Solutions for Transient Heat Conduction With Solid Body Motion and Convective Boundary Conditions

The analytical solution for the problem of transient thermal conduction with solid body movement is developed for a parallelepiped with convective boundary conditions. An effective transformation scheme is used to eliminate the flow terms. The solution uses Green’s functions containing convolution-type integrals, which involve integration over a dummy time, referred to as “cotime.” Two types of Green’s functions are used: one for short cotimes comes from the Laplace transform and the other for long cotimes from the method of separation of variables. A primary advantage of this method is that it incorporates internal verification of the numerical results by varying the partition time between the short and long components. In some cases, the long-time solution requires a zeroth term in the summation, which does not occur when solid body motion is not present. The existence of this zeroth term depends on the magnitude of the heat transfer coefficient associated with the convective boundary condition. An example is given for a two-dimensional case involving both prescribed temperature and convective boundary conditions. Comprehensive tables are also provided for the nine possible combinations of boundary conditions in each dimension.