Computerized Design of Multicomponent Distillation Columns Using the UNIFAC Group Contribution Method for Calculation of Activity Coefficients Aage Fredenslund," Jurgen Gmehling,2Michael L. Mlchelsen,' Peter Rasmussen,' and John M. Prau~nitz"~ lnstituttet for Kemiteknik, The Technical University of Denmark, DK-2800 Lyngby, Denmark, Abteilung Chemietechnik, Lehrstuhi Technische Chemie B, University of Dortmund, 46 Dortmund 50. West Germany, and Department of Chemical Engineering, University of California, Berkeley, California 94720
The UNIFAC group-contribution method for estimating activity coefficients provides the process engineer with a rapid and reliable method for predicting equilibrium conditions required in distillation-column design.
Introduction A large part of chemical engineering design is concerned with separation operations. Many of these are diffusional operations of the phase-contacting type, and distillation, absorption, and extraction are the most common. For rational design of such separation processes, we require quantitative information on phase equilibria in multicomponent mixtures. Satisfactory experimental equilibrium data are only seldom available for the particular conditions of temperature, pressure, and composition required in a particular design problem. It is therefore necessary to interpolate or extrapolate existing mixture data or, when suitable data are lacking, to estimate the desired equilibria from some appropriate correlation. A very useful correlation for this purpose, UNIFAC, was recently proposed. It is the aim of this work (1)to review the essential ideas of IJNIFAC, to report extensive modifications and extensions of the UNIFAC parameters required for its use, and to present illustrative examples that indicate the extent and reliability of UNIFAC for making quantitative estimates of phase equilibria; and (2) to show how UNIFAC, coupled with material- and energy balances, may be used to design multicomponent distillation columns. The UNIFAC Method UNIFAC is an abbreviation that indicates Universal quasichemical Functional Group Activity Coefficients. It is based on the quasichemical theory of liquid solutions proposed by Guggenheim (1952), generalized by Abrams and Prausnitz (1975), and applied to functional groups within the molecules (rather than the molecules themselves) by Fredenslund et al. (1975). In process design, the required phase equilibrium information is commonly expressed by K factors
where y , is the mole fraction of component i in the vapor phase and x , is the mole fraction of component i in the liquid phase. Using standard thermodynamics Tlfl O K, = -
PlP
(2)
where y, is the liquid-phase activity coefficient, f L O is the standard-state fugacity, pL is the vapor-phase fugacity coefficient, and P is the total pressure. For condensable compo-
' Technical University of Denmark. University of Dortmund. University of California. 450
Ind. Eng. Chem., Process Des. Dev., Vol. 16, No. 4, 1977
nents as considered here, fin is the fugacity of pure liquid i a t system temperature T and pressure P ; as shown elsewhere (Prausnitz, 1969) it is calculated from (3) where, for pure liquid i, Pis is the saturation (vapor) pressure, pisis the fugacity coefficient a t saturation, and u, is the molar liquid volume, all a t temperature T . Only pure-component data are required to evaluate f i n . The fugacity coefficients cpi (in the mixture) and pIS(pure i a t saturation) are found from vapor-phase volumetric properties. Normally, a t the low pressures considered here, these fugacity coefficients do not deviate much from unity. While various methods are available to calculate fugacity coefficients, we have here used the method of Hayden and O'Connell(l975) as briefly summarized in Appendix C. T o determine K factors, the most difficult-to-estimate quantity is the activity coefficient y. UNIFAC provides a method for estimating activity coefficients in nonelectrolyte liquid mixtures remote from critical conditions. T o use this method, no experimental data are required for the particular mixture of interest. In addition to the temperature and composition of the system, it is necessary only to know the molecular structure of every component in the mixture and the necessary group parameters. A large number of group-interaction parameters is given here. The large advantage of a group-contribution method is that it enables systematic interpolation and extrapolation of vapor-liquid equilibrium data simultaneously for many chemically related mixtures. Most important, it provides a reasonable method for predicting properties of mixtures where no mixture data at all are available. For such mixtures it is not necessary to measure the intermolecular interactions because these can be calculated whenever appropriate group-interaction parameters are known. These, however, are found from experimental data containing not the same molecules as those in the mixture of interest, but containing the same groups. The main advantage, then, is a form of "molecular scale-up." While there are many thousands of liquid nonelectrolyte mixtures of interest in chemical technology, these mixtures can be constituted from a much smaller (perhaps 50 or 100) number of functional groups. Langmuir (1925) suggested that physical (van der Waals) interactions between polyfunctional molecules may be estimated by summing interactions between the functional groups that constitute the molecules. Typical functional groups might
Table I. Group Volume and Surface Area Parameters Group name0
Rk
0.9011 0.6744 0.4469 0.2195 1.3454
Sample group assignment
Qk
0.848 0.540 0.228
Butane: 2 CH,, 2 CH, i-Butane: 3 CH,, 1 CH 0.000 2,2-Dimethylpropane 4 CH,, 1 C CH,=CH 1.176 1-Hexene: 1 CH,, 3 CH, 1 CH,=CH CH=CH 1.1167 2-Hexene: 2 CH,, 2 CH,, 0.867 “C=C” 1 CH=CH CH=C 0.8886 0.676 2-Methylbutene-2: 3 CH,, 1 CH=C 1.1173 2-Methylbutene-1: 2 CH,, 1 CH,, 1 CH,=C 0.988 0.5313 Benzene: 6 ACH 0.400 “ ACH” 0.3652 0.120 Styrene: 1 CH,=CH, 5 ACH, 1 AC ACCH, 1.2663 0.968 Toluene: 5 ACH, 1 ACCH, “ACCH,” ACCH, 1.0396 0.660 Ethylbenzene: 1 CH,, 5 ACH, 1 ACCH, 0.8121 0.348 Cumene: 2 CH,, 5 ACH, 1 ACCH 1.8788 1-Propanol: 1 CH,, 1 CH,CH,OH 1.664 1.8780 2-Butanol: 1 CH,, 1 CH,, 1 CHOHCH, 1.660 “CCOH” CHOHCH, 1.6513 1.352 3-Octanol: 2 CH,, 4 CH,, 1 CHOHCH, CH,CH,OH 2.1055 1.972 Ethanol: 1 CH,CH,OH CHCH,OH 1.6513 1.352 Isobutanol: 2 CH,, 1 CHCH,OH CH,OH 1.4311 1.432 Methanol: 1 CH,OH 0.92 1.40 Water: 1 H,O H,O ACOH 0.8952 0.680 Phenol: 5 ACH, 1 ACOH CH,CO 1.6724 1.488 Ketone group is 2nd carbon; 2-Butanone: 1 CH,, 1 CH,, 1 CH,CO “CH ,C0” CH,CO 1.4457 1.180 Ketone group is any other carbon; 3-Pentanone: 2 CH,, 1 CH,, 1 CH,CO CHO 0.9980 0.948 Acetaldehyde: 1 CH,, 1 CHO CH,COO 1.9031 1.728 Butyl acetate: 1 CH,, 3 CH,, 1 CH,COO “COOC” CH,COO 1.6764 1.420 Butyl propanoate: 2 CH,, 3 CH,, 1 CH,COO 1.1450 1.088 Dimethyl ether: 1 CH,, 1 CH,O 0.9183 0.780 Diethyl ether: 2 CH,, 1 CH,, 1 CH,O “CH,O” 0.6908 0.468 Diisopropyl ether: 4 CH,, 1 CH, 1 CH-0 0.9183 Tetrahydrofuran: 3 CH,, 1 FCH,O (1.1) CH,NH, 1.5959 1.544 Methylamine: 1 CH,NH, “CNH,” 1.3692 1.236 n-Propylamine: 1 CH,, 1 CH,, 1 CH,NH, 1.1417 0.924 Isopropylamine: 2 CH,, 1 CHNH, CH,NH 1.4337 1.244 Dimethylamine: 1 CH,, 1 CH,NH “CNH” CH,NH 1.2070 0.936 Diethylamine: 2 CH,, 1 CH,, 1 CH,NH CHNH 0.9795 0.624 Diisopropylamine: 4 CH,, 1 CH, 1 CHNH ’ ACNH, 1.0600 0.816 Aniline: 5 ACH, 1 ACNH, CH,CN 1.8701 1.724 Acetonitrile: 1 CH,CN “CCN” CH,CN 1.6434 1.416 Propionitrile: 1 CH,, 1 CH,CN COOH 1.3013 1.224 Acetic acid: 1 CH,, 1 COOH “COOH” HCOOH 1.5280 1.532 Formic acid: 1 HCOOH 1.4654 1.264 Butylchloride: 1 CH,, 2 CH,, 1 CH,Cl “CC1” 1.2380 0.952 Isopropyl chloride: 2 CH,, 1 CHCl 0.7910 0.7 24 tert-Butyl chloride: 3 CH,, 1 CC1 2.2564 Dichloromethane: 1 CH,Cl, 1.988 “ CCl, ” CHC1, 2.0606 1.684 1,l-Dichloroethane: 1 CH,, 1 CHC1, 1.8016 1.448 2,2-Dichloropropane: 2 CH,, 1 CCl, CHC1, 2.8700 2.410 Chloroform: 1 CHCl, CCl ,” 2.6401 2.184 l,l,l-Trichloroethane: 1 CH,, 1 CC1, cc1, 3.3900 2.910 Carbon tetrachloride: 1 CC1, ACCl 1.1562 0.844 Chlorobenzene: 5 ACH, 1 ACCl CH,NO, 2.0086 1.868 Nitromethane: 1 CH,NO, “CNO, ” CH,NO, 1.7818 1.560 1-Nitropropane: 1 CH,, 1 CH,, 1 CH,NO, CHNO, 1.5544 1.248 2-Nitropropane: 2 CH,, 1 CHNO, ACNO, 1.4199 1.104 Nitrobenzene: 5 ACH, 1 ACNO, 2.057 (1.65) Carbon disulfide: 1 CS, CS 2 a Note: Subgroups (for example CH,=CH and CH=CH) within the same main group (here C=C) have different values for R k and Q k but identical group interaction parameters, a w n . b Not t o be used with oxygen-containing hydrocarbons or amines . “CH,”
{ {
3
1‘
be CH3, OH, C1, N02, “ 2 , COOH, etc. Langmuir stated that if it is possible t o characterize quantitatively the physical interactions between such groups, it should then be possible t o estimate intermolecular interactions and finally, with a suit-
able model, t o estimate thermodynamic properties, in particular, activity coefficients. UNIFAC provides such a model. Langmuir’s suggestion received little attention until Wilson Ind. Eng. Chem., Process Des. Dev., Vol. 16, No. 4, 1977
451
and Deal (1962), Derr and Deal (1969), and Ronc and Ratcliff (1975) proposed the ASOG model, which is similar to UNIFAC in principle but not in detail. From the user’s point of view, UNIFAC provides three advantages: (1)flexibility, because UNIFAC, provides a priori estimates of sizes and surface areas; (2) simplicity, because UNIFAC parameters, unlike ASOG parameters, are essentially independent of temperature for the temperature range considered here, typically 30-125 OC; and finally (3) due to results reported in this work, UNIFAC parameters are now available for a considerably larger number of functional groups than ASOG parameters; therefore, the range of applicability for UNIFAC is larger. All group-contribution methods are necessarily approximations because any group within a molecule is not completely independent of the other groups within that molecule. But it is precisely this independence which is the essential basis of every group-contribution method. We can allow for interdependence of groups within a molecule by our definition of what atoms constitute a group. Increasing distinction of groups, however, also increases the number of group interactions that must be characterized. Ultimately, if we carry group distinction to the limit, we recover the individual molecules. In that event, the advantage of the group-contribution method is lost. Judgment and experience must tell us how to define functional groups so as to achieve a compromise between accuracy of prediction and engineering utility.
Advances in UNIFAC Prediction of Activity Coefficients The UNIFAC group-contribution method is summarized in Appendix A. Fredenslund et al. (1975) showed that the UNIFAC method may be used to predict fluid-phase equilibria in systems containing hydrocarbons (saturated, unsaturated, and aromatic), alcohols, water, ketones, esters, ethers, chlorinated hydrocarbons, amines, and nitriles. The original work considered 25 different functional groups. Since then, work on the UNIFAC method has continued in an effort to extend reliability and applicability for separation-process design. We have now determined UNIFAC group-interaction parameters for more than 50 different groups. The basis of the UNIFAC method now covers 70% of all published vapor-liquid equilibrium data for nonelectrolyte mixtures a t low to moderate pressures. The revised table of group-interaction parameters is based on vapor-liquid equilibrium data for roughly 2500 binary systems. The data were chosen from a data set of more than 4000 binary systems stored on magnetic tape and collected a t the University of Dortmund (Gmehling and Onken, 1977). These data are, where appropriate, supplemented with data from Wichterle et al. (1973). The availability of this large data base has enabled us to obtain more reliable values for group-interaction parameters than those published originally. I t has also made it possible better to ascertain which choice of functional groups results in the best correlation. The determination of group-interaction parameters from the data base proceeds as follows. (1) For thermodynamically consistent data (Christiansen and Fredenslund, 1975), activity coefficients are calculated from isothermal P-x, or isobaric T-x,-y, data. Vapor-phase nonidealities are accounted for as shown by Hayden and O’Connell (1975). (2) A pair of group-interaction parameters, e.g., U C O O H , C H ~ and U C H ~ , C O O Hare , obtained from a collection of all available, consistent, relevant experimental data, in this case for systems with alkanes (including isomers) and organic acids. The simplex method (Nelder and Mead, 1965) is used to fit the 452
Ind. Eng. Chem., Process Des. Dev., Vol. 16, No. 4, 1977
A
1 ATM.
0
.2
l
.4
.6
Mole Fraction W a t e r , x
.%
1.0
H 20
Figure 1. Experimental and calculated activity coefficients for organic acid-water systems exhibit both positive and negative deviations from Raoult’s law: calculated, -; experimental: formic acid (1) 0-water (4) 0 , Ito and Yoshida (1963); acetic acid (2) 0-water (4) m, Sebastiani et al. (1957); propionic acid (3) A-Water (4)A ,Ito and Yoshida (1963).
model to the experimental activity coefficients. The values of U C O O H , C H and ~ UCH~,COOHmust be known prior to obtaining U H ~ O , C O O Hand U C O O H , H ~ Ofrom data on systems containing organic acids and water. In some cases it is advantageous to evaluate several pairs of parameters, e.g., U C H ~ , C O O H , U C O O H , C H ~U , C O O H , H ~ O and , ~H~O,COOH simultaneously , from the sum of the two sets of data. The new UNIFAC parameters are given in Tables I and 11. Organic acids, secondary alcohols, methanol, most of the chlorinated hydrocarbons, carbon disulfide, and compounds with the nitro group were not included previously. The remaining parts of these tables indicate the revised parameters. Note that the ketone, ester, ether, secondary amines, and alcohol groups are redefined by the addition of another methylene group. Prediction of Activity Coefficients. A detailed account of the agreement between experimental and calculated activity coefficients for all of the groups included in UNIFAC will be published elsewhere (Fredenslund et al., 1977). In this section we only give a few examples of activity coefficients predicted by the UNIFAC method. Tables 111-VI show some of the results from the data correlation. We show typical experimental and calculated activity coefficients in the dilute regions for several of the binary systems. Tables 111-VI clearly show UNIFAC’s ability to represent a wide range of mixtures with only few parameters. For example, activity coefficients at infinite dilution for binary systems of alkanes and aliphatic ethers with one, two, or three ether groups or dioxane or tetrahydrofuran (Table 111) are all correlated extremely well using only two parameters. For those systems, the infinite-dilution activity coefficients range from roughly 1.1to 3.5. Similarly, Table IV shows that no more than two parameters are needed to represent the phase equilibria of alkanes with ketones ranging from acetone to nonanone. Tables V and VI show excellent agreement between predicted and observed equilibria in, respectively, alkane-alcohol and alkane-amine systems, Figure 1 shows results for systems containing water and the lower organic acids. Again, only two
h
- 1
I
1
I
I
Ind. Eng. Chem., Process Des. Dev., Vol. 16, No. 4, 1977
453
Table 111. Experimental and Calculated Activity Coefficients for Alkane-Ether Systems Component 1 n-Propyl ether n-Heptane Dimethoxyethane Heptane Bis(2-methoxyethyl) ether n-Decane Dioxane n-Hexane Dioxane n-Octane Dioxane Cyclohexane Tetrahydrofuran Cyclohexane
T ,K 343.2 343.2 343.2 343.2 393.2 393.2 353.2 353.2 353.2 353.2 353.2 353.2 298.2 298.2
Component 2 n-Heptane n-Propyl ether n-Heptane Dimethoxyethane n-Decane Bis(2-methoxyethyl) ether n-Hexane Dioxane n-Octane Dioxane Cyclohexane Dioxane Cyclohexane Tetrahydrofuran
x1
yl(expt1)
yl(ca1cd)
1.09 1.08 2.32
1.19 1.19 2.16 2.45 2.94 3.46 2.13 3.11 2.14 2.93 2.49 2.58 1.75 1.71
0 0 0 0 0 0
2.11
2.62 3.06 2.67 2.94 2.33 3.32 2.87 2.55 1.77 1.70
0.024 0.030 0.021 0.046 0 0 0.1 0.1
Table IV. Experimental and Calculated Activity Coefficients for Alkane-Ketone Systems Component 1
Component 2
T ,K
Acetone n-Hexane 2-Butanone n-Heptane 3-Pentanone n-Heptane 5-Nonanone n-Hexane
n-Hexane Acetone n -Heptane 2-Butanone n -Heptane 3-Pentanone n-Hexane 5-Nonanone
318.2 318.2 366.9 351.9 353.2 353.2 333.2 333.2
x1 0.1 0.1
0.027 0.027 0.056 0.051 0.1 0.1
yl(expt1)
71(calcd)
3.37 3.40 3.53 3.15 2.32 2.24 1.68 1.45
3.82 3.76 3.29 3.53 2.04 1.98 1.69 1.50
Table V. Experimental and Predicted Activity Coefficients for Alkane-Alcohol Systems" Component 1
0
Component 2
2-Butanol n-Hexane 2-Butanol n-Hexane 1-Octanol n-Heptane n-Heptane 2-Octanol n-Heptane 3-Octanol 4-Octanol n-Heptane n-Heptane 1-0ctanol n-Heptane 2-Octanol n-Heptane 3-Octanol n-Heptane 4-Octanol 1-Propanol Decane Decane 1-Propanol Ethanol n-Hexane n -Hexane Ethanol Methanol n-Hexane Methanol n-Hexane NOTE: Of the systems shown here only ethanol-hexane
T ,K
x1
0.094 333.15 0.069 333.15 0.1 293.15 0.1 293.15 0.1 293.15 0.1 293.15 0.1 293.15 0.1 293.15 0.1 293.15 0.1 293.15 0.0503 363.15 0.214 363.15 0.1 298.15 0.1 298.15 0.138 333.75 0.054 333.05 was included in the data base.
yl(expt1)
yl(calcd)
3.34 5.17 2.49 2.30 2.24 2.26 4.17 3.69 3.42 3.41 5.01
3.27 6.41 2.31 2.30 1.93 1.93 3.88 3.87 2.81 2.81 6.11 3.49 5.35 8.01 6.46 13.06
2.82
5.92 7.13 6.08 12.21
Table VI. Experimental and Calculated Activity Coefficients for Alkane-Primary Amine Systems Component 1 Methylamine n-Nonane Ethylamine n-Hexane n-Propylamine n-Hexane n-Butylamine n-Hexane n -Hexylamine n-Hexane
Component 2 n -Nonane Methylamine n -Hexane Ethylamine n-Hexane n-Propylamine n-Hexane n-Butylamine n-Hexane n-Hexylamine
T ,K 293.2 293.2 293.2 293.2 293.2 293.2 333.2 333.2 333.2 333.2
parameters are required to represent the interaction between COOH and H 2 0 groups for the systems shown. Since data for systems including formic acid are scarce, it is not certain whether the formic acid-water prediction shown in Figure 1 is fortuitous. As more data become available, it may be advantageow to include formic acid as a special group in the same manner as that for methanol. 454
Ind. Eng. Chem., Process Des. Dev., Vol. 16, No. 4, 1977
X1
0 0 0 0 0 0
0.128 0.104 0.123 0.094
yl(expt1) 3.55 7.90 2.95 2.64 2.60 2.56 1.66 1.77 1.55 1.49
yl(calcd) 3.72 7.97 2.62 2.80 2.47 2.34 1.68 1.67 1.49 1.42
Although no ternary (or higher) systems were included in the data base, predicted multicomponent equilibria are in good agreement with experiment. Table VI1 shows predictions for the system chloroform-methanol-ethyl acetate. This system is unusual, because it exhibits both large positive and negative deviations from Raoult's law. Table VI11 shows predicted and experimental activity coefficients for the system
Table VII. Experimental and Calculated Activity Coefficients a t 1 Atm for the System Chloroform(1)-Methanol(2)Ethyl Acetate(3) (Nagata, 1962) Y2
Y1 x1
X2
0.075 0.030 0.608 0.540 0.527 a
0.077 0.898 0.234 0.075 0.219
Y3
Exptl
UNIFAC
Exptl
UNIFAC
Exptl
UNIFAC
0.541 1.82 1.09 0.982 1.00
0.612 2.10 1.08 0.972 1.01
2.32 1.01 2.11 3.50 2.15
2.47 1.01 2.17 3.81 2.26
0.985 2.41 0.570 0.611 0.701
0.999 2.24 0.637 0.588 0.739
The average absolute deviation between yl(exptl)and y,(calcd) is 0.011mole fraction.
Table VIII. Calculated Activity Coefficients for the System 1,2-Dichloroethane(l)-l-Propanol(2)-Toluene(3)Acetone(4) a t 700 mmHe (Ma et al.. 1969) Fitted with Binarv Data (Wilson Eauationk Predicted bv the UNIFAC Method Exptl activity coefficients x1
x2
13
Y1
Y2
Y3
Y4
0.613
0.232
0.056
1.17
1.84
1.26
0.79
0.515
0.168
0.097
1.09
2.07
1.32
0.87
0.307
0.335
0.281
1.19
1.60
1.34
0.90
0.307
0.207
0.419
1.09
2.07
1.18
0.99
0.202
0.512
0.251
1.40
1.28
1.56
0.97
0.251
0.267
0.252
1.12
1.67
1.38
0.96
0.064
0.039
0.861
0.98
3.35
1.02
1.43
0.063
0.089
0.072
0.94
1.75
1.78
0.97
Calcd activity coefficients Yl Y2 Y3 1.15 1.83 1.34 1.96 1.15 1.10 2.09 1.32 1.09 1.05 2.16 1.15 1.51 1.33 1.26 1.63 1.26 1.16 1.95 1.18 1.15 1.05 2.14 1.11 1.50 1.22 1.53 1.29 1.51 1.38 1.15 1.58 1.33 1.73 1.26 1.10 3.80 1.01 1.11 3.57 1.01 0.98 0.87 1.54 1.27 1.90 1.52 0.93
Y4
0.71 0.91 0.80 0.95 0.82 1.01 0.84 1.01 0.91 1.10 0.88 1.02 1.26 1.27 0.99 1.01
Wn Ub W U W
u
W
u
W
u
W
u u w u
W
Mean Deviations between Experimental and Calculated Vapor-Phase Mole Fractions Component
Wilson fitted
UNIFAC method predicted
Raoult’s law predicted
1
0.011
0.009
0.030
2
0.010
0.006
0.087
Mean deviation = 5, =
( 2 I): k=l
0.007 3 0.004 4 0.014 0.016 W = Wilson’s equation. U = UNIFAC method. 1,2-dichloroethane-l-propanol-toluene-acetone (Ma et al., 1969). The authors give Wilson parameters based on the data, and activity coefficients calculated from these parameters are also shown. The UNIFAC method and the Wilson equation (with the parameters of Ma et al.) give overall mean deviations in vapor-phase mole fractions of 0.01;by contrast, using Raoult’s law does not result in satisfactory predictions. Systems Containing Branched Hydrocarbons and O t h e r Isomers. Fredenslund e t al. (1975)considered the effect of isomerization to only a limited extent. Since then much experience has been gained in this respect and, with the new UNIFAC parameters, the phase behavior of systems containing isomers may usually be predicted with good results. Table V shows that secondary and primary alcohols are represented very well using the same set of parameters; similarly, Figure 2 shows good agreement for predicted and calculated activity coefficients for the 2,2,3-trimethylbutane-benzene system. This favorable result is particularly gratifying because no branched alkanes were included in the data base for determining the methylene-aromatic hydrocarbon group interaction parameters. Liquid-Liquid Equilibria. The present set of UNIFAC parameters is, where possible, based on vapor-liquid equilibrium data only. For systems such as alkanes with water it
0.036 0.022
h(expt1) - y,.k(calcd)
ilr = no. of data points
was necessary to include liquid-liquid equilibrium data in the data base. Although the parameters are almost exclusively based on vapor-liquid equilibria, the present version of the UNIFAC model can predict liquid phase-splitting as indicated in Figure 3. For design of a liquid-liquid extraction cascade, one needs to know the relative distribution of the solute(s) between the two liquid phases. Since the solute mole fractions are often small, the relative error in the predicted mole fractions-and hence in the solute distribution ratios-may be large. This problem is encountered by all methods for calculating activity coefficients and the UNIFAC model is no exception. Therefore, UNIFAC is usually not suitable for accurate liquid-liquid extraction calculations. However, UNIFAC often predicts phase-splitting with sufficient accuracy for designing heterogeneous-azeotropic-distillationcolumns. Multicomponent Distillation Design Calculations The multicomponent column-design procedure used here is a modification of the successive approximation method described by Naphtali and Sandholm (1971).The distillation column configuration is shown in Figure 4.The general design equations, which include material- and energy balances and Murphree efficiencies coupled with vapor-liquid equilibrium Ind. Eng. Chem., Process Des. Dev., Vol. 16,No. 4, 1977
455
- UNIFAC 0
Distillate r-bondense-
OBSERVED
w Reflux
Feed Fn f n , i
XI
Figure 2. Activity coefficients for the 2,2,3-trimethylbutane(l)benzene(2) system at 1 atm (Harrisonand Berg, 1946). This binary system was not included in the data base.
1
2
1
- UNIFAC Figure 4. Distillation column configuration.
X1
A T = 0.0175 Figure 3. Experimental and calculated vapor phase mole fractions and phase-splitin the system n-heptane(l)-acetonitrile(2) (Werner and Schuberth, 1966). relationships, are given in Appendix B. A simplified set of design equations, based on the assumption of constant molal overflow, is also given. The calculation scheme is shown schematically in Figure 5. Input to the calculations is as follows. (1) Identification of Components to Be Separated. Antoine constants, giving the pure-component vapor pressures, and enthalpies of vaporization must be known for each component. The latter are not needed for the simplified version. (2) Parameters for Determination of Thermodynamic Properties. If all the necessary UNIFAC group-interaction parameters are available, the activity coefficient of every component i, infinitely dilute in component j for all possible i-j pairs in the multicomponent system, is calculated using the UNIFAC method. As explained in Appendix A, these calculations are carried out a t two temperatures T I and T2, typically the saturation temperatures of the least and most volatile components. The activity coefficients a t infinite dilution form the basis for estimating the UNIQUAC parameters for the molecular energy interactions. The linearized temperature dependence, eq 8A, is used rather than the exponential form, eq 7A. The calculated infinite-dilution activity coefficients a t the temperatures T I and T2 enable the determination of Aji (0) and A;i(I).The linear temperature depen456
Ind. Eng. Chern., Process Des. Dev., Vol. 16,No. 4, 1977
dence is introduced to save computer time. This introduces no significant error. Second virial coefficients are used to calculate vapor-phase fugacity coefficients. We use the method of Hayden and O’Connell(l975, see Appendix C) to estimate the virial coefficients for all possible binary interactions a t temperatures T I and Tz and interpolate the virial coefficients linearly in the temperature. (3) Specification of Separation Problem. The user must specify the number of actual stages, stage efficiencies, feedand sidestream locations, feed compositions, flow rates, and thermal states, total distillate flow rate, sidestream phase conditions and flow rates, reflux ratio, and column pressure. Above it was stated that UNIFAC is used to obtain UNIQUAC parameters. Using UNIQUAC parameters rather than UNIFAC parameters directly is based on the following. (1) The UNIQUAC method, which uses only molecular compositions, is faster than the UNIFAC method, which uses both molecular and group compositions. (2) Tests for systems where experimental data are available show that activity coefficients calculated from the UNIQUAC model and estimated limiting activity coefficients are, on the average, as accurate as those calculated directly using the UNIFAC method. (3) If experimental information is available for some of the constituent binaries, UNIQUAC parameters based on these data may be incorporated directly into the calculations. The column calculations can now proceed. The scheme outlined in Figure 5 is that of the simplified version of the program (constant molal overflow). The procedure for the full version is analogous. The total molar flow rates for each stage are calculated based on the separation problem specification. (In the case of the full version, they can only be estimated a t this point). As an initial estimate, a linear temperature profile is assumed, and the phase compositions on each stage are equated to the nearest feed composition. This gives the zeroth estimate of the independent variables l,,,, u,,,,, and T,, and hence the zeroth values of the test functions. The Newton-Raphson
Define mixture to be separated: Antoine constants Ai Bi Ci Group counts vk‘i) Temperature range T1 to T2
---
RAOULT
S
LAW
* Calculate activity coefficients at infinite dilution, y?, at T1 ard Ti using UNIFAC. This also gives ri, q i , and1Ei
to
UlilQLlAC
eqcation. T..
J l
I
A(?’ J1
This gives A ! ? ’ J 1 + A!f).T
and
A!:):
J1
m
0 4 .
~
Specify separation problem: No. of stages, feed stages, r e f l L x ratio, feed compositions and no. of moles per hour, amount of b o t t o m s product, sidestreams, and column pressure
I
0.3
\
.c
\
a 0
. P -I
Calc-late total liquid and vapor molar fiow rates on each stage: Lr and V,. Estimate initial temperature- and car.cer,trat ior. profiles b
for each stage n and each component i , using .Calculate J.I~I.Q C A C , t h e equilibrium ratio at the last estimated tem-
peratdye and concentration i K . . t h e ; o r Zn,: ~ i T hrescect to.!f;uid f l o w rates and temperature
Bottom
Determine the derivaphase component molar
5
7
9
11
S t a g e Number
13
15
17
19
d TOP
Figure 6. Liquid-phase concentration profiles for 1-propanol and toluene with cyclohexane and ethanol in a simple distillation column.
1 I -
3
Solve tne ccrrpor.ent naterial balances for each stage with ‘he new values o f t i l e stripping factors C n,i ‘JnKn,i‘Ln I
Are new the previously estimated values? Yes, within suecified limits Print resulting distillate and bottoms compositions and column temperature- and concentration profiles
Figure 5. Procedure for distillation column calculations (simplified version).
iteration method, utilizing simultaneous convergence of all independent variables, is now used to obtain the true values of l,,,, u,,,~,and T,,. Simultaneous convergence is more efficient than a sequential convergence method when the volatilities of the components depend both on temperature and compositions. Convergence is normally attained within 4-9 iterations. The computer time per iteration is proportional to the number of stages. In the simplified version (constant molal overflow), four components and 20 stages typically require 0.4 s per iteration on an IBM 370/165 computer using a Fortran G or Watfiv compiler. About 50%of this time is used for algebraic manipulations, and 50%is used for evaluating activity coefficients. The full Naphtali-Sandholm procedure requires the same number of iterations and typically 2.0 s per iteration for the above example. Similar calculations, where the Wilson equation is used instead of the UNIQUAC equation, require slightly less computer time. The parameters in the Wilson equation (or any other two-parameter expression for activity coefficients) may be calculated from the UNIFAC parameters in the same manner as that used to determine UNIQUAC parameters. Results of Distillation Calculations (Simplified Version). Distillation-column design calculations assuming constant molal overflow and ideal vapor phase were carried out for three different systems: cyclohexane(1)-ethanol(2) -l-propanol(3)-toluene(4); 1,2-dichloroethane(l)-l-propanol(2)-toluene(3)-acetone(4); and methylcyclohexane(1)toluene(2)-phenol(3). The first system was chosen because it exhibits large deviations from ideality and is thus an ex-
ample of systems where estimation of activity coefficients is absolutely essential; the second system was chosen because of the availability of extensive experimental multicomponent vapor-liquid equilibrium data (Table VIII); the third system is an example of extractive distillation. For the first two systems, feed is introduced on stage 10, there are 20 stages, and there are no sidestreams. The column configuration for the extractive-distillation example is given in Figure 7. The System Cyclohexane( 1)-Ethanol(2)-1-Propanol(3)-Toluene(4). Hydrocarbon-alcohol systems are known to deviate greatly from ideality; minimum-boiling azeotropes are found in many cases. In this system, no fewer than four binary azeotropes exist. Predictions of equilibrium ratios in the ternary system of components 2,3, and 4, using the UNIFAC method, indicate that no ternary azeotropes exist in this system. For this system we choose as feed composition x 1 = 0.1, x p = 0.4,x 3 = 0.4, x 4 = 0.1. The reflux ratio is 2.25, and 60 mol out of 100 mol in the feed are specified as distillate. The main results of the column calculations are shown in Table IX. The toluene-ethanol azeotrope has a marked effect on the results. Although toluene is the heaviest-boiling component, i t goes into the distillate almost quantitatively. The relative content of toluene and ethanol in the distillate corresponds closely to that of the azeotrope. It boils a t 77”C, 20°C lower than the normal boiling point of 1-propanol, which forms 99% of the bottoms product. A similar calculation, where the UNIQUAC model is replaced by the Wilson equation gives essentially the same results. Raoult’s law, on the other hand, gives completely erroneous results. The calculated propanol and toluene concentration profiles are shown in Figure 6. Computations performed with the full version of the column design procedure indicate that the assumptions of constant molal overflow and ideal vapor phase are justified in this case. The System 1,2-Dichloroethane( 1)-1-Propanol(2)Toluene(3)-Acetone(4). Predicted activity coefficients using the UNIFAC method are given in Table VIII, which also shows experimental activity coefficients. Essential results of the column calculations are given in Table IX together with those based on Raoult’s law and those using the Wilson equation instead of the UNIQUAC equation. Ind. Eng. Chem., Process Des. Dev., Vol. 16,No. 4,1977
457
c
-
Feed stage 12 ?12=76.74
d=:2.21- mles/h
I
I
21
moleslh
=o 1
stages
;o 2
x -1.0
ZC'4P:TER TIYE: 5 . 4 s e c a n t-le 191: ? 7 0 / 1 C 5
p = l atn
ST3RA3E SEQLIRE
.%e+ stage 7 _.c Fy23.26 mles/h xl=0.5
x 2--0 ' 5 x,=o cb
=W.:f
no:espb
T h e f r a m e d nunoe?s a r e c a l c u l i t e 3
FeiUltS
S t a s e YU7ber
Figure 7. Column configuration and liquid phase concentration profiles for extractive distillation of methylcyclohexane(1)and toluene(2) with phenol(3). Table IX. Component Flow Rates of Distillate and Bottoms. Comparison of Calculations Based on UNIFAC. Raoult's Law and Wilson's Equation. Simplified Version; 20 Stages; Feed Stage: 10; Pressure: 1 atm; No Sidestreams Feed," mol of comp. i
UNIQUAC Distillate, Bottoms, molof mol of comp. i comp. i
60
2.25
Table X. Component Flow Rates of Distillate and Bottoms. Comparison of Calculations Based on UNIFAC and Raoult's Law. Full Version." Ethanol( 1)I-Propanol(2)'-Water(3)-Acetic Acid(4) Component 1
2 3 4
Mol in distillate
Mol in bottoms product
Method
24.8 14.3 10.4 0.4
0.2 10.7 14.5 24.6
Raoult's law
Full NaphtaliSandholm with UNIFAC UReflux ratio: 2.0;distillate: 50 molh; number of stages: 15;feed stage: number 7;feed: (1):25 mol/h; (2):25 mol/h; (3):25 mol/h; (4):25 mol/h. 1 2 3 4
22.4 9.2 18.4 0.0
2.6 15.7 6.6 25.0
In the example given here, the feed contains mainly components 1 and 2, and the reflux ratio is 4. Separation between these two components is poor due t o the 1,2-dichloroethane1-propanol azeotrope (boiling point 68 "C) going into the distillate. The UNIQUAC results agree well with those using the Wilson equation. However, Raoult's law predicts a different (and erroneous) distribution ratio for components 1and 2. The System Methylcyclohexane( l)-Toluene(2)-Phenol(3). Methylcyclohexane and toluene are difficult t o separate by simple distillation. Smith (1963) presents a stage458
Raoult's law Distillate, Bottoms, mol of mol of comp. i comp. i
WilsonC Distillate, Bottoms, mol of mol of comp. i comp. i
Cyclohexane(l)-Ethanol(2)-l-Propanol(3)-Toluene(4) (1):10.0 (1): 0.0 (1): 9.7 (1): 0.3 (1):10.0 (2):39.7 (2): 0.3 (2):39.9 (2): 0.1 (2):39.9 (3): 0.3 (3):39.7 (3):10.3 (3):29.7 (3): 0.7 (4): 9.9 (4): 0.1 (4): 0.1 (4): 9.9 (4): 9.3 1,2-Dichloroethane(l)-l-Propanol( 2)-Toluene(3)-Acetone(4) (1):45 50 4 (1): 34.2 (1): 10.8 (1):42.3 (1): 2.7 (1): 35.5 (2):34.2 (2): 2.6 (2):42.4 (2): 9.6 (2):45 (2):10.8 (3): 5 (3): 0.0 (3): 5.0 (3): 0.0 (3): 5.0 (3): 0.0 (4): 5 (4): 5.0 (4): 0.0 (4): 5.0 (4): 0.0 (4): 4.9 Specified by user. Parameters estimated with UNIFAC method. c Parameters fitted with binary data.
(1): 10 (2):40 (3):40 (4):10
a
Distillate," mol Refluxa total ratio
Ind. Eng. Chem., Process Des. Dev., Vol. 16, No. 4, 1977
(1): 0.0 (2): 0.1 (3):39.3 (4): 0.7 (I): 9.5 (2):35.3 (3): 5.0 (4): 0.1
to-stage solution of a n extractive-distillation process for the above two components using phenol as a solvent. Using UNIFAC coupled with the simplified version of the distillation program, we obtain the same results as those presented by Smith. The process and resulting concentration profiles are shown in Figure 7.The solution of this problem required seven iterations and 3.4 s on the IBM 370/165 computer. Results of Distillation Calculations (Full Version). If excess enthalpy effects cannot be neglected, if stage efficiencies are to be included, or if the vapor phase is strongly nonideal, it is necessary to use the full Naphtali-Sandholm procedure. (Slight vapor phase nonidealities can be included in the simplified version by inserting the vapor-phase fugacity coefficient in the stripping factor (see Appendix B,eq 7B)and subsequently neglecting derivatives of the test functions with respect to vapor-phase composition). Systems containing organic acids are examples of systems where the full Naphtali-Sandholm procedure must be used. Liquid-phase nonidealities are, as before, accounted for via the UNIFAC correlation, and vapor-phase nonidealities, including organic-acid dimerization, are determined as explained in Appendix C. The results shown in Tables X and X I are for the case of a simple distillation column with one feed stream at stage 7 , no sidestreams, and 15 stages. Figure 8 shows results for a more complex column. The System Ethanol(l)-l-Propano1(2)-Water(3)Acetic Acid(4). For a n even feed of 25 mol/h of each component to a column as specified in Table X, one might expect a split between 1-propanol and water, such that most of the 1-propanol goes into the distillate, most of the water into the bottoms product, Raoult's law gives this result. However, as
Table XI.Component Flow Rates of Distillate and Bottoms. Comparison of Calculations Based on UNIFAC and Raoult’s Law. Full Version.n Benzene( 1)-1Propanol(2)-Toluene(3)-Acetic Acid(4) Component Distillate
I
1 2
I1
I11
3 4 1 2 3 4 1 2
3 4 IV
1
2 3 4
24.95 9.77 5.39 9.89 25.00 9.43 10.00 5.57 25.00 6.95 9.99 8.06 25.00 6.96 9.99 8.05
Bottom
Method
0.05 0.23 4.61 45.11 0.00 0.57 0.00 49.43
Raoult’s law
0.00
3.05 0.01
46.94
0.00
3.04 0.01
46.95
I \I
1
1
lf7i = 1
yi = 1
Constant molal flow (oi
1
stages
n n , =O.7
I
Sidestream (Saturated stage 9 Liquij:
,
=1
yi from UNIFAC
Constant molal flow Pi f 1 yi from UNIFAC Full NaphtaliSandholm Pi f 1 yi from UNIFAC
Reflux ratio: 2.0; distillate: 50 mol/h; number of stages: 15; feed stage: number 7; feed: (1): 25 mol/h; (2): 10 mol/h; (3): 10 mol/h; (4): 55 mol/h.
F,:5L
mcles/h
X i 4 ’ J T E F , TI:.iE:
17.4 s e c on t h e IBK :70/155
STORAGE R E Q U I R E M E N T : e 8 K
Bytes.
-he framed n u n t e r s a r e c a l c d i a t e d results.
the UNIFAC correlation correctly predicts, water has a much higher activity in acetic acid than 1-propanol has, and hence most of the 1-propanol goes into the bottoms product. In these calculations, we have neglected the effect of the relatively slow esterification of the alcohols with acetic acid. Figure 8 shows a summary of results for a complex column, where Murphree stage efficiencies of 0.7 were included. This required five iterations and 17.4 s computer time on the IBM 370/165 computer. The System Benzene(l)-l-Propano1(2)-Toluene(3)Acetic Acid (4).Table XI shows results for the case of an uneven feed: 25 mol/h benzene, 10 mol/h each of 1-propanol and toluene, and 55 mol/h of acetic acid. These results show the effects of successively including corrections for liquidphase nonideality (II), vapor-phase nonideality (III), and excess enthalpy effects (IV). I t is essential to include corrections for both liquid and vapor-phase nonideality, whereas the inclusion of enthalpy balances has only limited effect on the calculations. For most of the systems we have studied, the inclusion of enthalpy balances has little effect on the product compositions. They may, however, be of much importance in the determination of internal column flow rates. Conclusions This work has indicated that UNIFAC provides a powerful tool for the chemical design engineer. As outlined and illustrated here, it is possible to construct a completely computerized procedure for process design of a multicomponent distillation column, including estimation of phase equilibria. The UNIFAC method for calculating activity coefficients interfaces easily with iteration techniques for distillationcolumn computations. While the engineering utility of UNIFAC has been demonstrated, its utility inevitably rests on the quantity and quality of UNIFAC group-interaction parameters. These, in turn, depend on the quantity and quality of experimental phase-equilibrium data. The essence of UNIFAC is that it interprets and correlates such data to enable predictions of phase equilibria in previously studied mixtures under new conditions and in mixtures that have not been studied a t all experimentally.
Figure 8. Distillation results for the system ethanol(1)-1-propanol(2)-water(3)-aceticacid(4). Omissions in the group-parameter table clearly indicate where new experimental data are needed. As new and better phase-equilibrium data become available, the UNIFAC method, appropriately revised, will increase its accuracy of predictions and, more important, increase its range of applicability. However, even a t this incomplete state of development, it is evident that UNIFAC may be useful in at least some of those commonly encountered industrial situations where required data are lacking and where, for commercial reasons, a reasonable design is needed quickly. In chemical engineering process design there is always a need for good data, seasoned judgment, and sound experience. UNIFAC is no substitute for these requirements but it can contribute toward meeting their demands. A description and print-out of the UNIFAC programs for estimating activity coefficients and the computer programs which interface UNIFAC with distillation-column design is given by Fredenslund et al. (1977). Acknowledgment The authors thank Mr. Vagn Hansen for carrying out the “full version” of the Naphtali-Sandholm column calculations, Mr. Thomas Anderson for his assistance in reducing to practice the correlation indicated in Appendix C, the National Science Foundation and Statens teknisk-videnskabelige Forskningsraad for partial financial support, and Professor Ulfert Onken (University of Dortmund) for his active interest in this work. Appendix A T h e UNIQUAC a n d UNIFAC Models. Equations giving the activity coefficients as functions of composition and temperature are here stated very briefly, both according to the molecular UNIQUAC model (Abrams and Prausnitz, 1975) and according to the group-contribution method, the UNIFAC model (Fredenslund et al., 1975). Both models have a combinatorial contribution to the activity coefficients, essentially due to differences in size and shape of the molecules, Ind. Eng. Chem., Process Des. Dev., Vol. 16, No. 4, 1977
459
and a residual contribution, essentially due to energetic interactions In yi =
In y i c combinatorial
+
In y i R residual
(1A)
group-composition variable, e k , is now the group fraction of group k in pure fluid i. 111. Enthalpy of Mixing. One may obtain enthalpies of mixing from the UNIQUAC and UNIFAC models. From classical thermodynamics, it follows that
I. Combinatorial Part. The combinatorial contribution is the same in the two models
t
li = - (ri 2
- q i ) - (ri - 1); t = 10
Substituting eq l A , 2A, and 6A into eq 13A we obtain
Pure-component parameters ri and qi are, respectively, measures of molecular van der Waals volumes and molecular surface areas. They are calculated as the sum of the groupvolume and group-area parameters, R k and Qk r; =
k
Uk (i)Rk;
qi =
Uk
k
(’)&
(44
where V k ( I ) , always an integer, is the number of groups of type h in molecule i. Group parameters R k and Qk are obtained from van der Waals group volumes and surface areas, vk and Ah, given by Bondi (1968) Rk
= Vk/15.17;
Qk
= Akl(2.5 X lo9)
(5A)
11. Residual Part. A. UNIQUAC Equation. (6.4) (7-4) where u,, is a characteristic energy for the j-i interaction. For a small temperature range, the effect of temperature on T,, can be approximated by a linear form T,,
= A,, (O)
+ A,, ( l )
*
T
The parameter T,, is given by eq 8A. Equation 14A provides only a crude estimate when, as done here, parameters (U,, - U,,) and (ulL- u,,) are determined from vapor-liquid equilibrium data and are assumed to be independent of temperature. Appendix B Equations for Multicomponent Distillation Design. The distillation column configuration is shown in Figure 4. Distillation columns with up to 50 (actual) stages, up to ten components, a partial condenser, and any number of feed- and sidestreams may be considered. In our “operating column analysis” (King, 1971, p 520), the user must specify: (a) the number of actual stages, (b) stage efficiencies, (c) feed- and sidestream locations, (d) feed compositions, flow rates, and thermal states, (e) distillate flow rate, (f) sidestream phase conditions and flow rates, (g) reflux ratio, and (h) column pressure. The nomenclature for an arbitrary stage n ,which includes the possibility of feed- and sidestreams, is given below.
(8.4)
where A,,(O) and A,, ( l )are coefficients related to (u,, - u,,). B. UNIFAC Method (Group-Contribution). In y,R =
k
Uk(’)[lnr k all groups
- In r k “ ) ]
(9.4)
I’k is the group residual activity coefficient, and r k (I)is the residual activity coefficient of group k in a reference solution containing only molecules of type i.
E Um(i)X, 8, = ’- Q m X m EQ,X,’ n
x m =
CC U k ( i ) x i
(1W
i k
X , is the fraction of group m in the mixture. Qnm
= exp[-(a,,/T)I
*( 12A)
Parameter anm characterizes the interaction between groups n and m. For each group-group interaction, there are two parameters: anm # amn. No ternary (or higher) parameters are needed to describe multicomponent equilibria. Equations 10A-12A also hold for In rk ( i ) , except that the 480
Ind. Eng. Chem., Process Des. Dev., Vol. 16, No. 4, 1977
hn
Ln
vfl- 1
ln,i
un- I ,i
Hn- 1
Subscript n: flow from stage n , n = 1 , 2 , . . . , N ; subscript i: component i, i = 1 , 2 , . . . M ; H = vapor phase enthalpy; h = liquid phase enthalpy; H I = feed enthalpy; V = total vapor flow; u = component vapor flow; L = total liquid flow; 1 = component liquid flow; F = total feed; f = component feed; SL= liquid sidestream; Sv = vapor sidestream. General Relations w i t h Enthalpy Balances. For stage n one obtains the following set of dependent relationships (test functions F k ( f l , i ) which ) must be satisfied.
Component Balances. (Total: N X M relations)
~,,i -
~,-1,j
- In+l,j
- fn,j
=0
(1B)
Enthalpy Balances. (Total: N relations)
The assumption of equality of heats of vaporization and zero excess enthalpy is tantamount to the assumption of “constant molal overflow”. I t is not necessary to assume ideal vapor phase a t this point. However, in the solution of eq 9B it is convenient to assume that {,i does not depend upon the vapor phase composition. Newton-Raphson Iteration. Solving eq 4B or 9B means finding the set of values of the independent variables, x , which makes the set of test functions become equal to zero
F(x) = 0
H , - Hn-l - hn+l- Hf,, = 0 (2B) Equilibrium Conditions with Murphree Stage Efficiencies, v,,,. (Total: N X M relations). F.?(n,i)=
vn,iKn,lVnL,i/Ln - on,, + (1 - vn,i)un-1,iVn/Vn-1 = 0
+
which contains N X (2M 1) elements, and which may be solved for equally many unknowns
(t}
This equation is used to estimate x(new). When (x(new) - x(Old)) is sufficiently small, the correct set of values of x has been found, and the iteration stops. The variations between subsequent iterations are limited as follows: (1) negative component molar flow rates are equated to zero; (2) component flow rates exceeding L, are equated to L,; (3) the maximum change in component molar flow rates is O.25Ln, and (4)the maximum change in the temperature a t each stage, T,, is 10K. The limits of 0.2515, and 10K are chosen arbitrarily. The Jacobi matrix, (&Flax),is here very large, often of the order 200 X 200. The evaluation is greatly facilitated by the fact that the conditions on stage n only is influenced directly by the conditions on stages n 1and n - 1. As a result, the Jacobian becomes block-tridiagonal in structure, which permits rapid solution by block elimination. The derivatives of test functions with respect to temperature are found analytically, those with respect to component flow rates numerically.
+
where the vector 1 contains all the elements l,,,, v all elements and T all elements T,. Once all I,,,, u ~ ,and ~ , Tn’sare known, the product compositions and product flow rates and the concentration- and temperature profiles in the column follow readily. Simplified Version (Constant Molal Overflow). If one assumes q n I = 1, Pn,i = 1, and
F(new)(x(new)) = F(old)(X(old))
(3B)
The above relationships comprise a vector of test functions
x=
In Newton-Raphson iterations, a new set of values of the test functions, FcneW), are generated from a previous estimate in the following fashion
AHnM= 0, all n and i, and s l w = S 2 w .. . . = u M v a P
eq 1B and 3B may be combined as follows
Appendix C. Determination of Pure-Component Properties and Vapor-Phase Nonideality Pure-Component Reference Fugacity. The UNIFAC correlation applies to vapor-liquid equilibria a t normal pressures. The pure-component reference fugacity is calculated from eq 3 in the text. Antoine constants are used to calculate P,”,and values for pis are obtained from the correlation by Hayden and O’Conne11 (197~4,which includes components deviating strongly from ideality due to chemical interactions. Vapor-Phase Nonideality. A t normal pressures, vaporphase fugacity coefficients of pure components and components in a mixture may be calculated from second virial coefficients, B,,
where P,,,s is the pure-component vapor pressure of component i on stage n . One other set of independent relations is in this case given by the mass balance on each stage F5(n)=
Ln
- ln,1 - ln,2 - . . . - l
Equations 6B and 8B comprise N X ( M
n ,= ~
0
(8B)
+ I ) relations
M is the number of components in the system. In Hayden and O’Connell’s generalized correlation for predicting second virial coefficients, the overall second virial coefficient is considered as a sum of contributions form free pairs of molecules, metastable pairs, physically bound pairs. and chemically bound pairs
containing equally many independent variables (10B)
and equations are presented for each contribution. For assoInd. Eng. Chem., Process Des. Dev., Vol. 16, No. 4 , 1977
461
ciating components such as carboxylic acids, the dimerization equilibrium constant, K p , is obtained from
K
= P
- B b o u n d -k Bmetastable -k Bchern. RT
(4C)
The Hayden-O'Connell generalized correlation for Bi; is applicable to most systems where the UNIFAC method is presently used. Input to the correlation consists of: (a) purecomponent information: critical temperature and pressure, Thompson's mean radius of gyration, dipole moment, and an association factor, which depends only upon the type of associating group (hydroxyl, amine, carboxylic acid, etc.); (b) mixture information: solvation factor; (c) the system temperature. Hayden and O'Connell give the input parameters, including association and solvation factors, for a large number of systems. For systems containing carboxylic acids, we have chosen to neglect Bfreeand use the values of K p given by Tamir and Wisniak (1975) and Prausnitz (1969, p 138) to represent the vapor-phase nonideality. Nomenclature
anm
= UNIFAC binary interaction parameter (related to
qnrn
1
A;; = UNIQUAC parameter, see eq 8A Bi - second virial coefficient for i-j interaction
fid
pure-component reference fugacity of component i molar feed rate of component i on stage n molar feed rate on stage n enthalpy of vapor from stage n enthalpy of liquid from stage n AHM = enthalpy of mixing M v a p = enthalpy of vaporization K,,i = equilibrium ratio of component i on stage n 1,,i = liquid molar flow rate of component i from stage n 1, = pure-component constant defined in eq 3A L , = total liquid molar flow rate from stage n Pis = pure-component vapor pressure of component i qi = pure-component area parameter of component i Qk = group area parameter for group k ri = pure-component volume parameter of component i R k = group volume parameter of group k R = gas constant SnV = vapor-phase sidestream from stage n S , L = liquid-phase sidestream from stage n T = temperature u,; = UNIQUAC binary interaction parameter (related to ~ , and i A .i) V , = totai vapor molar flow rate from stage n = F, = H, = h, = fn,i
462
Ind. Eng. Chem.. Process Des. Dev., Vol. 16, No. 4, 1977
un,i =
vapor-phase molar flow rate of component i from stage
n u; = molar liquid volume of component i liquid-phase mole fraction of component i Xk = group fraction of group k yi = vapor-phase mole fraction of component i z = lattice coordination number, here equal to 10 xi =
Greek Letters y I = activity coefficient of component i
rk
k activity coefficient of group k in pure component i pi = fugacity coefficient of component i @i = segment fraction of component i U k (i) = number of groups of kind k in molecular species i 0; = area fraction of component i (3k = area fraction of group k T,> = UNIQUAC parameter, see eq 7A qnrn= UNIFAC parameter defined in eq 12A = stripping factor of component i on stage n = activity coefficient of group
rk( i ) =
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Receiced for reuieu: November 5. 1976 Accepted April 8, 1977
