123VOLUME V / INSTRUMENTS

2.6.1 Introduction

The term Equation Of State (EOS) indicates a relationshipbetween the variables P (pressure), T (temperature) and v(molar volume) that defines the thermodynamic state of asystem. This relationship can be generically represented inthe following way:

[1]

From this equation, it is possible to calculate thevolumetric properties of a fluid once the values of pressureand temperature are known. Moreover, using rigorousrelationships derived from thermodynamics, it is alsopossible to calculate other properties that cannot bemeasured directly, such as energy and entropy, that dependon the state of the system. In this case, EOS is used in orderto evaluate the quantities representing the deviations fromthe ideal gas behaviour (or residual quantities) byintegration, based on variations of thermodynamic propertiestogether with variations of volume and pressure. Forexample, for enthalpy the following relationship can beobtained:

[2]

or

[3]

where h* represents the enthalpy of ideal gas at the sametemperature and pressure conditions. Similar relationshipscan be obtained for other thermodynamic quantities such asentropy, Gibbs free energy, or Helmholtz free energy. Thelatter is represented by the equation:

[4]

These relationships can also be used in reverse. An equationof state can be generated by any other equation expressingone of the residual quantities as a function of temperatureand volume (or pressure). For example, the followingexpression of the residual Helmholtz energy:

[5]

corresponds to the well known van der Waals equation ofstate (see Section 2.6.3) and is obtained by deriving the latterwith respect to the volume (T�constant) and by applying therelationship:

[6]

where s is the molar entropy. By means of the partitionfunction and using statistical thermodynamic considerations,it is possible to determine the value of residual quantities(usually the residual Helmholtz energy) through which it ispossible to derive the EOS.

EOS can also be used to evaluate the properties ofheterogeneous systems composed of fluid phases inequilibrium. Phase equilibria modelling represents the mostcommon application of EOS, adopted both to correlateexperimental data, and to predict data in regions where nomeasurement exists or where measurements are particularlydifficult to obtain. The ideal EOS model should use physicalproperties that are easily measurable in order to predictphase equilibria as well as the other properties inherent in alltemperature and pressure conditions. Moreover, this modelshould be developed in conformation with theoreticalconsiderations. Unfortunately, a model with thesecharacteristics does not exist and a single equation of statecannot describe – in all temperature and pressure conditions –the behaviour of substances having very different molecularcharacteristics with respect to both their dimensions andtheir functional nature. This is the reason that spurned thedevelopment of several different equations of state.

In recent years, excellent reviews on the subject haveappeared (Anderko, 1990; Sandler et al., 1994; Wei andSadus, 2000; Valderrama, 2003). IUPAC (InternationalUnion of Pure and Applied Chemistry) published a bookdescribing the most important equations of state and theiruse for both pure components and mixtures (Sengers et al.,

da sdT PdvH = − −

a a RT v bv

av

− = −−

−* ln

( ) lna P RTv

dv RT PvRTH res

v

= −

−∞

∫

T PT

Pv

= ∂∂

−

+ −∞∫v

dv Pv RT

h T P h T P h T Pres , , ,( ) = ( ) − ( ) =∗

v T vT P

( ) ( ) ( )= − ∂

∂

∫

0

P

dP

h T,P h T,P h T,Pres ( ) = ( ) − ( ) =∗

f P T v( , , ) = 0

2.6

Equations of state

2000). In light of these works, different possible EOSclassifications have emerged, even if in reality the existenceof a strong interconnection among all the equations of statedeveloped in the last fifty years has been stressed, since a‘new’ equation of state is often simply a partial modificationof an already existing model.

The properties used to classify EOS include the degreeof a polynomial in terms of volume expansion obtained bydeveloping the equation of state, and the individual nature ofinteractions. According to the first criterion, the equationsare empirically divided between cubic and non-cubicequations of state. According to the more quantitative secondcriterion, the equations of state can be subdivided into threeclasses: empirical equations, theoretical equations andsemiempirical (or semitheoretical) equations (Nezbeda,2001). The first class consists of an arbitrary functioncontaining a high number of parameters, the numericalvalues of which are obtained by correlating experimentaldata. These equations are obviously very accurate when theymust reproduce correlated experimental data, but theirapplicability is limited to the description of the properties ofthe systems for which the parameters were obtained.Furthermore, they cannot be easily used to describe thebehaviour of mixtures. Theoretical equations, on the otherhand, are very general in essence and can be used to evaluatethermodynamic properties of entire families of chemicalcompounds, as they can be characterized by the sameintermolecular potential. These equations have thedisadvantage of having a validity related to the specificmodel of potential of interaction used and very often this isonly an approximated representation of reality. As aconsequence, most of the EOS models have been derived bycombining the two approaches: the equation is obtained onthe basis of theoretical considerations and the parametersappearing in the model are treated as quantities to becalculated by correlating experimental data. Often, theseequations are also known as ‘molecular’ equations of state,and they also have predictive capabilities, as in the case ofequations based on perturbation theories, written byincrementally considering the terms which are supposed toreflect the contributions of the different types of interactions(Donohue and Prausnitz, 1978; Economou and Donohue,1992; Mueller and Gubbins, 1995).

According to Valderrama (2003), equations of state canbe subdivided into three large families: virial type equationsof state, van der Waals type equations of state in which thecontributions of attractive and repulsive forces arerecognized, and equations of state based on molecularconsiderations.

A review of the different EOS families will be discussedbelow beginning with expressions describing purecomponents, and subsequently, equations for mixtures.

2.6.2 Virial equations of state

Pure components The virial type equations of state expresses the

compressibility coefficient by expansion in a power series interms of density r:

[7]

where coefficients B, C are commonly called the second andthird virial coefficients.

The compressibility factor also can be expressed as adevelopment in terms of pressure

[8]

The numerical values of these coefficients have beenreported by Dymond (Dymond and Smith, 1980), or theycan be calculated from relationships developed using theprinciple of corresponding states as a starting point. Forexample, for apolar compounds the following relationshipproposed by Tsonopoulos (1974) is particularly useful:

[9]

with

[10]

In these equations, PC and TC represent critical pressureand temperature, respectively; TR represents reducedtemperature (the ratio between temperature and criticaltemperature); and w is the acentric factor defined by thefollowing relationship:

[11]

where PRV is the reduced vapour pressure (ratio of the

vapour pressure and the critical pressure) calculated at areduced temperature equal to 0.7.

Due to lack of sufficient experimental data, thedetermination of numerical values for the higher degreecoefficients is particularly difficult. Moreover, from atheoretical point of view, even their calculation based on theknowledge of molecular interactions is not very easy. Forthese reasons, the equation is commonly interrupted at thesecond coefficient, and therefore its validity is limited torelatively low pressures.

Virial expansion, however, was used as a basis todevelop empirical equations of state, valid for quite extendeddensity intervals. The Beattie and Bridgemann (1929)equation of state:

[12]

together with the Benedict-Webb-Rubin (BWR) equation(Benedict et al., 1940, 1942):

[13]

a+ +r6α ccTr

r r3

2

2 21

+( ) −( )γ γexp

P RT B RT ACT

bRT a= + − −

+ −( ) +r r r

0 0

0

2

2 3

aA bB RT cB RT

+ − −0 0022

3 02

4

+

r r

bB RT

P RT B RT A cRT

= + − −

+r r0 0 2

2

ω = − −log10 1PRV

0 0637 0 331 0 424 0 00812B

T TR R

( ) = − − −.. . .

TTR8

TR3

0 0121 0 0− −. . 0006078TR

BT TR R

020 1445 0 33 0 1385( ) = − − −.

. .

BPRT

B BC

C

= +( ) ( )0 1ω

zPvRT

B P C P= = + + +1 2' ' ...

z PvRT

B C= = + + +1 2r r ....

PHYSICAL AND CHEMICAL EQUILIBRIA

124 ENCYCLOPAEDIA OF HYDROCARBONS

The latter was the first virial type EOS capable ofrepresenting the volumetric properties and the liquid-vapourequilibrium for mixtures of industrial interest, even at highdensities. The numerical values of the constants A0, B0, a, b,c in equation [12] and those of A0, B0, C0, a, b, c, a, g inequation [13] are determined by PVT experimental data.Subsequently, the numerical values of the constants in theBWR equation of state were determined for a series ofcompounds (Cooper and Goldfrank, 1967; Orye, 1969; Perryet al., 1984). The success of the BWR equation subsequentlyresulted in the proposal of numerous other equations withseveral modifications; among which, special mention isgiven to the equations proposed by and by Starling Han(1972) and Starling (1973):

[14]

In order to calculate the numerical values of the 11constants appearing in equation [14], generalizedrelationships were proposed once the temperature,critical pressure, and the acentric factor of a substancewere known.

Adding three further constants to the BWR equationprovides a more accurate description of the lowtemperature behaviour (eq. [14]). Another BWR equationwas later modified by Nishiumi and Saito (1975) in orderto more correctly describe the behaviour of heavyhydrocarbons:

[15]

Still another modification was credited to Lee andKesler (1975), who developed an equation used especially inthe context of a corresponding states approach.

Equations of state and corresponding states The corresponding states theorem was used to

provide a general form to equations of state, reducing theneed to calculate an excessive number of parameters foreach occasion. In its original form, the two-parametercorresponding states theorem demonstrates that it ispossible to describe the volumetric behaviour of everyfluid by using the same functional form, as long as theparameters are expressed in reduced terms. By using thecritical coordinates (TC, vC, PC) as reduction parameters,the compressibility factor can be expressed in thefollowing manner:

[16]

where TR�T�TC, rR�r�rC e PR�P�PC. The z function can bederived from any equation of state. Molecular hypotheses onwhich equation [16] is based are very restrictive andconsequently, it is possible to quantitatively describe thevolumetric properties of very few fluids (rare gases). Thetheorem was expanded by introducing a third parameter(three-parameter corresponding states theorem). Byincluding the acentric factor as an additional parameter, thecompressibility factor can be expressed according to anequation of this type:

[17]

where z(0) is the compressibility factor of a fluidcharacterized by a zero value of the acentric factor and z(1)

represents a deviation function. Lee and Kesler (1975)proposed a modification of the previous equation in order touse the same equation to calculate both terms:

[18]

In this case, z(0) and z(r) are the compressibility factorsfor a simple reference fluid (with w�0) and for a secondfluid (with w�w(r)). A rare gas (w�0) and n-octane(w�0.3978) were used as reference fluids and thecompressibility factor is calculated via the equation:

[19]

The numerical values of the equation are reported inTable 1.

Extension to mixturesIn order to describe the behaviour of mixtures by

applying the equations previously used to explain thebehaviour of pure components, it is necessary to introducecomposition as an additional variable. This is done byadopting certain mixing rules for the calculation of theparameters appearing in the different relationships, thusallowing us to find the numerical values of the parameters‘averaged’ from those of the pure components in thefollowing general form for a generic parameter A:

[20]

where yi and yj are the mole fractions of the mixture, Aii andAjj are the values of parameter A for the generic purecomponents i and j, and the mixed term Aij is calculated viaan average, arithmetic equation:

[21] AA A

ijii jj=

+2

A y y Aiji

j ij= ∑∑

cTR

4+33

2 2 2r r rR R Rβ γ γ+( ) −( )exp

c+ −1

ccT

cT

ddTR R

RR

R2 3

3

2

1

2 5+

+ +

+r r

zP

Tb

bT

bT

bT

R

R R R R RR= = + − − −

+

rr1

1

2 3

2

4

3

z T P z TrR R R,

ω+( ) −( ) ( )0

,,

( )

PRr

( )ω

z T P z T PR R R R, , ,ω( ) = ( ) +( )0

z T P z T P z T PR R R R R R, , , ,ω ω( ) = ( ) + ( )( ) ( )0 1

z PvRT

z T z T PR R R R= = = ( )( , ) ,r

r r r3 8 17

3 2 21+ + +

+( ) −( )cT

gT

hT

γ γexp

+ + + +

a dT

eT

fT4 23

α rr6 +

bRT a+ − −− − −

+dT

eT

fT4 23

3r

P RT B RT ACT

DT

ET

= + − − + −

+r r0 0

0

2

0

3

0

4

2

+ +( )cT

r r3

3 21 γ exp −−( )γ r2

bRT a+ − −−

+ +

+dT

a dT

r r3 6α

P RT B RT ACT

DT

ET

= + − − + −

+r r0 0

0

2

0

3

0

4

2

EQUATIONS OF STATE

125VOLUME V / INSTRUMENTS

or geometric equation:

[22]

for the values of the respective pure components.Very often, in order to improve the descriptive capacities

of equations, a binary interaction parameter kij is introducedas in equations [21] and [22], the numerical value of whichis determined on the basis of experimental values for thebinary mixture ij. The values of this parameter areparticularly important for the calculation of partialproperties such as fugacity and therefore it is usuallyestimated based on the regression of experimentalequilibrium data. The most widely used mixing rules will bediscussed below with reference to the equations previouslywritten.

The virial type EOS, interrupted at the secondcoefficient, is the only EOS for which therelationship permitting the calculation of the secondcoefficient for mixtures can be justified andtheoretically derived:

[23]

The calculation of mixed terms Bij using theTsonopoulos relationship [9] requires defined combinationrules for the critical coordinates and the acentric factor. Inthe case of normal fluids, it is possible to use the followingrules (Prausnitz et al., 1999):

[24]

[25]

[26]

[27]

[28]

If the molecules of the mixture components do not haveexceedingly varying dimensions, the kij parameter can beassumed to be equal to zero (numerical values of theparameter were published by Chueh and Prausnitz, 1967 andby Tsonopoulos, 1979). The mixing rules for differentequations are reported by Sandler (Sandler et al., 1994). Inthe Lee-Kesler equation in particular, several mixing ruleswere proposed. The most widely used are those proposed byPlocker (Plocker et al., 1978):

[29]

[30]

[31]

[32]

[33]

Numerical values of the binary parameter have beenreported by Knapp (Knapp et al., 1982).

2.6.3 Cubic equations of state or the van der Waals EOS

Pure components The van der Waals equation of state

[34]

or, as expressed in terms of compressibility coefficients,

[35]

represents the first successful attempt at describing thecoexistence of a liquid phase and a vapour phase. Parametera is an index of the attractive forces between the moleculesand parameter b is the covolume occupied by the molecules.The values of parameters a and b can be easily calculatedonce the critical coordinates are known. An important aspectof the van der Waals equation is that it can be interpreted asif it comprised the sum of two terms accounting for therepulsive and attractive forces. Unfortunately, this equationis not accurate enough as already observed by van der Waals,himself. The compressibility factor is equal to 0.375 at thecritical point whereas its real value for differenthydrocarbons falls between 0.24 and 0.29. Furthermore, as aand b are constant with temperature, it is not possible to

zv

v ba

RTv=

−−

PRTv b

av

=−

− 2

ω ωm i ii

y= ∑

V y y VCm i j Cijji

= ∑∑

TV

y y V TCmCm

i j Cij Cijji

= ∑∑11 4

1 4

/

/

VV V

CijCi Cj=

+

1 3 1 33

2

/ /

T T T kCij Ci Cj ij= ( )1 2/

Pz RTVCij

Cij Cij

Cij

=

ωω ω

iji j=+2

zz z

CijCi Cj=

+2

VV V

CijCi Cj=

+

1 3 1 33

2

/ /

T T T kCij Ci Cj ij= ( ) −( )1 2

1/

B y y Bm i jji

ij= ∑∑

A A Aij ii jj= ( )1 2/

PHYSICAL AND CHEMICAL EQUILIBRIA

126 ENCYCLOPAEDIA OF HYDROCARBONS

Table 1. Numerical valuesof the Lee-Kesler equation

Coefficient Simple fluidReference

fluid

b1 0.1181193 0.2026579

b2 0.265728 0.331511

b3 0.154790 0.027655

b4 0.030323 0.203488

c1 0.0236744 0.0313385

c2 0.0186984 0.0503618

c3 0 0.016901

c4 0.042724 0.041577

d1�104 0.155488 0.487360

d2�104 0.653920 0.0740336

b 0.650167 1.226

g 0.060167 0.03754

correctly evaluate the vapour pressure of the liquids and theirvariations with temperature, itself. For this reason, severalmodifications were introduced to the original equation,mostly concerning the attractive term, whereas the repulsiveterm is left unaltered, thus giving birth to the so-calledfamily of cubic equations. Table 2 is a summary of theexpressions proposed by different authors with the mostimportant principles described below.

Redlich and Kwong (1949) modified the attractive termby introducing a different dependence on temperature andvolume. The equation provides a slightly more precise valueof the compressibility coefficient which is equal to 0.333,but the values for vapour pressure and density of the liquidsare still less than accurate.

Soave (1972) substantially modified the dependence ontemperature by replacing the a�T1.5 term with a generalizeddependence a(T ):

[36]

with

[37]

This modification and the equation derived from itknown as the SRK (Soave-Redlich-Kwong) equation,allowed scientists to simulate the vapour pressure of apolarsubstances with great accuracy, particularly for values above1 bar. Subsequently, several authors proposed other

functional forms for a(T ) in order to improve the ability ofpredicting the vapour pressure for different classes ofcompounds. Valderrama (2003) presents an exhaustivereview of the different forms proposed.

The a term’s dependence on temperature influences thecapability of providing accurate values for vapour pressure,whereas its dependence on volume determines the capabilityof providing the correct values for volumetric properties.This is the reason diverse forms are included in the attractiveterm (see Table 2), among which the PR equation, due toPeng and Robinson (1976), is worth mentioning as it doesnot introduce additional parameters and is able to obtain avalue of the critical compressibility factor equal to 0.307:

[38]

with

[39]

SRK and PR equations are widely used because they requirea low number of parameters (critical properties and theacentric factor). These equations, however, are limited by thefact that they use the same value of the criticalcompressibility factor for all substances and they do notprovide reliable values for liquid densities.

In order to solve the first problem, the number ofparameters in the equations of state was increased (see Table2): hence, equations with three parameters (Fuller, 1976;Harmens-Knapp, 1980; Schmidt-Wenzel, 1980; Patel-Teja,1982), four parameters (Trebble-Bishnoi, 1987) and fiveparameters (Adachi et al., 1986) were proposed.

Peneloux (Peneloux et al., 1982) proposed a simple buteffective method to improve descriptive capabilities withrespect to the volumetric properties of saturated liquids. Inthis method, the value of the molar volume is translated by at value; as a consequence, the values of v and b are replacedby v�t and b�t. This correction can be applied to allequations derived from the van der Waals equation. Thus, forexample, the SRK equation takes this form:

[40]

Introducing this translation does not modify thecalculation of equilibrium for pure components nor formixtures, as long as the t parameter, in the case of mixtures,is calculated through a law that is linear with respect to thecomposition.

It is known that cubic equations of state which havebeen derived from the van der Waals equation are generallycapable of providing a satisfactory description of fluidproperties even when using approximated expressions toevaluate the contribution of both the attractive andrepulsive forces. For this reason, several modifications tothe repulsive term were suggested beyond the attemptsalready described to correct the attractive term. Most ofthese modifications led to equations of state that were nolonger cubic. Table 3 includes a summary of theseproposed expressions, along with the original van derWaals expression for completeness. In these equations,h�b�4v (alternatively h�ps3�6v: where s is the rigid

PRTv b

a Tv t v b t

=−

−( )

+( ) + +( )2

+1 5422. ω −− ) −( )( 0 26922 12 0 5 2. .ω TR

a T R TP

C

C

( ) = + +( )0 45724 1 0 374642 2

. .

z vv b

va Tv v b b v b RT

att =−

−+( ) + −( )

( )

m = + −0 480 1 57 0 176 2. . .ω ω

a T R TP

m TC

CR( ) = + −( ) 0 4274 1 1

2 20 5 2

. .

EQUATIONS OF STATE

127VOLUME V / INSTRUMENTS

Table 2. Different forms for the attractive termin the cubic equations

Equation �zatt

van der Waals

Redlich-Kwong (1949)

Soave (1972)

Peng-Robinson (1976)

Fuller (1976)

Schmidt-Wenzel (1980)

Harmens-Knapp (1980)

Patel-Teja (1982)

Adachi(Adachi et al., 1986)

Trebble-Bishnoi (1987)

aRTv

av b RT+( ) 1 5.

a Tv b RT

( )+( )va T

v v b b v b RT( )

+( ) + −( )

a Tv cb RT

( )+( )va T

v ubv wb RT( )

+ +( )2 2

va Tv vcb c b RT

( )+ − −( )

2 21)

va Tv v b c v b RT

( )+( ) + −( )

va T v c Tv b v d T v e T RT

( ) − ( ) −( ) − ( ) + ( )

va Tv b c v bc d RT

( )+ +( ) − +( )

2 2

sphere diameter) is defined as the packing fraction and a isa non-sphericity parameter.

The expressions in Table 3 have been used incombination with those reported in Table 2, generatingdifferent possible equations of state. For example, Carnahanand Starling (1972) combined their repulsive term with theattractive term of the Redlich-Kwong equation therebygenerating the CSRK equation and obtaining good resultsparticularly for hydrocarbons, both in the calculation of thedensities of saturated liquids and that of equilibria even athigh pressures.

A different approach was adopted by Chen andKreglewski (1977) who paired the repulsive term proposedby Boublík and Nezbeda (1977) with an attractive termderived by Alder (Alder et al., 1972) through a power-seriescorrelation of data generated by computer simulation:

[41]

In this equation, the numerical values of the universalconstants Anm have been determined by correlating volumetricand internal energy data for Argon; e�k (K Boltzmannconstant) represents a characteristic energy of interaction andis considered independent of temperature; v0 is the rigidsphere volume and is a function of temperature. The equationproposed by Chen and Kreglewski (1977), known by theacronym BACK (Boublík-Alder-Chen-Kreglewski), assumesthat n varies from 1 to 4 and m from 1 to 9.

Soave (1990) proposed a fourth order equation,including a generic attractive term and a repulsive termobtained by an equation that approximates the Carnahan-Starling repulsive term:

[42]

All equations belonging to the extended van der Waalsfamily of equations which are not cubic equations definitelygive better results in the description of volumetric properties,particularly for dense fluids and in the supercritical region, butthere is a higher number of parameters. Moreover, many of

these equations no longer satisfy the critical constraints andtherefore they overestimate critical temperature and pressure.

Extension to mixtures The calculation of phase equilibria represents the most

typical application of equations of state especially for thosebelonging to the van der Waals equation family. In thesecalculations, it is assumed that the EOS used to calculate theproperties of pure fluids can also be used to describe themixtures, as long as the parameters related to the mixture aresuitably evaluated. Generally, this result is obtained byapplying the mixing rules to a van der Waals fluid for bothparameters a and b:

[43]

[44]

Furthermore, it is necessary to add the combination rulesfor parameters aij and bij. Generally, it is assumed:

[45]

[46]

where kij and lij are binary interaction parameters, and theirnumerical value is obtained though the correlation ofequilibrium experimental data. Several sources report thenumerical data of such parameters (Grabowski and Daubert,1978a; 1978b; 1979; Knapp et al., 1982). In many cases, lijis neglected and the behaviour of mixtures is described byusing a single binary parameter. These combination rules canalso be partially justified from a theoretical viewpointremembering that the combination rule for a is similar to therule for the molecular interaction potential, and that thevalue of b is given in equation [46] if the molecules can bedescribed as rigid spheres.

For the volume translation parameter, a linear mixingrule is used in order to avoid altering the calculation of phaseequilibria

[47]

Equations of state such as the van der Waals equation,especially when applied together with classical mixing rules,provide good results, primarily for mixtures of nonpolarcompounds such as mixtures in the oil industry. Moreover, itis difficult in these cases to accurately correlate liquid-vapour equilibrium data if it is not previously ensured thatthe vapour pressure data of the pure components are alsoaccurately described. A great limitation of the classic mixingrules is represented by the fact that it is difficult to representthe behaviour of asymmetric mixtures (in which thecomponents have very different vapour pressures) as well asmixtures containing polar compounds. For this reason,mixing rules have been proposed for the term a where aconcentration-dependent binary interaction coefficient isused (expressed by the molar fractions xi and xj) by means ofappropriate constants (Kij, Kji, mij) such as thePanagiotopoulos and Reid (1986) rule,

[48]

the Adachi and Sugie (1986) rule,

k K K K xij ij ij ji i= − −( )

t t xi ii

= ∑

bb b

lijii jj

ij=+

−2

1( )

a a a kij ii jj ij= −( )1

b b z zij i j= ∑∑

a a z zij i jji

= ∑∑

z cb

v bav

v d v e= +

−−

+( ) +( )1

z mAkT

vv

attnm

mn

n m

=

∑∑ ε 0

PHYSICAL AND CHEMICAL EQUILIBRIA

128 ENCYCLOPAEDIA OF HYDROCARBONS

Table 3. Different forms for the repulsive termin the cubic equations

Name zrep

van der Waals

Guggenheim(1965)

Carnahan-Starling (1969)

Scott

Convex hardbody (Boublík-Nezbeda, 1977)

11 4− η

11 4−( )η

11

2 3

3+ + −

−( )η η η

η

RT v bv v b

+( )−( )

1 3 2 3 3 1

1

2 2 2 3

3

+ −( ) + − +( ) −

−( )α η α α η α η

η

[49]

and the Sandoval rule (Sandoval et al., 1989),

[50]

A limitation of these rules is the fact that they do notprovide a correct quadratic dependence on the second virialcoefficient. Moreover, as demonstrated by Michelsen andKistenmacher (1990), they have an important deficiency inthe situation where a binary mixture is fictitiouslyconsidered as a pseudoternary mixture with two equalcomponents, the value of the a mixture parameter does notcoincide with that calculated for the binary mixture.

Many mixtures of industrial interest present a highdegree of nonideality and traditionally, their behaviour hasbeen described by using the activity coefficient approachand the excess free energy models. For this reason, Huronand Vidal (1979) derived mixing rules to calculate parametera of equations of state using the dependence on theconcentration of the excess free energy models. Thehypotheses on which the model is based are: the excess freeenergy gE, calculated by an activity coefficient model for theliquid phase and the excess free energy calculated by anequation of state, coincide at infinite pressure; the volume ofmixture at infinite pressure is equal to covolume; the excessvolume at infinite pressure is zero.

If a mixing rule for parameter b is used, the followingexpression for parameter a is obtained:

[51]

where constant L depends on the type of cubic EOS usedand, in the case of the Redlich-Kwong equation: L�ln 2; g�

E,is the molar excess Gibbs energy at infinite pressure. Todescribe its dependence on composition, it is possible to useone of the models proposed in publications, such as theWilson equation, the Non-Random Two-Liquid (NRTL)equation, or the UNIversal QUAsi-Chemical (UNIQUAC)equation (Prausnitz et al., 1999). Huron and Vidalrecommended the NRTL model:

[52]

In this equation, a and C are parameters that shouldbe determined by correlating equilibrium experimentaldata: if aij�0, the original van der Waals mixing rulesare obtained. The Wilson equations were also used withgood results in the description of systems with a highdegree of nonideality. These mixing rules are notsuitable to describe the behaviour of nonpolar mixturessuch as those composed of hydrocarbons, nor in generalto describe the liquid-vapour equilibrium at lowpressure.

In order to overcome these difficulties, severalmodifications have been proposed (Mollerup, 1986; Dahland Michelsen, 1990; Michelsen, 1990). The modificationsuggested by Michelsen (1990) is known as the ModifiedHuron-Vidal first order mixing rule (MHV1). It wasobtained by imposing the equality between excess free

energy gE calculated via a liquid phase activity coefficientmodel, and the excess free energy calculated via a zeropressure equation of state:

[53]

where a�a�bRT, ai�ai �biRT, b��xibi , and q1 is a constantfor which a value of �0.593 is suggested.

Subsequently, Dahl and Michelsen (1990) proposed anew mixing rule known as the ‘Modified Huron-Vidalsecond order mixing rule’ (MHV2):

[54]

with q1��0.478 and q2��0.0047. These new mixing rules are particularly convenient as

they can use the values determined by low pressureequilibrium data correlation as the numerical values of theexcess free energy models. In particular, the MHV2 rule,used in combination with the group contribution modelUNIFAC (UNIversal Functional group Activity Coefficient),was expanded in order to account for interactions withgaseous components (Larsen et al., 1987), giving excellentresults.

Mixing rules using the equality of the excess freeenergies at infinite pressure and those imposing suchequality at zero pressure are both unable to give a quadraticdependency for the second virial coefficient with respect tocomposition, as predicted by statistical mechanics. Wongand Sandler (1992), starting with the excess Helmholtz freeenergy, proposed new rules coherent with the quadraticdependence of the second virial coefficient:

[55]

[56]

[57]

The numerical value of constant L, such as that found inequation [51], depends on the equation of state used; a�

E isthe excess Helmoltz free energy at infinite pressure; kij is thebinary interaction parameter. This mixing rule wassubsequently modified in order to determine if the classicvan der Waals (Orbey and Sandler, 1995) mixing rule was alimited case. Whereas equations [55] and [56] were leftunaltered, equation [57] was replaced by the equation:

[58]

The results obtained by applying this mixing rule haveproved to be excellent both for the prediction of

b aRT

b b k a a

RTij

i j ij i j−

=+

−−( )

2

1

b aRT

kb

aRT

baRTij

iji

ij

j−

=−

−

+ −

1

2

bx x b a

RT

xa

RTbaRT

i jijji

ii

i

E

i

=

−

− +

∑∑

∑ ∞1Λ

ab

xab

ai

i

ii

E

= +∑ ∞

Λ

gRT

E0= + xx b

biii

ln∑

q x q xi ii

i ii

1 2

2 2α α α α−

+ −

=∑ ∑

α α= + −

∑ ∑x

qgRT

xbbi

ii

E

ii

i

1

1

0 ln

g xx

CRT

C

x CE

i

j jiji

jij

k kiki

∞ =−

−

∑ exp

exp

α

αRRTk

i

∑

∑

a b xab

gi

i

i

i

E

= −

∑ ∞

Λ

k K x K x K K x xij ij i ji j ij ji i j= + + +( ) − −( )0 5 1.

k K m x xij ij ij i j= + −( )

EQUATIONS OF STATE

129VOLUME V / INSTRUMENTS

liquid-vapour equilibria at high pressure, and for thedescription of systems containing polymers or verylong-chain hydrocarbons (Orbey and Sandler, 1994; 1998).

2.6.4 Equations of state based on molecular considerations

Pure components Statistical mechanics and molecular simulation have

largely contributed to the development of new equations ofstate with a stronger theoretical foundation. Some of thesewere already described, including those belonging to the vander Waals family of equations, since they can always bebrought back to the concept that the pressure or thecompressibility factor are determined by the sum of thecontributions of the attractive and repulsive forces. Theequations of state based on molecular considerations are drawnmostly from the results obtained by Carnahan and Starling byevaluation of the contribution due to repulsive forces (see Table3). However, they also consider the fact that most of themolecules of the compounds involved do not have a sphericalstructure but rather a more articulated structure, includingstructures such as polymers or high molecular-weighthydrocarbons, which are similar to chains. Moreover, it mustbe noted that the thermodynamic properties of molecules aremore complex than those which are simply derivable whenthey are considered as rigid bodies. In fact, they also depend onthe rotational and vibrational movements of molecules. In thisway, it is assumed that the interaction energy can be subdividedinto a part that is a function of density and another that isindependent of density. Based on these preliminaryconsiderations, Beret and Prausnitz (1975) developed thePerturbed Hard-Chain Theory (PHCT), later proposed byseveral other authors (Donohue and Prausnitz, 1978; Ikonomouand Donohue, 1986). This theory is based on the followingassumptions:• The dependence on the density of the external degrees of

freedom is equal to that of the translational degrees offreedom, expressed in the Carnahan-Starling equation.

• A chain-shaped molecule behaves like a set of segments,each of which can interact with the neighboringsegments: in this case a ‘square-well’ interactionpotential is assumed.

• An adjustable parameter c is introduced, with 3crepresenting the external degrees of freedom.The resulting equation is:

[59]

where the last term expresses the attraction forces describedaccording to the Alder power-series expansion (Alder et al.,1972); r indicates the number of segments in the molecule, his the packing fraction, and t is defined by the relationship:

[60]

where k is the Boltzman constant, T the absolutetemperature, e the intermolecular potential, q the surfacearea per molecule. Finally v̄ is defined by the relationship:

[61]

where v0 represents the packed molar volume, which can becalculated starting from the rigid segment diameter s:

[62]

where N is the Avogadro number. For each pure component,PHCT equations contain three parameters eq, rs3 and c, thenumerical value of which is determined by vapour pressureand density data.

Due to its complexity, this equation was subsequentlymodified by replacing the Carnahan-Starling or the Alderterms with simpler ones. It is especially worth mentioningthe modification proposed by Kim (Kim et al., 1986), knownas the Simplified Perturbed Hard-Chain Theory (SPHCT),where the Alder term is replaced:

[63]

where for Zm, representing the coordination number of achain site, the value 36 is assumed. The other symbols havethe same meaning as those used in equation [59] and definedby equations [60] and [61].

In order to increase the accuracy of the description oflow-density thermodynamic properties, Cotterman(Cotterman et al., 1986a, 1986b) separates the attractive partof the Helmholtz free energy into a low-density and ahigh-density contribution:

[64]

where aidg is the ideal gas contribution, aref is the Carnahan-Starling contribution written for chain-shaped molecules, asv

is the low pressure contribution that can be calculated basedon the second virial coefficient and adf is the dense fluidcontribution that can be calculated as the sum of thedispersion forces and polar forces expressed as a powerseries (Gubbins and Twu, 1978); and finally, F is aninterpolation function between the low- and high-densitycondition. This equation was applied to different types ofcompounds and a method allowing the calculation of theparameters for the pure components was developed, based onthe knowledge of Bondi volume (Gregorowicz et al., 1991).

The SAFT equation (Statistical Association Fluid Theory;Chapman et al. 1990, Huang and Radosz, 1990) represents anattempt to account for the contribution of the differentinteraction forces and particularly for the possibility of fluidscreating associations among several molecules. It is foundedon the theoretical results obtained by Wertheim (1984a,1984b, 1986a, 1986b). The compressibility factor is thenexpressed (Huang and Radosz, 1990) by

[65]

where, if the number of segments per chain is indicated bym, the rigid sphere term zrs is given by the relationship

[66] z mrs = −−( )

4 21

2

3η η

η

z z z z zrs disp chain assoc= + + + +1

a a a F a Faidg ref sv df= + + −( ) +1

cZ vm−00 2

0 2

1

1

e

e

ε

ε

kT

kTv v

−( )+ −( )

zc r c r r

r=

+ −( ) + −( )( ) − ( )−( )

−1 4 3 3 2

1

2 3

3

η η η

η

vNr0

3

2=

σ

vvv

= 0

τε

=ckTq

cmAn− mm

m nmn v τ∑∑

zc r c r r

r=

+ −( ) + −( )( ) − ( )−( )

−1 4 3 3 2

1

2 3

3

η η ηη

PHYSICAL AND CHEMICAL EQUILIBRIA

130 ENCYCLOPAEDIA OF HYDROCARBONS

where h�trmv0 is the reduced density, t is a constant equalto 0.74048, r is the molar density and v0 is the molar volumeof the segment in the maximum packing situation. Thedispersive term zdisp is still given by the Alder expressionwith the numerical values of the parameters Dij, reported byChen and Kreglewski (1977):

[67]

The terms v0 and u are functions of temperature according tothe equations:

[68]

[69]

On the basis of the results obtained by Chen andKreglewski, C�0.12 and the energy per segment (e�k) equalto 10 for all molecules, except for those with a smallhindrance such as methane.

The term due to the bonds among segments or the chainterm zchain is given by the expression

[70]

finally the association term zassoc is given by

[71]

where XA is the mole fraction of the molecules not bonded tosite A and the summation is extended to all the sites that cangenerate an association.

The SAFT equation requires the knowledge of threeparameters (m, v00 and u0/k) for each non-associatedcomponent two additional parameters are added to these forthe associating components. The equation was applied withexcellent results to several pure fluids and many mixtures(Huang and Radosz, 1991) with very different natures. Forfurther information on these equations, see the review byWei and Sadus (2000) and for a more in-depth description ofthe perturbation theory, see Prausnitz’s book (Prausnitz etal., 1999).

Extension to mixturesAll equations belonging to this class can be easily

extended to the description of mixtures with the introductionof suitable combination rules for the parameters of purecompounds. The main advantage of these equations is thatgenerally a reduced number of binary parameters is required(generally one) and furthermore, these are independent oftemperature. An exhaustive review of mixing andcombination rules is presented by Sandler and Orbey (2000).

It is also important to note that in all these equations,eventual binary parameter expresses the interactions betweensegments and therefore, its use is not limited to thepossibility of describing the properties of the specific system

taken under consideration, but rather those of entire familiesof compounds (Fermeglia and Kikic, 1993).

The SAFT equation can be easily extended to mixturesbased on the suggestions of Huang and Radosz (1991). Inreference to the expressions derived from statisticalmechanics, a mixing rule is required with the introduction ofa binary parameter for the dispersive term only.

2.6.5 Conclusions

The equations of state belonging to three different familieshave been discussed. The virial equations of state arerecommended to meet the need for high accuracy in thecalculation of thermodynamic properties or phase equilibria.However, their use is not very extensive, since they requirenumerous parameters and considerable thermodynamic datain order to evaluate those parameters. They are usually usedto calculate water vapour and light hydrocarbon mixtureproperties.

The cubic equations of state represent the most widelyused class for a large range of applications, in particular forthe simulation of the behaviour in oil reservoirs or inhydrocarbon treatment plants. The introduction of new mixingrules related to expressions of the excess free energy not onlyallowed scientists to use the enormous mass of low pressureliquid-vapour equilibrium data – collected in recent decades topredict equilibria within a wide temperature and pressurerange – but also helped improve the ability to describe theproperties of mixtures containing polar compounds.

The equations of state based on molecular considerationsand molecular simulation results were often developed usingrelatively simple approaches. In general, the Carnahan-Starling repulsive term, the Adler equation for attractiveterm evaluation, and the formalism of perturbed chains areused in order to calculate the properties of molecules with ahigh molecular hindrance. Only SAFT replaces theformalism of perturbed chains with a formalism thataccounts for chain formation and for possible molecularassociations. This equation is used especially when it isnecessary to describe the behaviour of complex moleculesand of mixtures composed of species with very differentmolecular characteristics. However, it is important torecognize that semiempirical van der Waals equations ofstate are still being used today for engineering calculations.

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Ireneo Kikic

Dipartimento di Ingegneria Chimica, dell’Ambientee delle Materie Prime

Università degli Studi di TriesteTrieste, Italy

EQUATIONS OF STATE

133VOLUME V / INSTRUMENTS