Combustion and Flame 184 (2017) 55–67 Contents lists available at ScienceDirect Combustion and Flame journal homepage: www.elsevier.com/locate/combustfame A hierarchical method for Bayesian inference of rate parameters from shock tube data: Application to the study of the reaction of hydroxyl with 2-methylfuran Daesang Kim a,b , Iman El Gharamti a,c , Mireille Hantouche d , Ahmed E. Elwardany b,f , Aamir Farooq b , Fabrizio Bisetti a,b,e,∗ , Omar Knio a,g a SRI Center for Uncertainty Quantification, King Abdullah University of Science and Technology, Thuwal, Saudi Arabia b Clean Combustion Research Center, Division of Physical Sciences and Engineering, King Abdullah University of Science and Technology, Thuwal, Saudi Arabia c Department of Applied Mechanics, Aalto University, Aalto, Finland d Division of Physical Science and Engineering, King Abdullah University of Science and Technology, Thuwal, Saudi Arabia e Department of Aerospace Engineering and Engineering Mechanics, University of Texas at Austin, Austin, TX 78712-1085, USA f Mechanical Engineering Department, Faculty of Engineering, Alexandria University, Alexandria 21544, Egypt g Department of Mechanical Engineering and Materials Science, Duke University, Durham, NC 27708, USA a r t i c l e i n f o Article history: Received 3 January 2017 Revised 2 June 2017 Accepted 3 June 2017 Keywords: Chemical kinetics Shock tube Bayesian inference Rate parameters Surrogate model Uncertainty quantification a b s t r a c t We developed a novel two-step hierarchical method for the Bayesian inference of the rate parameters of a target reaction from time-resolved concentration measurements in shock tubes. The method was applied to the calibration of the parameters of the reaction of hydroxyl with 2-methylfuran, which is studied ex- perimentally via absorption measurements of the OH radical’s concentration following shock-heating. In the first step of the approach, each shock tube experiment is treated independently to infer the poste- rior distribution of the rate constant and error hyper-parameter that best explains the OH signal. In the second step, these posterior distributions are sampled to calibrate the parameters appearing in the Ar- rhenius reaction model for the rate constant. Furthermore, the second step is modified and repeated in order to explore alternative rate constant models and to assess the effect of uncertainties in the reflected shock’s temperature. Comparisons of the estimates obtained via the proposed methodology against the common least squares approach are presented. The relative merits of the novel Bayesian framework are highlighted, especially with respect to the opportunity to utilize the posterior distributions of the param- eters in future uncertainty quantification studies. © 2017 The Combustion Institute. Published by Elsevier Inc. All rights reserved. 1. Introduction The parameters appearing in rate constants for chemical reactions are commonly estimated through least squares or optimization approaches based on shock-tube, flame-speed or stirred-reactor data obtained experimentally (e.g., [1–5]). These methodologies aim at calibrating the rate parameters so as to min- imize the misfit between the predictions and rates deduced from experimental measurements. The latter are also inferred quanti- ties, generally estimated by minimizing the difference between the measured and simulated transient concentrations of selected species. In such a framework, the calibration of the chemical rate ∗ Corresponding author at: Department of Aerospace Engineering and Engineer- ing Mechanics, University of Texas at Austin, Austin, TX 78712-1085, USA. E-mail addresses: fbisetti@utexas.edu, fbisetti@gmail.com (F. Bisetti). expressions is based on data that are themselves the result of an optimization. This makes it difficult to quantify the impact of experimental errors or uncertainties in experimental conditions. In the present work, we explore the application of a Bayesian inference formalism to calibrate the chemical rate expression for the reaction of 2-methylfuran with hydroxyl, based on the tran- sient concentration of OH measured during a recent campaign of shock tube experiments [6]. The effort is motivated by a desire to take advantage of the capabilities afforded by Bayesian methods, specifically in accommodating noisy observations and enabling se- quential learning and updating of prior information. Due to these and other capabilities, Bayesian methods have re- ceived growing interest in kinetic calibration studies, e.g., [7–16]. In many cases, the application of Bayesian inverse methods has capitalized on advances in techniques developed in the field of uncertainty quantification (UQ) [17,18]. For example, effective http://dx.doi.org/10.1016/j.combustflame.2017.06.002 0010-2180/© 2017 The Combustion Institute. Published by Elsevier Inc. All rights reserved.