Equation of State Mixing Rules That Incorporate Only the Residual

Following our recent approach, only the attractive part of the excess Helmholtz energy derived from the equation of state (EOS) is related to the exce...
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Ind. Eng. Chem. Res. 1998, 37, 2929-2935

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Equation of State Mixing Rules That Incorporate Only the Residual Part of the UNIQUAC Model Federico Brandani, Stefano Brandani, and Vincenzo Brandani* Dipartimento di Chimica, Ingegneria Chimica e Materiali, Universita` de L’Aquila, Monteluco di Roio, I-67100 L’Aquila, Italy

New mixing rules that incorporate the residual part of the UNIQUAC activity coefficient model and the correct quadratic concentration dependence of the second virial coefficient are presented. Following our recent approach, only the attractive part of the excess Helmholtz energy derived from the equation of state (EOS) is related to the excess Gibbs energy. It is recognized that, in the extension of this concept to the UNIQUAC model, the interpretation of the physical contributions of the combinatorial and residual parts must be considered. We conclude that the attractive forces are related only to the residual part of the UNIQUAC model. This allows the formulation of a class of mixing rules that are independent of the repulsive term in the EOS and are applicable to both cubic and noncubic EOS. The proposed mixing rules are applied to different low-pressure systems, exhibiting vapor-liquid and liquid-liquid equilibria, and to two near critical and supercritical systems. Introduction In the correlation and prediction of fluid-phase equilibria it is common to use an equation of state (EOS). In order to apply EOS to mixtures, it is necessary to formulate appropriate mixing rules for the parameters of the EOS. Huron and Vidal (1979) were the first to incorporate activity coefficient models in the formulation of mixing rules. Since then many contributions have appeared in the literature. The general approach may be summarized as follows: a fixed-pressure reference state is chosen, and the excess Helmholtz energy derived from the EOS is equated to the excess Gibbs energy of the activity coefficient models. Following Huron and Vidal (1979), Wong and Sandler (1992) use the infinitepressure reference state but include the correct zeropressure limit for the composition dependence of the second virial coefficient. Heidemann and Kokal (1990) and Michelsen and co-workers (Michelsen, 1990a,b; Dahl and Michelsen, 1990; Michelsen and Heidemann, 1996) choose the zero-pressure reference state. This approach leads to mixing rules for cubic EOS which are usually algebraically more complex than the previous ones, since the liquid volume at zero pressure needs to be evaluated. The main difficulty with these mixing rules is that at supercritical conditions they require an extrapolation to a hypothetical zero-pressure liquid. Therefore, they suffer the same difficulty of activity coefficient models applied to such mixtures. All of the above methods aim at reducing the EOS approach to the activity coefficient model in order to be able to use existing values for the binary interaction parameters. The infinite-pressure reference state approach has been shown to be inapplicable to noncubic EOS based on the hard-sphere model (Brandani and Brandani, 1996). The zero-pressure reference state approach is difficult to extend to noncubic EOS since the evaluation of the zero-pressure liquid volume cannot be given in an analytical form. * Author to whom correspondence should be addressed. Telephone: (+39)862 434219. Fax: (+39)862 434203. E-mail: [email protected].

We recently developed a new approach to the derivation of EOS mixing rules (Brandani et al., 1998) that incorporate activity coefficient models. In our approach, activity coefficient models are used as mixing rules, and therefore the EOS does not reduce exactly to the activity coefficient models. The objective of combining EOS and lattice fluid models can be accomplished considering the basic assumptions used in the derivation of these models, and in particular it is necessary to distinguish the contributions of attractive forces (Brandani et al., 1998). Incorporating the UNIQUAC Model in the Proposed Mixing Rules Activity coefficients models, for example, Wilson and NRTL, were derived in the 1960s (Wilson, 1964; Renon and Prausnitz, 1968) based on the work of Guggenheim (1952). The main objective of these models was to properly describe the low-pressure vapor-liquid equilibrium of highly nonideal solutions. For such systems the prevailing contributions are due to attractive forces. Furthermore, these models really represent the composition dependence of the excess Helmholtz functions at a fixed lattice density, since in the low-pressure region the density is assumed independent of pressure (Guggenheim, 1952). An approximation is then introduced to equate the excess Helmholtz function to the excess Gibbs energy in order to derive the low-pressure activity coefficients as pointed out by Wong and Sandler (1992). The UNIQUAC model (Abrams and Prausnitz, 1975) introduces an important modification. It includes two physical contributions: the combinatorial and the residual parts. The combinatorial part is an entropic term that takes into account the differences in size and shape of the molecules. The residual part is an enthalpic term that describes the energetic interactions between molecules and depends on the binary interaction parameters, which can be related to internal energies (Abrams and Prausnitz, 1975).

S0888-5885(97)00584-8 CCC: $15.00 © 1998 American Chemical Society Published on Web 03/26/1998

2930 Ind. Eng. Chem. Res., Vol. 37, No. 8, 1998

If we now consider the generic EOS given by

z ) 1 + zrep + zattr

(1)

components in the systems considered are given by Brandani et al. (1998). We derive the expressions of the quantities needed for the formulation of the mixing rules.

then the reduced Helmholtz energy is

A(T, F) - AIG(T, F) A ˜ ) ) RT rep η z dη + 0 η

∫

∫

∫

A(T, P) - AIG(T, P) ˜ attr - ln(z) (3) )A ˜ rep + A A ˜ res ) RT and the excess Helmholtz function is

A ˜ )

A ˜ Erep

+

A ˜ Eattr

(  B ) b(1 RT)

)

(10) (11)

+

∑wi ln(zi/zM)

E

(4)

E G ˜ residual (UNIQUAC)

(5)

˜ E(lattice fluid) ) G ˜ E(UNIQUAC) A ˜ Eattr ) A

(6)

)A ˜ (lattice fluid) )

and not

as suggested in Brandani et al. (1998). Equation 5 must be imposed at a constant reduced reference density. Since most EOS have two mixture parameters, we can also impose an additional constraint. As pointed out by Wong and Sandler (1992) the thermodynamically correct behavior at zero density is the quadratic composition dependence of the second virial coefficient:

BM ) lim

Ff0

∂zM ) ∂F

∑∑wiwjBij

(7)

Derivation of the Mixing Rules for a Noncubic EOS We demonstrate the application of these mixing rules to a noncubic EOS characterized by the CarnahanStarling repulsive term (Carnahan and Starling, 1969) and the attractive term from Brandani et al. (1992):

z ) 1 + zrep + zattr ) 4 4η - 2η2 η(1 + k1η + k2η2) (8) 1+ 3 RT (1 - η) where k1 ) -1.432 79 and k2 ) 3.970 55 and

η ) bF/4

E (UNIQUAC) ) G ˜ residual

∑wi ln γi,residual ) M i 4δ ∑wi RT RT

(

)

(12)

where

where w is either x or y. According to Brandani et al. (1998) the attractive term E at constant reduced density, A ˜ attr , is related to the excess Gibbs energy of the UNIQUAC model. If we now E contains only the interaction energies, it note that A ˜ attr is evident that it must correspond only to the enthalpic term of the UNIQUAC model, that is, the residual part. We therefore propose that in the formulation of EOS mixing rules the following equation should be applied,

A ˜ Eattr

4 1 1 η 1 + k1η + k2η2 RT 2 3

The proposed mixing rules are given by

and therefore

E

A ˜ attr ) -

z-1 dη ) 0 η attr η z dη ) A ˜ rep + A ˜ attr (2) 0 η η

(9)

This EOS is characterized by the volume parameter b and the energy parameter . The temperature dependence of the two parameters and the values for the pure

1 1 δ ) ηlattice 1 + k1ηlattice + k2η2lattice 2 3

(

)

(13)

and

(

BM ) bM 1 -

)

M ) RT

∑∑wiwjBij

(14)

where

1 Bij ) (Bii + Bjj)(1 - kij) 2

(15)

The formulation of mixing rules following the proposed simple procedure yields relationships for the mixture parameters which are easily derived.

M ) RT

E i (UNIQUAC) G ˜ residual wi RT 4δ

∑

BM

bM ) 1-

E G ˜ residual i (UNIQUAC) + wi RT 4δ

(16) (17)

∑

An approximate value for the reduced density of the lattice can be derived from the coordination number, which in lattice fluid models is typically 10 (Prausnitz et al., 1986), and the coordination number corresponding to closed packing, 12.

ηlattice )

10 10 η ) (0.74048) 12 cp 12

(18)

These mixing rules are characterized by three binary interaction parameters that must be determined from experimental binary equilibrium data. Phase Equilibria of Sample Systems We may now evaluate the results that are obtained when these new mixing rules are applied to phase equilibria of low-pressure and high-pressure sample systems. For the calculation of phase equilibria we have to impose the isofugacity condition

φ′iw′i ) φ′′iw′′i

(19)

where wi is the mole fraction in the two phases, prime and double prime, at equilibrium. The resulting system

Ind. Eng. Chem. Res., Vol. 37, No. 8, 1998 2931

Figure 1. (a) Solubility of benzene in water. (b) Solubility of water in benzene.

of nonlinear equations was solved using globally convergent algorithms reported by Press et al. (1995). For the EOS considered,

ln(φi) )

( ) ∂nA ˜ ∂ni

- ln zM ) A ˜ + (zM - 1) +

T,V,nj*i

[ (∑

(zM - 1) 2

(

)

jwjBij -1 + BM

)]

ln γi,residual M bM i BM RT 4δ RT

(

)(

-

)

i ln γi,residual M 1 1 4η 1 + k1η + k2η2 2 3 RT 4δ RT ln zM (20) where

A ˜ )

1 1 4η - 3η2 4M η 1 + k1η + k2η2 2 RT 2 3 (1 - η)

(

)

(21)

Low-Pressure Sample Systems. We have applied the new mixing rules to the constituent binaries of the ternary system benzene-water-ethanol in order to predict the ternary liquid-liquid equilibrium at 25 °C.

Figure 2. (a) Vapor-liquid equilibrium of the system benzeneethanol at 25 °C. (b) Plot of y versus x plot for the system benzeneethanol at 25 °C.

The liquid-liquid equilibrium of benzene-water from 25 to 70 °C is reported by Sorensen and Arlt (1980). Using a simple linear temperature dependence for the UNIQUAC activity coefficient model parameters and predicting k12 from the Hayden-O’Connell second virial coefficient correlation (Prausnitz et al., 1980) it was possible to correlate the experimental data over the entire temperature range. The predicted values for k12 ranged from 0.833, at 25 °C, to 0.753, at 70 °C. The binary interaction parameters are given by a12, K ) 1051.3 + 0.7249T, and a21, K ) -585.48 + 3.3916T. The comparison between calculated and experimental data is shown in Figure 1a,b. The experimental vapor-liquid equilibrium data for the system benzene-ethanol at 25 °C are reported by Smith and Robinson (1970) and were correlated by Gmehling and Onken (1977). The system is shown in Figure 2a,b. The deviations between calculated and experimental values are reported in Table 1, with the same values obtained by Gmehling and Onken (1977) using the UNIQUAC model. The two approaches for correlating the data are practically equivalent. Using the experimental data of Hall et al. (1979), similar results are obtained for the system ethanol-water at 25 °C, which is shown in Figure 3a,b. The deviations

2932 Ind. Eng. Chem. Res., Vol. 37, No. 8, 1998 Table 1. Comparison of EOS Approach and Direct Application of Activity Coefficient Models for the Low-Pressure Vapor-Liquid Systems benzene-ethanol EOS a12, K a21, K k12 AADP, Torr MDPa, Torr AADy MDya a

676.97 -171.39 0.3929 0.48 1.06 0.0032 0.0060

Gmehling and Onken (1977) 402.34 -58.08 0.50 1.29 0.0039 0.0073

ethanol-water EOS 238.9 -18.71 0.3367 1.04 1.67 0.0063 0.0162

Gmehling and Onken (1977) 134.14 -15.89 0.49 1.36 0.0070 0.0149

Maximum deviation (MD).

Figure 3. (a) Vapor-liquid equilibrium of the system ethanolwater at 25 °C. (b) Plot of y versus x plot for the system ethanolwater at 25 °C.

Figure 4. Water-rich phase: (a) benzene; (b) water; (c) ethanol; (s) experimental; (b) EOS; (9) UNIQUAC model.

between calculated and experimental values are reported in Table 1. From these results it appears that in the low-pressure region our approach, using only the residual part of the UNIQUAC model, leads to the capability of correlating binary fluid phase equilibria equivalent to that which may be obtained applying the UNIQUAC activity coefficient model. This result confirms the validity of the

approach, since correlating experimental data at low pressures using EOS should be equivalent to the direct application of the activity coefficient model. All of the binary interaction parameters having been obtained, it is possible to predict the ternary liquidliquid equilibrium at 25 °C. This should be compared with the prediction of the UNIQUAC activity coefficient

Ind. Eng. Chem. Res., Vol. 37, No. 8, 1998 2933 Table 2. Results of the Regression for the System Propane-Carbon Dioxide 579.92 - 1.6821T 4.5985 + 0.8963 T -0.2997 + 0.00196T

a12, K a21, K k12 temp, °F AADP, bar MDP, bar AADy MDy

40 0.10 0.19 0.0040 0.0086

70 0.05 0.35 0.0025 0.0058

100 0.26 0.70 0.0025 0.0050

130 0.39 0.69 0.0073 0.0232

160 0.58 2.04 0.0066 0.0161

Figure 6. Vapor-liquid equilibrium for the system propanecarbon dioxide at 40, 70, 100, 130 and 160 °F.

Figure 5. Benzene-rich phase: (a) benzene; (b) water; (c) ethanol; (s) experimental; (b) EOS; (9) UNIQUAC model.

model using the parameters determined from the same constituent binaries. Tie lines were calculated assuming as the feed composition the mean of the experimen-

tal mole fractions. The comparison of the tie line compositions is shown in Figure 4 for the water-rich phase and in Figure 5 for the benzene-rich phase (experimental data are taken from Bancroft and Hubard (1942)). Qualitatively the two approaches are similar. The benzene solubility in the water-rich phase is typically underestimated by both methods. This may be in part due to the fact that at 25 °C the solubility of benzene in water is very low and close to the experimental uncertainty. Therefore, the benzene-water binary interaction parameters, which are calculated from the mutual binary solubility, may have a large uncertainty. This appears to be a feasible explanation of the behavior observed in the benzene-rich phase. It should be noted that the prediction from the EOS approach gives better quantitative agreement with the experimental data than the original UNIQUAC activity coefficient model. This difference may be in part due to the correlation of the benzene-water binary interaction parameters from mutual solubility data in a temperature range rather than at a single temperature. The main conclusion which may be reached is that the proposed interpretation of the physical contributions used in the derivation of these new mixing rules appears to be correct. High-Pressure Sample Systems. We have tested the proposed mixing rules on two vapor-liquid equilibrium systems at near critical and supercritical conditions. The experimental data for the system propanecarbon dioxide are reported by Reamer et al. (1951). For simplicity we assumed a linear temperature dependence for all of the binary interaction parameters. The values of the parameters and the results obtained from the simultaneous regression of all of the equilibrium isotherms are reported in Table 2. The comparison

2934 Ind. Eng. Chem. Res., Vol. 37, No. 8, 1998 Table 3. Results of the Regression for the System Carbon Dioxide-Ethanol 230.28 - 0.16423T -48.788 - 0.4095T 0.70993 - 0.00125T

a12, K a21, K k12 temp, K AADP, bar MDP, bar AADy MDy

314.5 0.28 0.51 0.0010 0.0028

325.2 0.19 0.55 0.0022 0.0067

337.2 0.18 0.43 0.0028 0.0100

plete UNIQUAC activity coefficient model. The approach has been used to predict the ternary liquidliquid system benzene-water-ethanol. The predictions obtained using the UNIQUAC model with the parameters obtained from the same constituent binaries are qualitatively similar to those calculated from the EOS using the proposed mixing rules. These results confirm the validity of the assumption that only the residual part of the UNIQUAC equation should be equated to the attractive part of the EOS excess Helmholtz energy. These density-independent mixing rules can be used also for the accurate description of vapor-liquid equilibria at near critical and supercritical conditions, both for simple and for highly asymmetric systems. Acknowledgment The authors are grateful to MURST for financial support. Notation

Figure 7. Vapor-liquid equilibrium for the system carbon dioxide-ethanol at 314.5, 324.2, and 337.2 K.

between calculated and experimental data is shown in Figure 6. The correlation of supercritical data introduces no additional difficulty with the present approach, and the proposed mixing rules give a good representation of this simple system. In order to test the applicability of the proposed mixing rules, we have also considered a gas-alcohol system. The experimental data for the system carbon dioxide-ethanol are reported by Jennings et al. (1991). The values of the parameters and the results obtained from the simultaneous regression of all of the equilibrium isotherms are reported in Table 3. The comparison between calculated and experimental data is shown in Figure 7. Our approach gives an excellent representation of this strongly asymmetric system. Conclusions To use the UNIQUAC model as a mixing rule for EOS it is necessary to consider the physical meaning of the combinatorial and residual terms. We have shown that combining the attractive part of the excess Helmholtz energy and only the residual part of the excess Gibbs energy of the UNIQUAC model gives a simple relationship which may be used to derive EOS mixing rules. The second condition imposed on the mixture parameters is the theoretically correct quadratic concentration dependence of the second virial coefficient, as suggested by Wong and Sandler (1992). The present formulation leads to simple mixing rules, which have been applied to a noncubic EOS. For low-pressure systems, the correlation of experimental data using the proposed EOS approach yields results comparable to those obtained by Gmehling and Onken (1977) using the com-

aij ) binary interaction parameters A ) Helmholtz free energy b ) volume parameter of the EOS B ) second virial coefficient k12 ) binary parameter for the virial cross coefficient G ) Gibbs free energy P ) pressure R ) ideal gas constant T ) temperature w ) mole fraction x ) liquid-phase mole fraction y ) gas-phase mole fraction z ) compressibility factor Greek Letters γ ) activity coefficient δ ) defined in eq 15  ) energy parameter of the EOS φ ) fugacity coefficient η ) reduced density Superscripts and Subscripts attr ) attractive part of the EOS C ) at the critical point cp ) closed packing E ) excess of a thermodynamic function M ) mixture rep ) repulsive part of the EOS res ) residual part of a thermodynamic function ∼ ) reduced thermodynamic function

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Received for review August 22, 1997 Revised manuscript received November 5, 1997 Accepted November 7, 1997 IE970584M