Hard-sphere, approaches

Based on the ionic radii, nine of the alkali halides should not have the sodium chloride structure. However, only three, CsCl, CsBr, and Csl, do not have the sodium chloride structure. This means that the hard sphere approach to ionic arrangement is inadequate. It should be mentioned that it does predict the correct arrangement of ions in the majority of cases. It is a guide, not an infallible rule. One of the factors that is not included is related to the fact that the electron clouds of ions have some ability to be deformed. This electronic polarizability leads to additional forces of the types that were discussed in the previous chapter. Distorting the electron cloud of an anion leads to part of its electron density being drawn toward the cations surrounding it. In essence, there is some sharing of electron density as a result. Thus the bond has become partially covalent. 

FIGURE 7.8 X-ray (square) and neutron (circle) scattering data for DMDBTDMA in n-dodecane contacted with water. Lines correspond to the simultaneous fit to the experimental X-ray and neutron data with the Baxter sticky hard-sphere approach. [DMDBTDMA] = 0.5 M, [Monomers] = 0.28 M, aggregation number = 4.4, and U/kBT = -1.7. 

FIGURE 7.10 Pseudo-binary phase diagram of the water, HNO3/DMDBTDMA, dodecane system to identify the third-phase limit. Experimental points (circles) and theory (lines) obtained from Baxter sticky hard-sphere approach. The theoretical line is obtained with the experimental determination of the linear variation of the stickiness parameter tt1 versus [HN03]/[DMDBTDMA]. The different lines illustrate the impact of the error in this T-1 experimental linear law. (From C. Erlinger, L. Belloni, T. Zemb, and C. Madic, Langmuir, 15 2290-2300, 1999. With permission). 

Hoomans, B. P. B., Kuipers, J. A. M., and van Swaaij, W. P. M., Discrete particle simulation of a two-dimensional gas-fluidised bed Comparison between a soft sphere and a hard sphere approach. Submitted for publication (1998). 

Hoomans B.P.B et al (1996) Discrete particle simulation of bubble and slug formation in a two-dimensional gas-fluidised bed A hard-sphere approach. Chemical Engineering Science, 51, pp. 99-118... 

In a hard sphere approach, particles are assumed to interact through instantaneous binary collisions. This means particle interaction times are much smaller than the free flight time and therefore, hard particle simulations are event (collision) driven. For a comprehensive introduction to this type of simulation, the reader is referred to Allen and Tildesley (1990). Hoomans (2000) used this approach to simulate gas-solid flows in dense as well as fast-fluidized beds. There are three key parameters in such hard sphere models, namely coefficient of restitution, coefficient of dynamic friction and coefficient of tangential restitution. Coefficient of restitution is discussed later in this chapter. Detailed discussion of these three model parameters can be found in Hoomans (2000). 

We can obtain at least a qualitative picture of such an exchange by a modification of the hard-sphere approach. First, recall that in equations (2-1) and (2-2) we obtained expressions for the postcollision velocities of two hard spheres in terms of the precollision values. The change in kinetic energies for this process is... 

The structure of a sol or liquid suspension, as shown in Fig. 11.4(a), is the simplest colloid dispersion, described in detail in Chapter 10. Particles are independent from neighboring grains, swimming in the solvoit with Brownian movement. Each particle can collide with its neighbor but the collision is not adhesive and so the particles remain fuUy dispersed. Such a sol can be modeled by the hard sphere approach. For spherical particles of equal size and zero adhesion, the stracture is random at low concentrations. 

To test the validity of the hard sphere approach we first consider the equiatomic alloy, NaK, which is a relatively simple liquid alloy and for which X-ray and neutron diffraction data have been obtained (Henniger et aL, 1966). In addition data are available for the density and Xj over the whole range of composition (Abowitz and Gordon (1962)). As for the one component case, the final answer will depend on the choice of which in turn relates to the choice of Ri and R2. Ashcroft and Lekner (1966) found that for pure Na and pure K, a close fit with experiment was obtained with % = 0.45 so that Ri = 3.28A (Na) and R2 = 4.06A(K). Presumably these would alter somewhat in the alloy due to the change in electron density. However, in the absence of any theory of this effect, we shall retain these hard sphere diameters so that if the density is taken as 0.87 g cm which corresponds to 373 K, (alloy) = 0.451. With these assumptions for Ri and R2, Enderby and North (1968) calculated ajj(q) from the hard sphere PY theory. In Figure 7.15 we compare the experiment X-ray diffraction data with the theoretical curve derived from the calculated a j(q) and the published values of the x-ray form factors (Hanson et al (1964)). The measurement of agreement, particularly around the first peak, is encouraging and shows that for this alloy at least the PY hard-sphere description is a very useful first approximation. 

Fig. 4.5 Depletion thickness of ideal polymer chains around a sphere (solid curve) as a function of the polymer-to-sphere size ratio q. Solid curve (4.2), dashed line is the classical penetrable hard-sphere approach = Rg and the dotted curve follows the approximation 5/R = 0.938 see (4.24)...

In DPMs, each particle is tracked individually and all collisions are calculated, thus providing a more reliable and detailed representation of the fluidized bed. The model was introduced by Hogue and Newland (1994), Hoomans et al. (1996), and Tsuji et al. (1992), and it employs either a hard-sphere approach for dilute systems or a soft-sphere approach for dense fluidized beds. 

In this section, particle particle collision dynamics will be discussed. A hard sphere approach is used for the particle particle collision analysis. In this approach, it is assumed that collisions between spherical particles are binary and quasi-instantaneous, and further, that there is a sequence of collisions during each time step. The equations, which are similar to the stepwise molecular dynamic simulation, are used to locate the minimum flight time of particles before any collision. 

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