Revista Mexicana de Física 37 Suplemento / (199/) S72-S8l

Phase equilibria in asymmetric mixturesLuis Ponce.Ramírez, Mariano López de Haro

Instituto de Investigaciones en MaterialesUniversidad Nacional Autónoma de México

Apartado postal 70-360, Coyoocán, O~51OMéxico, D.F., México

Carlos Lira.GaleanaDivisión de Estudios Especiales, Subdirección de Tecnología de Explotación

Instituto Mexicano del Petróleo

Abstract. The simplified version af the Perturbed lIard.Chain Theory(SPHCT) is used lo compute the phase equilibria and Jlenry's constaRtaror mixtures in which the molecules are highly asyrnmetric both in shapeand in the intermolecular potential. Thesc mixtures ¡ndude binarysyslems 5uch as CO2/hydrocarbons and hydrocarbonsjhydrocarbons,which are of particular ¡nterest for the oH ¡ndustry. For this type ofmixtures a single parameter (which is calculated from available exper-imental data of liquid-vapor equilibrium) is introduced. The approachyields results which compare well with the experimental behavior re-ported in the literature.

PAes: 64.1O.+h; 05.70.-&

1. Introduction

Qne of the aspects tha.t has received much a.ttention recently in connection with thedesign of new technologies for the oH ¡ndustry, is the development of mok'Cular mod-els to represent the thermodynamic properties and phase equilibria in asymmctricmixtures. In particular, this modeling has been prominent in the thermodynamicdescription oC sorne processes such as supercritical extraction, which is rclatcd totbe separation processes associated with the handling of heavy oils, and the intrinsicmechanisms oí enhanced oH recovery.

A rigorous description of these mixtures is difficult due to, on the one hand, theasymmetric structure oC their molecules and on the other to the huge number oCinternal and external degrecs oC íreedom and to the coupling bctwccn inter andintramolecular intera.ctions, which induce long range correlations and molecularclustering phenomena. These effects affect the coordination numbcr and bcncc alsoexert an influence on the determination oí the configurational propcrties oC thesystem.

The Perturbed !Iard.ehain Tbeory (PHTC) originally proposed by Beret andPr.usn;tz (1) and by Donohue and Prausnitz [2,31 has becn suc<csfully applied intbe prediction oC the thermodynamic properties of a wealth of syslems oí industrial¡nterest. This theory is based 00 the perturbation thcory of molceules intcractingvia a square-well potential. The reference system that ane lakes (repulsive interac-

Phase equilibrio in asymmdric miztUrt8 573

tions) is the Carnahan-Starling equation for hard spheres, while for the attractiveinteraction the expression inferred by Alder el al. [IJ from molecular dynamicsdata is chosen. In order to apply the perturbation theory to complex molecules,two additionaJ assumptions are introduced, namely one assumes a specific form forthe intcractions betwccn the segments of such molecules and secondly the deviationsfrom ideality that occur due to the interactions during the rotational and vibrationalmotions (which are density dependent) are catered for by introducing Prigogine'sparameter c. The resulting equation in the PHCT contains then, three parameters inthe case of pure components and five if mixtures are considered. Notwithstandinglhe fact lhal lhi, lheory predicl, very well lhe lhermodynamic behavior of ,m.Umolecules, high molecular weight hydrocarbons and polymers, it fiust be pointed outthat its mathematieal complexity has been a serious obstacle for wider applications.

Therefore, it is not surprising to find that io 1986 Kiffi, Vima1chand, Donohueand Sandler [41 propo,cd a Simplified Perlurbed Hard-Chain Theory (SPIICT) inwhich the attractivc part of the PHCT is replaced by a simpler but theoretieallysound expression bascd 00 the local composition model due to Lee, Lombardo andSandler [5J, The SPIICT relain, lhe ,ame cbaracleri,lics of lhe PIICT bul il has lheadvantage of its mathematical simplicity.

This paper contemplates two aspects. 00 the one hand, in the first part we uselhe SPIICT for lhe prediction of lhe vapor-liquid equilibrium of 'y,lems of lhe lypeCO2/hydrocarbons of interest in the processes of miscible injection of gases used inenhanced oil rceovery. This ineludes the generation of the characteristic parametersrequired for the use of the SPIICT cquation for more thao 70 pure compooents. Onthe other hand, in the second parl of lhe paper the SPHCT is generalized to the caseof continuous and scmicontinuous mixtures and applied to systems of techoologicaJinterest.

2. Simplified perlurbed hard-chain lheory

According lo Kim et al. [411he residualllelmhollz free energy may be expressedas the sum of repulsive and attractive contributions, namely

(1)

For Af(,P one uses the same expression as in the PHCT i.e., the Carnahan-Starlingequation in its most general form

A". 4q - 3q'--=c'----RT (I-q)'

(2)

where r¡= TV. /v is the reduced density, T = ,f2-x /6, v. is the c1ose-packing volume(v. = NAs(f3/../i),NA is Avogadro's number, s is the number of segments for agiven molecule (s = 1 for monomers), (f is the hard-core diameter of a segment ande is Prigogine's flexibilily parameter.

574 Luis Ponce-Ramirez el al.

In the case of Al.u the expression given by Kim et al. [4] is used, j.t:.

Al.uRT = -eZm In(1 + qF) (3)

where F = Y/r, Y = exp(T' /2T) - 1, T is lhe absolule lemperalure and T' isthe characteristic temperature (T. = t.qjck),t. is the characteristic energy per unitexterna! surface area of a molecule, q is the normalized surlare area per moleculeand Zm is the maximum coordination number.

The SPHCT contains three parameters for each pure component: the close--packing volume v" the characteristic temperature T. and Prigogine's c parameter.Kim el al. (4) have eslimaled values for lhese paramelers in lhe case of 19 purecomponents by adjusting vapor pressure data, and liquid density data, taking Zm= 36. Recenlly Ponee, Tapia and Lira [6J have eslimaled lhe paramelers of lhismodel for 70 pure componenls. Table I shows lhe parameler values used for lhepure components oC the mixtures discussed in this work.

The thermodynamic properties are computed in the usual way from the Helm-holtz free energy expression. For instan ce, the equation oí state turns out to be

(6A"')Z=I+q --6q T

(4 )

In the case of mixtures, Kim et al. [4] propose the following mixing rules which areakin lo lhe I-Huid lheory

(e) = L x;C;

and

(cv.Y) = L XiXjCiv;il'ijij

where Xi is the mole fraction of component i, and

( '--q- )l'ij = exp 2:k~ - 1

'fhe cross terms aij and tij are given by

<T;; = !(<T; + <T;)

and

';; = (,;,;)1/2(1 - k;;)

(5)

(6)

(7.a)

(U)

(8)

(9)

PhlUt equilibrio in lUymmetric mixtures 575

dp/p DataCompound v. 1" e % Ab•. Avg. BOurce

pI/mol) [K] Dev.

n-Alkane.Etbane 0.027603 120.500 1.25860 2.2 •Propane 0.035839 142.481 1.39133 0.2 •Butane 0.042500 154.895 1.64541 0.6 I

Pentane 0.052493 164.806 1.84808 lA •Bexane 0.081198 175.035 1.95703 1.0 •Heptane 0.069805 180.601 2.19904 0.9 •Decan. 0.097884 186.811 3.01172 1.1 •Hexadeeane 0.155357 204.084 4.08465 0.1 •Eic08ane 0.185420 209.650 4.91080 0.5 •Dotricontane 0.302092 216.487 7.38780 0.3 e

Tetracontane 0.375950 217.190 9.19000 6.0 e

lIoalkanesbobutane 0.042480 169.410 1.20356 0.6 •

Napbtbene.Cyelobexane 0.059832 177.542 2.09347 0.5 d

Aromatic.Benzene 0.044483 205.530 1.63590 1.7 •

Al.k.ilbensene.Toluene 0.046138 218.653 1.71334 1.0 •n-Butylbenzene 0.072272 217.780 2.37800 0.8 dm-Xylene 0.037240 214.391 2.30922 0.6 •

Al.k.ylnapbtbll1ene.1.Methy1naphthalene 0.062685 261.328 2.23241 0.4 e

Olelin••Etbylene 0.028646 112.050 1.20762 0.3 •PropyleDe 0.031998 139.845 1.40201 0.8 •l-Buune 0.041751 151.598 1.65693 0.4 •l-Bexene 0.058507 168.876 2.09517 0.1 •l-Repune 0.072497 1n.463 2.19248 0.3 •

Otber Huid-Carbon Dioxide 0.014600 105.612 1.88677 0.7 •B)'dro«en Sulfide 0.015547 136.106 1.56686 0.3 ,.: N.B. Vargaftik (11); 1: T.R. D•• el al. (12]; c: API Projecl 44 [13]:d: M. Hirata el eL [14]; e: J.A. Bierlein and W.B. Kay [IS]'

TABLE I. Pure-Component Parameurt (or 5PHCT Equation.

576 Luu Ponce-Ramírez el al.

Takiog ioto accouot Eq. (9) and the defioitioo of T', Eq. (7.h) may be cast io. theform

[(T'T~)I/' [e.q.] 1/' ]

Y;j = exp , 2~ ~q; (1 - kij) - l. (10)

As may be easiJy secn, in the case of mixtures, hesides the parameters of the purecomponents (c, v- , T-) one requires the knowledge of Si, q¡ and kij. For n.alkancsKim <1al. [4J ouggeot that si aod qi may be obtaioed from the olopes of the graphoof v- and cT- os. number oC cubon atoms, namely

ciT.-q¡ = 62.5

v~s. - I

0- 0.008667

(ll.a)

(Il.b)

In view of the mixing rules slated aboye, the expression for lhe residual Helmholtzfree energy may be rewritten as

A'e. 4~_ 3~'NkT = (e) (1 _ ~J2 - (e)Zm 10(1 + ~(F»

where ~ = T(v')fv, aod

( 12)

(F) n'(cv'Y) (13)= Tn(e)n(v')

is an auxiliary function that allows one to write the thermodynamic equations formixtures in a similar form than those of pure components.

Also, lhe residual chemical potential tums out to be given by

with

6(F) (F) [ 1 1 6 ('( ') ei vi ]n 6n¡ = (cv'y);; 6n¡ n cv y - (e) - (v')

where

~ 6~.(n'(cv'y» = LXj(C¡v;¡Y;j + e,vij Y,;)o j

The fugacity coefficjent of component i, ,pi, is given by

p~"In q,¡ = Nía - In Z.

(15)

( 16)

(17)

Pha8e equilibrio in a8vmmetnc mizture6 577

...........",.•• g ••••••..., ...

Xco,

FIOURJ.; 1. Comparison of vapor.liquid equilibrium data for cl'I.lbondioxide-tetraJin at 461.95,543.55, 623.35 and 664.64 K with SPHCT model. Experimental data, SebMtian,II.M. [251.

Here, ,pi = Ji/x; where Ji is the Cugacity oC component i. FinalIy, Henry's constantCorsolute 2 in solvent 1 is def1ned by

H l. h21= Im-., r,-O x2

(18)

Subsequently we will use eqs. (1) to (17) for lhe pure components and lhe mixturesto describe different phase behaviors.

3. Phase equilibrium calculations

To calcula.te vapor.liquid equi1ibria, we require Coreach component that its Cugacityis the same in both phases. Such calculations a.re conveniently perCorme<!using Kfadors (equilibrium ratios) defined as

y; <PtK¡=-=--yXi tP¡

(21)

where 4>r and 4>Y are the Cugacity coefficients oC component i in the liquid and vaporphases, respectively. lo order to compute K Cactors CoraH components, we use theSPIICT equation which requires a siogle binary interactioo parameter tij.

Substilulion of Eq. (17) inlo Eq. (21) gives

(22)

wherc the chemical potentials are calculated using Eq. (14). The superscrit L andV denote liquid phase and vapor phase. respectively.

518 Luis Ponce.Ramírezel al.

.....•... ..,. ...• 0 c ••••••. .

e ••••••

x ce,

FrOURE 2. CompatisOll or vapor.liquid equilibrium data ror earboo diaride-toluene at 393.25,422.45,476.95, S02.75 and ,)42.85 K ",¡tb SPHCT model. Experimental data, Sebastian.II.M. [23).

FlOURE 3. Compatison el vapor-liquid equilibrium data ror earbon dkJxi<k--m-xyleneal 462.15,502.05, .>43.35 a.nd 582.55 K "itb SPHCT model. Experimental data, Sebastian, R.M.[23J and Ve•• , n.J. " .1. [24J.

For bubble-point and dew-point pressure ealeulations, which are required duringthe kij estirnation proeedure, a minirnization strategy was employed. Saturationmixture properties obey the relationships

m

x(P) = In L K;x; = Oi=l

for bubble point ealculations, whilem

y'X(P) = In " -!. = O

L"Ki=l I

(23)

(24)

is used for dew.point predictions, and m stands for the number of components in amixture.

Phase equilibria in asymmetric mixtures 579

'"""""~"."""" ,

•~.FIGURE 4. Comparlson or vapor.liquid equilibrium ror mixtures or carbon dioxide with high boil-

ing n-alkanes at 423.2 K with SPHCTmode!. Experimental data, Ca.'JCm,K.A.M. andR.L. Roh;n",n (9J.

The most reliable cstimate for the binary parametcr kij is obtained from multiplcrneasurernents, usually a set of vapor-liquid equilibrium data. The optimum param.eter is that which rninimizcs the objedive funetion S according to thc algorithmsuggested by Anderson and Prausnitz [7,81

s = ~ ( ( PI ~ P;')' + (y; _ yf)') . (25)

Here, i is an index for cach datum of bubble pressure (P-y) which is subjcd loconstraints (24). Thc superscrit e denotes an cxpcrimcntally measurcd valuc, andsuperscrit e denotes the cstimated true valuc corrcsponding to each mcasurcd point.

Table 11 contains thc results for the vapor.liquid equilibriuffi for 27 binarysystems at different tempcraturcs. As may be clearly seen from this table, thctheoretical results compare weH with experimental data. For cach syslcrn wc haveincluded thc value of the parameter kij. Figs. 1 to 4 illustrate a typical result forthe bchavior obtaincd for thcsc systcms.

In arder to prediet thc valucs of thc Henry's constants, wc uscd the purecomponent parametcrs and the binary interaction parameters obtaincd from thcvapor-liquid equilibrium.

Fig. 5 shows thc IIcnry's constant prcdictions of the spnCT mo,id for carbondioxide in heavy normal alkanes as a funetion of the number of carbon atoms, at twotemperatures (373.2 and 423.2 K). The binary interaetion paramcler was ealculaledfrom the vapor.liquid equilibrium curve. Agreement w¡th the expcrimC'ntal valucsoC Gassem and Robinson [9J is c1early shown.

Figs. 6 lo 8 show the general trends of the Henry's constant valucs corr('spon<iingto mixtures of hydrocarbons (el lo C9) in ¡.methylnapthalenc, n-cicosanc and n-hexadecane at three temperatures (325,400 and 475 K). Predictions oC the SPIICTrnodel are comparcd with values estimated by Chappelow and Prausnilz [10] in therange el at C", showing a very good agrccment.

580 Duis Ponce.Ramírez ti al.

Av. Ab8. DC\I, DataSalute T[KJ dP/P dY K11 sourcen.Alkancsr¡-Etllane 21000 -1.08 1.28 0.11520 a

230.00 -0.89 0.07 0.10300 a250.00 0.54 -0.'12 0.10830 a250.00 -0.26 -0.13 010560 b27000 0.09 -0.60 0.11770 b

n-Propane 34126 -0.30 -3.42 0.11410 ,327.59 -0.'11 -1.97 0.12260 ,31u93 1.24 -0.89 0.10700 ,

n.Bulane 410.93 1.68 -3.06 000630 ,377.59 -0.79 -2.07 010300 ,344.26 0.23 -0.38 008790 ,310.93 -0.72 -034 0.08920 ,283.15 0.36 0.25 0.08830 ,273.1.\ 1.28 0.30 0.7,\500 ,

n-('ent.ane 377,59 -320 0.26 0,10920 ,34,1.1.\ -6.10 0.03 0,10,110 ,

ll-llcxarH' 313,15 2.05 -0.'10 0.06730 ,298.15 004 -0.51 0,07760 ,

n.l!eplanc 177.21 L57 -0.73 0.0,\810 ,'1- ()('canc 58365 -0.03 -7,23 0.15980 d

501295 0,00 -2.42 0.09920 d51093 1.89 -0.33 0.11200 ,<176.95 0.03 052 0.07000 d46320 0.07 0.00 0.06135 dH.1.26 1.97 1.60 0.08120 ,410.93 0.07 058 0.08110 ,377.59 -034 0.23 0.07890 ,

lI-llf'xadNalH' 623.15 0.87 -1.55 0,10880 d5.12.85 0.07 0.11 0,05370 d,16305 0.00 -0.04 004770 d

n-Eicosanc 323.20 0.74 008050 ,373,20 0,18 006180 ,

n-OclaeicO!'lanc 423.20 4.04 0,040,10 ,373.20 -0.88 006380 ,

Il-Dotriacontane 348.15 0.35 0.00 0.07660 f373.15 1.20 0,00 0,06980 f398,15 -0.44 0.00 005810 f

n-Ilexat.riacon! ane 42320 0.14 0.05330 ,373,20 1.66 0.06710 ,

n-Tdratdracontanc 373,20 2,13 0.06850 ,423.20 -0.20 0.05600 ,

NRphthcl1f!MCyclohexanc 373.20 0,46 0,10920 9

423.20 0.53 0,12950 9348.20 0.52 0,10920 9

A Iky IlIapht halellc!Il.~tethynaphthalt'ne 703.55 -0.17 -3.72 -0.06820 h

623.55 -1.08 -0.37 -0,05720 h463.05 0.62 -0.05 -0,02620 h5,13.45 -0.48 0,00 -0.04100 h

AroIHatiC:IIBcnzenc 32320 0.01870 9

313.20 0.85 0.01870 9313.20 2,30 0.01870 ,

TAHI..E IJ. Valurs or binary interadjon parameters ror earban dioxidc-hydroca.rbon mixtures.

Phase equilibria in asymmetric mixtures 581

Av. Abs. Dev. DataSolute T[K] dP/P dY K" source

AlkylbenzencsToluene 442.45 -0.57 0.58 -0.01400

542.85 1.40 -1.14 -0.04760502.75 -2.06 -0.25 -0.03050476.95 -1.12 0.58 -0.02580393.25 0.48 0.00 -0.00650

n.Butylbenzene 293.15 6.53 004610 ,273.15 5.60 0.04980 ,

m-Xylene 323.15 -6.66 -0.02 -0.01830 j343.15 -5.17 -0.30 -0.01330 j582.55 -0.30 -6.23 -0.13100 j543.35 -0.27 -3.84 -0.08670502.05 0.87 -2.17 -0.06760462.15 0.12 -2.10 -0.05180

Naphthe-aromaticaTrans-Decalin 348.20 -0.27 0.05480 9Tetralin 664.65 1.17 -6.84 0.01980 I

623.35 0.18 -3.28 -0.00320 I543.55 -0.18 -0.06 -001570 I461.95 -0.23 0.00 -0.00440 I

IsoalkanesIsobutene 273.15 1.74 3.65 0.07100 k

310.39 -2.40 -0.40 0.08860 k

OlefineaI.Butene 273.15 -0.93 -1.04 0.04,\80 k

310.85 1.35 -0.98 0.06860 m318.15 -2.61 0.77 0.08730 m

Propylene 252.95 -0.95 0.79 0.04760 k273.15 -0.45 0.37 0.07120 k283.15 -0.24 -0.49 0.06180 k298.15 1.08 -0.44 0.06800 k

l-Hexene 303.15 1.74 -0.40 0.05260 j323.15 -1.23 -0.40 0.06750 j343.15 0.61 0.65 0.06750 j

I.Heptene 303.15 3.01 0.00 0.05950 j323.15 1.28 0.09 0.07570 j343.15 0.08 0.22 0.07960 j

Other fluidaIl,S 288.15 1.28 1.03 0.06150 n

293.15 1.42 1.10 0.06190 n313.15 0.86 0.26 0.06280 n323.15 0.69 0.45 0.06160 n333.15 -2.66 -0.55 0.06430 n338.15 0.61 -0.40 0.06210 n

a: Brown, T.S. d al.(16]; b: Dával08, J. d al. [17]; e: Knapp, 11. d al. [18]; d: Sebastian, II.M. etal. [19]; e: Gasem, K.A.M. and R.L. Rabinson [9]; f: Fall, D.J. and K.O. Luks [20]; g: Anderson,J.M. d al. [21]; h: Sebastian, II.M. el al. [22]; i: Sebastian, II.M. d al. [23); j: Vera, f!.J. ti al. [24);k: Sebastian, II.M. el al. [25); 1: Nagahama, K. et al. [26]; m: Hehrens, P.K. el al. [27); n: Bierlein,l.A. " al. [15J.

TABLE II. (Cont.) Values or binary interaction parameters ror carbon dioxide-hydrocarbon mix-tures.

582 Luis Ponce-Ramírezel al.

o t •••••• lO'

• ea, ••,. ,.,.0 •• " •••• ,o'" ••••••,

FIGURE5. Th~ logarithm oC H~nry', con,ta.nt (in unita of bar) ror carbon dtoxid~ in h~avy normala.lka.ness.t 373.2 &Dd423.2 K. Experimental d&ts., G~m, K.A.M. &DdR.L. RDbin.OO" (9).

...."".o ••••••

• .0 .• ,_ •••••

o , • , • • • • • • 'o " ••CUlO" "~"'I.

FIGURE 6. Th~ logarithm of "enry', conslant (in units oCblU")of n.s.lkanes in l-methylnaphths.leneal 325, 400 and 475 K. The lIenry', (On8t&ntof n-s.lkanes (pentane to nona.n~)obtainfflin thifi work a.nd tbe lIenry'8 con8tant from metha.ne lo buta.n~ from interpolatffl Cromlhe data of Chappelow a.nd Pra.u8nitz [10J.

4. Condusions

Predictions of phase equilibria and Hcnry's constant for asyrnmetric systems wcremade by using the SPIfCT rnodel. Rcsults show that this rnodcl can be readilycmployed to predict the phase behavior of complcx systems.

D~dicatory

This paper is dcd.icated to Professor Lropoldo Carda-Colín on thc occasion of hissixticth birthday.

Phale tqUilibria in alvmmetric miztuf'es 583

I • • • I • , I • " • '.

cUlO" ••~••••••

FIOURE 7. Tbe losa.rit.hm oC Henry's const.ant. (in uniu of bar) of n-alkanes in n-eicosane at. 325,-400 and -475 K. Tbe open simbols reprnent. t.he Henry's const.ant.of n.a.lkanes (pent.aRelo nonane) obt.a.ined in t.bis work and t.be 80Iid simbols represent. t.be lIenry's const.ant.from met.hane lo but.&neint.erpola.ted from t.he dala. of Chappe10w and Prausnit.z POI .

••' "'00'

"0 I • , • I • • • • " " '.

FIGURE 8. The logarit.hm oC Henry's const.&Ilt.(in unit.s ofbar) of n-alkanes in n.hexadeeane at. 325,400 and 475 K. Tbe open llimbols represent. t.he Henry's eon6lant. of n.alkancs (pent.anelo nonane) obt.a.ined in t.bis work and t.be 80Iid simbols represcnt the IIcnry's eonstantfrom metbane lo but.aRe int.erpolaled from the data of Chappelow and Prausn:t.z [10J.

Acknowledgements

This resea.rch was aided by funds the Programa Universitario de Energía, UNAM,

México.

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584 Luis Ponce-Ramirez el al.

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Resumen. La versión simplificada de la Teoría de Perturbación decadenas duras es utilizada para el cáJculo del equilibrio entre fases yconstantes de lIenry para mezclas asimptricas. Estas mezclas incluyenlos sistemas binarios CO'2/hidrocarburos e hidrocarburos/hidrocarburosque son de particular interés en la industria del petróleo. Para la re-presentación de estas mpzdas, sólo se rp<¡uiere un parámetro que escalculado a partir de loo datos expf'rimf'ntales del l'quilibrio líquido-vapor. Los resultados obtenidos mediante esta teoría se comparansatisfactoriamente con los de la literatura.