VAN DER WAALS MIXING RULES FOR CUBIC EQUATIONS OF STATE .
Chemical Engineering Science, Volume 41, No. 5, pp.1303-1309, 1986VAN DER WAALS MIXING RULES FOR CUBIC EQUATIONSOF STATE. APPLICATIONS FOR SUPERCRITICALFLUID EXTRACTION MODELLINGT. Y. KWAK and G. A. MANSOORIDepartment of Chemical Engineering, University of Illinois at Chicago, (M/C 063), Chicago, IL 60607-7052, USAAbstract - A new concept for the development of mixing rules for cubic equations of state consistent withstatistical-mechanical theory of the van der Waals mixing rules is introduced. Utility of this concept isillustrated by its application to the Redlich-Kwong (RK) and the Peng-Robinson (PR) equations of state. The resultingmixing rules for the Redlich-Kwong and the Peng-Robinson equations of state are tested through prediction ofsolubility of heavy solids in supercritical fluids. It is shown that the new mixing rules can predict supercriticalsolubilities more accurately than the original mixing rules of the Redlich-Kwong and Peng-Robinson equations ofstate.INTRODUCTIONThere has been extensive progress made in recent yearsin research towards the development of analyticstatistical-mechanical equations of state applicable forprocess design calculations (Alem and Mansoori,1984). However,cubic equations of state are still widelyused in chemical engineering practice for the calcu lation and prediction of properties of fluids and fluidmixtures (Renon, 1983). These equations of state aregenerally modifications of the van der Waals equationof state,RTaP --- 2(1)v- b vwhich was proposed by van der Waals in 1873.According to van der Waals, the extension of thisequation of state to mixtures requires replacement of aand b by the following composition-dependentexpressions:a""iJ L L x,x1a,1(2)(3)Equations (2) and (3) are called the van der Waalsmixing rules. In these equations, aiJ and biJ (i j) areparameters corresponding to pure component(i) whilea,j and blJ (i j) are called the unlike-interactionparameters. It is customary to relate the unlike interaction parameters to the pure-component par ameters by the following expression:(4)a,1 (1 -ki1 )(aiiaiJ)112bi) (b;; bjj )/2.(5)Extensive research on equations of state have in dicated that the van der Waals equation of state is notaccurate enough for the prediction of properties ofcompressed gases and liquids (Rowlinson andSwinton, 1982). This deficiency of the van der Waalsequation of state has initiated a great deal of researchon the development of other equations of state,through the use of principles of statistical mechanics orby empirical or semi-empirical means. The majority ofequations of state used in chemical engineering prac tice are of the second category and are mostly cubic inthe volume, like the van der Waals equation of stateitself, but contain other forms of the temperature andvolume dependencies. One such equation of statewhich has a simple form is the Redlich-Kwongequation of state (Redlich and Kwong, 1949):aRTP -------112v-bp RTv-ba(T)( 1)v(v b) b(v-b)2a(T) a(Tc ){l ,c(l -T:12 )}a(Tc ) (3.1)(6)v(v-b)"This equation of state has found widespread appli cations in chemical engineering calculations. Wewould also like to mention that there exist a largenumber of more sophisticated empirical versions of thevan der Waals and Redlich-Kwong modifications. Ofthis category of equations of state, the Peng-Robinsonequation of state (Peng and Robinson, 1976) In eq. (4), k11 is a fitting parameter which is known asthe coupling parameter. With eq. (5) substituted in eq.(3), the expression for b will reduce to the followingone-summation form:T 0.45724 R;T!/Pc" 0.37464 1.54226co -0.26992cob(8) 0.0778 RTc/Pc(8.1)2(9)(10)is widely used for thermodynamic propertycalculations.While there has been extensive activity in thedevelopment of new and more accurate empirical cubicequations of state, there has been little attention to thefact that the van der Waals mixing rules are used1303
1304erroneously in these equations, eqs (2) and (3). In otherwords, for the extension of applicability of a cubicequation of state it is not generally correct to use eqs (2)and (3.1) as the mixing rules without attention to thealgebraic form of the equation of state. In the presentpaper we introduce a new concept for the developmentof mixing rules for cubic equations of state in the spiritof the van der Waals mixing rules. This concept isbased on statistical-mechanical arguments and the factthat the van der Waals mixing rules are for constants ofan equation of state and not for any thermodynamicstate function which may appear in an equationof state. The resulting mixing rules for theRedlich-Kwong equation of state, which is a simplemodification of the van der Waals equation of state,and for the Peng-Robinson equation of state, which isan advanced modification of the van der Waalsequation of state, are tested through prediction ofsolubility of heavy solids in supercritical fluids.'JHEORY OF THE VAN DER WAALS MIXING RULESLeland and co-workers (1968a, b, 1969) were able tore-derive the van der Waals mixing rules with the use ofstatistical-mechanical theory of radial distributionfunctions. According to these investigators, for a fluidmixture with a pair intermolecular potential energyfunction between molecules of the mixture in the formu1;(r) BiJ f{r/t7iJ)(11)the following mixing rules will be derived:nnii"nIiu3 LLX;x;u&eu3 LLX;X1 i;t7fi.(12)(13)In these equations, e11 is the interaction energy par ameter between molecules i and j and U;; is theintermolecular interaction distance between thetwo molecules. Coefficients a and b of the van derWaals equation of state are proportional to e and qaccording to the following expressions: 1.1250 R T" v" oc N0 eu3b 0.3333v" oc N0 u3 a(14)(15)We can see that eqs (12) and (13) are identical witheqs [2] and [3], respectively. Statistical-mechanicalarguments used in deriving eqs (12) and (13) dictate thefollowing guidelines in using the van der Waals mixingrules:(1) The van der Waals mixing rules are for constants ofan equation of state.(2) Equation (12) is a mixing rule for the molecularvolume, and eq. (13) is a mixing rule for (molecularvolume) x (molecular energy). It happens that band a of the van der Waals equation of state areproportional to molecular volume and (molecularvolume) x (molecular energy), respectively, and, asa result, these mixing rules are used in the formwhich was originally proposed by van der Waals. (3) Knowing that uii (for i j), the unlike-interactiondiameter, for spherical molecules, is equal to(16)This gives the following expression for b;; ofspherical molecules:(17)Then for non-spherical molecules the expressionfor b1; will be(18)bij (1-l,;)[ (bf;1 3 b1/ )] With the use of these guidelines, we now derive thevan der Waals mixing rules for the two representativeequations of state. A similar procedure can be used forderiving the van der Waals mixing rules for otherequations of state.33Mixing rules for the Redlich-Kwong equation of stateThe Redlich-Kwong equation of state, eq. (6), canbe written in the following form:z PVRT vv-ba- RTu (v b)"(l9)In this equation of state, b has the dimension of a molarvolume,b 0.26vc oc N0 u3 Then the mixing rule for b will be the same as that forthe first van der Waals mixing rule, eq. (3). However,the mixing rule for a will be different from the secondvan der Waals mixing rule. Parameter a appearing inthe Redlich-Kwong equation of state has the dimen sions of R- 112 (molecular energy)3 12 (molecularvolume), that is,a 1.2828RT/· 5v" oc: N0 (e/k) 1 . 5 u3 As a result, the second van der Waals mixing rule, eq.(13), cannot be used directly for the a parameter of theRedlich-Kwong equation of state. However, since( R 1 '2ab 1 '2) has the dimensions of (molecular energy)x (molecular volume), the second van der Waalsmixing rule, eq. (13), can be written for this term.Finally, the van der Waals mixing rules for theRedlich-Kwong equation of state will be in thefollowing form:a ;X ; X; a r/3 bf/3}1.5 / ("{ ,.b L I;x1x1bij.iJ tnX 1 X1 b11)1/2(20)(3)These mixing rules, when combined with theRedlich-Kwong equation of state, will constitute theRedlich-Kwong equation of state for mixtures, con sistent with the statistical-mechanical basis of the vander Waals mixing rules.Mixing rules for the Peng-Robinson equation of stateIn order to separate thermodynamic variables fromT. Y. KWAK and G. A. MANSOORIVAN DER WAALS MIXING RULES FOR CUBIC EQUATIONS OF STATE.APPLICATIONS FOR SUPERCRITICAL FLUID EXTRACTION MODELLINGChem. Eng’g Sci. 41(5): 1303-1309, 1986
Van der Waals mixing rulesconstants of the Peng-Robinson equation of state, wewill insert eqs (8) and (9) into eq (7) and we will write itin the following form:a/RT dz -Y--u-b- 2 ,/(ad/RT)(u 6) (b/u)@(21)- 6)where a a( T,) (1 rc)* and d a( T,)h-*/RT,This form of the Peng-Robinson equation of statesuggests that there exist three independent constants inthis equation of state: a, b and d. Now, following theprescribed guidelines for the van der Waals mixingrules, the mixing rules for a, b and d of thePeng-Robinson equation of state will bea 25 xixjaij1 jb 2In order to calculate solubility from eq. (29), we needto choose an expression for the fugacity coefficient.Generally, for calculation of a fugacity coefficient anequation of state with appropriate mixing rules is usedin the following expression (Prausnitz, 1969):.RTln& 2 x,xjdij(24)(1 - kij)(aiiajj)“’(25)6, (1 - lij) { (b:i’3 bjj3)/2}’(261d, (1 -n )((di” d / )/2} .(27)These mixing rules, when joinedwith thePeng-Robinsonequation of state, eq. (7), will constitute the Peng-Robinsonequation of state of amixture, consistent with the statistical-mechanicalbasis of the van der Waals mixing rules.FOREXTRACTIONSUPERCRITICAL (W*/P)(l/ 2MY’(PFt/P)(1/4,)expZ(30){(-b 2Zx,b,)/(u-6)exp(@“/RT){@* P(31)In & In (u/(u - 6)) (2zxj 6, - b)/(u - 6) - In Z a((2Zxjbi,dP(In ((u 6)/u)- b)/(b*RT’.‘))- b/(u 6)) -((2Zx,a,,)/(bRT“‘))x In ((u 6)/u).(32)with the use of the correct version of the vander Waals mixing rules, eqs (3) and (20), in theRedlich-Kwong equation of state, the fugacity coefficient will assume the following form:Now-6)) (2Xx,6, - b)/(u - 6) - In Z (a(2Exj 6, - b)/(b*RTwhere I’ is the fugacity coefficient of the condensedphase at the saturation pressure Pi”‘, and C& is thevapour phase fugacity at pressure P. Provided weassume that uys* is independent of pressure, the aboveexpression is converted to the following form: RTlnFor the Redlich-Kwong equation of state, with eqs(2) and (3) as its mixing rules (as customary in theliterature), the following expression for the fugacitycoefficient is obtained:In&i In (Y/(uFLUID(28)Y,C(dPldni)r.,.,,, - (RTIY)ldVwhere 2 Pu/RT.In the case of the van der Waals equation of state, eq.(l), with eqs (2) and (3) as the mixing rules, thefollowing expression for the fugacity coetlicient of asolute in a supercritical gas will be derived:MODELLINGA serious test of mixture equations of state is shownto be their application for the prediction of solubilityof heavy solutes in supercritical fluids (Mansoori andEly, 1985). In the present paper, we apply the van derWaals, the Redlich-Kwong and the Peng-Robinsonequations of state for supercritical fluid extraction ofsolids and study the effect of choosing different mixingrules for predicting the solubility of solids in supercritical fluids.The solubility of a condensed phase, y,, in a vapourphase at supercritical conditions can be defined as:Y2V-22Zxjaii/uRT}.with the following interaction parameters:APPLICATIONm& RT/(u-b)(l/P)expk xixjbij1 iaij I-1 jd 51305- b/(u 6)) -I.‘)) (In ((u 6)/u)((3 “2(Zxjaf , 3b , 3)/b’f2- a2j3 (Zxj 6, )/b3/*)/(bRT‘.5))x In ((u 6)/u)(33)whereThe fugacity coefficient based on the PengRobinson equation of state with eqs (2) and (3) as themixing rules (as usually derived) has the followingform (Kurnik et al., 1981):In& ((2Zxjbij-b)/b)(Z-1)sln(Z-B)-(A/2J2B)(2Xxjaii/ax {ln((Z 2J2B)/(Z-2J2B))]-(2Xx1 6,-6)/b)(34)* and B bP/RT.where A aP/R’TWith the use of the correct version of the van derWaals mixing rules, eqs (22 (24),in the Peng- PF’)/RT}.Robinson equation of state, the following expression(29)T. Y. KWAK and G. A. MANSOORIVAN DER WAALS MIXING RULES FOR CUBIC EQUATIONS OF STATE.APPLICATIONS FOR SUPERCRITICAL FLUID EXTRACTION MODELLINGChem. Eng’g Sci. 41(5): 1303-1309, 1986
T. Y. KWAK and G. A. MANSOOR 1306for the fugacity coefficient is derived:In 4, ((2Xx,&- b)/b)(Z-- (A/(2 ,/2B))l)-ln(Z-B)((2Xx,4, 2RTZx,d,,-2,/(RT)(aEc,dij dZxiuii)/J(ad))/ *x (ln ((Z - (2Z jb,(1 J2)B)l(Z-- b)/b)(1 - J2)g)))(35)wherea* a RTd-2J(adRT)A a*P/(RT)B bRT/(RT).We now utilize the above expressions for fugacitiesin order to predict the solubility of solids in supercritical gases at different temperatures and pressures,and compare the results with the experimental data.RESULTSANDDISCUSSIONIn Fig. 1, the solubility of 2,3-dimethyl naphthalene(DMN) in supercritical carbon dioxide is plotted vs.pressure at 308 K along with predictions obtainedfrom the van der Waals equation of state. According to.Ithis figure, predictions by the van der Waals equationof state will improve when eq. (3), along with thecombining rule, eq. (17), is used as the mixing rule for binstead of eq. (3.1) as customarily used. This comparison, and similar comparisons reported elsewhere(Mansoori and Ely, 1985) for other solute-solventsystems, establish the superiority of the doublesummation mixing rule, eq. (3), for b over the singlesummation expression, eq. (3.1).In Fig. 2, the same experimental solubility data as inFig. I are compared with predictions using theRedlich-Kwong equation of state. According to thisfigure, the corrected van der Waals mixing rules for theRed&h-Kwongequation of state, eqs (3) and (20), areclearly superior to the mixing rules which are customarily used, eqs (2) and (3.1), for this equation ofstate. Similar observations are made for the predictionof solubilities of other solids in supercritical fluids,which are not reported here.The Peng-Robinsonequation of state with itscustomary mixing rules, eqs (2) and (3), is widely usedfor predicting the solubility of heavy solutes in supercritical gases and for petroleum reservoir fluid-phaseequilibrium calculations (Firooxabadi et al., 1978;Katz and Firoozabadi, 1978; Kurnik et al., 1981). InFig. 3 the same experimental solubility data as in Figs 1and 2 are compared with the predictions using thePeng-Robinsonequation of state with its originalmixing rules and with its corrected van der Waalsmixing rules. According to this figure, the correctedk . -0.3537“ tkj-0“‘” 326Co2 - 23ll 0308Kk,-DMN- 0.095318 K ly -0.090328K-O.ceSKt I50100150 !%?ilJRE2002502(BAR Fig. 1. Solubility of 2, 3-dimethyl naphthaleue(DMN) insupercriticalcarbon dioxide at 308 K vs. pressure.The soliddots arc the experimental dam (Kurnik et al., 1981). Thedashed lines are the results of the van der Waals equation ofstate with eqs (2) and (3.1) as the mixing rules. The solids linesare results of the van der Waals equation of state with eqs (2)and (3) as the mixing rules and eqs (4) and (17) as thecombining rules.Fig. 2. Solubility of 2, 3-dimethyl naphthalene (DMN) insupercritical carbon dioxide at 308, 318 and 328 K vs.pressure The solid dots are the experimental data (Kumik1981). The dashed lines are the results ofet al.the Redhch-Kwong equation of state with eqs (2) and (3.1) asthe mixing rules. The solid lines are the results of theRedlich-Kwong equation of state with eqs (3) and (20) asthe mixing rules and eqs (4) and (17) as the combining rules.T. Y. KWAK and G. A. MANSOORIVAN DER WAALS MIXING RULES FOR CUBIC EQUATIONS OF STATE.APPLICATIONS FOR SUPERCRITICAL FLUID EXTRACTION MODELLINGChem. Eng’g Sci. 41(5): 1303-1309, 1986
Van der Wsals mixing ruks1307van der Waals mixing rules of the Peng-Robinsonequation of state apparently do not improve solubilitypredictions over the original mixing rules. However,variation of solubility vs. pressure for the new mixingrules is more consistent with the experimental datathan the old mixing rules. Also considering the fact thatthe new mixing rules for the Peng-Robinson equationof state contain three adjustable parameters (ku, I, andm,) whereas the old mixing rules contain only twoadjustable parameters (k,and 1, ) makes the newmixing rules more attractive. A demonstration of thesuperiority of the new mixing rules for thePeng-Robinson equation of state is shown here in Figs4-9. According to these figures, when the unlikeinteraction adjustable parameters of the mixing rulesare fitted to the experimental data, the Peng-Robinsonequation of state with the corrected van der Waalsmixing rules can predict solubility of heavy solids insupercritical fluid more accurately than with theoriginal mixing rules over different ranges of temperature and pressure and for different solutes and supercritical solvents.A :318K-:328Kjc050loo150PRESSURE2002503(BAR)Fig. 4. Volubility of 2, 3-dimethyl naphthalene (DMN)in supercritical carbon dioxide as calculated by thePeng-Robinson equationof state usingeqs (2) and (3) as themixing rules and compared with the experimentaldata.llliCC523 DMN-23DMNTm308K050loo150PRESURE200(BAR)250.2 318Kn: 328K3(3050loo1502002503Fig. 3. Solubihty of 2, 3-dimethyl naphthalene(DMN) inPRESSUREt BAR 1supercriticalcarbondioxideat 308 K vs. pressure.The dashedlinesare the resultsof the Peng-Robinson equation of stateFig. 5. Solubigty of 2, 3-dimethyl naphthalene(DMN) inwitheqs (2) and (3) as the mixingrules.The solid linesare thecarbondioxideas eakukted by the Peng-Robinson equationresultsof the Peng-Robinson equation of state with eqs (3)of state with eqs (X -o-(4) as the mixing ruks and comparedand (20) as the mixingrules.Roth sets of linesarefor k, 0.with the experimtntaldata.T. Y. KWAK and G. A. MANSOORIVAN DER WAALS MIXING RULES FOR CUBIC EQUATIONS OF STATE.APPLICATIONS FOR SUPERCRITICAL FLUID EXTRACTION MODELLINGChem. Eng’g Sci. 41(5): 1303-1309, 1986
1308T. Y. KWAK and G. A. MANSOORI.328CO29f423DMN 0-2l-,sl,050150100.:3084 -3t8-:328’KKK2002507I ij x-02420“iiIii -(X24820 308 318I :3282SDMNkS amCI3PRESSURE(SARIFig. 6. Solubihty of 2, 3-dimethyl naphthalene (DMN) insupercritical ethylene as calculated by the Pcng-Robinsonequation of state with eqs (2) and (3) as the mixing rules andcompared with the experimental data0KKKI50loo150PRESSURE200(BAR)2503(Fig 8. Solubility of 2, 6-dimethyl naphthalene (DMN) incarbon dioxide as calculated by the Peng-Robinson equationof state with eqs (2) and (3) as the mixing rules and comparedwith the experimental data.5.c328 K328 Kw-23DMNk jj o.l8 Iii -0.242‘ij 0.3117I ijSO.2723mu 06602l t3D8K4’3l8Km:328K.Ai0loo150C,: 308X: 318-K: 328’K,033(20025015020025050looPRESSURE BAR)PRESSURE (BAR,Fig. 9. Solubility of 2. 6-dimethyl naphthalene (DMN) inFig. 7. Solubihty of 2,3-dimethyl naphthalene in supercriticalas calculatedbythedioxidesupercritical carbonethylene as calculated by the Peng-Robinson equation ofPeng-Robinson equation of state with eqs (22)-(24) as thestate with eqs (22)-(24) as the mixing rules and compared withmixing rules and compared with the experimental data.the experimental data.T. Y. KWAK and G. A. MANSOORI50VAN DER WAALS MIXING RULES FOR CUBIC EQUATIONS OF STATE.APPLICATIONS FOR SUPERCRITICAL FLUID EXTRACTION MODELLINGChem. Eng’g Sci. 41(5): 1303-1309, 1986
1309Van der Weals mixing rulesAcknowledgement-Thisresearch was supportedNational Science Foundation Grant CPE 8306808.bytheNOTATIONVariableA, Bin the Peng-Robinsonequationofmiistateparameter in the equation of stateparameterin the Peng-Robinsonequationof statebinary interaction nsonequation of �snumberpressure, BarRTuniversala, hdk,lijPeng-RobinsonUletters5uinteraction energyacentric ngthSubscriptsCenergymolar volume, cm”/molgas phase mole fr
mixing rules for the Redlich-Kwong and the Peng-Robinson equations of state are tested through prediction of solubility of heavy solids in supercritical fluids. It is shown that the new mixing rules can predict supercritical solubilities more accurately than the original mixing rules of the Redlich-Kwong and Peng-Robinson equations of state.
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