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Abstract
Thermodynamics is continuously spreading in the engineering practice, which is especially true for non-equilibrium models in continuum problems. Although there are concepts and approaches beyond the classical knowledge, which are known for decades, their mathematical properties, and consequences of the generalizations are less-known and are still of high interest in current researches. Therefore, we found it essential to collect the most important and still open mathematical questions that are related to different continuum thermodynamic approaches. First, we start with the example of Classical Irreversible Thermodynamics (CIT) in order to provide the basis for the more general and complex frameworks, such as the Non-Equilibrium Thermodynamics with Internal Variables (NET-IV) and Rational Extended Thermodynamics (RET). Here, we aim to present that each approach has its specific problems, such as how the initial and boundary conditions can be formulated, how the coefficients in the partial differential equations are connected to each other, and how it affects the appearance of nonlinearities. We present these properties and comparing the approach of NET-IV and RET to each other from these points of view. In the present work, we restrict ourselves on non-relativistic models.
Highlights
IntroductionThe framework of Classical Irreversible Thermodynamics (CIT) is well-understood for classical phenomena, such as diffusive heat conduction or coupled problems (diffusive heat and moisture transport, Peltier and Seebeck effects, etc.) [5,6,7]
The famous finding of Fourier, widely known as Fourier’s law in which the heat flux qi is proportional with the temperature gradient ∂i T, qi = −λ∂i T, (1)serves as the basis for almost all of the thermal problems in the engineering practice, where λ is known to be the thermal conductivity
Independent of how one generalizes Equation (1), its consequences must be understood before starting to choose and utilize a model in practical problems, beyond trying to model an experimental setting. This stands as a motivation for the present paper: discussing how the thermodynamic modeling process restricts the mathematical properties of the resulting partial differential equations, and how it aids our understanding of a physical phenomenon
Summary
The framework of Classical Irreversible Thermodynamics (CIT) is well-understood for classical phenomena, such as diffusive heat conduction or coupled problems (diffusive heat and moisture transport, Peltier and Seebeck effects, etc.) [5,6,7] Even this classical approach of CIT contains some interesting and less-known possibilities for modeling. We emphasize that the present paper aims to offer the possibility for ‘grab and use’ the generalized models for various situations for a larger community beyond the experts in thermodynamics, saving the time that spent on the derivation procedures Some aspects, such as the definition of boundary and initial conditions, remarkably differ in the generalized models. The comparison of GENERIC with other non-equilibrium thermodynamical frameworks is currently under investigation [26,27]
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