gao1992 equation of state

(1)

Fluid Phase Equilibria, 74 (1992) 85-93 Elsevier Science Publishers B.V., Amsterdam

85

A simple,correlation to evaluate binary interaction parameters of the Peng-Robinson equation of state:

binary light hydrocarbon systems

Guanghua Gao

Department of Chemical Engineering, Tsinghua University, 100084 Beijing (China)

Jean-Luc Daridon, Henri Saint-Guirons and Pierre Xans *

Laboratoire de Physique des Matt%aux Industriels, Centre Universitaire de Recherche Scien- tifique, Universitk de Pau et des Pays de I’Adour, Avenue de Wniversite, 64000 Pau (France)

Frangois Monte1

&xi& Nationale ELF-Aquitaine -C.S.T.C.S Avenue Larribau, 64018 Pau Cedex (France) (Received May 19, 1991; accepted in final form December 30, 1991)

ABSTRACT

Gao, G., Daridon, J.-L., Saint-Guirons, H., Xans, P. and Montel, F., 1992. A simple correlation to evaluate binary interaction parameters of the Peng-Robinson equation of state: binary light hydrocarbon systems. Fluid Phase Equilibria, 74: 85-93

A simple correlation to evaluate binary interaction parameters kij of the Peng-Robin- son equation of state for light hydrocarbons mixtures is proposed, as a function only of critical temperature and compressibility factor. The method yields results with an accuracy bordering on that obtained with methods requiring specific adjustments for each binary mixture coefficient kij.

INTRODUCTION

The Peng-Robinson equation of state (1976) is most frequently used in the petroleum industry and related fields. For the prediction of thermody-

Correspondence to: P. Xans, Laboratoire de Physique des MatCriaux Industriels, Centre Universitaire de Recherche Scientifique, Universitt de Pau et des Pays de l’Adour, Avenue de l’Universit6, 64000 Pau, France.

(2)

86 G. Gao et al. /Fluid Phase Equilibria 74 (1992) 85-93

namical properties of a pure component, it requires only the knowledge of the critical pressure, the critical temperature and the acentric factor. To improve performance of this equation when used in studies of binary mixtures, it is customary to introduce binary interaction parameters, k,,, in classical mixing rules. Values of kij for all possible binary pairs are determined by regressing vapour-ljquid equilibrium data for experimental mixtures, a procedure which removes the predictive power of this equation.

The aim of this paper is to propose a simple correlation to calculate binary interaction parameters kij used with the Peng-Robinson equation of state for a certain number of hydrocarbon mixtures (C,-C,,). This procedure requires only the additional knowledge of the critical volume, V,, of each of the components.

BINARY INTERACTION PARAMETERS

To calculate the vapour-liquid equilibria of the fluids studied, the Peng-Robinson equation of state (1976) is used.

p=-- RT a

v-b v(v + b) + b(v -b)

The two parameters a and b are expressed for pure components as a = 0.45724( R2T:/Pc)~

b = 0.07780( RT,/P,) where

(Y = [l + m(1 - T;/2)]2

m = 0.37464 + 1 S42260 - 0.26992d2

(1)

(2) (3)

(4) (5)

For the mixtures, a and b are replaced by a, and b, with the following mixing rules:

a, = C Cxixj(aiaj)“*(l - kij)

i j

b, = &bi

(6)

(7) kij is an empirical binary parameter introduced to improve performance for the restitution of thermodynamic properties of mixtures, but its evalua- tion, from all possible binary pairs, is often a problem.

(3)

G. Gao et al. /Fluid Phase Equilibria 74 (1992) 85-93 87

For most authors the values of kij are back-calculated by trial and error from data available in the literature. These computations lead to the best results but remove the predictive character of the procedure used. For hydrocarbon pairs it should be noted that the binary interaction parameters are generally very close to zero, so they are usually taken as zero by many authors. Thus, they recover the predictive property of the Peng-Robinson equation as regards pure compounds, but this approach leads to less accurate results.

To maintain the predictive character of this procedure, even in the case of a number of hydrocarbon mixtures (C,-C,,), we propose the following correlation:

1 -k,, = [2(T,iT,j)1’2/(T,j + Tcj)]=“’

where

‘cij = (‘ci + zcj)/2 (9)

Zci and Zcj are the critical compressibility factors of the two components.

Consequently the Peng-Robinson equation should be used without any adjustable parameter, since the mixture law requires only knowledge of the critical parameters PC, V, and T, of each component.

PREDICTION RESULTS AND DISCUSSION

We have estimated vapour-liquid equilibria (VLE) at high pressure using the binary interaction parameter of the Peng-Robinson equation of state calculated by eqn. (8) for a number of hydrocarbon systems which are made up of paraffins, cycloalkanes, alkenes and aromatic hydrocarbons CC,-C,,). A total of 252 isothermal VLE (P, T, x, y) data sets and 2320 VLE determinations (Knapp et al., 1982) are available and are listed in Table 1. The critical temperature, critical pressure, critical compressibility factor and acentric factor required to calculate vapour-liquid equilibria are given by Reid et al. (1977).

The predicted results are summarized in Table 1 and compared with calculated values employing kij back-calculated by trial and error from experimental data which are compiled by Knapp et al. (1982). As indicated in Table 1, the agreement between estimated and experimental bubble point pressure as well as vapour phase composition in equilibrium is satisfactory. In other words, the deviation between experimental and predicted values using the binary interaction parameter kij calculated by eqn.

(8) is not far from that employing optimized kij for individual isothermal VLE data.

(4)

TABLE 1 Comparison of predicted and calculated values of vapor-liquid equilibria for the different mixing rules System Datum Temp. Max. Our mixing rule Original mixing rule with ki, point range (K)

-, pressure AP (MPa) (AAD%) ?A%, kl, ZAD%, $4D%, ki, Methane - n-ethane -n-propane -n-butane - n-pentane - n-hexane - n-heptane -n-octane - n-nonane - n-decane -isobutane -cyclohexane -benzene -toluene Ethane -propane -n-pentane - n-hexane -n-heptane

7 250-250 6.7 0.72 1.11 0.0079 0.38 0.29 0.0070 106 130-213 6.5 2.72 0.10 0.0152 2.92 0.05 0.0119 124 294-394 13.2 3.38 3.26 0.0218 1.49 1.47 0.0244 66 310-444 10.9 2.14 3.34 0.0267 1.31 0.80 0.0241 16 310-410 1.7 1.39 0.62 0.0311 1.35 0.33 0.0244 104 277-510 24.8 2.76 2.64 0.0352 2.18 0.57 0.0300 56 223-423 27.3 4.84 1.55 0.0382 1.97 0.23 0.0496 131 223-423 32.3 3.23 0.54 0.0413 2.91 0.41 0.0474 156 310-510 36.1 2.95 0.87 0.0428 2.94 0.22 0.0411 41 310-377 11.5 2.53 3.42 0.0200 1.68 0.79 0.0256 95 294-444 28.2 4.27 2.36 0.0374 2.96 0.35 0.0384 17 339-339 33.0 3.24 1.95 0.0383 2.57 0.89 0.0304 26 422-543 25.2 7.58 7.78 0.0412 4.94 2.23 0.0970 84 255-366 5.2 0.62 0.96 0.0013 0.54 0.88 0.0011 74 277-444 6.8 1.55 2.82 0.0063 1.44 1.10 0.0078 7 298-298 3.5 2.53 0.44 0.0086 1.93 0.40 0.0041 62 338-449 8.8 2.65 1.69 0.0109 2.42 1.13 0.0033

a

Q 8

(5)

-n-octane 64 273-373 5.3 2.10 0.50 0.0129 1.44 0.59 0.0185 - n-decane 119 277-510 11.8 3.10 1.36 0.0160 3.01 0.84 0.0144 -cyclohexane 83 233-533 9.0 3.81 1.78 0.0121 3.10 1.16 0.0178 -benzene 80 273-553 8.2 5.10 0.42 0.0127 1.78 0.58 0.0322 Propane -n-butane -n-pentane -n-hexane -n-heptane - n-decane -n-isopentane -benzene

43 323-423 4.1 0.37 1.83 0.0007 0.29 0.71 0.0033 78 344-460 4.5 3.00 2.66 0.0019 0.86 1.34 0.0267 40 333-493 4.8 1.13 1.25 0.0034 0.91 0.97 0.0007 44 333-533 4.8 1.62 3.59 0.0048 1.34 2.13 0.0056 66 277-510 7.1 2.73 0.62 0.0086 3.47 0.17 0.0000 103 273-453 4.6 2.15 2.47 0.0017 1.61 1.66 0.0111 81 310-477 5.9 4.41 2.15 0.0060 2.61 0.77 0.0233 n-Butane -n-heptane - n-decane 41 355-538 4.0 0.87 0.99 0.0019 0.83 0.56 0.0033 75 310-510 4.9 1.52 0.71 0.0045 1.49 0.50 0.0078 Ethylene -ethane - n-heptane -benzene

17 263-293 4.8 0.65 3.49 0.0002 0.64 0.52 0.0089 86 216-516 9.7 1.62 0.80 0.0138 1.55 0.66 0.0144 9 348-348 9.1 5.48 0.77 0.0158 3.30 0.69 0.0311 Propylene -n-propane -1-butene 77 310-344 3.1 1.25 1.13 o.oooo 0.31 0.08 0.0063 42 277-410 4.7 0.55 0.54 0.0007 0.31 0.46 0.0004

(6)

90 G. Gao et al. /Fluid Phase Equilibria 74 (I 992) 85-93

METHANE (1) - N-NONANE (2)

~ q-7x7-J

Fig. 1. Equilibrium pressure-composition diagram for methane-n-nonane: -9 4,

calculated by eqn. (8); - - -, k,, = 0; o, experimental data (Shipman and Kohn, 1966).

METHANE (1) - CYCLOHEXANE (2)

T - 344.26 K 320

X(I), Y(I)

Fig. 2. Equilibrium pressure-composition diagram for methane-cyclohexane: p, calculated by eqn. (8); - - -, k,, = 0; o, experimental data (Reamer et al., 1958).

kii

(7)

G. Gao et al. /Fluid Phase Equilibria 74 (1992) 85-93 ⁹¹

METHANE (1) - BENZENE (2)

,tyxF

100

0

0.0 .2 .4 .6 .6 1.0 X(I), Y(I)

Fig. 3. Equilibrium pressure-composition diagram for methane-benzene: -, ki, calculated by eqn. (8); ---, k,, = 0; o, experimental data (Elbishlawi and Spencer, 1951).

It is clear that, for mixtures made up of similar components, the coefficients kij are close to zero and our correlation gives results similar to those obtained with kij equal to zero. However, for asymmetric mixtures, e.g. methane-n-nonane, the improvement can reach 16% for bubble point pressures and 7% for the vapour phase composition.

Figures l-3 compare VLE data for the methane-n-nonane, methane- cyclohexane and methane-benzene systems with predicted values using the kij parameters calculated by eqn. (8). For comparison, results predicted by using zero values of kij are also shown in the same figures. The former procedure gives predicted values on better agreement with the experimental data. However, for the limited region in the vicinity of the critical point all these procedures show discrepancies between experimental and calculated results.

Other studies have been carried out to solve this question on the basis of various kinds of equations of state. For example, Chueh and Prausnitz (1967) have estimated interaction parameters in mixture T, and V, calculations and used the Redlich-Kwong equation (1949) with certain alterations to calculate critical pressures; for any family of chemical components these parameters, upon suitable reduction, follow definite trends and thus can be used to generalize the process to multicomponent systems. These proce-

(8)

92 G. Gao et al. /Fluid Phase Equilibria 74 (1992) 85-93

dures involve much more elaborate calculations than ours, but we would nonetheless point out that they are particularly useful in the critical region.

CONCLUSION

This simple correlation to evaluate binary interaction parameters of the Peng-Robinson equation of state for light hydrocarbon mixtures was given in terms of critical temperature and compressibility factor of pure components. The proposed procedure yields slightly poorer results than those given by k,, back-calculated by trial and error from experimental data, but its accuracy is not far from that of the latter. On the other hand this correlation maintains the predictive character of the Peng-Robinson equation in the case of a number of light hydrocarbon mixtures.

LIST OF SYMBOLS

a, b

kij

P R T

T,

V x Y Z

parameters in eqn. (1) binary interaction parameter pressure

gas constant

absolute temperature reduced temperature volume

mole fraction of component mole fraction in vapor phase compressibility factor

Greek letter

0 acentric factor Subscripts

C critical property 4 j component lj cross-component

m mixture

(9)

G. Gao et al. /Fluid Phase Equilibria 74 (1992) 85-93 ⁹³

REFERENCES

Chueh, P.L. and Prausnitz, J.M., 1967. Vapor-liquid equilibria at high pressures: calcula- tion of critical temperatures, volumes and pressures of non polar mixtures. AIChE J., 13:

1107-1113.

Elbishlawi, M. and Spencer, J.R., 1951. Equilibrium relations of two methane-aromatic binary systems at 150°F. Ind. Eng. Chem., 43: 1811-1815.

Knapp, H., Doring, R., Gellrich, L., Pldcker, U. and Prausnitz, J.M., 1982. Vapor-liquid equilibria for mixtures of low boiling substances. Dechema, Frankfurt am Main, Ger- many.

Peng, D.Y. and Robinson D.B., 1976. A new two-constant equation of state. Ind. Eng.

Chem. Fundam., 15: 59-64.

Reamer, H.H., Sage, B.H. and Lacey, W.N., 1958. Phase equilibriums in hydrocarbon systems - volumetric and phase behavior of the methane-cyclohexane system. Ind. Eng.

Chem. Data Ser., 3: 240-245.

Redlich, 0. and Kwong, J.N.S., 1949. On the thermodynamics of solutions. V - An equation of state. Fugacities of gaseous solutions. Chem. Rev., 44: 233-244.

Reid, R.C., Prausnitz, J.M. and Sherwood, T.K., 1977. The properties of Gases and Liquids, 3rd edn. McGraw-Hill, New York.

Shipman, L.M. and Kohn, J.P., 1966. Heterogeneous phase and volumetric equilibrium in the methane-n-nonane system. J. Chem. Eng. Data, 11: 176-180.