The Shrinking Core Model

Consequently, we see that increasing the temperature fiom 400°C to 500°C increases the conversion by only 1.7% Both engineers would have benefited fiom a more thorough study of this chapter. 

For a packed catalyst bed, the temperature-dependence part of the mass transfer coefficient for a gas-phase reaction can be written as 

Depending on how one fixes or changes the molar feed rate, U may also depend on the feed temperature. As an engineer, it is extremely important that you reason out the effects cf changing conditions, as illustrated in the preced-Important concept ing two examples. 

External Diffusion Effects on Heterogeneous Reactions Chap. 11 

As the carbon continues to be removed from the porous catalyst pellet, the reactant gas must diffuse farther into the material as the reaction proceeds to reach the unreacted solid phase. Note that approximately 3 hours was required to remove all of the carbon from the pellets at these conditions. The regeneratiqn time can be reduced by increasing the gas-phase oxygen concentration and temperature. 

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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 ... 

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 ... 

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... 

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 ... 

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. 

Consider the following process for converting waste shredded fibers into a useful product. Fibers and fluid are fed continuously into a mixed flow reactor where they react according to the shrinking core model with the reaction step as rate controlling. Develop the performance expression for this operation as a function of the pertinent parameters and ignore elutri-ation. 

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). 

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],... 

The nucleation model represents the other extreme. Here, the dissociation of hydrogen is the slow step. Once a nucleus of reduced metal exists, it acts as a catalyst for further reduction, as it provides a site where H2 is dissociated. Atomic hydrogen diffuses to adjacent sites on the surface or into the lattice and reduces the oxide. As a result, the nuclei grow in three dimensions until the whole surface is reduced, after which further reduction takes place, as in the shrinking core model. The extent of reduction (see Fig. 2.3a) shows an induction period, but then increases rapidly and slows down again when the reduction enters the shrinking core regime. 

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... 

In contrast, work published in journals of pure chemistry is usually concerned with reaction at the boundaries and commonly only initial rates are measured so that complications due to diffusion can be neglected. The results and conclusions in such studies are naturally different from case to case and will be enumerated later. First, however the shrinking core model will be reviewed. 

Solution of the shrinking core model at zero time (t=0) depends only on two parameters the solubility of solute in SC CO2 and the external particle to fluid mass transfer coefficient Kq. Hence, knowing the solubility, measurements of the initial extraction rates allow to determine the values of K(j. Detailed discussion on the evaluated mass transfer coefficients are given in [7].These authors found that the overall mass transfer from particles to fluid depends upon both free and forced convection mechanism. Figure 2 illustrates a parity plot of die experimental values of Sh number (evaluated by zero-time solution of the shrinking core model) and the calculated Sh number (using an appropriate mass transfer correlation). 

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. 

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 ... 

This complex reaction sequence leads to equally complex decomposition kinetics. As discussed in Chapter 5, the shrinking core model is applicable to simple one-step thermal decompositions of the type... 

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