Methods that predict heat transfer rates in thermally developing flows, important in engineering design, are often compared with the classical Graetz problem. Surprisingly, numerical solutions to this problem generally do not give accurate results in the entrance region. This inaccuracy stems from the existence of a singularity at the tube inlet. By adopting a fundamental approach based upon singular perturbation theory, the heat transfer process in the tube entrance has been analyzed to bring out the asymptotic boundary layer structure of the generalized problem with non-Newtonian flows. Using a standard finite difference method with only 21 radial nodes, results within 0.3% of the exact solution to the Graetz problem (Newtonian limit of generalized power law fluid flows) are obtained. Compared with previous numerical solutions reported in the literature, these results are an order of magnitude improvement in the accuracy with an order of magnitude decrease in the required number of radial nodes. Also, the number of radial nodes does not have to be increased in the present method to maintain this high level of accuracy as the initial singularity is approached. Solutions for power law, non-Newtonian fluid flows are presented, and generalized correlations are given for predicting Nusselt numbers in both the thermal entrance region and fully developed flows with 0 < n ≤ ∞.