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Activity

Here we discuss the activity coefficient models available in TEA. For an in depth discussion of these models see the standard references. For the calculation of activity coefficients and their derivatives (for diffusion calculations) see also Kooijman and Taylor (1991).

Ideal

For an ideal system the activity coefficient of all species is unity, and thus, ln(γi) = 0.

Regular solution

The regular solution model is due to Scatchard and Hildebrand. It is probably the simplest model of liquid mixtures. The activity coefficient is given by:

$\displaystyle \theta_i$ $\textstyle =$ $\displaystyle V_i / \sum_k^c x_k V_k$
$\displaystyle \ln \gamma^*_i$ $\textstyle =$ $\displaystyle {V_i \over R T}\left[ \delta_i - \sum_j^c x_j \delta_j \theta_j \right]^2$
$\displaystyle \ln \gamma_i$ $\textstyle =$ $\displaystyle \ln \gamma^*_i$

where δi is called the solubility parameter and Vi the molal volume of compound i (both read from the PCD-file).

The Flory-Huggins corrected version is also available:

$\displaystyle \ln \gamma_i$ $\textstyle =$ $\displaystyle \ln \gamma_i^* + \ln \theta_i + 1 - \theta_i$

Margules

The "Three suffix" or two parameter form of the Margules equation is implemented in TEA:

\begin{displaymath}\ln \gamma_i =[ A_{ij} + 2 ( A_{ji} - A_{ij} ) x_i ] x_j^2\end{displaymath}

It can only be used for binary mixtures.

Van Laar

The Van Laar equation is

\begin{displaymath}\ln \gamma_i ={A_{ij} \over \left(1+{A_{ij} x_i \over A_{ji} x_j}\right)^2}\end{displaymath}

It can only be used for binary mixtures.

Wilson

The Wilson equation was proposed by G.M. Wilson in 1964. It is a "two parameter equation". That means that two interaction parameters per binary pair are needed to estimate the activity coefficients in a multi-compound mixture. For mixtures that do NOT form two liquids, the Wilson equation is, on average, the most accurate of the methods used to predict equilibria in multi-compound mixtures. However, for aqueous mixtures the NRTL model is usually superior.

$\displaystyle \Lambda_{ij}$ $\textstyle =$ $\displaystyle (V_j/V_i) \exp ( -(\lambda_{ij}-\lambda_{ii}) )$
$\displaystyle S_i$ $\textstyle =$ $\displaystyle \sum_{j=1}^c x_j \Lambda_{ij}$
$\displaystyle \ln \gamma_i$ $\textstyle =$ $\displaystyle 1 - \ln (S_i) - \sum_{k=1}^c x_k \Lambda_{ki} / S_k$

The two interaction parameters are (λij-λii) and (λji-λjj) per binary pair of compounds. If interaction parameters g are specified in temperature units (K), then

$ \lambda_{ij} - \lambda_{ii} = g_{ij} / T $

or if interaction parameters g are specified in energy units, then

$ \lambda_{ij} - \lambda_{ii} = g_{ij} / RT $

or if interaction parameters g are specified temperature invariant, then

$ \lambda_{ij} - \lambda_{ii} = g_{ij}  $

NRTL

The NRTL equation due to Renon and Prausnitz is a three parameter equation. Unlike the original Wilson equation, it could also be used for liquid-liquid equilibrium calculations.

$\displaystyle \tau_{ij}$ $\textstyle =$ $\displaystyle (\lambda_{ij}-\lambda_{ii}) $
$\displaystyle G_{ij}$ $\textstyle =$ $\displaystyle \exp(-\alpha_{ij} \tau_{ij})$
$\displaystyle S_i$ $\textstyle =$ $\displaystyle \sum_{j=1}^c x_j G_{ji}$
$\displaystyle C_i$ $\textstyle =$ $\displaystyle \sum_{j=1}^c x_j G_{ji} \tau_{ji}$
$\displaystyle \ln \gamma_i$ $\textstyle =$ $\displaystyle C_i / S_i +\sum_{k=1}^c x_k G_{ik} ( \tau_{ik} - C_k / S_k ) / S_k$

The interaction parameters are (\lambdaij-\lambdaii), (\lambdaji-\lambdajj) and αij per binary (only one α is required as αji = αij).

If interaction parameters g are specified in temperature units (K), then

$ \lambda_{ij} - \lambda_{ii} = g_{ij} / T $

or if interaction parameters g are specified in energy units, then

$ \lambda_{ij} - \lambda_{ii} = g_{ij} / RT $

or if interaction parameters g are specified in temperature invariant units, then

$ \lambda_{ij} - \lambda_{ii} = g_{ij}  $

In addition, a temperature dependent modification of gij can be introduced. If specified, the above equations are modified to, when specified in temperature units (K), then

$ \lambda_{ij} - \lambda_{ii} = \left( g_{ij} - g_{T,ij} \left( 1 - T / T_{ref} \right) \right) / T $

or if interaction parameters g are specified in energy units, then

$ \lambda_{ij} - \lambda_{ii} = \left( g_{ij} - g_{T,ij} \left( 1 - T / T_{ref} \right) \right) / RT  $

or if interaction parameters g are specified in temperature invariant units, then

$ \lambda_{ij} - \lambda_{ii} = \left( g_{ij} - g_{T,ij} \left( 1 - T / T_{ref} \right) \right)  $

The reference temperature in TEA is Tref = 298.15 K

UNIQUAC

UNIQUAC stands for Universal Quasi Chemical and is a very widely used model of liquid mixtures that reduces, with certain assumptions, to almost all of the other models mentioned in the list. Like the Wilson equation, it is a two parameter equation but is capable of predicting liquid-liquid equilibria as well as vapour-liquid equilibria. Two types of UNIQUAC models are available Original and q-prime. Original is to be used if you have obtained interaction parameters from DECHEMA. The q-prime (q') form of UNIQUAC is recommended for alcohol mixtures. An additional pure compound parameter, q', is needed. If q' equals the q value it reduces to the original method.

$\displaystyle r$ $\textstyle =$ $\displaystyle \sum_{i=1}^c x_i r_i$
$\displaystyle q$ $\textstyle =$ $\displaystyle \sum_{i=1}^c x_i q_i$
$\displaystyle \phi$ $\textstyle =$ $\displaystyle x_i r_i / r$
$\displaystyle \theta$ $\textstyle =$ $\displaystyle x_i r_i / r$
$\displaystyle \tau_{ji}$ $\textstyle =$ $\displaystyle \exp( -(\lambda_{ji}-\lambda_{ii})/RT)$
$\displaystyle S_i$ $\textstyle =$ $\displaystyle \sum_{j=1}^c \theta_j \tau_{ji}$
$\displaystyle \ln \gamma_i^c$ $\textstyle =$ $\displaystyle 1 + \left(1 - {z \over 2} q_i \right)\ln \left({ \phi_i \over x_i }\right) + {z \over 2} q_i \ln \left({ \theta_i \over x_i }\right)  -{r_i \over r} + {z \over 2} q\left({r_i \over r} - {q_i \over q} \right)$
$\displaystyle \ln \gamma_i^r$ $\textstyle =$ $\displaystyle q_i \left( 1 - \ln (S_i) -\sum_{k=1}^c {\theta_k \tau_{ik} \over S_k} \right)$
$\displaystyle \ln \gamma_i$ $\textstyle =$ $\displaystyle \ln \gamma_i^c + \ln \gamma_i^r$

The interaction parameters are (λij-λii) and (λji-λjj) per binary. The parameters ri and qi are read from the compound database (PCD file).

The coordination number z is taken as 10. In the q-prime mode, q' is used instead of q in the residual part ln γir, whereas the combinatorial part ln γir uses q.

If interaction parameters g are specified in temperature units (K), then

$ \lambda_{ij} - \lambda_{ii} = g_{ij} / R $

or if interaction parameters g are specified in energy units, then

$ \lambda_{ij} - \lambda_{ii} = g_{ij} / RT $

or if interaction parameters g are specified in temperature invariant units, then

$ \lambda_{ij} - \lambda_{ii} = g_{ij}  $

UNIFAC

UNIFAC is a group contribution method that is used to predict equilibria in systems for which NO experimental equilibrium data exist. The method is based on the UNIQUAC equation, but is completely predictive in the sense that it does not require interaction parameters. Instead, these are computed from group contributions of all the molecules in the mixture.

UNIFAC-VL is fitted to vapor-liquid equilibria. UNIFAC-LL is fitted to liquid-liquid equilibria.

ASOG

ASOG is a group contribution method similar to UNIFAC but based on the Wilson equation. It was developed before UNIFAC but is less widely used because of the comparative lack of fitted group interaction parameters.

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