The least action principle for heat conduction and its optimization application

Abstract

The least action principle is used in various disciplines including linear transport processes. However, non-equilibrium thermodynamic analyses of linear transport processes involve the dot product of the thermodynamic flux and the thermodynamic force which must be the entropy production rate in the linear phenomenological law for such processes; thus, Fourier’s heat conduction law cannot be derived based on the variation of the entropy production rate. A generalized linear phenomenological law for various types of linear transport processes, including heat conduction, mass diffusion, electric conduction and fluid flow in porous medium is introduced here where the dot product of a generalized flux and a generalized force is taken as the action to give the generalized linear phenomenological law. For heat conduction, the entransy dissipation rate, which is the dot product of the heat flux and the negative of the temperature gradient, is taken as the action, and the variation of the entransy dissipation rate then leads to Fourier’s heat conduction law with constant thermal conductivity. Hence, the action of heat conduction process is the entransy dissipation rate rather than the entropy production rate. A nonlinear constitutive relation for the heat conduction with temperature dependent thermal conductivity is then converted to a linear problem by introducing a generalized temperature, which gives a least generalized entransy dissipation principle for nonlinear heat conduction processes. Finally, the least entransy dissipation principle is applied to optimize a one-dimensional heat conduction problem without heat-work conversion as an example where the minimum entropy generation principle is not applicable.