The extended Graetz problem with Dirichlet wall boundary conditions

The extension of the Graetz problem to include axial conduction has been of interest in view of its application to a number of low Peclet number heat or mass transfer situations. Past efforts in dealing with this problem have been plagued with uncertainties arising from expansion in terms of “eigenfunctions” and “eigenvalues” belonging to a nonselfadjoint operator. The uncertainties spring from a lack of basis for the assumptions that no complex eigenvalues exist and that the calculated eigenvectors originate from a complete set. Other methods have been entirely numerical. The present work produces an entirelyanalytical solution to the Graetz problem for the Dirichlet boundary condition based on a selfadjoint formalism resulting from a decomposition of the convective diffusion equation into a pair of first order partial differential equations. Physically, the decomposition views the convective diffusion process as a pair of stipulations on how the temperature (or concentration) and theaxial energy (or mass) flow through a partial tube cross-section vary with radial and axial distances. The solution obtained is simple, and readily computed. To whom correspondence may be addressed