Models van Laar

This, the Wilson model, is more complex than the Van Laar model, but it does retain the two-parameter feature. The terminal activity coefficients are related to the parameters ... 

In the original equation of van Laar, the effective molar volume was assumed to be independent of composition this assumption implies zero volume-change of mixing at constant temperature and pressure. While this assumption is a good one for solutions of ordinary liquids at low pressures, it is poor for high-pressure solutions of gases in liquids which expand (dilate) sharply as the critical composition is approached. The dilated van Laar model therefore assumes that... 

From the dilated van Laar model, Chueh obtains two-parameter expressions for the activity coefficients. They are ... 

It is easily possible to introduce refinements into the dilated van Laar model which would further increase its accuracy for correlating activity coefficient data. However, such refinements unavoidably introduce additional adjustable parameters. Since typical experimental results of high-pressure vapor-liquid equilibria at any one temperature seldom justify more than two adjustable parameters (in addition to Henry s constant), it is probably not useful for engineering purposes to refine Chueh s model further, at least not for nonpolar or slightly polar systems. 

The dilated van Laar model is readily generalized to the multicomponent case, as discussed in detail elsewhere (C3, C4). The important technical advantage of the generalization is that it permits good estimates to be made of multicomponent phase behavior using only experimental data obtained for binary systems. For example, Fig. 14 presents a comparison of calculated and observed -factors for the methane-propane-n-pentane system at conditions close to the critical.7... 

While the dilated van Laar model gives a reliable representation of constant-pressure activity coefficients for nonpolar systems, the good agreement between calculated and experimental high-pressure phase behavior shown in Fig. 14 is primarily a result of good representation of the partial molar volumes, as discussed in Section IV. The essential part of any thermodynamic description of high-pressure vapor-liquid equilibria must depend,... 

This factoring of In f produces a scale-invariant part that serves to interpolate smoothly between the value of f" and f = 1.0. Two well-known special cases of the van Laar model are the regular solution model, which results by setting Iab = Ca > aHd the Vanselow model, which results by setting fAB = fBA = l.O.1 The Vanselow model thus corresponds to an ideal solid solution (f = fd = 1.0 cf. Section 3.3). 

Related Calculations. The constants for the binary Margules and Van Laar models for predicting activity coefficients (see Related Calculations under Example 3.4) are simply the natural logarithms of the infinite-dilution activity coefficients A t2 = I n y(XJ and /12,1 = I n y2XJ. 

These data were shown in Figure 4.1 and used as the basis for Example 4.1.3. Carry out your own check of the accuracy of the Vignes correlation using the NRTL model for the activity coefficients and the parameters in Example 4.1.3. Repeat the calculations using the Van Laar model with parameters 12 = 1.5922 and 21 = 0.9836. 

Figure 4.2.1. VLE correlation of the methane and pentane binary system at 310, 377, and 444 K with the Huron-Vidal original (HVO) mixing rule with the van Laar excess fiee-energy model and the PRS V equation of state. The van Laar model parameters used are = A12/A21 = 0.1201/0.1430. Points are experimental data from the DECHEMA Chemistry Data Series, Gmehiing and Onken 1977, Vol. 6, p. 445 data tiles for this system on the accompanying disk are C1C5310.DAT, C1C5377.DAT andClC5444.DAT.

Experience has shown that the choice of activity coefficient model to be coupled with the EOS has some effect on the predictive performance of both the WS and HVO models. For example, as the complexity of the mixture increases, the NRTL and the UNK UAC models, which have a temperature dependence of their parameters, usually give better results over a range of temperatures than the simple van Laar model. 

Table 4.4.1. Van Laar model parameters (A12/A21) of some binary mixtures for various approximate EOS-G mixing rules...

PIJ ARE DIMENSIONLESS KAPPA PARAMETERS OF THE VAN LAAR MODEL) type 1, 1 and press RETURN. 

AC-VLE FROM ACTIVITY COEFFICIENT MODELS THE VAN LAAR MODEL PARAMETERS E12, P21 ... 

This results in the selection of the van Laar model for the excess free-energy term in the HVO mixing rule.)... 

At this stage the program attempts to optimize the two model parameters of the van Laar model, and the intermediate results are continuously displayed on the screen in the form of an error bar, When the optimization is complete, a message displaying summary of results appears on the screen for inspection. Press RETURN to continue, The results given below appear on the screen.)... 

Equation (3.41) describes the van Laar model of solvent mixtures and is applicable only to mixtures of low-molecular-weight components with approximately the same molar volume. 

The failure of the van Laar model to give realistic predictions of the thermodynamic properties of polymer solutions arises from the assumption made in this model that the solvent and solute molecules are identical in size. However, Flory [1] and Huggins [2] proposed, independently, a modified lattice theory which takes into account the large differences in size between solvent and polymer molecules, in addition to Intermolecular interactions. 

Write van Laar model this way to avoid division by zero. 

For the simpler models, it is possible to show by simple mathematics that the model either does or does not permit a double azeotrope. For example, the van Laar model is... 

Now for the benzene-hexafluorobenzene system (7ex has an interior maximum and an interior minimum. That is, d( Px/dx] is zero twice in the region 0 < x, < 1. To see if the van Laar model permits this we examine... 

Now ceand P must be of same sign (otherwise we get the square root of a negative number). Also, since 0 < xx <1 and 0 < x2 < 1, only positive sign is allowed. Thus x2/xj = Joc/[ when dCT jdxj = 0. And only an interior maximum (if oc> 0 and / > 0) or an interior minimum (if ot< 0 and P< 0) can occur, but not both Therefore, van Laar model can not describe the observed behavior. 

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