generalization of binary interaction parameters of

FluidPhase Equilibria, 56 (1 990) 8 1-88 Elsevier Science Publishers B.V., Amsterdam

GENERALIZATION

OF BINARY INTERACTION

PENG-ROBINSON EQUATION CONTAINING HYDROGEN HIDEO NISHIUMI Chemical Engineering

81 Printed in The Netherlands

and HIRONOBU Laboratory,

PARAMETERS

OF

OF STATE FOR SYSTEMS

GOTOH

Hosei University,

Koganei,

Tokyo 184 (Japan)

ABSTRACT Binary interaction parameters of the Peng-Robinson equation of state for the systems containing hydrogen are successfully correlated by a unique function of temperature with the maximumvalue at about 460 K, represented by Equations (3) and (4). Absolute average deviation of mole fraction of vapor and liquid phases at flash calculation is 0.007 and 0.018 for optimized values at each temperature of a system and for the generalized correlation, respectively. This correlation covers systems of hydrogen-alkane, - cycloalkane, -alkene, - aromatic hydrocarbon and -somepolar compound. VLE calculation for 34 binary systems containing hydrogen showed excellent generalized correlation. The genearalized binary interaction parameters as a unique function of temperature will predict vaporliquid equilibria for the binary systems containing hydrogen without inert gas where little or no data is available.

INTRODUCTION The hydrogen recovery processes from coal liquefaction and the hydrotreating processes in fossil-fuel technology have increased the demand for prediction method of vapor-liquid equilibria of systems containing hydrogen. Using an extended BWR equation of state with our proposed mixing rules, binary interaction parameters for 34 binary systems containing hydrogen were found to be temperaturedependent (Nishiumi, 1983). The binary interaction for systems containig hydrogen could be correlated as a function of temperature and the molar critical volume of a component in pair with hydrogen ( Nishiumi and Fukushima, 1989 ). We succeeded in providing the prediction method for the systems containing hydrogen, as well as for the systems composed of nonpolar substances ( Nishiumi and Saito, and McCoubrey,1960).

0378-38

1 2/90/$03.50

1977)

© 1 990 Elsevier

on the basis of the Hudson-McCoubrey theory (Hudson

Science

Publishers

B.V.

82

Using the Peng-Robinson equation of state, Valderrama and Molina (1986) correlated the binary interaction parameters for systems containing hydrogen as a quadratic function of temperature. Valderrama and Reyes (1983) generalized binary interaction as a function of the form of Sij = «2 - fa/Tr2 where a2 and & are quadratic in the acentric factors. For the systems composed of nonpolar susbstances, we correlated the binary interaction parameters of the Peng-Robinson equation of state in terms of the ratio of critical molar volumes and in addition absolute difference between the acentric factors of each component (Nishiumi et al., 1988). The objective of this paper is to extend the correlation method to generalize the binary interaction of the Peng-Robinson equation of state for the systems containing hydrogen.

EQUATION OF STATE

The Peng-Robinson p

RT v-b

equation

of state ( Peng and Robinson,

1976

a(T) v(v+b)+b(v-b)

) (1)

is based on the three-parameter corresponding states principle. Constants a and 6 in Eq.(l) are functions of TC,PC and w. The constant a of a mixture is related to the a values of each component, a,- and Oj, ay = mij^/aidj

v2)

where my is unity when i = j. In an original paper ( Peng and Robinson, 1976 ), a binary interaction parameter my is expressed as (1 - Sij). We, however, adopted my in this work, because Eq.(2) is anticipated to be affected by the Hudson- McCoubrey theory (1960). Equating the London theory of dipersion forces to the attractive term of the Lennard-Jones potential model, they derived that my for a system composed of nonpolar substances is a function of the molar critical volume ratio, VcijVcj. To predict various kinds of thermodynamic properties , we chose V^Vcj as a variable of my. We succeeded in applying this idea to the systems composed of nonpolar substances or those containing hydrogen using the BWR equation of state (Nishiumi et al., 1977, 1989), and to the systems composed of nonpolar substances using the Peng-Robinson equation of state (Nishiumi et al., 1988). It is very interesting to investigate whether we can apply this idea to the binary systems containing hydrogen using the Peng-Robinson equation of state.

GENERALIZATION

OF BINARY INTERACTION

PARAMETERS

Wemainly used the values compiled by Reid, Prausnitz and Shewood (1977) for the critical properties and the acentric factor of pure components. An optimum value of mtj is obtained

83 by minimizing the objective function £ (I ar,-)Ca/ - x^exp\ + | yiiCat - yifxp |) to be best fitted to vapor-liquid equilibrium data of a binary system composed of normal fluids below 25 MPa. Experimental data on binary vapor-liquid equilibria are the same as in a previous paper (Nishiumi and Saito, 1977). Best fitted values of m,j were determined for 34 binary sytems composed of hydrogen-Ci ~ Cie alkane, -alkene, -aromatic hydrocarbon, -inert gas, -polar compound. Binary interaction for systems containing hydrogen was found to be correlated as a function of temperature as well known (Valderrama et al., 1986 for the Peng- Robinson equation of state: Nishiumi et al., 1983, 1988 for the BWR equation of state). The optimum m;j(= 1 8^} values were plotted aginst temperature. Soon it proved to be a simpler problem than at first anticipated. Surprisingly, we have found that all points for the 34 systems were expressed by a unique curve with maxium at about 460K as shown in Figure 1. It means that binary interaction for the systems containing hydrogen using the Peng-Robinson equation of state can be generalized. We can predict vaporliquid equilibria for unknown binary systems containing hydrogen only knowing by the critical properties and the acentric factor of a component in pair with hydrogen. We do not need to use the correlatin based on the Hudson- McCoubrey theory. The correlated results are as follows, rriij

=

1.224-0.00440r+3.251

= 56.98-0.1655T+1.199x

x 1(T5T2

forT

< 461.75K

(3)

1(T4T2

forT

> 461.75K

(4)

The reason why optimum m,-j values approach an asymptote at about 460 K seems that the objective function becomes insensitive because the mole fraction of hydrogen in the liquid phase is small, and that in the vapor phase is close to unity around the characteristic temperature for systems tested in this work. In this correlation, we could ignore the effect brought by m,-j values greater than 10, because these points are very much insensitive as shown in Table 1.

RESULTS OF VAPOR-LIQUID

EQUILIBRIUM

CALCULATION

Comparison between the generalized binary interaction and the values optimized at each temperature wasmade by using vapor-liquid equilibrium data for 34 binary systems containing hydrogen at pressure below 25 MPa and the results are summarized in Table 2. Figures 2-7 show comparison of vapor-liquid equilibrium data on the hydrogen-bicyclohexyl, ethylene, - diphenylmethane, - tetralin, - carbon dioxide, and - nitrogen systems with values predicted using the m,-j values calculated from Equations (3) and (4). These figures indicate excellent correlation except near a critical point ( Figure 7 ) where an equation of state usually fails. However, our generalized results give poor prediction of composition in the liquid phase for the hydrogen- carbon dioxide, - nitrogen, and - argon systems.

84

-2

Figure 1: Generalization of ra;j(= Robinson equation of state

1 - ij P R T v x y

6a u S ubscript c »',j *J

constant in Eq.(l) constant in Eq.(l) (=1 - Sij ) binary pressure gas constant temperature molar volume mole fraction of a mole fraction of a binary interaction acentric factor

interaction

parameter defined by Eq.(2)

liquid phase vapor phase parameter in original

paper (Peng et al., 1976)

critical property component cross-component

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H

udson, G. H. and J. C. McCoubrey, 1960. Intermolecular Forces Between Unlike Complete Form of the Combining Rules, Trans.Faraday Soc., 5:561-766

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from MixMix-

'