Development of FDO Patterns in the BZ Reaction

Steve Scott

University of Leeds

Acknowledgements

• Jonnie Bamforth (Leeds)

• Rita Tóth (Debrecen)

• Vilmos Gáspár (Debrecen)

• British Council/Hungarian Academy of Science

• ESF REACTOR programme

Flow Distributed Oscillations

• patterns without differential diffusion or flow

• Very simple reactor configuration:

plug-flow tubular reactor fed from CSTR

• reaction run under conditions so it is oscillatory in batch, but steady-state in CSTR

Kuznetsov, Andresen, Mosekilde, Dewel, Borckmans

Simple explanation

• CSTR ensures each “droplet” leaves with same “phase”

• Oscillations occur in each droplet at same time after leaving CSTR and, hence, at same place in PFR

Explains:

existence of stationary patterns

need for “oscillatory batch” reaction

BZ system with f = 0.17 cm s1

[BrO3] = 0.24 M, H+ = 0.15M

[MA] = 0.4 M, [Ferroin] = 7 104 M Images taken at 2 min intervals

wavelength = velocity period

Using simple analysis of Oregonator model, predict:

2/12/13 ]H[]BrO[

~

• Doesn’t explain

critical flow velocity

nonlinear dependence of wavelength on flow velocity

other responses observed, especially the dynamics of pattern development

Analysis

• Oregonator model:

Has a uniform steady state uss, vss

)(

)()1(

12

2

qu

qufvuu

x

u

x

u

t

uP

vux

v

x

v

t

vP

2

2

Perturbation: u = U + uss, v = V + vss

linearised equations

Seek solutions of the form

VjUjx

U

x

U

t

UP 12112

2

VjUjx

V

x

V

t

VP 22212

2

U Aei x t ( ) V Bei x t ( )

Dispersion relation

2 2 3 4 2 2

) ( 2 2 2 ) , ( Tr i i Tr DP P P

0 Tri P

Tr = j11 + j22

= j11j22 – j12j21

2221 422)( TrTriP

Absolute to Convective Instability

Look for zero group velocity, i.e. find =0such that

gives

so

Setting Im(0)) = 0 gives AC

( )

0

0

012 i P

2221

21

0 4()( TrTri P

Bifurcation to Stationary Patterns

Required condition is = 0 with Im() = 0

Setting = 0 yields

So Im() = 0 gives critical flow velocity

0)(2 2234 PPP iTrTri

Tr

TrcrP 2

4 2

,

Bifurcation Diagram

Initial Development of Stationary Pattern

• Oregonator model = 0.25f = 1.0q = 8 104

= 2

0.4 time units per frame

Space-time plot

Experimental verification

BZ system with f = 0.17 cm s1

[BrO3] = 0.2 M,

H+ = 0.15M[MA] = 0.4 M, [Ferroin] = 7 104 M

Adjustment of wavelength to change in flow velocity

Oregonator model as before,

• Pattern already established

• now change from 2.0 to 4.0

space-time plot

Nonlinear - response

= 0.25

= 0.5 = 0.8

= 0.25f = 1.0q = 8 104

= 1.5

0.4 time units per frame

Complex Pattern Development

space-time plot = 1.5

more complexity = 1.4

CDIMA reaction

Patterns

but unsteady

Lengyel-Epstein model

• = 0.5 = 5

0.12 time units per frame