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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