Core model

The flow in the vortex finder is assumed to consist of an outer annulus of nonviscous, axisymmetric, potential vortex flow with a uniform axial velocity surrounding a core of solid-body rotation with negligible axial velocity. The derivation here is similar to that presented by Lewellen (1971) and Smith,S Jr. (1962)—both have origins in the original work of Binnie and Hooking (1948). 

We begin our derivation of the core flow model with a flow balance relating the average axial flow velocity, in the annular region located between radii Rc and R, to the average velocity normally computed or measured out 

In addition, for the free-vortex region, the tangential velocity at radius Rx, which we have called vecs elsewhere in this book is, according to Eq. (2.1.2). 

If we now apply Bernoulli s Equation (Eq. 2.1.3) between point 1 located at the entrance to the cyclone, and point 2, located on the iimer surface of the vortex core cyhnder, there results 

Equation (4.3.11) is the pressure difference between the total pressure entering the cyclone (pi = p + pv f2), i.e., static plus dynamic pressure, and the static pressure p2 at the inner of the vortex core. Yet, one normally regards cyclone pressure drop to be the difference in static pressures only. Therefore, if we wish to compare the results of the core-flow model with measurements (or correlations of measurements) of cyclone delta p for flow out the vortex tube (as we do in Table 4.3.1 later in this chapter) this is permissible providing pv /2 for the gas entering the cyclone is small in comparison with the p v + 

---

It has not proved possible to develop general analytical hard-core models for liquid crystals, just as for nonnal liquids. Instead, computer simulations have played an important role in extending our understanding of the phase behaviour of hard particles. Frenkel and Mulder found that a system of hard ellipsoids can fonn a nematic phase for ratios L/D >2.5 (rods) or L/D <0.4 (discs) [73] however, such a system cannot fonn a smectic phase, as can be shown by a scaling... 

In the irreversible limit R < 0.1), the adsorption front within the particle approaches a shock transition separating an inner core into which the adsorbate has not yet penetrated from an outer layer in which the adsorbed phase concentration is uniform at the saturation value. The dynamics of this process is described approximately by the shrinldng-core model [Yagi and Kunii, Chem. Eng. (Japan), 19, 500 (1955)]. For an infinite fluid volume, the solution is ... 

Kihara20 used a core model in which the Lennard-Jones potential is assumed to hold for the shortest distance between the molecular cores instead of molecular centers. By use of linear, tetrahedral, and other shapes of cores, various molecules can be approximated. Thomaes,41 Rowlinson,35 Hamann, McManamey, and Pearse,14 Atoji and Lipscomb,1 Pitzer,30 and Balescu,4 have used other models of attracting centers and other mathemtical methods, but obtain similar conclusions. The primary effect is to steepen the potential curve so that in terms of inverse powers of the inter-... 

Hiwatari Y. The applicability of the soft core model of fluids to dynamical properties of simple liquids, Progr. Theor. Phys. 53, 915-28 (1975). 

FIGURE 5.10 (continued) (b) Shell-core model for clusters. (From Macknight, W.J., Taggart, W.P., and Stein, R.S., J. Polym. Symp., 45, 127, 1974. With permission of Wiley, New York.)... 

The soft-core model may be more convenient in molecular dynamics simulation, since a continuously differentiable potential is available to calculate the force. In the case of a hardcore potential, collision times of all atom pairs have to be monitored and used to control the time step. 

The shrinking core and the volume-reaction models have been examined to interpret the conversion-time data of combustion and steam gasification of the gingko nut shell char [4]. The shrinking core model provides the better agreement with the experimental data. With the shrinking core model, the relationship between [1-(1-X) ] and the reaction time t at 350°C -... 

C for the steam gasification is shown in Fig. 3 where the shrinking core model predicts the experimental data very well. 

Strictly speaking, the validity of the shrinking unreacted core model is limited to those fluid-solid reactions where the reactant solid is nonporous and the reaction occurs at a well-defined, sharp reaction interface. Because of the simplicity of the model it is tempting to attempt to apply it to reactions involving porous solids also, but this can lead to incorrect analyses of experimental data. In a porous solid the chemical reaction occurs over a diffuse zone rather than at a sharp interface, and the model can be made use of only in the case of diffusion-controlled reactions. 

In general, there is no analytical solution for the partial differential equations above, and numerical methods must be used. However, we can obtain analytical solutions for the simplified case represented by the shrinking-core model, Figure 9.1(a), as shown in Section 9.1.2.3. 

I.2.3.I. Isothermal spherical particle. The shrinking core model (SCM) for an isothermal spherical particle is illustrated in Figure 9.1(a) for a particular instant of time. It is also shown in Figure 9.2 at two different times to illustrate the effects of increasing time of reaction on the core size and on the concentration profiles. 

Figure 9.2 The shrinking-core model (SCM) for an isothermal spherical particle showing effects of increasing reaction time t...

In the use of the shrinking-core model for a gas-solid reaction, what information could be... 

Consider the reduction of relatively small spherical pellets of iron ore (assume p m = 20 mol L-1) by hydrogen at 900 K and 2 bar partial pressure, as represented by the shrinking-core model, and... 

A kinetics study was performed to examine the rate-controlling steps in a gas-solid reaction governed by the shrinking-core model ... 

Two models developed in Chapter 9 to describe the kinetics of such reactions are the shrinking-core model (SCM) and the shrinking-particle model (SPM). The SCM applies to particles of constant size during reaction, and we use it for illustrative purposes in this chapter. The results for three shapes of single solid particle are summarized in Table 9.1 in the form of the integrated time (t conversion (/B) relation, where B is the solid reactant in model reaction 9.1-1 ... 

Puska, Eija Karita. Nuclear reactor core modelling in multifunctional simulators. 1999. 67 p. + app. 73 p. 

Choudhary et al. [58] found reaction controlled kinetics with an activation energy of 56.6 kJ/mol for the leaching of skeletal nickel, similar to the leaching of skeletal copper. The kinetics did not fit Levenspiel s shrinking core model [57] but it should be noted that the leaching solution was agitated with a flat stirrer at 1500 rpm. 

The shrinking core models described by Levenspiel cater for both reaction- and diffusion-controlled systems. Referring to the literature, how do these systems differ and which of these models do skeletal catalysts fit during their preparation by leaching ... 

Figure 52 (a) Chiral aggregation cartoon and (b) hard-core model to explain chirality of aggregates.321 Adapted with permission from Nakashima, H. Fujiki, M. Koe, J. R. Motonaga, M. J. Am. Chem. Soc. 2001, 123, 1963-1969, 2001... 

In order to further experimentally probe the hard-core model of polysilane aggregate chirality, a series of poly-[(alkyl)alkylphenylsilylene]s, differing in phenyl ring-substituent position and chain length were synthesized, designed such that the polymer chain diameter, d, and helical pitch, p, in the hard-core model were varied.343... 

Yoshioka, T Motoki, T. and Okuwaki, A., Kinetics of hydrolysis of poly(ethylene terephthalate) powder in sulfuric acid by a modified shrinking-core model, Ind. Eng. Chem. Res., 40, 75-79 (2001). 

While Debye and HUckel recognized the short-range repulsive forces between ions by assuming a hard-core model, the statistical mechanical methods then available did not allow a full treatment of the effects of this hard core. Only the effect on the electrostatic energy was included—not the direct effect of the hard core on thermodynamic properties. 

---