Overview

THEORETICAL TREATMENT OF VAPOUR PRESSURE AND VAPOUR-LIQUID EQUILIBRIUM DATA

3.5 Correlation of Vapour-Liquid Equilibria

3.5.1 Overview

Chapter 3. Theoretical Treatment of Vapour Pressure and Vapour-Liquid Equilibrium Data

In the prediction of vapour pressures of nonelectrolyte organic compounds via the group contribution method has been achieved within our research group in the form of the work of Nannoolalet al. (2006).

variables of pressure, temperature, volume and phase composition and the nonideal behaviour ofthe phases can be expressed through auxiliary functions.

(3.47)

Consequently, the modelling of the experimental VLE data is based on the use of suitable mathematical relations that incorporate adjustable or empirical parameters to best satisfy the isofugacity criterion. In addition to the use of efficient algorithms and statistical techniques for the actual data reduction procedure, there are two other factors that are crucial for the accurate modelling of the data. The first relates to the accuracy of the data, which can be assessed through the use of thermodynamic consistency testing (only for P-T-x-y), to be dealt with later on, and the second concerns the selection of a suitable thermodynamic model or method for satisfying the isofugacity criterion to represent the thermodynamic behaviour of the system.

There have traditionally been two approaches or methods for the correlation of VLE data in the form of the direct or phi-phi (~i - ~i) method and the indirect or gamma-phi (Yi - ~i) method.

The methods differ in terms of the auxiliary functions i.e. the activity coefficient ( Yi ) and the fugacity coefficient _(~i) used to represent the nonideal behaviour of the vapour and liquid phases. The _~i - ~i method employs the use of an equation of state with reliable mixing rules to represent the vapour_(~i) and liquid phases _(~i) and the Yi - _~i method treats both phases separately i.e. an activity coefficient or G^Emodel for the liquid phase(Yi ) and an equation of state for the vapour phase _(~i). For the treatment of a binary mixture, the binary interaction parameters are incorporated into the mixing rules for the equation of state and into the GEmodel to characterize the specific interactions between pairs of components.

Raal and Muhlbauer (1998) describe the emergence of an additional approach to those above in the form of the modern direct method to address the limitations of the ~i ^- ~i approach in dealing with the higher density of the liquid phase and with complex polar systems. These incorporate the activity coefficient approach into the treatment of the component fugacities via the EOS mixing rules and will be discussed at a later stage.

In terms of the Yi - ~i model, there is a further distinction in the type of equation of state employed in the treatment of the vapour phase with regards to the pressure range. The n- parameter virial equation of state usually suffices as an adequate treatment for the vapour phase up to pressures of 1.5 and 5 MPa with the two and three-parameter truncated forms, respectively

Chapter 3. Theoretical Treatment of Vapour Pressure and Vapour-Liquid Equilibrium Data

(Smith and Van Ness, 1975). However, for higher pressures, the vapour phase nonideality is treated with more accurate cubic or complex equations of state. In the discussion of the Yi - ~i

method, preference is given to the low to moderate-pressure approach as this is relevant to the operating range of the author.

The use of the ~i ^- ~i and Yi - ~iapproaches, each with its own advantages and disadvantages, has been dealt with extensively by many researchers (Raal and Muhlbauer, 1998) and comprises the bulk of this chapter.

3.5.2 The Gamma-Phi(Yi - _~i)Method

3.5.2.1Overview

The Yi - ~ⁱ method, also known as the activity coefficient approach involves the use of two distinct auxiliary functions to separately represent non ideal behaviour of the liquid and vapour phases in the isofugacity expression.

With regards to the vapour phase, non ideal behaviour for a component in the mixture IS

expressed through the use the fugacity coefficient, which is defined below:

(3.48)

where ~~ is the fugacity coefficient of component i in the vapour phase , f~ is the fugacity of component i in the vapour phase, fi

Q is the standard-state fugacity of the vapour phase, P is the total system pressure and Yⁱ is the vapour phase composition. The fugacity coefficient depends on pressure, temperature and in a multicomponent mixture on the mole fractions of all the other components in the vapour phase ( y^{j ).}

For the liquid phase treatment in the in the Y^{i -} ~ⁱ approach, the nonideal behaviour is lumped into the activity coefficient term and defined in an analogous fashion to the fugacity coefficient, below as follows:

fL

I

(3.49)

In Equation (3.49), Yi IS the activity coefficient of component i, f i^L is the fugacity of component i in the liquid phase, ff is the standard-state fugacity of the liquid phase and xi is the liquid phase composition. The activity coefficient depends on temperature, pressure (to a lesser extent) and the component mole fraction in the liquid phase.

The pressure dependence of the activity coefficient at low to moderate pressures is very weak and is usually ignored, however, for higher pressures this dependence has to be accounted for (in the HPVLE Yi - ~i approach).Prausnitz et al. (1980) described the use of an adjusted activity coefficient,

A ^{pr) ,}

shown in Equation (3.50), as a correction for the pressure dependency of the activity coefficientYi , defined in Equation (3.49), for correlating VLE data with the isothermal form of the Gibbs-Duhem equation (where dP and dT are set equal to zero). The Yi value at the system pressure

(p)

can be related to the Yi value at an arbitary reference pressure Le.

(pr).

In this way, the adjusted activity coefficient is independent of the experimental pressure and can be used in both the isobaric and isothermal Gibbs-Duhem equation.

(3.50)

where Vj

L is the partial liquid molar volume of component i.

An examination of Equations (3.48) and (3.49) reveals that the only difference between the two equations is indeed the manner in which the reference state is defined i.e. the choice of reference fugacities. For the vapour phase, the reference fugacity(f_i^D) is that of the partial pressure of an ideal gas in a mixture (Py^{i )} and the fugacity coefficient evaluates to a value of unity for ideal gas conditions.

The normalization of the activity coefficient (specification of a state in which theYi value is unity) can either follow the symmetric or unsymmetric convention with regards to the state of the components presents. If both components are condensable (for LPVLE measurements), the symmetric convention can be used, as shown below:

Yi ~1 as xi ~1 (3.51 )

Chapter 3. Theoretical Treatment of Vapour Pressure and Vapour-Liquid Equilibrium Data

However, the presence of a supercritical or noncondensable component in the liquid solution necessitates an unsymmetric normalization convention with regards to the two components, where for the noncondensable component the following applies:

(3.52)

where Y~ denotes that the definition is appl icable to supercritical components. The presence of a supercritical component and the resulting normalization represented by Equation (3.52) in fact violates the theoretical framework of the Yi - ~i approach with regards to the treatment of the liquid phase.

In the event that the liquid phase contains only condensable components, a suitable standard state for the components is obtained through an integration ofthe expression in Equation (3.53).

RT In fⁱ

D

= fP[V

^{L _}RT] dP

P

J

^o ' P (3.53)

where R is the universal gas constant, T is the system temperature, P is the system pressure, fi

D is the standard-state fugacity for the liquid phase and V_i^Lis the partial molar volume for the component i. The above integral is then split into two parts i.e. from a pressure of zero to the saturation pressure and from the saturation pressure to the system pressure (the effect of compressing the liquid) in the form of Equation (3.54) and (3.55).

fsat RTln-'-

=

p

,

sat

i

o

^Pt"[-

^{V - -}' ^RT]P ^dP (3.54)

RT In fⁱ

D

= f

^P

[V

^{L _ RT] dP}

P P,sa, , P (3.55)

After rearrangement and simplification, the above two expressions evaluate to Equation (3.56).

f_,^O

= ~satpsatexp[

_{I ' Jp;sa,}

^fP

^V_RTⁱ

^L

^dP]

where p^jsat

is the saturated vapour pressure of the pure liquid i at the temperature T of the (3.56)

system, <j>~a, is the fugacity coefficient of the pure saturated vapour i at pta, and temperature T and V_j^Lis the liquid molar volume of pure liquid i at T (for a condensable component at the

L - L

system temperature Vi = Vi ).

The difficulty of the Yi - <j>i approach in the treatment of a supercritical component is clearly evident in the expression for the standard-state liquid fugacity, which is valid only for the symmetric normalization in Equation (3.51), as the existence of a pure supercritical liquid is a physical impossibility (Prausnitzet al., 1980). Even with the introduction of a hypothetical or imaginary standard state for a pure supercritical liquid, determination of liquid phase fugacities through extrapolation from imaginary vapour pressures is an unattractive solution. A further discussion of the treatment of supercritical components in phase equilibrium computations is available in works of Raal and Muhlbauer (1998) and Prausnitzet al. (1980).

The exponential of the bracketed term in Equation (3.56) is known as the Poynting correction and is generally insignificant at low to moderate pressures, but increases significantly with an increase in pressure or at lower temperatures (as the pressure dependency of the liquid molar volume increases). Generally, V_i^L is assumed to be independent of pressure so as to simplifY calculations, as in Equation (3.57).

(3.57)

The above is only justified when liquids are incompressible for small, isothermal changes in pressure. When the difference between the system pressure and saturation pressure of the liquid is not considerable, the Poynting correction is close to unity and is omitted and since the gas is essentially an ideal gas at saturation pressure (not applicable to carboxylic acids and strongly associating compounds), <j>~al

=

1. With these two assumptions,f

i

D

=

^{Ptal .}

Substituting Equation (3.57) into Equation (3.49), applying the isofugacity criterion in Equation (3.46) and simplifYing, yields the following working equation for the Yi - <Pi computation of vapour-liquid equilibria:

Y P<f>

=

xy. p^sa'

I I I J I

From Equation (3.58), a new simplifYing term i.e.<Dj, has been introduced.

(3.58)

Chapter 3. Theoretical Treatment of Vapour Pressure and Vapour-Liquid Equilibrium Data

The <Dj term can be quantified as follows:

<D.

=

(1L)ex

p

[V^j

L

(p

^{sat _}

p)~

1 ~fat RT ¹

J

^(3.59)

An examination of Equations (3.58) and (3.59) for the computation of the equilibrium condition yields an interrelation between the experimentally accessible variables and the auxiliary functions that are used to quantify the vapour phase and liquid phase nonidealities in the form of the fugacity coefficient and the activity coefficient, respectively. The term <Djcontains, in addition to the Poynting correction, the~fatand ~: terms. For the evaluation of the Poynting correction factor, a suitable equation for the volume of the saturated liquid phase as a function of temperature is required. The Rackett correlation, as modified by Spencer and Danner (1972) is recommended in various literature sources (Reid et al., 1987) for the estimation of liquid molar volumes.

3.5.2.2 Calculation of the vapour phase nonideality

As mentioned above, in the ^{Yj -} _~j approach it is the liquid phase nonideal behaviour that is correlated through the activity coefficient with the various models of the excess Gibbs energy as a function of composition. Consequently, all vapour phase non ideal behaviour i.e. the fugacity coefficients, have to be determined indirectly or in an independent way. To this end, an equation of state relating the PVT properties of the fluid is required to enable the suitable calculation of the vapour phase fugacities. Equations of state are often most conveniently written in terms of the compressibility factor

(Z),

which for an ideal gas, is defined as:

Z

=

Z

=

PV^rn

=

1

o RT (3.60)

The above defines the limiting value of Z as zero density is approached. The theoretically-based truncated virial equation of state is the preferred choice for the representation of the vapour phase nonideality in the ^{Yj -} ~j approach for pressures below 0.5 MPa (Abbott, 1986). The virial equation of state was derived from a statistical mechanical framework and consequently has a sound theoretical basis for the representation of properties of pure gases and mixtures. The equation, as its name implies, is actually an expression for the compressibility correction factor as a McLaurin power series as function of density (reciprocal molar volume) in the form of the

volume explicit Leiden version or in the pressure-explicit form, known as the Berlin form.

These two forms are shown below as Equations (3.61) and (3.62), respectively.

B C

Z

=

I += + - 2 + .

V V (3.61)

where V is the molar volume, B is the second virial coefficient and C is the third virial coefficient.

Z

=

I +B'P+ C'p²+ . (3.62)

The second (B' ) and third (C' ) virial coefficients in the above pressure-explicit expansion are related to those in the volume-explicit expansion in Equation (3.61) through Equations (3.63) and (3.64), shown below:

B'=~

RT (3.63)

(3.64)

It must be remembered that for the expressions in (3.63) and (3.62) to be exact, the two power series shown above in Equations (3.61) and (3.62) have to retain the form of mathematically infinite Taylor series. However, to simplify the calculations and due to the lack of reliable experimental or predicted higher virial coefficients, the virial equation of state is for the most part usually truncated at the second virial coefficient. This results in a discrepancy (Malanowski and Anderko, 1992) between the values obtained through the use of Equations (3.61) and (3.62), which is typically 5 %. More importantly, this truncation strictly renders the virial equation of state applicable at low to moderate pressures. The virial equation can in principle be used at higher pressures; however, a larger number of virial coefficients (knowledge of which is highly limited) are required to correctly represent the PVT properties of fluids.

The virial equation is also limited to the treatment of only a single phase as the compressibility correction factor is not differentiable at the phase boundary between the one-phase and the two- phase regions. This is clearly visualized in the isotherms of the virial equation (Walas, 1985), which do not possess the S-shape (a characteristic for a proper equation of state for both

Chapter 3. Theoretical Treatment of Vapour Pressure and Vapour-Liquid Equilibrium Data

phases). Consequently, even with a large number of terms, it cannot represent liquid phase behaviour or the coexistence of the liquid and vapour phase. The apparent lack of applicability of the virial equation of state to higher pressure ranges necessitates the use of more accurate equations of state such as the cubic-type in HPVLE computations with the Yi - ~i approach.

As a rough approximation of the pressure range of applicability of the virial equation of state, Prausnitzet al. (1980) recommend the following relation as an indication of the latter:

(3.65)

Despite the limitations of the truncated-virial equation of state, its justification for suitable use in the Yi -~i approach stems from its demonstrated superiority over other traditional equations of state(e.g. cubic) in the treatment of the vapour phase in the low-pressure region (Sandleret al., 1994). A significant attribute of the virial expansion for gases is the relationship between the virial coefficients and the intermolecular forces i.e. for an ideal gas, the intermolecular forces are zero and the compressibility correction factor is unity. For real gases, statistical mechanics can be employed to relate expressions for the virial coefficients to intermolecular forces.

The virial coefficients obtained from Equations (3.61) and (3.62) for pure substances are temperature-dependent only.

For the treatment of mixtures, the mixture second virial coefficient i.e. B . is a function of both

mix

the temperature and the concentration of all the mixture components and can be obtained from a statistical-mechanical rigourous concentration dependent mixing rule shown in Equation (3.66).

n n

B_mix

= L: 2:>;

^{yjB ij}

j

(3.66)

for the m vapour phase compositions of components i and J' and where B·· is the interaction

I)

virial coefficient or the cross coefficient that represents the bimolecular interaction between the moleculesi andj.

For a binary mixture, this reduces to:

(3.67)

where the B_JIand B₂₂ terms represent the pure component virial coefficients and the B_l2 term is the mixture cross coefficient; all of which are functions of temperature only.

To evaluate the fugacity coefficient terms i.e. _~~atand _~; in the <Dj term in Equation (3.59), the virial equation of state truncated after the second virial coefficient is substituted as shown below into the following:

From equation (3.60),

f ^{P[ V I]}

In~.

=

^{_ I - -} dP

I 0 RT P

Z _ 1

=

^BP

RT

(3.68)

(3.69)

Rearrangement and substitution of Equation (3.69) into Equation (3.68) yields the following:

r

^p

[Z-1 ~

In~j

= J

_o

pJP

^(3.70)

An evaluation ofthe above for the pure saturated vapour and for the mixture components in the vapour phase, allows for the working equations for ~fatand ~;to be obtained:

where

Bp sat In",sat

= __

^{j _}

'PI RT

ln~v =~(B

_{I RT 11} ^+y_J²

8)

_IJ

8_IJ

=

(2B -B· -B .. )_{IJ 11 .u}

(3.71 )

(3.72)

(3.73)

Chapter 3. Theoretical Treatment of Vapour Pressure and Vapour-Liquid Equilibrium Data

The values of the virial coefficients can be determined from experimental measurements in the form of PVT data, from statistical mechanics where the pair intermolecular energy is quantified through the use of potential functions or empirical or semi-theoretical correlations. There are very few literature sources for experimental virial coefficients, which also display considerable scatter due to the different experimental methods used to obtain them. It has also indeed been acknowledged by many researchers (Meng et al., 2004) that in the older PVT data no attempt to compensate for the effects of physical adsorption was made, producing values that were far too negative for subcritical conditions. The most popular compilations of virial coefficients (Walas, 1985) are those of Dymond and Smith (1980) and Kogan (1968).

Analytical expressions for the prediction or correlation of virial coefficients can essentially be divided into two approaches i.e. those that are based purely on the corresponding states or on the extended corresponding states method and those that incorporate statistical mechanical considerations in the calculation of the second virial coefficient. With regards to chemically reacting systems (beyond the scope of this study), a chemical theory of vapour imperfections (Marek, 1955; Prausnitz, 1969; Nothnagel et aI., 1973; Prausnitz et aI., 1980) is required to calculate fugacity coefficients based on the true equilibrium concentrations i.e. equilibrium constants.

The corresponding states principle in its original form was essentially a two-parameter relation with reduced temperature, Tr , and reduced pressure, Pr' as the independent variables and was consequently only applicable to the simple, monoatomic molecules, where the latter did not exhibit size-shape, polarity or association effects.

The work of Pitzer and Curl (1957) was to extend the corresponding states principle to include a third parameter, known as the acentric factor

(co)

to account for size-shape effects or the non- spherical nature of nonpolar molecules:

co =

-1.0 - log (Pr^sat

)T =07

_{, .} (3.74)

where Pr^sat is the reduced vapour pressure at a Tr

=

0.7, since for simple monatomic gases P_r^sat =1 at T_r =0.7.

There have been other "third parameters" that have also been proposed by a variety of researchers (Tarakad and Danner, 1977) to characterize the non-sphericity of molecules.

The correlation for the second virial coefficient proposed by Pitzeret al. (1957) was the following:

(3.75)

where theB(0)and B(I)terms are expressions that are functions of T_r since B is a function of temperature only. The correlation above served as the inspiration for later, more successful correlations (Walas, 1985) such as that of Abbott (Smith and Van Ness, 1975) and Tsonopoulos (1974). TheB(O)and B(I) terms for the correlations of Pitzer and Curl, Abbott and Tsonopoulos are presented in Appendix B. The correlations based on the original Pitzer-Curl correlation suffer from the limitations of being applicable only to nonpolar, non-associating gases and more elaborate treatments are required i.e. consideration of a fourth parameter or additional terms to extend the corresponding states principle to polar fluids.

Tsonopoulos (1974) correlated second virial coefficients for polar non-associating (ketones, acetaldehyde, acetonitrile, ethers) and polar associating compounds (alkanols, water) through the use of additional terms with empirical parameters in conjunction with the same terms for nonpolar gases (see Appendix B) to obtain the Pitzer-Tsonopoulos correlation:

(3.76)

The Tsonopoulos correlation provides a very simple, purely empirical approach for correlating complex polar non-associating and associating compounds and has limitations at high values of reduced temperatures, where erroneous results are obtained.

Another corresponding states correlation for polar fluids of note is that of Tarakad and Danner (1977), which is a four-parameter corresponding states method. The four parameters used, apart from the reduced temperature and pressure are the radius of gyration and an empirical polarity factor. The radius of gyration, R , was used as the third parameter to effectively represent the non-spherical nature of molecules i.e. size-shape effects, since the use of the acentric factor as the third parameter does not strictly preclude the effects of polarity on the virial coefficient i.e.

the acentric factor is influenced by the polar nature of the molecule. A correlation with distinct terms for the size-shape and polarity contributions allows for a true measure of the size-shape effects of polar compounds as a contribution to the second virial coefficient terms. The fourth parameter

(et> i)

was obtained in an empirical fashion by correlating the difference between