On the solution of a 2-D,parabolic, partial differential energy equation subjected to a nonlinear convective boundary condition via a simple solution for a uniform,Dirichlet boundary condition

Separate theoretical and numerical analyses have been conducted for the prediction of the mean bulk-and wall-temperatures of hot fluids flowing inside horizontal tubes. Heat transmission between the internal forced flow and the external free flow of the surrounding fluid occurs through the solid wall of the tube. The mathematical formulation of this problem is expressed in terms of a parabolic, partial differential equation with a temperature-dependent, nonlinear boundary condition of third kind. The aim of the article is to critically examine the thermal response of this kind of in-tube flows utilizing two different mathematical models: (a) a complete 2-D differential model and (b) a largely simplified 1-D lumped model. For the 1-D lumped model, streamwise-mean values for the internal Nusselt numbers and the circumferential-mean values for the external Nusselt numbers have been taken from standard correlations that appear in basic textbooks. The combination of both mean Nusselt numbers leads to the calculation of a mean, equivalent Nusselt number, which serves to regulate the thermal interaction between the perpendicular fluid streams. For the two models tested, the computed results consistently demonstrate that the simplistic 1-D lumped model provides accurate estimates of the mean bulk- and wall-temperatures, when compared with those computed with the rigorous 2-D differential model. The former is associated with hand calculations, whereas the latter inevitably necessitated a finite-difference methodology and a personal computer. © 1995 John Wiley & Sons, Inc.