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Research Article
Calculation of Critical Curves for Carbon Dioxide+n-Alkane Systems

Jinglin Yu, Shujun Wang, Yiling Tian and Wei Xu

Trends in Applied Sciences Research, 2006, 1(4), 317-326.

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 square-well potential, so three adjustable parameters are required. Good agreement was obtained between the experimental data, the literature data and the calculated values.

ASCI-ID: 95-36

(1)

where the Bx and Cx parameters are the second and third virial coefficients of a square-well fluid and βx represents the molecular volume of the fluid. Vm is the molar volume of the mixture and xi is the molar fraction of component i.

Each of the molecular terms can be given by

(2)

where Tc,i is the critical temperature of component i and m is a temperature-dependent 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 pVT-data (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 N0 is Avogadro’s constant.

The virial attraction terms can be given by the following:

(3)

Here k is Boltzmann’s constant, ω is the relative width of the square-well 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 square-well widths are ωij = ωiijj used for different particle combinations in the binary fluid mixtures, the usual Lorentz-Berthelot combination rules together with empirical parameters kε and kσ can be applied:

(4)

kε and kσ 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 ω-values 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 kε and kσ can be determined from experimental mixture data or can be predicted by analogy from existing values of related systems. It appears that kε and kσ remain constant or vary only modestly and systematically within certain groups of systems. The diameter σ and the depth, ε, of the square-well are derived from critical data of the pure partners.

The third virial coefficient is given by:

(5)

The auxiliary functions of the virial coefficients I11 to I33 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):

(6)

(7)

(8)

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, Vm and the mole fraction, xi, 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, Am, can be obtained. The equation contains a term for a residual free energy as well as terms for a reference state with chemical potentials μiθ and pressure pθ (McGlashan, 1979). From thermodynamics, Am, which is a function of T, Vm and xi, can be given by:

(9)

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 Heilig-Franck EOS are shown (Fig. 1-8). 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+n-alkanes belong to different types.

Fig. 1: Tc-xc, pc-xc and pc-Tc diagrams of the SC CO2+CH4 system

Fig. 2: Tc-xc, pc-xc and pc-Tc diagrams of the SC CO2+C2H6 system

Fig. 3: Tc-xc, pc-xc and pc-Tc diagrams of the SC CO2+C3H8 system

Fig. 4: Tc-xc, pc-xc and pc-Tc diagrams of the SC CO2+C4H10 system

Fig. 5: Tc-xc, pc-xc and pc-Tc diagrams of the SC CO2+C5H12 system

Type I binary p-T diagrams include, CO2+ethane (Fig. 2) and CO2+n-butane (Fig. 4). This is the simplest case in which the p-T projection of the three dimensional pressure-temperature-composition (p-T-x) diagram consists of two vapor-pressure 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 gas-liquid critical curve The molecules of CO2 and ethane have similar shapes and sizes. The quadrupole moment of carbon dioxide is much stronger than that of ethane.

Fig. 6: Tc-xc, pc-xc and pc-Tc diagrams of the SC CO2+C6H14 system

Fig. 7: Tc-xc, pc-xc and pc-Tc diagrams of the SC CO2+C7H16 system

Fig. 8: Tc-xc, pc-xc and pc-Tc diagrams of the SC CO2+C8H18 system

The binary p-T diagram of CO2+ethane belongs to the third subdivision of type I, in which the critical curve is convex and exhibits a minimum temperature in the p-T plane. The binary p-T diagram of CO2-butane belongs to the second subdivision of type I, in which the critical curve is convex and exhibits a maximum pressure in the p-T plane. According to the classification of Konynenburg and Scott (1980), the binary p-T diagrams of CO2+methane (Fig. 1), CO2+propane (Fig. 3), CO2+pentane (Fig. 5) and CO2+hexane (Fig. 6) should also belong to type I fluid phase behavior, although the binary mixture of CO2+hexane may exhibit a metastable immiscibility at low temperature.

Fig. 9: Vm, c-x diagram of the SC CO2+n-alkane systems. Δ∇□: experimental data

Fig. 10: pc-pc diagram of the SC CO2+n-alkane systems. Δ∇□: experimental data

Type II binary p-T diagrams include, CO2+n-heptane (Fig. 7) and CO2+n-octane (Fig. 8). As the mutual solubility of the components decreases, the Upper Critical Solution Temperatures (UCSTs) versus pressure (UCSTs-p) line shows liquid-liquid immiscibility. This line starts at the liquid-liquid-gas triple phase line. There is a point on the UCSTs-p 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 gas-liquid critical curve of the type II mixture is similar to that of type I. However, at relatively low temperatures, it has liquid-liquid immiscibility and the loci of UCSTs remain distinct from the gas-liquid critical curve.

Figure 1-8 show the calculated critical curves for SC CO2+n-alkanes (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, kε and kσ, for the eight SC CO2+n-alkane systems. Table 2 gives the critical constants for CO2 and the eight alkanes. The values of ω and kσ 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 kσ describes the deviations of σij from 1/2(σiijj) and the factor kε describes the deviations of εij from 1/2(εiijj). The value of kε decreases with the increase in alkane carbon number except for n-octane. Octane has a higher critical temperature and lower critical pressure and the binary p-T diagram of SC CO2+n-octane belongs to type II. Therefore, it requires a higher kε value.

Figure 9 and 10 give the Vm, c-xc and pc-Pc curves. They show the similar rules that with the increase in alkane carbon number the curves give regular change, especially from C4 to C8.

The parameters in Table 1 provide a useful basis to estimate the homogeneous regions and the two-phase behavior of binary systems and can be applied to calculate the three-dimensional phase equilibrium surfaces. Using the parameter values in Table 2, the Tc-xc, pc-xc and pc-Tc diagrams have been plotted. The changes in the adjustable parameters ω, kε and kσ are very small for the different alkanes. If other cosolvents with different polarity and molecular size are used, then the values of ω, kε and kσ would be more dissimilar. The calculated data are compared with the experimental data in Fig. 1-8. They show a reasonable correlation. Therefore, the Heilig-Franck equation is suitable to predict bimodal curves and critical curves for the eight SC CO2+n-alkane systems at higher temperatures and pressures.

Conclusions

The critical curves of the eight binary systems from SC CO2+methane to SC CO2+n-hexane at higher temperatures and pressures all belong to type I and the critical curves of the binary systems for SC CO2+n-heptane and SC CO2+n-octane 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, kε and kσ, for the eight SC CO2+n-alkane systems have been given. The value of kε decreases with the increase in alkane carbon number except for n-octane and the values of ω and kσ are 2.3 and 1, respectively.
The adjustable parameters ω, kε and kσ, for the eight SC CO2+n-alkane systems and the critical constants for CO2 and the alkane partners may give a useful basis to estimate the homogeneous regions and two-phase behavior of binary systems and can be applied to calculate the three-dimensional 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 CO2 it is 9.73%. The Heilig-Franck equation of state has been found to have good prediction and correlation with binary vapor-liquid equilibrium data of the carbon dioxide+alkane systems.
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