A practical extension of equations of state to polar systems

Fluid Phase Equilibria, 29 (1986) 457-464 Elsevier Science Publishers B.V., Amsterdam

457 - Printed

in The Netherlands

A PRACTICAL EXTENSION OF EQUATIONS OF STATE TO POLAR SYSTEMS

HALDUN OZKARDESH The H.W. Kellogg Co., Three Greenway Plaza Houston, TX

77046-0395 (U.S.A.)

ABSTRACT A procedure is described which extends the applicabilityof the commonly used equations of state to the phase equilibria of the apolar-polar as well as strictly polar systems. The method defines a "residual activity coefficient" and uses it as a correction factor to the equilbrium K-values computed from the equation of state. The residual activity coefficients can be correlated by any of the conventionalmodels. The capability of the method as applied to Soave RK equation of state is illustratedwith examples.

INTRODUCTION The industriallywidely used equations of state, such as Soave, Peng-Robinson.BWR are strictly applicable to hydrocarbons and slightly polar compounds. However when a system contains some highly polar or associating components such as alcohols, ketones, ethers, etc., these equations of state become inapplicableand one has to use an activity coefficient based method for VLE calculations. Unfortunately,such an approach requires the development of the model parameters for the selected activity coefficient equation for all the binary pairs present in the system, including hydrocarbon-hydrocarbonpairs. Considering that a lo-componentsystem has 45 binary pairs, this can become a very tedious effort unless one makes substantial simplifying assumptions, such as ideality of some of the pairs, which, in some cases, may not be entirely valid. An alternative to the activity coefficientmodel would be the use of the equation of state for the whole system incorporatinga series of interaction coefficients,kij*s. for the prominent H&polar and polar-polar pairs, as has been shown in a variety of publications. However, as

indicated

below, a proper representationof some of the binary pairs containing polar

0378-3812/86/$03.50

0 1986 Elsevier Science Publishers

B.V.

458

componentswould requiretwo kij's,i.e., kijf kji as well as composition - dependent kij's. This paper provides another alternative. PROPOSED METHOD When an equation of state is used for VLE computations, the equilibrium K-value of a component is defined as the ratio of the fugacity coefficients of the component in the liquid and the vapor phases: the activity coefficients, although not calculated separately, are contained in these fugacity coefficients. In the UNIQUAC (Abrams and Prausnitz, 1975) and UNTFAC (Fredenslundetal., 1977) models, the logarithm of the activity coefficient is assumed to be the sum of two contributions:a combinatorialpart, essentially due to the differences in size and shape of the molecules in the mixture and a residual part, essentially due to energy interactions. The liquid phase activity coefficients inherent in an equation of state such as SRK (Soave, 1972) are, in general, adequate for representing this sumnation of the two contributions in hydrocarbon systems, but seem to require additional +esidual" contributions for highly non-ideal systems. For such systems we propose to define the activity coefficient,1, with SRK for example, as In Y = In Y (SRK, kid=0) + In Y (residual)

(1)

Fig. 1 illustrates, for the n-butane-methanolsystem, the relative magnitude and shape of the observed (UNIFAC), SRK (ktj=O), and the observed residual activity coefficients,the latter being calculated as the ratio of the UNIFAC to SRK values. It can be shown that with SRK, when kid's are used and the effect of kid values on the liquid compressibility,z,is neglected In Yi(residual) = (A/B) [(2/a) z

qbpj)0*5k~jl

W+W))

(2)

where A, 8, and a are the standard notations used with SRK. The infinite dilution activity coefficlents,Ymgs, of a binary system can be calculated directly from the pure component SRK parameters: for the residual part the equation is then

ln vqfresidual) - 2 (A.$Bj)(a~/aj)0'5 'kijln(lt(Bj/zj) An equation of state such as SRK can not, and lo not. expected to handle highly non-ideal systems because of the strict constrains imposed by eqn. (3)

459

on the relationship between kid and residual r=; requiring kij#

kji.

Furthermore, eqn.(2) is not adequate for proper representation of the residual Y 's. as shown in Fig. 2.

The interaction coefficients k12 = 0.05

and k21 = 0.32 used in this figure were calculated from eqn.(3) for the observed residual infinite dilution activity coefficients of 1.88 and 5.895, respectively, and using the butane and methanol parameters listed in Table 1. A practical substitute to eqn.(t) is to calculate the residual activity coefficients by a conventional model. On the basis of the preceding discussions, then, the K-values, in the proposed method are calculated for all the components as follows: K = (KE,S) (Residuals )

(4)

where KE,S is the K-value computed from the equation of state with kid = 0.0.

The residuals is computed from an activity coefficient equation

utilizing binary constants. These constants are obtained by curve fitting the selected model, with the residual y Y

calculated as:

(residual) = K(observed) / (KE,S)

for the polar-polar and apolar-polar pairs. For apolar-apolar pairs, such as hydrocarbon binaries, the constant would automatically be taken as zero, since v(residua1) - 1.0 for all such pairs. In eqn. (4) it is assumed that the non-idealities in the gaseous phase are adequately described by the equation of state, even when polar components are present. Although any error introduced by this assumption is absorbed in the residual Y'S as implied by eqn.(5), in certain cases the use of an interaction coefficient (kfj) specifically for the vapor phase may be appropriate. Such a kid may also alleviate some anticipated problems when the proposed method is applied to mixtures at the critical conditions. Furthermore, it is assumed that the equation of state used by practicing engineers has the flexibility or the features which enable it to predict the properties of the pure polar components. Otherwise, a practical alternative is, of course, to override the standard acentric factor of the component with a value which induces the equation of state to give a reasonable vapor pressure prediction for the component at the system conditions. EVALUATION OF THE MODEL The model was tested by coupling the SRK (API version) with the three-suffix form of the Hargules activity coefficient equation shown below for

460

0

0.2 ,YT.YI

0.4 I”

0.6

0.6

1.0 IWAY,

L1O”ID. YOLE n

Fig 1. Activity coefficients in the n-butane-methanol Fig 2. Residual activity coefficients

1

0

I

I

0.2

04 I”71WL.

I

,* LIOUID. YDIA

system at 93.3OC

L

06 YOU I”.

0.6

Fig3. a,b n-butane-methanol

1.0

system at 93.3OC

TABLE I Physical

constants

T, K n-butane acetone chloroform methanol

P, atm

acentric

3;47 46:41

0.193

factor

-

F”

461

a binary: TnYt 8 X3*( Aid t 2x1 (Aji - A~J) 1

(6)

The Margules model was selected for the expediencyand convenienceprovided in the equality of its constants to the logarithmsof the infinite dilution activity coefficients,and therefore could be estimated very easily as:

Aij= lnv,"p (residual)=

Tnyy (observeddata) - lnv;

(SRK, kfj

n

0)

(7)

n-Butane - Methanol System This is one of the important binary systems involved in the MTBE process. There are no publishedVLE data on this system. Figure 3 compares VLE and relative volatilitydata computed for 93.3 'C (200 OF) using UNIFAC (i.e., UNIFAC for the liquid phase, Redlich-KwongE/S For the vapor phase), the SRK E/S (with kij - 0.0). and the proposedmethod (SRK t residualv ). It can be noted that SRK totally misses the azeotrope but with the residual correction reproducesthe UNIFAC based data reasonablywell. A slight inconsistencyin the predicted azeotropicpressure may be taken to be due to the fact that UNIFAC values are also estimated data. The residual Margules constants for the system were computed from Y- values shown in Figure 1. Acetone - Chloroform - Methanol System In early 1984, 6. Soave published a paper (Soave, 1984) in which he presented a modified Van der Waals (VdW) equation of state introducingnew mixing rules, with the liquid phase excess-freeenergy term expressed by a NRTL-like model. The acetone-chloroform-methanol system was the prime example for the demonstrationof the applicabilityof this new equation to polar systems. We, therefore,used the same system to test our proposed'method. For the experimentaldata on the binary pairs and the ternary mixture we used the same sources as Soave selected and, for expediency,we took the Nargules constants for the experimentaldata from the same source (Gmehlingand Onken). The infinite dilution activity coefficientscorrespondingto these constants and the similar data obtained for the SRK (ktj - 0) are tabulated in Table 2.

Also given are the 'residualYw’values calculated by the ratio

and used to obtainThe residual Margules constants applied for the VLE predictions. The physical constantsused in the SRK predictionsare shown in Table 1. The results for the binary pairs are presented in Figures 4 through 6. may be noted that, as shown in Figure 4, for the acetone-chloroformsystem

It

462

650 1

20

40

60

100 J

60

wGC,OYL. MOLE.

system at 1 atm., Observed

Fig. 4 a,b The acetone-chloroform

m

Fig 5 a,b The acetone-methanol

Fig 6 a,b The

o.so

system at I atm., Observed

chloroform-methanoi

data: Kajima

’

0.2 lE6iOYE

0.4 0.6 IW LIO”ID,

et al., (1968)

Y0i.E

0.6 iR.

data: Wchida et al. (1953)

system at 1 atm., Observed

data: Nagata (1962)

463

TABLE 2 :~~~~~:e(~~'Ut~Po:~~~~Y2S

oefficien s a d residual Mar ules constants for the -methanor (3r system at 1 a&mosphere

Acetone-Chloroform YY2 f&

Acetone-Hethanol Chloroform-Hethanol

Y21 ~Ij.&

;ir

;ji

;iis

g[

Hargules constants (In ym(residual)) j&

: :I:##

fgj : -I:T#T

ft$j: 8:%f43

(*) Converted from the Nargules constants given in Bmehling and Onken

TABLE 3 Bubble goint calculations for the acetone (1) - chloroform (2) - methanol (3) system t 1 atm.

t

Oc

-

Modified Vd;l**)

61.0 58.7 57.0 56.0 56.0 56.0 56.5 59.8 (*) Shishkin and Kotsyuba, 1955

(**) Soave, 1984

Prooosed Ke;hod

464

(which manifests negative deviations from the Raoult's Law), the SRK without the residual correction totally misses the azeotrope. For the other two binaries which manifest positive deviations form the Raoult's Law, the equation of state, per se, seems to predict the azeotropic composition correctly but with substantial errors in temperatures and relative volatilities at other concentrations. The proposed method reproduces the experimental behavior quantitatively,although only the infinite dilution Y 's were used in the data reduction. Table 3 shows the results for the ternary mixtures with 10 mole X of acetone in the liquid and compares the predicted temperatures and vapor mole fractions with the experimental data and with those data presented by Soave for his modified VdW predictions. The predicted data with the 'proposed method" are as good as those predicted by the .Hodified VdW" and compare as well with the experimental data. The relatively poor showings at high chloroform concentrationsmay be attributed to the inconsistencyof the experimental data, as Soave also concluded. CONCLUSION A practical and convenient technique is proposed for the computation of phase equilibrium of systems containing polar components. The method provides flexibility in that any equation of state can be coupled with any activity coefficient model and the required additional parameters would be only for those component pairs which can not be adequately represented by the equation of state itself. ACKNOWLEDGEMENT The author is grateful to the H.W. Kellogg Co. for the permission to publish this paper. REFERENCES Abrams, D.S. and Prausnitz, J.N., 1975. A.1.Ch.E.J. 21: 116-128 Fredenslund. A., Gmehling, J. and Rasmussen, P., 1977, Vaoor,Liauid Eauilibria Usina UNIFAC, Elsevier. Gmehling, J. and Onken, V., VLE Data Coll., DECHEBA. Chem. Data Series. Vol. 1, Parts 3 + 4, page 92 (acetone-chloroform):Part 2a, page 85 (acetone-methanol);Part 2a. page 19 (chloroform-methanol);Part 2a, page 609 (Ternary system) Kajima, K. et al, 1968. Kagaku Kogaku 32: 337 Nagata, 1. 1962. Chem. Engng. Data 7: 367 Shishkin, K. and Kotsyuba, A.A.,1955, Tr.Dnepropetr.Khim.Tekhn.Inst., 4: 18 Soave, G..1972. Chem Eng. Sci.. 27: 1197-1203. Soave, G.,1984. ChemEng. Sci., 39: 359-369. Uchida, S. et al 1953. Kagaku Kikai Chem. Engng. 17: 191