E • CONVECTIVE HEAT TRANSFER AND FRICTION 



Probably the first attempt to predict heat transfer from Eq. 2-5 and 

 2-6, or Eq. 2-3 and 2-4, was made by Reynolds [4, pp. 81-85]. He assumed 

 constant properties, that t/tw = q/q^, that the molecular shear stress and 

 heat transfer terms in the equations are negligible compared with the 

 turbulent terms, and a = I. With these assumptions, Eq. 2-5 can be 

 divided by Eq. 2-6 and integrated to give T* = u*. If the temperatures 

 and velocities are weighted in the same way to calculate bulk temper- 

 atures and velocities, T* = u*, or 



q^ = ^p(^- - ^b)rw ^2-7) 



Eq. 2-7, which relates the heat transfer to the shear stress, is usually 

 called Reynolds analogy. It applies reasonably well to gases, which have 

 Prandtl numbers close to one, but fails for liquids. In fact, Eq. 2-7 follows 

 from Eq. 2-5 and 2-6 if Pr = 1, even if the molecular terms are not 

 neglected. 



A number of refinements of Reynolds' original analysis to make it 

 more general are given by various authors. Prandtl and Taylor [5, pp. 

 110-113] introduced a laminar layer near the wall and obtained better 

 agreement with the data for fluids with Prandtl numbers close to 1. 

 Von Karmdn [6] added a buffer layer between the laminar layer and the 

 turbulent core and thus extended the analysis to somewhat higher Prandtl 

 numbers, i.e. to liquids. Further improvements in the theory are given in 

 [3,7,8,9,10,11,12,18,14,15]. 



It is desirable to obtain relations for the heat transfer which do not 

 contain the shear stress or friction factor, as does Eq. 2-7. This is especi- 

 ally true in the case of variable fluid properties, where the friction factors 

 may be no better known that the heat transfer coefficients. In order to 

 obtain such relations it is necessary to make an assumption for the eddy 

 diffusivity €„, in Eq. 2-5 and 2-6. 



E,3. Expressions for Eddy Diffusivity. Several assumptions to 

 relate the eddy diffusivity in Eq. 2-5 and 2-6 to the mean flow have been 

 made by various investigators [7,11,13,15,16,17,18]. Reasonable assump- 

 tions for the variation of e„ follow. For purposes of analysis, the flow is 

 divided into two portions termed the "region away from the wall" and 

 the "region close to the wall." 



Region away from wall. In the region away from the passage wall, 

 it is assumed that the turbulence at a point is a function mainly of local 

 conditions, that is, of the relative velocities in the vicinity of the point 

 [W, p. 351]. This is probably not a good assumption near the passage 

 center where considerable diffusion of the turbulence occurs [SO]. How- 

 ever, in that region the velocity or temperature gradients are so small 



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