Fluid Phase Equilibria 181 (2001) 127–146
Accurate vapour–liquid equilibrium calculations for complex
systems using the reaction Gibbs ensemble
Monte Carlo simulation method
Martin Lísal a,b , William R. Smith b,∗ , Ivo Nezbeda a,c
a
b
E. Hála Laboratory of Thermodynamics, Institute of Chemical Process Fundamentals,
Academy of Sciences, 16502 Prague 6, Czech Republic
Department of Mathematics and Statistics and, School of Engineering, College of Physical and Engineering Science,
University of Guelph, Guelph, Ont., Canada N1G 2W1
c
Department of Physics, J.E. Purkyně University, 40096 Ústí n. Lab., Czech Republic
Received 12 April 2000; accepted 06 March 2001
Abstract
The reaction Gibbs ensemble Monte Carlo (RGEMC) computer simulation method [J. Phys. Chem. B 103 (1999)
10496] is used to predict the vapour–liquid equilibrium (VLE) behaviour of binary mixtures involving water,
methanol, ethanol, carbon dioxide, and ethane. All these mixtures contain molecularly complex substances, and
accurately predicting their VLE behaviour is a considerable challenge for molecular-based approaches, as well as for
traditional engineering approaches. The substances are modelled as multi-site Lennard–Jones (LJ) plus Coulombic
potentials with standard mixing rules for unlike site interactions. No adjustable binary-interaction parameters and no
mixture experimental properties are used in the calculations; only readily-available pure-component vapour-pressure
data are required. The simulated VLE predictions are compared with experimental results and with those of two
typical semi-empirical macroscopic-level approaches. These latter are the UNIFAC liquid-state activity-coefficient
model combined with the simple truncated virial equation of state, and the hole quasi-chemical group contribution
equation of state. The agreement of the simulation results with the experimental data is generally good and also
comparable with and in some cases better than those of the macroscopic-level empirical approaches. © 2001 Elsevier
Science B.V. All rights reserved.
Keywords: VLE; Computer simulations; Mixtures; Water; Ethane; Carbon dioxide; Ethanol; Methanol
Abbreviations: B-EOS: simple truncated virial equation of state; EPM2: modified extended primitive model; HQGCM: hole
quasi-chemical group contribution method; EOS: equation of state; GEMC: Gibbs ensemble Monte Carlo; LJ: Lennard–Jones;
LLE: liquid–liquid equilibria; MTBE: methyl tert-butyl ether; NPT: constant pressure–constant temperature; NVT: constant volume–constant temperature; OPLS: optimised potentials for liquid simulations; PTxx : pressure–temperature composition; RGEMC: reaction Gibbs ensemble Monte Carlo; TIP4P: transferable intermolecular potential; UNIFAC: universal
quasi-chemical functional group activity coefficients; VLE: vapour–liquid equilibria
∗
Corresponding author. Tel.: +1-519-824-4120/ext. 2155; fax: +1-519-837-0221.
E-mail address: wsmith@msnet.mathstat.uoguelph.ca (W.R. Smith).
0378-3812/01/$20.00 © 2001 Elsevier Science B.V. All rights reserved.
PII: S 0 3 7 8 - 3 8 1 2 ( 0 1 ) 0 0 4 8 9 - 7
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M. Lı́sal et al. / Fluid Phase Equilibria 181 (2001) 127–146
1. Introduction
The main goal of vapour–liquid phase equilibrium (VLE) calculations is the prediction of the PTxx
behaviour of the mixture (where P is the system pressure, T the temperature, and x denotes the compositions of the coexisting phases). In chemical engineering and chemistry such calculations are traditionally carried out by means of semi-empirical thermodynamic equation of state (EOS) and/or liquid-state
activity-coefficient models [1,2]. In order to implement these approaches, one requires as input information accurate data concerning the vapour-pressure behaviour of each constituent pure fluid; in addition, a
mixture parameter appearing in the theory is typically evaluated by means of an experimental measurement on the mixture (the latter making such approaches essentially corrrelative, rather than predictive).
Using this information, the system behaviour is calculated using standard thermodynamic relations [1].
The accuracy of these approaches in predicting the experimental data varies, depending both on the system and on the particular approach used; furthermore, as with all empirically-based methods, the path to
further progress is not always clear.
An alternative, but much less well-developed, approach uses as input an intermolecular potential model
for the interactions of the species molecules. The phase equilibrium compositions and other properties
of the mixture are then directly calculated by means of a computer simulation method. Such methods include the Gibbs ensemble Monte Carlo (GEMC) method [3], the NPT + test particle method
[4], the Gibbs–Duhem integration method [5], and the grand canonical Monte Carlo combined with
histogram-reweighting method [6]. Molecular-based simulation approaches have the considerable advantage over empirically-based approaches in that predictions may be made in the absence of experimental
data of any kind, provided one can construct an intermolecular potential model for the system. The construction of reasonable such models is now a relatively straightforward task [7–9]. They are constructed
from potentials of molecular fragments (groups), whose intermolecular potentials are generally intended
to be transferable to fluids of arbitrary molecules containing such groups. This approach is philosophically
similar to that underlying the UNIFAC method [2,10], a widely-employed semi-empirical macroscopic
approach.
The accuracy of the molecular-based approaches for the calculation of phase equilibria (in particular,
vapour-pressure) is currently generally not competitive with that of the empirically-based methods, especially for mixtures of any degree of complexity [11]. This is illustrated by the results of de Pablo and
Prausnitz [12] and de Pablo et al. [13], who applied the GEMC approach to binary alkane mixtures, of
Gotlib et al. [14], Agrawal and Wallis [15] and Strauch and Cummings [16], who applied the GEMC
approach to binary mixtures of methanol + ethane, methanethiol + propane, and methanol + water, respectively, of Delhommelle et al. [17] and Rivera et al. [18], who applied the GEMC approach to binary
mixtures of hydrogen sulfide + alkane, carbon dioxide + alkane, and nitogen + butane, respectively, and
of Fischer et al., who applied the NPT + test particle method to binary mixtures of methane, ethane and
carbon dioxide [4], and to the ternary methane + ethane + carbon dioxide system [19].
One goal of the aforementioned [4,8,9,13,14,18,19] simulation studies has been to produce more
accurate effective two-body potentials that can reproduce experimental vapour-pressure data for pure
fluids and their mixtures (e.g. [20]). However, this goal may be unrealistic in general, since it is likely that
three- and higher-body potentials will ultimately be required to accurately calculate the fluid properties
entirely from first principles. Furthermore, the level of molecular detail required to accurately predict
the properties of a given fluid using such an approach may ultimately preclude the transferability of the
resulting potential model to other fluids and their mixtures.
M. Lı́sal et al. / Fluid Phase Equilibria 181 (2001) 127–146
129
With the goal of retaining the molecular-level approach employing simple transferable potential models,
and of improving its accuracy to the point that it may ultimately be a useful engineering design tool, we
have recently proposed a new method called the reaction Gibbs ensemble Monte Carlo (RGEMC) method
[21,22]. This method circumvents the problem of obtaining accurate two-body potentials by incorporating
the experimental pure-component vapour-pressure data into the mixture simulations as constraints in a
novel way. These constraints are also employed by the semi-empirical macroscopic methods; such data
are typically readily available or can be accurately estimated by empirical techniques [10].
The RGEMC approach is a combination of the GEMC [3] method for phase equilibrium and the
reaction ensemble Monte Carlo [23,24] method for combined reaction and phase equilibrium. Unlike the
empirical approaches, the RGEMC approach uses no mixture experimental data of any kind, and can thus
be regarded as a predictive method. The underlying basis of the RGEMC method arises from viewing
phase equilibrium as a special case of a chemical reaction [25].
The RGEMC method has been previously applied to predict the phase equilibria of the MTBE ternary
system and its binaries, both in the absence [21] and the presence [22] of chemical reactions. The main goal
of this paper is to test the ability of the method to accurately predict VLE data for a range of complex binary
mixtures which lie at the boundaries of the capabilities of existing empirical approaches. We compare
the RGEMC results with those of typically-employed empirical approaches and with experiment.
In the next section of this paper, we briefly summarise the RGEMC method. In the following section,
we describe the intermolecular potential models used for the species involved. In the subsequent section,
we discuss the details of the computer simulations. Subsequent sections discuss the results and present
conclusions.
2. The reaction Gibbs ensemble Monte Carlo (RGEMC) method
The RGEMC method is described in detail elsewhere [21,22]; here, we summarise only the main
points. The required conditions of vapour–liquid equilibrium (VLE) are implemented by performing a
combination of three simulation steps: particle displacements to sample the configuration space, volume
changes to maintain a constant pressure P , and inter-phase particle transfers to implement equality of
chemical potentials for species in different phases. The vapour and liquid phases are represented by two
separate simulation boxes; an arbitrary simulation box in the following will be denoted by the symbol α.
The transition probability k → l for a particle displacement in box α [26] is
PklD = min[1, exp(−β
Uklα )]
and the transition probability k → l for a volume change of box α [26] is
Vlα
V
α
α
α
α
Pkl = min 1, exp −β
Ukl − βP (Vl − Vk ) + N ln α
Vk
(1)
(2)
In Eqs. (1) and (2), Uklα = Ulα − Ukα is the change in configurational energy in box α, β = 1/(kB T ), kB
Boltzmann’s constant, V α the volume of box α, and N α the total number of molecules in
box α.
The inter-phase particle transfers involve choosing the donor and recipient boxes at random, then
randomly choosing a particle of species i regardless of its type which is to be transferred from the donor
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box, and then attempting to transfer it to a random position in the recipient box. The transition probability
k → l for transfer of a particle from the liquid box () into the vapour box (g) is
N V g
g
→g
Pi,kl = min 1, Γi exp −β
Ukl − β
Ukl + ln
(3)
(N g + 1)V and similarly, the transition probability k → l for the transfer of a particle from the vapour box (g) into
the liquid box () is Eq. (3) with g and interchanged and with Γi replaced by its reciprocal. Γi is the
pseudo-ideal-gas driving term given by
Γi =
sat
Pi,exp
(T )
(4)
sat
Pi,sim
(T )
sat
sat
(T ) is the experimental vapour pressure and Pi,sim
(T ) the simulation vapour-pressure of
where Pi,exp
species i, respectively. The incorporation of Γi is similar to the device used for empirical equations
of state such as the Soave–Redlich–Kwong equation (e.g. [2]) to fit a model parameter to experimental vapour-pressure data. Γi ≡ 1 corresponds to the usual GEMC transition probability
[27].
3. Intermolecular potential models
We used OPLS potentials for ethane, methanol and ethanol [7,28], the TIP4P potential for water [29], and
the EPM2 potential for carbon dioxide [30]. In addition to the OPLS potentials for ethane and methanol,
we also utilized a potential for ethane due to Fischer et al. [31], and a potential for methanol due to Van
Leeuwen and Smit [32]. The molecules are described by interaction sites located on the nuclei, in which
the CHn groups are treated as united atoms centred on the carbons. The interactions among the molecules
are represented by site–site potentials; the interaction between sites a and b in different molecules
is described by the Lennard–Jones (LJ) and Coulombic potentials in the reaction-field geometry [33]
as
RF − 1 rab 3
Aab Cab
qa qb e2
uab (rab ) = 12 − 6 +
1+
(5)
4π0 rab
2RF + 1 rc
rab
rab
where rab is the distance between atoms a and b in different molecules, qa and qb the partial charges on
these atoms, rc the cut-off radius, RF the dielectric constant (set to experimental values or infinity in the
case of our models), 0 the permittivity of free space, and e the unit charge. In Eq. (5), the parameters
Aab and Cab can be expressed in terms of LJ well depth a and size σa via
Aaa = 4a σa12 ,
Caa = 4a σa6
The OPLS combining rules for Aab and Cab
Aab = Aaa Abb ,
Cab = Caa Cbb
(6)
(7)
were used in all cases except the methanol + ethane system modelled with the Van Leeuwen and
Smit potential for methanol [32], and with the Fischer et al. potential for ethane [31]. In this case,
M. Lı́sal et al. / Fluid Phase Equilibria 181 (2001) 127–146
131
Table 1
The Lennard–Jones well depths εi and sizes σi , partial charges qi , internal rotational function V (Φ) of ethanol and geometries
of ethane, methanol, ethanol, water and carbon dioxide
Atom
ε/kB (K)
σ (Å)
q
Geometry
Ethane (OPLS model [7])
104.17
CH3
3.775
0.0
CH3 –CH3 : 1.530 Å
Ethane (Fischet et al. model [31])
139.81
CH3
3.512
0.0
CH3 –CH3 : 2.353 Å
Methanol (OPLS model [7,28])
O
85.55
H
0.0
CH3
104.17
3.070
0.500
3.775
−0.700
0.435
0.265
O–H: 0.945 Å
CH3 –O: 1.430 Å
CH3 –O–H: 108.5◦
Methanol (Van Leeuwen and Smit model [32])
O
86.5
3.03
H
0.0
0.0
CH3
105.2
3.74
−0.700
0.435
0.265
O–H: 0.9451 Å
CH3 –O: 1.4246 Å
CH3 –O–H: 108.53◦
Ethanol (OPLS model [7,28])
H
0.0
O
85.55
59.38
CH2
CH3
104.17
0.435
−0.700
0.265
0.0
O–H: 0.945 Å
CH2 –O: 1.430 Å
CH2 –CH3 : 1.530 Å
H–O–CH2 : 108.5◦
O–CH2 –CH3 : 108◦
0.0
3.070
3.905
3.775
V (Φ) = (V1 /2)(1 + cos Φ) + (V2 /2)(1 − cos 2Φ) + (V3 /2)(1 + cos 3Φ)
V1 /kB = 419.68 K; V2 /kB = −58.37 K; V3 /kB = 375.90 K
Water (TIP4P model [29])
O
78.020
H
0.0
M-site
Carbon dioxide (EPM2 model [30])
O
80.507
C
28.129
3.1536
0.0
0.0
0.52
−1.04
O–H: 0.9572 Å
H–O–H: 104.52◦
O–M-site: 0.15 Å
3.033
2.757
−0.3256
0.6512
C–O: 1.149 Å
the Lorentz–Berthelot mixing rules
√
ab = a b ,
σab = 21 (σa + σb )
(8)
were used, and Aab and Cab were evaluated from Eq. (6). The molecular bond lengths and angles were fixed
and internal rotation of ethanol was included. The values of the potential parameters, the internal rotational
potential function V (Φ) for ethanol, and the molecular bond lengths and angles are given in Table 1.
4. Computational details
The RGEMC method requires preliminary simulations of several points on the vapour-pressure curves
of each pure fluid using the GEMC method; these are then fitted to a convenient form such as the
Antoine equation for use within the subsequent mixture calculations. Apart from these preliminary
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pure-component calculations, the time taken for mixture simulations is essentially the same as that
required for conventional GEMC simulations.
We used NVT GEMC [34] and NVT RGEMC [21,22] simulations for the pure fluids, and NPT RGEMC
[21,22] simulation for the binary mixtures, to determine their vapour–liquid coexistence curves; N is the
number of particles and V is the volume. For all simulations, we used N = 512 molecules in cubic boxes
with the minimum image convention, periodic boundary conditions, and with cut-off radius equal to
one-half the box length. We also employed inner (hard-core) cut-offs equal to 0.8σab to avoid unrealistic
interactions, especially during particle transfers. The LJ long-range corrections for the configurational
energy and the pressure were included [26], assuming that the radial distribution functions are unity
beyond the cut-off radius.
The simulations were organised in cycles as follows. Each cycle consisted of three steps: nD displacement moves (translation, rotation and (where relevant) internal rotation, chosen with equal probability),
nV volume moves, and nT particle transfers. The types of moves were selected at random with fixed
probabilities, chosen so that the appropriate ratio of moves was obtained. In the cases of NVT GEMC
and NVT RGEMC simulations, nD :nV :nT was set at N:1:5500. In the case of NPT RGEMC simulations,
nD :nV :nT was set at N:2:5500. The acceptance ratios for translational and rotational moves, and for volume changes, were adjusted to approximately 30%. After an initial equilibration period of 1×104 –2×104
cycles, we generated (depending on the thermodynamic conditions) between 0.5 × 105 and 1 × 105 cycles
to accumulate averages of the desired quantities. The precisions of the simulated results were obtained
using block averages, with 500 cycles per block. In addition to ensemble averages of the quantities of
direct interest, we also monitored the convergence profiles of the thermodynamic quantities, in order to
keep the development of the system under careful control [35].
5. Results and discussion
We studied binary mixtures involving water, methanol, ethanol, carbon dioxide, and ethane. In Table 2,
we summarise several pure-component experimental properties of these fluids. Since the basis of the
RGEMC is to ensure agreement of the pure-component vapour-pressure data with experiment, initial
pure-fluid simulation results are required [21]. Some are available in the literature, but in other cases, we
performed new simulations.
Table 2
Pure-component properties for ethane, methanol, ethanol, water and carbon dioxidea
Component
M (g/mol)
Tc (K)
Pc (bar)
vc (cm3 /mol)
ω
Tb (K)
Tm (K)
Ethane
Methanol
Ethanol
Water
Carbon dioxide
30.0696
32.0422
46.0690
18.0153
44.0098
305.33
513.380
513.92
647.10
304.1282
48.718
82.158
61.32
220.64
73.773
145.56
113.830
167
55.95
94.12
0.09910
0.5560
0.6441
0.34430
0.22394
184.55
337.632
351.44
373.12
194.75
90.352
175.610
159.0
273.16
216.592
a
M: molecular weight; Tc : critical temperature; Pc : critical pressure; vc : molar critical volume; ω: Pitzer acentric factor; Tb :
normal boiling temperature; Tm : melting temperature. Data were taken from [36–40].
M. Lı́sal et al. / Fluid Phase Equilibria 181 (2001) 127–146
133
Table 3
Vapour–liquid equilibrium data for the OPLS models of ethane [7] and ethanol [7,28] obtained by the Gibbs ensemble Monte
Carlo simulations of this worka
T (K)
um (kJ/mol)
um (kJ/mol)
vm (cm3 /mol)
vm (cm3 /mol)
P sat (bar)
Ethane [7]
260.41
281.25
298.15
312.50
−0.746
−1.2111
−1.7717
−2.87220
−10.4710
−9.5412
−8.6718
−7.41235
1294101
749.8630
488.6454
354.8326
66.8461
72.5493
79.33171
86.97245
13.7481
22.98130
33.10155
44.14345
Ethanol [7,28]
400
450
475
500
−2.5473
−3.1161
−4.5321
−6.7043
−33.1131
−28.2046
−25.4644
−22.5173
53491782
1866244
1021161
543.01227
66.5364
74.09104
79.88153
88.72252
3.2834
15.45238
26.79525
43.29595
g
g
a
T : temperature; um : molar configurational energy; vm : molar volume; P sat : vapour-pressure. Superscripts g and denote
vapour and liquid phases, respectively. The simulation uncertainties are given in the last digits as subscripts.
5.1. Pure fluids
For the Fischer et al. potential model for ethane, and the OPLS and Van Leeuwen and Smit potential
models for methanol, such data already exist [21,32,41]. For the OPLS potential models of ethane and
ethanol, the TIP4P potential model of water, and the EPM2 potential model of carbon dioxide, GEMC
VLE data were calculated by us. Our new GEMC VLE simulation results for these fluids are given in
Tables 3 and 4.
Table 4
Vapour–liquid equilibrium data for the TIP4P model of water [29] and for the EPM2 model of carbon dioxide [30] obtained by
the Gibbs ensemble Monte Carlo simulations of this worka
T (K)
um (kJ/mol)
um (kJ/mol)
vm (cm3 /mol)
vm (cm3 /mol)
P sat (bar)
Water [29]
350
400
450
500
550
−0.3531
−1.1838
−2.6756
−5.4253
−8.1991
−38.2920
−35.3421
−32.3322
−29.0331
−24.3957
495208220
85131216
2325276
736.3554
264.3359
18.9413
20.0118
21.6831
24.3844
31.66174
0.56631008
3.541522
13.33165
38.29358
87.931054
Carbon dioxide [30]
238
−0.606
248
−0.828
258
−1.0710
268
−1.4214
278
−1.7921
288
−2.3224
−12.4414
−11.8113
−11.1614
−10.4916
−9.7520
−8.8425
1250110
874.4683
642.8387
462.1456
347.3396
255.3266
40.5136
42.1538
44.0948
46.4168
49.51105
54.26167
13.63105
19.35121
26.02116
34.82221
44.27278
56.22279
g
g
T : temperature; um : molar configurational energy; vm : molar volume; P sat : vapour-pressure. Superscripts g and denote
vapour and liquid phases, respectively. The simulation uncertainties are given in the last digits as subscripts.
a
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M. Lı́sal et al. / Fluid Phase Equilibria 181 (2001) 127–146
Table 5
Critical temperature, Tc , critical pressure, Pc , molar critical volume, vc , and constants A, B and C for the Antoine vapour-pressure
equation (Eq. (11)) of ethane, methanol and ethanol, water, and carbon dioxide obtained from the simulations (see text for details)
Component
Tc (K)
Pc (bar)
vc (cm3 /mol)
A
B
C
Ethane [7]
Ethane [31]
Methanol [7,28]
Methanol [32]
Ethanol [7,28]
Water [29]
Carbon dioxide [30]
328.4
326.5
495.6
512.0
531.2
588.2
302.5
61.80
60.93
51.3
86.9
72.6
149.3
76.4
165.7
148.8
118.0
115.7
162.6
57.14
92.96
9.1872229
10.468059
10.653432
13.319871
10.301614
11.774845
9.6252521
−1546.5436
−2341.8442
−3023.3917
−4533.7452
−2378.3006
−3563.6212
−1394.0101
−22.962853
38.008706
−45.393498
0
−135.94342
−61.71318
−38.901111
We estimated the critical temperatures Tc and critical densities ρc for all simulated pure-fluid models
from a least-squares fit to the rectilinear diameter law [42]
1
(ρ g
2
+ ρ ) = ρc + C1 (T − Tc )
(9)
and the critical scaling relation [42]
ρ − ρ g = C2 (Tc − T )1/3
(10)
For each fluid, the estimated critical temperature and the critical volume, vc = 1/ρc , are given in Table 5.
We fitted the vapour-pressure curves for all pure fluids to the Antoine equation [10]
ln P sat = A +
B
T +C
(bar, K)
(11)
The Antoine constants are also given in Table 5, together with the critical pressures Pc (Tc ) obtained from
the Antoine equation.
Comparison of the GEMC, RGEMC, and experimental VLE behaviour for all pure fluids is shown
in Figs. 1–3. The GEMC vapour pressures typically differ from the experimental values by about 10%.
The best agreement between the GEMC and the experimental vapour-pressures is obtained using the Van
Leeuwen and Smit model for methanol [32]; in this case, the simulated vapour-pressures agree within
their statistical uncertainties with the experimental data [37].
The agreement of the GEMC results with the experimental orthobaric densities ranges from very good to
excellent (the RGEMC density results do not differ from the GEMC results within the scale of the graphs).
The worst agreement is obtained using the OPLS model for methanol [7,28] and the best agreement is
obtained using the Van Leeuwen and Smit model for methanol [32], and for the EPM2 model of carbon
dioxide [30]; in these cases, the simulated orthobaric densities agree within their statistical uncertainties
with the experimental orthobaric densities [37,40]. Agreement between the simulated and experimental
critical points is only good, except for the Van Leeuwen and Smit model for methanol [32], and the EPM2
model for carbon dioxide [30], the agreement for both of which is excellent. This is due to fact that the
OPLS, TIP4P and Fischer et al. potential models for these fluids were adjusted to experimental data at
ambient conditions.
M. Lı́sal et al. / Fluid Phase Equilibria 181 (2001) 127–146
135
Fig. 1. Vapour–liquid coexistence curves for (a) ethane and (b) methanol. Marks are simulation results and the solid line represents
the experimental results [36,37]. Filled circles are our Gibbs ensemble Monte Carlo simulation results. Open circles for ethane
are Vrabec et al. NPT + test particle method results [41] and those for methanol are Van Leeuwen and Smit GEMC results
[32]. The dashed line through the simulation vapour pressures represents fits of the simulation data to the Antoine equation.
Filled diamonds in the vapour-pressure curves are our reaction Gibbs ensemble Monte Carlo (RGEMC) simulation results; for
methanol, we also plotted our previous RGEMC simulation results [21].
5.2. Binary mixtures
We performed NPT RGEMC simulations for two sequences of binary mixtures: of water with methanol
and with ethanol, and of methanol with carbon dioxide and with ethane. The former sequence studies the
effects of molecular size differences when both components are strongly polar, and the second mainly
studies the effects of the molecular size at a nonpolar component when mixed with a given polar component. We compared the RGEMC results with experimental data and with results obtained using two
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M. Lı́sal et al. / Fluid Phase Equilibria 181 (2001) 127–146
Fig. 2. Vapour–liquid coexistence curves for (a) ethanol and (b) water. Filled circles are Gibbs ensemble Monte Carlo simulation
results of this work and the solid line represents the experimental results [38,39]. The dashed line through the simulation
vapour-pressures represents fits of the simulation data to the Antoine equation. Filled diamonds in the vapour-pressure curves
are our Reaction Gibbs ensemble Monte Carlo simulation results.
macroscopic-level approaches. These are the UNIFAC liquid-state activity-coefficient model combined
with the simple truncated virial equation of state [10] calculated by us (referred to as UNIFAC + B-EOS)
and with published calculations by the hole quasi-chemical group contribution equation of state [43]
(referred to as HQGCM EOS).
5.2.1. Water + methanol at 373.15 K
We used the OPLS potential for methanol [7,28] and the TIP4P potential for water [29]. Our RGEMC
results are listed in Table 6. They are also shown together with the experimental [44], UNIFAC-B +
EOS, and HQGCM EOS [43] results in Fig. 4. Although each fluid component is complex, the mixture
M. Lı́sal et al. / Fluid Phase Equilibria 181 (2001) 127–146
137
Fig. 3. Vapour–liquid coexistence curve for carbon dioxide. Filled circles are the Gibbs ensemble Monte Carlo simulation results
of this work and the solid line represents the experimental results [40]. The dashed line through the simulation vapour-pressures
represents fits of the simulation data to the Antoine equation.
behaviour is nearly ideal. Fig. 4 shows that the mutual agreement among the experimental, RGEMC,
UNIFAC + B-EOS, and the HQGCM EOS results is generally very good, except for the HQGCM EOS
near pure methanol. This is due to the inaccurate prediction of the vapour-pressure of pure methanol by
this approach. We remark that the good performance of the UNIFAC approach for the system is due to
the fact that the method treats both methanol and water as unique groups, whose interaction parameters
are determined from experimental data.
5.2.2. Water + ethanol at 393.15 K
We used the OPLS potential for ethanol [7,28] and the TIP4P potential for water [29]. At 393.15 K,
the system, in contrast to water + methanol, exhibits an azeotrope at Paz = 4.35 bar at a mole fraction of
water equal to ∼0.15.
Table 6
Vapour–liquid equilibrium data for the methanol + water system at the temperature 373.15 K from the RGEMC simulations of
this worka
P (bar)
1.073152
1.5
2.0
2.5
3.0
3.545590
Methanol (1) + water (2)
x1
y1
um (kJ/mol)
um (kJ/mol)
vm (cm3 /mol)
vm (cm3 /mol)
0
0.0659167
0.2185197
0.4901107
0.759297
1
0
0.3360166
0.5544209
0.7254228
0.8593115
1
−0.5238
−0.678
−0.9419
−1.2324
−1.5120
−1.6672
−36.8518
−36.3146
−35.3737
−33.6042
−32.1538
−30.6524
282403831
19810687
14610381
11400535
9321382
77961059
19.3117
21.1151
25.0160
32.5250
40.1475
47.3452
g
g
Methanol and water are modelled by the OPLS [7,28] and the TIP4P [29] potentials, respectively. xi and yi : mole fractions
of liquid and vapour, respectively; P : pressure; um : molar configurational energy; vm : molar volume. Superscripts g and denote
the vapour and liquid phases, respectively. The simulation uncertainties are given in the last digits as subscripts.
a
138
M. Lı́sal et al. / Fluid Phase Equilibria 181 (2001) 127–146
Fig. 4. The pressure-composition diagram for the methanol + water system at the temperature 373.15 K. Open circles are
experimental data [44] and filled diamonds are the reaction Gibbs ensemble Monte Carlo simulation results of this work. The
solid line represents our calculated predictions using the UNIFAC method and the dashed line represents predictions using the
hole quasi-chemical group contribution equation of state [43].
Our RGEMC results are listed in Table 7. They are also shown together with the experimental [44] and
our UNIFAC-B+EOS results in Fig. 5. Fig. 5 shows excellent mutual agreement among the experimental,
RGEMC, and UNIFAC + B-EOS vapour compositions, and very good mutual agreement for the liquid
compositions. The RGEMC liquid compositions are only slightly lower than those from the experiments
and those of the UNIFAC-B + EOS approach. HQGCMC EOS results are not available for this system.
We also attempted to perform an RGEMC simulation in the vicinity of the azeotropic pressure. However,
due to large fluctuations in the course of the simulation, already apparent in the simulation at P = 4 bar,
the simulations did not converge. This is a confirmation that the system is near an azeotropic point. To
our knowledge, no simulation methodology exists for determining such points.
Table 7
Vapour–liquid equilibrium data for the ethanol + water system at the temperature from the RGEMC simulations of this worka
P (bar)
1.956231
2.5
3.0
3.5
4.0
4.332691
Ethanol (1) + water (2)
x1
y1
um (kJ/mol)
um (kJ/mol)
vm (cm3 /mol)
vm (cm3 /mol)
0
0.019280
0.0496160
0.1232306
0.3390523
1
0
0.2423327
0.3871338
0.4844245
0.5858674
1
−0.8241
−1.0022
−1.0919
−1.3524
−1.6643
−3.8573
−35.7220
−35.5131
−35.2344
−34.8071
−33.8287
−33.7035
155601913
12330598
10200417
8585376
7222673
4361354
19.8520
20.7571
22.2782
25.57147
36.05173
65.6149
g
g
Ethanol and water are modelled by the OPLS [7,28] and the TIP4P [29] potentials, respectively. xi and yi : mole fractions of
liquid and vapour, respectively; P : pressure; um ; molar configurational energy; vm : molar volume. Superscripts g and denote
the vapour and liquid phases, respectively. The simulation uncertainties are given in the last digits as subscripts.
a
M. Lı́sal et al. / Fluid Phase Equilibria 181 (2001) 127–146
139
Fig. 5. The pressure-composition diagram for the ethanol + water system at the temperature 393.15 K. Open circles are experimental data [44] and filled diamonds are the reaction Gibbs ensemble Monte Carlo simulation results of this work. The solid
line represents our calculated predictions using the UNIFAC method.
5.2.3. Methanol + carbon dioxide at 323.15 K
We used the OPLS potential for methanol [7,28] and the EPM2 potential for carbon dioxide [30]. Since
323.15 K is slightly supercritical for carbon dioxide, we used an extrapolated vapour-pressure [45] for
carbon dioxide in conjunction with the RGEMC method (cf. Eq. (4)). This vapour-pressure was obtained
using the expression
ln P sat = A0 +
A1
+ A2 Tr + A3 Tr3
Tr
(12)
where Tr = T /Tc [45].
Our RGEMC results for this system are listed in Table 8 and they are also shown together with the
experimental [46] and the HQGCM EOS [43] results in Fig. 6. We note that the UNIFAC approach cannot
be used for this system since group parameters for carbon dioxide do not exist. Fig. 6 shows very good
mutual agreement among the RGEMC, HQGCM EOS [43] and the experimental [43] vapour data. At
lower pressures, the agreement among all three results for the liquid curve is very good, and (except for
one data point) at higher pressures the RGEMC results agree better with the experimental results than
those of the HQGCM EOS approach (note also that the experimental liquid compositions do not lie on
a smooth curve). From Fig. 6, we can roughly estimate the critical pressure at this temperature. This
estimation gives Pc ≈ 95 and 100 bar from our RGEMC simulations and from the experimental results,
respectively. These values are both slightly lower than the critical pressure of Pc = 103 bar predicted by
the HQGCM EOS [43].
5.2.4. Methanol + ethane at 298.15 K
We used (i) the OPLS potentials for both methanol and ethane [7,28] (referred to as Model 1), and (ii)
the Van Leeuwen and Smit potential for methanol [32] and the Fischer et al. potential for ethane [31]
(referred to as Model 2). At T = 298.15 K, GEMC simulations using Model 2 [14] and calculations
using the HQGCM EOS [43] have been published.
140
M. Lı́sal et al. / Fluid Phase Equilibria 181 (2001) 127–146
Table 8
Vapour–liquid equilibrium data for the methanol + carbon dioxide system at the temperature 323.15 K from the RGEMC simulations of this work; methanol and carbon dioxide are modelled by the OPLS [7,28] and the EPM2 [30] potentials, respectivelya
P (bar)
0.403238
5
20
40
60
70
80
90
Methanol (1) + carbon dioxide (2)
x1
y1
um (kJ/mol)
um (kJ/mol)
vm (cm3 /mol)
vm (cm3 /mol)
1
0.977848
0.8957136
0.7918210
0.6360314
0.5427521
0.4419327
0.2862165
1
0.120723
0.026183
0.015555
0.012743
0.013744
0.015573
0.017278
−0.5119
−0.2714
−0.497
−0.993
−1.646
−2.0511
−2.5420
−3.2750
−34.1224
−33.4842
−31.4154
−28.7369
−24.4497
−21.7089
−18.80102
−14.2755
331202975
523094
124322
571.9115
341.495
269.9120
216.8143
164.9294
43.4935
43.6059
43.9051
44.3053
45.8676
47.6989
50.05154
55.97222
g
g
xi and yi : mole fractions of liquid and vapour, respectively; P : pressure; um : molar configurational energy; vm : molar
volume. Superscripts g and denote the vapour and liquid phases, respectively. The simulation uncertainties are given in the last
digits as subscripts.
a
At 298.15 K, the system experimentally exhibits VLE at pressures below a three-phase pressure, Pt ,
and above Pt exhibits VLE at overall compositions near that of pure ethane, and liquid–liquid equilibrium
(LLE) over the mid-composition range. According to [47], the three-phase line coexistence properties are
I
II
I
II
Pt = 41.07 bar, xethane
= 0.3705, xethane
= 0.9128, vm
= 51.9 cm3 /mol, vm
= 79.8 cm3 /mol. According
I
I
to [48], the LLE on the three-phase line occur at Pt = 41.28 bar, xethane = 0.3528, vm
= 51.61 cm3 /mol.
Our VLE RGEMC results obtained using the Models 1 and 2 potentials are listed in Table 9. They are
also shown, together with the experimental [48], GEMC [14], UNIFAC-B + EOS and HQGCM EOS [43]
results, in Fig. 7.
Fig. 6. The pressure-composition diagram for the methanol + carbon dioxide system at the temperature 323.15 K. Open circles
are experimental data [46] and filled diamonds are the reaction Gibbs ensemble Monte Carlo simulation results of this work.
The dashed line represents predictions using the hole quasi-chemical group contribution equation of state [43].
M. Lı́sal et al. / Fluid Phase Equilibria 181 (2001) 127–146
141
Table 9
Vapour–liquid equilibrium data for the methanol + ethane system at the temperature 298.15 K from the RGEMC simulations of
this worka
P (bar)
Methanol (1) + ethane (2)
x1
With OPLS potentialb
0.18443
1
10
0.963486
20
0.9171160
25
0.8989216
30
0.8831248
35
0.8387257
42.05113
0
y2
um (kJ/mol)
um (kJ/mol)
vm (cm3 /mol)
vm (cm3 /mol)
1
0.014251
0.006840
0.006041
0.005132
0.004732
0
−0.112
−0.381
−0.843
−1.133
−1.496
−2.0015
−3.0220
−35.0343
−34.6737
−33.3354
−32.8965
−32.4860
−31.1078
−7.7321
1348001279
228727
102818
766.9192
582.1203
433.6289
383.6627
42.0537
42.9148
44.1758
44.5368
44.9084
46.1792
82.84123
−37.5741
−36.9748
−35.7485
−35.0472
−34.2471
−33.0257
−8.7419
1562001839
228229
102719
767.0190
588.3211
445.0276
386.7513
40.8757
41.4863
42.72103
43.3189
44.0680
45.3170
79.21163
g
With Van Leeuwen and Smit potentialc and Fischer et al. potentiald
0.15751
1
1
−0.123
10
0.968590
0.013061
−0.402
20
0.9272246
0.007947
−0.883
25
0.9047204
0.005634
−1.183
30
0.8813217
0.005846
−1.546
35
0.8411186
0.005633
−2.0213
41.87134
0
0
−3.1423
g
xi and yi : mole fractions of liquid and vapour, respectively; P : pressure; um : molar configurational energy; vm : molar
volume. Superscripts g and denote the vapour and liquid phases, respectively. The simulation uncertainties are given in the last
digits as subscripts.
b
For methanol and ethane [7,28].
c
For methanol [32].
d
For ethane [31].
a
At P = 35 bar and lower pressures, the RGEMC simulations exhibit VLE behaviour. At P = 37 bar
(the highest pressure previously simulated using the GEMC method [14]), the vapour box took on a
liquid-like density. This indicates that the system is near a three-phase line, and that the pressure of
this line, Pt , occurs at a lower pressure in comparison with the experimental value [47,48]. We remark
that prediction of three-phase equilibria using our general approach is feasible, but it requires a third
simulation box [49]; such simulations are beyond the scope of this paper. Close inspection of the previous
GEMC simulation [14] at P = 37 bar reveals that simulated density in the vapour box appears more
liquid-like than vapour-like. It is interesting to note that the HQGCM EOS predicts, as is also the case for
the simulations, a value of Pt that is lower than the experimental value [47,48], giving P = 38.2 bar. We
also preliminarily attempted to perform RGEMC simulations for prediction of the VLE located near pure
ethane overall compositions. However, these simulations failed because the liquid box vaporised due to
small differences in coexisting densities.
Above Pt , we also performed NPT GEMC simulations of the LLE behaviour. We used only the Model
2 potential because it gives a better description of the pure-component orthobaric densities in comparison
with experiments [36,37]. Our LLE GEMC results are listed in Table 10 and they are also plotted in Fig. 7.
From Fig. 7, we can roughly estimate the LL critical pressure at this temperature. This estimation gives
Pc ∼ 77 bar at a roughly equimolar composition of the mixture.
142
M. Lı́sal et al. / Fluid Phase Equilibria 181 (2001) 127–146
Fig. 7. The pressure-composition diagram for the methanol + ethane system at the temperature 298.15 K. Open circles are
experimental data [48], open boxes are the Gibbs ensemble Monte Carlo simulation results [14], and filled diamonds and boxes
are the reaction Gibbs ensemble Monte Carlo simulation results of this work for the Models 1 and 2 potentials, respectively. The
filled boxes at P ≥ 37 bar correspond to the Gibbs ensemble Monte Carlo simulation results of this work for the Model 2 potential.
The solid line represents our calculated predictions using the UNIFAC method and the dashed line represents predictions using
the hole quasi-chemical group contribution equation of state [43].
Fig. 7 shows that (i) our RGEMC results using the Models 1 and 2 potentials are identical within
their statistical uncertainties, (ii) the RGEMC results using both potentials agree within their statistical
uncertainties with the experimental as well as with the HQGCM EOS results up to P ≈ 25 bar, and both
RGEMC and HQGCM EOS results are slightly incorrect at higher pressures, (iii) the existing GEMC
results [14] are rather scattered and they deviate from the experimental results [48], especially at low
pressures, (iv) the UNIFAC + B-EOS approach is unable to accurately describe the VLE behavior of
Table 10
Liquid–liquid equilibrium data for the methanol + ethane system at the temperature 298.15 K from the GEMC simulations of
this work a
P (bar)
40
45
50
60
70
75
Methanol (1) + ethane (2)
x1I
x1II
uIm (kJ/mol)
uIIm (kJ/mol)
vmI (cm3 /mol)
vmII (cm3 /mol)
0.8121270
0.7513207
0.7333214
0.6839225
0.6173243
0.5763279
0.2018101
0.2312173
0.2428131
0.2919206
0.3606377
0.4490155
−32.2786
−30.4662
−30.0167
−28.5460
−26.7174
−25.4387
−14.4249
−15.2862
−15.6849
−17.2664
−19.19102
−21.7348
46.0097
48.0385
48.5674
50.1490
52.10105
53.86128
71.82294
69.49217
69.08224
65.50192
61.46185
58.79100
a
The system is modelled with the Van Leeuwen and Smit potential for methanol [32] and the Fischer et al. potential for ethane
[31]. x: mole fractions of the liquid phases; P : pressure; um : molar configurational energy; vm : the molar volume. Superscripts
I and II denote the methanol-rich and ethane-rich liquid phases, respectively. The simulation uncertainties are given in the last
digits as subscripts.
M. Lı́sal et al. / Fluid Phase Equilibria 181 (2001) 127–146
143
this system; this is likely because the UNIFAC CH3 parameters used to represent ethane were derived by
extrapolation from data for higher linear alkanes, and (v) neither empirical approach is able to predict the
three-phase line or the LE and VLE behavior above the three-phase line pressure.
6. Conclusions
We have employed molecular-level RGEMC simulations to calculate Pxy phase equilibrium data for
complex binary systems at representative temperatures involving water, methanol, ethanol, carbon dioxide, and ethane. We compared our results with experimental results, and with those calculated using two
semi-empirical engineering approaches: the UNIFAC method combined with the simple truncated virial
equation of state [10] (UNIFAC + B-EOS) and the hole quasi-chemical group contribution equation of
state [43] (HQGCM EOS).
The sequence (water + methanol, water + ethanol) displays differences in phase behaviour due to
molecular size when both components are strongly polar. The behaviour of the former system is nearly
ideal, whereas the latter system exhibits azeotropy. The UNIFAC + B-EOS approach captures the behaviour of the former system quantitatively, due to the fact that both water and methanol are treated as
groups in the approach. The RGEMC results are only slightly less accurate. The HQGCM EOS results are
inaccurate, due primarily to the fact that the approach poorly predicts the pure methanol vapour-pressure.
For the water + ethanol system, the RGEMC and the UNIFAC + B-EOS results are both equally accurate,
agreeing well with the experimental data. No HQGCM EOS results are available for this system for
comparison.
The sequence (methanol + carbon dioxide, methanol + ethane) primarily displays differences in phase
behaviour due to molecular size when one of the components is strongly polar. The behaviour of the former
system is quite nonideal, and that of the latter is more so, exhibiting three-phase behaviour. The RGEMC
results and those obtained using the HQGCM EOS approach are of similar accuracy, both agreeing
reasonably well with the experimental data, with the RGEMC results being slightly more accurate. No
UNIFAC + B-EOS results exist for this system, since no group parameters exist for carbon dioxide. For
the highly nonideal system methanol + ethane, the three-phase behaviour is predicted by the RGEMC
approach, but by neither of the empirical approaches. At pressures below the three-phase pressure, Pt , the
RGEMC and HQGCM EOS results agree equally well with the experimental data. The UNIFAC+B-EOS
results are poor for this range of pressures.
Both the RGEMC and the empirical approaches require pure-component vapour-pressure data for their
implementation. (When such experimental data is unavailable, empirical approaches may be utilized.)
The RGEMC approach has the advantage over empirical approaches in that, unlike them, it requires no
experimental mixture data. The result is that the former approaches are essentially correlative in nature
whereas the RGEMC approach is predictive. Another advantage is that, for the (wide range of) systems
studied here, the overall accuracy of the RGEMC approach is similar to or better than that of the most
accurate of the empirical methods tested; in addition, the accuracy of each empirical method depends on
the particular system.
The RGEMC approach lies intermediate between first-principles molecular-based simulation methods
and the empirical approaches. Due to its inherent computational complexity, the RGEMC method cannot
compete with the empirical approaches for routine chemical engineering implementation in software
such as process simulators. However, it can play a useful role in providing reasonably accurate predictions for hitherto unstudied systems for which no mixture data is available for incorporation within the
144
M. Lı́sal et al. / Fluid Phase Equilibria 181 (2001) 127–146
empirical approaches. In addition, RGEMC calculations may be used to provide mixture data for such
incorporation.
List of symbols
A, B, C
A0 , A1 , A2 , A3
Aab
Cab
C1
C2
e
kB
M
n
N
P
P sat
Pkl
q
rab
rc
T
uab
um
U
vm
V
V (Φ)
V1 , V2 , V3
x
x
y
coefficients of the Antoine equation
coefficients of Eq. (12)
OPLS potential parameter (J m12 )
OPLS potential parameter (J m6 )
coefficient of Eq. (9)
coefficient of Eq. (10)
unit charge (1.602 × 10−19 C)
Boltzmann’s constant (1.380658 × 10−23 J/K)
molecular weight (kg/mol)
number of moves
total number of molecules
pressure (Pa)
vapour-pressure (Pa)
transition probability k → l
partial charge on an atom
distance between atoms a and b in different molecules (m)
cut-off radius (m)
temperature (K)
site–site potential (J)
molar configurational energy (J/mol)
configurational energy (J)
molar volume (m3 /mol)
volume of simulation box (m3 )
internal rotational potential function (J)
coefficients of V (Φ)
mole fraction of liquid phase
compositions of coexisting phases
mole fraction of vapour phase
Greek letters
β
Γ
ε
0
RF
ρ
σ
Φ
ω
β = 1/(kB T ) (1/J)
pseudo-ideal-gas driving term
change in a quantity
Lennard–Jones well depth (J)
permittivity of free space (8.8542 × 10−12 C2 /N m−2 )
dielectric constant
molar density (mol/m3 )
Lennard–Jones size (m)
dihedral angle (rad)
Pitzer acentic factor
M. Lı́sal et al. / Fluid Phase Equilibria 181 (2001) 127–146
Subscripts
a, b
az
b
c
exp
D
i
k
l
m
r
t
T
V
index of atoms
azeotrope
normal boiling
critical
experimental
displacement
species
old configuration
new configuration
melting
reduced
three-phase
transfer
volume change
Superscripts
α
g
I
II
arbitrary simulation box
vapour phase
liquid phase
methanol-rich phase
ethane-rich phase
145
Acknowledgements
This research was supported by the Grant Agency of the Czech Republic under Grant No. 203/98/1446,
by the Grant Agency of Academy of Sciences of the Czech Republic under Grant No. A-4072712, and
by the Natural Sciences and Engineering Research Council of Canada under Grant No. OGP1041.
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