Ah

Ka

b

c

a

ARR1A

KCEHPV

1

fflfWouFicr

copweet

s3

0d

J. of Supercritical Fluids 55 (2010) 594–602

Contents lists available at ScienceDirect

The Journal of Supercritical Fluids

journa l homepage: www.e lsev ier .com/ locate /supf lu

n isomorphic Peng–Robinson equation for phase-equilibria properties ofydrocarbon mixtures in the critical region

h.S. Abdulkadirovaa, C.J. Petersb,c, J.V. Sengersa,c,∗, M.A. Anisimova,c

Institute for Physical Science and Technology, University of Maryland, College Park, MD 20742, USAChemical Engineering Department, Petroleum Institute, P.O. Box 2533, Abu Dhabi, United Arab EmiratesDepartment of Chemical and Biomolecular Engineering, University of Maryland, College Park, MD 20742, USA

r t i c l e i n f o

rticle history:eceived 14 July 2010eceived in revised form3 September 2010

a b s t r a c t

The principle of isomorphism of critical phenomena asserts that the equation of state of fluid mixturesin the critical region has the same form as that of the pure components, provided that the mixture isnot considered at a fixed composition but at a fixed value of a constant field variable related to thechemical potentials of the components of the mixtures. This principle has been successfully applied by

ccepted 14 September 2010

eywords:ritical phenomenaquation of stateydrocarbon mixtures

various investigators to formulate nonclassical equations of state for fluid mixtures in the critical region.In this paper we show how the same principle can be applied to simple classical equations of state usingthe Peng–Robinson equation as a representative example. The isomorphic Peng–Robinson equation thusobtained is applied to represent phase equilibria properties of some binary mixtures of methane, butaneand decane.

eng–Robinson equationapor–liquid equilibria

. Introduction

After more than a century since van der Waals extended hisamous equation of state to the description of binary and ternaryuid mixtures [1,2], his ideas have remained an important guidance

or modeling vapor–liquid equilibria. Cubic and generalized van deraals equations are still widely used in practice [3–7]. An example

f a popular cubic equation is the Peng–Robinson equation [8] oftensed in industry, especially for refinery and reservoir simulations.luid phase behavior in the near-critical region is of key importancen many process technologies at high pressures, such as supercriti-al fluid extraction, exploitation of rich gas condensate/volatile oileservoirs, and gas-injected enhanced oil recovery.

However, there are two problems with the application of classi-al equations of state to characterize the thermodynamic propertiesf fluids and fluid mixtures near critical points. The first well-knownroblem is that classical equations neglect density fluctuations

hich become large near the critical point. A second and perhaps

ven more conceptual problem is encountered when such classicalquations are used for mixtures near critical points. When a mix-ure with a fixed composition is represented by a classical equation,

∗ Corresponding author at: Institute for Physical Science and Technology, Univer-ity of Maryland, College Park, MD 20742, USA. Tel.: +1 301 405 4805; fax: +1 30114 9404.

E-mail address: sengers@umd.edu (J.V. Sengers).

896-8446/$ – see front matter © 2010 Elsevier B.V. All rights reserved.oi:10.1016/j.supflu.2010.09.021

© 2010 Elsevier B.V. All rights reserved.

the critical point of the classical equation is located at the top ofthe coexistence curve implied by the classical equation, while inreality the critical point is no longer located at the top of the dew-bubble curve. As a result, the critical point of the classical equationno longer corresponds to a real critical point but becomes a pseu-docritical point. It is the latter issue that will be addressed in thepresent publication.

Van der Waals understood this problem and saw the solution inthe proper application of the equilibrium and stability conditionswhich need to be reformulated upon the increase of number ofthermodynamic variables [9]. In fact, the extension of the moderntheory of critical phenomena to mixtures is based on a principleof isomorphism of critical phenomena [10–13] which utilizes thevan der Waals formulation of stability conditions. This principleasserts that the equation of state of a mixture in the critical regionis isomorphic to that of a one-component fluid provided that themixture is not considered at a fixed composition, but at a fixedvalue of a suitable chosen field variable. For binary mixtures with asimple phase diagram in which the critical points of the two compo-nents are connected by a continuous critical locus, an appropriatefield variable � is commonly taken to be a function of the chemicalpotentials �1 and �2 of the two components [14–16]:

� = 11 + exp[(�1 − �2)/RT]

, (1)

where T is the temperature and R is the universal molar gas con-stant. The advantage of the definition of the field variable is that

Kh.S. Abdulkadirova et al. / J. of Superc

Fig. 1. Vapor–liquid phase boundaries for mixtures of methane and ethane in thetemperature–density plane as calculated from an equation of state formulated byPovodyrev et al. [17,18]. The dashed curves represent vapor–liquid coexistencecct

0c

wesrvroeobaacflccm

oflepeawir(g

sooHcso

urves at various constant values of the field variable �, including the coexistenceurves of pure methane (� = 0) and pure ethane (� = 1). The solid curve representshe dew-bubble curve of the equimolar mixture (x = 0.5).

≤ � ≤ 1, where the values � = 0 and � = 1 correspond to the pureomponents.

To illustrate the principle of isomorphism of critical behaviore show in Fig. 1 phase boundaries for mixtures of methane and

thane in the critical region calculated from a scaled equation oftate developed by Povodyrev et al. [17,18]. The dashed curves rep-esent vapor–liquid phase boundaries of the mixtures at variousalues of the field variable �. The curves with � = 0.0 and � = 1.0epresent the coexistence curves of pure methane (� = x = 0) andf pure ethane (� = x = 1), where x represents the mole fraction ofthane. The solid curve in Fig. 1 represents the phase boundaryf the equimolar mixture, (x = 0.5), commonly referred to as dew-ubble curve [16]. While the coexistence curves at constant � arell similar to the coexistence curves of the pure components (� = 0nd � = 1), the dew-bubble curve is qualitatively different from theoexistence curves of the pure components. While the classical one-uid model, which assumes that the mixture at fixed compositionan be described by an equation of state similar to that of a one-omponent fluid, breaks down in the critical region, the one-fluidodel is valid when considered at constant �.Many investigators have applied the principle of isomorphism

f critical phenomena to characterize thermodynamic properties ofuid mixtures in the critical region in the context of nonanalyticalquations of state [16,19–49]. However, the principle of isomor-hism of critical behavior should also be applicable to classicalquations of state. Hence, it is of interest to investigate whetherpplication of the isomorphism principle to simple cubic equationsill lead to an improved capability of such equations for represent-

ng the thermodynamic properties of fluid mixtures in the criticalegion. We shall do this by adopting the classical Peng–RobinsonPR) equation as an example. An earlier attempt towards a similaroal was made by Fox and Storvick [50–52].

We shall proceed as follows. In Section 2 we consider the clas-ical PR equation and the corresponding Helmholtz energy forne-component fluids made dimensionless in a form suitable forur analysis. We shall then formulate a corresponding isomorphic

elmholtz energy for fluid mixtures in Section 3. In Section 4 weonsider a so-called critical-line condition to simplify the relation-hip between the isomorphic field variable � and the concentrationf the mixture. In Section 5 we then apply the isomorphic PR equa-

ritical Fluids 55 (2010) 594–602 595

tion to some mixtures of alkanes in the critical region. Concludingremarks are presented in Section 6.

2. Peng–Robinson equation for pure fluids

The original PR equation for the pressure P has the followingform in terms of the molar density �:

P = RT�

1 − b�− a(T)�2

1 + b� + b�(1 − b�). (2)

The system-dependent parameters a(T) and b are given by [8]:

a(T) = 0.45724 [1 + k(1 − T0.5r )]

2R2T2

c

Pc, (3)

b = 0.0778RTc

Pc, (4)

k = 0.37464 + 1.54226ω − 0.26992ω2, (5)

where Tr = T/Tc is a reduced temperature, ω the acentric factor andwhere Tc and Pc are the critical temperature and pressure, respec-tively. Eq. (2) yields for the compression factor at the critical point:

Z0 = Pc

RTc�0= 0.30745. (6)

In the classical PR equation one conventionally adopts an incorrectclassical critical density �0 = Pc/RTcZ0 which differs from the actualexperimentally observed value of the critical density �c.

Expressing the critical parameters through the coefficientsa = a(Tc) and b one obtains:

Tc = 0.17015a

Rb, Pc = 0.01324

a

b2, �0 = 0.25305

1b

. (7)

All thermodynamic properties and the system-dependent param-eters can be made dimensionless with the aid of the criticalparameters [53]:

� = �

�0, T = −Tc

T, P = PTc

PcT, � = �0Tc�

PcT, b = b�0,

a = a�0

RTc. (8)

In terms of dimensionless variables, the PR equation becomes:

P(�, T) = 1Z0

[�

1 − b�− a(T)�2

1 + 2b� − b�2

], (9)

where the temperature-dependent function a is:

a(T) = −Z1T

[1 + k

(1 −

(− 1

T

)0.5)]2

(10)

with:

Z1 = 0.45724Z0

= 1.4874, (11)

and:

b = 0.0778Z0

= 0.25308. (12)

An equation of state does not specify uniquely the temperaturedependence of caloric properties like the specific heat capacities.In the theory of critical phenomena it is advantageous to considerthe Helmholtz-energy density, i.e., the molar Helmholtz energy A

divided by the molar volume V [54]. The reduced Helmholtz-energydensity à = �ATc/TPc is related to the pressure, density and chemicalpotential by:

A = −P + ��, (13)

5 uperc

a

d

w

�

ig

�

Wt

A

)

Tititi

�

Ttcb

aA

A

w

A

wpd0tfl

P

a

�

w

�

Wac

P

�

w

P

96 Kh.S. Abdulkadirova et al. / J. of S

nd it satisfies a differential relation of the form:

A = −UdT + �d�, (14)

here U = �UPc is the reduced energy density [55,56]. Since:

˜ ≡(

∂�

∂�

)T

= �

(∂�

∂�

)T

, (15)

t follows that the chemical potential � can be obtained by inte-rating the equation of state for P(�, T):

˜ =∫

1�

(∂P

∂�

)T

d�. (16)

e thus obtain for the Helmholtz-energy density associated withhe PR equation:

˜(�, T) = �

Z0

{ln

(�

1 − b�

)+ a√

2arctanh

(1 − b�√

2

)}+ ��0(T).

(17

he function �0(T) in Eq. (17) arises from the integration constantn Eq. (16). It is not specified by the PR equation, but it does yield aemperature-dependent contribution to the specific heat capacitiesndependent of the density. In practice, one represents �0(T) by aruncated Taylor series expansion around the critical temperaturen terms of �T = (T − Tc)/T:

˜ 0(T) = �00 + �0

1�T + · · ·. (18)

he first two coefficients, �00 and �0

1, in this expansion determinehe zero points of entropy and energy. Hence, they do not affect thealculation of any of the thermodynamic properties and they cane given arbitrary values for one-component fluids.

It is convenient to separate the Helmholtz-energy density intocritical contribution �A and a regular background contribution

˜reg [55]:

˜ = �A + Areg (19)

ith:

˜ reg = −P0(T) + ��0(T), (20)

here P0(T) and �0(T) represent the pressure P and the chemicalotential � at the critical density � = �0. The contribution �A isefined so as to satisfy the following two conditions: first, �A =at the critical point and, second, �−1 = (∂2�A/∂�2)T , since � is

he physical quantity directly related to the strength of the criticaluctuations [54]. From Eqs. (9), (16) and (17) we find:

˜0(T) = 1Z0

[1

1 − b− a(T)

1 + 2b − b2

], (21)

nd:

˜ 0(T) = ��0(T) + �0(T) (22)

ith:

�0(T) = P0(T) + 1Z0

[− ln(1 − b) + a(T)√

2barctanh

(1 − b√

2

)].(23)

e note that P(�, T), P0(T) and ��0(T) only depend on temper-ture through a(T). If we expand P0(T) and ��0(T) around theritical temperature in terms of �T we obtain:

˜0(T) = P00 + P01�T, (24)

�0(T) = �00 + �01�T, (25)

ith:

˜00 = 1, (26)

ritical Fluids 55 (2010) 594–602

P01 = Z1(1 + k)Z0(1 + 2b − b2)

, (27)

�00 = 1 + 1Z0

[− ln(1 − b) + Z1√

2barctanh

(1 − b√

2

)], (28)

�01 = P01 − Z1(1 + k)

Z0

√2b

arctanh

(1 − b√

2

). (29)

The resulting expression for the critical part �A of thePeng–Robinson equation is then readily obtained as:

�A(��, �T) = A(�, T) − ��0(T) + P0(T) (30)

as a function of �� = � − 1 and �T = T + 1. �A is the critical contri-bution to the Helmholtz-energy density. It is this part which needsto be renormalized if one wants to incorporate the effects of thecritical fluctuations [55,57]. In the classical approximation adoptedhere the compact expression (17) for A and the decomposed expres-sion (19) for A remain equivalent.

3. Isomorphic Peng–Robinson equation for fluid mixtures

For mixtures the critical parameters Pc, Tc and �c depend on thecomposition. To deal with this additional complication we find itconvenient to adopt for mixtures a modified set of thermodynamicvariables [14,19]:

T = 1RT

, P = P

RT, �1 = �1

RT, �2 = �2

RT, A = �A

RT, U = �U.

(31)

The choice of �1 and �2 as the field variables is not convenientin practice, since �1 or �2 diverge in the pure component limits.Instead one defines two fields, h and �, that are related to the activ-ities e�1 and e�2 by:

h = ln(e� + e�2 ), � = 11 + e(�1−�2) , (32)

so that:

�1 = h + ln(1 − �), �2 = h + ln �. (33)

The pressure P considered as a function of T , h, and � satisfies thedifferential relation:

dP = −UdT + �dh + wd� (34)

with:

w = �(x − �)�(1 − �)

, (35)

where x is the mole fraction of component 2, so that �x is the partialdensity of component 2 (solute). A modified isomorphic Helmholtz-energy density Â, defined as:

Aiso(T , �, �) = �h − P, (36)

which satisfies the differential relation:

dAiso = UdT + hd� − wd� (37)

is more useful, since it has a measurable property (overall molardensity �) as a canonical variable. On comparing Eq. (37) with Eq.(14) we see that for the mixture h is the ordering field, like � for theone-component fluid, which is conjugate to the density as the orderparameter. The isomorphic field variable � is often called the hidden

field (i.e., irrelevant field) and w is the density variable conjugateto �. The mole fraction x of the solute is related to � by:

x = � − �(1 − �)�

(∂Aiso

∂�

)T ,�

. (38)

uperc

Tc

�

T�c(

A

waasw

k

IcifpbPpactmxp

(

d

w

w

Wa

4

yotsdcvne

P

Kh.S. Abdulkadirova et al. / J. of S

o implement the principle of isomorphism of critical behavior weonsider Aiso as a function of:

T =[

T − Tc(�)T

], �� = [� − �0(�)]

�0(�). (39)

he isomorphism principle states that for a mixture at constant, Âiso has the same dependence on �T and �� as A of a one-omponent fluid. Since in the one-component limit Aiso → A =P/RTc)A, we may write:

ˆ iso(�T, ��, �) = Pc

RTc(�)A(�T, ��, �), (40)

here à is again given by the Peng–Robinson Eq. (17). First ofll, Âiso depends on � through the critical parameters Pc(�), Tc(�)nd �c(�). Furthermore, A(�T, �, �) depends on � through theystem-dependent parameter k(�) in the definition (10) of a(T) forhich we adopt a quadratic mixing rule:

(�) = k1(1 − �) + k2� + kmix�(1 − �). (41)

n Eq. (41) k1 and k2 represent the values of k for the two pureomponents through the values ω1 and ω2 of the two componentsn Eq. (5), while kmix will be treated as an adjustable coefficientor the mixtures. With Eq. (10) the � dependence of the criticalarameters and of the coefficients in the expansion (18) for theackground contribution a(T), as well as that of the coefficients

˜01 and �01 in Eqs. (24) and (25), is now completely specified. Arocedure for handling the � dependence of the critical parametersnd of the coefficients in the expansion (18) for the backgroundontribution �0(T) to the chemical potential will be presented inhe subsequent section. The variable � varies from 0 to 1 when the

ole fraction x varies from 0 to 1. However, unlike the mole fraction, the field variable � will have the same value in the two coexistinghases.

The isomorphic Helmholtz-energy density Âiso, as defined by Eq.40), satisfies the differential relation:

Aiso = Ud�T + h�c(�)d�� − wd� (42)

ith:

˜ =(

∂Aiso

∂�

)�T,��

. (43)

e note that (∂Aiso/∂�)T ,� in Eq. (38) and (∂Aiso/∂�)�T,�� in Eq. (43)re related by:(

∂Aiso

∂�

)T ,�

=(

∂Aiso

∂�

)�T,��

− 1T

dTc(�)d�

(∂Aiso

∂�T

)��,�

− �

�20(�)

d�c

d�

(∂Aiso

∂��

)�T,�

. (44)

. Relationship between mole fraction x and field variable �

The isomorphic Peng–Robinson equation, defined by Eq. (40),ields the thermodynamic properties of mixtures as a functionf �T, ��, and�. In order to represent experimentally measuredhermodynamic-property data, we need to evaluate the relation-hip between the mole fraction x and the field variable �. Eq. (32)oes not specify this relationship uniquely, since it depends on thehoices made for the zero points of energy and entropy through the

alues of the coefficients �0

0 and �01 in Eq. (18) for the two compo-

ents [14]. It is convenient to specify this transformation so that �quals x on the critical locus:

c(�) = Pc(x), Tc(�) = Tc(x), �0(�) = �0(x). (45)

ritical Fluids 55 (2010) 594–602 597

Condition (45) is commonly referred to as “critical line condi-tion” (CLC) [15–17,58–60]. From Eq. (38) we conclude that the CLCrequires that at the critical locus:(

∂Aiso

∂�

)T ,�

=(

∂Areg

∂�

)T ,�

= 0, (46)

where:

Areg = Pc(�)RTc(�)

Areg (47)

with Ãreg given by Eq. (20). On the critical locus �A and its firstderivatives are zero, so that only the regular background contri-bution to the Helmholtz-energy density appears in the conditiongiven by Eq. (46). We note again that the coefficients in the expan-sions (18), (24), and (25) for P0(T) and �0(T) in Eq. (20) depend onk and, hence, on �, except for P00 and �00. To formulate the CLCexplicitly, we need to substitute the �-dependent expression (20)for Areg into Eqs. (47) and (46) and use Eq. (44). As earlier shown byPovodyrev et al. [19], condition (46) is satisfied if we demand that:

d�00

d�= − 1

Tc(�)dTc

d�[−P01(�) + �01(�) + �0

1(�)] + RTc(�)Pc(�)

d

d�

(Pc

RTc

)(48)

The � dependence of P01(�) and �01(�) is specified through Eqs. (27),(29) and (41). In addition we try to specify �0

0(�) and �01(�) so as to

satisfy Eq. (48). As mentioned in Section 2, the coefficients �00(�) and

�01(�) can be assigned arbitrary values for � = 0 and � = 1; however,

for intermediate � values �00(�) and �0

1(�) are no longer arbitrary,since they affect such quantities as the enthalpy of mixing [58,60].There are a number of ways to implement the CLC. The simplestway is to select �0

0(�) and �01(�) such that:

�01(�) = P01(�) − �01(�) (49)

and:

d�00(�)

d�= RTc(�)

Pc(�)d

d�

(Pc(�)

RTc(�)

). (50)

We refer to Eqs. (49) and (50) as CLC1, which is equivalent to theassumption that the internal energy U can be taken to be zero every-where on the critical locus [58]. CLC1 has the advantage that theisomorphic Peng–Robinson equation of state (IPREOS) will containonly one (adjustable) mixing parameter, namely kmix in Eq. (41).This simple one-parameter version of the IPREOS turns out to givea realistic representation of dew-bubble curves in a P–T diagramas shown in the Appendix. However, to retain more flexibility, inparticular to represent vapor–liquid curves in a P–x diagram, weretain the complete Eq. (48) and satisfy it for d�0

0(�)/d� by adoptinga quadratic interpolating formula for �0

1(�):

�01(�) = �0

1,1(1 − �) + �01,2� + �0

1,mix�(1 − �), (51)

using �01,1, �0

1,2and�01,mix as adjustable coefficients for the mix-

ture in addition to kmix. We refer to this more flexible version of theCLC as CLC2. We remark that for the calculation of pressure, spe-cific heat capacities and speed of sound, one does not need �0(�),but only its first and second derivatives with respect to � which

can be readily calculated from Eq. (48). However, if we were tocalculate excess enthalpies we would need �0(�) to be obtainedby integrating Eq. (48) [19,40]. Having satisfied the CLC, we canreplace dPc(�)/d�, dTc(�)/d�, and d�0(�)/d�, by dPc(x)/dx, dTc(x)/dx,and d�0(x)/dx, respectively.

598 Kh.S. Abdulkadirova et al. / J. of Supercritical Fluids 55 (2010) 594–602

Table 1System-dependent parameters (critical temperature, Tc; critical pressure, Pc; criticaldensity, �c; molecular weight, M; acentric factor, ω) for the pure fluids.

Methanea Butaneb Decanec

Tc (K) 190.564 425.20 617.6594Pc (MPa) 4.5992 3.796 2.120�c (kg/m3) 162.66 227.85 235.9635M (g/mol) 16.0428 58.125 142.489ω 0.008 0.193 0.489

a Refs. [61–63].b Refs. [63,64].c Ref. [65].

Table 2Coefficients Tj and Pj in Eqs. (52) and (53) for the critical parameters ofmethane + butane.

Tj (K) Pj (MPa mol/kJ)

T1 = 320.211 P1 = 0.04449T2 = −296.156 P2 = −0.18963T = 847.537 P = 0.36798

5

tapiatolateoi

tfflTt

5

o

T

wc(apo

0.0 0.2 0.4 0.6 0.8 1.0

0.001

0.002

0.003

0.004

0.005

0.006

0.0 0.2 0.4 0.6 0.8 1.0

200

250

300

350

400

450a

b

P c / (

RT c)

, MP

a m

ol /

kJ

x, mol.fr. n-butane

T c , K

3 3

T4 = −1388.105 P4 = −0.34442T5 = 722.760 P5 = 0.12188

. Applications

As an illustration we apply the isomorphic Peng–Robinson equa-ion of state (IPREOS) to represent experimental vapor–liquid P–Tnd P–x data for mixtures of methane, butane, and decane. In therevious section we considered two versions of an IPREOS, depend-

ng on the actual implementation of the critical line condition tossure that � equals x on the critical locus: one simple version ofhe IPREOS with CLC1, given by Eqs. (49) and (50), containing onlyne adjustable parameter kmix and a more general version with theess restrictive CLC2, given by Eqs. (48) and (51) containing threedditional adjustable parameters through Eq. (51). We prefer to usehe IPREOS with CLC2, since it yields a better representation of thexperimental data over a wider range of temperatures. The resultsbtained by adopting the simple IPREOS with CLC1 are presentedn the Appendix.

To implement the IPREOS in practice, we need equations forhe critical temperature Tc(x) and the critical pressure Pc(x) as aunction of the composition. Like in the case of one-componentuids, the critical density is then taken as �0(x) = Z−1

0 Pc(x)/RTc(x).he information needed to specify the Peng–Robinson equation forhe pure fluids is presented in Table 1.

.1. Methane + butane

In practice we represent the critical temperature and pressuref this mixture by polynomials of the form [17,19]:

c(x) = Tc1(1 − x) + Tc2x + (T1 + T2x + T3x2 + T4x3 + T5x4)x(1 − x),

(52)

Pc(x)RTc(x)

= Pc1

RTc1(1 − x) + Pc2

RTc2x + (P1 + P2x + P3x2 + P4x3

+ P5x4)x(1 − x), (53)

here x is the mole fraction of butane and where Tci and Pci are the

ritical temperatures and pressures of methane (i = 1) and of butanei = 2). We have determined the coefficients Tj and Pj (j = 1–5) fromfit of Eqs. (52) and (53) to experimental data for the critical tem-eratures and pressures for the mixture [66,67]. The values thusbtained are presented in Table 2. A comparison of the values calcu-

Fig. 2. Tc(x) and Pc(x)/RTc(x) for mixtures of methane + butane as a function of themole fraction x of butane. The curves represent the values calculated from Eqs. (52)and (53). The symbols indicate experimental data obtained by Sage et al. [66].

lated from Eqs. (52) and (53) with the experimental values reportedby Sage et al. [66] is shown in Fig. 2.

In applying the IPREOS with CLC2 to mixtures of methane (C1)and butane (C4) we adopted the following values for the coefficientsin Eqs. (41) and (51):

kmix = −1.7, �01,1 = 14.30, �0

1,2 = 12.828, �mix = −5.045,

(54)

deduced from a fit to experimental P–x data measured by Sage etal. [66]. In Fig. 3 we show P–x curves at the temperatures corre-sponding to the experimental data of Sage et al. In Fig. 4 we showP–T curves for three selected mole fractions x of butane. Note that,in order to make a comparison of the P–T curves with experimentalvalues, we had to extract such data from the experimental P–x data.We conclude that the IPREOS yields a reasonable representation ofthe phase-equilibria data except for a range corresponding to smallP and small x.

5.2. Methane + decane

The system methane + decane exhibits a phase diagram desig-nated as Type V in the classification of Scott and Van Konynenburg[68,69], so that the vapor–liquid critical locus does not exist for allvalues of the mole fraction x of decane. The vapor–liquid criticallocus corresponds to [67,70] (Fig. 5).

x ≥ xmin = 0.095, (55)

so our calculations are also limited to this concentration range.For a description of the critical temperature and pressure we haveadopted the following equations:

Tc(x) = (Tmin − Tc2xmin)(1 − x)(1 − xmin)

+ Tc2x + T1(1 − x)(1 + t4x)

exp[− t2

(x − xmin)t3

](56)

Pc(x)RTc(x)

=[

Pmin(x)RTmin(x)

− Pc2(x)RTc2(x)

xmin

](1 − x)

(1 − xmin)+ Pc2(x)

RTc2(x)x

+ P1(1 − x)(1 + p4x) exp[

p2

(x − xmin)p3

], (57)

Kh.S. Abdulkadirova et al. / J. of Supercritical Fluids 55 (2010) 594–602 599

0.0 0.2 0.4 0.6 0.8 1.0

0

2

4

6

8

10

12

14

394.26 K

360.93 K

327.59 K

P, M

Pa

x, mol. fr. n-butane

FtIa

wdtx[ve

Fbve

Table 3Values of the coefficients in Eqs. (56) and (57) for the critical parameters ofmethane + butane.

T1 = 825.89 (K)P1 = −0.03905 (MPa mol/kJ)t2 = p2 = 0.70t3 = p3 = 0.30t4 = p4 = −0.27313

0.0 0.2 0.4 0.6 0.8 1.0

0.000

0.005

0.010

0.015

0.0200.0 0.2 0.4 0.6 0.8 1.0

300

400

500

600

700

P c / (R

T c), M

Pa

mol

/ kJ

x, mol.fr. n-decane

Reamer et al. Lin et al.

T c , K

ig. 3. Vapor–liquid phase boundaries in terms of P versus x at constant T for mix-ures of methane and butane. The curves represent values calculated from thePREOS (with CLC2). The symbols indicate experimental data obtained by Sage etl. [66].

here Tc2 and Pc2 are the critical temperature and pressure ofecane, while Tmin = 277.6 K and Pmin = 36.13 MP are the criticalemperature and pressure corresponding to the cutoff composition

= xmin = 0.095. The experimental values reported by Reamer et al.

70] correspond to mole fractions below x = 0.3. To obtain additionalalues for the critical parameters we also extrapolated near-criticalxperimental data reported by Lin et al. [71]. The values obtained

300 350 400

0

2

4

6

8

10

12

14

0.9 C4

0.5 C4

0.2 C4

P, M

Pa

T, K

ig. 4. Pressure temperature dew-bubble curves for mixtures of methane andutane. The curves represent values calculated from the IPREOS (with CLC2) atarious values of the mole fraction x of butane (C4). The symbols indicate valuesxtracted from the experimental P–x data obtained by Sage et al. [66].

Fig. 5. Tc(x) and Pc(x)/RTc(x) for mixtures of methane and decane as a function ofthe mole fraction x of decane. The curves represent values calculated from Eqs. (56)and (57). The symbols indicate experimental values [70,71].

for the coefficients in Eqs. (56) and (57) for Tc(x) and Pc(x)/RTc(x)are presented in Table 3. The actual values of Tc(x) and Pc(x)/RTc(x)are shown in Fig. 5.

It turns out that the experimental P–T data reported by Reameret al. [70] and by Lin et al. [71] are not consistent with each other.Specifically, the mole fractions of methane obtained by Lin et al. inthe liquid phase differ by about 4% from those obtained by Reameret al. [70]. From a comparison of the two data sets we have con-cluded that the bubble pressures reported by Lin et al. [71] are toolow compared to those reported by Reamer et al. [70]. Neverthe-less we are able to predict the phase-equilibria data for mixtures ofmethane and decane, at least qualitatively, in terms of one mixingparameter kmix = −0.97 as shown in the Appendix. Here we showin Fig. 6 P–x curves calculated from the IPREOS with CLC2 adoptingthe following values for the coefficients in Eqs. (41) and (51):

kmix=0.96, �1,1 = −45.08, �1,2= − 17.96, �mix = 17.91,

(58)

It should be noted that the values for the critical parameters ofthis mixture are not very accurate, which affects the predictivecapability of the IPREOS.

6. Discussion

The principle of isomorphic thermodynamic behavior of mix-tures asserts that mixtures including the critical region can be

described by a one-fluid model provided that the one-fluid model istaken at constant chemical potentials and not at constant composi-tions. In this paper we have shown how one can apply this principlein terms of the Peng–Robinson equation of state. For an accuraterepresentation of the thermodynamic properties of the mixture

600 Kh.S. Abdulkadirova et al. / J. of Supercritical Fluids 55 (2010) 594–602

0.0 0.2 0.4 0.6 0.8 1.0

0

5

10

15

20

25

542 K

511 K

477 K

P, M

Pa

x, mol. fr. n-decane

Fmco

ntcpflai

A

wTD

AP

PgoowEti�toataf

180 210 240 270 300 330 360 390 420 450

0

2

4

6

8

10

12

14 0.2 C4

0.5 C4

0.8 C4P, M

Pa

T, K

Fig. A1. Pressure–temperature dew-bubble curves for mixtures of methane andbutane. The solid curves represent values calculated from the simple version ofthe IPREOS with kmix = −1.7 for various values of the mole fraction of butane (C4).

diagram, we prefer to use the more general version of the IPREOSadopted in this paper.

450

500

550

600

650

0 5 10 15 20 25 30

0.9 C10

0.8 C10

0.5 C10

0.2 C10

P, MPa

T, K

ig. 6. Vapor–liquid phase boundaries in terms of P versus x at constant T forixtures of methane and decane (x > xmin = 0.095). The curves represent values

alculated from the IPREOS (with CLC2). The symbols indicate experimental databtained by Reamer et al. [70] at T = 477 K and T = 511 K and Lin et al. [71] at T = 542 K.

ear the critical point, one would need an additional transforma-ion so as to incorporate asymptotic scaling laws with nonclassicalritical exponents [36,57,72–75]. However, even by applying therinciple of isomorphic thermodynamic behavior based on a one-uid model field to a classical equation of state, one is able to getlready a reasonable description of phase-equilibria properties, asllustrated for mixtures of methane, butane, and decane.

cknowledgments

The research of Khapissat S. Abdulkadirova was performedhile she was a postdoctoral fellow at the Laboratory of Applied

hermodynamics and Phase Equilibria of the Technical Universityelft in The Netherlands.

ppendix A. A simpler version of the isomorphiceng–Robinson equation

As was shown in Section 4, a simple version of the isomorphiceng–Robinson equation of state is obtained by adopting CLC1, asiven by Eqs. (49) and (50). The advantage of this simple versionf the IPREOS is that it specifies an equation for mixtures in termsf only one mixing parameter, namely kmix in Eq. (41). In practicee have used a more general IPREOS by adopting CLC2, as given by

qs. (48) and (51). This more general IPREOS contains three addi-ional adjustable parameters, namely �0

1,1, �01,2 and �mix in Eq. (51),

n part to compensate for incorrect values of the critical density0(x) implied by the Peng–Robinson equation and its effect upon

he calculation of phase boundaries in a P–x diagram. However,

ne may not have sufficient experimental data to determine fourdjustable coefficients for the mixtures. In that case one may wanto use the simpler version of the IPREOS. Hence, it is interesting tolso consider results obtained with this simple IPREOS, in particularor dew-bubble curves in a P–T diagram.

The dashed curve shows the critical locus. The symbols indicate experimental dataobtained by Sage et al. [66].

In Fig. A1 we present a comparison between the dew-bubblecurves for mixtures of methane and butane, calculated from thesimple version of the IPREOS with kmix = −1.7, and the experimen-tal data obtained by Sage et al. [66]. For comparison we also showthe dew-bubble curves calculated from the simple version of theIPREOS without using any mixing parameter i.e., kmix = 0. In Fig. A2we present a comparison between the dew bubble curves calcu-lated from the simplified version of the IPREOS with kmix = −0.97and experimental data obtained by Reamer et al. [70] and Lin et al.[71]. We conclude that the simplified IPREOS gives a realistic esti-mate of the pressure–temperature dew-bubble curves. However,for a calculation of the vapor–liquid phase boundaries in the P–x

Fig. A2. Pressure–temperature dew-bubble curves for mixtures of methane anddecane. The curves represent values calculated from the simple version of the IPREOSwith kmix = −0.97 for various values of the mole fraction of decane (C10). The dashedcurve shows the critical locus. The symbols indicate experimental data obtained byReamer et al. [70].

uperc

R

[

[

[

[

[

[

[

[

[

[

[

[

[

[

[

[

[

[

[

[

[

[

[

[

[

[

[

[

[

[

[

[

[

[[

[

[

[

[

[

[

[

[

[

[

[

[

[

[

[

[

[

[

[

Kh.S. Abdulkadirova et al. / J. of S

eferences

[1] A. Ya Kipnis, B.E. Yavelov, J.S. Rowlinson, Van der Waals and Molecular Science,Clarendon Press, Oxford, 1996.

[2] Johanna Levelt Sengers, How Fluids Unmix: Discoveries by the School of vander Waals and Kamerlingh Onnes, Edita KNAW, Amsterdam, 2002.

[3] A. Anderko, Equation of state methods for the modeling of phase equilibria,Fluid Phase Equilibria 61 (1990) 145–225.

[4] H. Orbey, S.I. Sandler, Modeling Vapor–Liquid Equilibria, Cambridge UniversityPress, 1998.

[5] A. Anderko, Cubic and generalized van der Waals equations, in: J.V. Sengers, R.F.Kayser, C.J. Peters, H.J. White Jr. (Eds.), Equations of State for Fluids and FluidMixtures, IUPAC Experimental Thermodynamics, vol. V, Elsevier, Amsterdam,2000, pp. 75–126.

[6] J.O. Valderrama, The state of the cubic equations of state, Industrial & Engineer-ing Chemistry Research 42 (2003) 1603–1618.

[7] I.G. Economou, Cubic and generalized van der Waals equations of state, in: A.H.Goodwin, J.V. Sengers, C.J. Peters (Eds.), Applied Thermodynamics of Fluids,IUPAC Experimental Thermodynamics, vol. VIII, Royal Society of Chemistry,London, 2010 (Chapter 4).

[8] D.-Y. Peng, D.B. Robinson, A new two-constant equation of state, Industrial &Engineering Chemistry Fundamentals 15 (1976) 59–64.

[9] J.D. van der Waals, Ph. Kohnstamm, Lehrbuch der Thermodynamik II, JohannAmbrosius Barth, Leipzig, 1912.

10] R.B. Griffiths, J.C. Wheeler, Critical points in multicomponent systems, PhysicalReview A 2 (1970) 1047–1064.

11] W.F. Saam, Thermodynamics of binary systems near the liquid–gas criticalpoint, Physical Review A 2 (1970) 1461–1466.

12] M.A. Anisimov, A.V. Voronel, E.E. Gorodetskii, Isomorphism of critical phenom-ena, Soviet Physics JETP 33 (1971) 605–612.

13] M.A. Anisimov, E.E. Gorodetskii, V.D. Kulikov, J.V. Sengers, Crossover betweenvapor–liquid and consolute critical phenomena, Physical Review E 51 (1995)1199–1215.

14] S.S. Leung, R.B. Griffiths, Thermodynamic properties near the liquid–vapor crit-ical line in mixtures of He3 and He4, Physical Review A 8 (1973) 2760–2783.

15] M.R. Moldover, J.S. Gallagher, Critical points of mixtures: an analogy with purefluids, AIChE Journal 24 (1978) 267–278.

16] J.C. Rainwater, Vapor–liquid equilibrium and the modified Leung–Griffithsmodel, in: T.J Bruno, J.F. Ely (Eds.), Supercritical Fluid Technology, CRC Press,Boca Raton, FL, 1991, pp. 57–162.

17] A.A. Povodyrev, G.X. Jin, S.B. Kiselev, J.V. Sengers, Crossover equation of statefor the thermodynamic properties of mixtures of methane and ethane in thecritical region, International Journal of Thermophysics 17 (1996) 909–944.

18] A. Kostrowicka Wyczalkowska, M.A. Anisimov, J.V. Sengers, Y.C. Kim, Impurityeffects on the two-phase isochoric hear capacity of fluids near the critical point,Journal of Chemical Physics 116 (2002) 4202–4211.

19] G.X. Jin, S. Tang, J.V. Sengers, Global thermodynamic behavior of fluid mixturesin the critical region, Physical Review E 47 (1993) 388–402.

20] J.C. Rainwater, F.R. Williamson, Vapor–liquid equilibrium of near-critical binaryalkane mixtures, International Journal of Thermophysics 7 (1986) 65–74.

21] J.C. Rainwater, J.J. Lynch, The modified Leung–Griffiths model for vapor–liquidequilibria application to polar fluid mixtures, Fluid Phase Equilibria 51 (1989)91–101.

22] J.C. Rainwater, Asymptotic expansions for constant-composition dew-bubblecurves near the critical locus, International Journal of Thermophysics 10 (1989)357–368.

23] V.G. Niesen, J.C. Rainwater, Critical locus, (vapor + liquid) equilibria, and coex-isting densities of (carbon dioxide + propane) at temperatures from 311 K to361 K, Journal of Chemical Thermodynamics 22 (1990) 777–795.

24] J.J. Lynch, J.C. Rainwater, The modified Leung–Griffiths model of vapor–liquidequilibrium: extended scaling and binary mixtures of dissimilar fluids, FluidPhase Equilibria 75 (1992) 23–37.

25] J.C. Rainwater, D.G. Friend, Calculation of enthalpy and entropy differences ofnear-critical binary mixtures with the modified Leung–Griffiths model, Journalof Chemical Physics 98 (1993) 2298–2307.

26] D.H. Smith, J.J. Lynch, Modeling of fluid phase equilibrium of multicomponenthydrocarbon mixtures in the critical region, Fluid Phase Equilibria 98 (1994)35–48.

27] J.J. Lynch, J.C. Rainwater, L.J. Van Poolen, D.H. Smith, Prediction of fluidphase equilibrium of ternary mixtures in the critical region and the mod-ified Leung–Griffiths theory, Journal of Chemical Physics 96 (1997) 2253–2260.

28] M.Yu. Belyakov, S.B. Kiselev, J.C. Rainwater, Crossover Leung–Griffiths modeland the phase behavior of aqueous ionic solutions, Journal of Chemical Physics107 (1997) 3085–3097.

29] S.B. Kiselev, Prediction of the thermodynamic properties and the phase behav-ior of binary mixtures in the extended critical region, Fluid Phase Equilibria 128(1997) 1–28.

30] S.B. Kiselev, J.C. Rainwater, Extended law of corresponding states and thermo-dynamic properties of binary mixtures in and beyond the critical region, Fluid

Phase Equilibria 141 (1997) 129–154.

31] L. Lue, J.M. Prausnitz, Thermodynamics of fluid mixtures near to and far fromthe critical region, AIChE Journal 44 (1998) 1455–1466.

32] S.B. Kiselev, J.C. Rainwater, Enthalpies, excess volumes, and specific heats ofcritical and supercritical binary mixtures, Journal of Chemical Physics 109(1998) 643–657.

[

[

ritical Fluids 55 (2010) 594–602 601

33] S.B. Kiselev, M.Y. Belyakov, J.C. Rainwater, Crossover Leung–Griffiths model andthe phase behavior of binary mixtures with and without chemical reaction,Fluid Phase Equilibria 150–151 (1998) 439–449.

34] S.B. Kiselev, J.C. Rainwater, M.L. Huber, Binary mixtures in and beyond the crit-ical region: thermodynamic properties, Fluid Phase Equilibria 150–151 (1998)469–478.

35] S.B. Kiselev, M.L. Huber, Thermodynamic properties of R32 + R134a andR125 + R32 mixtures in and beyond the critical region, International Journalof Refrigeration 21 (1998) 64–76.

36] S.B. Kiselev, D.G. Friend, Cubic crossover equation of state for mixtures, FluidPhase Equilibria 162 (1999) 51–82.

37] J. Jiang, J.M. Prausnitz, Critical temperatures and pressures for hydrocarbonmixtures from an equation of state with renormalization-group corrections,Fluid Phase Equilibria 169 (2000) 127–147.

38] S.B. Kiselev, J.F. Ely, Simplified crossover SAFT equation of state for pure fluidsand fluid mixtures, Fluid Phase Equilibria 174 (2000) 93–113.

39] J. Jiang, J.M. Prausnitz, Phase equilibria for chain-fluid mixtures near to and farfrom the critical region, AIChE Journal 46 (2000) 2525–2536.

40] Kh.S. Abdulkadirova, A. Kostrowicka Wyczalkowska, M.A. Anisimov, J.V. Sen-gers, Thermodynamic properties of mixtures of H2O and D2O in the criticalregion, Journal of Chemical Physics 116 (2002) 4597–4610.

41] S.B. Kiselev, J.F. Ely, Generalized corresponding states model for bulk and inter-facial properties in pure fluids and fluid mixtures, Journal of Chemical Physics119 (2003) 8645–8662.

42] J. Cai, J.M. Prausnitz, Thermodynamics for fluid mixtures near to and far fromthe vapor–liquid critical point, Fluid Phase Equilibria 219 (2004) 205–217.

43] F. Llovell, J. Pàmies, L.F. Vega, Journal of Chemical Physics 121 (2004) 10715.44] L. Sun, H. Zhao, S.B. Kiselev, C. McCabe, Predicting mixture phase equilibria and

critical behavior using the SAFT-VRX approach, Journal of Physical ChemistryB 109 (2005) 9047–9058.

45] J. Mi, C. Zhong, Y.-G. Li, Renormailzation group theory for fluids including crit-ical region. II. Binary mixtures, Chemical Physics 312 (2005) 31–38.

46] J. Mi, C. Zhong, Y.-G. Li, Y. Tang, Predicting of global VLE for mixtures withimproved renormalization group theory, AIChE Journal 52 (2006) 342–353.

47] F. Llovell, L.F. Vega, Global fluid phase equilibria and critical phenomena ofselected mixtures using the crossover soft-SAFT equation, Journal of PhysicalChemistry B 110 (2006) 1350–1362.

48] S.B. Kiselev, J.F. Ely, S.P. Tan, H. Adidharma, M. Radosz, HRX-SAFT equationof state for fluid mixtures: application to binary mixtures of carbon dioxide,water, and methanol, Industrial & Engineering Chemistry Research 45 (2006)3981–3990.

49] A. Belkadi, M.K. Hadj-Kali, F. Llovell, V. Gerbaud, L.F. Vega, Soft-SAFT modelingof vapor–liquid equilibria of nitriles and their mixtures, Fluid Phase Equilibria289 (2010) 191.

50] J.R. Fox, Development of a field-space corresponding-states method for fluidsand fluid mixtures, Fluid Phase Equilibria 37 (1987) 123–140.

51] J.R. Fox, T.S. Storvick, A field-space conformal solution method, InternationalJournal of Thermophysics 11 (1990) 49–59.

52] T.S. Storvick, J.R. Fox, A field-space conformal-solution method: binaryvapor–liquid phase behavior, International Journal of Thermophysics 11 (1990)61–72.

53] J.V. Sengers, J.M.H. Levelt Sengers, Thermodynamic behavior of fluids near thecritical point, Annual Reviews of Physical Chemistry 37 (1986) 189–222.

54] J.V. Sengers, J.M.H. Levelt Sengers, Critical penomena in classical fluids, in: C.A.Croxton (Ed.), Progress in Liquid Physics, Wiley, New York, 1978, pp. 103–174.

55] Z.Y. Chen, A. Abbaci, S. Tang, J.V. Sengers, Crossover from singular critical toregular classical thermodynamic behavior in fluids, Physical Review A 8 (1990)4470–4484.

56] M.A. Anisimov, J.V. Sengers, Critical and crossover phenomena in fluids andfluid mixtures, in: E. Kiran, P.G. Debenedetti, C.J. Peters (Eds.), SupercriticalFluids, Kluwer, Dordrecht, 2000, pp. 89–121.

57] A. Kostrowicka Wyczalkowska, M.A. Anisimov, J.V. Sengers, Critical fluctuationsand the equation of state of Van der Waals, Physica A 134 (2004) 482–512.

58] M.A. Anisimov, J.V. Sengers, On the choice of a hidden field variable near thecritical point of mixtures, Physics Letters A 172 (1992) 114–118.

59] J.C. Rainwater, D.G. Friend, Composition dependence of a field variable alongthe binary fluid mixture critical locus, Physics Letters A 191 (1994) 431–437.

60] M.A. Anisimov, E.E. Gorodetskii, V.D. Kulikov, A.A. Povodyrev, J.V. Sengers, Ageneral isomorphism approach to thermodynamic and transport properties ofbinary fluid mixtures near critical points, Physica A 220 (1995) 277–324, 223(1996) 272.

61] R. Kleinrahm, W. Wagner, Measurement and correlation of the equilibrium liq-uid and vapour densities and the vapour pressure along the coexistence curveof methane, Journal of Chemical Thermodynamics 18 (1986) 739–760.

62] U. Setzmann, W. Wagner, A new equation of state and tables of thermodynamicproperties for methane covering the range from the melting line to 625 K atpressures up to 1000 MPa, Journal of Physical and Chemical Reference Data 20(1991) 1061–1155.

63] J.M. Smith, H.C. Van Ness, Introduction to Chemical Engineering Thermody-

namics, McGraw-Hill, New York, 1987.

64] B.A. Younglove, J.F. Ely, Thermophysical properties of fluids. II. Methane, ethane,propane, isobutane, and normal butane, Journal of Physical and Chemical Ref-erence Data 16 (1987) 577–798.

65] R.C. Reid, J.M. Prausnitz, B.E. Poling, The Properties of Gases and Liquids,McGraw-Hill, New York, 1987.

6 uperc

[

[

[

[

[

[

[

[7–23.

02 Kh.S. Abdulkadirova et al. / J. of S

66] B.H. Sage, B.L. Hicks, W.N. Lacey, Phase equilibria in hydrocarbon systems: Themethane-n-butane system in the two-phase region, Industrial and EngineeringChemistry 32 (1940) 1085–1092.

67] C.P. Hicks, C.L. Young, Gas–liquid critical properties of binary mixtures, Chem-ical Reviews 75 (1975) 119–175.

68] P.H. Van Konynenburg, R.L. Scott, Critical lines and phase diagrams in binary vander Waals mixtures, Philosophical Transactions of the Royal Society of London

298 (1980) 495–540.

69] A. Boltz, U.K. Deiters, C.J. Peters, Th.W. de Loos, Nomenclature for phase dia-grams with particular reference to vapour–liquid and liquid–liquid equilibria,Pure & Applied Chemistry 70 (1998) 2233–2257.

70] H.H. Reamer, R.H. Olds, B.H. Sage, W.N. Lacey, Phase equilibria in hydrocarbonsystems, Industrial and Engineering Chemistry 34 (1942) 1526–1531.

[

[

ritical Fluids 55 (2010) 594–602

71] H.M. Lin, H.M. Sebastian, S.S. Simnick, K.-C. Chao, Gas–liquid equilibrium inbinary mixtures of methane with n-decane, benzene, and toluene, Journal ofChemical Engineering Data 24 (1979) 146–149.

72] A. van Pelt, J.V. Sengers, Thermodynamic properties of 1,1-difluoroethane(R152a) in the critical region, Journal of Supercritical Fluids 8 (1995) 81–99.

73] S.B. Kiselev, Cubic crossover equation of state, Fluid Phase Equilibria 147 (1998)

74] A. Kostrowicka Wyczalkowska, M.A. Anisimov, J.V. Sengers, Global crossoverequation of state of a van der Waals fluid, Fluid Phase Equilibria 158–160 (1999)523–535.

75] M.Yu. Belyakov, E.E. Gorodetskii, Approach to equation of state for fluids, Inter-national Journal of Thermophysics 27 (2006) 1387–1405.