External Gradients for Diffusion and Reaction in a Spherical Pellet

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One does not necessarily know the concentration at the surface of a catalyst pellet. One does know the bulk fluid concentration. Therefore the most general boundary conditions to use in solving the equation of continuity for a first-order reaction

are

and

As with the cases where the surface concentration is specified as the external boundary condition, there are limited situations for which an analytical solution can be found. Recognizing this we seek a numerical solution and will employ the collocation method.

In a separate Learning Module Numerical Approximation to Reaction with Diffusion - First Order Pellet the boundary condition (at Collocation Point 5) was given by

As explained more fully in the handout Collocation Primer, we now will simply replace this line of code with

For a catalyst pellet at steady-state the rate of diffusion into the pellet must equal the rate of reaction in the pellet. The collocation solution provides the gradient at the surface and this enables us to compute the value of R_Ap. In dimensional form

Recasting this into Matlab/Octave syntax, and letting the variable npts be the number of collocation points

We have selected a first-order reaction to examine here since there is an analytical solution and we can compare the numerical solution against the analytical solution. As you run the simulation the program will generate a curve that displays the concentrations at the collocation points and analytical solution, which can be used to compute the rate with

The programs were designed to examine the impact of the size of the Biot number relative to the Thiele modulus. For this reason, if you make the Thiele modulus too large by selecting a rate constant greater than 100 1/s, more collocation points are needed than the program has set. Hence, limit your values for the rate constant.

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THE QUESTIONS

1. The program was written for the catalyst properties given in text Example 7.1. If you input a value of k = 2.8 1/s, you will have a Φ = 2.0. Then by selecting different values for the external mass transfer coefficient k_m (cm/s) you can examine how the surface and bulk fluid concentrations begin to deviate. Set k_m = 0.1, 1.0 and 100. Notice how the rate changes and the surface concentration differs from the bulk value. Thinking about what the Biot Number represents, does this make sense to you?
2. Work at the extreme of a large Thiele modulus by setting k = 100. Study the impact of the same values on k_m and the Biot Number on the profiles and the differences in the bulk and surface concentration. What does this suggest to you about the interplay of internal and external gradients?
3. Make Φ take on values less than one and prove to yourself that even in this regime external gradients can be present. Remember this the next time you assume the bulk fluid and surface concentrations are equal.