COMBUSTION CHEMISTRY AND MODEL REDUCTION · 2018-12-15 · • Combustion chemistry studies how...

COMBUSTION CHEMISTRY AND MODEL REDUCTIONYi YangUniversity of Melbourne

Combustion Summer School17 December 2018, Sydney, Australia

Copyright ©2018 by Yi Yang. This material is not to be sold, reproduced or distributed without prior written permission of the owner, Yi Yang.

• Combustion chemistry studies how much and how fast energy is released from the chemical reactions in combustion.

Example

• Thermochemistry: heating value, flame temperature, chemical equilibrium …

• Chemical kinetics: reaction rate, reaction mechanism, model development …

• Chemical kinetics is focus.

Introduction

4 2 2 2 2 22( 3.76 ) 2 7.52CH O N CO H O N E+ + = + + + ∆

1

Overview of the Topic

Combustion chemistry

multi-reaction system

elementary reaction

2

4 2 2 22 2CH O CO H O+ = +

4 2 3 2CH O CH HO+ → +

2CO OH CO H+ → +

2H O OH O+ → +

CO oxidation

H2 oxidation

CH4 oxidation

3

Overview of the Topic

Combustion chemistry

Structure of This Lecture

Theory

Application

4

Reduced model

Structure of This Lecture

• Review of basics• Gas Phase, non-catalytic combustion. Ideal gas laws apply

• Elementary level investigation• Kinetic experiments on elementary reaction• Transition state theory

• Mechanism level investigation• Overview of hydrocarbon combustion chemistry• Fundamental combustion experiments• Model construction

• Model reduction• Sensitivity and uncertainty quantification• Removal of redundant species and reactions• Time scale analysis

5

Thermochemistry• State change• 1st and 2nd laws applied to a system with chemical reaction• ΔH – cp (LHV, Tad …)• ΔG – Kp• Relatively well known in comparison with chemical kinetics

Review

6

• Rate of state change• Elementary reaction and global reaction

• Chain reactions • Free radicals

-O-O- H-H

• Chain sequence

• Reaction rate equation (law of mass action)

Chemical Kinetics

+ +2 2 2

kf

kr

H O HO H + →2 2 212

GkH O H O

− = −1 1 1 12 2 2 2[ ]

[ ] [ ] [ ] [ ]f rd H k H O k HO H

dt− =2 2 2

[ ][ ] [ ]n mG

d H k H Odt

C

7

• Essence of chemical reaction• Boltzmann distribution of molecular energy• Arrhenius Law

• A-factor: total possible collisions per unit molar concentratione.g. half chances to break a primary C-H bond with ethane than with neo-pentane

C-C vs.

• Activation energy: energy barrier to be overcome for successful collision

• A simplified description of reaction dynamics

Reaction Rate

= −exp( )b au

Ek AT

R T

Energy Activation energy

Energy released Reactant

Product

Progress of Reaction 8

Progress of Reaction

Reactant

Product

Energy released

Activation energy

Energy

Reaction Rate

• Dependence of reaction rate on temperature and pressure

• Applicable to elementary reactions in general• May not apply to multi-reaction system

, , reaction rate,[ ] , reaction rate

( ), ( )

T kp Xk k T k k p

↑ ↑ ↑

↑ ↑ ↑= ≠

9

H2 explosion has negative pressure dependence!

(Glassman & Yetter, Combustion 2008)

Multiple-Reaction System

• - net production rate • N such terms in species equations – big burden for computation

+ +

+ +

+ +

+ + +

1

1

2

2

3

3

4

4

2 2 2

2

2 2

2 2

......

f

r

f

r

f

r

f

r

k

k

k

k

k

k

k

k

H O HO H

H O OH O

OH H H O H

H O M HO M

= + −

− −

+ −

− −

1 2 2 1 2

2 2 2

3 2 3 2

4 2 4 2

[ ] / ( [ ][ ] [ ][ ])

( [ ][ ] [ ][ ])

( [ ][ ] [ ][ ])

( [ ][ ][ ] [ ][ ])

......

f r

f r

f r

f r

d H dt k H O k HO H

k H O k OH O

k OH H k H O H

k H O M k HO M

υ υ= =∑ ∑' "

1 1

N N

ij i ij ii i

f

r

k

kX X ω υ υ

=

= −∑ " '1

( )M

i ij ij jj

qυ υ

= =

= −∏ ∏1 1

' "N N

j fj i rj ii i

ij ijq k c k c

for ith species, jth reaction

ω i

10

How to Study Combustion Chemistry

• Elementary level• Kinetic experiment – direct measurement of rate coefficients for

elementary reactions• Reaction dynamics theory – quantum mechanical calculations of

elementary reaction rates

• Mechanism level• Fundamental combustion experiments – measurement of multi-reaction

systems (indirect)• Kinetic modelling – physical model + chemical model

11

Elementary Level Investigation

12

Kinetic Experiment of Elementary Reactions

• Challenges – fast reaction, short lived species, complex mixture, high temperature (high pressure)

• Method 1 – Generate clean source of radical, then measure radical decay in the presence of excess substrate – pseudo first order kinetics

• Method 2 – React in a constant environment where background reaction supplies the radical needed.

2RH OH R H O+ = +Example

13

• Working principle of Method 1

2 6

0 0

[ ] [ ][ ]

[ ][ ]

[ ]

ln[ ] ln[ ] ( )

OH

OH

t OH

d OH k OH C Hdt

k OHd OH k dt

OH

OH OH k t t

− =

≈

− =

= − −

2 6 2 2 5OH C H H O C H+ → +

C2H6 in large excess

14


(Tully et al. J. Phys. Chem., 1986)

• N2O decomposed to produce OH

• OH decay detected with PLIF at 308 nm

2 2

2

** 2

N O h N OO H O OH

ν+ → ++ →

2 6 2 2 5OH C H H O C H+ → +

15


(Tully et al., J Phys Chem 1986)18 2.06

3

8.51 10 exp( 430 / ) [cm /molecule / s]k T T−= × −

0 0ln[ ] ln[ ] ( )t OHOH OH k t t= − −

ln ln ln /a uk A b T E R T= + −

Tully et al. (open circles and line)

16


• Method 2 – React in a constant kinetic environment (University of Hull approach)

• Small amount of hydrocarbon (0.1%) added to slow reacting H2/O2mixtures in a closed reactor

• Background chemistry unaffected by added fuel

• RH reacts with radicals (e.g. OH) produced by the H2/O2 mixture• Background chemistry is known and constant.

• Relative reaction rates are determined between H2 and RH • e.g. OH + RH vs. OH + H2• Absolute rate of OH + RH can then be obtained with the knowledge of OH +

H2 rate.17


• Method 2 – React in a constant kinetic environment (University of Hull’s approach)

2 6 2 2 5OH C H H O C H+ → +

2 2

2 6 2 2 5

OH H H O H

OH C H H O C H

+ → +

+ → +

1

21

R

R

21 2

2 621 2 6

[ ] [ ][ ]

[ ] [ ][ ]

d H k OH Hdt

d C H k OH C Hdt

− =

− =

]𝑑𝑑[𝐶𝐶2𝐻𝐻6]𝑑𝑑[𝐻𝐻2

=]𝑘𝑘21[𝐶𝐶2𝐻𝐻6

]𝑘𝑘1[𝐻𝐻2

18


• Other reactions consuming H2and RH assumed negligible

Hull Slow Reacting H2/O2 Experiment

(Baldwin et al. Trans. Faraday Soc. 1970)

x = xH2 y = xO2, 0.1% C2H6

19

• k21 can be obtained with k1 known.

(Baldwin & Walker. J Chem. Soc. Fara. Trans.1 1978)

Hull Slow Reacting H2/O2 Experiment

2 6 2 2 5OH C H H O C H+ → +

20

(Tully et al. J Phys Chem, 1986)

ln ln ln /a uk A b T E R T= + −

Baldwin et al.

Tully et al.

Comparison of Hull and Tully Results

21

Transition State Theory (TST)• Reactants undergo an intermediate state – transition state before

forming the product

+ → → + [ ]A BC A B C AB C

(Robertson et al. Low temp. combust. autoign., 1997)22

Application of TST

• Fall-off behaviour – Rate coefficients of unimolecular reactions show dependence on pressure, i.e. k = k(p)

[ *]Apply QSSA to [ *], 0

[ ][ ] [ *][ ] [ *] 0[ ][ ][ *][ ]

[ ] [ ][ ] [ ][ ] [ ]

a d r

a

d r

r a r auni

d r d r

d AAdt

k A M k A M k Ak A MAk M k

k k M k k Md P A kdt k M k k M k

=

− − =

=+

= ⇒ =+ +

(Robertson et al. Low temp. combust. autoign., 1997)

[ ] [ *]rd P k A

dt=

→+ ←

→

*

*

a

d

r

kk

k

A M A

A P

Lindemann mechanism

→unikA P[ ] [ ]uni

d P k Adt

=

Global reaction

23

• Dependence of reaction rate coefficient on pressure

Pressure

kuni Low Pressure limit

high Pressure limit

“fall-off” region

unik k∞=

0[ ]unik k M=

p , [M] , ,

p 0, [M] 0, [ ],

r auni uni

d

uni a uni

k kk k constk

k k M k p

→∞ →∞ = =

→ → = ∝

Application of TST

[ ][ ]r a

unid r

k k Mkk M k

=+

24

How to Model “Fall-Off” Behaviour?

• Lindemann approach

•

• falls off too fast

• Troe formulism (mostly used)•

•

• need to be specified to determine F.

12loglog 1 log

(log )r

centerr

P cF Fn d P c

− + = + − +

0

0

[ ][ ] 1

runi

r

Pk k Mk F k Fk k M P

∞∞

∞

= = + +

0[ ] /rP k M k∞=

1,F =

0.4 0.67 log0.75 1.27 log0.14

center

center

c Fn Fd

= − −

= −=

**

*** *(1 )exp exp expcenterT T TF a a

T T T = − − + − + −

*** * **, , ,a T T T

,exp abu

Ek A T

R T∞ ∞∞

∞

= −

0 ,000 exp

ab

u

Ek A T

R T

= −

25

1F ≠

H2O2(+M)OH+OH(+M) 2.951E+14 0.000 4.843E+04 LOW / 1.2020E+17 0.0000E+00 4.5500E+04 / TROE / 5.0000E-01 1.0000E-30 1.0000E+30 1.0000E+10 /

,, , aA b E∞ ∞ ∞

*** * **, , , a T T T

0 0 ,0, , aA b E

26

How to Model “Fall-Off” Behaviour?

**

*** *(1 )exp exp expcenterT T TF a a

T T T = − − + − + −

,exp abu

Ek A T

R T∞ ∞∞

∞

= −

0,00

0 expab

u

Ek A T

R T

= −

More Theoretical Treatment of TST

• Consider • Energy of the activated molecule is not a fixed value but is determined by a

probability density function • Reaction takes place in multiple routes with the possibility described by a PDF

• This means • Reaction rate coefficient is not a constant• Reaction does not happen in one or two steps.

• That is, determining the rate coefficient should consider all possible pathways that the molecule - of which the energy is a PDF - can undertake.

27

http://computational-chemistry.com/en/blog/transition-state/ 28



• The problem is now described by a Master Equation

• Problem is to solve the microcanonical rate coefficient, k(E)• Theory (e.g. RRKM for unimolecular decomposition) + potential energy surface • Solved by ab initio method, which solves Schrödinger equation of electronic

wavefunction• k(E) and overall k are obtained from first principles. Arrhenius law is not

assumed.

0Probability loss Probability loss Probability gain due to transition due to collision due to reactionfrom all other energy (E') to E

( , ) ( / ') ( ', ) ' ( , ) ( ) ( , )E t P E E E t dE E t k E E ttρ ω ρ ωρ ρ

∞∂= − −

∂ ∫

29

Mechanism Level Investigation

30

Typical Alkane Oxidation Chemistry

• Different chemistry at high temperature & low temperature

• Pressure is important in deciding reaction regimes.

2H O OH O+ → +

2 low temp branchingRO QOOH→ →

NTC regime

(Morley & Pilling, Low temp combust autoign, 1997) 31

High Temperature Chemistry

• Relevant to flame• Fuel decomposes readily to small species.

• β-scission

• Small species control oxidation rate. • Fuel structure less critical.

3 2 2 2 2 2 3

3 2 2 2 2 2 2

-scission3 2 2 2 2 2 2

-scission3 2 2 2 2

HCH CH CH CH CH CH CH

CH CH CH CH CH CH CH

CH CH CH CH CH CH CH

CH CH CH CH CH

β α

ββ

ββ

−− − − − − − →

− − − − − −

→ − − − − + =

→ − − + =

-scission3 2 2 CH CH CH

β→ + =

32

Low Temperature Chemistry

• Relevant to autoignition (knock, diesel ignition delay, HCCI, …)

• Fuel decomposition difficult to happen.

• Oxidation rate controlled by fuel-like species

• Fuel structure important

high temp chemistry

Low temp chemistry

(Westbrook, Combust Flame 2009)

33

Ketohydroperoxide – chain branching agent for low temperature oxidation

Low temperature chain branching


34

Fuel structure is important for low temperature chemistry

• Difficult to proceed the pathways as in n-heptane oxidation• Alternative pathways have high energy barriers


35

• Low temperature chain branching produces modest temperature increase (< 200 K).

• Main combustion not resulted because ROO decompose at higher temperature

• Further reactions produce low reactivity species (olefins, HO2, cyclic ethers etc.)

• Temperature increase overall reactivity decrease: negative temperature dependence.

• HO2 and H2O2 accumulate during NTC

• Further increasing temperature and pressure causes H2O2 to decompose and triggers hot ignition and main combustion.

Negative Temperature Coefficient Behaviour

36

2ROO R O→ +

2 2 2 2 2HO HO H O O+ → +

2 2H O M OH OH M+ → + +

2 2R O olefin HO+ → +

propane explosion limit

Evidence of NTC Behaviour

37

Oscillatory cool flame in acetaldehyde oxidation

(Glassman & Yetter, Combustion 2008)(Griffith & Sykes, Proc Royal Soc A 1989)

• Also low temperature heat release, first stage ignition etc. in compression ignition processes.

Fundamental Experiments for Studying Combustion Chemistry• Ignition delay measurement

• Shock tube• Rapid compression machine (RCM)

• Species concentration measurement• Closed reactor• Flow reactor• Jet stirred reactor (JSR)

• Flame measurement• Laminar flame speed• Flame structure

• Alternative method • Motored engine

• No perfect experiment – all fundamental experiments attempt to reproduce certain aspect of practical process.

38

Goldsborough et al. Prog Energy Combust Sci 2017

• Laminar flame experiment < 10 bar

• Different time scales

Fundamental Experiments for Studying Combustion Chemistry

39

• Target• Determine accumulated results from a system of elementary reactions• Chemical kinetics is derived (or validated) via modelling.

• Common challenge • Mixture preparation• Thermal/compositional inhomogeneity • Temperature measurement• All needed for accurate modelling

• Common technique – dilution

Fundamental Experiments for Studying Combustion Chemistry

40

Working Principle of Closed Reactor

• Constant volume, homogeneous, closed, reactors • Theory

• Challenge: mixture preparation, temperature and composition homogeneity

• Hull slow reacting H2/O2 approach for elementary reaction investigation.

• Not used for mechanism level investigations, but as the model for simulating shock tube and RCM experiments.

,

( / ) ( )

[ ]( )

u i i ii i

i P i ui

Q V R T hdTdt X c R

ω ω+ −=

−

∑ ∑∑

ω= [ ]i id X

dt

41

Ignition Delay Measurement

• Shock tube and RCM• Single-shot, transient process. • Similar principle to closed reactor with fast mixture preparation

• Single outcome from complex process.• Species concentrations are usually difficult to measure.

42

Fuel/oxidizermixture

Shock wave ormoving piston

2( , , , , %)fuel T P Oτ φ

Working Principle of Shock Tube

• Measure ignition delay at higher temperatures• Step change in temperature – rapid heating• Short ignition delay (≤ 5 ms, little heat loss)

• Challenge: boundary effect – development of turbulent boundary

43

Available test time @ P5, T5

Tsang & Lifshitz, Ann. Rev Phys Chem, 1990

Working Principle of Shock Tube

44

Working Principle of RCM

• One stroke process: compress and wait.• Measure ignition delay at low temperatures (close to engine autoignition)• Slow compression cf. shock tube, long ignition delay (10 – 100 ms)

(Chen et al. Combust Flame 2017)

ln1

C

o

T CT

o

pdTT p

γγ

= −

∫45

(Sung et al. Prog Energy Combust Sci 2014)

Working Principle of RCM

Challenge • Heat loss• Boundary vortex • Reaction during compression

• Treated in modelling

46

Typical Results from Shock Tube and RCM

• Ignition delay measured based on pressure & OH emissions.

• Modelled with closed homogeneous reactor

• Variation made to take into account “facility effect”

(Sarathy et al. Proc Combust Inst 2015)

ln ( ln ) au

EkR T

τ

− ∝

47

Species Concentration Measurement

• Jet Stirred Reactor and Flow Reactor• Steady flow process •• More details about reaction chemistry • For more comprehensive model validation.

( , , , , )i resdY fuel T P φ τ

48

Working Principle of JSR

• Concept: continuous-stirred tank reactor• Steady flow• Simple conservation equations

ω = − , ,( )i i i out i inMWV m Y Y

= −∑ ∑

i i i iout in

Q Yh Yhm

m Yi,in hi,in

m Yi,out hi,out

CVQ

Control surface

Control volume

Yi T P V

49

Working Principle of JSR

Challenge • Mixture preparation: instantaneous & complete mixing • Residence time distribution

τ ρ= /resident V m

Prob

abili

ty

residence time(Dagaut et al. J Phys E 1986)

50

Typical Results of JSR

• Sample collected at reactor exit. • Product analysis – gas

chromatography and more advanced.

• Temp measurement challenge• Modelled with CSTR

7 14 3C H O

n-heptane oxidation

Ketohydroperoxide detected for the 1st time!

(Herbinet et al. Combust Flame 2012)

(1 bar, res time 2s)

51

n-C7 ketohydroperoxide

Low temperature chain branching


52

Working Principle of Flow Reactor

• Concept: plug flow reactor• Steady, turbulent flow • Negligible diffusion along flow• Flow distance reaction time

Flow

∆x

x

CVFlow

∆x

x

Flow

∆x

x

CV

"

0i i x xi P P P

h dY ddT Qdx c dx c dx mc

υ υ Ρ+ + + =∑

0i i ix

dY MWdx

ωρυ

− =

53

Challenge• Mixture preparation: where

reaction starts?• Preheating vs. premixing• Fast mixing

0 100 200 300 400 500 600 700 800 900 1000

Reactor distance from mixer (mm)

0.066

0.068

0.07

0.072

0.074

0.076

0.078

0.08

0.082

Mol

e fra

ctio

n of

CO

2

CO2

(Lu et al. Meas Sci Tech 2017)

Working Principle of Flow Reactor

54

Typical Results of Flow Reactor

• Sample collected along flow.• Product analysis – gas chromatography

and more advanced.• Modelled with PFR• Turbulent vs. laminar flow reactor

(Yuan et al, Combust Flame 2019)

iso-octane oxidation(900 K, 10 bar, total res time 0.27s)

550 600 650 700 750 800 850 900

0

1000

2000

3000

4000

5000

6000

CO

mol

e fra

ctio

n (p

pm)

NC7H16 (a) exp.sim.LLNL

sim.NUI

550 600 650 700 750 800 850 900

Flow reactor nominal temperature (K)

(Mehl, PCI 2011)(Zhang, C&F, 2016)

(10 bar, res time 0.21s)

55

How to Build a Chemical Kinetic Model?

• Thermochemistry data • List of species••

• Reaction mechanism • Framework of reaction pathways• Arrhenius parameters of elementary reactions

• Transport data • Viscosity, thermal conductivity, diffusivity • For reactors where transport is important

• Mostly hand made to date

, ( )p ic T, , , ...dH dU dS dG

56

How to Build a Comprehensive, Detailed, Reaction Mechanism?

• Framework of reaction pathways• Rate rules – from modeller’s knowledge

• Arrhenius parameters of elementary reaction• Kinetic experiments• Ab initio calculation• Analogy from similar reactions• Can be of high uncertainty

57

Rate Rules for Building a Reaction Mechanism

High temperature oxidation

low temperature oxidation (high pressure)

(Westbrook et al, Combust Flame 2009)

58

Chemical Kinetic Models Developed To Date

• H2• Hydrocarbons

• C1-C4• C5 above• Alkanes• Olefins• Cycloalkanes• Aromatics

• Biofuels• Alcohols• FAME

…• Surrogate fuels

(Lu & Law, Prog Energy Combust Sci 2009)

59

How to Validate a Chemical Kinetic Model?

• Fundamental combustion experiments• Ignition delay• Species concentration• Laminar flame speed

• Validation outcome determined by both physical model of the process and chemistry involved (thermo, mech, transport).

• Validation is against accumulative results • Good agreement with combustion experiments do not mean the model must

be correct.

60

Validation of Kinetic Models with Fundamental Combustion Experiments

• Often one model is validated against one set of (own) experiment• “reasonable” “good” “excellent”…

• Comprehensive models (validated against most previous experiments) are rare.

• Predictability of models is often poor.

• Optimization of models within its uncertainty• What is the uncertainty? – uncertainty quantification (UQ)• What is most effective tweak? – sensitivity analysis (SA)

61

Model Reduction

62

Sensitivity Analysis• Local sensitivity – how a small change of individual parameters (rate

coefficients, thermochemical properties, reaction conditions etc) affects model outcome (ignition delay, species concentration, flame speed, etc.)

• e.g. sensitivity of species concentrations to rate coefficients can be written as

• normalized form

• Solution • brute-force method• decoupled direct method (ddm) (used by Chemkin and other solvers)

1 1

1

1

...

...

mi

jn n

n

c ck k

ck

c ck k

∂ ∂ ∂ ∂ ∂∂

= = ∂ ∂ ∂ ∂

∂ ∂

ck

i = 1, …, n speciesj = 1, …, m reactions

lnln

j i i

i j j

k c cc k k

∂ ∂=

∂ ∂

63

• Global sensitivity – how simultaneous, small changes of all parameters affect the model outcome?

• Require sampling in the parameter uncertainty space (hypercube)• Monte Carlo method etc.

• More computational expensive. Less commonly used.

64

Sensitivity Analysis

Uncertainty Quantification

• Origin and propagation• Could be estimated from local sensitivity

• only accurate for small range around the nominal values• Hypercube of uncertainty• Deterministic or probabilistic treatment of uncertainty (Global

approach)• Uncertainty constraining by experiments

65

2

2 2

1( ) ( )

mi

i jj j

cc kk

σ σ=

∂= ∂ ∑

Model Reduction

Target• Reduce computation time• Reduce stiffness in computation

Steps• Eliminate unimportant species and reactions• Lump similar species and reactions• Time scale analysis to decouple short-lived species and fast reactions

from the rest

66

Elimination Unimportant Species & Reactions

• Eliminate species first, then reactions.

• How to identify them?• Jacobian analysis• Directed Relation Graph (DRG)

67


Jacobian analysis• From the local sensitivity analysis, a fractional change in the

concentration of an important species (cd) can be caused by a fractional change in the concentration of a given species (ci)

• Sum the impact on all important species for species i

• Species with an impact smaller than the defined threshold can be eliminated.

lnln

d

i

cc

∂∂

2lnln

d

d i

cc

∂ ∂

∑

68

Jacobian analysis• Similarly, impact of a fractional change of a given rate coefficient (kj) on

the concentrations of important species (cd) can be summed:

• Reactions with an impact smaller than the defined threshold can be eliminated

• Jacobian analysis is rigorous but computationally inefficient.

2lnln

d

d j

ck

∂ ∂

∑

69


Directed Relation Graph (Lu & Law) • Select A as an important species, e.g. fuel• A depends on B, B & D interdepend• B & D must be kept • C, E & F can be eliminated.

(Lu & Law, Prog Energy Combust Sci, 2009)

70


Directed Relation Graph • Relative error by eliminating B

chemical coefficient of A in reaction jnet reaction rate of j=1 if reaction j involves B=0 otherwise

• To keep B, rAB must be larger than the user defined error threshold

ABr ε>71


,

,

A j j Bjj

ABA j j

j

v qr

v q

δ≡∑∑

,A jv

jq

Bjδ

(Lu PECS 2009)

Further development of DRG method

• DRG with error propagation (DRGEP, Pepiot & Pitsch)

• More aggressive error propagation

• DRG aided sensitivity analysis (DRGASA, Sankaran et al.)

• DRG + sensitivity Jacobian analysis• Right figure: LLNL n-heptane

mechanism (561 species, 2539 reactions reduced to 78 species 359 reactions)

72


(Lu PECS 2009)

Lumping

• Isomers have similar thermal and transport properties; many also have similar reaction pathways and reaction rates.

• Lumping reduces the number of species, reactions, diffusion coefficients.

• Lumping is currently done via modeler’s knowledge, automatic lumping not yet available.

73

(Lu PECS 2009)

Time Scale Analysis• Kinetic model after eliminating

redundant species & reactions, and species lumping, can be still too large to be used with CFD.

• Short-lived species and fast reactions can be identified and treated separately.

74(Maas & Pope Combust Flame 1992)

75

Concepts• Quasi steady state approximation• Partial equilibrium approximation

Approaches• Jacobian analysis• Computational Singular Perturbation (CSP)• Intrinsic Low Dimension Manifold (ILDM)

Time Scale Analysis

Quasi steady state approximation (QSSA)• Net production rate of short lived species can be approximated to be zero.

• QSSA Example

• Apply QSSA to [N]

0iQSS

dcdt

=

( ) ( )i jc f c j i= ≠

2

2

1

2

k

k

O N NO N

N O NO O

+ → +

+ → += −1 2 2 2

[ ] [ ][ ] [ ][ ]d N k O N k N Odt

1 2

2 2

[ ][ ][ ]

[ ]ssk O NNk O

=

76

=[ ]

0ssd N

dt

Time Scale Analysis

Partial equilibrium approximation• Fast reactions reach local equilibrium while slower reactions are progressing.

• Apply PE

A B C DA B C Dν ν ν ν+ +

ν νν

ν ν∆= = 0

[ ] [ ] ( / )[ ] [ ]

C D

A BC P uC DK K P R TA B

0exp( / )p T uK G R T= −∆

77

Time Scale Analysis

Partial equilibrium approximation• Example

much faster than

• Apply PE

78

+ ++ ++ +

2

2

2 2

R5 R2

R3

H O OH OO H OH HOH H H O H

2H OH M H O M+ + +

= = = 25 2 32 2 2

[ ][ ][ ][ ] [ ][ ]( ) ( ) ( )[ ][ ] [ ][ ] [ ][ ]C C C

H O HOH O OH HK T K T K TH O O H OH H

1/232 2 2

5 2 3 22

[ ][ ][ ] ( )

[ ]C C CO HH K K KH O

=

Time Scale Analysis

How to tell which species to apply QSSA and which reaction to apply PEA?• Modeller’s knowledge of chemical kinetics

• Jacobian analysis

• Perturbation

where

• Each eigenvalue of the Jacobian matrix is associated with a different time scale in the solution• larger lambda, smaller time scale

( )ddt

=c f c

1 21 2

...i i i i nnfx fx fx

c f f fc c ct c c c

∂∆ ∂ ∂ ∂= ∆ + ∆ + ∆ ∂ ∂ ∂ ∂

1 21 2( ) ... n

tt ti nc t C e C e C e

λλ λ∆ = + + +

fxi i ic c c= + ∆

( )( ) t tt e∆ = Jc A

79

Time Scale Analysis

J ∂=∂fc

Computational singular perturbation (CSP) (Lam & Goussis)• CSP method automatic orders of time scales by rewriting the reaction rate

with a new set of basis vectors so that reactions can be grouped into different time scales (modes).

for m reactions for n species

• Fast modes with short time scales are exhausted during iteration.

• CSP does this without knowledge of chemistry.• CSP is a local analysis and needs to be performed at various time points along

the process.• CSP can be used for other purposes, e.g. removing redundant species and

reactions but is expensive for computation.

1

m

j jj

d qdt =

=∑c v1

ni

ii

ddt =

⇒ =∑c a d

80

Time Scale Analysis

Intrinsic Low-Dimensional Manifold (ILDM) (Maas & Pope)

• Concept of slow manifold in combustion processes

• Numerous parameters and processes in the state space of combustion reaction

• Each variable represents a trajectory in the space

• Certain trajectories are slower and more dominant than others

(Maas & Pope Combust Flame 1992) 81

Time Scale AnalysisTrajectories in the state space for a CO/H2/air system projected into the CO2/H2O plane.

(Maas & Pope Combust Flame 1992) 82

Time Scale Analysis

Two-dimensional manifold for H radical Two-dimensional manifold for OH radical

• Intrinsic Low-Dimensional Manifold (ILDM) • ILDM method searches for slow manifolds to reduce the dimensions of

problem.• The number of dimension is prescribed, based on time scale analysis similar

to CSP (fast and slow modes).• Primary purpose of ILDM is for CFD uses. Lookup table generated for the low

dimensions. • Not a method to generate reduced kinetic model. Species/reactions

associated with the low dimensional manifold do not compose a validated chemical model of the reaction.

83

Time Scale Analysis

A Flow Chart of Model Reduction

84(Lu & Law, Prog Energy Combust Sci 2009)

Concluding Remarks

• Experiment vs. modelling• Physical understanding (chemical insight) is important • Comprehensive, detailed kinetic models are important.

85

Combustion Chemistry and Model ReductionIntroductionOverview of the TopicOverview of the TopicStructure of This LectureStructure of This LectureReviewChemical KineticsReaction RateReaction RateMultiple-Reaction SystemHow to Study Combustion ChemistryElementary Level InvestigationKinetic Experiment of Elementary ReactionsKinetic Experiment of Elementary ReactionsKinetic Experiment of Elementary ReactionsKinetic Experiment of Elementary ReactionsKinetic Experiment of Elementary ReactionsKinetic Experiment of Elementary ReactionsHull Slow Reacting H2/O2 Experiment Hull Slow Reacting H2/O2 Experiment Comparison of Hull and Tully ResultsTransition State Theory (TST)Application of TSTApplication of TSTHow to Model “Fall-Off” Behaviour?How to Model “Fall-Off” Behaviour?More Theoretical Treatment of TSTMore Theoretical Treatment of TSTMore Theoretical Treatment of TSTMechanism Level InvestigationTypical Alkane Oxidation Chemistry High Temperature ChemistryLow Temperature ChemistryLow Temperature ChemistryLow Temperature ChemistryNegative Temperature Coefficient BehaviourEvidence of NTC BehaviourFundamental Experiments for Studying Combustion ChemistryFundamental Experiments for Studying Combustion ChemistryFundamental Experiments for Studying Combustion ChemistryWorking Principle of Closed ReactorIgnition Delay MeasurementWorking Principle of Shock TubeWorking Principle of Shock TubeWorking Principle of RCMWorking Principle of RCMTypical Results from Shock Tube and RCMSpecies Concentration MeasurementWorking Principle of JSRWorking Principle of JSRTypical Results of JSRLow Temperature ChemistryWorking Principle of Flow ReactorSlide Number 55Typical Results of Flow ReactorHow to Build a Chemical Kinetic Model?How to Build a Comprehensive, Detailed, Reaction Mechanism?Rate Rules for Building a Reaction MechanismChemical Kinetic Models Developed To DateHow to Validate a Chemical Kinetic Model?Validation of Kinetic Models with Fundamental Combustion ExperimentsModel ReductionSensitivity AnalysisSensitivity AnalysisUncertainty QuantificationModel ReductionElimination Unimportant Species & ReactionsElimination Unimportant Species & ReactionsElimination Unimportant Species & ReactionsElimination Unimportant Species & ReactionsElimination Unimportant Species & ReactionsElimination Unimportant Species & ReactionsLumpingTime Scale AnalysisTime Scale AnalysisTime Scale AnalysisTime Scale AnalysisTime Scale AnalysisTime Scale AnalysisTime Scale AnalysisTime Scale AnalysisTime Scale AnalysisTime Scale AnalysisA Flow Chart of Model ReductionConcluding Remarks

COMBUSTION CHEMISTRY AND MODEL REDUCTION · 2018-12-15 · • Combustion chemistry studies how...

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