E • CONVECTIVE HEAT TRANSFER AND FRICTION 



An analysis by Einstein and Li [£4] assumed that the laminar sub- 

 layer is basically unsteady; that is, the sublayer grows until it becomes 

 unstable and then collapses, the cycle repeating itself. This model gave a 

 velocity profile agreement with data for the region close to the wall. 



Temperature distributions. For extending the analysis to heat trans- 

 fer an assumption must be made for a, the ratio of eddy diffusivities for 

 heat transfer to momentum transfer, in Eq. 2-6. The relation for a has 

 not been clearly established. It is a fact that analyses based on an a of one 

 agree closely with experiment [6,14]. However, some attempts to measure 

 a directly indicate values which, in general, are somewhat greater than 

 one [25; 26, pp. 122-126], except in the case of low Peclet or Prandtl 

 numbers where values of a less than one may occur [14; 27, pp. 405-409; 

 28]. The direct measurement of a is difficult, especially in the important 

 region close to the wall, because it involves the measurement of velocity 

 and temperature gradients. Reichardt [3] proposed the hypothesis that 

 a is one at the wall and increases as the distance from the wall increases. 

 For turbulent flow the important changes of velocity and temperature 

 take place close to the wall for Prandtl numbers on the order of one or 

 greater, so that the assumption of a = 1, in general, gives good results. 

 It is of interest that the Prandtl mixing-length theory [16], which assumes 

 that an eddy travels a given distance and then suddenly mixes with the 

 fluid and transfers its heat and momentum, gives a value of a equal to 1 . 

 Although the actual turbulence may be more complicated than indicated 

 by that theory, it does indicate that a value of a on the order of one is 

 not unreasonable. 



As was the case with the shear stress, the variation of heat transfer 

 per unit area with distance from the wall has but a slight effect on the 

 temperature distribution, except for liquid metals [14, Fig. 11]. With 

 q/q^ = a = \, and constant fluid properties, Eq. 2-6 can be integrated 

 numerically for the region close to the wall (e from Eq. 3-2) to obtain a 

 relation between T* and y* [15]. For the region away from the wall 

 iy* > 26), where the molecular shear stress and heat transfer are neg- 

 lected, it is easily shown from Eq. 2-5 and 2-6 that 



T* - T* = u* - wf (3-3) 



where T^ and wf are the values of T* and u* at yf = 26 (Fig. E,4a). 



Calculated temperature distributions are shown in Fig. E,4b on log- 

 log coordinates. The temperature parameter T* is plotted against y* for 

 various Prandtl numbers. The curves indicate that the temperature dis- 

 tributions become flatter over most of the passage cross section as the 

 Prandtl number increases. From Eq. 2-6, dT*/dy* = Pr at or very near 

 the wall, so that the slopes of the curves at the wall increase with Prandtl 

 number. The slopes of the curves in Fig. E,4b near the wall appear 



< 294 ) 



