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A simple group contribution equation of state for fluid mixtures
Chemical
Enyineering
Science,
Vol.
44,
No.
11, pp.
2703-2710,
1989.
OOO%2509/89
Printed in Great Britain.
0
A SIMPLE
GROUP
CONTRIBUTION EQUATION FOR FLUID MIXTURES
1989
$3.00+0.00
Pergamon
Press
plc
OF STATE
GUS K. GEORGETONt and AMYN S. TEJA* School of Chemical Engineering, Georgia Institute of Technology, Atlanta, GA 30332-0100,
U.S.A.
(Receiued 22 July 1988: accepted 22 February 1989) Abstract-A new group contribution equation of state based on the PateI-Teja (PT) equation is presented in this paper. The new equation retains the functional form (cubic in volume) of the PT equation, but uses group additiviry mixing rules for the equation of stale constants. Group interaction parameters in the new equation were regressed from a limited amount of mixture data. The equation of state was then used to predict VLE for a number of binary and multicomponent mixtures. The predictions were shown to be as good as, or better than, predictions using the PT equation with conventional mixing rules.
INTRODUCTION
The prediction of phase equilibria is most commonly accomplished using equations of state or group contribution methods. Equations of state have the ability to represent the volumetric properties of pure fluids over wide ranges of temperature and pressure, and can be extended to mixtures by the use of mixing rules for their constants. A major limitation of the equation of state approach, however, is that these mixing rules must often incorporate experimental information for each mixture. Group contribution methods, on the other hand, are able to represent a large number of substances (and mixtures) by a much smaller number of structural units, and therefore require only a limited experimental data base. However, these methods are usually restricted to small ranges of temperature and pressure. An equation of state based on group considerations offers one way to combine the advantages of the two approaches. Several such equations have been proposed recently (Skjold-Jorgensen, 1984; Majeed et al., 1984; Jin et al., 1986). As with conventional equations of state, the applicability and accuracy of the new group contribution equations depends strongly on the “mixing rules” used to obtain the constants. A particularly simple way to obtain a group contribution equation of state is to use group additivity to estimate the constants in a conventional equation of state. Jin et a[. (1986) recently used this approach to extend the Perturbed Soft Chain Theory (PSCT) equation of state. The resulting GPSCT equation retained all the features of the PCST, including the simple (van der Waals) mixing rules. We have recently proposed (Georgeton and Teja, 1988) a group contribution equation of state based on the Simplified Perturbed Hard Chain Theory (SPHCT) which uses similar ideas, with the important difference that interactions between groups in different molecules are
obtained from mixture data. Although this has some similarity to the introduction of binary interaction parameters in conventional mixing rules, a large number of binary mixtures can be represented by a much smaller number of groups. The basic assumption is that groups interact in the same way in different surroundings. Hence, only a limited mixture data base is necessary to obtain the group interaction parameters. The SPHCT equation can only be used for nonpolar fluids and therefore, the GSPHCT equation of state was limited in its application to such fluids. We describe below an extension of our work to the cubic equation of state proposed by Pate1 and Teja (1982). The choice of the equation of state is arbitrary and a matter of convenience. The method developed, however, can be used with other equations of state.
THE PATEL-TEJA
EQUATION
OF STATE
The cubic equation of state proposed by Pate1 and Teja (1982) is outlined briefly below. This equation has been shown to work well in the calculation of phase equilibria and thermodynamic properties of nonpolar and polar fluids (Georgeton et al., 1986) The Patel-Teja (PT) equation has the form PI----
aCTI
RT v-b
u(u + b) + c(u -b)
(1)
where a[T]
=Cz,
R2 7-T
----CT1 p,
b&&F
c RTc c=R,p,
(2) (3)
(4) (5)
‘Present address: DuPont Company, Savannah River Plant, Aiken, SC 29808, U.S.A. ‘Author to whom correspondence should be addressed. CES 44:11-s
(6)
Gus
2704
K. GEORGETON
and R,, R,, Q, are constants, obtained by the application of the critical constraints to the equation of state (Pate1 and Teja, 1982). The equation contains four substance-specific constants-the critical temperature T,, the critical pressure P,, the critical compressibility L&,and the slope F of the vapor pressure function. For nonpolar fluids, F and c, can be related to the acentric factor (Pate1 and Teja, 1982) and the PT equation uses the same input information as other cubic equations (i.e. T,, P, and w). These correlations do not, however, hold for polar fluids. We have therefore developed separate correlations for classes of polar fluids (Georgeton et al., 1986) based on the effective carbon number (ECN) concept and the properties of the n-alkanes. These correlations are summarized in Table 1. Equation (Ia) in Table 1 gives the relationship between carbon number and boiling point for the n-alkanes. It can be used to calculate the effective carbon number (ECN) of any nonpolar substance given the normal boiling point of that substance. Amborse and Sprake (1970) and Willman and Teja (1985) have shown that the properties of nonpolar substances such as benzene can be obtained from the properties of an nalkane having the same number of carbon atoms as the ECN of benzene. It should be added here that the ECN of benzene is not necessarily integral. Equations (IbHIc) in Table 1 relate the constants F and 5, of the PT equation to the ECN of any nonpolar substance. The relationships between T,, P, and ECN have been presented elsewhere by Willman and Teja (1985). Since polar fluids do not follow the n-alkane correlations, separate correlations were developed for F and [, for classes of polar fluids (e.g. alcohols). These correlations were derived in terms of the differences between F and c, of the polar substances and those of their n-alkane homomorphs. (The homomorph of a polar compound is assumed to be the n-alkane obtained when the polar functional group is replaced by a CH, group.) Equations (Id) and (Ie) Table 1 represent the correlations for the correction terms + and 8 defined by (7)
where the subscripts denote the properties of the polar fluid and its homomorph. As an example, the PT equation of state constants for 2-butanol would be. calculated as follows:
Table W
ECN 3’ k”a15 8?s.lc
1. Correlation X
Th ECN ECN ECN, ECN,
and AMYN
(9 The
S.
TEJA
homomorph
(ii) (iii)
(iv) (v)
amix =
bmi,=
is found
by
re-
(9)
xi xjaij
j=l
-f
xi bi
(10)
xici
(11)
i=l nc
c,ix=
c
i=l
where the summations are over the nc components. In conventional equation of state calculations the cross-interaction term aij in eq. (9) is obtained from the combining rule: aij = kij&
(121
where kij is an adjustable binary interaction parameter. kij is generally obtained by optimization of mixture VLE data, and is often specific to the data set from which it is determined. Values of ki, obtained from experimental data are often temperature-dependent and sometimes composition dependent. It should be noted that the geometric mean rule for aii was proposed by van der Waals for his equation of state. However, the constant a in the van der Waals equation is temperature-independent whereas that in the PT equation is temperature dependent. Further-
A + BX + CX2
B
-0.11630
-0.019376
1.1596 x lo--’
-1.5491
x 10-7
-0.23378 0.21964 - 2.5033 - 0.94677
-0.42953 -0.075188 - 1.4914 -00.44082
3.7077 5.1818 3.2209 6.6923
- 1.9514 -2.4477 3.2067 1.3956
x x x x
D
10-2 10-3 lo-’ lo-’
f$ F i=l
A
x x x x
2-butanol
This approach therefore allows the constants of the PT equation to be obtained from a knowledge only of the boiling point of the substance of interest. [The relationship between T,, P, and ECN presented by Willman and Teja (1985) can be used to obtain T,, P, of 2-butanol. Alternatively, the actual T, and P, of 2butanol can be used in the calculations.] Equation (1) is extended to mixtures using the van der Waals mixing rules:
of the constants of the PT equation of state. W= c
of
the OH group by a CH, group. Thus. the homomorph of 2-butanol is 2-methyl butane, which has a normal boiling point of 301 K. The ECN of 2-methyl butane is calculated to be 4.756 using eq. (Ia). F and c, for Z-methyl butane can now be calculated using eq. (Ib) and (1~). (FH = 0.7304 and &, = 0.307.) + and 8 of 2-butanol can now be calculated using eq. (Id) and (Ie). (JI = 0.7011 and 8 = 0.0906.) Finally, F and [, for 2-butanol can be calculated using eqs (7) and (8) leading to F= 1.2425 and <, = 0.307. placing
+ DX’
E
10-a lo-=’ 10m3 1W3
1.3513 x lo-” 3.8572 4.5070 -2.3365 -8.4516
x x x x
lo-’ 10-e 10m4 lo-’
+ EX4
+ FX1tz F
Equation
0.19102
Ia Ib IC ld le
1.0815 0.16208 4.2844 1.2355
Group contribution equation of state for fluid mixtures more, an assumption imphcrt in the geometric mean rule is that the temperature dependence of the i-j pair is the same as that of the i-i and j-j pairs. For dissimilar molecules, this is obviously not true as is evidenced by reported ki, values which deviate from unity and change with temperature. The group contribution equation of state developed in this work retains the van der Waals mixing rules [eqs (9)--(11)] in the calculation of the properties of mixtures. However, the cross-interaction term ajj is obtained from the unlike group-interactions in mixtures and not from eq. (12), as is explained further below.
DEVELOPMENT PATEL
OF
THE
-TEJA
GROUP
(GPT)
CONTRIBUTION
EQUATION
The first step in the development of the GPT equation, as shown in the development of GPCST equation (Jin et al., 1986) and GSPHCT equation (Georgeton and Teja, 1988), is the inclusion of group additivity rules for the constants of the equation. The PT equation contains four constants T,, P,, <, and F. Group additivity rules for T, and P, have been proposed in the literature (e.g. Lydersen, 1955). However, there is less justification for such rules for F and possibly for c,. On the other hand all four constants T,, P,, (, and F can be expressed in terms of the ECN and therefore of the normal boiling point Tb. There is some evidence in the literature that r, is a function of the groups which comprise the molecule of interest. A group additivity rule for the ECN is therefore proposed as fohows:
(15) of group k when it is in where Ed,,, is the contribution the presence of a group m, and XNp’ is the total number of groups in the molecule i. Similarly, for the case when i # j ECNij is expressed as follows:
Nf' Nz' ‘=’
ECNii=
;
E,_
a~,=’
(,z, Nf) +,$, N:r’>
(16)
The denominators in eqs (15) and (16) may be viewed as normalization factors to account for the number of groups present in the system. The evaluation of group parameters using these additivity rules is described below. Additional details are also given in a previous paper (Georgeton and Teja, 1988). EVALUATION
OF
GROUP
PARAMETERS
The evaluation of the group parameters was carried out in two parts: (i) the .slrLparameters were first obtained from pure component values, and (ii) the E,_,, parameters were obtained from mixture data. In both
cases, the parameters time. To demonstrate
2705
of one group
the approach,
were evaluated the evaluation
at a of the
Ear parameters will be described for CH,-CH, and CH,-CH, interactions. The CH,-CH, group interactions were evaluated from the ECN of ethane. Ethane consists of two CH, groups and has a carbon number (or ECN) of 2. Application of eq. (15) therefore yields an +,. ,CHs of 1.0. This value can now be used to using the known carbon number (or obtain ECH,ICH~ ECN) of a molecule which contains both CH, and CH, groups. n-Pentane was used in the present study. n-Pentane has an ECN of 5 and contains 2 CH, groups and 3 CH, groups. Application of eq. (15) of 1.0 once again. Note that the yields an +H2/CH2 evaluation now requires a knowledge of the E~~,,,.-.-~ interaction in the n-pentane molecule. This interaction was obtained using the geometric mean rule e;c*, = ./z
(17)
where the asterisk denotes an unlike interaction in a pure component. This procedure was followed in the evaluation of seven group interactions (CH,, CH,, CH,, CO,, ACH, AC and OH) in the present study. Values of the group interactions (Ebb)are given in Table 2. Note that CH, and CO, were treated as single groups and their group mteractrons Ebb were obtained directly from their ECNs (or boiling point in the case of CO,). The group parameters in Table 2 were obtained using only one compound per group. For repeating groups that appear in homologous series, e.g. ACH in aromatics, several compounds should generally be used to determine the contribution of the repeating group. Thus, instead of using n-pentane to evaluate the CH,/CH, could have hexadecane.
interaction as described above, we used say n-pentane, n-decane and nThis would average out the effect of
position of the group in the chain. Alternatively, some type of normalization could be used. However, in this study, only one substance was used in the evaluation of each group parameter. The interaction parameters for unlike groups were determined as follows. The Q,,, values when k # m were obtained by optimizing bubble point pressures of one mixture containing the groups k and m. Thus for example the unlike CH,-CH, interaction (~~u,,~u~) was obtained using bubble point data for the Table 2. Group constants for the GPT equation Compound Group CH, CH, CH, ACH AC CO, OH
Ekk 1 1 1 1.063 1.193 2.223 5.180
used
for evaluation ethane n-pentane methane (assumed) benzene m-xylene carbon dioxide I-butanol
2706
Gus
K. GEORGETON and AMYN S. TEJA
propane-n-decane system. This system includes CH,-CH, and CH,-CH, interactions. CH,-CH,, The like-interactions (elk) were obtained from pure component ECNs as described previously. The unlike interactions (sXmrk # m) were treated as adjustable parameters in the correlation of bubble point pressures. By analogy with the temperature dependence of kij, a temperature dependence for Ed,,,was assumed as follows:
(13) This form was chosen so that Ed, would asymptotically reach a limiting value at higher temperatures. Also, Q,,, was set equal to E,~ and the optimization of EL2 and Ev,’ was constrained in order to obtain positive values. In addition, the value of E$z for k = m was set equal to zero so that like-group contributions would be temperature independent.
Values of the parameters EiO,’and EL: for each pair of groups were obtained by optimizing deviations in bubble point pressure for binary mixtures containing these groups. For each pair of groups, a representative binary mixture data set was chosen covering as wide range of temperature and pressure as possible. The binaries used to obtain the group parameters and the average absolute deviations in bubble point pressures and vapor compositions associated with each optimization are given in Table 3. The complete parameter matrix for the groups in the GPT equation is given in Table 4. It should be noted that all EL:’values were set equal to zero as described above, with the exception of the value for the OH-OH interaction. It was found necessary to set E&?,,,, = 2025.0 to account for the effects of hydrogen bonding on the vapor pressure (and hence ECN) of alcohols. Bubble point pressures of alcohol mixtures could be correlated well using this value for E&&OH (Table 3).
Table 3. VLE correlations using the GPT equation of state
Groups
System conditions (T/K
CH,-CH, CH,-CH, CH,-CH, CH,-ACH CH,-ACH CH,-ACH ACH-AC CH,-AC AC-CH, CH,-AC CH,-CO, CH,-CO, CH,-CO, CO,-ACH CO,-AC CH,-OH CH,-OH OH-OH OH-ACH OH-AC
and P/MPa)
propane-n-decane (278< T< 511, P<6.89) methanexthane (191 < T< 283, P < 5.47) methane-nlheptane (311 < T
AP%’ GPT PT
GPT
AY’
PT
2.86
3.30
0.0034
0.0051
1.35
1.29
0.0075
0.0064
2.58
2.50
0.0120
0.0094
2.30
1.76
0.0055
0.0054
1.55
4.21
0.0153
0.0175
2.69
5.25
0.0340
0.027 1
0.86
0.77
0.0050
0.0035
0.45
0.88
0.0115
0.0126
0.70
1.18
0.0099
0.0096
2.46
4.73
0.0186
0.0265
1.64
1.41
0.0188
0.0140
7.01
0.55
0.0199
0.0059
2.34
2.04
0.0099
0.0087
2.17
1.87
0.0017
0.0016
5.96
5.41
0.0072
0.0056
11.0
12.8
0.0930
0.0860
4.67
4.34
0.0123
0.0109
2.01
2.06
0.0098
0.0036
4.20
5.8 1
0.0430
0.0423
17.8
3.82
0.0663
0.0148
+AP% = average absolute percent deviation in bubble point pressures. $Ay = average absolute deviation in vapor mole fractions.
Group contribution equation of state for fluid mixtures Table 4. Optimized
group parameters for the GPT
2707
equation OH
CH,
Cl-l*
CH,
ACH
AC
CH, CH, CH, ACH AC CO, OH
1.000 1.225 0.7510 1.061 0.4563 0.9401 3.262
1.225 1.000 0.8325 0.9308 1.057 0.8962 0.1806
0.7510 0.8325 1.000 0.4650 1.124 0.8874
1.061 0.9308 0.4650 1.063 1.126 1.212 1.433
0.4563 1.057 1.124 1.126 1.192 0.9622 0.0141
0.9401 0.8962 0.8874 1.212 0.9622 2.223
E:fJ
CH,
CH,
CH,
ACH
AC
CO,
10.916 15.000 11.614
14.608 14.776 17.065
30.669 36.000
4.866 0 17.893 0.0156
20.548 17.893 0 -
35.284 0.0156
b;z
CH, z?
0 27.878. 14.183
14.183 28.292 0
27.878 28.292 0
21.641 183.33 11.717
AC& AC CO, OH
21.641 10.9 16 14.608 30.669
11.717 11.614 17.065 36.000
183.33 15.ocul 14.776
0 4.866 20.548 35.284
The group interaction constants given in Table 4 can be used to obtain the ECNs and hence the constants of the PT equation of state for any pure substance or mixture. The alkane correlations in Table 1 must be used when the components of interest
CO,
3.262 0.1806 1.433 0.0141 5.180 OH
2025.0
10 0
PraLmne
0
n-Deco”e Group
PT
Fmtel-Te,a _-_---_____
iL
are non-polar and the alcohol correlations via the homomorph concept when the components are alcohols. In the case of a mixture of a nonpolar substance and an alcohol, the ECN of the homomorph corresponding to the nonpolar-polar interaction is required. This is obtained by subtracting 2.7 from the value obtained using eq. (16). The value of 2.7 was obtained by considering the ECN of n-butanol, which is 7.7. The homomorph of n-butanol is n-pentane with an ECN of 5, which is 2.7 less than the actual ECN of the polar n-butanol. Hence 2.7 is subtracted from the ECN of the polar-polar and polar-nonpolar interactions calculated using eq. (16). This gives the ECN of the homomorph. The equation of state constants can now be calculated using the equations presented in Table 1 as described previously.
0
I
I
I
I
I
I
0
I
2
3
4
5
Pressure(MPd
RESULTS
Fig.
1.
Experimental and correlated K-values propane-n-decane system at 378 K.
for
the
A comparison of calculated VLE for the PT and GPT (group contributions version of PT) equations is shown in Figs 1 and 2. The calculated values from the PT equation were obtained with optimal binary interaction coefficients (kii). The propane-n-decane system shown in Fig. 1 is representative of the results obtained for the systems used to determine the group parameters. Both the PT and GPT equations correlate the data extremely well. In general the deviations between correlation and experiment are very similar for the two equations. In these comparisons, the GPT equation may be considered to be equivalent to the PT equation with a temperature dependent binary interaction parameter. The addition of an adjustable parameter would be expected to improve the representation of VLE. However, the improvement is marginal for systems such as methane-benzene (shown in Fig.
2), and it is sometimes worse as in the case of the CO,-ethane system (Table 3). The GPT equation was also used to predict VLE for binary and multicomponent systems, and comparison with experimental values are shown in Table 5. Also shown are the comparisons of predictions with the PT equation (kii = 1). It is obvious that the GPT equation can predict VLE as well as the PT equation for many mixtures and therefore it can be used for systems where no data or substance-specific equation of state constants are available. For nonpolar systems, deviations between calculated and experimental K-values using the GPT equation in the correlative and predictive modes were of the same magnitude (as can be seen from Tables 3 and
2708
Gus K. GEORGETON and AMYN
5). Figure 3 shows VLE predictions for the methane-n-pentane system. Both the PT and GPT equations predict the bubble point pressure and the K-values very well, with no binary interaction parameters. In order to be useful in calculating mixture data, the GPT must at the very least give results similar to the PT equation with kij = 1. This can be seen in Fig. 4 for the methane-1-methylnaphthalene system. The PT equation predicts pressures and Kvalues very well as does the GPT equation. The multicomponent systems shown in Figs 5 and 6 are also well predicted by both equations. A more severe test of the GPT equation is the modeling of systems containing nonhydrocarbon or polar compounds. VLE for several mixtures containing CO, were predicted with the GPT and the PT equations, and the GPT was shown to represent the
0 Methone ” Benzene GW”DPT Potel-Te,o --------10
I K
I5 Pressure
Fig.
2.
22
29
36
(MPo)
Experimental and correlated K-values methane-benzene system at 339 K. Table 5. Comparison
Ethane/n-decane Ethane/n-pentane Ethane/n-hexane Ethane/n-octane Propane/n-hexane Propane/n-heptane n-Butane/n-decane n-Butane/n-hexane n-Pentanejn-octane Methane/propane Methane/n-pentane Propane/benzene n-Pentanelbenzene Benzene/n-octane Benzene/toluene m-Xylene/n-decane n-Heptane/toluene n-Octane/p-xylene Toluene/n-octane Toluene/n-decane Methane/toluene Methane/l-methyl napthalene CO,/propane CO,/n-pentane CO,/n-decane 1-Propanol/I-butanol 1-Propanol/n-decane Ethanol/n-heptane l-Propanol/heptane n-Hexane/l-propanol n-Heptane/l-butanol n-Hexane/l-butanol Ethanol/l-propanol Ethanol/l-pentanol 1-Propanol/l-pentanol Benzene/l-propanol Benzene/l-butanol Toluene/l-butanol l-Butanol/p-xylene
for
the
experimental data much better than the PT equation. Figure 7 shows the representative mixture CO,propane. The PT equation failed to converge at all points in the VLE calculations, presumably due to ki j = 1. The GPT equation however, follows the trend in the data very well. Thus parameters obtained from the
of VLE predictions using the GPT and PT equations of state
system
Ethanol/n-decane
S. TEJA
T range (K)
311-511 278444 339 298-373 373453 3333533 311-511 358484 292434 294-361 278444 311478 3 lo-347 355-388 354-382 374394 372-383 399410 384398 374-394 422-543 464-704 278-344 278-378 278-511 373-388 368 313-363 352433 348-362 339-355 367430 342-38 1 323-367 353403 373-386 318-368 351413 363-387 3899399
GPT 3.83 6.96 14.0 9.13 4.26 3.29 4.35 2.29 5.38 4:56 1.93 5.72 1.40 3.32 2.76 2.65 1.20 2.10 11.6 5.60 4.13 5.00 8.35 7.90 3.73 5.55 12.2 17.9 24.0 15.0 17.3 19.2 20.4 9.54 9.22 7.79 10.6 10.5 16.5 16.5
PT
3.15 1.83 19.9 2.14 1.18 1.68 1.47 0.82 0.92 2.82 1.96 7.34 4.93 1.19 1.05 2.66 1.97 3.10 1.78 6.34 6.70 6.18 20.4 27.3 25.7 1.22 17.4 24.4 26.8 23.3 22.2 24.0 20.5 2.84 5.61 20.1 19.6 21.1 21.0 19.8
GPT
0.0107 0.0131 0.0143 0.0071 0.0139 0.0240 0.0051 0.0100 0.0077 0.0024 0.0130 0.0124 0.0049 0.0072 0.0041 0.0227 0.0037 0.0097 0.0473 0.0072 0.0297 0.0178 0.0197 0.0156 0.0072 0.0248 0.0224 0.0970 0.0308 0.0655 0.0707 0.0798 0.0573 0.0452 0.0270 0.1448 0.0617 0.027 1 0.0593 0.0567
PT
0.0108 0.0150 0.0222 0.0073 0.0141 0.0236 0.0053 0.0060 0.0025 0.0012 0.0065 0.0146 0.0131 0.0124 0.0018 0.0164 0.0049 0.0098 0.0030 0.0109 0.0277 0.0120 0.0512 0.0407 0.0082 0.0094 0.0429 0.1285 0.1207 0.0949 0.0877 0.0976 0.0570 0.0186 0.0115 0.1245 0.0987 0.0627 0.0752 0.0680
Group contribution equation of state for fluid mixtures IOC
2709
I,
c I
Melhone
0
n- Pentan. Group --------
9
PT
Fate\-
Tel0
I<3
K
K P
0
9
0
I 4
I
I
I
I
7
10
I3
16
moues PT
1
___
Potei____
Tel0 -___
3
Pressure(Mb) Fig.
3.
Experimental and predicted K-values methane-n-pentane system at 328 K.
for
the
Fig.
5.
Experimental and predicted K-values component mixture at 3 11 K.
IO
Grow
&
PT
for
a
five
_-
; I
K
o
K
v
n - Decone Group
PT
Potel7ej.a _ - _ _ _ __ _ _ _ _ _
I
I
I
I
I
6
I I
16
21
26
Pressure
Fig.
4.
Experimental
and
K-values for system at 624 K.
predicted
I
I
I
I
( I
I5
I9
23
Pressure
(MPo)
methane-1-methylnaphthalene
I 7
the
CO,ethane and CO,-n-heptane systems can be used to predict other systems containing CO,, and with greater accuracy than the PT equation. This supports the assumption that groups tend to behave more consistently in different mixtures than do molecules. This is further supported by predictions of systems containing OH groups, i.e. alcohols. As can be seen from Fig. 8, which shows the l-propanol-n-decane binary, both equations do not predict the phase behavior well. However, the GPT predictions are
Fig.
6.
(MPd
Experimental and predicted K-values component mixture at 360 K.
for
a seven
better on average than the PT predictions. Other polar-nonpolar systems exhibit approximately the same improvement. The results are summarized in Table 5.
CONCLUSIONS
A method for developing group contribution equations of state from conventional equations of state has been presented in this work. The method is based on
2710
Gus
K. GEORGETON
and AMYN
S. TEJA
hence the normal boiling point) for nonpolar and polar fluids. New group contribution combining rules were then proposed for the effective carbon number of
I(
the interaction between two groups. The combining rules allow the group interaction parameters to be regressed from a limited amount of mixture data. The GPT equation of state was used to predict VLE for a number of binary and multicomponent mixtures.
2
4
3
Pressure
Fig.
5
6
(MPo)
7. Experimental
and predicted K-values dioxide-propane system at 3 11 K.
0 0
for carbon
I-Prapanol n - DecDn.2 Group
Potet________-_-
P-r Tel0
The predictions were generally shown to be as good as, or better than, predictions using the conventional PT equation. The implicit assumption that groups tend to interact in a more consistent manner in different mixtures than do molecules was shown to be reasonable. For each pair of group interactions, only one data set was used to regress the group parameters. VLE for all other mixtures containing some or all of the groups were therefore predicted. An advantage of the general group contribution technique developed in this work is that, in principle, it can be applied to other equations of state. In this way, an equation better suited to a particular application can easily be extended to complex mixtures. The functional form of the basic equation of state is retained and the new combining. rules for the ECN (and hence the cross-interaction parameters of the equation of state) overcomes some of the limitations of conventional mixing rules.
thanks the Georgia Mining and Mineral Resources Institute for the award of a Fellowship during the course of this work.
AcknowZedgement--GKG
REFERENCES
D. and Sprake, C. H. S., 1970, Thermodynamic properties of organic oxygen compounds XXV. Vapor pressures and normal boiling temperatures of aliphatic alcohols. J. Chem. Thermodyn. 2, 631-645. Georgeton, G. K. and Teja, A. S., 1988. A group contribution equation of state based on the simplified perturbed hard chain theory. Ind. Engng Chem. Res. 27, 657-664. Georgeton, G. K., Smith R. L. and Teja, A. S., 1986, Applications of cubic equation of state to polar fluids. Ambrose,
A.C.S.
Fig. 8. Experimental and predicted K-values for the lpropanol-n-decane system at 368 K. group additivity rules for the parameters of the equation of state, but uses mixture data in the evaluation of the unlike group interactions. The method was used successfully to obtain the group contribution versions of two representative equations of state-ne derived from the partition function (the Simplified Perturbed Hard Chain equation) and one empirical (the Patel-Teja equation). The Group Contribution Simplified Perturbed Hard Chain Theory (GSPHCT) was presented in an earlier paper (Georgeton and Teja, 1988). The Group Contribution Patel-Teja (GPT) equation is presented in this work. The constants of the Patel-Teja equation were correlated with the effective carbon number (and
Symp.
Ser. 300,
434-454.
Jin, G., Walsh, J. M., and Donahue, M. D., 1986, A group contribution correlation for predicting thermodynamic properties with the perturbed soft chain theory. Fluid Phase Equif. 36, 123-146. Lydersen, A. L., 1955, Estimation of the critical properties of organic compounds by the method of group contribution. University of Wisconsin College of Engineering, Engineering Experimental Station Report 3, April 1955, Madison, WI. Majeed, A. I., Wagner, J. and Erbar, J. H., 1984, Prediction of thermodynamic properties using the PFGC equation of state. Pmr. Summer Computer Simulation Con$, Boston, Mass. 1984, Vol. 1, pp. 537-542. Patel, N. C. and Teja, A. S., 1982, A new cubic equation of state for fluids and fluid mixtures. Chem. Engng Sci. 37, 443473. Skjold-Jorgensen, S., 1984, Gas solubility calculations II: application of a new group contribution equation of state. Fluid Phase
Equil. 16, 317-351.
Willman, E. T. and Teja, A. S., 1985, Method for the prediction ofpure-component vapor pressures in the range 1 kPa to the critical pressure. lnd. Engng Chem. Process Des. Dev. 24, 1033-1036.
Enyineering
Science,
Vol.
44,
No.
11, pp.
2703-2710,
1989.
OOO%2509/89
Printed in Great Britain.
0
A SIMPLE
GROUP
CONTRIBUTION EQUATION FOR FLUID MIXTURES
1989
$3.00+0.00
Pergamon
Press
plc
OF STATE
GUS K. GEORGETONt and AMYN S. TEJA* School of Chemical Engineering, Georgia Institute of Technology, Atlanta, GA 30332-0100,
U.S.A.
(Receiued 22 July 1988: accepted 22 February 1989) Abstract-A new group contribution equation of state based on the PateI-Teja (PT) equation is presented in this paper. The new equation retains the functional form (cubic in volume) of the PT equation, but uses group additiviry mixing rules for the equation of stale constants. Group interaction parameters in the new equation were regressed from a limited amount of mixture data. The equation of state was then used to predict VLE for a number of binary and multicomponent mixtures. The predictions were shown to be as good as, or better than, predictions using the PT equation with conventional mixing rules.
INTRODUCTION
The prediction of phase equilibria is most commonly accomplished using equations of state or group contribution methods. Equations of state have the ability to represent the volumetric properties of pure fluids over wide ranges of temperature and pressure, and can be extended to mixtures by the use of mixing rules for their constants. A major limitation of the equation of state approach, however, is that these mixing rules must often incorporate experimental information for each mixture. Group contribution methods, on the other hand, are able to represent a large number of substances (and mixtures) by a much smaller number of structural units, and therefore require only a limited experimental data base. However, these methods are usually restricted to small ranges of temperature and pressure. An equation of state based on group considerations offers one way to combine the advantages of the two approaches. Several such equations have been proposed recently (Skjold-Jorgensen, 1984; Majeed et al., 1984; Jin et al., 1986). As with conventional equations of state, the applicability and accuracy of the new group contribution equations depends strongly on the “mixing rules” used to obtain the constants. A particularly simple way to obtain a group contribution equation of state is to use group additivity to estimate the constants in a conventional equation of state. Jin et a[. (1986) recently used this approach to extend the Perturbed Soft Chain Theory (PSCT) equation of state. The resulting GPSCT equation retained all the features of the PCST, including the simple (van der Waals) mixing rules. We have recently proposed (Georgeton and Teja, 1988) a group contribution equation of state based on the Simplified Perturbed Hard Chain Theory (SPHCT) which uses similar ideas, with the important difference that interactions between groups in different molecules are
obtained from mixture data. Although this has some similarity to the introduction of binary interaction parameters in conventional mixing rules, a large number of binary mixtures can be represented by a much smaller number of groups. The basic assumption is that groups interact in the same way in different surroundings. Hence, only a limited mixture data base is necessary to obtain the group interaction parameters. The SPHCT equation can only be used for nonpolar fluids and therefore, the GSPHCT equation of state was limited in its application to such fluids. We describe below an extension of our work to the cubic equation of state proposed by Pate1 and Teja (1982). The choice of the equation of state is arbitrary and a matter of convenience. The method developed, however, can be used with other equations of state.
THE PATEL-TEJA
EQUATION
OF STATE
The cubic equation of state proposed by Pate1 and Teja (1982) is outlined briefly below. This equation has been shown to work well in the calculation of phase equilibria and thermodynamic properties of nonpolar and polar fluids (Georgeton et al., 1986) The Patel-Teja (PT) equation has the form PI----
aCTI
RT v-b
u(u + b) + c(u -b)
(1)
where a[T]
=Cz,
R2 7-T
----CT1 p,
b&&F
c RTc c=R,p,
(2) (3)
(4) (5)
‘Present address: DuPont Company, Savannah River Plant, Aiken, SC 29808, U.S.A. ‘Author to whom correspondence should be addressed. CES 44:11-s
(6)
Gus
2704
K. GEORGETON
and R,, R,, Q, are constants, obtained by the application of the critical constraints to the equation of state (Pate1 and Teja, 1982). The equation contains four substance-specific constants-the critical temperature T,, the critical pressure P,, the critical compressibility L&,and the slope F of the vapor pressure function. For nonpolar fluids, F and c, can be related to the acentric factor (Pate1 and Teja, 1982) and the PT equation uses the same input information as other cubic equations (i.e. T,, P, and w). These correlations do not, however, hold for polar fluids. We have therefore developed separate correlations for classes of polar fluids (Georgeton et al., 1986) based on the effective carbon number (ECN) concept and the properties of the n-alkanes. These correlations are summarized in Table 1. Equation (Ia) in Table 1 gives the relationship between carbon number and boiling point for the n-alkanes. It can be used to calculate the effective carbon number (ECN) of any nonpolar substance given the normal boiling point of that substance. Amborse and Sprake (1970) and Willman and Teja (1985) have shown that the properties of nonpolar substances such as benzene can be obtained from the properties of an nalkane having the same number of carbon atoms as the ECN of benzene. It should be added here that the ECN of benzene is not necessarily integral. Equations (IbHIc) in Table 1 relate the constants F and 5, of the PT equation to the ECN of any nonpolar substance. The relationships between T,, P, and ECN have been presented elsewhere by Willman and Teja (1985). Since polar fluids do not follow the n-alkane correlations, separate correlations were developed for F and [, for classes of polar fluids (e.g. alcohols). These correlations were derived in terms of the differences between F and c, of the polar substances and those of their n-alkane homomorphs. (The homomorph of a polar compound is assumed to be the n-alkane obtained when the polar functional group is replaced by a CH, group.) Equations (Id) and (Ie) Table 1 represent the correlations for the correction terms + and 8 defined by (7)
where the subscripts denote the properties of the polar fluid and its homomorph. As an example, the PT equation of state constants for 2-butanol would be. calculated as follows:
Table W
ECN 3’ k”a15 8?s.lc
1. Correlation X
Th ECN ECN ECN, ECN,
and AMYN
(9 The
S.
TEJA
homomorph
(ii) (iii)
(iv) (v)
amix =
bmi,=
is found
by
re-
(9)
xi xjaij
j=l
-f
xi bi
(10)
xici
(11)
i=l nc
c,ix=
c
i=l
where the summations are over the nc components. In conventional equation of state calculations the cross-interaction term aij in eq. (9) is obtained from the combining rule: aij = kij&
(121
where kij is an adjustable binary interaction parameter. kij is generally obtained by optimization of mixture VLE data, and is often specific to the data set from which it is determined. Values of ki, obtained from experimental data are often temperature-dependent and sometimes composition dependent. It should be noted that the geometric mean rule for aii was proposed by van der Waals for his equation of state. However, the constant a in the van der Waals equation is temperature-independent whereas that in the PT equation is temperature dependent. Further-
A + BX + CX2
B
-0.11630
-0.019376
1.1596 x lo--’
-1.5491
x 10-7
-0.23378 0.21964 - 2.5033 - 0.94677
-0.42953 -0.075188 - 1.4914 -00.44082
3.7077 5.1818 3.2209 6.6923
- 1.9514 -2.4477 3.2067 1.3956
x x x x
D
10-2 10-3 lo-’ lo-’
f$ F i=l
A
x x x x
2-butanol
This approach therefore allows the constants of the PT equation to be obtained from a knowledge only of the boiling point of the substance of interest. [The relationship between T,, P, and ECN presented by Willman and Teja (1985) can be used to obtain T,, P, of 2-butanol. Alternatively, the actual T, and P, of 2butanol can be used in the calculations.] Equation (1) is extended to mixtures using the van der Waals mixing rules:
of the constants of the PT equation of state. W= c
of
the OH group by a CH, group. Thus. the homomorph of 2-butanol is 2-methyl butane, which has a normal boiling point of 301 K. The ECN of 2-methyl butane is calculated to be 4.756 using eq. (Ia). F and c, for Z-methyl butane can now be calculated using eq. (Ib) and (1~). (FH = 0.7304 and &, = 0.307.) + and 8 of 2-butanol can now be calculated using eq. (Id) and (Ie). (JI = 0.7011 and 8 = 0.0906.) Finally, F and [, for 2-butanol can be calculated using eqs (7) and (8) leading to F= 1.2425 and <, = 0.307. placing
+ DX’
E
10-a lo-=’ 10m3 1W3
1.3513 x lo-” 3.8572 4.5070 -2.3365 -8.4516
x x x x
lo-’ 10-e 10m4 lo-’
+ EX4
+ FX1tz F
Equation
0.19102
Ia Ib IC ld le
1.0815 0.16208 4.2844 1.2355
Group contribution equation of state for fluid mixtures more, an assumption imphcrt in the geometric mean rule is that the temperature dependence of the i-j pair is the same as that of the i-i and j-j pairs. For dissimilar molecules, this is obviously not true as is evidenced by reported ki, values which deviate from unity and change with temperature. The group contribution equation of state developed in this work retains the van der Waals mixing rules [eqs (9)--(11)] in the calculation of the properties of mixtures. However, the cross-interaction term ajj is obtained from the unlike group-interactions in mixtures and not from eq. (12), as is explained further below.
DEVELOPMENT PATEL
OF
THE
-TEJA
GROUP
(GPT)
CONTRIBUTION
EQUATION
The first step in the development of the GPT equation, as shown in the development of GPCST equation (Jin et al., 1986) and GSPHCT equation (Georgeton and Teja, 1988), is the inclusion of group additivity rules for the constants of the equation. The PT equation contains four constants T,, P,, <, and F. Group additivity rules for T, and P, have been proposed in the literature (e.g. Lydersen, 1955). However, there is less justification for such rules for F and possibly for c,. On the other hand all four constants T,, P,, (, and F can be expressed in terms of the ECN and therefore of the normal boiling point Tb. There is some evidence in the literature that r, is a function of the groups which comprise the molecule of interest. A group additivity rule for the ECN is therefore proposed as fohows:
(15) of group k when it is in where Ed,,, is the contribution the presence of a group m, and XNp’ is the total number of groups in the molecule i. Similarly, for the case when i # j ECNij is expressed as follows:
Nf' Nz' ‘=’
ECNii=
;
E,_
a~,=’
(,z, Nf) +,$, N:r’>
(16)
The denominators in eqs (15) and (16) may be viewed as normalization factors to account for the number of groups present in the system. The evaluation of group parameters using these additivity rules is described below. Additional details are also given in a previous paper (Georgeton and Teja, 1988). EVALUATION
OF
GROUP
PARAMETERS
The evaluation of the group parameters was carried out in two parts: (i) the .slrLparameters were first obtained from pure component values, and (ii) the E,_,, parameters were obtained from mixture data. In both
cases, the parameters time. To demonstrate
2705
of one group
the approach,
were evaluated the evaluation
at a of the
Ear parameters will be described for CH,-CH, and CH,-CH, interactions. The CH,-CH, group interactions were evaluated from the ECN of ethane. Ethane consists of two CH, groups and has a carbon number (or ECN) of 2. Application of eq. (15) therefore yields an +,. ,CHs of 1.0. This value can now be used to using the known carbon number (or obtain ECH,ICH~ ECN) of a molecule which contains both CH, and CH, groups. n-Pentane was used in the present study. n-Pentane has an ECN of 5 and contains 2 CH, groups and 3 CH, groups. Application of eq. (15) of 1.0 once again. Note that the yields an +H2/CH2 evaluation now requires a knowledge of the E~~,,,.-.-~ interaction in the n-pentane molecule. This interaction was obtained using the geometric mean rule e;c*, = ./z
(17)
where the asterisk denotes an unlike interaction in a pure component. This procedure was followed in the evaluation of seven group interactions (CH,, CH,, CH,, CO,, ACH, AC and OH) in the present study. Values of the group interactions (Ebb)are given in Table 2. Note that CH, and CO, were treated as single groups and their group mteractrons Ebb were obtained directly from their ECNs (or boiling point in the case of CO,). The group parameters in Table 2 were obtained using only one compound per group. For repeating groups that appear in homologous series, e.g. ACH in aromatics, several compounds should generally be used to determine the contribution of the repeating group. Thus, instead of using n-pentane to evaluate the CH,/CH, could have hexadecane.
interaction as described above, we used say n-pentane, n-decane and nThis would average out the effect of
position of the group in the chain. Alternatively, some type of normalization could be used. However, in this study, only one substance was used in the evaluation of each group parameter. The interaction parameters for unlike groups were determined as follows. The Q,,, values when k # m were obtained by optimizing bubble point pressures of one mixture containing the groups k and m. Thus for example the unlike CH,-CH, interaction (~~u,,~u~) was obtained using bubble point data for the Table 2. Group constants for the GPT equation Compound Group CH, CH, CH, ACH AC CO, OH
Ekk 1 1 1 1.063 1.193 2.223 5.180
used
for evaluation ethane n-pentane methane (assumed) benzene m-xylene carbon dioxide I-butanol
2706
Gus
K. GEORGETON and AMYN S. TEJA
propane-n-decane system. This system includes CH,-CH, and CH,-CH, interactions. CH,-CH,, The like-interactions (elk) were obtained from pure component ECNs as described previously. The unlike interactions (sXmrk # m) were treated as adjustable parameters in the correlation of bubble point pressures. By analogy with the temperature dependence of kij, a temperature dependence for Ed,,,was assumed as follows:
(13) This form was chosen so that Ed, would asymptotically reach a limiting value at higher temperatures. Also, Q,,, was set equal to E,~ and the optimization of EL2 and Ev,’ was constrained in order to obtain positive values. In addition, the value of E$z for k = m was set equal to zero so that like-group contributions would be temperature independent.
Values of the parameters EiO,’and EL: for each pair of groups were obtained by optimizing deviations in bubble point pressure for binary mixtures containing these groups. For each pair of groups, a representative binary mixture data set was chosen covering as wide range of temperature and pressure as possible. The binaries used to obtain the group parameters and the average absolute deviations in bubble point pressures and vapor compositions associated with each optimization are given in Table 3. The complete parameter matrix for the groups in the GPT equation is given in Table 4. It should be noted that all EL:’values were set equal to zero as described above, with the exception of the value for the OH-OH interaction. It was found necessary to set E&?,,,, = 2025.0 to account for the effects of hydrogen bonding on the vapor pressure (and hence ECN) of alcohols. Bubble point pressures of alcohol mixtures could be correlated well using this value for E&&OH (Table 3).
Table 3. VLE correlations using the GPT equation of state
Groups
System conditions (T/K
CH,-CH, CH,-CH, CH,-CH, CH,-ACH CH,-ACH CH,-ACH ACH-AC CH,-AC AC-CH, CH,-AC CH,-CO, CH,-CO, CH,-CO, CO,-ACH CO,-AC CH,-OH CH,-OH OH-OH OH-ACH OH-AC
and P/MPa)
propane-n-decane (278< T< 511, P<6.89) methanexthane (191 < T< 283, P < 5.47) methane-nlheptane (311 < T
AP%’ GPT PT
GPT
AY’
PT
2.86
3.30
0.0034
0.0051
1.35
1.29
0.0075
0.0064
2.58
2.50
0.0120
0.0094
2.30
1.76
0.0055
0.0054
1.55
4.21
0.0153
0.0175
2.69
5.25
0.0340
0.027 1
0.86
0.77
0.0050
0.0035
0.45
0.88
0.0115
0.0126
0.70
1.18
0.0099
0.0096
2.46
4.73
0.0186
0.0265
1.64
1.41
0.0188
0.0140
7.01
0.55
0.0199
0.0059
2.34
2.04
0.0099
0.0087
2.17
1.87
0.0017
0.0016
5.96
5.41
0.0072
0.0056
11.0
12.8
0.0930
0.0860
4.67
4.34
0.0123
0.0109
2.01
2.06
0.0098
0.0036
4.20
5.8 1
0.0430
0.0423
17.8
3.82
0.0663
0.0148
+AP% = average absolute percent deviation in bubble point pressures. $Ay = average absolute deviation in vapor mole fractions.
Group contribution equation of state for fluid mixtures Table 4. Optimized
group parameters for the GPT
2707
equation OH
CH,
Cl-l*
CH,
ACH
AC
CH, CH, CH, ACH AC CO, OH
1.000 1.225 0.7510 1.061 0.4563 0.9401 3.262
1.225 1.000 0.8325 0.9308 1.057 0.8962 0.1806
0.7510 0.8325 1.000 0.4650 1.124 0.8874
1.061 0.9308 0.4650 1.063 1.126 1.212 1.433
0.4563 1.057 1.124 1.126 1.192 0.9622 0.0141
0.9401 0.8962 0.8874 1.212 0.9622 2.223
E:fJ
CH,
CH,
CH,
ACH
AC
CO,
10.916 15.000 11.614
14.608 14.776 17.065
30.669 36.000
4.866 0 17.893 0.0156
20.548 17.893 0 -
35.284 0.0156
b;z
CH, z?
0 27.878. 14.183
14.183 28.292 0
27.878 28.292 0
21.641 183.33 11.717
AC& AC CO, OH
21.641 10.9 16 14.608 30.669
11.717 11.614 17.065 36.000
183.33 15.ocul 14.776
0 4.866 20.548 35.284
The group interaction constants given in Table 4 can be used to obtain the ECNs and hence the constants of the PT equation of state for any pure substance or mixture. The alkane correlations in Table 1 must be used when the components of interest
CO,
3.262 0.1806 1.433 0.0141 5.180 OH
2025.0
10 0
PraLmne
0
n-Deco”e Group
PT
Fmtel-Te,a _-_---_____
iL
are non-polar and the alcohol correlations via the homomorph concept when the components are alcohols. In the case of a mixture of a nonpolar substance and an alcohol, the ECN of the homomorph corresponding to the nonpolar-polar interaction is required. This is obtained by subtracting 2.7 from the value obtained using eq. (16). The value of 2.7 was obtained by considering the ECN of n-butanol, which is 7.7. The homomorph of n-butanol is n-pentane with an ECN of 5, which is 2.7 less than the actual ECN of the polar n-butanol. Hence 2.7 is subtracted from the ECN of the polar-polar and polar-nonpolar interactions calculated using eq. (16). This gives the ECN of the homomorph. The equation of state constants can now be calculated using the equations presented in Table 1 as described previously.
0
I
I
I
I
I
I
0
I
2
3
4
5
Pressure(MPd
RESULTS
Fig.
1.
Experimental and correlated K-values propane-n-decane system at 378 K.
for
the
A comparison of calculated VLE for the PT and GPT (group contributions version of PT) equations is shown in Figs 1 and 2. The calculated values from the PT equation were obtained with optimal binary interaction coefficients (kii). The propane-n-decane system shown in Fig. 1 is representative of the results obtained for the systems used to determine the group parameters. Both the PT and GPT equations correlate the data extremely well. In general the deviations between correlation and experiment are very similar for the two equations. In these comparisons, the GPT equation may be considered to be equivalent to the PT equation with a temperature dependent binary interaction parameter. The addition of an adjustable parameter would be expected to improve the representation of VLE. However, the improvement is marginal for systems such as methane-benzene (shown in Fig.
2), and it is sometimes worse as in the case of the CO,-ethane system (Table 3). The GPT equation was also used to predict VLE for binary and multicomponent systems, and comparison with experimental values are shown in Table 5. Also shown are the comparisons of predictions with the PT equation (kii = 1). It is obvious that the GPT equation can predict VLE as well as the PT equation for many mixtures and therefore it can be used for systems where no data or substance-specific equation of state constants are available. For nonpolar systems, deviations between calculated and experimental K-values using the GPT equation in the correlative and predictive modes were of the same magnitude (as can be seen from Tables 3 and
2708
Gus K. GEORGETON and AMYN
5). Figure 3 shows VLE predictions for the methane-n-pentane system. Both the PT and GPT equations predict the bubble point pressure and the K-values very well, with no binary interaction parameters. In order to be useful in calculating mixture data, the GPT must at the very least give results similar to the PT equation with kij = 1. This can be seen in Fig. 4 for the methane-1-methylnaphthalene system. The PT equation predicts pressures and Kvalues very well as does the GPT equation. The multicomponent systems shown in Figs 5 and 6 are also well predicted by both equations. A more severe test of the GPT equation is the modeling of systems containing nonhydrocarbon or polar compounds. VLE for several mixtures containing CO, were predicted with the GPT and the PT equations, and the GPT was shown to represent the
0 Methone ” Benzene GW”DPT Potel-Te,o --------10
I K
I5 Pressure
Fig.
2.
22
29
36
(MPo)
Experimental and correlated K-values methane-benzene system at 339 K. Table 5. Comparison
Ethane/n-decane Ethane/n-pentane Ethane/n-hexane Ethane/n-octane Propane/n-hexane Propane/n-heptane n-Butane/n-decane n-Butane/n-hexane n-Pentanejn-octane Methane/propane Methane/n-pentane Propane/benzene n-Pentanelbenzene Benzene/n-octane Benzene/toluene m-Xylene/n-decane n-Heptane/toluene n-Octane/p-xylene Toluene/n-octane Toluene/n-decane Methane/toluene Methane/l-methyl napthalene CO,/propane CO,/n-pentane CO,/n-decane 1-Propanol/I-butanol 1-Propanol/n-decane Ethanol/n-heptane l-Propanol/heptane n-Hexane/l-propanol n-Heptane/l-butanol n-Hexane/l-butanol Ethanol/l-propanol Ethanol/l-pentanol 1-Propanol/l-pentanol Benzene/l-propanol Benzene/l-butanol Toluene/l-butanol l-Butanol/p-xylene
for
the
experimental data much better than the PT equation. Figure 7 shows the representative mixture CO,propane. The PT equation failed to converge at all points in the VLE calculations, presumably due to ki j = 1. The GPT equation however, follows the trend in the data very well. Thus parameters obtained from the
of VLE predictions using the GPT and PT equations of state
system
Ethanol/n-decane
S. TEJA
T range (K)
311-511 278444 339 298-373 373453 3333533 311-511 358484 292434 294-361 278444 311478 3 lo-347 355-388 354-382 374394 372-383 399410 384398 374-394 422-543 464-704 278-344 278-378 278-511 373-388 368 313-363 352433 348-362 339-355 367430 342-38 1 323-367 353403 373-386 318-368 351413 363-387 3899399
GPT 3.83 6.96 14.0 9.13 4.26 3.29 4.35 2.29 5.38 4:56 1.93 5.72 1.40 3.32 2.76 2.65 1.20 2.10 11.6 5.60 4.13 5.00 8.35 7.90 3.73 5.55 12.2 17.9 24.0 15.0 17.3 19.2 20.4 9.54 9.22 7.79 10.6 10.5 16.5 16.5
PT
3.15 1.83 19.9 2.14 1.18 1.68 1.47 0.82 0.92 2.82 1.96 7.34 4.93 1.19 1.05 2.66 1.97 3.10 1.78 6.34 6.70 6.18 20.4 27.3 25.7 1.22 17.4 24.4 26.8 23.3 22.2 24.0 20.5 2.84 5.61 20.1 19.6 21.1 21.0 19.8
GPT
0.0107 0.0131 0.0143 0.0071 0.0139 0.0240 0.0051 0.0100 0.0077 0.0024 0.0130 0.0124 0.0049 0.0072 0.0041 0.0227 0.0037 0.0097 0.0473 0.0072 0.0297 0.0178 0.0197 0.0156 0.0072 0.0248 0.0224 0.0970 0.0308 0.0655 0.0707 0.0798 0.0573 0.0452 0.0270 0.1448 0.0617 0.027 1 0.0593 0.0567
PT
0.0108 0.0150 0.0222 0.0073 0.0141 0.0236 0.0053 0.0060 0.0025 0.0012 0.0065 0.0146 0.0131 0.0124 0.0018 0.0164 0.0049 0.0098 0.0030 0.0109 0.0277 0.0120 0.0512 0.0407 0.0082 0.0094 0.0429 0.1285 0.1207 0.0949 0.0877 0.0976 0.0570 0.0186 0.0115 0.1245 0.0987 0.0627 0.0752 0.0680
Group contribution equation of state for fluid mixtures IOC
2709
I,
c I
Melhone
0
n- Pentan. Group --------
9
PT
Fate\-
Tel0
I<3
K
K P
0
9
0
I 4
I
I
I
I
7
10
I3
16
moues PT
1
___
Potei____
Tel0 -___
3
Pressure(Mb) Fig.
3.
Experimental and predicted K-values methane-n-pentane system at 328 K.
for
the
Fig.
5.
Experimental and predicted K-values component mixture at 3 11 K.
IO
Grow
&
PT
for
a
five
_-
; I
K
o
K
v
n - Decone Group
PT
Potel7ej.a _ - _ _ _ __ _ _ _ _ _
I
I
I
I
I
6
I I
16
21
26
Pressure
Fig.
4.
Experimental
and
K-values for system at 624 K.
predicted
I
I
I
I
( I
I5
I9
23
Pressure
(MPo)
methane-1-methylnaphthalene
I 7
the
CO,ethane and CO,-n-heptane systems can be used to predict other systems containing CO,, and with greater accuracy than the PT equation. This supports the assumption that groups tend to behave more consistently in different mixtures than do molecules. This is further supported by predictions of systems containing OH groups, i.e. alcohols. As can be seen from Fig. 8, which shows the l-propanol-n-decane binary, both equations do not predict the phase behavior well. However, the GPT predictions are
Fig.
6.
(MPd
Experimental and predicted K-values component mixture at 360 K.
for
a seven
better on average than the PT predictions. Other polar-nonpolar systems exhibit approximately the same improvement. The results are summarized in Table 5.
CONCLUSIONS
A method for developing group contribution equations of state from conventional equations of state has been presented in this work. The method is based on
2710
Gus
K. GEORGETON
and AMYN
S. TEJA
hence the normal boiling point) for nonpolar and polar fluids. New group contribution combining rules were then proposed for the effective carbon number of
I(
the interaction between two groups. The combining rules allow the group interaction parameters to be regressed from a limited amount of mixture data. The GPT equation of state was used to predict VLE for a number of binary and multicomponent mixtures.
2
4
3
Pressure
Fig.
5
6
(MPo)
7. Experimental
and predicted K-values dioxide-propane system at 3 11 K.
0 0
for carbon
I-Prapanol n - DecDn.2 Group
Potet________-_-
P-r Tel0
The predictions were generally shown to be as good as, or better than, predictions using the conventional PT equation. The implicit assumption that groups tend to interact in a more consistent manner in different mixtures than do molecules was shown to be reasonable. For each pair of group interactions, only one data set was used to regress the group parameters. VLE for all other mixtures containing some or all of the groups were therefore predicted. An advantage of the general group contribution technique developed in this work is that, in principle, it can be applied to other equations of state. In this way, an equation better suited to a particular application can easily be extended to complex mixtures. The functional form of the basic equation of state is retained and the new combining. rules for the ECN (and hence the cross-interaction parameters of the equation of state) overcomes some of the limitations of conventional mixing rules.
thanks the Georgia Mining and Mineral Resources Institute for the award of a Fellowship during the course of this work.
AcknowZedgement--GKG
REFERENCES
D. and Sprake, C. H. S., 1970, Thermodynamic properties of organic oxygen compounds XXV. Vapor pressures and normal boiling temperatures of aliphatic alcohols. J. Chem. Thermodyn. 2, 631-645. Georgeton, G. K. and Teja, A. S., 1988. A group contribution equation of state based on the simplified perturbed hard chain theory. Ind. Engng Chem. Res. 27, 657-664. Georgeton, G. K., Smith R. L. and Teja, A. S., 1986, Applications of cubic equation of state to polar fluids. Ambrose,
A.C.S.
Fig. 8. Experimental and predicted K-values for the lpropanol-n-decane system at 368 K. group additivity rules for the parameters of the equation of state, but uses mixture data in the evaluation of the unlike group interactions. The method was used successfully to obtain the group contribution versions of two representative equations of state-ne derived from the partition function (the Simplified Perturbed Hard Chain equation) and one empirical (the Patel-Teja equation). The Group Contribution Simplified Perturbed Hard Chain Theory (GSPHCT) was presented in an earlier paper (Georgeton and Teja, 1988). The Group Contribution Patel-Teja (GPT) equation is presented in this work. The constants of the Patel-Teja equation were correlated with the effective carbon number (and
Symp.
Ser. 300,
434-454.
Jin, G., Walsh, J. M., and Donahue, M. D., 1986, A group contribution correlation for predicting thermodynamic properties with the perturbed soft chain theory. Fluid Phase Equif. 36, 123-146. Lydersen, A. L., 1955, Estimation of the critical properties of organic compounds by the method of group contribution. University of Wisconsin College of Engineering, Engineering Experimental Station Report 3, April 1955, Madison, WI. Majeed, A. I., Wagner, J. and Erbar, J. H., 1984, Prediction of thermodynamic properties using the PFGC equation of state. Pmr. Summer Computer Simulation Con$, Boston, Mass. 1984, Vol. 1, pp. 537-542. Patel, N. C. and Teja, A. S., 1982, A new cubic equation of state for fluids and fluid mixtures. Chem. Engng Sci. 37, 443473. Skjold-Jorgensen, S., 1984, Gas solubility calculations II: application of a new group contribution equation of state. Fluid Phase
Equil. 16, 317-351.
Willman, E. T. and Teja, A. S., 1985, Method for the prediction ofpure-component vapor pressures in the range 1 kPa to the critical pressure. lnd. Engng Chem. Process Des. Dev. 24, 1033-1036.