Solution multiplicity in the catalytic pellet reactor

1. Solution multiplicity in the catalytic pellet reactor LPPD seminar Kedar Kulkarni 04/12/2007 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) Gain useful knowledge about the system. What are the best-fit transport and kinetic properties for the distributed system?b) Avoid accidents (e.g estimated highest temperature in the reactor is lower than the actual)- Causes of multiplicitya) Multiplicity in inversion solution: (i)Multiple data sets (ii) Erroneous datasetsb) Multiplicity in state-variable (concentration) profiles:Inherent characteristics of the system (non-linear coupled differential equations) lead to multiplicity in solution

3. Outline PROGRESS: • Multiplicity in catalytic pellet profiles: a) Brief background of coupled bulk + pellet - kinetics b) Use of methods that identify ONE initial-guess dependent solution (Background + results): i) The Weisz- Hicks method ii) The method of Orthogonal Collocation over finite elements (OCFE) c) Use of methods that identify ALL possible solutions (Background + results): The Global terrain method coupled with the method of Orthogonal Collocation over finite elements (OCFE) • Multiplicity in catalytic pellet profiles vis-à-vis the bulk reactor: • Distribution of pellet profiles along the reactor for different ranges of γ, b and f (DIFFUSION + CONVECTION) • Causes for multiplicity • Possible methods to handle challenges OUTLINE OF THE PAPER: Only figures • Conclusions and Future work

4. Cooling Outlet Multiscale Model B A Tubular Reactor Cooling inlet Packed Catalytic Pellet Bed Catalyst Pellet Micro Pores of Catalyst Catalytic Pellet Reactor Bulk model Darcy’s law Mass and energy balance Pellet model

5. Brief background of coupled bulk + pellet - kinetics • The bulk contains spherical pellets • A Heterogeneous first order reaction A  B goes on in the bulk Mass/energy balance in the BULK Mass/energy balance in the CATALYTIC PELLET BC’s: BC’s: (simplest case) De – Effective Diffusivity of reactant A in the pelletke – Effective thermal conductivity in the pellet hs – Mass transfer coefficient (Biot number) DA – Bulk Diffusivity of reactant Akbulk – Bulk thermal conductivity

6. Derivation of h (internal effectiveness factor )

7. Brief background of coupled bulk + pellet - kinetics Under the assumptions of constant pellet properties (heat of the reaction, De and ke) and that the reaction constant ‘k’ is a function of temperature alone we can write: (A) where: where: BC’s:

8. Multiplicity of pellet profiles (Weisz - Hicks): For certain values of γ, β and ϕthere are multiple pellet concentration profiles that satisfy the mass and energy balances and the boundary conditions: For γ = 30 and β = 0.6: - ϕ = 0.07 (2 solutions) - ϕ = 0.2 (3 solutions) - ϕ = 0.7 (1 solution) η ϕ

9. Use of methods that identify ONE initial-guess dependent solution o obtain pellet profiles (I – Weisz-Hicks method) “Shooting” + Bisection method to solve for y(0)=y0: Task: Given: γknown, βknown and ϕknown to find the y0 that will solve the IVP-equivalent of eq (A) Steps: • Solve eq (A) with some y0 and integrate till y(x)=1 • Let x’=x at which y(x)=1 • Set ϕmodel = x’ (explanation on next slide) • The optimization problem to be solved is: (A) Solution using Bisection Method!

10. R R’=aR Explanation Enforce the co-ordinate transformation on x in (A): (BVP) (IVP) Now If we integrate (IVP) until: For the BVP: Thus: and

11. x1 x2 … xn n nodes Use of methods that identify ONE initial-guess dependent solution o obtain pellet profiles (I - OCFE) Simple Collocation (choice of nodes arbitrary): Spherical catalytic pellet where OCFE: Nodes in each element are the roots of the Jacobian polynomial Polynomial · · · · · · · · · · · · Nodes · ´ ´ ´ ´ ´ ´ ´ ´ ´ ´ ´ ´ ´ element 1 element i+1 element m element i x = 0 x = 1

12. Use of methods that identify ONE initial-guess dependent solution o obtain pellet profiles (I - OCFE) Equations: ‘n’ collocation nodes (x1 = 0) and (xn = 1) n equations in n unknowns We are solving for the nodal y values (yi = y(xi))

13. Solution from S-curve FE collocation solution Use of methods that identify ONE initial-guess dependent solution o obtain pellet profiles (I – results) y0 = [10-200, 10-100, 10-50, 10-25, 10-11,…,0.9976, 0.9984, 0.999] ONE SOLUTION 44 points used in the search-space for bisection + shooting method hS = 225.6511 hF =197.3701

14. Solution from S-curve FE collocation solution Use of methods that identify ONE initial-guess dependent solution o obtain pellet profiles (I – results) TWO SOLUTIONS hS1= 682.1231 hS2= 378.27 hF1= 1474.9 hF2= 608.78

15. Solution from S-curve FE collocation solution Use of methods that identify ONE initial-guess dependent solution o obtain pellet profiles (I – results) THREE SOLUTIONS hS2= 1.0638 hF2= 1.1 hS2= 9.8791 hF2= 46.8 hS1= 513.9283 hF1= 1541.3

16. Equations Feasible region Starting point (1.1, 2.0) Use of methods that identify ALL possible solutions (Background): The global terrain method (Lucia and Feng, 2002): A method to obtain all stationary points of a non-linear system of equations Contours of FTF

17. Formulation for the Jacobian and the Hessian Equations for the nodal y values (yi = y(xi)) Where:

18. Use of methods that identify ALL possible solutions (Results): ONLY ONE SOLUTION EXPECTED! 3 collocation nodes (0,0.5,1): ´ γ = 30 b = 0.2 f =0.7 S2 ´ S3 S3 ´ y1 S1 S2 S1 3 solutions obtained! y2 5 collocation nodes (0,0.25,0.5, 0.75,1): γ = 30 b = 0.2 f =0.7 S1 2 solutions obtained! S2

19. Explanation of error BAD RESOLUTION: x=0:0.1:1 r = 1.0e-012*[-0.0002 0.0007 -0.0001 -0.0044 -0.2238] => r*r’=5.0106e-026  0 6 NODES => 5 EQUATIONS => 5 RESIDUALS

20. Explanation of error BETTER RESOLUTION: x=0:0.005:1

21. THREE SOLUTIONS ORDER 3, ELEMENTS 4 or 5 (GTM + FE COLLOCATION)

22. Solution from S-curve GTM + FE collocation solution Starting solution A (top) Time: 21.2 min ORDER 3, ELEMENTS 4

23. Solution from S-curve GTM + FE collocation solution Starting solution B (middle) Time: 216.24 min (3.6 hrs) ORDER 3, ELEMENTS 4

24. Solution from S-curve GTM + FE collocation solution Starting solution C (bottom) Time: 22.41 min ORDER 3, ELEMENTS 4

25. Solution from S-curve GTM + FE collocation solution Starting solution B (middle) Time: 86.5462 min (1.44 hrs) ORDER 3, ELEMENTS 5

26. TWO SOLUTIONS ORDER 3, ELEMENTS 5 (GTM + FE COLLOCATION)

27. Solution from S-curve GTM + FE collocation solution Starting solution B (LOWER ONE) Time: 323.0963 min (5.3849 hrs) ORDER 3, ELEMENTS 5

28. ONE SOLUTION ORDER 3, ELEMENTS 6 (GTM + FE COLLOCATION)

29. Solution from S-curve GTM + FE collocation solution Time:112.7334 min (1.8789 HRS)

30. CATALYTIC PELLET REACTOR - SIMULATION DIFFUSION + CONVECTION

31. All volumes go to the same solution (A) u0 = 0.001 m/s (The profiles for 0.01 and 0.1 m/s are almost the same Ti = 773 K) r z

32. All volumes go to the same solution (A) u0 = 0.001 m/s (The profiles for 0.01 and 0.1 m/s are almost the same Ti = 300 K) r z

33. Causes for multiplicity • 1) Concentration multiplicity (isothermal) • non-monotonic kinetics • Some diffusion resistance associated with the pellet (internal (De) OR external (hs)) • 2) Thermal multiplicity (exothermic reactions) • As reactant is consumed, rate decreases • But since temperature increases, rate increases due to Arrhenius term • Conflicting effects on rate lead to hysteresis • 3) Froment and Bischoff B B A

34. Future Challenges • The parameters Gamma, Beta and Phi vary locally, hence solution of each volume is different • Possible solutions: • Interpolation: Multidimensional interplolation • in different regions • MEAN volume: A volume that has average values of these parameters

35. Thank you!