Shrinking particle model

The following mass balance at steady state is valid for component A  

This means that the flux through the interface equals the reaction in the interface, expressed in terms of mathematics accordingly (mass balance)  

The reaction rate R is defined as a function of the surface concentrations in the system, R = R(c ). For an arbitrary component reacting in the gas phase, Equation 8.76 is generalized to 

For general reaction kinetics, the unknown surface concentrations must be solved using a balance equation of a nature similar to Equation 8.77 the flux Ni is always dependent on the surface concentration c,  

For a solid component B, Equation 8.18 can be used as such. A requirement for solving Equation 8.18 is that the reaction rates are expressed through bulk-phase concentrations, c . The unknown surface concentrations, c , can be solved analytically in the balance equation in certain special cases, that is, first-order reactions. 

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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 ... 

Figure 56 above describes the phenomenology of the char combustion regime (III). The concept of the shrinking core or shrinking particle model is usually applied in mathematical modelling of char combustion in regime (III). 

The limitation of such a model to first-order reaction rates is not as restricting as it seems. In fact, many reactions might at least be considered as of pseudo -first order, which means that they behave macroscopically like first-order reactions. This is the case for diluted fluids and for non-catalytic gas/solid reactions such as the so-called shrinking core or shrinking particle model. Other examples are electrochemical reactions [106],... 

The solution of liquid extract out of solid particles is described according to a shrinking particle model by convective mass transfer and diffusion in the particle structure. 

The shrinking-particle model only presumes reactions on the outer surface ... 

Figure 6.4. (a) Fraction of forsterite reacted away as a function of time. (b)The reacted fraction data fit to the shrinking particle model.The slope of the line is kp. 

Geochemists are more familiar with the closely related particle lifetime model of Lasaga (1998b), which is based on the same assumptions as the shrinking particle model. This section shows how these models are related. 

Taking the cube root of both sides of this equation followed by rearrangement produces the shrinking particle model derived earlier as Eq. (6.51). Equation (6.63) can also be rearranged and expressed in terms of... 

For the case where abundant nuclei are present in the initial stages of the transformation, which is termed site saturation, the extended volume of one sphere of product phase is modeled by changing the sign for the rate constant in the shrinking particle model (Chapter 6) to make it a growing particle model. The linear growth rate of each sphere of the product phase is G (m/sec) and its volume is V (m ). 

Only gases are formed, the chemical reaction is fast, and the effective rate is determined by external mass transfer to the surface of the shrinking particle (model 4 in Figure 4.6.1). 

Equation 8.18 relates the particle radius to the surface reaction rate in a general way. The surface concentration, c , is highly dependent on the conditions on the reactive surface. Let us now consider two extreme cases a particle with a porous product layer (ash layer model) and the shrinking particle model. 

The reaction kinetics is of first order with respect to A. The solid particles are spherical and equal sized. The shrinking particle model can be applied on the particles consisting exclusively of B. 

In the case of the random pore model, the dimensionless parameter can be used as fitting parameter or can be calculated from analysis data. Figure 3.3d shows the experimental data (points) and the conversion curves if the different conversion functions are used. In this example, the shrinking particle model represents the data best. Consequently, the acquired data should be reported, including the determined kinetic parameters, the partial pressure reaction order, the conditions applied to produce the char and carry out the experiment, and finally, the identified particle model that best describes the conversion. 

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