A second degree approximation for the calculation of the transfer function coefficients for heat conduction through walls

Calculation of the conduction transfer function coefficients using a state space representation requires the transient governing differential equation to be discretized in space by the use of finite difference or finite element methods, in order to obtain a set of first order differential equations. The use of FEM to discretize the media gives an additional advantage due to it is possible to use a higher order approximation of the dependent variable, which gives us a better accuracy with less elements. In this paper, the transient heat flow problem is tackled using a quadratic finite element. The variational formulation for the governing differential equation is developed, the Ritz approximation to construct the finite element formulation is used and the approximation functions are presented using a normalized local coordinate system for elements with three equally spaced nodes for the one-dimensional problem. The 2D transient problem is presented using a rectangular 8 node element. Results with 1, 2 and 3 three-node elements are compared with the ASHRAE conduction transfer functions for the 3, 5, 6, 8 and 32 wall groups and a 2D-example is given.