Solution multiplicity in catalytic pellet reactor

1. Solution multiplicity in catalytic pellet reactor LPPD seminar Kedar Kulkarni 02/15/2007 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 systemb) Avoid accidents (e.g estimated highest temperature in the reactor is lower than the actual)- Causes of multiplicitya) Inherent characteristics of the system (non-linear coupled differential equations) lead to multiplicity in state-variable (concentration) profilesb) Multiple erroneous datasets lead to multiplicity in inversion solution

3. Outline • Review of pellet kinetics: a) Brief theory of coupled differential equations b) The “shooting” method and results c) Use of orthogonal collocation over finite elements • The use of GTM to obtain all solutions automatically: a) Existing 3 nodes code b) Formulation for ‘m’ collocation nodes in ‘n’ finite elements • Contour maps in the bulk-parameter space: • 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 Darcy’s law Mass and energy balance Pellet model

5. Review of pellet kinetics • Bulk contains pellets (e.g spherical, cylindrical etc.) • Heterogeneous first order reaction A  B Mass balance over pellet (eq 1) Energy balance over pellet (eq 2) BC’s: DA and kb are bulk diffusivity and bulk thermal conductivity respectively (eq 3) Rearranging eq 1 and 2: CAs and Ts are surface concentration and temperature Integrating: (eq 4)

6. Review of pellet kinetics Reaction const as a function of T: (eq 5) Using eq 4 and 5: where: Thus, for a spherical catalytic pellet eq 1 becomes (Weisz and Hicks): (eq 6) where:

7. The “shooting” method: • Eq 6 is a BVP. Use the following method to convert it into an IVP a) Solve eq 6 with some y(0) and intergrate till y(x)=1 b) Choose a=(1/x’) where x’=x at which y(x)=1 c) Choose ϕ0 = x’ d) Calculate η as • Characteristics of this method: Input:γ, β, ϕ0 Output:y(x) Actual problem: Reformulation using shooting: Input:γ, β, y(0) Output:y(x), ϕ0

8. The “shooting” method (Weisz and Hicks): η η ϕ0 ϕ0 η η ϕ0 ϕ0

9. Obtaining pellet profiles for different ϕ0: Choose γ = 30 and β = 0.6 Choose: - ϕ0 = 0.07 (2 solutions) - ϕ0 = 0.2 (3 solutions) - ϕ0 = 0.7 (1 solution) η ϕ0

10. Obtaining pellet profiles for different ϕ0 (shooting) y y x x y x

11. x1 x2 … xn n nodes Polynomials · · · · · · · · · · · · ´ ´ ´ ´ ´ ´ ´ ´ ´ ´ ´ tf Collocation points Element NS Element i Element i+1 Element 1 Simple Collocation and Orthogonal collocation over finite elements (OCFE) Simple Collocation: Spherical catalytic pellet OCFE: x = 0 x = 1

12. Simple Collocation and Orthogonal collocation over finite elements (OCFE) Equations: Let us assume there are ‘n’ collocation nodes totally n equations in n unknowns

13. Obtaining pellet profiles for different ϕ0 (OCFE) y y x x y x

14. A quick comparison: Same order of magnitude

15. The use of Global Terrain Method Basic concept of Global Terrain Method (Lucia and Feng,2002) A method to find all physically meaningful solutions and singular points for a given (non) linear system of equations (F=0) Based on intelligent movement along the valleys and ridges of the least-squares function of the system (FTF) The task : tracing out lines that ‘connect’ the stationary points of FTF. Mathematical background Valleys and ridges in the terrain of FTF could be represented as the solutions (V) to: Applying KKT conditions to the above optimization problem we get the following optimization problem Thus solutions or stationary points are obtained as solutions to an eigen-value problem Initial movement It can be calculated from M or H using Lanzcos or some other eigenvalue-eigenvector technique (Sridhar and Lucia) Direction Downhill: Eigendirection of negative Eigenvalue Uphill: Eigendirection of positive Eigenvalue V = opt gTg such that FTF = L, for all L єL F: a vector function, g = 2JTF, J: Jacobian matrix, L: the level-set of all contours Hi : The Hessian for the i th function

16. Global Terrain Method (example) Equations Feasible region Starting point (1.1, 2.0) 3D space of case 1

17. Multiplicity of concentration trajectories using simple collocation (3 nodes) (N=3) • Different concentration and temperature trajectories satisfy the same set of transport equations and boundary conditions 3 equations in 3 unknowns

18. Use of global terrain methods to handle multiplicity • S1 and S3 are minima • S2 and S4 are saddle points • S5 is a minimum outside physically meaningful bounds y1 y2

19. Formulation for the Jacobian and the Hessian Where:

20. Conclusions and future work • OCFE and ‘shooting’ can be used to obtain multiple pellet profiles • GTM can be used to obtain all possible pellet profiles automatically • The use of OCFE may enhance the ability of GTM to obtain all solutions • Future Work: • Use OCFE with GTM and validate existing results • The use of effectiveness factor obtained to model the bulk equations • Solving for multiple bulk-properties using GTM • List of figures for the paper

21. Thank you!