An analytical derivation of source term dependent, 2-D `generalized Poisson conduction shape factorsʹ

This paper presents derivations for new “generalized Poisson conduction shape factors” (GPCSFs), that are more general than the commonly used conduction shape factors since they include the effects of both source terms and arbitrary, spatially variable boundary temperatures. The GPCSFs are derived for a single, circular vessel eccentrically imbedded in a uniformly heated, homogeneous, circular tissue matrix. Two GPCSF formulations are presented, as based on the difference between the average vessel wall temperature and; (1) the average tissue boundary temperature (Stb), or (2) the average tissue matrix temperature (Stm). The results show first, that the presence of a source term significantly affects the GPCSFs for all vessel radii and eccentricities. Second, unlike standard shape factors with uniform boundary temperatures, the GPCSFs are a function of both the magnitude and distribution of the boundary temperatures; however, these effects are significant only for large vessel radii and eccentricities. Finally, when the vessel cools the tissue, the average tissue matrix temperature GPCSFs are much less sensitive to the source term than are the tissue boundary temperature GPCSFs.