Bayesian parameter inference for PICA devolatilization pyrolysis at high heating rates

Bayesian parameter inference for PICA devolatilization pyrolysis at high heating rates

Joffrey Coheur

Promotors:M. Arnst1_{, P. Chatelain}2_{, T. Magin}4

Collaborators:P. Schrooyen3_{, K. Hillewaert}3_{, A. Turchi}4_,

F. Panerai5_{, J. Meurisse}5_{, N. N. Mansour.}5 1_{Aerospace and Mechanical Engineering, Universit´e de Li`ege}

2_{Institute of Mechanics, Materials and Civil Engineering, Universit´e Catholique de Louvain} 3_{Cenaero, Gosselies}

4_{Aeronautics and Aerospace Department, von Karman Institute for Fluid Dynamics} 5_{NASA Ames Research Center}

3rd International Conference on Uncertainty Quantification in Computational Sciences and Engineering

Motivation: atmospheric entry

Thermal protection systems (TPS)

Dragon capsule (Space X)

Space debris

Ablative materials for thermal protection systems

I PICA: Phenolic-Impregnated Carbon Ablator

Mars Science Laboratory thermal protection system (NASA)

Porous thermal protection material [Helber, 2016] Bottom view (after burn) Lawson et al. Fibers + resin 2 / 22

Modeling the pyrolysis of ablative thermal protection materials

Aerodynamic simulation (CooLFluiD, VKI) Aerothermal simulation (Argo, Cenaero) [Coheur, Schrooyen, 2017] High T°

gas flow Pyrolysis

gas Ablation

I CFD codes require accurate models for ablation [Lachaud, 2014; Schrooyen, 2016], e.g. conservation of mass species

∂ghρiig

∂t +∇ · (ghρiighuig) =∇ · hJii +h ˙ω pyro

i i

I Objectives: deduce h ˙ωipyroi from dedicated pyrolysis experiments

Table of Contents

Introduction

Characterization of physico-chemical parameters relevant to pyrolysis reactions Experiments

Modeling Challenges

Non-linear posterior and highly-correlated parameters Zero-variance data in the dataset

Application to complex pyrolysis model Conclusions

Table of Contents

Introduction

Characterization of physico-chemical parameters relevant to pyrolysis reactions Experiments

Modeling Challenges

Non-linear posterior and highly-correlated parameters Zero-variance data in the dataset

Application to complex pyrolysis model Conclusions

Pyrolysis experiments [Wong et al., 2015; Bessire and Minton, 2017]

Phenolic sample (50 mg) Pyrolysis gases Heated wall 400 600 800 1000 1200 0 0.2 0.4 0.6 0.8 Temperature, T Mass yields, mg H2 CO CO2 CH4 H2O 400 600 800 1000 1200 30 40 50 Temperature, K Sample mass, mg 4 / 22

Table of Contents

Introduction

Characterization of physico-chemical parameters relevant to pyrolysis reactions Experiments

Modeling Challenges

Non-linear posterior and highly-correlated parameters Zero-variance data in the dataset

Application to complex pyrolysis model Conclusions

Phenomenological laws for pyrolysis mass loss and gas species production

I Pyrolysis decomposition of ablative materials follows successive reaction rates [Goldstein, 1969; Trick, 1997] 400 600 800 1000 1200 0 2 4 6 8 ·10−3 Temperature, K ∂ ξj /∂ T j = 1 j = 2 400 600 800 1000 1200 0.4 0.6 0.8 1 Temperature, K m / m0 h ˙ωpyroi i= Np X j Fij∂ξj ∂Tτ m0 ∂ξj ∂T = (1− ξj) nj Aj τ exp  − Ej RT 

I ξj: advancement of reaction of the fictitious resin component j

Phenomenological laws for pyrolysis mass loss and gas species production

I Pyrolysis decomposition of ablative materials follows successive reaction rates [Goldstein, 1969; Trick, 1997] 400 600 800 1000 1200 0 2 4 6 8 ·10−3 Temperature, K ∂ ξj /∂ T j = 1 j = 2 400 600 800 1000 1200 0.4 0.6 0.8 1 Temperature, K m / m0 m = m0− Np X j Fjξjm0 ∂ξj ∂T = (1− ξj) nj Aj τ exp  − Ej RT 

Bayesian inference for parameter calibration

Objective: Infer on a finite set of parameters p∈ Rq _from

I A set of measurements dobs_={d

i ∈ R, i = 1, . . . , nobs}

I A model that predicts the measurements

using Bayesian rule for improving our knowledge on p

π(p_|dobs) = π(d obs_|p)π 0(p) R Rpπ(d obs_|p)π 0(p)dp (Bayes’ theorem)

I Choice for the prior π0(p): uniform pdf (bounded support)

I Choice for the likelihood π(dobs|p): additive Gaussian noise π(dobs_{|p) =} 1 Πi(2πσi2)n/2 exp − ns X i=1 n X k=1 dobs ik − ηi(xk, p) 2 2σi2 ! 6 / 22

Benchmark: two-equation model

200 400 600 800 1,000 1,200 1,400 0 0.5 1 1.5 ·10 −3 Temperature, K Pro duction rate, s − 1 Reaction I Reaction II

Synthetic data using: σI= 0.149 s−1

σII = 0.5382 s−1

A E m F

Reaction I 541218 74046.7 9.08 0.11 Reaction II 5360000 103680 5.07 0.35

Random-Walk Metropolis-Hastings: bivariate posterior PDFs

0.4 0.6 0.8 1 1.2 1.4 ·105 0 0.5 1 1.5 2 ·10 8 E A 4 6 8 10 12 14 0 0.5 1 1.5 2 ·10 8 m A 0 0.1 0.2 0.3 0.4 0 0.5 1 1.5 2 ·10 8 F A 4 6 8 10 12 14 0.4 0.6 0.8 1 1.2 1.4 ·10 5 m E 0 0.1 0.2 0.3 0.4 0.4 0.6 0.8 1 1.2 1.4 ·10 5 F E 0 0.1 0.2 0.3 0.4 4 6 8 10 12 14 F m 8 / 22

Challenges for the inference of pyrolysis kinetic parameters

I Non-linear posterior and highly-correlated parameters: random-walk Metropolis-Hastings may fail or be very slow, tuning of the proposal covariance hard for high dimensional problems

• Reparametrization of parameters space adapted to the physical model

• Gradient-based algorithms (Itˆo-SDE)

I Fitting unimportant data may biased the results

• Hyperparameter choice

Table of Contents

Introduction

Characterization of physico-chemical parameters relevant to pyrolysis reactions Experiments

Modeling Challenges

Non-linear posterior and highly-correlated parameters Zero-variance data in the dataset

Application to complex pyrolysis model Conclusions

Reparametrization of parameters space adapted to the physical model

I More complex models become prohibitive in terms of number of iterations required I Need to improve the mixing of the Markov chains

Non-linear transformation of the parameter space ˜ Ai = ln Ai − Ei/(RTi), ˜ Ei = Ei/Ei, ˜ mi = mi/ ¯mi, ˜ Fi = Fij/Fij. Reaction rate: ki = exp ˜Ai + ˜EiT˜i(t) 

Local reciprocal temperature: ˜

Benchmark: two-equation model

200 400 600 800 1,000 1,200 1,400 0 0.5 1 1.5 ·10 −3 Temperature, K Pro duction rate, s − 1 Reaction I Reaction II

Synthetic data using: σI= 0.149 s−1 σII = 0.5382 s−1 A E m F Reaction I 541218 74046.7 9.08 0.11 Reaction II 5360000 103680 5.07 0.35 11 / 22

Bivariate posterior PDFs (initial parameter space)

0.4 0.6 0.8 1 1.2 1.4 ·105 0 0.5 1 1.5 2 ·10 8 E A 4 6 8 10 12 14 0 0.5 1 1.5 2 ·10 8 m A 0 0.1 0.2 0.3 0.4 0 0.5 1 1.5 2 ·10 8 F A 4 6 8 10 12 14 0.4 0.6 0.8 1 1.2 1.4 ·10 5 m E 0 0.1 0.2 0.3 0.4 0.4 0.6 0.8 1 1.2 1.4 ·10 5 F E 0 0.1 0.2 0.3 0.4 4 6 8 10 12 14 F m

Bivariate posterior PDFs (rescaled parameter space)

0.8 1 1.2 −2.6 −2.4 −2.2−2 −1.8 −1.6 ˜ E ˜ A 0.6 0.8 1 1.2 1.4 1.6 −2.6 −2.4 −2.2−2 −1.8 −1.6 ˜ m ˜ A 0.9 1 1.1 1.2 −2.6 −2.4 −2.2−2 −1.8 −1.6 ˜ F ˜ A 0.6 0.8 1 1.2 1.4 1.6 0.8 1 1.2 ˜ m ˜ E 0.9 1 1.1 1.2 0.8 1 1.2 ˜ F ˜ E 0.9 1 1.1 1.2 0.6 0.8 1 1.2 1.4 1.6 ˜ F ˜m 13 / 22

Itˆ

o Stochastic Differential Equation based MCMC method

We use the Itˆo Stochastic Differential Equation (ISDE) introduced by Soize (2008) dΞ = [ bC ]Hdt dH = −∇ξφ(Ξ)dt− 1 2ζ0Hdt + pζ0LCˆ −T dW with φ =− log(π(dobs_{|p)), [ bC ]: approximation of the covariance matrix, ζ}

0: free parameter

(Arnst, Soize (2019)). Discretization in time using a Stormer-Verlet method (Soize, Ghanem (2016)) Ξ(`+12) = Ξ(`)+4t 2 [ bC ]H(`) H(`+1)₌ 1−b 1+bH(`)−1+b4t ∇ξφ  Ξ(`+12)  + √ζ0 1+bLCb −T ∆W(`+1) Ξ(`+1)= Ξ(`+12) + 4t 2 [ bC ]H(`+1)

Table of Contents

Introduction

Characterization of physico-chemical parameters relevant to pyrolysis reactions Experiments

Modeling Challenges

Non-linear posterior and highly-correlated parameters Zero-variance data in the dataset

Application to complex pyrolysis model Conclusions

Zero variance data in the data-set

I Hyperparameter choice π(dobs_{|p) =} 1 Πi(2πσi2)n/2 exp − ns X i=1 n X k=1 dobs ik − ηi(xk, p) 2 2σi2 !

I Taking all the data might lead to inaccurate results, although zero variance data (before activation temperature of after reaction) are obtained experimentally

I Feedback on the experiments: fit only important features (activation temperature, maximum production peaks)

Illustration using experimental data: two-reaction model with one species

I Observable (data) d_Hobs_2k: mass yields of species H2 at each temperature. Np= 2.

dH2k = Np X j FH2,jm0  ξ_j(k)− ξj(k−1)  , ∂ξ_j(k) ∂t =  1− ξ(k)j nj Ajexp  − Ej RTk  , ξ_j(0)= 0. 400 600 800 1000 1200 1400 0 2 4 ·10−6 Temperature, K Pro duction rate, s − 1

I Simple point-mass model (0D), k = 1, . . . , nT

I p={A1, E1, n1, FH2,1, A2, E2, n2, FH2,2}

I Initial guess obtained from genetic algorithm

Comparison of the different approaches

I Comparison of posterior samples obtained from the Markov chains (length = 1e4) using Itˆo-SDE (blue dots) and random-walk Metropolis-Hastings in the reparametrized parameter space (orange dots) and in the initial parameter space (green dots)

0.9 1 1.1 1.2 1.3 1.4 1.5 ·105 0 0.5 1 ·107 E A

Table of Contents

Introduction

Characterization of physico-chemical parameters relevant to pyrolysis reactions Experiments

Modeling Challenges

Non-linear posterior and highly-correlated parameters Zero-variance data in the dataset

Application to complex pyrolysis model Conclusions

Application to pyrolysis experiments [Bessire and Minton, 2015]

Proposed five-equation model (38 unknown parameters):

R1(s) F1,1H2O + F1,2CO2+ F1,3(1-propanol)

k1

R2(s) _FF2,1H2O + F2,2CO + F2,3CO2 + F2,4Phenol + F2,5Cresol + 2,6(Dimethyl Phenol) + F2,7(Trimethyl Phenol)

k2

R3(s) _FF3,1CH4 + F3,2H2O + F3,3CO + F3,4H2+ F3,5CO2+ 3,6Xylene + F3,7Benzene + F3,8Toluene

k3

R4(s) k4 F4,1H2

Propagation results: production rate curves

400 600 800 1,000 1,200 1,400 0 2 4 6 ·10−6 Temperature, K Pro duction rate, s − 1 H2O H2 400 600 800 1,000 1,200 1,400 0 1 2 3 ·10−6 Temperature, K Pro duction rate, s − 1 CH4 CO 19 / 22

Propagation results: production rate curves

400 600 800 1,000 1,200 1,400 0 1 2 3 ·10−7 Temperature, K Pro duction rate, s − 1 Phenol Dimethyl phenol Trimethyl phenol 400 600 800 1,000 1,200 1,400 0 1 2 3 4 ·10−7 Temperature, K Pro duction rate, s − 1 CO2 Cresol

Propagation results: production rate curves

400 600 800 1,000 1,200 1,400 0 2 4 ·10−8 Temperature, K Pro duction rate, s − 1 1-propanol Benzene 400 600 800 1,000 1,200 1,400 0 2 4 6 8 ·10−8 Temperature, K Pro duction rate, s − 1 Xylene Toluene 19 / 22

Table of Contents

Introduction

Characterization of physico-chemical parameters relevant to pyrolysis reactions Experiments

Modeling Challenges

Non-linear posterior and highly-correlated parameters Zero-variance data in the dataset

Application to complex pyrolysis model Conclusions

Conclusions

I Application of a simple model to simulate pyrolysis experiments.

I Inference on the parameters of pyrolysis reactions using Bayesian approach.

I Efficient method to infer on kinetic parameters and characterize their uncertainties. However, lack of indentifiability for A and E .

I Rescaling the parameter space to their posterior distribution based on model features reduce significantly the tuning of the proposal covariance

I Ito-SDE method enables no proposal tuning and samples efficiently the posterior distribution, but requires the computation of gradients

I Simulations of gas production rate and mass loss with error bars lead to good agreement with experimental curves.

Future work

I Apply Itˆo-SDE method to the full set of experimental data

I Efficient computation of model gradients, e.g. adjoint-based

I Use more general models, e.g. include competitive reaction schemes (F. Torres, J. Blondeau), include more data (different heating rates)→ model inference

I Results interpretation and their used in CFD codes; uncertainty propagation for the validation of CFD codes

Acknowledgments

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)

I Scientific stay at NASA ARC was supported by the Wallonie-Bruxelles International grant WBI.World

I F. Torres (VKI), B. Helber (VKI), J. Lachaud (I2M Bordeaux), H.-W. Wong (UMassachusetts Lowell), George Bellas-C. (VKI), J.-B. Scoggins (VKI)

Bayesian parameter inference for PICA devolatilization pyrolysis at high heating rates

Joffrey Coheur

Promotors:M. Arnst1_{, P. Chatelain}2_{, T. Magin}4

Collaborators:P. Schrooyen3_{, K. Hillewaert}3_{, A. Turchi}4_,

F. Panerai5_{, J. Meurisse}5_{, N. N. Mansour.}5 1_{Aerospace and Mechanical Engineering, Universit´e de Li`ege}

2_{Institute of Mechanics, Materials and Civil Engineering, Universit´e Catholique de Louvain} 3_{Cenaero, Gosselies}

4_{Aeronautics and Aerospace Department, von Karman Institute for Fluid Dynamics} 5_{NASA Ames Research Center}

3rd International Conference on Uncertainty Quantification in Computational Sciences and Engineering

Additional Information

Table of Contents

Bibliography

Mathematical model Model Selection

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 probabilistic programming 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. In 47th 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 in atmospheric 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 Competition in 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 from phenol-formaldehyde resin pyrolysis. Polymer Degradation and Stability, 112:122–131, 2015

I C. Soize, Construction of probability distributions in high dimension using the maximum entropy principle: Applications to stochastic processes, random fields and random matrices, International Journal for Numerical Methods in Engineering 76 (2008).

I C. Soize, R. Ghanem, Data-driven probability concentration and sampling on manifold, Journal of Computational Physics 321 (2016) 242–258

Simple model for simulating pyrolysis experiments

For point-mass material, conduction is assumed to be instantaneous and T is uniform inside the material (lumped capacitance model, Bi = Lch

k << 1). I T = Ti = cte in ∆t = [0, tf] ξ(t) = 1−  (1− n)(Ci− A exp  −_RTE i  t) 1/(n−1) , with Ci= (1_{− ξ(0))}1−n 1_{− n}

I Tout = τ t, τ heating rate

ξ(T ) = 1₋  (1_{− n)}  −A_τT exp  −E RT  −A_τE_REi  −_RTE  + C 1−n1 C = (1− ξ0) 1_{− n} 1−n +A τT0exp  −E RT0  + Ei  −E RT0  EA τ , where Ei(x)≡ Z −x −∞ exp(v ) v dv

For more complex temperature evolution, integration should be performed numerically. When Bi > 1 (Lc, h%, or k &) more complex model should be used (Argo).

Numerical set-up: 3 reactants model with 5 species

I Observable (data) dobs

ik with i ={H2,CO,CO2,CH4,H2O}: mass yields at each

temperature iteration k. Np= 3. dik = Np X j Fijm0  ξ_j(k)− ξj(k−1)  , ∂ξ_j(k) ∂t =  1− ξ(k)j nj Ajexp  − Ej RTk  , ξ_j(0)= 0. 400 600 800 1000 1200 0 0.2 0.4 0.6 0.8 Temperature, T Mass yields, 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

Posterior predictive checks: 3 reactants model with 5 species

300 400 500 600 700 800 900 1000 1100 1200 1300 0 0.2 0.4 0.6 0.8 Temperature, K Mass yields, mg

Posterior predictive checks: 3 equations, 5 species

300 400 500 600 700 800 900 1000 1100 1200 70 80 90 100 Temperature, K Sample Mass, % 28 / 22

How to select an appropriate model?

I Principle of Parsimony (Occam’s razor): “Shave away all that is unnecessary” Accuracy >< complexity I Kullback-Leibler information I (f , g ) = Z f (x) log  f (x) g (x|θ)  dx I Information criteria AIC =−2 log(L(y|ˆθ)) + 2K BIC =−2 log(L(y|ˆθ)) + K log n

Illustration with the two-equation benchmark

I Approximate model: one-equation model

400 600 800 1000 1200 1400 0 1 2 ·10−4 Temperature, K Pro duction rate 30 / 22

Illustration with the two-equation benchmark

I Approximate model: two-equation model

400 600 800 1000 1200 1400 0 1 2 ·10−4 Temperature, K Pro duction rate

Illustration with the two-equation benchmark

I Approximate model: three-equation model

400 600 800 1000 1200 1400 0 1 2 ·10−4 Temperature, K Pro duction rate 30 / 22

Illustration with the two-equation benchmark

I Computation of information criteria:

Model log-like Nparams Nsamples AIC AICc BIC

1 reaction −851.59 4 101 1711.172 1711.588 1711.188

2 reactions −46.08 8 101 108.1501 109.7153 108.1847

3 reactions −46.51 12 101 117.0258 120.5712 117.1078 3 reactions (2) −46.27 12 101 116.54 120.0855 116.592