K. U. Kholmatov,
F. J.
Keil
Journal of the University of Chemical
Technology
and
Metallurgy, 46, 3, 2011, 267-274
MONTE CARLO SIMULATIONS OF PHASE EQUILIBRIA OF BINARY MIXTURES
CONTAINING METHANE, ETHANE, n-PENTANE, NITROGEN, AND OXYGEN
COMPARISON WITH EXPERIMENTAL MEASUREMENTS AND PREDICTIONS
OF PENG-ROBINSON EOS AND PERTURBED-CHAIN SAFT
K. U. Kholmatov, F. J. Keil
Hamburg University of Technology (TUHH),
Institute of Chemical Reaction Engineering,
Eissendorfer str. 38, 21073, Hamburg, Germany
E-mail: [email protected], [email protected]
Received 07 June 2011
Accepted 30 June 2011
ABSTRACT
The Gibbs ensemble Monte Carlo computer simulation method was applied to predict vapour-liquid equilibria of
the binary systems methane - ethane, methane - n-pentane, ethane - n-pentane, and nitrogen - oxygen at various temperatures. The simulated vapour-liquid equilibria are compared with experimental data, results of Peng-Robinson EOS and
Perturbed-Chain SAFT. The simulated results are in good agreement with experiments.
Keywords: Monte Carlo, simulation, Gibbs ensemble, phase equilibria, hydrocarbons, nitrogen, oxygen, PerturbedChain SAFT, Peng-Robinson EOS.
INTRODUCTION
Thermodynamics of vapor-liquid equilibria plays
an important role in many chemical processes associated with phase separation. To study this thermodynamic
behavior the accurate knowledge of phase equilibria for
each of the pure components and the multicomponent
mixtures, which is usually obtained from experimental
observations, is required for the design of separation processes. Of particular interest are the phase equilibria of
mixtures containing alkanes, nitrogen and oxygen, which
are typically found in the petrochemical industry. Because of special requirements of laboratory equipment
and the limited time or financing of experiments, the
obtained experimental data are generally discrete and limited. For calculation of thermodynamic properties traditionally different empirical or semitheoretical models such
as the well-known Peng-Robinson (PR) equation of state
(EOS) [1], the Redlich-Kwong equation [2], the RedlichKwong equation modified by Soave [3] are used. These
models can reproduce data with good accuracy in special systems and conditions. However, most of these
models need a range of experimental data to fit the model
parameters.
Computer simulations based on molecular
modeling are a promising approach to study phase equilibria, in particular the phase coexistence properties of
many industrially relevant fluids [4]. The remarkable advantage of molecular modeling over the empirical or
semiempirical models is that this approach, in principle,
allows of predicting vapor-liquid phase equilibria (VLE)
of a fluid system under any thermodynamic conditions.
Different methods have been proposed, such as the Gibbs
Ensemble Monte Carlo method (GEMC) [5, 6], the
Gibbs-Duhem integration method [7, 8], Histogram
Reweighting Grand Canonical Monte Carlo [9, 10] and
the NPT + test particle method [11, 12].
METHODOLOGY AND MODEL
Gibbs Ensemble Monte Carlo simulations
Gibbs Ensemble Monte Carlo simulations are
performed using two simulation boxes, one for the liquid and the other for the gas phase, where volume and
particle number exchanges are executed for the two
phases of the system. For this ensemble three kinds of
moves are performed in Monte Carlo steps, which are
random molecule displacements inside the boxes,
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Journal of the University of Chemical Technology and Metallurgy, 46, 3, 2011
changes of the volumes, random transfer of molecules
between the two boxes.
The sampling distribution of the Gibbs ensemble
(constant NVT version) is defined by
model. In our study the Lennard-Jones (LJ) 12-6 potential model was used for dispersion interactions, which
has the following form:
(4)
(1)
where, rij is the distance between interaction sites i and
j, u is the potential energy between molecules i and j, εij
is the energy parameter of the interaction, and óij is the
LJ size. We used the united atom approach for modeling methane as one-center, ethane as two-center and npentane as five-center molecules. Here the center of the
unit is located at the center of carbon atom.
The Lorentz-Berthelot mixing rules
where rN are the coordinates of particles of the system,
VI and VII are the volumes of boxes I and II, respectively, â is the reciprocal temperature (â = 1/kBT, where
kB is Boltzmann’s constant), and E is the configurational
energy.
The Metropolis acceptance rule is used [13] for
the random displacement of particles. The acceptance
rule for Monte Carlo move from configuration j to j+1
is given as min(1, Pj → j+1), where
(5)
were used to determine the interaction between unlike
sites.
We used the TraPPE (transferable potential for
phase equilibria) force field by Martin and Siepmann
[14] for methane, ethane, and n-pentane. For nitrogen
the TraPPE model of Potoff and Siepmann [15], for
oxygen force field introduced by Hansen et al. [16] have
been used.
The Lennard-Jones parameters employed in our
study are listed in Table 1.
From Monte Carlo simulations in the Gibbs ensemble we determined data for the pure components
and mixture phase diagrams.
(2)
For NPT Gibbs Ensemble Monte Carlo where
number of particles, pressure, and temperature are constant the sampling distribution is expressed as:
(3)
where NIá is the number of molecules of species á in
region I, and P is the imposed system pressure. In this
case VI and VII are independent.
Perturbed-chain statistical associating fluid theory
Additionally a version of the perturbed-chain statistical associating fluid theory (PC-SAFT) proposed by
Gross and Sadowski [17] is used to investigate the phase
equilibria for the binary mixtures. The SAFT equation
of state and its many modifications are a theoretically
Molecular Model and Simulation Details
For molecular simulation the interactions between
molecules are usually described by a suitable potential
Table 1. Force field parameters
Site
CH4
CH3
CH2
N (in N 2)
COM a (in N2)
O (in O 2)
COM a (in O2)
a
268
º/kB [K]
148.000
98.000
46.000
36.000
0.0
49.048
0.0
COM, center of mass.
ó [Å]
3.730
3.750
3.950
3.310
0.0
3.013
0.0
q [e]
0.0
0.0
0.0
-0.482
+0.946
-0.123
+0.246
K. U. Kholmatov, F. J. Keil
derived model, based on perturbation theory. A fluid is
assumed to consist of equal-sized hard spheres. Then a
dispersive potential is added to account for attraction
between the spheres, e.g. a Lennard-Jones potential.
Each sphere is given two or more sticky spots, which
enable the formation of a chain. Specific interaction
sites in the chain enable their association through some
attractive interaction, e.g. hydrogen bonding. Each of
these steps contributes to the Helmholtz energy.
The pure-compound parameters, m (segment
number), ó, and ε (potential parameters) used in this
work for methane, ethane and n-pentane were taken from
the work of Gross and Sadowski [17].
RESULTS AND DISCUSSION
Pure components
Simulation results for the temperature-density coexistence curve of pure methane, ethane, n-pentane, nitrogen and oxygen are compared to experimental data from
literature in Fig. 1. The simulations have virtually generated the same results as experimental data reported for
methane by Setzmann and Wagner [18], for ethane by
Friend et al. [19], for n-pentane by Smith and Srivastava
[20], and for nitrogen and oxygen by NIST [21].
The critical temperature was determined by fitting simulated saturated liquid and vapor densities to
the density scaling law for the critical temperature [22]:
(6)
where A1 is fitting constant, and â is the critical exponent.
The critical density was obtained from the law of
rectilinear diameters [23]:
(7)
where A2 is a fitting constant.
The constants â and A were fitted to experimental temperature - density data.
Mixtures
Results from our molecular simulations are compared to experimental data, to results from PengRobinson EOS and to the simulation results from other
authors. The parameterisation of Peng-Robinson EOS
requires the critical temperature, Tc, the critical pres-
sure, Pc, and the acentric factor, ù, of the pure fluids.
These data were taken from Poling et al. [32]. As the
Peng-Robinson EOS does not yield, in all cases, very
satisfactory results as a predictive tool, additional calculations were performed with more sophisticated equation of state - the Perturbed-Chain SAFT.
Methane - ethane
Methane and ethane are the most important hydrocarbons of natural gas. Many industrial processes
like processing of natural gas require knowledge of phase
equilibria properties of the binary system methane ethane. Simulation results for the pressure-composition
(P-xy) diagrams of the system methane - ethane at 160,
172.04, 180, 199.93 and 250 K are presented in Fig. 2
together with experimental data reported by Miller et
al. [24], Wichterle and Kobayashi [26], Wei et al. [26],
and predictions of Peng-Robinson EOS. As indicated
by the position of the squares to the circles, the Gibbs
Ensemble Monte Carlo simulation results for methane
- ethane, are found to be in good agreement with experiment at the five temperatures considered in this
study. We can observe a perfect agreement between simulation, experimental and Peng-Robinson EOS results
for liquid compositions at the five temperatures for all
pressures. The vapor mole fraction of methane for the
lower pressures tends to be underpredicted by simulations. However, the best agreement for vapor compositions in the middle and high pressure range is obtained
at 160, 172.04, 180 and 199.93 K. For the temperature
of 250 K coincidence between experiment and simulation is also quite good.
Methane - n-pentane
Simulation results for the pressure-composition
(P-xy) diagrams of the system methane - n-pentane at
310.89, 344.45, and 360.07 K are presented in Fig. 3
together with experimental data reported by Reiff et al.
[27], predictions of Peng-Robinson EOS and PerturbedChain SAFT. As indicated by the position of the squares
to the circles, the Gibbs Ensemble Monte Carlo simulation results for methane - n-pentane, are found to be
in relatively good agreement with experiment at the three
temperatures considered in this study. The best agreement for liquid compositions is obtained at all temperatures studied for all pressures.
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Journal of the University of Chemical Technology and Metallurgy, 46, 3, 2011
Ethane - n-pentane
Simulation results for the pressure-composition (P-xy) diagrams of the system ethane - n-pentane at 277.59, 310.93 and 344.26 K are presented in
Fig. 4 together with experimental data reported by
Reamer et al. [28], predictions of Peng-Robinson EOS
and Perturbed-Chain SAFT, and simulated results
from Serbanovic et al. [29]. As indicated by the position of the squares to the circles, the Gibbs Ensemble
Monte Carlo simulation results for ethane - n-pentane, are found to be in relatively good agreement
with experiment at the three temperatures considered
in this study. We can observe a good agreement between simulation, experimental, Peng-Robinson EOS
and Perturbed-Chain SAFT results for liquid compositions at 344.26 K for all pressures.
The results of the present work for the liquid
mole fraction of ethane at 310.93 K are in much better
Fig. 1. Vapor-liquid coexistence curve for methane, ethane, n-pentane, nitrogen and oxygen. Open symbols represent the
simulation results. The solid line represents experimental data for methane from Setzmann and Wagner [18], for ethane from
Friend et al. [19], for n-pentane by Smith and Srivastava [20], and for nitrogen and oxygen by NIST [21].
270
K. U. Kholmatov, F. J. Keil
agreement with the experimental and Peng-Robinson
EOS results in contrast to the simulation results from
Serbanovic et al. [29] where Gibbs ensemble Monte
Carlo simulation method has been applied and optimised
potential for the liquid simulating (OPLS) model by
Jorgensen et al. [30] for the system ethane - n-pentane
has been used.
Nitrogen - oxygen
Knowledge of phase equilibria of the binary system nitrogen - oxygen is essential for many industrial
processes such as air liquefaction and air separation.
Simulation results for the pressure-composition (P-xy)
diagrams of the system nitrogen - oxygen at 85, 105 and
125 K are presented in Fig. 5 together with experimen-
Fig. 2. Pressure-composition diagram for the methane - ethane system at 160, 172.04, 180, 199.93 and 250 K. Comparison of
our simulated results (squares) with the experimental data (circles) from Miller et al. [24], Wichterle and Kobayashi [25], Wei
et al. [26], and predictions of Peng-Robinson EOS.
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Journal of the University of Chemical Technology and Metallurgy, 46, 3, 2011
tal data reported by Dodge [31], and predictions of PengRobinson EOS. We can observe an excellent agreement
between simulation, experimental and Peng-Robinson
EOS results for liquid and vapor compositions at 85,
105 and 125 K for all pressures.
pentane are compared with experimental data at about
344 K. Additionally, the system methane - ethane has been
compared for 250 K. As expected the vapor pressure is far
higher for the system consisting of one shorter alkane.
CONCLUSIONS
Comparison of Peng-Robinson EOS results
In Fig. 6 results obtained by the Peng-Robinson
EOS of the mixtures methane - n-pentane and ethane - n-
The Gibbs Ensemble Monte Carlo was used to
study the vapor-liquid equilibria properties of methane,
Fig. 3. Pressure-composition diagram for the methane n-pentane system at 310.89, 344.45, and 360.07 K.
Comparison of our simulated results (squares) with the
experimental data (circles) from Reiff et al. [27], and
predictions of Perturbed-Chain SAFT and PengRobinson EOS.
Fig. 4. Pressure-composition diagram for the ethane - npentane system at 277.59, 310.93 and 344.26 K. Comparison
of our simulated results (squares) with the experimental data
(circles) from Reamer et al. [28], predictions of PerturbedChain SAFT and Peng-Robinson EOS, and simulated results
from Serbanovic et al. [29].
272
K. U. Kholmatov, F. J. Keil
Fig. 6. Pressure-composition diagram for the methane ethane system at 250 K, the methane - n-pentane system at
344.45 K, and the ethane - n-pentane system at 344.26 K.
Comparison of the experimental data (open symbols) for
the methane - ethane system at 250 K from Wei et al. [26],
for the methane - n-pentane system at 344.45 K from Reiff et
al. [27], for the ethane - n-pentane system at 344.26 K from
Reamer et al. [28] with predictions of Peng-Robinson EOS.
SAFT. Comparison of the simulated results with experimental data demonstrated that Gibbs Ensemble
Monte Carlo simulations can be used to predict vaporliquid equilibria with accuracy close to experiments.
Acknowledgements
Financial support from Deutscher Akademischer
Austausch Dienst is gratefully acknowledged.
Fig. 5. Pressure-composition diagram for the nitrogen - oxygen
system at 85, 105 and 125 K. Comparison of our simulated
results (squares) with the experimental data (circles) from
Dodge [31], and predictions of Peng-Robinson EOS.
ethane, n-pentane, nitrogen, and oxygen. Gibbs Ensemble
Monte Carlo simulations of the phase equilibria of binary mixtures containing methane, ethane, n-pentane,
nitrogen, and oxygen were performed on the basis of
the investigation of pure components. Simulation results for the temperature-density coexistence curve of
pure methane, ethane, n-pentane, nitrogen and oxygen
as well as the pressure-composition diagrams of the systems methane - ethane, methane - n-pentane, ethane n-pentane, nitrogen - oxygen at several different temperatures are compared to experimental data, and predictions of Peng-Robinson EOS and Perturbed-Chain
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