MATHEMATICAL ANALYSISOF MULTICOMPONENT FREE-DIFFUSION EXPERIMENTS
3305
Mathematical Analysis of Multicomponent Free-Diffusion Experiments
by J. L. Duda and J. S. Vrentas Process Fundamentals Research Laboratory, The Dow Chemhd Company, Midland, Michigan (Received March 1 , 1966)
An exact procedure is developed which permits the calculation of the (N - 1)2independent diffusion coefficients of an N-component system from appropriate free-diffusion experiments. In this development, the coefficients are considered to be arbitrary functions of concentration, and the effect of volume changes due to mixing of the diffusing species is included. It is shown that the number of experiments necessary’for the determination of the diffusion coefficients at several concentrations can be minimized by the use of this new procedure. In order to realize this economy in experiments it is necessary to obtain accurate concentration and concentration gradient distribution data over large concentration ranges in multicomponent systems. The special case of no volume change on mixing is also considered.
Introduction I n recent years, there has been increasing interest in the measurement of diffusion coefficients in multicomponent systems. The majority of such studies has been concerned with the analysis of free-diffusion experiments employing ternary liquid systems. Free-diffusion experiments utilizing in situ methods of measuring concentration or concentration gradient distributions possess distbct advantages over other methods for the study of multicomponent diffusion.2 Consequently, in what follows we have chosen to restrict our analysis of diffusion experiments to those of the freediffusion type. The interpretation of any multicomponent diffusion experiment is greatly complicated by the fact that interactions bet,ween the individual species must be included in any comprehensive analysis. In an N-component system, there are thus (N - 1)2 independent diffusion coefficients that must be determined for complete description of the diffusion process. In general, the data from one free-diffusion experiment can be used to generate only N - 1 equationsSfor the ( N - 1)2coefficients. Consequently, in the general case where the diffusion coefficients are functions of composition, it is necessary to perform N - 1 independent experiments which have a common point of composition in their diffusionfields if all ( N - 1)2diffusion coefficients at this common composition point are to be evaluated. For m compositions, it is evident that m(N - 1) experiments
generating m(N - 1)2equations are needed to determine 1)2 different values of the diffusion coeffithe m(N cients. For a ternary system, it follows that two experiments which have a common composition point in their diffusion fields are needed to evaluate the four independent coefficients at this point. The common procedure previously has been to conduct two experiments with the same mean concentration but with different terminal compositions. In practice, the terminal compositions of these two experiments are set very close to the mean composition so that the diffusion coefficients can be considered constant over the concentration range of interest and so that any volume change on mixing can be essentially eliminated.4 An added benefit of performing experiments over small concentration ranges when the common interferometric methods are employed is that the refractive index at any point in the free-dif-
-
(1) Typical studies which give references to previous work: G. Reinfelds and L. J. Gosting, J . Phys. Chem., 68, 2464 (1964); P. J. Dunlop and L. J. Gosting, ibid., 68, 3874 (1964). (2) Reviews which discuss various methods of measuring diffusion coefficients: L. J. Gosting, Aduan. Protein Chem., 11, 429 (1956); A. L. Geddes in “Physical Methods of Organic Chemistry,” Vol. I, A. Weissberger, Ed., Interscience Publishers, Inc., New York, N. Y., 1949. (3) These are simply the N - 1 species continuity equations. (4) J. G. Kirkwood, R. L. Baldwin, P. J. Dunlop, L. J. Gosting, and G. Kegeles, J. Chem. Phys., 3 3 , 1505 (1960),have shown that volume change on mixing effects tend t o vanish as the terminal composition differences become small.
Volume 69, Number 10 October 1966
3306
fusion field can be assumed to be a linear function of the concentrations a t this point. Since most experimental data have been collected over small concentration ranges, the methods which have been developed for the analysis of ternary free-diff usion experiments are specifically for the case of constant diffusion coefficients and no volume change on m i ~ i n g . ~ These ,~ procedures require a minimum of two experiments for the evaluation of the complete set of four independent diffusion coefficients at each composition point in the three-component system. The required number of experiments could be significantly reduced, however, if the individual experiments were conducted over large concentration ranges and if these experiments were chosen with some degree of discretion. The key to this experimental economy is choosing the terminal compositions of a given experiment such that its diffusion path intersects the diffusion paths of as many of the other experiments as possible. For the sake of illustration, we consider diffusion in a ternary system of components A, B, and C whose composition at any point in a free-diffusion field can be represented as a point on a triangular diagram such as shown in Figure 1. If a solution of composition represented by point K on this diagram is brought into contact with the composition represented by point L in a free-diffusion cell, the compositions in the resulting diffusion field will lie on a line connecting these terminal points. In general, this line formed from the locus of compositions within the free-diffusion field will not be straight and for illustrative purposes has been arbitrarily drawn as shown. Similarly, the other lines in Figure 1 represent the diffusion paths of other free-diffusion experiments with different terminal compositions. I n principle, if the concentration and concentration gradient distributions are known as functions of time for each of these experiments, then the complete set of independent diffusion coefficients can be determined a t each of the compositions represented by the points of intersection of these concentration profiles. In general, one cannot expect a diffusion path to intersect another diffusion path more than once. Therefore, if we have m diffusion paths generated from m independent experiments and if each of these paths intersects all of the other m - 1 paths once, there will be a total of m(m - 1)/2 distinct intersection points. I n summary then for a ternary system, it is possible to determine the diffusion coefficients a t m/2 compositions with m experiments (where m must be an even number here) by the conventional method whereas it is, in principle, possible to evaluate coefficients a t m(m - 1)/2 compositions by the new procedure. It should The Journal of Phyakal Chemhtrtry
J. L. DUDAAND J. S. VRENTAS
E
Figure 1. A typical diagram showing the intersections of several free-diff usion paths in a ternary system.
be pointed out that m(m - 1)/2 is the maximum number of distinct intersections if all the diffusion paths cross all the other diffusion paths. Figure 1shows how eight experiments on a ternary system can be designed to give 16 intersections and, ultimately, the diffusion coefficients a t 16 compositions. This number is less than the predicted maximum value of 28 but is certainly a great improvement over the four sets of coefficients that could be obtained from eight conventional-type experiments. The eight experiments represented in Figure 1 have been chosen to cover most of the ternary diagram as well as to ensure the intersection of diffusion paths, and, consequenbly, the number of intersections is less than optimal. The extension of this economy of experiments to systems of four or more components is not straightforward since we must first guarantee that N - 1 experimental diffusion paths intersect a t the same composition point1 before planning experiments to maximize the number of points of intersection. This is largely an experimental problem that can perhaps be overcome. Before the aforementioned economy in experiments can be realized, there are two other obstacles that must be overcome. First, it is necessary to develop methods of accurately measuring concentration or concentration gradient distributions over large Concentration ranges in free-diffusion cells. Accuracy of these measured ( 5 ) H. Fujita and L. J. Gosting, J . Am. Chem. Sac., 7 8 , 1099 (1956). (6) H.Fujita and L. J. Gosting, J . Phys. Chem., 64, 1256 (1960). (7) For example, for a four-component system, this requires that three concentration curves in three-dimensional space intersect a t the same composition point.
3307
MATHEMATICAL ANALYSISOF MULTICOMPONENT FREE-DIFFUSION EXPERIMENTS
quantities is especially important here because, in part, point values of the concentration profiles are employed rather than integrated values as in the conventional methods. One difficulty of obtaining data over large concentration ranges in multicomponent systems is the uncertainty in deducing concentration distributions from refractive index distributions. In addition, care must be taken to avoid convective mixing resulting from adverse density gradients which can develop in multicomponent free-diffusion processes.E Secondly, methods of analyzing such data must be developed. The purpose of this paper is to develop a method of analyzing mu1ticomponent free-diffusion experiments which will include the effects of variable diffusion coefficients and the effect of volume changes,during mixing of the diffusing specie^.^ This development will be based on the premise that accurate concentration and concentration gradient distribution data are available. It is hoped that this development will serve as an incentive for the experimentalists to perfect methods of obtaining such data. The following analysis for the general case will be developed in part in matrix notation for a system of N components. The special case of variable diffusion coefficients with no volume change on mixing will also be considered.
Formulation of General Case Development of Basic Equations. To effect a detailed analysis of data from a free-diffusion cell, we must consider the conservation laws which describe the diffusion process in a continuum. If we limit our development to an isothermal system of N nonreacting components, the equations of change which mathematically describe the physical behavior of the systemlo are the total continuity equation
the N
-
thermore, by restricting the analysis to purely viscous, nonpolar fluids, we can write the thermal equation of state in the form P =
P(P1, P2,
’ ’
’, PN-1,
P)
(4)
A complete analysis of any isothermal diffusion process necessarily requires the inclusion of all of the above equations. However, in certain flow fields it is possible to effect great simplifications by uncoupling the equations of motion from the species continuity equations. We shall now establish that free diffusion is one class of flows for which such simplifications are possible. For a one-dimensional free-diffusion process, there is one-directional motion with all variables changing in that direction only. Furthermore, free-diff usion cells are usually constructed so that the flows are best described relative to rectangular Cartesian coordinate axes and so that the effects of external fields can be neglected. Consequently, eq. 1 and 2 can be expressed as (5)
Now, introduction of the mass fraction W I
=
PI P
(7)
into eq. 6 and utilization of eq. 5 give the following modified form of the species continuity equations bar
p-
bt
b.iI + pv-bwI + bx -= 0 ax
In addition, eq. 5 can also be written as
1 species continuity equations
Also, for a free-diffusion process the pressure variations throughout the system will be small, and, to a high degree of accuracy, we can assume that the total density and the three equations of motion . .
N
I=1
N
N
E
I= 1
’IF‘
E
+I = 1
T,ii’j
(3)
-
Here. D i and ”i:- denote comDonents of the mass-average velocityll and the mass diffusion flux of componentI relative to the mass average velocity, respectively. Fur-
(8) Some aspects of gravitational stability in three-component systems are discussed by R. P. Wendt, J. Phys. Chem., 66, 1740 (1962). (9) It has been shown elsewhere (J. L. Duda and J. S.Vrentas, to be published in I n d . Eng. Chem. Fundamentals) that volume change on mixing effects must be taken into account if accurate values of diffusion coefficients are t o be determined for binary systems. (10) C. Truesdell and R. A. Toupin in “Handbuch der Physik,” Band I I I A S. Flogge, Ed., Springer-Verlag, Berlin, 1960. (11) R. B. Bird, W. E. Stewart, and E. N. Lightfoot, “Transport Phenomena,” John Wiley and Sons, Inc., New Tork, N. Y., 1960, p. 497 E.
Volume 69. N u m b e r 10 October 1965
3308
J. L. DUDAAND J. S. VRENTAS
of the system is affected negligibly by the pressure. Consequently, we can rewrite eq. 4 as p = p(w1, u2,
. , ., ON-I)
where
(10)
from which we deduce that
and where, in general
N-1
(11)
Finally, if each of the species continuity equations represented by eq. 8 is multiplied by (dp/bwI),, and the resulting equations are summed over N - 1 components, there results
DIK
= DIK(w1,
~
2
., . ., w N - 1 )
In the above development, we have neglected the effect of the pressure gradient on the diffusion flux. Finally, we establish the boundary conditions which represent the physical behavior of a free-diffusion cell. First of all, the following set of boundary conditions is valid for each species I
x0
(204 (204
In addition, diffusion cells are usually constructed or can easily be modified so that the fluid is constrained at one of the infinite boundaries. Consequently, we can set the velocity there equal to zero, and the appropriate boundary condition becomes, for example
v(t, - m )
It is evident from eq. 8, 10, and 14 that, if the velocity is specified at one point in the diffusion field, it is possible to investigate the diffusion process independently of the equations of motion. We have thus essentially uncoupled the species continuity equations from the momentum equations. For complete specification of eq. 8 and 14, we must, of course, substitute proper expressions for the mass diffusion fluxes of each of the components in the system. If we assume that the fluids we are considering are isotropic, it can be shown from nonequilibrium thermodynamics12 that the proper linear constitutive equation for the mass diffusion flux of component I relative to the mass-average velocity can be expressed as
(19)
=
0
(21)
It should be pointed out that a very complex problem results if neither of the infinite boundaries is con&rained. Solution for the D I K . We proceed now to solve eq. 8, 10, 14, 17, 20, and 21 for the ( N - 1)2diffusion coefficients D I K in terms of quantities which can be obtained experimentally from a free-diffusion cell. It is possible to simplify the problem considerably by converting the partial differential equations to ordinary differential equations by introduction of the familiar Boltzmann transformation X q=2t’l’
Application of eq. 22 gives (15) where pJ is the specific chemical potential of component J and the LIJterms are the usual phenomenological coefficients. Now, since P J = P J ( W , w2,
. . ., WN-1, PI
for eq. 8 and 17
(16)
at constant temperature we can equivalently write eq. 15 as N-1
=
- PK =Cl
The Journal of Physical C h m i s t r y
(12) S. R. de Groot and P. Mazur, “Non-Equilibrium Thermodynamics,” North-Holland Publishing Co., Amsterdam, 1962, p. 33 ff.
MATHEMATICAL ANALYSIS OF MULTICOMPONENT FREE-DIFFUSION EXPERIMENTS
3309
-2 lim ( q p ) = lim aI
for eq. 14 and 17, and
q-+
wd-m) wI( a)
= =
(25a)
WIO
(25b)
WIm
for the boundary conditions. Integration of eq. 24 using eq. 25c and substitution of the resulting expression into eq. 23 yield
(33)
- 0
and, since we can infer an equivalent expression for cyK, it follows then that lim ( a I -
(254
=0
vfi(-m)
q-c
- 0
=
cyK)
0
7-c -m
(34)
Utilization of this boundary condition in the above firstorder differential equation leads to the immediate conclusion that eq. 32 can be satisfied only if ffI
- aK
=
0
(35)
everywhere in the diffusion field. Thus, we conclude that ff1
where
= ff2 = . .
.
-
aN-1
(36)
and eq. 30 can therefore be shown to reduce to (27) (37) and where use has been made of the fact that all gradients vanish at the infinite limit. Equation 26 represents a set of N - 1 coupled Volterra integro-differential equations which are to be solved for the variables of interest, the N - 1 & . 1 3 Division of eq. 26 by p dw,/dq and differentiation with respect to q lead to
It remains now only to solve each differentia,l equation of this set separately for the N - 1 tI. Integration of eq. 37 from q = - co to any point in the diffusion field gives aI
- lim 7-b
aI =
(38)
-m
which upon integration by parts becomes
1 dP dfr dw, - P2
(
+
- N-l(
")
P2 J z l bwJ
dr drl / d n )
dSJ
_ .
-1-
(28)
d5I dw, - lim dq dq q - + - m
-2~7 (29)
d?l
P
J=l
bwJ
+ lim ( 2 ~ 7 )+ l > s d p q+
-m
(39)
Substitution of eq. 33 into eq. 39 and further integration14 yield
and, consequently, eq. 28 can be converted to
dLJ]aI
=
WRdq
However, from eq. 10 we can write
- 2 p = - -d-f f I
(zpg)
51
= -L>pvdoI
+ SYr~>vdpdw 010
(40)
+
WRd?
If the double integral in this expression is integrated by parts, we obtain the desired result
where
If we now subtract from eq. 30 the equivalent equation for component K , we discover that
From eq. 26 it is evident that
We shall now proceed to the final stage of the development, the derivation of explicit expressions for the ( N - 1)2diffusion coefficients at one composition from ( N - l)z equations of the type represented by eq. 41. As mentioned previously, since each free-diffusion experi(13) Once the 51 are known, we can then determine the Drx since the diffusion coefficients appear only in the €1 variable. (14) Note here that [ I ( - m ) = 0 because all gradients are zero at
,,=
-m.
Volume 69,Number 10 October 1966
33 10
J. L. DUDAAND J. S. VRENTAS
ment necessarily yields only N - 1 equations similar in form to eq. 41, we must conduct N - 1 such experiments (possessing a common composition point) if we are to generate the ( N - 1)2required independent equations. For each component, the N - 1 equations of the form of eq. 41 derived from the N - 1 experiments can be expressed compactly in matrix notation. For example, for component I we can write
{BII = [TI@), (42) where { B )and I { D ]I are ( N - 1) X 1 column matrices defined by
and where the (N given by
- 1) X (N - 1) square matrix [TI is
................................
Premultiplication of eq. 46 by [T*]-' gives an explicit expression for the column of diffusion coefficients
ID) = [T*]-'(B)
(48)
where the inverse of [T*]is simply
o
rf~1-1
...
o 1
The diagonal form of [T*]allows us to solve the set of 1) linear equations by inverting one ( N - 1) X ( N matrix rather than an (N - l)zX ( N - l)zmatrix. Equation 48 is the final desired result. From eq. 43, 44,45, and 49, it is clear that eq. 48 can be used to determine the values of all of the diffusion coefficients at one composition from mass fraction distribution datai6 (for each component and for N - 1 experiments) and from the multicomponent density-mass fraction relationship. The former information can, of course, be obtained from freediffusion experiments whereas the density-composition data must be obtained from an entirely independent experiment. Binary and Ternary Spstems. We now deduce expressions for two- and three-component systems from the general results derived above. For a binary system we need consider only eq. 41 which for component one reduces to
-
This equation was derived previously e l s e ~ h e r e .For ~ a three-component system eq. 48 produces the following results after inversion of the matrix [TI
I n the above equations, a Greek letter subscript denotes the number of the experiment. There exists, of course, an equation analogous to eq. 42 for each of the N - 1 components. By combining these N - 1 matrix equations, we arrive at the matrix equation
tBj = [T*I(Dj (46) Here, { B ) is an ( N - 1)2 X 1 column matrix with submatrices defined by eq. 43 as elements. { D )is an ( N - 1)2 X 1 column matrix with submatrices defined by eq. 44 as elements, and [T*]is the ( N - 1)2 X ( N - l)z square matrix defined by [TI 0 ... [T*] = 0 [TI * . . (47) ...................
1'0 0 The Joumal of Phyeical Chemistry
...
1J
[TI
Ii =
(-I P p q d p 1 + w' P J>vdp)
(56)
PI0
a
(15) All variables, gradients of variables, and integrals must, of course, be evaluated at the intersection composition.
3311
MATHEMATICAL ANALYSIS OF MULTICOMPONENT FREE-DIFFUSION EXPERIMENTS
The quantities 13 and I4 are defined similarly for component two. Finally, it should be noted that completely equivalent expressions can, of course, be obtained if the velocity at the positive infinity boundary is set equal to zero. However, integration of the experimental data must always begin at the infinite boundary at which the velocity vanishes.
In addition, since N
Cjl=O I=l we can modify eq. 62 to give the result
Now, eq. 17 can equivalently be written as
Special Case of No Volume Change on Mixing
N- 1
I n this section, we obtain expressions for the determination of diffusion coefficients from experiments for which volume change on mixing effects are completely negligible but the concentration dependence of the diffusion coefficients is not. We first derive explicit relationships between the diffusion coefficients relative to the mass-fixed frame of reference (the D I K introduced above) and the diffusion coefficients based on the volume-fixed frame of reference (these will be denoted by D I K ) . The latter set of coefficients is the one most often measured by the experimentalist since it can be determined for systems which exhibit no volume change on mixing without the need for auxiliary data. We shall then develop a procedure for determining the D I K for the special case outlined above. Finally, we shall derive an alternate form of eq. 41 which illustrates the correction terms that must be included to describe the effects of significant volume changes on mixing. Relationships between the D I K and the Dig. The velocity of each component in the mixture relative to stationary coordinate axes can be expressed as PIVI
= prv
+
j I
=
PI^ 4- 71
(58)
where 0 is the volume-average velocityll and 3; is the mass diffusion flux of component I relative to the volume-average velocity. Multiplication of eq. 58 by the partial specific volume PI and summation over all components give N 0 - 0 s
j J p J
J= 1
(59)
where we have utilized the well-known identities N
N
C I=l
=
0
(61)
Substitution of eq. 59 into eq. 58 yields the relationship between the two types of diffusion fluxes
N- 1
where we concern ourselves only with the ordinary diffusion contribution to the mass flux. Furthermore, it can be shown that
Substitution of eq. 65 and 66 into eq. 64 gives after some rearrangement N-1 JI
=
c
-s-
1
bP s
DIS
ax
where
(68) In an analogous manner, it is possible to obtain the inverse relationship
P
