lib.EoS.Cubic module

Library with the implemented cubic equation of state

Furthemore in a separated file it’s the common functionality

Other equations don’t implemented

Cubic

There are many cubic EoS don’t implemented

Clausius (1881)

Historical equation of state

\[\begin{split}\begin{array}[t]{l} P = \frac{RT}{V-b}-\frac{a}{T \left(V+c\right)^2}\\ \Omega_a = 27/64\\ \Omega_b = Z_c-0.25\\ \Omega_c = 3/8-Z_c\\ \end{array}\end{split}\]

Clausius, R., “Ueber das Verhaiten der Kohlensaure in Bezug auf Druck, Volumen und Temperatur”, Ann. Phys. Chem. 9 (1880) 337-359

Wilson (1964)

Modified RK temperature dependence

\[\begin{split}\begin{array}[t]{l} P = \frac{RT}{V-b}-\frac{a}{V\left(V+b\right)}\\ \Omega_a = \Omega_{ac}\left(1+m\left(\frac{1}{T_r}-1\right)\right)T_r\\ m = 1.57 + 1.62\omega\\ \end{array}\end{split}\]

Wilson, G.M. Vapor-Liquid Equilibria, Correlations by Means of a Modified Redlich-Kwong Equation of State. Adv. Cryog. Eng. 9(D2) (1964) 168-176.

HPW (1976)

Hederer, Peter, Wenzel equation of state, published like PR in 1976.

\[\begin{split}P = \frac{RT}{V-b}-\frac{aT^{\alpha}}{V\left(V+b\right)}\\\end{split}\]

α, a and b are compound specific parameters, the paper give correlations for parameter for several homologous series like alkanes, alkenes or alkynes, but there isn’t any general correlation.

Hederer, H., Peter, S., Wenzel, H. Calculation of Thermodynamic Properties from a Modified Redlich-Kwong Equation of State. Chem. Eng. J., 11 (1976) 183-190, http://dx.doi.org/10.1016/0300-9467(76)80039-1

vdW Adachi (1984)

Adachi modification to van der Waals original correlation using a logaritmic temperature dependence for a

\[\begin{split}\begin{array}[t]{l} P = \frac{RT}{V-b}-\frac{a(T)}{V^2}\\ a(T) = \frac{27}{64}10^{m\left(1-T_r\right)}\\ m = 0.228165 + 0.791981\omega - 0.648552*\omega^2 + 0.654505*\omega^3\\ b = 0.125\frac{RT_c}{P_c}\\ \end{array}\end{split}\]

Valid only by VLE calculation, not applicable in a general use

Adachi, Y., Lu, B.C.-Y. Simplest Equation of State for Vapor-Liquid Equilibrium calculation: a Modification of the van der Walls Equation. AIChE J. 30(6) (1984) 991-993, http://dx.doi.org/10.1002/aic.690300619

Usdin-McAuliffe (1976)

Incomplete study to improve SRK72 EoS liquid density prediction

\[\begin{split}\begin{array}[t]{l} P = \frac{RT}{V-b}-\frac{a(T)}{V\left(V+d\right)}\\ a = \frac{R^2T_c^2}{A_cP_c}\\ b = \frac{RT_c}{B_cP_c}\\ d = \frac{RT_c}{D_cP_c}\\ \alpha^{0.5} = 1 + m\left(1-\sqrt{T_r}\right)\\ \end{array}\end{split}\]

If T_r ≤ 0.7:

\[m = 0.48049 + 4.516\omega Z_c^* + \left(0.67713\left(\omega-0.35\right) -0.02\right)x\left(T_r-0.7\right)\]

If 0.7 < T_r ≤ 1.0:

\[m = 0.48049 + 4.516\omega Z_c^* + \left(37.7846\omega{Z_c^*}^3+0.78662 \right)\left(T_r-0.7\right)^2\]

\(D_c\) is the most positive root of

\[D_c^3 + D_c^2\left(6Z_c-1\right) + D_c\left(4Z_c-1\right)3Z_c + \left(8Z_c-3\right)Z_c^2 = 0\]

\(B_c\) and \(A_c\) are calculated known \(D_c\) from

\[\begin{split}\begin{array}[t]{l} B_c = D_c + 3Z_c - 1\\ A_c = Z_c^3/B_c \end{array}\end{split}\]

Usdin, E., McAuliffe, J.C. A One Parameter Family of Equations of State. Chem. Eng. Sci., 31(ll) (1976) 1077-1084, http://dx.doi.org/10.1016/0009-2509(76)87030-3

Toghiani-Vismanath (1986)

Equation intended to polar substances using the associating paramter of Halm and Stiel, χ. Not implemented because this parameter is not available in database.

\[\begin{split}\begin{array}[t]{l} P = \frac{RT}{V-b}-\frac{a\alpha\left(T_r\right)}{V^2+bV+cbV-cb^2}\\ a_c = \Omega_a\frac{R^2T_c^2}{P_c}\\ b_c = \Omega_b\frac{RT_c}{P_c}\\ c_c = \frac{1-3\zeta_c}{\zeta_c \eta_c}\\ \Omega_a = \left(1-\zeta_c\left(1-\eta_c\right)\right)^3\\ \Omega_b = \eta_c \zeta_c\\ \alpha^{0.5} = 1 + m\left(1-\sqrt{T_r}\right)\\ \end{array}\end{split}\]

The paper define the following generalization for parameters

\[\begin{split}\begin{array}[t]{l} m = A_1 + A_2\omega + A_3\omega^2 + A_4\chi + A_5\chi^2 + A_6\omega \chi + A_7\omega_3 + A_8\chi^3\\ \zeta_c = B_1 + B_2\omega + B_3\omega^2 + B_4\chi + B_5\chi^2 + B_6\omega \chi + B_7\omega_3 + B_8\chi^3\\ \end{array}\end{split}\]

with the following parameters

m

\(\zeta_c\)

A₁ = 0.441926073

B₁ = 0.324020789

A₂ = 1.342755128

B₂ = -0.056675895

A₃ = -0.328431972

B₃ = -0.001268996

A₄ = -15.020572758

B₄ = -3.259131762

A₅ = -2.226936391

B₅ = 1.399475880

A₆ = 21.112706213

B₆ = 4.769592801

A₇ = 0.015 079204

B₇ = 0.003431898

A₈ = 234.965900518

B₈ = 59.450596850

\(\eta_c\) may be determined using the relation

\[\eta_c^3 + \eta_c^2\left(\frac{2}{\zeta_c}-3\right) + 3\eta_c - 1 = 0\]

The polar factor of Halm and Stiel is defined using acentric factor as

\[\chi = \log{P_r|_{T_r=0.6}} + 1.7\omega + 1.552\]

so it could be calculated from compound with vapor pressure data available.

Toghiani, H., Viswanath, D.S. A Cubic Equation of State for Polar and Apolar Fluids. Ind. Eng. Chem. Proc. Des. Dev. 25(2) (1986) 531-536, http://dx.doi.org/10.1021/i200033a032

Harmens-Knapp (1980)

Developed only with data of normal fluids, alkanes and several nonpolar gases so not appropiate for general use

\[\begin{split}\begin{array}[t]{l} P = \frac{RT}{V-b}-\frac{a\alpha}{V^2+cbV-\left(c-1\right)b^2}\\ a = \Omega_a\frac{R^2T_c^2}{P_c}\\ b = \Omega_b\frac{RT_c}{P_c}\\ c = 1 + \frac{1-3\zeta}{\beta \zeta}\\ \Omega_a = 1-3\zeta+3\zeta^2+\beta\zeta\left(3-6\zeta+\beta\zeta\right)\\ \Omega_b = \beta \zeta\\ \beta = 0.10770 + 0.76405\zeta - 1.24282\zeta^2 + 0.96210\zeta^3\\ \zeta = 0.3211 - 0.08\omega + 0.0384\omega^2\\ \alpha = \left(1+A\left(1-\sqrt{T_r}\right)-B\left(1-\frac{1}{T_r}\right) \right)^2\\ \end{array}\end{split}\]

The paramters of α for ω ≤ 0.2

\[\begin{split}\begin{array}[t]{l} A = 0.5 + 0.27767\omega + 2.17225\omega^2\\ B = -0.022 + 0.338\omega - 0.845\omega^2\\ \end{array}\end{split}\]

If ω > 0.2

\[\begin{split}\begin{array}[t]{l} A = 0.41311 + 1.14657\omega\\ B = 0.0118\\ \end{array}\end{split}\]

For temperatures greater than critical the temperature dependence is

\[\alpha = 1 - \left(0.6258+1.5227\omega\right)\ln{T_r} + \left(0.1533+0.41\omega\right)\left(\ln{T_r}\right)^2\]

Harmens, A., Knapp, H. Three-Parameter Cubic Equation of State for Normal Substances. Ind. Eng. Chem. Fundam. 19(3) (1980) 291-294, http://dx.doi.org/10.1021/i160075a010

Fuller (1976)

Fuller modified the SRK to improve liquid densities prediction accuracy.

\[\begin{split}\begin{array}[t]{l} P = \frac{RT}{V-b}-\frac{a}{V\left(V+cb\right)}\\ a = \frac{\Omega_a R^2T_c^2 \alpha}{P_c}\\ b = \Omega_b \frac{RT_c}{P_c}\\ c = \frac{1}{\beta}\\ \left(\sqrt{\frac{1}{\beta}-\frac{3}{4}}-\frac{3}{2}\right)\\ \Omega_a = \frac{\left(1+c\beta\right)^2\Omega_b} {\beta\left(1-\beta\right)^2\left(2+c\beta\right)}\\ \Omega_b = \beta \frac{\left(1-\beta\right)\left(2+c\beta\right) - \left(1+c\beta\right)}{\left(2+c\beta\right)\left(1-\beta\right)^2}\\ \alpha^{0.5} = 1 + q\left(1-Tr^{0.5}\right)\\ q = \left(\frac{\beta}{0.26}\right)^{1/4}m\\ m = 0.48 + 1.574\omega - 0.176\omega^2\\ \beta = \beta_c + \left(\beta_o-\beta_c\right)\left(\frac{2} {1+e^{\theta\left(T_r-1\right)}}-1\right)\\ \frac{\beta_o}{\beta_c} = 7.788 - 36.8316Z_c + 50.7061Z_c^2\\ \theta = 10.9356+0.0285\bar{P}\\ \end{array}\end{split}\]

βc can be calculated from critical properties:

\[Z_c = \frac{P_cV_c}{RT_c} = \frac{\left(1-\beta_c\right)\left(2+c_c \beta_c\right)-\left(1+c_c\beta_c\right)} {\left(2+c_c\beta_c\right)\left(1-\beta_c\right)^2}\]

This equation isn’t implemented because need the parachor parameter for earch compound, not available in database and calculable by any group contribution method.

Fuller, G.G. A Modified Redlich-Kwong-Soave Equation of State Capable of Representing the Liquid State. Ind. Eng. Chem. Fundam. 15(4) (1976) 254-257, http://dx.doi.org/10.1021/i160060a005

Mathias (1983)

Modification of temperature dependence of α for SRK

\[\begin{split}\begin{array}[t]{l} P = \frac{RT}{V-b}-\frac{a(T)}{V\left(V+b\right)}\\ a(T) = 0.42747\frac{R^2T_c^2}{P_c}\alpha\\ b = 0.08664\frac{RT_c}{P_c}\\ \alpha = 1 + m\left(1-T_r^0.5\right) - p\left(1-T_r\right)\left(0.7-T_r \right)\\ m = 0.48508 + 1.55191\omega - 0.15613\omega^2\\ \end{array}\end{split}\]

This method need a compound specific parameter, p, a polar parameter.

Mathias, P.M. A Versatile Phase Equilibrium Equation of State. Ind. Eng. Chem. Process Des. Dev. 22(3) (1983) 385-391, http://dx.doi.org/10.1021/i200022a008.

Martin (1979)

\[\begin{split}\begin{array}[t]{l} P = \frac{RT}{V-b}-\frac{a(T)}{\left(V+c\right)^2}\\ a(T) = \frac{27}{64} \frac{R^2T_c^2}{P_c}T_r^{-n}\\ b = \left(0.857Z_c-0.1674\right)\frac{RT_c}{P_c}\\ c = \left(-0.857Z_c+0.2924\right)\frac{RT_c}{P_c}\\ \end{array}\end{split}\]

This equation require a compound specific parameters, the temperature exponent n, determined by equating the slope of the critical isochore to the slope of the vapour pressure curve at the critical point.

Martin, J.J. Cubic Equations of State-Which?. Ind. Eng. Chem. Fundam. 18(2) (1979) 81-97, http://dx.doi.org/10.1021/i160070a001.

Rogalski (1990)

Modified version of the volume-corrected Peng-Robinson equation state

\[\begin{split}\begin{array}[t]{l} P = \frac{RT}{V-b}-\frac{a}{\left(V+4.82843b\right)}\\ \end{array}\end{split}\]

This method has several alpha temperature dependence calculation by chemical type of compounds, furthermore the calculation of pseudovolume is calculated with a group contribution method.

Rogalski, M., Carrier, B., Solimando, R., Péneloux, A. Correlation and Prediction of Physical Properties of Hydrocarbons with the Modified Peng-Robinson Equation of State. 2. Representation of the Vapor Pressures and of the Molar Volumes. Ind. Eng. Chem. Res. 29(4) (1990) 659-666, http://dx.doi.org/10.1021/ie00100a026.

Raimondi (1980)

Modification of temperature dependence of α for SRK

\[\begin{split}\begin{array}[t]{l} P = \frac{RT}{V-b}-\frac{a(T)}{V\left(V+b\right)}\\ a(T) = 0.42747\frac{R^2T_c^2}{P_c}\alpha\\ b = 0.08664\frac{RT_c}{P_c}\\ \alpha = 1 + \mu(\omega)\left(1-T_r^0.5\right) + \delta\left(T_r\right) \nu(\omega, \chi) \left(1-\sqrt{T_r/0.7}\right)^2\\ \mu(\omega) = m_o + m_1\omega + m_2\omega^2 + m_3\omega^3 + m_4\omega^4 + m_5\omega^5\\ \delta\left(T_r\right) = 1 for T_r < 0.7\\ \delta\left(T_r\right) = 0 for T_r ≥ 0.7\\ \nu(\omega, \chi) = \nu_0(\omega) + \chi(\omega)\\ \nu_0(\omega) = r_0^0 + r_1^0\omega + r_2^0\omega^2 + r_3^0\omega^3\\ \nu_1(\omega) = r_0^1 + r_1^1\omega + r_2^1\omega^2 + r_3^1\omega^3\\ \end{array}\end{split}\]

where mi and ri are fixed numerical coefficient originate from regression analysis.

m

\(r^0\)

\(r^1\)

m₀ = 0.99717930

r₀ = 0.033445

r₀ = 4.6970402

m₁ = 3.39879325

r₁ = -1.2353988

r₁ = -0.38872809

m₂ = -0.00727715

r₂ = -0.36104233

r₂ = 0.20135167

m₃ = -0.03785315

r₃ = 0.16525837

r₃ = -0.068628773

m₄ = -0.03426992

m₅ = 0.05631978

This method isn’t implemented, because it needs another compound specific parameter, χ, it represents a second parameter to improve the experimental vapour pressure fitting, like the acentric factor but defined at Tr=0.5.

Raimondi, L., A Modified Redlich-Kwong Equation of State for Vapour-Liquid Equilibrium Calculations. Chem. Eng. Sci. 35(6) (1980) 1269-1275, http://dx.doi.org/10.1016/0009-2509(80)85119-0

Gibbons-Laughton (1984)

Modification of temperature dependence of α for SRK

\[\begin{split}\alpha = 1 + X\left(T_r-1\right) + Y\left(\sqrt{T_r}-1\right)\\\end{split}\]

X and Y are compound specific properties chosen by minimising the error in the complete vapour pressure curve.

Gibbons, R.M., Laughton, A.P. An Equation of State for Polar and Non-polar Substances and Mixtures. J. Chem. Soc., Faraday Trans. 2 80(9) (1984) 1019-1038, http://dx.doi.org/10.1039/F29848001019.

Ishikawa-Chung-Lu (1980)

Mixture of hard sphere model with RK atractive term.

\[\begin{split}\begin{array}[t]{l} P = \frac{RT}{V}\frac{\left(2V+b\right)}{\left(2V-b\right)} - \frac{a(T)}{T^{0.5}V\left(V+b\right)}\\ a(T) = \Omega_a\frac{R^2T_c^2}{P_c}\alpha\\ b = \Omega_b\frac{RT_c}{P_c}\\ \Omega_a = \sum_{i} a_iT_r^i\\ \Omega_b = \sum_{i} b_iT_r^i\\ m = 0.48508 + 1.55191\omega - 0.15613\omega^2\\ \end{array}\end{split}\]

The pure component parameter are only available for 22 compound in paper, eight for each compound for a accuracy not better than other equations with less parameters, furthermore only with parameter available for alkanes and several inorganic gases.

Ishikawa, T., Chung, W.K., Lu, B.C.-Y. A Cubic Perturbed, Hard Sphere Equation of State for Thermodynamic Properties and Vapor-Liquid Equilibrium Calculations. AIChE J. 26(3) (1980) 372-378, https://doi.org/10.1002/aic.690260307.

vdW711 (1989)

van der Waals volume translation equation with a modified α temperature dependence.

\[\begin{split}\begin{array}[t]{l} P = \frac{RT}{V+t-b}-\frac{a_c\alpha}{\left(V+t\right)^2}\\ a_c = \frac{27}{64}\frac{R^2T_c^2}{P_c}\\ b = \frac{1}{8}\frac{RT_c}{P_c}\\ t = t_o + \left(t_c-t_o\right)\exp{\beta \left(1-T_r\right)}\\ t_o = \frac{RT_c}{P_c} \left(0.03901+0.0.04451\omega-0.02274\omega^2\right)\\ t_c = \frac{RT_c}{P_c}\left(\frac{3}{8}-Z_c\right)\\ Z_c = 0.2890 - 0.0701\omega - 0.0207\omega^2\\ \beta = -7.35356-24.5176\omega+9.19829\omega^2\\ \alpha = \left(1+m\left(1-\sqrt{T_r}\right)\right)^2\\ m = 0.48553 + 1.62400\omega - 0.21884\omega^2\\ \end{array}\end{split}\]

Androulakis defined a enhanced α temperature dependence with three compound specific parameters:

\[\alpha = 1 + d_1\left(1-T_r^{2/3}\right) + d_2\left(1-T_r^{2/3}\right)^2 + d_3\left(1-T_r^{2/3}\right)^3\]

Watson, P., Cascella, M., May, D., Salerno, S., Tassios, D. Prediction of Vapor Pressure and Saturated Molar Volumes with a Simple Cubic Equation of State: Part II: The van der Waals- 711 EOS. Fluid Phase Equilibria, 27 (1986) 35-52, http://doi.org/10.1016/0378-3812(86)87039-x

Androulakis, I.P., Kalospiros, N.S., Tassios, D.P. Thermophysical Properties of Pure Polar and Nonpolar Compounds with a Modified vdW-711 Equation of State. Fluid Phase Equilibria, 45 (1989) 135-163, http://doi.org/10.1016/0378-3812(89)80254-7.

Schmidt-Wenzel (1980)

Generalized form of van der Waals cubic equation of state.

\[\begin{split}\begin{array}[t]{l} P = \frac{RT}{V-b}-\frac{a}{V^2+ubV+wb^2}\\ u = 1-w\\ w = -3\omega\\ a = \Omega_a\alpha\frac{R^2T_c^2}{P_c}\\ b = \Omega_b\frac{RT_c}{P_c}\\ \Omega_a = \left(-\xi_c\left(1-\beta_c\right)\right)^3\\ \Omega_b = \beta_c\xi_c\\ \alpha = 1+k\left(1-T_r^{0.5}\right)\\ k = k_o + \frac{\left(5T_r-3k_o-1\right)^2}{70}\\ k_o = 0.465 + 1.347\omega - 0.528\omega^2\\ \end{array}\end{split}\]

βc can be calculated solving the cubic equation:

\[\left(6\omega+1\right)\beta_c^3+3\beta_c^2+3\beta_c-1=0\]

Schmidt, G., Wenzel, H. A Modified van der Waals Type Equation of State. Chem. Eng. Sci. 35(7) (1980) 1503-1512, http://doi.org/10.1016/0009-2509(80)80044-3.

Lee-Erbar-Edmister (1973)

Modified version of Grayson-Streed-Chao-Seader equation with a modified non-cubic equation for the gas phase, this EoS use interaction parameters for mixtures.

\[\begin{split}P = \frac{RT}{V-b} - \frac{a}{V\left(V-b\right)} + \frac{bc}{V\left(V-b\right)\left(V+b\right)} {}\\\end{split}\]

with all three parameters as functions of reduced temperature, critical temperature, critical pressure and acentric factor:

\[\begin{split}\begin{array}[t]{l} a_i = \frac{R^2T_{ci}^2}{P_{ci}} \left[\left(0.246105+0.02869\omega_i \right) - \left(0.037472+0.149687\omega_i\right)T_{ri} \\ {} + \frac{\left(0.16406+0.023727\omega_i\right)}{T_{ri}} + \frac{\left(0.04937+0.132433\omega_i\right)}{T_{ri}^2}\right]\\ b_i = \frac{RT_{ci}}{P_{ci}} \left(0.086313+0.002\omega_i\right)\\ c_i = \frac{R^2T_{ci}^2}{P_{ci}} \left[\frac{\left(0.451169+0.00948\omega_i \right)}{\sqrt{T_{ri}}} \frac{\left(0.387082+0.078842\omega_i\right)}{T_{ri}^2}\right]\\ \end{array}\end{split}\]

and a different correlation for liquid fugacity coefficient

\[\begin{split}\begin{align*} \ln \nu_i = A_1 + \frac{A_2}{T_r} + A_3\ln T_r + A_4T_r + A_5T_r^2 + A_6T_r^7\\ {} + \left(A_7 + \frac{A_8}{T_r} + A_9\ln T_r + A_{10}T_r^2 + A_{11}T_r^7\right)P_r\\ {} + A_{12}T_r^3P_r^2 + \left[\left(1-T_r\right)\left(A_{13}+\frac{A_{14}} {T_r} + A_{15}T_r\right)\\ {} + A_{16}\frac{P_r}{T_r} + A_{17}T_rP_r^2\right]\omega - \ln P_r\\ \end{align*}\end{split}\]

17 coeeficient, with many set of values for different compounds.

The activity coefficient for liquid phase use a modified Scatchard-Hildebrand versión with as many as 4 interaction parameters.

\[\begin{split}\begin{array}[t]{l} \ln{\gamma_i} = \frac{V_i^L}{RT}\left(\sum_j B_{ij}\Phi_j - \frac{1}{2} \sum_j \sum_m B_{jm}\Phi_j\Phi_m\right)\\ B_{ij} = \left(\delta_i-\delta_j\right)^2 + 2l_{ij}\delta_i\delta_j\\ \end{array}\end{split}\]

Its little improved performance does not justify the increased complexity of the correlation.

Lee, B-I, Erbar, J.H., Edmister, W.C. Prediction of Thermodynamnic Properties for Low Temperature Hydrocarbon Process Calculations. AIChE J. 19(2) (1973) 349-356, http://doi.org/10.1002/aic.690190221.

Robinson-Chao (1971)

Modified version of Grayson-Streed-Chao-Seader equation using a Redlich-Kwong cubic equation modified by Chueh-Prausnitz

\[\begin{split}\begin{array}[t]{l} P = \frac{RT}{V-b} - \frac{a}{T^{0.5} V\left(V+b\right)}\\ a_i = \frac{\Omega_a R^2 T_{ci}^{2.5}}{P_{ci}}\\ b_i = \frac{\Omega_b R T_{ci}}{P_{ci}}\\ a = \sum_i \sum_j y_i y_j a_{ij}\\ b = \sum_i y_i b_i\\ a_{ii} = \frac{\Omega_{ai} R^2 T_{ci}^{2.5}}{P_{ci}}\\ a_{ij} = \frac{\left(\Omega_{ai}+\omega_{aj}\right) R^2 T_{cij}^{2.5}} {P_{cij}}\\ P_{cij} = \frac{z_{cij}RT_{cij}}{v_{cij}}\\ v_{cij}^{1/3} = \frac{1}{2}\left(v_{ci}^{1/3}+v_{cj}^{1/3}\right)\\ z_{cij} = 0.291 - 0.08 \left(\frac{\omega_i+\omega_j}{2}\right)\\ T_{cij} = \sqrt{T_{ci}T_{cj}}\left(1-k_{ij}\right)\\ \end{array}\end{split}\]

Forthemore using different mixing rules for a parameter, the Ω parameters are compound specific.

The liquid fugacity has modified expresion

\[\begin{split}\begin{align*} \log \nu = \log nu^o + \omega \log nu^1\\ \log \nu^o = B_o + B_1P_r + B_2P_r^2-\log P_r\\ \log \nu^1 = \log \nu_{0.6}^1 + \left(P_r-0.6\right) \frac{\partial\log\nu^1}{\partial P_r} \end{align*}\end{split}\]

B₀, B₁ and B₂ are function of Tr, with different dependences at different values of Tr.

Robinson, R.L, Chao, K.-C. A Correlation of Vaporization Equilibrium Ratios for Gas Processing Systems. Ind. Eng. Chem. Process Des. Develop. 10(2) (1971) 221-229, http://doi.org/10.1021/i260038a015. Chueh, P.L., Prausnitz, J.M. Vapor-Liquid Equilibria at Hith Pressures. Vapor- Phase Fugacity Coefficients in Nonpolar and Quantum-Gas Mixtures. Ind. Eng. Chem. Fundam. 6(4) (1967) 492-498, http://doi.org/10.1021/i160024a003.

Others

Carrier 1988 Heyen 1980 Adachi Lu 1984

Wu-Stiel

This correlation combine Lee-Kesler correlation for nonpolar fluids with an accurate equation of state for water used as reference substance for polar fluids.

Wu, G.Z.A., Stiel, L.I. A Generalized Equation of State for the Thermodynamic Properties of Polar Fluids. AIChE J. 31(10) (1985) 1632-1644, http://dx.doi.org/10.1002/aic.690311007.

Virial

Beattie-Bridgeman (1928) Benedict-Webb-Rubin (1940) Benedict 1951: fugacity correlations Cooper-Goldfrank (1967) BWR Coefficients Orye (1969) BWRO Morsy (1970) BWRM Starling (1973) BWRS

Low T modification Bloomer-Rao (1952) Motard-Organick (1960) Barner-Schreiner (1966) Starling (1971) Orye (1969)

Critical region Eubank-Fort (1969) Hirschfelder (1958)

More terms modifications Strobridge (1962) Bender (1970) Morsy (1970) Starling (1971) Jacobsen-Stewart (1973) Lee-Kesler (1975) Nishiumi-Saito (1975) Schmidt-Wagner (1985) AGA Starling (1991) Angus (1972, 1976, 1978, 1979 1980, 1985) Angus-de Reuck (1976)

Starling-Han (1972) Nishiumi (1980)

Lee-Kesler (1975) Joffe (1976) Plocker (1978) Oellrich (1981) Yu (1982) Wu-Stiel (1985)

Yamada

Yamada create a generalized version of BWR equation with 24 parameters (BWR24) with the parameters dependence of acentric factor

\[\begin{split}\begin{align*} Z = 1 + \left(B_o-\frac{A_o}{T_R}-\frac{C_o}{T_R}\right)\frac{1}{V_R} \\ {} + \left(a_\frac{b_R}{T_R}\right)\frac{1}{V_R^2} + \frac{\alfa_R}{V_R^5 T_R} \\ {} + \frac{c_R}{T_R^3 V_R^2}\left(1+\frac{\gamma_R}{V_R^2}\right) \exp\left(-\frac{\gamma_R}{V_R^2}\right) \\ \end{align*}\end{split}\]

The equation don’t use the critical volume as reducing value for volume, use

\[V_{sc} = \frac{V_{0.6}}{0.3862-0.0866\omega}\]

where V_0.6 is the the liquid density at a reduced temperature of 0.6.

The fitting of parameters use data points only for argon, methane, ethane, n-butane, n-pentane, n-heptane and nitrogen, so the generalization is poor and no include polar compounds, in fact the author recommend use the correlation only for compounds with acentric factor not larger than 0.35.

The range of application of equation is only below reduced density of 1.8. The paper give other correlation with 44 parameters to try to fit a wider range of densities, but using same databank

\[\begin{split}\begin{array} Z = 1 + \left(B_o+\frac{B_1}{T_R}+\frac{B_2}{T_R}+\frac{B_3}{T_R^3}\right) \frac{1}{V_R} \\ {} + \left[C_o+\frac{C_1}{T_R}+\frac{C_2}{T_R^3}\left(1+\frac{C_4}{V_R^2} \right) \exp \left(-\frac{C_4}{V_R^2\right)\right] \frac{1}{V_R^2} \\ {} + \left(D_o+\frac{D_1}{T_R}\right)\frac{1}{V_R^3} \\ {} + \left[E_o+\frac{E_1}{T_R} \left(1+\frac{E_2}{V_R^4}\right) \exp \left(-\frac{E_2}{V_R^4}\right)\right] \frac{1}{V_R^4} \\ {} + \left(F_o+\frac{F_1}{T_R}\right) \frac{1}{V_R^5} \\ \end{align*}\end{split}\]

Yamada, T. An Improved Generalized Equation of State. AIChE J. 19(2) (1973) 286-291, http://dx.doi.org/10.1002/aic.690190212.

Nishiumi (1980)

Trying to extend the BWRS equation of state to polar compounds Nishiumi developed this EoS adding 3 additional polar parameters

\[\begin{split}\begin{align*} Z = 1 + \left(B_o - \frac{A_o}{T_r} - \frac{C_o}{T_r^3} - \frac{D_o}{Tr^4} \frac{E_o+\Psi_E}{T_r^5}\right) \rho_r \\ {} + \left(b - \frac{a}{T_r} - \frac{d}{Tr_2} - \frac{e}{T_r^5} - \frac{f}{T_r^{24}}\right) \rho_r^2 \\ {} + \alpha \left(\frac{a}{T_r} + \frac{d}{T_r^2} + \frac{e}{T_r^5} + \frac{f}{T_r^{24}}\right) \rho_r^5 \\ {} + \left(\frac{c}{T_r^3} + \frac{g}{T_r^9} + \frac{h}{T_r^{18}} + y(T_r) \right) \rho_r^2 \left(1+\gamma\rho_r^2\right)\exp{-\gamma\rho_r^2} \\ \end{align*}\end{split}\]

The aditioanl parameter are:

\[\begin{split}\begin{array}[t]{l} y(T_r) = \frac{s_1}{T_r^{s_2}}\\ \Psi_E = 2.83 \frac{\mu^2}{T_c V_c} \end{array}\end{split}\]

This equation is only applicable to pure compound, don’t define mixing rules to apply to mixtures.

Nishiumi, H. An Improved Generalized BWR Equation of State with Three Polar Parameters Applicable to Polar Substances. J. Chem Eng. Japan 13(3) (1980) 178-183, http://dx.doi.org/10.1252/jcej.13.178.

References