The Use of Global Methods In The Evaluation of Non Linear Chemical Kinetic Models

1. Energy and Resources Research Institute Faculty of Engineering The Use of Global Methods In The Evaluation of Non Linear Chemical Kinetic Models Alison S. Tomlin

2. Use of complex kinetic mechanisms • Many examples of areas where complex kinetic mechanisms are used in engineering and environmental design and control: • design of efficient, clean combustion devices • safety applications for range of fuels and hydrocarbons • atmospheric response to pollution control measures. • In practical applications, complex kinetics linked to detailed models of fluid flow and other physical processes. • Experiments designed to isolate kinetics as much as possible from other physical effects for evaluation of mechanisms: • simple flow geometries such as flat flames • photochemical reactor studies for atmospheric chemistry • micro gravity experiments for ignition studies.

3. Evaluation of kinetic mechanisms • Comparison of model with experiment for simple to complex scenarios. - Then what? • If agreement is it for the right reasons? How much confidence can we place in simulations? • If lack of agreement then how do we find the contributing causes? • Sensitivity and uncertainty analysis can help to answer these questions but particular issues for complex chemical systems: • - models often highly nonlinear and computationally expensive • - mechanisms are often large with very many parameters • - isolating the kinetics in experiments not always straightforward • - tempting to optimise the mechanism by fitting parameters to specific sets of experiments – can we really isolate parameters well enough to do this? • - need strong feedback loop between model evaluation and methods for model improvement.

4. The tyranny of parameters? • Some examples of kinetic mechanisms currently in use: • The Master Chemical Mechanism for tropospheric chemistry simulations: 13,500 reaction rates, many of which are estimated. • Mechanisms for alkane combustion: propane, 122 species in 1137 reactions,cyclo-hexane, 499 species in 3348 reactions etc. • Both reaction rates and thermo-chemistry could be uncertain leading to huge number of potentially uncertain input parameters. • How do we deal with this level of complexity? • global screening methods • automatic methods for identifying sensitivities of only important parameters using small number of computer simulations • -HDMR + optimisation techniques.

5. Screening methods • Methods such as the Morris method can be used for screening out unimportant parameters before more complex methods such as Monte Carlo simulations, FAST, HDMR are used. • Often the parameter space to be investigated is enormous: • - large no. of parameters n • - large uncertainty ranges. • Need to be careful where non linear responses are possible that an appropriate Morris sample is used. • Updated Morris method from Campolongo could be useful. • Question: • Screening methods can be expensive – order n x Morris runs. • Are they always necessary for systems with large no. parameters?

6. Morris method • Assesses role of each parameter “one at a time” within different parts of the parameter space. • Computational cost of the order n – no of parameters. • Steps: • 1. Establish uncertainty limits for each parameter under investigation. • 2. Carry out control simulation by randomly selecting parameter values from within range. Carry out r runs • 3. Change randomly selected factor kjby amount . All other parameters remain the same. Re-run simulation. Repeat n times

7. Assessing results Elementary effect of parameter kjon variable ci given by: Mean effect of factor kjon variable ci : Variance of effect:

8. Monte Carlo (MC) simulations • Conceptually straightforward. • Based on random or quasi random sampling of input parameter space. • Perform many simulations until output mean/variances converge. • No. of necessary runs depends on number of important parameters • - unlike Morris, stratified sampling methods e.g. Latin Hypercube, MC methods may not increase in cost with input space dimension. • Cost may still be prohibitive. • Interpretation of results for large input space: • Scatter plots used for each parameter to see overall effect. Large scatter often obscures mean effect of individual parameter. • Linear effects easily shown using Pearson correlations. • Non-linear effects can be incorporated using variable transformations and rank correlation (Spearman correlations) but not straightforward. • Large sample sizes required to perform e.g. Sobol’s method.

9. Scatter plots • The examples show possible nonlinearities and large scatter – obscuring overall sensitivity to parameter.

10. High Dimensional Model Representations (HDMR) • Developed to provide detailed mapping of the input variable space to selected outputs. • Mapping useful for an overall analysis of the model. • Output is expressed as a finite hierarchical function expansion: • Usually second-order expression provides satisfactory results. • Model replacement built using quasi random sample and approximation of component functions by orthonormal polynomials. • Model replacement can be used to generate full Monte Carlo statistics. • 1st & 2nd order sensitivity indices easily calculated from component functions.

11. Component functions

12. Ignition delay studies • Accurate prediction of ignition phenomena is important for several reasons and for a wide range of fuels. • Safety applications such as minimum auto-ignition temperature for various fuel mixtures. • Study of cool flame phenomena and subsequent ignition. • Engine applications. • Ignition studies also provide useful sets of data for the evaluation of chemical mechanisms across wide temperature and pressure ranges. • Low temperature chemistry important for ignition.

13. CO/H2 ignition in air at high pressures • How would the ignition delay of hydrogen + air at high pressure be affected if there was an increasing extent of substitution of hydrogen by carbon monoxide? • Experimental studies carried out in a rapid compression machine (RCM) at high pressures (Mittal and Sung). • Simulations expected to be straightforward since chemistry of H2/CO combustion one of best categorised systems (?!).

14. CO/H2 ignition in air at high pressures

15. Example of the predicted ignition delay for stoichiometric H2 + CO combustion 1007K and 30 bar The numerical data are sufficiently flawed to give a qualitatively incorrect interpretation RCM studies, Mittal and Sung RCM studies from Mittal and Sung Simulation Simulation

16. Morris analysis of ignition delays for 0.8CO + H2 at 50 bar and 1040 K76 irreversible reactions (Davis, 2005)

17. log A in primary data set – from experiment The experimental value for tign is predicted in only a few cases regardless of all other parameter values Experimental tign at 50 bar and 1040 K Significance of the parameter values for CO + HO2 on the predicted ignition delay • Monte Carlo Analysis (13000 simulations), where • log A is used to reflect variations of log k The sensitivity to log A is much reduced for log A < -11

18. Only a few parameters drive output uncertainty Blue – parameter fixed at previous value in primary data set. Green – parameter fixed at remodelled value from Klippenstein. Still large variability for “well known” scheme!

19. Predictions of ignition delay at 30 bar and 1007 k based on parameter values for CO + HO2 derived from master equation calculations (S.J. Klippenstein) SJK : Two barrier function k = 1.149 x 10-17 T1.609 e(-8805/T)

20. 32kft Z Pullup 1.8g Reduced- Gravity 24kft KC-135 Reduced-gravity experiments,Pearlman and coworkers Parabolic Maneuvers KC-135A • Reduced-Gravity Test Time ~ 20-23s • g-level ~  5x10-2 gearth • Ra reduced from 105 -106at 1g to 103-104 at reduced g

21. Propane ignition and micro gravity (MG) studies • Micro-gravity studies mitigate 3D complexities associated with buoyant convection on low temperature reactions, cool flames and auto-ignition fronts. • Can be used as a test bed for low-temperature chemical kinetic models. • Experiments carried out in quartz reactor on KC-135. Propane used as fuel. 1:1 C3H8:O2 30-80 kPa • 593 – 623K Cool flame Ignition

22. 2 stage ignition boundary 51.6, 64.2, 78.6 kPa exp simul Cool flame boundary 20, 25, 30, 35 kPa initial pressures Simulations using EXGAS propane mechanism – 1:1 C3H8:O2 Dashed lines – MG experiments. Solid lines – simulations. 60 species, 450 reaction reduced mechanism employed in 0-D closed vessel for uncertainty analysis. Master equation model of C3H8/O2 system by Taatjes et al., incorporated. τ2 τ1 Scheme appears to be too reactive.

23. Input Uncertainties • Reaction rates represented as Arrhenius expressions: • ATne(-E/RT) • Reversible reactions used and therefore heats of formation provide reverse rates via equilibrium constant. • Uncertainties in heats of formation Hof and A factors used. • Uniform distributions used since many of parameters estimates and probability distributions not available. • Morris analysis first used to rank parameter importance and screen unimportant parameters. • Outputs: time to cool flame, τ1 • cool flame temperature T1 • time from cool flame to ignition, τ2

24. Morris Analysis: time to cool flame Heats of formation Reaction rates Strong nonlinear effects indicated by high standard deviations. Strong influence of heats of formation of a few key species.

25. Morris analysis: cool flame temperature Controlled by same set of important species and reactions. Lower nonlinearities.

26. Morris analysis: time to ignition Time to ignition, highly nonlinear response.

27. Morris runs took a long time to stabilise Sensitivity to parameter changes sign depending on position in parameter space. More efficient Morris method may reduce screening burden. Perturbation for each Morris run to the cool flame temperature caused by variation of DHof for n‑C3H6OOH.

28. Monte Carlo scatter plots for ΔHof for n-O2C3H6OOH Note log scale for times to cool flame and ignition, indicating nonlinear response.

29. Effects of adjusting DHofOptimisation is rarely unique! exp press. record 0 n-O2C3H6OOH n-C3H7O2 n-O2C3H6OOH: 7.5kJmol-1 n-C3H7O2 : -5kJmol-1 i-C3H7O2 : 5kJmol-1

30. Important reactions and species for future ab initio studies • Rate Coefficients: • C3H8 + OH ↔ i-C3H7 + H2O (75) • C3H8 + OH ↔ n-C3H7 + H2O (76) • O2C3H6OOH ↔ C3H5OOOH + OH (67) • C3H8 + HO2 ↔ n-C3H7 + H2O2 (78) • 2HO2 ↔ H2O2 + O2 (371) Thermochemistry: n-O2C3H6OOH C2H5O2 i-C3H7O2 C2H5OOH n-C3H7O2 n‑C3H6OOH

31. Flame studies • Low pressure flat flame studies often used for evaluation of mechanisms at high temperatures. • Flat flame allows 1-D simulations to be performed using • standard codes such as CHEMKIN. • Comparisons of species profiles for • given temperature profiles.

32. Evaluation of sulphur chemistry via low pressure flame experiments • Sampling/GC used to evaluate concentrations of major C/H/O species. • Laser Induced Fluorescence (LIF) used for determine relative concentrations of NO,NS,SO2 for flames of different stoichiometries i.e. fuel/air ratios. • Flat flame modelled with a 1D model using PREMIX  complex chemistry included. • - Major C/H/O species well represented within experimental uncertainty. • - Attempted to model the relative change in NO emission on the addition of varying amounts of SO2 for flames of varying stoichiometries ( = 0.7-1.6).

33. In rich flame the addition of sulphur enhances NO production. Model over predicts enhancement. In lean flame the addition of sulphur reduces NO emission. Model under predicts reduction.

34. Model Scenario • Effect of large uncertainties in SOx scheme explored. • Uncertain parameters: 153 reaction rates and 24 heats of formation (calculated by NASA polynomials). • Kinetic evaluations used where available but many parameters estimated or based on single experimental or modelling study. • Wide uncertainty ranges used. • HDMR used for a variety of input scenarios • - see presentation of Ziehn • Method successfully coped with very large parameter space. • Extensions to HDMR developed based on optimisation of polynomial order and automatic exclusion of unimportant component functions. • Screening method not required – first and second order sensitivities computed with only 1024 model runs.

35. Sensitivity Indices

36. Nonlinearities • Second order interactions small. • However, significant nonlinearities for some parameters.

37. Nonlinearities • Component functions give automatic way of visualising nonlinear response to parameters.

38. Conclusions 1 • Nonlinear chemical kinetics provide challenging systems for the application of global sensitivity methods due to large, often highly uncertain parameter sets, nonlinear responses. • Often only a small number of parameters drive output uncertainty. • Global sensitivity methods provide essential step in model evaluation by identifying this parameter set. • Further ab initio studies can then be focussed on key parameters improving model performance. • Optimisation rarely unique from single experiment. • Each experiment can however, help to narrow down feasible range for parameters.

39. Conclusions 2 • Screening methods can be used to but in some cases require large no. of model simulations. • An extended HDMR based method has proven effective in application to a large complex model – providing up to second order sensitivities with relatively low computational cost. • Second order interactions were found to be relatively unimportant, although strong nonlinearities in first order responses were found. • Application of HDMR to further systems required to explore general importance of higher order interactions. HDMR software demonstration available in poster session!

40. Acknowledgements • Thanks to: • James Benson • Nick Dixon • John Griffiths • Kevin Hughes • Tilo Ziehn • This work has been supported financially by • EPSRC (GR/R76172/01(P), GR/S58904/01(P)) • European Union (EESD-ESD-3 (JO 2000/C 324/09), Proposal No EVG1- 2001-00098 – SAFEKINEX ) • Maria Goeppert Mayer scholarship from Argonne National Laboratories.