Validity of linear phenomenological equations

If a system is not far from global equilibrium, linear phenomenological equations represent the transport and rate processes involving small thermodynamic driving forces. Consider a simple transport process of heat conduction. The rate of entropy production is 

The corresponding linear relation between the heat flow and the thermodynamic force is 

Equation (3.299) is identical to Fourier s law of heat conduction, k = LqJT2. The validity of Eq. (3.299) is the same as the validity of Fourier s law, and the equation is valid when the relative variation of temperature is small within the mean free path distance A in the case of gases 

Since this condition is satisfied for most systems, the linear phenomenological equations are satisfactory approximations for transport processes. 

For an elementary chemical reaction, the local entropy production and the linear phenomenological equation are 

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Within the region of validity of linear phenomenological equations, the theorem of minimal entropy generation at steady state is a general stability criterion. The static head is the natural steady state where the net ATP flow vanishes and a minimum of 1 occurs along the loci of the static-head states... 

In Part One, we have discussed theory and related experiments concerning steady states in the range of validity of linear phenomenological equations. We shall call such states as linear steady states. We come across stable steady states even beyond the domain of validity of linear non-equilibrium thermodynamics (LNT) where the flux equations are non-linear. 

The DSC can only measure a true total enthalpy change for a chemical or physical process when the specimen size and the scanning rate are such that the deviation of the sample from equilibrium remains in the range where the assumption of linear phenomenological equations is valid and when integration is carried out over the total range of temperature where the reaction or process may occur. 

Considerable amount of work has been done on both charged and uncharged membranes [46-53]. We shall discuss some of the recent data in the context of the conclusion of the thermodynamic theory developed in an earlier section. It may be noted that Eqs. (4.14) and (4.15) are used as axioms in the development of the phenomenological theory. The only justification for these is that these are consistent with the theory developed in the subsequent section. A direct test of the validity of the linear phenomenological equation has been performed recently. For pyrex sinter with water as permeant, (7)totd. (J) 4,=o have been measured individually. It is found... 

The linear phenomenological equations are valid within the nonequilibrium thermodynamics postulates made in 3.5.3. The summary of the assumptions are ... 

Data on the electro-osmosis of water through a membrane of cellulose acetate have been analysed in terms of non-equilibrium thermodynamics. The linear phenomenological equations were found to be valid for the system. The efficiencies of energy conversion for both electro-osmosis (electrical energy into mechanical work) and streaming potential (mechanical work into electrical energy) were calculated. 

The theory treating near-equilibrium phenomena is called the linear nonequilibrium thermodynamics. It is based on the local equilibrium assumption in the system and phenomenological equations that linearly relate forces and flows of the processes of interest. Application of classical thermodynamics to nonequilibrium systems is valid for systems not too far from equilibrium. This condition does not prove excessively restrictive as many systems and phenomena can be found within the vicinity of equilibrium. Therefore equations for property changes between equilibrium states, such as the Gibbs relationship, can be utilized to express the entropy generation in nonequilibrium systems in terms of variables that are used in the transport and rate processes. The second law analysis determines the thermodynamic optimality of a physical process by determining the rate of entropy generation due to the irreversible process in the system for a required task. 

The question of a mechanically normal behaviour has an analogous bearing upon the range of validity of the linear phenomenological flow equations of irreversible thermodynamics. Somewhere between a normal fluid and a gel there must exist systems which are still fluid but where an elastic relaxation dissipation process invalidates the main assumptions used in both approaches, or makes them incomplete. In such systems, Pick s laws may no longer be vahd. ... 

Despite the uncertainty as to the validity of (16) it would be of interest to use phenomenological transfer functions to discuss the non-linear behavior of reactors in accordance with (16). In Appendix A we treat also an assumption which in some sense represents the opposite extreme from equation (16). 

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. 

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