent of Mach number and 

 Reynolds number. Now, it is expected that the turbulent Prandtl num- 

 ber should also be independent of Mach number and Reynolds number. 

 Hence, if Eq. 11-17 is correct, there should be one value of Prt such that 

 r — const. Indeed, it is seen in Fig. F,llb that for r = const, the turbu- 

 lent Prandtl number must be 0.86, using incompressible flow friction 

 coefficients. Furthermore, it is of great interest to observe that the corre- 

 sponding recovery factor has a theoretical value of 0.88, which checks 

 well with experiment. A laminar Prandtl number of 0.71, corresponding 

 to 100°F for air, was used in the calculations. Squire's PH is also indi- 

 cated in Fig. F,llb. 



Since the turbulent Prandtl number is a molar characteristic of the 

 turbulent motion itself, it appears that the value Prt = 0.86 should apply 

 to all fluids. 



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