Carnahan-Starling term

The application of this equation of state to mixtures requires the replacement of the Carnahan-Starling term with the expression reported by Mansoori et and more complicated chain and association terms. In the SAFT equation, and other equations of this type, each of the terms has its own theoretically-based mixing rule that is different from the mixing rules for other terms in the same equation. For example, the mean attractive energy associated with the first rder perturbation is treated by Galindo et al SAFT and the related methods can be considered molecular-based equations of state for associating fluids. SAFT is reviewed in Chapter 8 and by Muller and Gubbins. ... 

Finally, to correctly evaluate the non-ideal contribution to the free energy of the nanospheres, the weighted-density Carnahan-Starling term is added. (For the reasoning behind its inclusion in the overall free energy, see Reference [32].) The weighted particle density is calculated via ... 

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) ... 

The next step is to provide a closure for the pair correlation function appearing in the collision source and collisional-flux terms. For moderately dense flows, the collision frequency for finite-size particles is known to be larger than that found using the Boltzmann Stofizahlansatz (Carnahan Starling, 1969 Enksog, 1921). In order to account for this effect, the pair correlation function can be modeled as the product of two single-particle velocity distribution functions and a radial distribution function ... 

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]... 

At least at low order, this expansion is reproduced to good accuracy by the well-known Carnahan-Starling equation [6], z = l + q + rf — r/ )/(l — 7), which yields 6 = n -(-n—2 for n> 1.) The expansion in terms of the packing fraction rj is related to the more familiar virial expansion in terms of the particle density p,... 

The Carnahan-Starling formulation is used for A and the segment-segment dispersion A p is described using a fourth-order perturbation term [62] the contribution of chain formation as well as the association term is accounted for based on the work of Wertheim [63]. 

It was later modified to include an attractive contribution, " regarding the latter as a small perturbation. It is therefore tempting to modify the Carnahan-Starling equation of state in such a way that it would become applicable to deformable fluid droplets. Unfortunately the perturbation approach is not relevant to the case of deformable fluid droplets. This becomes clear if one writes the perturbation term, Pqs, for the osmotic pressure ... 

Further improvement has been aceomplished by the use of a more realistic equation with respect to the molecular interaction such as the Carnahan-Starling-van der Waals type of equation of state for the description of the solvent [28], However, the cubic equations as well as the Carnahan-Starling kind of equations are not accurate in the critical region [27], The computational method preserved here is based on a Carnahan-Starling-van der Waals kind of equation but expected by a perturbation term that corrects the pVT-behavior in the critical region. This approach is expected to be a promising tool for the correlation of solubility behavior even up to pressures above 100 MPa. 

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. 

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 ... 

Because the particle lattice is assumed to be static, the osmotic pressure term due to particle conformation for the disordered suspension derived by Carnahan and Starling [25] and used by Dickinson [26] and Evans and Napper [27] must be added to the previous equation ... 

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. 

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... 

The perturbed-hard-ehain (PHC) theory developed by Prausnitz and coworkers in the late 1970s was the first successful application of thermodynamic perturbation theory to polymer systems. Sinee Wertheim s perturbation theory of polymerization was formulated about 10 years later, PHC theory combines results fi om hard-sphere equations of simple liquids with the eoneept of density-dependent external degrees of fi eedom in the Prigogine-Flory-Patterson model for taking into account the chain character of real polymeric fluids. For the hard-sphere reference equation the result derived by Carnahan and Starling was applied, as this expression is a good approximation for low-molecular hard-sphere fluids. For the attractive perturbation term, a modified Alder s fourth-order perturbation result for square-well fluids was chosen. Its constants were refitted to the thermodynamic equilibrium data of pure methane. The final equation of state reads ... 

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