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

Abstract

Parallel theoretical and numerical analyses have been conducted for the predic tion of the mean bulk temperatures of hot fluids flowing inside circular tubes. Heat exchange between an internal forced flow and an external, normal forced flow of a cold fluid occurs through the tube wall. The formal mathematical formulation of this physical problem is expressed 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 flows implementing two different mathematical models: a 2-D differential-based model and a 1-D lumped model. The key element for the latter is that streamwise-mean, internal Nusselt numbers and circumferential-mean, external Nusselt numbers are taken from standard correlations equations based on a Dirichlet boundary condition at the tube wall. The combination of these Nusselt numbers leads to the calculation of a mean, equivalent Nusselt number, which serves to regulate the thermal interaction 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 algebraic expression, provides accurate estimates of the mean bulk temperature distribution when compared with the exact ones computed with the mathematical 2-D distributed model and a personal computer.