Generalized Solution for 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 non-homogeneous boundary condition over part of the surface. The solution accommodates three kinds of boundary conditions: prescribed temperature, prescribed heat flux and convective. Also, the possibility of combining prescribed heat flux and convective heating/cooling on the same boundary is addressed. The means of dealing with these conditions involves adjusting the convection coefficient. Large Biot numbers such as 1010 effectively produce a prescribed-temperature 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 (SOV) can be inefficient at the heated surface. The solution method builds upon previous work done in generating analytical solutions in two-dimensional problems with partial heating. But the generalized solution proposed here provides a minimum of 512 individual solutions with one analytical formulation. An indexed numbering system is used in order to organize these individual solutions. Heating along a portion of a non-homogeneous surface is addressed as part of the formulation in order to maximize the generality of the solution. From the basic building block developed here with heating on one face of a parallelepiped, the use of superposition would allow the solution to be expanded into a limitless number of applications.