AN ALGEBRAIC-SOLUTION FOR A 2-D PARTIAL-DIFFERENTIAL ENERGY EQUATION WITH A ROBIN BOUNDARY-CONDITION GENERATED FROM THE SIMPLER SOLUTION FOR A DIRICHLET BOUNDARY-CONDITION

Parallel theoretical and numerical analyses have been conducted for th e prediction of the mean bulk temperatures of hot fluids flowing insid e circular tubes. Heat exchange between an internal forced how and an external, normal forced flow of a cold fluid occurs through the tube w all. The formal mathematical formulation of this physical problem is e xpressed in terms of a 2-D, partial differential equation of parabolic type with a Robin boundary condition at the tube wall. The aim of the paper is to critically examine the thermal response of the internal f lows implementing two different mathematical models: a 2-D differentia l-based model and a 1-D lumped model. The key element for the latter i s that streamwise-mean, internal Nusselt numbers and circumferential-m ean, external Nusselt numbers are taken from standard correlations equ ations based on a Dirichlet boundary condition at the tube wall. The c ombination of these Nusselt numbers leads to the calculation of a mean , equivalent Nusselt number, which serves to regulate the thermal inte raction between the two perpendicular, unmixed fluid streams, one hot and the other cold. The computed results consistently demonstrate that the 1-D lumped model, associated with hand calculations of an algebra ic expression, provides accurate estimates of the mean bulk temperatur e distribution when compared with the exact ones computed with the mat hematical 2-D distributed model and a personal computer.