Time dependent heat conduction

Abstract

In Chaps. 9 and 10 we have considered heat conduction problems with only one independent variable. Then, in Chap. 11, we have shown how simple heatflow problems can be solved using shell energy balances. In this Chapter, we discuss more complex cases, where the temperature distribution depends on more than one variable, generally one spatial coordinate and time. In Sect. 12.1 the energy equation for flow systems is derived using an Eulerian approach, showing that at the end we obtain the same result as in Chap. 6, where a Lagrangian approach was adopted. Then, we study heat conduction in a semi-infinite slab, due to either a sudden heating of the wall (Sect. 12.2) or to a heat pulse (Sect. 12.3). Unsteady heat conduction in a finite slab is considered in Sect. 12.4, using the separation of variable approach and obtaining at the end a solution in the form of an eigenvalue expansion. Finally, we consider the steady transport of heat in a pipe, where, as independent variable, time is replaced by the axial coordinate. This problem is studied either using an overall shell balance approach (Sect. 12.5) or solving the full energy equation (Sect. 12.6).