Irreversible Thermodynamics of Heterogeneous Systems - American

J. Phys. Chem. 1987, 91, 1184-1199

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1184

systems studied here. This is somehow not an unexpected result the LB rule. Curve 3 corresponds to a mixture in which both the since mixtures of hard spheres do not show any liquid-liquid phase differences between the reduced temperatures and the departures transition. Even though mixtures of very anisotropic hard convex from the LB rule are larger than in curve 3; however, the introduction of differences in size strongly prevents concentration bodies present this kind of transitions, it it unlikely that anisotropy fluctuations, and so the tendency to homocoordination in the of shape can explain the Scc(0) values of the systems shown in Figure 6. mixture. Figures 6 and 7 show that not far from the liquid-liquid As pointed out by Ashcrofts5 immiscibility is driven by both phase transition not only packing effects but also attractive forces potential energy and entropy considerations. While the long-range are important in determining the distribution of the molecules in portions of the interaction potentials are important for the former, the solution. It remains to be known if perturbation theories the entropy is mainly determined by the correlations resulting from describe such a distribution under those conditions. It also remains the short-range interactions. In this way Joarder and S i l b e ~ t ~ ~ unclear what is the quantitative influence of nonrandomness on have shown that it is possible to obtain values of Scc(0) larger the thermodynamic properties of the mixtures. In the normal than that of the ideal mixture when the attractive part of the liquid range, far away from the phase separation, it seems that interaction potential is taken into account within a mean field packing effects account for any influence of this type. Patterson theory. Recently Kojima et calculated the G , integrals for et aL5*have recently shown that the W-shape of the CpEcurves mixtures of Lennard-Jones spheres within the Percus-Yevick of some liquid mixtures of complex molecules might be due to approximation. We have calculated the Scc(0) for some of the their proximity to an UCST. As we have seen, this situation cases studied by Kojima et al., the results being shown in Figure corresponds to sharp peaks in ScC(O),and thus to a tendency to 7 . As can be observed, increasing the difference in interaction homocoordination. Both the strong positive contributions to CpE energies between the pure components increases the concentration and to S,(O) are related to the divergences of the partial structure fluctuations above the ideal mixture for mixtures of equal-sized factors a,@) near the UCST. An analogous situation must exist spheres, assuming the Lorentz-Berthelot (LB) combination rule. in dilute solutions near the critical point of the solvent (typical Curve 4 corresponds to equal-sized spheres for which the reduced of supercritical extraction experiments), and thus noticeable temperatures of the pure components are intermediate cases bedifferences between local and bulk compositions could be expected in this case. tween curves l and 2, but there is a noticeable departure from (55) Ashcroft, N. W. In Liquid Metals, Vol. 1, Evans, R.; Greenwood, D. A,, Eds.; The Institute of Physics: Bristol, 1977; Institute of Physics Conference Series No. 30. (56) Joarder, R. N.; Silbert, M. Chem. Phys. 1985, 95, 357. (57) Kojima, K.: Kato, T.; Nomura, H . J . Solurion Chem. 1984, 13, 151.

Registry No. Ar, 7440-37-1; Kr, 7439-90-9; Xe, 7440-63-3; N, 7727-37-9; 0, 7782-44-7; C,H,, 74-84-0; NO, 10102-43-9; CIH,, 7485-1; CH,, 74-82-8; CF,, 75-73-0; HBr, 10035-10-6; HC1, 7647-01-0. ~~

(58) Patterson, D.; Costas, M.; CBceres, M., unpublished results.

Irreversible Thermodynamics of Heterogeneous Systems Byung Chan Eu Department of Chemistry, McGill University, Montreal, Quebec H3A 2K6,Canada (Received: February 19, 1986; In Final Form: June 6, 1986)

Global macroscopic evolution equations are derived for heterogeneous systems from the local field equations which were obtained previously on the basis of the kinetic theory. The global equations may be applied to systems removed far from equilibrium. Based on the theory, the equilibrium conditions are studied and the Gibbs phase rule is recovered for a reacting, heterogeneous system. Thermodynamic stability of heterogeneous systems is also analyzed. As an illustration of application of the theory, a calculation is presented in which the efficiency of an irreversible Carnot cycle is obtained. Its efficiency is one-half the efficiency of the reversible Carnot cycle: Teff = (1/2)(1 - T,/T,).

I. Introduction Macroscopic processes are often studied in a single phase system which is assumed to be so large that the boundaries do not affect the processes of interest. This is of course a simplification of the reality, but it is quite reasonable for many situations we face in nature. However, many natural phenomena occur in a condition that involves a number of phases or subsystems, either homogenous or inhomogeneous, and hence the entire system is globally heterogeneous. Examples are numerous: substances consisting of heterogeneous phases; biological systems made up with cells of various kinds that are separated by membranes and intercellular fluids and communicate with each other through chemical and physical means; mechanical engines consisting of reactors, pistons, and reservoirs, or more generally energy converters; electrical cells; an array of electronic logical units, etc. Therefore, when looked at from the standpoint of such heterogeneous systems, a single phase consideration of irreversible processes is clearly a special case of a more general theory of heterogeneous systems, which should be a global theory by necessity. It is the aim of this paper 0022-3654/87/2091-1184%01.50/0

to formulate a theory of irreversible thermodynamics of processes in heterogeneous systems and to study some of its general consequences. If the system as a whole is away from equilibrium, there necessarily exists a process in the system which arises owing to the spatial and temporal inhomogeneities in thermodynamic intensive properties, and as a result the system is generally driven to its steady or equilibrium state while at the same time energy and matter are transported within the phases (or subsystems) and across the phase boundaries. The steady state may coincide with the thermodynamic equilibrium state, but the coincidence is not generally assured since the system may tend to move away from equilibrium if there exists more steady states than one. In this work we take an approach in which global evolution equations for macroscopic variables are derived from the local continuum mechanics equations for macroscopic variables which strictly obey the thermodynamic laws. It is then possible to study the approach to equilibrium by the system and in particular the equilibrium conditions which will turn out to be the Gibbs phase 0 1987 American Chemical Society


Irreversible Thermodynamics of Heterogeneous Systems rule.'$* The evolution equations derived also provide a mathematical means to investigate the stability of the steady states. This approach to study of equilibrium and stability is different from that in the Gibbs theory that is based on virtual variations of entropy away from equilibrium, since the present approach makes use of the dynamical equations for macroscopic variables characterizing the macroscopic thermodynamic states. It therefore is not necessary to introduce the device of virtual variation as is in the theory of Gibbs,' which lacks dynamical evolution equations for macroscopic variables. Since equilibrium thermodynamics deals with thermodynamic states in which the intensive variables are homogeneous in space, there does not appear any compelling necessity to make distinction between the local and global behaviors of thermodynamic properties. When we apply equilibrium thermodynamics to macroscopic systems our thinking is generally on the global basis and theories formulated on such basis do not connect directy with local theories. On the other hand, macroscopic processes in the large are affected by local inhomogeneities and cannot be satisfactorily understood without the help of molecular theory. Since the local theories can be founded) on the molecular picture (the kinetic theory), there must be at some point in the development of theory a way to connect the two distinctive viewpoints. In the present formulation we convert the local irreversible thermodynamic formalism to the global formalism and thereby attempt to bridge the two modes of thinking. This is of course the reverse of the procedure taken for obtaining local field equations in irreversible thermodynamics. Nevertheless, there are some new aspects in the present study that are not available in the literature. For example, since the local field equations can be derived) from a suitable kinetic equation, they can come equipped with molecular (statistical) formulas for various transport coefficients, etc. Therefore by connecting global equations to the local field equations, we provide a mathematical means to investigate global irreversible thermodynamics in terms of statistical mechanical transport coefficients that phenomenological irreversible thermodynamics for systems in the large take as empirical parameters. Laying the foundation for this practical aspect is one of the aims of this article. Another new aspect is in the inclusion of fluxes in the large (global fluxes) when the energy and entropy changes are considered for phases in the large. The inclusion of global fluxes amounts in effect to a generalization of the equilibrium thermodynamic formalism and should facilitate study of phenomena occurring (in a global system) in states removed far from equilibrium. The present formulation is so aimed at providing the starting point for such studies in the future. In section I1 are presented various local conservation equations for conserved variables and evolution equations for fluxes and any other macroscopic observables. In section I11 we then consider a multiphase system and derive various conservation laws and evolution equations for the system in the large. The global entropy balance equation is also derived. Then in section IV the global equilibrium conditions are examined and a phase rule is derived therefrom for a chemically reacting fluid from the conditions for equilibrium. In section V we consider the thermodynamic stability of the system and in section VI we calculate, as an illustration of the theory developed, the efficiency of an irreversible Carnot engine under a set of simplifying assumptions. Section VI1 is for discussions and conclusions. 11. Local Conservation Laws and Evolution Equations

We assume that the system consists of w phases in which there are r chemically reacting species Bi, i = 1, 2 , ..., r. It is also (1) Gibbs, J. W.The Scientific Papers o f J . W.Gibbs; Dover: New York, 1961;Vol. I . (2) See, for example, Kirkwood, J. G.,Oppenheim, I. Chemical Thermodynamics; McGraw-Hill: New York, 1961. (3) For the kinetic theory derivations of equations appearing in the phenomenological formulation of irreversible processes, see Eu, B. C. J. Chem. Phys. 1980, 73, 2958. 1981, 74, 6362. 1981, 74, 6376. 1985, 82, 4283. Generalized Gibbs Equations in Irreversible Thermodynamics in Recent Developments in Nonequilibrium Thermodynamics, Casas-Vazquez, J., Jou, D., Lebon, G. Springer: Berlin, 1984.

The Journal of Physical Chemistry, Vol. 91, No. 5, 1987 1185 assumed that there are n chemical reactions. We will denote the rate of reaction k in phase a by RZ and the associated stoichiometric coefficient times the mass of species i by u t which is counted positive for the products and negative for the reactants. The phases will be distinguished by the superscript a. The following additional notations are useful for carrying out the present study:

Jf

mass density in phase a mass density of species i in phase a fluid velocity in phase a internal energy density in phase a external force per unit mass of species i in phase a mass flux of species i in phase a

;p7

stress tensor of species i in phase a

PU PP

ii"

'"

5

FQ"

s

PP A?

2 e:. h;

c+

Z=lC

heat !lux of species i in phase a

CLIQp

pressure : 0

in phase a

c,

(1/3) Tr - pp; excess trace part of Pp c 1 / 2 ) ( 5 + $) - ( 1 / 3 ) U T r F Qp - hPJf enthalpy density per unit mass of i in phase a

The following collective symbols for fluxes are useful: @f")* I p&l(d": @fib I &. 1 , @{2)" I A* @f""

a?)" = 2, etc.

I &a;

The basic approach taken in the present study is that the macroscopic process of interest in each phase is described by a set of field equations as postulated in the phenomenological theory of irreversible processes formulated by the present author: for example. In this theory we postulate a set of conservation equations and flux evolution equations. The sybmols defined above are those used in the previous papers) on statistical mechanical studies of irreversible thermodynamics by the present author, and the explicit forms for evolution eq 2.5 can be found in those papers just cited.3 The local forms of the conservation equationsSand the evolution equations for phase a are as follows: mass:

concentrations:

a at p?

n

= -div

(3+ ppii") + k=C v ~ k R f 1

i = l , 2 ,..., r; a = l , 2 ,..., w

momentum: c,

at

p"ii" =

-div (P+ p " i i " P )

+ i=l

p p e

(2.3)

+ L3.e

(2.4)

energy:

a p " p = -div -

at fluxes:

c,

(@

+ p V i i " ) - P:Grad

i= 1

where Z?" and Af')" are quantities depending on the conserved variables pa, 2,e", and the fluxes themselves. We will call Z?)" the convective term and A?" the dissipative term. The convective term includes the thermodynamic force driving the flux and the dissipative term is related to the energy dissipation due to the flux in question. It is generally a nonlinear function of fluxes. In the phenomenological approach4 the convective and the dissipative terms must be suitably postulated but in such a way that the (4) Eu, B. C. Ann. Phys. 1982, 140, 341. (5) (a) Leigh, D.C. Nonlinear Continuum Mechanics; McGraw-Hill: New York, 1968. (b) Eringen, A. C., Ed. Continuum Physics; Academic: New York, 1975,Vol. V.

Eu

1186 The Journal of Physical Chemistry, Vol. 91, No. 5, 1987

second law of thermodynamics is satisfied. This condition is expressed as follows: Let us assume that there exists the entropy density of phase CY which is a piecewise differentiable function of conserved variables as well as fluxes-nonconserved variables. We will denote it by Sa.It then can be shown that the second law is satisfied in phase CY if A,@)uare such that /

i C(asa/a&j.)a) o

3

i = l a=O

o

(C1)

Here A:')" stands for the reaction rate for i; see (2.13). One of the aims of the present study is to find global macroscopic equations for the entire system such that the thermodynamic laws are satisfied. In fact, the assumption of piecewise differentiability implies that there exists a differential form (one-form) for 8": r

r

T" dSa = deu + p" du" - C j i g dcp + i= 1

/

i=l a=l

w)" O d&J')* (2.6)

where


1 T" = (asU/a€U) pa/^ = (asa/aua)

- p p / ~ = (asa/acy)

P ) ~ /= P (asa/a&j")a)

where the nonclassical entropy flux 2"is determined by the equation (subject to an appropriate boundary condition)

+ @.I"'*0 xj"'") = 0

by distance ALQ.

assumed to be valid in phase CY ( a = 1, 2, ..., o)which will form the basis of the analysis of irreversible processes in a heterogeneous system. 111. Global Equations It is now possible to obtain the global balance and evolution equations for a heterogeneous system. To facilitate the derivations, we equip ourselves with the following mathematical tools. A . Mathematical Preparation. Suppose there is a volume enclosed by a boundary R which moves at a velocity ii,. The positive direction of the outward normal vector of the bounding surface (boundary) is taken from the inside to the outside of the system of interest. The volume may then be defined as the integral

(2.7)

When the eq 2.1-2.5 are used in eq 2.6 and if the entropy flux is defined by4

+ (1 / T")(ZI")" 0

dRa Figure 1. The_surfaceelement dQ' has moved in the normal direction

V(t) =

df

"(')

(3.1)

The integral is taken over the space enclosed by the boundary. The rate of change in V may be calculated as follows: d V - lim -[V(t 1 _ + At) - V(t)] dt At-0 At

=

(2.9)

V(f+At)

1

['

lim At-0 At

1 = lim At-0 At

then the entropy balance equation is given by

d7-

sn

sV(') d':]

dfi.Ai,

(2.10) where r (TU

where

/

=

$0)"

0 A?)" 3 0

(2.11)

Gn =

i = l a=O

&/P

(2.12)

= RY

(2.13)

= W)"/T"( a = 1, 2, ..., I )

(2.14)

*O)a

*)a

=

r

-dp = Cvgifig (i = 1 , 2, ..., n)

lim (A%/At) A t 4

with Ai, denoting the displacement of the boundary in the direction of the normal vector to the surface; see Figure 1. A similar calculation may be performed for an arbitrary macroscopic variable. Let us denote it by M(t) which may be expressed in terms of its density as follows:

(2.15)

M(t) =

k= 1

s'

d': A(?,t)

Its rate of change is then calculated as follows:

dM = dt

lim [M(t

At-0

=

lim At-0

=

lim At-0

[s

+ At) - M(t)]

V(r

+ At)

d': A(':,t

s""

d': A(':,t)]

lv(r+A') d7 + At) t)] + At-0lim [ 1 A(?, 1 d7 [A(':,t

v(t+Ar)

where the subscript T means that the temperature is held fixed in the differentiation. The affinity .A; of the ith reaction in phase CY is a measure of displacement from chemical equilibrium at which it vanishes. The equations presented above are the local equations

+ At) -

di

- A(':,

t) -

'(')

A(':,t)]

The Journal of Physical Chemistry, Vol. 91, No. 5, 1987 1187

Irreversible Thermodynamics of Heterogeneous Systems If there is a field equation for the local variable

a

of mass change in the following manner:

+ E,

Jn = -?.(iiJn) at

then by inserting it into the last integral above, we obtain the equation

if the boundary conditions (3.4) are imposed, and

(3.3) for the global rate of change in M(t). This is the master formula for equations of change for various global macroscopic variables. B. Boundary Conditions for Velocities. The boundary !lmay be movable as a whole or only a part of it may move. It also may let particles through or may not. Therefore it is appropriate to consider !l that is a union of movable boundary v and fixed boundary w :

n

= XvgRgP if RS is uniform in v*

(3.10)

k= I

They indicate respectively the rate of change in mass due to diffusion across the boundary of phase a and that due to chemical reactions occurring in the phase. Note that the reaction rate term may be split as indicated in (3.8). Thus the total rate of mass change in (3.5) consists of diffusive and reactive components:

!l=vuw

(3.5')

We assume that at v the fluid sticks so that whether v is permeable

or not


ii=ii,#O

atv

(3.4a)

Since the mass must be conserved in a reaction, there holds the condition

(3.4b)

zvy&= 0

while at w

n

ii = ii, = 0 if ii, = 0 if

w

w

is impermeable

is permeable

(3.4c)

These boundary conditions are quite sensible physically. If these boundary conditions are applied to (3.3) the evolution equation for M(t) takes the form d V - M(t) = d i E,,, if w is impermeable (3.3') dt

=1"di

E, -

So

d 3 4 if w is permeable

(3.3")

where k = 1, 2, ..., n and a = 1, 2, ..., w . Moreover, since the mass also conserves for the whole system, there holds the relation (3.12) This is the mass conservation law for the whole heterogeneous system. The two equations (3.1 1) and (3.12) mean that the total mass change due to diffusions across the boundary is also conserved:

C. Equation of Change in Mass. The mass of species i in phase a may be written as

My =

J v pdF #(F,

diMy

w

CZ,t-

-0

(3.13)

a=l i = l

That is, the gain in mass in one phase through diffusion across the interfacial boundary results in the corresponding loss to the rest of the system. If (3.5') is summed over i alone, there results the equation

t)

By using (2.2) and (3.3), we then find

-=-

.. .

dt

(3.5)

diMa dt

(3.14)

where

Since the diffusion flux is defined by

3 = pp(iip - i i a )

(3.1 1)

i= I

(3.6)

the first integral on the right of (3.5) may be rewritten as (3.7) in terms of the diffusion flux relative to the moving boundary instead of the center of mass of the fluid moving with velocity 6". If the concentrations and the temperature are sufficiently uniform in space so as not to change the reaction rates significantly, then the reaction rates may be assumed uniform in space except at the boundaries where reaction sites, e.g., enzymes or catalysts, may be present. In the event that the reactions at the boundaries are important the reaction contribution in (3.5) may be split into two parts; one for the bulk excluding the boundaries and the other the boundary contribution:

where 8 P denotes the volume of an infinitesimally thin layer at the boundary and G the rest of the volume vtl=r%+avtl

It is sometimes convenient to express the terms in (3.5) with rates

r

MU = C M y i= 1

Equation 3.5 or 3.5' is the global mass balance equation for species i in phase a. D. Equation of Change in Momentum. Define the total momentum of phase a by the volume integral as follows:

Application of (3.3) yields its rate of change in the form

(3.16) for which we have used the local momentum balance eq 2.3. If the external forces are uniform over the phase, then the volume integral in (3.16) may be written as follows: (3.17) It must be noted that fi is the force per u a t mass of i. It is convenient to split the stress tensor P" into tae pressure pu, the excess trace part Aa, and the traceless part P":

Eu

1188 The Journal of Physical Chemistry, Vol. 91, No. 5, 1987 (3.18)

where

-pu

A" = (1/3) Tr

k

(3.19)

+ pf)- ( 1 / 3 ) F T r

= (1/2)(p

(3.20)

c-,

and U is the second rank unit tensor. Here it is assumed the stress tensor is symmetric. Hence there is no antisymmetric part in (3.18). The three components of the stress tensor may be further decomposed into species components if the system consists of a mixture:

p" = c p 4

and its equation of change is easily obtained by applying (3.3) and the local internal energy balance eq 2.4:

s"

s'

+ d3 i2-e(3.24) i= The first term on the right involving & accounts for the heat d3 ?:Grad

ii"

1

transported across the boundary Q". Therefore it may be written as

i= I r

A" =

neglected when their efficiencies are calculated. E . Equation of Change in Energy. The internal energy of phase cy is defined by the formula

EA?

(3.21)

i= I

(3.25)

r o

0

P=Cpp Downloaded by STOCKHOLM UNIV on September 7, 2015 | http://pubs.acs.org Publication Date: February 1, 1987 | doi: 10.1021/j100289a032

i= 1

When a mixture is the object of study it is necessary to express the total pressure, the total stress tensor, and the total heat flux in terms of their species components although from the phenomenological standpoint the components are generally not possible to measure separately. The same situation is encountered even in equilibrium thermodynamics since, for example, the equation of state for a mixture must be expressed in terms of virial coefficients of components and the cross coefficients which are generally not possible to measure directly. This difficulty is avoided with the help of a molecular theory. As in equilibrium thermodynamics of mixtures, it is necessary to have a molecule theory to find the partial properties in irreversible thermodynamics of mixtures. It must be recognized that the partial properties in (3.21) contain implicitly the contributions from the inoeractions between different species. For example, in the case of Pp we may write 0

pp

0

where q" sgnds for the heat content in cy. When the decomposition (3.18) of P" is used, the volume integral involving the stress tensor may be written as

s

+ 1'd3 (A"U + P"):Grad -

)p

d3 p" div ii"

0

ii"

If the pressure is uniform in phase cy, then we may write the first integral as follows:

and if the boundary condition

ii" = ii;

at

is imposed, it follows that

s'"

0

+ CPKj

= Qi

di; p" div ii" = p"

j

(3.26)

Y

+

sue

d W p " E" (3.27)

where kvij are the contributions from the virial tensor arising from the interactiojs (potential energies) between molecules of species is the kinetic contribution arising from the moi and j , and PKi mentum transfer by molecules of species i. These components are computable, in principle, if statistical mechanics is employed. If the area of the moving boundary is A" and if the pressure is uniform over A", then

for which we have used (3.2). The second term on the right accounts for the energy transported per unit time across the permeable boundary due to diffusion. By using this result and the definition of thermodynamic forces, we may write the work due to the stress in the form

where 2" is a unit vector defined by

and the energy balance equation in phase

cy

in the form

It is then convenient to define 2"= A%". It is a _vectorin the direction of the momentum in the present case of p". We may therefore write (3.16) in the form where

ha is the enthalpy density per unit mass of phase a h" =

s'd3

ip??

(3.16')

e"

+ pave

If the definition of mass change

0

If A" and P vanish or are negligible on 0" and the boundary is impermeable, the surface integral vanishes, and we obtain

This equation may be used for some calculations related to heat engines6for which the mechanical energy degradation is commonly

(6) For some recent studies on thermal engines and their efficiencies see Andresen, B.; Berry, R. S.; Nitzan, A,; Salamon, P. Phys. Reu. A 1977, 1 5 , 2086. Andresen, B.; Berry, R. S.;Ondrechen, M. J.; Salamon, P. Acc. Chem. Res. 1984, 17, 266. Escher, C.; Ross, J. J . Chem. Phys. 1985, 82, 2453. Escher, C.; Kloczkowski, A.; Ross, J. J . Chem. Phys. 1985, 82, 2457.

Irreversible Thermodynamics of Heterogeneous Systems is used in the surface integral in (3.29) under the assumption that then the integral may be the enthalpy density is uniform on d, written in a more transparent form

The Journal of Physical Chemistry, Vol. 91, No. 5, 1987 1189 Therefore, if the heat flux is in the x direction the surface element d0" in question is perpendicular to the unit vector in the x direction since

E

(dfi".v, =

(d02a)bSbx = (d!?),

b=x,y,z

With this we cast the energy balance equation for phase a in the form dE" dt

dq" dt

dV" dt

-

diM" dt

+ E; + Wr

- = - - P a - + h" -

(3.29')

Let us now imagine a parallelepiped of a substance, e.g., iron, whose two ends form the boundaries of interest Qy and Q;,the remaining faces being assumed adiabatic. We consider a heat flow from Qp to 0;. If the temperatures are uniform on the boundaries, but the two boundaries are at different temperatures, the surface integrals may be written as

where -P 3" I

$'

d?

+

(Aa~(2)"

wy = $'

d7.

&:x(l)O)

$]?.e

(3.30a)

where 2" the present case. Assume

F(Qp)

(3.30b)

i= 1

It must be stressed that (3.29) or (3.29') holds if the pressure is uniform in phase a. If not, (3.27) does not hold and the pressure-volume work term must be replaced by the integral Downloaded by STOCKHOLM UNIV on September 7, 2015 | http://pubs.acs.org Publication Date: February 1, 1987 | doi: 10.1021/j100289a032

rj3la= Xi[In ~"(anp)- In ~ " ( n g ) ] ~ % ~ is the vector defined in (3.22). It is parallel to rl(3)a in

A similar comment applies to the enthalpy density term as well, since it also requires a uniform pressure; see (3.28). The last two integrals in (3.29) represent the dissipation of mechanical energy into heat due to diffusions under the influence of the external forces, e.g., an external electric field, and due to bulk and shear viscosities. In the case of an electric field the external force contribution is the Coulomb heating. It is perhaps useful to remark that the concept of dissipation is relative since it depends on the circumstances in which the mechanism responsible for dissipation of energy is used. For example, if we are interested in generating heat with an electric current, the energy generated is not regarded as a dissipation, but it is a dissipation when looked at from the standpoint of the electric current that is responsible for the heat generated, because the kinetic energy of the current ends up being reduced owing to the generation of heat. Since energy must be conserved for the whole system which may be regarded as isolated, the summation of (3.29') over all phases yields

dE" E-=O dt

=

P(@)+ AT (AT 2 0)

Then approximately

-xj2

rj3)"N

This is simply the Fourier law of heat conduction for phase a in the lar e Similar interpretations hold for other fluxes. The global flux If3; will evolve in time if AT changes in time. By differentiating (3.32) with time and using (3.3) and the evolution equation for fluxes, we find d -

=

dt

-$

wa

+ $" di: (Z?" + A?)")

diJ"-p"(ii" - ii8)a.j")"

Z?)" = XP)"/gj")g+ AZI")"

(3.34)

where g?O is a function of conserved variables. For example3 &)a

= 1/2pp;

gj2b = 3/5pp;

g("" = 1/p$p;

g p= 1/cp (3.35)

Thus if g?)" are uniform across phase a and in the case where x?)" can be written as a gradient or a divergence = - grad

~ ( " ' 0

= - div

rf")"I $ 'd? p"&p)"

(3.33)

The convective terms Z.j")"invariably contain the thermodynamic force driving the flux in question. It therefore is convenient to make it explicit by writing

(3.31)

as an expression for the first law of thermodynamics for a multiphase, heterogeneous system that is isolated as a whole. F. Equation of Change in Fluxes. If there is a flux of matter or energy across a phase, its evolution must be considered. To facilitate it, we define the global flux by the following formula

(2"= ~ " 2 " )

AT/P(Q;)

df")*= - div ($(")" 3

(3.36a) (3.36b)

,$?)a

it is possible to write the volume integral of the convective term in the form

$'

di: Z?)" =

+ $'

d~.~&)"/g.j")"

(3.32)

di: AZj")" for (3.36a) (3.37a)

The integration over the volume does not change the tensorial rank of ry)".The physical significance of this volume integral may be understood with the following example of thermal conduction. The local formula of heat flux to the lowest orier of approximation is proportional to the temperature gradient V In F according to the Fourier law: a(3)a=

In T*

Let us assume that the thermal conductivity is a constant. Substitution of this into (3.32) yields

=

dk$j")"/&)"

+ $'

d? AZ(")" for (3.36b) (3.37 b)

+

Moreover, if 0" = Qp Q: and if the values of are different at 07 and 0;as is usually the case-for example, the temperatures at the two ends of a parallelepiped may be different if there is a temperature gradient-then the surface integral may be written as

s"

dfi".v&)*/gj")"

= [&)*(Qp) - @,?a(Qq)]i"/g(")"

(3.38a) for which we have used the identity

where the direction of r?" is from the face 07 to the face Q;. In the case of 4,'")" being a tensor, Le., the surface integral in (3.36b), the formula gets more involved, but if the parallelepiped

Eu


of a substance is considered in Cartesian coordinates, there holds a formula similar to (3.38a):

s'"

d+.#,(+/gy)a

=

2%[&)a(

Q;.) -

,#,jU)a(Q,)]

/gjn)" (3.38b)

We thus see that the global flux is driven, as expected, by the difference in A&)": = $,l")"(Qf) - 4{")*(Qp)

If the boundary condition (3.26) is satisfied by the velocities the flux evolution equation takes the form

Sa(?) =

s'

(3.44)

d i pasa

where S a is the entropy density described by (2.10). Its rate of change is then

dS" = dt

1'-di%[*

+ (5" - ii8)paSa] +

s

VQ

d7 ua(7, t) (3.45)

where 2 and 8are respectively defined by (2.8) and (2.1 1). We now note that the entropy density may be given in the form r

r

Sa =

[ea

+ paua - CPPCP+ I=1

/

Cw'" 0 &ia)a]/T"

(3.46)

i=l PI

In the following calculation we shall assume

Zn)*/F = uniform in P = A$j")"/gj")*

s'

+ d7 (AZ?).

[in the case of (3.38)] (3.39)


=

for (3.36a)

=

for (3.36b)

&)"

to make the entropy change come out in a differential form. We = 0 in V remark that this assumption does not mean that since is proportional to and can be a nonvanishing e.g., constant in V if the thermodynamic forces are constant in P, the temperature gradient is a nonvanishing constant over the volume of phase a. This certainly is experimentally realizable, especially, at a steady state. Now, substitution of the formulas for 2 and 8, the assumption (3.47), and use of (3.38) and (3.46) yield the rate of change in Sa in the following form

p)a

+ A?)")

where $?)a

(3.47)

and

A$[")" = AaA&Ia

for (3.38a)

= 2".A&)"

for (3.38b)

dSa

dt = - J a n d+.[C(&

- &&J/T"

+

The lowest order approximation to the dissipation term is linear in $?)a: where various symbols so far undefined are

A{")"= -ZRFb)a@Ib)a Jb

-

=a

where Rfb)*are coefficients obeying the Onsager reciprocal relations. These coefficients may be treated as phenomenological coefficients or calculated3 by means of a suitable kinetic equation such as the Boltzmann equation or its generalization. They are functions of density and temperature and may be regarded as being independent of position, viz., uniform over the phase if the latter variables (conserved varaibles) are slowly varying in space. In that case we may write (3.40)

r

/

CC

= S'di

0X(~)"/T~

(3.49)

i = l n=O

@jO)a

= RP

(3.50)

-#)"

= .24p

(3.51)

Note that 5"accounts for the dissipation of energy due to the intrinsic irreversible processes occurring in phase a. There will be additional dissipations due to the transport of matter and energy across the phase boundaries as will ee seen later. If y e make use of (3.7) and (3.25) and recall that Q" is the sum of ef,the rate of entropy change can be put in a more insightful form

Let us assume that AZ?)u = 0 to the lowest order approximation3 which is valid near equilibrium. Then, if (3.38) is true, the evolution equation may be written as

Since at the steady state

The heat change term in (3.52) may be eliminated by using the energy balance eq 3.29'. We then find the rate of entropy change in the form

the steady flux is determined by the linear relations 1 R t b ) " r j b ) a = -A$p)"/d")a

(3.42)

jb

which is simply a set of force-flux relations for phase a in the large. we find By solving (3.42) for the flux in the large

rp)" = -C(R-l)jTb)

A$,jb)m/g(b)"

(3.43)

Jb

The coefficients determine the transport coefficients for the global processes in phase a . The way this result is obtained shows the limitations implied by global "flux-force" relations such as (3.43). G. Entropy Balance Equation for Phase a . The entropy of phase a is defined by the formula

r

/

1 i = l a-1

0

dI'j")" + AE" dt

(3.53)

The Journal of Physical Chemistry, Vol. 91, No. 5, 1987 1191

Irreversible Thermodynamics of Heterogeneous Systems

e)= -E@j4)%9Tp4 r

r

C@'j4'".(Xj4'" i= I

(3.55)

i= I

The total rate of entropy change for the whole heterogeneous system is obtained by summing (3.63) over all phases:

r

= -Cay j=l

d,M';

dt

[see (3.10)]

dSa

dS

since C;=l&14)a = 0; see also (2.16). We may also cast the entropy production due to chemical reaction as follows:

-dt = a=l E x

Er

(3.56)

c,yp a -d r p /

i-1 a = l

which combines with the chemical potential term-the third term on the right of (3.53)-to give the sum total for the mass change


Equations 3.56 of course presumes the uniformity of chemical potentials in P.Under these conditions and the conditions imposed for the derivation of (3.54) we find

dt

This is the result we set out to derive for a heterogeneous system in the large. It is in effect the entropy balance equation from which the entropy flux and the entropy production may be identified. Especially the entropy production is now due to the transport of energy and matter across the phase boundaries, and the equilibrium conditions can be obtained as the necessary and sufficient conditions for the entropy production to vanish. If either Jp"= 0 or

Combining these results, we finally obtain dSa-_ dt

then the usual equilibrium formula for entropy change in phase a follows: (3.55)

If there exist potentials for the external forces fi =

(3.65)

-e$#

Thus (3.64) may be regarded as an extension of the equilibrium formula (3.65) to a situation where global fluxes are driven by the presence of differences in intensive variables-including the velocity-at the boundaries that separate a phase from the neighboring phases; see section IIIF.

then we may define the mean potential energy by i= I

and the total potential energy of phase a by

IV. Equilibrium Conditions and the Phase Rule

This definition motivates the following decomposition of

p: (3.60)

i= I

We then calculate the rate of change in dt

p with

(3.60): (3.61)

j-1

We now define a new internal energy and chemical potential respectively

E$ = E" figi =

p4

+ + $4

The rate of entropy change derived in the previous section can be used to obtain equilibrium conditions for the system to reach thermodynamic equilibrium. To facilitate this investigation, it is convenient to assume that the entropy change is entirely due to the internal transfers of energy and matter across the phase boundaries since the whole system may be considered isolated. The boundaries may be fixed without loss of generality except for the case of mechanical equilibrium related to the pressurevolume work. Since it is possible to examine the phases pairwise to obtain equilibrium conditions, it is sufficient to consider a two-phase system. We therefore consider the rate of entropy change for w = 2:

(3.62a) (31.62b)

Equation 3.61 and the definitions of the quantities above enable us to write (3.57) in the form By the energy conservation law-the

first law

we may write for the rate of entropy chance in phase a. If the external force is an electric field, E; is the electrochemical energy and fi;, the electrochemical potential for species i in phase a.

Since the total volume is also constant, we may also write

Eu


The rate of change in mass (dMy/dt) consists of two basically distinctive parts; namely, the mass change due to the mass exchange between the two phases and the mass change due to chemical reactions as was indicated before: dMy

diM7

dt

dt

-=-

d,My

+- dt

Since the diffusional mass changes are exactly balanced we have diMi

diM;

dt

dt

-= --

and consequently we may write the chemical potential term in (4.1) in the form


(4.4) Here we have used (3.10) and the definitiion of affinity. Now substituting the above results into (4.1) produces the entropy change in the form

continuously driving the system away from equilibrium. Typical examples would be an oscillating electric field or an oscillating magnetic field. If the intensive variables are uniform throughout the whole system on account of the conditions (4.6), it then is easy to see that r ( 2 ) a = rj3b

=

rj4b

(4.10a)

=0

for all i and a since they are proportional to the differences in are of the intensive variables at the boundaries. However, rj1Ia a different nature since it is related to the shear rate (rate of strain tensor), which involves a different kind of variable from the intensive variables such as temperature, pressure, and chemical potentials. The variable in question is the velocity of the fluid, and the phases will not suffer a stress and the system will be in equilibrium with regard to stress if there is no variation in velocity within a phase and across the phases owing to the absence of stress. This implies that

al).= 0

for all i and a

(4.10b)

This is a mathematical expression for the absence of stress in the system and thus an internal mechanical equilibrium. The conditions (4.8)-(4.9) do not appear in the theory of phase equilibrium by Gibbs since the assumptions are made from the beginning that the phases are in internal equilibrium and the system is free of external forces. Especially, the conditions (4.7) reduce the number of independent chemical potentials and thereby result in a significant modification of the phase rule holding for nonreactive systems. Since the equilibrium conditions can be obtained for the whole system by considering the phases pairwise, we finally reach the conclusion that TI =

rz = .,. = T"

p1 = p2 =

... = p" (4.6')

In (4.5) there are two basically different classes of terms: the terms in the square brackets consist of either intrinsic nonequilibrium processes such as chemical reactions and variations in the fluxes in the large (global fluxes) or extrinsic processes such as variations in the external forces; and the remainder consists of changes due to variations in conserved variables, i.e., energy, volume, and mass. Since (dE$/dt), (dV'/dt), and (diM!/dt) are arbitrary, the equilibrium will be achieved if z-1 = p; p1 = p2; fibi = figi (i = 1, 2, ..., r ) (4.6) in addition to the conditions

Ay=O

a = 1 , 2 ; j = l , 2 ,.,., n

(4.7)

fi& = pir =

... = fi;r

Ay = 0 a = 1, 2, ...) 0; j = 1, 2, ...) n

J$+=O

a = l , 2 ,..., I ; a = 1 , 2 ,..., w ; i = l , 2,..., r (4.9')

In addition to the condition (4.7') for equilbrium, if the mole numbers are denoted by n:, then there can hold certain relations among them: g,(n?, n?, ...) n;) = 0

d -rrj")"=O dt

a = 1 , 2 ; a = l , 2 ,..., I; i = l , 2 ,..., r (4.9a)

(4.7')

(4.1 1 )

where g, denotes a functional relation. An important example is the electroneutrality condition for an electrolytic solution. Assume that there are s independent relations of such kind. These relations are independent of phases as long as the same number of species is present for all phases. There are ( r 2)(w - 1) conditions in (4.6') while there are ( r + 1)w independent intensive variables. The chemical equilibrium conditions in (4.7') are also independent of phases; thus there are n such conditions. Therefore the number f of free intensive variables are' f = r 2 - w - (s n) (4.12)

+

or J p a

=0

(4.9b)

Since the condition (4.9b) also implies (4.9a) by virtue of p)a being proportional to r?" to the lowest order approximation, we shall retain (4.9b) for our discussions in what follows. The conditions (4.6) are the usual equilibrium conditions'$* originally obtained for nonreactive systems by Gibbs while (4.7) is the condition for chemical equilibrium. This latter condition makes the equilibrium condition (4.6) slightly more general and perhaps a little more useful since chemical reactions are generally excluded from consideration in the existing literature when the phase rule is considered, although Gibbs discussed it, albeit differently from the present work, in his famous work.' Equation 4.8 is the condition required to prevent the external force from

+

+

For example, if there are two components in two phases and one reaction occurring in both phases and if s = 0, thenf= 1, whereas if n = 0 as well, thenf= 2. The chemical reaction thus reduces the thermodynamic degrees of freedom by one in this case. The phase rule (4.12) of course holds when all phases consist of the ( 7 ) This result appears to have been obtained first by Miinster, A. Classical Thermodynamics; Wiley: New York, 1970.

The Journal of Physical Chemistry, Vol. 91, No. 5, 1987

Irreversible Thermodynamics of Heterogeneous Systems same variety of species and the same reactions occur in all phases in addition to the usual assumptions on the nature of the boundaries. Provided the conditions (4.7')-(4.9') hold, the conditions (4.6') are also sufficient for equilibrium since the conserved variables do not change in time for the whole system if the system is in equilibrium. This can be easily seen if the equilibrium conditions are applied to (4.1) and the conservation laws are applied thereafter. We therefore see that the conditions (4.6!)-(4.9') are the necessary and sufficient conditions for equilibrium.

V. Thermodynamic Stability It is possible to examine some aspects of thermodynamic stability of the system in the large consisting of heterogeneous phases if the entropy eq 3.64 is used. For the discussions in the present section we shall assume that the external force does not change in time, i.e.

holds where

s p = ( a 2 s / a y 7 ayjfl))

for all a and i, so that we have

(5.8)

the usual stability theory follows from the conditions (5.4)-(5.6) except that the theory is for a global heterogeneous sytem and the globalfluxes are taken into account. We pursue the theory a little further by considering the time derivative of 6's. Since we have

(as/aq) J P / F= (as/ap) 1/ P =

- P ~ ~ / T D=L( a s / a M ' j )

X ~ / P=

(as/arp)

(5.9)

the first variation from an arbitrary state is given by

6s =

d+f/dt = 0

1193

a

1[ s q + p

a w-

r

5 pgi

i= 1

P

1

+ i=l acpa o srp] =l


(5.10) and consequently (5.7) and (5.9) yield L

Lest the results of the present discussion are applied beyond the limitations of the basic equations used, it is important to recall that there are some assumptions involved with (5.1) regarding the intensive variables; see section 111. Equation 5.1 implies that the global entropy S is a function of E$, P, My, and I'?)", i = 1, 2, ..., r; a = 1, 2, ...; a = 1, 2, ..., 1. We will abbreviate these variables with a single symbol y p ) , k = 1, 2, ..., q = 1, 2, ..., w where q denotes the number of independent variables in the phase as indicated. We denote the equilibrium entropy by S, which is the value of the entropy at the thermodynamic state satisfying the equilibrium conditions (4.6'), (4.7'), and (4.9'). Then the quantity of interest is

As = s - s,

60

(5.3)

Let us now observe a mathematical theorem2 regarding the variation of a function f(x): Ifflx) 3 0 and its first 2n order variations vanish, then 62n+1f(x) = 0 where n = 1, 2, ... . Thus in that event the first nonvanishing variation is 62n+2flx). Since by the equilibrium conditions

6s = 0 at constant E$, V, M = CaciW, the first nonvanishing variation is the second variation of S. When this second variation is convex downward, i.e. 62s

0

(5.6)

is satisfied, the equilibrium is unstable. These inequalities are the basis of the stability theory by Gibbs.' Since the relation

(5.17) see (3.9) and (3.10). Equation 5.15 means that the mass variation term in (5.13) splits into two components. To comprehend the implication of (5.13), we consider its particular cases when there are only two phases. A . 6P = 6 M p = 6I?j"Ia = 0. In this case we have

Eu


(5.18)

we have d dt

- 62s < 0 Since d dt

- 6E;

in that case. Thus we have the time derivative of ti2S that is positive if the system in the large is stable thermodynamically and negative if it is unstable thermodynamically. C. 6E; = 6P = 6I'?)* = 0. The rate of change in 62S in this case is given by

d +6Ei1= 0 dt

we may write (5.18) in the form

dS2S=[ 6 ( + ) - 6 ( - ! + ) ] dt If we set TI1 =

Thus if 6T

;6E$

T' + 6T, then In this discussion we shall assume that the reaction rates are uniform. Since the mass changes as pointed out in (5.14) and (5.15) have two components, one due to diffusion across the boundaries and the other due to chemical reactions, (5.27) may be written as

> 0 and if G((d/dT)E$) > 0, then d dt


- 62s > 0

(5.19)

This is precisely the case when the system is thermodynamically stable since if the temperature of phase I1 gets higher than that of phase I the energy (heat) will be transferred to phase I, that is

6(:

E;)

>0

(5.20)

or E)!

6(

0

(5.21)

then d

- 62s < 0

(5.22)

dt

Since (5.21) cannot be expected of a thermodynamically stable system, we may conclude that (5.22) is a manifestation of thermodynamic instability. B . 6ET = 6 M p = 6I'j"* = 0 . In this case we have d dt If 71 = as

(5.26)

dt

T" and plI = p1+ 6p, then the equation may be written

(5.28) where Ry = R y P The two components, however, are not completely independent of each other and hence cannot be considered separately. The first term on the right of (5.28) is basically dependent on the boundary conditions and changes in these boundary terms can regulate the reaction rates and vice versa since the mass balance must be maintained for the whole system. Glansdorff and Prigogine9 considered

(5.29) in the case of a single phase although they referred to boundary conditions that maintain the system far from equilibrium. They claim that for h2S < 0 -d6 2 S = - E 6 dt j=1

():

- 6Rj>0

and thus 62S is a Lyapounov function. We see that if the system is finite and boundary conditions play a role in keeping the system away from equilibrium the second variation in entropy, and even d S itself, cannot be independent of the boundary contributions such as the first term on the right of (5.28). Consequently, it is not really possible to infer the inequality from (5.28) unless the system is so large that the boundary contributions can be neglected. For a finite system the boundary contributions in (5.28) can upset the inequality (5.30) and it is possible to have cases where d dt

- 62s < 0 For a thermodynamically stable system if 6p > 0, then 6 ( - p ) < O

(5.23)

(5.30)

(5.31)

since it is within the bounds of experimental manipulation to change the boundary contributions. It is also possible to have the inequality

and as a consequence d dt

- 62s > 0

(5.24)

The inequality (5.23) is again in accordance with the Le Chatelier principle. Since for a thermodynamically unstable system

but owing to the boundary terms the system may end up to exhibit the inequality (8) For a local stability theory based on the local field equations presented in this paper, see Eu, B. C . J . Chem. Phys. 1981, 74, 2998. (9) Glausdorff, P.; Prigogine, I. Thermodynamic Theory of Structure, Stability and Flucfuafions:Wiley: New York, 1971.

Irreversible Thermodynamics of Heterogeneous Systems d dt

- 62s > 0

(5.32)


Therefore, although it is expected that (5.32) holds in general for a thermodynamically stable reacting system, there is no reason that two contributions in (5.28) satisfy the inequality separately. The Glansdorff-Prigogine criterion of stability should be considered valid only for a closed single-phase system and the boundary contributions destroy the Lyapounov property that has held well for processes other than chemical reactions so far, i.e., the processes considered in subsections VA and VB. D. 6E; = 6'v" = 6M: = 0. The imposition of the equilibrium conditions (4.6') and (4.7') implies that there are no global fluxes arising from the differences in temperature, pressures, and = 0 for a = 2, 3, chemical potentials. We therefore find 2). 4 and for all i and a. There remains only the shear stress to consider. Let us imagine a situation in which two phases of a fluid move in opposite directions or one phase rotates or slides on another phase of a smooth surface. Then there is a shear stress applied between them, and we would like to consider the global stability condition for the system. It is sufficient to consider a two-phase system consisting of a single component. The rate of change in 62S is then given by (5.33) where we have dropped the subscript for the species from the symbols. Since it is always possible to imagine that one phase is at rest, if there is no internal stress as is the case when (5.1) is assumed to hold, we may put, say 6$')2 =0 and thus consider the equation

Now to the first order 6$1)1

= +(I)

6r(l)1

where S(I)is a constant independent of 6I'(I)l. If the system is stable with respect to stress, then for 6I'(I) > 0 (5.35) since the system will evolve toward a state of a reduced stress when a stress is applied to it. Therefore, for a stable system we must have d - 62s > 0 (5.36) dt which is a manifestation of thermodynamic stability of the system with respect to stress. This inequality, however, is not always guaranteed since the variational evolution equation for 6I'(l)' can be nonlinear with respect to 6I'(l)I and there can be cases when -6r(I)IF ( 6 W l ) < o

(5.37)

where F is defined by (5.38) The inequality (5.37) can occur if F(6I'(l)') has multiple roots, say, three and one of them is unstable. In that case a variation from the unstable root will result in (5.37) and we may associate the inequality d - 62s < 0 (5.39) dt with instability. In conclusion, we have shown that the second variation a2S in the global entropy may be regarded as a Lyapounov function for processes other than chemical reactions that occur under the

The Journal of Physical Chemistry, Vol. 91, No. 5, 1987 1195 RESERVOIR

I

111

TI or

T2

w3

;

GAS

T

... ...

...

-

............ : : : : : : : : ; x3 ... ...

...

... ...

PISTON

. . AP . . . . j. . . . . . . . . . . . . . . . . . . . . . . . ... . CY LlNDER

I1

RESERVOIR

Figure 2. A cylinder connected to heat reservoirs. The walls except for w j are adiabatic at all times while w j can be made adiabatic or diath-

ermal. constraints of boundary conditions, and in the former case it is possible to formulate a thermodynamic stability theory of global systems on the basis of 62S and its rate of change in time. But such a formulation is generally not possible for reacting open systems.

VI. Efficiency of an Irreversible Carnot Cycle To illustrate application of the theory presented, we calculate the efficiency of an irreversible Carnot cycle by means of the formalism. The Carnot cycle is rendered irreversible by the presence of processes occurring irreversibly in a finite time interval, but it is still an idealization of real cycles, which is necessary in order to reduce the computational complexity involved with calculation for real cycles. Although idealized, the example represents a significant departure from the reversible Carnot cycle commonly considered in thermodynamics. Its important features are also different from the cycles considered in finite-time thermodynamid although the ultimate aim of the calculation is similar to what finite-time thermodynamics aims, Le., the efficiency of irreversible engines among other questions. The methodology used is also different. We will consider the following idealized irreversible cycles as an example for an approximation to what happens in a real cycle. As shown in Figure 2, the system (engine) is heterogeneous, consisting of two (infinite) reservoirs and a cylinder fitted with a frictionless piston. The reservoirs may not have to be infinite in size, but the essential result will not be impaired by the assumption of infinite reservoirs. The walls of the system are assumed to be adiabatic except for one face (w3)which can be made adiabatic or diathermal at will by a suitable device. The working temperature T of the working fluid is between the reservoir temperatures T , and T2. As in the reversible Carnot cycle there are four steps, each of which occurs in a finite time interval: Operation I: By absorbing heat from reservoir T I (subsystem I) at a finite rate the fluid (gas) expands. This step requires shutting d2 off with d, open. On completion of this step the field reaches the temperature T 1same as the reservoir temperature. The time span is T~ and the pressure and the volume are respectively p 2 and v,. Operation 11: The boundary w 3 is made adiabatic and the fluid is adiabatically expanded until its temperature becomes T again. The time span for this step is 72 and the pressure and the volume are respectively p 3 and V,. Operation 111: The boundary w 3 is made diathermal and dl is shut off while d, is open. Then by emitting heat to reservoir T2 (subsystem 11) at a finite rate the fluid is compressed to V, at p4. At the end of the process the temperature of the fluid is T2. The time span for this step is T ~ . Operation IV: Now the boundary w 3 is again made adiabatic and the fluid is further compressed to p1 and VI at which point the fluid reaches the temperature T,the working temperature of the fluid. The time span for this step is r4.

Eu


for this, we rewrite the divergence of @ in the following manner:

The total time required for a cycle to finish is therefore T = T~ T~ + T~ which is assumed to be finite. The above processes and the states involved can be schematically represented by the diagram T,

+ +

(6.2) Combining the second term on the right of (6.2) with EO;and remembering that the fluid sticks with the boundary, we may then write (6.1) in the form

kY


p4.v4, r2

Note that Q,is the amount of heat absorbed by the subsystem 111 while Q2 is the amount of heat emitted by it. In connection with the above model we make the following remark: Once the temperature of reservoirs and the working temperature is chosen for a given working material, the operation times T~ are fixed by the nature of the system and therefore they are not free parameters. In our analysis we will take some upperbound values to ensure each operation to reach the assigned physical state of the system as indicated. To facilitate the calculations we make the following assumptions: (1) The fluid is nonviscous so that the shear and bulk viscosity are equal to zero. Therefore there is no viscous energy dissipation and thus viscous heating. As was briefly mentioned earlier in this paper, dissipation is termed to mean a transformation of one kind of energy into another that is not available to the specific kind of work performed. Therefore, if a pressurevolume work is the work in question, then the heat generated by the accompanying viscous heating effect is a dissipation, but if a heat generation is the work in question, the pressure-volume "work" is in fact a dissipation because the latter tends to reduce the heat generated for a given input of energy. Therefore the concept of dissipation may be deemed opposite to that of availability used in thermodynamics. (2) The temperature relaxation time is much shorter than T ~ T ~ T3, , T.,, so that the temperature of the fluid is uniform throughout the subsystem 111 except for the thin interfacial region of width AI at the boundary w3, when operations I and 111 are carried out. This assumption implies that the energy dissipation essentially occurs in the interfacial region where there exists a temperature gradient between the boundary w j and the fluid. This assumption may be removed at the cost of complication in calculation if one computes the temperature profiles along the axis of the cylinder. (3) Subsystems I and 11, Le., the reservoirs, have an infinite heat content. This assumption makes it necessary to consider only the subsystem 111. We stress that these assumptions are not mandatory in principle, but useful to make the calculation of energy dissipation feasible without unnecessarily great computational effort. They are in fact not needed at all when the formal expression for the efficiency of a cycle is calculated but will be needed when the efficiency is explicitly calculated in an analytical form in terms of the parameters characterizing the cycle. The global energy balance equation for operation is given by

Let us divide the cylinder up into infinitesimal cylinders of equal length where the temperature may be regarded as uniform to a good approximation. Then we find

where Qui stands for the boundaries of the ith infinitesimal cylinder. Therefore the heat loss per unit time by the ith cylinder is the heat gain by the (i+l)th cylinder, etc. Since there is only one infinitesimal cylinder adjacent to w3, we see that there is only one term left in (6.5): ,

Since the second term on the right of (6.3) is in fact the work per unit time, we define (6.7) When the results above are combined, we finally obtain the energy balance equation of the form dE" dt

-

deG dW" +-+E" dt dt

Since there is a cycle of operations, the energy balance equation for a cycle is obtained if (6.8) is integrated over time and summed over a:

The left-hand side vanishes by the first law of thermodynamics: (6.10)

(6.1) where fl denotes the boundary w 3 and a3,and EO;stands for the viscous dissipation which may be put equal to zero if the assumption on no viscous dissipation is imposed. We will keep it for the moment for the sake of generality. If the fluid sticks at the boundary, then 'i = 'ioand the second term on the right of (6.1) vanishes. Since the temperature is generally nonuniform in the axial direction of the cylinder owing to the heat transfers involved during operations I and 111, there is a dissipation of heat (entropy production) which tends to reduce the work. To account

The second term on the right is the work over a cycle: (6.1 1) The last term is the total dissipation of energy which is always positive by the Carnot theorem:

E = x S T n d tZ"(t) "

0

0

(6.12)


Irreversible Thermodynamics of Heterogeneous Systems The heat transfer term on the right of (6.9) may be written as follows:

FA"

deqg deqf, deqB' dt = AT'dt dt + A" dt dt

dt

= -Qi

+ Q2

(6.13)

1197

where K ( T ~=) X(T,)A/AI with A denoting the cross section of the cylinder. To obtain (6.22) we have used the following approximation for the volume integrals in (6.20):

The sign assignment is in agreement with the convention adopted in this paper and with the sign convention on work. Since operations I1 and IV are adiabatic, there is no contribution to the heat transfer term from them whereas the heat transfer is -Q, in the case of operation I by the sign convention adopted in this paper and it is Q2 in the case of operation 111. Thus combining (6.10)-(6.13) into (6.9), we obtain W = Q l - Q2 - E

(6.15)

This calculation is rigorous under the assumption that the temperature is uniform in the rest of the length excluding Al, provided AI is sufficiently small. On the other hand, under the sign convention adopted

The derivation of this formula for efficiency does not require the assumptions stated previously. It is a rigorous statement, but an approximate form for qeffwill be obtained if qca is calculated under the assumptions mentioned before. Since

(6.23)

(6.14)

and thus the efficiency of the cycle is


qeff

= W/QI = 1 - Q d Q l - B/QI

(6.16)

E30

the efficiency Teff of the irreversible Carnot cycle is less than that of a reversible Carnot cycle that operates on absorption of Q, and emission of Q2: 7% = 1 -

(6.17)

QdQi

if the linear law is taken for heat transfer in a manner consistent with (6.21). Since we may write

where

That is (6.18)

qeff

This is precisely the content.of Carnot's theorem on the efficiency of reversible cycles and (6.16) means that a reversible work is a maximum work; cf. (6.14). More rigorous proof of this statement is given elsewhere.I0 The quantities appearing in (6.15) can be calculated in the following manner if the various assumptions are exploited. For the purpose we first take the constitutive relation for heat flux which is commonly used in linear irreversible thermodynamics:

e = -A?

In T = -X?T

(6.19)

x = X/T

For the cycle under consideration the energy dissipation

B

is

(6.23) describes the temperature evolution. It is easily found from the solutions of (6.23): T(t) = TI

YI = K(Tl)/CI,

AI [TI - T(t)]

Y2

= K(TZ)/CIII

which are the temperature relaxation parameters. They generally depend on temperature. With this result we can calculate Ql, Q2, and E:

...

& = - QTI) -

(6.24)

where T is the initial temperature and

Q2 =

Since it is assumed that there exists a significant temperature difference between the boundary w3 and the working material (gas) only in a thin interfacial region of width AI, we may write

- 7,) exP(-Ylt)

T ( t ) = T2 + ( T - T2) exp(--v2f)

Q1 =

(6.20)

- (TI

B

6" dt dt

ded

= CI(Tl - T )

s," dt d,qf,I' dt = CIII(T- T2)

= (CI/2Tl)(Tl - T)'

+ (C111/2T2)(T-

T2)23 0

(6.25a) (6.25b) (6.25~)

where

2; = CI[1 - exP(--vlrl)l

Gn = CI11[1 - exp(-y273)1

for a = I

It is convenient to define the following abbreviation:

0=

CI[1/CI

With the results above, we find the efficiency Therefore we find

(6.22) (10)

Eu, B. C.unpublished.

The working temperature T i s arbitrary. It is preferable to optimize the efficiency of the cycle. The desired optimization can be achieved if the availability is maximized or, to put it another

1198


Equation 6.26 appears to be singular in T at first glance but it is not so, since 6 must tend to zero at the same rate as the rate of T - T I . The physical reason for this is in the fact that as T tends to TI it takes longer to transfer heat Q1 from the reservoir to the fluid. In other words, 0/(1 - T / T , ) = finite and similarly 8/(T/T2 - 1) = finite. This is easy to see from (6.25a,b) and the definition of 8. Careless use of (6.26) can lead to a nonsensical result. Equation 6.28 does not have such an apparent singularity problem. We remark that the method of derivation of (6.28) is entirely different from that for (6.29). It is useful to stress the difference in the methods of calculation for (6.28) and (6.29). For (6.28) we have calculated the dissipation Z accompanying the irreversible processes in the irreversible Carnot cycle whereas for (6.19) the definition of efficiency is the same as that of the reversible cycle since

TABLE I: Comparison of we- and wra TI/ T , 0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1.O

‘ICA

‘Icff

1 .oo

0.50 0.45 0.40 0.35 0.30 0.25 0.20 0.15 0.10 0.05 0

0.68 0.55 0.45 0.37 0.29 0.23 0.16 0.11 0.05 0


TABLE 11: Efficiency of Real Heat Engines (Plants) heat plant T,, K TI, K ?OM, ?’E Teffl West Thurrock (U.K.) -295 838 36 32

coal-fired steam plant (ref 13) CANDU PHW nuclear reactor (ref 14) Larderello (Italy) geothermal steam plant (ref 15)

TCA,

40

298

573

30

24

28

353

523

16

16.1

17.5

way, if the dissipation is minimized. It is easy to find the value of T minimizing E : T = (1

+ 0)T2/(0 + T2/TI)

(6.27)

On substitution of (6.27) into (6.26), there follows the efficiency for the optimized working temperature: (6.28) which is exactly one-half the efficiency of the ideal Carnot cycle. Curzon and Ahlbornl’ considered an irreversible Carnot cycle similar to the present one except that there are two working temperatures instead of one as in the present case. By using a method different from the one used here, they calculated the power output and the efficiency of the cycle. Their efficiency formula is given by TCA

=

- (T2/T1)”2

(6.29)

The numerical values for efficiency given by (6.28) and (6.29) are quite similar except for small temperature ratios as shown in Table I. In Table I1 we comapre the predictions by (6.28) and (6.29) with the observed performance of rea! heat engines. The efficiencies predicted by (6.28) are surprisingly close to the efficiencies of real heat engines as is the case with (6.29). Formula (6.29) shows that the efficiency of an irreversible cycle reaches that of the reversible cycle whereas (6.28) indicates that an irreversible cycle never attains the reversible cycle value owing to the dissipatiort of available energy. Clearly, the latter values are more in line with the Carnot theorem. In fact, fluid dynamic calculations based on nonlinear transport coefficientsI2 indicate that heat transfer is not enhanced by making the temperature difference large; on the contrary, it becomes more inhibited as the temperature gap increases owing to ensuing nonlinear effects. Therefore, taking a large temperature gap would not improve the efficiency of a cycle. The factor of 1/2 in (6.28) arises from the presence of irreversible heat transfers in the cycle. The parameter 0 depends on the relaxation times and operation time spans T~ and T~ in addition to the heat capacities. Since heat capacities are approximately constant, we may approximate 8 by the form 6 = [ 1 - exp(-rz.3)1/[1

= T273/?171 =1

- exP(-rl7l)l

if Y I T I , 7 2 7 3

Irreversible Thermodynamics of Heterogeneous Systems - American

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