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Ind. Eng. Chem. Fundam. 1904, 23,374-377
proportional to the phase composition, and consequently this common eigenvector is only found for equilibrium phases of identical composition, i.e., an azeotropic mixture. In this situation the equilibrium is neutral (transfer of material of the azeotropic composition does not change the Gibbs energy) whereas all systems with equilibrium phases of different composition yield positive G. The only if, i.e., that positive semidefiniteness of both G matrices is a prerequisite for a positive definite H, is incorrect for systems with a number of components c larger than 2. The error committed by the authors is that the vector 6, defined in eq 73, is treated as if its elements were independent. The vector has 2c - 2 elements, but from its definition only c can be chosen freely. Positive definiteness of the final matrix H3 is therefore a sufficient but not a necessary condition for positive definiteness of H. Thus, a two-phase “equilibrium” with one phase (intrinsically) unstable can well represent a constrained minimum in the Gibbs energy. Apparently the authors confuse the existence of such a minimum with stability of the equilibrium state. Stability of the individual phases is an additional requirement for a stable equilibrium.
In section 6 new “facts”, indeed, are presented. The azeotropic case, for which the last c - 1 elements of the 6 vector are identically zero, ought to demonstrate to the authors that their earlier proof is incorrect. Instead they believe that the resulting “strange” situation is specific to azeotropic distillation and conclude that “two-phase azeotropic equilibrium might be stable if one of the phases is materially unstable providing the other phase is so strongly stable that the composite matrix is positive definite”. This statement is incorrect.
Literature Cited Fournier, R. L.; Boston, J. F. Chem. Eng. Commun. 1981, 8 , 305-326. Gautam, R.; Seider, W. D. AIChEJ. 1979, 25, 999-1006. Michelsen, M. L. Fluid Phase Equilib. 1982, 9, 1-19. S0rensen, J. M.; Ark W. “Liquid-liquid Equllibrium Data Collection. Binary Systems. DECHEMA Chemistry Data Serles”, Vol. V, Part 1. DECHEMA, Frankfurt, 1979. Van Dongen, D. B.; Doherty, M. F.; Haight, J. R. Ind. Eng. Chem. Fundam. 1983, 22, 472-485.
Znstituttet for Kemiteknik Bygning 229, DTH DK-2800 Lyngby, Denmark
Michael L. Michelsen
Response to Comments on “Material StabllNy of Multicomponent Mixtures and the Multiplicity of Solutions to Phase-Equilibrium Equations. 1. Nonreacting Mixtures” Sir: In his letter, Dr. Michelsen raises three main points which we will respond to individually below. (1) Interpretation of Material Stability Our paper is concerned exclusively with local stability theory. This was stated clearly in the introductory section titled, “Preliminary Definitions”. In numerous places throughout the paper we reinforce this with such statements as: “Again,we m u m e that all three phases resulting from the phase split differ only slightly in composition from that of the original phase” (Van Dongen et al., 1983 p 476), and “...which guarantee that the calculations predict a state that is at least a local minimum of G and hence stable with respect to small perturbations” (p 472). In order to avoid repeated use of the rather clumsy compound adjective “materially stable” (mixture) we frequently dropped the word “materially” in the latter part of the paper. We expected the reader to interpret the meaning of the word “stable” in context. For instance, we begin section 4 emphasizing that the analysis is local (“phases differing only slightly in composition”) and end by saying, “We therefore conclude that if the matrix g is positive definite, then the phase is stable with respect to....” The meaning of “stable” here can only be interpreted as “locally stable”; the nature of the analysis precludes any other possibility. In case readers consider us peculiar for our above usage, let us quote what Prigogine and Defay (1965, p 209) had to say on this subject. “Very often we describe both stable and meta-stable phases as stable, since both have certain properties in common which distinguish them from unstable phases”. Again, the implication is that the reader should interpret the word in context. In order to remove any uncertainty, let us repeat here what we mean by the stability or instability of a phase. There are two types of situations which can be encountered when discussing whether a phase will split up into two (or more) phases: microscopic fluctuations and macroscopic fluctuations. If an arbitraty microscopic fluctuation brings
about a reduction in the total Gibbs free energy of the mixture, then we can be certain that the original phase will spontaneously split up into two or more phases. The original phase is called materially unstable and such phases can be detected using local analysis based upon Taylor series expansions. A materially unstable phase will initially split into two (or more) nonequilibrium phases which continuously change composition until they are ultimately in phase equilibrium. The time-scale and size-scale for the composition changes during the nonequilibrium transient is not a matter for thermodynamics. However, such details can be predicted by the nonequilibrium theory (i.e., diffusion theory) of Cahn and Hilliard (1958). If a mixture is stable with respect to microscopic fluctuations then we call it materially stable (i.e., locally stable). Such mixtures may or may not be stable with respect to macroscopic fluctuations. A materially stable mixture which is unstable with respect to macroscopic fluctuations is said to be metastable. Metastable mixtures will survive as a homogeneous phase provided heterogeneous nucleation sites are absent. Such mixtures exist in nature and in the laboratory (e.g., see Reid (1978) on superheat limit temperatures in pure fluids and mixtures). We generally refer to a materially stable mixture which is also stable with respect to macroscopic fluctuations as being absolutely stable. Such mixtures must exist as a homogeneous phase. Simple convexity criteria can be developed which make it easy to distinguish between materially stable and materially unstable mixtures. It is harder (but by no means impossible or impractical) to distinguish between metastable and absolutely stable mixtures. In our paper we point out the metastable region in Figures 1, 3, 12, and 14. In addition, we emphasize the precarious nature of a metastable phase on p 474, “Homogeneous liquids can exist in this region but only under carefully controlled conditions”,and again on p. 483, “If an experimental setup could be constructed so as to handle the liquid phase with extreme care in order to
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Ind. Eng. Chem. Fundam., Vol. 23, No. 3, 1984
prevent heterogeneous nucleation ....” Dr. Michelsen says, “In the paper by Van Dongen et al., only intrinsically unstable mixtures are considered unstable, whereas metastable mixtures are treated as stable” (i.e., absolutely stable). This is not true.
(2) Relevance of Material Stability Theory and Spinodals Dr. Michelsen indicates that local stability theory is not relevant. The following list of applications is presented to rebut this point. (i) Polymer-Polymer and Polymer-Solvent Phase Equilibria. When these systems phase separate, it is difficult to measure the binodal curve experimentally. Most experimental procedures reported to date measure the so-called “cloud point curve“ which is taken to be the spinodal, not the binodal. Therefore, when comparing solution theories with experimental data, it is usual to plot the spinodal curve generated by the theory alongside the data measured in the experiments. See, for example, Koningsveld et al. (1974), who state (p 16), “...and if additional information on the AG function itself were obtained, for instance by measuring critical points and spinodals. Spinodals in particular could be very informative ....” One of the many references we could cite is an experimental paper by Derham et al. (1974). They say, “Pulse-induced critical scattering (PICS) is a new, fast technique for determining thermodynamic and kinetic parameters of polymer solutions. ...Spinodal and critical loci for the system polystyrenecyclohexe are measured. ...The evidence suggests that spinodal loci are governed mostly by M,....” This paper has an in-depth discussion on the importance of measuring spinodal curves and relating them to theory. (ii) Spinodal Decomposition and Compatibility of Polymer Blends. The landmark papers by Cahn and Hilliard (1958, 1961, 1965) in the late 1950’s and early 1960’s introduced the mechanism of phase separation which has come to be known as spinodal decomposition. This method of phase separation occurs for certain materials when the overall composition of the fluid mixture lies inside the spinodal region. The g matrix actually appears in the modified convective-diffusion equation describing this mechanism. Materials for which spinodal decomposition is important include: glasses, metal alloys, and polymer blends (see Kwei and Wang, 1978). Of this list, the compatibility of polymer blends is probably the most important application in chemical engineering. In recent years the factors affecting polymer compatibility have been deduced by Sanchez and co-workers (e.g., Lacombe and Sanchez, 1976; Sanchez and Lacombe, 1978; Sanchez, 1978,1980). Their method relies on the use of local stability inequalities. In particular, they study the spinodal curve rather than the binodal curve, “The general character of the liquid-liquid phase diagram can be deduced by studying the spinodal” (Lacombe and Sanchez, 1976, p 2571). We think this technique yields impressive results. (iii) Superheat Limit Temperatures in Pure Fluids and Mixtures. It is well-known that metastable fluids can be made to exist close to their limit of material stability (i-e.,the spinodal). This phenomenon is thought to have important application for the understanding of explosive formation of vapors in some industrial accidents (e.g., see Reid, 1978a, 197813, 1978~). (iv) Compact Mathematical Transformations between Intensive and Extensive Derivatives. In the above applications, it is most convenient to write the local stability conditions and the spinodal condition in terms
375
of intensive variables because equations of state are always written in terms of molar volumes and mole fractions rather than total volume and mole numbers. In our paper, we derive a simple relationship between the derivatives (a2G/dnian.)Tp,n, and the more convenient derivatives (a2g/ axiaxjlTp,,t and (api/ax j ) T p J t , viz. 1 G = -(X - I)Tg(X- I) (1) n and p =
(I - X)Tg
(2) These relationships show how material stability conditions in terms of G get transformed into the corresponding conditions in terms of g and p. Exactly the same reasoning leads to similar relationships between the extensive material stability conditions at constant U, V ,n ; T, V ,n ,etc., and the corresponding (and more convenient) intensive conditions. We did not publish these latter relationships because we thought the extensions were obvious. In their classic book, Prigogine and Defay (1965, p 252-253) considered it relevant to prove that the spinodal condition for ternary mixtures at constant T, P, n can be written as either det G = 0 (3) or det g = 0 (4) By long-hand multiplication of the determinants, they show that for ternary mixtures detG=
(x)
2
detg
The extension of this result to mixtures with four or more components is simply impractical by direct multiplication. In addition, the voluminous algebra gives no hint of the simple underlying transformations between the various derivatives. Our eq 1and 2 above not only give the exact transformations involved but also give the determinants for multicomponent mixtures by inspection, no algebra involved, viz. det p = x , det g (6) XC
det G = det p ).&c-l
(7)
hence x,2 det G = nC-’ det g
Using our formalism, similar relationships can easily be established in any other representation (e.g., U, V, n , etc.). (v) The Importance of Global Stability Checks. Section 7 of our paper gives some of the most convincing evidence to date justifying the need to perform global stability checks in process calculations. We show that solution models will predict multiple equilibria when an insufficient number of liquid phases are considered in the problem. These multiple equilibria have a simple interpretation in terms of G surfaces, phase planes, and most importantly, the underlying phase diagram. Some of these equilibria are even locally stable and so cannot be “thrown out” using local convexity arguments. When multiplicities arise, all the equilibria are unstable with respect to macroscopic fluctuations and hence none of them are of interest in chemical process calculations (e.g., distillation). Therefore, when dealing with highly nonideal mixtures it is essential to avoid accepting an equilibrium calculation
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without performing a global stability check. The consequences of accepting spurious equilibria are potentially disasterous and we show that they are bound to exist if liquid heterogeneities are ignored or undetected. On page 484 of our paper we warn against this in several places, e.g., "Therefore, the character of the phase plane and the interpretation of it depends critically on the types of phases present and great care should be taken to ensure that spurious phase planes are not accepted as true representations of the system's behavior." (3) Two-Phase Equilibria We now address the last point raised by Dr. Michelsen, that of mutual stability of a two-phase equilibrium. Let us first define the problem, together with the meaning of "mutual stability". Suppose that a two-phase multicomponent mixture is placed in a closed container and held a t constant T and P. The mixture in the container will eventually attain a state which is a global minimum in G. This process may involve the formation of new phases or the disappearance of old ones. However, we will constrain the problem by saying that the final state consists of just the two original phases and that all components are present in each phase. If such a state exists, it will be at least a local stationary minimum in G. It may or may not be a global stationary minimum in G depending on whether the total Gibbs free energy can be reduced even further by the formation of new phases. A two-phase equilibrium state which is a local minimum in G is called a mutually stable equilibrium state. Physi d l y realistic two-phase equilibria are mutually stable and materially stable (in process calculations we need to strengthen this to absolutely stable) in each individual phase. Say we have located a stationary point in G by solving the stationary conditions
(i = 1, 2, ..., c) (9) Then several important questions arise: (i) How can we establish whether the equilibrium is mutually stable (i.e., a local minimum in G)? Note, we will refer to every stationary point in G as an equilibrium. (ii) Having identified a local minimum in G, does this guarantee that the phases involved are individually materially stable? (iii) If each of the phases in a two-phase equilibrium is materially stable, does this guarantee that the equilibrium is a local minimum in G? The answers to some of these questions are not obvious and indeed, they are hardly discussed in most thermodynamic books. Even the classic book by Callen (1960) treats only the single-component case. This is a peculiar situation, especially when one considers the frequency with which chemical engineers solve eq 9. Our paper is an attempt to raise these issues and to show that multiple mutually stable equilibria can be displayed by conventional solution models. In fact, the mutually stable equilibria in our examples are even materially stable in each phase. This is most easily seen by inspecting Figure 18. The questions raised above are answered by studying the variation in G about a stationary point, i.e. /.$
= p?I
This is eq 57 in our paper. This quadratic form is most unyielding since each matrix GI and Gn is singular. As an example, H might be
Each of these matrices is symmetric and singular; never-
theless in this example H is strictly positive definite. When attempting to establish whether a stationary point of G is a minimum, maximum, or saddle we must look at the sign-definiteness of the composite matrix H.This is most conveniently done by studying the matrix H2 in our paper (eq 69 and 70) which already has the Gibbs-Duhem equations incorporated into it. In fact, it was the form of H2 which led us further with the analysis. For binary mixtures, H is positive definite if and only if Wl?
+ Pl? > 0
(12)
and (11111
+ 1111II) ( W 2 2 + P z 2 9 - (1112 + 11129 > 0 (13)
Using the Gibbs-Duhem equations to eliminate p12, p12n, from the second inequality leads to
p22, w2J1
PllI
+ P1lI1 > 0
~ 1 1 /.tllI1(NII '
- N1")2 > 0
(14) (15)
Provided N,' # Nl", a necessary and sufficient condition for H to be positive definite is that pllI
> 0 and pllI1 > 0
(16)
i.e., each phase is materially stable. When N,' = N?I the mixture exhibits an azeotrope and det H = 0. This algebraic procedure becomes extremely difficult to carry out even for ternary mixtures (det H multiplies out into 132 terms, each of which contains 5 quantities such as 2 1111'1122p221N2N2u).Therefore, we are forced to seek vector-matrix transformations. The transformations given in our paper are all correct. The quadratic form for 6G given by eq 81 is identical with that given by eq 72 and 70. All of these are identical with the expression for 6G which would be obtained by multiplying out eq 57 (i.e., eq 10, above) and then substituting in the Gibbs-Duhem equations. Equation 81, i.e.
has the convenient property that 6G is expressed solely in terms of the material stability matrices for each phase. This representation of 6G is new, as are the transformations leading to it. We conclude that if GI is positive definite and G" is positive definite then 6G > 0 and the two-phase equilibrium is a local minimum in G. It is not obvious whether 6G > 0 implies GI and G" are each positive definite. However, as we will show below, any other sitution is unlikely. To begin, we will identify the elements of a1 and a2. a=
le
(18)
It is shown that r
Therefore
The elements vl, 712 ..., qc are'completely arbitrary variations (related to the 4 variations by eq 71 in the paper) and the quantities (N? - N?I), i = 1,2, ..., c-1 are constants characteristic of the stationary point (i.e., equilibrium) under discussion. Let us now look at the properties of eq 17 above for binary and ternary mixtures.
Ind. Eng. Chem. Fundam. 1084, 23,377-379
Binary Mixtures. Multiplying out eq 17 gives 1 6G = ;IPl,'?lZ + P1l"hl + Wll - Nl")1/2)21 (21) Since q1 and q2 are completely arbitrary, we see that 6G > 0 if and only if pl,' > 0 and plln > 0. This agrees with the earlier result in eq 16. Ternary Mixtures. Let us now suppose that we have found a local minimum in G such that GI is positive definite and Gn is indefinite. Consider the variation ql = q2 = 0,q3 # 0. Such a variation isolates the Gn matrix. Thus
where we let a = (N> - Nln),b = (PI; - N,"). By hypothesis, Gn is indefinite and a and b are either positive or negative constants. Under these conditions the quadratic form in eq 22 will not generally be greater than zero. This contradicts the original proposition that 6G > 0 for all possible variations. Nevertheless, it is conceivable that the pijn's happen to have values which make this quadratic form positive for the particular values of a and b prevailing a t the equilibrium point under examination. Therefore, Dr. Michelsen is correct when he says that it is possible to fiid a two-phase minimum in G with one of the phases materially unstable. We do not think this is very likely, in general, and it presumably corresponds to a very special event on a ternary phase diagram. It certainly is not a commonplace event. Finally, we will consider the case of an azeotropic equilibrium. For such equilibria, the molar Gibbs free energy of the liquid is equal to that of the vapor C
C
gL = C p F x i ; gv = Z p 7 y i ; and x i = yi (i = i=l
1=1
1,2, ..., c); piL = pi" (i = 1, 2, ...,C) (23) and hence the G surface for the overall system displays a linear trough (as opposed to a banana shaped trough) with a horizontal bottom. This is precisely why det H = 0 for azeotropic mixtures. For such mixtures we show that 6G takes on a special form, namely
377
This is even true for binary mixtures as can be verified by multiplying out the quadratic form in eq 10 above and then using the Gibbs-Duhem equations to get Hence, for azeotropic equilibria it is possible to locate a minimum in G with one of the phases materially unstable. Moreover, this is a much more likely event that the corresponding situation for nonazeotropic mixtures. The reason for this is evident from the way GI and Gnappear together in eq 24 but individually in eq 17. More than all this, our paper not only develops the appropriate mathematics to discover this phenomenon, but it also gives an interpretation on a binary phase diagram. The spurious homogeneous azeotrope pin-pointed in Figure 18 in our paper is typical of the situation discussed above. At this point, pllL < 0; pllv > 0, and as we say in the paper, such an equilibrium is indeed spurious. Literature Cited Cahn, J. W.; Hilllard, J. E. J . Chem. phvs. 1958, 28, 258-267. Cahn, J. W. Acta Metall. 1981, 9 ,795-801. Cahn, J. W. J . Ct".RIP. 1985, 42, 93-99. Callen, H. B. "Thermodynamics"; Wiley: New York, 1960. Derham. K. W.; Goldsbrough, J.; Gordon. M. Pwe Appl. Chem. 1974, 38. 97-116. Konlngsveld, R.; Kleintjens, L. A.; Schoffeleers, H. M. Pure Appl. Chem. 1974, 39, 1-32. Kwel, T. K.; Wang, T. T. "Phase Separation Behavior of Polymer-Polymer Mixtures", In "Polymer Blends", Voi. I, Paul, D. R.; Newman, S., Ed.; Academlc Press: New York, 1978. Lacombe, R. H.; Sanchez, I. C. J . Wys. Chem. 1978, 80, 2568-2550. Prigoglne, I.; Defay, R. "Chemical Thermodynamics" (Translated by Everett, D. H.), Longmans: London, 1965. Reid, R. C. Chem. Eng. Educ. 1978a, Spring Issue, 80; i978b, Summer Issue, 108; 1978c, Fall Issue, 194. Sanchez, I . C.; Lacombe, R. H. Mewomobcules 1978, 11, 1145-1156. Sanchez, I. C. "Statlstlcal Thermodynamlcs of Polymer Blends", Chapter 3 in "Polymer Blends", Vol. I , Paul, D. R.; Newman, S., Ed., Academic Press: New York, 1978. Sanchez, I . C. J . Mecromol. Scl.-phys. 1980, 617(3), 565-589. Van Dongen, D. B.; Doherty, M. F.; Halght, J. R. I d . Eng. Chem. Fundam. 1983, 22, 472-485.
Department of Chemical Engineering University of Massachusetts Amherst, Massachusetts 01003
Michael F. Doherty* David B.Van Dongen James R. Haight
Comments on "Effect of Vapor Efflux from a Spherical Particle on Heat Transfer from a Hot Gas" Sir: In a recent paper Kalson (1983) concluded that the Ackermann factor
for correction of the heat transfer from a solid body to a gas stream to account for finite rates of mass transfer, depends on the geometry of the body. He obtained significantly different results for the flat plate and spherical geometries. This apparent dependence of ZH on the geometry considered is more than a little disquieting and we show below that this is a direct consequence of his definition of the heat transfer coefficients in the two cases h = kH/6 (2) where 6 is the "film" thickness. It is the object of the present communicationto show that if an alternative, more physically meaningful, definition of the heat transfer coefficient is employed, taking proper account of the 01964313/84/1023-0377$01.50/0
curvature of the interface, and proper nondimensionalized parameters are used, identical values of the Ackermann correction factors are obtained for planar, cylindrical, and spherical geometries. The results are generalizable to the consideration of the effect of mass transfer on the mass transfer coefficients in nonideal multicomponent fluid mixtures, showing the general applicability of the form of the Ackermann correction factor, eq 1. In view of our objective to generalize the Ackermann theory, we use a consistent nomenclature, different fro that adopted by Kalson (1983), and employ molar units in place of mass units. An entirely parallel treatment holds for mass units. For steady-state transport in nonreacting mixtures, the equations of continuity of mass and energy reduce to d(YEr) d(r"Nir) - 0 (i = 1,2, ..., n); -(3) dr dr - 0 where v = 0, 1, and 2 respectively stand for planar, cylin0 1984 American Chemical Society
