Ind. Eng. Chem. Res. 1988,27, 657-664
657
Nataraj, S. Ph.D. Thesis, Purdue University, West Lafayette, IN, 1985. Nicholas, R. A.; Fox, J. B. J. Chromatogr. 1969,43,61. Rieke, R. D. Sep. Sci. Technol. 1984,19,261. Rod, V. Chem. Eng. J. 1984,29,77. Sussman, M. V. Chem. Technol. 1976,6,260. Sussman, M. V.; Rathore, R. N. S. Chromatographia 1975,8, 55. Sussman, M. V.; Astill, K. N.; Rombach, R.; Cerullo, A.; Chen, S. S. Ind. Eng. Chem. Fundam. 1972,Il,181. Turina, S.; Marjanovic-Krajovan, S.; Kostomaj, T. 2.Anal. Chem. 1962,189,100. Wankat, P. C. Separ. Sci. 1972a, 7, 233. Wankat, P. C. Separ. Sci. 1972b,7, 345. Wankat, P. C. AIChE J. 1977,23,859. Wankat, P. C. Sep. Sci. Technol. 1984,29,801. Wankat, P. C. Large Scale Adsorption and Chromatography;CRC: Boca Raton, FL, 1986. Wankat, P. C.; Middleton, A. R.; Hudson, B. L. Ind. Eng. Chem. Fundam. 1976,15,309. Zakaria, M.; Gonnord, M. F.; Guiochon, G. J. Chromatogr. 1983,271, 127. Zhukovitskii, A. A. In Gas Chromatography;Scott, R. P. W., Ed.; Butterworths: London, 1960; pp 293-300.
Baniel, A. M.; Blumberg, R.; Hajdu, K. U.S.Patent 4 275 234, June 23, 1981. Barker, P. E. In Preparative Gae Chromatography; Zlatkis, A., Pretorius, V., Eds.; Wiley-Interscience: New York, 1971; Chapter 10. Bowen, H. H. “Sorption Processes”. In Chemical Engineering; Coulson, J. M., Richardson, J. F., Eds.; Pergamon: Oxford, 1971; VOl. 3, pp 475-574. Canon, R. M.; Begovich, J. M.; Sisson, W. G. Sep. Sci. Technol. 1980, 15, 655. Frey, D. D. Sep. Sci. Technol. 1982-1983,17(13-14), 1485. Gidaspow, D.; Onischak, M. Can. J. Chem. Eng. 1973,51, 337. Greminger, D. C.; Burns, G. P.; Lynn, S.; Hanson, D. N.; King, C. J. Ind. Eng. Chem. Process Des. Dev. 1982,21,51. Hudson, B.; Wankat, P. C. Sep. Sci. 1973,8,599. Hughes, M. A.; Parker, N. In Chemical Separations; King, C. J., Navratil, J. D., Eds.; Litarvan Literature: Denver, 1986; Vol. I, pp 261-275. King, C. J. “Separation Processes Based upon Reversible Chemical Complexation”. Proc. Sym. Sep. Tech. National Taiwan Institute of Technology: Taipei, May 16-18, 1983; pp 139-143, 156. Levashova, L. B.; Darienko, E. P.; Degtyarev, V. F. J. Gen. Chem. USSR (Engl. Transl.) 1955,25,1025. Martin, A. J. P. Discuss. Faraday SOC.1949,7, 332. Meltzer, H. L. Fed. Proc. 1956,15,128. Meltzer, H.L. J. Biol. Chem. 1958,233,1327. Meltzer, H. L.; Buchler, J.; Frank, Z. Anal. Chem. 1965,37, 721.
Received for review June 19, 1987 Accepted December 2, 1987
GENERAL RESEARCH A Group Contribution Equation of State Based on the Simplified Perturbed Hard Chain Theory Gus K. Georgeton and Amyn S. Teja* School of Chemical Engineering, Georgia Institute of Technology, Atlanta, Georgia 30332-0100
A group contribution equation of state based on the simplified perturbed hard chain theory is proposed in this work. T h e resulting GSPHCT equation of state can successfully predict binary and multicomponent phase equilibria, using group interaction parameters obtained from a limited amount of mixture information. T h e method appears to incorporate the advantages of both the simplified perturbed hard chain theory and of the group contribution concept and can easily be extended to other equations of state.
A knowledge of phase equilibria over wide ranges of pressure and temperature is required in the design of separation processes. It is often necessary to predict such information using limited experimental data-and sometimes using no experimental data at all. In practice, this is often achieved via equation of state methods or via group contribution methods. Group contribution methods such as ASOG (Derr and Deal, 1969) or UNIFAC (Fredenslund et al., 1975) are particularly attractive when experimental data are lacking. However, these methods are limited to the prediction of low-pressure phase equilibria over narrow ranges of temperature (typically 273-423 K). In contrast, equation of state methods are, in principle, applicable over wide ranges of pressure and temperature. However, these methods require data for the pure components and for the mixture. The application of the group contribution concept to equations of state has the potential to overcome the limitations of both types of methods and is therefore 0888-5885/88/2627-0657$01.50/0
receiving increasing attention in the literature. A group contribution equation of state based on the simplified perturbed hard chain theory is proposed in this work. The perturbed hard chain theory (PHCT) was proposed by Prausnitz and co-workers (Beret and Prausnitz, 1975a,b; Donohue and Prausnitz, 1978) for the calculation of the fluid-phase properties of a wide variety of substances ranging in complexity from methane to polyethylene. Their equation of state, however, is complex because they used an expression for the attractive term based on the molecular dynamic studies of Alder et al. (1972). More recently, Kim et al. (1986) developed the simplified perturbed hard chain theory (SPHCT) by replacing the attractive term in the PHCT by a term obtained from a lattice model of Lee et al. (1985). The resulting equation of state is simple to use and is valid at all densities for molecules as diverse as methane and polyethylene. Moreover, the extension of the SPHCT to 0 1988 American Chemical Society
658 Ind. Eng. Chem. Res., Vol. 27, No. 4, 1988
mixtures requires relatively simple mixing rules as was shown by Kim et al. (1986) who proposed a set of mixing rules based on the segmental interaction of chainlike molecules. The success of the resulting equation for the calculation of pure component and mixture properties (PVT, phase equilibria) has led us to propose a group contribution version of the SPHCT. Since a large number of pure components and mixtures may be characterized by a small number of functional groups, this new development should reduce the need for an extensive data base of pure component constants and, in addition, for mixture data. Furthermore, the method proposed here for the SPHCT is, in principle, applicable to other simple equations of state. T h e Simplified Perturbed Hard Chain Equation of State Prausnitz and co-workers (Beret and Prausnitz, 1975a,b; Donohue and Prausnitz, 1978) have proposed the following generalized van der Waals partition function for chainlike molecules:
where N is the number of molecules in volume V at temperature T , A is the de Broglie wavelength, V , is the free volume available to the molecules, is the potential energy due to attractive forces, k is Boltzmann's constant, and 3c is the total number of external degrees of freedom. The quantity 3c arises from the factoring of the rotational and vibrational partition function into external and internal contributions and of the assumption that the internal contributions depend only on the temperature and therefore do not affect the equation of state. Using the Carnahan-Starling equation of state for hard spheres to obtain the free volume and an extension of the lattice model of Lee et al. (1985) to obtain the potential energy of attraction, Kim et al. (1986) derived the following equation of state from the generalized van der Waals partition function: (2)
where 47u*/u
Zw =
(1 -
- 2(7u*/u)2 TU*/U)3
z) 1
Y = exp( 2ck T
-
(3)
(4)
In eq 2-5, N Ais Avogadro's number, 7 is a constant equal to 0.7405,ZMis the maximum coordination number chosen to be 36 in the work of Kim et al., s is the number of segments in the molecule, q is the external surface area of the molecule, t is the characteristic energy per unit external surface area, and u is the molecular diameter. The equation of state thus contains three parameters, t q , us, and c, which are characteristic of the molecule (the products tq and us are considered here to be single parameters since these quantities always occur together). Because €4, us,and c are molecular parameters, it is plausible to assume that they may be obtained from the properties of the groups which comprise the molecule.
Table 1. Substance-Dependent Constants for the SPHCT Equation of State (Kim et al., 1986) compd W c , cal 1o1Oa, m C methane 1.1246 3.5719 1.0000 ethane 2.0944 4.0165 1.2586 4.3825 1.3941 propane 2.7415 1.6329 3.4909 4.6405 n-butane 4.9795 1.8393 n-pentane 4.1817 n-hexane 4.7681 5.2559 1.9649 5.4777 2.1990 n-heptane 5.4758 5.6072 2.3697 n-octane 6.1302 5.8904 2.6541 n-nonane 6.8909 6.1293 3.0246 n-decane 7.7947 n-tridecane 9.7235 6.6109 3.5703 7.2868 4.0036 n-hexadecane 11.269 4.2466 12.145 7.2090 n-heptadecane 7.5791 4.9108 n-eicosane 14.213 1.6359 4.6416 4.7094 benzene 1.8866 2.7506 3.2485 carbon dioxide 3.5407 1.9019 sulfur dioxide 3.9405 4.5516 2.4094 methyl acetate 5.3292 5.0016 2.4370 diethyl ether 4.8557
The major advantage of a group contribution method such as UNIFAC is, of course, its application to the calculation of the properties of mixtures. Kim et al. (1986) extended the SPHCT to mixtures of molecules using the following one-fluid mixing rules: (6)
( u * ) = cxiuii*
(c) = (cu*
(7)
CXiC[I
Y ) = ccxixjcipij*(ex.(
=) 2ciikT
- 1)
(8)
with the cross terms given by
The terms in angular brackets ( ) represent mixture properties. The extension of these mixing rules to solutions of groups is described below. Group Contribution Equation Based on SPHCT P u r e Components. As outlined in the previous section, the SPHCT equation contains three substance-dependent parameters: t, the intermolecular interaction energy per unit surface area; u, the diameter of a segment; and c, one third of the external degrees of freedom of the molecule. Associated with t and u are q, the external surface area of the molecule, and s, the number of segments in the molecule, respectively. The resulting terms, tq and us, always appear together and each can be treated as a single parameter. This can also be interpreted as setting q = s = 1,i.e., assuming that the molecule or group consists of one segment of unit area (see Kim et al. (1986)). The three constants, e, u, apd c, in the SPHCT equation of state have been determined for several compounds, and the values for these constants appear in Table I. These values for the n-alkanes are plotted versus the number of carbon atoms in the molecule in Figure 1. The near linear behavior of this correlation indicates that each segment of an n-alkane contributes almost uniformly to the molecular parameters, which implies that the hard core volume, characteristic energy, and the number of external degrees of freedom may be estimated by a combination of segment, or group, values of these parameters. The effectiveness of this type of group contribution technique has been
Ind. Eng. Chem. Res., Vol. 27, No. 4, 1988 659 Table 11. Group Constants for the GSPHCT Equation of State compd used group 1oz1Ckk lolook ck for evaluation 3.188 0.6293 ethane CHB 1.047 2.719 0.1766 n-hexane CH2 0.6629 3.572 1.000 methane CHI 1.125 ACH 0.7736 2.592 0.2727 benzene 3.248 1.887 carbon dioxide COZ 2.750 Table 111. Comparisons between Experimental and Calculated Values Using the GSPHCT Eauation of State % AAD pressure or substance T range, K P range, MPa vol propane 253-363 0.2-3.7 6.7 393-563 0.1-2.3 n-octane 9.7 473-613 0.2-2.0 n-decane 7.2 400-500 0.1 polyethylene 3.4
5
0
10
16
iI
20
c
0
I
I
I
5
10
15
0
6
0
20
Number of Carbons Figure 1. Pure component parameters for n-alkanes as a function of the number of carbon atoms in the n-alkane.
documented by Jin et al. (1986) for their GPSCT (Group Contribution Perturbed Soft Chain Theory) equation. Following the work of Jin et al. (1986), we have chosen to calculate the molecular Constants (tii,oii,and cii) for pure component i from group contributions as follows: Cii
= xNk(i)Ckk
(11)
= xNk(’)Skk3
(12)
CNk(i)Nm(i)EkmSkm3
(13)
k
Qi:
k
E 11..g. 11.3 =
k m
PROPANE N-OCTANE N-DECANE Qrouo SPHCT
0
with
In eq 11-15, Nk(i)represents the number of groups of type k in molecule i, and C k k , S k k , and Ekk are the constants of the equation of state for group k. Having defined group “combining rules” to estimate the molecular parameters, we can then regress the group constants for several representative groups from the pure component values. For several of the groups considered in this work, compounds containing only that group were identified, e.g., ethane, which contains 2CH3 groups. In some cases, the group was assumed to be identical with the molecule, as in the case of CH,. The parameters for these groups were
200
SO0
400
,
500
E00
’
Temperature (K) Figure 2. Comparison of experimental and calculated vapor pressures for several n-alkanes.
calculated from a knowledge of the molecular value and the number of groups. In some instances, a group did not exist by itself in a component. This is true of the CH2 group, characteristic of linear alkanes. The constants for this type of group were obtained by using a component containing one or more previously defined groups. A similar evaluation of one group at a time was also used in the determination of group parameters for UNIFAC (Fredenslund et al., 1975). The combining equations were written with only one unknown group and solved to give values for the unknown group parameters. Size and shape parameters for groups and energy parameters for the interactions of the same types of group were determined in this fashion using a knowledge only of a few equation of state constants. In this work we identified five groups (in order of regression)-CH3, CH2 (linear), CH4,ACH (aromatic), C02. Of these, only the CH2group did not entirely comprise a component and was evaluated using a compound containing a previously defined group. The calculated group constants and the compounds used to determine the values are shown in Table 11. It should be noted that only one compound (n-hexane) was used to determine the group constants for CH2. In general, several compounds should be used to determine the contribution of the repeating group in a homologous series such as the n-alkanes. It is possible that, as the
660 Ind. Eng. Chem. Res., Vol. 27, No. 4,1988
io0
410
440
,do
480
560
Temp (K) Figure 3. Comparison of experimental and calculated liquid densities for polyethylene.
number of repeating groups increases, the contribution of each group could depend on its position. Thus, normalization terms may have to be added, or an average value of the group parameter over several molecular lengths may be necessary. Jin et al. (1986) successfully employed the former method t~ obtain group parameters in their GPSCT equation. In our work, however, we have not used such an adjustment to account for the varying number of groups, and we have used data for n-hexane to estimate the CH2parameter values. Using these parameters, we have predicted the pure component vapor pressures of propane, n-octane, and n-decane. Figure 2 shows these predicted vapor pressures together with the experimental data from API Project 44. The reasonable comparison of the calculated vapor pressures with the actual data (see Table 111) indicates that our determination of average CH2parameter values can be used with confidence to predict vapor pressures to be used in VLE calculations. An additional consideration is the validity of the (repeating) group parameters for molecules containing a large number of these groups. This is of considerable importance in the application of the method to polymers. Average values of group parameters might not give an accurate representation of long molecules, and molecular parameters used to regress the necessary group information would most likely not be available. Many researchers have noted, however, that molecular parameters attain a constant value, or reach a state of constant change, as the chain length increases. This would imply that the contributions of the groups to the molecular parameters also attain a limiting value, and this limiting value could be used in predictions for large molecules. In order t o examine the applicability of such limiting values of parameters to polymers, the GSPHCT was used to calculate liquid volumes for polyethylene. The limiting value for uCH, was assumed to be that regressed from eicosane. Only the v parameter was obtained from the properties of eicosane because of its relation to the size of the molecule and thus to the volume of the fluid. A molecular weight of 26 OOO was assumed for polyethylene, and “experimental” liquid volumes were calculated by using the corresponding states correlation reported by Maloney and Prausnitz (1974) for polyethylene. (This correlation agreed with the experimental data for polyethylene of a number of workers to *1%.) Volumes were predicted with an AAD of 3.4% (Table 111) and are shown in Figure 3. It should be noted that group parameters calculated from pure component data may, in principle, be used to predict phase equilibria in mixtures. This was demonstrated by Jin et al. (1986) using their GPSCT method. However, their calculationswere limited to a small number
of mixtures, and it is likely that the properties of other mixtures would not be predicted with the same level of success. When conventional equations of state are used to correlate phase equilibria in mixtures, it is usual to introduce mixture information via binary interaction parameters in order to improve agreement between calculation and experiment. In an analogous manner, group parameters were obtained from a limited amount of mixture data as described in the next section. Mixtures. In conventional equation of state methods, mixture data are often regressed to obtain binary interaction parameters. These adjustable parameters greatly improve the fit of binary equilibrium data and may be used for the prediction of phase equilibria at other conditions and for multicomponent mixtures. However, these parameters have sometimes proved to be very specific to the conditions of the data set used in the regression. In addition, they must also be adjusted to improve the correlation of data for multicomponent systems. In order to introduce mixture information in the group contribution method, two group interaction parameters for unlike group interactions are proposed here. The underlying assumption is that groups will have a greater tendency to interact consistently in different surroundings than would molecules. The behavior of groups would thus be less sensitive to pressure, temperature, class of compounds in the mixture, and other factors. Thus, only a small amount of data would be needed to obtain the interaction between two groups, and the interaction would be assumed to be the same in all systems where the groups appear. The group interaction parameters were introduced as follows. The mixing rules of Kim et al. (1986) given by eq 6-8 were used to extend the group contribution equation of state to mixtures. These mixing rules require the molecular parameters eij and vi) The like parameters were obtained by using the group contribution equations (11)-(13), whereas the unlike parameters were obtained by using
where i and j refer to components i and j , and k and m refer to groups of type k and m. Skm is obtained by using eq 15. Note that eq 17 is also applicable when i = j (when it becomes identical with eq 13) and that it contains two parameters, Ekmand Emk,for interactions between groups m and k. In contrast to the pure component case when Emk = Ekm = (EmmEkk)1/2, Ekm and Emk were treated as adjustable parameters in mixtures. This has some parallel with the use of binary interaction parameters in equation of state methods (see, for example, the work of Panagiotopoulos and Reid (1986) in which two binary interaction parameters are used to describe deviations from the geometric mean rule). However, groups m and k in mixtures could arise from a single component, so that this approach also has some similarites with n-fluid theories, since each pure component in a mixture is replaced by some “reference component” (with Eii containing mixture information) and the “reference components” then combine according to the van der Waals one-fluid model. In our work, the adjustable parameters, Ekmand Emkr for each pair of groups were obtained by optimizing deviations in bubble point pressure and vapor composition for binary mixtures containing these groups. For each pair of groups, a representative binary mixture data set was chosen, covering as wide a range of temperature and pressure as possible. The binaries used to obtain the cross
Ind. Eng. Chem. Res., Vol. 27, No. 4, 1988 661 Table IV. VLE Correlation Using the GSPHCT Eauation of State4 binary system (conditions, T, K, and P, atm) groups n-butane-n-decane (344 < T < 511, P < 47) CH3, CHz methane-ethane (144 < T < 200, P < 50) CH3, CH4 ethane-benzene (313 < T < 513, P < 81) CH3, ACH carbon dioxide-ethane (223 < T < 293, P < 62) CH3, COz methane-n-pentane (278 < T < 383, P < 169) CH2, CHI CHZ, ACH n-hexane-benzene (298 < T < 350, P < 1.0) carbon dioxide-n-pentane (278 < T < 378, P < 95) CHz, CO2 methane-benzene (339 < T < 501, P < 326) CHI, ACH methane-carbon dioxide (200 < T < 272, P < 78) CHI, COz ACH, COZ carbon dioxide-benzene (298 < T < 313, P < 76)
%AP
AY
5.28 3.18 2.58 2.63 2.83 10.2 5.44 4.68 8.87 3.89
0.0071 0.0038 0.0161 0.0104 0.0174 0.0474 0.0281 0.0379 0.0125 0.0036
"All data from Knapp et al. (1982).
Table V. Group Interaction Parameters for the GSPHCT Equation of State 1OZ1Ek, group CH3 CHZ CH4 ACH COZ
CH3
CHZ
CH4
ACH
COZ
10'0Sk
Ck
1.047 0.2411 0.8045 0.2541 1.411
1.331 0.6629 1.858 1.006 1.576
1.361 -0.5176 1.125 -0.5628 1.313
1.377 0.5324 1.954 0.7736 0.6433
1.884 1.100 2.173 2.179 2.750
3.188 2.719 3.572 2.592 3.248
0.6293 0.1766 1.000 0.2727 1.887
4
:: METHANE 3
BENZENE
G r o w SPHCT .........................
a P)
a -:
3
? :
I
j N-BUTANE N-DECANE
0 0
...........
......................... I
1
8
15
* i d 0
22
,
28
I
36
,,.'
1
43
Pressure (atm) Figure 4. Comparison of experimental and correlated K values for the butane-n-decane system at 444.3 K.
terms for each pair of groups, and the deviations associated with each optimization, are given in Table IV. The complete parameter matrix and group constants for the group contribution SPHCT for mixtures are given in Table V. Results and Discussion A comparison of calculated VLE for the SPHCT and group SPHCT (GSPHCT) is shown in Figures 4 and 5. For the n-butane-n-decane system, the SPHCT equation correctly predicted two phases for all data points but underpredicted the experimental K values and the bubble point pressure. With the introduction of the mixture data in the GSPHCT, however, the experimental points could be correlated with greater accuracy. For the more nonideal methane-benzene system (Figure 5), the SPHCT did not predict two phases above 130 atm, whereas GSPHCT extended the two-phase region to 300 atm and more precisely reproduced the experimental K values. For the methane-benzene system, the maximum pressure was 326 atm. The GSPHCT correlated this wide range very well with only 4.68% average deviation in bubble point pressures and only 0.038 average deviation in K values. This reinforces the conclusion of Kim et al. (1986) that the SPHCT
7
73
138
205
271
337
403
Pressure (atm)
Figure 5. Comparison of experimental and predicted K values for the methane-benzene system at 338.7 K.
can be used for wide ranges of conditions. In addition, the inclusion of mixture data in the form of adjustable parameters further extends the applicable range of conditions. It should be noted that the improvement realized by using group interaction coefficients obtained from mixture data is not unexpected. Many researchers have found that both pure component and mixture data are necessary to accurately reproduce multicomponent system behavior. The major achievement of the GSPHCT is, however, its ability to predict phase equilibria in mixtures for which there are not experimental data. This is discussed below. A severe test of the model is its ability to predict VLE for binary and multicomponent systems using the group parameters obtained from the regression of data on other binary systems. A comparison of the results presented in Tables IV and VI shows that average deviations between calculation and experiment are of the same magnitude for both correlation and prediction. The propane-n-hexane system shown in Figure 6 and the methane-n-heptane system in Figure 7 are representative of VLE calculations using the GSPHCT. Also shown in Figures 6 and 7 is a comparison with calculations using the SPHCT with no
662 Ind. Eng. Chem. Res., Vol. 27, No. 4, 1988
e, PROPANE 3 N-HEXANE Group SPHCT ......SPHCT .........
CARBON DIOXIDE PROPANE QtouD SPHCT ........SPHCT ....................... .I
7
I 13
I
I
20
27
0
I
I
34
41
48
56
Pressure (atm) Figure 6. Comparison of experimental and predicted K values for the propane-n-hexane system at 433.2 K.
i;
4
10
28
19
37
: :
METHANE N-HEPTANE Group SPHCT ............PHCT .............
46
86
48
3
3
0
.>
130
172
214
-
256
Pressure (atm) Figure 7. Comparison of experimental and predicted K values for the methane-n-heptane system at 344.3 K.
mixture information. The SPHCT was not able to completely predict the two-phase region for these and several of the correlated binaries. The ability of the GSPHCT to more closely approximate the two-phase regions supports the assumption that groups will tend to interact in a consistent manner regardless of the system in which they appear. This is further demonstrated in the case of the C02-propane system shown in Figure 8. The GSPCHT was able to estimate the two-phase region well for this system. The GSPHCT was also used to predict multicomponent VLE for systems containing the groups CH3, CH2, and CH,. The deviations in calculated pressure and K values were of the same order as the deviations for the correlated binaries. Figure 9 is representative of the multicomponent systems studied. The SPHCT without binary interaction parameters was unable to correctly predict the two-phase region due to the lack of mixture information. The effect of using the group interaction parameters outside the temperature and pressure range of correlation is shown in Table VI. For these systems, the CH3-CH2 interactions were obtained in the range 344-511 K and pressures up to 47 atm. The range of conditions for sys-
64
56
Pressure (atm) Figure 8. Comparison of experimental and predicted K values for the carbon dioxide-propane system at 294.3 K. METHANE ETHANE PROPANE N-PENTANE
1 s
37
60
io1
138
186
197
2 !O
Pressure (atm) Figure 9. Comparison of experimental and predicted K values for a multicomponent mixture at 366.5 K.
tems 2 and 3 only partially falls into this range, and as the overlap area decreases, the deviations from the experimental values increase. A temperature dependence for the group interaction parameters and a normalization of sizes would reduce this uncertainty due to extrapolation. These enhancements are discussed further in the following section.
Further Improvements Several improvements can be made to the GSPHCT to further enhance the prediction of phase equilibria. Improvements in predictions may be possible by using normalized group constants, temperature dependence of group constants, and /or regression of specific types of data. In applications where members of homologous series are present, one or more groups may be repeated very often, e.g., the CH2 group in an n-alkane. The contribution of each CH2in a shorter molecule may very well be different from the contribution of CH2groups in a longer molecule. Thus, some form of normalization is necessary to prevent
Ind. Eng. Chem. Res., Vol. 27, No. 4, 1988 663 Table VI. V L E Predictions Using the GSPHCT Equation of State T, K P, atm 373-453 17-48 1. propane n-hexane 278-411 3-61 2. ethane n-pentane 4-52 298-373 3. ethane n-octane 393-473 20-48 4. propane n-heptane 14-204 methane 311-444 5. n-heptane propane benzene carbon dioxide propane methane propane n-butane methane ethane propane n -p entane n-hexane n-decane methane ethane propane n-pentane n-heptane n-decane methane ethane propane n-butane n-pentane
6. 7. 8. 9.
10.
11.
Table VII. System T,K 378 411 444
9.23 5.27 1.28
AY 0.0019 0.0024 0.0033
T,K 478 511
2.53 8.52 18.8 4.86 5.59
378-444
5-58
3.17
278-328
10-61
3.34
311
50-120
5.48
339-394
7-204
8.66
366
17-206
3.36
311
35-118
2.67
VLE Calculations for the n -Butane-n -Decane
%AP
%AP
%AP 2.90 7.04
AY 0.0099 0.0207
an over- or underestimation of a particular parameter. The q and s terms in the SPHCT can be used in a similar manner to normalize group interactions and group size. Several values may be obtained for different sized molecules and these values correlated, thus giving a more accurate description of the molecular constants. Another possible improvement in the method would be to introduce temperature dependence for the group interaction parameters. Table VI1 presents the results of our calculations for the n-butane-n-decane system. Data at all temperatures were regressed to obtain the CH3-CH2 group parameters. As can be seen in Table VII, deviations in the bubble point pressure are largest at the extremes of the temperature range and smallest toward the middle of the range. This suggests that average values for the CH,-CH2 interaction are obtained when the entire range of data is used in the regression. Temperature-dependent group interactions would help to eliminate the disparity at the temperature extremes. The smaller "end effects" would lead to more confident extrapolation outside this range, a problem realized above in the prediction of multicomponent equilibria. Conclusions We have shown that a group contribution version of the simplified perturbed hard chain theory (termed the
AY 0.0225 0.0225 0.0325 0.0325 0.0070 0.0070 0.0221 0.0221 0.0108 0.0108 0.0159 0.0159 0.0276 0.0276 0.0156 0.0031 0.0125 0.0184 0.0062 0.0013 0.0074 0.0039 0.0075 0.0048 0.0066 0.0012 0.0043 0.0021 0.011 0.0076 0.0076 0.0035 0.0053 0.0064
ref Knapp et al., 1982 Knapp et al., 1982 Knapp et al., 1982 Knapp et al., 1982 Knapp et al., 1982 Knapp et al., 1982 Knapp et al., 1982 Rigas et al., 1959 Vairogs et al. 1971
Yarborough, 1972
Hanson and Brown, 1945
GSPHCT equation of state in this work) can successfully predict binary and multicomponent phase equilibria, using group interaction parameters obtained from a limited amount of information. The method appears to incorporate the advantages of both the simplified perturbed hard chain theory and of the group contribution concept and supports our initial assumption that pairs of groups interact in a uniform manner in different mixtures. Because of the availability of only a limited number of pure component constants for the SPHCT equation of state, we have derived group interaction constants for only a small number of groups. The method we have developed is, however, applicable to other equations of state and we plan to demonstrate its application to simple cubic equations (for which constants are available for many substances) in another paper. Acknowledgment
G.K.G. thanks the Georgia Mining and Mineral Resources Institute for financial support in the form of a fellowship during this project. Literature Cited Alder, B. J.; Young, D. A.; Mark, M. A. J. Chem. Phys. 1972, 56, 3013. Beret, S.; Prausnitz, J. M. AZChE J. 1975a, 21, 1123. Beret, S.; Prausnitz, J. M. Macromolecules 1975b, 8, 878. Derr, E. L.; Deal, C. H. Znst. Chem. Eng. Symp. Ser. 1969,3(32),40. Donohue, M. D.; Prausnitz, J. M. AZChE J. 1978,24, 849. Fredenslund, A.; Jones, R. L.; Prausnitz, J. M. AZChE J. 1975,21, 1086. Hanson, G. H.; Brown, G. G. Znd. Eng. Chem. 1945,37, 821. Jin, G.; Walsh, J. M.; Donohue, M. D. Fluid Phase Equilib. 1986,31, 123.
Ind. Eng. Chem. Res. 1988,27, 664-671
664
Kim, C. H.; Vimalchand, P.; Donohue, M. D.; Sandler, S. I. AZChE J. 1986,32, 1726. Knapp, H.; Doring, R.; Oellrich, L.; Plocker, U.; Prausnitz, J. M. VapopLiquid Equilibria for Mixtures of Low Boiling Substances; DECHEMA Chemistry Data Series; DECHEMA: Frankfurt, 1982;Vol. VI. Lee, K. H.; Lombardo, M.; Sandler, S. I. Fluid Phase Equilib. 1985, 21, 177. Maloney, D. P.;Prausnitz, J. M. J. Appl. Polymer Sei. 1974,18, 2703.
Panagiotopoulos, A. Z.; Reid, R. C. ACS Symp. Ser. 1986,300,571. Rigas, T.J.;Mason, D. F.; Thodos, G . J . Chem. Eng. Data 1969,4, 201. Vairogs, J.; Klekers, A. J.; Edmister, W. C. AZChE J. 1971,17, 308. Yarborough, L. J . Chem. Eng. Data 1972,17,129. Received f o r review June 17,1987 Revised manuscript received November 3, 1987 Accepted November 20, 1987
Thermodynamic Excess Properties in Binary Fluid Mixtures Keshawa P. Shukla, Ariel A. Chialvo, and James M. Haile* Department of Chemical Engineering, Clemson University, Clemson, South Carolina 29634
This paper illustrates how a plot of the excess Gibbs free energy versus excess enthalpy can be used to organize our knowledge of the nonideal solution behavior in binary liquid mixtures. Such a plot is particularly suited to presenting temperature effects, and we therefore use the plot to show (a) how temperature changes influence the strength and kind of nonideal solution behavior and (b) how temperature changes promote various kinds of liquid-liquid immiscibility. We also show that such a plot can be used to quantify how changes in intermolecular forces affect nonideal solution behavior. T o accomplish this, statistical mechanical perturbation theory calculations of the excess free energy and excess enthalpy were performed on a number of Lennard-Jones mixtures. The calculations were performed at three state conditions and over a range of size ratios 1 I g g g / g u I 2 and over a range ~ 4. The results demonstrate how energetic and entropic effects can of energy ratios 0.2 I E B B / E I either compete or cooperate to produce particular kinds of nonidealities. 1. Introduction
The myriad varieties of thermodynamic and phase equilibrium behavior displayed by fluid mixtures present a significant challenge to the organizational concepts of classical and statistical thermodynamics. Attempts to organize the observed equilibrium behavior of fluid mixtures have been couched in both molecular and thermodynamic terms. Molecular schemes classify mixtures according to the kinds of intermolecular forces acting among the constituent molecules (see, e.g., Rowlinson and Swinton (1982)). Thermodynamic schemes focus on a progression in the behavior of particular macroscopic properties. For example, a popular thermodynamic classification scheme is that of Scott and van Konynenburg (1970, 1980) who divide binary mixtures into six primary classes, according to the behavior of mixture critical lines in pressure-temperature space. Another system of thermodynamic classification can be based upon the thermodynamic excess properties. For a generic, extensive thermodynamic property M , the excess property ME is defined by M E = M - Mis (1) where M is the mixture property at a chosen temperature, pressure, and composition and Misis the ideal solution value of M at the same state condition. The appeal of (1) is that M" contains ideal gas contributions plus entropic contributions due to the distinguishability of molecular species, so that the excess property ME contains only contributions originating from differences in intermolecular interactions. As far as we are aware, the first systematic classification of binary mixtures based on excess properties was made by Malesinski (1965). Subsequently, the approach has evolved along the lines of fiding the most informative pair of excess properties to plot on a set of two-dimensional, orthogonal axes (Kauer et al., 1966). This evolution culminated in the work of Gaube and his colleagues (Gaube and Koenen, 1979; Kohler and Gaube, 1980; Koenen and 0888-5885/88/2627-0664$01.50/0
Gaube, 1982) who collected from the literature excess property data for about 200 equimolar binary mixtures and presented the data on a plot of g E versus hE. Here g E is the excess Gibbs free energy and hE is the excess enthalpy. The gE-hE diagram embodies, in a particularly informative way, the 12 classes of binary mixtures identified by Malesinski. In this paper we attempt to provide a comprehensive summary of the kinds of mixture behavior that can be deduced from such a diagram. In the next section we describe the g E-hEdiagram, including the mixture characteristics represented by its principal features. We then show the kinds of trajectories that a particular mixture can follow on the diagram in response to a change in temperature. The analysis of temperature effects leads to consideration of liquid-liquid immiscibility and the g E-hEdiagram proves particularly suited to describing the behavior of binary mixtures that exhibit either closed solubility loops or miscibility gaps. In the last section we present gehE diagrams for mixtures of Lennard-Jones (U) atoms, for which we have calculated the excess properties via statistical mechanical perturbation theory. The diegrams resulting from those calculations provide new information on how changes in the LJ size and energy parameters affect excess properties in simple mixtures. 2. The g E-hE Diagram Classical thermodynamics gives the following fundamental relations among the excess properties for closed systems: gE = hE - TsE (2) sE = - ( a g E / a n p ,
(3)
hE = - T ~( a ( g E / T ) / a n p X (4) cpE= ( a h E / a q p X= T ( a S E / a n p , = -T ( a 2 g E / a T 2 ) , (5) In these equations Tis the absolute temperature, s is the 0 1988 American Chemical Society
