Bifurcation Analysis of Bifurcation Analysis of Discontinuous Periodically Forced Reactors Discontinuous Periodically Forced Reactors

Bifurcation Analysis of Bifurcation Analysis of

Discontinuous Periodically Forced Reactors Discontinuous Periodically Forced Reactors

Erasmo Mancusi

Erasmo Mancusi

, Lucia Russo

, and Gaetano Continillo

a

Dipartimento di Ingegneria, Università del Sannio, Benevento, Italy

b

Dipartimento di Ingegneria Chimica, Università Federico II, Naples,

Italy

Rome Rome

Naples Naples

Benevento Benevento Venice

Venice Florence

Florence

Milan

Milan

Outline Outline

•General mathematical properties of the dynamical models of periodically forced reactive systems

•Numerical technique

•Examples of computational non-linear analysis of catalytic combustors:

1. Lumped parameter model (low-dimensional system)

2. Distributed parameter model (high-dimensional system)

Example: Two cascaded CSTR with Example: Two cascaded CSTR with

Periodically Inverted Feed Periodically Inverted Feed

V

V

V

V

R  P

Model equations Model equations

   

     

 

 

-1

-1

1 exp

1

1 e

0 0 mo

xp 1

d 2 1 ( )

1 mod 2 1

1, 2

.

i n i

i i i

i

i n i

i i i H i

i

d Da

dt

d Da B

g t g t

if t g

d

t t i

if t

 

         

 

      

 



 

 

 



   



    



      









   

 

, , ,

2

 



  

 





d f t

dt t T

x F x x

x x

Periodically forced RFR reactors are a particular class of non-autonomous dynamical systems.

f(t) is a periodic forcing function with minimum period T

   

   

, ,

, , ' 0 '

 

   

f t f t T

f t f t T T T

x x

x x

The system is T -Periodic

General Properties of a

General Properties of a Reversed Flow Reversed Flow Reactor

Reactor Model Model

0  2 3

f(t)

The vector field is invariant under the application of a reflection matrix together with a time shift

 ^{, ,}  ^  ^{, , } 

is a

t t

  

Reflection GF x

Ma F Gx

G trix

The RFR mathematical model is spatio-temporal symmetric

General Properties of a

General Properties of a Reversed Flow Reversed Flow Reactor

Reactor Model Model

Symmetric solutions :Z

Symmetric solutions :Z

symmetry symmetry

 ^

0.1 0.2 0.3

2

0.1 0.2 0.3

dimensonless time

60 65 70 75 80



0.1 0.2

0.3

a b

Each reactor experiences the same oscillation but shifted of  in time.

   

2

t

t

    

 

0.0 0.5 1.0

 

0.1 0.2 0.3

 

0.1 0.2 0.3

 

0.0 0.5 1.0

G

dimensionless time

50 60 70 80

  

0.0 0.5 1.0

dimensionless time

50 60 70 80

  

0.0 0.5 1.0

a b

c ^d

Asymmetric solutions and multiplicity : Z

Asymmetric solutions and multiplicity : Z

symmetry symmetry

Multiplicity is due to the system symmetry

Non-linear analysis Non-linear analysis

Typical representations used are:

• Solution diagrams

time

x

• Time waveform diagrams

x₁

x₂

• Phase portraits

Da

• Bifurcation diagrams 

Non-linear analysis Non-linear analysis

• Bifurcation analysis Bifurcation analysis

• Techniques Techniques

Design Control

Simulation (brute force) Simulation (brute force)

Continuation algorithms Continuation algorithms

Lengthy

Unstable regimes not detectable

Stability

Stability analysis analysis

Construction of periodic Construction of periodic and multiperiodic

and multiperiodic branches

branches

Local

Local bifurcations bifurcations

Application of Continuation Algorithms to a Application of Continuation Algorithms to a

RFR Model: Problems RFR Model: Problems

• The system is periodic and thus non-autonomous

• It is impossible to transform the system into an autonomous one by adding a state variable, because the forcing input is periodic but time discontinuous

Standard Software “ability”

(AUTO, Content,…)

• Continuous autonomous systems

• Non-autonomous systems with C

vector field where the forcing input is solution of an autonomous system

• Discrete autonomous systems

The Poincaré Map The Poincaré Map

The Poincarè map is related to a Poincaré section transverse to the orbit of the underlying continuous time system.

T

Autonomous systems:

The Poincarè map is local

Periodic systems:

The Poincarè map is global

x

=P(x

,)

Continuous Periodic Systems are dynamically Continuous Periodic Systems are dynamically

equivalent to Discrete Poincaré Maps equivalent to Discrete Poincaré Maps

Continuous-time system

Periodic regimes

k -Periodic regimes Quasi-Periodic regimes

Chaotic regimes

Poincaré map

Fixed Points of P

Fixed Points of P

Closed Curves Fractal Sets

( , ) F t



x  x x

 P x ( , )



N -S b ifu rc a tio n s

The Poincaré Map for a RFR The Poincaré Map for a RFR

For a RFR system the Poincaré map can be expressed as:

 

  

2

   

       T

P G G G

where is the evolution operator of the unforced system. The Poincaré map is the second-iterate of another map H



 



  

P H H G

P and H are composed by diffeomorphisms.

This property overcomes the problems related to the discontinuities of

the vector field in the continuous time system

The Poincaré Map for a RFR The Poincaré Map for a RFR

Continuation algorithms can then be applied to the discrete-time

“equivalent” system

Given the mentioned symmetry property, most information can be obtained

by studying the bifurcations of the discrete-time system with reference to

the H map instead of the P map. This may save up to half of the computing

time.

The software AUTO can trace the fixed point locus of a discrete map given an initial point of the locus.

AUTO requires a functional representation of the discrete

time system. Since no explicit expression is available for the map, it must be provided via numerical computation.

NUMERICAL INTEGRATOR SUBROUTINE FUNC

OTHER SUBROUTINES

AUTO

u f(u)

The external integrator (VODE) performs an accurate computation of the time integral from 0 ^to 2 

Numerical approach

Numerical approach

Example: Two Cooled CSTR with Example: Two Cooled CSTR with

Periodically Inverted Feed Periodically Inverted Feed

0 R  P   H

r kC 

 

o

exp E

k k   RT

State Variables , , ,

A B A B

   

 _Conversion

Dimensionless temperature



Parameters Da, 

V₁

V₁ V₂

V₂

Da Damköhler number

dimensionless heat

transfer coefficient

Frequency Locking Frequency Locking

Re() Im()

|| =1 1 1



q p

If q/p (rotation number) is rational a (q,p)- periodic resonant solution exist If q/p (rotation number) is irrational a quasi-periodic regime exist

1,2



 2 q

e p





 

At any Neimark-Sacker bifurcation two Floquet multipliers are:

Arnold Tongues Arnold Tongues

Da

0.133 0.134 0.135 0.136



0.76 0.80 0.84 0.88 0.92

Solution diagram of (2,5)-resonant solution Bifurcation Diagram

It is worthwhile mentioning that, according to the theory, the Arnold Tongues approach a point on the Neimark-Sacker curve where the multipliers of the original fixed point have the representation:

 2

= 2π = 144°

5

V

V

V

V

A Reversed Flow Reactor: Discontinuous A Reversed Flow Reactor: Discontinuous

Periodically Forced Operation

Periodically Forced Operation

T

, T

: temperatures at left and right ends of the reactor

300 400 500 600 700 800 900

dimensionless time

0 500 1000 1500 2000

T

T

[K]

Symmetric solution: Z

Symmetric solution: Z

symmetry symmetry (RFR) (RFR)

500 600 700 800 900 1000

dimensionless time

0 500 1000 1500 2000

T

T

[K]

Asymmetric solution: Z

Asymmetric solution: Z

symmetry symmetry (RFR) (RFR)

T

, T

: temperatures at left and right ends of the reactor

Solution diagram: catalytic combustor

P a rtico la re

Periodo d'inversione

0.0 0.1 0.2 0.3



G ,O U T

-0.5 0.0 0.5 1.0 1.5 2.0

2.5

D ia g ra m m a d e lle so lu zio n i

Periodo d'inversione

200 400 600 800 1000 1200 1400 1600 1800

a b

c d

e f

 

 









Resonant Regime Solutions

Resonant Regime Solutions

Conclusions Conclusions

It is possible to reconstruct systematically the regime behavior of discontinuous periodically-forced systems, by applying robust continuation algorithms to an associated discrete-time system, properly constructed and implemented numerically starting from a Poincaré section.

The approach proposed permits to identify bifurcations and to

automatically trace solution branches stemming from pitchfork

bifurcations. This constitutes a complete bifurcational study

otherwise unfeasible for the underlying discontinuous periodically-

forced non-autonomous system.