Surrogate models and optimal design of experiments for chemical - PowerPoint PPT Presentation

Surrogate models and optimal design of experiments for chemical kinetics applications . Bisetti, A. Farooq, D. Kim, O. Knio, Q. Long, R. Tempone | January 7, 2015 F CLEAN COMBUSTION RESEARCH CENTER www.kaust.edu.sa King Abdullah University of Science and Technology | Reactive Flow Modeling Laboratory

Fossil fuels sustain worldwide energy demand through 2030 "B66B?2$L?1 Energy growth is mainly related to '. 0121345617 S GDP 89:614; '* <=>;? Largest growth in Non-OECD @47 countries '% AB6 Pollutants (CO 2 emissions, C?46 + particulates, etc.) act as a constraint / The role of combustion ( Combustion technology enables more & efficient and cleaner processes: power, '.,& '+'& '+*& '++& %&(& transportation, industrial S$T2:69>17$5B?M9167 Source: BP Energy Outlook 2030 (2011) Jan 7, 2014 2/28 F. Bisetti, A. Farooq, D. Kim, O. Knio, Q. Long, R. Tempone – Surrogate models and optimal design of experiments for chemical kinetics applications

The complex network of reactions... Hydrocarbon fuels (C n H m O p ) and oxygen react to produce H 2 O and CO 2 (and traces of pollutants...) with intense energy release (“heat”) This process occurs through thousands of intermediate species and reactions (e.g., H + O 2 − − ⇀ − OH + O) ↽ − Overall rate of reaction controls: (a) heat release, pressure rise, ...; (b) stability and robustness of combustion process; (c) pollutant emissions Chemical kinetic models The formulation of a (complex) network of reactions involving many chemical species is key to the accurate and faithful description of combustion processes ⇒ Reaction are characterized by reaction rate parameters, which are the model parameters we are concerned with in this talk Jan 7, 2014 3/28 F. Bisetti, A. Farooq, D. Kim, O. Knio, Q. Long, R. Tempone – Surrogate models and optimal design of experiments for chemical kinetics applications

A single reaction and its description Consider the chain branching reaction H + O 2 − − − OH + O − ⇀ ↽ The rate at which this reaction occurs (mol cm − 3 s − 1 ) is given by the following expression q = k f C H C O 2 − k b C OH C O (1) where k f , b are the forward and backward rate “constants” and C i the species concentration (mol cm − 3 ). The “Arrhenius form” is customarily used to reflect T dependence k f = k f ( T ) = A T b exp ( − E / R T ) (2) k b = K eq ( T ) k f (3) A (pre-exponential factor), b (temperature exponent), and E (activation energy) are the model parameters θ = ( A , b , E ) . Jan 7, 2014 4/28 F. Bisetti, A. Farooq, D. Kim, O. Knio, Q. Long, R. Tempone – Surrogate models and optimal design of experiments for chemical kinetics applications

A fundamental tool in combustion: shock tube pressure diaphragm sensors end driver driven section wall (inert gas) (test mixture) optical detector T 0 , p 0 shock reflected shock Adapted from Petersen Combust Sci Tech 2009. Allows to bring a gas to p 0 and T 0 instantaneously via a (reflected) shock Time evolution of spatially homogeneous mixture is modeled as system of ODEs Ideal for kinetics studies when complemented with advanced diagnostics: τ ign , X OH (t), X CH 4 ( t ) , etc. Jan 7, 2014 5/28 F. Bisetti, A. Farooq, D. Kim, O. Knio, Q. Long, R. Tempone – Surrogate models and optimal design of experiments for chemical kinetics applications

Seeking to measure reaction rate parameters Goal. Apply a framework of optimal experimental design to the characterization of elementary reactions of interest to Prof. Farooq H + O 2 − − ⇀ − O + OH ↽ − Methylfuran + OH − − → products Key features Bayesian inference approach A measure of information gain A fast estimate of the expected information gain Polynomial Chaos expansion as surrogates for the model Sparse adaptive pseudo-spectral projection to build surrogates Jan 7, 2014 6/28 F. Bisetti, A. Farooq, D. Kim, O. Knio, Q. Long, R. Tempone – Surrogate models and optimal design of experiments for chemical kinetics applications

Measuring the rate of H + O 2 − − O + OH (I) ⇀ ↽ Data gathering A shock tube is filled with H 2 , O 2 , Ar (inert diluent) in prescribed proportions The mixture is shock-heated at p 0 and T 0 and water H 2 O is formed The concentration of H 2 O is measured by time-resolved tunable diode laser absorption The ODE system is solved adjusting the rate constant until a good fit is found Water conc. [ppm] 2500 Best fit k 1 2000 Best fit k 1 x 1.1 1500 Best fit k 1 x 0.9 1000 Experiment 500 (a) 0 0.0 0.2 0.4 0.6 0.8 1.0 Time [ms] Jan 7, 2014 7/28 F. Bisetti, A. Farooq, D. Kim, O. Knio, Q. Long, R. Tempone – Surrogate models and optimal design of experiments for chemical kinetics applications

Measuring the rate of H + O 2 − − O + OH (II) ⇀ ↽ Selecting the composition $ Typically tests are run for very 1 1 diluted fuel/rich mixtures: e.g., O 2 Sensitivity Index at 0.1%, H 2 at 0.9% and balancing Sens. Index Ar 0.5 0.5 For these mixture compositions , H + O 2 = O + OH the peak rate of water formation is very sensitive to H + O 2 − − ⇀ − O + OH ↽ − 0 0 1 5 10 11 20 1 5 10 15 20 Reaction ID Reaction ID Note. The selection of “optimal mixture composition” is a result of years of experience and considers issues of signal-to-noise ratio, detectability thresholds, and time resolution Jan 7, 2014 8/28 F. Bisetti, A. Farooq, D. Kim, O. Knio, Q. Long, R. Tempone – Surrogate models and optimal design of experiments for chemical kinetics applications

Measuring the rate of H + O 2 − − O + OH (III) ⇀ ↽ Obtaining A , b , and E Tests are performed for various values 3333 K 2000 K 1429 K 1111 K 13 2x10 of T 0 K 13 10 For each test, the “best fit” k is found by -1 ] solving the ODE system with perturbed -1 sec 3 mol 12 10 values of k k 1 [cm The pairs ( k , T 0 ) i are “best fit” to the Hong et al. (2010) Masten et al. (1990) Arrhenius expression Pirraglia et al. (1989) 11 10 Hong et al. (2010) 10 4x10 0.3 0.5 0.7 0.9 1000/ T [1/K] Quoting from Hong et al.: “The rate is given as k = ( 1 . 04 ± 0 . 03 ) × 10 14 exp [( 7705 ± 40 ) / T ] cm 3 mol − 1 s − 1 over the temperature range 1100 to 3370 K based on forward propagation of uncertainties on other reactions, accuracy of experimental diagnostics, knowledge of T 0 , etc.” Jan 7, 2014 9/28 F. Bisetti, A. Farooq, D. Kim, O. Knio, Q. Long, R. Tempone – Surrogate models and optimal design of experiments for chemical kinetics applications

Bayesian experimental design We begin with a model for the experiment y = G ( θ , ζ ) + ǫ (4) y : observable (i.e, maximum rate of increase of [H 2 O]) θ : model parameters (Arrhenius coefficients of target reaction, A , b , and E ) ζ : experimental design variables (e.g., temperature and mixture composition) G : combustion simulation model (approximated by surrogate). ǫ : measurement noise, assumed to be Gaussian. The evaluation of G requires the solution of an initial value problem featuring a system of ODEs of relative large size Jan 7, 2014 10/28 F. Bisetti, A. Farooq, D. Kim, O. Knio, Q. Long, R. Tempone – Surrogate models and optimal design of experiments for chemical kinetics applications

Information gain metric We adopt the Kullback-Leibler divergence as information gain � log p ( θ | y ) D KL = p ( θ ) p ( θ | y ) d θ . (5) Θ with the expected information gain, � � log p ( θ | y ) I = E [ D KL ] = p ( θ ) p ( θ | y ) d θ p ( y ) d y . (6) Θ Y where p ( θ ) : prior p ( θ | y ) : Bayesian posterior Jan 7, 2014 11/28 F. Bisetti, A. Farooq, D. Kim, O. Knio, Q. Long, R. Tempone – Surrogate models and optimal design of experiments for chemical kinetics applications

Computing the expectation of D KL The computation of I = E [ D KL ] is challenging as it must be computationally inexpensive if seeking to maximize I as a function of experimental conditions ζ . Double loop Monte Carlo Laplace approximation to D KL We use a Laplace approximation (Long et al. 2013, 2014) Analytical integration against the posterior measure The calculation of I is reduced to a single loop Requires derivatives of the model (Hessian matrix) Because it’s fast, it enables searches for the optimized experimental designs Jan 7, 2014 12/28 F. Bisetti, A. Farooq, D. Kim, O. Knio, Q. Long, R. Tempone – Surrogate models and optimal design of experiments for chemical kinetics applications

Surrogates Fast model surrogates for the estimations of D KL and I are required and truncated Polynomial Chaos expansion (PCE) are used to construct such surrogates N � G ( ξ ) ≈ c k Ψ k ( ξ ) (7) k = 0 ξ = ( ξ 1 , ..., ξ α ) : a vector of germs which maps model parameters θ and design variables ζ . G ( ξ ) : Quantities of interest (QoI). { Ψ k } : a set of multi-dimensional Legendre polynomials. c k = � G , Ψ k � / � Ψ k , Ψ k � obtained via sparse adaptive pseudo-spectral projection (Winokur et al. 2013) The surrogates provide derivatives needed for Laplace approximation Jan 7, 2014 13/28 F. Bisetti, A. Farooq, D. Kim, O. Knio, Q. Long, R. Tempone – Surrogate models and optimal design of experiments for chemical kinetics applications