Shrinking-sphere model

The carbon loading is found to be proportional to coal feed rate and particle burning time. It is a variable of major importance because it determines the carbon loss from the bed and also has an important impact on the emissions of nitrogen oxides. Estimates of the carbon loading can be obtained by evaluating t from the relation for a shrinking sphere model of a diffusion-limited oxidation of carbon. It is readily shown that, for particles of density pc and initial diameter d ... 

FIGURE SJ2 Shrinking sphere model. From Levenspiel [1], copyright 1972 by John l ey Sons, Inc. Reprinted by permission of John Wiley S[Pg.140]

As with the shrinking core model, boundary layer mass and heat transfer fluxes are applicable as well as the surface reaction flux. The fluxes are combined in a way similar to that of the shrinking core model to give the results in Table 5.3 for a. shrinking sphere model. When this model is applicable, the partide morphology dianges drastically during reaction from particles to flakes of partides. 

The ratio of apparent density versus fractional bum-off is also plotted in Figure 5 and closely follows the shrinking sphere model. The initial apparent density, was measured experimentally. The initial shape factor, < ), was obtained from the following equations. 

The fractional burn-off includes the mass loss due to attrition, but this may not alter above conclusions since mass loss due to attrition is very small compared to that due to combustion. Furthermore, attrition is predicted by the shrinking sphere model. 

The char particles burn at a constant density in accordance with a shrinking sphere model. 

The above rate laws are for coupons of well-defined geometry, however they can be adapted to powders as well. Jander (Hulbert, 1969) first developed a shrinking sphere model for this purpose. First assume the radius of each particle, R, decreases linearly with time ... 

If the initiation step, the activation of H2, is fast, as may be the case on noble metal oxides or highly defective oxide surfaces, the shrinking core or contracting sphere model applies (see Figure 2.3). The essence of this model is that nuclei of reduced metal atoms form rapidly over the entire surface of the particle and grow into a shell of reduced metal. Further reduction is limited by the transport of lattice oxygen out of the particle. The extent of reduction increases rapidly initially, but slows down as the metal shell grows. 

Figure 2.3 Left, reduction models. In the shrinking core or contracting sphere model the rate of reduction is initially fast and decreases progressively due to diffusion limitations. The nucleation model applies when the initial reaction of the oxide with molecular hydrogen is difficult. Once metal nuclei are available for the dissociation of hydrogen, reduction proceeds at a higher rate until the system comes into the shrinking core regime. Right the reduction rate depends on the concentration of unreduced sample (1-a) as f(a) see Expressions (2-5) and (2-6).

In other cases, the product D flakes off the surface of particle B, because there is a large difference in the molar volume of reactant B and product D. This type of reaction is modeled as a shrinking sphere as seen in Figure 5.2 [1]. 

Kechaou N., Roques M., 1990, A model for convective drying of nonporous shrinking spheres, IDS90, 1, 369-373. 

For the case where abundant nuclei are present in the initial stages of the transformation, which is termed site saturation, the extended volume of one sphere of product phase is modeled by changing the sign for the rate constant in the shrinking particle model (Chapter 6) to make it a growing particle model. The linear growth rate of each sphere of the product phase is G (m/sec) and its volume is V (m ). 

Elaborations upon the accessible surface include the contact, re-entrant, and molecular surfaces. The contact surface is that part of the van der Waals surface that can be touched by a probe sphere, without that sphere experiencing overlap with any other atom. The re-entrant surface is made up of the inward-facing part of the probe sphere as it touches more than one atom. Together, the contact and re-entrant surfaces form a continuous envelope, which is called the molecular surface. This surface resembles shrink wrap placed around a hard sphere model. 

TABLE 8.4 Evaporative Conversion, Xb, versus Time for Shrinking Liquid Core Model Sphere Xg = 1 - r/Raf... 

By allowing one of the spheres to shrink to a point and assuming its motion to be similar to a gas molecule s, the coagulation problem is rendered slightly more tractable. In the case of only surface interaction, the problem reduces to that of condensation or sorption. Conversely, if longer range potentials, which are not necessarily monotonic such as the electrostatic interaction with a polarizable aerosol particle are considered, the system can be taken to model one of two important aspects of realistic two-particle interactions. This aspect is the role of particle size relative to gas density in the determination of an interactant s collision or repulsion. The other aspect which cannot yet be addressed, is that of attraction due to mutual shielding discussed above. 

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