A procedure for determining the joint uncertainty of Arrhenius parameters across multiple combustion reactions of interest is demonstrated. This approach is capable of constructing the joint distribution of the Arrhenius parameters arising from the uncertain measurements performed in specific target experiments without having direct access to the underlying experimental data. The method involves constructing an ensemble of hypothetical data sets with summary statistics consistent with the available information reported by the experimentalists, followed by a fitting procedure that learns the structure of the joint parameter density across reactions using this consistent hypothetical data as evidence. The procedure is formalized in a Bayesian statistical framework, employing maximum-entropy and approximate Bayesian computation methods and utilizing efficient Markov chain Monte Carlo techniques to explore data and parameter spaces in a nested algorithm. We demonstrate the application of the method in the context of experiments designed to measure the rates of selected chain reactions in the H2-O2 system and highlight the utility of this approach for revealing the critical correlations between the parameters within a single reaction and across reactions, as well as for maximizing consistency when utilizing rate parameter information in predictive combustion modeling of systems of interest.
This study evaluates the applicability of the Octane Index (OI) framework under conventional spark ignition (SI) and “beyond Research Octane Number (RON)” conditions using nine fuels operated under stoichiometric, knock-limited conditions in a direct injection spark ignition (DISI) engine, supported by Monte Carlo-type simulations which interrogate the effects of measurement uncertainty. Of the nine tested fuels, three fuels are “Tier III” fuel blends, meaning that they are blends of molecules which have passed two levels of screening, and have been evaluated to be ready for tests in research engines. These molecules have been blended into a four-component gasoline surrogate at varying volume fractions in order to achieve a RON rating of 98. The molecules under consideration are isobutanol, 2-butanol, and diisobutylene (which is a mixture of two isomers of octene). The remaining six fuels were research-grade gasolines of varying formulations. The DISI research engine was used to measure knock limits at heated and unheated intake temperature conditions, as well as throttled and boosted intake pressures, all at an engine speed of 1400 rpm. The tested knock-limited operating conditions conceptually exist both between the Motor Octane Number (MON) and RON conditions, as well as “beyond RON” conditions (conditions which are conceptually at lower temperatures, higher pressures, or longer residence times than the RON condition). In addition to directly assessing the performance of the Tier III blends relative to other gasolines, the OI framework was evaluated with considerations of experimental uncertainty in the knock-limited combustion phasing (KL-CA50) measurements, as well as RON and MON test uncertainties. The OI was found to hold to the first order, explaining more than 80% of the knock-limited behavior, although the remaining variation in fuel performance from OI behavior was found to be beyond the likely experimental uncertainties. This indicates that the effects of specific fuel components on knock which are not captured by RON and MON ratings, and complicating the assessment of a given fuel by RON and MON ratings alone.
Prescribing uncertainty measures to rate expressions is crucial for performing useful predictive combustion computations. Raw experimental measurement data, and its associated noise and uncertainty, is typically unavailable for most reported investigations of elementary reaction rates, making the direct derivation of the desired joint uncertainty structure of the parameters in rate expressions difficult. To approximate this uncertainty structure we construct an inference procedure, relying on maximum entropy and approximate Bayesian computation methods, and using a two-level nested Markov Chain Monte Carlo algorithm, to arrive at a joint density on rate parameters and missing data. This method employs the reported context of a specific experiment to construct a set of hypothetical experimental data profiles consistent with the reported statistics of the data, in the form of error bars on rate constants at the experimental temperatures. Bayesian inference can then be performed using these consistent data sets as evidence to determine the joint posterior density on the rate parameters for any choice of chemical model. The method is also used to demonstrate the combination of missing data from different experiments for the generation of consensus rate expressions using these multiple sources of experimental evidence.
The thermal decomposition of H2O2 is an important process in hydrocarbon combustion playing a particularly crucial role in providing a source of radicals at high pressure where it controls the 3rd explosion limit in the H2-O2 system, and also as a branching reaction in intermediatetemperature hydrocarbon oxidation. As such, understanding the uncertainty in the rate expression for this reaction is crucial for predictive combustion computations. Raw experimental measurement data, and its associated noise and uncertainty, is typically unreported in most investigations of elementary reaction rates, making the direct derivation of the joint uncertainty structure of the parameters in rate expressions difficult. To overcome this, we employ a statistical inference procedure, relying on maximum entropy and approximate Bayesian computation methods, and using a two-level nested Markov Chain Monte Carlo algorithm, to arrive at a posterior density on rate parameters for a selected case of laser absorption measurements in a shock tube study, subject to the constraints imposed by the reported experimental statistics. The procedure constructs a set of H2O2 concentration decay profiles consistent with these reported statistics. These consistent data sets are then used to determine the joint posterior density on the rate parameters through straightforward Bayesian inference. Broadly, the method also provides a framework for the replication and comparison of missing data from different experiments, based on reported statistics, for the generation of consensus rate expressions.
The thermal decomposition of H2O2 is an important process in hydrocarbon combustion playing a particularly crucial role in providing a source of radicals at high pressure where it controls the 3rd explosion limit in the H2-O2 system, and also as a branching reaction in intermediatetemperature hydrocarbon oxidation. As such, understanding the uncertainty in the rate expression for this reaction is crucial for predictive combustion computations. Raw experimental measurement data, and its associated noise and uncertainty, is typically unreported in most investigations of elementary reaction rates, making the direct derivation of the joint uncertainty structure of the parameters in rate expressions difficult. To overcome this, we employ a statistical inference procedure, relying on maximum entropy and approximate Bayesian computation methods, and using a two-level nested Markov Chain Monte Carlo algorithm, to arrive at a posterior density on rate parameters for a selected case of laser absorption measurements in a shock tube study, subject to the constraints imposed by the reported experimental statistics. The procedure constructs a set of H2O2 concentration decay profiles consistent with these reported statistics. These consistent data sets are then used to determine the joint posterior density on the rate parameters through straightforward Bayesian inference. Broadly, the method also provides a framework for the replication and comparison of missing data from different experiments, based on reported statistics, for the generation of consensus rate expressions.
The reaction of OH with H2 is a crucial chain-propagating step in the H2-O2 system thus making the specification of its rate, and its uncertainty, important for predicting the high-temperature combustion of hydrocarbons. In order to obtain an uncertain representation of this reaction rate in the absence of actual experimental data, we perform an inference procedure employing maximum entropy and approximate Bayesian computation methods to discover hypothetical data from a target shock-tube experiment designed to measure the reverse reaction rate. This method attempts to invert the fitting procedure from noisy measurement data to parameters, with associated uncertainty specifications, to arrive at candidate noisy data sets consistent with these reported parameters and their uncertainties. The uncertainty structure of the Arrhenius parameters is obtained by fitting each hypothetical data set in a Bayesian framework and pooling the resulting joint parameter posterior densities to arrive at a consensus density. We highlight the advantages of working with a data-centric representation of the experimental uncertainty with regards to model choice and consistency, and the ability for combining experimental evidence from multiple sources. Finally, we demonstrate the utility of knowledge of the joint Arrhenius parameter density for performing predictive modeling of combustion systems of interest.