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>

UNIFAC

Molecular volume parameter for component i (k: subgroup index) :

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

Molecular surface area parameter for component i (k: subgroup index):

<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 component 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 k in component i:

<math> \theta_k^{(i)} = \frac{X_k^{(i)} Q_k^{(i)}}{\sum_l X_l^{(i)} Q_l^{(i)}} </math>

<math>\theta</math> is a matrix where the columns make up the components in the mixture and the rows are made up by the subgroups.

Local surface area fraction for j around i (the dot-product is performed of every single subgroup-row of matrix <math>\theta</math> with the columns of matrix <math>\tau</math>):

<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>

Interaction parameters between identical main groups become 0.

The indexes n and m refer to subgroups. Thus, parameters a,b and c of different subgroups belonging to the same main group are identical. A subgroup to maingroup lookup has to be made when generating the data matrices for a, b and c.

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 interaction 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.

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/.

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 of the PSRK equation for component i in a mixture:

<math> ln \varphi_i = \frac{b_i}{b} (\frac{P*v}{R*T} - 1) - ln \frac{P*(v-b)}{R*T} - ( \frac{1}{q_1} * ln \gamma_i + \frac{a_i}{RTb_i} + \frac{1}{q_1}(ln \frac{b}{b_i} + \frac{b_i}{b} - 1)) ln \frac{v+b}{v} </math>

with <math> q_1 = -0.64663 </math>

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