-
Bayesian Inference on Uncertain Kinetic Parameters for the
Pyrolysis ofComposite Ablators
Joffrey Coheur
M. Arnst1, P. Chatelain2, P. Schrooyen3, T. Magin4
1Aerospace and Mechanical Engineering, Université de
Liège2Institute of Mechanics, Materials and Civil Engineering,
Université Catholique de Louvain
3Cenaero, Gosselies4Aeronautics and Aerospace Department, von
Karman Institute for Fluid Dynamics
SIAM Conference on Uncertainty QuantificationGarden Grove,
California, USA
April 16, 2018
-
Motivation: atmospheric entry
Thermal protection systems (TPS)
Dragon capsule (Space X)
Space debris
ATV-1 Jules Verne (ESA)
1 / 19
-
Ablative materials for thermal protection systems
Mars Science Laboratory thermalprotection system (NASA)
Porous thermalprotection material
[Helber, 2016]
Bottom view(after burn)
Lawson et al.
Fibers+ resin
2 / 19
-
Modeling the pyrolysis of ablative thermal protection
materials
Aerodynamic simulation(CooLFluiD, VKI)
Aerothermal simulation(Argo, Cenaero)
[Coheur, 2017]
High T°
gas flowPyrolysis
gas Ablation
I CFD codes requires accurate model for ablation [Lachaud, 2014;
Schrooyen, 2016], e.g.conservation of mass species
∂�g 〈ρi 〉g∂t
+∇ · (�g 〈ρi 〉g 〈u〉g ) = ∇ · 〈Ji 〉+ 〈ω̇pyroi 〉
I The governing laws for the pyrolysis gas production 〈ω̇pyroi 〉
must be derived from recentexperimental studies and their
associated uncertainties must be quantified
3 / 19
-
Table of Contents
Introduction
Characterization of physico-chemical parameters relevant to
pyrolysis reactionsDescription of the problemFormulation of the
problemImplementationApplication to pyrolysis experiments
Conclusions
-
Table of Contents
Introduction
Characterization of physico-chemical parameters relevant to
pyrolysis reactionsDescription of the problemFormulation of the
problemImplementationApplication to pyrolysis experiments
Conclusions
-
Pyrolysis experiments [Wong et al., 2015]
Phenolic sample(50 mg)
Pyrolysisgases
Heatedwall
400 600 800 1000 12000
0.2
0.4
0.6
0.8
Temperature, T
Mas
syi
eld
s,m
g
H2 CO CO2 CH4 H2O
400 600 800 1000 1200
30
40
50
Temperature, K
Sam
ple
mas
s,m
g
4 / 19
-
Phenomenological laws for pyrolysis mass loss and gas species
production
I Previous pyrolysis experiments showed that the pyrolysis
decomposition of ablativematerials follows successive reaction
rates [Goldstein, 1969; Trick, 1997]
400 600 800 1000 12000
2
4
6
8·10−3
Temperature, K
∂ξ j/∂T
j = 1j = 2
400 600 800 1000 12000.4
0.6
0.8
1
Temperature, K
m/m
0
〈ω̇pyroi 〉 =Np∑j
Fij∂ξj∂T
τm0
∂ξj∂T
= (1− ξj)njAjτ
exp
(− EjRT
)
I ξj : advancement of reaction of the fictitious resin component
j
5 / 19
-
Phenomenological laws for pyrolysis mass loss and gas species
production
I Previous pyrolysis experiments showed that the pyrolysis
decomposition of ablativematerials follows successive reaction
rates [Goldstein, 1969; Trick, 1997]
400 600 800 1000 12000
2
4
6
8·10−3
Temperature, K
∂ξ j/∂
T
j = 1j = 2
400 600 800 1000 12000.4
0.6
0.8
1
Temperature, K
m/m
0 m = m0 −Np∑j
Fjξjm0
∂ξj∂T
= (1− ξj)njAjτ
exp
(− EjRT
)
I ξj : advancement of reaction of the fictitious resin component
j
5 / 19
-
Table of Contents
Introduction
Characterization of physico-chemical parameters relevant to
pyrolysis reactionsDescription of the problemFormulation of the
problemImplementationApplication to pyrolysis experiments
Conclusions
-
Mathematical description
I Forward problem: di = η(xi ,p)
modelcode output
I Inverse problem: Find p knowing the observations dobsi
associated with errors εi
dobsi = η(xi ,p) + εi
Deterministic inverse problemWhat is the best estimate of p?
p̃ = arg minp∈Pad
||dobs − d||
Statistical inverse problemWhat is the plausibility about p?
π(p|dobs) = π(dobs|p)π0(p)∫
Rp π(dobs|p)π0(p)dp
6 / 19
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Pyrolysis experiments: the forward problem
Experimental observations dobsik
400 600 800 1000 12000
0.2
0.4
0.6
0.8
Mas
syi
eld
s,m
g
400 600 800 1000 1200
30
40
50
Temperature, K
Sam
ple
mas
s,m
g
Code outputs
I Collected mass of species i
dik =
∫ tktk−1〈ω̇pyroi 〉|T=Tkdt
=
Np∑j
Fijm0(ξ
(tf )j − ξ
(ti )j
)I Mass of the sample
mk = m0 −ns∑i=1
k∑l=1
dil
7 / 19
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Bayesian inference: the inverse problem
π(p|d) = π(d|p)π0(p)∫Rp π(d|p)π0(p)dp
(Bayes’ theorem)
I Choice for the prior π0(p): uniform pdf (bounded support)
I Choice for the likelihood π(dobs|p)
π(dobs|p) = 1Πi (2πσi 2)n/2
exp
(−
ns∑i=1
n∑k=1
[dobsik − ηi (xk ,p)
]22σi 2
)
8 / 19
-
Table of Contents
Introduction
Characterization of physico-chemical parameters relevant to
pyrolysis reactionsDescription of the problemFormulation of the
problemImplementationApplication to pyrolysis experiments
Conclusions
-
Markov Chain Monte Carlo methods
'
&
$
%
1. Initialize p02. Iteration i :
a) Draw a new value p∗ from proposal distribution J(·|pi−1). J
specifies p∗ basedon previous value pi−1, and J(·|pi−1) should not
be interpreted as a conditionaldensity.
b) Construct acceptance probability
α(p∗|pi−1) = min(
1,π(p∗|d)J(·|pi−1)π(pi−1|d)J(pi−1|·)
)(1)
c) Sample a Uniform(0,1) random variable U. Set
pi =
{p∗, if U ≤ α(p∗|pi−1)pi−1, otherwise
(2)
�� ��Metropolis-Hastings algorithm
9 / 19
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In practice ...
I Choice for the proposal J(p∗|pi−1)→ random walk metropolis
J(p∗|pi−1) =1√
(2π)mdetΣexp
(−1
2
(p∗ − pi−1
)TΣ−1
(p∗ − pi−1
))
Σ =
σ21 ρ12σ1σ2 · · · ρ1mσ1σm
ρ21σ2σ1 σ22 ρ2mσ2σm
......
. . ....
ρm1σmσ1 ρm2σmσ2 · · · σ2m
I We assume Σ to be a diagonal matrix
I Using log(A) and log(E ) values increases the mixing
10 / 19
-
Table of Contents
Introduction
Characterization of physico-chemical parameters relevant to
pyrolysis reactionsDescription of the problemFormulation of the
problemImplementationApplication to pyrolysis experiments
Conclusions
-
Numerical set-up: one reactant model with one species
I Observable (data) dobsH2k : mass yields of species H2 at each
temperature. Np = 1.
dH2k = FH2m0(ξ(k) − ξ(k−1)
),
∂ξ(k)
∂t=(
1− ξ(k))n
A exp
(− ERTk
),
ξ(0) = 0. 400 600 800 1000 12000
0.1
0.2
0.3
0.4
Temperature, T
Massyields,mg
I Simple point-mass model (0D), k = 1, . . . , nT
I p = {A,E , n,FH2}
11 / 19
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Application to pyrolysis experiments: Markov-Chain
2.5 5 7.5 100
2
4
6·105
Iteration number ·105
A
2.5 5 7.5 101
1.2
1.4
1.6
·105
Iteration number ·105
E
2.5 5 7.5 10
2
3
4
Iteration number ·105
n
2.5 5 7.5 10
3.6
3.8
4
4.2
·10−2
Iteration number ·105F
12 / 19
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Bivariate posterior PDFs
1 1.2 1.4 1.64
6
8
10
12
E · 105
log(A
)
2 2.5 3 3.51
1.2
1.4
1.6
n
E·1
05
3.6 3.8 4 4.2
2
2.5
3
3.5
F · 10−2
n
13 / 19
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Posterior predictive checks assuming 1 reactant (4
parameters)
300 500 700 900 1100 13000
0.1
0.2
0.3
0.4
Temperature, K
Massyield,mg
300 500 700 900 1100 1300
96
98
100
Temperature, K
Sam
ple
mass,%
14 / 19
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Numerical set-up: 3 reactants model with 5 species
I Observable (data) dobsik with i = {H2,CO,CO2,CH4,H2O}: mass
yields at eachtemperature iteration k . Np = 3.
dik =
Np∑j
Fijm0(ξ
(k)j − ξ
(k−1)j
),
∂ξ(k)j
∂t=(
1− ξ(k)j)nj
Aj exp
(− EjRTk
),
ξ(0)j = 0.
400 600 800 1000 12000
0.2
0.4
0.6
0.8
Temperature, T
Massyields,mg
I Simple point-mass model (0D), k = 1, . . . , nT
p = { {A3,E3, n3,FH2O,3, } R1 → H2O{A2,E2,
n2,FCO,2,FCH4,2,FH2O,2} , R2 → CO + CH4 + H2O{A1,E1,
n1,FCO,1,FCO2,1,FH2,1} } R3 → CO + CO2 + H2 15 / 19
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Posterior predictive checks: 3 reactants model with 5
species
300 400 500 600 700 800 900 1000 1100 1200 13000
0.2
0.4
0.6
0.8
Temperature, K
Massyield,mg
16 / 19
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Posterior predictive checks: 3 equations, 5 species
300 400 500 600 700 800 900 1000 1100 120070
80
90
100
Temperature, K
Sam
ple
Mass,%
17 / 19
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Table of Contents
Introduction
Characterization of physico-chemical parameters relevant to
pyrolysis reactionsDescription of the problemFormulation of the
problemImplementationApplication to pyrolysis experiments
Conclusions
-
Conclusions and further work
Conclusions:
I Application of a simple model to simulate pyrolysis
experiments
I Inference on parameter laws using Bayesian approach
I Efficient method to find kinetic parameters and characterize
their uncertainties
I Simulation of mass yields with error bars using the simulated
posterior samples. Goodagreement with experimental curves.
Further work:
I Improve algorithms to allow the use of more elaborated models.
Adaptive MCMC has ahigh rejection rate. Use Hamiltonian Monte
Carlo?
I Inference on reaction mechanisms together with parameter
inference [Galagali andMarzouk, 2016]
I Results interpretation and their used in CFD codes
18 / 19
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Acknowledgments
I K. Hillewaert (Cenaero)
I A. Turchi, B. Helber (von Karman Institute for Fluid
Dynamics)
I H.-W. Wong (University of Massachusetts Lowell)
I F. Panerai, N. Mansour (NASA Ames)
I This work is supported by the Fund for Research Training in
Industry and Agriculture(FRIA) provided by the Belgian Fund for
Scientific Research (F.R.S-FNRS)
19 / 19
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Bayesian Inference on Uncertain Kinetic Parameters for the
Pyrolysis ofComposite Ablators
Joffrey Coheur
M. Arnst1, P. Chatelain2, P. Schrooyen3, T. Magin4
1Aerospace and Mechanical Engineering, Université de
Liège2Institute of Mechanics, Materials and Civil Engineering,
Université Catholique de Louvain
3Cenaero, Gosselies4Aeronautics and Aerospace Department, von
Karman Institute for Fluid Dynamics
SIAM Conference on Uncertainty QuantificationGarden Grove,
California, USA
April 16, 2018
-
Additional Information
20 / 19
-
Table of Contents
Bibliography
Mathematical model
-
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21 / 19
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Simple model for simulating pyrolysis experiments
For point-mass material, conduction is assumed to be
instantaneous and T is uniform insidethe material (lumped
capacitance model, Bi = Lchk 1 (Lc , h↗, or k ↘) more complex model
should be used (Argo).
22 / 19
IntroductionCharacterization of physico-chemical parameters
relevant to pyrolysis reactionsDescription of the
problemFormulation of the problemImplementationApplication to
pyrolysis experiments
ConclusionsAppendixAdditional
InformationBibliographyMathematical model