The Coupling of Conduction With Forced Convection in Graetz Problems

This paper presents an analytical solution of the energy equation for a coupled conduction-forced convection heat transfer problem in ducts. In order to compare the thermal field in plane and circular ducts, the solution is obtained in a form describing both flows. The method is based on an asymptotic expansion of the Laplace transform of the temperature and on an application of the stationary-phase method; it enabled us to write the solution, for any value of the coupling parameter p, in terms of confluent hypergeometric functions. Some simple and accurate expressions of the interface temperature, temperature at the axis, bulk temperature, and Nusselt number are given for small and high values of p. The accuracy of the results is proved by a comparison with those obtained through an expansion in terms of 120 eigenfunctions.