Pergamon Int. Comm. Heat Mass Transfer, Vol. 27, No. 7, pp. 955-964, 2000 Copyright © 2000 Elsevier Science Ltd Printed in the USA. All rights reserved 0735-1933/00IS-see front matter PII: S0735-1933(00)00175-5 AN UPPER BOUND ESTIMATE FOR A CLASS OF CONDUCTION HEAT TRANSFER PROBLEMS WITH NONLINEAR BOUNDARY CONDITIONS Rogtrio Martins Saldanha da Gama Laborat6rio Nacional de Computaq~o Cientffica Petrtpolis, RJ 25651-070, Brazil (Communicated by J.P. Hartnett and W.J. Minkowycz) ABSTRACT This work is concerned with the previous establishment of an upper bound for the solution of a .large class of steady-state heat transfer problems, represented by a partial differential equation subjected to nonlinear boundary conditions. Within this class we have the conduction/radiation heat transfer problems in gray bodies. It is presented a general upper bound estimate obtained with the aid of a (conveniently "chosen) auxiliary function and of a given (bounded) set. The employed procedure is based on the partial differential inequality div(k grad(T-W))>0 and its features. © z000 ElsevierScience Ltd Introduction Most of the realistic descriptions of energy transfer processes in rigid opaque bodies are characterized by a nonlinear relationship between the normal heat flux and the temperature on the boundary. In other words, these problems are described by a partial differential equation subjected to a nonlinear boundary condition [1,2]. Sometimes, the simulation of a complex nonlinear heat transfer problem is carried out only for verifying if the maximum temperature remains lower than a given bound. This simulation becomes unnecessary if an upper bound estimate for the solution is already available. On the other hand, the previous knowledge of an upper bound is necessary for using some specific solution techniques as, for instance, the one presented in reference [3], where steady-state nonlinear conduction/radiation heat transfer problems are simulated by means of the solution of very simple linear problems. 955