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

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

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

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

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

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

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

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

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

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

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

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

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

Bibliography

I M. Bruns. Inferring and propagatin kinetic parameter uncertainty for condensed phase burning models. Fire Technology, 52:93–120, 2016.

I B. Carpenter, A. Gelman, M. D. Hoffman, D. Lee, D. Goodrich, M. Betancourt, M. Brubaker, J. Guo, P. Li, and A. Riddell. Stan: A probabilisticprogramming language. Journal of Statistical Software, 76(1), 2017. DOI 10.18637/jss.v076.i01

I J. Coheur, A. Turchi, P. Schrooyen and T. Magin. Development of a unified model for flow-material interaction applied to porous charring ablators. In47th AIAA Thermophysics Conference, AIAA Aviation Forum 2017.

I N. Galagali and Y. Marzouk. Bayesian inference of chemical kinetic kinetics model from proposed reactions. Chemical Engineering Science,123:170–190, 2015.

I H. E. Goldstein. Pyrolysis Kinetics of Nylon 6-6, Phenolic Resin, and Their Composites. Journal of Macromolecular Science: Part A - Chemistry,3:649–673, 1969.

I B. Helber, A. Turchi, J. Scoggins, A. Hubin and T. Magin. Experimental investigation of ablation and pyrolysis processes of carbon-phenolic ablators inatmospheric entry plasmas. International Journal of Heat and Mass Transfer, 100:810–824, 2016.

I J. Lachaud and N. N. Mansour. Porous-material analysis toolbox based on Open-FOAM and applications. Journal of Thermophysics and Heat Transfer,28(2):191–202, 2014.

I P. Schrooyen, K. Hillewaert, T. E. Magin, P. Chatelain. Fully Implicit Discontinuous Galerkin Solver to Study Surface and Volume Ablation Competitionin Atmospheric Entry Flows. International Journal of Heat and Mass Transfer, 103:108–124, 2016.

I K. A. Trick, T. E. Saliba, and S. S. Sandhu. A kinetic model of the pyrolysis of phenolic resin in a carbon/phenolic composite. Carbon, 35:393–401,1997.

I H.-W. Wong, J. Peck, R. Bonomi, J. Assif, F. Panerai, G. Reinish, J. Lachaud and N. Mansour. Quantitative determination of species production fromphenol-formaldehyde resin pyrolysis. Polymer Degradation and Stability, 112:122–131, 2015

21 / 19

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