School of somethingFACULTY OF OTHER

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

Alison S. Tomlin

Energy and Resources Research Institute

Faculty of Engineering

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.

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.

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.

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?

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.3. Change randomly selected factor kj by amount . All other parameters remain the same. Re-run simulation.

Repeat n times

Carry out r runs

Assessing results

Elementary effect of parameter kj on variable ci given by:

Mean effect of factor kj on variable ci :

Variance of effect:

r

dd

r

l

lij

ij

1

)1(

)(

)(

2

1 1

2

2

rr

ddr

d

r

l

r

l

lij

lij

ij

)k(),,,...,()k( ,...,111 imjjji

ij

ckkkkkcd

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.

Scatter plots

The examples show possible nonlinearities and large scatter – obscuring overall sensitivity to parameter.

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.

)x,...,x,xx,xf)(xfff n21nji

jiij

n

iii

1

12...n1

0 (f...)()(x

Component functions

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.

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 (?!).

CO/H2 ignition in air at high pressures

-10 -5 0 5 10 15 200

10

20

30

40

andsimulation

experiment

experiment

simulation

experiment

simulation

0.5CO + 0.5H20.5N2 + 0.5H2

non-reactive

(experiment and simulation)

p /

ba

r

t / ms

Example of the predicted ignition delay for stoichiometric H2 + CO combustion 1007K and 30 bar

0 20 40 60 80 1000

5

10

15

20

25

30

t ign /

ms

% CO in H2

Simulation

RCM studies,Mittal and Sung

The numerical data are sufficiently flawed to give a qualitatively incorrect interpretation

RCM studies from

Mittal and Sung

Simulation

Morris analysis of ignition delays for 0.8CO + H2 at 50 bar and 1040 K

76 irreversible reactions (Davis, 2005)

-12.0 -11.5 -11.0 -10.5 -10.00

2

4

6

8

10

12

t/m

s

log(A/molecule-1 cm3 s-1)

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

log A in primarydata set – from experiment

Experimentaltign at 50 barand 1040 K

The experimental value for tign is predicted in only a few cases regardless of all other parametervalues

The sensitivity to log A is much reduced for log A < -11

0.0 2.5 5.0 7.5 10.0 12.5 15.00.0

0.2

0.4

0.6

0.8

1.0

No

rma

lize

d f

req

ue

nc

y

tign

/ ms

Predicted distribution when only 9 important rate parameters identified by the Morris analysishave uncertainty ranges assigned. All other parameters take fixed values.

Open circles: uncertainty on all rate parameters

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!

0 20 40 60 80 1000

5

10

15

20

25

30

RCM data

Existing data

SJK2

t ign /

ms

% CO in H2

s.d at +40%

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)

KC-135 Reduced-gravity experiments,Pearlman and coworkers

Reduced-Gravity Test Time ~ 20-23s g-level ~ 5x10-2 gearth

Ra reduced from 105 -106 at 1g to 103-104 at reduced g

Parabolic Maneuvers

KC-135A

-2

-1.5

-1

-0.5

0

0.5

1

0 20 40 60 80 100 120 140

Acc

eler

atio

n (

g/g

ea

rth)

Time (s)

X

Y

Z

ReducedGravity

Reduced-Gravity

Pullup1.8g

32kft

24kft

Z

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

Simulations using EXGAS propane mechanism – 1:1 C3H8:O2

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.

Dashed lines – MG experiments.

Solid lines – simulations.

0 4 8 12 16 200

20

40

60

80

100

p/kP

a

t/s

20, 25, 30, 35 kPa initial pressures

51.6, 64.2, 78.6 kPa

Scheme appears to be too reactive.

520 540 560 580 6000

10

20

30

40

50

60

70

80

90

Pre

ssur

e/kP

a

T/K

2 stage ignition boundary

Cool flame boundary

exp

simul

τ2τ1

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

Morris Analysis: time to cool flame

0 2 4 6 8 10 12 14 160

5

10

15

20

25

30

Sta

ndar

d D

evia

tion

1 absolute mean perturbation/s

O2

OOH

n-C3H

7O

2

i-C3H

7O

2

. OOH

Heats of formation Reaction rates

0 2 4 6 8 100

2

4

6

8

10

12

14

1676

75

67

51

7892

406

26

93

44

101

97

50

422

401

77

98

371

3

Sta

ndar

d D

evia

tion

1 absolute mean perturbation/s

Strong nonlinear effects indicated by high standard deviations.Strong influence of heats of formation of a few key species.

Morris analysis: cool flame temperature

0 20 40 600

5

10

15

20

25

30

76

75

67

371

97

51

18

35544

62

422

24592

26

3

Sta

ndar

d D

evia

tion

T1 absolute mean perturbation/K

0 20 40 60 80 100 120 140 1600

5

10

15

20

25

30

35

40

45

50

55

60

Sta

ndar

d D

evia

tion

T1 absolute mean perturbation/K

O2

OOHOOH.

C2H

5O

2

C2H

5OOH

n-C3H

7O

2

i-C3H

7O

2

Controlled by same set of important species and reactions.

Lower nonlinearities.

Morris analysis: time to ignition

0 5 10 15 200

10

20

30

40

50

60

70

80

90

100

Sta

ndar

d D

evia

tion

2 absolute mean perturbation/s

O2

OOH

C2H

5O

2

n-C3H

7O

2

i-C3H

7O

2

0 2 4 6 8 100

10

20

30

40

50

60

70

67

78

371

9732625

44

42292

93

51373

18

62

75

77

76

Sta

ndar

d D

evia

tion

2 absolute mean perturbation/s

Time to ignition, highly nonlinear response.

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.

0 2 0 0 4 0 0 6 0 0 8 0 0 1 0 0 0

- 2 0 0

- 1 5 0

- 1 0 0

- 5 0

0

5 0

Coo

l fla

me

pert

urba

tion/

K

M o r r i s r u n n u m b e r

Perturbation for each Morris run to the cool flame temperature caused by variation of Ho

f for n‑C3H6OOH.

Monte Carlo scatter plots for ΔHof

for n-O2C3H6OOH

-15 -10 -5 0 5 10 151E-3

0.01

0.1

1

10

100

2/s

Change to Hf/kJ mol-1

Note log scale for times to cool flame and ignition, indicating nonlinear response.

-15 -10 -5 0 5 10 15

600

650

700

750

800

850

900

950

T1/K

Change to Hf/kJ mol-1

-15 -10 -5 0 5 10 15

1

10

100

1/s

Change to Hf/kJ mol-1

Effects of adjusting Hof

Optimisation is rarely unique!

0 2 4 6

600

700

800

900

1000

1100

T/K

t/s

0 -5 -10

0 5 10 15 20

600

800

1000

0

20

40

60

80

100

T/K

t/s

p/k

Pa

n-O2C3H6OOH n-C3H7O2 n-O2C3H6OOH: 7.5kJmol-1

n-C3H7O2 : -5kJmol-1

i-C3H7O2 : 5kJmol-1

exp press. record

0

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

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.

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).

= 0.7

-10

-9

-8

-7

-6

-5

-4

-3

-2

-1

0

0 0.1 0.2 0.3 0.4 0.5 0.6

% SO2

% N

O c

ha

ng

esimulated

experiment

= 1.6

0

10

20

30

40

50

60

70

0 0.1 0.2 0.3 0.4 0.5 0.6

% SO2

% N

O in

cre

as

e simulatedexperiment

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.

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.

Sensitivity Indices

Nonlinearities

• Second order interactions small.

• However, significant nonlinearities for some parameters.

Nonlinearities

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

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.

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!

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.