Mixing Rules

6. Equations Of State And Excess Energy Models

6.3. Mixing Rules

44 | P a g e polar components. The MC alpha function contains three constants regressed from vapour pressure data. It is consequently better at predicting vapour pressures than the original function and is the function of choice in this work. It is defined as (Mathias & Copeman 1983):

𝛼𝑖(𝑇) = [1 + 𝑐_1,𝑖(1 − √𝑇𝑟,𝑖) + 𝑐2,𝑖(1 − √𝑇𝑟,𝑖)²+ 𝑐3,𝑖(1 − √𝑇𝑟,𝑖)³]²

6.25

where 𝑐1,𝑖, 𝑐2,𝑖, 𝑐3,𝑖 are the MC parameters which are regressed from pure-component vapour pressure data of species i. The function can correlate data accurately from the triple point to the critical temperature. In the supercritical region it presents abnormal behaviour where it increases with increasing temperature (Twu et al. 2002). Vapour pressures were only measured below the critical point in this work, a region where the MC alpha function can accurately represent the data. The MC alpha function was used in this work.

In mixtures the parameters 𝑎 and 𝑏 are also concentration dependent and this gives rise to mixing rules (Valderrama & Zavaleta 2005). The fugacity coefficient of a component i in the mixture then becomes:

𝑙𝑛𝜑̂_𝑖 = 𝑏_𝑖

𝑏_𝑚(𝑍 − 1) − ln(𝑍 − 𝐵_𝑚) − 𝐴_𝑚 2√2𝐵𝑚

[2 ∑^𝑛_𝑗=1𝑧_𝑗𝑎_𝑖𝑗 𝑎_𝑚 − 𝑏_𝑖

𝑏_𝑚] 𝑙𝑛 (𝑍 + (1 + √2)𝐵_𝑚 𝑍 + (1 − √2)𝐵𝑚

)

6.26

where 𝑎𝑚 and 𝑏𝑚 form the mixing rules.

45 | P a g e model to experimental data. Thus only when the classical mixing rules fail should one use the more advanced multiple parameter models.

The van der Waals One Fluid Classical Mixing Rules

The one-fluid mixing rules were developed from the virial equation of state. There is a relation between the parameters of a cubic equation of state and the virial coefficients of the virial equation of state. The relation is that the low density composition dependence of a cubic EoS can be the same as the virial expansion if the cubic EoS parameters satisfy the van der Waals one-fluid mixing rule (Naidoo 2004).

From the virial expansion it can then be deduced that the second virial coefficient boundary condition is that the composition dependence must be quadratic in nature. The mixing rules are given by (Peng &

Robinson 1976):

𝑎_𝑚= ∑ ∑ 𝑥_𝑖𝑥_𝑗𝑎_𝑖𝑗

𝑗 𝑖

6.27

𝑏_𝑚= ∑ ∑ 𝑥_𝑖

𝑗

𝑥_𝑗𝑏_𝑖𝑗

𝑖

6.28

Equations 6.27 and 6.28 represent interactions between the species in the mixture. The cross parameters are linked to pure fluid parameters by what are known as the combining rules:

𝑎_𝑖𝑗 = √𝑎𝑖𝑎_𝑗(1 − 𝑘_𝑖𝑗)

6.29

𝑏_𝑖𝑗=1

2(𝑏_𝑖+ 𝑏_𝑗)(1 − 𝑙_𝑖𝑗)

6.30

The parameter 𝑘𝑖𝑗 is an adjustable binary interaction parameter that must be regressed from experimental phase equilibrium data. The parameter 𝑙𝑖𝑗 is set to zero for non-polar and most simple mixtures (Naidoo 2004). It was set to zero in this work.

46 | P a g e G^E mixing rules

For mixtures of greater complexity Huron & Vidal (1979) proposed that excess Gibbs energy models and EoS be used together. The key idea was to match the excess Gibbs free energy of an EoS with that of an activity coefficient model at a specified state. Two boundary conditions were satisfied. The first is that at low densities the composition dependence must be quadratic in nature. The second comes from requiring that the excess Gibbs energy from an EoS at liquid-like (high) densities be equivalent to that from an activity coefficient model (Sandler 2006). This is done to solve for the mixing rule 𝑎𝑚 in the vdW type equations of state using the equation (Voutsas et al. 2004):

𝐺_𝐸𝑂𝑆^𝐸 (𝑥_𝑖, 𝑇, 𝑃^𝑟𝑒𝑓) = 𝐺_𝐴𝐶^𝐸 (𝑥_𝑖, 𝑇, 𝑃 = 0)

6.31

Huron & Vidal (1979) matched the excess Gibbs free energy (G^E) of an activity coefficient to that of the EoS at infinite pressure such that 𝑃^𝑟𝑒𝑓in the above equation is infinite pressure. An EoS and the excess Gibbs free energy are related at infinite pressure by:

𝐺^𝐸= 𝐴^𝐸+ 𝑃𝑣^𝐸

6.32

It follows that the excess molar volume 𝑣^𝐸 must approach zero if G^E is to remain finite at infinite pressure. At liquid densities, while the excess Helmholtz free energy 𝐴^𝐸 is insensitive to pressure, 𝐺^𝐸 diverges as 𝑃 → ∞ because of the 𝑃𝑣^𝐸 term. Thus most G^E models assume that 𝑣^𝐸= 0 whereas the G^E of an EoS is a function of pressure. As a consequence limiting values for pressure (𝑃 → ∞, 𝑃 → 0) are used to obtain a G^E mixing rule (Fischer & Gmehling 1996). It is also useful to use 𝐴^𝐸 rather than 𝐺^𝐸. The Huron and Vidal mixing rule had some shortcomings (Sandler et al. 1994) among which it did not account for the pressure effects as the G^E value is different at low pressure and at infinite pressure. The Huron-Vidal mixing rule does not satisfy the second virial boundary condition. The resulting model could not directly utilise existing activity coefficient parameters correlated from low pressure data. To correct this Mollerup (1986) and Dahl & Michelsen (1990) proposed matching G^E of an activity coefficient to that of an EoS at low pressure. The attractive parameter is evaluated at low or zero pressure as well as maintaining that excess molar volume is zero. The zero-pressure approach has the disadvantage that the EoS zero pressure liquid exists at low temperatures therefore an extrapolation has to be made to obtain a hypothetical zero pressure liquid at higher temperatures (Voutsas et al. 2004).

Modifications to extend the applicability of this approach were done and these include the modified Huron-Vidal first order (MHV1) and second order (MHV2) and the Predictive Soave-Redlich-Kwong

47 | P a g e mixing rules. These models improved correlative and predictive abilities but still do not satisfy the second virial coefficient boundary condition. Wong and Sandler (1992) developed the most capable mixing rule. The Wong-Sandler mixing rule is consistent with experimental data at the high density limit and satisfies the second virial coefficient boundary condition at the low-density limit (Wong &

Sandler 1992). This makes it one of the most wide-ranging mixing rules available. Wong et al. (1992), Sandler (1994), Coutsikos et al. (1995) and Satyro & Trebble (1998) have all discussed the capabilities of the Wong-Sandler mixing rules.

The Wong-Sandler (WS) Mixing Rules

The Wong-Sandler mixing rules have the advantage that they are capable of modelling highly non-ideal systems. Of all excess energy mixing rules, WS mixing rules are the most widely used and are used in this work. The WS mixing rules, however, use the excess Helmholtz free-energy A^E at infinite pressure rather than G^E (Valderrama & Zavaleta 2005) and the advantage is that A^E is not as strongly dependent on pressure as G^E and the assumption that 𝑣^𝐸= 0 is not necessary (Nelson 2012). The insensitivity to pressure means that low pressure activity coefficient model parameters are found to be useful. The WS mixing rules are summarised as follows (Wong & Sandler 1992):

𝑏_𝑚=

∑ ∑ 𝑥_𝑖𝑥_𝑗(𝑏 − 𝑎 𝑅𝑇)_𝑖𝑗

𝑛𝑗 𝑛𝑖

1 −∑ 𝑥^𝑛_{𝑖 𝑖}𝑎_𝑖

𝑏_𝑖𝑅𝑇 −𝐴_∞^𝐸(𝑥) 𝜀𝑅𝑇

6.33

(𝑏 − 𝑎 𝑅𝑇)

𝑖𝑗=1

2[𝑏_𝑖+ 𝑏_𝑗] −√𝑎𝑖𝑎_𝑗

𝑅𝑇 (1 − 𝑘_𝑖𝑗)

6.34

𝑎_𝑚 = 𝑏_𝑚[∑ 𝑥^𝑛_{𝑖 𝑖}𝑎_𝑖

𝑏_𝑖 +𝐴^𝐸_∞(𝑥) 𝜀 ]

6.35

For the PR EoS 𝜀 = 1

⁄√2𝑙𝑛 (√2 − 1)

.

The adjustable binary interaction parameter 𝑘𝑖𝑗 is included in the combining rule in equation 6.34 and therefore the equation of state. The excess Helmholtz free- energy of the EoS is matched with that of an activity coefficient model to solve for the attractive parameter 𝑎𝑚 by these equations:

48 | P a g e 𝐴_𝐸𝑂𝑆^𝐸 (𝑥_𝑖, 𝑇, 𝑃 → ∞) = 𝐴_𝐴𝐶^𝐸 (𝑥_𝑖, 𝑇, 𝑃 → ∞)

6.36

The excess Helmholtz free-energy at infinite pressure, 𝐴∞𝐸(𝑥) and the excess Gibbs free energy at low pressure 𝐺^𝐸 are approximated by assuming that

𝐴^𝐸_∞(𝑥) ≈ 𝐴₀^𝐸(𝑥) ≈ 𝐺₀^𝐸(𝑥) = 𝐺^𝐸

6.37

This equality is possible because of the insensitivity to pressure of A^E. The excess Gibbs free energy and hence the excess Helmholtz free energy at infinite pressure is calculated by an activity coefficient model such as the Van Laar model, Wilson model or the Non-Random Two Liquid (NRTL) activity coefficient model. The NRTL model was used to calculate the excess Gibbs free energy in this work.

Wilson and NRTL models make use of the local composition concept which provides a convenient method for introducing non-randomness in a liquid mixture (Renon & Prausnitz 1969). The Wilson model however has some disadvantages. In particular it is not applicable to mixtures with only partial liquid miscibility. Moreover, the NRTL provides good representation of a wide variety of mixtures including highly non-ideal mixtures with partial miscibility. The NRTL was also used by authors who worked with fluorocarbons (Valtz et al. 2007; Ramjugernath et al. 2009; Subramoney et al. 2010; Nandi et al. 2013) and excellent results were obtained. It should be noted however, that due to the chemical nature of the components used in this work, similar results can be expected for the NRTL, Wilson and Van Laar models.

6.4. The Non-Random, Two Liquid Activity Coefficient Model (Renon & Prausnitz