On The Model Reduction for Chemical and Physical Kinetics · On The Model Reduction for Chemical...
Transcript of 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
3 Entropy Prod. Ana.
4 n-heptane/aircomplex dynamics
5 Conclusion
LAVLaboratorium für Aerothermochemie und VerbrennungssystemeAerothermochemistry and Combustion Systems Laboratory
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
3 Entropy Prod. Ana.
4 n-heptane/aircomplex dynamics
5 Conclusion
LAVLaboratorium für Aerothermochemie und VerbrennungssystemeAerothermochemistry and Combustion Systems Laboratory
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
1 IntroductionMotivation
Chem. Kin. Model Reduction
Thesis Outline
2 gRRM
3 Entropy Prod. Ana.
4 n-heptane/aircomplex dynamics
5 Conclusion
LAVLaboratorium für Aerothermochemie und VerbrennungssystemeAerothermochemistry and Combustion Systems Laboratory
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 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
I Stiffness/ Non-linearity
3 / 44
1 IntroductionMotivation
Chem. Kin. Model Reduction
Thesis Outline
2 gRRM
3 Entropy Prod. Ana.
4 n-heptane/aircomplex dynamics
5 Conclusion
LAVLaboratorium für Aerothermochemie und VerbrennungssystemeAerothermochemistry and Combustion Systems Laboratory
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
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
1 IntroductionMotivation
Chem. Kin. Model Reduction
Thesis Outline
2 gRRM
3 Entropy Prod. Ana.
4 n-heptane/aircomplex dynamics
5 Conclusion
LAVLaboratorium für Aerothermochemie und VerbrennungssystemeAerothermochemistry and Combustion Systems Laboratory
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)
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1 IntroductionMotivation
Chem. Kin. Model Reduction
Thesis Outline
2 gRRM
3 Entropy Prod. Ana.
4 n-heptane/aircomplex dynamics
5 Conclusion
LAVLaboratorium für Aerothermochemie und VerbrennungssystemeAerothermochemistry and Combustion Systems Laboratory
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
1 IntroductionMotivation
Chem. Kin. Model Reduction
Thesis Outline
2 gRRM
3 Entropy Prod. Ana.
4 n-heptane/aircomplex dynamics
5 Conclusion
LAVLaboratorium für Aerothermochemie und VerbrennungssystemeAerothermochemistry and Combustion Systems Laboratory
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
1 IntroductionMotivation
Chem. Kin. Model Reduction
Thesis Outline
2 gRRM
3 Entropy Prod. Ana.
4 n-heptane/aircomplex dynamics
5 Conclusion
LAVLaboratorium für Aerothermochemie und VerbrennungssystemeAerothermochemistry and Combustion Systems Laboratory
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
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1 IntroductionMotivation
Chem. Kin. Model Reduction
Thesis Outline
2 gRRM
3 Entropy Prod. Ana.
4 n-heptane/aircomplex dynamics
5 Conclusion
LAVLaboratorium für Aerothermochemie und VerbrennungssystemeAerothermochemistry and Combustion Systems Laboratory
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
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1 Introduction
2 gRRMSIM Def.
MIM
Constructing Manifold
Using Manifold
3 Entropy Prod. Ana.
4 n-heptane/aircomplex dynamics
5 ConclusionII. The global Relaxation RedistributionMethod
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1 Introduction
2 gRRMSIM Def.
MIM
Constructing Manifold
Using Manifold
3 Entropy Prod. Ana.
4 n-heptane/aircomplex dynamics
5 Conclusion
LAVLaboratorium für Aerothermochemie und VerbrennungssystemeAerothermochemistry and Combustion Systems Laboratory
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
Constructing Manifold
Using Manifold
3 Entropy Prod. Ana.
4 n-heptane/aircomplex dynamics
5 Conclusion
LAVLaboratorium für Aerothermochemie und VerbrennungssystemeAerothermochemistry and Combustion Systems Laboratory
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
Constructing Manifold
Using Manifold
3 Entropy Prod. Ana.
4 n-heptane/aircomplex dynamics
5 Conclusion
LAVLaboratorium für Aerothermochemie und VerbrennungssystemeAerothermochemistry and Combustion Systems Laboratory
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
Constructing Manifold
Using Manifold
3 Entropy Prod. Ana.
4 n-heptane/aircomplex dynamics
5 Conclusion
LAVLaboratorium für Aerothermochemie und VerbrennungssystemeAerothermochemistry and Combustion Systems Laboratory
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
Constructing Manifold
Using Manifold
3 Entropy Prod. Ana.
4 n-heptane/aircomplex dynamics
5 Conclusion
LAVLaboratorium für Aerothermochemie und VerbrennungssystemeAerothermochemistry and Combustion Systems Laboratory
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
Constructing Manifold
Using Manifold
3 Entropy Prod. Ana.
4 n-heptane/aircomplex dynamics
5 Conclusion
LAVLaboratorium für Aerothermochemie und VerbrennungssystemeAerothermochemistry and Combustion Systems Laboratory
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
Constructing Manifold
Using Manifold
3 Entropy Prod. Ana.
4 n-heptane/aircomplex dynamics
5 Conclusion
LAVLaboratorium für Aerothermochemie und VerbrennungssystemeAerothermochemistry and Combustion Systems Laboratory
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
Constructing Manifold
Using Manifold
3 Entropy Prod. Ana.
4 n-heptane/aircomplex dynamics
5 Conclusion
LAVLaboratorium für Aerothermochemie und VerbrennungssystemeAerothermochemistry and Combustion Systems Laboratory
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 0.5 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
Constructing Manifold
Using Manifold
3 Entropy Prod. Ana.
4 n-heptane/aircomplex dynamics
5 Conclusion
LAVLaboratorium für Aerothermochemie und VerbrennungssystemeAerothermochemistry and Combustion Systems Laboratory
Multiscale Dissipation
Davis-Skodje System (J. Chem. Phys. 1999):dxdt =−x
dydt =−γy+ (γ−1)x+γx2
(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
Constructing Manifold
Using Manifold
3 Entropy Prod. Ana.
4 n-heptane/aircomplex dynamics
5 Conclusion
LAVLaboratorium für Aerothermochemie und VerbrennungssystemeAerothermochemistry and Combustion Systems Laboratory
Multiscale Dissipation
Davis-Skodje System (J. Chem. Phys. 1999):dxdt =−x
dydt =−γy+ (γ−1)x+γx2
(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
Constructing Manifold
Using Manifold
3 Entropy Prod. Ana.
4 n-heptane/aircomplex dynamics
5 Conclusion
LAVLaboratorium für Aerothermochemie und VerbrennungssystemeAerothermochemistry and Combustion Systems Laboratory
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
Constructing Manifold
Using Manifold
3 Entropy Prod. Ana.
4 n-heptane/aircomplex dynamics
5 Conclusion
LAVLaboratorium für Aerothermochemie und VerbrennungssystemeAerothermochemistry and Combustion Systems Laboratory
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
Constructing Manifold
Using Manifold
3 Entropy Prod. Ana.
4 n-heptane/aircomplex dynamics
5 Conclusion
LAVLaboratorium für Aerothermochemie und VerbrennungssystemeAerothermochemistry and Combustion Systems Laboratory
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
Constructing Manifold
Using Manifold
3 Entropy Prod. Ana.
4 n-heptane/aircomplex dynamics
5 Conclusion
LAVLaboratorium für Aerothermochemie und VerbrennungssystemeAerothermochemistry and Combustion Systems Laboratory
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
Constructing Manifold
Using Manifold
3 Entropy Prod. Ana.
4 n-heptane/aircomplex dynamics
5 Conclusion
LAVLaboratorium für Aerothermochemie und VerbrennungssystemeAerothermochemistry and Combustion Systems Laboratory
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
Constructing Manifold
Using Manifold
3 Entropy Prod. Ana.
4 n-heptane/aircomplex dynamics
5 Conclusion
LAVLaboratorium für Aerothermochemie und VerbrennungssystemeAerothermochemistry and Combustion Systems Laboratory
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
Constructing Manifold
Using Manifold
3 Entropy Prod. Ana.
4 n-heptane/aircomplex dynamics
5 Conclusion
LAVLaboratorium für Aerothermochemie und VerbrennungssystemeAerothermochemistry and Combustion Systems Laboratory
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
Constructing Manifold
Using Manifold
3 Entropy Prod. Ana.
4 n-heptane/aircomplex dynamics
5 Conclusion
LAVLaboratorium für Aerothermochemie und VerbrennungssystemeAerothermochemistry and Combustion Systems Laboratory
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
Constructing Manifold
Using Manifold
3 Entropy Prod. Ana.
4 n-heptane/aircomplex dynamics
5 Conclusion
LAVLaboratorium für Aerothermochemie und VerbrennungssystemeAerothermochemistry and Combustion Systems Laboratory
RRM Manifold ConstructionSingular perturbed system*
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
*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
Constructing Manifold
Using Manifold
3 Entropy Prod. Ana.
4 n-heptane/aircomplex dynamics
5 Conclusion
LAVLaboratorium für Aerothermochemie und VerbrennungssystemeAerothermochemistry and Combustion Systems Laboratory
RRM Manifold ConstructionSingular perturbed system*
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
*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
Constructing Manifold
Dimensionality
gRRM
RCCE
Using Manifold
3 Entropy Prod. Ana.
4 n-heptane/aircomplex dynamics
5 Conclusion
LAVLaboratorium für Aerothermochemie und VerbrennungssystemeAerothermochemistry and Combustion Systems Laboratory
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
Constructing Manifold
Dimensionality
gRRM
RCCE
Using Manifold
3 Entropy Prod. Ana.
4 n-heptane/aircomplex dynamics
5 Conclusion
LAVLaboratorium für Aerothermochemie und VerbrennungssystemeAerothermochemistry and Combustion Systems Laboratory
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 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
Constructing Manifold
Dimensionality
gRRM
RCCE
Using Manifold
3 Entropy Prod. Ana.
4 n-heptane/aircomplex dynamics
5 Conclusion
LAVLaboratorium für Aerothermochemie und VerbrennungssystemeAerothermochemistry and Combustion Systems Laboratory
Dimensionality issues
=⇒ 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
Constructing Manifold
Dimensionality
gRRM
RCCE
Using Manifold
3 Entropy Prod. Ana.
4 n-heptane/aircomplex dynamics
5 Conclusion
LAVLaboratorium für Aerothermochemie und VerbrennungssystemeAerothermochemistry and Combustion Systems Laboratory
Dimensionality issues
=⇒ 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
Constructing Manifold
Dimensionality
gRRM
RCCE
Using Manifold
3 Entropy Prod. Ana.
4 n-heptane/aircomplex dynamics
5 Conclusion
LAVLaboratorium für Aerothermochemie und VerbrennungssystemeAerothermochemistry and Combustion Systems Laboratory
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
Constructing Manifold
Dimensionality
gRRM
RCCE
Using Manifold
3 Entropy Prod. Ana.
4 n-heptane/aircomplex dynamics
5 Conclusion
LAVLaboratorium für Aerothermochemie und VerbrennungssystemeAerothermochemistry and Combustion Systems Laboratory
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
Constructing Manifold
Dimensionality
gRRM
RCCE
Using Manifold
3 Entropy Prod. Ana.
4 n-heptane/aircomplex dynamics
5 Conclusion
LAVLaboratorium für Aerothermochemie und VerbrennungssystemeAerothermochemistry and Combustion Systems Laboratory
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
Constructing Manifold
Dimensionality
gRRM
RCCE
Using Manifold
3 Entropy Prod. Ana.
4 n-heptane/aircomplex dynamics
5 Conclusion
LAVLaboratorium für Aerothermochemie und VerbrennungssystemeAerothermochemistry and Combustion Systems Laboratory
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
Constructing Manifold
Dimensionality
gRRM
RCCE
Using Manifold
3 Entropy Prod. Ana.
4 n-heptane/aircomplex dynamics
5 Conclusion
LAVLaboratorium für Aerothermochemie und VerbrennungssystemeAerothermochemistry and Combustion Systems Laboratory
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
Constructing Manifold
Dimensionality
gRRM
RCCE
Using Manifold
3 Entropy Prod. Ana.
4 n-heptane/aircomplex dynamics
5 Conclusion
LAVLaboratorium für Aerothermochemie und VerbrennungssystemeAerothermochemistry and Combustion Systems Laboratory
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
Constructing Manifold
Dimensionality
gRRM
RCCE
Using Manifold
3 Entropy Prod. Ana.
4 n-heptane/aircomplex dynamics
5 Conclusion
LAVLaboratorium für Aerothermochemie und VerbrennungssystemeAerothermochemistry and Combustion Systems Laboratory
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
Constructing Manifold
Dimensionality
gRRM
RCCE
Using Manifold
3 Entropy Prod. Ana.
4 n-heptane/aircomplex dynamics
5 Conclusion
LAVLaboratorium für Aerothermochemie und VerbrennungssystemeAerothermochemistry and Combustion Systems Laboratory
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
Constructing Manifold
Dimensionality
gRRM
RCCE
Using Manifold
3 Entropy Prod. Ana.
4 n-heptane/aircomplex dynamics
5 Conclusion
LAVLaboratorium für Aerothermochemie und VerbrennungssystemeAerothermochemistry and Combustion Systems Laboratory
gRRM for H2/air combustion
ns = 9 Species, T0 = 1500K, P = 1atm, φ = 1.0
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
�, fresh mixture; ? , equilibrium point; − detailed kinetics path, Coloredsurfaces, Manifolds
18 / 44
1 Introduction
2 gRRMSIM Def.
MIM
Constructing Manifold
Dimensionality
gRRM
RCCE
Using Manifold
3 Entropy Prod. Ana.
4 n-heptane/aircomplex dynamics
5 Conclusion
LAVLaboratorium für Aerothermochemie und VerbrennungssystemeAerothermochemistry and Combustion Systems Laboratory
gRRM for H2/air combustion
ns = 9 Species, T0 = 1500K, P = 1atm, φ = 1.0
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
�, fresh mixture; ? , equilibrium point; − detailed kinetics path, Coloredsurfaces, Manifolds
18 / 44
1 Introduction
2 gRRMSIM Def.
MIM
Constructing Manifold
Dimensionality
gRRM
RCCE
Using Manifold
3 Entropy Prod. Ana.
4 n-heptane/aircomplex dynamics
5 Conclusion
LAVLaboratorium für Aerothermochemie und VerbrennungssystemeAerothermochemistry and Combustion Systems Laboratory
gRRM for H2/air combustion
ns = 9 Species, T0 = 1500K, P = 1atm, φ = 1.0
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
�, fresh mixture; ? , equilibrium point; − detailed kinetics path, Coloredsurfaces, Manifolds
18 / 44
1 Introduction
2 gRRMSIM Def.
MIM
Constructing Manifold
Using Manifold
T0 = 1500K
T0 = 1000K
3 Entropy Prod. Ana.
4 n-heptane/aircomplex dynamics
5 Conclusion
LAVLaboratorium für Aerothermochemie und VerbrennungssystemeAerothermochemistry and Combustion Systems Laboratory
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
Constructing Manifold
Using Manifold
T0 = 1500K
T0 = 1000K
3 Entropy Prod. Ana.
4 n-heptane/aircomplex dynamics
5 Conclusion
LAVLaboratorium für Aerothermochemie und VerbrennungssystemeAerothermochemistry and Combustion Systems Laboratory
H2/air auto-ignition
Adiabatic, constant pressure reactorT0 = 1000K, P = 1atm, φ = 1.0
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
Constructing Manifold
Using Manifold
T0 = 1500K
T0 = 1000K
3 Entropy Prod. Ana.
4 n-heptane/aircomplex dynamics
5 Conclusion
LAVLaboratorium für Aerothermochemie und VerbrennungssystemeAerothermochemistry and Combustion Systems Laboratory
H2/air auto-ignition
Adiabatic, constant pressure reactorT0 = 1000K, P = 1atm, φ = 1.0
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.
4 n-heptane/aircomplex dynamics
5 Conclusion
III. Entropy production analysis formechanism reduction
22 / 44
1 Introduction
2 gRRM
3 Entropy Prod. Ana.Entropy Production
Skeletal Mechanism Gen.
4 n-heptane/aircomplex dynamics
5 Conclusion
LAVLaboratorium für Aerothermochemie und VerbrennungssystemeAerothermochemistry and Combustion Systems Laboratory
Entropy Production
23 / 44
1 Introduction
2 gRRM
3 Entropy Prod. Ana.Entropy Production
Skeletal Mechanism Gen.
4 n-heptane/aircomplex dynamics
5 Conclusion
LAVLaboratorium für Aerothermochemie und VerbrennungssystemeAerothermochemistry and Combustion Systems Laboratory
Entropy Production
23 / 44
1 Introduction
2 gRRM
3 Entropy Prod. Ana.Entropy Production
Skeletal Mechanism Gen.
4 n-heptane/aircomplex dynamics
5 Conclusion
LAVLaboratorium für Aerothermochemie und VerbrennungssystemeAerothermochemistry and Combustion Systems Laboratory
Entropy Production
23 / 44
1 Introduction
2 gRRM
3 Entropy Prod. Ana.Entropy Production
Skeletal Mechanism Gen.
4 n-heptane/aircomplex dynamics
5 Conclusion
LAVLaboratorium für Aerothermochemie und VerbrennungssystemeAerothermochemistry and Combustion Systems Laboratory
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
3 Entropy Prod. Ana.Entropy Production
Skeletal Mechanism Gen.
4 n-heptane/aircomplex dynamics
5 Conclusion
LAVLaboratorium für Aerothermochemie und VerbrennungssystemeAerothermochemistry and Combustion Systems Laboratory
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
3 Entropy Prod. Ana.Entropy Production
Skeletal Mechanism Gen.
nC7 H16 /air Ske. Mech.
Validation of Ske. Mech.
4 n-heptane/aircomplex dynamics
5 Conclusion
LAVLaboratorium für Aerothermochemie und VerbrennungssystemeAerothermochemistry and Combustion Systems Laboratory
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
3 Entropy Prod. Ana.Entropy Production
Skeletal Mechanism Gen.
nC7 H16 /air Ske. Mech.
Validation of Ske. Mech.
4 n-heptane/aircomplex dynamics
5 Conclusion
LAVLaboratorium für Aerothermochemie und VerbrennungssystemeAerothermochemistry and Combustion Systems Laboratory
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
3 Entropy Prod. Ana.Entropy Production
Skeletal Mechanism Gen.
nC7 H16 /air Ske. Mech.
Validation of Ske. Mech.
4 n-heptane/aircomplex dynamics
5 Conclusion
LAVLaboratorium für Aerothermochemie und VerbrennungssystemeAerothermochemistry and Combustion Systems Laboratory
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
3 Entropy Prod. Ana.Entropy Production
Skeletal Mechanism Gen.
nC7 H16 /air Ske. Mech.
Validation of Ske. Mech.
4 n-heptane/aircomplex dynamics
5 Conclusion
LAVLaboratorium für Aerothermochemie und VerbrennungssystemeAerothermochemistry and Combustion Systems Laboratory
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
3 Entropy Prod. Ana.Entropy Production
Skeletal Mechanism Gen.
nC7 H16 /air Ske. Mech.
Validation of Ske. Mech.
4 n-heptane/aircomplex dynamics
5 Conclusion
LAVLaboratorium für Aerothermochemie und VerbrennungssystemeAerothermochemistry and Combustion Systems Laboratory
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
Kooshkbaghi, M., et al., (2014). Combust. Flame, 161(6), 1507-1515.
28 / 44
1 Introduction
2 gRRM
3 Entropy Prod. Ana.Entropy Production
Skeletal Mechanism Gen.
nC7 H16 /air Ske. Mech.
Validation of Ske. Mech.
4 n-heptane/aircomplex dynamics
5 Conclusion
LAVLaboratorium für Aerothermochemie und VerbrennungssystemeAerothermochemistry and Combustion Systems Laboratory
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
Kooshkbaghi, M., et al., (2014). Combust. Flame, 161(6), 1507-1515.
29 / 44
1 Introduction
2 gRRM
3 Entropy Prod. Ana.Entropy Production
Skeletal Mechanism Gen.
nC7 H16 /air Ske. Mech.
Validation of Ske. Mech.
4 n-heptane/aircomplex dynamics
5 Conclusion
LAVLaboratorium für Aerothermochemie und VerbrennungssystemeAerothermochemistry and Combustion Systems Laboratory
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
3 Entropy Prod. Ana.Entropy Production
Skeletal Mechanism Gen.
nC7 H16 /air Ske. Mech.
Validation of Ske. Mech.
4 n-heptane/aircomplex dynamics
5 Conclusion
LAVLaboratorium für Aerothermochemie und VerbrennungssystemeAerothermochemistry and Combustion Systems Laboratory
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
3 Entropy Prod. Ana.
4 n-heptane/aircomplex dynamics
PSR Setup
Validation of Ske. Mech.
1P Cont.
Multi-P Cont.
5 ConclusionIV. n-heptane/air complex dynamics
32 / 44
1 Introduction
2 gRRM
3 Entropy Prod. Ana.
4 n-heptane/aircomplex dynamicsPSR Setup
Validation of Ske. Mech.
1P Cont.
Multi-P Cont.
5 Conclusion
LAVLaboratorium für Aerothermochemie und VerbrennungssystemeAerothermochemistry and Combustion Systems Laboratory
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
3 Entropy Prod. Ana.
4 n-heptane/aircomplex dynamicsPSR Setup
Validation of Ske. Mech.
1P Cont.
Multi-P Cont.
5 Conclusion
LAVLaboratorium für Aerothermochemie und VerbrennungssystemeAerothermochemistry and Combustion Systems Laboratory
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
3 Entropy Prod. Ana.
4 n-heptane/aircomplex dynamicsPSR Setup
Validation of Ske. Mech.
1P Cont.
T vs τ
T vs φ
T vs Q̇loss
Multi-P Cont.
5 Conclusion
LAVLaboratorium für Aerothermochemie und VerbrennungssystemeAerothermochemistry and Combustion Systems Laboratory
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
3 Entropy Prod. Ana.
4 n-heptane/aircomplex dynamicsPSR Setup
Validation of Ske. Mech.
1P Cont.
T vs τ
T vs φ
T vs Q̇loss
Multi-P Cont.
5 Conclusion
LAVLaboratorium für Aerothermochemie und VerbrennungssystemeAerothermochemistry and Combustion Systems Laboratory
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
3 Entropy Prod. Ana.
4 n-heptane/aircomplex dynamicsPSR Setup
Validation of Ske. Mech.
1P Cont.
T vs τ
T vs φ
T vs Q̇loss
Multi-P Cont.
5 Conclusion
LAVLaboratorium für Aerothermochemie und VerbrennungssystemeAerothermochemistry and Combustion Systems Laboratory
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
3 Entropy Prod. Ana.
4 n-heptane/aircomplex dynamicsPSR Setup
Validation of Ske. Mech.
1P Cont.
T vs τ
T vs φ
T vs Q̇loss
Multi-P Cont.
5 Conclusion
LAVLaboratorium für Aerothermochemie und VerbrennungssystemeAerothermochemistry and Combustion Systems Laboratory
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
3 Entropy Prod. Ana.
4 n-heptane/aircomplex dynamicsPSR Setup
Validation of Ske. Mech.
1P Cont.
T vs τ
T vs φ
T vs Q̇loss
Multi-P Cont.
5 Conclusion
LAVLaboratorium für Aerothermochemie und VerbrennungssystemeAerothermochemistry and Combustion Systems Laboratory
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
3 Entropy Prod. Ana.
4 n-heptane/aircomplex dynamicsPSR Setup
Validation of Ske. Mech.
1P Cont.
Multi-P Cont.
5 Conclusion
LAVLaboratorium für Aerothermochemie und VerbrennungssystemeAerothermochemistry and Combustion Systems Laboratory
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
3 Entropy Prod. Ana.
4 n-heptane/aircomplex dynamicsPSR Setup
Validation of Ske. Mech.
1P Cont.
Multi-P Cont.
5 Conclusion
LAVLaboratorium für Aerothermochemie und VerbrennungssystemeAerothermochemistry and Combustion Systems Laboratory
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
3 Entropy Prod. Ana.
4 n-heptane/aircomplex dynamics
5 ConclusionSummary
Directions for future work
Publications
Acknowledgments
LAVLaboratorium für Aerothermochemie und VerbrennungssystemeAerothermochemistry and Combustion Systems Laboratory
Summary
41 / 44
1 Introduction
2 gRRM
3 Entropy Prod. Ana.
4 n-heptane/aircomplex dynamics
5 ConclusionSummary
Directions for future work
Publications
Acknowledgments
LAVLaboratorium für Aerothermochemie und VerbrennungssystemeAerothermochemistry and Combustion Systems Laboratory
Directions for future work
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
3 Entropy Prod. Ana.
4 n-heptane/aircomplex dynamics
5 ConclusionSummary
Directions for future work
Publications
Acknowledgments
LAVLaboratorium für Aerothermochemie und VerbrennungssystemeAerothermochemistry and Combustion Systems Laboratory
Directions for future work
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
3 Entropy Prod. Ana.
4 n-heptane/aircomplex dynamics
5 ConclusionSummary
Directions for future work
Publications
Acknowledgments
LAVLaboratorium für Aerothermochemie und VerbrennungssystemeAerothermochemistry and Combustion Systems Laboratory
Directions for future work
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 |
)
42 / 44
1 Introduction
2 gRRM
3 Entropy Prod. Ana.
4 n-heptane/aircomplex dynamics
5 ConclusionSummary
Directions for future work
Publications
Acknowledgments
LAVLaboratorium für Aerothermochemie und VerbrennungssystemeAerothermochemistry and Combustion Systems Laboratory
Directions for future work
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 |
)
Temperature contoursDetailed (D35)
Reduced (R20)
42 / 44
1 Introduction
2 gRRM
3 Entropy Prod. Ana.
4 n-heptane/aircomplex dynamics
5 ConclusionSummary
Directions for future work
Publications
Acknowledgments
LAVLaboratorium für Aerothermochemie und VerbrennungssystemeAerothermochemistry and Combustion Systems Laboratory
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
3 Entropy Prod. Ana.
4 n-heptane/aircomplex dynamics
5 ConclusionSummary
Directions for future work
Publications
Acknowledgments
LAVLaboratorium für Aerothermochemie und VerbrennungssystemeAerothermochemistry and Combustion Systems Laboratory
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