## Brazilian Journal of Chemical Engineering

##
*Print version* ISSN 0104-6632*On-line version* ISSN 1678-4383

### Braz. J. Chem. Eng. vol.17 n.4-7 São Paulo Dec. 2000

#### http://dx.doi.org/10.1590/S0104-66322000000400040

**CALCULATION OF MIXTURE CRITICAL DIAGRAMS USING AN EQUATION OF STATE BASED ON THE LATTICE FLUID THEORY **

**S.Mattedi ^{1*}, F.W.Tavares^{2} and M.Castier^{2 }**

^{1}DEQ, Escola Politécnica, Universidade Federal da Bahia,

R. Aristides Novis 2, Federaçao, 40.210-630, Salvador - BA, Brazil

E-mail: silvana@ufba.br

^{2}Escola de Química, Universidade Federal do Rio de Janeiro,

Caixa Postal 68542, Rio de Janeiro - RJ, 21949-900, Brazil

*(Received: November 8, 1999 ; Accepted: April 7, 2000)*

Abstract- A modified form of the Hicks and Young algorithm was used with the Mattedi-Tavares-Castier lattice equation of state (MTC lattice EOS) to calculate critical points of binary mixtures that exhibit several types of critical behavior. Several qualitative aspects of the critical curves, such as maxima and minima in critical pressure, and minima in critical temperature, could be predicted using the MTC lattice EOS. These results were in agreement with experimental information available in the literature, illustrating the flexibility of the functional form of the MTC lattice EOS. We observed however that the MTC lattice EOS failed to predict maxima in pressure for two of the studied systems: ethane + ethanol and methane + n-hexane. We also observed that the agreement between the calculated and experimental critical properties was at most semi-quantitative in some examples. Despite these limitations, in many ways similar to those of other EOS in common use when applied to critical point calculations, we can conclude that the MTC lattice EOS has the ability to predict several types of critical curves of complex shape.

Keywords: critical points, equations of state, lattices, mixtures

**INTRODUCTION**

Calculations of mixture critical points represent a severe test for equations of state (EOS) because of the complex types of phase diagrams that are experimentally observed for several mixtures (van Konynenburg and Scott, 1980; van Pelt, 1992). For example, the projection of the gas-liquid critical line of a binary mixture of similar components in the temperature-pressure plane is often continuous and connects the two pure component critical points. However, for dissimilar components, the gas-liquid critical line can have branches that terminate at a critical endpoint (where there are two critical phases in equilibrium with a non-critical phase) or go to infinite pressure. Moreover, for several systems, liquid-liquid critical lines are also observed, and these can also go to very high pressures. Therefore, verifying which types of critical behavior can be predicted by a given EOS is of both theoretical and practical interest. The objective of this paper is to analyze the performance of a recently developed EOS (Mattedi et al., 1998a) that combines the Staverman-Guggenheim athermal contribution and a local composition residual contribution, when used for critical point calculations. Because of the assumptions in the development of this EOS, it can be considered as a generalization of the UNIQUAC/UNIFAC models, in order to account for volumetric effects. This equation has been successfully used to model vapor-liquid equilibrium of pure components, mixtures, and polymer systems. A group-contribution version of this equation has also been derived and parameters have been determined for alkane and alkanol groups. The results obtained show that the EOS is able to correlate vapor pressure of pure alkanes and alcohols, and to predict the vapor pressure of pure heavy alkanes. Vapor-liquid equilibrium diagrams of binary mixtures containing alkane and alcohols were satisfactorily calculated over wide temperature ranges. Good predictive results were also obtained for highly asymmetric mixtures (small + large molecules) and azeotropic systems. This EOS was also applied to polymer+solvent systems (Mattedi et al., 1998b; Lyrio et al., 1999), and it was observed that the equation is able to correlate pure polymers accurately and to represent satisfactorily the vapor-liquid equilibrium of solutions containing polymer and solvents of different polarities.

In this paper, we compare the predictions of the EOS model with experimental critical data of binary mixtures. In order to search comprehensively for the critical states of mixtures exhibiting complex behavior, including the possibility of multiple critical points at a specified composition, a modified version of the Hicks and Young (1977) algorithm was used. For the pure components, the model parameters were determined from vapor pressure data. While it is usual to force cubic EOS to satisfy pure component critical temperature and pressure, here with the MTC lattice EOS, this constraint was not imposed. Therefore, the calculated pure component critical points represent EOS predictions. For mixtures, the model parameters were estimated from vapor-liquid equilibrium (VLE) data, only some of which at high pressure, but no experimental critical information was directly used in the fitting procedure. Therefore, the calculations represent predictions rather than correlations of the critical curves.

The critical points were calculated as described earlier (Castier and Sandler, 1997) using a modified version of the Hicks and Young (1977) algorithm to determine possible multiple critical points at any given composition and refining the solutions with the Heidemann and Khalil (1980) procedure. Only critical points that passed the local and global stability tests are reported here.

**CRITICAL POINT CALCULATIONS**

For the calculation of critical points in mixtures, the most used method is that of Heidemann and Khalil (1980), who derived the criticality conditions using a series expansion of the Helmholtz free energy (*A*). It follows from their formulation that the quadratic (*q*) and cubic (*c*) forms of the expansion should be both equal to zero at a critical point, i.e.:

(1) |

(2) |

for any perturbations D*n*_{i}, D*n _{j}*, and D

*n*. Stability requires that the Hessian matrix

_{k}_{}of the quadratic form, with elements:

(3) |

be positive semi-definite. Therefore, the zero of the quadratic form must be detected by a zero value of the smallest eigenvalue of this matrix. The initialization strategies proposed by Heidemann and Khalil (1980) for the calculation of gas-liquid critical points are very satisfactory but their procedure is less satisfactory for the calculation of liquid-liquid critical points. More important, the system of equations describing mixture critical conditions may have a single solution, several solutions or no solution at all. For this reason, to trace the critical lines, we used a modified version of the Hicks and Young (1977) which is capable of computing multiple critical points at a given composition. In this algorithm, one searches a region in the temperature versus molar volume plane following the locus along which the smallest eigenvalue of the matrix of the quadratic form is equal to zero, while monitoring the sign of the cubic form. As this locus is followed, whenever the cubic form changes sign, a critical point is bracketed, and an excellent initial estimate is obtained to refine the solution using the Heidemann and Khalil method. Several entry and exit points may exist in the search region having a zero value for the quadratic form, and all of them must be traced in the search for critical points.

Solution of the criticality conditions is a necessary but not sufficient condition for the identification of a critical point. At a critical point, the first non-zero term of the Helmholtz free energy expansion should be even in order and positive. In our implementation, this test is limited to evaluating the fourth-order term of this expansion using a four-point difference formula. We also performed a global stability test (Michelsen, 1982) in the calculated critical points. All critical points reported here passed the local and global stability tests. However, it should be mentioned that the global stability test was only implemented to verify the possibility of additional fluid phases; the possibility of forming solid phases was not tested in our implementation.

Thermodynamic Model: the MTC Lattice EOS

In the development of the model, we assumed that a lattice of coordination number *Z* (set equal to 10), containing *M* cells of fixed volume *V* *, represents a fluid of total volume *V*. In group contribution form, the EOS is:

(4) |

where * z* is the compressibility factor, is the number of groups of type

*a*in a molecule of type

*i*,

*Q*is the surface area of a group of type

^{ a}*a*, and Y is a constant of the lattice structure, set to 1 (Mattedi et al., 1994). The average number of segments occupied by a molecule in the lattice (

*r*), the mean number of nearest neighbors (

*Zq*) and the reduced molar volume () are given by:

(5) |

(6) |

(7) |

(8) |

(9) |

Here, *R ^{ a}* and

*V*are the group contributions to the number of segments and hard core volume, respectively;

^{ a}*v*is the molar hard core volume parameter of a group of type

^{a}*a*. We also define:

(10) |

(11) |

(12) |

where *u ^{ma}* is the interaction energy between groups

*m*and

*a*. The fugacity coefficient derived from the EOS is:

(13) |

We assumed that *v ^{a}* and

*u*are given by (Chen and Kreglewski, 1977):

^{ba}
(14) |

(15) |

In summary, our group contribution EOS has five parameters for each group (, *Q ^{a}*, ,

*A*and

^{a}*B*) and two parameters for interactions between unlike groups (and

^{aa}*B*).

^{ba}In this work, the lattice coordination number *Z* is set equal to 10, as usually done in the derivation of lattice-based thermodynamic models. The empirical characteristic constant **Y** is set equal to 1 and a value of 5 cm^{3}/mol was used for the cell volume on a molar basis (*v**) as suggested in previous work (Mattedi et al., 1998a and Mattedi et al., 1998b).

Expressions for the derivatives of the fugacities with respect to mole numbers at constant temperature and total volume required in the critical point calculations were determined and automatically programmed using *Thermath* (Castier, 1999). These expressions can be obtained from the authors on request.

**FORMULATION**

**Parameter Fitting**

Although the EOS is written in a group contribution form, in this work we used a molecular approach, and so each compound was considered as a group. Pure component parameters were fitted using the Simplex as proposed by Nelder and Mead (Press et al., 1989) to minimize the objective function:

(16) |

where V*P* is the vapor pressure, *PC* is the critical pressure and *TC* is the critical temperature. The subscripts *exp* and *cal* mean experimental and calculated values and *npt* refers to the number of experimental data points. In eq. 16, a large weight was given to the deviations in critical properties, to better represent the critical behavior of the pure components. The critical temperature and pressure were calculated forcing the EOS to obey the critical conditions, which can be represented by:

(17) |

The vapor pressure experimental data used were taken from Boublík et al*.* (1984) and Danner (1992). Pure component critical data were taken from Reid et al*.* (1987). Interaction parameters were fitted in order to minimize square deviations from experimental bubble pressures for selected binary VLE data. The data used were mainly in the moderate to high pressure region, because of our interest in critical point calculations.

**RESULTS**

The fitted EOS parameters for pure compounds and the relative deviations from experimental vapor pressure, critical temperature and critical pressure are shown in Table1. The results indicate that the EOS satisfactorily correlates vapor pressures in a large reduced temperature range, and is also able to represent critical points of pure compounds.

Interaction parameters and bubble pressure deviations for the studied binary systems are shown in Table 2. The MTC lattice EOS was able to correlate VLE for the studied systems in the moderate to high pressure region.

Figures 1 to 10 present the critical behavior predicted by the MTC lattice EOS for the systems shown in Table 2. In these figures, the critical behavior calculated by Castier and Sandler (1997), using the Peng-Robinson EOS coupled with the NRTL model using the Wong-Sandler mixing rule (PR+WS+NRTL), is also shown. It should be noted that the PR+WS+NRTL model has four adjustable parameters per binary, whereas the MTC lattice EOS has only two.

For the system methane + ethanol (Fig. 1), the MTC lattice EOS correctly predicts that the gas-liquid critical line does not connect the pure component critical points, but the critical line proceeds to very high pressures, in disagreement with experimental data and the curve obtained using the PR+WS+NRTL model. However, it should be noted that the correct representation of the temperature-pressure projection of the critical line is a very difficult test for the models. For instance, it is interesting to observe that the molar volume-mole fraction projections (Fig. 2) for these two models have a similar trend.

In Fig. 3, we present results for the system ethane + ethanol. As in the previous example, the MTC lattice EOS correctly predicts that the critical line does not connect the pure component critical points. Because of difficulties with numerical precision in the phase stability test, we can not guarantee whether the calculated critical branch that starts at the critical point of pure ethanol goes continuously to very high pressures or terminates at a critical endpoint. In any circumstance, the critical line proceeds to high pressures, in disagreement with experimental data. The calculated critical branch that starts at the critical point of pure ethane terminates at critical endpoints for both the PR+WS+NRTL and MTC lattice models.

The results for the systems n-butane + ethanol and n-pentane + ethanol are shown in Figs. 4 and 5, respectively. For these mixtures, the gas-liquid critical line is continuous between the two pure component critical points, and the MTC lattice EOS correctly predicted this behavior. The system propane + methanol (Fig. 6) is particularly interesting. The experimental information indicates that there is a continuous gas-liquid critical line between the two pure components. The PR+WS+NRTL model fails in this aspect, predicting the existence of critical endpoints along the gas-liquid critical line. On the other hand, the MTC lattice EOS does predict a continuous gas-liquid critical line, but also the existence of a liquid-liquid critical line that extends to very high pressures. We could not find experimental information about the existence of such a critical branch in this system.

The calculated gas-liquid critical line for the systems n-hexane + methanol and n-pentane + acetone are shown in Figs. 7 and 8, respectively. This critical line for these systems is continuous between the two pure component critical points and has a minimum in temperature. The MTC lattice EOS correctly captured this qualitative behavior, even though a minimum in pressure was also predicted for the n-pentane + acetone system, but it was not experimentally observed.

Results for the mixture water + n-dodecane are shown in Fig. 9. The discontinuity in the gas-liquid critical line and the existence of a high pressure critical branch that starts at the critical point of water were correctly predicted by the MTC lattice EOS. Even though satisfactory quantitative agreement with the experimental critical data was achieved only in a limited region, it is a remarkable fact that the EOS predicts such a complex critical behavior. As a final example, we consider the methane + n-hexane system (Fig. 10), that also has a rather complex critical behavior, even though it only contains hydrocarbons. For this system, the MTC lattice EOS failed to predict a maximum in pressure and the gas-liquid critical line proceeded to very high pressures.

**CONCLUSIONS**

A modified form of the Hicks and Young (1977) algorithm and the MTC lattice EOS were used to calculate critical points for selected binary mixtures of complex behavior. In comparison with experimental critical data, the MTC lattice EOS correctly predicted whether the gas-liquid critical line would be continuous or discontinuous between the two pure component critical points in some of the systems tested. In the calculations of the pressure-temperature projections, it was possible to observe situations in which maxima and minima in pressure were calculated, as well as a situation in which a minimum in temperature occurred, illustrating the flexibility of the functional form of the MTC lattice EOS. We observed however that the MTC lattice EOS failed to predict maxima in pressure for two of the studied systems: ethane + ethanol and methane + n-hexane. We also observed that the agreement between the calculated and experimental critical properties was at most semi-quantitative in some examples. Despite these limitations, in many ways similar to those of other EOS in common use when applied to critical point calculations, we can conclude that the MTC lattice EOS has the ability to predict several types of critical curves of complex shape.

**NOMENCLATURE**

temperature dependence parameter of the hard core volume for group of type a | |

temperature dependence parameter of the interaction energy between groups of types a and b | |

l | bulkiness factor |

n_{c} | number of components |

n_{g} | number of groups |

npt | number of experimental data points |

P | pressure |

PC | critical pressure |

surface area of group of type a | |

R | universal gas constant |

number of segments occupied by each group of type a | |

r | number of segments |

area fraction of group of type a in a void free basis | |

T | temperature |

TC | critical temperature |

TR | reduced temperature () |

molar interaction energy between groups b and a | |

temperature independent molar interaction energy between groups b and a | |

v | molar volume |

reduced molar volume | |

v* | molar volume of lattice cells |

molar hard core volume parameter of a group of type a | |

V* | volume of a lattice cell |

V | total volume |

hard core volume of group of type a | |

temperature independent molar hard core volume of group of type a | |

VP | vapor pressure |

x | molar fraction |

Z | lattice coordination number |

z | compressibility factor |

Zq | mean number of nearest neighbors |

*Greek Letters*

DBP/BP | mean absolute relative deviation between experimental and calculated bubble pressure |

DPC/PC | absolute relative deviation between experimental and calculated critical pressure |

DTC/TC | absolute relative deviation between experimental and calculated critical temperature |

DVP/VP | mean absolute relative deviation between experimental and calculated vapor pressure |

DY | mean absolute deviation between experimental and calculated vapor mole fraction of component 1 |

number of groups a present at the i-th molecule | |

r | molar density |

fugacity coefficient of component i in the mixture | |

Y | empirical universal constant of the lattice structure |

**ACKNOWLEDGEMENTS**

M.C. acknowledges the hospitality of Prof. José Tojo Suárez, from the University of Vigo (Spain), where part of this work was done. M.C. and F.W.T. are grateful for the financial support of CNPq/Brazil and PRONEX (Grant no. 124/96). S.M. is grateful for the financial support of the PRODOC program from CADCT/SEPLANTEC/Bahia.

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*To whom correspondence should be addressed