Diffusion through the product

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... 

Another important example of a solid transformation where growth requires diffusion through the product film is the transformation of solid spheres. The principles of this process are similar to those for planar films, but now the concentration profile is not linear, and the expression one obtains for the transformation and the solid conversion is more complex. 

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 ... 

This vapor acts as a driving force for formaldehyde diffusion from the wood cel I towards the product surface, and for emission from the finished wood product. An internal vapor pressure of 20 Torr would approximately correspond to a formaj ehyde air concentration of about 1 ppm at 25 t, a load factor of I m and a ventilation rate of 1 ach. However, as emission continues and depletes the methylene glycol concentration in the wood moisture, the dissociation of hemiacetals will set in and add to the formaldehyde source. The bottleneck in the formaldehyde transport will be diffusion through the product towards the product surface. This process depends on the permeability of the product which, in turn, depends on diffusion... 

Diffusion through Product Controlling For rapid chemical reactions at the interface and a lo.w-D, diffusion through the product iayer m.av. deter-inine the rate, even at smalTcon If this is the case, = 0, 2 is large, and Eq. (14-19) becomes. 

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. 

The cobalt cations and the electrons diffuse through the product spinel to the other side. So does oxygen through the gas phase. 

According to Jost, the reactants AX and BY are separated from one another by the product phases AY and BX. A layer of BX grows to cover the reactant AX, and a layer of AY grows to cover the reactant BY. This situation results from the fact that only the cations are mobile. In order that the reaction can proceed, it is necessary that there be a slight solubility of A" ions in BX and of B" " ions in AY, since these ions must diffuse through the product phases. In order that this problem may be treated quantitatively, it is, in principle, necessary that the partial pressures of X2 and Y2 in the surrounding gas phase, as well as P and T, be fixed. The activity, gradients of the cations in the product phases are then uniquely determined, and if the transport coefficients are known, the increase in thickness can be easily calculated, since the cation fluxes at the phase boundary BX/AY must be coupled. 

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

All solid state reactions, whether alloying or oxidation/reduction reactions, involve the formation of one or more product phases between the reactants. Without mechanical alloying, the reactant phases become separated and the reaction rate is determined by the contact area and the diffusion path length through the product phases. Diffusion through the product phases is invariably the rate-controlling process, and consequently high temperatures are required for flie reaction to occur at a measurable rate. 

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. 

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