Temperature-Dependent Feature Sensitivity Analysis for Combustion Modeling

Sensitivity analysis is one of the most widely used tools in kinetic modeling. Typically, it is performed by perturbing the A-factors of the individual reaction rate coefficients and monitoring the effect of these perturbations on the observables of interest. However, the sensitivity coefficients obtained in this manner do not contain any information on possible temperature dependent effects. Yet, in many combustion processes, especially in premixed flames, the system undergoes substantial temperature changes, and the relative importance of individual reaction rates may vary significantly within the flame. An extension of conventional sensitivity analysis developed in the present work provides the means of identifying the temperatures at which individual reaction rate coefficients are most important as a function of input parameters and specific experimental conditions. The obtained information is demonstrated to be of critical relevance in optimizing complex reaction schemes against multiple experimental targets. Applications of the presented approach are not limited to sensitivities with respect to reaction rate coefficients; the method can be used for any temperaturedependent property of interest (such as binary diffusion coefficients). This application is also demonstrated in this paper. Introduction Modeling of combustion phenomena requires knowledge of reaction rates, diffusion coefficients, and other physical/chemical properties of the reactants as input parameters, and produces the observables (such as species concentration and temperature profiles, laminar flame speed) as the output, with the input and the output connected by dynamic equations. Application of sensitivity analysis, a useful and efficient approach to examine the relationship of the underlying input and output, has been adopted in kinetic modeling sometime ago (e.g., [1,2]). Since sensitivity analysis methods were introduced to combustion research (e.g., [3-5]), they have widely used either to aid in qualitatively understanding the observed kinetic behavior or to assist in improving or optimizing kinetic models. Sensitivity analyses are usually performed on the reaction rates, specifically the A-factors of the corresponding rate coefficients, by perturbing them with a small constant, thus essentially perturbing the specific rate constant over the entire temperature range of the calculation (Fig. 1). However, in many combustion processes, especially in flames, the system transformation inherently occurs over a very wide range of temperature. Given the fact that reaction rates generally exhibit widely differing and nonlinear dependence on temperature, the relative sensitivities of the reactions are also expected to vary with temperature. Sensitivity methods that vary only A-factors cannot provide information on temperature dependence of the sensitivity spectrum. Free-radical diffusion rates can be as critical as elementary reaction kinetics in accurately predicting a combustion process, and the main model input parameters (binary diffusion coefficients) are also temperature-dependent. Although the importance of species diffusion in flame kinetics has long been recognized, sensitivity analysis with respect to diffusion coefficients was not commonly performed until very recently [6]. Mishra et al. [6] applied sensitivity analysis to a burner-stabilized flame to qualitatively describe the influence of diffusion coefficients on the predicted flame structure. More recent efforts by Yang et al. [7] and Middha et al. [8] have focused on the effects of binary diffusion coefficients on laminar flame speed. In these works, sensitivity analysis was performed in a similar manner as for the reaction rates, i.e. no temperature-dependent information was obtained. In this work, temperature-dependent sensitivity coefficients with respect to elementary reaction rate coefficients and binary diffusion coefficients for premixed flames were obtained by applying a small temperature-dependent perturbation to the property of interest. The perturbation was taken to be a Gaussian function of temperature, and the center of the Gaussian profile was varied with the assigned temperature mean. By gradually moving the center of this Gaussian window, new, temperature-dependent sensitivity coefficients are obtained. A similar idea has recently been suggested by Rumminger et al. [9]. Rumminger et al. used a rectangular-shaped perturbation (which they called a “band”) in premixed flame simulations to identify the temperature ranges where the flame suppression by inhibitors is expected to be most effective. A series of tests was performed for atmospheric CH4/air laminar premixed free flames, and, amongst the particular tests, Rumminger et al. [9] reported temperaturedependent responses of predicted laminar flame speed to the perturbations applied to two important reactions, H + O2 = OH + O (R1) and CO + OH = CO2 + H. (R2). Here, we propose a more rigorous methodology and apply it in a systematic manner to a number of cases relevant to detailed reaction model development, with broader implications with respect to combustion modeling in general. The sensitivity parameters for several elementary reactions, such as H + O2 = OH + O, (R1), CO + OH = CO2 + H, (R2), HCO + M = CO + H + M, (R3), HCO + O2 = CO + HO2 (R4), and H + O2 (+ M) = HO2 (+ M) (R5) were obtained for H2, H2/CO, CH3OH, and C3H8 kinetic flame models. The temperature-dependent sensitivity of binary diffusion coefficients of H-atom with He and H2O were also determined for selected conditions. The obtained results were analyzed to identify the physical/chemical mechanisms responsible for the observed behavior. The practical examples through which we demonstrate the utility of the present methodology for development of comprehensive kinetic models are discussed in detail below. Methodology and Implementation To develop a more general formulation, we consider a dynamic system which yields an observable F(x,α) given a vector of variables x and vector of parameters α. Elementary sensitivity gradient (coefficient) Si with respect to the parameter αi for the observable F is defined as i i F S α ∂ ∂ = , (1) In combustion modeling, it is more common to use logarithmic sensitivity gradients (or normalized sensitivity coefficients),