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Solution multiplicity in the catalytic pellet reactor

Solution multiplicity in the catalytic pellet reactor. LPPD seminar Kedar Kulkarni 04/05/2007 Advisor: Prof. Andreas A. Linninger Laboratory for Product and Process Design , Department of Chemical Engineering, University of Illinois, Chicago, IL 60607, U.S.A.

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Solution multiplicity in the catalytic pellet reactor

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  1. Solution multiplicity in the catalytic pellet reactor LPPD seminar Kedar Kulkarni 04/05/2007 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 allowable values for the transport and kinetic properties ?b) Avoid accidents (e.g estimated highest temperature in the reactor is lower than the actual)- Causes of multiplicitya) Multiplicity in inversion solution:Multiple erroneous datasetsb) Multiplicity in state-variable (concentration) profiles:Inherent characteristics of the system (non-linear coupled differential equations) lead to

  3. Outline PROGRESS: • Multiplicity in catalytic pellet profiles: a) Brief background of coupled bulk + pellet - kinetics b) Use of LOCAL METHODS (Background + results): i) The Weisz- Hicks method ii) The method of Orthogonal Collocation over finite elements (OCFE) c) Use of GLOBAL methods (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 OUTLINE OF THE PAPER: Only figures • Conclusions and Future work

  4. Cooling Outlet Multiscale Model B A Tubular Reactor Cooling inlet Packed Catalytical 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 DA – Bulk Diffusivity of reactant Akbulk – Bulk thermal conductivity

  6. 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:

  7. 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

  8. Use of LOCAL METHODS to obtain pellet profiles (I - theory) “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’ • The optimization problem to be solved is: (A) Solution using Bisection Method!

  9. Use of LOCAL METHODS to obtain pellet profiles (I - results) “Shooting” + Bisection method to solve for the initial condition y x y y x x

  10. x1 x2 … xn n nodes Polynomials · · · · · · · · · · · · ´ ´ ´ ´ ´ ´ ´ ´ ´ ´ ´ tf Collocation points Element NS Element i Element i+1 Element 1 Use of LOCAL METHODS to obtain pellet profiles (II - theory) Simple Collocation: Spherical catalytic pellet OCFE: x = 0 x = 1

  11. Use of LOCAL METHODS to obtain pellet profiles (II - theory) Equations: ‘n’ collocation nodes We are solving for the nodal y values (yi = y(xi)) n equations in n unknowns

  12. Use of LOCAL METHODS to obtain pellet profiles (II – results) y y x x y x

  13. Equations Feasible region Starting point (1.1, 2.0) Use of GLOBAL METHODS to obtain pellet profiles 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

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

  15. Use of GLOBAL METHODS to obtain pellet profiles (results) 3 collocation nodes (0,0.5,1): γ = 30 b = 0.2 f =0.7 y1 1 solution expected! y2 5 collocation nodes (0,0.25,0.5, 0.75,1): γ = 30 b = 0.2 f =0.7 1 solution expected!

  16. Use of GLOBAL METHODS to obtain pellet profiles (results) SIMPLE COLLOCATION Expected 2 and obtained 1! Expected 1 and obtained 1! Expected 3 and obtained 3!

  17. Use of GLOBAL METHODS to obtain pellet profiles (results) Expected 1 and obtained 1! Expected 1 and obtained 2! F-E COLLOCATION Expected 3 and obtained 1! Expected 3 and obtained 3!

  18. Multiple pellet profiles in the Catalytic Pellet Reactor r = R A A A A A A A A A r = 0 z = L z = 0 9 internal volumes γ 2.1 b  0.0 f =0.4 A B All the pellets go to A! C

  19. Multiple pellet profiles in the Catalytic Pellet Reactor r = R B A A A A C B A A r = 0 z = 0 z = L γ 30.1 b  0.5 f  0.2 9 internal volumes A The volumes near the entrance go to B or C! B C

  20. Outline of the paper (Only figures) Introduction: • Multiplicity in inversion solution due to multiple erroneous datasets. • State-multiplicity (multiple pellet profiles) due to non-linear coupled differential equation system. k y D x (a) (b)

  21. Outline of the paper (Only figures) Methodology: 1) Methodologies to handle inversion and state-multiplicity – GTM and Niche-GA 2) Methodology to convert PDE-constraints into algebraic equations (b) (a) Figure 2: Motivating example to demonstrate the use of (a) the global terrain method and (b) reduced space hybrid evolutionary methods to obtain all stationary points of a non-linear system of equations.

  22. Outline of the paper (Only figures) Methodology: 1) Methodologies to handle inversion and state-multiplicity – GTM and Niche-GA (b) (a) Figure 3: Flow-chart of the sequence of operations for (a) the reduced space GTM and (b) the reduced space hybrid evolutionary methods along with the full-space gradient-based and evolutionary methods

  23. Outline of the paper (Only figures) Methodology: 2) Methodology to convert PDE-constraints into algebraic equations Figure 4: Use of the Finite Volume Method (FVM) to convert partial differential transport equations in to algebraic equations.

  24. Outline of the paper (Only figures) Case studies: Case study – I: The test transport reactor (b) (a) Figure 5: Surface contours for (a) the reactant A and (b) the product B in the hypothetical test-transport problem.

  25. Outline of the paper (Only figures) Case studies: Case study – I: Results using reduced space GTM Table 1: Table summarizing the co-ordinates of the stationary-points and their type for the test-transport problem. Figure 6: Contours of the least squares TKIP function for the test-transport problem in the space of the reaction constant and the diffusivity. The reduced-space global terrain method predicts three stationary points S1, S2 and S3.

  26. Outline of the paper (Only figures) Case studies: Case study – I: Results using reduced space hybrid evolutionary methods S1 S2 Iteration 100 Iteration 50 Iteration 1 Figure 7: Successive iterations in the solution of the TKIP for the test-transport problem depict two minima (S1 and S2) in the space of the reaction constant and the diffusivity.

  27. Outline of the paper (Only figures) Case studies: Case study – II: State-multiplicity in the catalytic pellet reactor Figure 8: Schematic depicting the multi-scale model in the catalytic pellet reactor.

  28. Outline of the paper (Only figures) Case studies: Case study – II: Results using LOCAL methods – “shooting” and FE collocation (a) (b) (c) (e) (d) (f) Figure 9: Demonstration of state multiplicity using local methods. (a), (b) and (c) show one, two and three solutions respectively obtained using “shooting”. (d), (e) and (f) show the same solutions obtained using FE collocation

  29. Outline of the paper (Only figures) Case studies: Case study – II: Results using GLOBAL methods – GTM with FE collocation (a) (b) (c) Figure 10: Demonstration of state multiplicity using global methods. (a), (b) and (c) show one, two and three solutions obtained continuously using GTM + FE collocation

  30. Outline of the paper (Only figures) Case studies: Case study – II: Multiplicity in pellet profiles along the reactor γ 30.1 b  0.5 f  0.2 γ 2.1 b  0.0 f =0.4 (a) (b) Figure 11: Multiplicity in pellet profiles along the length of the reactor depending on the average values of γ, b, and f.

  31. Future work • The idea of a MEAN volume ? – A volume that has all the pellets in it with γ, b, and f computed as an average of all the volumes • Find out if there is a pattern in the way the pellet profiles distribute along the reactor – or are they chaotic ? • Error surfaces for the bulk state-variables (C and T) in the domain of the bulk parameters

  32. Thank you!

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