Perform the order-of \(-\epsilon^{2}\) approximation for the problem of diffusion with reaction in a catalyst of variable activity. Compare the flux with that obtained from the BVP4C solver in MATLAB. Perform computations for \(\phi\) of 10 , 20, and 30.The results are 9.7313. 19.7416, and 29.745, respectively.

Now consider the problem of varying catalyst activity with the maximum activity at the center. The diffusion-reaction model is now given by

\[\epsilon \frac{\partial^{2} c}{\partial x^{2}}=c(1-x)\]

Perform a perturbation analysis for the problem. Use MAPLE or other programs to solve the \(c_{0}, c_{1}\), etc. problems. You will note that the \(c_{0}\) problem is an Airy function. Derive an approximate formula for the effectiveness factor.

Verify the following results obtained from BVP4C:

Note that \(\epsilon\) is equal to \(1 / \phi^{2}\). The flux, \(p\), is at the surface of the catalyst.

Transcribed Image Text:

20 3.3837 5.3714 Thiele, Flux, p 10 30 7.0384 40 8.5259