E • CONVECTIVE HEAT TRANSFER AND FRICTION 



on an annulus of fluid of differential length of dx [4-7]. This gives, for a 

 tube, the equation shown: 



q^r^dx = d ^J^^' CpTpu{r^ - y)dy~^ - CpT.d ^j^^' pu{r^ - y)dij~^ (3-10) 



where the differentials of the integrals indicate changes in the x direction. 

 The temperature outside the thermal boundary layer T^ does not vary 

 with X inasmuch as it is assumed that no heat penetrates the region out- 

 side the boundary layer. For uniform heat flux at the wall, and constant 

 properties, Eq. 3-10 can then be integrated to give, in dimensionless form, 



This equation gives x/D as a function of 5*, or dimensionless boundary 

 layer thickness, and of r*, which is a kind of Reynolds number. A similar 

 equation can be obtained for the growth of the flow, or velocity, bound- 

 ary layer, whose thickness in general differs from that of the thermal 

 boundary layer [47]. In order to solve these equations, the relations be- 

 tween u* and y* and T* and y* must be known. These relations were 

 already obtained from Eq. 2-5, 2-6, 3-1, and 3-2, and are plotted in Fig. 

 E,4a and E,4b. Those values of w* and T* are to be used ior y* ^ 5* and 

 y* ^ 5* respectively. Outside the flow and thermal boundary layers, u* 

 and T* are assumed uniform. 



The relations for the Nusselt and Reynolds numbers given by Eq. 

 3-4, 3-5, 3-6, and 3-7 can be used in the entrance region as well as for 

 fully developed flow. It is necessary, of course, that T* = T*{8h) for 

 y* ^ dt and u* = u*{8u) for y* ^ 6*. If the velocity profile is fully 

 developed, 5* = r*. 



Fig. E,4f shows predicted values of local Nusselt number over the 

 fully developed value plotted against x/D for Prandtl numbers of 1, 10, 

 and 100, and for various Reynolds numbers. The curves are for a uniform 

 wall heat flux and a fully developed velocity distribution (5„ = r^). A 

 fully developed velocity distribution at the thermal entrance could be 

 obtained experimentally by placing a long unheated length of tubing 

 ahead of the heated section. 



In general the Nusselt numbers, or heat transfer coefficients, very 

 nearly reach their fully developed values in an entrance length less than 

 10 diameters. These entrance lengths for turbulent heat transfer are much 

 shorter than those for the laminar case, apparently because of the rapid 

 radial diffusion of heat by turbulence. The entrance effect also decreases 

 as Prandtl number increases. This trend is opposite to that for laminar 

 flow [49]. The heat diffuses through the fluid more slowly at the higher 

 Prandtl numbers in the case of laminar flow, so that the thermal bound- 

 ary layer develops more slowly. The same phenomenon also tends to in- 



< 300 ) 



