Shrinking unreacted core model

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. 

For the noncatalytic reaction of particles with surrounding fluid, we consider two simple idealized models, the progressive-conversion model and the shrinking unreacted-core model. 

Particle diameter as a function of the conversion measured particle radius a homogeneous reaction model b shrinking unreacted core model c local volumetric rate model... 

Shrinking Unreacted Core Model (Rate Determined by Diffusion Through Product Layer)... 

The reaction system studied includes reactions R1 and R2 only. Gaseous reduction of a single iron ore particle proceeds following the shrink unreacted core model. Gas film resistance on mass transfer around the particle could be ignored since ore fines are under bubbling state. The shrink unreacted core model is used for expressing the reaction rates of R1 and R2 and they are expressed as Eqs.( 3-4). 

Therefore, one should use the shrinking unreacted-core model with the utmost care, especially in extrapolating results beyond the range of parameters employed in obtaining experimental data. 

In Chapter 3 we discussed gas-solid reactions involving nonporous solids and showed how different reaction steps that are coupled in series interact with each other. In many gas-solid reactions the solid is porous, allowing diffusion and chemical reaction to occur simultaneously throughout the solid therefore, the reaction occurs in a diffuse zone rather than at a sharp boundary. The reaction of a porous solid with a gas has not been investigated as extensively as that of a nonporous solid due to the difficulties in analyzing the experimental data. Furthermore, even the analysis of the results of experiments on a porous solid has often been based on the shrinking unreacted-core model. 

Since many solid reactants have some initial porosity and the simple shrinking unreacted-core model is often inapplicable to such systems, there have been recent efforts to find valid models for these reaction systems. A review of these will be presented. 

Most of the studies on the nonisothermal effects in gas-solid reactions have been based on the shrinking unreacted-core model. Ishida et al. [64]... 

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 ... 

To date, there have been several unsuccessful attempts to fit these results to a simple model—for example, one based on a shrinking unreacted core or on reaction of a porous solid. The apparent role of water in the mechanism suggests that sulfur dioxide may be oxidized to sulfur trioxide on the surface and that sulfur trioxide diffuses through a product layer to react with calcium carbonate. This concept would be consistent with the similar kinetics observed for half- and fully calcined stone since the rate-determining step would presumably be the same in either case. This view is supported by the observation that reactivity in a fluidized bed decreases somewhat above about 850 °C because the thermodynamics of sulfur dioxide oxidation become less favorable. On the other hand, Borgwardt s observations with fully calcined stone (1) suggest that the decreased reactivity is caused by hard-burning of the stone. 

Heterogeneous Model with Shrinking Unreacted Core ... 

HETEROGENEOUS MODEL WITH SHRINKING UNREACTED CORE... 

Figure 9.2(a) or (b) shows the essence of the SCM, as discussed in outline in Section 9.1.2.1, for a partially reacted particle. There is a sharp boundary (the reaction surface) between the nonporous unreacted core of solid B and the porous outer shell of solid product (sometimes referred to as the ash layer, even though the ash is desired product). Outside the particle, there is a gas film reflecting the resistance to mass transfer of A from the bulk gas to the exterior surface of the particle. As time increases, the reaction surface moves progressively toward the center of the particle that is, the unreacted core of B shrinks (hence the name). The SCM is an idealized model, since the boundary between reacted and unreacted zones would tend to be blurred, which could be revealed by slicing the particle and examining the cross-section. If this... 

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. 

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... 

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.

The unreacted core shrinking model gives rise to the size reduction function of the following form (13) ... 

To illustrate the principles of the shrinking core model, we shall consider the removal of carbon from the catalyst particle just discussed. In Figure 11-15 a core of unreacted carbon is contained between r = 0 and r = R. Carbon has been removed from the porous matrix between r = R arid r = R. Oxygen diffuses from the outer radius Ro to the radius R, where it reacts with carbon to form carbon dioxide, which then diffuses out of the porous matrix. The reaction... 

The particle thus consists of an unreacted inner core surrounded by an outer ash layer (provided the particle contains ash). This concept is the basis of the well known shrinking core model it yields some practical conversion (X)-time (t) relations.[ 51 ] Assuming a steady-state and isothermal conditions the shrinking core model gives for spherical particles [ 52j ... 

The number of active sites per unit surface of B will presumably be constant, and the rate proportional to the total number of sites. Hence the rate should be proportional to the surface area of the unreacted core. A feature of the shrinking-core model is that this area is known and is equal to 47rr - for a spherical core. This may not be a realistic area for reaction in real situations, but it is the characteristic of the model that permits mathematical analysis of the process. 

Model 4. This is a variation of Model 3 in which the counter-ion from the solution does not permeate beyond the portion of particle which has been converted to the exchanging ionic form. The boundary of the unreacted core reduces the time such that this is called the shrinking core model. It is this... 

The mathematical description of such a situation would comprise the continuity equations for the fluid and solid reactants encountered in Sec. 4.3 for the unreacted shrinking-core model and a heat balance that assumes pseudo steady state in the shell and an integral averaged temperature in the core up to the frcmL... 

The final argument for the existence of a monolayo of CaO stems from the sulphation model. The sulphation reaction has been described by a shrinking core model. The caldum aluminate dispersed on the pore walls is assumed to react immediately when SO3 arrives. Due to this first chemical reaction a skin of reacted sorbent is formed, while the inner core remains unreacted. Taking external mass transfer into account, we obtained for the inverse of the rate of change in sulphation conversion [6]... 

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,... 

With these we enlist the two fundamental approaches to the noncatalytic gas-solid reaction systems The shrinking core model and volume reaction model. In the volnme reaction model, the solid is porous, the fluid easily diffuses in or ont of the solid, such that the reaction can take place homogeneously everywhere in the solid. On the other hand, with the shrinking core model (SCM), also called the sharp interface model (SIM), there is a sharp interface between the unreacted core and reacted shell of the particles. 

The type of process where a solid is converted to a product with a smaller volume (a) by the action of a gas (eq. (6.14)) lends itself best to a general analysis. There is the well known "shrinking core" model, that describes the conversion of a massive solid reactant into a porous product by the action of a gaseous reactant. (Yagi and Kunii, 1952, Levenspiel, 1980). As the reaction process, the massive unreacted core will shrink and a layer of porous reaction product ("ash") will cover the core. Reactants have to dif fuse through the porous "ash" to reach the surface of the massive core, where the reaction takes place. In principle, the rate of the process is determined by diffusion through the porous layer, that becomes thicker on conversion, and reaction at the core si ace, that becomes smaller. 

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