SOME NON-LINEAR RECIPROCAL RELATIONS

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PERIODICA POLYTECHNICA SER CHEM. ENG. VOL. 42. NO. 1. PP. 65-68 (J998)

SOME NON-LINEAR RECIPROCAL RELATIONS

.J6zsef VERH.'\S t Technical University of Budapest

H-1.521 Budapest, Hungary email: Verhas§!phy.bme.hu ReceiYed: March 30, 1997

Abstract

The validity of the anti-symmetric reciprocal relation can be proved for non-linear constitutive equations generally if it holds in the linear approximation. The constitutive equations may be formulated so that the coefficients of the main effects are positive.

Keywords: non-linear reciprocal relation, anti-symmetry.

1. Introduction

A number of practically important applications of non-equilibrium thermodynamics leads to a two-variable problem; with hvo thermodynamic fluxes and two forces [1 6]. Gsually, this is the situation when studying the ma- chines of energy transformation, e.g., heat engines [7 la], electric motors or generators, etc. If the reversible limit case has importance it is very con- venient to choose the independent variables so that the reciprocal relation in ;:he realm of linearity ^ISanti-symmetry;

(1 ) The inequalities

Lll

>

0 and (2)

express the second law and the inequalities turn into equalities in the reversible limit. It is very useful to know that the constitutive equations can be put into the customary form [1. 2, 11-13],

11

=

LllXl

+

L12X2,

12

=

^L2l...\1

+

L22...\2 , (3)

even III the non-linear regime and the relations (1) and (2) for the L coef- ficient&- (depending on the independent variables Xl and ...\2 ) c&.'1 be pre- served. The non-linear generalization of Onsager's reciprocal relations by

)This work was motivated by the EC project CARNET and has been supported by the Hungarian National Scientific Research Fund, OTKA (1949, T-17000) and the EC (Contract No: ERBCIPDCT 940005)

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66 J. VERH.4S

HURLEY - GARROD [14, 15], and VERH . .\S [16] ensures the validity of the inequalities (2) only for an open set around the equilibrium.

Non-linear thermodynamic modelling needs some further support replacing the linear approachability of continuously differentiable functions [17]. When looking for this support the above facts (to be prove1) are helpful.

2. A Lemma

The proof is based on the lemma:

If a continuously differentiable multi-variable function F(x 1) :1'2, ... , xn) is zero if all the independent variables are zero, it can be given in the form

where aI, a2, . .. ,an are continuous functions of Xl, 1:2, ... , x n ·

The functions aI, a2 , ... ,an are not determined uniquely.

The proof of the lemma is based on Lagrange's mean value theorem.

The auxiliary function

satisfies the equalities

G(O)

=

⁰ ^and ^G(l)

=

F(:q, .1'2, ... , Xn)

from which

F(X1, X2, ... , xn)

=

^G(l)

=

^{G(l) -} ^G(O)

=

^dG

I

1 - 0 d~ 0<';<1

of I ^of I ^of I

=

^OX1 0<';<1 Xl

+

^OX2 0<';<1 x2

+ ... +

^OXn0<';<1 xn

follows.

3. The Sketch of the Proof

Assume a t\vo-variable problem for which the reciprocal relation in the approximation is anti-symmetric;

h=1₁(X1 , X2), 12

=

^h(X1 , X2)

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SOME NON·LINE.4.R RECIPROCAL RELATIONS

with

oh

^I

+

⁰¹² ¹ ^-

°

OX2 X_j=X₂=O oX 1 X_j=X₂=O - .

Applying the lemma, we can write

with

h =

A.(Xl, X2)Xl

+

B(Xl, X2)X2,

h =

C(Xl, X2)X 1

+

D(Xl' X 2)X2,

B(O,O)

+

C(O,O)

=

0.

Applying the lemma again, we get

Having eliminated C, the entropy inequality reads

(Ts

=

(A.

+

EX2)Xf

+

(FXl

+

D)X:j ~

° .

If one of the two terms (say the second) is negative the inequality

(A.

+

^EX2)Xf

>

^-(FXl

+

^D)Xi

67

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holds out of equilibrium. It makes possible to choose a function H so that the expression H (X 1, X 2)X

i

xi is between the two sides;

(A.

+

^EX2)xl

>

Hxlxi

>

-(FXl

+

^{D)xi ,}

from which the inequalities

(A.

+

EX2 - Hxi)xl

>

0, (FXl

+

^D

+

^Hxl)xi

> °

follow. Eqs. (3.3) can be cast into the form

h =

^(A.

+

EX2 - HXi)Xl

+

(+B - EX1

+

HXIX2)X2 ,

12

=

^(-B

+

EXl - HXIX2)Xl

+

(D

+

FXl

+

HXl)X2 Introducing the notations

Lll

=

^A.

+

EX2 - HX'#.

L'11

=

^-B

+

EXl - HX1Xz ,

L12

=

^{B - EX}¹

+

H X lX2 , L22 = D

+

^{F Xl}

+

^{H X}

f ,

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we obtain the usual form as given in equations (3) together with the reciprocal relation (1) and the inequalities (2) for sufficiently smooth constitutive equations.

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68 J. VERHAS

References

[1] DE GROOT, S. R. (19.51): Thermodynamics of Irreversible Processes, North-Holland Pub!. Co., Amsterdam.

[2] DE GROOT, S. R. - MAZUR, P. (1962): Non-equilibrium Thermodynamics, North- Holland Pub!. Co., Amsterdam.

[3] GYARMATI, 1. (1970): Non-Equilibrium Thermodynamics, Springer, Berlin.

[4] Jou, D. CASAS-VASQGEZ .J. - LEBON, G. (1993): Extended Irreversible Thermo- dynamics, Springer, New York, Berlin. .

[.5] MEIXNER, J. - REIK, H. G. (1959): Thermodynamik der Irreversiblen Processe Hand- buch der Physik. Vo!. III/2, pp. 413, Springer, Berlin.

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128.

[10] DE Voss, A. (1992): Endoreversible Thermodynamics of Solar Energy Conversion, Oxford University Press. Oxford.

[11] ONSAGER, L. (1931): Reciprocal Relations in Irreversible Processes 1. Phys. Rev., Vo!. 37, pp. 40·5-426.

[12} ONSAGER, L. (1931): Reciprocal Relations in Irreversible Processes II. Phys. Rev., Vo!. 38, pp. 226.5-22/9.

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[l^el] GARROD, C. Hl:RLEY .. J. (1983); Symmetry Relations for the Conductivity Tensor.

Phys. Rev., Vol. A 27. pp. 1-187-1490.

[15] HURLEY,.J. GARROD. C. (1982); Generalization of the Onsager Reciprocity The- orem. Phys. Rev. Left .. \'01. 48. pp. 1.57.5-1.577.

[16J VERf-!.~S .. J. (1983); An Extension of the Governing Principle of Dissipative Pfi:lCE~SSI2S

to :\on-linear Constitutive Equations. Ann. d. Phys., Vol. 7/40, pp. 189-193.

[17J :\1'fFU. B. (191)>':<): A :\on-linear Extension of the Local Form of Gyarmati's Govel'niI1g~

Principle of Dissipath'e Processes. Acta Phys. Hung .. Vol. 6:3, pp. 13-16.

SOME NON-LINEAR RECIPROCAL RELATIONS

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