Activity-Coefficient Models¶

For an activity-coefficient model the liquid logarithmic fugacity coefficient and its derivatives are

\[ \begin{aligned}\ln\varphi_{i}^{L} & =\ln\gamma_{i}+\ln\frac{P_{i}^{\mathrm{sat}}}{P}+\ln\mathrm{Poy}_{i}\\ \frac{\partial\ln\varphi_{i}^{L}}{\partial T} & =\frac{\partial\ln\gamma_{i}}{\partial T}+\frac{d\ln P_{i}^{\mathrm{sat}}}{dT},\qquad\frac{\partial\ln\varphi_{i}^{L}}{\partial n_{j}}=\frac{\partial\ln\gamma_{i}}{\partial n_{j}} \end{aligned} \]

with the vapor-pressure term taken from the compound correlation. The activity-coefficient derivatives follow. Composition derivatives are given with respect to \(x_{k}\) and are converted to the mole-number basis by the projection of the Scope and Notation section.

NRTL¶

Writing the NRTL activity coefficient with the column sums

\[ \begin{aligned}\ln\gamma_{i} & =\frac{S_{1i}}{S_{2i}}+\sum_{j}\frac{x_{j}G_{ij}}{D_{j}}\!\left(\tau_{ij}-E_{j}\right)\\ S_{1i} & =\sum_{j}x_{j}\tau_{ji}G_{ji},\quad S_{2i}=\sum_{j}x_{j}G_{ji},\quad D_{j}=\sum_{k}x_{k}G_{kj}\\ N_{j} & =\sum_{m}x_{m}\tau_{mj}G_{mj},\quad E_{j}=N_{j}/D_{j} \end{aligned} \]

\[ \begin{aligned}\tau_{ij} & =\frac{A_{ij}+B_{ij}T+C_{ij}T^{2}}{RT},\qquad G_{ij}=e^{-\alpha_{ij}\tau_{ij}}\\ \frac{\partial\tau_{ij}}{\partial T} & =\frac{B_{ij}+2C_{ij}T}{RT}-\frac{A_{ij}+B_{ij}T+C_{ij}T^{2}}{RT^{2}}\\ \frac{\partial G_{ij}}{\partial T} & =-\alpha_{ij}G_{ij}\frac{\partial\tau_{ij}}{\partial T} \end{aligned} \]

The temperature derivative (composition fixed) is

\[ \begin{aligned}\frac{\partial\ln\gamma_{i}}{\partial T} & =\frac{S_{1i}'S_{2i}-S_{1i}S_{2i}'}{S_{2i}^{2}}\\ & \quad{}+\sum_{j}x_{j}\!\left[\frac{G_{ij}'D_{j}-G_{ij}D_{j}'}{D_{j}^{2}}(\tau_{ij}-E_{j})+\frac{G_{ij}}{D_{j}}(\tau_{ij}'-E_{j}')\right] \end{aligned} \]

where a prime denotes \(\partial/\partial T\) , \(S_{1i}'=\sum_{j}x_{j}(\tau_{ji}'G_{ji}+\tau_{ji}G_{ji}')\) , \(S_{2i}'=\sum_{j}x_{j}G_{ji}'\) , \(D_{j}'=\sum_{k}x_{k}G_{kj}'\) , \(N_{j}'=\sum_{m}x_{m}(\tau_{mj}'G_{mj}+\tau_{mj}G_{mj}')\) and \(E_{j}'=(N_{j}'D_{j}-N_{j}D_{j}')/D_{j}^{2}\) . The composition derivative (temperature fixed, so \(\tau\) and \(G\) constant) is

\[ \begin{aligned}\frac{\partial\ln\gamma_{i}}{\partial x_{k}} & =\frac{G_{ki}}{S_{2i}}\!\left(\tau_{ki}-\frac{S_{1i}}{S_{2i}}\right)+\frac{G_{ik}}{D_{k}}(\tau_{ik}-E_{k})\\ & \quad{}-\sum_{j}x_{j}\frac{G_{ij}G_{kj}}{D_{j}^{2}}\!\left[(\tau_{ij}-E_{j})+(\tau_{kj}-E_{j})\right] \end{aligned} \]

UNIQUAC¶

UNIQUAC splits into a combinatorial part, which depends on composition only through \(S_{r}=\sum_{k}r_{k}x_{k}\) and \(S_{q}=\sum_{k}q_{k}x_{k}\) , and a residual part built from \(\tau_{ji}=e^{-a_{ji}/T}\) (with \(z=10\) and \(l_{i}=\tfrac{z}{2}(r_{i}-q_{i})-(r_{i}-1)\) ):

\[ \begin{aligned}\ln\gamma_{i}^{C} & =\ln\frac{r_{i}}{S_{r}}+\frac{z}{2}q_{i}\ln\frac{q_{i}S_{r}}{r_{i}S_{q}}+l_{i}-\frac{r_{i}}{S_{r}}\sum_{j}x_{j}l_{j}\\ \ln\gamma_{i}^{R} & =q_{i}\!\left[1-\ln S_{i}-\sum_{j}\frac{\theta_{j}\tau_{ij}}{S_{j}}\right] \end{aligned} \]

with \(\theta_{i}=q_{i}x_{i}/S_{q}\) and \(S_{i}=\sum_{j}\theta_{j}\tau_{ji}\) . Only the residual part depends on temperature, through \(\partial\tau_{ji}/\partial T=\tau_{ji}\,a_{ji}/T^{2}\) , giving

\[ \begin{aligned}\frac{\partial\ln\gamma_{i}^{R}}{\partial T} & =q_{i}\!\left[-\frac{S_{i}'}{S_{i}}-\sum_{j}\theta_{j}\frac{\tau_{ij}'S_{j}-\tau_{ij}S_{j}'}{S_{j}^{2}}\right],\qquad S_{i}'=\sum_{j}\theta_{j}\frac{\partial\tau_{ji}}{\partial T}\end{aligned} \]

The composition derivatives are

\[ \begin{aligned}\frac{\partial\ln\gamma_{i}^{C}}{\partial x_{k}} & =-\frac{r_{k}}{S_{r}}+\frac{z}{2}q_{i}\!\left(\frac{r_{k}}{S_{r}}-\frac{q_{k}}{S_{q}}\right)+\frac{r_{i}r_{k}}{S_{r}^{2}}\sum_{j}x_{j}l_{j}-\frac{r_{i}}{S_{r}}l_{k}\end{aligned} \]

\[ \begin{aligned}\frac{\partial\ln\gamma_{i}^{R}}{\partial x_{k}} & =-\frac{q_{i}}{S_{i}}\frac{\partial S_{i}}{\partial x_{k}}-q_{i}\sum_{j}\tau_{ij}\frac{(\partial\theta_{j}/\partial x_{k})S_{j}-\theta_{j}(\partial S_{j}/\partial x_{k})}{S_{j}^{2}}\\ \frac{\partial\theta_{j}}{\partial x_{k}} & =\frac{q_{j}\delta_{jk}-\theta_{j}q_{k}}{S_{q}},\qquad\frac{\partial S_{i}}{\partial x_{k}}=\sum_{j}\frac{\partial\theta_{j}}{\partial x_{k}}\tau_{ji} \end{aligned} \]

Wilson¶

With \(s_{i}=\sum_{j}x_{j}\Lambda_{ij}\) and \(\Lambda_{ij}=(V_{j}/V_{i})\,e^{-a_{ij}/(RT)}\) (molar volumes at a fixed reference temperature, so \(\partial\Lambda_{ij}/\partial T=\Lambda_{ij}\,a_{ij}/(RT^{2})\) ):

\[ \ln\gamma_{i}=1-\ln s_{i}-\sum_{k}\frac{x_{k}\Lambda_{ki}}{s_{k}} \]

\[ \begin{aligned}\frac{\partial\ln\gamma_{i}}{\partial T} & =-\frac{s_{i}'}{s_{i}}-\sum_{k}x_{k}\frac{\Lambda_{ki}'s_{k}-\Lambda_{ki}s_{k}'}{s_{k}^{2}},\qquad s_{i}'=\sum_{j}x_{j}\frac{\partial\Lambda_{ij}}{\partial T}\end{aligned} \]

\[ \frac{\partial\ln\gamma_{i}}{\partial x_{m}}=-\frac{\Lambda_{im}}{s_{i}}-\frac{\Lambda_{mi}}{s_{m}}+\sum_{k}x_{k}\frac{\Lambda_{ki}\Lambda_{km}}{s_{k}^{2}} \]

Group-Contribution Models (UNIFAC, Modified UNIFAC)¶

UNIFAC is a combinatorial plus a residual (solution-of-groups) part, \(\ln\gamma_{i}=\ln\gamma_{i}^{C}+\ln\gamma_{i}^{R}\) . With \(J_{i}=r_{i}/\sum_{k}x_{k}r_{k}\) and \(L_{i}=q_{i}/\sum_{k}x_{k}q_{k}\) the combinatorial part is

\[ \ln\gamma_{i}^{C}=1-J_{i}+\ln J_{i}-5q_{i}\!\left(1-\frac{J_{i}}{L_{i}}+\ln\frac{J_{i}}{L_{i}}\right) \]

The residual part uses the group surface fraction \(\Theta_{m}=\Big(\sum_{i}x_{i}q_{i}\nu_{m}^{(i)}\Big)/\sum_{k}x_{k}q_{k}\) , the per-component group sum \(\beta_{im}=\sum_{k}\nu_{k}^{(i)}\tau_{km}\) and \(s_{m}=\sum_{l}\Theta_{l}\tau_{lm}\) , where \(\nu_{k}^{(i)}\) is the number of groups of type \(k\) in molecule \(i\) :

\[ \ln\gamma_{i}^{R}=q_{i}\!\left[1-\sum_{m}\!\left(\frac{\Theta_{m}\beta_{im}}{s_{m}}-\nu_{m}^{(i)}\ln\frac{\beta_{im}}{s_{m}}\right)\right] \]

Only \(\tau_{km}\) depends on temperature. For the original UNIFAC \(\tau_{km}=e^{-a_{km}/T}\) and \(d\tau_{km}/dT=-\tau_{km}\ln\tau_{km}/T\) ; the combinatorial part is temperature-independent. With \(\beta_{im}'=\sum_{k}\nu_{k}^{(i)}\,d\tau_{km}/dT\) and \(s_{m}'=\sum_{l}\Theta_{l}\,d\tau_{lm}/dT\) ,

\[ \frac{\partial\ln\gamma_{i}^{R}}{\partial T}=-q_{i}\sum_{m}\!\left[\Theta_{m}\frac{\beta_{im}'s_{m}-\beta_{im}s_{m}'}{s_{m}^{2}}-\nu_{m}^{(i)}\!\left(\frac{\beta_{im}'}{\beta_{im}}-\frac{s_{m}'}{s_{m}}\right)\right] \]

For the composition derivative the combinatorial ratios and the group fraction vary through

\[ \begin{aligned}\frac{\partial J_{i}}{\partial x_{p}} & =-\frac{r_{i}r_{p}}{\left(\sum_{k}x_{k}r_{k}\right)^{2}},\qquad\frac{\partial L_{i}}{\partial x_{p}}=-\frac{q_{i}q_{p}}{\left(\sum_{k}x_{k}q_{k}\right)^{2}}\\ \frac{\partial\Theta_{m}}{\partial x_{p}} & =\frac{q_{p}\left(\nu_{m}^{(p)}-\Theta_{m}\right)}{\sum_{k}x_{k}q_{k}},\qquad\frac{\partial s_{m}}{\partial x_{p}}=\sum_{l}\frac{\partial\Theta_{l}}{\partial x_{p}}\tau_{lm} \end{aligned} \]

( \(\beta_{im}\) is composition-independent). Writing \(R_{i}=J_{i}/L_{i}\) with \(\partial R_{i}/\partial x_{p}=\big[(\partial J_{i}/\partial x_{p})L_{i}-J_{i}(\partial L_{i}/\partial x_{p})\big]/L_{i}^{2}\) , the combinatorial and residual composition derivatives are

\[ \begin{aligned}\frac{\partial\ln\gamma_{i}^{C}}{\partial x_{p}} & =-\frac{\partial J_{i}}{\partial x_{p}}+\frac{1}{J_{i}}\frac{\partial J_{i}}{\partial x_{p}}-5q_{i}\!\left(-\frac{\partial R_{i}}{\partial x_{p}}+\frac{1}{R_{i}}\frac{\partial R_{i}}{\partial x_{p}}\right)\\ \frac{\partial\ln\gamma_{i}^{R}}{\partial x_{p}} & =-q_{i}\sum_{m}\!\left[\frac{\partial\Theta_{m}}{\partial x_{p}}\frac{\beta_{im}}{s_{m}}-\Theta_{m}\frac{\beta_{im}}{s_{m}^{2}}\frac{\partial s_{m}}{\partial x_{p}}+\nu_{m}^{(i)}\frac{1}{s_{m}}\frac{\partial s_{m}}{\partial x_{p}}\right] \end{aligned} \]

The two parts are summed and projected onto the mole-number basis. For Modified UNIFAC (Dortmund and NIST) the combinatorial volume term uses \(r_{i}^{3/4}\) in \(J_{i}\) , and the group interaction is temperature-dependent, \(\Psi_{km}=\exp\!\big[-(a_{km}+b_{km}T+c_{km}T^{2})/T\big]\) , with

\[ \frac{d\ln\Psi_{km}}{dT}=\frac{a_{km}-c_{km}T^{2}}{T^{2}} \]

everything else being identical.