E,4 • ANALYSIS FOR CONSTANT FLUID PROPERTIES 



transfer in an entrance region was made by Latzko [4-3]. His analysis was 

 for a Prandtl number of one and was based on an assumed ^-power ve- 

 locity profile and the Blasius resistance formula. More recent develop- 

 ments are given in [15,44,4^A6A'^]- 



For analyzing heat transfer and flow in an entrance region, integral 

 methods will be utilized here. The usual boundary layer assumptions used 

 with integral methods are made; that is, it is assumed that the effects of 

 heat transfer and friction are confined to fluid layers close to the surface 

 (thermal and flow boundary layers, respectively). The temperature and 

 velocity distributions outside the boundary layers are assumed uniform, 

 and the temperature and total pressure are constant along the length of 



0.01 



u 0.00, 



Q. 



^ 0.0001 



0.00001 



0.1 



1.0 



1000 



10,000 



10 100 



Pr = Cp\i/k 



Fig. E,4e. Comparison of various analyses. Reynolds number = 10,000. 



the passage for the region outside the boundary layers. More exact analy- 

 ses [48] indicate that these assumptions are vahd, even for laminar flow, 

 except in the region at a distance from the entrance where the boundary 

 layer fills a large portion of the tube. In that region, however, the Nusselt 

 number and friction factors have values very close to the values for fully 

 developed flow.^ 



In order to obtain a relation between distance along the passage x 

 and thermal boundary layer thickness 5^, we can write an energy balance 



* A recent analysis by E. M. Sparrow, T. M. Hallman, and R. Siegel [85] indicates, 

 however, that the entrance lengths computed by this method may be too short, 

 although the entrance length is still on the order of 10 diameters. The difference is 

 apparently due to the assumption that the temperature is constant along the length 

 of the passage for the region outside the boundary layer. That should still be a good 

 assumption when the boundary layer is very thin. 



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