Biochemical Engineering Fundamentals, Part 1 - Bailey J., Ollis D. 1989

Application of enzyme-catalyzed reactions
Kinetics of reactions catalyzed by immobilized enzymes
Simultaneous mass transfer resistance in the boundary layer and within the catalyst particle

In the two previous sections, we examined special cases of the mutual influence of mass transfer and chemical reaction, which are schematically depicted in Fig. 4.16. In the general case, however, the substrate must first pass through a boundary layer or stagnant film and then diffuse into the catalyst particle, where its conversion takes place. One of the objectives in analyzing this most general situation is to find ways to apply the simplified models discussed above. In other words, we need to know when internal mass transfer resistance dominates, in which cases external resistance plays the determining role, and when both types of resistance must be taken into account.

It is appropriate to begin addressing these tasks by studying a simple model of an immobilized enzyme catalyst with plate geometry and first-order intrinsic reaction kinetics. In this case, the steady-state material balance equation for the substrate inside the particle can be written as follows:

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with the Symmetry condition for the centerline

No substrate accumulates on the outer surface of the plate (x = L). Transport through this surface into the catalyst particle by diffusion equals the transport in the opposite direction through the boundary layer; therefore, the boundary condition is expressed by the following equation:

Solving equations (4.46) through (4.48) simultaneously makes it possible to determine the effectiveness factor ηs, which in our case can be written in the following form:

In this expression, the Thiele modulus ø has the same physical meaning as before [see equation (4.37); for plate geometry V_р/A_р = L]. Here we have introduced a new important parameter, Bi, called the Biot number and defined as

Equation (4.49) is more conveniently transformed into another form:

Here η (η = tgø/ø) is the catalyst effectiveness factor in the absence of transport resistance in the boundary layer. The reciprocal of the effectiveness factor can be viewed as a measure of the resistance to substrate conversion caused by mass transport requirements. Thus, equation (4.51) reflects the well-known rule stating that the total resistance is equal to the sum of the individual resistances.

Equation (4.51) allows us to answer the question of under what conditions one of the resistances can be neglected. If

then the resistance of the external boundary layer can be neglected. If, on the other hand, kL/k_s is much greater than unity, the internal resistance of the catalyst particle can be disregarded. In all intermediate cases, both resistances must be accounted for. Note that at large values of ø, an inverse proportionality is observed between η_s and ø². An alternative criterion, which does not require knowledge of the intrinsic rate constant, is based on plotting η_s versus Ф for various Bi values; for Bi≥100, the Effect of External resistance is negligible.

The analysis principles discussed are also applicable to immobilized Enzymes with other particle geometries and different intrinsic kinetics; these cases are covered in detail in the literature cited at the end of the chapter. We shall limit ourselves to noting that the evaluation of which mass transfer types exert a major influence on the overall process rate is always based on a comparative Study of the intrinsic reaction rate, diffusion effects within the catalyst particle, and external mass transfer.