Chaos and Controlin Combustion

Steve Scott

School of Chemistry

University of Leeds

Outline

• Review of H2 and CO combustion

• Use of flow reactors

• Oscillatory ignition

• Mechanistic comments

• Complex oscillations

• Chaos

• Control of Chaos

The H2 + O2 reaction

The classic example of a branched chain reaction

simplest combustion reaction etc.

s lo w re ac tio n

th ird lim it

sec o n dlim it

firs t lim it

ig n itio n

am b ie n t te m p e ra tu re , T /Ka

pres

sure

, p/to

rr4 0 0

5

7 0 0 8 0 0

H2 + O2 branching cycle

H + O2

OH+O

H2

H2

H2O + H

H+OH

H2

H2O + H

Overall: H + 3H2 + O2 3H + 2H2O

rb = 2 kb [H][O2]

rds

Mechanism at 2nd limit

balance between chain branching and gas-phase (termolecular) termination

H + O2 3 H rb = 2kb[H][O2]

H + O2 + M HO2 + M rt = kt[H][O2][M]

Then:

where

is the net branching factor.

< 0: evolve to low steady state

> 0: exponential growth

]O][H[]H[

2 itbi rrrrdt

d

]M[2 tb kk

Condition for limit

Critical condition is = 0

2 kb = kt [M]

crRTEE

t

b peA

ART tb /)(2

T

p = 0

< 0

> 0s lo w re ac tio n

ig n itio n

Studies in flow reactors

• Continuous-flow, well-stirred reactor (CSTR)

• Also shows p-Ta ignition limits

• Study in vicinity of 2nd limit

p-Ta diagram for H2 + O2 in CSTR

0

1 0

2 0

3 0

4 0

5 0

p/to

rr

am b ien t tem p e ra tu re T /Ka

6 5 0 7 0 0 7 5 0 8 0 0

s lo w re ac tio ns tea d y ig n iteds ta te

o sc illa to ryig n itio n

tres = 8 s

Oscillatory ignition

How does oscillation vary with experimental operating conditions?

“Limit cycles”

Oscillation in time corresponds to “lapping” on limit cycle

Extinction at low Ta

tres = 2 s

“SNIPER” bifurcation

More complex behaviour

different oscillations at same operating conditions: birhythmicity

Mixed-mode oscillations

H2-rich systems

Why do oscillations occur?• Need to consider “third body efficiencies”

remember ignition limit condition

2 kb = kt [M]this assumes all species have same ability to

stabilise HO2-speciesin fact, different species have different

efficiencies: aO2 ~ 0.3, aH2O ~ 6

so: overall efficiency of reacting mixture changes with composition

Allow for this in following way:

In ignition region: > 0, based on reactant composition.

After “ignition”, composition now has H2 and O2 replaced by H2O, so overall efficiency is increased, such that for this composition f < 0.

H2O outflow and H2+O2 inflow causes to increase again – next ignition can develop.

)/(

]OH[]O[]H[2

OHOHOOHH

2OH

2O

2H

22222

2

222

RTpxaxaxk

kkkk

tott

tttb

Explains:

oscillatory nature and importance of flow;

period varies with Ta – through kb;

upper Ta limit to oscillatory region ( > 0 even for “ignited composition”;

extinction of oscillations at ignition limit.

Doesn’t explain:

complex oscillations.

Need to include: a few more reactions + temperature effects

CO + O2 in closed vessels

• shows p-Ta ignition limit

• chemiluminescent reaction (CO2*) “glow”

• can get “steady glow” and “oscillatory glow” – the lighthouse effect (Ashmore & Norrish, Linnett)

• very sensitive to trace quantities of H-containing species

CO + O2 in a CSTR• p-T ignition limit diagram shows region of

“oscillatory ignition”

Complex oscillations

Record data under steady operation

Next-maximum map

examplechaotic trace

next-maximumMap

Extent of chaotic region for system with p = 19 mmHg.

parameter lower boundary upper boundary value used

Temperaturea (K) 786 ( 2) 791 ( 2) 789

O2 flowb (sccm) 4.0 ( 0.1) 9.0 ( 0.15) 5.6

CO flowc (sccm) 6.9 ( 0.5) 7.4 ( 0.2) 7.14

sccm = standard cubic centimetre per minute; awith = 5.6 sccm and fCO = 7.14 sccm; bwith T = 789 K and fCO = 7.14 sccm; cwith T = 789 K and = 5.6 sccm.

fO2

fO2

A quick guide to maps

0.0 0.2 0.4 0.6 0.8 1.00.0

0.2

0.4

0.6

0.8

1.0

Xn+1

Xn

xn+1 = A xn (1 – xn) 1 < A < 4

A = 2 A = 2.5 A = 3.2

0 0.1 0.1 0.1

1 0.18 0.23 0.29

2 0.30 0.44 0.66

3 0.42 0.61 0.72

4 0.49 0.59 0.64

5 0.5 0.6 0.74

lots 0.5 0.6 0.51

lots + 1 0.5 0.6 0.80

iteration of the map

Perturbing the map

fixed point shifts

targeting the fixed point

need to determine : location of fixed point of unperturbed systemslope of map in vicinity of fixed pointshift in fixed point as system is perturbed

Ott, Grebogi, Yorke 1990; Petrov, Peng, Showalter 1991

experimental strategy

From the experimental time series:

• collect enough data to plot the map

• fit the data to get the fixed point and the slope in its region

• perturb one of the experimental parameters

• determine the new map – fit to find shift in fixed point

control constant

Can calculate a “control constant” g

where m is the slope of the map and dxF/df is the rate of change of the fixed point with some experimental parameter

df

dx

m

mg F)1(

Note: m and dxF/df can be measured experimentally

Calculate appropriate perturbation

g

xx

dfdxm

mf

F

/1

If we observe system and it comes “near to” the fixed point of the map : x = x xF

Can calculate the appropriate perturbation to the operating conditions

Exploiting the map Chaos control

Map varies with the exptl conditions

Control of Chaosby suitable,very smallamplitudedynamicperturbationscan controlchaos

perturbationsdetermined from Experiment

Davies et al., J. Phys. Chem. A: 16/11/00

0

2

4

6

8

10

12

0 200 400 600

time, t /s

Flo

w / s

cc

m

Chaotic region

(C)

0.0

2.0

4.0

6.0

8.0

0 200 400 600

time, t /s

Pe

ak

PM

T s

ign

al / m

VControl off

CTT

(B)

some unexpected features

0.0

1.0

2.0

3.0

4.0

5.0

6.0

0 100 200 300 400 500 600

time, t /s

Pe

ak

PM

T s

ign

al / m

V

Control off

CTT

0.0

1.0

2.0

3.0

4.0

5.0

6.0

7.0

0 100 200 300 400 500 600

time, t /s

Pe

ak

PM

T s

ign

al / m

V

Control offCTT

0.0

1.0

2.0

3.0

4.0

5.0

6.0

7.0

0 100 200 300 400 500 600

time, t /s

Pe

ak

PM

T s

ign

al /

mV

CTT Control off

0.0

1.0

2.0

3.0

4.0

5.0

6.0

0 100 200 300 400 500 600

time, t /s

Pe

ak

PM

T s

ign

al /

mV

Control offCTT

control transient time depends on how long perturbation is applied for

optimal control occursfor perturbation applied foronly 25% of oscillatory period

Conclusions• Oscillations, including complex oscillations and

even chaotic evolution, arise naturally in chemical reactions as a consequence of “normal” mechanisms with “feedback”

• Chaos occurs for a range of experimental conditions.

• Chaotic systems can be “controlled” using simple experimental strategies

• These need no information regarding the chemical mechanisms and we can determine all the parameters necessary from experiments even if only one signal can be measured

Acknowledgements

Barry Johnson

Matt Davies, Mark Tinsley, Peter Halford-Maw

Istvan Kiss, Vilmos Gaspar (Debrecen)

British Council – Hungarian Academy

ESF Scientific Programme REACTOR