H. J. M. Honley
Institute for Materials Research National Bureau of Standards U.S. Department of Commerce Boulder, Colorado 80302
Thermodynamics of Transport Phenomena in Membrane Systems
Few theories have the power, scope, and versatility of classical thermodynamics. Yet thermodynamics does not discuss the dynamics of a natural process and so it cannot describe the large and important class of phenomena called transport processes, the rate processes which have their origin in the fundamental tendency of any nonequilibrium system to always approach equilibrium. Nevertheless, thermodynamic concepts can be applied to transport phenomena if the basic foundations of classical thermodynamics are re-examined and extended. This has been done and the theory known as nonequilibrinm thermodynamics or irreversible thermodynamics has been developed. While noneauilibrium thermodvnamics is well established as a complementary theory to thermodynamics and well understood theoretically for so-called mtinuous systems, it is less understood, but more widely applied, for so-called discontinuous systems. These terms have the following meaning: a system is called continuous if the intensive state variables (e.g., temperature and density) are continuous functions of the space co-ordinates and of time. Conversely a system is called discontinuous if the state variables are not continuous functions of the space coordinates but vary discontinuously a t certain regions in the system due to the presence of discrete boundaries. In thermodynamic terms, a system with state variables which are continuous functions of the space coordinates is said to consist of a single nonuniform phase. If this is not so the system is said to consist of several nonuniform phases and to be heterogenons.' A system which contains a membrane is discontinuous or heterogeneous. We wish to discuss some aspects of the thermodynamics of transport phenomena in systems with a membrane, and in so doing will examine and compare the application of nonequilibrium thermodynamics to discontinuous systems as opposed to the application of nonequilibrium thermodynamics to continuous system: a difference students often find confusing. In the fimt two sections we review the application of
' In thermodynamics, a system is homogeneous (or consists of one ~ h a s e if ) it is physically and chemimlly uniform throughout. A nonequilibrium system is never uniform throughout and, in a literal sense, therefore consists of an infinite number of phases. But this is in codict with the usual concept of a multiphase system: one usually associates a multiphare syctem with the idea of discontinuities or boundaries between certain ngions, and calls each region a separate phase. We conform to common practice and define a continuous system as a nonuniform single phaae, and a. discontinuous system as heterogeneous or multiphase. 2 The flux of thermal energy consists of two contributions: (a) "pure" heat conduction caused by molecular interaction and ( b ) thermal energy transported by the diffusion of molecules.
thermodynamics to discontinuous system and to continuous systems respectively, as found in the major texts (1-5). In subsequent sections we discuss the applications to membrane systems. The reader may skip the first two sections if he is only interested in the latter topic. Discontinuous System% Introduction to Nonequilibrium Thermodynamics
I n many texts (8, 3, 5) a simple model is used to explain discontinuous systems and, more importantly, is used as a starting point for discussions on the thermodynamic approach to transport phenomena. Figure 1
Figure 1. The model constructed b r purposes of dircurrion; two reservoirs, I and I1 are connected by a membranelll.
depicts this simple model which is a system composed of two reservoirs I and II connected by a membrane III. The memhrane can be a porous plug, capillary, rubber partition, or the like. The model is convenient because it represents a common experimental arrangement; experimentally one considers the transport phenomena which take place when a number of the state variables are constrained at constant, but different, values in I and II. To maintain this difference, state variable gradients have to be present in the membrane subsystem III, so III, at least, is in a nonequilihrium state. As the system tries to approach equilibrium (and thereby to eliminate the gradients), each gradient will give rise to a conjugate flux of some entity associated with it. For example a temperature gradient will be responsible for a thermal energy flux2 throughout the system. I n nonequilibrium thermodynamics, gradients are given the symbol X, fluxes the symbol J and the general expression is written down
for n gradients, assuming a linear relation between J and X. The rate constants L'x are the transport coe5cients. I n the nomenclature of nonequilibrium thermodynamics, however, Lu are known as phenomenological coefficients while X is known as a thermoVolume 44, Number 12, December 1967
/
717
dynamic force. One has to remember that eqn. (1) is a general expression for many long-familiar phenomena and that the subscripts are dummy indices: for instance let Xt be a temperature gradient, grad T, and let Ji be the conjugated thermal energy flux which we call q, we could have
and chemical potential of component k respectively, and the symbol A represents a difference of a quantity between I and 11,i.e., AT TI - T,,. The differences AT/T2, A(pn/T) are identified as gradients (forces) and dl Ur/dt, dMkl/dt, etc., as fluxes and defined by
--
J, Ji
which is Fourier's Law with X the coefficient of thermal conductivity. Equally familiar is Ohm's Law
where dS is the total change of entropy, d 8 the entropy change of the system with its surroundings and d,S the entropy change caused by irreversible reactions in the system. By the second law diS/dt is zero at equilibrium. Referring to Figure 1; dS/dt can be determined if it is assumed that the whole system is closed and that I and II are so large compared to 111that all irreversible behavior occurs in III. If this is so, classical thermodynamics can be applied to I and II separately. Specifically, most texts consider the case in which the temperature is constrained, i.e., is fixed a t T I and TI,in I and 11 respectively, and a gas of n components is distributed throughout the system. Provided that the pressure of the gas is low enough (see the section on Membrane Systems below for what is meant by "low enough"), it is experimentallyfound that there is a couplmg between an energy and matter flow across 111-this phenomenon is called thermal transpiration. When the rate of creation of irreversible entropy is calculated (the details depend on the background material introduced in any given text) it is found to be of the form (2),
where dtUX/dt and dMBZ/dt represent the rate of change in I of energy exchanged between I and 11,and of component k respectively. The quantities for I and 11 are related from the conservation of energy and matter; i.e., d, U'/dt = -dt Urr/dt and dhlnl/dt = -dM,"/dt. The terms T, im are the mean temperature 71 8 / Journal o f Chemical Education
-diU1/dt
=
diUU/dt
=
-dMd/dt
=
dMxl'/dt
X, = ' A T / T 1 Xi
with E = -grad 4, where I is the current density, c the conductivity, E the electric field intensity, and4 the electrostatic potential. Note that eqn. (1) states that J , depends not only on its conjugate force X t but also depends on X I (i # k) provided that the determinant Lu exists. The linear relation between Jtand X1 is called coupling, an example of which is given in the next paragraph. Equation (1) is quite general and is a useful empirical relation, hut J and X are defined thermodynamically by considering the rate of creation of irreversible entropy, d,S/dt, for the system in question. This term come from the second law expression for the change of total entropy
=
=
(3)
A(dT)
This latter force can be written in experimentauy measurable quantities,
where ht is the specific enthalpy of component i, Ap is the pressure diierence across 111at the mean pressure p, and R is the gas constant. Equation (2) has the form
and using the linear law we have,
Equations (5) and (6) represent a coupled phenomenon because it is seen that the flux of energy, J,, depends not only on its conjugate force X , hut also on X,. Conversely the flux of matter of species i depends on X I (k = 1,2,. . . ,n,i# k ) and on X, as well as on its conjugate
x,.
To complete the above formalism another statement is made. It is stated that the coupling mechanism is symmetric, i.e., that
This is a statement of the famous Onsager reciprocal relations. We have more to say on the relations in the next section. The conventional way to look at a discontinuous system is thus to look at the system as a whole. No information is forthcoming on the irreversible processes occurring inside the membrane, only the effectof these processes on I and 11is noted. In many cases this total treatment is quite sufficient and is straightforward, convenient, and practical. However, recent work on membrane systems requires a more sophisticated outlook, especially for biological problems in which a membrane often plays a dominant role ( 6 , 7 ) . Equations (5) and (6), or their equivalents, can also be misleading. For instance, eqns. (5) and (6) suggest that coupling between the energy and matter transport processes mill occur, whereas this is not always so: Coupling does not take place if n = 1 (onecomponent gas in the system) and if the pressure is high. It is possible, in this situation to have an energy flow if AT # 0, and a matter flow occurs if a pressure difference is applied across III. However, this matter flow obeys the law of viscous
flow and we have no provision for viscosity in eqn. (3). In other words the equation is not complete. To clear up these points we look a t the membrane system from the continuous viewpoint and show that the continuous and discontinuous approaches are compatible. Continuous Systems
It is best to introduce the fnndamentals of nonequilibrium thermodynamics for an isotropic fluid continuous system by macroscopic postulates. We do this below, referring the reader to the literature for details (1, 4). These macroscopic postulates have to be proven by experiment. Later, if required, they can be verified by statistical mechanical arguments, but by insisting on a prior experimental verification, the diiculties associated with the application of statistical mechanics to nonequilibrium systems can be avoided. All experimental evidence available indicates that the postulates are valid (1, 8).
J,, Jz and n fluxes of thermal energy, matter and momentum respectively, and the gradient terms are forces. The derivation of eqn. (9) requires the following definitions. The density:
where pn is the density of species k. The center of mass velocity:
where vl: is the velocity of k. The velocities are the average velocities in dV a t r with respect to a fixed external co-ordinate system. The diffusionflux of k: We also require the relation:
Local Equilibrium and Entropy Production: Postulate 1
In the continuous treatment the state variables or parameters have local values and are a function of position r and time t [e.g., the density p written in full is p(r,t)]. The vector r locates the position of an element of volume dV, typical of the fluid as a whole, which is small compared to the dimensions of the vessel containing the fluid but large compared to the mean free path of the molecular constituents. Now the local value of a parameter is the average of the parameter in the volume element. It is assumed that the variation of the parameter in dV is negligible and the volume element is said to be in loud equilibrium. Postulate 1 states that equilibrium thermodynamic functions of state can be applied to each of these elements provided that the thermodynamic entities have their local values. Accepting the postulate we can thus write the Gibbs relation for the element (in specific quantities),
"
Tds
=
du
+ pdv - C
Linear Relations: Postulate 2
I t is postulated that a linear relation between the fluxes and forces is valid, i.e., that eqn. (1) is valid for the system. The Onsager Relations: Posfulate 3
It is postulated that if a linear relation [eqn. (I)] is written for the fluxes and forces and if the fluxes and forces are those appearing in eqn. (9), the matrix of the coefficients of eqn. (1) is ~yrnmetric.~We have introduced the relations by eqn. (7). Next we wish to introduce, again with the minimum of explanation, three useful working theorems. Curie's theorem. Equation (9) can be split into segments of a given teusorial character, i.e., with
ptdzh
(8)
k-1
Here s, u, and v are the specific entropy, energy, and volume respectively; p. is the chemical potential of species k, x,,is the mass fraction of k and T the temperature. We should write in full s(r,t), u(r,t), T(r,t) etc., but will omit writing the dependence on r and t for brevity. When eqn. (8) is differentiated with respect to time and use made of the conservation equations of mass, momentum, and energy we obtain a generalization of eqn. (4) for the rate of creation of entropy. We write down the result for a system of n chemical species which is in a nonequilibrium state due to the presence of gradients of concentration, local velocity and temperature. We do not consider here the iduence of external forces or of chemical reactions. The result is (1)
Here u is the rate of creation of entropy per unit volume,
The J's and X's are to be identified with the corresponding entities of eqn. (9) and the subscripts refer to the tensorial rank of the entities. Curie's theorem states that entities of a given tensorial rank do not couple with entities of another tensorial rank4. Linear laws can therefore be written down for u1 and u2,viz,
where L,* and d? are the appropriate scalar phenomenological coefficients. Linear transform of j'luzes and forces. A linear transform of the fluxes and forces can be made to simplify 8 We do not wish to elahborate on the Onsager rel&ms. See Chap. 4. Ref. (4), 4 This is not a complete statement, see Ref. (4), Chap. 4, and Ref. (9).
Volume 44, Number 12, December 1967
/
71 9
the formalism of a given experimental situation. This transformation leaves the entropy production invariant and does not impair the validity of the Onsager relations. These transformations are important, for instance we see in eqns. (4) and (9) that the force conjugate to the diffusive flow contains a gradient of temperature factor. It is mathematically convenient to remove the factor. This can be done by using the identity h
T grad ( p r / T ) = (grad p r ) ~- 2 grad T T
(11)
and defining a new energy flux J,'
Equation (9) then becomes, 1
o = -- J
T2
' . grad T - 1T k p"l
Jx
. (grad P I ) T -
The subscript T on the gradient indicates that the operation is to be carried out at constant temper* ture. Prigogine's theorem. At mechanical equilibrium any reference velocity can be used to define the isothermal diffusion flnx of species k and leave the entropy production invariant. Thus instead of eqn. (10c) we may define J n by J I = PX(%
- v*)
(loco)
where v' is an arbitrary velocity such as the velocity of another component or the velocity of an external reference. A system is in mechanical equilibrium if the pressure is uniform throughout provided external forces are absent. If external forces are present then they are exactly balanced by the gradient of pressure. After this somewhat lengthy introduction we next apply nonequilibrium thermodynamics of continuous systems to a system with a membrane and show that the results are compatible with those of the first section.
Membrane Systems
While it is known that a linear relation frequently represents the flux-force relation for many membrane systems, it is not immediately obvious that the remainder of the postulates can be applied to a discontinuous system. Or, to be more specific, it is not obvious whether Postulate 1 is valid for a membrane system because (while local equilibrium is a reasonable concept to conceive for a continuum) local equilibrium requires further thought to visualize for, say, the system in which a rarefied gas is flowing through a membrane. However, we will refer to the system in Figure 1 to show that the concept of local equilibrium is still valid for this situation, Once this is done we can then write down the thermodynamic continuum equations. We consider the simple example where the system of Figure 1 contains a single gaseous component ( n = l ) and subsystem III is a membrane of porous material. Chemical reactions or the influence of an external force will not be considered. We still assume that I and II 720 / Journal of Chemical Education
have constant temperatures of TIand TI, respectively. This example is used for discussion purposes; the ideas developed are perfectly general. Three experimental situations are associated with this example. The situations are classified in terms of the gas pressure. We call the gas pressure high if the mean free path of the gas is small compared to a characteristic mean pore radius of III. The pressure is low if the mean free path of the gas is large compared to the radius, and the pressure is intermediate if the mean free path is of the same order of magnitude as the radius: Case (a) the gas pressure is high. Case (b) the gas pressure is low. Case (c) the pressure is intermediate. Usually either the experimental situation of case (a) or (b) is encountered in practice. Case ( e ) is more complicated to understand theoretically and experimentally. In particular there is evidence that a macroscopic linear law does not adequately represent the flux-force rela tion for the processes taking place under the conditions of (c) (10). If this is so, the structure of nonequilibrium thermodynamics has to be reexamined, and we do not wish to discuss this here. The discussion is therefore applicable to case (a) or (b) only. As we have stated, the results obtained for experimental conditions which give rise to case (a) or (b) are different. I n case (a) there is no diffusive flow and coupling between the energy flow (arising from the difference in temperature between the reservoirs) and a matter flow is not observed. A matter flow takes place if a pressure difference is applied across III but the flow rate obeys the laws of viscous flow. In case (b) there is coupling between the energy and matter flows and the matter flow does not obey the laws of viscous flow; we then have the phenomenon of thermal transpiration. Since our object is to show that local equilibrium is a valid concept, we have to demonstrate that 111 can be subdivided into volume elements with dimensions of the order of a few means free ~ a t leneths h which are homogeneous and represent III as a whole. Case (a). The mean free path of the gas is small compared to the pore size: the statistical mechanism involves the mechanisms of the gas/gas collisions. Hence, to all intensive purposes, the membrane merelv behaves as a boundary. Figure 2 represents III for case (a). Membrane particles are shown by open circles and the "path" of the gas by the shading. To logically select a representative volume element, one must ignore the particles and select the element in the gas phase. An example is given by the crossFigure 2. Diogromotic represento- Shading in the figure, open circler depict tion of case (01: of the membrane, the Thus, although there are rhoding depicts the gas, and the two phases in 111, (the gas cross-shading depicts a reprerentoand the particles) the live volume element.
-
particle phase is not expected to contribute in a thermodynamic sense and so, thermodynamically, III is a one-component single nonuniform phase or a continuum. The phenomenological postulates are valid when applied to this continuum. Case (b). The mean free path of the gas is large compared to the pore size: Figure 3 depicts III for this case. The open circles denote the particles as before, but because the gas is rarefied, a possible path is shown by a line. To establish a volume element in local equilibrium component. This is r m onable because the statistical mechanism predominately involves gas/particle collisions as opposed to gas/gas collisions for case (a). The shading represents a typical volume element. The system thus approximates a thermodynamic twocomponent mixture or solution. Provided that the membrane structure is homogeneous, III approximates a two-component single nonuniform phase (or a continuum) and the phenomenological postulates are valid. By arguing in this manner, the entire discontinuous system (i.e., I, II, and I I I ) is very similar to the continuum discussed in the section on Continuous Systems, athough of course, it consists of several nonuniform phases and is therefore heterogeneous overaL6 We assume that the postulates and theorems of continuous systems are valid and can be applied to each phase of a discontinuous system, provided that the state variables can be averaged in a manner suitable for macroscopic analysis. picts the gas path and the shading o reprerentolive volume element.
Enlropy Production in a Membrane
S i c e we assume that the system can be divided into elements of volume which are in local equilibrium, the equation for the entropy production in the membrane can be written down. We remain with the example discussed in the section on Membrane Systems and further specifically assume that the membrane is stationary and that the membrane/gas system is an isotropic medium in mechanical eq~ilibrium.~ The entropy equation is the same as eqn. (9) [or eqn. (13) in terms of the flux J,']. But the summation of the term
6 Subsystem III is frequently a capillary: it is rather di5oult to equate the ideas on local equilibrium to a capillsry. However, provided that allowance is made for the geometric facton involved, a capillary system behaves very much like s. porous membrane system. This point is discussed with respect to the "Dusty Gas" theory, see Ref. (11). 6 See Ref. (7) for the removal of these restrictions
runs from 1 to n, where n is the number of components including the membrane. Since we restrict ourselves to a single gaseous component, we designate the gas as component 1 and the membrane as component n. For convenience we write eqn. (13) [or eqn. (9) if desired] as
J.
. (grad k
I:
)
~ --
P:grad u
(14)
and examine the modifications to it [or to eqn. (9)] for case (a) and case (b) of the preceedii section. Case (a). In the limit in which the representative volume element consists entirely of the gaseous component, p, = 0. Thus from eqns. (10c) and (10d) we have n-1
Jn=O
and k
C = Ib = O
The diffusion contribution in the entropy production therefore vanishes. Also in the limiting case when p, = 0, we have that p + pl from eqn. (10a). Therefore grad u = grad VI. Thus Lim c = 0-0
1 0:grad VI --T1' J ' . grad T -T
(15)
The gas flowing through the membrane consequently obeys the laws of viscous flow, hut there is no coupling between the two terms because this is forbidden by Curie's principle. I t is standard procedure to show that for a capillary of radius a, and length 1, the second term on the right hand side of eqn. (15) yields, with the linear law, Poiseuille's formula for viscous flow (13). For instance, the viscous flow caused by a pressure drop dp across 1 is given by the equation
where JVi. is the flow and the terms q, p, T and R represent the gas viscosity, the mean pressure, the temperature and the gas constant respectively. Case (b). The subsystem III is a thermodynamic two-component mixture so neither pl or p. is zero. However p, >> pl, hence p -+ p. Because of this, the last term on the right hand side of eqn. (14) vanishes in the limit which follows from eqn. (lob). Thus
The gas flow through the membrane is the result of a diffusive mechanism, and coupling between the matter flow and heat flow is allowed by Curie's principle. Equations (15) and (16) can clearly be generalized for any number of components provided the limiting conditions are valid. We make two remarks, the first concerns the fact that we have to decide from experiment whether the membrane is an effective component or otherwise, i.e., whether to put p, = 0 or p, # 0 in the averaging scheme. We point out that this decision is not peculiar to discontinuous systems; this decision is made for all physical experiments, but, in most cases, the effect of the Volume 44, Number 12, December 1967
/
721
membrane (or container in more generd terms) is negligible and can be ignored. The second and more important remark is that we now have a control parameter p for a discontinuous system which permits us to decide on the nature, viscous or diffusive, of the flow process in the membrane. There was no control in the equations of the first section; remember that it is implied there that the matter flow is always diffusive, and if n = 1 (one-component gas), Jl exists. This is obviously not always so if the mean free path is small compared to the membrane radius.
where sr is the specific entropy of component k. With eqn. (21) and the identity
eqn. (20) becomes 1 -"T
u =
n-1
J '"
x
. grad T --T k - 1 JP . T grad (px/T) -
$ P: grad u
(23)
where
Diffusion Flux With the Membrane as a Reference
We have shown in the previous section that the discontinuous and continuous outlooks are compatible. In this section we discuss a further feature which should be considered when analyzing a discontinuous process. The diffusion fluxes measured in the laboratory are not those defined with respect to the local center of mass but rather are defined with respect to an external laboratory co-ordinate system, and the membrane itself is usually conveniently chosen as the reference (7). The problem is analogous to that of multi-component diffusion in a liquid in which it is far more convenient to measure the diffusion flux of a species with respect to the solvent than to measure the flux with respect to the local center of mass. However, we have Prigogine's theorem which permits us to change the reference velocity for Jx without altering the entropy production. But Prigogine's theorem is only proved if the entropy production is of the form given by eqn. (13). We must examine the more general relation [eqn. (9)] if a relationship between this equation and eqns. (5) and (6) is to be established. A new flux Jbmisdefined with respect to the velocity of the membrane vmby Jk"
=
and we have used eqn. (19). If, however, the relation eqn. (11) is substituted into eqn. (23), we obtain 1 Ja' . grad T - -TB
=
x
"-1
. (grad p x ) ~-
JP
Tk=l
1 P: grad u T
(25)
a result which would have followed immediately on application of Prigogine's theorem to eqn. (13). Let us now refer once again to the simple example of two components (Fig. 3). The physics of the heat and matter flow process through a porous medium has not changed by the redefinition of the diffusion flux, but we have an interesting and convenient result for case (b) because Lim JP
(k = 1, n)
= Jk
(26d
n-P"
therefore Lim Jp = J.
P*(V~ -V)
(26b)
0-rn
where k = 1, 2, . . . , n, with the membrane designated as the nth component. We also have, with eqn. (W,
Thus, Lim u
=
--1 J . grad T - -
P-pn
T4
'
"-1
x
T k - ~
J*
. T grad ( d T )
(27)
. (grad PI)T
(28)
Because the membrane is stationary we have Jxm =
ptvt
(17)
P-PI
and Jx = J P
- mu
Note that J"'" = 0
because the membrane has been designated as component n. Substituting eqn. (18) into eqn. (9), we have
" uT
pr
gad ( d T )
k-1
To simplify this expression the general form of the Gibbs-Duhem relation is required:
722
Limn =
/
Journal o f Chemical Education
--1 J ' . grad T -TP
x
n-1
T k - l
JI
I n other words the choice of the frame of reference for the diffusion flux does not influence the final working equations. Simplified Expression for the Entropy Production for Case (b)
Finally, to reconcile the notations of the fimt and fourth sections we write the gradients as differences. This can be done because in practically every experimental case the vector fluxes passing through the membrane will flow perpendicular to the membrane surface and will have the same value at all points on the surface. The vector notation, therefore, can be dispensed with and the gradients replaced by differentials and the fluxes by flows. For example the entropy production due to vectorial processes at a point in the membrane c m be written, for the x direction (Fig. I),
Further simplification follows if it is assumed that the flows are constant in the membrane. If the flows are constant, we can integrate across the membrane:
summation from k to n in eqn. (32) now includes the membrane as component n. We have thus reconciled the discontinuous and continuous approach to the phenomenon of thermal transpiration. Acknowledgment
We are grateful for valuable discussions with Dr. D. C. Mikulecky. Literature Cited
or, alternately,
(1) DE GROO', S. R., AND MAZUR, P., "Nonequilibrium Thermodynamics," North-Holland, Amsterdam, 1963. (2) DE GROOT,S. R.,"Thermodynamics of Irreversible Processes," North-Holland, Amsterdam, 1963. I., "Introduction to the Thermodynamics of (3) PRIGOGINE, Irreversible Processes," Thomas Press, Springf~eld,Ill.,
The integration is carried out between the limits of 0 and I where 1 is the length of the membrane (Fig. 1). The result [writing it for eqn. (31) only] is
-
p, and consequently We have assumed the limit of p omitted the superscript m from J , and Jx. The differentials are thus replaced by differences A, and v is now the entropy production for the membrane as a whole. (We refer to the literature (6) for discussions on the validity of the required assumption that the values of T and pk (which can actually be measured) are the same :IS the values a t the surface of the membrane.] It is seen that we can return eqns. (5) and (6) hut the
105s.
(4) F ~ r r s ,D. D., "Nonequilibrium Thermodynmnios," McCraw-Hill Book Co., Inc., New York, 1962. (5) DENsloa, K. G., "The Thermodynamics of the Steady State," Methuen, London, 1958. (6) KATCAAI^SKY, A., AND CURRAN, P. F., "Nonequilibrium Thermodynamics in Biophysics," Harvard University Press, Cambridge, Mass., 1965. (7) MIKULECKY, D.C., AND CAPLAN, 5. R., J . Phvs. Chem.,70, 2 1114 I l O f--,. iR\~ -"-" (8) MILLER,D. G., C h m . Reu., 60, 15 (1960); Am. J . Physics, 24, 555 (1956). (9) WEI, J., Ind. Eng. Chm., 58, 55 (1966). (10) HANLEY, H. J. M., AND STEELE,W. A., Tram. Faraday Sac., 61,2661 (1965). 1111 MASON.E. A.. EVANS.R. B. 111. AND WATSON.G. M.. J . &ern. phis.,38, isas (19631.' L. D., AND LIFSAITZ,E. M., "Fluid Mechsnics," (12) LANDAU, Pergamon Press, Inc., Oxford, 1963. ".\
. .
Volume 44, Number 7 2, December 1967
/
723
