UNIFAC

UNIFAC (UNIQUAC Functional-group Activity Coefficients) is an Excess Gibbs free energy model used to estimate activity coefficients in non-ideal mixtures. It is particularly useful for predicting the phase behavior of chemical mixtures, including liquid-liquid equilibrium (LLE) and vapor-liquid equilibrium (VLE). In this model the Excess Gibbs free energy is calculated from the contribution of a combinatorial term and a residual term.

$$\frac{G^E}{RT} = \frac{G^{E,r}}{RT} + \frac{G^{E,c}}{RT}$$

Being:

• Combinatorial: Accounts for the size and shape of the molecules.

• Residual: Accounts for the energy interactions between different functional groups.

Each substance of a mixture modeled with UNIFAC must be represented by a list a functional groups and other list with the ocurrence of each functional group on the substance. The list of the functional groups culd be accesed on the DDBST web page: https://www.ddbst.com/published-parameters-unifac.html

Combinatorial term derivatives

Calculate the UNIFAC combinatorial term of reduced Gibbs excess energy. The subroutine uses the Flory-Huggins and Staverman-Guggenheim combinatory terms as follows:

Flory-Huggins

$$G^{E,FH} = RT \left(\sum_i^{NC} n_i \, \text{ln} \, r_i - n \, \text{ln} \, \sum_j^{NC} n_j r_j + n \, \text{ln} \, n \right)$$

$$\frac{dG^{E,FH}}{dn_i} = RT \left(\text{ln} \, r_i - \text{ln} \, \sum_j^{NC} n_j r_j + \text{ln} \, n + 1 - \frac{n r_i}{\displaystyle \sum_j^{NC} n_j r_j} \right)$$

$$\frac{d^2G^{E,FH}}{dn_i dn_j} = RT \left(- \frac{r_i + r_j}{\displaystyle \sum_l^{NC} n_l r_l} + \frac{1}{n} + \frac{n r_i r_j}{\displaystyle \left(\sum_l^{NC} n_l r_l \right)^2} \right)$$

Staverman-Guggenheim

$$\frac{G^{E,SG}}{RT} = \frac{z}{2} \sum_i^{NC} n_i q_i \left(\text{ln} \frac{q_i}{r_i} - \text{ln} \, \sum_j^{NC} n_j q_j + \text{ln} \, \sum_j^{NC} n_j r_j \right)$$

$$\frac{1}{RT}\frac{dG^{E,SG}}{dn_i} = \frac{z}{2} q_i \left( - \text{ln} \, \left( \frac{r_i \sum_j^{NC} n_j q_j}{\displaystyle q_i \sum_j^{NC} n_j r_j} \right) - 1 + \frac{\displaystyle r_i \sum_j^{NC} n_j q_j}{\displaystyle q_i \sum_j^{NC} n_j r_j} \right)$$

$$\frac{1}{RT}\frac{d^2G^{E,SG}}{dn_i dn_j} = \frac{z}{2} \left(- \frac{q_i q_j}{\displaystyle \sum_l^{NC} n_lq_l} + \frac{q_i r_j + q_j r_i}{\displaystyle \sum_l^{NC} n_l r_l} - \frac{\displaystyle r_i r_j \sum_l^{NC} n_l q_l} {\left(\displaystyle \sum_l^{NC} n_l r_l \right)^2} \right)$$

Fredenslund et al. (UNIFAC)

$$\frac{G^{E,\text{UNIFAC}}}{RT} = \frac{G^{E,FH}}{RT} + \frac{G^{E,SG}}{RT}$$

Residual terms derivatives

Evaluate the UNIFAC residual term. The residual Gibbs excess energy and its derivatives are evaluated as:

$$\frac{G^{E,R}}{RT} = - \sum_i^{NC} n_i \sum_k^{NG} v_k^i Q_k (\Lambda_k - \Lambda_k^i)$$

With:

$$\Lambda_k = \text{ln} \, \sum_{j}^{NG} \Theta_j E_{jk}$$

$$\Lambda_k^i = \text{ln} \, \sum_{j}^{NG} \Theta_j^i E_{jk}$$

$$E_{jk} = \text{exp} \left(- \frac{U_{jk}}{RT} \right)$$

$$\Theta_j = \frac{Q_j \displaystyle \sum_{l}^{NC} n_l v_j^l} {\displaystyle \sum_{k}^{NC} n_k \sum_{m}^{NG} v_m^l Q_m}$$

$$\Theta_j^i = \frac{Q_j v_j^i}{\displaystyle \sum_k^{NG} v_k^i Q_k}$$

In the UNIFAC model, the $\Theta_j^i$ values are calculated assuming that the molecule “i” is pure, hence only the subgroups of the molecule “i” must be considered for the calculation. On the other hand, for the $\Theta_j$ values, all the system’s subgroups are considered.

The compositional derivatives:

$$\frac{1}{R T} \frac{\partial G^{E,R}}{\partial n_\alpha} = - \sum_k^{\mathrm{NG}} v_k^\alpha Q_k \left(\Lambda_k - \Lambda_k^\alpha \right) - \sum_i^{\mathrm{NC}} n_i \sum_k^{\mathrm{NG}} v_k^i Q_k \frac{\partial \Lambda_k}{\partial n_\alpha}$$

$$\frac{1}{R T} \frac{\partial^2 G^{E,R}}{\partial n_ \alpha \partial n_\beta} = -\sum_k^{\mathrm{NG}} Q_k \left(v_k^\alpha \frac{\partial \Lambda_k}{\partial n_\beta} + v_k^\beta \frac{\partial \Lambda_k}{\partial n_\alpha}\right) - \sum_k^{\mathrm{NG}} \left(\sum_i^{\mathrm{NC}} n_i v_k^i\right) Q_k \frac{\partial^2 \Lambda_k}{\partial n_\alpha \partial n_\beta}$$

With:

$$\frac{\partial \Lambda_k}{\partial n_\alpha} = \frac{\sum_j^{\mathrm{NG}} v_j^\alpha Q_j E_{j k}} {\sum_l^{\mathrm{NC}} n_l \sum_j^{\mathrm{NG}} v_j^l Q_j E_{j k}} - \frac{\sum_m^{\mathrm{NG}} v_m^\alpha Q_m} {\sum_l^{\mathrm{NC}} n_l \sum_m^{\mathrm{NG}} v_m^l Q_m}$$

$$\frac{\partial^2 \Lambda_k}{\partial n_\alpha \partial n_\beta} = - \frac{\left(\sum_j^{\mathrm{NG}} v_j^\alpha Q_j E_{j k}\right) \left(\sum_j^{\mathrm{NG}} v_j^\beta Q_j E_{j k}\right)} {\left(\sum_l^{\mathrm{NC}} n_l \sum_j^{\mathrm{NG}} v_j^l Q_j E_{j k}\right)^2} + \frac{\left(\sum_m^{\mathrm{NG}} v_m^\alpha Q_m\right)\left(\sum_m^{\mathrm{NG}} v_m^\beta Q_m\right)} {\left(\sum_l^{\mathrm{NC}} n_l \sum_m^{\mathrm{NG}} v_m^l Q_m\right)^2}$$

The temperature derivatives:

$$\frac{\partial\left(\frac{G^{E, R}}{R T}\right)}{\partial T} = -\sum_i^{\mathrm{NC}} n_i \sum_k^{\mathrm{NG}} v_k^i Q_k \left(\frac{\partial \Lambda_k}{\partial T} -\frac{\partial \Lambda_k^i}{\partial T}\right)$$

$$\frac{\partial^2\left(\frac{G^{E,R}}{R T}\right)}{\partial T^2} = -\sum_i^{\mathrm{NC}} n_i \sum_k^{\mathrm{NG}} v_k^i Q_k \left(\frac{\partial^2 \Lambda_k}{\partial T^2} - \frac{\partial^2 \Lambda_k^i}{\partial T^2}\right)$$

With:

$$\frac{\partial \Lambda_k}{\partial T} = \frac{\sum_{j}^{NG} \Theta_j \frac{d E_{jk}}{dT}} {\sum_{j}^{NG} \Theta_j E_{jk}}$$

$$\frac{\partial \Lambda_k^i}{\partial T} = \frac{\sum_{j}^{NG} \Theta_j^i \frac{d E_{jk}}{dT}} {\sum_{j}^{NG} \Theta_j^i E_{jk}}$$

$$\frac{\partial^2 \Lambda_k}{\partial T^2} = \frac{\sum_{j}^{NG} \Theta_j \frac{d^2 E_{jk}}{dT^2}} {\sum_{j}^{NG} \Theta_j E_{jk}} - \left(\frac{\partial \Lambda_k}{\partial T} \right)^2$$

$$\frac{\partial^2 \Lambda_k^i}{\partial T^2} = \frac{\sum_{j}^{NG} \Theta_j^i \frac{d^2 E_{jk}}{dT^2}} {\sum_{j}^{NG} \Theta_j^i E_{jk}} - \left(\frac{\partial \Lambda_k^i}{\partial T} \right)^2$$

Temperature-compositional cross derivative:

$$\frac{\partial \left(\frac{G^{E, R}}{R T} \right)} {\partial n_\alpha \partial T}= -\sum_k^{\mathrm{NG}} v_k^\alpha Q_k \left(\frac{\partial \Lambda_k} {\partial T} - \frac{\partial \Lambda_k^\alpha}{\partial T}\right) -\sum_k^{\mathrm{NG}} \left(\sum_i^{\mathrm{NC}} n_i v_k^i \right) Q_k \frac{\partial^2 \Lambda_k}{\partial n_\alpha \partial T}$$

With:

$$\frac{\partial^2 \Lambda_k}{\partial n_\alpha \partial T} = \frac{\sum_j^{\mathrm{NG}} v_j^\alpha Q_j \frac{\partial \tilde{E}_{j k}}{\partial T}}{\sum_l^{\mathrm{NC}} n_l \sum_j^{\mathrm{NG}} v_j^l Q_j \tilde{E}_{j k}} - \frac{\left(\sum_j^{\mathrm{NG}} v_j^\alpha Q_j \tilde{E}_{j k}\right) \left(\sum_l^{\mathrm{NC}} n_l \sum_j^{\mathrm{NG}} v_j^l Q_j \frac{\partial \tilde{E}_{j k}}{\partial T}\right)} {\left(\sum_l^{\mathrm{NC}} n_l \sum_j^{\mathrm{NG}} v_j^l Q_j \tilde{E}_{j k}\right)^2}$$

Examples

Calculating activity coefficients

We can instantiate a UNIFAC model with a mixture ethanol-water and evaluate the logarithm of activity coefficients of the model for a 0.5 mole fraction of each, and a temperature of 298.0 K.

use yaeos__constants, only: pr
use yaeos__models_ge_group_contribution_unifac, only: Groups, UNIFAC, setup_unifac

! Variables declarations
type(UNIFAC) :: model
type(Groups) :: molecules(2)
real(pr) :: ln_gammas(2)

! Variables instances
! Ethanol definition [CH3, CH2, OH]
molecules(1)%groups_ids = [1, 2, 14] ! Subgroups ids
molecules(1)%number_of_groups = [1, 1, 1] ! Subgroups occurrences

! Water definition [H2O]
molecules(2)%groups_ids = [16]
molecules(2)%number_of_groups = [1]

! Model setup
model = setup_unifac(molecules)

! Calculate ln_gammas
call model%ln_activity_coefficient([0.5_pr, 0.5_pr], 298.0_pr, ln_gammas)

print *, ln_gammas

You will obtain:

>>> 0.18534142000449058    0.40331395945417559

References

1. Dortmund Data Bank Software & Separation Technology
2. Fredenslund, A., Jones, R. L., & Prausnitz, J. M. (1975). Group‐contribution estimation of activity coefficients in nonideal liquid mixtures. AIChE Journal, 21(6), 1086–1099. https://doi.org/10.1002/aic.690210607
3. Skjold-Jorgensen, S., Kolbe, B., Gmehling, J., & Rasmussen, P. (1979). Vapor-Liquid Equilibria by UNIFAC Group Contribution. Revision and Extension. Industrial & Engineering Chemistry Process Design and Development, 18(4), 714–722. https://doi.org/10.1021/i260072a024
4. Gmehling, J., Rasmussen, P., & Fredenslund, A. (1982). Vapor-liquid equilibriums by UNIFAC group contribution. Revision and extension. 2. Industrial & Engineering Chemistry Process Design and Development, 21(1), 118–127. https://doi.org/10.1021/i200016a021
5. Macedo, E. A., Weidlich, U., Gmehling, J., & Rasmussen, P. (1983). Vapor-liquid equilibriums by UNIFAC group contribution. Revision and extension.
6. Industrial & Engineering Chemistry Process Design and Development, 22(4), 676–678. https://doi.org/10.1021/i200023a023
7. Tiegs, D., Rasmussen, P., Gmehling, J., & Fredenslund, A. (1987). Vapor-liquid equilibria by UNIFAC group contribution. 4. Revision and extension. Industrial & Engineering Chemistry Research, 26(1), 159–161. https://doi.org/10.1021/ie00061a030
8. Hansen, H. K., Rasmussen, P., Fredenslund, A., Schiller, M., & Gmehling, J. (1991). Vapor-liquid equilibria by UNIFAC group contribution. 5. Revision and extension. Industrial & Engineering Chemistry Research, 30 (10), 2352–2355. https://doi.org/10.1021/ie00058a017
9. Wittig, R., Lohmann, J., & Gmehling, J. (2003). Vapor−Liquid Equilibria by UNIFAC Group Contribution. 6. Revision and Extension. Industrial & Engineering Chemistry Research, 42(1), 183–188. https://doi.org/10.1021/ie020506l
10. SINTEF - Thermopack