User:WilfriedC/Playground/NRTL

The Non-Random Two Liquid model (short NRTL equation) is a activity coefficient model that correlates the activity coefficients $$\gamma$$ with the composition of a  mixture of chemical compounds, expressed by mole fractions $$x$$.

Equations
For a binary mixture the following equations are used:


 * $$\ln\ \gamma_1=x^2_2\left[\tau_{21}\left(\frac{G_{21}}{x_1+x_2 G_{21}}\right)^2 +\frac{\tau_{12} G_{12}} {(x_2+x_1 G_{12})^2 }\right]$$


 * $$\ln\ \gamma_2=x^2_1\left[\tau_{12}\left(\frac{G_{12}}{x_2+x_1 G_{12}}\right)^2 +\frac{\tau_{21} G_{21}} {(x_1+x_2 G_{21})^2 }\right]$$

with


 * $$\ln\ G_{12}=-\alpha_{12}\ \tau_{12}$$

and


 * $$\ln\ G_{21}=-\alpha_{12}\ \tau_{21}$$

$$\tau_{12}$$ and  $$\tau_{21}$$ as well as $$\alpha_{12}$$ are fittable parameters. In most cases the parameters $$\tau$$


 * $$\tau_{12}=\frac{\Delta g_{12}}{RT}$$

and


 * $$\tau_{21}=\frac{\Delta g_{21}}{RT}$$

are scaled with the gas constant and the temperature and then the parameters $$\Delta g_{12}$$ and $$\Delta g_{21}$$ are fitted.

Temperature dependend parameters
If activity coefficients are available over a larger temperature range (maybe derived from both vapor-liquid and solid-liquid equilibria) temperature-dependend parameters can be introduced.

Two different approaches are used:


 * $$\tau_{ij}=f(T)=a_{ij}+\frac{b_{ij}}{T}+c_{ij}\ \ln\ T+d_{ij}T$$


 * $$\Delta g_{ij}=f(T)=a_{ij}+b_{ij}\cdot T +c_{ij}T^{2}$$

Single terms can be omitted. E. g., the logarithmic term is only used if liquid-liquid equilibria (miscibility gap) have to be described.

Parameter determination
The NRTL parameters are fitted to activity coefficients that have been derived from experimentally determined phase equilibrium data (vapor-liquid, liquid-liquid, solid-liquid) as well as from heats of mixing. The source of the experimental data are often factual data banks like the Dortmund Data Bank. Other options are direct experimental work and predicted activity coefficients with UNIFAC and similar models.