Ind. Eng. Chem. Res. 1999, 38, 2515-2522
2515
REVIEWS Multicomponent Diffusion C. F. Curtiss† and R. Byron Bird*,‡ Theoretical Chemistry Institute, Department of Chemistry, and Chemical Engineering Department and Rheology Research Center, University of WisconsinsMadison, Madison, Wisconsin 53706
Historically there have been two major formulations for the mass-flux relations in multicomponent diffusion: (1) a generalization of Fick’s law in which the mass flux is written as a linear combination of concentration gradients and (2) a generalization of Maxwell’s expression in which the concentration gradient is given as a linear combination of the mass fluxes. The thermodynamics of irreversible processes has made it possible to generalize these expressions to include thermal, pressure, and forced diffusion. Associated with these two formulations, there are various definitions for the multicomponent diffusivities. Here the interrelations among the variously defined diffusivities are given, some connections with molecular theories are made, and several neglected publications are cited. 1. Introduction
F books1-9
The appearance of many dealing with multicomponent diffusion reflects a resurgent interest among workers in separations science, combustion, and biological processes, who have to deal with diffusion in systems with three or more chemical species. In these books, and in other works as well, several references10-15 on multicomponent diffusion have apparently been overlooked, and we feel that it is appropriate to call attention to these references and show how they are related to other works in the field. Our own immediate interest in reexamining the subject of multicomponent diffusion results from our wish to improve and extend our recently published studies on diffusion in liquid mixtures containing polymers.16,17 This molecular theory led to expressions for the mass fluxes of the Maxwell-Stefan form but predicted that the mass fluxes would be dependent on the velocity gradients in the system. Such a dependence is not allowed in the classical irreversible thermodynamics, where only linear flux-force relations are studied. However, our work is incomplete in that we did not account fully for the interspecies interactions, so that our final mass-flux equations contained concentration, rather than activity, driving forces. That is, our results were not consistent with the thermodynamics of irreversible processes for linear systems. In connection with remedying this difficulty, we have taken a detour into investigating some aspects of the description of multicomponent diffusing systems in the linear limit. The starting point for the present discussion is Jaumann’s18 entropy-balance equation: * Author to whom all correspondence should be addressed. Telephone: 608-262-5920. Fax: 608-262-5434. E-mail: bird@ engr.wisc.edu. † Chemistry Department, Institute for Theoretical Chemistry. ‡ Chemical Engineering Department and Rheology Research Center.
DS ˆ ) -(∇‚s) + gS Dt
(1.1)
in which F is the density of the fluid mixture, S ˆ is the entropy per unit mass, s is the entropy-flux vector, and gS is the rate of entropy production per unit volume. The operator D/Dt ) ∂/∂t + (v‚∇) is the “substantial derivative”. We next use (i) the equations of change for mass, momentum, and energy (see MTGL,19 p 698, or ref 20, p 560) and (ii) a differential relation from equilibrium h i/Mi) dωi; in thermodynamics dU ˆ ) T dS ˆ - p dV ˆ + Σi(G the latter ωi ) Fi/F is the mass fraction of species i, Fi is the mass concentration of species i, T is the absolute temperature, U ˆ and V ˆ are the internal energy and volume per unit mass, Mi is the molecular weight, and G h i is the partial molar Gibbs free energy (also called the “chemical potential”) of species i. By using these relations, one obtains explicit expressions for s and gS for a mixture made up of species i ) 1, 2, 3, ..., v (cf. MTGL, p 702, or ref 21, p 24):
s)
(
gS ) - q‚
1
1
(
T
v
q-
ji ∑ i)1M i
)
(1.2)
) ( [ ( ) ]) ( )∑ v
∇T -
T2
G hi
∑ i)1
ji‚ ∇
hi 1 G
T Mi
-
1
T
gi
-
v hi 1 G 1 τ: ∇v ri (1.3) T i)1T Mi
In eq 1.3, which is a sum of products of fluxes and forces, q is the heat-flux vector, ji is the mass-flux vector, gi is the force per unit mass acting on the ith species, τ is the viscous contribution to the stress tensor, and ri is the mass rate of production of species i by chemical reactions. The mass-flux vector ji is the flux of mass per unit area per unit time with respect to the mass-average v ji ) 0; that is, there are velocity v, and as a result Σi)1 only v - 1 independent mass fluxes.
10.1021/ie9901123 CCC: $18.00 © 1999 American Chemical Society Published on Web 06/12/1999
2516 Ind. Eng. Chem. Res., Vol. 38, No. 7, 1999
The forces that go with the fluxes ji are not all independent either, because of the Gibbs-Duhem equation, which, in the entropy representation,22 is U d(1/ h i/T) ) 0; here p is the pressure, T) + V d(p/T) - ΣiNi d(G h i is the Ni is the number of moles of species i, and H partial molar enthalpy. After division by the volume, this equation can be put into the form needed here: v
( ) hi 1 G
H hi 1
v
1 - ∇p + Fi ∇T ) 0 2 T i)1 Mi T
Fi∇ ∑ TM i)1
∑
i
( ) ([( )
Fi
ji
v
1
∇T -
T2
∑ i)1 F
hi 1 G
‚ Fi∇
-
T Mi
i
H hi 1
Fi 1
∇p +
FT
])
H hi 1 1 1 Fi v ∇T -Fi ∇T - Figi + Fj gj Mi T 2 Mi T 2 T T F j)1
∑
v
q(h) ) -R00∇ ln T -
R0 j(Lj - Lk) ∑ j)1
(1.10)
j*k v
(1.4)
in which Fi is the mass concentration of species i. Guided by the form of eq 1.4, we now add some extra terms in the second term on the right side of eq 1.3 in such a way that this term is unaltered; the singleunderlined terms obviously cancel each other, and the double-underlined terms do not contribute to the sum, because Σiji ) 0. Then the rate of entropy production becomes
gS ) - q‚
and k denotes an arbitrarily selected species. Because each vector flux must depend linearly on all of the vector forces in the system, we have
Rij(Lj - Lk) ∑ j)1
ji ) -Ri0∇ ln T -
j*k
in which the Rij are the phenomenological coefficients. Because there are v - 1 independent mass fluxes ji (i * k) and v - 1 independent diffusional forces di (i * k), the coefficients Rij must be symmetric according to the Onsager reciprocal relations. Because eqs 1.10 and 1.11 depend on singling out species k as special, it is advantageous to rewrite them to include the i ) k terms by defining v
Rkj ) + ... (1.5)
Here + ... indicates the contributions from viscous dissipation and chemical reactions (these being of no further interest in this discussion), and F ) ΣiFi is the total mass density of the fluid mixture. We now let c ) Σici ) Σi(Fi/Mi) be the total molar concentration, and we define a set of forces (cRT/Fi)di (with dimensions length/ time2) to go with the ji, in such a way that Σidi ) 0 (here R is the gas constant). When this is done, it is then clear that the number of independent driving forces di will be the same as the number of independent mass fluxes ji. We now see that the first three terms in the brackets in eq 1.5 sum to zero according to the Gibbs-Duhem relation in the entropy representation and also that the last two terms, involving the external forces, sum to zero. The remaining term in the brackets we combine with the q term. This means that eqs 1.2 and 1.3 become
i * k (1.11)
∑ i)1
v
Rij
Rjk ) Rkj ) -
and
i*k
Rji ∑ i)1
(1.12)
i*k
for j ) 0, 1, 2, 3, ..., v. That is, the Rij are symmetric and also satisfy the relations ΣiRij ) ΣiRji ) 0, with the sums being over all i. Now eqs 1.10 and 1.11 may be rewritten as v
q(h) ) -R00∇ ln T v
ji ) -Ri0∇ ln T - Fi
∑ j)1
cRTRij FiFj
∑ j)1
cRTR0j
dj
Fj
dj
(1.13)
i ) 1, 2, 3, ..., v (1.14)
The coefficients Ri0 and R0i are equal according to the Onsager reciprocal relations; they are the generalized thermal diffusion coefficients, henceforth written as DTi (with dimensions mass/(length)(time)), and they have the property that v
s)
1 T
v
(h)
q
+
ji ∑ i)1M v
(h)
TgS ) -(q ‚∇ ln T) -
S hi
(1.6)
i
(
ji‚ ∑ i)1
cRT Fi
)
di + ... (1.7)
where v
q(h) ) q cRT Fi
( )
di ) T∇
hi 1 G
T Mi
H hi
ji ∑ i)1M
H hi
(1.15)
according to the relations in eq 1.12. The mass transport (the Ri0 terms) resulting from the thermal gradient force is called the Soret effect, and the heat transport (the R0j terms) owing to the diffusive forces is called the Dufour effect. 2. Multicomponent Diffusivities
(1.8)
i
+
DTi ) 0 ∑ i)1
1 1v ∇ ln T - ∇p - gi + Fjgj Mi F F j)1 (1.9)
∑
The diffusional driving forces di have the dimensions of reciprocal length. Because Σiji ) 0, the last term in eq 1.7 may be v rewritten as -Σi)1 (ji‚(Li - Lk)), where Li ) (cRT/Fi)di
There are many definitions for multicomponent diffusivities, and we discuss two of them here: a. Zero-Diagonal-Diffusivity Definition. This choice for multicomponent diffusivities, called Dij (with units length2/time), was proposed in 1949 by Curtiss and Hirschfelder23 in their extension of the ChapmanEnskog dilute-gas kinetic theory to multicomponent systems. It was then incorporated into the discussion of the thermodynamics of irreversible processes in Chapter 11 of MTGL (see eqs 11.2-34 and 11.2-36) and elsewhere.20
Ind. Eng. Chem. Res., Vol. 38, No. 7, 1999 2517
Many relations for multicomponent diffusion may be written in simpler form by using the quantitites D ˆ ij ) (ωiωi/xixj)Dij, rather than the Dij defined on p 715 of MTGL. These D ˆ ij are related to Rij (with dimensions of (mass)(time)/length3) by
D ˆ ij ) -
Rij ) -
(
cRT Rij
F cRT
F
(
Fj
v
+
)
Rik
∑ k)1 F k*i
i
∑ FkDˆ ik k)1 k*i
(2.2)
The D ˆ ij thus defined have the properties that (see Note to p 718 on p 1205 of MTGL)
D ˆ ii ) 0
i ) 1, 2, 3, ..., v i, j ) 1, 2, 3, ..., v
(2.4)
v
D ˆ ijdj ∑ j)1
i ) 1, 2, 3, ..., v
(2.5)
b. Symmetric-Diffusivity Definition. The lack of symmetry of the Dij of MTGL (or, equivalently, the D ˆ ij) was regarded by many as disadvantageous. In response to this objection, Curtiss10 made an alternative proposal in 1968 (which was further studied by Condiff13 the following year), namely
cRTRij D ˜ ij ) FiFj length2/time
and which have
(2.7)
(3.1)
Then eq 2.9, when written for species i and k, gives
D ˜ ij dj ∑ j)1
j ) 1, 2, 3, ..., v
(2.8)
The D ˜ ij matrix is singular, and there are 1/2v(v - 1) independent diffusivities D ˜ ij. They are also the coefficients in a positive definite quadratic form, since, according to the second law of thermodynamics, the entropy is never decreasing. In terms of these symmetric diffusivities, the mass-flux expression becomes v
D ˜ ijdj ∑ j)1
i ) 1, 2, 3, ..., v
The interrelations among the D ˜ ij and the D ˆ ij are
(2.9)
(3.2)
and v
D ˜ kj dj ∑ j)1
V′k ) -
(3.3)
When each of these is multiplied by C ˜ ik (with k * i and with dimensions time/length2) and one equation is subtracted from the other, we get N
C ˜ ik(D ˜ kj - D ˜ ij)dj ∑ j)1
C ˜ ik(V′k - V′i) ) -
(3.4)
Next the sum on k is performed
C ˜ ik(V′k - V′i) ) -∑ ∑C ˜ ik(D ˜ kj - D ˜ ij) dj ∑ k*i j k*i
[
]
i ) 1, 2, 3, ..., v (3.5)
If the quantity within brackets is taken to be equal to -δij + ωi, then when eq 2.8 is used, the right side becomes simply +di. Later (in eq 4.6) it is shown that a set of C ˜ ik can be found for which this equality holds. Now eq 3.5 becomes
( ) jk
v
ji ) -DTi ∇ ln T - Fi
FiV′i ) ji + DTi ∇ ln T
(2.6)
i, j ) 1, 2, 3, ..., v
ωiD ˜ ij ) 0 ∑ i)1
Instead of giving the mass-flux vector ji in terms of the driving forces di, it is sometimes convenient to express the di as a linear function of the ji. This can be done by following the procedure suggested by Merk.11,12 Merk used the MTGL definitions of the diffusivities in eq 2.1, whereas here we use the definition of eq 2.6. First, we define a modified diffusion velocity V′i by
v
ji ) -DTi ∇ ln T - F
˜ ji D ˜ ij ) D
(2.11)
V′i ) -
and they form the elements of a nonsingular matrix. Because of the restrictions in eqs 2.3 and 2.4, there are 1/ v(v - 1) independent D ˆ ij. It should be noted that the 2 D ˆ ij are not symmetric. For this choice of the multicomponent diffusivities, the mass-flux vectors become
which have dimensions of the following properties:
ωkD ˆ ik ∑ k*i
These expressions follow directly from eqs 11.2-36 and 37 of MTGL. Note that in eq 2.5 there is a F in front of the summation sign, whereas in eq 2.9 there is a Fi.
(2.3)
v
∑ (Dˆ ki - Dˆ kj) ) 0 k)1
˜ ij ) -D ˆ ij + ωiD
(2.10)
3. Transformation to the Maxwell-Stefan Form
)
v
FjD ˆ ij - ωj
(2.1)
D ˆ ij ) -ωi(D ˜ ij - D ˜ ii)
C ˜ ik ∑ F k*i
k
-
ji
Fi
) di -
( )
C ˜ ik ∑ k*i
DTk Fk
-
DTi
(∇ ln T) Fi i ) 1, 2, 3, ..., v (3.6)
These are the inverted versions of eq 2.9 and are called the generalized Maxwell-Stefan equations. A diffusion equation of this form was first suggested by Maxwell24 for binary dilute gas mixtures and then extended to multicomponent dilute gas mixtures by Stefan;25 their equations were for ordinary diffusion only and did not include thermal, pressure, and forced diffusion. Curtiss and Hirschfelder23 in their kinetic theory of multicomponent gas mixtures also obtained eq 3.6. Sometimes the notation C ˜ ik ) xixk/Dik is used, where xi are mole fractions. The Dik (with dimensions length2/time) are referred to as the Maxwell-Stefan diffusivities.
2518 Ind. Eng. Chem. Res., Vol. 38, No. 7, 1999
We can now use eq 3.6 to eliminate the dj from eq 1.13 to get
q
(h)
[
(
)]
DTk DTj cRT T ) - R00 + Dj C ˜ jk ∇ ln T Fk Fj j)1k)1 Fj v
v
∑∑
( )
jk jj cRT T Dj C ˜ jk Fk Fj j)1k)1 Fj v
v
∑∑
(3.7)
The coefficient of ∇ ln T in eq 3.7 is, by common agreement, taken to be the thermal conductivity λ times T. This means that the heat flux vector q can be written as the sum of three terms: the conductive heat flux, the diffusive heat flux, and a contribution originating from the Dufour effect
q ) q(c) + q(d) + q(x) ) v H v v hi jk jj cRT T -λ∇T + ji Dj C ˜ jk ) Fk Fj i)1Mi j)1k)1 Fj
∑
[
v
-λ∇T +
( ) ∑( )]
∑∑
H hi
∑ i)1 M
+
cRT
v
DTi C ˜ ij
Fi
j)1
Fi
i
-
˜ ji DTj C Fj
and was able to prove symmetry for a ternary system but not for a system with more than three components. We now get an expression for the C ˜ ik in terms of the D ˜ ij. We write eq 4.1 omitting the equation for j ) i, we write eq 2.8 with the summation index i in the latter replaced by k, and then we eliminate the mass fraction ωi by using the fact that the mass fractions must sum to unity. We thus get the following two sets of equations:
C ˜ ik(Bi)kj ) ωi ∑ k*i
s)
T
v
(c)
q
+
S hi
1
ji + q ∑ T i)1M i
v
-λ∇ ln T +
[
S hi
∑ i-1 M
i
+
(x)
ωk(Bi)kj ) -D ˜ ij ∑ k*i
v
∑
Fi j)1
(j * i)
(4.3)
˜ kj - D ˜ ij, that is, the kj component of in which (Bi)kj ) D a matrix called Bi, which is of order (v - 1) × (v - 1). If we now multiply eqs 4.2 and 4.3 by (B-1 i )jn, elements of the inverse of Bi, and sum on j, we get
(B-1 ∑ i )jn j*i
(4.4)
D ˜ ij(B-1 ∑ i )jn j*i
(4.5)
C ˜ in ) ωi
ji (3.8) and
ωn ) -
Because (B-1 i ) ) (adj Bi)/(det Bi), eqs 4.4 and 4.5 can be combined to give
)
cR
(4.2)
and
Then, because q(h) ) q - q(d) ) q(c) + q(x), the entropy flux given in eq 1.6 is
1
(j * i)
(
DTi C ˜ ij Fi
-
)]
DTj C ˜ ji Fj
ji (3.9)
That is, the entropy flux is the conductive heat flux divided by the temperature, plus the diffusive transport of entropy, plus a contribution originating from the Dufour effect. We conclude by noting that our C ˜ ik are the same as Cikxixk of Merk11,12 and also the same as βikωiωk of Condiff.13 4. Expressions for the C ˜ ik in Terms of the D ˜ ij In going from eq 3.5 to eq 3.6, we stated tentatively that
C ˜ ik(D ˜ kj - D ˜ ij) ) -δij + ωi ∑ k*i
C ˜ ik ) -ωiωk
D ˜ ij(adj Bi)jk ∑ j*i
i, k ) 1, 2, ..., v (4.6)
This demonstrates that the condition introduced at eq 3.5 is permissible. It does not, however, seem to be easy to make any statements concerning the symmetry of the C ˜ ik from this solution, and therefore an alternate approach had to be found. 5. Symmetry of the C ˜ ik In eq 4.1 the C ˜ ii are not defined, but this equation can be written as
C ˜ ikD ˜ kj - ∑C ˜ ik ∑ k*i k*i
( )
(4.1)
But is this the choice unique? Apparently so, for (i) if ωi had been omitted, then we would not get an identity when eq 4.1 is multiplied by ωj and summed on all j and when eq 2.8 is used and (ii) if ωi were replaced by Aωi, then multiplication by ωj and summing on j does not give an identity. Note that eq 4.1 appears as eq 34 of ref 9, from the kinetic theory of dilute monatomic gas mixtures. Equation 4.1 then provides a relation among the C ˜ ik and the D ˜ ij. Because the diagonal elements C ˜ ii do not appear in this equation, there are v(v - 1) independent D ˜ ij in eq 4.1. Therefore, because there are 1/2v(v - 1) independent C ˜ ik, there must be some relations among the C ˜ ik. Merk11,12 surmised that his Cik were symmetric
(adj Bi)jk ∑ j*i
D ˜ ij ) -δij + ωi
(5.1)
This suggests that it might be useful to define the C ˜ ii as follows:
C ˜ ik ∑ k*i
C ˜ ii ) -
(5.2)
so that
∑ C˜ ik ) 0
(5.3)
∑ C˜ ikD˜ kj ) -δij + ωi
(5.4)
all k
Then eq 5.1 becomes
all k
Ind. Eng. Chem. Res., Vol. 38, No. 7, 1999 2519
We now multiply by ωj and use eq 2.11 to get
ωlD ˆ jl ∑ C˜ ik -Dˆ jk + ∑ l*j
(
all k
)
) -δij ωj + ωiωj
(5.5)
This cannot be solved directly for the C ˜ ik, because the matrix of the D ˜ ik is singular. Because of eq 5.3, eq 5.5 can be simplified to
-
∑
C ˜ ik D ˆ jk ) -δijωj + ωiωj
(5.6)
all k
Next define C ˆ ij as the elements of a matrix inverse to the matrix of the D ˆ jk, which is nonsingular, as pointed out after eq 2.4:
∑j Cˆ lj Dˆ jk ) δlk
(5.7)
Then when eq 5.6 is multiplied by C ˆ lj and summed on j, we get
(
ˆ ji C ˜ ij ) ωi C
∑k ωkCˆ jk
)
(5.8)
˜ lm ) ωlωmD ˜ ml ωmωlD
∑p ωp Dˆ lp
) (
(5.9)
∑p ωp Dˆ mp
(5.10)
∑l ωlCˆ il -Dˆ ml + ∑p ωp Dˆ mp
(5.11)
(
)
Next multiply by C ˆ jm and sum on m to get
(
ˆ ji ωi C
C ˆ jmωm ∑ m
) (
) ωj C ˆ ij -
(39)
D ˜ 12 ) D ˜ 21 ) -(ω1ω2/C ˜ 12)
(38) Ternary
D ˜ 11 )
C ˆ imωm ∑ m
)
(5.12)
Relations among the D ˜ ij and the C ˜ ij are given in eq 4.1. However, these relations are not easily solved to ˜ ij. Equation 2.8 give the D ˜ ij explicitly in terms of the C can be used to rewrite eq 4.1 as follows (cf. eq 35) of ref 10):
)
C ˜ ik + ∑ ∑C˜ in D˜ kj ) -δij + ωi ω n*i k*i i
(40) (41)
Quaternary
D ˜ 11 ) [ω42(C ˜ 12C ˜ 13 + C ˜ 12C ˜ 23 + C ˜ 13C ˜ 23) + ω32(C ˜ 12C ˜ 14 +
D ˜ 12 ) [ω42(C ˜ 12C ˜ 13 + C ˜ 12C ˜ 23 + C ˜ 13C ˜ 23) + ω32(C ˜ 12C ˜ 14 + C ˜ 12C ˜ 24 + C ˜ 14C ˜ 24) - ω2(ω1 + ω3 + ω4)(C ˜ 13C ˜ 14 + C ˜ 13C ˜ 34 + C ˜ 14C ˜ 34) - ω1(ω2 + ω3 + ω4)(C ˜ 23C ˜ 24 + C ˜ 23C ˜ 34 + C ˜ 24C ˜ 34) (ω1 + ω3)(ω2 + ω4)C ˜ 13C ˜ 24 - (ω1 + ω4)(ω2 + ω3)C ˜ 14C ˜ 23 +
∆4 ) C ˜ 12C ˜ 13C ˜ 14 + C ˜ 12C ˜ 13C ˜ 24 + C ˜ 12C ˜ 13C ˜ 34 + C ˜ 12C ˜ 14C ˜ 23 + C ˜ 12C ˜ 14C ˜ 34 + C ˜ 12C ˜ 23C ˜ 24 + C ˜ 12C ˜ 23C ˜ 34 + C ˜ 12C ˜ 24C ˜ 34 + ˜ 14C ˜ 23 + C ˜ 13C ˜ 14C ˜ 24 + C ˜ 13C ˜ 23C ˜ 24 + C ˜ 13C ˜ 23C ˜ 34 + C ˜ 13C C ˜ 13C ˜ 24C ˜ 34 + C ˜ 14C ˜ 23C ˜ 24 + C ˜ 14C ˜ 23C ˜ 34 + C ˜ 14C ˜ 24C ˜ 34 [15.3]
the sum on k by omitting the term with k ) j and add a compensating term:
RikD ˜ kj + Rij D ˜ jj ) ωi ∑ k*i
i, j ) 1, 2, 3, ..., v (6.1)
Solving for the D ˜ ij using these equations involves inverting a v × v determinant. In what follows we show how to develop a formula which involves a determinant of lower order by 1. We designate the quantity in parentheses in eq 6.1 by Rik and omit the equation with j ) i. We also rewrite
j*i
(6.2)
k*j
Then use of eq 2.8 allows us to eliminate the D ˜ jj to get
[Rik(1 - δik) - Rij (ωk/ωj)]D ˜ kj ) ωi ∑ k*j
6. Determining the D ˜ ij from the C ˜ ij
(
+ (ω2 + ω3)2C ˜ 23]/∆3
∆3 ) C ˜ 12C ˜ 13 + C ˜ 12C ˜ 23 + C ˜ 13C ˜ 23
(5.13)
which is the symmetry relation that Merk11 was unable to verify.
ωk
+
ω22C ˜ 13
D ˜ 12 ) [ω32C ˜ 12 - ω2(ω1 + ω3)C ˜ 13 - ω1(ω2 + ω3)C ˜ 23]/∆3
which, according to eq 5.8 is just
˜ ji C ˜ ij ) C
[ω32C ˜ 12
(ω3 + ω4)2C ˜ 12C ˜ 34]/∆4 [15.2]
Now multiply by C ˆ il and sum on l to get
ωm(-δim + ωi) )
˜ 12) D ˜ 22 ) (ω12/C
(ω2 + ω3 + ω4)2(C ˜ 23C ˜ 24 + C ˜ 23C ˜ 34 + C ˜ 24C ˜ 34)]/∆4 [15.1]
)
) ωl -D ˆ ml +
(37)
D ˜ 11 )
(ω3 + ω4)2C ˜ 12C ˜ 34 + (ω2 + ω4)2C ˜ 13C ˜ 24 + (ω2 + ω3)2C ˜ 14C ˜ 23 +
We then use eq 2.13 to get
(
Binary
(ω22/C ˜ 12)
C ˜ 12C ˜ 24 + C ˜ 14C ˜ 24) + ω22(C ˜ 13C ˜ 14 + C ˜ 13C ˜ 34 + C ˜ 14C ˜ 34) +
which is a relation between the C ˜ ij and the C ˆ ij. Because the D ˜ ij are symmetric,
ωm -D ˆ lm +
Table 1. Summary of Expressions for the D ˜ ij in Terms of the C ˜ ij (Numbers in Parentheses Are the Corresponding Equations in Reference 10; Numbers in Brackets Are the Corresponding Equations in Reference 13)
j * i (6.3)
The matrix of quantities in the brackets is of order (v 1) × (v - 1). Next we multiply eq 5.3 by ωiωj and rewrite the equation in terms of the C ˜ ik. This gives
∑ k*j
[
{
ωj (1 - δik) (ωi + ωk)C ˜ ik + ωk
{
˜ ij + ωj ωk (ωi + ωj)C
}]
C ˜ in ∑ k*i n*j
}
C ˜ in ∑ k*i n*k
D ˜ kj ) ωi2ωj
-
j * i (6.4)
This can be written compactly as follows
∑ ∑[{δnk(1 - δik) - δik}ωj - δnjωk]C˜ inD˜ kj ) ωiωj k*j n*i
j * i (6.5)
2520 Ind. Eng. Chem. Res., Vol. 38, No. 7, 1999 Table 2. Summary of Expressions for the C ˜ ij in Terms of the D ˜ ij
When eq 7.4 is summed on i and use is made of the fact that Σidi ) 0, we get
Binary v
1 C ˜ 12 ) -ω1ω2 D ˜ 12
or
1 1 C ˜ 12 ) ω22 ) ω12 D ˜ 11 D ˜ 22
∇p )
[ciRT∇ ln ai + ci(H hi - H h i°)(∇ ln T)] ≡ ∑ i)1 v
∇pi ∑ i)1
Ternary
(D ˜ 12 + D ˜ 33 - D ˜ 13 - D ˜ 23)
C ˜ 12 ) -ω1ω2
D ˜ 12D ˜ 33 - D ˜ 13D ˜ 23 Quaternary
C ˜ 12 ) -ω1ω2[D ˜ 12(D ˜ 33 + D ˜ 44) + D ˜ 13(D ˜ 24 + D ˜ 34) + D ˜ 14(D ˜ 23 + D ˜ 34) + D ˜ 34(D ˜ 23 + D ˜ 24) + D ˜ 33D ˜ 44 - 2D ˜ 12D ˜ 34 D ˜ 13(D ˜ 23 + D ˜ 44) - D ˜ 14(D ˜ 24 + D ˜ 33) - D ˜ 23D ˜ 44 - D ˜ 24D ˜ 33 - D ˜ 342]/ [D ˜ 12D ˜ 33D ˜ 44 + (D ˜ 13D ˜ 24 + D ˜ 14D ˜ 23)D ˜ 34 - D ˜ 12D ˜ 342 - D ˜ 13D ˜ 23D ˜ 44 ˜ 24D ˜ 33] D ˜ 14D
From a computational point of view, this result is more ˜ ij. useful than eq 6.1 for getting the D ˜ ij in terms of the C The expressions in Table 1 were generated using a computer program based on eq 6.5. The inverse expressions given in Table 2 were obtained from eq 4.6. 7. Further Comments on the Generalized Driving Force di We now return to eq 1.9 and rewrite the expression for di in a form that will be more useful in describing the interdiffusion in dense gases and liquids. We begin by changing to molar concentrations, defined by ci ) Fi/ Mi:
()
cRTdi ) ciT∇
G hi T
+ ciH h i∇ ln T - ωi∇p - Figi + ωi
Fj gj ∑ j)1
(7.1)
The quantities G h i, H h i, and p are functions of the state of a fluid element, which may be described by the temperature T and the set of molar concentrations ci, at each position r and at each time t. We now define the activity ai(T,c1,c2,...,cv) for species i by
h i°(T) + RT ln ai G hi ) G
()
v
cRTdi ) ∇pi - ωi∇p - Figi + ωi
( )
G hi G h i° ) T∇ + RT∇ ln ai ) ∇G h i° - G h i°∇ ln T + T T RT∇ ln ai ) -(TS h i° + G h i°)∇ ln T + RT∇ ln ai (7.3)
h i°/dT)∇T ) -S h i°∇T has in which the relation ∇G h i° ) (dG been used. Substitution of this expression into eq 7.1 gives
hi - H h i°)∇ ln T - ωi∇p cRTdi ) ciRT∇ ln ai + ci(H v
Fj gj ∑ j)1
Figi + ωi
(7.4)
Fj gj ∑ j)1
(7.6)
The first two terms on the right side describe the effects of the intermolecular forces, and the last two terms describe the effects of the external forces. If the only external forces are gravity forces, then the last two terms cancel one another. Equation 7.6 is a generalization of Maxwell’s seminal idea that concentration gradients result from the friction between the molecules of different species. Equation 7.6 can be put into another form by considering the ai to be a function of p, T, and all the xi (excluding xv); for a solvent containing several solutes, it might be convenient to label the solvent as species v. Then use of the chain rule of partial differentiation and standard formulas from equilibrium thermodynamics gives
ciRT∇ ln ai ) cRT
∑ j)1
( ) ∂ ln ai ∂ ln xj
∇xj + ciV h i∇p -
T,p,x
hi - H h i°)∇ ln T (7.7) ci(H in which the subscript x stands for all the xk except for k ) j and k ) v. When this expression is substituted into eq 7.4, one finds that (cf. eq 11.2-29 of MTGL)
(7.2)
in which G h °i(T) is a function of the temperature defined by the “standard state”, for example, the free energy of 1 mol of pure i at T and some specified pressure. Then
T∇
As indicated, this serves to define a “partial pressure” pi (within an additive constant). For ideal gas mixtures, the quantity pi can be shown to be consistent with the usual definition of partial pressure in such systems. The final expression for the generalized driving force di is then
v-1
v
(7.5)
v-1
cRTdi ) cRT
∑ j)1
( ) ∂ ln ai ∂ ln xj
∇xj + (ciV h i - ωi)∇p -
T,p,x v
Figi + ωi
Fj gj ∑ j)1
(7.8)
This form displays clearly the driving forces contributing to concentration diffusion, pressure diffusion, and forced diffusion. The thermal diffusion driving force is described separately by the terms containing the DTi . 8. Concluding Remarks We can use eq 7.6, along with eqs 2.9 and 3.6, to rewrite the mass-flux expressions in the Fick form and in the Maxwell-Stefan form as follows:
Ind. Eng. Chem. Res., Vol. 38, No. 7, 1999 2521
ji ) -DTi ∇ ln T v Fi v D ˜ ij ∇pj - ωj∇p - Fjgj + ωj Fkgk (8.1) cRT j)1 k)1
∑
(
jk
ji
( )
C ˜ ik ∑ F k*i
cRT
k
-
Fi
)
∑
v
Fj gj ∑ j)1
) ∇pi - ωi∇p - Figi + ωi
( )
C ˜ ik ∑ k*i
cRT
DTk Fk
-
DTi Fi
(∇ ln T) (8.2)
These two expressions contain the same information and are interrelated according to eqs 4.6 and 6.5, which give the connection between the multicomponent diffusivities D ˜ ij and the multicomponent “inverse diffusivities” C ˜ ij. For low-density gases, ∇pi ) ∇(ciRT), and the standard results are obtained. For polymeric liquids, an equation of the form of eq 8.2 has been found from a molecular theory,16,17 but with the pressure p and partial pressure pi replaced by the total stress tensor π and the partial stress tensor πi. The dependence of the mass flux on the velocity gradients then enters via the stress tensor and the partial stress tensors, along with additional dependence on ∇T and ∇ci. The main thrust of this paper has been to summarize key results obtainable from continuum considerations. In conclusion, we summarize what is presently known about the multicomponent diffusivities from several molecular theories: (i) From the Curtiss-Hirschfelder theory23 of dilute monatomic gas mixtures, it is known that C ˜ ik ≈ xixk/ Dik, in which Dik is the diffusivity of the pair i-k in a two-component system and depends only on the intermolecular force for the binary pair of gases. This relation is only approximately true (probably good within about 3%), because it is obtained by using the lowest-order approximation in the expansion of the distribution function in terms of the Sonine polynomials. In this approximation, the Dik are known to be independent of the concentration and inversely proportional to the pressure. When multicomponent diffusion problems are solved to get the xi as functions of position and time in dilute gas mixtures, it is definitely preferable to use eq 8.2 with the concentration-independent Dik (available from binary diffusion experiments) than to use eq 8.1 and the D ˜ ij, which are strongly dependent on the concentration as is evident from Table 1 (see also ref 20, eqs 18.4-19 and ref 25, p 977, eq 15). (ii) In the Bearman-Kirkwood27,28 kinetic theory of monatomic liquid mixtures, the Maxwell-Stefan equations appear in their eq 5.8, although they are not referred to as such. In these equations there appear friction coefficients ζij (which are concentration depend˜ ik ) nζikxixk/kT. ent), which are related to our C ˜ ik by C Bearman and Kirkwood give a molecular expression for ζij in eq 5.9 of their paper. From the latter it can be seen that the friction coefficient is a property of the pair of molecules i-j at the level of approximation in their treatment and therefore can be determined approximately from a binary diffusion experiment. In order to use the Maxwell-Stefan expressions for liquids, one, of course, has to have data on activities and partial molar enthalpies. (iii) The Curtiss-Bird16,17 kinetic theory of polymeric liquid mixtures, in the linear limit, also yields the Maxwell-Stefan equations, containing a set of quanti-
˜ ik used here (specifically ties Zik closely related to the C C ˜ ik ) FZikωiωk/nkT). These Zik involve only the friction tensor associated with the ikth pair of molecules in the mixture. In the approximation that this friction tensor depends only on the properties of the two species, Zik are measurable in a binary diffusion experiment. It should be pointed out that in our work we introduced assumptions that resulted in an inadequate treatment of the interpolymer forces (assumption 1 on p 9907 of ref 17 and the postulate in eq 22 of ref 16). When these assumptions are removed, the final expression (eq 28 of ref 16 and eq C16 of ref 17) will contain the full stress tensor π and the partial stress tensors πi where the pressure and partial pressures appear in eq 8.2. This point will be elaborated on in a future publication. Acknowledgment The authors wish to thank Professor L. A. Belfiore for reading the manuscript and suggesting improvements. Literature Cited (1) Cussler, E. L. Multicomponent Diffusion; Elsevier: Amsterdam, The Netherlands, 1976. (2) Cussler, E. L. DiffusionsMass Transfer in Fluid Systems; Cambridge University Press: Cambridge, U.K., 1997; Chapter 7. (3) Wesselingh, J. A.; Krishna, R. Mass Transfer; Ellis Horwood: New York, 1990. (4) Taylor, R.; Krishna, R. Multicomponent Mass Transfer; Wiley-Interscience: New York, 1993. (5) Kuiken, G. D. C. Thermodynamics of Irreversible Processess Applications to Diffusion and Rheology; Wiley: New York,1994; Chapter 6. (6) Rutten, Ph. W. M. Diffusion in Liquids; Delft University Press: Delft, The Netherlands, 1992. (7) Lightfoot, E. N. Transport Phenomena in Living Systems; Wiley: New York, 1974; Chapter III. (8) Chapman, S.; Cowling, T. G. The Mathematical Theory of Non-Uniform Gases, 3rd ed.; Cambridge University Press: Cambridge, U.K., 1970; Chapter 18. (9) Slattery, J. C. Advanced Transport Phenomena; Oxford University Press: Oxford, U.K., 1999. (10) Curtiss, C. F. Symmetric gaseous diffusion coefficients. J. Chem. Phys. 1968, 49, 2917-2919. (11) Merk, H. J. Mass-transfer in laminar boundary layers by forced convection. Ph.D. Thesis, Technische Universiteit Delft, 1957, Hoofdstuk I. (12) Merk, H. J. The macroscopic equations for simultaneous heat and mass transfer in isotropic, continuous and closed systems. Appl. Sci. Res. 1959, A8, 73-99. (13) Condiff, D. W. On symmetric multicomponent diffusion coefficients. J. Chem. Phys. 1969, 51, 4209-4212. (14) Truesdell, C. A. Sulla basi della termomeccanica. Atti Accad. Naz. Lincei, Cl. Sci. Fis., Mat. Nat., Rend., Ser. 8 1957, 22, 33-38, 158-166; contains extensive references to earlier works. (15) Kirkwood, J. G.; Crawford, B., Jr. The macroscopic equations of transport. J. Phys. Chem. 1952, 56, 1048-1051; references to earlier works are cited here. (16) Curtiss, C. F.; Bird, R. B. Multicomponent diffusion in polymeric liquids. Proc. Natl. Acad. Sci. USA 1996, 93, 74407445. (17) Curtiss, C. F.; Bird, R. B. Fokker-Planck equation for the one-molecule distribution function in polymer mixtures and its solution. J. Chem. Phys. 1997, 106, 9899-9921; Appendix C, Part 3. (18) Jaumann, G. A. J. Closed system of physical and chemical differential laws. Wien. Akad. Sitzungsberichte (Math.-Naturw. Klasse) 1911, 120, 2a, 385-530. (19) Hirschfelder, J. O.; Curtiss, C. F.; Bird, R. B. Molecular Theory of Gases and Liquids; Wiley: New York, 1954; corrected
2522 Ind. Eng. Chem. Res., Vol. 38, No. 7, 1999 printing with notes added 1964, Chapter 7 (kinetic theory), Chapter 11 (continuum theory); this is referred to in the text as “MTGL”. (20) Bird, R. B.; Stewart, W. E.; Lightfoot, E. N. Transport Phenomena; Wiley: New York, 1960; Chapter 18. (21) de Groot, S. R.; Mazur, P. Non-Equilibrium Thermodynamics; North-Holland: Amsterdam, The Netherlands, 1962; multicomponent diffusion is discussed in section 5 of Chapter XI. (22) Callen, H. B. Thermodynamics and an Introduction to Thermostatistics, 2nd ed.; Wiley: New York, 1985. (23) Curtiss, C. F.; Hirschfelder, J. O. Transport properties of multicomponent gas mixtures. J. Chem. Phys. 1949, 17, 550-555. (24) Maxwell, J. C. Illustrations of the dynamical theory of gases. Philos. Mag. 1860, XIX, 19-32; 1868, XX, 21-32; 1868, XX, 33-36. (25) Stefan, J. On the equilibrium and movement of gas mixtures, in particular diffusion. Sitzungsber. Kais. Akad. Wiss.
Wien 1871, LXIII (2), 63-124. Stefan, J. On the dynamical theory of diffusion of gases. Sitzungsber. Kais. Akad. Wiss. Wien 1872 , LXV (2), 323-363. (26) Hougen, O. A.; Watson, K. M. Chemical Process Principles; Wiley: New York, 1947; Chapter XX. (27) Bearman, R. J.; Kirkwood, J. G. The statistical mechanics of transport processes. XI. Equations of transport in multicomponent systems. J. Chem. Phys. 1958, 28, 136-145. (28) Bearman, R. J. On the molecular basis of some current theories of diffusion. J. Phys. Chem. 1961, 65, 1961-1968.
Received for review February 11, 1999 Revised manuscript received April 16, 1999 Accepted April 23, 1999 IE9901123
