Chapter 8, Problem 8.15P

Consider the laminar thermal boundary layer development near the entrance of the tube shown in Figure 8A. When the hydrodynamic boundary layer is thin relative to the tube diameter, the inviscid flow region has a uniform velocity that is approximately equal to the mean velocity. Hence the boundary layer development is similar to what would occur for a flat plate.
(a) Beginning with Equation 7.23, derive an expression for the local Nusseltnumber $N_{uD}$ , as a function of the Prandtl number Prand the inverse Graetz number $G{z}_D^{-1}$ . Plot the expression using the coordinates shown in Figure 8.10afor $\mathrm{Pr}=0.7$ .
(b) Beginning with Equation 7.30, derive an expression for the average Nusseltnumber ${\overline{Nu}}_D$ , as a function of the Prandtl number Pr and the Inverse Graetz number $G{z}_D^{-1}$ . Compare your results with the Nussek number for the combined entrancelength in the limit of small x.