Ash layer

Diffusion through a product layer can be treated like a film resistance. The surface concentration is measured inside the ash layer at the unbumed surface of the particle. If the ash thickness is constant and as 0, then the rate has the form of Equation (11.48). The ash thickness will probably increase with time, and this will cause the rate constant applicable to a single particle to gradually decline with time. 

The oldest of these sedimentation rates, Region I (150 g m 2 yr-1 or 0.063 cm yr-1) represents natural or the pre-cultural sedimentation rates before 1889. Near this location in the lake a volcanic ash layer overlain by 480 cm of sediment has been found, Gould and Budinger [15]. Radiocarbon dating of... 

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... 

Equation 9.1-17 is the continuity equation for unsteady-state diffusion of A through the ash layer it is unsteady-state because cA = cA(r, a To simplify its treatment further, we assume that the (changing) concentration gradient for A through the ash layer is established rapidly relative to movement of the reaction surface (of the core). This means that for an instantaneous snapshot, as depicted in Figure 9.3, we may treat the diffusion as steady-state diffusion for a fixed value of rc i.e., cA = cA(r). The partial differential emiatm. 

Equations 9.1-28 and -29 both give rise to special cases in which either one term (i.e., one rate process) dominates or two terms dominate. For example, if De is small compared with either kAg or kj, this means that ash-layer diffusion is the rate-determining or controlling step. The value of f or q is then determined entirely by the second term in each equation. Furthermore, since each term in each equation refers only to one rate process, we may write, for the overall case, the additive relation ... 

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,... 

An important difference between a shrinking particle reacting to form only gaseous product(s) and a constant-size particle reacting so that a product layer surrounds a shrinking core is that, in the former case, there is no product or ash layer, and hence no ash-layer diffusion resistance for A. Thus, only two rate processes, gas-film mass transfer of A, and reaction of A and B, need to be taken into account. 

For the reaction and assumptions in Example 22-1, except that reaction-rate control replaces ash-layer-diffusion control, suppose the feed contains 25% of particles of size R for which t = 1.5 h, 35% of particles of size 2R, and 40% of particles of size 3R. What residence time of solid particles, fB, is required for /B = 0.80 ... 

The performance of a reactor for a gas-solid reaction (A(g) + bB(s) -> products) is to be analyzed based on the following model solids in BMF, uniform gas composition, and no overhead loss of solid as a result of entrainment. Calculate the fractional conversion of B (fB) based on the following information and assumptions T = 800 K, pA = 2 bar the particles are cylindrical with a radius of 0.5 mm from a batch-reactor study, the time for 100% conversion of 2-mm particles is 40 min at 600 K and pA = 1 bar. Compare results for /b assuming (a) gas-film (mass-transfer) control (b) surface-reaction control and (c) ash-layer diffusion control. The solid flow rate is 1000 kg min-1, and the solid holdup (WB) in the reactor is 20,000 kg. Assume also that the SCM is valid, and the surface reaction is first-order with respect to A. 

As shown in Example 22-3, for solid particles of the same size in BMF, the form of the reactor model resulting from equation 22.2-13 depends on the kinetics model used for a single particle. For the SCM, this, in turn, depends on particle shape and the relative magnitudes of gas-film mass transfer resistance, ash-layer diffusion resistance and surface reaction rate. In some cases, as illustrated for cylindrical particles in Example 22-3(a) and (b), the reactor model can be expressed in explicit analytical form additional results are given for spherical particles by Levenspiel(1972, pp. 384-5). In other f l cases, it is convenient or even necessary, as in Example 22-3(c), to use a numerical pro-... 

For gas-film mass transfer control, we use equation 22.2-16a for reaction control, we use equation 22.2-18 and for ash-layer diffusion control, we integrate equation 22.2-13 numerically in conjunction with 22.2-19, as described in Example 22-3(c). The results generated by the E-Z Solve software (file ex22-4.msp) are shown in Figure 22.4. 

B = 0.80, t, which is a measure of the size of reactor, is about 1.7 min for ash-layer control, 9.5 min for reaction control, and 14.5 min for gas-film control. The relatively favorable behavior for ash-layer diffusion control in this example reflects primarily the low value of (1.67 min versus 6.67 min for the other two cases) imposed. 

If the results cited above are put another way, from the point of view of determining fB for a given f (or reactor size), a mean residence time of 1.7 min gives fB = 0.80 for ash-layer control, as noted, but only 0.37 for reaction control, and only 0.23 for gas-film control. 

The cases considered thus far have all been based upon the premise that one process, ash-layer diffusion, surface reaction, or gas-film mass transfer, is rate controlling. However, in some cases, more than one process affects the overall kinetics for the conversion of the solid. This has two implications ... 

Repeat problem 22-8(a), if the rate is controlled by ash-layer diffusion... 

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 25.6 Representation of a reacting particle when diffusion through the ash layer is the controlling resistance.

For convenience, let the flux of A within the ash layer be expressed by Fick s law for equimolar counterdiffusion, though other forms of this diffusion equation will give the same result. Then, noting that both (2a dCJdr are positive, we have... 

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... 

Step 3. Diffusion of reaction products from the surface of the solid through the gas film back into the main body of gas. Note that the ash layer is absent and does not contribute any resistance. 

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. 

---