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An Extension of the Governing Principle of Dissipative Processes to Nonlinear Constitutive Equations.

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Annalen der Physik. 7. Folge, Band 40, Heft 4/5, 1983, S. 189-193
J. A. Barth, Leipzig
An Extension of the Governing Principle of Dissipative
Processes t o Nonlinear Constitutive Equations
By. J. V E R ~ S
Institute of Physics, Technical University, Budapest, Hungary
Abstract. A generalisatipn of Onsager’e reciprocal relations has been proved by phenomenological methods. It is pointed out that Gyarmati’s variational principle is valid in case of nonlinear
constitutive equations, too.
Eine Ausdehnung des Leit-Prinzips der dissipativen Vorgiinge auf nichtlineare
Material-Cileichungen
Inhaltsubersicht. Eine Verallgemeinerung der Onaagerschen Reziprozitiitsbeziehungenwurde
durch phhnomenologieche Methoden bestiitigt. Es wurde gezeigt, daB das Gyamatische Variationsprinzip auch fur den Fall von nichtlinearen Material-Gleiuhungengiiltig ist.
Introduction
Recently LENGYEL
and GYARMATI
[ 1, 21 have published a paper profoundly analysing the problems posed by the inconsistency of some nonlinear theories of thermodynamics with the classical theory of chemical kinetics. The nonlinear theories studied by
them are based on the generalized reciprocal relations (G.R.R.) proposed by GYARMATI
[3] and LI [4- 71
Though neither a theoretical nor an experimental verification is known, a number of
authors contributed to the original theory. With the works of GYARMATI
[8-lo], VERH ~ Ill],
S
EDELEN
[123, PEESNOV
1131 and KELLER[14] we have an elaborated nonlinear
theory of irreversible processes) the fidelity of which depends on that of the generalized
reciprocal relations. In their analysis, LENGYEL
and GYARMATI
have proved that the
generalized reciprocal relations are not consistent with chemical kinetics if chemical
affinities are assumed to be the thermodynamical forces, as it is customary in linear
theory. The principles of chemical kinetics have been well-tested since the works of
Guldberg and Waage, so we are forced to look for a new interpretation of the forces or
for a new generalization of Onsager’s reciprocal relations.
In this paper the latter choice is taken. A generalization of Onsager’s reciprocal
relations is proposed, the validity of which ranges over any constitutive equation having
continuous 2nd derivatives. The nonlinear reciprocal relations are proved with mathematical rigour. A nonlinear generalization of Gyarmati’s variational principle is based
on the new reciprocal relations. The Lagrangian is given in the usual form
9=a8-!P-@
but the Q, and !P functions are not potentials any more.
J. V E R H ~ ~ S
190
1. The Generalized Reciprocal Theorem
Consider the bilinear form of the entropy source strength
(T,
=2
i
XiJi
(1.1)
where J i stands for the i-th independent flux and X i for the i-th thermodynamical
force or, on the contrary, X i stands for a flux and J i for the corresponding force, according to Meixner’s transformation rules [16, 171. We should fix upon the notation that X i
being an independent variable in the constitutive equations is either the i-th flux or
the i-th force. The constitutive equations are chosen so that Onsager’s reciprocal relations should hold in the realm of linear approximation.
According to what has just been said, the constitutive equations read
x,, r,,
J~ = J((x,,
. . ., r,,.. .)
(1.2)
where X I , X,, ... are the independent variables present explicitly in (1.1)- for brevity,
they will be referred to as forces - and I‘,
... stand for any other independent variable if necessary. As
... variables are of little importance in our arguments,
they will not be dealt with from now on. The constitutive equations are supposed t o
have continuous 2nd derivatives with respect to the forces. This requirement leads to
Taylor’s formula with the remainder in Lagrange’s form for the real functions
r,, r,,
r,,
Ji(A) = JJAX,, AX,, . ..)
(1.3)
depending on the real variable A
where [E (0, A). If A = 1, the functions given by the formulae (1.3) or (1.4) reduces t o
the original constitutive functions (1.2). The linear equations are displayed by the form
(1.4) if the remainder is negligible. The coefficients of the linear term are identical with
the generalized conductivities of the linear theory, for which the validity of Onsager’s
reciprocal relations are presumed.
It is stressed here, that Onsager’s reciprocal relations are obtained not from Taylor’s
formula but from the linear theory of irreversible processes.
For convenience, the provisional notation
is introduced. The liik coefficients depend on the forces, as the actual value of [ is
determined by the forces and the structure of the functions (1.3). (If several values of 6
are possible, the most covenient one can be chosen.) Now, the const,itutive equations
read
The definitions (1.6) have the liik quantities satisfying the symmetry relations
I t3k
. . - 1.%kj.
The conductivity coefficients for the nonlinear theory are defined as
Li, = L!k f 2 (lij, f lkii - &)Xi.
j
(1.8)
(1.9)
Extension of the Principle of Dissipative Processes to Nonlinear Equations
191
The generalized reciprocal relations
Lid-= L,i
(1.10)
are the consequences of Onsager’s reciprocal relations (1.5) and the symmetry relations
(1.8). The constitutive equations get
as the last term on the right hand side is zero and the others are present in eq. (1.’7),too.
The outcome of the above arguments is summarizedinourstatement that constitutive
equations can be cast into the form
(1.12)
where even Likcoefficients themselves depend on the forces and they satisfy the generalized reciprocal relations (1.10) if the latter are valid for the linear approximation.
A change in the choice of the independent variables causes no trouble, as it is governed by a transformation being formally the same as that in the linear theory. It
means that Casimir’s reciprocal relations may occur.
It is quite obvious that Onsager-Casimir reciprocal relations
Lik =
Lk.
(1.13)
*
are valid and i t is of no importance what variables the Lik coefficients depend on.
A further important property of the-conductivity coefficients in the linear theory
is that they make a positive quadratic form:
2 L&aiak2 0.
(1.14)
i,k
They do so in the nonlinear theory as well, because the positivity of a quadratic form is
guaranteed by the positive value of the principal minors depending continuously on the
coefficients and which are also continuous functions of the forces. It means that the
quadratic form in question is positive in a realm much larger than that of the linear
theory.
2. The G.P.D.P. for Nonlinear Constitutive Equations
The present generalization of Onsager’s reciprocal relations gives way to the construction of the Lagrangian for the G.P.D.P. in the traditional form
where the Riknumbers are the elements of the reciprocal of the Lik matrix. ( 2RiiLjk=
j
Sik.)As the quadratic forms belonging t o the Lik or Rik coefficients are positive, the Lagrangian should be negative in case the constitutive equations (1.12) are not satisfied.
On the other hand, the Lagrangian is zero if and only if the constitutive equations hold.
The absolute maximum of the Lagrangian is a necessary and sufficient condition for
the X - s and J-s belonging t o a real material process.
Removing the parentheses from (2.1), we get the simple form
9=Ca-!P-@
where o, is given by (1.1) and
1
!P = - 2 LikXiXk
i.k
(2.2)
J. V E R H ~
192
and
The fornis are strikingly similar to those of the linear theory. The only difference is
that the Lik and R , coefficients are not constant but they depend on the forces; consequently, the y function is not a quadratic one and its derivatives with respect to the
forces do not equal the fluxes. Really,
thus the w function ceases being a potential. This circumstance does not touch the
validity and versatility of Gyarmati’s variational principle. The integration of the local
form leads to the Governing Principle of Dissipative Processes :
J(u,- !P - @)dV = maximum
(2.5)
or
JJ (a, - Y - @)dV dt
= maximum.
The G.P.D.P. is not restricted to the field of linear constitutive equations, but it is
almost totally general a8 the requirement on the existence of continuous 2nd derivatives
is not a strong restriction from physical point of view.
References
[l] LENOYEL,S. ; GYARMATI,
I. : Nonlinear Thermodynamical Studies of Homogeneous Chemica
Kinetic System. Period. Polytech. Chem. Eng. 25 (1981) 63-99.
[2] LENOYEL,S.; GYARMATI,I.: On the Thermodynamics of Elementary Chemical Reactiom
in Homogeneous Systems. J. Chem. Phys. 75 (1981) 2384-%389.
GYARBUTI,I.: On the Phenomenological Basis of Irreversible Thermodynamics. Period. Politechn. 5 (1961) 219--243; 6 (1961) 321-339.
LI, J. M. C. : Thermodynamics for Nonisothermal Systems. The Classical Formulation. J. Chem.
Phys. 29 (1958) 747-764.
LI, J. M. C.: Persistency, Pseudo-Entropy and Thermokinetic Potential. Phys. Rev. 127 (1962)
1784-1786.
LI, J. M. C.: Stable Steady State and the Thermokinetic Potential. J. Chem. Phys. 37 (1962)
1692-1595.
LI, J. M. C. : Thermodynamics for Nonequilibrium Systems. The Principle of Macroscopic
Separability and the Thermokinetic Potential. J. Appl. Phys. 33 (1962) 616--624.
GYARMATI,
I.: On the Fundamentals of Thermodynamics. Acta Chimica Hung. 30 (1962)
147-206.
GYARMATI,
I. : Non-equilibrium Thermodynamics. Berlin-Heidelberg-New York: Springer 1970.
GYARMATI,
1.: On the “Governing Principle of Dissipative Processes” and its Extension to Nonlinear Problems. Ann. Physik (Leipz.) 23 (1969) 353-378.
V E R H ~ SJ., : The Construction of Dissipation Potentials for Non-linear Problems and the Application of Gyarmati’s Principle to Plastic Flow. Z. Phys. Chem. 249 (1972) 119-122.
EDELEN,D. G. B.: Generalized Onsager Fluxes and Forces: A Nonlinear Phenomenological
Theory. Z. Phys. Chem. (Neue Folge) 80 (1974) 37-53.
PRESNOV,E. V. : Potential Character of the Evolution Criteria in Thermodynamics of Nonequilibrium Processes. (In russian) Ross. J. Phys. Chem. 47/1(1973) 3902-2904.
Extension of the Principle of Dissipative Processes to Nonlinear Equations
193
[14] KELLER,J. U.: A General Antisymmetry Property of the Constitutive Equation of Non-Equilibrium Thermodynamics. Int. J. Eng. Sci. 1 7 (1979) 715-723.
[t5] HURLEY,
H.; GARROD,C.: Generalization of the Onsager Reciprocity Theorem. Phys. Rev.
Letters 48 (1982) 1575-1577.
[ l G ] MEIXXER,
J . : Zur Thermodynamik der irreversiblen Prozesso in Gasen mit chemisch reagierenden, dissoziierenden und anregbaren Komponenten. Ann.Physik (Leipz.)43 (1943) 244- 270.
[t7] MEIXNER,
J. : Consistency of the Onsager-Casimir Reciprocal Relations. Adv. Mol. Relax.
Process 6 (1973) 319-331.
Bei der Redaktion eingegangen am 15. April 1983.
Anschr. d. Verf.: Dr. JOSEFV E R ~ S
Institute of Physics
Technical University
H-1521 Budapest (Hungary)
Budafoki ut. 8
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