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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.

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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!


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