Solution multiplicity in the catalytic pellet reactor

1. Solution multiplicity in the catalytic pellet reactor LPPD seminar Kedar Kulkarni 04/19/2007 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. Motivation: Why investigate multiplicity in solutions? • Multiplicity in pellet concentration profiles and/or inversion problems:a) Avoid accidents (e.g estimated highest temperature in the reactor is lower than the actual)b) Gain knowledge about the system-constants. What are the best-fit transport and kinetic properties for the distributed system?- Causes of multiplicitya) Multiplicity in inversion solution: (i)Multiple data sets (ii) Erroneous datasets b) Multiplicity in state-variable (concentration) profiles:Inherent characteristics of the system (non-linear coupled differential equations) lead to multiplicity in solution

3. Outline Theory: • Polynomial approximation and Method of weighted residuals (MWR) - Use of different weights • Efficient collocation: The use of Orthogonal polynomial roots as collocation points • Orthogonal collocation over finite elements (OCFE) • Two-dimensional interpolation (2D Lagrangian polynomials) OCFE: • Formulation • Some results Global Terrain Method + OCFE: • Formulation • Some results • Discussion of challenges • Conclusions and Future work

4. Polynomial approximation and MWR Why is it used ? • Interpolation: Given data (xi, yi), we want to find y-values for intermediate points • Solutions of a systems of equations: • Non-linear set of equations • Solution of ODE/PDE’s What constitutes polynomial approximation ? yN(x): Approximating Nth order polynomial y(x): Analytical solution (say, it is known) Task: To identify the coeffs of yN(x) such that the quantity yN(x) y x x x y(x) b a x is minimized

5. Polynomial approximation and MWR Use of the residual function • In general, analytical solution is unknown • For example, let yN(x) be the approximate solution to: BC’s: Define the Residual function as: ‘a’ is the vector of polynomial coeffs

6. Polynomial approximation and MWR Task re-visited: To identify the coeffs of yN(x) such that the quantity (N + 1) eqns, (N + 1) unknowns Assumptions: 1) Without any loss of generality 2) We assign different weights corresponding to each coeff Different choice of weights gives rise to different methods: • Collocation method • Subdomain method • Galerkin method • Least squares method

7. Polynomial approximation and MWR a) Collocation method: (N + 1) points/nodes The residual is zero at each node; this implies: • It passes through all given points (xi,yi) (INTERPOLATION) • It satisfies the ODE/PDE exactly at all the nodes (ODE/PDE solution) b) Subdomain method: c) Galerkin method: d) Least squares method:

8. The use of Orthogonal polynomial roots as collocation points 1) These methods involve computation of integrals - Difficult to evaluate analytically - Use quadrature to approximate the integral 2) So far the choice of xj was arbitrary; we can make specific choices: Principle: If a weight function is of the form there exists an OPTIMAL choice of quadrature points, so the highest power term of the power series expansion of the residual is computed ‘more correctly’. These points are the roots of which satisfies the property:

9. Proof that best approximate Galerkin method is Collocation with roots of orthogonal polynomial (OP) as collocation points Consider the following yN(x) for the cylindrical-pellet problem (1st order reaction): Galerkin weight: Equations for the coeffs: Consider: and Thus: Using quadrature over M roots of OP: Choose M = N Thus:

10. Orthogonal collocation over finite elements We demonstrated that: - The best approximate Galerkin method is Collocation with roots of orthogonal polynomial (OP) as collocation points - Collocation with roots of orthogonal polynomial (OP) as collocation points is more efficient than one with arbitrary nodes Problems with collocation (with/without use of OP root-nodes): - Not suitable for curves that have ‘corners’ or bends -- Need more collocation points -- Larger problems/polynomials -- Oscillatory solution Higher order polynomial Lower order polynomial y y x x x x x x x x

11. Orthogonal collocation over finite elements Polynomial · · · · · · · · · · · · Nodes · ´ ´ ´ ´ ´ ´ ´ ´ ´ ´ ´ ´ ´ element 1 element i+1 element m element i x = 0 x = 1 Possible ways to handle this: • Divide the entire domain into elements • Nodes in each element are roots of lower order orthogonal polynomials m – elements n – number of nodes in each element

12. Two dimensional interpolation (2D Lagrangian polynomials) 1D Lagrangian polynomials: n – number of nodes 2D Lagrangian polynomials: y x

13. Use of OCFE to solve the spherical-catalytic pellet problem Let there be ‘m’ elements and ‘n’ nodes in each element: m*n equations in m*n unknowns!

14. THREE SOLUTIONS USING FE COLLOCATION

15. Solution from S-curve FE collocation solution FE COLLOCATION CO AND C1 CONTINUOUS POLYNOMIAL

16. Solution from S-curve FE collocation solution FE COLLOCATION ONLY CO CONTINUOUS POLYNOMIAL

17. Solution from S-curve FE collocation solution FE COLLOCATION ONLY C1 CONTINUOUS POLYNOMIAL

18. Solution from S-curve FE collocation solution FE COLLOCATION

19. Solution from S-curve FE collocation solution FE COLLOCATION

20. Solution from S-curve FE collocation solution FE COLLOCATION

21. Solution from S-curve FE collocation solution FE COLLOCATION

22. Solution from S-curve FE collocation solution FE COLLOCATION

23. Solution from S-curve FE collocation solution FE COLLOCATION

24. Solution from S-curve FE collocation solution FE COLLOCATION

25. Solution from S-curve FE collocation solution FE COLLOCATION

26. Solution from S-curve FE collocation solution FE COLLOCATION FINAL

27. TWO SOLUTIONS USING FE COLLOCATION

28. Solution from S-curve FE collocation solution FE COLLOCATION

29. Solution from S-curve FE collocation solution FE COLLOCATION

30. Solution from S-curve FE collocation solution FE COLLOCATION FINAL

31. ONE SOLUTION USING FE COLLOCATION

32. Solution from S-curve FE collocation solution FE COLLOCATION

33. Solution from S-curve FE collocation solution FE COLLOCATION CO

34. Solution from S-curve FE collocation solution FE COLLOCATION CO AND C1

35. Solution from S-curve FE collocation solution FE COLLOCATION FINAL: CO IS BETTER!

36. ONE SOLUTION USING GTM + FE COLLOCATION

37. Solution from S-curve FE collocation solution Time: 35.5951 min FINAL

38. Solution from S-curve FE collocation solution Time: 13.3018 min FINAL

39. GTM + FE collocation Only CO continuity: n*(m-1)+1 eqns

40. TWO SOLUTIONS USING GTM + FE COLLOCATION

41. Solution from S-curve FE collocation solution Start solution: B Time: 234.4404 min (3.9073 hrs) FINAL

42. Solution from S-curve FE collocation solution Start solution: A Time:211.7250 min (3.5288 hrs) FINAL

43. THREE SOLUTIONS USING GTM + FE COLLOCATION

44. Solution from S-curve FE collocation solution Start solution: B Time: 265.9350 min (4.4322 hrs)

45. Solution from S-curve FE collocation solution Start solution: C Time: 308.7170 min (5.1453 hrs)

46. Conclusions and challenges: • Orthogonal collocation over finite elements (OCFE) gives solutions that match with the ones given by Weisz-Hicks- The GTM + FE collocation method identifies only one solution inspite of multiple possible. Possible reasons:a) The original GTM uses Newton update for a square systemSince our system is non-square (only CO continuous), we use the Gauss-Newton update b) The saddle point condition is too stringent to allow identification of existing saddles

47. THANK YOU!