
Research Article


Calculation of Critical Curves for Carbon Dioxide+nAlkane Systems 

Jinglin Yu,
Shujun Wang ,
Yiling Tian
and
Wei Xu



ABSTRACT

The critical curves of eight binary systems from carbon dioxide+methane to carbon dioxide+octane at temperatures from 200 to 570 K and pressures from 2.5 to 14.7 MPa have bee calculated. The critical pressures, the critical temperatures, the critical mole fractions, the critical molar volumes and the critical densities are obtained by using an Equation of State (EOS), which consists of a hard body repulsion term and an additive perturbation term. The latter term accounts for the attractive molecular interactions and uses a squarewell potential, so three adjustable parameters are required. Good agreement was obtained between the experimental data, the literature data and the calculated values.





Introduction High pressure vaporliquid equilibria of carbon dioxide+hydrocarbon systems have been widely investigated. Since the mid1980s, supercritical (SC) CO_{2}+lower molecular weight alkanes or alcohols systems have been of interest because of their importance of SC fluids and cosolvent pairs in the separation of biomaterials (Fornari, 1990; Dohrn, 1995; Christov and Dohrn, 2002). The calculation of critical curves for binary mixtures is very important in the study of phase equilibria. Critical states of mixtures are of interest for a number of reasons. They delineate the homogeneous and heterogeneous regions. They are also used in numerous correlations for the properties of mixtures. They are intricately connected with retrograde phenomena. Measurements of critical points are very difficult. Therefore, they are often determined from thermodynamic models using phase equilibrium parameters. A robust method for this task is a powerful tool for the calculation of critical loci. Early theoretical discussions by van der Waals on critical phenomena in pure and mixed fluids were instrumental in encouraging many researchers in the latter part of the last century to undertake experimental work in this field (Van der Waal, 1900). Scott and Van Konynenburg (1970) have classified critical curves using the van der Waals equation for nonpolar components. Schneider (1978) and Rowlinson and Swinton (1978) have published a comprehensive discussion on the classification of critical curves for binary systems. Although there is a great deal of literatures describing the measurements of critical points, the calculation of critical curves for SC CO_{2}+lower molecular nalkanes has not been reported in detail. Especially, the data of critical molar volume and critical densities are absent. Neichel and Franck (1996) have calculated the critical curves of several binary systems by means of the HeiligFranck Equation of State (EOS). Tian et al. (2003) have calculated the critical curves for supercritical CO_{2}+1alkanol systems. Many equations of state have been proposed and applied in the past. In this study, the HeiligFranck EOS is employed in which only three adjustable parameters are required and the experimental data are not required.
Equation of State and Thermodynamic Relations
To predict the gasliquid critical curves of binary mixtures under high
temperatures and pressures, an appropriate equation of state (HeiligFranck
EOS) was derived from a perturbation method by using a squarewell molecular
interaction potential. Employing a hard body repulsion term and an additive
perturbation term, the pressure is given by
where the B_{x} and C_{x} parameters are the second and third virial coefficients of a squarewell fluid and β_{x} represents the molecular volume of the fluid. V_{m} is the molar volume of the mixture and x_{i} is the molar fraction of component i. Each of the molecular terms can be given by where T_{c,i} is the critical temperature of component i and m is a temperaturedependent exponent. The applicability of this relationship is limited to a region of relatively high temperatures. The value of the characteristic exponent, m, can be estimated from the general properties of molecular interaction or from adjustments of experimental pVTdata (Christoforakos and Franck, 1986). For the calculations in the present study m = 10 has been chosen (Wu et al., 1990). σ is the sphere diameter and N_{0} is Avogadro’s constant. The virial attraction terms can be given by the following: Here k is Boltzmann’s constant, ω is the relative width of the squarewell in units of σ, ω is its depth and σ is its core diameter. The selection of parameters for mixed interactions is a central problem. If identical relative squarewell widths are ω_{ij} = ω_{ii}+ω_{jj} used for different particle combinations in the binary fluid mixtures, the usual LorentzBerthelot combination rules together with empirical parameters k_{ε} and k_{σ} can be applied: k_{ε} and k_{σ} are binary mixing coefficients. The relative width of the potential well, ω, can be set at values between 1.5 and 2.5 or derived from vapor pressure curves. The ωvalues decrease with the increase of the molecular polarity and can be correlated with an acentric factor (Christoforakos and Franck, 1986). These are adjustable parameters defined by combination rules. The factors k_{ε} and k_{σ} can be determined from experimental mixture data or can be predicted by analogy from existing values of related systems. It appears that k_{ε} and k_{σ} remain constant or vary only modestly and systematically within certain groups of systems. The diameter σ and the depth, ε, of the squarewell are derived from critical data of the pure partners. The third virial coefficient is given by: The auxiliary functions of the virial coefficients I_{11} to I_{33} have been given by Hirschfelder et al. (1964). To determine the critical phenomena of binary systems, the stability criteria formulated with the Helmholtz energy A have to be observed. The critical points of mixtures are obtained when all of the physical properties of two coexisting phases are identical. This is obtained when the following conditions are satisfied simultaneously (Sadus, 1992b): where A, T and V denote the Helmholtz function, temperature and volume, respectively. The conditions W = 0 and X = 0 express the relationships between the temperature T, the molar volume, V_{m} and the mole fraction, x_{i}, of the critical point. The condition Y>0 guarantees the thermodynamic stability of the calculated critical point. The analytical determination of the critical curve is possible only when relatively simple expressions for the molar Helmholtz function of the fluid mixture, A_{m}, can be obtained. The equation contains a term for a residual free energy as well as terms for a reference state with chemical potentials μ_{i}^{θ} and pressure p^{θ} (McGlashan, 1979). From thermodynamics, A_{m}, which is a function of T, V_{m} and x_{i}, can be given by: Shmonov et al. 1993) and Deiters et al. (1993) have given more detailed descriptions. Results and Discussion
The critical curve is of great importance for characterizing the real behavior
of mixtures. Critical curves of binary mixtures are usually classified into
six principal types (Konynenburg and Scott, 1980). The shapes of critical curves
are very sensitive to the molecular size and interactions of the components.
In this section, the results for eight binary mixtures calculated by the HeiligFranck
EOS are shown (Fig. 18). A systematic classification
of binary mixtures has been proposed by van Konynenburg and Scott (Konynenburg
and Scott, 1980). In this classification different binary mixtures of carbon
dioxide+nalkanes belong to different types.

Fig. 1: 
T_{c}x_{c}, p_{c}x_{c}
and p_{c}T_{c} diagrams of the SC CO_{2}+CH_{4}
system 

Fig. 2: 
T_{c}x_{c}, p_{c}x_{c}
and p_{c}T_{c} diagrams of the SC CO_{2}+C_{2}H_{6}
system 

Fig. 3: 
T_{c}x_{c}, p_{c}x_{c}
and p_{c}T_{c} diagrams of the SC CO_{2}+C_{3}H_{8}
system 

Fig. 4: 
T_{c}x_{c}, p_{c}x_{c}
and p_{c}T_{c} diagrams of the SC CO_{2}+C_{4}H_{10}
system 

Fig. 5: 
T_{c}x_{c}, p_{c}x_{c}
and p_{c}T_{c} diagrams of the SC CO_{2}+C_{5}H_{12}
system 
Type I binary pT diagrams include, CO_{2}+ethane (Fig.
2) and CO_{2}+nbutane (Fig. 4). This is the simplest
case in which the pT projection of the three dimensional pressuretemperaturecomposition
(pTx) diagram consists of two vaporpressure curves for the pure components
and a critical line. Type I can be further divided into five subdivisions according
to the shape of the continuous gasliquid critical curve The molecules of CO_{2}
and ethane have similar shapes and sizes. The quadrupole moment of carbon dioxide
is much stronger than that of ethane.

Fig. 6: 
T_{c}x_{c}, p_{c}x_{c}
and p_{c}T_{c} diagrams of the SC CO_{2}+C_{6}H_{14}
system 

Fig. 7: 
T_{c}x_{c}, p_{c}x_{c}
and p_{c}T_{c} diagrams of the SC CO_{2}+C_{7}H_{16}
system 

Fig. 8: 
T_{c}x_{c}, p_{c}x_{c}
and p_{c}T_{c} diagrams of the SC CO_{2}+C_{8}H_{18}
system 
The binary pT diagram of CO_{2}+ethane belongs to the third subdivision
of type I, in which the critical curve is convex and exhibits a minimum temperature
in the pT plane. The binary pT diagram of CO_{2}butane belongs to
the second subdivision of type I, in which the critical curve is convex and
exhibits a maximum pressure in the pT plane. According to the classification
of Konynenburg and Scott (1980), the binary pT diagrams of CO_{2}+methane
(Fig. 1), CO_{2}+propane (Fig. 3),
CO_{2}+pentane (Fig. 5) and CO_{2}+hexane
(Fig. 6) should also belong to type I fluid phase behavior,
although the binary mixture of CO_{2}+hexane may exhibit a metastable
immiscibility at low temperature.

Fig. 9: 
V_{m, c}x diagram of the SC CO_{2}+nalkane
systems. Δ∇□: experimental data 

Fig. 10: 
p_{c}p_{c} diagram of the SC CO_{2}+nalkane
systems. Δ∇□: experimental data 
Type II binary pT diagrams include, CO_{2}+nheptane (Fig. 7) and CO_{2}+noctane (Fig. 8). As the mutual solubility of the components decreases, the Upper Critical Solution Temperatures (UCSTs) versus pressure (UCSTsp) line shows liquidliquid immiscibility. This line starts at the liquidliquidgas triple phase line. There is a point on the UCSTsp diagram, where the heavy component can be precipitated by a small temperature increase, a small pressure decrease, a large temperature decrease, or an extremely pronounced pressure increase. At higher temperatures, the gasliquid critical curve of the type II mixture is similar to that of type I. However, at relatively low temperatures, it has liquidliquid immiscibility and the loci of UCSTs remain distinct from the gasliquid critical curve.
Figure 18 show the calculated critical
curves for SC CO_{2}+nalkanes (from methane to octane) in comparison
with experimental curves from the literature (Mraw et al., 1978; Horstmann
et al., 2000; Roof and Baron, 1967; Freitas et al., 2004; Chen
et al., 2003; Liu et al., 2003; Choi and Yeo, 1998; Kalra et
al., 1978) and our experimental data.
Table 1: 
The interaction parameters for the binary systems 

Table 2: 
The critical properties of the pure substances 

Our calculated results have excellent agreement with the experimental data.
Table 1 gives a compilation of the adjustable parameters,
ω, k_{ε} and k_{σ}, for the eight SC CO_{2}+nalkane
systems. Table 2 gives the critical constants for CO_{2}
and the eight alkanes. The values of ω and k_{σ} for each
of the systems are 2.3 and 1, respectively. The value of ω is determined
by the molecular polarity. Therefore, ω is constant for the nonpolar alkanes.
The factor k_{σ} describes the deviations of σ_{ij}
from 1/2(σ_{ii}+σ_{jj}) and the factor k_{ε}
describes the deviations of ε_{ij} from 1/2(ε_{ii}+ε_{jj}).
The value of k_{ε} decreases with the increase in alkane carbon
number except for noctane. Octane has a higher critical temperature and lower
critical pressure and the binary pT diagram of SC CO_{2}+noctane belongs
to type II. Therefore, it requires a higher k_{ε} value.
Figure 9 and 10 give the V_{m, c}x_{c}
and p_{c}P_{c} curves. They show the similar rules that with
the increase in alkane carbon number the curves give regular change, especially
from C_{4} to C_{8}.
The parameters in Table 1 provide a useful basis to estimate
the homogeneous regions and the twophase behavior of binary systems and can
be applied to calculate the threedimensional phase equilibrium surfaces. Using
the parameter values in Table 2, the T_{c}x_{c},
p_{c}x_{c} and p_{c}T_{c} diagrams have been
plotted. The changes in the adjustable parameters ω, k_{ε}
and k_{σ} are very small for the different alkanes. If other cosolvents
with different polarity and molecular size are used, then the values of ω,
k_{ε} and k_{σ} would be more dissimilar. The calculated
data are compared with the experimental data in Fig. 18.
They show a reasonable correlation. Therefore, the HeiligFranck equation is
suitable to predict bimodal curves and critical curves for the eight SC CO_{2}+nalkane
systems at higher temperatures and pressures.
Conclusions
• 
The critical curves of the eight binary systems from SC CO_{2}+methane
to SC CO_{2}+nhexane at higher temperatures and pressures all belong
to type I and the critical curves of the binary systems for SC CO_{2}+nheptane
and SC CO_{2}+noctane belong to type II. 
• 
The critical molar volumes and densities are obtained with the equation
of state by Heilig and Franck. The complete critical curves of carbon dioxide
and low molecular alkanes afford more data to researching on the fundamental
chemistry and chemical engineering. 
• 
The adjustable parameters, ω, k_{ε} and k_{σ},
for the eight SC CO_{2}+nalkane systems have been given. The value
of k_{ε} decreases with the increase in alkane carbon number
except for noctane and the values of ω and k_{σ} are
2.3 and 1, respectively. 
• 
The adjustable parameters ω, k_{ε} and k_{σ},
for the eight SC CO_{2}+nalkane systems and the critical constants
for CO_{2} and the alkane partners may give a useful basis to estimate
the homogeneous regions and twophase behavior of binary systems and can
be applied to calculate the threedimensional phase equilibrium surface. 
• 
The calculated critical curves in these systems are in good agreement
with experimental data. The greatest relative error for pressure is 5.69%
and for the mole fractions of CO_{2} it is 9.73%. The HeiligFranck
equation of state has been found to have good prediction and correlation
with binary vaporliquid equilibrium data of the carbon dioxide+alkane systems. 

REFERENCES 
Chen, J.W., W.Z. Wu and B.X. Han, 2003. Phase behavior, densities and isothermal compressibility of CO2+pentane and CO2+acetone systems in various phase regions. J. Chem. Eng. Data, 48: 15441548.
Choi, E.J. and S.D. Yeo, 1998. Critical properties for carbon dioxide+nalkane mixtures using a variablevolume view cell. J. Chem. Eng. Data, 43: 714716.
Christoforakos, M. and E.U. Franck, 1986. An equation of state for binary fluid mixtures to hightemperatures and highpressures. Ber. Bunsenges. Phys. Chem., 90: 780789.
Christov, M. and R. Dohrn, 2002. Highpressure fluid phase equilibria experimental methods and systems investigated (19941999), Fluid Phase Equilib., 202: 153218.
Deiters, U.K., M. Neichel and E.U. Franck, 1993. Prediction of the thermodynamic properties of hydrogenoxygen mixtures from 80 to 373 K and to 100 MPa. Ber. BunsenGes. Phys. Chem., 97: 649657.
Dohrn, R., 1995. Highpressure fluidphase equilibria: Experimental methods and systems investigated (19881993). Fluid Phase Equilib., 106: 213282.
Fornari, R.E., 1990. High pressure fluid phase equilibria: Experimental methods and systems investigated (19781987). Fluid Phase Equilib., 57: 119.
Freitas, L., G. Platt and N. Henderson, 2004. Novel approach for the calculation of critical points in binary mixtures using global optimization. Fluid Phase Equilib., 225: 2937.
Hirschfelder, J.C., C.F. Curtiss and R.B. Bird, 1964. Molecular Theory of Gases and Liquids. John Wiley and Sons, New York
Horstmann, S., K. Fischer and J. Gmehling, 2000. Experimental determination of the critical line for (carbon dioxide+ethane) and calculation of various thermodynamic properties for (carbon dioxide+nalkane) using the PSRK model. J. Chem. Therm., 32: 451464.
Kalra, H., H. Kubota and D.B. Robinson, 1978. Equilibrium phase properties of the carbon dioxidenheptane system. J. Chem. Eng. Data, 23: 317321.
Konynenburg, P.H.V. and R.L. Scott, 1980. Critical lines and phase equilibria in binary van der waals mixtures. Philos. Trans. R. Soc., 298: 495540.
Liu, J.G., Z.F. Qin and G.Y. Wang, 2003. Critical properties of binary and ternary mixtures of hexane+methanol, hexane+carbon dioxide, methanol+carbon dioxide and hexane+carbon dioxide+methanol. J. Chem. Eng. Data, 48: 16101613.
McGlashan, M.L., 1979. Chemical Thermodynamics. Academic Press, London
Mraw, S.C., S.C. Hwang and R. Kobayashi, 1978. Vaporliquid equilibrium of the CH4CO2 system at low temperature. J. Chem. Eng. Data, 23: 135139.
Neichel, M. and E.U. Franck, 1996. Critical curves and phase equilibria of waternalkane binary systems to high pressures and temperatures. J. Supercrit. Fluids, 9: 6974.
Roof, J.G. and J.D. Baron, 1967. Critical loci of binary mixtures of propane with methane, carbon dioxide and nitrogen. J. Chem. Eng. Data, 12: 292293.
Rowlinson, J.S. and F.L. Swinton, 1978. Liquids and Liquid Mixtures. 3rd Edn., Butterworth Society, London
Sadus, R.J., 1992. Highpressure Phase Behavior of Multicomponent Fluid Mixtures. Elsevier, Amsterdam
Sadus, R.J., 1992. Novel critical transitions in ternary fluids mixtures. J. Phys. Chem., 96: 51975202.
Schneider, G.M., 1978. A Special Periodical Report. The Chemical Society, London
Shmonov, V.M., R.J. Sadus and E.U. Franck, 1993. Highpressure phase equilibria and supercritical pVT data of the binary water+methane mixture to 723 K and 200 MPa. J. Phys. Chem., 97: 90549059.
Scott, R.L. and P.H. Van Konynenburg, 1970. Static properties of solutions. Discuss Faraday Soc., 49: 8797.
Tian, Y.L., Li Chen and M.W. Li, 2003. Calculation of gasliquid critical curves for carbon dioxide1alkanol binary systems. J. Phys. Chem. A, 107: 30763080.
Wu, G., M. Heilig, H. Lentz and E.U. Franck, 1990. High pressure phase equilibria of the waterargon system. Ber. BunsenGes. Phys. Chem., 94: 2427.



