A macroscopic derivation of Onsager's relations

Energy Conversion.

Vol. 11, pp. 91-100.

Pergamon Press, 1971.

Printed in Great Britain

A Macroscopic Derivation of Onsager’ Relationst s Rm Bs

EVANS$!j

(Received 3 February 1971)

Introduction Based on observations by Seebeck (1821) and Peltier (1834), it was found by Thomson [I] in 1854 that the flow of electricity through a conductor is not independent of the flow of thermal energy through it. Instead, these two flows behave as though they were coupled. In 1931, Onsager [2] developed relations which predict not only this coupling but also the coupling of the flows of many other quantities. Onsager’s relations were studied extensively [3-91 and in 1961 they were presented by Tribus [lo, 1l] from a different point of view. Tribus derived certain thermostatic analogs of these relations, and he proposed that the actual explanation of coupling phenomena might lie in these analogs. In this paper, it is shown that the Tribus proposal leads to a macroscopic theorem for non-oscillatory processes. The Onsager relations for such processes in isotropic media result as a corollary.

After the manner of Tribus, we may define static coefficients P as follows: & 211=

z&;

ae

an

an

z21r--;

%2=ab;

aa

g22

=

3%’

Inserting these identities into Equations (1) and (2) and dividing these equations by dx, we obtain,11 de _-=&da+& dx dx dn = &

2 2 dx

da 221

db

Y& +

222

TX.

(3) (4)

Equations (3) and (4) are thermostatic relationships. In the manner of Tribus, we may now introduce the following thermo-dynamic analogs: da -k=L11-

dx

da -iz=L21-++22-.

db -tLlZ-&

db

dx

The Tribus Proposal

dx

Let us consider systems whose intensive states are completely determined by their pressure, temperature and composition-such effects as electricity, magnetism, gravity, etc. being neglected for the moment. Let it further be stipulated (for this simplified case) that the concentrations of all components except one is to remain fixed at each point. Let n denote the concentration (i.e. amount per unit volume) of this one component whose concentration is variable. For this simplified case, the concentration n is completely determined by two intensive variables a and b so that we may write n = n(a, b). For example, a could represent the temperature while b could represent the Gibbs chemical potential. In a similar manner, the concentration e of energy (i.e. energy per unit volume) is determined by a and b so that we may write e = e(a, b). The total differentials of the functions e = e(a, b) and n = n(u, b) may now be written:

Here, & and fi denote the rates of flow per unit area of energy and matter respectively in the x direction. The dynamic coefficients Lu, LIZ, L21, L22 are defined by Equations (5) and (6). Tribus proposed that there might be a simple relationship between each dynamic coefficient L and its corresponding static coefficient-!?. For non-oscillatory processes, this may be shown to be the case.

de =$da+;db

Non-Oscillatory Processes Consider a system, one part of which is initially hotter than the other. Let energy flow between the parts until equilibrium is reached. If the flow of energy should oscillate back and forth as the system approaches equilibrium, we will say that this process is oscillatory. Similarly, the process will be called oscillatory if the flow of matter or of any other conserved quantity7 should oscillate back and forth. In addition to the oscillation of conserved quantities, we will also consider the oscillation of potential work. As an

(1)

dn=;da+;db.

I/ Differentials such as de and dn in Equations (1) and (2) are usually considered to represent changes with respect to time. However, from the standpoint of information theory, these differentials may refer equally well to increments of space. Regarding these differentials to occur in one direction x, we may divide equations (1) and (2) by dr to obtain Equations (3) and (4). 7 The conserved quantities considered here are energy, matter (i.e. chemical and nuclear species or components), electric charge and momentum.

t The work in this paper was supported by a grant from the National Science Foundation. 3 Research Associate, Thayer School of Engineering, Dartmouth College, Hanover, New Hampshire. 0 Now at: School of Mechanical Engineering, Georgia Institute of Technology, Atlanta, Georgia 30332. 91

R B. EVANS

92

example of potential work, consider the gross kinetic energy of a moving fluid. This kinetic energy represents a potential of the fluid to do work-say by means of a paddle wheel. In order to illustrate the difference between potential work and energy, we make the following observation : it may sometimes happen that there is a flux of thermal energy in the opposite direction whose magnitude is such that the total energy flow & is zero (i.e. the vector sum of the flows of thermal and gross kinetic energy is zero). This would represent a situation where there is a flow of potential work even though k = 0. It may also be noted that when the fluid comes to rest via friction, the gross kinetic energy disappears (so that potential work is not conserved), even though the total energy of the fluid is conserved. Let the symbol E denote the potential work per unit volume, while i denotes the rate of flow of potential work per unit in the x direction. We may now define a nonoscillatory process as follows :

the receiving part will have to return to the other part of the medium before equilibrium can be attained. Hence the direction of energy flow will have to be reversed at least once. This represents an oscillation of the type which the criteria (7) and (8) prohibit. Before considering the criterion (7), let us first apply the criteria (8), which prohibit the oscillation of conserved quantities. The Effect of the Non-Oscillation

Lll_

da

dx

= - Ly,j db

4?lZdb dx’

_Yp11-=

Wdx >

o

(7)

h -

cWdx >

o

i = 1,2,. . .m

fiz

where Ar represents the rate of flow of the ith conserved quantity per unit area in the x directiont while in the absence of external force fields, ng denotes the concentration of this quantity (i.e. the amount of this quantity per unit volume).$ In the case of the simple systems considered in the previous section, the flows t and li are represented by Al and liz respectively in the criteria (8), so that for nonoscillatory processes,

deldx, -

dnldx , __

dx

o

(10)

Combining equations (9) and (lo), we have under this condition, da db da db Lll~lZ-= L12811--. (11) dx dx dx dx The gradients da/dx and db/dx are left in Equation (11) in order to avoid division by zero (for example Pip12might be zero so that da/dx = 0 if de/dx = 0). Let us make the usual assumption that the flow & is linear-that is, b is taken to be a linear function of the gradients-or in other words, the dynamic coefficients LII and LIZ are taken to be independent of the gradients da/dx and db/dx. Upon this assumption, Equation (11) must remain valid for all values of de/dx. Allowing de/dx to vary, the gradients da/dx and db/dx are arbitrary and may be cancelled from Equation (11). Thus for linear flow in a non-oscillatory process, we always have, Lll912

o

L

(9)

dx

da

Definition

A process involving flows of m conserved quantities will be called ‘non-oscillatory’ only if the following inequalities are always obeyed:

of Conserved Quantities

Let us set de/dx = 0 by adjusting the magnitudes of da/dx and db/dx in Equation (3). From the criterion - [(de/dx)/b] > 0, we see that e must also be zero (for any finite value of & would yield - [(de/dx)/ti] = 0 in violation of - [(de/dx)/&] > 0). Hence the right sides of equations (3) and (5) are both zero under this condition so that,

= Ll2311.

(12)

Let us multiply Equations (3) and (5) by L11 and 211, respectively, to obtain, .

?i

As an illustration of the criteria (7) and (8), consider a process in which de/dx = 0 throughout a medium, but in which there is nevertheless a finite flow of energy 6 so that - [(de/dx)/&] = 0. Since de/dx = 0 at equilibrium, the energy concentration of the receiving part of the medium will rise above its equilibrium value. The excess energy of

h-

de dx

da dx

= Lll~ll-

- 211t = Lll211-

db dx

+ L11-912 -

(13)

db + L&zpll -. dx dx

(14)

da

In view of Equation (12), the right sides of equations (13) and (14) are equal so that the left sides are also equal, and since from Equation (8a) the quantity k/(de/dx) is finite, we thereby have,

t For anisotropicmedia,Equations(7)and (8)mustalso be written for the y and z directions.The conservedquantitiesconsideredhere are energy,matter (i.e.chemicaland nuclearspeciesor components), L11= electriccharge and momentum. $ In the presence of external force fields, ni is defined to be the effective concentration of the ith conserved quantity-where the A similar multiplication of effective concentration is defined to be the actual concentration and 212 respectively yields, minusthe amount by whichthe equilibriumconcentrationis affected by the external force. fields. For example, the concentrations of LIZ = oxygenand nitrogenin the atmospheredecreasewithelevationunder equilibriumconditions,due to the presenceof the gravitationalfield.

’ e

de/dx

211.

Equations (3) and (5) by LIZ

~ e

deJdx

912.

(16)

A Macroscopic Derivation of Onsager’s Relations

Equations (15) and (16) suggest the definition of a total diffusivity D, by,

have from Equation (23) (since dV = 0 for a stationary boundary) : TS = t -

(17) Using this definition, Equations (15) and (16) become,

93

pi.

(24)

Also from Equation (23) with the volume V held constant we may write (letting s = S/V, e = E/V, n = N/V),

Lii = De911

(18)

L12

(19)

Tds=de-pdn. =

De9’12.

The material flow li may be treated in exactly the same manner with respect to Equations (4) and (6) for whence we find, L21=

Dd-5’21

L22 = D&22

(20) (21)

where

D+L.

dnldx

(22)

From Equations (18)-(21), it is seen that for linear flow? in a non-oscillatory process, the total diffusivities De and Dla are independent of the magnitudes of the applied gradients da/dx and db/dx. Thus the effect of the non-oscillation of the conserved quantities, energy and matter, is to yield total diffusivities for energy and matter which are independent of the applied gradients in linear flow. In Appendix 2, this result is generalized to apply to multi-component systems and also to the conserved quantities, momentum and electric charge. Let us now apply the criterion (7), which prohibits the oscillation of potential work. The Effect of the Non-Oscillation of Potential Work In order to implement the criterion (7), we will introduce entropy into our formulation. Let us continue to consider the simple systems for which Equations (1) and (2) apply. For such systems at equilibrium, the Gibbs equation is [IO-131, dE=TdS-PdV+pdN

Since Equation (25) involves only intensive quantities, it remains valid in general, quite independently of whether or not the volume V is constant-a point which is demonstrated in detail in Appendix 1. In order to apply the criterion (7), we must consider the formulation of potential work. In Ref. [14], it is proven (via techniques developed by Tribus,t Cox [15-171 and Jaynes [17, 181) that the only consistent measure of potential work is the quantity ‘exergy’ which has emerged in the recent literature [19,20]. For the simple system being considered here, the exergy function d is given by [14, 19,201, C=E+P,V-ToS--ON.

E s energy ; T = absolute temperature;

S I entropy; P z pressure ; V z volume;

p = Gibbs chemical potential; N = total amount of the one component whose concentration may vary. Let us consider the flow of energy and matter into such a system across a stationary boundary of area A in time dt. Letting S z dS/A dt, i = dE/A dt and ri = dN/A dt, we t The term ‘linear flow’ which was introduced on page 92 implies that the dynamic coefficients L are independent of the magnitudes of the applied gradients.

(26)

The subscript ‘0’denotes the state of maximum entropy of the system plus its given environment. The quantities PO, TO and ~0 are constant with respect to all variations of the system [14, 191, so that differentiation of Equation (26) gives, db=dE+PodV-TodS--odN.

(27)

Let us consider the same flow of energy and matter which was considered in obtaining Equation (24) from Equation (23). Letting i = d&/A dt while recalling that dV = 0, Equation (27) yields for this situation, < = in- Toi - poiz.

(28)

Substitution of Equation (24) into Equation (28) gives the following expression for the flow Z of potential work: TTo ~=------_~+__nn. T

(23)

where

(25)

Top -

Tpo .

T

It may be of interest to note in passing that the ratio To)/T is the familiar Carnot efficiency for the work obtainable from a how of thermal energy-while the ratio (Top - Tpo)/T reduces for isothermal processes where T = TO to the familiar work expression, p = ~0, for the maximum work obtainable from an isothermal flow of matter. Let us now divide Equation (26) by the volume V to obtain (letting E = 8/V),

(T -

E =: e + PO -

TOS -

pan.

(30)

Differentiation of Equation (30) yields, de := de - TO ds - /*o dn.

(31)

We may substitute Equation (25) into Equation (31) and divide the result by dx (in the manner discussed in the

t The development by Tribus is given in reference 35.

R. B. EVANS

94

footnote on p. 91) to obtain, dc dx

T-

_=--

T

TQ de + TOP - TPO dn xi’ dx T

non-oscillatory processes in simple systems are a corollary of Theorem 1. (32)

Substitution of the identities (17) and (22) into Equation (29) yields, T- To D de _ ToP - TPO D d!! ; = _ ___ ____ T n dx’ T ’ dx

(33)

Multiplying Equation (32) by De and adding the result to Equation (33), we obtain, i f De ;

= Top ; Tp0 (D, - Dn) ;.

(34)

If we set da/dx = 0 [by adjusting the magnitudes of de/dx and dn/dx in Equation (32)], then from the criterion (7), E’must also be zero under this condition. Since D, is finite [from Equations (17) and @a)], it follows that the left side of Equation (34) is zero when de/dx = 0. Thus for dr/dx = 0 and dn/dx # 0, Equation (34) reduces to,

??e T

(D,

-

Dn)

r

(36)

In summary, the non-oscillation of both potential work and conserved quantities requires that the total diffusivities for energy and matter be equal. In view of Equation (36), one may condense equations (18)-(21) into the following form : Lrj = D&‘ij

i= j=

1,2 1,2.

The conjugate potentials which enter into Onsager’s relations for the flows &and li of the conserved quantities E and N are given by aS/aE and aS/aN respectively [4, 6,9] [where aS/aE and aS/aN are taken from the fundamental equation of state S = S(E, V, N) which applies to the simple systems being considered here]. Letting p z iW/aE while a 5 aS/aN, it is seen from Equation (23) that,

fk;

Letting - p and - (Y replace the variables a and b respectively in Equations (5) and (6), these two equations become,

0.

In view of Equations (18)-(21) the total diffusivities De and D, are independent of the magnitudes of the gradients de/dx and dnldx (for linear flow in a non-oscillatory process). Thus Equation (35) must remain valid for all values of deldx and dnldx. Since Top - Tpo/T is not zero,? it follows that for linear flow in any non-oscillatory process, DE = D,.

Onsager’s Relations for Non-Oscillatory Processes in Simple Systems

(37)

A =

LZl da + Lza !fkY

dx

dx’

(41)

The relationship LIZ = LZI which often holds with respect to Equations (40) and (41) is the Onsager relation for the simple systems being considered here. If ,512is not equal to Ls1, then Onsager’s relations are said not be to obeyed in the systems under considerati0n.t The thermostatic analog of Equations (40) and (41) results from letting - p and - 01replace the variables a and b respectively in Equations (3) and (4): de - =_$d!? +-41,,* dx dx dx

(42)

- dn - = $Psr ds + Zpzs* dx dx dx

(43)

The derivation of Equation (37) constitutes the proof of the following theorem of non-oscillatory processes : Theorem 1 For linear flow in any non-oscillatory process in a simple system,$ each dynamic coefficient LO is related to its corresponding static coefficient =Ypl5 by Ltj = DeYdj, where De is the total diffusivity for energy. It may now be shown that the Onsager relations for

(44) The nature of the thermostatic coefficients Y in Equations (42) and (43) may be seen via the following form of the Gibbs-Duhem relationship which is derived as Equation (Al. 7) of Appendix 1: - d@p) = e d/I + n dor.

t For certain values of T and p, the ratio Top - Tpo/T may be zero. Should such values of T and p occur, we may always consider the system with respect to an enlarged environment for which the magnitudes of TO and po are altered-thus yielding a non-zero value for Top - Tpo/T. $ For the purpose of this theorem, a simple system is defined as one in which two intensive variables, a and b, are suEicient to determine the intensive state of the system at any given location. In such a system there can be only two flows of conserved quantities-one of these conserved quantities always being the energy.

(45)

t Onsager’s relations are sometimes erroneously presented as though they could hold even if LIZ # LZIwith respect to Equations (40) and (41)-it being erroneously stated that a selection of new ‘fluxes’ and ‘potentials’ for which the new dynamic coefficients LIZ’ and L21’ are equal is a valid example of Onsager’s relations. AS pointed out by Coleman and Truesdell [9], such a choice of ‘fluxes’ and ‘potentials’ violates the conditions of Onsager’s theorem. In any legitimate transformation (such as the transformation to the familiar entropy flux [21,22]) any symmetry or anti-symmetry of the L matrix must be preserved [9].

95

A Macroscopic Derivation of Onsager’s Relations

Since d(/3Bp)= F

d/3 + y

dor,

it is seen that e = - a(,$)/?+ and IZ= - &!~JJ)/&Y. Thus in view of Equations (44) and (45), the static coefficients .z’ in Equations (42) and (43) are given (after Tribus [lo, 111)by,

Here, the symbol ‘V’ represents the familiar gradient operator. In the case of Equations (3) and (4), the symbols ni, ns, ai and us represent e, n, a and b respectively. For m conserved quantities, the matrix [g] is an m by m matrix. Similarly, the general form of Equations (5) and (6) is, (- 2) = [L](Va).

(W

Here, iii denotes the vector flux of the ith conserved quantity. The following general form of Theorem 1 is demonstrated in Appendix 2: Theorem 2 It follows that 21s = Z’s1 in Equations (42) and (43), since C@)/&@ = ~@JJ)/@?&x from elementary calculus: 912

=

921.

(47)

This equality of 21s and -%‘a1is particularly transparent in the information theory approach, since a(&)/%@ and a(lsp)/&% are simply two different expressions for the co-variance (per unit volume) of energy and matter [lo, 111. In view of Theorem 1, the dynamic coefficients Liz and Lsi in Equations (40) and (41) must be equal for linear flow in any non-oscillatory process: Ll2 =

L2l.

For the linear flow of m conserved quantities in any non-oscillatory process in isotropic media, the matrix of dynamic coefficients [L] is related to the matrix of static coefficients [P] by [L] = D&f’], where De is the total diffusivity for energy. In order to show that Onsager’s relations (for nonoscillatory processes with linear flow in isotropic media) are a corollary of Theorem 2, we may write the following generalization of Equations (40) and (41) :

(3 = MY4 where ,=-!!

(48)

It follows that for non-oscillatory processes with linear flow in simple systems, the Onsager relations result as a corollary of Theorem 1.

(51)

T

while /Jr=-1 since Ni z E

Generalization In the preceding sections, it was assumed that N referred to a quantity of matter. We may, however, consider that N refers to either electric charge or transverse momentum, as well as to species of matter. If N is taken to represent the total electric charge, then the corresponding potential p represents the voltage. For this case, Equation (48) represents the familiar Onsager relation for an electrical conductor-from which the well known behavior of thermocouples may be derived [l 1,21-231. On the other hand, if N is taken to represent the linear momentum transverse to the x direction, then the corresponding potential TV represents the transverse velocity. For this case, the total diffusivity Dn represents the kinematic viscosity-as indicated in the following section. The generalization to the consideration of any number of conserved quantities in three dimensional flow in isotropic media is largely an exercise in algebra. This generalization, which is carried out in Appendix 2, contributes nothing new to the basic physics covered in the preceding section. The results of this generalization will be presented here in matrix form. Letting [Y] denote a square matrix of elements Yif while (2) denotes a column vector of elements Zt, the general form of Equations (3) and (4) is, @n) =

PTW

(49)

and ni = e. Onsager’s relations are said to hold if the matrix of dynamic coefficients [L] is symmetric in Equation (51) (i.e. if Ltj = 42). The corresponding generalizations of Equations (42) and (43) are, (- Vn) =

rww

(52)

where (53) As shown in Appendix 2, a more general form of the Gibbs-Duhem relationship [equation (45)] is, (54) Hence the generalization of Equation (47) is, gaj = PjI.

(55)

Thus the matrix of static coefficients [Z’] is symmetric in Equation (52). In view of Theorem 2, the matrix [L] of dynamic coefficients in Equation (51) must also be symmetric. Hence for linear flow in any non-oscillatory process in isotropic media, Onsager’s relations result as a corollary of Theorem 2.

96

R. B. EVANS

Discussion In the preceding sections, it has been demonstrated that a sufficient condition for Onsager’s relations to hold for linear flow in isotropic media is that the flow process be non-oscillatory. While this condition is sufficient, it may not always be necessary. For example, Onsager’s relations are known to hold when two ideal-gas phasest separated by a surface of discontinuity (such as a porous membrane) are allowed to interact. This interaction will be oscillatory since the energy flow may be non-representative (as pointed out by Tribus [ll] p. 601) from whence matter may flow into regions of higher density in violation of the criteria (8). The class of those processes which are non-oscillatory (in compliance with the criteria 7 and S), is rather narrow. For example, in the consideration of viscosity, Theorem 2 requires (as shown in Appendix 3) that the Prandtl number be equal to z(,/z(, (where z(, and c, are the heat capacities at constant pressure and constant volume respectively). For gases such as oxygen and nitrogen, the Prandtl number is on the order of 0.7-which indicates oscillatory transfers of thermal energy and momentum from convection as opposed to pure conduction. It may be noted in passing that perfect obedience of the familiar Reynold analogy also requires that the Prandtl number be z(,/c, (see Ref. [25], for example). As indicated in Appendix 3, it may be expected that many situations where the familiar analogies (such as the Reynolds analogy) between the transfers of thermal energy, mass and momentum [25,26] are obeyed may be relatively non-oscillatory-so that Theorem 2 may be valid to a good approximation in such situations. One process to which Theorem 2 appears to apply is the conduction of electricity and thermal energy in metals. Assuming Theorem 2 to hold for a Fermi-Dirac electron gas, one obtains (as shown in Appendix 3) a Lorenz number of 2.45 x 10-s Vs/“K2-a value which agrees well with experiment as shown by Kittel (Ref. [27], p. 242).

When the conjugate potentials which enter into Onsager’s relations are introduced [equations (38), (39) and (51)], the mtarix [Z’] of static coefficients is symmetric. For linear flow in any non-oscillatory process in isotropic media, this theorem requires that the matrix [L] of dynamic coefficients must also be symmetric. Hence Onsager’s relations for this case result as a corollary of this theorem. While this theorem shows that a non-oscillatory process is a sufficient condition for Onsager’s relations (for linear flow in isotropic media), it is not a necessary conditionas discussed above. The primary importance of this theorem is that it is derived from macroscopic considerations alone-and thus it contains no microscopic assumptions such as ‘microscopic reversibility’. This theorem may be expected to apply to many situations where the familiar analogies among the transfers of mass, momentum and thermal energy (such as the Reynolds analogy) are obeyed. As discussedintheprevious section, it appears to apply to the conduction of electricity and thermal energy in metals. The derivation of this theorem requires the ‘use of the concept of ‘exergy’ introduced on p. 93. The quantity ‘exergy’ has also been used in considerations of sea-water conversion [19,20,28-311. Based on 10 years’ study in this field, the author believes that the exergy and information theory treatments of thermodynamics may contain the key to a complete macroscopic explanation of Onsager’s relations, through an elucidation and clarification of the basic considerations ofquantummechanicsandre1ativity.t While the final results are not yet in, the writer has been favorably impressed by the information theory treatments of irreversible thermodynamics which have been put forward by Jaynes [32], Shannon [33] and Wallis [34]. References

For the linear flow of m conserved quantities in any non-oscillatory process in isotropic media, the matrix of dynamic coefficients [L] is related to the matrix of static coefficients [=Y] by [L] = &[-EP], where De is the total diffusivity for energy.

[l] Sir W. T. Thomson (Lord Kelvin), Collected Papers-I, pp. 232-291. University Press, Cambridge (1882). 121 L. Onsager, Phys. Rev. 37,405-426 (1931); Phys. Reo. 38, Ser. - i,2265-2279 (1931). 131 J. Meixner. Ann. Phvs. 5. 43. 244 (1943). i4] H. B. G. dasimir, Rev. &ok Phys. 17,‘343-350 (1948). IS] H. B. Callen, Phys. Rev. 73, 1349-1358 (1948). i6] S. R. de Groot, Thermodynamics of Irreversible Processes, Chant. I. Sec. 2. vv. 5-9. North-Holland. Amsterdam (1951). [7] J. b. Hirschfelder, C. F. Curtiss and k. B. Bird, ~olec&lar Theory of Gases andliquicis, Chapt. 11, Sec. 2a, pp. 705-708. John Wiley, New York (1954). 181 I. Priaoaine. Introduction to Thermodvnamics of Irreversible s >rocess&-Charles C. Thomas, Illinois (1955). ” [9] B. D. Coleman and C. Truesdell, J. them. Phys. 33, 28-31 (1960). [lo] M. Tribus, J. uppl. Mech. 28, l-8 (1961). [ll] M. Tribus, Thermostatics and Thermodynamics. Van Nostrand, Princeton (1961). [12] J. W. Gibbs, The ColIected Works, Vol. I, pp. 33-371. Yale University Press (1928). [13] K. Denbigh, The Principles of Chemical Equilibrium. Cambridge University Press, London (1954).

t The word ‘phase’ is used here in the sense of Gibbs, who introduced this word to refer to intensive states-each intensive state representina an example of a different phase (Gibbs 1121 P. 96). Gibbs alsosometimes-referred to sets of phases as them&es being phases from whence came such familiar terms as ‘gas phase’, ‘liquid phase’ and ‘phase space’ [24, 111.

t In this regard, it may be of interest to take note of some recent work under preparation by the writer. In this work the exergy concept is used to derive Newtonian mechanics via information theory. Exergy plays the role of the Lagrangian as it appears in Hamilton’s principle in this derivation.

Conclusions It has been shown that the Tribus proposal, which connects dynamic coefficients (conductances) with static coefficients (thermodynamic derivatives), leads to a new macroscopic theorem of thermodynamics. This theorem is as follows : Theorem

A MacroscopicDerivation of Oasager’s Relations [14] R. B. Evans, A Proof that Exergy is the Only Consistent Measure of Potential Work, a Thesis for the degree Doctor of Philosophy, Thayer School of Engineering, Dartmouth College, Hanover (1969). [15] R. T. Cox, Am. J. Phys. 14, l-13 (1946). [16] R. T. Cox, The Algebra of Probable Inference. Johns Hopkins Press, Baltimore (1961). [17] E. T. Janes, Am. J. Phys. 31,66-67 (1963). [18] E. T. Jaynes, Probability Theory in Science and Engineering, Colloquium Lectures in Pure and Applied Science, No. 4, Field Research Laboratory, Socony Mobil Oil Co., Dallas, Texas (1958). [19] R. B. Evans, Basic Relationships Among Entropy, Exergy, Energy and Availability (1963), in Principles of Desalination, Chapt. 2, Appendix A, edited by K. S. Spiegler. Academic Press, New York (1966). [20] R. B. Evans, A Contribution to the Theory of ThermoEconomics, Report No. 62-36, Department of Engineering, University of California (1962). [21] J. Lee and F. Sears, Thermodynamics. Addison-Wesley, Massachusetts (1956). [22] E. Obert, Concepts of Thermodynamics. McGraw-Hill, New York (1960). [23] G. N. Hatsopoulos and J. H. Keenan, J. appl. Mech. 25, 428-432 (1958). [241 _ _ J. W. Gibbs. The Collected Works, Vol. II. Yale University Press (1928). [25] R. E. Tyrebal, Mass-Transfer Operations. McGraw-Hill, New York (1955). [26] W. H. Giedt, Principles of Engineering Heat Transfer. Van Nostrand, Princeton (1957). [27] C. Kittel, Introduction to Solid State Physics, 2nd edition. John Wiley, New York (1960). [28] M. Tribus and R. B. Evans, Therrnoeconomic Design Under Conditions of Variable Price Structure, Proceedings of the First International Symposium on Water Desalination, Vol. 3, pp. 699-716. U.S. Government Printing Oflice, Washington, D.C. (1965). [29] M. Tribus, R. B. Evans and G. Grulich, The Use of Exergy and Thermoeconomics in the Design of Desalination Plants, Office of Saline Water, Research and Development Progress Report, Contract No. 14-01-001-928, U.S. Office of Technical Services (1966). [30] R. S. Silver, Water Supply by Distillation, a paper submitted to the Centre Belge D’etude et de Documentation des Eaux, 13th International Conference, Liege, Belgium (1960). [31] H. C. Simpson and R. S. Silver, Technology of Sea Water Desalination, a report read before an OSW sponsored conference on Sea Water Demineralization at Wood’s Hole, Massachusetts, (1961) Research Report No. 481, G. and J. Weir Ltd., Cathcart, Glasgow, Scotland (1961). [32] E. T. Jaynes and D. Scalapino, Non-Local Transport Theory, a talk presented at the first meeting of the Society for Natural Philosophy, Baltimore, Maryland (1963). [33] P. T. Shannon and M. R. Feinberg, An Analysis of SteadyState Phenomena From the Information Theory Viewpoint, a paper presented at the Thermo-dynamics Session of the A.1.Ch.E. meeting, San Juan, Puerto Rico (1963). [34] G. B. Wallis, Supplementary Notes for E.S. 52, Thayer School of Engineering, Dartmouth College, Hanover (1967). [35] M. Tribus, Rational Descriptions, Decisions and Designs, Pergamon Press, New York.

the familiar Gibbs-Duhem equation simple systems being considered here: VdP=

The well-known integrated form of Equation (23) is [lo-131, (Al. 1)

Differentiation of Equation (Al. 1) yields,

VdP + Ndp.

(A1.2)

Subtraction of Equation (23) from Equation (Al .2) yields

(Al. 3)

(Al .4)

IdP=sdT+ndp.

(Al. 5)

Differentiation of (Al .4) gives, de=Tds-+pdn-dP+sdT+ndp.

(A1.6)

Subtraction of (Al .5) from (Al .6) yields equation (25) -it being noted that the volume V was at no time assumed to be constant during this derivation: Tds = de - p dir.

(25)

We will also make use of the energy form of the GibbsDuhem equation which may be obtained by introducing the identities p = l/T and OL= - p/T into Equation (Al .5) [while using (Al .4) to eliminate s from (Al .5)]: - d@P) = e d/3 + n da.

(Al. 7)

Equation (Al .7) results directly in the information theory approach [IO, 111. Appendix 2 A generalization of Theorem 1

Let us following [Equation quantities

first limit our attention to systems where the generalized form of the Gibbs equation (23)] applies at equilibrium for m conserved : dE=TdS-PdV+

5

PidNt

(A2.1)

i==l

~1 =

-1

Nl = E.

(A2.la) (A2. lb)

As discussed on p. 95, NCmay refer to electric charge and linear momentum as well as to species of matter. For such systems, the intensive state is completely determined by m variables af so that the generalized form of Equations (3) and (4) is, i=

1,2,3,...m

(A2.2)

where nl - e i=

1,2,3,...m

j=l,2,3

,... m.

(A2.2a) (A2.2b)

In the case of Equations (3) and (4), the variables a and b correspond to the variables al and a2 respectively in Equations (A2.2). Similarly, the generalized form of Equations (5) and (6) is, -&=

dE=TdS-PdVfpdNfSdT

Ndp.

e=Ts-P++n

ant .Yi* = -%

A detailed derivation of Equation (25) from Equation (23)

-

SdT+

[IO-131 for the

Division of Equations (Al . 1) and (Al .3) by the volume V yields (letting s IE S/V, e = E/V, n z N/V),

Appendix 1

E=TS-PVfpN.

97

g &,$

i-1

i=

1,2,3,. . .m.

(A2.3)

Let us set all the gradients daf/dx equal to zero except two of them which we will call daz/dx and daj/dx. Applying the criterion (8), we obtain the following

R. B. EVANS

98

generalization of Equations (9) and (10) when dnc/dx = 0 : i = 1,2,3..

dat= - Lg!?

&

dx

’dx gtt-

dai dx

j#i,j= i=

da5

= -q&

de =de-Tads-

5 wodni

(A2.17)

i=2

.m

(A 2.4)

de -= dx

1,2,3...m

‘,7-o;;

+ 5 TOP*; Gtog

1,2,3...m

dm Dtdx (A2.19)

(A2.5) j#i,j=

1,2,3...m.

Hence, assuming linear flow, the generalized form of Equation (12) is, i = 1,2, 3,. . .m (A2.6) LtrYtJ = LiI-E”U j#i,j=l,2,3 ,... m. Multiplying Equations (A2.2) and (A2.3) by Lgf and Ptj respectively, one obtaines the following generalization of Equations (15) and (16): i = 1,2,3.. Lu = - ___ni YtJ dnc/dx

dr = x ..___. E’+ De __ m ToP’ ;*(D, dx i=2

!k-$&

Dl~_-dLL

(A2.8)

where Dl = De

since Pi1= t and Hence the generalized form of Equations (18~(21) is, i = 1,2,3..

.m (A2.9)

Lc5 = D&tl

j=

1,2,3...m.

Thus the effect of the non-oscillation of conserved quantities (criterion 8) is to yield total diffusivities Di which are independent of the applied gradients in linear flow. In order to assess the effect of the non-oscillation of potential work, we note that the generalized form of Equations (24) and (25) is, Ti=P-

5 pgts

(A2.10)

i=2

Tds = de -

(A2.11)

2 PLY dnt. i=2

;

ptNi

i = 2,3,4.

. .m

De i=2,3,4...m.

the following

(A2.21) (A2.22)

generalization

i = 1,2,3.,

.m

j = 1,2,3..

.m.

of

(A2.23)

[Ll = Wfl

(A2.24)

where [Y] denotes a square matrix of elements Ytj. The corresponding matrix forms of Equations (A2.2) and (A2.3) are, (A2.25)

@=[4pl($) (- 4 = WI

(g)

(A2.26)

where (Z) denotes a column vector of elements ZZ. In isotropic media, the matices [-?J and [L] apply to the y -+ and z directions as well as to the x direction. Letting & denote the vector flux of the ith conserved quantity while V denotes the familiar gradient operator, we thereby obtain the following three-dimensional generalizations of Equations (A2.25) and (A2.26) :

The general expression for exergy is [14, 19,201, &=E+PoV-ToS-

0

(A2.20)

- D,)g

Equation (A2.23) was obtained with respect to systems which obey Equation (A2.1). In the presence of external force fields, the effective concentration (footnote, p. 92) may be introduced into the more general expressions of Refs. [lo, 11, 19, 201 with the result that Equations (A2.2HA2.23) remain unchanged in form. We may thereby regard Equation (A2.23) to apply in general to linear flow in non-oscillatory processes. The matrix form of Equation (A2.23) is,

(A2.8a)

dni/dx = deldx.

=

Lrj = D&‘,f

1,2,3...m.

i= 1,2,3...m

Wdx

Dt)

We thereby obtain Equation (37) :

.m

We define the total diffusivities Dr by,

(De -

Dt=

(A2.7) j=

(A2.18)

i=2

(A2.12)

W = PXW

(A2.27)

(- 2) = [L](Va).

(A2.28)

i=2 so

that Equations (27)-(36) may be generalized as follows : dg = dE + PO dV - To ds -

5 /MI dN$ (A2.13)

The derivation of Equation (A2.24) in the context of Equations (A2.27) and (A2.28) constitutes the proof of the following generalized form of Theorem 1:

i-2 m E’ =

e -

ToS -

2 pi&

(A2.14)

i=2

T= To <=----&+ T

cm Tow - Ttao lit T i-2

(A2.15)

.._ E =

e + PO -

Tos -

Z pmnt is2

(A2.16)

Theorem 2 For the linear flow of m conserved quantities in any non-oscillatory process in isotropic media, the matrix of dynamic coefficients [L] is related to the matrix of static coefficients [2] by [L] = D&Z], where De is the total diffusivity for energy. In the case where equation (A2.1) applies, the following

A MacroscopicDerivation of Omager’s Relations generalization

of the Gibbs-Duhem

relationship [Equa-

99

A comparison of Equations (A3.5) and (A3.6) gives,

tion (45)] holds:

De=$

(A3.7)

(A2.29) where (A2.29a)

More general treatments of the Gibbs-Duhem are discussed in Refs [lo, 11, 19,201. Appendix

The conventional [25, 261,

thermal diffusivity De’ is defined by

(A2.29b)

(A3.8)

equation

where z(, is the constant pressure heat capacity. Hence, De’ = ;T De.

3

Now the Prandtl number Pr is defined by [25,26],

As pointed out on pp. 93 and 98, the total diffusivities D, and DZ are independent of the applied gradients for linear flow in non-oscillatory processes. Consequently Dr will in these circumstances coincide with the kinematic viscosity in the case where the ith flux is the flux of transverse linear momentum. Similarly, D( for mass transfer will coincide with the conventional diffusivity while in heat transfer D, will be closely related to the conventional thermal diffusivity. To see this, consider, for example, a flux of thermal energy which from equation (17) is given by,

d”

edx’

(A3.9)

P

On total di$%sivities in non-oscillatory processes

b=__D

21

(A3.1)

In the case of heat transfer, the energy flow t occurs in the absence of mass transfer so that fi = 0. Writing

(A3.10) where Dr is the kinematic viscosity (the ith flux being understood in this case to represent the flux of transverse momentum). In view of Equation (A3.9) we have, (A3.11) For linear flow in non-oscillatory processes, Dr - De [from Equation (A2.22)] so that for this case, P&g

2)

(A3.12)

The dimensionless Schmidt number SC is defined by [25,261 for the case of heat transfer, we have,

(A3.13) (A3.2)

The derivative (ae/ax)h=o may be expanded by

(~)b, =(%,(aa’;)ti=, so that

In view of Equation (22), the derivative (ae/aT)b=o may be written as (ae/aQ where the subscript n denotes that the derivative is evaluated at a constant concentration n. Hence, 2 = -

De ($)&$;o.

(243.4)

But (ae/aT), is simply n&, where c, is the constant volume heat capacity (per unit of matter). Thus, (A3.5) The thermal conductivity K is defined by, (A3.6)

where DZis the kinematic viscosity and 03 is the diffusivity for mass transfer. From equation (A2.22) Dr = 05 so that for linear flow in non-oscillatory processes involving viscosity, the Schmidt number SC must be unity. In cases not involving viscosity, it is only required (for linear flow in non-oscillatory processes) that the ratio SclPr be equal to e,/cp, since from Equations (A3.11) and (A3.13) we have, SC De c, - =--7 Pr Dj c,

(A3.14)

while from Equation (A2.22) De = Df. As pointed out in Ref. [25], the condition SclPr = c,/z(, is required for an ideal analogy between mass transfer and the pure conduction of heat. It may thereby be expected that Theorem 2 may be approximately valid under this condition. Similarly, Pr = z(,/c, is required for ideal obedience of the familiar Reynolds analogy-while the analogy between mass the momentum transfer requires SC = 1, It thereby may be expected that many situations where these analogs are obeyed may be relatively non-oscillatory-so that Theorem 2 may be valid to a good approximation in such situations. In the case of the conduction of thermal energy in

R. B. EVANS

100

metals for which the Fermi-Dirac electron gas applies, the heat capacity c, in equation (A3.7) is given by (Ref. [27], P. 258),

Now from p. 257 of Ref. [27], we have to a good approximation p = p” while from p. 578 of Ref. [ll], we have ~0 = ArW, where A is a constant, so that, 3n -an=-* + 2~0

(A3.15)

(A3.21)

where k z Boltzmann’s constant; ~0 = electron potential (or ‘Fermi level’) at absolute zero.

Hence the electrical conductivity (A3.20) 3n u = D,q2-. 2P0

Thus for this case, the thermal conductivity K is, from Equation (A3.7), K=

DenF.

(A3.16)

av

(A3.17)

where j is the current density and V is the voltage. Noting thatj = qli (where q is the charge per electron) while p = qV, Equation (A3.17) becomes,

~=_~!k q2

(A3.18)

ai

Lo+. Substitution of Equations (A3.23) gives,

*c-D

!!!a,

nap ai

(A3.19)

a = D,q2-. +

and (A3.22) into

0 q’

(A3.24)

If Theorem 2 applies, we have D, = Dn so that,

04

= 2.45 x 10s v”

“K2’

Comparing Equations (A3.18) and (A3.19), we have, an

(A3.16)

Lo=D,n2k2 &3

Lo=j

nax

(A3.23)

IIs k 2

But from Equation (22), j,=__D

(A3.22)

Now the Lorenz number Lo is defined by [27],

Let us now find for this case the electrical conductivity u which is defined by,

J=--a

CJis from Equation

(A3.20)

(A3.25)

This is the same value.of Lo found by Kittel [27], which indicates that D, and Dn are set equal in his theoretical assumptions.