Unreacted core

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. 

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... 

The basis for the analysis using the SCM is illustrated in Figure 9.3. The gas film, outer product (ash) layer, and unreacted core of B are three distinct regions. We derive the continuity equation for A by means of a material balance across a thin spherical shell in the ash layer at radial position r and with a thickness dr. The procedure is the same as that leading up to equation 9.1-5, except that there is no reaction term involving (- rA), since no reaction occurs in the ash layer. The result corresponding to equation 9.1-5 is... 

Referring to the concentration profiles for A in Figure 9.2, we realize that if there is no resistance to the transport of A in either the gas film or the ash layer, cA remains constant from the bulk gas to the surface of the unreacted core. That is,... 

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. 

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. 

Comparison of Models with Real Situations. In slicing and examining the cross section of partly reacted solid particles, we usually find unreacted solid material surrounded by a layer of ash. The boundary of this unreacted core may not... 

Step 2. Penetration and diffusion of A through the blanket of ash to the surface of the unreacted core. 

The decrease in volume or radius of unreacted core accompanying the disappearance of dN moles of solid reactant is then given by... 

Replacing Eq. 6 in 4 gives the rate of reaction in terms of the shrinking radius of unreacted core, or... 

The radius of unreacted core in terms of fractional time for complete conversion is obtained by combining Eqs. 8 and 9, or... 

Consider a partially reacted particle as shown in Fig. 25.6. Both reactant A and the boundary of the unreacted core move inward toward the center of the particle. But for GIS systems the shrinkage of the unreacted core is slower than the flow rate of A toward the unreacted core by a factor of about 1000, which is roughly the ratio of densities of solid to gas. Because of this it is reasonable for us to assume, in considering the concentration gradient of A in the ash layer at any time, that the unreacted core is stationary. 

Figure 25.7 illustrates concentration gradients within a particle when chemical reaction controls. Since the progress of the reaction is unaffected by the presence of any ash layer, the rate is proportional to the available surface of unreacted core. Thus, based on unit surface of unreacted core, the rate of reaction for the stoichiometry of Eqs. 1, 2, and 3 is... 

Combination of Resistances. The above conversion-time expressions assume that a single resistance controls throughout reaction of the particle. However, the relative importance of the gas film, ash layer, and reaction steps will vary as particle conversion progresses. For example, for a constant size particle the gas film resistance remains unchanged, the resistance to reaction increases as the surface of unreacted core decreases, while the ash layer resistance is nonexistent at the start because no ash is present, but becomes progressively more and more important as the ash layer builds up. In general, then, it may not be reasonable to consider that just one step controls throughout reaction. 

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... 

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...

Unreacted Core Model—Fast Chemical Reaction... 

Particles of a sulphide ore are to be roasted in a fluidised bed using excess air the particles may be assumed spherical and uniform in size. Laboratory experiments indicate that the oxidation proceeds by the unreacted core mechanism with the reaction rate proportional to the core area, the time for complete reaction of a single particle being 16 min at the temperature at which the bed will operate. The particles will be fed and withdrawn continuously from the bed at a steady rate of 6 tonnes of product per hour (1.67 kg/s). The solids hold-up in the bed at any time is 10 tonnes. 

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 Equation 8 the first two terms (x = 1 - E ) give the conversion for the unreacted core model. The remaining terms in this equation give the conversion in the inner core for the two stage model. The data at 1000 °C and 1040 °C showed that fractional conversions of the samples are approximately the same as the values predicted by the first two terms of Equation 8. This shows that at high temperatures unreacted core model becomes the controlling mechanism due to the increased concentration of CO2 in the pores and diffusion limitations. Experiments carried out at different temperatures also showed that the ratio of macropore... 

The overall process of coal combustion involves the following three steps in sequence transport oxygen from the bulk stream of gas to the outer surface of the particles, diffusion of oxygen through the ash layer to the reaction interface, and finally reaction with the combustible matter in the unreacted core. The overall flux of oxygen per unit surface area can be expressed as follows ... 

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