3.1.1 Introduction

Chapters 1 and 2 discuss steady-state problems, i.e., problems in which temperature depends on position (e.g, x and y) but does not change with time ( t). This chapter discusses transient conduction problems, where temperature depends on time.

3.1.2 The Lumped Capacitance Assumption

The simplest situation is zero-dimensional (0-D); that is, the temperature does not vary with position but only with time. This approximation is often referred to as the lumped capacitance assumption and it is appropriate for an object that is thin and conductive so that it can be assumed to be at a uniform temperature at any time. The lumped capacitance approximation is similar to the extended surface approximation that was discussed in Section 1.6. The resistance to conduction within the object is neglected as being small relative to the resistance to heat transfer from the surface of the object. Therefore, the lumped capacitance approximation is justified in the same way as the extended surface approximation, by defining an appropriate Biot number. For the case where only convection occurs from the surface of the object, the Biot number is:

where L _cond is the conduction length within the object, h is the average heat transfer coefficient, and k is the conductivity of the material. Note that Eq. (3-1) is the simplest possible Biot number; it is often the case that heat transfer from the surface of the object will be resisted by mechanisms...