On The Model Reduction for Chemical and Physical Kinetics · On The Model Reduction for Chemical...

1 Introduction

2 gRRM

3 Entropy Prod. Ana.

4 n-heptane/aircomplex dynamics

5 Conclusion

On The Model Reduction for Chemical andPhysical Kinetics

Mahdi Kooshkbaghi

Aerothermochemistry and Combustion Systems LaboratorySwiss Federal Institute of Technology

LAVLaboratorium für Aerothermochemie und VerbrennungssystemeAerothermochemistry and Combustion Systems Laboratory

PhD Thesis Presentation18th August 2015

1 / 44

1 Introduction

2 gRRM



5 Conclusion


Outline

Introduction and Motivation

The global Relaxation Redistribution Method

Entropy production analysis for mechanism reduction

n-heptane/air complex dynamics

Summary and future works

2 / 44

1 IntroductionMotivation

Chem. Kin. Model Reduction

Thesis Outline

2 gRRM



5 Conclusion


Challenges and Motivation

I Every Mesh Grid, Every Time Step1. Mass Conservation Equation2. Momentum Conservation Equations3. Energy Conservation Equation4. ns PDEs for temporal evolution of ns species

I Stiffness/ Non-linearity

3 / 44



Thesis Outline

2 gRRM



5 Conclusion




I Size of detailed chemical kinetics

Lu, T., & Law, C. K. (2009). Prog. Energ. Combust., 35(2), 192-215.

CH4 C7H16 C10H22 C12H26 C20H42-2Species 53 561 940 1282 7200

Reactions 325 2539 3878 5030 31400

Sizes of detailed reaction mechanisms forsample hydrocarbons


3 / 44



Thesis Outline

2 gRRM



5 Conclusion





Goussis, D. A., & Maas, U. (2011). In TurbulentCombustion Modeling (pp. 193-220). 0 0.05 0.1 0.15 0.2

t [s]

600

800

1000

1200

1400

1600

1800

2000

2200

2400

T [

K]

0 0.05 0.1 0.15 0.21×10-15

2×10-15

3×10-15

4×10-15

Tim

esca

le [

s]T

n-heptanen

s = 561 τ

fast

T0 = 650 K

P = 1atmφ = 1.0

3 / 44



Thesis Outline

2 gRRM



5 Conclusion


Computational Cost Reduction for Chemical Kinetics

A Time scale analysis:Describe chemistry using fewer variables.X QSSA: Bodenstein (1913)X CSP: Lam & Goussis (1989)X ILDM: Maas & Pope (1992)X MIM: Karlin & Gorban (1991)

X RRM: Kooshkbaghi et al. (2014)

B Conventional Reduction Methodology:Generate smaller skeletal mechanisms from the detailedmechanism by systematically removing unimportant speciesand reactions.X CSP: Massias et al. (1999)X DRG: Lu & Law (2005)X PFA: Sun et al. (2010)

X Entropy Production Analysis: Kooshkbaghi et al. (2010)

C Storage and Retrieval methodsX ISAT: Pope (1997)X PRISM: Tonse (2003)

4 / 44



Thesis Outline

2 gRRM



5 Conclusion


Thesis Outline

Const

ruct

ing l

ow

-dim

ensi

onal

m

anif

old

and a

ppli

cati

on

* Invariance equation

* Film extension of dynamics

* Global Relaxation Redistribution method

* Application in Hydrogen/air combustion

* Equilibrium and Quasi Equilibrium

* Spectral Quasi Equilibrium manifolds

* Application in Hydrogen/air, Syngas/air

and Methane/air combustion

Chem

ical

Kin

etic

sP

hysi

cal

Kin

etic

s

* Inଏnite-dimensional dynamical system

* Boltzmann equation

* Analytical solution of invariance condition

* non-perturbative reduced hydrodynamic manifold

Skel

etal

mec

han

ism

gen

erat

ion

and a

ppli

cati

on

* Entropy Production Analysis

* Most-contributing reactions

* Skeletal mechanism generation for

large fuels

* Complex dynamics of heavy hydrocarbon

* Bifurcation analysis

* Reactions supporting and opposing

critical behaviour

Ch. 2

Ch. 3

Ch. 6

Ch. 4

Ch. 5

Thesis in a nutshell

5 / 44

1 Introduction

2 gRRMSIM Def.

MIM

Constructing Manifold

Using Manifold



5 ConclusionII. The global Relaxation RedistributionMethod

6 / 44

1 Introduction

2 gRRMSIM Def.

MIM


Using Manifold



5 Conclusion


Concept of Slow Invariant Manifold

Classification of Systems

I Autonomous system

I Cauchy-Lipschitz

dNdt

= f(N)

f : Rns ⊃ S→ Rns

N ∈ S, t ∈ Tφ t

t∈T is called a flow where

Nt = φtN0

Neq is a unique fixed point.

I In dynamical system{T,S,φ t}, U ⊂ S is invariantmanifold (set) if N0 ∈ U then∀t: φ tN0 ∈ U

I Slow Invariant Manifold (SIM)7 / 44

1 Introduction

2 gRRMSIM Def.

MIM


Using Manifold



5 Conclusion


Concept of Slow Invariant Manifold

Classification of Systems

I Autonomous system

I Cauchy-Lipschitz

dNdt

= f(N)

f : Rns ⊃ S→ Rns

N ∈ S, t ∈ Tφ t

t∈T is called a flow where

Nt = φtN0

Neq is a unique fixed point.

I In dynamical system{T,S,φ t}, U ⊂ S is invariantmanifold (set) if N0 ∈ U then∀t: φ tN0 ∈ U

I Slow Invariant Manifold (SIM)

0 0.2 0.4 0.6 0.8 1

[S]

0

0.2

0.4

0.6

0.8

1

[ES

]

t = 0.1

t = 0.2

t = 0.4

t = 0.3

t = 1.0t = 2.0

t = 4.0

t = 6.0t =

10.0

t =

20.0

S + Ekf1−−⇀↽−−kr1

ES kcat−−→ E + P

k1f = k1r = kcat = 1

7 / 44

1 Introduction

2 gRRMSIM Def.

MIM


Using Manifold



5 Conclusion


Multiscale Dissipation

Davis-Skodje System (J. Chem. Phys. 1999):dxdt =−x

dydt =−γy+ (γ−1)x+γx2

(1+x)2

γ � 1,γ = 5

0 0.2 0.4 0.6 0.8 1−2

−1.5

−1

−0.5

0

0.5

1

1.5

2

y

x

8 / 44

1 Introduction

2 gRRMSIM Def.

MIM


Using Manifold



5 Conclusion





(1+x)2

γ � 1,γ = 5

0 0.5 1Ȃ3

Ȃ2

Ȃ1

0

1

2

3

y

x

xy

0.35 2.5 50

0.5

1

1.5

2

t

Appro

xim

ate

Locati

on w

here

Traje

cto

ry M

eet

Th

e S

low

Man

ifold

8 / 44

1 Introduction

2 gRRMSIM Def.

MIM


Using Manifold



5 Conclusion





(1+x)2

γ � 1,γ = 5

x = x0e−t

y =x0e−t

1+ x0e−t +��

�:0C(x0,y0)e−γt

yslow =x

1+ x

8 / 44

1 Introduction

2 gRRMSIM Def.

MIM


Using Manifold



5 Conclusion





(1+x)2

γ � 1=⇒ ySIM = x

1+x

0 0.2 0.4 0.6 0.8 1 1.2 1.4 1.50

0.2

0.4

0.6

0.8

1

y

x

Sample T rajectories

Sample T rajectories

SlowMotion

F astMotion

SIM 0 1 2 3 4 50.00.20.40.60.81.0

x

0 1 2 3 4 5time

0.00.51.01.52.0

y

DetailedReduced Model: y=x/1+x

8 / 44

1 Introduction

2 gRRMSIM Def.

MIM

RRM Algorithm


Using Manifold



5 Conclusion


Method of Invariant Manifold (MIM)

f(N(ξ )) = f(N(ξ ))‖TW+ f(N(ξ ))⊥TW

f(N(ξ ))‖TW= Pf(N(ξ ))

f(N(ξ ))⊥TW= ∆ = (I−P)f(N(ξ ))

Invariance Condition

∆ = 0, ξ ∈ Ξ

MIM: The slow invariant manifoldis the stable solution of the film

extension of dynamics:

dN(ξ )

dt= ∆

Gorban, A. N., & Karlin, I. V. (2004). Lect. Notes Phys., 660.

9 / 44

1 Introduction

2 gRRMSIM Def.

MIM

RRM Algorithm


Using Manifold



5 Conclusion


Relaxation Redistribution Method (RRM)

1. Find/Choose the slow parameterization variables

2. Construct the initial guess of slow manifold

3. Relax all the points on the initial manifold

4. Points moving toward local equilibrium manifold

5. Redistribute back to neutralize slow motionI Redistribution : Interpolation for interiorI Redistribution : Extrapolation for missing point

Chiavazzo, E., & Karlin, I. V. (2011). Phys. Rev. E., 83(3), 036706.Kooshkbaghi, M. et al., (2014). J. Chem. Phys., 141(4), 044102.

10 / 44

1 Introduction

2 gRRMSIM Def.

MIM

RRM Algorithm


Using Manifold



5 Conclusion


RRM Manifold ConstructionSingular perturbed system*

dxdt = 2− x− y

dydt = γ(

√x− y)

γ � 1

Relaxation Step

0 0.2 0.4 0.6 0.8 1−0.5

0

0.5

1

y

x

Initial grid

1 Step Relaxation, δt = 0.07

*Tsoumanis, A. C. et al., (2012). New J. Phys., 14(8), 083037.Kooshkbaghi, M. et al., (2014). J. Chem. Phys., 141(4), 044102.

11 / 44

1 Introduction

2 gRRMSIM Def.

MIM

RRM Algorithm


Using Manifold



5 Conclusion



dxdt = 2− x− y

dydt = γ(

√x− y)

γ � 1

Redistribution Step

0 0.2 0.4 0.6 0.8 1−0.5

0

0.5

1

y

x

Initial grid1 Step Relaxation, δt = 0.071 Step RRM


11 / 44

1 Introduction

2 gRRMSIM Def.

MIM

RRM Algorithm


Using Manifold



5 Conclusion



dxdt = 2− x− y

dydt = γ(

√x− y)

γ � 1

I ILDM manifold is neither invariant nor slow for 0≤ x. 0.7.

0 0.2 0.4 0.6 0.8 1

−0.5

0

0.5

1

y

x

Sample Trajectories

RRM ManifoldILDM

Trajectory initialize on the RRM manifold

Trajectory initialize on the ILDM manifold


11 / 44

1 Introduction

2 gRRMSIM Def.

MIM


Dimensionality

gRRM

RCCE

Using Manifold



5 Conclusion


Dimensionality issues

I Computational costI Manifolds are represented on a gridI Retrieving data of high dimensional tables, imposes

restrictions on the dimension

=⇒ Target : 2D/3D manifold

I Dimension of SIMsI SIMs usually limited to a small neighborhood around

equilibrium

=⇒ How to extend it further to cover the states all the wayto the fresh mixture?

Pope, S. B. (2013). Proc. Combust. Inst., 34(1), 1-31.

12 / 44

1 Introduction

2 gRRMSIM Def.

MIM


Dimensionality

gRRM

RCCE

Using Manifold



5 Conclusion



I Computational costI Manifolds are represented on a gridI Retrieving data of high dimensional tables, imposes

restrictions on the dimension

=⇒ Target : 2D/3D manifoldI Dimension of SIMs

I SIMs usually limited to a small neighborhood aroundequilibrium

=⇒ How to extend it further to cover the states all the wayto the fresh mixture?

Pope, S. B. (2013). Proc. Combust. Inst., 34(1), 1-31.

12 / 44

1 Introduction

2 gRRMSIM Def.

MIM


Dimensionality

gRRM

RCCE

Using Manifold



5 Conclusion



=⇒ How to extend it further to cover the states all the way tothe fresh mixture?I Construct the Slow invariant manifold and extend via

prolongation with linear extrapolation*

��

��

��

*Bykov, V., & Maas, U. (2007). Proc. Combust. Inst., 31(1), 465-472.

13 / 44

1 Introduction

2 gRRMSIM Def.

MIM


Dimensionality

gRRM

RCCE

Using Manifold



5 Conclusion



=⇒ How to extend it further to cover the states all the way tothe fresh mixture?I Construct the Slow invariant manifold and extend via

prolongation with linear extrapolation*��

��

��

I Construct the initial grid which covers the admissiblesolution space and refine it via RRM, (global RRM**)*Bykov, V., & Maas, U. (2007). Proc. Combust. Inst., 31(1), 465-472.**Kooshkbaghi, M. et al., (2014). J. Chem. Phys., 141(4),044102.

13 / 44

1 Introduction

2 gRRMSIM Def.

MIM


Dimensionality

gRRM

RCCE

Using Manifold



5 Conclusion


global RRM (gRRM)

1. Construct 2D initial grid (Ξ will be defined later)

2. Find the boundaries of initial grid

2.8 3 3.2 3.4 3.6 3.80

0.2

0.4

0.6

0.8

1

1.2

1.4

1.6

ξ 2

ξ1

Equilibrium

Fresh Mixture

3. Relax interior points

4. Redistribute back points on initial grid via interpolation of scatteredgrid

14 / 44

1 Introduction

2 gRRMSIM Def.

MIM


Dimensionality

gRRM

RCCE

Using Manifold



5 Conclusion


global RRM (gRRM)

The gRRM nd-dimensional manifold is:I The nd-dimensional SIM +I The extension of SIM to the far from equilibrium states

=⇒ The initial grid is important both for convergence andaccuracy of extension=⇒ In this work the initial grid is found based on notation ofQuasi Equilibrium Manifold (QEM)

15 / 44

1 Introduction

2 gRRMSIM Def.

MIM


Dimensionality

gRRM

RCCE

Using Manifold



5 Conclusion


global RRM (gRRM)


=⇒ The initial grid is important both for convergence andaccuracy of extension

=⇒ In this work the initial grid is found based on notation ofQuasi Equilibrium Manifold (QEM)

15 / 44

1 Introduction

2 gRRMSIM Def.

MIM


Dimensionality

gRRM

RCCE

Using Manifold



5 Conclusion


global RRM (gRRM)


=⇒ The initial grid is important both for convergence andaccuracy of extension=⇒ In this work the initial grid is found based on notation ofQuasi Equilibrium Manifold (QEM)

15 / 44

1 Introduction

2 gRRMSIM Def.

MIM


Dimensionality

gRRM

RCCE

Using Manifold



5 Conclusion


Initial Grid via QEM (CEM)

min Gs.t. BN = ξ

B = [E Bd] and ξ = [ξ eξ

d]

I ne×ns elemental constraints matrix, E

EN = ξe

ξe is specified by the initial composition

Eji : number of atoms of element j in species i

I nd×ns constraints matrix Bd

(Bd)N = ξd

ξd : slow parameters

Bd : rows define the linear combination of N as the slow constraints

16 / 44

1 Introduction

2 gRRMSIM Def.

MIM


Dimensionality

gRRM

RCCE

Using Manifold



5 Conclusion


Manifold parametrization Ξ / Proper choose for extension

How to Choose a good set of constraints Bd?

I Rate-Controlled Constrained-Equilibrium method (RCCE)

I constraints for H2/air combustion with ns = 9 species and nr = 21

elementary reactions*Reduce Parameter** H2 N2 H O OH O2 H2O HO2 H2O2

ξ1=Total Mole 1 1 1 1 1 1 1 1 1ξ2=Active Valence 0 0 1 2 1 0 0 0 0ξ3=Free Oxygen 0 0 0 1 1 0 1 0 0

Bd =

1 1 1 1 1 1 1 1 10 0 1 2 1 0 0 0 00 0 0 1 1 0 1 0 0

*Li, J. et al., (2004). Int. J. Chem. Kin., 36(10), 566-575.**Tang, Q., & Pope, S. B. (2004). Combust. Theory Model., 8(2), 255-279.

17 / 44

1 Introduction

2 gRRMSIM Def.

MIM


Dimensionality

gRRM

RCCE

Using Manifold



5 Conclusion


gRRM for H2/air combustion

ns = 9 Species, T0 = 1500K, P = 1atm, φ = 1.0

2.83

3.23.4

3.63.8

0

0.5

10

0.2

0.4

0.6

0.8

1

ξ1

RCCE

ξ2

NH

2

2.83

3.23.4

3.63.8

0

0.5

10

0.2

0.4

0.6

0.8

1

ξ1

RRM

ξ2

NH

2

ξ1=Total Mole, ξ2=Active ValenceSlight improvements in main species

�, fresh mixture; ? , equilibrium point; − detailed kinetics path, Coloredsurfaces, Manifolds

18 / 44

1 Introduction

2 gRRMSIM Def.

MIM


Dimensionality

gRRM

RCCE

Using Manifold



5 Conclusion




2.53

3.54

0

0.5

10

0.05

0.1

0.15

0.2

0.25

ξ1

RCCE

ξ2

NO

2.53

3.54

0

0.5

10

0.05

0.1

0.15

0.2

0.25

ξ1

RRM

ξ2

NO

ξ1=Total Mole, ξ2=Active ValenceSmall improvements in main radicals


18 / 44

1 Introduction

2 gRRMSIM Def.

MIM


Dimensionality

gRRM

RCCE

Using Manifold



5 Conclusion




2.8 3 3.2 3.4 3.6 3.8

0

0.5

10

0.05

0.1

0.15

0.2

ξ1

RCCE

ξ2

NOH

2.8 3 3.2 3.4 3.6 3.8

0

0.5

10

0.05

0.1

0.15

0.2

ξ1

RRM

ξ2

NOH

ξ1=Total Mole, ξ2=Active ValenceSmall improvements in main radicals


18 / 44

1 Introduction

2 gRRMSIM Def.

MIM


Dimensionality

gRRM

RCCE

Using Manifold



5 Conclusion




2.8 3 3.2 3.4 3.6 3.80

0.5

1

0

1

2

3

x 10−4

RCCE

ξ1ξ2

NHO

2

2.8 3 3.2 3.4 3.6 3.80

0.5

1

0

0.2

0.4

0.6

0.8

1

x 10−3

RRM

ξ1ξ2

NHO

2

ξ1=Total Mole, ξ2=Active ValenceLarge improvements for low-concentration radicals


18 / 44

1 Introduction

2 gRRMSIM Def.

MIM


Using Manifold

T0 = 1500K

T0 = 1000K



5 Conclusion


H2/air auto-ignition

Adiabatic, constant pressure reactorT0 = 1500K, P = 1atm, φ = 1.0

2D manifold results

0 0.5 1 1.5

x 10−4

1500

2000

2500

3000

T[K

]

t [s]

Detailed

RCCE TM+AV

RRM 2D

0 0.5 1 1.5

x 10−4

0

0.01

0.02

0.03

YH

2

t [s]0 0.5 1 1.5

x 10−4

0

2

4

6x 10

−3

YH

t [s]

0 0.5 1 1.5

x 10−4

−0.01

0

0.01

0.02

0.03

YO

t [s]0 0.5 1 1.5

x 10−4

0

0.01

0.02

0.03

YOH

t [s]0 0.5 1 1.5

x 10−4

0

0.05

0.1

0.15

0.2

0.25

YO

2

t [s]

0 0.5 1 1.5

x 10−4

0

0.05

0.1

0.15

0.2

YH

2O

t [s]0 0.5 1 1.5

x 10−4

0

2

4

6x 10

−5

YHO

2

t [s]0 0.5 1 1.5

x 10−4

0

0.5

1

1.5x 10

−5

YH

2O

2

t [s] 19 / 44

1 Introduction

2 gRRMSIM Def.

MIM


Using Manifold

T0 = 1500K

T0 = 1000K



5 Conclusion




2D manifold results

0 1 2 3 4

x 10−4

1000

1500

2000

2500

3000

T[K

]

t [s]

RCCE TM+AV

RRM 2D

Detailed

0 1 2 3 4

x 10−4

0

0.005

0.01

0.015

0.02

0.025

0.03

YH

2

t [s]0 1 2 3 4

x 10−4

0

1

2

3

4

5x 10

−3

YH

t [s]

0 1 2 3 4

x 10−4

0

0.005

0.01

0.015

YO

t [s]0 1 2 3 4

x 10−4

−0.005

0

0.005

0.01

0.015

0.02

0.025

YOH

t [s]0 1 2 3 4

x 10−4

0

0.05

0.1

0.15

0.2

0.25

YO

2

t [s]

0 1 2 3 4

x 10−4

0

0.05

0.1

0.15

0.2

0.25

YH

2O

t [s]0 1 2 3 4

x 10−4

0

0.5

1

1.5x 10

−4

YHO

2

t [s]0 1 2 3 4

x 10−4

10−8

10−7

10−6

10−5

10−4

YH

2O

2

t [s] 20 / 44

1 Introduction

2 gRRMSIM Def.

MIM


Using Manifold

T0 = 1500K

T0 = 1000K



5 Conclusion




3D manifold results

0 1 2 3 4

x 10−4

1000

1500

2000

2500

3000

T[K

]

t [s]

RCCE TM+AV+FO

RCCE TM+AV

RRM 2D

RRM 3D

Detailed

0 1 2 3 4

x 10−4

0

0.005

0.01

0.015

0.02

0.025

0.03

YH

2

t [s]0 1 2 3 4

x 10−4

0

1

2

3

4

5

6x 10

−3

YH

t [s]

0 1 2 3 4

x 10−4

0

0.005

0.01

0.015

YO

t [s]0 1 2 3 4

x 10−4

0

0.005

0.01

0.015

0.02

0.025

0.03

YOH

t [s]0 1 2 3 4

x 10−4

0

0.05

0.1

0.15

0.2

0.25

YO

2

t [s]

0 1 2 3 4

x 10−4

0

0.05

0.1

0.15

0.2

0.25

YH

2O

t [s]0 1 2 3 4

x 10−4

0

1

2x 10

−4

YHO

2

t [s]0 1 2 3 4

x 10−4

10−8

10−6

10−4

10−2

YH

2O

2

t [s] 21 / 44

1 Introduction

2 gRRM

3 Entropy Prod. Ana.Entropy Production

Skeletal Mechanism Gen.


5 Conclusion

III. Entropy production analysis formechanism reduction

22 / 44

1 Introduction

2 gRRM




5 Conclusion


Entropy Production

23 / 44

1 Introduction

2 gRRM




5 Conclusion


Entropy Production

ns

∑i=1

ν′ikNi

ns

∑i=1

ν′′ikNi, k = 1, · · · ,nr

qk = qfk −qrk = kfk

ns

∏i=1

[Ni]ν ′ik − krk

ns

∏i=1

[Ni]ν ′′ik , k = 1, · · · ,r

The entropy production per unit volume

1V

dSdt

= Rc

nr

∑k=1

(qfk −qrk )ln(

qfkqrk

)The relative contribution of each reaction in total entropy production at time

t

rk(t) =Rc(qfk −qrk )ln

(qfkqrk

)1V

dSdt

Threshold for contributionrk(t)≥ ε%

24 / 44

1 Introduction

2 gRRM




5 Conclusion


Most-Contributing Reactions

n-heptane LLNL2 Mechanism (ns = 561, nr = 2539)*T0 = 650 K, P = 1 atm, φ = 1

ε = 5%

0 0.02 0.04 0.06 0.08 0.1 0.12 0.14 0.16 0.18 0.2

t [s]

600

1000

1400

1800

2200

2600

3000

3400

T [

K]

(a) (b)

(c)

ch2cho+o

2=ch

2o+co+oh

nc7h

16+o

2=c

7h

15-1/2/3/4+ho

2

c7h

14ooh

1-4=c

7h

14o

1-4+oh

(d)

nc7h

16+oh=c

7h

15-2+ho

2

nc7ket

24=nc

3h

7cho+ch

3coch

2+oh

nc7ket

35=c

2h

5cho+c

2h

5coch

2+oh

nc7ket

42=ch

3cho+nc

3h

7coch

2+oh

o2c

2h

4oh =oh+2ch

2o

(d)hocho+h =h

2+co

2+h

hocho+oh =h2o+co

2+h

hocho+oh =h2o+co

2+h

(a)(a)

(b)

(c)

nc7h

16+oh=c

7h

15-3+ho

2

c7h

14ooh

2-5=c

7h

14o

2-5+oh

c7h

14ooh

3-6=c

7h

14o

2-5+oh

co+o = co2

h+oh= h2o

*Curran, H. J., et al., (1998). Combust. Flame, 114(1), 149-177.25 / 44

1 Introduction

2 gRRM



nC7 H16 /air Ske. Mech.

Validation of Ske. Mech.


5 Conclusion


Generate Skeletal Mechanism

Most-contributing reactions

nc7h

16+o

2=c

7h

15-1/2/3/4+ho

2

c7h

14ooh

1-4=c

7h

14o

1-4+oh

nc7h

16+oh=c

7h

15-2+ho

2

nc7ket

24=nc

3h

7cho+ch

3coch

2+oh

nc7ket

35=c

2h

5cho+c

2h

5coch

2+oh

nc7ket

42=ch

3cho+nc

3h

7coch

2+oh

o2c

2h

4oh =oh+2ch

2o

hocho+h =h2+co

2+h

hocho+oh =h2o+co

2+h

hocho+oh =h2o+co

2+h

nc7h

16+oh=c

7h

15-3+ho

2

c7h

14ooh

2-5=c

7h

14o

2-5+oh

c7h

14ooh

3-6=c

7h

14o

2-5+oh

co+o = co2

h+oh= h2o

ch2cho+o

2=ch

2o+co+oh

Important Species

nc7h16, o2, c7h15-1/2/3/4,ho2, oh, nc7ket24, nc7ket35,nc7ket42, · · ·

Skeletal Mechanism Generation

I Eliminate non-importantspecies

I Keep all elementary reactionsincluding important species

26 / 44

1 Introduction

2 gRRM






5 Conclusion


Skeletal Mechanism for n-heptane/air kinetics

I Detailed Mechanism (D561): ns = 561, nr = 2539I Sampled points took from autoignition in adiabatic

constant pressure reactorI 650≤ T0 ≤ 1400 KI 1≤ P≤ 20 atmI 0.5≤ φ ≤ 1.5

I ε = 0.2%−→ ns = 203, nr = 879 (R203)I ε = 0.6%−→ ns = 149, nr = 669 (R149)

Kooshkbaghi, M., et al., (2014). Combust. Flame, 161(6), 1507-1515.

27 / 44

1 Introduction

2 gRRM






5 Conclusion


Validation : Ignition Delay

0.8 1 1.2 1.4 1.6

1000/T0 [1/K]

10-5

10-4

10-3

10-2

10-1

100

τig

[s]

D561R203R149

φ = 0.5

φ = 1.0

φ = 1.5

φ = 0.5

φ = 1.0

φ = 1.5

P = 1 atm

P = 20 atm


28 / 44

1 Introduction

2 gRRM






5 Conclusion


Stiffness

0 0.05 0.1 0.15 0.2

t [s]

600

800

1000

1200

1400

1600

1800

2000

2200

2400

T [

K]

0 0.05 0.1 0.15 0.210

-15

10-14

10-13

10-12

10-11

10-10

10-9

10-8

Tim

esc

ale

[s]

D561

R203

R149

T τfast

τfast

τfast

T0 = 650 K

P = 1atmφ = 1.0


29 / 44

1 Introduction

2 gRRM






5 Conclusion


Validation : Single-Zone Engine Model

Tinlet = 650 K, Pinlet = 5 atm, φ = 0.8 at −40 oATDC, ω = 700 rpm

Crank Angle [0ATDC]

T[K

]

40 20 0 20 40 60 80 100

1000

1500

2000

2500

Crank Angle [0ATDC]

p[a

tm]

40 20 0 20 40 60 80 100

10

20

30

40

Crank Angle [0ATDC]

Mas

sF

ract

ion

40 20 0 20 40 60 80 10010

5

104

103

102

101

CO2

H2O

COnC

7H

16

Crank Angle [0ATDC]

Mas

sF

ract

ion

40 20 0 20 40 60 80 100

1010

108

106

104

102

OH

HO2

H2O

2

CH2O

HCO

Figure: Temperature (a), pressure (b) and selected speciesconcentration (c,d) profiles for the single-zone engine model. Thelean fresh mixture (φ = 0.8) is injected at −40 0ATDC with p0 = 5atm and T0 = 750 K (D561: solid line, R203: circles, R161: dashedline).

30 / 44

1 Introduction

2 gRRM






5 Conclusion


Validation : Laminar Premixed Flame

Tu = 650 K, P = 1 atm

0.7 0.8 0.9 1 1.1 1.2 1.3 1.4120

130

140

150

160

170

180

190

SL [

cm

/s]

0.7 0.8 0.9 1 1.1 1.2 1.3 1.4

φ

2200

2300

2400

2500

Tf [

K]

D561R203R149

T0 = 650 K

P = 1 atm

(a)

(b)

Kooshkbaghi, M., et al., (2014). Combust. Flame, 161(6), 1507-1515.31 / 44

1 Introduction

2 gRRM



PSR Setup


1P Cont.

Multi-P Cont.

5 ConclusionIV. n-heptane/air complex dynamics

32 / 44

1 Introduction

2 gRRM


4 n-heptane/aircomplex dynamicsPSR Setup


1P Cont.

Multi-P Cont.

5 Conclusion


PSR setup and hysteresis of temperature

dYk

dt=

Wkω̇

ρ+

Y0k −Yk

τ

dTdt

=−∑nsi=1 Wiω̇ihi

ρcp+

∑nsi=1 Y0

i (h0i −hi)

cpτ− Q̇loss

ρcp

Parametersτ: Residence Timep: Reactor Pressureφ : Inlet MixtureT0: Inlet TemperatureQ̇loss: Heat loss per unit volume

Typical S-shaped bifurcation diagramof a PSR

Numerical toolAUTO-07p: Continuation and

Bifurcation Software for ODEs +CHEMKIN III: Chemical kinetics data

33 / 44

1 Introduction

2 gRRM




1P Cont.

Multi-P Cont.

5 Conclusion


Validation Of Skeletal Mechanism

τ [s]

T[K

]

106

105

104

103

102

101

100

750

1000

1250

1500

1750

2000

2250

2500

2750

p = 1 atmp = 5 atmp = 20 atm

Dependence of reactor temperature on residence time of adiabatic PSRT0 = 650 K, φ = 1.0

D561 (solid lines) and R149 (open circles)

34 / 44

1 Introduction

2 gRRM




1P Cont.

T vs τ

T vs φ

T vs Q̇loss

Multi-P Cont.

5 Conclusion


One parameter continuationReactor Temperature vs Residence Time

T0 = 700 K, φ = 1.0, p = 1 atm

10-6

10-5

10-4

10-3

10-2

10-1

100

τ [s]

600

800

1000

1200

1400

1600

1800

2000

2200

2400

2600

T [

K]

TP4

TP1

TP2

HB1

HB2

TP3

S1

HB1HB

2

TP3

TP4

10-3

700

800

strongly burning

unstable branch

cool flameextinguished

Kooshkbaghi, M., et al. (2015). Combust. Flame.35 / 44

1 Introduction

2 gRRM




1P Cont.

T vs τ

T vs φ

T vs Q̇loss

Multi-P Cont.

5 Conclusion


One parameter continuationReactor Temperature vs Residence Time

T0 = 700 K, φ = 1.0, p = 1 atm

10-6

10-5

10-4

10-3

10-2

10-1

100

τ [s]

600

800

1000

1200

1400

1600

1800

2000

2200

2400

2600

T [

K]

TP4

TP1

TP2

HB1

HB2

TP3

S1

HB1HB

2

TP3

TP4

10-3

700

800

strongly burning

unstable branch

cool flameextinguished

Kooshkbaghi, M., et al. (2015). Combust. Flame.

0 0.005 0.01 0.015 0.02

time [s]

800

1000

1200

1400

1600

1800

T [

K]

TP1 - 1 K

0 0.1 0.2 0.3 0.4 0.5

time [s]

800

1000

1200

1400

1600

1800

2000

2200

2400

2600

T [

K]

TP2 + 1 K

0 0.005 0.01 0.015 0.02 0.025 0.03

time [s]

690

700

710

720

730

740

T [

K]

TP3 - 1 K

0 0.02 0.04 0.06 0.08 0.1 0.12

time [s]

550

600

650

700

750

800

T [

K]

0.2 0.205 0.21 0.215 0.22Y

O2

0

2×10-6

4×10-6

6×10-6

YO

H

S1

36 / 44

1 Introduction

2 gRRM




1P Cont.

T vs τ

T vs φ

T vs Q̇loss

Multi-P Cont.

5 Conclusion


One parameter continuationReactor Temperature vs Equivalence ratio

T0 = 700 K, φ = 1.0, p = 1 atm

10-6

10-5

10-4

10-3

10-2

10-1

100

Residence Time [s]

600

800

1000

1200

1400

1600

1800

2000

2200

2400

2600

T [

K]

T = 2219 K

T = 1296 K

T = 765 K

T0 = 700 K, τ = 10−3 s, p = 1 atm

0 0.5 1 1.5 2 2.5 3 3.5 4 4.5 5 5.5 6 6.5 φ

800

1000

1200

1400

1600

1800

2000

2200

2400

T [

K]

strongly-burning

unstable

unstable cool flame

HB1

T = 2219 K

T = 1296 K

T = 765 KHB

2oscillatory cool flame

TP2

TP1

For fixed residence time, the change of reactor temperature respect to the inlet mixturecomposition.

37 / 44

1 Introduction

2 gRRM




1P Cont.

T vs τ

T vs φ

T vs Q̇loss

Multi-P Cont.

5 Conclusion


One parameter continuationReactor Temperature vs Equivalence ratio

T0 = 700 K, τ = 10−3 s, p = 1 atm

0 0.5 1 1.5 2 2.5 3 3.5 4 4.5 5 5.5 6 6.5 φ

800

1000

1200

1400

1600

1800

2000

2200

2400

T [

K]

strongly-burning

unstable

unstable cool flame

HB1

T = 2219 K

T = 1296 K

T = 765 KHB

2oscillatory cool flame

TP2

TP1

0 1 2 3 4 5

800

1200

1600

2000

2400

T [

K]

0 2 4 6 8 10

800

1200

1600

2000

2400

0 3 6 9 12 15

800

1200

1600

2000

2400

T [

K]

0 2 4 6 8

800

1200

1600

2000

2400

0 0.5 1 1.5 2 2.5 3

φ

800

1200

1600

2000

2400

T [

K]

0.4 0.8 1.2 1.6 2

φ

800

1200

1600

2000

2400

τ = 1 s

τ = 10−1

s

τ = 10−3

s τ = 10−4

s

τ = 10−2

s

τ = 0.2 s

(a)

(e)

(c)

(f)

(d)

(b)

For fixed residence time, the change of reactor temperature respect to the inlet mixturecomposition.

37 / 44

1 Introduction

2 gRRM




1P Cont.

T vs τ

T vs φ

T vs Q̇loss

Multi-P Cont.

5 Conclusion


One parameter continuationReactor Temperature for non-adiabtic reactors

10-5

10-4

10-3

10-2

10-1

100

101

τ [s]

750

1000

1250

1500

1750

2000

2250

2500

T [

K]

TP1

HB2HB

3

TP2

6.60 6.90 7.20

10

80

11

20

11

60

TP2

HB1

1×10-3

2×10-3

700

750

800

HB3

HB2

TP3

0 0.5 1 1.5 21150

1155

1160

1165

T [

K]

τ = 6.6 [s]

0 0.5 1 1.5 21100

1105

1110

1115

1120

T [

K]

τ = 7.0 [s]

0 100 200 300 400

t [s]

1084

1086

1088

1090

1092

T [

K]

τ = 7.1605 [s]

Dependence of reactor temperature on residence time for non-adiabatic PSRp = 1 atm, T0 = 700 K, φ = 1 and Q̇loss = 0.1kJ/(s×m3)

38 / 44

1 Introduction

2 gRRM




1P Cont.

Multi-P Cont.

5 Conclusion


Multi-parameter continuationContinuation in (T0− τ) parameters

10-6

10-5

10-4

10-3

10-2

10-1

100

τ [s]

600

800

1000

1200

1400

1600

1800

T0 [

K]

BT

TPB2

TPB1

HBB

TPB3

CP1

(a)(b)

(c)

(d)

(e)

CP2

TPB4

10-3

10-2

600

650

700

750

HBB

TPB3

BT

TPB4

CP2

10-5

10-4

10-3

10-2

10-1

100

1000

1500

2000

250010

-610

-510

-410

-310

-210

-11000

1500

2000

2500

10-4

10-2

100

τ

500

1000

1500

2000

2500

10-6

10-5

10-4

10-3

10-2

10-1

100

1600

2000

2400

2800

39 / 44

1 Introduction

2 gRRM




1P Cont.

Multi-P Cont.

5 Conclusion


Multi-parameter continuationContinuation in (T0− τ−φ) parameters

10−610−5

10−410−3

10−210−1

100101

600800100012001400160018000.7

0.8

0.9

1

1.1

1.2

φ

τ [s]

T0[K]

φ

40 / 44

1 Introduction

2 gRRM



5 ConclusionSummary

Directions for future work

Publications

Acknowledgments


Summary

41 / 44

1 Introduction

2 gRRM



5 ConclusionSummary


Publications

Acknowledgments



I Entropy Production AnalysisGenerate Skeletal Mechanisms for Heavy Fuels

n-decane (n-C10H22) (p = 20 atm, φ = 1.0, T0 = 700 K)

0 0.002 0.004 0.006 0.008 0.01 0.012

time [s]

700

800

900

1000

1100

1200

1300

1400

1500

1600

1700

1800

1900

2000

T [

K]

LLNL mechanismn

s=2115, n

r=8157

ns=281, n

r=995

42 / 44

1 Introduction

2 gRRM



5 ConclusionSummary


Publications

Acknowledgments



I Entropy Production AnalysisDirect Numerical Simulations using skeletal mechanismsCH4/air premixed flame (p = 1 atm, φ = 0.9, T0 = 300 K, δf =

Tf−T0

max| dTdx |

)

0 0.5200

400

600

800

1000

1200

1400

1600

1800

2000

2200

T [

K]

-0.2 -0.1 0 0.1 0.2 0.3 0.4 0.5

X [cm]

0

0.005

0.01

0.015

0.02

0.025

0.03

0.035

0.04

0.045

0.05

Y

10YOH

YCO

1000YHCO

T

Detailed D35 (solid lines), Skeletal R20 (dashed lines)42 / 44

1 Introduction

2 gRRM



5 ConclusionSummary


Publications

Acknowledgments




Tf−T0

max| dTdx |

)

42 / 44

1 Introduction

2 gRRM



5 ConclusionSummary


Publications

Acknowledgments




Tf−T0

max| dTdx |

)

Temperature contoursDetailed (D35)

Reduced (R20)

42 / 44

1 Introduction

2 gRRM



5 ConclusionSummary


Publications

Acknowledgments


Publications

Journal Publications1. Kooshkbaghi, M., Frouzakis, C. E., Chiavazzo, E., Boulouchos, K., & Karlin,

I. V. (2014). The global relaxation redistribution method for reduction ofcombustion kinetics. J. Chem. Phys. 141(4), 044102.

2. Kooshkbaghi, M., Frouzakis, C. E., Boulouchos, K., & Karlin, I. V. (2014).Entropy production analysis for mechanism reduction. Combust. Flame, 161(6),1507-1515.

3. Karlin, I. V., Chikatamarla, S. S., & Kooshkbaghi, M. (2014). Non-perturbativehydrodynamic limits: A case study. Physica A, 403, 189-194.

4. Kooshkbaghi, M., Frouzakis, C. E., Boulouchos, K., & Karlin, I. V. (2015).n-Heptane/air combustion in perfectly stirred reactors: Dynamics, bifurcationsand dominant reactions at critical conditions. Combust. Flame.

5. Kooshkbaghi, M., Frouzakis, C. E., Boulouchos, K., & Karlin, I. V. (2015).

Spectral Quasi Equilibrium Manifold for Chemical Kinetics. In preparation for

J. Chem. Phys.

Conferences1. Kooshkbaghi, M., Frouzakis, C. E., Chiavazzo, E., Karlin, I. V., & Boulouchos,

K. (2013). IWMRRF, San Francisco, California, USA.

2. Kooshkbaghi, M., Boulouchos, K., Frouzakis, C. E., Karlin, I. V., &

Chiavazzo, E. (2013). IEA, 36th TLM, Stavanger, Norway.43 / 44

1 Introduction

2 gRRM



5 ConclusionSummary


Publications

Acknowledgments


Acknowledgments

I Prof. Konstantinos BoulouchosI Dr. Christos FrouzakisI Prof. Ilya KarlinI Prof. Yannis KevrekidisI My friends and colleagues at LAVI Swiss National Science FoundationI My family

44 / 44

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