Kinetic rate-controlled regime equations

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. 

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. 

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]

If Da = 1 is defined as the transition between diffusionally controlled and kinetically controlled regimes, an inverse relationship is observed between the particle diameter and the system pressure and temperature for a fixed Da. Thus, for a system to be kinetically controlled, combustion temperatures need to be low (or the particle size has to be very small, so that the diffusive time scales are short relative to the kinetic time scale). Often for small particle diameters, the particle loses so much heat, so rapidly, that extinction occurs. Thus, the particle temperature is nearly the same as the gas temperature and to maintain a steady-state burning rate in the kinetically controlled regime, the ambient temperatures need to be high enough to sustain reaction. The above equation also shows that large particles at high pressure likely experience diffusion-controlled combustion, and small particles at low pressures often lead to kinetically controlled combustion. 

Here tm is the mass-transfer time. Only under slow reaction kinetic control regime can intrinsic kinetics be derived directly from lab data. Otherwise the intrinsic kinetics have to be extracted from the observed rate by using the mass-transfer and diffusion-reaction equations, in a manner similar to those defined for catalytic gas-solid reactions. For instance, in the slow reaction regime,... 

The parameter 5- = k /k shows the detachment ability that compensates for electron losses dne to attachment. If 5- 1, the attachment inflnence is negligible and kinetic equation (4-21) becomes eqnivalent to one for non-electronegative gases. The kinetic equation inclndes the effective rate coefficients of ionization, kf = kj + g, and recombination, k f = kf + gk. Eqnation (4-21) describes electron density evolution to the steady-state magnitnde of the recombination-controlled regime ... 

The reaction is controlled by the chemical kinetics in this regime, and hence diffusional limitations are absent. Depending on the reaction, the rate equation can have any of many hyperbolic forms, such as those presented in Chapter 8. 

FIGURE 5.5 Summary of the key kinetic concepts associated with active gas corrosion under the surface reaction, diffusion, and mixed-control regimes, (a) Schematic iUusIration and corrosion rate equation for active gas corrosion under surface reaction control, (b) Schematic illustration and corrosion rate equation for active gas corrosion under reactant diffusion control. (c) Schematic illustration and corrosion rate equation for active gas corrosion under mixed control, (d) Illustration of the crossover from surface-reaction-conlrolled behavior to diffusion-controlled behavior with increasing temperature. The surface reaction rate constant (k ) is exponentially temperature activated, and hence the surface reaction rate tends to increase rapidly with temperature. On the other hand, the diffusion rate inereases only weakly with temperature. The slowest process determines the overall rate. 

Hyperbolic equations were used in Chapter 6 to represent reactions catalyzed by solid surfaces. They are referred to as LHHW models and they can be empirically extended to homogeneous catalysis in liquid phase reactions. The actual rate equation to be used for a given reaction will depend on the regime of that reaction. Methods of discerning the controlling regimes for catalytic gas-liquid reactions described in the gas-liquid chapter were based on simple power law kinetics. Extension of these methods to gas-Uquid reactions catalyzed by homogeneous catalysts involves no new principles, but the mathematics becomes more... 

In the kinetic control regime (where the overall effectiveness factor t = 1), the rate is directly proportional to the concentration of active sites, L, which is incorporated into the rate constant. In the regime of internal (pore) diffusion control, the rate becomes proportional to and when external diffusion controls the rate there is no influence of L, i.e., there is a zero-order dependence on L. This can be seen by examining equations 4.47 and 4.68. This observation led to the proposal by Koros and Nowak to test for mass transfer limitations by varying L [62]. This concept was subsequently developed further by Madon and Boudart to provide a test that could verify the absence of any heat and mass transfer effects as well as the absence of other complications such as poisoning, channeling and bypassing [63]. 

The rate equation is indistinguishable from that of an extraction process occurring in a kinetic regime, which is controlled by a slow, interfacial partition reaction ... 

From Table 11.17, it can be seen that the kinetic parameters are included in the rate equations for regimes 1 and 3, whereas the rate equations for regime 2 and 4 represent completely mass transfer-controlled operations. Therefore, for obtaining the kinetic parameters, it is important that the experiments in a stirred cell satisfy the conditions of either regime 1 or regime 3. A given stirred cell is characterized by vessel diameter (T), impeller diameter (U), and impeller design and location from the gas-liquid interface. 

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