Proper eigenvalue solution for the transient response of multidimensional heat transfer systems

Abstract

AbstractWe present a new procedure for the numerical calculation of the transient response of systems characterized by partial differential equations in several space variables and time. The procedure is based on: (i) spatially discretizing the system, (ii) deriving an equivalent circuit for each volume element, (iii) applying a time Laplace transformation and obtaining an admittance matrix, (iv) writing the expressions for the Laplace transforms of the desired variables according to Cramer's determinant rule, (v) interpreting the determinants as generalized eigenvalue problems, (vi) obtaining the proper eigenvalues of the system (poles and zeros), and (vii) finding the time‐dependent solutions by ‘trivial’ inverse Laplace transformations. Although the procedure proposed applies to many partial differential equations (fluid flow in porous media, conductive heat transfer with convection, pressure tests in oil wells, electrical conduction, etc.), in the paper we illustrate the procedure in detail for a conduction heat transfer system (nine nodes). The procedure presented is much more efficient numerically than the direct numerical integration (in space and time) of the ruling PDE, or than solutions which require ‘standard’ numerical inverse Laplace transformation after matrix inversion.