Enskog factor

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... 

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-... 

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. 

Lu et al, 2001 Rahaman et al, 2003). Ny is scaled by di ttittyg considering that Ny varies greatly with the Enskog factor. Three surfaces intersect at line 9i = 9j, about which both our results and Rahaman s results are symmetric. That is consistent with theoretical analysis as discussed above, while Huilin s result fails. Furthermore, we can observe that our surfaces are higher than the Huilin s prediction and lower than Rahaman s when 9i<9y This is expected as the Huilin s profile can only be applied in energy equipartition system, which may underestimate the collision frequencies, whereas Rahaman s hypothesis implies aU the collisions happen in a 2D plane, which obviously overestimates the collision frequency. 

Figure 10 plots the solids shear viscosity and its comparison with literature results. Different sources of data are scale by the Enskog factor to be viewed in the same figure. Unlike Rahaman s result where the viscosity has a minimum at 9 = 9y our result shows that the are stiU mainly determined by 9 and the maximum of viscosity appears at the maximum of 9, which demonstrates that the dilute viscosity increases with the increase of the granular temperature. 

Assuming that A << /o and that /o varies appreciably only over distances x L, it is easy to show that A//o —XjL, where A is the mean free path length i.e. /o is a good approximation if the characteristic wavelengths of p, T and u are all much greater than the mean free path. The exact solution / can then be expanded in powers of the factor X/L. This systematic expansion is called the CAia.pma.n-Enskog expansion, and is the subject of the next section. 

Theories based on the Enskog collision time (84) or other solid-like approaches do not have a strongly temperature-dependent frequency correlation time. But they do have a temperature-dependent factor resulting from the need to create the solvent fluctuations in the first place. Thus, all fast-modulation theories predict that the dephasing rate will go to zero at 0 K. 

The most important result of the model is the weak dependence of D/D on particle mass for Z > 1. First this suggests that the Enskog theory will provide a good estimate for heavy atoms in a molecular fluid at all densities. Second the Enskog diffusion coefficient itself is only weakly dependent on mass, namely, as [(1 -I- Z)/Z] and if this factor is used to estimate the mass... 

The superlinear density dep>endence of the simulation results for listed in Table IV is well accounted for by the estimate of the Enskog enhancement factors of the Br-Ar pair. For a packing fraction of 0.01, this factor is close to 1. At this density Nordholm et al. evaluate a Br2-Ar collision frequency from analysis of the time variation of the Br2 internal energy of 2.0 x 10 which is comparable to a value of 1.5 x 10 estimated from a collision diameter fferj.Ar = Br.Ar + The collision number Nyj for V-T transfer is kg/Z and is 65 at low density. We conclude that at low densities where collisions are well resolved, the model of this work and the simulations give similar collision numbers. 

This pressure tensor closure was derived by Gidaspow [22] in accordance with the Enskog theory presented by Chapman and Cowling [11], chap 16. That is, with the restitution coefficient e equal to one, the y factor substituted by go, and bp = Aad this relation corresponds to equation (16.34 — 2) in Chapman and Cowling [11]. 

Enskog proposed that since the gas is not in equilibrium we should multiply Eq. (142) by a factor [ (r,r+ak), which is the nonequilibrium pair correlation function for two particles in contact. Since the form of this function is not known, he made the following assumption The analytic dependence of (r, r+ak) on density is exactly the same as that of g(a), but the density to be used in computing if (r, r+afc) is the local density at a point midway between the two spheres, i.e., n(r+5ak). Since g(a) has the density expansion " ... 

Boltzmann-Enskog approximation consists of recollision terms that describe correlated binary collisions. The effects of these processes on the behavior of time correlation functions have not yet been fully studied because the calculations involved are considerably more complicated. The problem where the effects of recollisions have been most extensively investigated is that of the velocity autocorrelation function, which is a simpler function than the dynamic structure factor. From this problem we can already see the kind of analysis involved in treating correlated collisions. ... 

When a gas mixture is subjected to a temperature gradient, diffusion occurs and a concentration gradient is established. Enskog (1911) and Chapman (1916) deduced the transport equation, which relates separation factor, 9, to the temperature gradient. 

The surface concentration dependence of the lateral mobility of Fig. 7 was analyzed in terms of the free-volume theory of hard sphere liquids of Cohen and Turnbull [55, 56], as well as in view of the Enskog theory of dense gases [57] extended by Alder s molecular dynamics calculations to liquid densities [58]. The latter approach was particularly successful. It revealed that the lateral diffusion constant of the Fc amphiphiles does follow the expected linear dependence on the relative free area, Af/Ao, where Af = A — Ao, A = MMA, and Aq is the molecular area of a surfactant molecule. It also revealed that the slope of this dependence which is expected to inversely depend on the molecular mass of a diffusing particle, was more than 3 orders of magnitude smaller [54]. Clearly, this discrepancy is due to the effect of the viscous drag of the polar head groups in water, a factor not included in the Enskog theory. 

Other parameters, including the lattice sound speed Cs and weight factor fj, are lattice structure dependent. For example, for a typical D2Q9 (two dimensions and nine lattice velocities see Fig. 1) lattice structure, we have tQ = 4/9, ii 4 = 1/9, f5 8 = 1/36, and = A /3Afi, where Ax is the spatial distance between two nearest lattice nodes. Through the Chapman-Enskog expansion, one can recover the macroscopic continuity and momentum (Navier-Stokes) equations from the above-defined LBM dynamics ... 

This direct proportionality between the rough hard-sphere transport properties and the Enskog coefficients has formed the basis for many correlations of liquid transport properties (Easteal Woolf 1984 Li etal. 1986 Walker etal. 1988 Greiner-Schmid et al 1991 Harris etal 1993). For a successful data fit, with unique values for Vq and the proportionality factors, it is necessary to fit a minimum of two prt rties simultaneously, with the same Vq values. This is exemplified in the case of methane in Chapter 10. It is further shown in Chapter 10 that successful correlation of transport property data for nonspherical molecular liquids can be made, based on the assumption that transport properties for these fluids can also be directly related to the smooth hard-sphere values. 

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