Shrinking core models

For ease of solution, it is assumed that the spherical shape of the pellet is maintained throughout reaction and that the densities of the solid product and solid reactant are equal. Adopting the pseudo-steady state hypothesis implies that the intrinsic chemical reaction rate is very much greater than diffusional processes in the product layer and consequently the reaction is confined to a gradually receding interface between reactant core and product ash. Under these circumstances, the problem can be formulated in terms of pseudo-steady state diffusion through the product layer. The conservation equation for this zone will simply reflect that (in the pseudo-steady state) there will be no net change in diffusive flux so 

Solution of the entire pseudo-steady state problem (commonly referred to as the shrinking core model) is achieved by analytical integration of eqn. (53) and substitution of the result into eqn. (55), subsequently eliminating the unknown Ca by the use of eqn. (54). Substitution into eqn. (56) then gives the overall reaction rate in terms of CAg, and r. This result is not particuleirly useful, however, until the shrinking core radius, r, is related to time. Recalling the chemical stoichiometric relationship [eqn. (50)] the rate of consumption of A in terms of the core radius is 

The complexity of eqn. (58) arises because it was not assumed that any one of the three rate processes identified is rate determining. The relation between time and conversion is considerably simplified one of the rate processes dominates the overall process. When reactant gas flows through a fixed bed of solid particles, the gas film resistance to conversion is 

Industrial reactors employed for heterogeneous catalytic and gas—solid 

Consider a spherical solid particle of radius R containing the reactant B the particle is in contact with a gaseous stream containing the reactant A at concentration Cas- The stoichiometric equation for the reaction between A and B is 

With the progress of the reaction, the central unreacted core will shrink in size and hence this model is known as SCM. is the concentration of A on the surface of the solid particle and Cj, is the concentration of A on the surface of the imreacted core. The global rate expression is derived by taking into account three rate equations, namely, 

Rate of transfer of A (r ) from the bulk of the gas to the surface of the particle through the gas film resistance Rg 

Profile of within the single solid particle according to SCM. 

---

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. 

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 shrinking-core model [Yagi and Kunii, Chem. Eng. (Japan), 19, 500 (1955)]. For an infinite fluid volume, the solution is ... 

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 ... 

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 ... 

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). 

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

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.

SHRINKING-CORE MODEL FOR SPHERICAL PARTICLES OF UNCHANGING SIZE... 

Shrinking-Core Model for Spherical Particles of Unchanging Size 571... 

Table 25.1 Conversion-Time Expressions for Various Shapes of Particles, Shrinking-Core Model...

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 ... 

Assuming that reaction proceeds by the shrinking-core model calculate the time needed for complete conversion of a particle and the relative resistance of ash layer diffusion during this operation. 

Reaction proceeds according to the shrinking core model with reaction control and with time for complete conversion of particles of 1 hr. 

---