Abstract
A wide variety of multidimensional heat conduction problems of interest involve a first-order dissipation term with either homogeneous or nonhomogeneous boundary conditions. An important class of similar problems of practical interest quite often also occurs in the mass diffusion that involves a diffusion-impeded chemical reaction inside a porous catalyst pellet. Developing an efficient analytical solution strategy for such problems is important in view of the fact that different analytical solutions of the same problem can exhibit widely different convergence behavior depending on the solution methodology. In the present study, a three-dimensional rectangular parallelepiped is taken as a case study. The governing partial differential equation with nonhomogeneous boundary conditions is solved using two different approaches. An analytical solution is first obtained using the standard commonly used decomposition approach. The application of the particular solution approach is also detailed here. The two-dimensional problem is considered first, extending the same analysis later to the three-dimensional problem. Owing to relevance to the engineering design and optimization, expressions for the efficiency or the effectiveness factor are obtained in all cases, and a comparison is carried out. The particular solution approach offers much better convergence behavior than the decomposition approach and than others reported in the literature so much so that the computing time for the same level of accuracy in some cases is found to differ by several orders of magnitude.