New formulation of local-equilibrium thermodynamics

Nobuo Yoshida

doi:10.1063/1.5108793

Abstract

This study points out that the conventional separation of balance equations for energy in fluids into a flow and a source term is not thermodynamically valid. It corrects this inadequacy and thereby identifies an expression for the nonnegative rate of entropy production that is a sum of products of kinetic and thermodynamic quantities. Here, the kinetic and corresponding thermodynamic quantities are the flow of internal energy and the gradient of the reciprocal temperature; the flow of each component and the negative gradient of its chemical potential divided by the temperature; the rate of each chemical reaction and its affinity divided by the temperature; and the conversion of mechanical energy into internal energy and the reciprocal temperature. One of the deductions from this expression is that the internal energy of a fluid never turns into its mechanical energy on its own, a fact which had heretofore not been actually proved in general. On the other hand, the stability condition of local equilibrium leads to a nonpositive integral over the volume of a system under suitable boundary conditions. The integrand is found to be the same form as the entropy production expression, with each thermodynamic quantity replaced by its time derivative. It thus turns out that the thermodynamic quantities always vary temporally so as to lower the entropy production of the system. It can also be seen that if a particular kinetic and thermodynamic quantity pair alone is considered only locally, then the absolute value of the latter invariably lessens with time.

Highlights

Thermodynamics describes how systems of matter change their physical states
The resulting local-equilibrium thermodynamics has successfully been applied to various phenomena in physics, chemistry, biology, and engineering
To eliminate the surface integrals, we introduce boundary conditions that stipulate either that the temperature and chemical potentials be independent of time on the surface, i.e., the surface be in a stationary state or that the flows of heat and matter through the surface disappear

Summary

INTRODUCTION

Thermodynamics describes how systems of matter change their physical states. It was originally concerned with changes between equilibrium states. It is nowadays extended so as to include nonequilibrium situations Many such treatments are based on the so-called assumption of local equilibrium. It seems that the fundamental aspects of local-equilibrium thermodynamics are not fully appreciated This is partly because balance equations for energy in fluids had hitherto not been treated properly. We present the correct way of separation II is formed by a sum of products of kinetic and thermodynamic quantities, with the former including the corrected flow and source term for the internal energy The implications of this result are discussed. The theory developed there is modified according to this definition in Appendix D

ENTROPY PRODUCTION

ENERGY BALANCE EQUATIONS

RECONSIDERATION OF ENTROPY PRODUCTION

STABILITY CONDITION FOR LOCAL EQUILIBRIUM

COMPARISON WITH PRIOR WORK

CONCLUSIONS