Generalized Solution for Rectangular Three-Dimensional Transient Heat Conduction Problems with Partial Heating

The development of a generalized solution is presented for a three-dimensional transient heat conduction problem in a rectangular parallelepiped. To make the method as general as possible, one face of the body is subjected to a nonhomogeneous boundary condition over part of the surface. The solution accommodates three kinds of boundary conditions: prescribed temperature, prescribed heat flux, and convective. The means of dealing with these conditions involves adjusting the convection coefficient. Large Biot numbers such as 1010 effectively produce a prescribedtemperature boundary condition, and small ones such as 10−10 are used for generating an insulated boundary condition. This paper also presents three different methods to develop the computationally difficult steady-state component of the solution, as separation of variables can be inefficient at the heated surface. The generalized solution proposed here provides thousands of individual solutions with one analytical formulation. Boundary conditions of prescribed temperature, prescribed heat flux, and convection, as well as a semi-infinite boundary condition, can be simulated. Partial heating at the corner of a parallelepiped is featured as an example, which can be extended to any location on the heated surface using superposition, including subtractive superposition.