Predictive Soave-Redlich-Kwong (PSRK)

Generic cubic equation of state

<math> P = { { R T } \over { V - b } } - { { a(T) } \over { ( V + \epsilon b ) ( V + \sigma b ) } } </math>

Parameters for Soave-Redlich-Kwong equation are :

<math> \epsilon = 0</math> and <math> \sigma = 1</math>

Thus:

<math> P = { { R T } \over { V - b } } - { { a(T) } \over { V ( V + b ) } } </math>

PSRK mixing rule for calculating a(T) and b

Cohesion pressure (attractive parameter) :

<math>a(T) = b RT \left( \sum x_i { {a_{ii}(T)} \over {b_i R T} } + { \frac{ { \frac{g_0^E}{R T} } + \sum x_i \ln \left( \frac{b}{b_i} \right) }{ \ln \left( \frac{u}{u + 1} \right)} } \right)</math>

with

<math> u = 1.1 </math>,

<math> a_{ii}(T) = \Psi \frac{ \alpha_i (T_{r,i}) R^2 T_{C,i}^2 }{ P_{C,i} } </math>, and

<math> \Psi = 0.42748 </math>.

Excluded volume or "co-volume" (repulsive parameter):

<math> b = \sum x_i b_i </math>

where

<math> b_i = \Omega { \frac{ RT_{C,i} }{ P_{C,i} } } </math>, and

<math> \Omega = 0.08664 </math>

Different mixing rules can be applied and a list will be compiled here:

...

Mathias-Copeman equation

Fitting experimental data with Mathias-Copeman parameters <math> c_{1,i} </math>, <math> c_{2,i} </math> and <math> c_{3,i} </math>:

<math> \alpha_i (T_{r,i}) = \left[ 1 + c_{1,i} \left(1 - \sqrt{T_{r,i}} \right) + c_{2,i} \left(1 - \sqrt{T_{r,i}} \right)^2 + c_{3,i} \left(1 - \sqrt{T_{r,i}} \right)^3 \right]^2 </math>

General form if no experimental data available:

<math> c_{1,i} = 0.48 + 1.574 \omega_i - 0.176 \omega_i^2 </math>

<math> c_{2,i} = 0 </math>

<math> c_{3,i} = 0 </math>

Gibbs-Excess energy

<math> g^E = g_{c}^E + g_{r}^E</math>

<math> g_{c}^E = RT \sum x_i ln( {{\omega_i} \over {x_i}} ) </math>

<math> g_{r}^E = RT \sum x_i \frac{z}{2} q_i \ln \frac{\theta_{ii}}{\theta_i} </math>

<math> \Rightarrow g_{0}^E = R T_0 \sum x_i \left( \ln \frac{\omega_i}{x_i} + \frac{z}{2} + q_i \ln \frac{\theta_{ii}}{\theta_i} \right) </math>

modified UNIFAC

Molecular volume parameter for component i :

<math> r_i = \sum_k \nu_k^{(i)} * R_k </math>

Molecular surface area parameter for component i:

<math> q_i = \sum_k \nu_k^{(i)} * Q_k </math>

where <math>\nu</math> is the number of the particular subgroups which a component i can be divided into. <math>R_k</math> is the volume parameter and <math>Q_k</math> the surface area parameter for subgroup k. <math>R_k</math> and <math>Q_k</math> are tabulated parameters and provided by the UNIFAC Consortium (http://unifac.ddbst.de/). Each subgroup can be assigned to a main group.

Modified volume fraction [Kikic et al.; 1980]:

<math> \omega_i = {{x_i * r_i^{2/3}} \over {\sum_j x_j * r_j^{2/3}}} </math>

Group mole fraction of subgroup k in compound i :

<math> X_k^{(i)} = {{\sum_i \nu_k^{(i)} * x_i} \over {\sum_i \sum_l \nu_l^{(i)} * x_i}} </math>

Surface area fraction of subgroup! (not component) i: Indexing can get confusing here...better indexing needed

<math> \theta_i = { {X_i * Q_i} \over {\sum_k X_k * Q_k} } </math>

Local surface area fraction for j around i:

<math> \theta_{ji} = { {\theta_j * \tau_{ji}} \over {\sum_m \theta_m * \tau_{mi}} } </math>

where

<math> \tau_{mi} = \Psi_{nm} </math>

and <math>\tau_{mi}</math> is the Boltzmann factor and can be calculated by transposing the main group interaction parameter matrix:

<math> \Psi_{nm} = exp(- { {a_{nm} + b_{nm}*T + c_{nm}*T^2} \over {T} }) </math>

<math>a_{nm}</math>, <math>b_{nm}</math> and <math>c_{nm}</math> are the binary interaction parameters representing the interaction between the main groups where the following applies:

<math> a_{nm} \ne a_{mn} </math>; <math> b_{nm} \ne b_{mn} </math>; <math> c_{nm} \ne c_{mn} </math>

The binary interaction parameters are also tabulated parameters and provided by the UNIFAC Consortium (http://unifac.ddbst.de/). The original UNIFAC model uses only <math> a_{nm} \ne a_{mn} </math> as interactions parameters. Modified UNIFAC and PSRK include <math> b_{nm} \ne b_{mn} </math> and <math> c_{nm} \ne c_{mn} </math> for describing main group interactions. Very important: PSRK relies on the modified UNIFAC equations but the parameters are different and therefore a different parameter dataset is provided for PSRK calculations.

A Excel file written by Carl Lira (http://www.egr.msu.edu/~lira/) can help to understand UNIFAC calculations. Take a look at the ACTCOEFF.XLS file under http://www.egr.msu.edu/~lira/computer/EXCEL/. Beware of that not the modified UNIFAC method is applied in Lira's Excel sheet but the original UNIFAC method. PSRK relies on the modified UNIFAC method and uses PSRK specific parameters.

The UNIFAC consortium has published all parameters for the original UNIFAC model: http://www.ddbst.com/published-parameters-unifac.html

For obtaining the parameters for applying the modified UNIFAC model or the PSRK model one has to become a member of the UNIFAC consortium or has to have access to the appendix of the following paper: doi:10.1016/j.fluid.2004.11.002

These pages can also help to understand this topic:

http://www.pvv.org/~randhol/xlunifac/html/node9.html

http://www.aim.env.uea.ac.uk/aim/info/UNIFACgroups.html

Procedure for calculating vapor-liquid equilibria (VLE) (phi-phi approach)

Equilibrium condition :

<math> x_i * \varphi_i^L = y_i * \varphi_i^V </math>

Fugacity coefficient for component i in a mixture:

<math> ln \varphi_i = \frac{1}{R*T} \int_{\infty}^V [ \frac{RT}{V} - ( \frac{\partial P}{\partial n_i} )_{T, V, n_{j \ne i}} ] dV - ln z </math>

Fugacity coefficient for the liquid phase:

...

Fugacity coefficient for the vapor phase:

...

K-factor:

<math> K_i = { {y_i} \over {x_i} } = { {\varphi_i^L} \over {\varphi_i^V} } </math>

Sum of mole fractions:

<math> S = \sum y_i = \sum K_i * x_i </math>

Flow diagram for calculating isothermal VLE using PSRK:

Flow diagram of the phi-phi approach for calculating isothermal vapor-liquid equilibria (VLE) with the predictive Soave-Redlich-Kwong equation of state (EOS) File:Phi-phi approach PSRK.pdf

See also

Other cubic equations of state (EOS):

• PengRobinson EOS in FPROPS
• Redlich Kwong EOS in FPROPS