Thermal response factors to a 2nd order shaping function for the calculation of the 1D heat conduction in a multi-layered slab

• First implementation of the thermal response factor method using 2nd order holder.
• First implementation which shows the use of thermal response factors with more than one parameter.
• Stimulate new research or modification of the present methods.
• Recalls the attention about how to employ new shaping functions to achieve a energy-conservative, stable and accurate method for heat transfer.
• Shows that it is capable of computing at 1 h step reaching an accuracy equivalent to the 5 min step of the traditional Mitalas’ method.

A popular method to obtain the 1-D conduction heat transfer of multi-layered plane geometries is attributed to Stephenson and Mitalas (1971) and Mitalas (1968). It is applied in different forms known as; CTF (Conduction Transfer Functions) or RT (Response factors). Roughly his idea consists in sampling the temperature at each side of a wall at a certain fixed time step (usually one hour). Between sampling points, due to the lack of information, a linear profile for the evolution of those temperatures is imposed. A triangle shaping function is used to get such a piecewise linear profile. The method although is powerful, has passed through the years without questioning. The paper proposes an improvement by extending the idea of a shaping function. Instead of assuming a linear evolution, a specially designed second order (parabolic) evolution is enforced. Now, two parameters must be determined to define the temperature shape between sampling points at both sides of the wall; on one side, the new values of the temperatures and on the other, their acceleration within the time step. Contrary to Mitalas, now two equations are needed at each side to determine these new shapes; the heat power flux balance at both surfaces at the new sampling point (like in Mitalas’ method) plus the thermal energy balance along the period of time between sampling points. This turns the scheme into an energy conservative one. Finally it is shown that the accuracy obtained by the parabolic case at one hour of sampling rate is similar to the linear case with a sampling rate of five minutes.