Abstract
The paper discusses how the two thermodynamic properties, energy (U) and exergy (E), can be used to solve the problem of quantifying the entropy of non-equilibrium systems. Both energy and exergy are a priori concepts, and their formal dependence on thermodynamic state variables at equilibrium is known. Exploiting the results of a previous study, we first calculate the non-equilibrium exergy En-eq can be calculated for an arbitrary temperature distributions across a macroscopic body with an accuracy that depends only on the available information about the initial distribution: the analytical results confirm that En-eq exponentially relaxes to its equilibrium value. Using the Gyftopoulos-Beretta formalism, a non-equilibrium entropy Sn-eq(x,t) is then derived from En-eq(x,t) and U(x,t). It is finally shown that the non-equilibrium entropy generation between two states is always larger than its equilibrium (herein referred to as “classical”) counterpart. We conclude that every iso-energetic non-equilibrium state corresponds to an infinite set of non-equivalent states that can be ranked in terms of increasing entropy. Therefore, each point of the Gibbs plane corresponds therefore to a set of possible initial distributions: the non-equilibrium entropy is a multi-valued function that depends on the initial mass and energy distribution within the body. Though the concept cannot be directly extended to microscopic systems, it is argued that the present formulation is compatible with a possible reinterpretation of the existing non-equilibrium formulations, namely those of Tsallis and Grmela, and answers at least in part one of the objections set forth by Lieb and Yngvason. A systematic application of this paradigm is very convenient from a theoretical point of view and may be beneficial for meaningful future applications in the fields of nano-engineering and biological sciences.
Highlights
This paper presents a derivation of the entropy evolution in macroscopic systems initially out of equilibrium
Once it is accepted that exergy—under the above mentioned assumptions—is well-defined and measurable for systems not extremely remote from their equilibrium, as long as the energy of a system is known, a non-equilibrium entropy can be derived which reduces in the limit to the standard definition when the initial state of the system is an equilibrium one
The remainder of this paper presents a discussion about the evolution and the properties of the entropy function defined by Equation (17) related to irreversible changes inside the system [31]
Summary
This paper presents a derivation of the entropy evolution in macroscopic systems initially out of equilibrium. The derivation begins by showing that for any given macroscopic system it is possible, under certain very general assumptions, to calculate a “non-equilibrium exergy” by analytically solving the diffusion and the thermodynamic equations. The procedure is based only on physical and classical first-order thermodynamic reasoning (heat and mass diffusion laws), and no new axiom is invoked except for the local equilibrium assumption. It is presented here for a continuum, but an extension to finite ensembles of interacting homogeneous particles is possible.
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