Non-Random Two Liquid model

The Non-Random Two Liquid model [Renon H., Prausnitz J. M., "Local Compositions in Thermodynamic Excess Functions for Liquid Mixtures", AIChE J., 14(1), S.135-144, 1968] (short NRTL equation) is an 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 [Reid R. C., Prausnitz J. M., Poling B. E., The Properties of Gases & Liquids, 4. Edition, McGraw-Hill, 1988] are used:

:$ln\; gamma\_1=x^2\_2left\; [\; au\_\{21\}left(frac\{G\_\{21\{x\_1+x\_2\; G\_\{21\; ight)^2\; +frac\{\; au\_\{12\}\; G\_\{12\; \{(x\_2+x\_1\; G\_\{12\})^2\; \}\; ight]$

:$ln\; gamma\_2=x^2\_1left\; [\; au\_\{12\}left(frac\{G\_\{12\{x\_2+x\_1\; G\_\{12\; ight)^2\; +frac\{\; au\_\{21\}\; G\_\{21\; \{(x\_1+x\_2\; G\_\{21\})^2\; \}\; ight]$

with

:$ln\; G\_\{12\}=-alpha\_\{12\}\; au\_\{12\}$

and

:$ln\; G\_\{21\}=-alpha\_\{12\}\; au\_\{21\}$

$au\_\{12\}$ and $au\_\{21\}$ as well as $alpha\_\{12\}$ are fittable parameters. In most cases the parameters $au$

:$au\_\{12\}=frac\{Delta\; g\_\{12\{RT\}$

and

:$au\_\{21\}=frac\{Delta\; g\_\{21\{RT\}$

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

Temperature dependent parameters

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

Two different approaches are used:

:$au\_\{ij\}=f(T)=a\_\{ij\}+frac\{b\_\{ij\{T\}+c\_\{ij\}\; ln\; T+d\_\{ij\}T$

:$Delta\; g\_\{ij\}=f(T)=a\_\{ij\}+b\_\{ij\}cdot\; T\; +c\_\{ij\}T^\{2\}$

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.

Literature