Diffusive energy regime

Rayleigh quotient method has been used only in the spatial diffusion limited regime but not in the energy diffusion limited regime (see the next Seetion). 

Two important restrictions must be introduced to allow a general representation of the temperature and concentration dependence of the effective reaction rate in the diffusion controlled regime. The first concerns the restriction to simple reactions, i.e. which can be described by only one stoichiometric equation. Whenever several reactions occur simultaneously, it is obvious that the individual activation energies and reaction orders may be influenced quite differently by transport effects. Thus, how the coupled system in such a case finally will respond to a change of temperature or concentration cannot be specified in a generally valid form. 

However, when the view is restricted to simple, irreversible reactions obeying an nth order power rate law and, if additionally, isothermal conditions arc supposed, then—together with the results of Section 6.2.3—it can be easily understood how the effective activation energy and the effective reaction order will change during the transition from the kinetic regime to the diffusion controlled regime of the reaction. 

Here T1U is the so called one-dimensional TST estimate for the rate and is mainly determined by the one-dimensional potential of mean force w(q). The depopulation factor Y becomes much smaller than unity in the underdamped limit and is important when the rate is limited by the energy diffusion process. In the spatial-diffusion-limited regime, the depopulation factor Y is unity but the spatial diffusion factor becomes much smaller than unity. The major theme of this review is theoretical methods for estimating the depopulation and spatial diffusion factors. 

Evidently, if u, = 0 (cf. Eq. (53)), the motion of the unstable mode is decoupled from the rest of the stable modes. In this limit, the escape rate would be zero since the particle cannot escape from the well without receiving the necessary energy from its surrounding. The small parameter which identifies the energy-diffusion-limited regime is thus u,. 

A reaction in solution can be regarded as the outcome of two stages one is the encounter of two reactant species this is followed by their reaction in the second stage, if they acquire their activation energy. If the rate-determining step is the former, then the reaction i.s said to be diffusion-controlled. If the rate-determining step is the latter, then the reaction is activation controlled. For a reaction of the form A -1- B -> P that obeys the second-order rate law v = t2[A][B], in the diffusion-controlled regime,... 

Estimate the maximum temperature at the center of a catalytic pellet at large intrapellet Damkohler numbers in the diffusion-limited regime when the thermal energy generation parameter P = 2 (i.e., exothermic chemical reaction), the temperature on the external surface of the pellet is surface = 300 K, and the average pore size is greater than 1 tim. 

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