public subroutine Ge_residual(self, n, T, Ge, dGe_dn, dGe_dn2, dGe_dT, dGe_dT2, dGe_dTn)

UNIFAC residual term

Evaluate the UNIFAC residual therm

Description

Evaluate the UNIFAC residual therm. 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:

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

References

1. SINTEF - Thermopack

Arguments