Thermodynamics rational

In mosaic nonequilibrium thermodynamics formulations not all the flows are dependent on all the fiee-energy differences, mainly because only a subset of catalytic components affects each flow relation. In this respect, the models differ from classical nonequilibrium thermodynamics where all flows are a function of all forces. 

Rational thermodynamics provides a method for deriving the constitutive equations without assuming local equilibrium. In this formulation, absolute temperature and entropy do not have a precise physical interpretation. It is assumed that the system has a memory, and the behavior of the system at a given time is determined by the characteristic parameters of both the present and the past. However, the general expressions for the balance of mass, momentum, and energy are still used. 

Rational thermodynamics is formulated based on the following hypotheses (i) absolute temperature and entropy are not limited to near-equilibrium situations, (ii) it is assumed that systems have memories, their behavior at a given instant of time is determined by the history of the variables, and (iii) the second law of thermodynamics is expressed in mathematical terms by means of the Clausius-Duhem inequality. The balance equations were combined with the Clausius-Duhem inequality by means of arbitrary source terms, or by an approach based on Lagrange multipliers. 

The Clausius-Duhem equation is the fundamental inequality for a single-component system. The selection of the independent constitutive variables depends on the type of system being considered. A process is then described by solving the balance equations with the constitutive relations and the Clausius-Duhem inequality. 

Studies on thermodynamic restrictions on turbulence modeling show that the kinetic energy equation in a turbulent flow is a direct consequence of the first law of thermodynamics, and the turbulent dissipation rate is a thermodynamic internal variable. The principle of entropy generation, expressed in terms of the Clausius-Duhem and the Clausius-Planck inequalities, imposes restrictions on turbulence modeling. On the other hand, the turbulent dissipation rate as a thermodynamic internal variable ensmes that the mean internal dissipation will be positive and the thermodynamic modeling will be meaningful. 

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Truesdell, C. (1969). Thermodynamics of Diffusion. Rational Thermodynamics, Lecture 5. New York McGraw-Hill... 

The foundation of rational thermodynamics is the Clausius-Planck inequality defining the change of entropy between two equilibrium states, 1 and 2... 

In the rational thermodynamics formulations, the above equation becomes... 

C. Truesdell, Rational Thermodynamics, McGraw-Hill, New York, 1969. 

A theory of thermoviscoelasticity that includes the temperature dependence of the relaxation or retardation functions is necessarily nonlinear, and consequently the elastic-viscoelastic correspondence principle is not applicable. Nevertheless, a linear theory of thermoviscoelasticity can be developed in the framework of rational thermodynamics with further constitutive assumptions (Ref. 5, Chap. 3 see also Ref. 10). 

Truesdell, C. 1984. Thermodynamics of diffusion. In Rational thermodynamics, 2" -edition,... 

S. Passman, J. Nunziato, E. Walsh (1984). A theory of multiphase mixtures. In Rational Thermodynamics (Ed. C. Truesdell), pp. 286-325. Springer, Berlin. 

After 1960 two new approaches to the thermodynamics of irreversible processes emerged, rational thermodynamics and extended irreversible thermodynamics [12], The latter formulation was based on similar assumptions to that of Natanson. 

Models and their developments in this book are based on the method of rational thermodynamics which has substantially contributed to the present-day understanding of the bases of thermodynamics. 

Rational thermodynamics develops from critical revision of continuum mechanics [21-23, 48, 50-52], thanks to pioneer work of Coleman and NoU [46] concerning the new interpretation of the entropy inequahty (see also [53-61]). 

At the end of the discussion of rational thermodynamics we stress that in this theory we in fact study mathematical models (in this sense this theory is a part of mathematics) and only after their application in a real situation and with real material we can decide about the limits of their practical validity. Although practical application is out of scope of the theory developed here, it motivates the types of material models studied in this book and offered as various constitutive equations to be selected for particular application. Such applications motivate some concepts or procedures in the theory and also exclude some unusual properties of these models because the real materials are much more complicated to avoid, e.g., instabilities (manifested, e.g., by phase changes), we exclude zero values of some transport coeffl-eients or heat capacities. Such and similar regularity properties we add to constitutive equations and the resulting models we then denote as regular (see (3.232), (3.234), Rems, in Chap. 1, 2, 6, 8, and 9). 

Conversely the results that energy and entropy are the state functions permit us to formulate their constitutive equations in rational thermodynamics. These, together with balances (say of energy (1.5), (2.1), etc.) and entropy inequality (like (1.42),... 

Samohyl, I. Radonalnf termodynamika chemicky reagujfcfch smesf (Rational thermodynamics of chemicaUy reacting mixtures). Academia, Praha (1982)... 

In this chapter we discuss uniform systems, the properties of which change only in time. Similarly as in [1-8] our main aim here is to demonstrate the method of rational thermodynamics and application of its principles in simple material models. In other words, the main aim is pedagogical—to begin with simple issues, demonstrations, and examples. Nevertheless, even this chapter contains practical results which can be applied on many simple real systems. Among others the principal results of classical equilibrium thermodynamics will be obtained and this will be shown also for reacting mixtures and heterogeneous (multiphase) uniform systems. 

Rational thermodynamics proposes the construction of such equations by the use of constitutive principles [1,3] which generalize long experience accumulated during the past with proposals of special constitutive equations (cf. Sect. 1.1). Some of these constitutive principles more resemble rules or recommendations (e.g., principles of determinism or equipresence below) serving rather as a guidance. On the other hand, some constitutive principles (of admissibility below and objectivity in Sect. 3.5) seem to be sufficiently general and therefore in turn all eonstitutive equations should be in accord with them. ... 

To find the final forms of constimtive equations (2.6)-(2.9) we will ultimately use" the constitutive principle of admissibility (called also the dissipation or entropy principle [3, 8, 34, 35]). This principle was proposed by Coleman and Noll [36] and it opens a way to building up rational thermodynamics. To formulate this principle we define the admissible thermodynamic process as any thermodynamic process (2.3)-(2.5) fulfilling (2.1) which is consistent with (admissible by) the proposed constitutive equations (some model from (2.6)-(2.9) such a process is a sequence of states in the sense of Sect. 1.2). 

Summary. In this Section, the principles used in rational thermodynamics to derive constitutive equations modeling the behavior of specific (material) bodies (systems) were described. Four simple general models of behavior of fluids were proposed, (2.6)-(2.9), taking into account most of these principles. The entropy inequality was formulated for uniform systems and modified introducing the free energy, (2.12), to the final— reduced—form (2.13). The basic exposition of rational methodology is thus prepared for the application of the very thermodynamic principle in the following Section. 

Summary. Section 2.4 illustrates the extension of rational thermodynamics methodology on mixtures with chemical reaction(s) using a very simple model of two-component uniform mixture. The composition variable(s) enters the constitutive equations, cf. (2.76)-(2.79). In a uniform mixture, the classical chemical thermodynamics was obtained, i.e., its validity also in nonequilibrium covered by this model was demonstrated, cf. e.g., (2.82), (2.83), (2.85), (2.87), (2.88). Traditional quantities known from the equilibrium chemical thermodynamics may be thus introduced and used out of equilibrium—affinity by (2.89), chemical potential by (2.93) or (2.100), or Gibbs energy by (2.97). Gibbs and Gibbs-Duhem equations also remain valid,... 

Summary. The last Section illustrating the basics of rational thermodynamics shows how phase equilibria can be treated by this methodology. Constitutive equations should be modified to describe the effects of different phases, see (2.107)-(2.109). Traditional condition of phase equilibrium in terms of chemical potentials was derived, (2.116) or (2.129). 

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