Cubic EoS

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The following cubic EoS are available in TP-Cloud:

• Soave Redlich Kwong (SRK)
• Peng Robinson (PR)
• SRK78
• PR78
• Quantum-corrected SRK
• Quantum-corrected PR
• Translated-consistent Redlich Kwong (tc-RK)
• Translated-consistent Peng Robinson (tc-PR)
• Patel Teja (PT)
• Schmidt-Wensel (SW)

They are all compatible with a Péneloux-volume shift, and can be combined with the following mixing rules:

• Standard mixing rules
• Huron Vidal with a 0th, 1st or 2nd order polynomial
• Wong Sandler
• UNIFAC
• UMR
• PSRK

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The general formulation of cubic EoS

The pressure of cubic EoS is formulated as ¹:

$$P_{\text{cubic}} = \frac{RT}{v-b}-\frac{\tilde{a}}{(v-r_1b)(v-r_2b)}$$

where R is the Universal gas constant, T is the temperature, v is the molar volume, b is the co-volume, and $r_1$ and $r_2$ are constants that are different for different types of cubic EoS. Moreover:

$$\tilde{a}=a_c\alpha$$

where $a_c$ is a constant that ensures that the cubic EoS gives the correct critical temperature and pressure for pure components and $\alpha$ is usually a temperature dependent function that depends on the component. To improve the accuracy in representation of liquid phase densities, Péneloux volume shift is often used. Advanced mixing rules can be used to improve the accuracy of phase equilibrium predictions, also for associating components. The pressure is integrated to obtain the residual Helmholtz energy:

$$A_{\text{cubic}}^{\text{r}} = -\int_{\infty}^{V}\left(P_{\text{cubic}}-\frac{RT}{v}\right)dV$$

where the ideal gas pressure is subtracted, and superscript "r" means residual with respect to the properties of an ideal gas. Computing the integral analytically gives:

$$\frac{A_{\text{cubic}}^{\text{r}}}{RT} = -\ln{\left(1-\frac{b}{v}\right)}+\frac{\tilde{a}}{RTb\left(r_1-r_2\right)}\ln{\left(\frac{1-r_1b/v}{1-r_2b/v}\right)}$$

which can be differentiated with respect to temperature, volume and number of particles to obtain other thermodynamic properties. The constants $r_1$ and $r_2$ for the cubic EoS that are available in TP-Cloud are listed in the table below, where $c$ is a parameter and $w$ is the acentric factor. :

Péneloux shift

Cubic equations of state are usually accurate in prediction of gas-phase properties and phase equilibria, but less accurate in prediction of liquid-phase volumes. This can to some extent be alleviated by using a Péneloux volume shift ⁸. Solving the equation:

$$P_{\text{cubic}}(T_{\text{spec}},v_\text{cubic}) = P_{\text{spec}}$$

results in a predicted molar volume from the cubic EoS, $v_\text{cubic}$, which is shifted as follows:

$$v_\text{cubic,vt}=v_\text{cubic}-c$$

to arrive at a volume translated (subscript vt) molar volume. Since the molar volume of a gas is much larger than the liquid, the volume shift will mainly influence the liquid-phase. Since $c$ is a constant, or a temperature-dependent function, the volume shift does not influence phase-equilibrium compositions. The constant depends on the component and the cubic EoS.

$\alpha$-functions (SRK, PR, SRK78 and PR78)

The $\alpha$-function, its functional form and parameters is important for accurate representation of both phase equilibria with both one and several components. The $\alpha$-function of many EoS follows the formulation by Graboski and Daubert ⁹:

$$\alpha(T_r)=\left(1+m\left(1-T_r^{0.5}\right)\right)^2$$

where $T_r$ is the reduced temperature. The original SRK formulation uses:

$$m_\text{SRK}=0.480+1.574w-0.176w^2$$

whith a different expression for hydrogen. For PR, the expression is:

$$m_\text{PR}=0.37464+1.54226w-0.26992w^2$$

In 1978, these functions were improved, leading to what has been called the SRK78 ¹⁰:

$$m_\text{SRK78}=0.4797+1.576w-0.1925w^2+0.025w^3$$

and there was also made improvement to the EoS by Peng Robinson EoS in the same year, resulting in what is know as PR78 EoS ¹¹ with

$$m_\text{PR78}=m_\text{PR78} \qquad \text{if w}\leq 0.491$$

and

$$m_\text{PR78}=0.379642+1.48503w-0.164423w^2+0.016666w^3 \qquad \text{if w}> 0.491$$

The advantage of the above expressions is they all use the acentric factor of the fluid as input, which has been tabulated for many fluids. However, the most accurate predictions with cubic EoS are achieved using the 3-parameter Twu91 $\alpha$-function ¹²:

$$\alpha(T_r)=T_r^{N(M-1)}\exp{\left[L(1-T_r^{MN})\right]}$$

which uses the parameters $N$, $N$ and $L$, which can be fitted for each component.

Translation consistent cubic EoS (tc-PR and tc-RK)

In 2016 ¹³, Le Guennec and coworkers combined the best working components of the machinery behind cubic EoS and developed the translated-consistent cubic equations of state tc-PR and tc-RK. Making use of consistency tests, they fitted the parameters $N$, $N$ and $L$, in the Twu91 $\alpha$-function and a constant Peneoulx shift to reproduce the experimental saturated liquid volume at a reduced temperature of 0.8 for a large number of components. For non-associating and non-polar components, this is arguably the most accurate formulation of cubic EoS available at the moment. These cubic EoS and the fitted parameters are available in TP-Cloud.

Quantum-corrected cubic EoS

It was recognized by Soave already in 1972 ⁴ that cubic EoS worked less well for hydrogen. The strong quantum effects for fluids such as hydrogen, helium, neon and deuterium at low temperatures were incorporated by Aasen and coworkers in 2020 ¹⁴ by deriving a temperature-dependent co-volume, and refitting the Twu91 $\alpha$-functions of the fluid influenced by quantum effects. This has resulted in quantum-corrected SRK and quantum-corrected PR, which are available in TP-Cloud. An example is shown below, where the quantum cubic EoS for normal hydrogen (dashed line), gives nearly identical supercritical densities as the reference multiparameter EoS for hydrogen (solid line).

Mixing rules

There are several types of mixing rules that can be used to estimate the properties of the mixture on the basis of the pure-component parameters, and possibly additional binary parameters. The standard van der Waals mixing rules with a regressed interaction coefficient gives an excellent representation of phase-equilibria of non-associating and non-polar substances. The Huron-Vidal and Wong-Sandler mixing rules were developed to incorporate association into the cubic EoS framework. NRTL, UNIFAC, UMR and PSRK are exples of other mixing rules that are available in TP-Cloud. A description of these mixing rules and when they should be applied can be found below. For the co-volume, the standard mixing rule is:

$$b=\sum_{i=1}^{N_c}\sum_{j=1}^{N_c}x_ix_j\left[\frac{b_i^{1/s}+b_j^{1/s}}{2}\right]^s\left(1-l_{ij}\right)$$

where the cross-volume parameter, $l_{ij}$, is usually set to zero and the parameter $s$ is usually set to 1. The attractive parameter is usually mixed according to the following quadratic mixing rule:

$$\tilde{a}=\sum_{i=1}^{N_c}\sum_{j=1}^{N_c}x_ix_j\sqrt{\left(\tilde{a}_i\tilde{a}_j\right)}\left(1-k_{ij}\right)$$

where the unknown interaction coefficient, $k_{ij}$ is regressed such that the cubic EoS reproduces the vapor-liquid equilibrium between components $i$ and $j$. In the equation above, $x_i$ is the mole fraction of component $i$.

Excess Gibbs energy models can be included in cubic EoS as mixing rules by considering the infinite ($G_{\infty}^{\text{ex}}$) or zero ($G_{0}^{\text{ex}}$) pressure limits. The mixture attraction parameter $\tilde{a}$ of the cubic EoS then takes the following form in the infinite pressure limit:

$$\frac{\tilde{a}}{RTb}=\sum_ix_i\frac{\tilde{a}_i}{RTb_{i}} - \frac{1}{h_\infty}\frac{G_\infty^{\text{ex}}}{RT}.$$

where $h_\infty = \frac{1}{r_1 - r_2} \ln \frac{1-r_2}{1-r_1}$, and $r_1$ and $r_2$ are parameteres in the cubic EoS as explained above. In the zero pressure limit, the following expression is used for the mixture attraction parameter $\tilde{a}$:

$$\frac{\tilde{a}}{RTb} = \sum_ix_i\frac{\tilde{a}_i}{RTb_{i}} - \frac{1}{h_0}\left(\sum_ix_i\ln\frac{b}{b_{i}} + \frac{G_0^{\text{ex}}}{RT}\right),$$

where $h_0$ equals $-0.593$ for SRK and $-0.53$ for PR. A model for $G_{\infty}^{\text{ex}}$ or $G_{0}^{\text{ex}}$ thus yields one equation for the two mixture parameters $a$ and $b$, and one additional equation is therefore needed to complete the mixture rule.

Some of the most common excess Gibbs energy ($G_{\infty}^{\text{ex}}$) mixing rules are group contribution methods. The underlying idea of group contribution methods is to treat pure species as being composed of functional groups, and then model group-group interactions. If the group interaction energies are known from e.g. fits to experimental vapor-liquid equilibrium data, these parameters can be used to predict properties for any molecule comprised of known functional groups. Examples of such groups are $CH_2$ and $CH_3$, that can be thought of as monomers in a hydrocarbon polymer. A molecule too small to divide into groups is usually considered a group in itself; in particular, this is the case for both carbon dioxide and water.

UNIFAC mixing rules

The UNIFAC (UNIQUAC Functional-group Activity Coefficients) model ¹⁵ is a group contribution version of the UNIQUAC model ¹⁶. In the original UNIFAC model, the overall excess Gibbs energy is the sum of two terms: a combinatorial contribution describing the excess Gibbs energy arising from differences in molecular size and shape, and a residual term describing the excess Gibbs energy differences due to molecular interactions. The UNIFAC residual term, $G^r$, is in TP-Cloud used according to the definition by Fredenslund et al.¹⁵. Adding the combinatorial term $G^{\text{ex,c}}$, the overall excess Gibbs energy for the UNIFAC model becomes

$$\frac{G^{\text{ex}}}{RT} = \frac{G^r}{RT} + \frac{G^{\text{ex,c}}}{RT}.$$

The $G^{\text{ex,C}}$ term is derived on the basis of statistical mechanics. Other mixing rules such as UMR, and PSRK use different variants of the UNIFAC mixing rules.

Huron Vidal mixing rules

Huron and Vidal ¹⁷ derived an expression for the infinite pressure excess Gibbs energy of cubic EoS, and equated it with a modified NRTL model that contains the quadratic mixing rule as a special case. The modified NRTL model is given by

$$G_{\infty}^{\text{ex}}=RT\sum^{N_C}_{i=1} x_i\frac{\sum^{N_C}_{j=1}\tau_{ji}b_jx_jC_{ji}}{\sum^{N_C}_{k=1} b_kx_kC_{ki}}, \qquad \tau_{ji} = \frac{\Delta g_{ji}}{RT}.$$

Here, $\Delta g_{ji}/R$ is either a zeroth, first, or second order polynomial in temperature

$$\Delta g_{ji}/R=d_{ji}+e_{ji} \cdot T+f_{ji} \cdot T^2,$$

and $C_{ji}=\exp{(-\alpha_{ji}\tau_{ji})}$. In this work, we have used $\alpha_{ij} = \alpha_{ji} = 0.03$. Indeed, the model in the above equation is insensitive to the value of $\alpha_{ij}$. Depending on the degree of the polynomial $\tau_{ji}$, the models employ 2, 4 or 6 binary interaction parameters, which in TP-Cloud will be referred to as Huron-Vidal 0, Huron-Vidal 1, and Huron-Vidal 2, respectively, where the number refers to the degree of the polynomial.

Wong Sandler mixing rules

Wong and Sandler ¹⁸ derived an expression for the infinite pressure excess Helmholtz energy $A_{\infty}^{\text{ex}}$ of a cubic EoS, and equated it with the NRTL model, which after derivations gives:

$$b - \frac{a}{RT} = \sum_i \sum_j x_i x_j \left(b - \frac{a}{RT} \right)_{ij},$$

The Wong--Sandler rule uses the three binary interaction parameters $k_{ij}$, $e_{ij}$ and $e_{ji}$, given by

$$\Delta g_{ji}/R=e_{ji} \cdot T,$$

and

$$b_{ij}-\frac{a_{ij}}{RT} = \frac{\left( b_i-\frac{a_i}{RT} \right) + \left(b_j-\frac{a_j}{RT} \right)}{2}(1-k_{ij}).$$

PSRK mixing rules

PSRK (predictive SRK) uses the UNIFAC excess Gibbs mixing rule ¹⁹. PSRK gives reasonable predictions if the components are of similar sizes and lengths.

UMR mixing rules

The UNIFAC excess Gibbs mixing rule is also used as the basis for the "Universal Mixing Rule" (UMR) presented by Voutas and coworkers in 2004 ²⁰. If the mixing rule is combined with the PR cubic EoS, the combined model is denoted PR-UMR. PR-UMR improves the performance of the earlier PSRK model, which gives poor predictions in highly non-symmetric systems, containing both long chain molecules and short chain molecules. UMR applies the zero pressure limit when including the excess Gibbs energy in the cubic EoS, but ignores the logarithmic term. As a consequence, the expression for the attraction parameter $a$ becomes as for the infinite pressure limit, using $h_0$ instead of $h_\infty$.

UMR relies on the original temperature-independent UNIFAC parameters published by Hansen ²¹ and the Dortmund Data Bank ²². Furthermore, $s=2$ is usually used as covolume mixing parameter for this model formulation.

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References

Footnotes

1. Kontogeorgis, Georgios M., and Ioannis G. Economou. "Equations of state: From the ideas of van der Waals to association theories." The Journal of Supercritical Fluids 55.2 (2010): 421-437. ↩

2. van der Waals; J. D. (1873). Over de continuiteit van den gas- en vloeistoftoestand (On the Continuity of the Gaseous and Liquid States) (doctoral dissertation). Universiteit Leiden. ↩

3. Redlich-Kong, On the thermodynamics of solutions; an equation of state; fugacities of gaseous solutions. Chem. Rev., 44 (1949): p. 233. ↩

4. Soave, Giorgio. "Equilibrium constants from a modified Redlich-Kwong equation of state." Chemical engineering science 27.6 (1972): 1197-1203. ↩ ↩²

5. Peng, Ding-Yu, and Donald B. Robinson. "A new two-constant equation of state." Industrial & Engineering Chemistry Fundamentals 15.1 (1976): 59-64. ↩

6. Patel, Navin C., and Amyn S. Teja. "A new cubic equation of state for fluids and fluid mixtures." Chemical Engineering Science 37.3 (1982): 463-473. ↩

7. Schmidt, G., and H. Wenzel. "A modified van der Waals type equation of state." Chemical Engineering Science 35.7 (1980): 1503-1512. ↩

8. Péneloux, André, Evelyne Rauzy, and Richard Fréze. "A consistent correction for Redlich-Kwong-Soave volumes." Fluid phase equilibria 8.1 (1982): 7-23. ↩

9. Graboski, Michael S., and Thomas E. Daubert. "A modified Soave equation of state for phase equilibrium calculations. 3. Systems containing hydrogen." Industrial & Engineering Chemistry Process Design and Development 18.2 (1979): 300-306. ↩

10. Soave, Giorgio S. "Application of the redlich-kwong-soave equation of state to solid-liquid equilibria calculations." Chemical Engineering Science 34.2 (1979): 225-229. ↩

11. D.B. Robinson, D.Y. Peng, The characterization of the heptanes and heavier fractions for the GPA Peng–Robinson programs, Gas processors association, Research report RR-28, 1978 (Booklet only sold by the Gas Processors Association, GPA). ↩

12. Twu, Chorng H., et al. "A cubic equation of state with a new alpha function and a new mixing rule." Fluid Phase Equilibria 69 (1991): 33-50. ↩

13. Le Guennec, Yohann, Romain Privat, and Jean-Noël Jaubert. "Development of the translated-consistent tc-PR and tc-RK cubic equations of state for a safe and accurate prediction of volumetric, energetic and saturation properties of pure compounds in the sub-and super-critical domains." Fluid Phase Equilibria 429 (2016): 301-312. ↩

14. Aasen, Ailo, et al. "Accurate quantum-corrected cubic equations of state for helium, neon, hydrogen, deuterium and their mixtures." Fluid Phase Equilibria 524 (2020): 112790. ↩

15. Fredenslund, Aage, Russell L. Jones, and John M. Prausnitz. "Group‐contribution estimation of activity coefficients in nonideal liquid mixtures." AIChE Journal 21.6 (1975): 1086-1099. ↩ ↩²

16. Abrams, Denis S., and John M. Prausnitz. "Statistical thermodynamics of liquid mixtures: a new expression for the excess Gibbs energy of partly or completely miscible systems." AIChE journal 21.1 (1975): 116-128. ↩

17. Huron, Marie-José, and Jean Vidal. "New mixing rules in simple equations of state for representing vapour-liquid equilibria of strongly non-ideal mixtures." Fluid Phase Equilibria 3.4 (1979): 255-271. ↩

18. Wong, David Shan Hill, and Stanley I. Sandler. "A theoretically correct mixing rule for cubic equations of state." AIChE Journal 38.5 (1992): 671-680. ↩

19. Holderbaum, Thomas, and J. P. S. R. K. Gmehling. "PSRK: A group contribution equation of state based on UNIFAC." Fluid Phase Equilibria 70.2-3 (1991): 251-265. ↩

20. Voutsas, Epaminondas, Kostis Magoulas, and Dimitrios Tassios. "Universal mixing rule for cubic equations of state applicable to symmetric and asymmetric systems: Results with the Peng−Robinson equation of state." Industrial & engineering chemistry research 43.19 (2004): 6238-6246. ↩

21. Hansen, Henrik K., et al. "Vapor-liquid equilibria by UNIFAC group contribution. 5. Revision and extension." Industrial & Engineering Chemistry Research 30.10 (1991): 2352-2355. ↩

22. Wittig, Roland, Juergen Lohmann, and Juergen Gmehling. "Vapor−liquid equilibria by UNIFAC group contribution. 6. Revision and extension." Industrial & engineering chemistry research 42.1 (2003): 183-188. ↩