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.

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Solution multiplicity in catalytic pellet reactor

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.


Solution multiplicity in catalytic pellet reactor

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


Solution multiplicity in catalytic pellet reactor

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


Solution multiplicity in catalytic pellet reactor

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


Solution multiplicity in catalytic pellet reactor

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)


Solution multiplicity in catalytic pellet reactor

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:


Solution multiplicity in catalytic pellet reactor

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


Solution multiplicity in catalytic pellet reactor

The “shooting” method (Weisz and Hicks):

η

η

ϕ0

ϕ0

η

η

ϕ0

ϕ0


Solution multiplicity in catalytic pellet reactor

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


Solution multiplicity in catalytic pellet reactor

Obtaining pellet profiles for different ϕ0 (shooting)

y

y

x

x

y

x


Solution multiplicity in catalytic pellet reactor

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


Solution multiplicity in catalytic pellet reactor

Simple Collocation and Orthogonal collocation over finite elements (OCFE)

Equations:

Let us assume there are ‘n’ collocation nodes totally

n equations in n unknowns


Solution multiplicity in catalytic pellet reactor

Obtaining pellet profiles for different ϕ0 (OCFE)

y

y

x

x

y

x


Solution multiplicity in catalytic pellet reactor

A quick comparison:

Same order of magnitude


Solution multiplicity in catalytic pellet reactor

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


Solution multiplicity in catalytic pellet reactor

Global Terrain Method (example)

Equations

Feasible region

Starting point

(1.1, 2.0)

3D space of case 1


Solution multiplicity in catalytic pellet reactor

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


Solution multiplicity in catalytic pellet reactor

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


Solution multiplicity in catalytic pellet reactor

Formulation for the Jacobian and the Hessian

Where:


Solution multiplicity in catalytic pellet reactor

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


Solution multiplicity in catalytic pellet reactor

Thank you!


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