380
Ind. Eng. Chem. Process Des. Dev. 1983, 22, 380-385
It may be emphasized that the above inferences rest on the validity of the correlations used in the presence of vapor maldistribution. Care was excercised to check if the flow conditions are within the ranges of the correlations recommended in the Bubblenay Manual. When the flow conditions are too close to the dumping point, over a small fraction of ZLthe conditions were beyond the ranges of the correlations used. However, it is expected that this will not influence the results significantly. The likely effect of entrainment under these conditions has been explored in a companion paper. Conclusions A method of estimation of tray efficiency in the presence of vapor maldistribution is presented. It is shown that vapor maldistribution has a detrimental effect even in the case of perfectly mixed and plug models. However, tray efficiency is found to increase with the liquid flow-path length well up to the point of dumping as a result of increase in point efficiencies. A rational design of tray layout of large diameter columns should also consider the effect of vapor maldistribution on tray efficiency. Nomenclature A , B, C = constants in eq 2 b = constant by the equation y* = mx + b E- = Murphree tray efficiency E ~ ( w=)point efficiency at w Em = average point efficiency F(w) = G(w)p,’/Z G = average vapor flux, mol m-2 h-’ G(w) = vapor flux at w ,mol m-2 h-’ G(0) = vapor flux at the inlet weir, mol mb2h-’ h, = weir height, cm L = liquid flow rate, mol h-’ L(w) = liquid flow rate per unit width, mol m-l h-l m = slope of the equilibrium curve
Effect of Vapor
Ma
NL = number of liquid transfer units Nc = number of vapor transfer units Pe = Peclet number P, Q, R, S = constants defined by eq 7 R‘ = defined by eq 11 Rvd = vapor-distribution ratio V = vapor flow rate w = dimensionless distance from inlet weir, Z/ZL x = liquid mole fraction X = defined by eq 7 y = vapor mole fraction y(w) = vapor mole fraction at w Z = distance from inlet weir, m ZL = liquid flow-path length A(w) = liquid gradient at w A G = G(l) - G(0) y = mV/L pv = vapor density Subscripts AIChE = evaluated as per AIChE Bubble-Tray Manual n = tray n n - 1 = tray n - 1
Literature Cited Bobs, W. L. In Smith, B. D. “Design of Equilibrium Stage Processes”; McQraw-HIII: New York, 1963; Chapter 14. “Bubble-Tray Design Manual”; American Institute of Chemical Engineers: New York, 1956. Furzer. I. A. A I C M J . 1989, 15, 235. Holland, C. D. “Fundamentals end Modeling of Separation Processes”; Prentice-Hall Inc: Englewood Cliffs, NJ, 1975; pp 356. Holm, R. A. A I C M J . 1981, 7 , 346. Lockett. M. J.; Dhulesla, H. A. Chem. Eng. J . 1980, 19, 183. Lockett, M. J.; Satekwrdi, A. Chem. Eng. J . 1978, 1 1 , 111. Rao, K. K. M. Tech. b s l s I.I.T., Kanpur, India, 1976. Van Winkle. M. ”Distillation”, m a w - H i l i Book Co: New York, 1967; p 553.
Received for review May 27, 1981 Revised manuscript received October 21, 1982 Accepted December 16, 1982
rtbution and Entrainment on Tray Efficiency
Tadlnad Mohan, Krovvldl K. Rao,t and D. Prahlada Rao’ D e p a m n t of Chemical Eng/nwIng, Indian Institute of Technology Kanpur, Kanpw 2080 16, India
The well-known Colburn model used for correcting tray efficiency for entrainment gives an underestimate in the case of dispersed and plug flow models of the liquid phase. For these models, equations for tray efficiency have been obtained considering the variation of entrained liquid mole fractlon along the liquid flow path length. A parametric study of the composite effect of entralnment and vapor maldistributlon on tray efficiency has been presented. By chenging the length to width ratio of a rectangular tray of a fixed area, the dependence of tray efficiency on the liquid f k w path length has been studied. The results indicate that tray efflciency increases with the liquid flow path length well up to the point of liquid dumping.
In the design of large diameter trays the composite effect of entrainment and vapor maldistribution has to be considered. In a companion paper, we have reported studies of the effect of vapor maldistribution on tray efficiency. In this paper, the composite effect of entrainment and ?Departmentof Chemical Engineering,University of California, Davis, CA 95616. 0198-4305/83/1122-0380$01.50/0
vapor maldistribution has been studied. In general, the Colburn model is used for correcting tray efficiency for entrainment. In this model it is assumed that the mole fraction of the entrained liquid is the same as that of the liquid leaving the tray. This leads to an underestimation of the tray efficiency. The Lewis relation and the dispersed model equation for tray efficiency presented in the AIChE Bubble-Tray Manual (1958)have been extended to include the effect of entrainment. The effect of liquid flow path 0 1983 American Chemical Society
Ind. Eng. Chem. Process Des. Dev., Vol. 22, No. 3, 1983 381
length on tray efficiency has been reexamined. Modified Tray Efficiency The vapor, streaming out of the liquid pool of a tray, carries away the droplets of the liquid with which it was in contact. As a result, the mole fraction of the liquid droplets just above the liquid (froth) pool varies along the liquid flow path. However, it is visualized that the vapor as well as the entrained liquid droplets get perfectly mixed before they enter the tray above. Hence, the average mole fraction of the entrained liquid, x’,, is the average mole fraction of liquid on the tray n (or weighted average if entrainment varies along the length). Further, it is assumed that the liquid droplets brought in by the vapor are well-mixed with the liquid on the tray and their influx is equal to the liquid entrained at any given location. The mole fraction of the mixed stream of the vapor and liquid droplets, Y, just above the froth at any given location is y + ex y =l+e where x and y are the mole fractions of the liquid and vapor, respectively, and e is the entrainment. Then, the modified point efficiency, E bo, is defined as
or
y + ex - ex’,,-1 EbG = y* + ex - yn-l - ex’,,-1
y n + ex ,’ - yn-l - ex ’n-l -
-
Ebv =
1”’ +
ex - yn-l - ex’,,-1)dw y*, ex, -
The above equation can be rewritten as
(2b)
(34
or E L v = y*, + ex,
where A’ = mV/L’ and w is the dimensionless distance from the inlet weir. In arriving at eq 6 it is assumed that EbG is constant. It is shown later that EbG is, in fact, constant if EOG and e are constant. From the definition of E’MVit follows that
Substituting eq 6 in eq 7 we get
where y* is the mole fraction of the vapor which is in equilibrium with the liquid at that point and yn-l is the mole fraction of vapor leaving the tray n - 1. Analogous to the Murphree tray efficiency we can define a “modified tray efficiency” as
Yn - Yn-1 E’Mv = Y*, - Y,-l
where L’ = L + e V, and L and V are the liquid and vapor flow rates, respectively. Making use of the equilibrium relation 0, = mx* + b) and the definition EbG, eq 5 can be manipulated to yield e (1 + ; ) x - x*,-1- -x’,,-1 m
(3b)
where y*, is the mole fraction of the vapor which is in equilibrium with the liquid leaving the tray n. In contrast, the Colburn efficiency, E,, is defined as ~n - e(xn+l- x n ) - Yn-1 + e(x, - Xn-J E, = Y*n - Yn-1 + e(x, - xn-1) Tray Efficiency with Uniform Vapor Distribution In this section, the relations between E‘* and Elm are obtained for the dispersed and plug flow models of the liquid phase. In the case of perfectly mixed flow model, we can easily show that E’Mv/E’oc = 1 (4) and when there is no entrainment eq 4 reduces to Em/Em = 1. Though eq 4 apparently does not resemble the corrected tray efficiency, E,, obtained by Colburn (1936), it is shown later that both models predict the same enrichment of vapor. Plug Flow Model. Consider a differential slice of liquid (froth) covering its entire height and width, and of thickness dw in the lateral direction. The material balance equation for the slice can be written as (y + ex - ex’,,-1) dV = L’dx (5)
If we set e = 0, eq 8 reduces to the well-known Lewis relation. Dispersed Flow Model. For the dispersed flow model, the material balance equation for a differential slice can be shown to be 1 d2x - dx - ’V0 , + ex - Y,,J = 0 (9) Pe’dw2 dw L where Pe’is the Peclet number based on L‘. Making use of the definition of E’m and the equilibrium relation, we can rewrite eq 9 as -1- -d2x - - dx Pe’ dw2 dw
Using the boundary conditions dx/dw = 0 and x = x , at w = 1,the solution of eq 10 can be obtained as e (1 + ; ) x - x*,1-x’,,-1 m (1
+ ;)x,
e m
- x*,-1 - -x’,-1
Substituting eq 11 in eq 7 and on integration, we get
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Ind. Eng. Chem. Process Des. Dev., Vol. 22, No. 3, 1983
EhV -Etm
-
+ Pe?] + (7’ + Pe?[l + (9’+ Pe)/q’] 1 - exp[-($
On setting e = 0, eq 12 reduces to the corresponding relation presented in the AIChE Bubble-Tray Manual (1958). Tray Efficiency with Vapor Maldistribution In thisaection, Etm evaluation in the presence of vapor maldistribution and entrainment is presented. In the earlier paper it has been pointed out that the vapor distribution can be represented by G(w) = G(0) + AGw (13)
As a result of variation of vapor velocity and entrainment along the liquid flow path E’m varies, and it is found that E bc can be represented as E’w = A‘+ B% + C‘w2 (14) where A’, B’, and C’ate constants. For the perfectly mixed flow of the liquid, we can now show that
ELV = A’(G(0)
+ AG/2) + B‘(
+ AG/3) +
For the dispersed flow of the liquid, the differential material balance can be shown to be 1 d2X dX Pe’ dW2 dw m -(A’+ B‘w + CIw2)(G(0)+ Aw)
L‘
where (1
x=,
+ ;)x
(1 + ; ) x n
e
- %*,-I - ,x,,-,
- x*,-,
e
- -x;-1 m
Equations 16 and 7 are solved numerically to obtain X vs. w. Then, Elm was obtained by the numerical integration of the equation
E’MV= L$lXE&(~)(G(0) vo
+ AGw) dw
(17)
Setting Pe’ = m, Etm is similarly evaluated for the plug flow model. Relation between E’OG and EoG The relations Elm and Elm will be useful only if we can relate E’m to Em and e. By adding and substracting ex*,1 to both the numerator and the denominator of eq 2 we get - x*,,-~) (y - yn-l) + e(x -x*,,-~) E’% = (y* - yn-J + e(x - x*,J - e(x’n-l - x*,,-~) (18) Now, eq 18 can be rearranged as
A close look at the term e(x Ll - x*”-J/ Cy* - ynJ indicates that it is very much smaller than Eo@ Since the term appears in the numerator as well as in the denominator, it can be neglected. Therefore eq 19 reduces to e EOG + E’m = e 1+m Use of Ebfv in the Stage to Stage Calculations The use of E‘m appears to be more straightforward than the use of E, in stage to stage calculations. For binary systems, in place of the x-y diagram we can draw Y [= Cy* + ex)/(l + e)] vs. 3c. The operating linea with slopes L’/[V + eV] can be constructed. On such a diagram, EhVcan be used to find the number of stages in a manner similar to that of Em. The proposed method can be extended in a straightforward manner to multicomponent mixtures if the equilibrium relationships yi = mi+ bj (i refers to the component) can be assumed to hold for all the components. The values Emj and Em,ican be estimated as described earlier. The E’mi can be incorporated into a tridiagonal matrix formulation of material balance and equilibrium relations of multistage-wise process calculations, in a manner shown by King (1980), replacing L, V, K,and Em with L’,V(l + e), (K+ e)/(l + e) and respectively. The column vector in the matrix equation represents component i flow rates of the mixed vapor stream from tray to tray in the tower. It may be pointed out that the values mi and hence Xi may differ by several orders of magsitude from component to component in a mixture. As a result EOGs and even more significantly will differ from component to component. Hence, the modified tray efficiency may have to be used in a realistic analysis of multicomponent distillation. Results and Discussion Uniform Vapor Distribution. Upon varying the entrainment, e, as a parameter, E, and Elm are estimated for the Peclet number in the range 0 to m. The relative effect of entrainment on the enrichment of vapor, estimated by the proposed model and the Colburn model, could not be compared directly because Etm is based on Y* while E, is based on y*. Hence, for the sake of comparison, a new term, Em,h, which denotes the tray efficiency, corrected for entrainment, in terms of dry-vapor mole fractions, and another tray efficiency, E,,, for the Colburn model, similarly corrected for entrainment, are introduced. The outline of the methods of estimation of these dry-vapor efficiencies is given in the appendix. The details are given elsewhere (Mohan, 1981). The effect of entrainment on dry-vapor efficiencies is shown in Figure 1. It can be seen that the Colburn model overcorrects efficiency in the case of plug flow and dispersed flow models for it is assumed that the mole fraction of the liquid entrained into tray n is taken to be x,,.-~instead of x’,,-~ (note that x,, < x’,,-J. However, the two models yielded the same values of dry-vapor efficiencies for the perfectly mixed flow model because the liquid entrained into the tray n is taken to be xn-l in both the models. The variation of (Ew*/Em) vs. X’Em with the Peclet number as a parameter is shown in Figure 2. It can be seen that the effect of entrainment increases with X’EOG. There is a considerable deviation between the proposed model and the Colburn model at the higher values of X‘Em and Peclet numbers. The striking similarity of eq 9 with the equation for dispersed flow model without entrainment suggests that the curves Em/EOG vs. xEm in the AIChE Bubble-Tray
Ind. Eng. Chem. Process Des. Dev., Vol. 22, No. 3, 1983 383 35 AEOG ~ 1 . 9 Colburn model
__--
- Proposod mcdol 3c
2.
C
+
.
.o 0 >
.I
2c
06 0
005
01
015
02 e
025
03
035
Figure 1. Effect of entrainment on dry-vapor efficiency aa predicted by the Colburn and proposed model.
Figure 3. Tray efficiency curves accounting liquid mixing and entrainment.
10
20 I h‘ E ~ ( G 1 e / m )lav
30
+
7i
I
0
lo
1
I
20
30
AEOG
Figure 4. Tray efficiency in presence of vapor maldistribution and entrainment.
Figure 2. Variation of dry-vapor efficiency with X‘Ed.
Table I. Operating Conditions and the Tray Details
Manual (1958) can be generalized to include the effect of entrainment, and such a plot is shown in Figure 3. Setting e = 0, one can obtain Em from this figure. Vapor Maldistribution. The methods for the estimation of tray hydraulics and vapor maldistribution are described in the previous paper by the authors. The entrainment is expected to vary with w in the presence of vapor maldistribution. It is asaumed that the entrainment, e, depends on the prevailing local conditions of both the phases. The e vs. w data are estimated from the Fair and Mathews correlation using the local flow conditions. The tray efficiency, E h , are evaluated from eq 16 and 17. Keeping G(O)/Gand Pe as the parameters, the values of e and E’m/(EbG)avare evaluated for a circular tray. The operating conditions and the pertinent tray details are given in Table I. The results are shown in Figure 4. It should be noted that curve 1 (AIChE) in Figure 4 corresponds to the tray efficiency estimated by using the method outlined in the AIChE Bubble-Tray Manual
vapor flow rate liquid flow rate percent flooding m transport properties active area bubble cap weir height weir length pitch cap spacing
5.56 x lo4 m3 h-l 109 m3 h-l 90% 1.5 benzene-toluene system at 1atm 9 mz 4 in standard carbon steel bubble-cap (Bolles) 7.5 cm 0.7 0 times tower diameter triangular 25% of cap diameter
(1958). It can be seen that the maximum reduction in tray efficiency still occurred at about a Peclet number of 5. Effect of 2 on Elm. To examine the variation of tray efficiency with the liquid flow path length in the presence of vapor maldistribution and entrainment, varying the length/width ratio of a rectangular tray E‘m is evaluated. The operating conditions and the tray details are the same as those given in Table I. For a typical case the results
384
Ind. Eng. Chem. Process Des. Dev., Vol. 22, No. 3, 1983
Table 11. Variation of Tray Efficiency with Liquid Flow Path Length
m 1.6 2.5 3.3 4.1 4.9 6.2
ZL,
AIChE.
Pe 56 75 86 92 94 98
Rvd
0.31 0.50 0.69 0.93 1.30 1.90
Entrainment only.
G(O)/c 0.91 0.83 0.74 0.64 0.48 0.10
Maldistribution only.
are presented in Table 11. Even in this case, the tray efficiencies can be seen to increase with ZLand &d until dumping of the liquid is encountered. The reasons for this kind of behavior have been discussed in the previous paper. It can be seen that the detrimental effect of entrainment is smaller than that of vapor maldistribution. As ZLincreases, the vapor flow between the trays will tend toward the plug flow conditions. As a result, for the large diameter columns, in which the liquid flows in opposite directions on successive trays, the Murphree tray efficiency itself would be lower (Lewis, 1936) and the detrimental effect of entrainment on E'W could be more than in the case considered here. Hence, it is possible that Elm may attain a maximum before dumping is encountered. At the end, it may be reemphasised that the results obtained in this study heavily rest on the applicability of various empirical correlations used.
Conclusions It has been shown that the Colburn model overcorrects the tray efficiency for entrainment in the case of dispersed and plug flow models. For these models, tray efficiency equations were obtained considering the variation of the mole fraction of the entrained liquid along the liquid flow path. It is found that vapor maldistribution can lead to a greater reduction in tray efficiency than entrainment in large diameter trays. It is also found that higher liquid flow path length leads to higher tray efficiencies in spite of the detrimental effect of vapor maldistribution and entrainment. Appendix Estimation of Dry-Vapor Efficiencies. The Proposed Model. The evaluation of dry-vapor efficiencies requires dry-vapor mole fractions. These are evaluated as shown below. The material balance around tray n (the trays are numbered from the bottom end) and the condenser is VU,-, = L'x, + DxD (All where D is the flow rate of distillate and XD is its mole fraction. The modified tray efficiency for tray n is
In case of the mixed flow model for the liquid phase, Y,-l and x , can be evaluated directly from eq A1 and A2 for a known set of values of Y,, V , L', e, D,and xD, since x , = x',. But, in case of the other two models, Y,-l and x , can be evaluated provided x ',,is known. For given values of Y,-l and x,, the profile x vs. w can be evaluated from eq 6 or 11 depending on the liquid phase flow model, and x',, can be found from the relation
.',,=I v 'G(w)x dw
Using an iterative procedure, Yn-l,x,, and x',, can be evaluated. The dry-vapor mole fraction, y, (= Y,, - ex,)
EMVa
1.34 1.73 2.06 2.34 2.57 2.90
EMV.drv
1.32 1.70 2.03 2.30 2.51
b
EMVC
1.22 1.54 1.83 2.07 2.25
EMV.drv
d
1.20 1.52 1.80 2.02 2.17 2.35
Entrainment and maldistribution. can be found. Since y1and XD are known, the calculations
can be performed downward starting from the top tray to obtain the dry-vapor mole fractions of the trays, and dry-vapor efficiency can be found from the relation
The Colburn Model. The material balance around the tray n and the condenser can be written as VY',,-1 = Lx,+ DxD (A3) where Y'n-l = yn-l - e(xn- x,J. The Colburn efficiency for the tray n is
The values YL-l and xn-', and hence y,, can be found from eq A3 and A4 for a given set of values of Y',,, L, V , e, D, and x D The dry-vapor efficiencies can be found by adapting the procedure outlined above. Nomenclature A', B', C' = constants in eq 14 b = constant in equilibrium relation y* = mx + b D = distillate flow rate e = entrainment, mol of liquid/mol of vapor E, = tray efficiency corrected for entrainment, defined by Colburn E+,. = dry-vapor efficiency based on the Colburn model E- = Murphree tray efficiency E'- = modified tray efficiency E-,* = dry-vapor efficiency based on the proposed model Eoc = point efficiency E'OG = modified point efficiency 0 = average vapor velocity, mol h-' m-2 C(w) = vapor velocity at w ,mol h-' m-2 G(0) = vapor velocity at the inlet weir, mol h-' m-2 L = liquid flow rate in absence of entrainment L' = liquid flow rate in presence of entrainment, mol h-' m = slope of equilibrium line Pe' = modified Peclet number (based on L') Rvd = vapor distribution ratio V = vapor flow rate, mol h-l V ' = V + eV w = dimensionless distance from the inlet weir x = mole fraction of liquid x* = 0, - b ) / m x ' = average mole fraction of entrained liquid X = defined by eq 16 y = mole fraction of vapor Y = mole fraction of the mixed vapor stream defined by eq 1
Y ; = Y n - e(%,+,- xn) y* = mx + b Y* = Cy* + ex)/(l + e) 2, = liquid flow path length, m A' = mV/L' 9' = defined by eq 11 AG = G(1) - G(0) Subscripts n = tray n
Ind. Eng. Chem. Process Des. Dev. 1983, 22, 385-391
n - 1 = tray n - 1 i = component i D = distillate Literature Cited
385
King, C. J. “Separation Process”, 2nd ed.; McGraw-Hill, Inc; New York, 1981; p 472 Lewis, W. K. Ind. Eng. Chem. 1936, 28. 399. Mohan, T. M.Tech. Thesis, I.I.T., Kanpur, India, 1981.
“BubbleTray Design Manual”; American Institute of Chemical Engineers, New Ycfk, 1958. Colbwn, A. P. Ind. Eng. Chem. 1936, 28, 526.
Received for review May 27, 1981 Revised manuscript received October 21, 1982 Accepted December 16,1982
A Versatile Phase Equilibrium Equation of State Paul 1111. Mathlas’ Energy Laboretfny and Chemical Engineering Department, Massachusetts Instftute of T e c h n o w , C a m b r m , Massachusetts 02139
The Soave modification of the Redlich-Kwong equation has been very effective for conelating the phase equilibrium of systems containing nonpolar and slightly polar substances. In this work further modiflcations are introduced which retain the simplicity and robustness of the Soave equation but extend its application to systems containing highly polar substances such as water. Appllcations shown in the work include systems of interest in coal processing and some common systems containing water.
Introduction The increased use of computers for chemical process design has stirred great interest in the analytic representation of the phase behavior of multicomponent systems. This analytic description must, of course, be quantitatively accurate, but it should also possess the qualities of few and easily attainable correlation parameters, never predicting physically absurd results and being computationally robust and efficient. Very broadly, practical methods to represent phase behavior can be divided into activity coefficient and equation of state approaches. In the activity coefficient method an activity coefficient model, say the UNIQUAC equation, is used to represent the nonideality of the liquid phase($ while a different model, usually an equation of state, is used to describe the departure from the ideal gas reference state of the vapor phase. The latter approach uses the same equation of state to represent all coexisting phases. The equation of state approach has been effective in describing systems containing only nonpolar and slightly polar components, but the common consensus appears to be that the additional flexibility of activity coefficient models is necessary to correlate highly nonideal systems which contain polar substances. The activity coefficient approach has been found to correlate a wide variety of complicated phase behavior, but this has been at the expense of a large number of parameters which extrapolate poorly with temperature. Another-and more serious-disadvantage is that anomalous results are obtained in the critical region since different models are employed for vapor and liquid phases. On the other hand, the equation of state approach predicta critical regions quite naturally, and even relatively simple equations like those of Soave (1972) and Peng and Robinson (1976) are capable of satisfactory predictions of complicated critical region phase behavior (e.g., see Heidemann and Khalil, 1979). *Air Products and Chemicals, Inc., Box 538, Allentown, PA
18105. 0198-430518311122-0385$01.50/0
Thus it would be very useful to have an equation of state applicable to systems containing polar substances. One attempt to achieve this goal was that by Gmehling et al. (1979), who assumed a chemical equilibrium hypothesis. In their work, Gmehling et al. assumed that polar species form dimers, and by fitting data to determine a standard-state enthalpy and entropy of dimerization they correlated vapor-liquid equilibria for many polar mixtures. However, this approach has several disadvantages. Whiting and Prausnitz (1981) point out three of them. First, experimental evidence supports the existence of dimers only in rare cases. Second, application of chemical theory greatly increases the computational load since it requires the solution of chemical-as well as phaseequilibrium. Third, the number of pure-component and binary parameters increases because standard-state enthalpies and entropies of dimerization are required. An important disadvantage not cited by Whiting and Prausnitz (1981) concerns applications at severe conditions, say in the retrograde region or even approaching the critical point. Under these conditions even the use of relatively simple models like Soave (1972) have presented severe computational challenges requiring sophisticated solutions (Michelson, 1980; Asselineau et al., 1979; Mathias et al., 1981; and Chan and Boston, 1981). Models employing chemical theory may well be pathological to computational robustness. A remarkable success in the design of an equation of state for the correlation of fluid-phase equilibrium has been the simple idea of Soave (1972). He recognized that a prerequisite for the correlation of phase equilibria of mixtures is the correlation of the vapor pressures of pure substances. The Soave modification of the Redlich-Kwong (1949) equation has been very successful in correlating the phase behavior of multicomponent systems containing nonpolar and slightly polar substances. In this work, the Soave approach is used to extend the Redlich-KwongSoave equation to systems containing highly polar substances such as water and the alcohols. The work also utilizes the modification of the Soave equation suggested by Boston and Mathias (1980) to im@ 1983 American Chemical Society
