Rate-control regime

In the A sector (lower right), the deposition is controlled by surface-reaction kinetics as the rate-limiting step. In the B sector (upper left), the deposition is controlled by the mass-transport process and the growth rate is related linearly to the partial pressure of the silicon reactant in the carrier gas. Transition from one rate-control regime to the other is not sharp, but involves a transition zone where both are significant. The presence of a maximum in the curves in Area B would indicate the onset of gas-phase precipitation, where the substrate has become starved and the deposition rate decreased. 

Table 3. Apparent catalyst rate behavior depending on rate controlling regime (isothermal case).

Changes in the particle structure have a strong effect on combustion behavior, influencing the particle temperature, mass transfer and pore diffusion rates, and consequently the rate-control regime of the process ( 5 ) Tlie changes in size and density of particles that have a homogeneous pore structure and small pore sizes (relative to particle size) are related to fractional burn-off, u, by... 

FIGURE 18 Rate-controlling regimes in gas-solid reactions for an impervious solid m is the rate controlled by mass transfer of the oxidizer to the solid s is the rate controlled by surface reaction, (S) solid (B) boundary layer (b) bulk concentration of oxidizer (z) zero concentration of oxidizer (from Mulcahy and Smith [30]). 

It is rather straightforward to employ numerical methods and demonstrate that the effectiveness factor approaches unity in the reaction-rate-controlled regime, where A approaches zero. Analytical proof of this claim for first-order irreversible chemical kinetics in spherical catalysts requires algebraic manipulation of equation (20-57) and three applications of rHopital s rule to verify this universal trend for isothermal conditions in catalytic pellets of any shape. 

If 4 a( 7 = 0) = 1 as indicated in part (b), then diffusion does not hinder the ability of reactants to populate the central core of the catalyst. Furthermore, chemical reaction does not deplete reactant A because its molar density at the center of the catalyst is equivalent to that on the external surface. This situation occurs when A 0 and the catalyst operates in the reaction-rate-controlled regime. Hence, the effectiveness factor is unity under isothermal conditions. This result can be obtained mathematically from the integral expression for the effectiveness factor by setting = 0) = 1 — where s < 10 . ... 

Simulations are presented below in tabular and graphical forms when the temperature at the external surface of the pellet is constant at 350 K. The effective thermal conductivity of alumina catalysts is 1.6 x 10 J/cm s K. The chemical reaction is first-order and irreversible and the catalysts exhibit rectangular symmetry. Most important in Tables 27-5 to 27-8 and Figures 27-1 to 27-3, the diffusivity ratio a(0) varies with temperature in the mass transfer equation. This effect was neglected in Tables 27-1 to 27-4. Notice that in all of these tables (i.e., 27-1 to 27-8), numerical simulations reveal that the actual max exceeds I + fi, except when the intrapellet Damkohler number is small enough and 4 a( = 0) > 0 because the center of the catalyst is not reactant starved in the chemical-reaction-rate-controlled regime. 

Fig. 5.17 Temperature dependence of gas volume released during thermal decomposition of barium titanyl oxalate under heating rates (a) 50, (b) 100, (c) 300 °C/h, (d) rate controlled regime [283]...

Fig. 5.21 Specific surface area of nanocrystalline zirconia powder vs. heating rate at 100 % decomposed zirconyl hydroxide 1-rate controlled regime, 2,3-linear heating [300]...

Figure 5.1 Rate-controlling regimes for heterogeneous reactions [4].

The equations of combiaed diffusion and reaction, and their solutions, are analogous to those for gas absorption (qv) (47). It has been shown how the concentration profiles and rate-controlling steps change as the rate constant iacreases (48). When the reaction is very slow and the B-rich phase is essentially saturated with C, the mass-transfer rate is governed by the kinetics within the bulk of the B-rich phase. This is defined as regime 1. 

Presently, the quantitative theory of irreversible polymeranalogous reactions proceeding in a kinetically-controlled regime is well along in development [ 16,17]. Particularly simple results are achieved in the framework of the ideal model, the only kinetic parameter of which is constant k of the rate of elementary reaction A + Z -> B. In this model the sequence distribution in macromolecules will be just the same as that in a random copolymer with parameters P(Mi ) = X =p and P(M2) = X2 = 1 - p where p is the conversion of functional group A that exponentially depends on time t and initial concen-... 

When considering the macrokinetics of PAR described by equations (Eq. 17), it is reasonable to focus on two limiting regimes. The first of these, the kinetically-controlled regime, takes place provided the rate of diffusion of molecules Z appreciably exceeds that of the chemical reaction. In this case, a uniform concentration Z = Ze should be established all over the globule after time interval t R2/D. Subsequently, during the interval t 1 /kZe, which is considerably larger than f[Pg.152]

Thus, th in a kinetically controlled regime is described by a dl law. Furthermore, th is found to be inversely proportional to pressure (for a first-order reaction) under kinetically controlled combustion, and in contrast, independent of pressure under diffusionally controlled combustion (since D P-1). In the kinetically controlled regime, the burning rate depends exponentially upon temperature. 

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