Thermodynamic Models¶

The next sections will give an overview of different thermodynamic models ranging from implicit solvation models to Non-random Two-liquid (NRTL) and group contribution (GC) models. Many of these models found on common theory and can even be combined to obtain fine-tuned parameter sets.

Currently the Cebule engine supports the following models:

COSMO-SAC, COSMO-RS and PCM implicit solvation models
Non-random Two-liquid (NRTL) and Wilson models
Group contribution (GC) models including UNIQUAC, Original UNIFAC, UNIFAC (Dortmund), UNIFAC (Lyngby) and Predictive Soave-Redlich Kwong (PSRK)
PC-SAFT
Hybrid models: UNIFAC-VISCO, COSMO-NRTL

Implicit solvation models¶

COSMO-SAC¶

COSMO-RS¶

PCM¶

Non-random Two-liquid (NRTL) models¶

Wilson¶

NRTL¶

Group contribution models¶

UNIQUAC¶

Original UNIFAC¶

The Original UNIFAC model consists of the following equations :

\[ln \gamma_i = ln \gamma_i^C + ln \gamma_i^R\]

\[ln \gamma_i^C = 1 - V_i + ln V_i - 5 q_i(1 - \frac{V_i}{F_i} + ln \frac{V_i}{F_i}) \]

Volume/mole fraction ratio for component i (j: component index) in the mixture:

\[ V_i = \frac{r_i}{\sum_j r_j x_j} \]

Surface area/mole fraction ratio for component i (j: component index) in the mixture:

\[ F_i = \frac{q_i}{\sum_j q_j x_j} \]

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

\[ r_i = \sum_k \nu_k^{(i)} R_k \]

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

\[ q_i = \sum_k \nu_k^{(i)} Q_k \]

\[ ln \gamma_i^R = \sum_i \nu_k^{(i)} (ln \Gamma_k - ln \Gamma^{(i)}_k ) = \sum_i \nu_k^{(i)} \frac{ln \Gamma_k}{ln \Gamma^{(i)}_k} \]

For $ ln \Gamma_k $ group mole fractions ($X_k$ and surface area fractions ($$ \Theta_m $\() are calculated with the composition ($\)x_j$$) of the mixture.

For $$ ln \Gamma^{(i)}_k $$ group mole fractions ($\(X_k^{(i)}$\)) and surface area fractions ($$ \Theta_m $$) are calculated with $\(x^{(i)} = 1$\) for the pure component i.

\[ ln \Gamma_k = Q_k [ 1 - ln (\sum_m \Theta_m \Psi_{mk}) - \sum_m \frac{\Theta_m \Psi_{km}}{\sum_n \Theta_n \Psi_{nm}}] \]

\[ \Theta_m = \frac{Q_m X_m}{\sum_n Q_n X_n} \]

\[ X_m = \frac{\sum_j \nu_m^{(j)} x_j}{\sum_j \sum_n \nu_n^{(j)} x_j} \]

\[ \Psi_{nm} = exp(- { {a_{nm} } \over {T} }) \]

The UNIFAC consortium manages the parameter database for the original UNIFAC, modified UNIFAC and PSRK model: The UNIFAC Consortium

Modified UNIFAC (Dortmund)¶

The following modifications are made to the Original UNIFAC model :

Combinatorial part of liquid activity coefficient:

\[ ln \gamma^C_i = 1 - V'_i + ln V'_i - 5 q_i (1 - \frac{V_i}{F_i} + ln \frac{V_i}{F_i}) \]

Volume/mole fraction ratio:

\[ V_i' = \frac{r_i^{3/4}}{\sum_j r_j^{3/4} x_j} \]

Main group interaction parameter matrix:

\[ \Psi_{nm} = exp(- { {a_{nm} + b_{nm} T + c_{nm} T^2} \over {T} }) \]

Modified UNIFAC (Lyngby)¶

Predictive Soave-Redlich Kwong (PSRK)¶

Generic cubic equation of state¶

\[ P = { { R T } \over { v - b } } - { { a(T) } \over { ( v + \epsilon b ) ( v + \sigma b ) } } \]

Parameters for Soave-Redlich-Kwong equation are :

\[ \epsilon = 0$$ and $$ \sigma = 1\]

Thus:

\[ P = { { R T } \over { v - b } } - { { a(T) } \over { v ( v + b ) } } \]

PSRK mixing rule for calculating a(T) and b¶

Cohesion pressure (attractive parameter) :

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

$\(P^{ref}$\) = 1 atm

with

:$$ u = 1.1 $$,

:$$ a_{ii}(T) = \Psi \frac{ \alpha_i (T_{r,i}) R^2 T_{C,i}^2 }{ P_{C,i} } $$, and

:$$ \Psi = 0.42748 $$.

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

\[ b = \sum x_i b_i \]

where

:$$ b_i = \Omega { \frac{ RT_{C,i} }{ P_{C,i} } } $$, and

:$$ \Omega = 0.08664 $$

Mathias-Copeman equation¶

Fitting experimental data with Mathias-Copeman parameters $$ c_{1,i} $$, $$ c_{2,i} $$ and $$ c_{3,i} $$:

:$$ \alpha_i (T_{r,i}) = \left[ 1 + c_{1,i} \left(1 - \sqrt{T_{r,i}} \right) + c_{2,i} \left(1 - \sqrt{T_{r,i}} \right)^2 + c_{3,i} \left(1 - \sqrt{T_{r,i}} \right)^3 \right]^2 $$

General form if no experimental data available:

:$$ c_{1,i} = 0.48 + 1.574 \omega_i - 0.176 \omega_i^2 $$ :$$ c_{2,i} = 0 $$ :$$ c_{3,i} = 0 $$

Gibbs-Excess energy ===¶

\[ g^E = g_{c}^E + g_{r}^E\]

\[ g_{c}^E = RT \sum x_i ln( {{\omega_i} \over {x_i}} ) \]

\[ g_{r}^E = RT \sum x_i \frac{z}{2} q_i \ln \frac{\theta_{ii}}{\theta_i} \]

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

UNIFAC¶

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

:$$ r_i = \sum_k \nu_k^{(i)} R_k $$

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

:$$ q_i = \sum_k \nu_k^{(i)} Q_k $$

where $\(\nu$\) is the number of the particular subgroups which a component i can be divided into. $\(R_k$\) is the volume parameter and $\(Q_k$\) the surface area parameter for subgroup k. $\(R_k$\) and $\(Q_k$\) 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 :

:$$ \omega_i = {{x_i r_i^{2/3}} \over {\sum_j x_j r_j^{2/3}}} $$

Group mole fraction of subgroup k in component i :

:$$ X_k^{(i)} = {{\sum_i \nu_k^{(i)} x_i} \over {\sum_i \sum_l \nu_l^{(i)} x_i}} $$

Surface area fraction of subgroup k in component i:

:$$ \theta_k^{(i)} = \frac{X_k^{(i)} Q_k^{(i)}}{\sum_l X_l^{(i)} Q_l^{(i)}} $$

$\(\theta$\) 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 $\(\theta$\) with the columns of matrix $\(\tau$\)):

:$$ \theta_{ji} = { {\theta_j \tau_{ji}} \over {\sum_m \theta_m \tau_{mi}} } $$

where

:$$ \tau_{mi} = \Psi_{nm} $$

and $\(\tau_{mi}$\) is the Boltzmann factor and can be calculated by transposing the main group interaction parameter matrix:

:$$ \Psi_{nm} = exp(- { {a_{nm} + b_{nm} T + c_{nm} T^2} \over {T} }) $$

:$\(a_{nm}$\), $\(b_{nm}$\) and $\(c_{nm}$\) are the binary interaction parameters representing the interaction between the main groups where the following applies:

:$$ a_{nm} \ne a_{mn} $\(; :$\) b_{nm} \ne b_{mn} $\(; :$\) c_{nm} \ne c_{mn} $$

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 $$ a_{nm} \ne a_{mn} $$ as interaction parameters. Modified UNIFAC and PSRK include $$ b_{nm} \ne b_{mn} $$ and $$ c_{nm} \ne c_{mn} $$ for describing main group interactions.

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 (Dortmund)]] 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: https://doi.org/10.1016/j.fluid.2004.11.002

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

Equilibrium condition :

\[ x_i * \varphi_i^L = y_i * \varphi_i^V \]

Fugacity coefficient of the PSRK equation for component i in a mixture:

\[ ln \varphi_i = \frac{b_i}{b} (\frac{P*v}{R*T} - 1) - ln \frac{P*(v-b)}{R*T} - ( \frac{1}{q_1} * ln \gamma_i + \frac{a_i}{RTb_i} + \frac{1}{q_1}(ln \frac{b}{b_i} + \frac{b_i}{b} - 1)) ln \frac{v+b}{v} \]

with $$ q_1 = -0.64663 $$

K-factor:

\[ K_i = { {y_i} \over {x_i} } = { {\varphi_i^L} \over {\varphi_i^V} } \]

Sum of mole fractions:

\[ S = \sum y_i = \sum K_i * x_i \]

Flow diagram for calculating isothermal VLE using PSRK: