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.