1. Chaos and Control in Combustion Steve Scott School of Chemistry University of Leeds

2. Outline Review of H 2 and CO combustion Use of flow reactors Oscillatory ignition Mechanistic comments Complex oscillations Chaos Control of Chaos

3. The H 2 + O 2 reaction The classic example of a branched chain reaction simplest combustio n reaction etc.

4. H 2 + O 2 branching cycle H + O 2 OH + O H2H2 H2H2 H 2 O + H H + OH H2H2 H 2 O + H Overall: H + 3H 2 + O 2  3H + 2H 2 O r b = 2 k b [H][O 2 ] rds

5. Mechanism at 2 nd limit balance between chain branching and gas- phase (termolecular) termination H + O 2  3 Hr b = 2k b [H][O 2 ] H + O 2 + M  HO 2 + M r t = k t [H][O 2 ][M]

6. Then: where is the net branching factor.  < 0: evolve to low steady state  > 0: exponential growth

7. Condition for limit Critical condition is  = 0 2 k b = k t [M]

8. Studies in flow reactors Continuous-flow, well-stirred reactor (CSTR) Also shows p-T a ignition limits Study in vicinity of 2 nd limit

9. p-T a diagram for H 2 + O 2 in CSTR t res = 8 s

10. Oscillatory ignition

12. How does oscillation vary with experimental operating conditions?

13. “Limit cycles” Oscillation in time corresponds to “lapping” on limit cycle

14. Extinction at low T a t res = 2 s

15. “SNIPER” bifurcation

16. More complex behaviour different oscillations at same operating conditions: birhythmicity

18. Mixed-mode oscillations H 2 -rich systems

20. Why do oscillations occur? Need to consider “third body efficiencies” remember ignition limit condition 2 k b = k t [M] this assumes all species have same ability to stabilise HO 2 -species in fact, different species have different efficiencies: a O2 ~ 0.3, a H2O ~ 6 so: overall efficiency of reacting mixture changes with composition

21. Allow for this in following way: In ignition region:  > 0, based on reactant composition. After “ignition”, composition now has H 2 and O 2 replaced by H 2 O, so overall efficiency is increased, such that for this composition f < 0. H 2 O outflow and H 2 +O 2 inflow causes  to increase again – next ignition can develop.

22. Explains: oscillatory nature and importance of flow; period varies with T a – through k b ; upper T a 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

23. CO + O 2 in closed vessels shows p-T a ignition limit chemiluminescent reaction (CO 2 *) “glow” can get “steady glow” and “oscillatory glow” – the lighthouse effect (Ashmore & Norrish, Linnett) very sensitive to trace quantities of H- containing species

24. CO + O 2 in a CSTR p-T ignition limit diagram shows region of “oscillatory ignition”

25. Complex oscillations

26. Record data under steady operation

27. Next-maximum map example chaotic trace next- maximum Map

28. Extent of chaotic region for system with p = 19 mmHg. parameterlower boundary upper boundaryvalue used Temperature a (K) 786 (  2)791 (  2) 789 O 2 flow b (sccm) 4.0 (  0.1)9.0 (  0.15) 5.6 CO flow c (sccm) 6.9 (  0.5)7.4 (  0.2) 7.14 sccm = standard cubic centimetre per minute; a with = 5.6 sccm and f CO = 7.14 sccm; b with T = 789 K and f CO = 7.14 sccm; c with T = 789 K and = 5.6 sccm.

29. A quick guide to maps x n+1 = A x n (1 – x n ) 1 < A < 4

30. A = 2A = 2.5A = 3.2 00.1 10.180.230.29 20.300.440.66 30.420.610.72 40.490.590.64 50.50.60.74 lots0.50.60.51 lots + 10.50.60.80

31. iteration of the map

32. Perturbing the map fixed point shifts

33. targeting the fixed point need to determine : location of fixed point of unperturbed system slope of map in vicinity of fixed point shift in fixed point as system is perturbed Ott, Grebogi, Yorke 1990; Petrov, Peng, Showalter 1991

34. 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

35. control constant Can calculate a “control constant” g where m is the slope of the map and dx F /df is the rate of change of the fixed point with some experimental parameter Note: m and dx F /df can be measured experimentally

36. Calculate appropriate perturbation If we observe system and it comes “near to” the fixed point of the map :  x = x  x F Can calculate the appropriate perturbation to the operating conditions

37. Exploiting the map Chaos control Map varies with the exptl conditions

38. Control of Chaos by suitable, very small amplitude dynamic perturbations can control chaos perturbations determined from Experiment Davies et al., J. Phys. Chem. A: 16/11/00

40. some unexpected features control transient time depends on how long perturbation is applied for

41. optimal control occurs for perturbation applied for only 25% of oscillatory period

42. 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

43. Acknowledgements Barry Johnson Matt Davies, Mark Tinsley, Peter Halford-Maw Istvan Kiss, Vilmos Gaspar (Debrecen) British Council – Hungarian Academy ESF Scientific Programme REACTOR