Stepwise approximation

Figure 4.1. Finite-differencing a tubular reactor with the stepwise approximation of the...

Similar stepwise approximation of the weighting function h(t) with the discrete values h, . .., hy, and replacement of ytt ) by the observed values... 

This relation corresponds to the so-called stepwise approximation of the exponential term under the integral of eqn. (3) since it has been obtained under the assumption that this term is equal to 1 for the reagent pairs with the distances R > Rt and to 0 for those with R < Rt where... 

Fig. 1. The function 0(R,t) at v = 1015s 1 and a = 1 A at various times, t. The broken line corresponds to the stepwise approximation of 0(R,t)...

This formula was first derived in ref. 6 when calculating the kinetics of donor luminescence decay in the presence of the randomly, i.e. chaotically, located acceptors under the condition n N and on the assumption of the resonance exchange mechanism of energy transfer. Similar equations were later used for the analysis of experimental data on the kinetics of electron tunneling reactions obtained under conditions of the chaotic distribution of the reagents and at n < N. As a rule, only the first term of the exponent in eqn. (23) has been taken into account, which is equivalent to employing the previously mentioned (see Sect. 2.1) stepwise approximation of the function 0(R,t) = exp[- 1V(jR)(]. In this case, one obtains... 

Let us discuss what will be the influence on the kinetics of electron tunneling reactions of such factors as the more complicated, rather than the simple, exponential dependence of the tunneling probability W(R) on the mutual location of the reagents. To exercise such an analysis, it is necessary to consider in more detail the limits of applicability of the stepwise approximation of the function 9(R,t) = exp[ — W(/ )(], which was used in the previous section to derive the kinetic equations for electron tunneling reactions in the case of the exponential dependence of the eqn. (2) type for... 

For sufficiently large observation times the stepwise approximation of the function 0(R,t) in the form used in Sect. 2.1 is justified owing to the sharp dependence of the tunneling probability on the distance. As an example, in... 

It can be seen that the function obtained as a result of the stepwise approximation differs to a comparatively small extent from the precise function. Nevertheless, in some cases even this small difference can result in considerable errors in describing the process kinetics, especially at short observation times. 

A general theoretical analysis of the necessary and sufficient conditions for the applicability of the stepwise approximation for describing the kinetics of electron tunneling reactions has been made in ref. 13. As shown in this reference, the stepwise approximation can also be used for describing the kinetics of electron tunneling reactions at such dependences lF(i ), which are more complicated than a simple exponent. For determining the distance of tunneling Rt [i.e. the radius of the stepwise approximation of the function 0(R,t)] it appears expedient (see ref. 13) to use the expression... 

Substituting eqns. (38) and (2) for W i , tp) and W R) into this formula, integrating the expression obtained with respect to tp and R, and using the stepwise approximation, we find the following expression for the recombination luminescence intensity... 

The approximate form of the function f(i ) can be found also with the help of a simple method. Actually, under the conditions of the validity of the stepwise approximation of the function 0(R,t) = exp - lF(i )<, from ratio (3) of Chap.4 we have... 

The approximate form of the function N(R,0) can also be found with the help of a simpler method which is based on using the stepwise approximation of the function [1 - exp( - VF(jR)01 for W(R) described by eqn. (3). Indeed, from eqn. (19) it follows that... 

The conversion and selectivity of the reaction can be decisively influenced by the design and the operation of the heat transfer circuit. The most obvious, although technically most complex solution, is to arrange different heat transfer circuits so as to achieve a stepwise approximation of an optimum temperature profile. The purposeful utilization of the temperature change of the heat transfer medium flowing through the reactor is technically simpler, and will be discussed here in connection with cocurrent or countercurrent cooling of a fixed-bed reactor with an exothermic reaction. 

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