Construction of Short-Time Solutions in Heat Conduction

The concept of both penetration and deviation times for rectangular coordinates along with the principle of superposition for linear problems, and some adequate physical considerations due to thermal symmetries with respect to the middle plane of a slab to simulate homogeneous boundary conditions of the first and second kinds, allow short-time solutions to be constructed for a one-dimensional finite body from the well-known solution of a semi-infinite body. These solutions can be applied at the level of accuracy desired (one part in 10A, with A = 2, 3, …, 15) with respect to the maximum temperature rise (that always occurs at the active surface and at the time of interest) in place of the exact analytical solution to the problem of interest. This method readily applies to transient problems with boundary conditions of the first and second kinds. In particular, the proposed method is used to find short-time solutions applicable at different times and different locations within the body for four different flat plates cases. They are denoted X11, X12, X21, and X22 in the heat conduction numbering system. Increasing the number of terms of the solution extends its time range of applicability. In fact, applying the method infinite times yields the well-known short-time form of the exact analytical solution coming from the application of Laplace transform (but in a very simple way that does not require the knowledge of Laplace transform). A comparison with the large-time form of the solution is given to establish the partitioning time between short- and large-time solutions and, hence, to enhance the efficiency of the computational analytical solution.