Bilinearization Approach for Diffusion Problems Involving the Calculation of Dissipative Fields

Abstract

A straightforward procedure is developed to determine dissipation fields (e.g., exergy destruction in conduction, pumping power, Joule heating) from the linear boundary conditions applied on their associated linear diffusion problem (e.g., heat conduction, Darcy flow, electrical flow). The mathematical tool proposed in this article takes advantage of the quadratic (nonlinear) behavior that comes with dissipative fields equations. The first objective of this article is to build a mathematical formulation expressing any dissipation field as a combination of fundamental dissipative fields resulting from decomposed boundary conditions, similarly to linear problems where fundamental solutions may be added with the superposition principle. The second objective is to demonstrate the numerical advantage of this formulation when applied to heat transfer and fluid flow optimization problems. Application of the mathematical tool proposed in this article to boundary control problems provided significant computational time reductions.

The authors’ work is supported by the Natural Sciences and Engineering Research Council of Canada (NSERC).