373
Ind. Eng. Chem. Fundam. 1964, 23,373-374
Greek Symbols
j3 = a parameter determined from eq 7 or a measure of reaction
plane movement, cm/s1I2 0 = time of contact, s
Subscript 0 = initial Superscript * = at the interface Registry No. SOz, 7446-09-5;Ca(OH)z,1305-62-0.
Literature Cited B i b , I.; Bengetsson, S.; Farnkvist, K. Chem. Eng. Scl. 1972, 27, 1853. Boelter. L. M. K. Trans. Am. Inst. Chem. Eng. 1943, 39, 557. Cullen, E. J.; Davldson, J. F. Trans. Faraday Soc. 1962, 53, 113. Danckwerts. P. V. “Gas-Lbuid Reactbns”; Mcoraw-HiII: New York, 1970; p 81. Hobbler, T. “MassTransfer and Absorbers”; Pergamon Press: Oxford, 1968; p 481.
Matsuyama, Y. Mem. Fac. Eng. Kyoto Unlv. 1953, 15, 142. NiJsing, R. A. T. 0.; Hendriksz, R. H.; Kramers, H. Chem. Eng. Sci. 1959, 10, 88. Onda, K.; Kobayashl, T.; Fugine, M.; Takehashi, M. Chem. Eng. Scl. 1971, 26, 2009. Onda, K.; Sada, E.; Kobayashi, T.; Odaka, M. Kagaku Kogaku 1969, 33, 886. Rehm, T. R.; Moll, A. J.; Babb, A. L. AIChEJ. 1963, 9 , 780. Scriven, L. E.; Plgford, P. L. AIChEJ. 1959, 5 , 397. Sharma, M. M.; Danckwerts, P. V. Chem. Eng. Sci. 1963, 18, 729. Toor, H. L.; Raimondi, P. AIChE J. 1959, 5 , 86.
Department of Chemical Engineering and Applied Chemistry University of Toronto Toronto, Ontario, Canada M5S l A 4
Dasari Ram Babu’ G . Narsimhan’ Colin R. Phillips*
Received for review April 20, 1982 Accepted January 25, 1984 Research Laboratory, ylderabad, India. * Regional Bendei State University, Nigeria.
CORRESPONDENCE Comments on “Materlai Stabiilty of Multicomponent Mixtures and the Multiplicity of Solutions to Phase-Equllibrlum Equations. 1. Nonreacting Mixtures” Sir: A paper by Van Dongen et al. on “Material Stability of Multicomponent Mixtures and the Multiplicity of Solutions to Phase Equilibrium Equations” appeared in the November 1983 issue of this journal. The acceptance for publication of this work deserves some comments, since in my opinion the paper lacks relevance, uses substantial amounts of space for repeating known material, and is in error. The bulk of the paper is concerned with stability, and according to basic definitions mixtures of fixed overall composition at given temperature and pressure are classified as either stable, metastable, or intrinsically unstable. A stable mixture cannot decrease its Gibbs free energy. Metastable mixtures are unstable and form separate phases in virtually all situations of practical interest. The distinction between metastable mixtures and intrinsically unstable mixtures is that for the former the matrix of second composition derivatives of the Gibbs free energy has no negative eigenvalues (i.e., is positive semidefinite), whereas for intrinsically unstable mixtures at least one eigenvalue is negative. For metastable mixtures a split into phases with a macroscopic difference in composition is required to decrease the Gibbs energy, and for intrinsically unstable mixtures the decrease is obtainable with infinitesimal differences. In the paper by Van Dongen et al., only intrinsically unstable mixtures are considered unstable, whereas metastable mixtures are treated as stable. The inadequacy of this treatment is well demonstrated by a liquid mixture of water and benzene at 298 K. Using the UNIQUAC model with parameters from Serrensen and Arlt (1979, p 341) the mixture will form two liquid phases when the overall water concentration is in the range 0.003 < XHl0 < 0.9996. However, intrinsic instability is only observed in the range 0.43 < X H I O < 0.91. This result is of little value, since any algorithm for practical industrial application must be able to correctly handle metastable mixtures. Recent examples of such algorithms are those 01964373/04/1023-0373$01.50/0
of Gautam and Seider (1979),Fournier and Boston (1981), and Michelsen (1982). The authors of the present paper do not refer to these attempts but only state that a more complete stability analysis requires a tremendous effort in trial and error search. Admittedly, the relevance of any subject is open to discussion. A more detailed treatment of the criterion of intrinsic instability might be justified if “the extension of the theories to multicomponent mixtures is nontrivial and lacks careful development”,and if, as claimed, “several new facts about material stability and the spinodal curve have been found”. However, the authors’ “new facts” are often incorrect. Section 5 contains a 3-page “proof“ that a two-phase equilibrium will be stable if and only if the phases are individually stable with respect to phase splitting, and it is stated that “this is the first satisfactory proof“ of an often quoted result. The basic equation used by the authors is eq 58 H = G’ GI1 where H is the matrix of second derivatives of the total Gibbs energy with respect to the amount of material transferred between phases, and the G matrices represent matrices of second derivatives of the Gibbs energy with respect to mole numbers for the individual phases. The authors claim that H is positive definite if and only if the G matrices are both positive semidefinite. The if is trivially correct since the individual terms in the sum of quadratic forms, eq 59
+
1
+1
G = -eTG’c -eTG”c 2 2 are both nonnegative when the G matrices are both positive semidefinite. The sum can only be zero provided t is a common eigenvector with zero eigenvalue for both matrices. By virtue of the Gibbs-Duhem equation, both matrices have a zero eigenvalue eigenvector with elements 0 1984 American Chemical Society
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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
0196-4313/84/1023-0374$01.50/00 1984 American Chemical Society
