An exact solution of extended Graetz problem with axial heat conduction

Abstract

The convective heat transfer properties of a hydrodynamically, fully developed viscous flow in a circular tube are analyzed without using any simplified assumption such as low Reynolds numbers or Peclet numbers. The pipe is then subjected to a step-change in wall temperature. A straightforward approach using the Fourier transform technique is utilized to obtain analytical expressions for temperature distribution, heat flux, and Nusselt numbers. The effects of axial heat conduction are included in these expressions in both the upstream and the downstream directions for Peclet numbers ranging from 0 to ∞. By first taking the Fourier transform of the temperature field and expanding the coefficients of the transformed temperature in terms of the Peclet number, the energy equation with discontinuous wall temperature and longitudinal heat conduction is transformed into a set of ordinary differential equations. This resulting set of equations is then solved successively. The representative curves illustrating the variations of bulk temperature, heat flux, and Nusselt numbers with pertinent parameters are plotted. The asymptotic Nusselt numbers for small, as well as large Peclet numbers, is obtained as 3.6565, compared to the exact classical value of 3.6568. The significance of each curve is also discussed.