Modification of the Predictive Capability of the PRSV-2 Equation of State for Critical Volumes of Binary Mixtures

Full text

Page 1

Ž.
Fluid Phase Equilibria 157 1999 1–16
Modification of the predictive capability of the PRSV-2 equation of
state for critical volumes of binary mixtures
Samir I. Abu-Eishah)
Department of Chemical Engineering, UniÕersity of Bahrain, Isa Town, Bahrain
Received 5 October 1998; accepted 24 December 1998
Abstract
Ž.
The predictive capability of the Peng–Robinson–Stryjek–Vera PRSV-2 equation of state for critical
properties of binary mixtures showing continuous critical lines has been investigated. The procedure adopted by
Heidemann and Khalil and discussed by Abu-Eishah et al., in a previous paper, has been followed. The effect of
using the pure-component parameters of the PRSV-2 equation of state k , k
revised values of k , or giving zero values for these parameters have been investigated. The effect of using zero
1
values or optimized values for the binary interaction parameter on the PRSV-2 predictive capability of critical
properties have also been investigated. The standard and the average of the absolute relative deviations in
critical properties are included. The predicted critical temperature and pressure for the 20 nonpolar and 18 polar
systems studied here agree well with experimental data, and are always better than those predicted by the
group-contribution method. A correction has been introduced here to the critical volume predicted by PRSV-2
equation of state that makes the average deviations between the predicted and experimental values very close to
or even better than those predicted by the group-contribution method. q1999 Elsevier Science B.V. All rights
reserved.
Ž.
and k , new values of k ,
3121
Keywords: Binary mixtures; Critical properties; Equation of state: modified critical volume predictions
1. Introduction
Critical properties of a fluid or fluid mixtures are important for describing fluid phase behavior,
predicting physical properties, developing equations of state, and designing supercritical-fluid extrac-
tion processes, compression and refrigeration units 1 . For commonly used pure substances, these
w x
)Tel.: q973-782122; fax: q973-684844.
0378-3812r99r$ - see front matter q 1999 Elsevier Science B.V. All rights reserved.
Ž.
PII: S0378-3812 99 00010-2

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S.I. Abu-EishahrFluid Phase Equilibria 157 1999 1–16
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critical constants have been determined experimentally. Beside direct measurements, critical proper-
ties of mixtures are often estimated using various correlating methods. Li and Kiran 2 divided the
bases of the existing correlating methods into six different groups and gave references on each of
Ž .Ž .
them. These groups are a graphical approach, b equation of state approach, c excess property
Ž .Ž
approach, d conformal solution corresponding state principle approach, e thermodynamic poten-
Ž .
tial approach, and f group-contribution approach.
The prediction of true critical properties of multicomponent systems is an important aspect of the
general problem of predicting the overall phase behavior of a system. Knowledge of the critical
behavior of mixtures is important in order to determine the existing phase conditions or permissible
operating ranges in reactors and mass transfer equipment.
Fluid-property predictions and design calculations in the critical region are often the most difficult
to make, and a knowledge of the precise location of the critical point for the system under study is of
utmost importance. Prediction of critical properties is also important in modeling the phase behavior
exhibited by gas mixtures for the simulation of enhanced-oil recovery processes 3 .
Evaluation of critical points for multicomponent mixtures based on an equation of state has
attracted considerable attention in recent years. In this approach the second and third derivatives of
the molar free energy with respect to composition at constant temperature and pressure must be equal
to zero. Determination of the critical properties for mixtures involves a simultaneous solution of an
extended form of these derivatives and an equation of state. Several attempts have been made using
wx
the Redlich–Kwong equation of state 4,5 ; Peng–Robinson equation of state 6,7 ; Soave–Redlich–
wx
Kwong equation of state 8–12 ; Teja–Patel equation of state 7 ; Deiters’ as well as Guggenheim
wx
equations of state 13 ; simplified perturbed hard-chain-theory equation of state 14 . More details
about the capability of these equations are summarized elsewhere 15 .
wx
The Heidemann–Khalil method 10 , compared to the rigorous Peng–Robinson method 6 , is far
superior; it is much more efficient, requires less computational effort, and does not need the
evaluation of a very large number of high order determinants. Also the partial derivatives required
using the Helmholtz free energy concept are much more rapidly evaluated than those using the Gibbs
wx
free energy concept 11 .
wx
In a previous work 25 , the Peng–Robinson–Stryjek–Vera PRSV-2 equation of state have been
used to predict the critical properties of several polar and non-polar mixtures and a correction, based
on the ratio between predicted and experimental critical volume, has been introduced there. The
correction has been used there to modify the critical volume predicted by the PRSV-2 equation of
state but that correction is limited to where experimental critical volumes are available. Here a general
correction to the critical volume predicted by the PRSV-2 equation of state is introduced where it can
be used whether experimental critical volumes are available or not.
w x
Ž .
.Ž .
w x
wx
w x
wx
wx
w x
Ž.
2. Application of Heidemann–Khalil approach to the PRSV-2 equation of state
The PRSV–2 equation of state has been successfully used for vapor–liquid equilibrium calcula-
tions over a wide range of temperatures and yields a good representation of the saturation pressure of
pure compounds even at low reduced temperatures 16,17 . Therefore, the PRSV-2 equation of state
has been chosen here to test its ability to predict the critical properties of binary mixtures following
wx

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S.I. Abu-EishahrFluid Phase Equilibria 157 1999 1–16
3
wx
the Heidemann and Khalil approach 10 . The Peng–Robinson equation of state in its modified form
is given in Appendix A.
The optimization procedure of k
is based on the minimization of the sum of the squares of the
ij
relative errors in the critical temperature for a given set of data. The effect of using a two-parameter
Margules-type mixing rules on the predictive capability of the PRSV-2 equation of state has been
wx
studied elsewhere 15 .
If the Helmholtz free energy is expanded around some test point T , V , n , n , ..., n
according to the approach of Heidemann and Khalil 10 , one gets
1
2
AyA y
EArEn Dn s
E ArEn E
Ý Ý Ýž
o
ii
2
iji
1
q
Ý Ý Ýž
3!
kj
The stability of the test point is assured if the quadratic term in Eq. 1 is positive-definite, i.e.,
equals to zero. At such a point, the stability is determined by the properties of the cubic and
Ž .
higher-order terms in Eq. 1 . If we note that
EArEn
sRTln f
ii
T,Õ,n an
ji
then the derivative elements in the quadratic and cubic terms in Eq. 1 at constant T, P, and n a n ,
become
?
ž/
jiij
and
½5
ž/
kjiikj
That is, the first and second partial derivatives of fugacity with respect to the number of moles of the
constituents j and k have to be evaluated. The expressions that have been reached for these
derivatives, using the PRSV-2 equation of state with conventional mixing rules, are summarized
wx
elsewhere 15 .
The necessary condition for a point to lie on the limit of stability is that the matrix Q with elements
q sE ArEn En
sRT E ln f rEn
T,Õ
T,Õ
i j jiij
should have a zero determinant, i.e.,
det Q s0
Ž .
and the cubic term in Eq. 1 must vanish, i.e.,
NNN
22
Cs
n E ln f rEn En Dn Dn Dn s0
½5
Ý Ý Ý
Tikjijk
kji
Ž .Ž .
The resulting two-nonlinear Eqs.6and 7have been solved simultaneously for the critical
temperature and volume. The critical pressure is then calculated from the PRSV-2 equation of state
itself. A correction is introduced to the calculated critical volume as follows. The compressibility
Ž
factor of the mixture, Z, is calculated using Eq. A11 in Appendix A, at the calculated T and P .
Ž.
oo1 2o No
wx
NNN
Dn Dn
i
Ž.
/
j nij
NNN
24
E ArEn En En Dn Dn Dn qO Dn
/
kji
1
Ž.Ž .
ijk
i
Ž .
<
2
Ž.Ž .
Ž .
ji
E2ArEn En sRT E ln f rEn
3
4
Ž .
E3ArEn En En sRT E2ln f rEn En
4 Ž .
2
<<
5
?4
Ž .
6
Ž .Ž .
7 Ž .
.
cc

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S.I. Abu-EishahrFluid Phase Equilibria 157 1999 1–16
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Fig. 1. Schematic flowchart for the calculation procedure of the critical properties of a mixture.
Then the corrected value of the critical volume is calculated as the value of Õ calculated from the
PRSV-2 equation of state minus one-half the value of ZRT rP .
The flowchart of the computational procedure described by Heidemann and Khalil 10 and
followed in this work is shown in Fig. 1, and discussed elsewhere 15 . The experimental critical data
w
used in this work are those of Hicks and Young 20 and Vandana and Teja 21 for the water–acetic
acid system. The pure-component properties have been taken from Stryjek and Vera 16,17 and
wx
Proust and Vera 18 . Revised k parameters for alkanes have been taken from Proust et al. 19 . The
1
wx
data of Reid et al. 22 have been used for the pure-component properties that are not available in
wx
Refs. 16–18 .
c
cc
wx
wx
xwx
wx
wx
3. Results and discussion
wx
The algorithm described in Fig. 1 and discussed earlier 15 has been applied to the PRSV-2
equation of state to predict the critical properties of 38 different binary mixtures. Among these
systems are paraffins, aromatics, alcohols, ethers, water, acetic acid, hydrogen sulfide, sulfur dioxide,
carbon dioxide, nitrogen, oxygen, and ammonia. The pure-component properties of the systems
studied in this work are listed in Table 1. The calculated optimum values of the conventional

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S.I. Abu-EishahrFluid Phase Equilibria 157 1999 1–16
5
Table 1
Pure-component properties used in this work
Compound
Ž .
K
Ž.Ž.
TP
kPa
vkkk
Õ
mlrmol
72.5
93.9
122.2
80.9
98.5
89.5
73.4
57.1
99.0
148
203
255
304
370
432
492
548
603
780
130
308
259
316
118
167
274
209
171
239
280
178
221
335
cc123c
NH
CO
SO
HCl
H S
2
N
2
O
2
H O
2
Methane
Ethane
Propane
Butane
Pentane
Hexane
Heptane
Octane
Nonane
Decane
Tridecane
Ethylene
Cyclohexane
Benzene
Toluene
Methanol
Ethanol
1,Butanol
Acetone
Acetic acid
Chloroform
Diethyl ether
Dimethyl ether
Methyl ethyl ether
Hexafluorobenzene
405.6
304.2
430.8
324.6
373.2
126.2
154.8
647.3
190.6
305.4
369.8
425.2
469.7
507.3
540.1
568.8
594.6
617.5
675.8
282.4
553.6
562.2
591.8
512.6
513.9
563.0
508.1
592.7
536.6
466.7
400.1
437.8
516.7
11289.5
7382.4
7883.1
8308.6
8940.0
3400.0
5090.0
22089.8
4595.0
4879.8
4249.5
3796.6
3369.0
3012.4
2735.8
2486.5
2287.9
2103.5
1722.5
5042.0
4075.0
4898.0
4106.0
8095.8
6148.0
4412.7
4696.0
5786.0
5471.6
3640.0
5240.0
4410.0
3273.0
0.25170
0.22500
0.25100
0.12610
0.10000
0.03726
0.02128
0.34380
0.01045
0.09781
0.15416
0.20096
0.25143
0.30075
0.35022
0.39822
0.44517
0.49052
0.62264
0.08652
0.20877
0.20929
0.26323
0.56533
0.64439
0.59020
0.30667
0.45940
0.21600
0.28100
0.18909
0.23479
0.39610
0.00100
0.04285
0.03962
0.01989
0.03160
0.01996
0.01512
y0.06635
y0.00159
0.02669
0.03136
0.03443
0.03946
0.05104
0.04648
0.04464
0.04104
0.04510
0.04157
0.04191
0.07023
0.07019
0.03849
y0.16816
y0.03374
0.33431
y0.00888
y0.19724
0.02899
0.05004
0.05717
0.16948
0.02752
y0.1265
0.0
NA
y0.0036
NA
0.3162
y0.0090
0.0199
0.1521
0.1358
0.2757
0.6767
0.3940
0.8634
0.9331
0.6214
0.6621
0.8549
0.9387
NA
0.6146
0.7939
0.5261
y1.3400
y2.6846
y1.1743
0.2871
0.8136
NA
NA
y0.1211
0.0515
0.8172
0.510
0.0
NA
0.310
NA
0.535
0.490
0.443
0.517
0.424
0.447
0.461
0.457
0.460
0.496
0.509
0.519
0.527
0.528
NA
0.530
0.523
0.510
0.588
0.592
0.642
0.537
0.541
NA
NA
0.481
0.768
0.565
3
2
2
NAsNot available.
binary-interaction parameters for the systems studied are listed in Table 2. The criteria used to
compare predicted and experimental critical properties are the standard deviation, SD, and the average
of the absolute relative deviations, AD, defined below.
0.5
M
Ý
i
M
Ý
i
2
SDs
exp. value-calc. value r My18
Ž.Ž. Ž .
<<
ADs
exp. value-calc. valuerexp. value 100rM
9 Ž .
where M is the number of points in a given set of data.

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S.I. Abu-EishahrFluid Phase Equilibria 157 1999 1–16
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Table 2
Optimized values of the conventional binary interaction parameter, k12
Mixture
Mk12
y0.018230
0.115592
0.134531
0.044056
0.047723
0.05
0.031610
0.094116
0.033767
0.05
0.063486
0.029111
0.073827
0.139330
0.022619
0.147287
y0.001203
0.020555
y0.016720
0.086103
0.059211
y0.027815
y0.042001
0.124243
0.006808
0.086253
0.086272
y0.008858
y0.009107
y0.051866
y0.089880
y0.154690
y0.002443
0.050966
0.086653
y0.111223
y0.038320
y0.080425
N –O
2
CO –H S
2
CO –propane
2
Methane–N
Methane–H S
Methane–CO
Methane–ethane
Ethane–H S
Ethane–butane
Ethane–heptane
Propane–H S
Propane–octane
Propane–HCl
Butane–CO
Butane–heptane
Butane–ammonia
Hexane–heptane
Hexane–decane
Hexane–cyclohexane
Hexane–1,butanol
Heptane–ethylene
Nonane–cyclohexane
Tridecane–cyclohexane
Cyclohexane–ethanol
Acetone–benzene
Benzene–methanol
Benzene–ethanol
Benzene–toluene
Methanol–1,butanol
Ethanol–water
Ammonia–water
Water–acetic acid
Ethylene–chloroforma
1,Butanol–diethyl ether
Hexafluorobenzene–decane
SO –dimethyl ether
2
SO –diethyl ether
2
SO –methyl ethyl ether
2
7
8
5
2
2
12
6
6
8
6
12
10
7
7
5
5
5
6
9
9
9
8
8
9
9
6
15
8
7
9
4
8
7
8
4
4
10
4
4
5
2
a
2a
2
a
2
a
2
a
2
a
a
a
aOscillatory convergence.
3.1. PredictiÕe capability of the PRSV equation of state
It should be first noticed that all systems studied have continuous critical curves, and the critical
locus of each of these systems exhibits a critical temperature that varies monotonically with
composition. A point of minimum critical temperature has been noticed on the critical loci of the

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S.I. Abu-EishahrFluid Phase Equilibria 157 1999 1–16
7
propane–H S, and butane–ammonia systems, while a point of maximum critical temperature has
2
been noticed on the critical loci of the SO –dimethyl ether and SO –methyl ethyl ether systems. A
2
minimum or maximum critical temperature is an indication of the formation of a positive or a
negative azeotrope, respectively, i.e., an azeotrope with a minimum or a maximum in its boiling
wx
temperature 23 .
Although the algorithm of solution used here uses the k’s of the PRSV-2 equation of state as
starting values for the prediction of the critical properties, it should be mentioned that these k’s have
been originally estimated for temperature ranges below or up to the critical point of the pure
Ž.
compounds involved i.e., T s-1 . This does not match, in most cases, with the reduced tempera-
r
ture ranges involved in the calculation of the mixtures critical properties T is mostly greater than
.
1.0 . This might explain the oscillatory convergence of some systems such as methane–H S,
methane–CO , methane–ethane, ethane–heptane, propane–octane, butane–heptane, heptane–ethyl-
2
ene, and ethylene–chloroform when the T’s involved are far above 1.0. This oscillatory convergence
r
has been pointed out to it in the tables of the results and in Table 2. This mathematical oscillation has
been reduced or even eliminated for some systems when a two-parameter Margules mixing-rules type
wx
is applied 15 . See Table 3 for the heptane–ethylene system as an example.
2
Ž
r
2
Table 3
On the predictive capability of the PRSV-2 equation of state
Mixture
T
c
Ž .
SD K
P
Õ
SD mlrmol
cc
Ž .Ž. Ž .Ž. Ž .
AD %SD bar AD %AD %
Heptane– ethylene
Two-parameter Margules mixing rules type k sy0.127265, k sy0.06653
2.000.40
One-parameter conventional mixing rules type k s0.06484
6.691.10
Ž.
1221
7.04 7.7427.607.41
Ž.
12
3.503.12 14.465.54
Methanol-1,butanol
k’s are used only for T s-0.7 k sy0.009107
r
0.19
k’s are used for the whole T range k sy0.024356
0.47
Ž.
12
0.03
Ž
0.06
5.03 6.0557.5428.26
.
r12
5.11 6.1657.7128.35
()
Butane– heptane original k and reÕised k : k s0.022619
1
Original k
0.98
1
Revised k
0.81
1
1 12
0.16
0.13
0.76
0.61
1.67
1.31
75.96
75.31
21.61
21.41
SO – diethyl ether
2
Using Proust and Vera k for each component in the mixture k sy0.038515
1
0.41 0.06
Using k s0 for each component in the mixture k sy0.039002
1
0.430.07
Ž.
12
2.77 4.1735.35 15.42
Ž.
12
2.894.3339.2617.04
Ammonia–water
Using k’ss0 only for T s-0.7 k sy0.0926
r
1.91
Using all k’s only for T s-0.7 k sy0.0928
r
1.87
Ž.
12
0.2819.87 8.03––
Ž.
12
0.2818.25 7.38––

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S.I. Abu-EishahrFluid Phase Equilibria 157 1999 1–16
8
When the Stryjek and Vera k values are applied for the whole T range, the results are usually
better than when these k’s are only used for T s-0.7, as recommended by Stryjek and Vera for
r
wx
systems containing water or alcohols 17 . See Table 3 for the methanol-1,butanol system as an
example.
It has been noticed that using revised values of k for alkanes 19 rather than those given in Ref.
wx
16 has no effect at all on the predicted critical properties and the value of the optimum k
as the k’s are applied for T s-0.7. But when the k’s are used for the whole T range, the results
r
are better using the revised k
values. This has been tried for methane–ethane, ethane–butane,
1
butane–heptane, hexane–heptane, and hexane–decane systems. See Table 3 for the butane–heptane
system as an example.
Using the values of k given by Proust and Vera 18 instead of giving them zero values, produced
1
better results for the systems involved when used for the whole range of T , but the results are exactly
the same when T s-0.7. Examples are heptane–ethylene, ethylene–chloroform, SO –DEE, SO –
r
DME, SO –MEE, 1,butanol–DEE, ethane–H S, and propane–H S. See Table 3 for the SO –diethyl
22
ether system as an example.
It has been noticed that using the pure compound parameters given in Refs. 16–19 or giving them
zero values, has no significant effect neither on the optimized k
Ž
properties as long as the k’s are only used for T s-0.7. See Table 3 for the ammonia–water
Ž.
system no Õ data are available .
c
Using more experimental critical data points, or using data from different sources, has sometimes,
significant effect on the optimized k
values. This has been demonstrated for several systems, e.g.,
12
methane–H S, methane–ethane, propane–HCl, butane–heptane, and ethanol–water see Table 4 .
2
The k
data used in this work are only those listed in Table 2.
12
It should be noted that the method used in this work predicts exact critical temperature and pressure
Ž
for the pure compounds i.e., at x s0 and at x s1 . But the predictive capability of the PRSV-2
21
equation of state itself for critical volumes is very poor.
It should also be noted that the computation program used in this work does not work for mole
fractions equals to zero, i.e., x s0 or x s0. This is due to floating point division by zero, so the
12
value of x in this case is usually replaced by a very small number, say 10y4or 10y5.
r
wx
1
as long
12
r
wx
r
22
22
wx
value nor on the predicted critical
12
r
Ž.
.
Table 4
Effect of number of data points on optimized k12
Mixture
Mk12
Methane–H S6 0.047723
y0.005386
0.031610
0.016826
0.073827
0.077408
0.022619
0.018492
y0.051866
y0.064235
2
13
8
12
5
10
5
14
8
16
Methane–ethane
Propane–HCl
Butane–heptane
Ethanol–water

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S.I. Abu-EishahrFluid Phase Equilibria 157 1999 1–16
9
Using the conventional one-binary interaction parameter type, Tables 5 and 6 show the standard
deviation, SD, and the average absolute relative deviation, AD, in critical temperature and pressure
for 20 nonpolar systems. The average standard deviation for all these systems is 1.89 K, with a
maximum SD of about 7.80 K for the critical temperature. For the critical pressure the average SD is
about 1.70 bar, with a maximum SD of about 9.87 bar. Note that the uncertainties in the experimental
values are typically about 0.6 K for the critical temperature and 0.2 bar for the critical pressure 24 .
wx
Table 5
Standard and average of the absolute relative deviations in critical properties for nonpolar mixtures using PRSV-2 equation
of statea
Mixture
MT
c
Ž .
SD K
P
c
Ž .Ž. Ž .
AD %SD barAD %
b
Methane–CO
Methane–N
Methane–ethane
Hexane–heptane
Hexane–decane
Hexane–cylohexane
Nonane–cyclohexane
Tridecane–cyclohexane
Nitrogen–oxygen
Overall average:
Optimized k
k s0
12
6 6.06
0.53
0.94
0.13
0.90
0.26
0.11
0.78
0.24
1.94
0.26
0.28
0.02
0.12
0.04
0.01
0.10
0.14
3.53
0.85
1.06
0.25
0.71
0.23
0.69
1.70
0.38
3.26
1.29
1.21
0.78
2.24
0.59
1.93
4.84
0.61
2
12
8
9
9
9
9
9
7
2
1.11
2.62
0.32
0.65
1.04
1.18
1.86
2.13
12
aMsNumber of data points.
bOscillatory convergence.
Table 6
Standard and average of the absolute relative deviations in critical properties for some non-polar mixtures using the PRSV-2
Ž
equation of state before and after critical volume correction Õ sÕ yÕ r2
Mixture
MTP
cc
Ž .Ž .Ž.
SD KAD % SD barAD %
.
cpp0
Õ Before correction
c
Ž
SD mlrmol
9.08
8.86
16.00
27.80
40.12
19.90
14.78
53.93
21.57
75.31
53.15
Õ After correction
c
Ž
SD mlrmol
8.91
7.35
8.77
5.46
14.46
7.91
13.35
21.57
7.60
16.73
18.44
Ž .. Ž .. Ž .
AD % AD %
CO –H S
2
Methane–H S
Ethane–H S
Ethane–butane
Ethane–heptane
Propane–H S
Propane–CO
Propane–octane
Butane–CO
Butane–heptane
Heptane–ethylene
Overall average:
Optimized k
k s0
12
8 1.06
7.80
0.29
0.70
3.95
0.68
1.17
2.98
1.70
0.81
6.69
0.27
2.41
0.07
0.13
0.78
0.16
0.29
0.41
0.41
0.13
1.10
2.29
9.87
0.34
0.77
2.04
1.40
0.98
0.85
1.82
0.61
3.50
2.13
7.69
0.25
1.02
2.31
2.07
0.99
1.52
1.77
1.31
3.12
8.74
7.67
11.44
14.21
12.00
12.37
8.16
12.00
11.61
21.41
14.23
8.70
7.56
6.81
2.62
5.54
5.26
7.62
6.41
3.96
4.17
8.17
2
a
13
6
12
10
8
5
7
5
5
8
2
2
a
2
2
a
2
a
2.53
6.94
0.56
1.60
2.23
3.08
2.21
3.50
30.96
33.05
12.17
13.37
11.87
11.29
6.08
5.62
12
aOscillatory convergence.

Page 10

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S.I. Abu-EishahrFluid Phase Equilibria 157 1999 1–16
10
On the other hand, the average AD for all the nonpolar systems listed in Tables 5 and 6 is about
Ž
0.45% for the critical temperature with a maximum of 2.41% and about 2.05% for the critical
Ž.
pressure with a maximum of 7.69% . Tables 5 and 6 also show that using zero interaction parameter,
the average SD and AD in the critical temperature and pressure are 5.0 K and 1.17%, and 2.23 bar
and 2.88%, respectively.
For the 18 polar systems studied, the SD and AD in critical properties are shown in Tables 7 and 8
using conventional mixing rules. The average SD in the critical temperature for all of those systems is
Ž.
1.64 K with a maximum of 5.44 K . The value of SD in the critical pressure for the systems listed in
Tables 7 and 8 where critical pressure data are available 12 systems is 5.02 bar with a maximum of
.
about 18.25 bar for the ammonia–water system . On the other hand, the average AD in the critical
Ž
temperature for these systems is 0.18% with a maximum of 0.83% , and that in the critical pressure
Ž.
is 4.82% with a maximum of about 10.34% . Tables 5 and 6 also show that using zero interaction
parameter, the average SD and AD in the critical temperature and pressure are 7.8 K and 1.4%, and
5.7 bar and 4.7%, respectively.
Table 9 shows a comparison between the calculated SD and AD for PRSV-2 equation of state with
conventional one-parameter type and those of the work of Li and Kiran 2 which are based on the
group-contribution method. The method of Li and Kiran does not need experimentally adjusted
interaction parameters to predict the critical properties. It is clear that the prediction of the critical
temperature and pressure is always much better on the basis of the PRSV-2 equation of state. For the
systems listed in Table 9, the overall average of SD and AD using the PRSV-2 equation of state and
the group-contribution method are as follows: 1.42 K, 8.66 K, 0.24% and 1.68% for the critical
temperature, and 2.03 bar, 8.91 bar, 2.53% and 10.45% for the critical pressure.
.
Ž.Ž
.
w x
3.2. Modification of the predicted critical Õolumes
Ž.
The predictive capability of the PRSV-2 or any equation of state for critical volumes is not as
good as its capability of prediction of critical temperature or critical pressure. The group-contribution
Table 7
Standard and average of the absolute relative deviations in critical properties for polar mixtures using PRSV-2 equation of
state
MixtureM
T
c
Ž .
SD K
P
c
Ž .Ž. Ž .
AD %SD bar AD %
aa
Acetone–benzene
Hexafluorobenzene–decane
Cyclohexane–ethanol
Ethanol–water
Hexane–1,butanol
Benzene–toluene
Propane–HCl
Ammonia–water
Overall average:
Optimized k
12
k s0
12
15
10
6
16
8
9
10
7
0.49, 1.00
2.01, 10.41
2.60, 15.43
1.60, 6.74
1.12, 9.33
0.32, 1.24
1.04, 6.02
1.87, 9.80
0.07, 0.16
0.32, 1.72
0.38, 2.62
0.21, 1.06
0.17, 1.56
0.04, 0.20
0.25, 1.56
0.28, 1.46
–
–
–
–
–
0.09, 0.07
4.11, 4.18
18.25, 26.63
–
–
–
–
–
0.16, 0.13
5.74, 5.57
7.38, 9.66
1.38
7.50
0.22
1.29
7.48
10.29
4.43
5.12
aNo experimental P data are available to compare with.
c

Page 11

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S.I. Abu-EishahrFluid Phase Equilibria 157 1999 1–16
11
Table 8
Standard and average of the absolute relative deviations in critical properties for some polar mixtures using the PRSV-2
Ž
equation of state before and after critical volume correction Õ sÕ yÕ r2
Mixture
MTP
cc
Ž . Ž .Ž.
SD KAD % SD barAD %
.
cpp0
Õ Before correction
c
Ž
SD mlrmol
57.54
50.48
49.77
14.18
32.24
56.25
29.40
35.35
30.96
84.24
Õ After correction
c
Ž
SD mlrmol
17.94
11.55
18.21
14.26
6.26
7.84
3.74
4.42
5.66
46.39
Ž ..Ž .. Ž .
AD % AD %
Methanol–1,butanol
Benzene–ethanol
Benzene–methanol
Ethylene–chloroform
Butane–ammonia
1,Butanol–diethyl ether
SO –dimethyl ether
2
SO –diethyl ether
2
SO –methyl ethyl ether
2
Water–acetic acid
Overall average:
Optimized k
k s0
12
4
7
8
4
6
4
4
4
5
8
0.19
5.44
0.87
4.21
1.92
0.73
0.58
0.41
0.95
0.97
0.03
0.70
0.13
0.83
0.42
0.11
0.10
0.06
0.18
0.11
5.03
3.31
8.86
6.61
6.75
0.81
1.75
2.77
0.39
b
–
6.05
4.67
10.34
4.96
7.98
1.52
2.34
4.17
0.52
b
–
28.26
24.83
28.82
10.28
23.10
18.37
17.53
15.42
16.02
54.54
8.83
4.87
9.48
7.41
4.94
2.46
1.57
1.37
2.30
30.20
a
a
1.84
8.00
0.33
1.51
4.20
4.16
4.95
4.61
39.50
40.74
20.35
21.03
9.95
9.97
4.79
4.94
12
aOscillatory convergence.
bNo P data available.
c
Table 9
Standard and average of the absolute relative deviations in critical properties using PRSV-2 equation of state with critical
Ž.
volume correction Õ sÕ yÕ r2 and Li and Kiran work
cp p0
Mixture
MT
c
Ž .Ž .
SD K AD %
a
P
Õ
SD mlrmol
8.77, 4.29
7.91, 6.35
18.44, 9.85
17.94, 11.30
7.84, 13.90
4.42, 15.10
5.65, 6.81
10.14, 9.66
cc
Ž. Ž .Ž.Ž .
SD barAD % AD %
Ethane–H S
Propane–H S
Heptane–ethylene
Methanol–1,butanol
1,Butanol–diethyl ether
SO –diethyl ether
2
SO –methyl ethyl ether
2
Overall average
6
7
8
4
4
4
5
0.29, 5.91
0.68, 19.90
6.69, 15.60
0.19, 7.38
0.73, 1.18
0.41, 9.20
0.95, 1.48
1.42, 8.66
0.07, 1.49
0.16, 4.45
1.10, 2.87
0.03, 1.03
0.11, 0.13
0.06, 1.52
0.18, 0.24
0.24, 1.68
0.34, 7.48
1.40, 18.60
3.50, 6.19
5.03, 17.70
0.81, 1.05
2.77, 6.65
0.39, 4.70
2.03, 8.91
0.25, 9.57
2.07, 23.90
3.12, 5.34
6.05, 17.60
1.52, 1.82
4.17, 8.59
0.53, 6.31
2.53, 10.45
6.81, 2.68
5.26, 3.64
8.17, 3.37
8.83, 4.81
2.46, 3.27
1.37, 5.60
2.30, 2.98
5.03, 3.76
2
2
b
a
Ž.
First set of data: PRSV-2 prediction optimized k
Ž
method Li and Kiran work .
bOscillatory convergence.
after critical volume correction; second set of data: group-contribution
12
.
w x
method 2 is superior in predicting the critical volume over the PRSV-2 and any other known cubic
equation of state. The poor representation of the mixture critical volumes by cubic equations of state
is well known and has been widely attributed to the fact that the pure-component critical compressibil-
ity calculated from the equation of state is, in general, not equal to the experimental compressibility of
wx
most fluids 23 . The above statement is confirmed when one compares the pure-component critical

Page 12

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S.I. Abu-EishahrFluid Phase Equilibria 157 1999 1–16
12
Ž.
compressibility predicted by the Peng–Robinson equation of state 0.3074 to the range of the
experimental compressibilities of the pure compounds used in this study 0.224–0.29 .
Since the error between experimental critical volume and that predicted by the PRSV-2 equation of
ww xx
state is too large see for example, Ref. 15 , a correction has been introduced in this work to correct
for the predicted critical volume. This correction is based on calculating the critical volume predicted
by the PRSV-2 equation of state, Õ , and the value of the critical volume calculated based on the
p
predicted T , P and Z by the PRSV-2 equation of state itself, i.e., Õ sZ RT rP . Then a modified
ccc
value of the predicted critical volume is calculated as Õ sÕ -Õ r2. When this method is applied on
the systems listed in Tables 6 and 8, the results are as follows. For the 11 nonpolar systems listed in
Table 6, the average SD in critical volumes has fallen from 30.96 mlrmol before correction to 11.87
mlrmole after correction, while the average AD has decreased from 12.17% to 6.08%. For the 10
polar systems listed in Table 8, the average SD has fallen from 39.50 mlrmol to 9.95 mlrmol, while
the average AD has decreased from 20.35% to 4.79%.
It should be mentioned here that the proposed correction for the predicted critical volume works
good for all the systems studied except for the water–acetic acid mixture. For the latter system, the
values of SD and AD have been decreased only by about 50% but still the deviation from
experimental data of Õ is high. This might be attributed to the high polarity of the water–acetic acid
c
w
system. This problem has been solved elsewhere 25 and the values of SD and AD have been reduced
to 1.8 mlrmol and 0.8%, respectively.
For the sake of comparison with the group-contribution method, the critical volume correction has
been introduced to the systems listed in Table 9 and the results become very close to those predicted
by the group-contribution method. The respective average values of SD and AD in critical volume
before correction were 39.02 mlrmol and 16.82% 15 and become 10.14 mlrmol and 5.03% after
.
correction compared to 9.66 mlrmol and 3.76% for the Li and Kiran work 2 .
Lastly, for the carbon dioxide–butane system, Table 10 shows the point-by-point values of the
critical properties using PRSV-2 equation of state and the values of the critical volume after using the
above mentioned correction. It is clear that the SD and AD for the critical volume have decreased
Ž
sharply from 21.57 mlrmol and 11.61% before correction to 7.60 mlrmol and 3.96% after
.
correction .
Ž.
p0ccc
cpp0
x
wx
Ž
w x
.Ž
Table 10
Predicted critical properties for the carbon dioxide 1 –butane 2 system using PRSV-2 equation of state with conventional
Ž.
mixing rules k s0.13933 with and without critical volume correction
12
Ž .Ž.
xT
K
P
bar
1cc
Exp.PRSV-2 Exp.PRSV-2
Ž .Ž .
Ž.
Õ
Exp.
mlrmole
c
a
PRSV-2PRSV-2
0.1694
0.3334
0.4984
0.6740
0.8273
325.93
351.71
377.21
398.76
412.26
SD s1.70
T
AD s0.41
T
323.95
352.88
379.04
399.95
413.51
79.08
81.71
75.37
62.81
51.10
SD s1.82
P
AD s1.77
P
75.76
81.15
74.91
61.94
50.10
104.90
131.70
162.30
192.90
217.20
SD s21.57
V
AD s11.61
V
125.17
143.15
173.72
211.52
246.22
c
SD s7.60
V
c
AD s3.96
V
107.48
125.17
152.81
184.82
212.07
aCorrection form Õ sÕ yÕ r2.
cpp0

Page 13

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S.I. Abu-EishahrFluid Phase Equilibria 157 1999 1–16
13
4. Conclusions
Ž.Ž.
The predictive capability of the Peng–Robinson–Stryjek–Vera PRSV-2 equation of state 1986
for critical properties of binary mixtures has been investigated. The PRSV-2 equation of state has
been used to predict the critical properties of mixtures on the basis of the algorithm adopted by
wxw
Heidemann and Khalil 10 and explained earlier 15 . For the 20 nonpolar system and the 18 polar
systems listed in Tables 5–8, the overall standard deviation using the optimized conventional
binary-interaction parameter are respectively, 1.89 and 1.70 K in the critical temperature, and 1.64
and 5.02 bar in the critical pressure. Better predictions are always obtained in both critical
temperature and pressure compared to those obtained by the group-contribution method.
The group-contribution method always gives less deviation in the predicted critical volume than
any other equation of state. A correction has been introduced in this work to the critical volume
predicted by the PRSV-2 of state which has the form: Õ sÕ yÕ r2. By using this correction, the
overall standard deviation, SD, in critical volume for 21 systems studied in this work has decreased
from 35 mlrmol to 11.0 mlrmol while the average of the absolute deviation, AD, in critical volume
has decreased from 16.7 to 5.5%. By using such a correction, the overall SD and AD for the systems
w x
studied by Li and Kiran 2 have decreased from 39.02 mlrmol and 16.82% before correction to
Ž.
10.14 mlrmol and 5.03% after correction , compared to 9.66 mlrmol and 3.76% predicted by the
group-contribution method used by Li and Kiran.
x
cpp0
Ž.
5. Nomenclature
a
A
A
AD
b
B
C
CX
Det
f
k
M
n
N
P
Q
QX
q
R
SD
T
Attraction parameter in the Peng-Robinson equation of state
Ž.2
aPr RT
Helmholtz free energy
Average of the absolute relative deviations defined by Eq. 8
Repulsion parameter in the Peng–Robinson equation of state
bPrRT
Ž .
Cubic term defined by Eq. 7
Derivative of the cubic term C
Determinant
Fugacity
Binary interaction parameter
Number of data points
Number of moles
Number of components
Ž.
Pressure bar
Matrix of the quadratic terms in Eq. 1
Derivative of the Q matrix
Elements of the Q matrix, defined by Eq. 5
Universal gas constant
Standard deviation, defined by Eq. 9
Ž .
Absolute temperature K
Ž .
Ž .
Ž .
Ž .

Page 14

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S.I. Abu-EishahrFluid Phase Equilibria 157 1999 1–16
14
Ž.
V
Õ
x
Z
Volume ml
Molar volume mlrmol
Mole fraction
Compressibility factor
Ž.
Greek letters
D
e
k
k
o
k , k , k
12
v
Difference in property
Error tolerance
Function of reduced temperature and acentric factor, Eq. A5
Function of acentric factor, Eq. A6
Pure-compound parameters in PRSV-2 equation of state
Acentric factor
Ž.
Ž.
3
Subscripts
c
i, j, k, l, m, n
m
o
p
r
T
Critical property
Component number
Mixture property
Initial state
Predicted property
Reduced property
Total property
Appendix A
The Peng–Robinson–Stryjek–Vera equation of state in its modified form is given by
PsRTr Õyb ya T r Õ Õqb qb Õyb
2
asa 0.457235 RT
rP
cc
bs0.077796 RT rP
cc
Ž
r
ksk q k qk k yT
1yT
o123rr
k s0.378893q1.4897153vy0.17131848v2q0.0196544v3
o
w
As recommended by Stryjek and Vera 16,17 , Eq. A5 has been used in this work for T s-0.7,
Ž.
while Eq. A6 has been used for T )0.7.
r
For mixtures, the following mixing rules have been applied:
A1
?4
Ž. Ž .
.
Ž.Ž.Ž
Ž
Ž
.
.
.
A2
Ž
A3
2
0.5
as 1qk 1yT
A4
Ž
Ž
Ž
.
.
.
.
0.5 0.5
1qT
0.7yT
A5
Ž.Ž.
Ž. Ž.
rr
A6
x
Ž.
r
NN
as
x x a
i
A7
Ž.
Ý Ý
i
j ij
j
N
bs
x b
i
A8
Ž.
Ý
i
i

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S.I. Abu-EishahrFluid Phase Equilibria 157 1999 1–16
15
For the cross parameter, a , the following expressions have been used:
ij
Ž . a Zero interaction parameter form
0.5
a s a a
ij
A9
Ž.Ž.
ii jj
Ž . b Conventional one-binary interaction parameter form
0.5
a s a a
ij
1yk
A10
Ž.Ž.
Ž.
ii jj ij
where k
In terms of the compressibility factor, Z, Eq. 1 can be written as
is the binary interaction parameter between components i and j.
ij
Ž .
Z3y 1yB Z2q Ay3B2y2B Zy AByB2yB3s0
Ž.2
where AsaPr RT
, BsbPrRT, and ZsPÕrRT.
The fugacity of component i for the PRSV-2 equation of state, as a function of temperature,
volume, and mole numbers, is given by
A11
Ž.Ž.Ž.Ž.
'
'
b
b
A
2Ýx a
a
b
b
Zq 1q 2 B
Zq 1y 2 B
Ž
Ž
.
.
ii iji
ln f s
Zy1 yln
ZyB rx P yy
ln
?4
Ž.Ž.
ii
ž/
'
ž/
2 2 B
mmm
A12
Ž.
where x is the mole fraction of component i in the mixture.
i
References
w x
1 J.F.P. Gomes, Hydrocarbon Process 9 1988 110–112.
w x
2 L. Li, E. Kiran, Chem. Eng. Commun. 94 1990 131–141.
w x
3 N.R. Nagarajan, A.S. Cullick, A. Griewank, Fluid Phase Equilibria 62 1991 211–223.
w x
4 J. Joffe, D. Zudkevitch, Chem. Eng. Prog. Symp. Ser. 63 1967 43–51.
w x
5 R.R. Spear, R.L. Robinson, K.C. Chao, I&EC Fund. 8 1969 2–8.
w x
Ž
6 D.Y. Peng, D.B. Robinson, AIChE J. 23 1977 137–144.
w x
7 A.S. Teja, R.L. Smith, S.I. Sandler, Fluid Phase Equilibria 14 1983 265–272.
w x
Ž.
8 M.-J. Huron, Chem. Eng. Sci. 31 1976 837–839.
w x
9 M.-J. Huron, G.-N. Dufour, J. Vidal, Fluid Phase Equilibria 1 1977 247–265.
x
10 R.A. Heidemann, A. Khalil, AIChE J. 26 1980 769–779.
wx
11 M.L. Michelsen, R.A. Heidemann, AIChE J. 27 1981 521–523.
wx
12 M.L. Michelsen, Fluid Phase Equilibria 4 1980 1–10.
wx
13 D.E. Mainwaring, R.J. Sadus, C.L. Young, Chem. Eng. Sci. 43 1988 459–466.
wx
14 F. Garcia-Sanchez, J. Ruiz-Cortina, C. Lira-Galeana, L. Ponce-Ramirez, Fluid Phase Equilibria 81 1992 39–84.
wx
15 S.I. Abu-Eishah, N.A. Darwish, I.H. Aljundi, Int. J. Thermophys. 19 1998 239–258.
wx
Ž.
16 R. Stryjek, J.H. Vera, Can. J. Chem. Eng. 64 1986 334–340.
wx
Ž.
17 R. Stryjek, J.H. Vera, Can. J. Chem. Eng. 64 1986 820–826.
wx
Ž.
18 P. Proust, J.H. Vera, Can. J. Chem. Eng. 67 1989 170–173.
wx
19 P. Proust, E. Meyer, J.H. Vera, Can. J. Chem. Eng. 71 1993 292–298.
wx
Ž.
20 C.P. Hicks, C.L. Young, Chem. Rev. 75 1975 119–175.
Ž.
Ž.
Ž.
Ž.
Ž.
.
Ž.
Ž.
w
Ž.
Ž.
Ž.
Ž.
Ž.
Ž.
Ž.