Application of a lumped model to solids with linearly temperature-dependent thermal conductivity

The application of a simple lumped model to unsteady cooling (or heating) processes in solids involving heat convection is limited by the value of the Biot number, Bi. For Bi < 0.1, assuming constant thermal properties, the lumped model approximates the exact solutions with only a small error. In this paper we study the lumped model for a 1-D rectangular solid, when thermal conductivity depends linearly on temperature, a type of dependence very common in metals and alloys at a wide range of working temperatures. From the study, new limits for the Biot are deduced as a function of a sole dimensionless parameter defined from the extreme values of thermal conductivity. The Biot limits depend on the thermal process (heating or cooling) and on the type of temperature dependence-positive or negative. (c) 2005 Elsevier Inc. All rights reserved.