Carnahan-Starling model

However, without much error. Equations 4.5 also can be applied to less concentrated systems, if other representations are used for the osmotic pressure correction function and Enskog factor. In particular, the approximate Carnahan-Starling model [26], which is widely used to describe particulate systems, results in alternative expressions... 

We might just as easily have made use of some other statistical theory of dense gases and liquids to describe G(0) and /(0). However, the Carnahan - Starling model seems preferable since it does not involve unwieldy numerical calculation and leads to simple analytical expressions. In expressing these functions for low-concentration systems, it is also possible to use the standard technique of virial expansions. 

On the other hand, applicability of the Carnahan-Starling model (as well as applicability of other approximate statistical models of the same kind) to particulate systems of very high concentrations appears to be questionable, to say the least. In particular, this possible inapplicability of this model can be due to its failure to account for spontaneous origination of ordered crystalline phase patterns at < ) = 0.55 - 0.59. This model also fails to describe a sharp increase in pressure and dynamic viscosity for the dispersed phase when on the verge of the closed-packed state. In contast to this, the Enskog model leads, however empirically, to physically correct conclusions that both pressure and viscosity tend to infinity as ()) approaches the value attributed to the state of close packing. 

In compliance with the discussion in Section 8, we choose the Carnahan-Starling model to define concentrational fluctuation variance according to Equation 7.4. With the help of Equation 4.9, this same model can be employed to express the osmotic pressure correction function and the Enskog factor for practically all suspension concentrations which lie beyond a narrow concentration range adjoining the closed-... 

Figure 4. Neutral stability curves for fluidized beds as foilow from the Carnahan-Starling model at Vi = 1 and different values of IgSc (figures at the curves) in the limiting regimes of constant and varying fiuctuation temperature (solid and dashed curves, respectively) dotted curves correspond to the osmotic pressure correction function calculated with the help of the Enskog model.

The first parameter appears as a result of quasi-viscous stresses in the dispersed phase affecting the development of initial plane waves. In fact, this parameter characterizes an influence on fluidized bed stability caused by dispersed phase viscosity. The occurrence of the second parameter is due to the restriction imposed from below on permissible wave numbers for these plane waves. Actually, the second parameter descibes a so-called scaling effect of the bed dimensions on bed stability. The curves in Figure 4 correspond to the Carnahan-Starling model, save for the dotted ones which have been drawn when using Equation 4.8 to represent the osmotic pressure correction function and the Enskog factor. 

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]... 

Fig. 19. Simulation results for both the soft-sphere model (squares) and the hard-sphere model (the crosses), compared with the Carnahan-Starling equation (solid-line). At the start of the simulation, the particles are arranged in a FCC configuration. Spring stiffness is K = 70,000, granular temperature is 9 = 1.0, and coefficient of normal restitution is e = 1.0. The system is driven by rescaling.

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. 

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 ... 

Note that this assumption simply transforms the problem of modeling the pair correlation function into the new problem of modeling o-The usual model for go assumes that the radial distribution function depends neither explicitly on the collision angle (i.e. on X12) nor explicitly on x. The former amounts to assuming that the particle with velocity V2 has no preferential spatial direction relative to the particle with velocity vi. The radial distribution function can then be modeled as a function of the disperse-phase volume fraction. For example, a typical model is (Carnahan Starling, 1969)... 

As follows from the curves in Figure 1, the models by Enskog and by Carnahan-Starling lead to somewhat different results, and the problem of a proper choice between these models arises. The theoretical curves for fluctuation temperature are compared with the experimental data of Carlos and Richardson [42] in Figure 2. These experiments were conducted with metallic balls 8.9 nun in diameter fluidized by dimetilphtalate. The maximal fluctuation temperature was experimentally observed at / = 0.32 - 0.34, which agrees well with our theoretical prediction. On the whole, the agreement between the presented theory and experiments looks quite satisfactory. 

Figure 1. Dimensionless fluctuation temperature for fluidized beds of small (1) and large (2) spherical particles according to the Carnahan-Starling and Enskog models (solid and dashed curves, respectively) u° is the terminal fall velocity of a single particle (j). = 0.6.

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)...

One drawback of the MF1V2 model is the inability of UNIFAC to predict (vapour + liquid) equilibria (VLB) and (liquid + liquid) equilibria (LLE) conditions using the same set of group-interaction parameters. In general, cubic equations of state do not provide precise predictions of the phase equilibria when the mixture is asymmetric in size that is attributed to the large differences in the pure-component co-volumes. The Carnahan -Starling equation for hard spheres is a more realistic model for the repulsive contribution than that proposed by van der Waals. Mansoori et al. proposed an equation for mixtures of hard spheres that has been found to correlate the phase behaviour of non-polar mixtures with large molecular size differences. 

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... 

Nitta et. al. ( 7) extended the group interaction model to thermodynamic properties of pure polar and non-polar liquids and their solutions, including energy of vaporization, pvT relations, excess properties and activity coefficients. The model is based on the cell theory with a cell partition function derived from the Carnahan-Starling equation of state for hard spheres. The lattice energy is made up of group interaction contributions. 

The form of the function efr ( ) is different in different versions of the smoothed-density approximation proposed by Somo-za and Tarazona [71, 72] and by Poniwier-ski and Sluckin [69, 73]. The density functional model of Somoza and Tarazona is based on the reference system of parallel hard ellipsoids that can be mapped into hard spheres. In the Poniwierski and Sluckin theory the effective weight function is determined by the Maier function for hard sphe-rocylinders and the expression for Ayr (p) is obtained from the Carnahan-Starling ex-... 

In the chain-of-rotators (COR) equation proposed by Chien, Greenkom, and Chao [17], a chain molecule is modeled as joined spheres in which adjacent covalent bonds can rotate. Rotation takes place because the chain is not straight any two adjacent bonds are joined at an angle like the radials from the center of a pyramid to the four comers. Chien et al. found the pressure of rotation by comparing the pressure of Boublik and Nezbeda s hard dumbbell fluid [18] with the pressure of Carnahan and Starling s hard spheres. The rotational pressure found is... 

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 ... 

The expressions for gijidij ) and A used in original S AFT as well as in PC-S AFT are based on the work of Boublik [27] and Mansoori et al. [28], who derived them for mixtures of hard spheres. For pure substances, these expressions become identical to the simpler one proposed by Carnahan and Starling [11]. Kouskoumvekaki et al. [29] also applied these simplified expressions to mixtures (simplified PC-SAFT) and obtained, in most cases, almost similar modeling results. Because for pure substances the expressions for gij dij ) and A become identical for PC-SAFT and simplified PC-SAFT, the pure-component parameters for PC-SAFT and simplified PC-SAFT are identical. 

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