Diffusion in Ternary Mixtures with and without Phase Boundaries
Mass transfer through an interface between two phases in a ternary mixture is analyzed by a steady-state method, using a “two-film’’ mechanism, with diffusion in each film described by Stefan-Maxwell relations. In certain circumstances the interfacial compositions, and hence the complete solution, can be obtained by a simple geometric construction in the triangle diagram. The possibility of emulsion formation by a mechanism suggested by Ruschak and Miller is discussed.
Introduction Of fundamental importance in the theory of mass transfer is the question whether two given liquids, brought into contact in some well-defined way, will form a single phase or two phases separated by an interface. In both cases there will be diffusive mass transport unless the liquids have identical compositions, but if an interface is formed two separate diffusion problems must be treated and linked with the equilibrium conditions a t the interface. The problem of determining the interfacial compositions and the diffusional fluxes is simple in the case of a binary mixture, and its solution can be found in elementary textbooks which treat the “two-film” theory of mass transfer. When there are more than two components the problem is more difficult but the ternary case has been treated by several authors including Ruschak and Miller (1972). Ruschak and Miller considered two semi-infinite phases separated by a plane interface, with compositions fixed and specified on planes infinitely distant from the interface on either side. In this situation diffusion is accompanied, in general, by a net transfer of material across the interfacial plane, which therefore moves as diffusion proceeds. It is then necessary to solve partial differential equations for unsteady diffusion in each phase and to impose matching conditions so that the fluxes are continuous across the moving interface. This determines simultaneously the fluxes, the composition profiles in both phases, and the motion of the interface. Unfortunately, the interfacial velocity is not constant, nor does it approach any constant value asymptotically, so the problem cannot be reduced to a steady-state problem by choosing axes which move with the interface. Nevertheless, Ruschak and Miller were able to solve the transient equations and make interesting predictions regarding the circumstances under which an interface can exist and the possibility of interfacial emulsions. It is the purpose of this note to present a reformulation of the problem which retains the interesting physical features of Ruschak and Miller’s treatment but permits a straightforward steady-state solution. Furthermore, with certain simplifying assumptions, it will be shown that the problem can be solved by a simple geometric construction in the ternary phase diagram.
essary as the diffusion fluxes can be found without determining the pressure difference explicitly. To be specific, let us seek conditions for a stationary interface at a position distant 1 from the left-hand end of the tube and 1‘ from the right-hand end, as indicated in Figure 1. Then the parts of the diffusion tube on the two sides of the interface are analogous to the two “stagnant” films in the Whitman picture of mass transfer through an interface. The boundary conditions are determined by the compositions of the streams flowing past the ends of the diffusion tube, and these are specified by the mass fracions of the three species present, which will be denoted by x i 0 in the case of the lefthand stream and yio for the right hand stream. Correspondingly the compositions a t general points to the left and right of the interface will be denoted by x i and yi, respectively. Diffusion will be described by the requirement that the mass fluxes ni satisfy relations of the Stefan-Maxwell form, so that
The “Two-Film” Problem
with general solution
Consider the situation depicted in Figure 1, in which different mixtures of the same three substances flow past opposite ends of a tube, through which they may interdiffuse. In these circumstances, depending on the compositions of the mixtures, a phase boundary may or may not form within the tube. If such a boundary exists it is possible to maintain it at rest by applying a suitable pressure difference between the phases and, unless the diffusion tube is very narrow, the required pressure difference will be small. It can be calculated by methods analogous to those used by Kramers and Kistemaker (1943) in treating equimolar counterdiffusion in a binary mixture, but for the present purpose this is not nec304
Ind. Eng. Chem.,Fundam., Vol. 16, No. 2, 1977
where p denotes the density of the mixture, Bi; are the binary pair diffusion coefficients, and z is an axial coordinate, which will be measured from an origin in the interface. For simplicity we will neglect the dependence of p and the coefficients Bij on composition. Of course, there is scant evidence that the Stefan-Maxwell relations give an accurate description of diffusion in condensed phases, but they provide the simplest consistent generalization of Fick’s equation to three component mixtures. Since continuity demands that the fluxes ni be independent of z , eq 1 all have constant coefficients and can be integrated explicitly. A particularly simple situation arises if we further assume that all the binary pair diffusion coefficients are the same Bij
=
(all ij)
when the eq 1reduce to ni
- xin = - p a )
dxi dz
-
(3)
(4)
+ +
where n = n l n2 n3 and Ai is a constant of integration. Ai can be found from the boundary condition xi = x i 0 at z = -1, and (4) then reduces to
(5) giving the composition profile in the fluid to the left of the interface in Figure 1. To the right of the interface the corresponding solution is
2
1Figure 1. Diffusion through an interface.
where p' is the density of the mixture to the right of the interface and D' is the common value of the diffusion coefficients D'ij in this mixture. The interfacial compositions xi(O), yi(0) are then obtained by setting z = 0 in (5) and (6)
The four interfacial mole fractions x1(0),x2(0), y1(0),y2(0) and the three fluxes nl, n2, n3 must satisfy the four independent eq 7 and 8, together with three conditions of phase equilibrium. Thus, there are seven equations to match the seven unknowns, and the problem is solved, in principle. However, the algebraic reduction of these equations is considerably simplified if we restrict attention to situations in which pD/1 = p ' W / l ' . This is the case, to a good approximation, when the interface is centered in the diffusion tube and the densities and diffusion coefficients are similar for the two phases. Whether or not this simplification is quantitatively accurate, we shall show that it permits a simple geometric interpretation of eq 7 and 8 which gives an excellent qualitative "feel" for the circumstances in which an interface will form in the diffusion tube. Accepting this simplification nl, n2, and n3 may be replaced by three equivalent variables [I, &, and q , defined as
Figure 2. Geometric construction of two-phase solution.
further than this. Suppose a direction is associated with each of these line segments by considering the uectors x(0)- yo and y(0) - xo. Then (14) requires that these vectors should be parallel or antiparallel. However, since it follows from the definition of q in eq 9 that q is a positive quantity, eq 15 ensures that they are, in fact, antiparallel. It is now easy to show that this simple geometric requirement, together with the equilibrium conditions at the interface, determines the solution completely. The equilibrium condition is represented in the triangle diagram by a curve bounding the two-phase region, together with a set of tie lines connecting pairs of points representing mixtures at equilibrium, as indicated in Figure 2. The points xo and yo in this diagram represent the compositions of the two flowing streams, while x(0) and y(0) represent the interfacial compositions. With the tie lines as drawn, it is seen that the antiparallel condition for the vectors x(0) - yo and y(0) - xo determines a unique tie line, and hence the interfacial compositions. Furthermore, setting i = 1and 2 successively in (5) and eliminating exp(n(z l ) / p a ) ) between the resulting equations, we obtain
+
X I
"10
Then eq 7 and 8 can be written in the form
Y d O ) = Y2oq +
t2(1
-9)
(13)
and it is a straightforward matter to eliminate the three variables f l , f 2 , and q from these four equations, with the result
"l(0) - Y l O - Yl(0) - " I O "do) - Y20 YZ(0) - "20
(14)
Furthermore, when this condition is satisfied (15) Geometric Construction of the Solution Equations 14 and 15 have a simple geometric significance. Suppose the composition of the mixture is represented by a point in a triangle diagram, in the usual way, and introduce the symbol x to denote the point with coordinates ("1, xz). Then (14) is simply the condition that the line segment joining the points x(0)and yo should be parallel to a second line segment joining the points y(0) and XO.It is possible to go a little
- nlln - x2 - ndn - n h x20 - nzln
(16)
which shows that the composition profile in the fluid to the left of the interface is represented by a straight line path in the triangle diagram. This is indicated by a broken line in Figure 2, and a second broken line shows the corresponding diffusion path in the fluid to the right of the interface, which must also be a straight line. It remains only to determine the fluxes n l , n2, and n3, and these can be found from eq 10-13. When eq 14 is satisfied, only three of these four equations are independent, and they may be solved for the three unknowns { I , (2, and q. nl, n2, and n then follow from eq 9 and the solution is complete. It must be emphasized once again that this simple form of the equations, and hence the geometric construction which follows, is valid only when: (a) all the binary pair diffusion coefficients take a common value, a)to the left of the interface and B' to the right of the interface, and (b) pal1 = p'B'I1'. Interfaces and Emulsions Let us now consider the conditions under which an interface will form. In seeking a tie line which satisfies the antiparallel condition for given xo and yo, it is clear that the search may be restricted to those which intersect the straight line segment joining xo to yo. Usually just one member of this set will be found for which the vectors x(0) - yo and y(0) - xo are antiparallel, though it is conceivable that the disposition of the tie lines may be such that more than one satisfies this condition. On the other hand, if the line segment joining xo to yo does not pass through the two-phase region, it is not interInd. Eng.
Chem., Fundam., Vol. 16, No. 2, 1977 305
2
Figure 3. Single-phaseand supersaturated diffusion paths.
L
Figure 4. )iffusion through a sequence of interfaces. sected by any tie line, and correspondingly there can be no solution with a phase interface. This situation is represented by the stream compositions A and B in Figure 3, and the corresponding solution is single-phase diffusion along a composition path represented by the line segment joining A to B. (In the case of a single phase, the fluxes are not completely determined unless the pressure difference between the flowing streams is also specified.) Thus points A and B in Figure 3 represent a typical situation in which there is single-phase diffusion without an interface, while Figure 2 shows a typical situation in which an interface forms. There is also a third possibility, typified by the stream compositions C and D in Figure 3. In this case the line segment CD passes through the two-phase region, but there is no tie line for which the antiparallel condition is satisfied. Then a two-phase solution is not possible, nor is there a single phase solution unless a considerable degree of supersaturation is accepted where the line CD traverses the two-phase region. This is one of the situations in which Ruschak and Miller suggest emulsification may occur, and it is interesting to pursue this possibility a little further using the present approach. Consider the possibility of a solution with not one, but several interfaces, as indicated in Figure 4. Then a t each interface separately the antiparallel condition must be satisfied; for example the vectors xz - y1 and y2 - x3 must be antiparallel. With fixed stream compositions xo and yo, the appropriate tie lines representing the interfacial compositions can then be located by a trial and error procedure, starting with a guess a t the position of y1. All subsequent interfacial compositions can then be determined sequentially using the antiparallel rule, as indicated in Figure 5, and y1 must be adjusted until the final vector y4 - yo is antiparallel to x4 - y3. The diffusion paths in the five separate regions are represented by the broken line segments, and it is seen that the maximum degree of supersaturation is substantially less than it would be for a single-phase solution represented by the straight line joining xo to yo. Furthermore, the maximum degree of supersaturation can be reduced to any desired extent simply by increasing the number of interfaces. This sequence of narrow regions separated by numerous
306 Ind. Eng. Chem., Fundam., Vol. 16, No. 2, 1977
2
Figure 5. Geometric construction of multiphase solution. interfaces can be regarded as a simple one-dimensional analog of an emulsion, so a physically acceptable solution, without an excessive degree of supersaturation,can always be obtained by interposing a sufficiently finely divided “emulsion” of this sort in the diffusion tube. Whether or not such a structure (or, more properly, its three-dimensional analog) will actually form in preference to the single-phase diffusion path depends, of course, on whether appropriate condensation nuclei are present. Nomenclature A, = integration constant B,Bf = common values of DIJ,B‘,,, respectively B,l,B‘rl = binary pair diffusion coefficients in fluids to left and right of interface, respectively 1,l’ = lengths of diffusion tube to left and right of interface, respectively n = n l n2 n3 n, = mass flux of substance i along diffusion tube q = exp(-nl/pD) x L = mass fraction of substance i in fluid to left of interface x L 0 = mass fraction of substance i in left-hand flowing stream x , (0) = mass fraction of substance i immediately to the left of interface xo = set of mass fractions (x10,XZO, X30) x(0) = set of mass fractions (xl(O), x 2 ( 0 ) , xs(0)) y, = mass fraction of substance i in fluid to right of interface yl0 = mass fraction of substance i in right-hand flowing stream yL(0)= mass fraction of substance i immediately to right of interface yo = set of mass fractions (YIO, y20, y30) y(0) = set of mass fractions (y1(0),yz(O), y3(0)) z = axial coordinate in diffusion tube, measured from interface
+ +
Greek Letters = n,/n p,p‘ = densities of fluids to the left and right of interface, respectively
b
Literature Cited Kramers, H. A., Kistemaker. J., Physica, 10, 699 (1943). Ruschak. K. J., Miller, C. A.. lnd. Eng. Chem., Fundam., 11, 534 (1972).
Department of Chemical Engineering Rice University Houston, Texas 77001
Roy Jackson
Received for review March 25, 1976 Accepted September 10,1976
