Update on solution multiplicity in catalytic pellet reactor

1. Update on solution multiplicity in catalytic pellet reactor LPPD seminar Kedar Kulkarni 02/22/2007 Advisor: Prof. Andreas A. Linninger Laboratory for Product and Process Design, Department of Chemical Engineering, University of Illinois, Chicago, IL 60607, U.S.A.

2. Update on 3 issues: • How the Weisz-Hicks (shooting) method to solve the PDE for pellet-concentration works. - some examples and case studies- Use of different local methods to solve for the correct initial concentration y(0) that gives the desired- Comparison of solution using GTM+3 collocation nodes with the solution by reading off the Weisz-Hicks curve. ϕ0

3. How the Weisz-Hicks method works PDE: BCs: where: The Weisz-Hicks method: Converts BVP into IVP PDE: BCs:

4. How the Weisz-Hicks method works Thus: (*) The Algorithm: a) Choose a value K=K0 and y(0)=y0 and solve: until: Now, let Now, from (*) Thus,

5. How the Weisz-Hicks method works b) Since: So, in effect you solve the original problem with a different depending on the choice of K0 and y0 ϕ0 Some pellet results:

6. Some pellet results For the same y0, one gets DIFF profiles depending on choice of K0!!

7. Example problem Analytical solution: Weisz-Hicks formulation:

8. Some results Analytical vs Weisz-Hicks for different y0’s Analytical vs Weisz-Hicks for one y0 For the same y0, one gets SAME profiles for different choices of K0!!

9. Use of different methods to obtain correct y0: Comments: • Newton-method fails due to zero derivative very often • This was for gamma, beta and phi that admit only one profile • These methods can obtain 2 or 3 y(0)’s depending on correct choice of the interval • When tried obtaining results for 3 profiles, Bisection method was the most robust

10. Comparison of solution obtained by GTM with Weisz-Hicks: Comments: • One solution by GTM (S3) matched with the one suggested by Wiesz-Hicks • The residual for the Weisz-Hicks solution is higher than S1 given by GTM

11. Thank you!