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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.*

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 B_{x} and C_{x} parameters are the second and third virial coefficients of a square-well 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

(2) |

where T_{c,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 N_{0} 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 ω_{ij} = ω_{ii}+ω_{jj} used for different particle combinations in the binary fluid mixtures, the usual Lorentz-Berthelot combination rules together with empirical parameters k_{ε} and k_{σ} can be applied:

(4) |

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 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 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 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 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):

(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, 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:

(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: | 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 p-T diagrams include, CO_{2}+ethane (Fig.
2) and CO_{2}+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 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 p-T 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 p-T plane. The binary p-T 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 p-T plane. According to the classification
of Konynenburg and Scott (1980), the binary p-T 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}+n-alkane
systems. Δ∇□: experimental data |

Fig. 10: | p_{c}-p_{c} diagram of the SC CO_{2}+n-alkane
systems. Δ∇□: experimental data |

Type II binary p-T diagrams include, CO_{2}+n-heptane (Fig. 7) and CO_{2}+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 CO_{2}+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, k_{ε} and k_{σ}, for the eight SC CO_{2}+n-alkane
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 n-octane. Octane has a higher critical temperature and lower
critical pressure and the binary p-T diagram of SC CO_{2}+n-octane 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 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 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. 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 CO_{2}+n-alkane
systems at higher temperatures and pressures.

**Conclusions**

• | The critical curves of the eight binary systems from SC CO_{2}+methane
to SC CO_{2}+n-hexane at higher temperatures and pressures all belong
to type I and the critical curves of the binary systems for SC CO_{2}+n-heptane
and SC CO_{2}+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, k_{ε} and k_{σ},
for the eight SC CO_{2}+n-alkane systems have been given. The value
of k_{ε} decreases with the increase in alkane carbon number
except for n-octane 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}+n-alkane systems and the critical constants
for CO_{2} 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 CO_{2} 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. |