Shrinking core mechanism

F, and are geometric factors for grains and particle, respectively, and have the values 1 for slabs, 2 for cylinders, and 3 for sj ieres. The conversion x in the grain attained by a shrinking-core mechanism is written... 

Joshi et al. (30) proposed reactor models based on the shrinking core mechanism. Since the particles take part in the reaction their role was evaluated based on the residence time distribution. For extremely fine pyrite particles, (< 100 ym), it has been shown (31) that the RTD of the solid and liquid phases can be asstimed to be identical and the RTD of the solid phase is given by the diffusion-sedimentation model. Various rate controlling steps that were considered are (1) gas-liquid mass transfer (2) liquid-solid mass transfer (3) ash diffusion (4) chemical reaction and, (5) intraparticle diffusional resistance (for particles encased in the coal matrix). 

The time for diffusion (without reaction) to increase the average concentration within a fiber wall to one-half of its ultimate value, with the fiber treated as a fully collapsed slab of thickness equal to two fiber wall widths, gives a 0.19. For a fiber wall thickness of a = 4.0 x 10 m and an effective diffusion coefficient of D = 1.3 X 10 m /s, Xd = 0.0023 s. The ozonation of pulp fibers follows a shrinking core mechanism, limited by the rate of ozone diffusion through the fiber wall. Exposed fibers are completely penetrated and reacted in seconds under typical operating conditions (Bennington et al., 1999). 

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 ... 

The shrinking core reaction mode is not necessarily limited to non-porous unreacted solids. With a fast reaction in a porous solid, diffusion into the core and chemical reaction occur in parallel. The mechanism of the process is very similar to the mechanism of the catalysed gas-solid reactions where the Thiele modulus is large, the effectiveness factor is small and the reaction is confined to a thin zone (Section 3.3.1). This combination of reaction with core diffusion gives rise to a reaction zone which, although not infinitely thin but diffuse, nevertheless advances into the core at a steady rate. 

Single pellet reduction. The reduction occurs at high temperatures and a shrinking core model is appropriate as confirmed experimentally(5). Removal of oxygen occurs at the advancing interface while the water gas shift reaction occurs in the outer layer of reduced iron. The mechanism of the water gas shift reaction is thought to be (10,11) between adsorbed oxygen and CO on the active surface of the product iron, i.e ... 

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). 

For these and similar systems the original source, resin or solution, of the counter-ion being chemically consumed and the nature of the co-ion greatly influence the observed kinetics. The association-dissociation of weakly functional resins is of particular practical interest since in these instances a reactive and non-reactive core respectively forms within the resin which shrinks towards the bead centre as exchange proceeds. This Shrinking Core or Shell Progressive mechanism is usually particle diffusion controlled and explains why exchange on weakly functional resins is invariably flow-rate sensitive under column operation. 

Non-catalytic reactions involving two phases are common in the mineral industry. Reactions such as the roasting of ores or the oxidation of solids are carried out on a massive scale but the rates of these processes are often controlled by physical, not chemical, effects. Reactant or product diffusion is the main rate controlling factor in many cases. As a result, mechanisms of reaction become models of reaction with consideration of factors such as external diffusion film control or the shrinking core yielding the various models. Matters are further complicated by considerations regarding particle shape and external fluid flow regimes. 

Figure 15.7a shows reaction at the interface between a solid reactant (B) and a product (5) after the fluid has diffused through an inwardly advancing shell of the product (ash). There is no reaction in either the ash layer or in the body of the reactant B (the core), but only at the surface of B. This is the shrinking core or sharp interface model and represents perhaps the most common mechanism of gas-solid reactions for nonporous solids. The overall reaction can be controlled... 

An unusual variation of the shrinking core model is the expanding core model in which the reaction starts at the center (see Doraiswamy and Gokarn, 1988). The product— ash core instead of the reactant core in normal behavior— moves outward till it completely envelops the particle. The controlling mechanisms described for the shrinking core model apply equally here, but the equations are different. We shall discuss this at some length later in this section while describing a model for the disproportionation of potassium terephthalate. 

Solids play different roles in the different processes. In direct coal liquefaction, a part of the solid is dissolved in liquid (mainly in the preheater) and a part (i.e. mineral matter) may act as a catalyst for the hydrogenation reactions. In Fischer-Tropsch slurry processes, solids are catalysts. Finally, in chemical cleaning of coal, only a part of solid (i.e. sulfur) takes part in the reaction following the shrinking core diffusion/ reaction mechanism. The role of solids in the design and scaleup of the reactors for the three processes is therefore different. 

Ruether (29) examined the case of oxydesulfurization for completely backmixed stirred tanks in series assuming the diffusion-controlled mechanism. The reaction in the particles was described by the shrinking core model. The results obtained on the conversion as a function of residence time were shown for various number of reactors in series. The procedure to calculate the conversion for a system having a distributed particle size... 

While the actual mechanism of the LiFeP04/FeP04 transformation remains a subject of debate, it seems clear that the shrinking core model does not apply to the primary particles themselves. The model may, however, still be applicable on a larger scale, for example, to the secondary particles comprised of agglomerates. 

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