E • CONVECTIVE HEAT TRANSFER AND FRICTION 

 or 



n^uy 



e„ = n''uy{\ - e"^ (3-2) 



where n is an experimental constant. ^ 



Eq. 3-2 becomes, in dimensionless form, 



p '^w 



= n2w*^*(l - e""'^";^"^"*''*) (3-20 



Mw/Pv 



Eq. 3-1 and 3-2 for e„ can be considered as reasonable first approxi- 

 mations. Whether or not these approximations are adequate can, at pres- 

 ent, be determined only by experiment. 



E,4. Analysis for Constant Fluid Properties. Velocity distribution 

 data for flow without heat transfer have been used to evaluate the con- 

 stants K and n in Eq. 3-1 and 3-2. 



Velocity distributions. Eq. 2-3, with Eq. 3-1 or 3-2, was integrated 

 numerically or analytically for constant properties for the regions close to 

 and at a distance from the wall in [15] and [22]. The integration was 

 carried out for both a constant and a linearly varying shear stress (r = 

 at passage center) with similar results for Reynolds numbers > 10,000, 

 so that the effect of variation of shear stress is neglected in most of the 

 following calculations.^ In the region at a distance from the wall the 

 molecular shear stress is neglected because it is small compared with 

 the turbulent shear stress [llf., Fig. 12]. The familiar Karman-Prandtl 

 logarithmic equation is obtained in the region away from the wall : 



M* = - In ^ + M* 



where y* is the lowest value of y* for which the equation applies and 

 uf is the value of u* at yf. In obtaining this equation, one integration 

 constant was set equal to by using the usual condition that du*/dy* = 

 00 for y* = [18]. This assumption can be avoided by including the 

 molecular shear stress and heat transfer in the region away from the wall 

 and evaluating the constant by assuming a continuous velocity derivative 

 at y* [14, Fig. 12]. This assumption gives essentially the same numerical 

 results as that made above. 



^ The quantity in parenthesis in Eq. 3-2, which represents the effect of kinematic 

 viscosity on «„, becomes important only for heat or mass transfer at Prandtl or 

 Schmidt numbers appreciably greater than one. For Prandtl or Schmidt numbers on 

 the order of one or less, or for velocity profiles, «« = n'^uy (the value of n differs 

 from that in Eq. 3-2) is a good approximation for the region close to the wall [13]. 



* The variation of shear stress is neglected only for the purpose of simplifying the 

 calculations, and as shown in [14, Fig. 11], this neglect has little or no effect on the 

 results, even for variable properties. 



( 292 ) 



