Znd. Eng. Chem. Res. 1996,34, 3195-3202
3195
Calculation of Residue Curve Maps for Mixtures with Multiple Equilibrium Chemical Reactions Sophie Ung and Michael F. Doherty* Department of Chemical Engineering, University of Massachusetts, Amherst, Massachusetts 01003
We derive the differential equations governing the simple distillation of mixtures with multiple equilibrium chemical reactions. The results are most conveniently expressed in terms of transformed composition coordinates, which provide a reduction of dimensionality consistent with the Gibbs phase rule. The resulting residue curve maps give an easy visualization of the combined phase and chemical equilibria, and from them it is possible to determine the presence of reactive azeotropes, as well as nonreactive azeotropes that survive the reactions. Three examples are described: an ideal mixture based on the chemistry of formaldehyde water, the synthesis of MTBE in the presence of inert n-butane, and the alkylation of m-xylene in the presence of inert p-xylene.
+
Introduction One of the lasting contributions of Transport Phenomena is the systematic framework that it provides for tackling complex transport problems in a way that transcends the special features of each new situation encountered. Prior to its publication each class of transport problem tended t o be treated in isolation. Until recently the same situation prevailed in the nonideal separations literature. Each problem has its own particular folklore, but the total body of knowledge does not provide a vehicle to solve the next problem unless it happens t o be almost identical to one already encountered. Equally important, this body of knowledge does not provide a systematic way of inventing new alternative separation systems even for mixtures which belong t o the existing knowledge base. In the spirit of Transport Phenomena there has been an effort during the past decade to provide a more systematic framework for understanding the separation of complex mixtures based on geometric methods, beginning with residue curve maps and leading to more advanced concepts such as fured-point branches and bifurcation diagrams. This paper is intended to be a contribution t o the development of systematic methods for understanding complex separations, and is dedicated to the authors of Transport Phenomena. Residue curve maps, which are obtained from the study of simple distillation (open evaporation) processes, are a useful tool for representing phase diagrams. A residue curve map contains the same information as the phase diagram for a mixture. It indicates the presence of nonreactive azeotropes that survive the reactions, the presence of reactive azeotropes, and the presence of distillation boundaries for continuous distillation at infinite reflux. Such maps, therefore, are a useful way of representing phase equilibria and provide a link between the intrinsic thermodynamic behavior of the mixture and its limiting separation characteristics in both continuous and batch columns (Malone and Doherty, 1995). Calculation of residue curve maps for mixtures with a single reaction has been developed by Barbosa and Doherty (1988),who derived the set of autonomous ordinary differential equations describing the dynamics of homogeneous reactive simple distillation using a set of transformed composition variables. In this paper we derive the equations that describe reactive simple distillation of mixtures with multiple equilibrium chemi-
cal reactions and calculate residue curve maps for a selection of examples. With this approach we are able to represent the phase equilibrium and limiting separation characteristics of quite complex mixtures in a relatively simple way.
Derivation of the Equations In reactive simple distillation or open evaporation, a liquid is vaporized a t constant pressure while the vapor is continuously removed and all the equilibrium reactions are proceeding simultaneously. We assume that there are a set of c reacting components and Z inert components in the mixture for a total of C components. The stoichiometriccoefficientsare specified by the usual convention, vir < 0 if component i is a reactant in reaction r, vir > 0 if component i is a product in reaction r, and vir = 0 if component i does not participate in reaction r (or if component i is an inert). As the liquid is vaporized, the most volatile components generally evaporate first, and the composition of the liquid changes with time. The locus of the liquid compositions determines the residue curve map. As the R equilibrium reactions are simultaneously occurring in the system, we can write the material balance for component i as W x j ) = -vy, dt
dc + viT dt
i = 1, ..., c
(1)
where H is the molar liquid holdup in the still, V the molar flow rate of the vapor evaporating, and 6 is the column vector of the R molar extents of reaction. Among the C equations, we can set apart the overall mass balance, as well as R equations for R reference components that will be used to eliminate the extents of reaction. The R reference components are labeled (without loss of generality) from (C - R 1)to C. We are left with an equivalent set of C equations:
+
-d(Hxi) - -vyi + viT& dt
dt
i = 1,..., C - R - 1 (3)
0888-5885/95/2634-3195$09.0QIQ 0 1995 American Chemical Society
3196 Ind. Eng. Chem. Res., Vol. 34, No. 10, 1995
components, we find (4) where YRef is the column vector of dimension R , y h f = (yc-R+1, ..., ydT, of the vapor mole fractions for the R reference components. Notice that there is no material balance for the component labeled (C- R);this balance has been replaced by the overall balance, eq 2. We define a e f as the square matrix of stoichiometric coefficients for the R reference components in the R reactions. The components are ordered by rows and the reactions by columns: '(C-R+l)l
or
v ( ~ - ~ + i ) ~
vir
...
i 'CR
Provided that the matrix R e f is invertible, which is guaranteed by choosing a suitable set of reference components as described by Ung and Doherty (1995a), (also see the Appendix), we invert the set of R eqs 4 and obtain
which we can, in turn, substitute into eqs 2 and 3 giving:
Finally,
and
i H -h + - x .d=H dt dt
(14)
i = 1,..., C - R - 1 (7)
We recognize the presence of the transformed composition variables Xi and Yi,defined by the equations
Equation 6 becomes
and
Yi=
and eqs 7 give
yi
- vT(v&f)-lY&f
1 - v;OT(v&f)-lY&f
i = 1, ..., C - R
(16)
derived in Ung and Doherty (1995a,b). The transformed composition variables satisfy the following constraints:
cxa=l
C-R
i = 1, ..., C - R - 1 (9)
i=l
Rearranging eq 9 further gives and
c
C-R
Y,=l
(18)
i=l
which is ensured by choosing appropriate reference
These variables have the same dimensionality as the number of composition degrees of freedom determined by the Gibbs phase rule for an isobaric reactive system, namely (C- R - 1). Defining a warped time, z,by
Ind. Eng. Chem. Res., Vol. 34, No. 10,1995 3197
A2
we substitute eqs 15, 16, and 19 into eq 14 and find
dXildt = X i- Yi i
1, ..., C - R
-1
(20)
For the sake of physical consistency we desire the warped time to go in the same direction as real time, and this imposes a second criterion on the choice of reference components. We describe the relationship between the warped time, z, and the real time, t , and give the second criterion in the Appendix. In the new coordinate space of transformed composition variables, the simple distillation equation appears in a simpler form, and has exactly the same functional form as the simple distillation equation for non-reacting mixtures in terms of mole fractions. The singular points of eq 20 occur where X = Y,which correspond to all the reactive azeotropes in the mixture, as well as the surviving nonreactive azeotropes and pure components (Ung and Doherty, 1995~).In order to solve eq 20 a phase and chemical equilibrium algorithm is needed to relate Y to X. We use the algorithm described in Appendix B of Ung and Doherty (1995a) to calculate y from X. The corresponding value of Y is obtained from eq 16. Each residue curve in the map is characterized by specifying an initial value for (C - R - 1)transformed liquid composition variables, and eqs (20) are solved using Gear's method as implemented in the IMSL subroutine DIVPAG. The representation of residue curve maps in the transformed composition coordinates is a powerful way of describing the very complex phase equilibrium in these systems. It provides a visualization of the phase behavior which could not be done easily in mole fraction coordinates.
Example 1: Ideal Mixture Based on the Chemistry of Formaldehyde-Water Systems We compute the residue curve map for an ideal mixture based on the chemistry of formaldehyde-water mixtures (see Ung and Doherty (1995a1, Table 1, for thermodynamic data). Component A1 reacts with component A2 to form component &. Component A3 forms an oligomer that grows as a molecule of component A3 is added to the chain, and a molecule of A2 is eliminated (i.e., condensation polymerization). Here, we will only consider the first polymerizationreaction. This confines the problem to two reactions which can be written as A, +&-A3
(21a)
2.&-%+&
(21b)
The degrees of freedom for the isobaric reactive system are C - R - 1 = 4 - 2 - 1 = 1. The system thus has one degree of freedom, and in transformed composition variables the residue curve lies on the straight line segment bounded by zero and unity. This provides a very simple visualization of the information computed. In order to give more structure to the solution, we will represent the residue curve in the mole fraction coordinate space. In this case, it is possible to do so because there are only four components and the mole fraction space is represented by a tetrahedron. The calculations, however, are performed using the trans-
A, Figure 1. Residue curve map in mole fraction coordinates for the ideal mixture based on formaldehyde-water chemistry at P = 1 atm. A1 A2 -&; 2& - & f A2.
+
formed composition variables, solving eqs 20, and backcalculating the corresponding mole fractions. Using the procedure described in an earlier paper (Ung and Doherty, 1995a) to determine the expressions for the transformed composition variables we have (22)
(23) The residue curve map contains exactly the same information as the phase diagrams since the calculations draw only on the thermodynamic phase and reaction equilibrium data. In this particular case, since the isobaric system has one degree of freedom, the residue curve map will be a single curve enclosed in the mole fraction tetrahedron and the residue curve map (see Figure 1)will be identical to the reactive phase diagram (see Ung and Doherty, 1995a). Each point on this curve is at a Werent equilibrium temperature, and at one extreme of the curve we obtain pure A1 while at the other end we have pure A2. The arrow on the curve points in the direction of increasing time, which is also the direction of increasing temperature.
Example 2: MTBE Reaction System with Inert Component Present Methyl tert-butyl ether (MTBE) is produced by reacting methanol with isobutene in the presence of inert Cis. For simplicity, we follow the experimental study of DeGarmo et al. (1992), who used n-butane as a representative inert component in place of the mixed Cis. The reactive system is described as isobutene (AI) reacting with methanol (A21 to produce the desired product MTBE (Ad, in the presence of an inert, nbutane (Ad:
+
isobutene (1) methanol (2)
.-MTBE
(3)
3198 Ind. Eng. Chem. Res., Vol. 34,No. 10,1995
equation, saturated vapor pressures are calculated from the Antoine equation, and the reaction equilibrium constant is an exponential function of temperature. Since there are only four components, we can still visualize the residue curves in the mole fraction space (see Figure 2). In the mole fraction tetrahedron, the residue curves are confined to be on a two-dimensional surface. If we now represent the residue curves in the transformed composition coordinates (Figure 31, they will still be enclosed in a two-dimensional space, (same degrees of freedom),but this time it is a planar triangle; i.e., the solution space in transformed composition variables for this particular reactive mixture is a tiangle. The transformed composition variables are given by
Methanol (2)
.
I
Nonreactive azeotrope
u Isobutene (1)
MTBE (3)
-
Figure 2. Residue curves in the mole fraction tetrahedron for the reactive mixture isobutene (1) methanol (2) MTBE (3) with n-butane (4) as inert. P = 11 atm.
+
The degrees of freedom for the isobaric reactive system a r e F = C - R - 1= 4 - 1- 1=2. The thermodynamic models and data used in the calculations are reported in Table 3, Ung and Doherty (1995a). The liquid-phase activity coefficients are calculated from the Wilson
+x3 x,= 1
(25)
x2 + x 3 X ' --1+x3
(26)
X -- x4
(27)
x1
+x3
4-1+x3
We choose XI and XZ as the independent variables. Figure 3 represents the residue curve map for the MTBE system at 1 atm pressure, and Figure 4 shows Enlargement (Detail of bottom comer)
Methanol (64.79C) 1.o
0.8
0.6 x2
0.4
0.2
Non-reactive Azeotrope (-0.73OC)\
0.0 0.0
0.2
0.4
0.6
0.8
N-butane X1 (-0.5OC) Figure 3. Residue curves in transformed composition variables for the reactive system isobutene as inert. P = 1 atm.
1.o
Isobutene (-6.9OC) + methanol MTBE with n-butane
-
Ind. Eng. Chem. Res., Vol. 34, No. 10,1995 3199 Enlargement (Detail of bottom comer)
0.0
0.2
0.4
0.6
0.8
N-butane X1 (80.88OC) Figure 4. Residue curves in transformed composition variables for the reactive system isobutene as inert. P = 10 atm.
the same system a t 10 atm. Each corner of the triangle represents a pure component, i.e., methanol, n-butane, isobutene. The left side of the triangle represents the binary nonreactive mixture n-butane methanol and the base is the nonreactive edge representing the binary mixture n-butane isobutene. The hypotenuse of the triangle is the reactive edge methanol-isobuteneMTBE where each point on the hypotenuse is a t a different equilibrium composition. The interior of the triangle represents the reactive mixture of all four components at chemical and phase equilibrium. The residue curves enable us to identify nonreactive azeotropes that survive the reactions as well as reactive azeotropes that appear because of reaction. Here, we can locate the presence of a nonreactive azeotrope between n-butane+methanol that survived the reaction, located where all the residue curves converge on the lefthand side of the triangle. We can also determine the presence of a pseudo-reactive azeotrope located halfway up the hypotenuse. It is called "pseudo" because the residue curves are heavily curved toward it, but do not exactly converge to the point. This upseudo" reactive azeotrope acts as a severe tangent pinch, and in the context of continuous equilibrium reactive distillation it is difficult (essentially impossible) to pass this point. It is composed of an equilibrium amount of MTBE, isobutene and methanol, at high conversion, and it is an intermediate boiling reactive mixture (see the phase diagram shown in Figure 5, taken from Ung and Doherty (1995a)). Topologically, this point acts as a
1.o
Isobutene (71.85OC) + methanol MTBE with n-butane
-
+
+
-
Figure 5. Phase diagram in transformed composition variables
+
and temperature for the reactive system isobutene methanol MTBE with n-butane as inert. P = 1 atm. (Reprinted with permission from Ung and Doherty (1995).Copyright 1994 Elsevier Science Ltd.)
saddle on one side, and as a stable node on the other side of the map. Finally, in the enlargement of Figure 3 it is not possible to tell from our calculations whether the n-butane vertex is a nonelementary half-saddle
3200 Ind. Eng. Chem. Res., Vol. 34, No. 10, 1995
singular point, or whether there is additional fine structure to the map, as occurs a t higher pressures (see Figure 4). It is interesting to observe what happens to these reactive and nonreactive azeotropes as the pressure is changed. We increase the pressure from 1 to 10 atm (Figure 4) and find that the composition of the azeotropes change with pressure. The nonreactive azeotrope of n-butane methanol moves up in composition (i.e., richer in methanol), whereas the pseudo-reactive azeotrope "opens up" as the residue curves do not converge to it quite as tightly as in the low-pressure case. In addition, a quaternary saddle reactive azeotrope becomes clearly visible in the neighborhood of the nbutane vertex. Finally, we note that the information obtained in Figures 2 and 4 is the same (except for a slight change in pressure) but it is easier to determine the structure of the map and the separation feasibilities in transformed composition coordinate space (i.e., Figure 4) than in mole fraction space (Figure 2).
Benzene (5)
Mefa-xylene (2)
(78.8"O 0.2
0.0
(1 38.7'C)
x3
0.4
0.8
0.6
1.0
+
Example 3: Alkylation of Xylenes with Di-tert-butylbenzene We now study a six-component mixture undergoing two equilibrium reactions. m- andp-xylene are difficult to separate by nonreactive distillation because they have close boiling points. Extractive distillation cannot easily be used, because adding a third component does not shift their relative volatilities. However, Saito et al. (1971) have suggested that the separation is feasible when using reactive distillation. This takes advantage of the fact that in an alkylation process m-xylene is selectively reacted whereas p-xylene is left unreacted. The alkylation reactions are described as follows:
+
di-tert-butylbenzene m-xylene tert-butyl-m-xylene tert-butylbenzene
+ m-xylene
-
-
+ tert-butylbenzene
tert-butyl-m-xylene
+ benzene
Figure 6. b s i d u e curve map in transformed composition variables for the xylene mixture without p-xylene at P = 1 atm. The ternary reactive edge consists of components 1,4, and 5,where 4 is tert-butylbenzene (bp 168.2 "C).
transformed composition variables are determined by first choosing two reference components. Components A1 and A2 are suitable choices, because the matrix of their stoichiometriccoefficients is invertible. We therefore have
or
Y2,
where di-tert-butylbenzene is component AI, m-xylene is component A2, tert-butyl-rn-xylene is component &, tert-butylbenzene is component &, and benzene is component &. p-Xylene is component & and does not appear in the set of reactions because it acts as an inert. The presence of p-xylene, does however, influence the reactive vapor-liquid phase diagram. When p-xylene is present, the degrees of freedom for the isobaric system given by the Gibbs phase rule for reactive systems is three, giving rise to a three-dimensional residue curve map. When there is nop-xylene, the number of degrees of freedom decreases to two and the residue curve map is two-dimensional. For the sake of simplicity, we first compute the residue curve map for the reactive system without the presence of inert p-xylene. In our calculations we treat the mixture as ideal in both phases and use the model and parameters reported in Ung and Doherty (1995a). The system has two degrees of freedom, and the residue curve map can be completely represented in a two-dimensional transformed composition space. The
= -1
Y22 =
-1
and
The expressions for the transformed composition variables are
among which only two variables are independent due to eq 17. We chooseX3 andX5 as independent variables. In this set of coordinates the composition space is confined within a trapezoid. Figure 6 shows the residue curve map in transformed composition space. Each corner of the trapezoid represents a component that does not undergo any dissociation reaction. The top edge of
Ind. Eng. Chem. Res., Vol. 34, No. 10, 1995 3201 Pam-xylene (6) (137.JOC)
x6
t
Mebxykne (2)
Benzene (5)
( 138.7"C)
(78.8'C)
Figure 7. Three-dimensional residue curve map in transformed composition variables for the xylene mixture with p-xylene at P = 1 atm. The ternary reactive edge consists of components 1, 4, and 5,where 4 is tert-butylbenzene (bp 168.2"C). Component 1 is di-tert-butylbenzene (bp 232.8 "C).
the trapezoid represents the binary nonreactive mixture of benzene (45) and m-xylene (Ad, the right-hand-side edge represents the binary nonreactive mixture of rn-xylene (Ad and tert-butyl m-xylene (&); the slanted bottom edge is the nonreactive binary edge of tert-butylm-xylene (&) and di-tert-butylbenzene(AI). Finally, the remaining edge represents the ternary reactive mixture of benzene, di-tert-butylbenzene, and tert-butylbenzene, where tert-butylbenzene (Ad undergoes dissociation into benzene (&) and di-tert-butylbenzene (Ad. We assume that the system reaches thermodynamic equilibrium instantly, and therefore & can never be obtained pure because it instantly dissociates. Thus, there are four singular points to the simple distillation equation located a t the corners of the trapezoid: component A1 is a stable node, AZ and & are saddles, and A5 is an unstable node. We deduce from this residue curve map that there are no reactive azeotropes and no distillation boundaries in the mixture. Let us now consider the case where p-xylene is present. Adding one inert to the mixture increases the degrees of freedom from two to three. We determine the solution space in transformed composition variables using the same procedure as earlier. We keep the same reference components A1 and A2, and the expressions for X3,Xq, and x6 are identical to eqs 30-32. In addition, there is an expression for x6
When the solution space in transformed composition variables is mapped out, the presence of inert A6 adds a vertex above the planar trapezoid. This creates a three dimensional object shown in Figure 7, where the base is the trapezoid found when nop-xylene is present. All the edges linking the pure component vertices to the inert are nonreactive edges, whereas the volume inside the three-dimensional object characterizes a reactive mixture of all six components. Since p-xylene is an intermediate boiler, it is a saddle and the other vertices keep the same stability. In Figure 7, the residue curve map shows that there are no reactive azeotropes and no distillation boundaries.
Conclusion Our approach provides a way of expressing the differential equations for simple distillation with multiple equilibrium chemical reactions in terms of new composition variables. The resulting model has a familiar structure that has many properties in common with the model for nonreactive mixtures e.g., singular points occur a t pure components and azeotropes. We have performed calculations and plotted residue curve maps for three reactive mixtures containing up to six components and two equilibrium reactions. Two examples have simple structure, showing no azeotropes either with or without reaction. "he MTBE example, however, has a richer structure. Two of the three nonreactive binary azeotropes do not survive the reaction, and additional azeotropes appear because of reaction. These features are easy to visualize on a residue curve map. The stable and unstable nodes in the map provide candidate products (bottoms and distillate, respectively) from continuous equilibrium reactive distillation columns. This information provides the basis for a column sequencing strategy for equilibrium reactive distillation systems. Results on this problem will be reported separately.
Acknowledgment This research was supported by the National Science Foundation (Grant No. CTS-9113717). We are grateful for the drafting services of Ms.Pamela Stephan.
Nomenclature c = total number of reacting components C = total number of reacting and inert components I = total number of inert components H = molar liquid holdup in the still P = pressure R = total number of independent reactions t = time V = molar flow rate of the vapor ?&f = square matrix of dimension (R, R)of the stoichiometric coefficients for the R reference components in the R reactions xkf = column vector of mole fractions for the R reference components in the liquid phase xi = mole fraction of component i in the liquid phase Xi = transformed composition variable for component i in the liquid phase X = vector of the (C - R - 1) independent liquid transformed composition variables yi = mole fraction of component i in the vapor phase Yi = transformed composition variables in the vapor phase Y = vector of the (C - R - 1) independent vapor transformed composition variables YRef = column vector of the mole fractions for the R reference components in the vapor phase Greek Symbols E , = molar extent of reaction r E = column vector of the R molar extents of reaction vir= stoichiometric coefficient of component i in reaction r v: = row vector of the stoichiometric coefficients for component i in each reaction V T ~= sum of all stoichiometric coefficients for reaction r v & , ~ = row vector of the sum of the stoichiometric coefficients for each reaction z = warped time Subscripts C = total number of components
3202 Ind. Eng. Chem. Res., Vol. 34, No. 10, 1995
that (1 - v;oT('Z&f)-lxhf) is never zero. Since (1 v&T('Z&f)-lx&f)* 0, it must be either strictly negative or strictly positive for all values of xhdt). There is an advantage to selecting the reference components so that (1 - V;oT(%f)-'XRef) > 0 and (1- v;oT(%ef)-lyRef) > 0 since this implies (by eq (A2))
i = components r = reactions Ref = reference components T, TOT = total Superscripts
T = transpose of matrix or vector -1 = inverse of matrix
Appendix. Relationship between t and r During the derivation of the simple distillation equation, we obtained eq 19, which relates the real time t t o a warped time z, as follows:
dzldt > 0 (A71 which guarantees that the warped time increases in the same direction as real time. Therefore, we find that z variesrnonotonically from 0 t o as t varies from 0 to t, where t,, is the time a t which no liquid remains in the still, i.e., H(t& = 0. The criteria (1 - Y;OT('Z&f)-'X&f) > 0 and (1 v;m(%f)-lyhf) > 0 are guaranteed by selecting the reference components so that
+-
(1) and (2)
or
Vhf is invertible T YTOT(
v&f)-' is a row vector containing
negative or zero entries The overall mass balance (eq 8) provides the relationship between the molar liquid holdup H and the molar vapor flow rate V. Rearranging this equation gives
All the examples in this and other papers that we have written on multireaction systems (see Ung and Doherty, 1995a-c) satisfy conditions A8.
Literature Cited
Substituting into eq A2 we obtain
which may be written
Then, defining t such that z = 0 at t = 0, we obtain a relationship between the real time t and the warped time z by integrating eq A5:
I
t = In
]
H(o)[1- v:OT( vhf)-'x&ko)l H(t)[1 - &T( v&f)-'X~&)l
Barbosa, D.; Doherty, M. F. The Simple Distillation of Homogeneous Reactive Mixtures. Chem. Eng. Sci. 1988,43,541-550. DeGarmo, J . L.; Parulekar, V. N.; Pinjala, V. Consider Reactive Distillation. Chem. Eng. Prog. 1992,88 (3),43-50. IMSL. FORTRAN Subroutines for Mathematical Applications; Houston, Tx,1987. Malone, M. F.; Doherty, M. F. Separation System Synthesis for Nonideal Liquid Mixtures. In Foundations of Computer-Aided Process Design; Biegkr, L. T., Doherty, M. F., Eds.; AIChE: New York, 1995. Saito, S.; Michishita, T.; Maeda, S. Separation of m- andp-xylene Mixture by Distillation Accompanied by Chemical Reaction. J. Chem. Eng. Jpn. 1971,4,37-43. Ung, S . ; Doherty, M. F. Vapor-Liquid Phase Equilibrium In Systems With Multiple Chemical Reactions. Chem. Eng. Sci. 199Sa,50, 23-48. Ung, S.;Doherty, M. F. Theory of Phase Behavior in MultiReaction Systems. Chem. Eng. Sci. 1996b,in press. Ung, S.; Doherty, M. F. Necessary and Sufficient Conditions for Reactive Azeotropes in Multi-Reaction Mixtures. AIChE J . 1996c,in press.
Received for review April 11, 1994 Revised manuscript received February 7, 1995 Accepted February 17, 1995@
(A6)
where H(0)is the initial liquid holdup in the still and xhet(0)is the initial liquid composition of the R reference components. Equation A6 is well-defined since the rules for choosing reference components are that 'Z&f is invertible and
IE940239M
@
Abstract published in Advance ACS Abstracts, August 15,
1995.
