Variational principle for entropy in electrochemical transport phenomena

Pergamon

Int. J. Engng Sci. Vol. 34, No. 5. pp. 549-5611,1996

0020-7225(95)00105-0

Copyright(~) 1996Elsevier Science Ltd Printed in Great Britain. All rights reserved 0020-7225/96 $15.00+ 0.(Xl

V A R I A T I O N A L PRINCIPLE FOR E N T R O P Y IN ELECTROCHEMICAL TRANSPORT PHENOMENA S. S I E N I U T Y C Z Institute of Chemical and Process Engineering, Warsaw Technological University, Warsaw, Poland S. K. R A T K J E Division of Physical Chemistry, Norwegian Institute of Technology, The University of Trondheim, Trondheim, Norway (Communicated by E. S. ~UHUBi) Abstract--Transport of energy, matter and electric charge is analyzed in the framework of a quasilinear variational formalism. A new extremum principle extending that of Onsager to a nonstationary quasilinear regime is applied to electrochemical transport. The principle is set in physical space-time rather than in three-dimensional space and, as such, it substantiates the role of the entropy rather than the entropy production. For the prescribed state and/or fluxes at the system boundary the principle implies a least possible growth of entropy under the constraints imposed by conservation laws, which proves that the entropy plays a role in thermodynamics similar to that of action in mechanics. One can use the principle to derive nontruncated sets of the phenomenological equations, equations of change and bulk overvoltage properties in complex systems. The paper prepares a background to dynamical formulations for electrochemical processes. The nonequilibrium temperatures and chemical potentials are interpreted in terms of the Lagrangian multipliers of the variational principle for the entropy. These quantities converge to the classical thermodynamic intensities when the local equilibrium is attained in the system. Copyright © 1996 Elsevier Science Ltd

1. I N T R O D U C T I O N W e consider c o u p l e d transfer of the heat, mass and electric charge in the bulk of an electrochemical cell or electrolyzer u n d e r the assumption that the system is in mechanical equilibrium. T h e system is a m u l t i c o m p o n e n t one with electrochemical reactions at the electrodes (which we do not consider here), and various transport p h e n o m e n a in the bulk. T h e m a c r o s c o p i c m o t i o n of the system is neglected by the choice of the vanishing barycentric frame and the constancy of the system density, p, consistent with the mechanical equilibrium assumption. This assumption is actually a simplification which allows us to single out the transfer processes of the heat, mass and electricity f r o m the total ( m o r e complex) t h e r m o h y d r o d y n a m i c behavior. U n d e r this simplification the total mass density, p, and the pressure, P, are the constant p a r a m e t e r s rather than the state variables, and a reference frame where the whole system rests can easily be established. Yet, such a simplification is sufficient when a large n u m b e r of cells is e n c o u n t e r e d in practice. Since O n s a g e r ' s syntheses and his original e x t r e m u m principle [1-4] there have been n u m e r o u s attempts to construct variational and e x t r e m u m principles for irreversible processes, aimed, in particular, on nonlinear generalizations of the theory. Prigogine's [5] reformulation has e m p h a s i z e d the role of the e n t r o p y p r o d u c t i o n rather than that of the dissipative potentials. H a s e g a w a [6] has f o u n d an interesting c o n n e c t i o n of these potentials with variational F o k k e r - P l a n c k equations of nonlinear q u a n t u m fluids described by the probability density and action as basic variables. Essex [7] has n o t e d a potential of m i n i m u m e n t r o p y t h e o r e m s to yield the nonlinear balance equations for radiative transfer. Fabrizio et aL [8] has designed associated m i n i m u m principles applicable to viscoelastic systems. L e b o n and D a u b y [9] have f o u n d a nonlinear formulation for c o m p l e x heat transport in dielectric crystals described by e x t e n d e d t h e r m o d y n a m i c s . H o w e v e r , it has b e c o m e clear that nonlinear formulations are quite exceptional and no general strategy has b e e n w o r k e d out which could allow one to 549

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systematically handle nonlinear nonstationary models of irreversible processes. The well known Gyarmati's principle [10-13] and some related approaches [14] while of a considerable generality admitting generalizations to quasi-linear processes, belong to the class of the restricted variational principles of Rosen's type [15] or local potential type [16]. In such formulations some variables and/or derivatives are subjectively "frozen", in order to produce a relevant result. This was not encountered in those older results of minimum entropy dissipation [17] or energy dissipation [18] which, while restricted to linear processes, were at least based on objective concepts. The difficulties have been representative for various partial differential equations of physics [19] where the irreversibility manifests by presence of odd time derivatives. They have been especially strong in situations when the irreversible phenomena accompany the dynamical ones. In this paper, an extremum principle following directly from the entropy balance is formulated for irreversible electrochemical transport. This claim needs substantiation in view of the existence problem, which is always serious when irreversibility enters into the issue. As it is already well recognized, the system of differential equations admits a variational formulation if and only if it is self-adjoint [20-27]. This means that a set of stringent conditions involving the partial derivatives of the related differential operator must be identically specified. It is also well known that the typical equations of irreversible processes are, as a rule, not self-adjoint [24, 27, 28], so that one can doubt the existence of the variational formulation in our case. A relevant substantiation is following: while the equations of irreversible processes do not admit variational formulation in the state space spanned on their own dependent variables (this is the case where the non-self-adjointness applies), an alternative variational formulation in the extended space spanned on these state variables and certain new variables, called state adjoints, is always possible [27]. This gives rise to the so-called composite variational principles involving functionals which are scalar products of the original differential equations and new or adjoint variables [20,27,28]. A common property of such formulations is that the physical conservation laws (for energy, m o m e n t u m and species) are given constraints rather than outcomes of the variational analysis, these laws being of fundamental significance for thermodynamics in any case, reversible or not [29]. Actually a majority of successful recent variational formulations for irreversible continua are of this sort [30-36]. The role of the additional or adjoint variables has also been appreciated in various action-based descriptions of reversible continua [37-41]. There, the conservation laws can be derived as new, independent results. Attempts towards unification of the two sorts of variational descriptions have lead to construction of an irreversible formalism incorporating the second law, energy conservation, and Onsager's dissipative potentials in the framework of a generalized action principle [42]. This approach yields an extended thermodynamics and a fundamental equation for the nonequilibrium energy in sense of Gibbs and Callen [29]. A crucial role of new variables has already been firmly established. Some older variational principles, which didn't introduce new variables, had to operate with an exponential factor explicitly containing time, but they have turned out to be too restrictive [43, 44]. Synthetic information has been summarized in reviews [45], including those on extended thermodynamics [46, 47]. Possible objections to new variational formulations caused by the presence of new or adjoint variables as those which may be, perhaps, "unphysical", are removed early, especially when problems in irreversible thermodynamics are in question. This occurs because, for the standard set of the balance equations (including balances of the energy, momentum and species) and the entropy performance criterion, the new or adjoint variables are simply the nonequilibrium thermodynamic intensities. For a stable local equilibrium they converge to the well known equilibrium intensities: temperature reciprocal, Planck potentials, etc. [48]. Then, all system states become constrained to reside on the Gibbs' equilibrium manifold (where the standard Gibbs fundamental equation holds) and both state and adjoint variables are linked by this equation.

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551

However, at disequilibrium the state and adjoint variables are no longer dependent. They both span an extended space of the (dependent) thermodynamic variables. As seen from an earlier analysis [48] and from equations (26)-(28) of the present paper, any genuine description of disequilibrium (without local equilibrium assumption) must consistently use a differential model comprising both state variables and their thermodynamic adjoints. We show here that such a physical picture is consistent with a generalized variational principle of minimum dissipation which simplifies to that of Onsager in a steady state situation. For a given dissipation intensity qb+ qJ expressed as a function of the extended state (C, J) and its derivatives, the entropy function, which is generated by motions along the extremal paths of the variational solution, is, as a rule, a kinetic entropy which may depend on both the classical (static) variables C and fluxes J. This entropy is then a nonequilibrium (extended) entropy. The distinction between the static (Gibbsian) and the kinetic (Onsagerian, evolution related) entropy manifests in such framework in an explicit way. It explains some difficulties in describing disequilibria in terms of the static quantities exclusively. The distinction, and the related difference between the thermodynamic and kinetic transfer potentials (u and w), is not a consequence of the approximate nature of the Onsager's dissipation model, but a general physical property, the one which has a nontrivial significance for the nonequilibrium thermodynamic theory. While we restrict here to the case of an electrochemical system without convection, we feel that development of related principles for more general situations (as, e.g. those involving convection) will help to understand many complex evolutions and, in particular, the origin of thermo-hydrodynamic instabilities. These more complex cases should be a task of further effort.

2. LEAST-ENTROPY GENERATION IN CONTINUOUS SYSTEMS It has been shown for conservative systems [48] that the evolution of an isolated system with gradients in temperature T and concentrations ci follows the path which minimizes the final entropy of the system, under the system constraints which include the conservation laws. In short, the entropy grows, but its growth is the least possible one under the system constraints. The principle emphasizes the role of the entropy itself rather than the entropy production. By implying the least increase of the entropy during a transient relaxation to equilibrium in any isolated system, the principle shows that the entropy plays a role in thermodynamics similar to that of action in mechanics. The main practical value of the principle lies in that it allows for derivation of the equations of change and the phenomenological equations in a system way, under the given dissipation potentials and the well known information contained in the conservation laws. Here the principle is extended to the electrochemical systems. A quasilinear extension is derived under the assumptions of negligible convection and absence of viscosity. The extension is interesting because it shows explicitly the role of the electric work in the energy and entropy formulas. The present paper is complementary to the work [48] in the sense that the previous work has been directed towards the variational formulation of the second law in the conservative wave (hyperbolic) heat and mass transfer. On the other hand the present paper analyzes the different context of the (parabolic) electrochemical systems which are, in fact, nonconservative due to the work of the external electric field. The potential property of this work is what makes the extension of the variational principle possible. The principle, however, ceases its validity whenever the role of the magnetic effects become essential. Other new features are: an unconventional discussion of the local equilibrium as an effect of the reduction of the variational variables and the discovery that the thermodynamic state and thermodynamic adjoints are variables independent of each other off the Gibb's equilibrium manifold. As distinguished from [48], the present work intentionally uses the definition of state in terms of

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S. S 1 E N I U T Y C Z and S. K. R A T K J E

thermostatic intensities rather than densities, to show explicitly that the consequence of the independence mentioned above is the existence of two different temperatures. They may be called the Gibbsian (thermodynamic) and Onsagerian (kinetic) temperatures and they coincide only at the local equilibrium. Analogous effects pertain to the chemical potentials. The present extension uses an effective energy flux Je which includes the transfer of the electric energy. The electric charge conservation yields the divergence property for the electric work which is essential in this respect, d W / d t = i .E = - i • grad d~ = -div(i~). This property is valid when the role of the electromagnetic vector potential A in the electric field E can be neglected.

3. B A L A N C E E Q U A T I O N S AND I R R E V E R S I B L E T H E R M O D Y N A M I C S Under the assumptions of negligible convection, absence of viscosity terms, and electroneutrality of the solution the conservation equations for mass, electric current, and energy are c~Ci

--+ 3t

V" Ji = 0 V-i = 0

Oe v

--+ 3t

(1) (2)

V . J. = - i . V~h.

(3)

Here, ci and e,, are the molar concentrations and internal energy per unit volume, and Ji, J,, and i are respectively the flux densities of independent component fluxes, internal energy flux, and electric current. The components are neutral [49-52], set according to the Gibbs phase rule [53,52]. As clarified by Sundheim [49] this setting leads to the independent thermodynamic fluxes. The description of electrochemical systems in terms of neutral components should be distinguished from the conventional ionic description [17, 54]. The visible similarity of the electrochemical model developed here to the standard model of the simultaneous heat and mass transfer (without electric current) is the consequence of using merely component description instead of the conventional ionic description. This similarity may be misleading in the sense that the specificity of the electrochemical effects may seem to be lost (or at least restricted) in a component-based model. However, this is not the case; the role of the electric current i in the component-based model is quite nontrivial and the model, equations (1)-(28), describes ionic systems as precisely as the conventional ionic model as long as the charge separation effects can be neglected. Therefore, the model is always applicable to the transport in the bulk, where the electroneutrality holds. A formal proof of the equivalence between the two descriptions is available [49]. On the other hand, use of the component description has several important operational advantages that have been appreciated in full only recently. These are: omission of the troublesome thermodynamic properties of ions, use of the pure chemical (not electrochemical) potentials, working with sets of independent thermodynamic fluxes, explicit electric current in place of partial ionic flows, use of operational electric potentials, and an easy, natural description of electro-mechanical couplings [50-52]. Recent thermo-electrochemistry uses the component formalism quite commonly and successfully, in particular to highly nonisothermal cells and electrolysers with fused salts [51, 52]. For i = ~ = 0 all properties of the classical thermal diffusion systems are recovered. The energy equation (3) contains the electric power term - i • V~b and the divergence of the energy flux J, = J,~ + ~ hiJi where Jq is the heat flux. Introducing the effective energy flux J~ = J. + ~ i = Jq + ~ h~J~ + d~i I

(4)

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553

and using the equation of the charge conservatism in equation (3) the energy equation is transformed into the sourceless form with the electric work incorporated: ae~ + V- J, = 0.

(5)

at

Thus, all balance constraints assume the same formal structure of the vanishing space-time divergences and can be written in a common matrix notation [17] aC --+V.J=0. at

(6)

We use this equation in the case when J is the matrix of independent fluxes

J

= (Je,

J,, J2 . . . . . J,,

(7)

I, i) y

(the superscript T means transpose of the matrix) and for the corresponding column vector of densities C, C = (eo, cl, c2 . . . . . c,,_ 1,0) T-

(8)

The nth mass flux J,, has been eliminated by using the condition ~ J/M~ = 0 for i = 1, 2 . . . . . n. the last component of C vanishes because of the electroneutrality. Following Forland et al. [51] the entropy production or can be written in two alternative forms:

asv --+V.J,=tr=J,.VT-'-

J " V(#iT ~ ) - T - l i " V~6

a~t

i=l

= -T-J(Jq" V ln T + ~ Ji" Vrl~i + i. Vch)

(9)

i=l

with J, being the entropy flux . z

z I

~,~,j 1

and t2k = ~,,MkMS l - I~k. In terms of the conserved fluxes (7) the entropy production and the entropy flux take the form o = Je" V T - ' - ~ a~. V ( b t g T - ' ) - i . V(cbT-') i=1 n-

1

= J e . V T - I + ~] J / . V ( ~ i T - 1 ) - i . V ( c h T

~)

(11)

i=1

and

1

These equations contain the vector of the transport potentials u, the matrix of independent

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S. SIENIUTYCZ and S. K. RATKJE

fluxes J, equation (7), and the associated vector of the independent thermodynamic forces X = Vu:

u=(T

~,~,T-',~2T

X - = V u = (VT -1, V ( ~ I T

1. . . . . Fz,, ~r ' , - q b T

1), V(/.~2 T ,) . . . . .

')

~7(/..~,,_1T 1), _ V ( O T - I ) ) T .

(13) (14)

In the matrix notation introduced the conserved currents obey the simple relations J,,=J'u~u'L'Vu

cr = J . V u ~ V u - L- Vu.

(15) (16)

(The arrows indicate merely the corresponding formulas obeyed on the extremal surfaces of the governing functional (21); clearly, the related constitutive equations are not a starting formula but rather an outcome of our variational formulation.) For the steady systems which are one-dimensional in space, the integration of the entropy production formula over the total system volume yields simply crV = J . (u, - uz)A,

(17)

where V is the volume and A the cross sectional area of any one-dimensional system. While the state space of any genuine nonequilibrium problem is spanned on both the classical variables and the fluxes, the local equilibrium (Onsagerian) limit involves the entropy potential depending on the classical variables only. The limiting Gibbs equation for the entropy so = ps of an incompressible system with the mass density p = ~ Mici takes the classical form dso = u . dC

(18)

describing the potential nature of the entropy at the local equilibrium limit and defining the corresponding transport potentials u, equation (13), in terms of this (static) entropy. All relations allow an easy algebra. It may be proved that the analogous matrix equations hold also for the original, dependent fluxes, that include J,,, and the related tilde-free potentials when the condition of constant density is relaxed. This has a role for a generalization of the present problem for the case including convection. However, we omit here a discussion of this issue. The positiveness of the entropy production in equation (11) implies the following matrix phenomenological equation: S = L ( u ) . Vu,

(19)

where, L is the Onsager matrix of the phenomenological coefficients. We admit here that L depends on the thermodynamic state (represented by the vector C). This pertains to the so-called quasilinear description [12] which is more realistic than the standard linear theory. We will show that nonetheless the quasilinear kinetic equations follow from a variational principle of the minimum entropy given conservation laws, equation (6).

4. E N T R O P Y F U N C T I O N A L , E X T R E M U M C O N D I T I O N S AND LINK WITH ONSAGER'S CRITERION

The variational principle, equation (21) below, describes the second law in terms of the two Onsagerian dissipation functions. The first is flux dependent and the second is force

Variation principle for entropy

555

dependent. Use of both functions is necessary for validity of the variational formulation. The conservation laws (6) are built into the entropy functional with the help of the vector of the Lagrangian multipliers w = (wo, wj, w2 . . . . . w,,_~, wn). This vector constitutes the adjoint of the vector differential operator appearing on the left hand side of equation (6). The inclusion of constraints does not change the numerical value of the entropy functional but influences its extremum properties which then correspond to the free (unconstrained) extremum. An important physical meaning of w follows from formal properties of the Lagrangian multipliers, which describe the change of the optimization criterion with change in the value of the constraint. The e x t r e m u m value of w is the vector of the kinetic conjugates of the extensities C, equations (6) and (8), corresponding with the entropy function generated by the functional (21). On the extremal surfaces of the entropy functional (21) the vector w coincides with the transport potential vector (13) in the limiting situation of the local equilibrium Weq = u = ( T - ' , / 2 , T - ' , / 2 2 T - ' . . . . . / 2 , r -I, -~bT ').

(20)

The coincidence does not occur for any extremal nonequilibrium solution and therefore, w and u are generally two distinctive sorts of the field variables in the entropy "functional extremized. As long as no constraint (25) linking w and u is imposed, they constitute two fields independent of each other. In particular, they do not coincide when arbitrary (nonOnsagerian) dissipation potentials are used. They may be interpreted, respectively, as the kinetic (Onsagerian) and thermodynamic (Gibbsian) intensities which coincide in a stable extremal process. Any kinetic intensity is the Lagrangian multiplier of the related conservation law, whereas any Gibbsian intensity is the appropriate partial derivative of the entropy with respect to the adjoint extensity. On extremals w = u meaning that the extremal Lagrangian multipliers coincide with the components of the entropy gradient in the state space of 6",.. In a limiting local equilibrium situation wi converge to the static (equilibrium) intensities (20), otherwise they converge to certain nonequilibrium intensities that are still the partials of an extended entropy although they then depend on both Ci and J~ (extended thermodynamics). To simplify the notation single integral symbols are used to designate multiple integrals requiring the integration with respect to the space and time variables. The governing functional describes the second law between the two fixed times tl and a subsequent t2: S(t2) =

min S ( t l ) +

- J , ( J , u)" dA dt I,

0,

where $,(J, u) is simply the product J . u, in accord with equation (15). The derivation of this functional structure from an error criterion has been omitted here as it is fully analogous to the one given in an earlier work for the conservative thermal processes [48]. Briefly, equation (21) states that S ( t 2 ) = m i n ( S ( f i ) + s p r ° d - S C X C h ) . Here, S pr°d and S c×~h are the production and exchange components of the kinetic entropy whose change is obtained on the extremal surface of the functional ( S ( t l ) + S pr°J - S~XCh),equation (21), in course of its minimization. In order for the functional to have a minimum it is required that the first variation 6 ( S ( t j ) + S pr°dS ~xch) = 0, for all free variations ¢5u, 6J, 6Vu, 6VJ, and 6w including 6 ( S ( t l ) + S vr°d -- S cxch) = 0 for those 6u and 6J which vanish on the system boundary. Therefore the Euler-Lagrange equations of variational calculus, variational derivatives 6 A / 6 q ~ = 0 or

0A ---V. Oq

(22)

VA

0

0A - - - 0 ,

OVq

at O(Oq/~t)

556

S. SIENIUTYCZ and S. K. RATKJE

where A is the integrand of the functional extremized, should hold for arbitrary (preassigned or free) variations of the generalized state q~ = (u, J and w independent of each other) and for arbitrary (fixed or variable) region of the space-time. The four-dimensional space-time integral of the entropy production is the only quantity needed to obtain the partial differential equations governing the process under the fixed boundary conditions, for the prescribed state variables on the boundary surface A. These boundary conditions are then represented by any given functions u(r A, t) and J(r A, t). For the problems when some state variables qJ are free at the system boundaries the boundary conditions should be determined; then, the variation of the surface functional of equation (21) serves as the basis for these boundary conditions. For our model this leads to the phenomenological equation J = L(u) • X in the form of the so-called natural transversality condition describing vanishing of the normal component of the vector J - L • X on the boundary surface A. An experiment with the basic structure (21) is useful. For the constant state and flows at the system boundaries the surface term in equation (21) can be ignored and the entropy functional can be "gauged" by transforming it into an equivalent form yielding the same field equations. This is done by subtracting from equation (21) the space-time divergence OCw/Ot + div(Jw). As is well known from varational calculus, subtracting such divergences does not change the extremum properties of the original functional, in particular the Euler-Lagrange equation (22) still applies. The so-transformed functional shows an explicit correspondence of the present theory to the classical Onsager's criterion [1], equation (24) below. A modified version of the gauged functional (21) reads S+=min

L(u):VuVu+~L(u) "JJ-C(u)

-J'Vw+~b:(w-u)(w-u)

dvdt.

0w

(23)

The surface term has been omitted here, as it does not influence the form of the partial differential equations obtained. To preserve the explicit convergence of the kinetic intensities w to thermodynamic ones u, a term ½b: (w - u)(w - u) has been introduced under the integral of S T. Here, b is the new Lagrangian multiplier field, and the whole term may be interpreted as the entropy production of a relaxation process leading w to u. Note that the introduction of b makes w the state variable rather than a Lagrangian multiplier. The extremum condition of S v with respect to b is w = u, which preserves an explicit convergence of w towards u on a set of constrained paths corresponding to local equilibrium. Other extremum conditions for S rr, equation (23), are the same as for equation (21). Onsager [1] proposed the "quasivariational" principle whereby the phenomenological laws follow from the restricted variation (frozen u) of the expression

i( 1

rain , ~L l : J J - J ' V u

) dV

(24)

with respect to the fluxes J. This yields a -- L • Vu as the only outcome; no information about the thermodynamic densities C or intensities u is obtained. Approaches related to extensions involving the conservation laws and equations of change were pursued later in many research groups (Prigogine [5]; de G r o o t and Mazur [17]; Glansdorff and Prigogine [16]; Gyarmati [11]), and coworkers of these authors (see the Introduction and References). Onsager's functional (24) can be obtained from our functional (23) in the steady state case. In the steady-state situation the derivative Ow/at vanish. Then, when only J is varied and w converges to u on the extremal surfaces, Onsager's functional (24) and the kinetic equation J - - - L , Vu follow from equation (23) as the steady-state formulae. In that sense Onsager's formulation is limited to the steady-state. In the unsteady situation equation (23) yields, as the

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557

Euler-Lagrange equations (22) with respect to the variables u , J and w, a more general outcome. It is a quasilinear set representing (at w = u) the model of the unsteady transfer of heat, mass and electric charge. Indeed, the Euler-Lagrange equation of the functional (23) with respect to h is

u=w,

(25)

whereas those for w, J and u (at w -- u) are respectively 0C(u) + V . J = 0

(26)

Ot

L(u) -~. J = Vw 0W

~(u) . . . .

Ot

V . ( L . Vu),

(27) (28)

where ~ = aC/au is the thermodynamic capacity matrix or the entropy hessian. The extremum conditions for the functional (21) are equations (26)-(28) only. (The conservation laws are, of course, only recovered, the new results are the quasilinear phenomenological equations and the equations of change.)

5. DISCUSSION

Consider the extremum conditions (26)-(28) which hold for both functionals (21) and (23). The first equation is, of course, the set of the conservation laws in the system whereas the second is the quasilinear kinetics with the state dependent phenomenological matrix L(n). The last equation is the Fourier-Kirchhoff type matrix equation of change linking the fields of the temperature, chemical potentials and electrical potential. An equation could also be obtained by eliminating the Lagrangian multiplier w from the set, which is a valid result linking the state u (or C) and the flux J, independent of the balance equation. For the special case of the pure heat transfer problem, one could thus eliminate the temperature variable from the first and the second Fourier's laws of heat conduction (in spite of the fact that such elimination does not add anything new to these t w o classical equations). In the general situation equations (27) and (28) represent the generalization of these classical laws for the simultaneous transfer of heat, mass and electric charge, and the equation obtained by eliminating of w is a sort of kinetic equation in terms of the extended state C (or u) and J. In practice, however, one deals as a rule with the two equations (27) and (28) rather than with their combination as above. Thus, the theory developed deals with the nontruncated field equations and generalizes Onsager's variational approach for the quasilinear, nonstationary electrochemical systems. This quasilinear extension pertains to the situation when the Onsager phenomenological coefficients can depend on the thermodynamic state of the system (T and ci) yet the phenomenological equations remaining linear with respect to the forces (gradients). This, nonetheless, can comprise a quite broad class of systems. The nonlinear relations between the extensive and intensive thermodynamic quantities, which occur at equilibrium, are met in the present extension due to the quasilinearity assumption which allows one to apply in the governing functionals the exact (state dependent) functions L(u) and, especially, ~:(u). When Onsagerian forms of the dissipative potentials are applied, as in equations (21) and (23), the local equilibrium limit can be obtained from equations (26)-(28) even without extra ES 34:5-E

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condition (25). It is observed that equation (28) follows in this local equilibrium limit from equations (26) and (27), i.e. the last equations become dependent. This is well known from the practice of heat transfer equations, for example. However, in any genuine disequilibrium situation, where both dissipation potentials are unequal and/or differ from those of Onsager, the variables w and u cgpseq(c)/oC may differ. The distinction between w and u is explicit in both equations (21) and (23) and the stationarity conditions, equations (26)-(28). Equation (21) differs from equation (23) in respect that the first admits the local equilibrium implicitly, only due to the particular structure of the physical problem, whereas the second forces the local equilibrium in an explicit way. Thus, should one omit the b term in equation (23) and use more complex dissipation functions, the coincidence of w and u would not be secured any longer, and an evolution through a set of disequilibrium states, with w # u, would be admitted. This may refer, in particular, to any system where a growing unstable behavior is observed. Since it is only for the specific form of the dissipative potentials that w coincides with u, we feel that the local equilibrium constraint (25) is unnecessary in more general cases. (Yet, it is a reasonable extra restriction when one relies on use of the Onsager's potentials, which, ultimately, have such a restriction incorporated in their derivation.) Given conservation laws the compatibility with equilibrium can be attained without any extra constraints imposed on the kinetics [48,55]. The possibility of independent w and u in a general case proves that any orthodoxy treating the entropy or any other thermodynamic function as function of the classical state C only is too restrictive. The entropy becomes a function of the extended state (u, J . . . . ) in agreement with Truesdell's equipresence principle [56,57] and extended thermodynamics [46]. This conclusion can be a suitable argument for generalization of the integrand of equation (21) to highly nonequilibrium phenomena, where the classical meaning of the temperature and other intensities is lost, in general when heat transfer or change of local thermodynamic variables occurs at a rate comparable to the internal relaxation of the system. The distinction between w and u variables could, in particular, play a role in explaining the origin and development of thermo-hydrodynamic instabilities by, for example, tracing the growing differences between u and w on unstable trajectories. The formulation given in this work is physically acceptable insofar as it shows that any coupled transfer of heat, mass and electricity operates so that the increase of the generated entropy is a minimum, subject to the conservation constraints defining the class of the varied processes. Our embedding of the process in four-dimensional space-time have resulted in the following benefits: physical insight to the related functionals in terms of the entropy state function rather than entropy production, removal of subjectively "frozen" fields, and the nontruncated nonlinear field equations. For a general open system the minimal final entropy of the system is achieved by minimizing the difference between the total entropy generated within the system and that exchanged by the system, equation (21). For an isolated distributed system the principle implies the least possible increase of the system entropy between any two successive configurations. For steady-state processes the principle implies the least possible entropy output for any input constant in time. In this case it goes over into the well known Onsagerian principle which yields the phenomenological equations by minimizing the difference between the flux-based dissipation function and the bilinear expression J • X for the entropy production. The present principle is integral, yet yielding all differential (local) phenomenology, it is valid for both stationary and transient situations, and it does admit variations of all possible physical fields, for known thermodynamic and transport properties which may be nonlinear. The point crucial for this successful formulation is that the entropy produced has been expressed as the sum of the two dissipation functions and a fourdimensional functional over a region in space-time, has been used. Such a functional replaces succesfully the three-dimensional integrals over space only, considered in earlier works, which could only describe the properties of the entropy production and not the behavior of the entropy state function itself. =

Variation principle for entropy

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The existence of the two distinct temperatures that may be assigned to a nonequilibrium fluid has also been shown in the statistical way, in Karkheck's analysis of the Boltzmann kinetic equation obtained from the maximum entropy formalism [58]. Also the Hamilton's action approach [40-42] (which is principally different from the entropy approach used here) distinguishes sharply between the static and kinetic intensities. Finally, such a distinction appears explicit in extended irreversible thermodynamics [46] and nonequilibrium molecular dynamics [59]. However, at the time of writing, no unifying theory exists which could show the equivalence between the kinetic temperatures in all these cases. This should be a task of further investigations. Acknowledgements--This work was supported by a grant from SINTEF. The first author acknowledges the kind hospitality of the Laboratory of Physical Chemistry, NTH, University of Trondheim, Norway.

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(Received 5 May 1995; accepted 4 July 1995)