Internal Mass Transfer

Enzymes are typically immobilized on the internal surfaces of porous supports or entrapped in matrices through which the substrate can diffuse. Therefore, it is important to evaluate the concentration profile within the pellet, to estimate the observed rate of substrate disappearance.

We begin with first considering the effective diffusion coefficient (D_eff) and noting that the diffusion rates of all species through the support are subject to the following influences:

1. Particle porosity (e_p) : some of the particle cross section is occupied by solid and hence not available for diffusive transport.

2. Tortuosity factor (t) : the pore network is entangled and diffusion occurs only in frequently changing directions.

3. Hindrance factor (H) : accounts for the steric interactions between the solute and the pore wall and enhanced drag on the solute resulting from the presence of the wall.

D_eff is then given by :

where:

D_s : Bulk diffusion coefficient

H : may be calculated by the “centerline approximation” as:

g : ratio of solute radius to pore radius

A steady state material balance yields the following differential equation for the system:

subject to:

Click here to view a complete derivation of the above result.

The dimensionless form of the above differential equation yields the following concentration profile for a spherical catalyst pellet:

The dimensionless term f is called the Thiele modulus and is the ratio of the intrinsic chemical reaction rate in the absence of mass transfer limitation to the rate of diffusion through the particle. The physical interpretation of f² is analogous to the Damkohler number.

We then define the internal effectiveness factor (h_I) as:

In terms of the above dimensionless parameters:

Therefore in general,

As b approaches 0, h_I converges to that of the corresponding first order reaction.

Analogous to our earlier discussion on h_E and its relation to the Damkohler number, we define an observable Thiele modulus, F, as:

At S=S₀, from equation (16):

Thus;

Therefore,

Note that, the curve of h_I versus F is far less sensitive to b than the analogous h_I(f) curves. This can be summarized as follows:

Therefore, once F is known, a reasonable estimate of h_I can be obtained without having the exact value of b. Clearly, the precise knowledge of b is necessary to use the h_I(f) curves.

Page created by Asif Ladiwala and Shripad Gokhale.

Last modified : 13 December, 2000.