Solid-core model

Figure 9. Schematic illustration of particle disintegration and the postulated morphological models SCM (solid core model),...

The simplest of the diffusive models proposed, the Solid core model, is based on a spherical catalyst particle with a spherical shell of polymer growing around it. 

SCM solid-core model SIR solvent impregnated resin... 

Several models have been proposed to account for both the fingmentation of catalyst particles and the diffusional resistance. The simplest is the solid-core model, in which the polymer is assumed to grow around a solid catalyst particle without any breakage. We show the model in Figure 5.12, in which the catalyst... 

The microparticle diffusion is treated in the same way as in the solid-core model, and it is assumed that each of these microparticles grows independent of each other according to the existing local monomer concentration. To write the mole balance for the monomer in the macroparticle in spherical coordinates, let us define Di as the effective diffusion coefficient for the macroparticle, ri as the radial length, and R Mi, as the rate of consumption of monomer at The governing equation for the macroparticle can be easily derived as... 

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

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 ... 

Fig. 4. Gas—solid reaction (shrinking core model), that...

Fig. 3. Comparison between prediction of typical shell and core model for partial reaction of a solid by a liquid or gas leading to solution or gasification with an unattacked solid residue.

Shrinking-Core Model (SCM). Here we visualize that reaction occurs first at the outer skin of the particle. The zone of reaction then moves into the solid, leaving behind completely converted material and inert solid. We refer to these as ash. Thus, at any time there exists an unreacted core of material which shrinks in size during reaction, as shown in Fig. 25.3. 

Figure 25.3 According to the shrinking-core model, reaction proceeds at a narrow front which moves into the solid particle. Reactant is completely converted as the front passes by.

Limitations of the Shrinking Core Model. The assumptions of this model may not match reality precisely. For example, reaction may occur along a diffuse front rather than along a sharp interface between ash and fresh solid, thus giving behavior intermediate between the shrinking core and the continuous reaction models. This problem is considered by Wen (1968), and Ishida and Wen (1971). 

Despite these complications Wen (1968) and Ishida et al. (1971), on the basis of studies of numerous systems, conclude that the shrinking core model is the best simple representation for the majority of reacting gas-solid systems. 

A batch of solids of uniform size is treated by gas in a uniform environment. Solid is converted to give a nonflaking product according to the shrinking-core model. Conversion is about for a reaction time of 1 h, conversion is complete in two hours. What mechanism is rate controlling ... 

In the shrinking core model a film of initial thickness transforms with an unreacted core of thickness l. The initial volume of a planar solid film is... 

Shrinking core model The shrinking core model has been derived for noncatalytic solid-fluid reactions (Levenspiel, 1972). However, it has been successfully used for specific ion-exchange systems—those using synthetic ion exchangers, mainly chelating resins (Cortina et al, 1998 Juang, 1999). 

Figure 4.16 Shrinking core model the reaction proceeds at a narrow front, which moves deeper into the solid particle as time passes.

The shrinking-core model (SCM) is used in some cases to describe the kinetics of solid and semi-solids-extraction with a supercritical fluid [22,49,53] despite the facts that the seed geometry may be quite irregular, and that internal walls may strongly affect the diffusion. As will be seen with the SCM, the extraction depends on a few parameters. For plug-flow, the transport parameters are the solid-to-fluid mass-transfer coefficient and the intra-particle diffusivity. A third parameter appears when disperse-plug-flow is considered [39,53],... 

Most of the gas-solid reactions that have been studied appear to proceed by the shrinking core reaction mode. In the simplest type of unreacted core model it is assumed that there is a non-porous unreacted solid with the reaction taking place in an infinitely thin zone separating the core from a completely reacted product as shown in Fig. 3.36 for a spherical particle. Considering a reaction between a gaseous reactant A and a solid B and assuming that a coherent porous solid product is formed, five consecutive steps may be distinguished in the overall process ... 

Fig. 3.36. Unreacted core model, impermeable solid, showing gas phase reactant...

Fig. 35. Dashed line Potential energy function for a valence electron in the impenetrable core model. Rc = core radius. Solid line Pseudopotential, after Austin and Heine 14 )...

G. Uhde, U. Hoffmann, Noncatalytic Gas-Solid Reactions Modelling of Simultaneous Reaction and Formation of Surface with a Nonisothermal Crackling Core Model, Chem. 

It is shown that the mechanism of gas-solid noncatalytic reactions can be understood better by following the variations in pore structure of the solid during the reaction. By the investigation of the pore structures of the limestone particles at different extents of calcination, it has been shown that the mechanism of this particular system can be successfully represented by a two stage zone reaction model below 1000 °C. It has also been observed that the mechanism changes from zone reaction to unreacted core model at higher temperatures. 

The mechanism of many of the noncatalytic fluid-solid reactions can be described by a model in between unreacted core and homogeneous reactions models. Ishida and Wen (9) formulated such a model using the zone reaction concept of Ausman and Watson (10). In this model the reaction is not restricted to the surface of the core as in the unreacted core model but occurs homogeneously within a retreating core of reactant. Wen and Ishida (11) combined the grain concept with the zone reaction model and analyzed the reaction of SO2 with CaO particles. In the study conducted by Mantri, Gokarn and Doraiswamy (12) the concept of finite reaction zone model was further developed. 

In the second case, as shown in Fig. 2, the entire surface of the reactant particle is covered with a thin layer of the solid product very soon after contacting the reactant gas and the reaction boundary advances inward as reaction proceeds. This model has been known as the shrinking core model [2, 3] and has been used by many investigators. Since the functional dependence of S on t behaves well in this model, provided that the assumption of smooth advancement of the reaction boundary without changing its shape is valid, consideration of a variety of forms of r can conveniently be included and the kinetics can be described up to the completion of reaction. Moreover, even in the first case, in which the... 

Figure 5.10 Representation of the unreacted-core gas/solid reaction model for a particle of unchanging size. As reaction time progresses from left to right in the figure, the reaction surface recedes into the particle, the unreacted core shrinks, and the ash layer (containing the reaction product) increases in thickness.

A batch of spherical monodisperse silicon metal particles is treated in a uniform ammonia gas. The solid is converted to SigN4 with the same particle morpholep, according to the shrinking core model. Conversion is seven-ei ths complete after 1 hr and totally complete after 2 hr. What is the rate determining step ... 

Core radii for the empty-core model, in A. These are listed also in the Solid State Table. 

---