Carnahan—Starling fluid

Carnahan-Starling fluid 67 Cavitation 38 Chain contraction 21 Chain increment method 16 Chemical potential 202 Chi parameter 236, 242 Closure 220-225, 236 atomic 220... 

Now, let us consider a model in which the association site is located at a distance slightly larger than the hard-core diameter a. The excess free energy for a hard sphere fluid is given by the Carnahan-Starling equation [113]... 

The results for the chemical potential determination are collected in Table 1 [172]. The nonreactive parts of the system contain a single-component hard-sphere fluid and the excess chemical potential is evaluated by using the test particle method. Evidently, the quantity should agree well with the value from the Carnahan-Starling equation of state [113]... 

Zero-sepertion Values of B(r) Calculated from Several Approximations for HS Fluid, Compared to Carnahan-Starling EOS Data... 

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) ... 

Figure 7.12 Excess chemical potential of the hard-sphere fluid as a function of density. The open and filled circles correspond to the predictions of the primitive quasi-chemical theory and the self-consistent molecular field theory, respectively. The solid and dashed lines are the scaled-particle (Percus-Yevick compressibility) theory and the Carnahan-Starling equation of state, respectively (Pratt and Ashbaugh, 2003).

Here p(r) is the smoothed density and A is the thermal de Broglie wavelength. The repulsive part of the Helmholtz free energy is usually calculated by the Carnahan-Starling equation derived for the hard sphere fluid [80] ... 

Johnston KP, Eckert CA. An analytical carnahan-starling van der Waals model for solubility of hydrocarbon solids in supercritical fluids. AIChE J 1981 27 773. 

Figure 4.8 Phase diagram for a pure substance composed of hard spheres. The fluid-phase Z was computed from the Carnahan-Starling equation (4.5.4) the solid-phase Z was taken from the computer simulation data of Alder et al. [14]. The broken horizontal line at Zt = 6.124 connects fluid (T = 0.494) and solid (t = 0.545) phases that can coexist in equilibrium, as computed by Hoover and Ree [12].

By combining the Carnahan-Starling equation with the integral forms in 4.4.2, we can evaluate the residual properties of a pure hard-sphere fluid. The results are [15]... 

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]... 

Kraska, T. and Deiters, U. 1992) Systematic investigation of the phase behavior in binary fluid mixtures, n. Calculations based on the Carnahan-Starling-Redlich-Kwong equation of state J. Chem. Pl s. 96, 539-547. 

Fig. 3.1 The pressure of hard spheres. The curves are the Carnahan—Starling exjaession (3.1) for a fluid < 0.494) and the cell model result (3.12) for an fee crystal (solid curves, (j) > 0.545). The closed symbols are Monte Carlo computer simulation results [13]. The two open symbols cmrespond to the fluid-solid coexistence from simulation [11], the dotted line is the themetical result (see Sect. 3.2.3)...

Theory and computer simulation provides information in addition to the virial equation of state that can be used to develop mixing and combining rules. The equation of state for the pure hard-sphere fluid can be represented by the equation of state of Carnahan-Starling ... 

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. ... 

Similarly to the fluid-fluid intermolecular potential, we split the solid-fluid intermolecular potential into repulsive hard-sphere and attractive interactions. Here Fhs Ps P is the excess free energy of the solid-fluid HS mixture, for which we employ Rosenfeld fundamental m ure functional [26] with the recent modifications that mve an accurate Carnahan-Starling equation of state in the bulk limit [27,28] r-r ) is the attractive part of the solid-fluid intermolecular potential. Since the iM>lid-soIid attraction interaction is not included, the solid is effectively modeled as a compound of... 

The Carnahan-Starling equation of state is obtained from the virial expansion of a hard sphere fluid by curve-fitting and extrapolation. Table 1 compares values of the compressibility factor, Z, and the excess entropy, S, of a hard sphere fluid obtained from the Percus-Yevick equation and the Carnahan-Starling equation. 

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 ... 

III. 1.1. The equation of stale. Carnahan and Starling (26) have proposed a simple and accurate equation of state for hard-sphere fluids as a function of the volume fraction ij, which is given by... 

The fluid phase for a hard-sphere system is stable up to rj = 0.49, at which point a solid phase with rj = 0.55 is predicted to coexist in equilibrium with the fluid phase. Carnahan and Starling [30] have proposed the following simple and accurate equation of state ... 

For concentrated suspensions of hard spheres, the radial distribution function for the fluid phase is generated from the solution to the Percus-Yevick [37] equation using a Heaviside step function mviltiplied by a nearest neighbor geometric function for a disordered fluid. TTie result is a function for the compressibility derived by Carnahan and Starling [25] ... 

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