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Application of the CORGC equation of state with a new mixing rule to polar and nonpolar fluids
Fluid Phase Equilibria, 91 (1993) 239-263 Elsevier Science Publishers B.V., Amsterdam
239
Application of the CORGC equation of state with a new mixing rule to polar and nonpolar fluids Mohammad
Zia-Razzaz
’
and Mahmood
Moshfeghian
*
Department of Chemical Engineering, Shiraz University, Shiraz, Iran (Received December 22, 1992; accepted in final form May 3, 1993)
ABSTRACT Mohammad Ziz-Razzaz and Mahmood Moshfeghian, 1993. Application of the CORGC equation of state with a new mixing rule to polar and nonpolar fluids. Fluid Phase Equilibria, 91: 239-263. This paper presents a new group contribution equation of state (EOS) applicable to both polar and nonpolar compounds. This EOS is based on the chain-of-rotator group contribution (CORGC) equation considering an NRTL type local composition mixing rule. The parameters for 37 organic and inorganic groups have been calculated. Several pure components and mixtures have been studied including polar and multicomponent systems. Comparison between the predicted and experimental thermodynamic properties are also presented. Keywords: theory; equation of state; group contributions;
polar; nonpolar.
INTRODUCTION
Separation units such as distillation towers or extraction cascades are of great importance in almost any chemical plant. The maximum efficiency of such units is often limited by the thermodynamic equilibrium compositions of the phases concerned. Phase equilibrium information for a variety of mixtures over wide ranges of temperature and pressure is therefore of great importance. The number of mixtures of interest to, for example, the petrochemical industry is enormous, even when the scope of components is restricted to low molecular mass (e.g. a carbon number of less than 25). Hence, it is desirable to be able to predict phase equilibria based on as little experimental information as possible or at best without any experimental data at all. The equations of state (EOSs) are the commonest tool for predicting thermodynamic properties of pure and mixed components. Old EOSs like * Corresponding author. ’ Graduate student. 037%3812/93/%06.00 0
1993 - Elsevier Science Publishers B.V. All rights reserved
240
M. Zia-Razzaz
and M. Moshfeghian 1 Fluid Phase Equilibria 91 (1993) 239-263
Redlich-Kwong (RK), Soave-Redlich-Kwong (SRK) and PengRobinson (PR) are written for nonpolar components in the gas phase and then generalized to liquid phase and slightly polar components. Predictions of thermodynamic properties in the liquid phase can be improved by using activity coefficient equations based on the group contribution models. Some well-known equations of this kind are NRTL and UNIFAC. The new generation of EOSs has been made by combination of both group contribution and free volume models. This kind of EOS is efficient in predicting thermodynamic properties of both gases and liquids, and polar and nonpolar systems. Parameter from group contribution (PFGC), group contribution equation of state (GC-EOS) and chain-of-rotator group contribution (CORGC) are members of this generation of EOSs. CORGC is the latest group contribution EOS developed by Pults and Chao (1989). They obtained parameters for 20 nonpolar and slightly polar groups. For polar compounds, they also added a corrective term to the EOS. The results of this polar version of the equation (CORGCP) are not so good. Recently, Dehghani (1991) and Shariat et al. (1993) revised the parameters of CORGC and extended its application to polar and halogenated systems. In this work a new local composition model has been combined with CORGC. The parameters for 37 groups have been calculated and the capabilities of this equation in predicting thermodynamic properties of pure and mixed, polar and nonpolar systems have been evaluated.
PROPOSED
EQUATIONS
The Helmholtz free energy form of the proposed EOS is formulated eqn. (1):
1
AR 4y-3y2 RT= (1 -y)’
c(a - 1) (4+ cr)y - 3y2 + 2 (1 -Y)’
+(l
+cr)ln(l
-y)
-
a
in
ln(l+4y) bRT
(1) This equation is a contribution of three terms, considered for the translational, rotational and attractive energy of molecules. The translational part of the equation is the same as the Carnahan-Starling hard sphere equation (Carnahan and Starling, 1972) :
(> AR
RT
=
f”
4y - 3y2 (1
-Y12
The rotational
(2)
part is taken from the Boublik and Nezbeda (1977) hard
M. Zia-Razzaz
241
and M. Moshfeghian / Fluid Phase Equilibria 91 (1993) 239-263
dumbbell equation extended by Chien et al. (1983) to hard chain fluids: =c(a: - 1) (4+a)y 2
-3y*
(1 -Y)’
1+
For attractive terms a Redlich-Kwong
(1 + a)ln(l -y)
-
type equation has been applied:
a ln(l+4y) -AR bRT ( RT >att = -
(4)
In these equations, y = b/4V is the reduced density. Pressure- and other thermodynamic functions can be derived from the Helmholtz energy, by using appropriate mathematical manipulations: (5)
IRT
(6)
T,n
The attractive part of the pressure expressed as follows:
patt =-
and compressibility
factor
is then
a
(7)
V( V + b)
zatt = _
where the parameter a’ is the first derivative of a with respect to volume at constant temperature and composition:
(9) Finally, the proposed equation can be expressed as a summation attractive and repulsive contributions: zJ+Y+Y2-Y3
+[c(I-l),2][3y+3;>_$+l)y3]
(1 -VI' a -RT(V+b)
of the
+
(10)
The following mixing rules are used in the proposed equation. In these relations the parameters b,, c,,, and q,,, are used for the group m, and b, c and Q for the fluid covolume, equivalent degree of freedom and the normalized area, respectively.
242
M. Zia-Razzaz NC
and M. Moshfeghian / Fluid Phase Equilibria 91 (1993) 239-263
NG
b = ~x~&,,&~
c
=
Q=
i
m
NC
NG
(11) (12)
~x,~vimcm i m NC
NG
I
m
~x,&rnqm
(13)
NC
(14) y = b/4V
(15) -
(16) (17)
=
RTQ ym 8m(H2,H6,
(18)
where H2,, H5, and H6, are auxiliary variables as defined in the Appendix. The parameters b, and a,,,,, are temperature dependent: b, = bz exp( - T/T?)
(19)
amn= az,(T/T$,)
(20)
- o~18135 exp( - T/T:,)
To distinguish between the original CORGC EOS and the proposed one, we have called our version MCORGC. GROUP PARAMETERS
The proposed EOS contains six parameters per group bz, Tz, cm, qm, atm and T&. The optimized values of these parameters are presented in Table 1 for 37 groups. In addition, for each pair of groups three interaction parameters are needed: the attraction parameter a&, and its characteristic temperature T&,, and the nonrandom parameter CI,,. Based on the original CORGC, the group interaction parameters are assumed to be symmetrical, i.e. CI,, = anrn, a,,, = an,,, and Tz, = T,*,. The optimized values of these parameters are presented in Table 2. The normalized area parameter qm is calculated using van der Waals areas normalized by a base parameter q,,, = 10 for methane.
M. Zia-Razzaz
and M. Moshfeghian
/ Fluid Phase Equilibria 91 (1993) 239-263
243
TABLE 1 Group parameters for the MCORGC
EOS
Group ID
Group name
T, (K)
1 2 3 4 5 6 7 8 9 10
al-CH, al-CH,-al :CH>c: (CH)ar ( C) ar-al ar_CH, ar-CH (C) ar-ar
11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37
(CH,)cyc (6 ring) (CHJCYC (5 ring) =CH, =CHHZ I% CH4 C,He CO* C,Hz, H,S MeOH -OH (1”) -OH (2”) -OH (3”) CH&O-CH&OCH3-O-oHz0 NH, CC&F, CC&FH CCIF, CF,H CF, CFCI, C,Cl,F,
b: ( cm3/mol-‘)
a&/1000
Tk (K)
52.91 61.40 33.75 34.07 72.53 32.48 10.00 41.75
321.25 973.40 1340.00 1823.80 683.59 7431.30 284.64 4000.00 50.00 644.34
0.412 1.024 - 0.448 -2.671 0.627 0.200 0.353 0.162 - 1.062 0.708
7.31 4.66 4.66 0.72 3.45 1.03 7.31 4.66 0.73 4.66
46.19
492.94
1.389
4.66
101.32
449.58
36.74 35.22 39.68 78.34 80.46 132.97 92.43 102.78 81.63 150.40 50.30 68.99 61.38 83.72 75.02 57.44 45.62 78.55 85.32 139.40 150.61 138.74 108.23 125.25 200.47 226.98
241.72 355.70 55.91 139.32 220.45 345.26 347.00 328.91 367.43 597.75 612.12 205.83 419.69 3042.60 1834.60 1597.70 7338.60 686.51 529.98 511.62 564.74 343.45 425.52 218.60 540.99 463.96
19.237 22.546 2.116 - 1.346 0.879 -4.520 -6.881 4.173 10.183 - 13.084 1.357 10.996 1.292 10.195 -6.472 -2.014 -4.925 - 14.399 - 14.068 14.938 4.277 8.331 -4.101 7.135 3.394 10.947
6.48 3.72 4.76 8.48 10.00 14.60 11.10 10.70 10.70 12.40 10.34 10.34 10.34 12.83 10.45 9.38 6.72 12.07 10.00 10.00 10.00 10.00 10.00 10.00 10.00 10.00
270.08 323.89 9.838 45.617 49.653 66.860 94.261 95.270 109.54 268.40 606.45 243.99 817.37 113.42 91.30 130.42 157.68 132.62 123.15 245.81 308.26 184.89 144.25 129.38 413.97 474.62
190.94 344.64 248.12 148.19 250.47 331.25 277.22 337.47 383.63 366.33 131.27 230.81 110.69 841.81 502.74 208.17 189.49 528.26 393.49 484.91 479.74 322.88 312.79 204.44 479.38 408.46
62.50 43.40
74.030 80.226 - 167.1 - 28942 121.59 271.23 77.076 32.194 3820.2 78.638
287.96
472.60 268.03 303.84 577.58 1273.5 396.78 852.16 196.51 573.69
244
M. Zia-Razzaz and M. Moshfeghian 1 Fluid Phase Equilibria 91 (1993) 239-263
TABLE 2 Interaction parameters for dissimilar groups of the MCORGC Group n 2 3 4 5 6 8 9 12 13 15 16 17 18 20 22 23 24 25 26 27 28 3 5 6 8 12 13 15 16 17 18 20 22 25 26 27 28 15 23 24 6 7 8 9
Group m 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 3 3 4 5 5 5 5
EOS
a:,/1000 ( cm6 bar mol12) 57.678 49.946 246.91 70.344 22.12 63.657 50.161 38.154 23.563 95.169 66.997 67.517 28.671 71.38 156.88 129.87 150.6 82.724 63.664 105.71 114.53 145.5 96.356 169.3 28.737 146.08 89.322 10.289 62.095 69.982 95.809 65.772 95.712 25.549 32.853 51.516 82.642 -70.353 49.946 524.42 254.72 79.826 83.532 247.69
626.26 746.9 1074.4 557.14 644.38 765.86 3909.5 661.43 462.73 145.95 226.47 328.82 675.67 436.56 371.31 285.42 277.56 586.04 726.27 577.53 492 185.78 526.06 1015.4 1711.9 521.43 479.2 7949.3 465.01 421.56 272.31 577.44 466.61 1860.8 974.49 2049.7 364.44 19999 328.98 265.91 939.25 493.83 630.56 1007.7
0.000 0.000 0.000 0.000
0.000 0.000 0.000 0.022 0.266 -0.981 1.211 -64.8 4.825 - 1.875 -0.235 0.299 - 0.228 - 0.294 -0.991 - 0.689 -1.120 0.000 0.000 0.000 0.000 -0.544 - 10.000 - 0.422 - 2.805 -0.934 500.000 -51.980 - 1.020 24.210 6.221 - 4.047 - 15.020 23.300 1.309 -0.070 0.000 0.000 0.000 0.000
M. Zia-Razzaz
and M. Moshfeghian
/ Fluid Phase Equilibria 91 (1993) 239-263
245
TABLE 2 (continued) Group n
Group m
a&/1000 ( cm6 bar mol -‘)
16 7 8 9 13 16 17 18 20 17 18 18 20 29
5 6 6 6 12 15 15 15 15 16 16 17 17 21
69.148 199.46 55.289 10000 58.605 43.449 41.504 50.476 108.19 54.612 61.251 68.703 82.78 222.93
a WI”
399.37 519.97 4000 129.55 584.47 195.66 229.87 242.85 150.51 302.22 261.53 299.38 346.67 381.14
- 0.464 0.000 0.000 0.000 -0.314 -0.137 -0.130 -0.561 -0.719 0.000 -0.142 1.471 0.205 0.300
The remaining parameters are determined from pure component binary mixture VLE data as described in the following sections. OPTIMIZATION
and
OF GROUP PARAMETERS
A computer package using a subroutine based on a nonlinear regression algorithm was developed to optimize the group parameters (Zia-Razzaz, 1993). This subroutine minimizes the error objective function by using the method of Marquardt (1963). In order to optimize pure group parameters, a vast pure component VLE data bank has been used. The predicted and experimental data for each set of parameters are then compared, and the summation of the error weighted squares minimized. The following objective function is used for pure component VLE data:
(22) where Wi are the weighting factors for the relative errors of vapor pressure, vapor volume, liquid volume and latent heat of vaporization, respectively. In the above equation, Psat, V”, V’ and Hvap represent vapor pressure, vapor volume, liquid volume and heat of vaporization, respectively. The subscripts E and C stand for experimental and calculated values, respectively. Most group interaction parameters have been calculated using binary mixture VLE data. For these systems, errors in predicting vaporization
246
M. Zia-Razzaz
and M. Moshfeghian 1 Fluid Phase Equilibria 91 (1993) 239-263
equilibrium ratios are used as a measure of parameter following objective function is used for binary mixtures:
accuracy.
The
(23) where Ki and K2 are the VLE ratios (K values) for components 1 and 2, W, and W, are weighting factors, and the subscripts E and C represent experimental and calculated quantities, respectively. In this case all predicted properties have been calculated at constant temperature and pressure. This method is more efficient than the classical equilibrium calculation method, which requires about ten equilibrium calculations for each point and consuming about ten times as much computer time. However, for the final results and comparisons, the properties are calculated at the equilibrium condition using flash calculations at constant temperature and pressure. RESULTS
The capabilities of the proposed EOS to predict VLE properties of pure and mixture systems were examined. About 70 pure compounds and 30 binary systems were studied. The calculated properties are usually as accurate as experimental values and the order of errors (especially for pure compounds) is very low. The proposed equation proves to be a useful tool for the prediction of a number of pure fluid thermodynamic properties. Saturated thermodynamic properties of a vast set of substances were studied. The summary results for 21 pure compounds are given in Table 3, and for a few compounds the results are shown graphically in Figs. l-6. These are typical of the results obtained in this study. Vapor pressure is the most important pure VLE property to be predicted by an EOS, since a good representation of vapor pressure will give good results in the mixture VLE calculations. The results of vapor pressure prediction for several components are shown in Figs. 1 and 2. The total average absolute error for 70 compounds studied by Zia-Razzaz ( 1992) is 1.47%, and for most compounds this value is less than unity. The ability of an EOS to predict liquid densities is a measurement of its accuracy. The results of saturated liquid densities calculated for a number of compounds are shown in Figs. 3 and 4. The average absolute error of V’ prediction is 0.98% for 42 studied substances (Zia-Razzaz, 1992). At this moment, no EOS is known with the same capability of predicting liquid densities.
M. Zia-Razzaz TABLE
and M. Moshfeghian / Fluid Phase Equilibria 91 (1993) 239-263
247
3
Pure component
vapor-liquid
saturation
Substance
No. of points
Jyw
F”
CA
29
0.44
n-GH,,
30
n-C&L,
(AAD%
of MCORGC
from experimental
data)
F’
P”“’
T range (K) T,. range (K)
Data source
0.53
0.95
0.33
Perry et al. (1984)
1.01
1.65
1.35
0.94
17
2.25
2.75
1.76
0.70
n-G&
7
2.11
1.19
1.81
0.55
n-C,H,,
8
2.73
2.50
1.92
1.07
40
1.05
1.47
0.76
0.86
9
2.28
2.92
1.42
1.90
GW&
12
0.48
-
0.25
0.65
CF,
15
0.73
1.99
1.36
1.00
CClF,H
27
0.41
0.43
0.79
0.44
CClF3
11
0.83
0.90
2.39
0.74
W&F,
39
1.10
0.95
0.76
1.12
CC&F
39
0.68
0.69
0.89
0.25
N,
11
3.04
2.19
0.92
0.86
NH,
32
1.88
1.64
0.78
1.09
CO,
46
0.94
1.00
0.93
0.43
H2S
25
1.16
1.42
1.07
0.77
Hz0
51
2.98
3.63
2.25
1.70
CH,OH
24
5.37
1.17
0.91
1.68
C,H,OH
9
6.77
6.69
2.93
3.51
10
9.03
5.87
4.11
1.38
144.26-299.82 0.54-0.98 255.37-416.48 0.60-0.98 309.19-462.00 0.66-0.98 371.60-500.00 0.69-0.93 398.80-535.00 0.70-0.94 322.04-533.15 0.57-0.95 383.78-575.00 0.65-0.97 303.15-413.15 0.49-0.67 133.15-210.93 0.59-0.93 144.26-288.71 0.48-0.96 144.26-294.26 0.48-0.97 183.15-394.26 0440.94 250.93-460.93 0.53-0.98 77.04 124.82 0.61-0.99 199.82-377.59 0.49-0.93 216.48-299.82 0.71-0.99 222.04-355.37 0.59-0.95 338.71-616.48 0.52-0.95 273.15-483.15 0.53-0.94 351.45-503.00 0.68-0.97 371.20-533.10 0.70-0.99
G-b
CH,GH,
n-C,H,OH
FRPI - Fluid Properties
Research
Inc.
Perry et al. ( 1984) Perry et al. (1984) FPRI
(1980)
FPRI
(1980)
Perry et al. ( 1984) Perry et al. (1984) Perry et al. (1984) ASHRAE
(1982)
ASHRAE
(1982)
ASHRAE
(1982)
ASHRAE
(1982)
ASHRAE
(1982)
FPRI
( 1980)
FPRI
( 1980)
Perry et al. (1984) Perry et al. (1984) Perry et al. (1984) Perry et al. (1984) Perry et al. (1984) Perry et al. (1984)
248
M. Zia-Razzaz
and M. Moshfeghian 1 Fluid Phase Equilibria 91 (1993) 239-263
300
400
Temperature
500
[K]
Fig. 1. Vapor pressure of some highly polar solutions.
1
+7~~~-17-m-m-rr-~~~ 350
400
--met
500
450
Temperature
5
[K]
Fig. 2. Vapor pressure of primary alcohols; data from Fluid Properties Research Inc. (1980).
Saturated vapor volume is another VLE property that can be calculated by an EOS. The average absolute error of V” prediction for 40 studied compounds is 1.28% ( Zia-Razzaz, 1992). The latent heat of vaporization of a pure compound can be calculated by subtracting coexistence vapor and liquid residual enthalpies. The latent
M. Zia-Razzaz
200
and M. Moshfeghian
II1III,,,IIII1\, 220
7”“” 240
III,IIIIIIIIV 260
1 Fluid Phase Equilibria 91 (1993) 239-263
III IIIIII*,!III!I 260 300
Temperature
‘“I 320
340
249
rl
: 16 0
[K]
Fig. 3. Saturated liquid volumes of CO2 and H,S.
40 100
150
200
250
300
Temperature
350
400
4 i0
[K]
Fig. 4. Saturated liquid volume of refrigerants.
heats of several compounds have been calculated and are shown in Figs. 5 and 6. The average absolute deviation for latent heat prediction of 39 studied compounds is 1.65% (Zia-Razzaz, 1992). Equilibrium calculations have been carried out for 30 binary systems including polar, nonpolar and light gas components (Zia-Razzaz, 1992).
M. Zia-Razzaz
250
200
220
and M. Moshfeghian 1 Fluid Phase Equilibria 91 (1993) 239-263
240
260
300
280
320
Temperature Fig. 5. Heats of vaporization
Oj
ITIII8III,,
100
*
150
3 8’#““‘)“‘t
250
IIIII,IIIIIIIII,II11I,II
300
Temperature Fig. 6. Heats of vaporization
:
of CO, and H,S.
I1I1III,3IIII!III,
200
340
[K]
350
400
4
[K]
of refrigerants.
Typical results for mixture flash calculations are tabulated in Table 4. Figures 7 and 8 show the comparison between calculated and experimental equilibria for a few typical systems. One isobar for the methanol-water system has been studied, as shown in Fig. 7. The AAD%s for K value prediction at 1.013 bar are 1.96 and 2.56
M, Zia-Razzaz
and M. Moshfeghian / Fluid Phase Equilibria 91 (1993) 239-263
251
TABLE 4 Vaporization data)
equilibrium ratios for binary systems (AAD% of MCORGC from experimental
Mixture camp. (1:2)
No. of points
Temp. range (K)
Pressure range (KPa)
K,
(AAD%)
;AD%)
Data source
CH, + C3H, CH, + r&H,, CH, + n-C,&, CH, + n-C,,H,, CH, + CO, N, + CH, C,H, + n-CmH,, C2H, + H$ C,H, + GH, C,H, + r&H,4 C,H, + CO, N, + C,Hh N, + H,S N, + Cd% N, + r&H,, CO, + n-CJ-4, CO2 + n-Cd-I,, CH,OH + H,O
56 15 21 12 198 118 39 28 32 29 185 43 52 41 31 26 40 21 18 17
130.3-213.7 310.9-410.9 423.1-583.0 462.2-703.3 173.3-283.1 121.0-183.1 310.9-510.9 199.9-283.1 144.2-255.3 338.7-449.8 222.0-298.1 138.7-194.2 256.4-344.2 173.1-353.1 277.4-377.5 227.9-283.1 277.6-377.5 337.6-373.1 313.5-332.5 327.7-327.7
1.72-55.16 4.45- 17.24 27.6-116.1 20.3-50.7 6.89-85.20 2.79-49.64 3.45-68.95 0.99-29.87 0.03-14.12 6.89-75.84 6.24-66.30 3.45-134.1 17.3-207.0 13.8-137.9 2.50-141.0 0.33-39.37 2.28-96.32 1.01-1.01 0.14-0.74 0.06-0.23
6.05 5.06 9.38 9.19 5.96 4.18 11.08 5.75 5.84 7.39 2.47 12.21 8.45 12.63 12.74 5.77 5.67 1.96 8.46 4.97
9.76 4.41 7.21 5.84 6.91 4.65 14.75 5.99 9.98 7.23 3.35 18.78 3.63 14.84 9.64 8.81 7.32 2.56 8.63 4.68
27 11 17 20 5, 29 26 13 15 8 31 9, 12 26 2 25 15 14 2 16 21 19
21
398.8-409.3
1.01-1.01
3.65
6.39
n-C,H,, + C,H,C,HS n-C,H,, + CzH,H,
19
for methanol and water, respectively, an excellent result for an EOS. Figure 8 shows the binary mixture of ethyl benzene and n-heptane at 327.76 K. The AAD%s of K value prediction of this system are 4.97 and 4.68, respectively. The parameters used for this system are calculated from pure component VLE data. Good representation of this nonideal aromatic + paraffin system shows further capability of the proposed EOS. For most EOSs, accurate representation of binary systems is an indication of good prediction of multicomponent mixtures. In order to evaluate this capability of the proposed equation, a multicomponent mixture was studied, containing N *, CO2 and six paraffins. The predicted and experimental K values are compared in Fig. 9, which shows a fairly good ability of this EOS to predict complicated multicomponent systems.
252
M. Zia-Razzaz
and M. Moshfeghian / Fluid Phase Equilibria 91 (1993) 239-263
D
Fig. 7. T-x-y
COMPARISON
diagram of the water + methanol system at 1.013 bar.
WITH OTHER EOSs
The saturation state properties of a number of compounds have been calculated using SRK, PR, PFGC and the MCORGC. The comparison results in terms of average absolute deviation are shown in Table 5. As can be seen, almost all predictions of the proposed CORGC are much more accurate than those of the other equations.
M. Zia-Razzaz
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1 Fluid Phase Equilibria 91 (1993) 239-263
253
0.10
0.05
O.o0_1,,~..~..~,~,.......,,........,..,......,...rr 0.0
Fig. 8. P-x-y
02
0.4
n-Heptane
0.0
mole
0-B
diagram of the n-heptane + ethylbenzene
OTHER THERMODYNAMIC
0
fraction
system at 327.76 K.
PROPERTIES
In order to demonstrate the ability of the proposed EOS to predict single phase properties of compounds, superheated and sub-cooled ethane has been studied. The enthalpy and volume at three temperatures (300, 400 and 500 K) have been calculated. Figure 10 shows the pressure-enthalpy diagram for ethane. As can be seen, deviations in the saturated states and 300 K isotherm are in the range of experimental data accuracy. At higher
1.00 1.18 1.00 2.38 1.37 1.00 3.33 1.21 1.93 1.80
1.28 1.17 1.22 1.55 1.81 1.97 4.91 1.05 1.03 1.09
1.71
Methane Ethane Propane n-Butane i-Butane Nitrogen Ammonia Water Ethylene Propylene
Average
1.62
PR
SRK
1.90
1.60 1.03 1.06 1.33 1.12 1.00 1.91 2.67 4.25 3.03
PFGC
pressure
Vapor
Substance
0.98
0.69 0.33 0.35 0.94 1.11 0.86 1.09 1.70 0.37 2.34
MCORGC
1.38
0.94 0.34 1.05 0.94 0.75 2.38 5.41 0.71 0.68 0.57
SRK
1.88
2.12 1.21 0.83 1.94 1.71 2.10 2.90 2.18 2.24 1.56
PR
2.83
3.05 0.50 2.32 2.88 2.47 1.30 2.01 2.81 6.11 4.86
PFGC
Vapor vohime
1.79
1.60 0.53 0.88 1.65 1.42 2.19 1.64 3.63 1.46 2.93
MCORGC
10.8
5.58 7.07 8.08 8.79 9.40 22.6 25.4 3.62 6.25 6.61
SRK
8.70
9.23 6.95 6.07 4.86 5.61 11.8 15.8 10.6 1.35 6.55
PR
Liquid volume
Comparison of saturated properties calculated by four EOSS in terms of AAD%
TABLE 5
3.24
4.89 3.68 2.91 1.62 2.79 5.04 4.53 1.59 2.63 2.71
PFGC
1.24
0.98 0.95 1.57 1.35 1.98 0.92 0.78 2.25 0.63 0.99
MCORGC
2.65
3.92 1.58 2.60 1.57 3.16 3.10 4.18 2.07 1.67 1.29
SRK
2.18
3.09 1.52 1.99 1.22 2.71 2.71 2.79 1.70 1.87 1.31
PR
Latent heat
2.90
3.75 0.78 2.12 2.18 6.87 0.66 1.79 2.83 4.51 4.15
PFGC
1.87
1.73 0.44 1.16 1.01 2.64 3.04 1.88 2.98 0.83 2.97
MCORGC
M. Zia-Razzaz
0.001
f
and M. Moshfeghian / Fluid Phase Equilibria 91 (1993) 239-263
*
*
I
l
.
I
255
t
loo
1
P?~~~sure
[bar]
Fig. 9. Vaporization equilibrium ratios of a multicomponent data from Yarbrough and Vogel ( 1967).
mixture at 366 K; experimental
temperatures there are some minute differences between predicted and experimental enthalpies. The calculated deviation is almost constant for each isotherm and does not change significantly with increasing pressure. So, it seems that the observed deviations might be due to the poor accuracy of the ideal gas state enthalpy prediction at these temperatures, not the residual part as calculated by MCORGC.
M. Zia-Razzaz and M. Moshfeghian / Fluid Phase Equilibria 91 (1993) 239-263
\ \ \
I I I
n\
P
I
I
1
I
\
I
\
I
\
I
\
b 300
K ;
400 KI
500 KI I
I
0:
Al \ \
P
I I I I
I
0
9
I
1 I I
I
I I I
I
I
I I
I al
0 I \
I I
I
I
I I
I
I
I
I
I
I
I
I
I I + I
Fig. 10. Pressure-enthalpy
I \
I
a, I 4
0 I I I
diagram for ethane (Perry et al., 1984).
Figure 11 shows the pressure-volume behavior of ethane at three temperatures (300, 400 and 500 K) as well as the saturation states. With the exception of sub-cooled region and critical point, there is no significant deviation between calculated and experimental volumes. Normal boiling point is another thermodynamic property that can be predicted by an EOS. The normal boiling points of almost all substances can be found in the literature, while complete VLE data for many compounds are not available. So, comparing predicted and experimental boiling
M. Zia-Razzaz and M. Moshfeghian / Fluid Phase Equilibria 91 (1993) 239-263
\
I
\ \
1
\ \
I
\ \
1w
\
\
‘+
4 I
‘1
’
\
’
‘0 I 1300
\
‘b
i
I
’ ,500
\ K
\
T
1
\ \
\
257
IO
\ \
’ ‘4
K
\ a
2 al
5 i
rl:
Fig. 11. Pressure-volume
diagram for ethane (Perry et al., 1984).
points of pure compounds can be a good measure of the accuracy of an EOS. There are some compounds which are not used in the parameter optimization such as the paraffins n-C, rHZ4 to n-C20H42. The error in normal boiling point prediction of these compounds is less than 0.8 K and almost the same as for their lighter homologs (Zia-Razzaz, 1992). This proves the accuracy of this EOS modeling and shows that the optimized parameters can be used for compounds not included in the optimization data base.
258
M. Zia-Razzaz and M. Moshfeghian 1 Fluid Phase Equilibria 91 (1993) 239-263
CONCLUSIONS
A new group contribution EOS, MCORGC, based on the CORGC and an NRTL type local composition mixing rule applicable to both polar and nonpolar systems has been presented. Its parameters for 37 groups have been determined and reported. Its ability to predict the thermodynamics properties of pure and multicomponent systems has been tested against reported experimental data and very good agreements have been obtained. In addition, its ability and accuracy have been compared with other widely used EOSs and superior results have been obtained.
ACKNOWLEDGMENT
The authors wish to express their appreciation to the School of Engineering for use of computer facilities and the financial support of the Vice-Chancellor for Research of Shiraz University.
LIST OF SYMBOLS
a amn aT
a’
al ar A A att AR
AAD AAD% b bi b, br
ci Gl CYC c Catt
CORGC
attraction energy parameter of an EOS (cm6 bar mol12) attraction energy between groups m and n (cm” bar mol - ’ K- ‘) (aa/aT) (cm6 bar mole2 Ke2) (aa/av> (cm5 mol12 K-l) aliphatic group aromatic group Helmholtz free energy (J mol - ‘) attraction part of the Helmholtz free energy (J mol- ‘) residual Helmholtz free energy (J mol- ‘) average absolute deviation percentage average absolute deviation covolume ( cm3 mol - ‘) covolume of molecule i ( cm3 mol - ‘) covolume of group m ( cm3 bar mol - ‘) (ab/t?T) (cm’ mol-’ K-‘) equivalent degree of freedom of molecule i equivalent degree of freedom of group m cyclic group equivalent degree of freedom attractive constant chain-of-rotator group contribution equation of state
M. Zia-Razzaz and M. Moshfeghian / Fluid Phase Equilibria 91 (1993) 239-263
VLE J%9w,, w,, w, xi yb/4V yi Z Z att
equation of state group contribution equation of state residual enthalpy (J mol - ‘) vaporization enthalpy (KJ mol - ‘) auxiliary parameters of the modified CORGC equation branched molecule vaporization equilibrium ratio number of moles (mol) normal (straight-chain) molecule number of components number of groups number of data point nonrandom two liquid theory pressure (bar) attraction part of pressure (bar) saturated pressure (bar) parameters from group contribution Peng-Robinson equation normalized molecular surface area of molecule i normalized surface area of group m normalized molecular surface area of fluid universal gas constant (83.14 cm3 bar mol-’ K-l) residual entropy (J mol - ’ K - ‘) Soave-Redlich-Kwong equation temperature (K) covolume parameter temperature dependency (K) energy parameter temperature dependency (K) volume ( cm3 mol - ‘) liquid volume ( cm3 mol - ‘) vapor volume (cm3 mol - ‘) vapor-liquid equilibria weighting factors for the error objective functions liquid mole fraction of component i dimensionless density vapor mole fraction of component i compressibility coefficient attraction part of 2
Greek letters a amn 0,
dumbbell geometric constant, nonrandomness parameter area fraction of group m
EOS GC-EOS HR H “aP H2,,H3,,... ; n ;c NG NP
NRTL P P att P sat
PFGC PR 41 9m
Q
R SR
SRK T TZI TLI V V’ V”
1.037
259
260
M. Zia-Razzaz
and M. Moshfeghian 1 Fluid Phase Equilibria 91 (1993) 239-263
number of group m in molecule i fugacity coefficient local composition parameter Subscripts C i
r m II T
critical point properties molecular properties of component i reduced property group properties of group m group properties of group n derivation with respect to temperature
Superscripts
att fv I rot R V
attractive contribution free volume contribution liquid properties rotational contribution residual properties vapor properties
REFERENCES ASHRAE, 1982. Handbook of Fundamentals, American Society of Heating Refrigerants and Air Conditioning Engineer, New York. Besserer, G.J. and Robinson, D.B., 1973. J. Chem. Eng. Data, 18: 416. Boublik, T. and Nezbeda, I., 1977. Equation of state for hard dumbbells. Chem. Phys. Lett., 46: 315-316. Carnahan, N.F. and Starling, K.E., 1972. Intermolecular repulsions and the equation of state for fluids. AIChE J., 18: 1184-1189. Cheung, S. and Wang, D.I., 1964. Solubility of volatile gases in hydrocarbon solvents at cryogenic temperatures. Ind. Eng. Chem., Fundam., 3: 3550. Chien, C.H., Greenkorn, R.A. and Chao, K.C., 1983. Chain-of-rotators equation of state. AIChE J., 29: 560-571. Dehghani, F., 1991. M.Sc. Thesis. Chemical Engineering Department, Shiraz University, Iran. Djordjevich, L. and Budenholzer, R.A., 1970. J. Chem. Eng. Data, 15: 1. Fluid Properties Research Inc., 1980. Oklahoma State University, Stillwater, OK. Fredenslund, A. and Mollerup, J., 1974. Measurement and prediction of equilibrium ratios for the C,H, + CO, system. Faraday Trans. 1, 70: 1653. Geiseler, G. and Koehler, H., 1968. Ber. Bunsenges. Phys. Chem., 72: 697. Gun, R.D., Mcketa, J.J. and Ata, N., 1974. AIChE J., 20: 347. Hakuta, T., Nagahama, K. and Suda, S., 1970. Binary vapor-liquid equilibria of C02-CIH, hydrocarbons. Kagaku Kogaku, 34(9): 953 (in Japanese). Karla, H., Krishnan, T.R. and Robinson, D.B., 1976. J. Chem. Eng. Data, 21: 222. Karla, H., Robinson, D.B. and Besserer, G.J., 1977. J. Chem. Eng. Data, 25: 216.
M. Zia-Razzaz
and M. Moshfeghian
1 Fluid Phase Equilibria 91 (1993) 239-263
261
Kojima, K., Tochigi, K., Seki, H. and Watase, K., 1968. Kagaku Kogaku, 32: 149. Lin, H.M., Sebastian, H.M., Simnick, J.J. and Chao, K.C., 1979. Ind. Eng. Chem. Data, 24: 146. Marquardt, D.M., 1963. An algorithm for least squares estimation of nonlinear parameters. J. Sot. Ind. Appl. Math., 11: 431-441. Mentzer R.A., Greenkorn, R.A. and Chao, K.C., 1982. J. Chem. Thermodyn., 14: 817. Molin, H., Sebastian, H.M. and Chao, K.C., 1980. J. Chem. Eng. Data, 25: 254. Perry, R.H., Green, D.W. and Maloney, J.O., 1984. Perry’s Chemical Engineering Handbook, 6th edn., McGraw-Hill, New York. Pults, J.D., 1988. The Chain-of-Rotators Equation of State, Ph.D. Thesis. Purdu University. Schindler, D.L., Swift G.W. and Kurata, F., 1966. Hydrocarbon Processing, 45: 205. Shariat, M.H., Dehghani, F., and Moshfeghian, M., 1993. Extension and evaluation of the chain-rotator group contribution equation of state for prediction of thermodynamic properties of polar and non-polar compounds. Fluid Phase Equilibria, 85: 19-40. Stryjek, R., Chaplier, P.S. and Kobayashi, R.J., 1974. Chem. Eng. Data, 19: 334. Wichtrle, I. and Koyabashi, R., 1972. Chem. Eng. Data, 17: 9. Yarbrough, L. and Vogel, L., 1981. Chem. Eng. Prog., Symp. Ser., 63: 1. Yorizan, M., Sadamoto, S., Yyoshimurra, S., Masuoka, H., Shiki, N., Kimura, T. and Toyama, A., 1968. Vapor-liquid equilibrium at low temperatures. Kagaku Kogaku, 32(3): 257. Zia-Razzaz, M., 1992. M.Sc. Thesis. Chemical Engineering Department, Shiraz University, Iran. Zias, E.J. and Silberbergs, I.H., 1965. J. Chem. Eng. Data, 15: 253.
APPENDIX:
FUGACITY
AND ENTHALPY
EXPRESSIONS
Fugacity expression
The fugacity coefficient expression for an EOS can be obtained by differentiating the residual Hehnholtz energy with respect to component number of moles: - In Z T,V+ The resulting expression for the proposed equation is as follows:
In 4i =
In A=
4y - 3y2 C,(a - 1) (4+a)y (1 _y)2 + 2 (1
-3Y2+(I
+cr)ln(l _y)
-Y)’
+ C(a
A, _ I-
1
-
+
l)( 3y31 -Y)3 3cry2 (CI l)y’>
1
( >
(A3
diZj 1T,Kn,
(A3)
+
where
+
(Al)
Aj+b abi ln( 1 + 4y) - In Z
a(n2a) L
262
M. Zia-Razzaz
Ai = -RTV
and M. Moshfeghian / Fluid Phase Equilibria 91 (1993) 239-263
y VimqmH2, + em{ Qi[H5, + H2m( 1 - H6m)]
+ H3im ’ H2mH7im)
(A4)
bi = y v,bm m
Ci =
(A5)
T V,Cpj
(A6)
m
Qi = y
YinUrn
m
(A7)
(A9) k
H4,
=y
6&k,,,
(AlO)
k
H5,
=f
f6,
=
k
y
Bkzkmakm(akm
-
%&km@/RTV)2/H4m
(All)
8kzkrnak&krn
-
amm)(QlRWlH4m
(A19
k
(Al3)
k
Entropy
and en thalpy expressions
The residual entropy and enthalpy functions can be obtained by differentiating the residual Hehnholtz energy with respect to temperature: ARIRT=
1
4y - 3y2 c(cI - 1) (4 + a)y - 3y2 (1 _y)2 + 2 (1
+(l
+cc)ln(l -y)
-
-Y)'
a ln(l+4y) bRT
(Al4)
(Al%
M. Zia-Razzaz
and M. Moshfeghian
1 Fluid Phase Equilibria 91 (1993) 239-263
C(a - 1)(3 + 3ay - (a + l)y2) 2(1 -Y13
263
1 (A161
HR/RT=(Z-
1) +gT+$
HR/RT
_
=
_(z
-RTb(l
(Al?)
‘co!- 1)(32;131;;cc1 + ‘)f)
1)
4a ]--$(ff-~-+(l+4y) +4y)
(A181
where b T=dT
db ’
(A19
(AV
(A221 ak=
(A231
(
ahn = -amn ++ mn
0.18135 T >
(A24)
(A27)
239
Application of the CORGC equation of state with a new mixing rule to polar and nonpolar fluids Mohammad
Zia-Razzaz
’
and Mahmood
Moshfeghian
*
Department of Chemical Engineering, Shiraz University, Shiraz, Iran (Received December 22, 1992; accepted in final form May 3, 1993)
ABSTRACT Mohammad Ziz-Razzaz and Mahmood Moshfeghian, 1993. Application of the CORGC equation of state with a new mixing rule to polar and nonpolar fluids. Fluid Phase Equilibria, 91: 239-263. This paper presents a new group contribution equation of state (EOS) applicable to both polar and nonpolar compounds. This EOS is based on the chain-of-rotator group contribution (CORGC) equation considering an NRTL type local composition mixing rule. The parameters for 37 organic and inorganic groups have been calculated. Several pure components and mixtures have been studied including polar and multicomponent systems. Comparison between the predicted and experimental thermodynamic properties are also presented. Keywords: theory; equation of state; group contributions;
polar; nonpolar.
INTRODUCTION
Separation units such as distillation towers or extraction cascades are of great importance in almost any chemical plant. The maximum efficiency of such units is often limited by the thermodynamic equilibrium compositions of the phases concerned. Phase equilibrium information for a variety of mixtures over wide ranges of temperature and pressure is therefore of great importance. The number of mixtures of interest to, for example, the petrochemical industry is enormous, even when the scope of components is restricted to low molecular mass (e.g. a carbon number of less than 25). Hence, it is desirable to be able to predict phase equilibria based on as little experimental information as possible or at best without any experimental data at all. The equations of state (EOSs) are the commonest tool for predicting thermodynamic properties of pure and mixed components. Old EOSs like * Corresponding author. ’ Graduate student. 037%3812/93/%06.00 0
1993 - Elsevier Science Publishers B.V. All rights reserved
240
M. Zia-Razzaz
and M. Moshfeghian 1 Fluid Phase Equilibria 91 (1993) 239-263
Redlich-Kwong (RK), Soave-Redlich-Kwong (SRK) and PengRobinson (PR) are written for nonpolar components in the gas phase and then generalized to liquid phase and slightly polar components. Predictions of thermodynamic properties in the liquid phase can be improved by using activity coefficient equations based on the group contribution models. Some well-known equations of this kind are NRTL and UNIFAC. The new generation of EOSs has been made by combination of both group contribution and free volume models. This kind of EOS is efficient in predicting thermodynamic properties of both gases and liquids, and polar and nonpolar systems. Parameter from group contribution (PFGC), group contribution equation of state (GC-EOS) and chain-of-rotator group contribution (CORGC) are members of this generation of EOSs. CORGC is the latest group contribution EOS developed by Pults and Chao (1989). They obtained parameters for 20 nonpolar and slightly polar groups. For polar compounds, they also added a corrective term to the EOS. The results of this polar version of the equation (CORGCP) are not so good. Recently, Dehghani (1991) and Shariat et al. (1993) revised the parameters of CORGC and extended its application to polar and halogenated systems. In this work a new local composition model has been combined with CORGC. The parameters for 37 groups have been calculated and the capabilities of this equation in predicting thermodynamic properties of pure and mixed, polar and nonpolar systems have been evaluated.
PROPOSED
EQUATIONS
The Helmholtz free energy form of the proposed EOS is formulated eqn. (1):
1
AR 4y-3y2 RT= (1 -y)’
c(a - 1) (4+ cr)y - 3y2 + 2 (1 -Y)’
+(l
+cr)ln(l
-y)
-
a
in
ln(l+4y) bRT
(1) This equation is a contribution of three terms, considered for the translational, rotational and attractive energy of molecules. The translational part of the equation is the same as the Carnahan-Starling hard sphere equation (Carnahan and Starling, 1972) :
(> AR
RT
=
f”
4y - 3y2 (1
-Y12
The rotational
(2)
part is taken from the Boublik and Nezbeda (1977) hard
M. Zia-Razzaz
241
and M. Moshfeghian / Fluid Phase Equilibria 91 (1993) 239-263
dumbbell equation extended by Chien et al. (1983) to hard chain fluids: =c(a: - 1) (4+a)y 2
-3y*
(1 -Y)’
1+
For attractive terms a Redlich-Kwong
(1 + a)ln(l -y)
-
type equation has been applied:
a ln(l+4y) -AR bRT ( RT >att = -
(4)
In these equations, y = b/4V is the reduced density. Pressure- and other thermodynamic functions can be derived from the Helmholtz energy, by using appropriate mathematical manipulations: (5)
IRT
(6)
T,n
The attractive part of the pressure expressed as follows:
patt =-
and compressibility
factor
is then
a
(7)
V( V + b)
zatt = _
where the parameter a’ is the first derivative of a with respect to volume at constant temperature and composition:
(9) Finally, the proposed equation can be expressed as a summation attractive and repulsive contributions: zJ+Y+Y2-Y3
+[c(I-l),2][3y+3;>_$+l)y3]
(1 -VI' a -RT(V+b)
of the
+
(10)
The following mixing rules are used in the proposed equation. In these relations the parameters b,, c,,, and q,,, are used for the group m, and b, c and Q for the fluid covolume, equivalent degree of freedom and the normalized area, respectively.
242
M. Zia-Razzaz NC
and M. Moshfeghian / Fluid Phase Equilibria 91 (1993) 239-263
NG
b = ~x~&,,&~
c
=
Q=
i
m
NC
NG
(11) (12)
~x,~vimcm i m NC
NG
I
m
~x,&rnqm
(13)
NC
(14) y = b/4V
(15) -
(16) (17)
=
RTQ ym 8m(H2,H6,
(18)
where H2,, H5, and H6, are auxiliary variables as defined in the Appendix. The parameters b, and a,,,,, are temperature dependent: b, = bz exp( - T/T?)
(19)
amn= az,(T/T$,)
(20)
- o~18135 exp( - T/T:,)
To distinguish between the original CORGC EOS and the proposed one, we have called our version MCORGC. GROUP PARAMETERS
The proposed EOS contains six parameters per group bz, Tz, cm, qm, atm and T&. The optimized values of these parameters are presented in Table 1 for 37 groups. In addition, for each pair of groups three interaction parameters are needed: the attraction parameter a&, and its characteristic temperature T&,, and the nonrandom parameter CI,,. Based on the original CORGC, the group interaction parameters are assumed to be symmetrical, i.e. CI,, = anrn, a,,, = an,,, and Tz, = T,*,. The optimized values of these parameters are presented in Table 2. The normalized area parameter qm is calculated using van der Waals areas normalized by a base parameter q,,, = 10 for methane.
M. Zia-Razzaz
and M. Moshfeghian
/ Fluid Phase Equilibria 91 (1993) 239-263
243
TABLE 1 Group parameters for the MCORGC
EOS
Group ID
Group name
T, (K)
1 2 3 4 5 6 7 8 9 10
al-CH, al-CH,-al :CH>c: (CH)ar ( C) ar-al ar_CH, ar-CH (C) ar-ar
11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37
(CH,)cyc (6 ring) (CHJCYC (5 ring) =CH, =CHHZ I% CH4 C,He CO* C,Hz, H,S MeOH -OH (1”) -OH (2”) -OH (3”) CH&O-CH&OCH3-O-oHz0 NH, CC&F, CC&FH CCIF, CF,H CF, CFCI, C,Cl,F,
b: ( cm3/mol-‘)
a&/1000
Tk (K)
52.91 61.40 33.75 34.07 72.53 32.48 10.00 41.75
321.25 973.40 1340.00 1823.80 683.59 7431.30 284.64 4000.00 50.00 644.34
0.412 1.024 - 0.448 -2.671 0.627 0.200 0.353 0.162 - 1.062 0.708
7.31 4.66 4.66 0.72 3.45 1.03 7.31 4.66 0.73 4.66
46.19
492.94
1.389
4.66
101.32
449.58
36.74 35.22 39.68 78.34 80.46 132.97 92.43 102.78 81.63 150.40 50.30 68.99 61.38 83.72 75.02 57.44 45.62 78.55 85.32 139.40 150.61 138.74 108.23 125.25 200.47 226.98
241.72 355.70 55.91 139.32 220.45 345.26 347.00 328.91 367.43 597.75 612.12 205.83 419.69 3042.60 1834.60 1597.70 7338.60 686.51 529.98 511.62 564.74 343.45 425.52 218.60 540.99 463.96
19.237 22.546 2.116 - 1.346 0.879 -4.520 -6.881 4.173 10.183 - 13.084 1.357 10.996 1.292 10.195 -6.472 -2.014 -4.925 - 14.399 - 14.068 14.938 4.277 8.331 -4.101 7.135 3.394 10.947
6.48 3.72 4.76 8.48 10.00 14.60 11.10 10.70 10.70 12.40 10.34 10.34 10.34 12.83 10.45 9.38 6.72 12.07 10.00 10.00 10.00 10.00 10.00 10.00 10.00 10.00
270.08 323.89 9.838 45.617 49.653 66.860 94.261 95.270 109.54 268.40 606.45 243.99 817.37 113.42 91.30 130.42 157.68 132.62 123.15 245.81 308.26 184.89 144.25 129.38 413.97 474.62
190.94 344.64 248.12 148.19 250.47 331.25 277.22 337.47 383.63 366.33 131.27 230.81 110.69 841.81 502.74 208.17 189.49 528.26 393.49 484.91 479.74 322.88 312.79 204.44 479.38 408.46
62.50 43.40
74.030 80.226 - 167.1 - 28942 121.59 271.23 77.076 32.194 3820.2 78.638
287.96
472.60 268.03 303.84 577.58 1273.5 396.78 852.16 196.51 573.69
244
M. Zia-Razzaz and M. Moshfeghian 1 Fluid Phase Equilibria 91 (1993) 239-263
TABLE 2 Interaction parameters for dissimilar groups of the MCORGC Group n 2 3 4 5 6 8 9 12 13 15 16 17 18 20 22 23 24 25 26 27 28 3 5 6 8 12 13 15 16 17 18 20 22 25 26 27 28 15 23 24 6 7 8 9
Group m 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 3 3 4 5 5 5 5
EOS
a:,/1000 ( cm6 bar mol12) 57.678 49.946 246.91 70.344 22.12 63.657 50.161 38.154 23.563 95.169 66.997 67.517 28.671 71.38 156.88 129.87 150.6 82.724 63.664 105.71 114.53 145.5 96.356 169.3 28.737 146.08 89.322 10.289 62.095 69.982 95.809 65.772 95.712 25.549 32.853 51.516 82.642 -70.353 49.946 524.42 254.72 79.826 83.532 247.69
626.26 746.9 1074.4 557.14 644.38 765.86 3909.5 661.43 462.73 145.95 226.47 328.82 675.67 436.56 371.31 285.42 277.56 586.04 726.27 577.53 492 185.78 526.06 1015.4 1711.9 521.43 479.2 7949.3 465.01 421.56 272.31 577.44 466.61 1860.8 974.49 2049.7 364.44 19999 328.98 265.91 939.25 493.83 630.56 1007.7
0.000 0.000 0.000 0.000
0.000 0.000 0.000 0.022 0.266 -0.981 1.211 -64.8 4.825 - 1.875 -0.235 0.299 - 0.228 - 0.294 -0.991 - 0.689 -1.120 0.000 0.000 0.000 0.000 -0.544 - 10.000 - 0.422 - 2.805 -0.934 500.000 -51.980 - 1.020 24.210 6.221 - 4.047 - 15.020 23.300 1.309 -0.070 0.000 0.000 0.000 0.000
M. Zia-Razzaz
and M. Moshfeghian
/ Fluid Phase Equilibria 91 (1993) 239-263
245
TABLE 2 (continued) Group n
Group m
a&/1000 ( cm6 bar mol -‘)
16 7 8 9 13 16 17 18 20 17 18 18 20 29
5 6 6 6 12 15 15 15 15 16 16 17 17 21
69.148 199.46 55.289 10000 58.605 43.449 41.504 50.476 108.19 54.612 61.251 68.703 82.78 222.93
a WI”
399.37 519.97 4000 129.55 584.47 195.66 229.87 242.85 150.51 302.22 261.53 299.38 346.67 381.14
- 0.464 0.000 0.000 0.000 -0.314 -0.137 -0.130 -0.561 -0.719 0.000 -0.142 1.471 0.205 0.300
The remaining parameters are determined from pure component binary mixture VLE data as described in the following sections. OPTIMIZATION
and
OF GROUP PARAMETERS
A computer package using a subroutine based on a nonlinear regression algorithm was developed to optimize the group parameters (Zia-Razzaz, 1993). This subroutine minimizes the error objective function by using the method of Marquardt (1963). In order to optimize pure group parameters, a vast pure component VLE data bank has been used. The predicted and experimental data for each set of parameters are then compared, and the summation of the error weighted squares minimized. The following objective function is used for pure component VLE data:
(22) where Wi are the weighting factors for the relative errors of vapor pressure, vapor volume, liquid volume and latent heat of vaporization, respectively. In the above equation, Psat, V”, V’ and Hvap represent vapor pressure, vapor volume, liquid volume and heat of vaporization, respectively. The subscripts E and C stand for experimental and calculated values, respectively. Most group interaction parameters have been calculated using binary mixture VLE data. For these systems, errors in predicting vaporization
246
M. Zia-Razzaz
and M. Moshfeghian 1 Fluid Phase Equilibria 91 (1993) 239-263
equilibrium ratios are used as a measure of parameter following objective function is used for binary mixtures:
accuracy.
The
(23) where Ki and K2 are the VLE ratios (K values) for components 1 and 2, W, and W, are weighting factors, and the subscripts E and C represent experimental and calculated quantities, respectively. In this case all predicted properties have been calculated at constant temperature and pressure. This method is more efficient than the classical equilibrium calculation method, which requires about ten equilibrium calculations for each point and consuming about ten times as much computer time. However, for the final results and comparisons, the properties are calculated at the equilibrium condition using flash calculations at constant temperature and pressure. RESULTS
The capabilities of the proposed EOS to predict VLE properties of pure and mixture systems were examined. About 70 pure compounds and 30 binary systems were studied. The calculated properties are usually as accurate as experimental values and the order of errors (especially for pure compounds) is very low. The proposed equation proves to be a useful tool for the prediction of a number of pure fluid thermodynamic properties. Saturated thermodynamic properties of a vast set of substances were studied. The summary results for 21 pure compounds are given in Table 3, and for a few compounds the results are shown graphically in Figs. l-6. These are typical of the results obtained in this study. Vapor pressure is the most important pure VLE property to be predicted by an EOS, since a good representation of vapor pressure will give good results in the mixture VLE calculations. The results of vapor pressure prediction for several components are shown in Figs. 1 and 2. The total average absolute error for 70 compounds studied by Zia-Razzaz ( 1992) is 1.47%, and for most compounds this value is less than unity. The ability of an EOS to predict liquid densities is a measurement of its accuracy. The results of saturated liquid densities calculated for a number of compounds are shown in Figs. 3 and 4. The average absolute error of V’ prediction is 0.98% for 42 studied substances (Zia-Razzaz, 1992). At this moment, no EOS is known with the same capability of predicting liquid densities.
M. Zia-Razzaz TABLE
and M. Moshfeghian / Fluid Phase Equilibria 91 (1993) 239-263
247
3
Pure component
vapor-liquid
saturation
Substance
No. of points
Jyw
F”
CA
29
0.44
n-GH,,
30
n-C&L,
(AAD%
of MCORGC
from experimental
data)
F’
P”“’
T range (K) T,. range (K)
Data source
0.53
0.95
0.33
Perry et al. (1984)
1.01
1.65
1.35
0.94
17
2.25
2.75
1.76
0.70
n-G&
7
2.11
1.19
1.81
0.55
n-C,H,,
8
2.73
2.50
1.92
1.07
40
1.05
1.47
0.76
0.86
9
2.28
2.92
1.42
1.90
GW&
12
0.48
-
0.25
0.65
CF,
15
0.73
1.99
1.36
1.00
CClF,H
27
0.41
0.43
0.79
0.44
CClF3
11
0.83
0.90
2.39
0.74
W&F,
39
1.10
0.95
0.76
1.12
CC&F
39
0.68
0.69
0.89
0.25
N,
11
3.04
2.19
0.92
0.86
NH,
32
1.88
1.64
0.78
1.09
CO,
46
0.94
1.00
0.93
0.43
H2S
25
1.16
1.42
1.07
0.77
Hz0
51
2.98
3.63
2.25
1.70
CH,OH
24
5.37
1.17
0.91
1.68
C,H,OH
9
6.77
6.69
2.93
3.51
10
9.03
5.87
4.11
1.38
144.26-299.82 0.54-0.98 255.37-416.48 0.60-0.98 309.19-462.00 0.66-0.98 371.60-500.00 0.69-0.93 398.80-535.00 0.70-0.94 322.04-533.15 0.57-0.95 383.78-575.00 0.65-0.97 303.15-413.15 0.49-0.67 133.15-210.93 0.59-0.93 144.26-288.71 0.48-0.96 144.26-294.26 0.48-0.97 183.15-394.26 0440.94 250.93-460.93 0.53-0.98 77.04 124.82 0.61-0.99 199.82-377.59 0.49-0.93 216.48-299.82 0.71-0.99 222.04-355.37 0.59-0.95 338.71-616.48 0.52-0.95 273.15-483.15 0.53-0.94 351.45-503.00 0.68-0.97 371.20-533.10 0.70-0.99
G-b
CH,GH,
n-C,H,OH
FRPI - Fluid Properties
Research
Inc.
Perry et al. ( 1984) Perry et al. (1984) FPRI
(1980)
FPRI
(1980)
Perry et al. ( 1984) Perry et al. (1984) Perry et al. (1984) ASHRAE
(1982)
ASHRAE
(1982)
ASHRAE
(1982)
ASHRAE
(1982)
ASHRAE
(1982)
FPRI
( 1980)
FPRI
( 1980)
Perry et al. (1984) Perry et al. (1984) Perry et al. (1984) Perry et al. (1984) Perry et al. (1984) Perry et al. (1984)
248
M. Zia-Razzaz
and M. Moshfeghian 1 Fluid Phase Equilibria 91 (1993) 239-263
300
400
Temperature
500
[K]
Fig. 1. Vapor pressure of some highly polar solutions.
1
+7~~~-17-m-m-rr-~~~ 350
400
--met
500
450
Temperature
5
[K]
Fig. 2. Vapor pressure of primary alcohols; data from Fluid Properties Research Inc. (1980).
Saturated vapor volume is another VLE property that can be calculated by an EOS. The average absolute error of V” prediction for 40 studied compounds is 1.28% ( Zia-Razzaz, 1992). The latent heat of vaporization of a pure compound can be calculated by subtracting coexistence vapor and liquid residual enthalpies. The latent
M. Zia-Razzaz
200
and M. Moshfeghian
II1III,,,IIII1\, 220
7”“” 240
III,IIIIIIIIV 260
1 Fluid Phase Equilibria 91 (1993) 239-263
III IIIIII*,!III!I 260 300
Temperature
‘“I 320
340
249
rl
: 16 0
[K]
Fig. 3. Saturated liquid volumes of CO2 and H,S.
40 100
150
200
250
300
Temperature
350
400
4 i0
[K]
Fig. 4. Saturated liquid volume of refrigerants.
heats of several compounds have been calculated and are shown in Figs. 5 and 6. The average absolute deviation for latent heat prediction of 39 studied compounds is 1.65% (Zia-Razzaz, 1992). Equilibrium calculations have been carried out for 30 binary systems including polar, nonpolar and light gas components (Zia-Razzaz, 1992).
M. Zia-Razzaz
250
200
220
and M. Moshfeghian 1 Fluid Phase Equilibria 91 (1993) 239-263
240
260
300
280
320
Temperature Fig. 5. Heats of vaporization
Oj
ITIII8III,,
100
*
150
3 8’#““‘)“‘t
250
IIIII,IIIIIIIII,II11I,II
300
Temperature Fig. 6. Heats of vaporization
:
of CO, and H,S.
I1I1III,3IIII!III,
200
340
[K]
350
400
4
[K]
of refrigerants.
Typical results for mixture flash calculations are tabulated in Table 4. Figures 7 and 8 show the comparison between calculated and experimental equilibria for a few typical systems. One isobar for the methanol-water system has been studied, as shown in Fig. 7. The AAD%s for K value prediction at 1.013 bar are 1.96 and 2.56
M, Zia-Razzaz
and M. Moshfeghian / Fluid Phase Equilibria 91 (1993) 239-263
251
TABLE 4 Vaporization data)
equilibrium ratios for binary systems (AAD% of MCORGC from experimental
Mixture camp. (1:2)
No. of points
Temp. range (K)
Pressure range (KPa)
K,
(AAD%)
;AD%)
Data source
CH, + C3H, CH, + r&H,, CH, + n-C,&, CH, + n-C,,H,, CH, + CO, N, + CH, C,H, + n-CmH,, C2H, + H$ C,H, + GH, C,H, + r&H,4 C,H, + CO, N, + C,Hh N, + H,S N, + Cd% N, + r&H,, CO, + n-CJ-4, CO2 + n-Cd-I,, CH,OH + H,O
56 15 21 12 198 118 39 28 32 29 185 43 52 41 31 26 40 21 18 17
130.3-213.7 310.9-410.9 423.1-583.0 462.2-703.3 173.3-283.1 121.0-183.1 310.9-510.9 199.9-283.1 144.2-255.3 338.7-449.8 222.0-298.1 138.7-194.2 256.4-344.2 173.1-353.1 277.4-377.5 227.9-283.1 277.6-377.5 337.6-373.1 313.5-332.5 327.7-327.7
1.72-55.16 4.45- 17.24 27.6-116.1 20.3-50.7 6.89-85.20 2.79-49.64 3.45-68.95 0.99-29.87 0.03-14.12 6.89-75.84 6.24-66.30 3.45-134.1 17.3-207.0 13.8-137.9 2.50-141.0 0.33-39.37 2.28-96.32 1.01-1.01 0.14-0.74 0.06-0.23
6.05 5.06 9.38 9.19 5.96 4.18 11.08 5.75 5.84 7.39 2.47 12.21 8.45 12.63 12.74 5.77 5.67 1.96 8.46 4.97
9.76 4.41 7.21 5.84 6.91 4.65 14.75 5.99 9.98 7.23 3.35 18.78 3.63 14.84 9.64 8.81 7.32 2.56 8.63 4.68
27 11 17 20 5, 29 26 13 15 8 31 9, 12 26 2 25 15 14 2 16 21 19
21
398.8-409.3
1.01-1.01
3.65
6.39
n-C,H,, + C,H,C,HS n-C,H,, + CzH,H,
19
for methanol and water, respectively, an excellent result for an EOS. Figure 8 shows the binary mixture of ethyl benzene and n-heptane at 327.76 K. The AAD%s of K value prediction of this system are 4.97 and 4.68, respectively. The parameters used for this system are calculated from pure component VLE data. Good representation of this nonideal aromatic + paraffin system shows further capability of the proposed EOS. For most EOSs, accurate representation of binary systems is an indication of good prediction of multicomponent mixtures. In order to evaluate this capability of the proposed equation, a multicomponent mixture was studied, containing N *, CO2 and six paraffins. The predicted and experimental K values are compared in Fig. 9, which shows a fairly good ability of this EOS to predict complicated multicomponent systems.
252
M. Zia-Razzaz
and M. Moshfeghian / Fluid Phase Equilibria 91 (1993) 239-263
D
Fig. 7. T-x-y
COMPARISON
diagram of the water + methanol system at 1.013 bar.
WITH OTHER EOSs
The saturation state properties of a number of compounds have been calculated using SRK, PR, PFGC and the MCORGC. The comparison results in terms of average absolute deviation are shown in Table 5. As can be seen, almost all predictions of the proposed CORGC are much more accurate than those of the other equations.
M. Zia-Razzaz
and M. Moshfeghian
1 Fluid Phase Equilibria 91 (1993) 239-263
253
0.10
0.05
O.o0_1,,~..~..~,~,.......,,........,..,......,...rr 0.0
Fig. 8. P-x-y
02
0.4
n-Heptane
0.0
mole
0-B
diagram of the n-heptane + ethylbenzene
OTHER THERMODYNAMIC
0
fraction
system at 327.76 K.
PROPERTIES
In order to demonstrate the ability of the proposed EOS to predict single phase properties of compounds, superheated and sub-cooled ethane has been studied. The enthalpy and volume at three temperatures (300, 400 and 500 K) have been calculated. Figure 10 shows the pressure-enthalpy diagram for ethane. As can be seen, deviations in the saturated states and 300 K isotherm are in the range of experimental data accuracy. At higher
1.00 1.18 1.00 2.38 1.37 1.00 3.33 1.21 1.93 1.80
1.28 1.17 1.22 1.55 1.81 1.97 4.91 1.05 1.03 1.09
1.71
Methane Ethane Propane n-Butane i-Butane Nitrogen Ammonia Water Ethylene Propylene
Average
1.62
PR
SRK
1.90
1.60 1.03 1.06 1.33 1.12 1.00 1.91 2.67 4.25 3.03
PFGC
pressure
Vapor
Substance
0.98
0.69 0.33 0.35 0.94 1.11 0.86 1.09 1.70 0.37 2.34
MCORGC
1.38
0.94 0.34 1.05 0.94 0.75 2.38 5.41 0.71 0.68 0.57
SRK
1.88
2.12 1.21 0.83 1.94 1.71 2.10 2.90 2.18 2.24 1.56
PR
2.83
3.05 0.50 2.32 2.88 2.47 1.30 2.01 2.81 6.11 4.86
PFGC
Vapor vohime
1.79
1.60 0.53 0.88 1.65 1.42 2.19 1.64 3.63 1.46 2.93
MCORGC
10.8
5.58 7.07 8.08 8.79 9.40 22.6 25.4 3.62 6.25 6.61
SRK
8.70
9.23 6.95 6.07 4.86 5.61 11.8 15.8 10.6 1.35 6.55
PR
Liquid volume
Comparison of saturated properties calculated by four EOSS in terms of AAD%
TABLE 5
3.24
4.89 3.68 2.91 1.62 2.79 5.04 4.53 1.59 2.63 2.71
PFGC
1.24
0.98 0.95 1.57 1.35 1.98 0.92 0.78 2.25 0.63 0.99
MCORGC
2.65
3.92 1.58 2.60 1.57 3.16 3.10 4.18 2.07 1.67 1.29
SRK
2.18
3.09 1.52 1.99 1.22 2.71 2.71 2.79 1.70 1.87 1.31
PR
Latent heat
2.90
3.75 0.78 2.12 2.18 6.87 0.66 1.79 2.83 4.51 4.15
PFGC
1.87
1.73 0.44 1.16 1.01 2.64 3.04 1.88 2.98 0.83 2.97
MCORGC
M. Zia-Razzaz
0.001
f
and M. Moshfeghian / Fluid Phase Equilibria 91 (1993) 239-263
*
*
I
l
.
I
255
t
loo
1
P?~~~sure
[bar]
Fig. 9. Vaporization equilibrium ratios of a multicomponent data from Yarbrough and Vogel ( 1967).
mixture at 366 K; experimental
temperatures there are some minute differences between predicted and experimental enthalpies. The calculated deviation is almost constant for each isotherm and does not change significantly with increasing pressure. So, it seems that the observed deviations might be due to the poor accuracy of the ideal gas state enthalpy prediction at these temperatures, not the residual part as calculated by MCORGC.
M. Zia-Razzaz and M. Moshfeghian / Fluid Phase Equilibria 91 (1993) 239-263
\ \ \
I I I
n\
P
I
I
1
I
\
I
\
I
\
I
\
b 300
K ;
400 KI
500 KI I
I
0:
Al \ \
P
I I I I
I
0
9
I
1 I I
I
I I I
I
I
I I
I al
0 I \
I I
I
I
I I
I
I
I
I
I
I
I
I
I I + I
Fig. 10. Pressure-enthalpy
I \
I
a, I 4
0 I I I
diagram for ethane (Perry et al., 1984).
Figure 11 shows the pressure-volume behavior of ethane at three temperatures (300, 400 and 500 K) as well as the saturation states. With the exception of sub-cooled region and critical point, there is no significant deviation between calculated and experimental volumes. Normal boiling point is another thermodynamic property that can be predicted by an EOS. The normal boiling points of almost all substances can be found in the literature, while complete VLE data for many compounds are not available. So, comparing predicted and experimental boiling
M. Zia-Razzaz and M. Moshfeghian / Fluid Phase Equilibria 91 (1993) 239-263
\
I
\ \
1
\ \
I
\ \
1w
\
\
‘+
4 I
‘1
’
\
’
‘0 I 1300
\
‘b
i
I
’ ,500
\ K
\
T
1
\ \
\
257
IO
\ \
’ ‘4
K
\ a
2 al
5 i
rl:
Fig. 11. Pressure-volume
diagram for ethane (Perry et al., 1984).
points of pure compounds can be a good measure of the accuracy of an EOS. There are some compounds which are not used in the parameter optimization such as the paraffins n-C, rHZ4 to n-C20H42. The error in normal boiling point prediction of these compounds is less than 0.8 K and almost the same as for their lighter homologs (Zia-Razzaz, 1992). This proves the accuracy of this EOS modeling and shows that the optimized parameters can be used for compounds not included in the optimization data base.
258
M. Zia-Razzaz and M. Moshfeghian 1 Fluid Phase Equilibria 91 (1993) 239-263
CONCLUSIONS
A new group contribution EOS, MCORGC, based on the CORGC and an NRTL type local composition mixing rule applicable to both polar and nonpolar systems has been presented. Its parameters for 37 groups have been determined and reported. Its ability to predict the thermodynamics properties of pure and multicomponent systems has been tested against reported experimental data and very good agreements have been obtained. In addition, its ability and accuracy have been compared with other widely used EOSs and superior results have been obtained.
ACKNOWLEDGMENT
The authors wish to express their appreciation to the School of Engineering for use of computer facilities and the financial support of the Vice-Chancellor for Research of Shiraz University.
LIST OF SYMBOLS
a amn aT
a’
al ar A A att AR
AAD AAD% b bi b, br
ci Gl CYC c Catt
CORGC
attraction energy parameter of an EOS (cm6 bar mol12) attraction energy between groups m and n (cm” bar mol - ’ K- ‘) (aa/aT) (cm6 bar mole2 Ke2) (aa/av> (cm5 mol12 K-l) aliphatic group aromatic group Helmholtz free energy (J mol - ‘) attraction part of the Helmholtz free energy (J mol- ‘) residual Helmholtz free energy (J mol- ‘) average absolute deviation percentage average absolute deviation covolume ( cm3 mol - ‘) covolume of molecule i ( cm3 mol - ‘) covolume of group m ( cm3 bar mol - ‘) (ab/t?T) (cm’ mol-’ K-‘) equivalent degree of freedom of molecule i equivalent degree of freedom of group m cyclic group equivalent degree of freedom attractive constant chain-of-rotator group contribution equation of state
M. Zia-Razzaz and M. Moshfeghian / Fluid Phase Equilibria 91 (1993) 239-263
VLE J%9w,, w,, w, xi yb/4V yi Z Z att
equation of state group contribution equation of state residual enthalpy (J mol - ‘) vaporization enthalpy (KJ mol - ‘) auxiliary parameters of the modified CORGC equation branched molecule vaporization equilibrium ratio number of moles (mol) normal (straight-chain) molecule number of components number of groups number of data point nonrandom two liquid theory pressure (bar) attraction part of pressure (bar) saturated pressure (bar) parameters from group contribution Peng-Robinson equation normalized molecular surface area of molecule i normalized surface area of group m normalized molecular surface area of fluid universal gas constant (83.14 cm3 bar mol-’ K-l) residual entropy (J mol - ’ K - ‘) Soave-Redlich-Kwong equation temperature (K) covolume parameter temperature dependency (K) energy parameter temperature dependency (K) volume ( cm3 mol - ‘) liquid volume ( cm3 mol - ‘) vapor volume (cm3 mol - ‘) vapor-liquid equilibria weighting factors for the error objective functions liquid mole fraction of component i dimensionless density vapor mole fraction of component i compressibility coefficient attraction part of 2
Greek letters a amn 0,
dumbbell geometric constant, nonrandomness parameter area fraction of group m
EOS GC-EOS HR H “aP H2,,H3,,... ; n ;c NG NP
NRTL P P att P sat
PFGC PR 41 9m
Q
R SR
SRK T TZI TLI V V’ V”
1.037
259
260
M. Zia-Razzaz
and M. Moshfeghian 1 Fluid Phase Equilibria 91 (1993) 239-263
number of group m in molecule i fugacity coefficient local composition parameter Subscripts C i
r m II T
critical point properties molecular properties of component i reduced property group properties of group m group properties of group n derivation with respect to temperature
Superscripts
att fv I rot R V
attractive contribution free volume contribution liquid properties rotational contribution residual properties vapor properties
REFERENCES ASHRAE, 1982. Handbook of Fundamentals, American Society of Heating Refrigerants and Air Conditioning Engineer, New York. Besserer, G.J. and Robinson, D.B., 1973. J. Chem. Eng. Data, 18: 416. Boublik, T. and Nezbeda, I., 1977. Equation of state for hard dumbbells. Chem. Phys. Lett., 46: 315-316. Carnahan, N.F. and Starling, K.E., 1972. Intermolecular repulsions and the equation of state for fluids. AIChE J., 18: 1184-1189. Cheung, S. and Wang, D.I., 1964. Solubility of volatile gases in hydrocarbon solvents at cryogenic temperatures. Ind. Eng. Chem., Fundam., 3: 3550. Chien, C.H., Greenkorn, R.A. and Chao, K.C., 1983. Chain-of-rotators equation of state. AIChE J., 29: 560-571. Dehghani, F., 1991. M.Sc. Thesis. Chemical Engineering Department, Shiraz University, Iran. Djordjevich, L. and Budenholzer, R.A., 1970. J. Chem. Eng. Data, 15: 1. Fluid Properties Research Inc., 1980. Oklahoma State University, Stillwater, OK. Fredenslund, A. and Mollerup, J., 1974. Measurement and prediction of equilibrium ratios for the C,H, + CO, system. Faraday Trans. 1, 70: 1653. Geiseler, G. and Koehler, H., 1968. Ber. Bunsenges. Phys. Chem., 72: 697. Gun, R.D., Mcketa, J.J. and Ata, N., 1974. AIChE J., 20: 347. Hakuta, T., Nagahama, K. and Suda, S., 1970. Binary vapor-liquid equilibria of C02-CIH, hydrocarbons. Kagaku Kogaku, 34(9): 953 (in Japanese). Karla, H., Krishnan, T.R. and Robinson, D.B., 1976. J. Chem. Eng. Data, 21: 222. Karla, H., Robinson, D.B. and Besserer, G.J., 1977. J. Chem. Eng. Data, 25: 216.
M. Zia-Razzaz
and M. Moshfeghian
1 Fluid Phase Equilibria 91 (1993) 239-263
261
Kojima, K., Tochigi, K., Seki, H. and Watase, K., 1968. Kagaku Kogaku, 32: 149. Lin, H.M., Sebastian, H.M., Simnick, J.J. and Chao, K.C., 1979. Ind. Eng. Chem. Data, 24: 146. Marquardt, D.M., 1963. An algorithm for least squares estimation of nonlinear parameters. J. Sot. Ind. Appl. Math., 11: 431-441. Mentzer R.A., Greenkorn, R.A. and Chao, K.C., 1982. J. Chem. Thermodyn., 14: 817. Molin, H., Sebastian, H.M. and Chao, K.C., 1980. J. Chem. Eng. Data, 25: 254. Perry, R.H., Green, D.W. and Maloney, J.O., 1984. Perry’s Chemical Engineering Handbook, 6th edn., McGraw-Hill, New York. Pults, J.D., 1988. The Chain-of-Rotators Equation of State, Ph.D. Thesis. Purdu University. Schindler, D.L., Swift G.W. and Kurata, F., 1966. Hydrocarbon Processing, 45: 205. Shariat, M.H., Dehghani, F., and Moshfeghian, M., 1993. Extension and evaluation of the chain-rotator group contribution equation of state for prediction of thermodynamic properties of polar and non-polar compounds. Fluid Phase Equilibria, 85: 19-40. Stryjek, R., Chaplier, P.S. and Kobayashi, R.J., 1974. Chem. Eng. Data, 19: 334. Wichtrle, I. and Koyabashi, R., 1972. Chem. Eng. Data, 17: 9. Yarbrough, L. and Vogel, L., 1981. Chem. Eng. Prog., Symp. Ser., 63: 1. Yorizan, M., Sadamoto, S., Yyoshimurra, S., Masuoka, H., Shiki, N., Kimura, T. and Toyama, A., 1968. Vapor-liquid equilibrium at low temperatures. Kagaku Kogaku, 32(3): 257. Zia-Razzaz, M., 1992. M.Sc. Thesis. Chemical Engineering Department, Shiraz University, Iran. Zias, E.J. and Silberbergs, I.H., 1965. J. Chem. Eng. Data, 15: 253.
APPENDIX:
FUGACITY
AND ENTHALPY
EXPRESSIONS
Fugacity expression
The fugacity coefficient expression for an EOS can be obtained by differentiating the residual Hehnholtz energy with respect to component number of moles: - In Z T,V+ The resulting expression for the proposed equation is as follows:
In 4i =
In A=
4y - 3y2 C,(a - 1) (4+a)y (1 _y)2 + 2 (1
-3Y2+(I
+cr)ln(l _y)
-Y)’
+ C(a
A, _ I-
1
-
+
l)( 3y31 -Y)3 3cry2 (CI l)y’>
1
( >
(A3
diZj 1T,Kn,
(A3)
+
where
+
(Al)
Aj+b abi ln( 1 + 4y) - In Z
a(n2a) L
262
M. Zia-Razzaz
Ai = -RTV
and M. Moshfeghian / Fluid Phase Equilibria 91 (1993) 239-263
y VimqmH2, + em{ Qi[H5, + H2m( 1 - H6m)]
+ H3im ’ H2mH7im)
(A4)
bi = y v,bm m
Ci =
(A5)
T V,Cpj
(A6)
m
Qi = y
YinUrn
m
(A7)
(A9) k
H4,
=y
6&k,,,
(AlO)
k
H5,
=f
f6,
=
k
y
Bkzkmakm(akm
-
%&km@/RTV)2/H4m
(All)
8kzkrnak&krn
-
amm)(QlRWlH4m
(A19
k
(Al3)
k
Entropy
and en thalpy expressions
The residual entropy and enthalpy functions can be obtained by differentiating the residual Hehnholtz energy with respect to temperature: ARIRT=
1
4y - 3y2 c(cI - 1) (4 + a)y - 3y2 (1 _y)2 + 2 (1
+(l
+cc)ln(l -y)
-
-Y)'
a ln(l+4y) bRT
(Al4)
(Al%
M. Zia-Razzaz
and M. Moshfeghian
1 Fluid Phase Equilibria 91 (1993) 239-263
C(a - 1)(3 + 3ay - (a + l)y2) 2(1 -Y13
263
1 (A161
HR/RT=(Z-
1) +gT+$
HR/RT
_
=
_(z
-RTb(l
(Al?)
‘co!- 1)(32;131;;cc1 + ‘)f)
1)
4a ]--$(ff-~-+(l+4y) +4y)
(A181
where b T=dT
db ’
(A19
(AV
(A221 ak=
(A231
(
ahn = -amn ++ mn
0.18135 T >
(A24)
(A27)