Carnahan-Starling repulsion term

In order to combine a good description of the pVT behavior in the critical region and a mathematical function that does not require a numerical iteration, a semi-empirical approach for an equation of state has recently been developed [21,37]. This approach is based on a classical equation of state consisting of the Carnahan-Starling repulsion term [38] and a van der Waals-like attraction term [39]. Such an equation usually shows deviations from experimental data in the critical region. The correlation of the deviations is accomplished by a perturbation term that describes the deviation of a local density from the average density. 

Li et al. [189] assumed that a pair of deformable droplets has the shape of truncated sphere separated by a planar film and used the improved Carnahan-Starling equation to describe the repulsion term as ... 

In eq 3.1, the activity coefficients appear as a result of the hard-sphere repulsions among the droplets. Since the calculations focus on the most populous aggregates, the hard-sphere repulsions will be expressed in terms of a single droplet size corresponding to the most populous aggregates. One can derive expressions for the activity coefficients y ko of a component k in the continuous phase O starting from an equation for the osmotic pressure of a hard-sphere fluid,3-4 such as that based on the Carnahan—Starling equation of state (see Appendix B for the derivation) ... 

It is also assumed that, in the case of n-alkane mixtures, the ky-values are independent of the chain length of component j. Dimitrelis and Prausnitz[8] showed that there is a systematic deviation from the Carnahan and Starling[9] repulsive term as the difference in molecular size between two molecules increase. It is thus expected that the value of the interaction parameter ly will be related to the difference in size between the two molecules. It is assumed that the value of ly will approach a constant value when this difference becomes large. The interaction parameters for propane and n-butane were found by fitting this equation of state to the data mentioned above. The parameters are shown in table 2 ... 

This investigation follows our efforts previously made on the modification of the RK equation of state (3,30). The repulsive term of the RK equation was retained with the anticipation that the original terms would be preserved as part of the new equation. This practice may be subject to modifications in future endeavors. The repulsive term may be replaced by a more suitable term such as that proposed by Carnahan and Starling... 

We now consider modifications to the repulsive term in the van der Waals equation. Although the van der Waals hard-sphere term is correct at low densities. Figure 4.13 shows that it quickly becomes erroneous as the density is increased the excluded volume is not constant, but depends on density in some complicated way. Therefore we can improve the equation of state by using the Carnahan-Starling form (4.5.4) for Z/jg. Our modified Redlich-Kwong (mRK) equation of state is then [29]... 

Many generalizations of van der Waals ideas have been proposed for improving the prediction of the fluid phase behavior in a wide range of thermodynamic states and to extend the description to molecular fluids and mixtures [79]. Longuet-Higgins and Widom [80] suggested that the repulsive term Po be replaced with the accurate expressions for the pressure of hard spheres elaborated in the theory of liquids [81], such as the Carnahan-Starling approximation,[82]... 

Unfortunately, almost no empirical equations of state contain an adequate description of the hard-sphere fluid. The VDW term, (V — Nb) 1, seems to be a part of most, if not all, the popular empirical equations of state. Carnahan and Starling (14), Gubbins (30), Prausnitz (31), and no doubt, others have called for the replacement of this term in chemical-engineering practice. It is time that their advice was followed. The VDW term (V — Nb)"1 is an inadequate description of a singlecomponent, hard-sphere fluid. It is even less suitable to describe mixtures or fluids composed of nonspherical molecules. Fortunately, the replacement of (V — Nb)"1 is easily accomplished since, in all the popular empirical equations of state, the contributions of the repulsive and attractive forces are separated clearly. Moreover, the form of the term representing the contribution of the attractive forces generally is written in a theoretically reasonable form. The replacement of (V — Nb)"1 by a more reliable expression results not only in an intellectually more satisfying equation but usually in a more reliable one also. 

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