Classical Irreversible Thermodynamics Research Articles

Numerical Analysis of Non-Fourier Model-Based Bio-Heat Transfer in the Laser-irradiated Axisymmetric Living Tissue.

The current work is related to the numerical investigation of non-Fourier heat transfer inside the short-pulsed laser-irradiated axisymmetric soft tissue phantom. It utilizes the modified discrete ordinate method to solve the transient radiative transfer equation (TRTE) for determining the intensity field. The laser energy absorbed by the soft tissue phantom behaves like a source in the Fourier/non-Fourier heat conduction model based-bio-heat transfer equation (BHTE), which is solved by employing the finite volume method (FVM) to determine the temperature distribution. Despite the prevalent use of non-Fourier BHTE for this purpose, a second law analysis is considered crucial to detect any potential anomalies. Equilibrium entropy production rates (EPR) are initially computed based on classical irreversible thermodynamics (CIT), which may yield negative values, possibly contravening the second law. Consequently, the EPR based on CIT is adjusted using the extended irreversible thermodynamics (EIT) hypothesis to ensure positivity. After that, the current research findings are compared with the results from the literature, and found good agreement between them. Then, the independent study is performed to select the optimum grid size, control angle size, and time step. A comparative analysis of results between the traditional Fourier and non-Fourier models has been performed. The impact of different parameters on the temperature fields and EPRs, is discussed. The effect of the optical properties of the inhomogeneity on the temperature distribution has been investigated. This study may help to enhance the effectiveness of the laser-based photo-thermal therapy.

Derivations of the stress-strain relations for viscoanelastic media and the heat equation in irreversible thermodynamic with internal variables

Abstract By using a procedure of classical irreversible thermodynamics with internal variable (CIT-IV), some possible interactions among heat conduction and viscous-anelastic flows for rheological media are studied. In particular, we introduce as internal variables a second rank tensor ε α β ( 1 ) \varepsilon _{\alpha \beta }^{(1)} that is contribution to inelastic strain and a vector ξα that influences the thermal transport phenomena and we derive the phenomenological equations for these variables in the anisotropic and isotropic cases. The stress-strain equations, the general flows and the temperature equation in visco-anelastic processes are obtained and when the medium is isotropic, we obtain that the total heat flux J (q) can be split in two parts: a first contribution J (0), governed by Fourier law, and a second contribution J (1), obeying Maxwell-Cattaneo-Vernotte (M-C-V) equation.

Minimum entropy production in inhomogeneous thermoelectric materials

Due to their potential applications in energy production based on waste heat, direct solar radiation or other energy sources, semiconductor materials have for years attracted the attention of theoretical and experimental researchers. The focus has been on improving the performance of thermoelectric devices through several strategies and special interest has been placed on materials with spatially inhomogeneous transport properties. Inhomogeneity can be achieved in various ways, all of them leading, to a greater or lesser extent, to an improvement of the thermoelectric performance. In this paper, general linear heat and electric charge transport processes in inhomogeneous materials are addressed. The guiding idea followed here is that there exists a relationship between inhomogeneity (structuring), minimum entropy production and performance which may be fruitfully exploited for designing more efficient thermoelectric semiconductor devices. We first show that the stationary states of such materials are minimum global entropy production states. This constitutes an extension of the validity of Prigogine’s minimum entropy principle. The heat and charge transport equations obtained within the framework of classical irreversible thermodynamics are solved to find the stationary profiles of temperature and self-consistent electric potential in a one-dimensional model of a silicon–germanium alloy subjected to an external temperature difference. This allows us to assess the effect of the spatial inhomogeneity on the thermoelectric performance. We find that, regardless of the value of the applied temperature difference, the system may efficiently operate in a regime of minimum entropy production and high efficiency.

Classical irreversible thermodynamics versus extended irreversible thermodynamics. The role of the continuity equation

Classical irreversible thermodynamics versus extended irreversible thermodynamics. The role of the continuity equation

Thermodynamic stability analysis of non‐Fourier model‐based bio‐heat transfer in laser‐irradiated cylindrical‐shaped multilayered biological tissue

AbstractThe present research focuses on examining the thermic response of living tissue in the form of a triple‐layered cylindrical structure when subjected to laser light and the compatibility analysis of non‐Fourier heat transfer with thermodynamics second law. The temperature field in the triple‐layered cylindrical living tissue subjected to laser light is determined by numerically solving the transient radiative transfer equation in conjunction with the dual phase lag (DPL) based bioheat equation. Once the temperature field is known, the equilibrium and nonequilibrium entropy production rate (EPR) is calculated based on the hypothesis of classical irreversible thermodynamics and extended irreversible thermodynamics, respectively. The present results are verified against the data from the literature and found a good match between them. A comparative analysis of the Fourier and non‐Fourier models is accomplished. The equilibrium and nonequilibrium EPR values for the Fourier model are found to be positive. While the equilibrium EPR is negative for non‐Fourier heat conduction and does not satisfy the thermodynamics second law, nonequilibrium EPR is always a positive value for Fourier, DPL, and hyperbolic models and satisfies thermodynamics second law. It has been investigated how thermal relaxation times affect the temperature field and EPRs in tissue are subjected to laser light.

A second-order constitutive theory for polyatomic gases: theory and applications

In the classical irreversible thermodynamics (CIT) framework, the Navier–Stokes–Fourier constitutive equations are obtained so as to satisfy the entropy inequality, by and large assuming that the entropy flux is equal to the heat flux over the temperature. This article is focused on the derivation of second-order constitutive equations for polyatomic gases; it takes the basis of CIT, but most importantly, allows up to quadratic nonlinearities in the entropy flux. Mathematical similarities between the proposed model and the classic Stokes–Laplace equations are exploited so as to construct analytic/semi-analytic solutions for the slow rarefied gas flow over different shapes. A set of second-order boundary conditions are formulated such that the model's prediction for the drag force is in excellent agreement with the experimental data over the whole range of Knudsen numbers. We have also computed the normal shock structure in nitrogen for Mach ${Ma} \lesssim 4$ . A very good agreement was observed with the kinetic theory, as well as with the experimental data.

How Flexible Is the Concept of Local Thermodynamic Equilibrium?

It has been demonstrated by using generalized phenomenological irreversible thermodynamic theory (GPITT) that by replacing the conventional composition variables {xk} by the quantum level composition variables {x˜k,j} corresponding to the nonequilibrium population of the quantum states, the resultant description remains well within the local thermodynamic equilibrium (LTE) domain. The next attempt is to replace the quantum level composition variables by their respective macroscopic manifestations as variables. For example, these manifestations are, say, the observance of fluorescence and phosphorescence, existence of physical fluxes, and ability to register various spectra (microwave, IR, UV-VIS, ESR, NMR, etc.). This exercise results in a framework that resembles with the thermodynamics with internal variables (TIV), which too is obtained as a framework within the LTE domain. This TIV-type framework is easily transformed to an extended irreversible thermodynamics (EIT) type framework, which uses physical fluxes as additional variables. The GPITT in EIT version is also obtained well within the LTE domain. Thus, GPITT becomes a complete version of classical irreversible thermodynamics (CIT). It is demonstrated that LTE is much more flexible than what CIT impresses upon. This conclusion is based on the realization that the spatial uniformity for each tiny pocket (cell) of a spatially non-uniform system remains intact while developing GPITT and obviously in its other versions.

ON THE HEAT DISSIPATION FUNCTION FOR MAGNETIC RELAXATION PHENOMENA IN ANISOTROPIC MEDIA

Using the methods of classical irreversible thermodynamics with internal variables, the heat dissipation function for magnetizable ani­sotropic media, in which phenomena of magnetic relaxation occur, is derived. It is assumed that if different types of irreversible microscopic phenomena give rise to magnetic relaxation, it is possible to describe these microscopic phenomena splitting the total specific magnetization in two irreversible parts and introducing one of these partial specific magnetizations as internal variable in the thermodynamic state space. It is seen that, when the theory is linearized, the heat dissipation func­tion is due to the electric conduction, magnetic relaxation, viscous, magnetic irreversible phenomena. This is the case of complex media, where different kinds of molecules have different magnetic susceptibili­ties and relaxation times, present magnetic relaxation phenomena and contribute to the total magnetization. These situations arise in nuclear magnetic resonance in medicine and biology and in other fields of the applied sciences. Also, the heat conduction equation for these media is worked out and the special cases of anisotropic Snoek media and anisotropic De-Groot-Mazur media are treated.

Riemann solvers for phase transition in a compressible sharp-interface method

In this paper, we consider Riemann solvers with phase transition effects based on the Euler–Fourier equation system. One exact and two approximate solutions of the two-phase Riemann problem are obtained by modelling the phase transition process via the theory of classical irreversible thermodynamics. Closure is obtained by appropriate Onsager coefficients for evaporation and condensation. We use the proposed Riemann solvers in a sharp-interface level-set ghost fluid method to couple the individual phases with each other. The proposed sharp-interface method is validated against molecular dynamics data of evaporating Lennard–Jones truncated and shifted fluid. We further study the effects of phase transition on a shock-drop interaction with the novel approximate Riemann solvers.

A multiscale thermodynamic generalization of Maxwell-Stefan diffusion equations and of the dusty gas model

Despite the fact that the theory of mixtures has been part of non-equilibrium thermodynamics and engineering for a long time, it is far from complete. While it is well formulated and tested in the case of mechanical equilibrium (where only diffusion-like processes take place), the question how to properly describe homogeneous mixtures that flow with multiple independent velocities that still possess some inertia (before mechanical equilibrium is reached) is still open. Moreover, the mixtures can have several temperatures before the temperatures relax to a common value. In this paper, we derive a theory of mixtures from Hamiltonian mechanics in interaction with electromagnetic fields. The resulting evolution equations are then reduced to the case with only one momentum (classical irreversible thermodynamics), providing a generalization of the Maxwell-Stefan diffusion equations. Then, we reduce that description to the mechanical equilibrium (no momentum) and derive a non-isothermal variant of the dusty gas model. These reduced equations are solved numerically, and we illustrate the results on efficiency analysis, showing where in a concentration cell efficiency is lost. Finally, the theory of mixtures identifies the temperature difference between constituents as a possible new source of the Soret coefficient. For the sake of clarity, we restrict the presentation to the case of binary mixtures; the generalization is straightforward.

An overview of entropy principle

A brief overview of the development from classical linear irreversible thermodynamics to the modern rational thermodynamics with Coleman–Noll and Müller–Liu procedures is presented, emphasizing the basic assumptions and formulation details. The major arguments concerned are the improvement of physical assumptions and mathematical formulation differences. Extended thermodynamics is also briefly commented.

Stochastic dynamics, large deviation principle, and nonequilibrium thermodynamics.

By examining the deterministic limit of a general ε-dependent generator for Markovian dynamics, which includes the continuous Fokker-Planck equations and discrete chemical master equations as two special cases, the intrinsic connections among mesoscopic stochastic dynamics, deterministic ordinary differential equations or partial differential equations, large deviation rate functions, and macroscopic thermodynamic potentials are established. Our result not only solves the long-lasting question of the origin of the entropy function in classical irreversible thermodynamics, but also reveals an emergent feature that arises automatically during the deterministic limit, through its large deviation rate function, with both time-reversible dynamics equipped with a Hamiltonian function and time-irreversible dynamics equipped with an entropy function.

A non-equilibrium thermodynamic approach to study the behaviour of magnetizable media

In this didactic paper we formulate a linear theory for magnetizable media in the framework of classical irreversible thermodynamics (CIT) and using its general methods we investigate the behaviour of these media and the possibility of cross effects among mgnetic phenomena, heat conduction, electric conduction and mechanical phenomena. In Galilean approximation all the processes occurring inside these materials are governed by two groups of laws: Maxwell equations and the balance equations. The entropy production is analyzed following the classical thermodynamic procedures and the constitutive equations in the linear anisotropic and isotropic case are derived. The obtained results have applications in several fields of applied sciences, like medicine, biology, physics, chemistry and others.

Review of Extended Irreversible Thermodynamics

In non-equilibrium thermodynamics, Extended irreversible thermodynamics (EIT), together with classical irreversible thermodynamics (CIT) and rational thermodynamics (RT) has been among the mainstream ofresearch. Critical analysis of EIT has been discussed in this paper.

Port-Hamiltonian Modeling of Thermofluid Systems and Object-Oriented Implementation With Modelica I: Thermodynamic Part

In this paper, we present the physical foundations and the development of the thermodynamic part of a Modelica library with the fundamental components for modeling thermofluid systems. We have chosen Modelica because it is an object-oriented modeling language that allows an elegant design of the library, with a top-down conception that starts from very general components where we model the thermodynamic properties common to all simple substances and descend by inheritance to model the properties of each particular substance. To model the behavior of each component, we have used: classical thermodynamics to define the equilibrium states, the local equilibrium hypothesis of Classical Irreversible Thermodynamics to model the changes of state, and the port-Hamiltonian approach to obtain the equations of the system dynamics. With this formulation, we implement the thermodynamic behavior of ideal gases (including monatomic gases as a particular case), the 2073 substances defined for the CEA (Chemical Equilibrium with Applications) NASA Glenn computer program, the IAPWS Formulation 1995 for the Thermodynamic Properties of Water Substance for General and Scientific Use, and the Syltherm 800 HTF (Heat Transfer Fluid). We also define graphical symbols for each library component that facilitate modeling complex systems with simple drag-and-drop manipulations, component connection, and parameter selection. These symbols are a slightly modified version of those used in bond graphs to facilitate their reading and the representation of the structure of complex systems. We also show the modeling, simulation, and comparison for accuracy, performance, and scalability of some thermodynamic systems implemented with the Modelica Standard Library (MSL) and the proposed library.

Anomalies of Lévy-based thermal transport from the Lévy-Fokker-Planck equation

<abstract> Lévy-type behaviors are widely involved in anomalous thermal transport, yet generic investigations based on the mathematical descriptions of the confined Lévy flights are still lacking. In the frameworks of classical irreversible thermodynamics and Boltzmann-Gibbs statistical mechanics, the Lévy-Fokker-Planck equation is connected to near-equilibrium thermal transport. In this work, we show that thermal transport dominated by the confined Lévy flights will be paired with an anomaly, namely that the local effective thermal conductivity is nonlocal. It is demonstrated that the near-equilibrium assumption is not unconditionally valid, which relies on several thermodynamic restrictions expressed by the probability density function (PDF). It is illustrated that the Lévy-Fokker-Planck equation based on the Caputo operator will give rise to two signatures of anomalous thermal transport, the power-law size-dependence of the global effective thermal conductivity and nonlinear boundary asymptotics of the stationary temperature profile. These anomalies are interrelated with each other, and their quantitative relations can be considered as criteria for Lévy-based thermal transport. </abstract>

On extended thermodynamics: From classical to the relativistic regime

The recent monumental detection of gravitational waves by LIGO, the subsequent detection by the LIGO/VIRGO observatories of a binary neutron star merger seen in the gravitational wave signal [Formula: see text], the first photo of the event horizon of the supermassive black hole at the center of Andromeda galaxy released by the EHT telescope and the ongoing experiments on Relativistic Heavy Ion Collisions at the BNL and at the CERN, demonstrate that we are witnessing the second golden era of observational relativistic gravity. These new observational breakthroughs, although in the long run would influence our views regarding this Kosmos, in the short run, they suggest that relativistic dissipative fluids (or magnetofluids) and relativistic continuous media play an important role in astrophysical-and also subnuclear-scales. This realization brings into the frontiers of current research theories of irreversible thermodynamics of relativistic continuous media. Motivated by these considerations, we summarize the progress that has been made in the last few decades in the field of nonequilibrium thermodynamics of relativistic continuous media. For coherence and completeness purposes, we begin with a brief description of the balance laws for classical (Newtonian) continuous media and introduce the classical irreversible thermodynamics (CIT) and point out the role of the local-equilibrium postulate within this theory. Tangentially, we touch the program of rational thermodynamics (RT), the Clausius–Duhem inequality, the theory of constitutive relations and the emergence of the entropy principle in the description of continuous media. We discuss at some length, theories of non equilibrium thermodynamics that sprang out of a fundamental paper written by Müller in 1967, with emphasis on the principles of extended irreversible thermodynamics (EIT) and the rational extended irreversible thermodynamics (REIT). Subsequently, after a brief introduction to the equilibrium thermodynamics of relativistic fluids, we discuss the Israel–Stewart transient (or causal) thermodynamics and its main features. Moreover, we introduce the Liu–Müller–Ruggeri theory describing relativistic fluids. We analyze the structure and compare this theory to the class of dissipative relativistic fluid theories of divergent type developed in the late 1990 by Pennisi, Geroch and Lindblom. As far as theories of nonequilibrium thermodynamics of classical media are concerned, it is fair to state that substantial progress has been made and many predictions of the extended theories have been placed under experimental scrutiny. However, at the relativistic level, the situation is different. Although the efforts aiming to the development of a sensible theory (or theories) of nonequilibrium thermodynamics of relativistic fluids (or continuous media) spans less than a half-century, and even though enormous steps in the right direction have been taken, nevertheless as we shall see in this review, still a successful theory of relativistic dissipation is lacking.

The statistical foundation of entropy in extended irreversible thermodynamics

In the theory of extended irreversible thermodynamics (EIT), the flux-dependent entropy function plays a key role; it is a fundamental distinction between EIT and the usual flux-independent entropy function adopted by classical irreversible thermodynamics (CIT). However, its existence, as a prerequisite for EIT, and its statistical origin have never been justified. In this work, by studying the macroscopic limit of an ϵ-dependent Langevin dynamics, which admits a large deviations (LD) principle, we show that the stationary LD rate functions of probability density p ϵ (x, t) and joint probability density actually turn out to be the desired flux-independent entropy function in CIT and flux-dependent entropy function in EIT respectively. The difference of the two entropy functions is determined by the time resolution for Brownian motions times a Lagrangian, the latter arises from the LD Hamilton–Jacobi equation and can be used for constructing conserved Lagrangian/Hamiltonian dynamics.

Open Mathematical Aspects of Continuum Thermodynamics: Hyperbolicity, Boundaries and Nonlinearities

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.

Relationships between rational extended thermodynamics and extended irreversible thermodynamics.

We consider a few conceptual questions on extended thermodynamics, with the aim to contribute to a higher contact between rational extended thermodynamics and extended irreversible thermodynamics. Both theories take a number of fluxes as independent variables, but they differ in the formalism being used to deal with the exploitation of the second principle (rational thermodynamics in the first one and classical irreversible thermodynamics in the second one). Rational extended thermodynamics is more restricted in the range of systems to be analysed, but it is able to obtain a wider number of restrictions and deeper specifications from the second law. By contrast, extended irreversible thermodynamics is more phenomenological, its mathematical formalism is more elementary, but it may deal with a wider diversity of systems although with less detail. Further comparison and dialogue between both branches of extended thermodynamics would be useful for a fuller deployment and deepening of extended thermodynamics. Besides these two approaches, one should also consider the Hamiltonian approach, formalisms with internal variables, and more microscopic approaches, based on kinetic theory or on non-equilibrium ensemble formalisms. This article is part of the theme issue 'Fundamental aspects of nonequilibrium thermodynamics'.