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Solution multiplicity in catalytic pellet reactor

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Solution multiplicity in catalytic pellet reactor

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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.

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

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

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

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)

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:

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

The “shooting” method (Weisz and Hicks):

η

η

ϕ0

ϕ0

η

η

ϕ0

ϕ0

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

Obtaining pellet profiles for different ϕ0 (shooting)

y

y

x

x

y

x

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

Simple Collocation and Orthogonal collocation over finite elements (OCFE)

Equations:

Let us assume there are ‘n’ collocation nodes totally

n equations in n unknowns

Obtaining pellet profiles for different ϕ0 (OCFE)

y

y

x

x

y

x

A quick comparison:

Same order of magnitude

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

Global Terrain Method (example)

Equations

Feasible region

Starting point

(1.1, 2.0)

3D space of case 1

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

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

Formulation for the Jacobian and the Hessian

Where:

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

Thank you!