I n d . Eng. Chem. Res. 1987,26, 2036-2042
2036
Prediction of Liquid-Liquid Equilibria with UNIFAC: A Critical Evaluation Parag A. Guptet and Ronald P. Danner* Department of Chemical Engineering, The Pennsylvania State University, University Park, Pennsylvania 16802
T h e UNIFAC-LLE method has been comprehensively tested by using a large LLE data base. Evaluations were carried out for binary and ternary systems. The ternary system evaluations were grouped according t o those used in the parameter development and those not used. T h e ability of the UNIFAC-LLE method to predict phase changes with temperature was also investigated. UNIFAC is at Dresent the only generalized method available for predicting LLE data. %he UNIFAC-LLE parameters were developed by Magnussen et al. (1981). These parameters are distinct from those in the VLEUNIFAC matrix, although for a few group-group interactions the VLE parameters of Skjold-Jerrgensen et al. (1979) are used. UNIFAC-LLE group interaction parameters are available for 32 different groups representing hydrocarbons, water, alcohols, organic acids, halogenated hydrocarbons, nitriles, etc. The UNIFAC-LLE method is not applicable to components with normal boiling points below 300 K, to strong electrolytes, or to polymers. The temperature range is restricted to between 10 and 40 "C. Extrapolation outside this temperature range is not recommended since LLE parameters are strongly temperature sensitive. In this work, the UNIFAC-LLE model was comprehensively tested by using a data base compiled at the Technical University of Denmark. This data base contains data for 1130 binary, 1011 ternary, and 28 quaternary systems. Evaluations for T e r n a r y Systems The 1011 ternary systems available in the LLE data base are in the temperature range 0-150 "C. It is estimated that Magnussen (1980) used about 400 of these ternary systems in the development of the UNIFAC-LLE parameter matrix. Most of the systems selected for data reduction had temperatures in the range 20-30 "C. Furthermore, each system selected was isothermal. The UNIFAC-LLE parameters were obtained by generating a binodal (twophase) curve with current estimates of the parameters and then modifying these parameters by minimizing the sum of squares of the deviations in the mole fractions. The objective function used was
F = lOO[C min cc(xi;k- f i j k ) 2 / 6 m ] ' / 2 (1) k
i l
where x = experimental mole fraction; f = calculated mole fraction; i = 1-3 (components);j = 1,2 (phases); k = 1,2..., m (tie lines); m = number of tie lines. The calculated mole fractions were obtained by interpolating in a grid of tie lines such that the inner double summation in eq 1 was minimized. This procedure was first proposed by Varhegyi and Eon (1977). This procedure is a true least-squares method, unlike the traditional approaches based on the isofugacity criterion. For the present work, 600 ternary systems were selected for evaluation. The remaining ternary systems were excluded either because the temperature was outside the
* T o whom
correspondence should be addressed.
Presently at the Union Carbide Corporation, South Charleston,
WV 25303.
range of the UNIFAC-LLE model or because the systems included a compound not available in the DIPPR Data Compilation Project (Daubert and Danner, 1986). An examination of the 600 selected systems revealed that 227 of these were used by Magnussen for developing the UNIFAC-LLE parameter matrix; the remaining 373 systems were not used. Flash calculations were performed on the 600 ternary systems. The feed composition was selected as the midpoint composition of the experimental tie line. Absolute root mean square (rms) deviations in all phase compositions and percent rms deviations in the solute distribution ratio were determined. Deviations in other quantities (solvent distribution ratios and selectivities) were also determined. Errors in these quantities obtained with the UNIFAC model were so high, however, that they were not tabulated. All the evaluation results were grouped by family. This kind of representation permits meaningful conclusions to be drawn from the large mass of numerical data. Ternary family groupings are difficult to define because of the relatively large number of possible groupings. Ternary chemical groupings which contained at least five systems were identified separately. The remaining ternary groupings were lumped under a miscellaneous heading. Triangular coordinates are used to represent ternary data as shown in Figure 1. Each apex of the triangle represents a pure component. The distance from a point within the triangle to the side opposite the apex represents the mole fraction of the component in the mixture. Serrensen (1980) has classified ternary LLE systems as depicted in Figure 1. Ninety-five percent of the observed systems were of type 1 or 2. Table I contains the evaluation results for the 227 systems which had been used previously by Magnussen in the parameter development. These systems are represented by 13 chemical families. The mole-fraction deviations given correspond to each component in the listed order. Thus, for the first family (hydrocarbon-alcohol-water), x1 corresponds to the deviation in the hydrocarbon component, x 2 to the deviation in the alcohol, and x 3 to water. The superscripts I and I1 refer to the phases. The percent deviations in K2 correspond to the deviation in the distribution ratio of the solute. The solute is the common component of the two miscible binaries for a type 1 system and the component whose distribution ratio at infinite dilution is closest to unity for a type 2 system. Thus, from Table I, it is not possible to identify which component is the solute, unless only type 1 systems are involved in a particularly family. In the first hydrocarbon-alcohol-water family, there are only type 1 systems. It is possible, in this case, to identify K2 as the distribution ratio of the alcohol. In the third family (aromatic-alcohol-water), there are two type 2 systems, and for these
0888-588518712626-2036$01.50/0 0 1987 American Chemical Society
Ind. Eng. Chem. Res., Vol. 26, No. 10, 1987 2037 Table 1. Flash Calculations: Errors for Ternary LLE Systems Used in UNIFAC-LLE Parameter Estimationa no. of systems phase compositions abs dev no. of temp of total no. (rms X 100) family components systems range, K type 1 type 2 of pts xI1 xlll x 2 x;* xgI xJ1 1
2
3
4
5
6
% dev, K 2
298.15 ~.
hydrocarbon alcohol water
17
298.15
17
0
127
7.8
9.0
18.6
9.1
23.7
2.7
50
alcohol alcohol water
10
293.15 303.15
10
0
72
5.7
0.9
4.1
3.1
6.7
3.7
415
aromatic alcohol water
11
298.15 303.15
9
2
119
2.5
4.0
9.3
5.8
11.1
2.4
320
oxygen containing alcohol water
29
293.15 310.95
20
9
192
3.2
1.2
3.6
2.7
5.5
3.5
1053
C1 or N containing alcohol water
17
273.15 298.15
14
3
105
6.0
3.0
8.1
5.7
8.5
4.8
442
6
298.15 304.15
5
1
34
3.7
0.6
2.4
2.3
2.4
2.2
41
39
289.15 497.75
33
6
217
6.4
1.6
3.1
1.9
8.0
2.7
102
30
293.15 304.15
22
8
229
2.0
1.6
2.1
1.6
3.5
2.0
73
9
298.15 304.15
9
0
66
2.5
0.5
1.5
1.3
3.3
1.6
31
11
283.15 313.15
10
1
79
1.5
0.7
1.7
2.6
1.5
2.5
67
6
297.15 304.15
4
2
43
2.1
1.9
1.8
2.1
3.1
3.1
33
10
293.15 303.15
10
0
68
3.6
2.6
2.4
2.1
4.7
2.6
51
32
318.15 291.15
28
4
157
4.0
2.7
1.4
1.4
4.4
3.4
25
hydrocarbon X
water 7
8
oxygen containing oxygen containing water C1 containing X
water 9
X
N containing water 10
11
aromatic oxygen containing water hydrocarbon X
glycol 12
13
hydrocarbon alcohol glycol X X X
a x represents any other component.
TYPE 0
TYPE 1
?
2
TYPE 3
TYPE 3 A
Figure 1. Types of ternary liquid-liquid equilibria (Ssrensen, 1980).
systems, either the aromatic or alcohol component could be the solute.
Table I1 contains the evaluation results for the 373 systems not used by Magnussen in the parameter development. The systems are grouped according to the same families, and all the comments given in the preceding paragraph for Table I apply equally to Table 11. There are, however, two additional families (hydrocarbon-xalcohol and aromatic-acid-water) which had a sufficient number of systems to warrant inclusion. The hydrocarbon-alcohol-glycol family is not represented in this table. Table I11 contains the evaluation results obtained directly from Magnussen’s work. That is, these errors are the deviations reported by Magussen (1980) as part of the parameter estimation effort. As discussed before, the parameter estimation was based on the Varhegyi and Eon (1977) method. Since the systems represented in Table I11 were used in the parameter estimation procedure, these deviations should be compared with those in Table I. Familes 8 and 9 of Table I have been combined in Table 111. There are 283 systems in Table I11 as opposed to 227 in Table I. Tables I and I11 represent the performance of the UNIFAC-LLE method by calculating errors in two different ways. These tables are indicative of the performance
2038 Ind. Eng. Chem. Res., Vol. 26, No. 10, 1987 Table 11. Flash Calculations: Errors for Ternary LLE Systems Not Used in UNIFAC-LLEParameter Estimation”
family 1
2b
3
4
5
6
18
temp range, K 273.15 331.15
alcohol alcohol water
41
aromatic alcohol water
components hydrocarbon alcohol water
no. of systems
no. of systems of type 1 type 2
total no. of pts
xlI
phase compositions abs dev (rms X 100) xlI1 x21 x;I x? xJ1
70 dev, K 2
18
0
88
11.5
9.3
14.9
10.4
19.4
4.9
65
273.15 363.15
35
5
229
5.2
1.0
3.8
1.9
8.4
2.7
213
36
288.15 337.15
32
4
324
4.3
4.3
10.6
6.4
13.7
3.4
240
oxygen containing alcohol water
45
273.15 353.15
34
11
298
5.1
2.0
6.1
3.0
10.5
4.2
488
C1 or N containing alcohol water
11
293.15 340.15
10
1
97
4.5
6.0
9.5
10.3
9.5
9.6
1634
21
293.15 343.15
17
4
120
3.4
1.3
5.5
6.9
6.5
4.6
38
50
283.14 343.15
47
3
320
7.7
1.3
3.6
1.8
10.3
2.9
296
25
273.15 333.15
20
5
155
2.8
2.3
3.3
2.0
5.4
2.6
36
N containing water aromatic oxygen containing water hydrocarbon
15
281.75 369.85
11
4
84
7.1
1.9
6.9
2.2
13.0
1.6
1321
10
283.15 333.15
8
2
113
1.1
0.2
2.1
2.0
2.2
2.2
53
X
22
293.15 353.15
20
2
101
7.8
1.4
7.8
9.1
2.8
9.6
82
22
318.15 273.15
20
2
111
8.4
3.4
4.9
4.7
8.7
7.1
52
10
278.15 313.15
9
1
55
6.9
6.5
1.9
2.4
7.0
6.5
49
47
273.15 363.15
44
3
319
3.2
3.2
2.9
3.5
5.0
3.5
48
hydrocarbon X
water 7
8
oxygen containing oxygen containing water C1 containing X
water 9
10
11
X
glycol 13
X
X X
14
hydrocarbon X
alcohol 15
O x
aromatic acid water
represents any other component. *Contains one type 0 system.
of the method for systems used in the development of its parameters. The errors in Table I11 are lower than those in Table I. This happens because the errors in Table I were obtained by using flash calculations with a specified tie line passing through the midpoint composition of the experimental tie line. The results obtained by Magnussen (Table 111), on the other hand, are derived by selecting the tie line that minimizes the errors in the compositions. Since this procedure is not restricted to a specific tie line, it produces smaller errors. The errors obtained by using the UNIFAC-LLE model will vary according to which procedure is used for the calculations. Generally, the UNIFAC-LLE model is used to obtain phase compositions with flash calculations; thus, the errors of Table I would usually be more indicative of the deviations to be expected in practical applications. When comparing the results of Tables I and 11, one would expect lower errors in Table I, since these systems were used in the parameter estimation work. A close examination of these tables, however, reveals that there is no large difference in the quality of the predictions.
Overall, the errors in Table I are slightly lower than those in Table 11. Nevertheless, there is no significant difference which could lead to the speculation that UNIFAC is prejudiced toward the systems from which its parameters were determined. The lack of significant difference in the quality of predictions in the two sets of systems reflects the intrinsic strength of the UNIFAC-LLE model and the success of the group-contribution approach. Examination of Tables 1-111 reveals that the absolute rms errors (X100) in the phase compositions are usually small and rarely exceed 10. This is indicative of the relative accuracy of UNIFAC for the prediction of mole fractions and representation of binodal curves. The low errors in phase compositions are a direct consequence of the use of a concentration objective function (eq I) and the binodal curve generation parameter estimation technique employed by Magnussen. It is also evident that errors in K 2 are relatively high. In some cases, these exceed 1000%. This usually means that the slope of the predicted tie lines is not in agreement with the slope of experimental tie lines. A large error in
Ind. Eng. Chem. Res., Vol. 26, No. 10, 1987 2039 Table 111. E r r o r s for T e r n a r y LLE Systems: Errors Obtained from UNIFAC Parameter Estimation (Magnussen, 1980) for Systems Used in UNIFAC-LLE Parameter Developmenta phase compositions abs dev no. of systems of no. (rms X 100) family components no. of systems type 1 type 2 of pts xI1 x;' X$ x$I X? xtl % dev, K2 2 1.1 3.3 2.1 3.0 2.3 1.6 37 18 166 1 hydrocarbon 20 alcohol water 1.1 1.3 2.7 1.6 106 0 74 0.6 2.3 10 10 2 alcohol alcohol water 1.2 1.8 1.5 1.3 1.3 54 4 154 0.7 14 10 3 aromatic alcohol water 2.1 1.4 98 15 329 1.4 0.9 1.7 0.9 49 34 4 oxygen containing alcohol water 2.3 1.7 2.0 1.5 50 21 3 159 1.9 1.3 24 5 C1 of N containing alcohol water 2.2 1.9 68 4 2.0 0.2 2.1 1.9 10 6 65 hydrocarbon 6 X
water
7
83
oxygen containing oxygen containing water
42
30
12
252
3.1
0.8
3.8
2.0
5.2
2.3
43
C1 or N containing
45
33
12
360
1.7
1.1
2.1
1.6
2.5
1.4
82
12
10
2
91
1.4
0.9
1.6
1.5
1.7
1.2
39
3
3
0
16
2.3
3.6
1.1
1.7
2.4
3.1
43
9
8
1
61
1.8
1.0
3.0
1.8
2.8
1.7
49
45
33
12
249
1.1
1.0
1.4
1.6
1.8
1.4
23
X
water 10
aromatic oxygen containing water
11
hydrocarbon X
glycol 12
hydrocarbon alcohol glycol
13
X X X
ax
represents any other component.
K 2 can also be caused by an incorrectly predicted twophase region-either too large or too small compared to the experimental two-phase region. We have observed that the two-phase region predicted by UNIFAC LLE is usually larger than the experimental region. This would, however, also lead to large deviations in the phase compositions. The large deviations in K z are also caused by very small solute distribution ratios; i.e., relatively small absolute errors lead to very large percentage errors. Analysis of Type 3 and Type 3A Ternary Systems The LLE data base contains 10 ternary type 3 and type 3A systems. Eight of these 10 systems have components which are represented in the DIPPR Data Compilation Project (Daubert and Danner, 1986). These eight systems were separately analyzed by using the UNIFAC-LLE method. A type 3 ternary system consists of a three-phase region at a given temperature (see Figure 1). In addition to the three-phase region, a type 3 system also has three twophase regions and three one-phase regions. Thus, a type 3 system contains three immiscible binaries. A type 3A system also contains three immiscible binaries, but unlike a type 3 system, it exhibits two two-phase regions and two one-phase regions. Usually a system which exhibits type 3A behavior at one temperature can exhibit a type 3 be-
2
A '
TYPE 1
3
2
'
TYPE 2
Figure 2. Three-phase formation in type 0, 1, and 2 systems.
havior at a higher temperature. Three-phase formation can also occur in type 0, 1,and 2 systems. Figure 2 depicts this type of behavior. A three-phase type 0 system has one three-phase region, three two-phase regions, and one one-phase region. Such a system consists of three miscible binaries. Similarly, a three-phase type 1 system has one three-phase region, three two-phase regions, and one one-phase region. Unlike a three-phase type 0 system, however, a type 1 system has one immiscible and two miscible binaries. Examples of three-phase type 1 systems are butylcellosolve-n-decanewater and butylcellosolve-n-nonane-water at 25 OC. A three-phase type 2 system has one three-phase region,
2040 Ind. Eng. Chem. Res., Vol. 26, No. 10, 1987 Table IV. Analysis of Type 3 and Type 3A Systems: UNIFAC Method phase compositions abs dev (rms x 100) upper phase 3 1 2 3 1
no. of pts 8
type of system
1
left phase 2
2
3
3
5.7
7.9
3.4
0.4
9.4
9.6
0.0
2.9
2.9
11
3
4.1
5.6
2.5
0.8
9.6
9.9
0.0
2.8
2.8
heptane aniline water heptane aniline water
2
3
0.6
0.3
0.3
1.2
11.1
9.8
0.0
0.5
0.5
1
3
2.0
2.0
0.0
1.2
0.6
0.6
0.1
0.6
0.7
1-dodecanol nitromethane 1,Z-ethanediol
8
3
19.8
10.8
16.6
0.6
52.1
22.6
1.0
36.9
51.2
system nonanol nitromethane water nonanol nitromethane water
three two-phase regions, and two one-phase regions. Such a system contains two immiscible binaries and one miscible binary. Examples of a three-phase type 2 system are hexylcarbitol-n-dodecane-water and triethylene glycol monooctyl ether-n-dodecane-water at 25 "C. The LLE data base did not contain data for three-phase type 0, 1, or 2 systems. Such systems are relatively rare and were not considered in this work. Type 3 and 3A systems were separately analyzed to investigate the possibility of three-phase formation. Flash calculations were performed on these systems. The midpoint of the experimental tie lines was used as the feed composition. Flash calculations were performed by using the LLECAL program of Magnussen (1980). This program employs a differential phase stability test. The program has been comprehensively tested by Magnussen, and it was found to always yield the correct number of phases. Thus, any errors in the number of phases obtained in the present analysis are entirely attributable to limitations in the UNIFAC model. It was found that six of the eight systems tested produced fairly accurate predictions. Five of these systems are type 3 systems, and the last one is a type 3A system. The absolute rms deviations (X100) in the various quantities for the type 3 systems are given in Table IV. The remaining two systems were incorrectly predicted by the UNIFAC-LLE model. The 1-hexanol-nitromethane-water system exhibits type 3A behavior at 23 "C. At this temperature, UNIFAC LLE correctly predicts a type 3A system. This system is included in the Table IV evaluation results. At 21 "C, however, the same system exhibits type 3 behavior. At this temperature, the UNIFAC-LLE model still predicts type 3A behavior. The system nonanol-nitromethanewater exhibits type 3 behavior a t 20 and 23 "C. At these temperatures, the UNIFAC-LLE model correctly predicts a type 3 system. These two systems are included in the error analysis in Table IV. At a temperature of 45.1 "C, however, the system exhibits type 3A behavior, whereas UNIFAC LLE still predicts type 3 behavior. One of the possible explanations for this is that the temperature 45.1 "C is outside the range of the UNIFAC-LLE model (10-40 "C). The above analysis seems to indicate that UNIFAC is reasonably accurate in predicting the correct number of phases. Considering the simplifying assumption of the solution-of-groups concept, the correct predictions by UNIFAC for six out of a possible eight systems are quite
I
0
UCST
XI
!
UCST
__ XI
TYPE c
I
1 .o
TYPE b
TYPE a
0
right phase
1 .o
0
XI
1 .o
TYPE d
Figure 3. Types of binary liquid-liquid equilibria (Smensen et al., 1979).
good. Also,. we have.never observed any wrong phase predictions for type 0, 1, or 2 systems. It does appear, however, that rapid changes with system temperature (as for the 1-hexanol-nitromethane-water system) are not correctly predicted by UNIFAC LLE. Binary Systems Figure 3 shows the primary types of binary LLE diagrams identified by Serrensen et al. (1979). These are not the only possible types; however, types b and d make up approximately 90% of the systems investigated. For type b systems, temperatures above the upper critical solution temperature, UCST, will yield only one phase. For type d systems, the liquids are partially immiscible a t all temperatures and the critical solution temperatures are missing. Data for 1130 binary systems are included in the LLE data base. Some of these systems were used by Magnussen in the estimation of UNIFAC-LLE parameters. Binary LLE data are nonisothermal. Since only isothermal LLE data were used in the parameter-estimation work, only one tie line was used for each binary system. hlagnussen et
Ind. Eng. Chem. Res., Vol. 26, No. 10, 1987 2041 Table V. Mutual Solubility Calculations: Errors for Binary Systems
no. of systems
no. of pts
phase composition abs dev (rms x 100)
4
38
11.3
37
444
6.8
5
76
11.9
13
129
0.2
water C1 containing
5
49
0.7
water ester water ether water hydrocarbon
16
182
3.1
11
125
1.6
4
22
0.0
water ketone
18
165
5.9
water N containing
7
82
11.4
hydrocarbon aldehyde hydrocarbon N containing
6
60
6.2
27
338
20.6
acid N containing
5
63
22.3
alcohol aromatic
9
104
7.0
alcohol hydrocarbon
9
101
11.1
alcohol N containing
7
102
16.0
components water acid water alcohol water aldehyde water aromatic
al. (1981) have not commented on the accuracy of the UNIFAC-LLE method for binary data. Thus,in this work, the performance of the UNIFAC-LLE model for binary systems was investigated. No tie-line data are contained in the LLE data base. At a given temperature, only one phase composition is reported. Occasionally, the same temperature is later repeated, and the composition in the second phase is then available. In general, however, this is not the case, and no tie-line information can be obtained. Mutual solubility calculations were carried out on 183 binary systems, consisting of 2080 data points. Since tie-line information is not available, it is not immediately clear which predicted phase is to be matched with the given experimental one. This difficulty was solved by determining the differences between the experimental solubility and the two predicted solubilities. When the absolute values of these differences were compared, it was assumed that the smaller absolute difference corresponded to the correct phase. This approach is likely to fail in the vicinity of the critical solution points. Since 53% of the systems do not possess either an upper or lower critical solution point, and since even for systems with critical solution points, the data do not always extend to this point, this was not a serious problem. In this analysis, only systems with five or more data points were considered. The systems selected were in the temperature range of the UNIFAC-LLE model. Since it was not possible to determine which phase the experi-
mental data point represents, the deviations have all been averaged to give a single average error over the entire solubility curve. Initial guesses for the mutual solubility calculations were assigned zero values. Except for some highly nonideal systems (typically containing carboxylic acids), convergence was obtained in each case. The mutual solubilities were computed by using a Newton-Raphson solution of the isoactivity criterion. The errors were grouped according to family and are presented in Table V. The deviation reported is the absolute error in the phase composition. Overall, the errors for binary systems are larger than those for ternary systems. This is because binary data are nonisothermal and testing the model for such data tests its temperature dependence. The highest errors are found for systems containing alcohols, acids, and water. The very low errors for the water-hydrocarbon and water-aromatic families are the result of the low mutual solubilities of these systems. Conclusions The UNIFAC-LLE group-contribution method has been comprehensively tested with a large low-pressure data base. The UNIFAC model was found to yield the same quality of predictions for systems not used in the development of the parameters as for those used in the development. This underscores the success of the group-contribution approach for LLE. The UNIFAC-LLE model was found to yield quantitative estimations of phase compositions. The representation of distribution ratios and selectivities was not always good. This is caused by the use of a phasecomposition objective function in the parameter estimation. The UNIFAC-LLE model was tested with type 3 and type 3A ternary systems. The model predicted the number of phases correctly for six out of eight such systems. The model, however, is unable to handle changes in phase behavior over small temperature ranges. For the more common cases of type 1 and type 2 systems, no incorrect predictions of phases were found. The model was also tested with nonisothermal binary data. The errors for such systems were larger than those for (isothermal) ternary systems. The larger deviations are to be expected, since the quantitative representation of binary LLE data require four UNIQUAC parameters (to account for the temperature dependence). Acknowledgment The Design Institute for Physical Property Data of the American Institute of Chemical Engineers provided financial support of the work. We express our gratitude to Prof. Aa. Fredenslund of the Technical University of Denmark for providing the LLE data base used in the evaluations, as well as the complete work sheets and printouts of the Magnussen parameter-estimation work. Nomenclature F = objective function m = number of tie lines Xr,k = experimental mole fraction of component i in phase j for tie line k x , j k = calculated mole fraction Literature Cited Daubert, T. E.; Danner, R.P. Data Compilation Tables of Properties of Pure Compounds; American Institute of Chemical Engineers: New York, (extant) 1986. Magnussen, T. Ph.D. Thesis, The Technical University of Denmark, 1980.
Ind. Eng. Chem. Res. 1987,26, 2042-2047
2042
Magnussen, T.; Rasmussen, P.; Fredenslund, Aa. Ind. Eng. Chem. Process Des. Deu. 1981, 20, 331. Skjold-Jorgensen, S.; Kolbe, B.; Gmehling, J.; Rasmussen, P. Ind. Eng. Chem. Process Des. Dev. 1979, 18, 714. Ssrensen, J. M. Ph.D. Thesis, The Technical University of Denmark, 1980.
S~rensen,J. M.; Magnussen, T.; Rasmussen, P.; Fredenslund, Aa. Fluid Phase Equilib. 1979, 2, 297. Varhegyi, G.; Eon, C. H. Ind. Eng. Chem. Fundam. 1977, 16, 2.
Received for review April 18, 1986 Accepted July 7 , 1987
Further Work on Multicomponent Adsorption Equilibria Calculations Based on the Ideal Adsorbed Solution Theory Hee Moon* and Chi Tien Department of Chemical Engineering and Materials Science, Syracuse University, Syracuse, New York 13244-1240
It was found that the procedures developed previously for calculating multicomponent adsorption equilibria can be readily extended to the case where the pure component isotherm data are represented by the Langmuir expression with a first-order correction term. In contrast to the recent published work of O’Brien and Myers which considered some aspects of the same problem, the present method requires, for any N-component mixture, the solution of only one or two nonlinear algebraic equations instead of the N equations necessary with the O’Brien-Myers method. The saving in computation time becomes significant as the number (N) increases. O’Brien and Myers (1985) recently suggested a method for calculating multicomponent gas adsorption equilibria from pure component data based on the ideal adsorbed solution (IAS) theory. Their starting point is that the single-component isotherm data can be expressed by the Langmuir expression with a first-order correction, or
where the superscript o denotes the pure component state and the subscript i, the ith component. np is therefore the amount of the ith component adsorbed per unit mass of adsorbent, and m, is the maximum (saturation) value of n:. qi is defined as qi =
Kiplo
(2)
where Pp is the partial pressure of the ith component and Ki, a constant. Thus, there are three parameters (Ki, mi, and ai) to be specified when eq 1 is used to represent the pure component isotherm data. When an N-component gas mixture is in equilibrium with a particular adsorbent, the equilibrium state is defined by the concentrations in the adsorbed phase (nL,i = 1,2, ..., N) and the concentrations in the gas phase (P, = Py,, i = 1, 2, ...,N>. O’Brien and Myers developed their method (known as FASTIAS) to calculate the concentrations of one phase given the concentration of the other phase. The two cases (calculating ni with given Pi and vice versa) the researchers considered, however, do not cover all the situations of interest in adsorption calculation. As Wang and Tien (1982) and Larsen and Tien (1984) pointed out in their work on fixed-bed and batch adsorption processes, if the intraparticle diffusion is described by the homogeneous diffusion model or the lumped parameter model (also known as the linear driving force model), one
* Present
address: Department of Chemical Engineering, Chonnam National University, Kwangju 505, Korea.
is required to calculate the concentrations of both phases under the condition
Pi + Aini = Bi for i = 1, 2, ..., N (3) where Ai and Bi are constants. It should also be mentioned that several earlier studies concerned the calculation of adsorption equilibria based on the IAS theory. Tien and co-workers (Wang and Tien, 1982; Larsen and Tien, 1984; Tien, 1986) have shown that for liquid-phase adsorption, if one expresses the singlecomponent isotherm data either by the Freundlich expression or by the Freundlich expression on a piecewise basis, the required calculation is reduced to solving two nonlinear algebraic equations for any multicomponent systems. In other words, an increase in the number of components does not have a significant effect on computation. The purposes of the present work are 2-fold. First, the so-called FASTIAS procedure developed by O’Brien and Myers is extended to cover the situation described by eq 3. Second, to simplify computation, the FASTIAS procedure is reformulated according to the method developed previously by Tien and co-workers. Sample calculations illustrate how this reformulation simplifies computation.
FASTIAS Procedure The IAS theory, upon which the FASTIAS procedure is based, defines the adsorption equilibrium between a gas mixture of N components and a specific adsorbent by a system of equations: Pyi = Pprixi i = 1, 2, ..., N (4)
nt = [&/n:]-l
(6)
1
ni = ntxi i = 1, 2, ..., N (7) where x iand yiare the mole fractions of the ith component
0888-5885/87/2626-2042$01.50/0 0 1987 American Chemical Society
