Diffusion through the product layer

A parabolic rate law will also be obtained if part or even all, of the diffusion through the product layer is by grain boundary diffusion rather than diffusion through the volume of each grain. The volume diffusion coefficient is quite simply defined as the phenomenological coefficient in Fick s laws. The grain boundary diffusion must be described by a product, DbS, where S is the grain... 

For ease of solution, it is assumed that the spherical shape of the pellet is maintained throughout reaction and that the densities of the solid product and solid reactant are equal. Adopting the pseudo-steady state hypothesis implies that the intrinsic chemical reaction rate is very much greater than diffusional processes in the product layer and consequently the reaction is confined to a gradually receding interface between reactant core and product ash. Under these circumstances, the problem can be formulated in terms of pseudo-steady state diffusion through the product layer. The conservation equation for this zone will simply reflect that (in the pseudo-steady state) there will be no net change in diffusive flux so... 

In the grain model, it is assumed that the CaO consists of spherical grains of uniform size distributed in a porous matrix. The rate of reaction is controlled by the diffusion of SO2 through the porous matrix and the product CaSO layer formed on each grain (11). Allowance can be made for a finite rate of the CaO/SC reaction (12). The models have been found to describe experimental data for many limestones (13) by adjusting the constants in the model, most notably the diffusivity through the product layer. 

Eq. (11) shows the relationship between the three reaction resistances. The first term represents the external mass transfer resistance, the second the resistance associated with diffusion through the product layer, and the third the chemical reaction resistance at the reactant-product interface. 

When diffusion through the product layer controls the global rate, D a/ s mA Rd Dqa/ s ... 

A slow stage in which pores in the original calcium oxide have been filled or plugged by calcium carbonate. Thus, access to the unreacted calcium oxide requires diffusion through the product layer of carbonate. 

Consider the formation of NiCr204 from spherical particles of NiO and Cr203 when the reaction rate is controlled by diffusion through the product layer. 

The process typically yields SIM behavior and is controlled either by heat or gas diffusion through the product layer. For heat transfer through the product layer controlling, the interface stays isothermal and the... 

The rate of transport of the gaseous reactant A by diffusion through the product layer, which is established and increases during the conversion of the solid, is given by ... 

Figure 4.6.2 Concentration profiles of gaseous reactant Aina non-porous spherical particle and formation of a solid porous product, if film diffusion and diffusion through the product layer influence the rate simply hatched solid reactant, crossed hatched solid product. Adapted from Baerns et al. (2006).

For example, for a conversion of only 6%, Eq. (4.6.20) already yields a reaction time that is 10% higher than without considering diffusion through the product layer [see Eq. (4.6.19b) without the term 5-15(1 — Xb) Thus with increas-... 

Case III reaction and external mass transfer are fast compared to diffusion through the product layer (k and fi DA,e (lrp model 2 in Figure 4.6.1) ... 

Figure 4.6.4 shows the plot of Xb versus (/% for the boarder cases discussed before (i) absence of any diffusional resistances, (ii) control by diffusion through the product layer, and (iii) control by external diffusion. These curves are helpful in analyzing experimental data, as the shape of the curve indicates which case we probably have. For comparison the time is normalized with the final time ffi for complete conversion of the solid. The correlations are ... 

Figure 4.6.7 Conversion of a porous solid with a gas for different cases (i) absence of diffusional resistances, Eq. (4.6.70), (ii) control by diffusion through product layer, Eq. (4.6.71), and (iii) both the chemical reaction and the diffusion through the product layer determine the effective rate for a value of the Thiele modulus =...

When diffusion through the product layer is much slower than the diffusion through the gas film—which is quite probable—the Biot number for the mass transfer of component A approaches infinity, that is, BIam = oo, and Equation 8.69 can be simplified to... 

Diffusion through the fluid film and the product layer Diffusion through the product layer Diffusion through the fluid film... 

Depending on whether diffusion through the product layer or diffusion through the gas film is the rate-determining step, different limiting cases for Equation 8.141 are obtained. These limiting cases were already mentioned in Section 8.2.2. 

When diffusion through the product layer is so rapid that the reactants cannot combine fast enough at the reaction interface to establish equilibrium the solid-state reaction is phase-boundary controlled. The product layer is discontinuous when the molar volume of the product phase is considerably different from that of the reactant upon which it is growing. According to Laidler ( ), when a discontinuous product phase occurs the rate-determining step may be the chemical process occurring at the phase boundary. Under these circumstances the rate is determined by the available interface area, and such processes are referred to as topochemical. 

Gaseous reaction products diffuse through the product layer from the reaction zone to the surface of the particle. 

We shall obtain the solution for spherical particles. (The solutions for infinite slabs and long cylinders can be obtained similarly.) The rate of diffusion through the product layer of a spherical particle is given by... 

To obtain the general rate expression, we can start from the governing differential equation for diffusion through the product layer with interfacial chemical reaction and external mass transport providing the boundary conditions. Since each step occurs in series and is independent of the others, we can make use of the results already obtained in the previous two sections. 

First, we will consider the case where external mass transport is fast. The overall rate of reaction can be expressed in terms of the rate of chemical reaction or the rate of diffusion through the product layer. 

The reason for the greater error in a noncatalytic system is that the gaseous reactant must first diffuse through the product layer to reach the reaction interface. The concentration at the interface is smaller than in the bulk, making the rate of chemical reaction more sensitive to the reaction order. (Generalization is made in terms of the bulk concentration.)... 

For a system that undergoes an exothermic reaction and is controlled by the diffusion through the product layer, Luss and Amundson [33] have calculated the maximum temperature rise in the solid as affected by the various parameters. 

Reaction of Porous Pellets Controlled by Diffusion through the Product Layer... 

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