Irreversible thermodynamics reciprocity

Although irreversible thermodynamics neatly defines the driving forces behind associated flows, so far it has not told us about the relationship between these two properties. Such relations have been obtained from experiment, and famous empirical laws have been established like those of Fourier for heat conduction, Fick for simple binary material diffusion, and Ohm for electrical conductance. These laws are linear relations between force and associated flow rates that, close to equilibrium, seem to be valid. The heat conductivity, diffusion coefficient, and electrical conductivity, or reciprocal resistance, are well-known proportionality constants and as they have been obtained from experiment, they are called phenomenological coefficients Li /... 

In the absence of gradients of salt concentration and temperature, flows of water and electric current in bentonite clay are coupled through a set of linear phenomenological equations, derived from the theory of irreversible thermodynamics (Katchalsky and Curran, 1967), making use of Onsager s Reciprocal Relations (Groenevelt, 1971) ... 

Formally, it will be even necessary to make corrections already in the starting flux equations. The detailed formulation of linear irreversible thermodynamics also includes coupling terms (cross terms) obeying the Onsager reciprocity relation. They take into account that the flux of a defect k may also depend on the gradient of the electrochemical potential of other defects. This concept has been worked out, in particular, for the case of the ambipolar transport of ions and electrons.230... 

Onsager s reciprocal relations of irreversible thermodynamics [27-30] imply that if temperature gradients give rise to diffusion velocities (thermal diffusion), then concentration gradients must produce a heat flux. This reciprocal cross-transport process, known as the Dufour effect, provides another additive contribution to q. It is conventional to express the concentration gradients in terms of differences in diffusion velocities by using the diffusion equation, after which it is found that the Dufour heat flux is [5]. 

Equation 3.3.7 expresses the Onsager reciprocal relations (ORR), named after Lars Onsager who first established the principles of irreversible thermodynamics (Onsager, 1931). The ORR have been the subject of many journal papers receiving support as well as criticism, the latter from, in particular, Coleman and Truesdell (1960) and Truesdell (1969). We shall assume the validity of the ORR in the development that follows. 

Thermodynamics for non-equilibrium processes is referred to as irreversible thermodynamics. The scientific field of irreversible thermodynamics was established during the early 1900 s. There are three major reasons why irreversible thermodynamics is important for non-equilibrium systems. In the first place special attention is paid to the validity of the classical thermodynamic relations outside equilibrium (i.e., simple systems). In the second place the theory gives a description of the coupled transport processes (i.e., the Onsager reciprocal relations). In the third place the theory quantifies the entropy that is produced during transport. Irreversible thermodynamics can also be used to assess the second law efficiency of how valuable energy resources are exploited. 

It is understood that these relations are derived adopting several relations from irreversible thermodynamics, e.g., the second law of thermodynamics, the Gibbs-Duhem relation, the linear law and the Onsager reciprocal relations [39, 22, 62, 18, 5]. 

In irreversible thermodynamics Onsager reciprocity relations are (usually) postulated which in our context (4.514), (4.515) are... 

We should also mention that the normal solution of the Boltzmann equation discussed here, together with the //-theorem discussed in the previous section, can be used to provide a derivation of the principles of nonequilibrium thermodynamics. For mixtures, one can show that the various diffusion coefficients that occur in the Navier-Stokes equations can be expressed in a form where Onsager reciprocal relations are satisfied. However, both for mixtures and for pure gases the relation between the normal solution and irreversible thermodynamics only holds if one does not go beyond in the -expansion of the distribution function. ... 

In the crack layer theory (75,76) the movement of the crack and surrounding damage is decomposed into elementary movements translation, rotation, isotropic expansion, and distortion. In this way, the damage surrounding the tip can evolve in a general sense. These elementary movements become thermodynamic fluxes. The reciprocal forces contain an active part (energy release rates associated with each movement) Aj, and a resistive part (an energy barrier) Rt. Within the context of classical irreversible thermodynamics, the entropy of the system. Si, can be expressed in terms of a bilinear form of forces and fluxes shown in equation 27. 

In order to identify EPHs of the cell or electrode reactions from the experimental information, there had been two principal approaches of treatments. One was based on the heat balance under the steady state or quasi-stationary conditions [6,11, 31]. This treatment considered all heat effects including the characteristic Peltier heat and the heat dissipation due to polarization or irreversibility of electrode processes such as the so-call heats of transfer of ions and electron, the Joule heat, the heat conductivity and the convection. Another was to apply the irreversible thermodynamics and the Onsager s reciprocal relations [8, 32, 33], on which the heat flux due to temperature gradient, the component fluxes due to concentration gradient and the electric current density due to potential gradient and some active components transfer are simply assumed to be directly proportional to these driving forces. Of course, there also were other methods, for instance, the numerical simulation with a finite element program for the complex heat and mass flow at the heated electrode was also used [34]. 

It has been mentioned above that two methods, the heat balance under the steady state or quasi-stationary conditions, and the irreversible thermodynamics and Onsager s reciprocal relations, had been used to treat the heat effects in the electrochemical reactions. Although these methods can determine EPH of electrode reaction imder some assumption, they are helpless to answer those problems presented in introduction. 

Note that the contribution to the stress from the nematic potential is independent of the deformation rate and is therefore elastic. The coefficients i are known as the Leslie viscosities. (The factors of V2 in the equations do not appear in the original literature because of different definitions of D and 52.) The Onsager reciprocal relations from irreversible thermodynamics require that 2 + 3 = 6 - 5. Conservation of angular momentum must also be satisfied by the director, which takes the form... 

The Maxwell-Stefan dififusivity D, , obeys the Onsager reciprocal relation of irreversible thermodynamics, i.e.,... 

L. Onsager (Yale) discovery of the reciprocity relations bearing his name, which are fundamental for the thermodynamics of irreversible processes. 

There are three different approaches to a thermodynamic theory of continuum that can be distinguished. These approaches differ from each other by the fundamental postulates on which the theory is based. All of them are characterized by the same fundamental requirement that the results should be obtained without having recourse to statistical or kinetic theories. None of these approaches is concerned with the atomic structure of the material. Therefore, they represent a pure phenomenological approach. The principal postulates of the first approach, usually called the classical thermodynamics of irreversible processes, are documented. The principle of local state is assumed to be valid. The equation of entropy balance is assumed to involve a term expressing the entropy production which can be represented as a sum of products of fluxes and forces. This term is zero for a state of equilibrium and positive for an irreversible process. The fluxes are function of forces, not necessarily linear. However, the reciprocity relations concern only coefficients of the linear terms of the series expansions. Using methods of this approach, a thermodynamic description of elastic, rheologic and plastic materials was obtained. 

The usual emphasis on equilibrium thermodynamics is somewhat inappropriate in view of the fact that all chemical and biological processes are rate-dependent and far from equilibrium. The theory of non-equilibrium or irreversible processes is based on Onsager s reciprocity theorem. Formulation of the theory requires the introduction of concepts and parameters related to dynamically variable systems. In particular, parameters that describe a mechanism that drives the process and another parameter that follows the response of the systems. The driving parameter will be referred to as an affinity and the response as a flux. Such quantities may be defined on the premise that all action ceases once equilibrium is established. 

Fortunately, several simplifications can be made (Nye, 1957). Transport phenomena, for example, are processes whereby systems transition from a state of nonequilibrium to a state of equilibrium. Thus, they fall within the realm of irreversible or nonequilibrium thermodynamics. Onsager s theorem, which is central to nonequilibrium thermodynamics, dictates that as a consequence of time-reversible symmetry, the off-diagonal elements of a transport property tensor are symmetrical (i.e., xy = X/,-). This is known as a reciprocal relation. The Norwegian physical chemist Lars Onsager (1903-1976) was awarded the 1968 Nobel Prize in Chemistry for reciprocal relations. Thus, the tensor above can be rewritten as... 

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