A tangent linear approximation of the ignition delay time. II: Sensitivity to thermochemical parameters
Introduction
Sensitivity analyses are frequently employed to identify the key parameters contributing to the variability in quantities of interest (QoIs) predicted by a chemical reaction model. In particular, these methods have been applied to guide model reduction and refinement. A prominent example of the application of sensitivity tools in the context of ignition simulations concerns the characterization of the ignition delay time (, defined precisely later) to local perturbations in chemical rate parameters.
In contrast to the extensive body of work focusing on the impact of rate parameters, a relatively lower number of studies have addressed the role of thermodynamic properties on mixture reactivity [1], [2], [3], [4], [5], [6], [7], [8]. This is a primary focus of the present work.
Traditionally, sensitivity analyses have relied on a (so-called brute-force) finite-difference methodology based on perturbing one-at-a-time the input parameters and performing independent simulations of the system to determine the perturbed QoI. When independent parameters are considered, this requires independent simulations if first-order differences are used, or in the case of centered differences.
Recently, however, several approaches [9], [10], [11], [12] have been conducted that focus on exploring alternative means of estimating sensitivities that can specifically reduce the cost associated with the brute force approach. These include methodologies based on tangent linear approximations [9], [12], [13], [14], adjoint techniques [10], as well as optimization approaches [11]. Applications in Ji et al. [9], Lemke et al. [10], Gururajan and Egolfopoulos [11], Almohammadi et al. [12] have particularly focused on estimating ignition delay time sensitivities to rate parameters. These works have shown that the underlying techniques offer accurate and efficient means of estimating sensitivities and that, in high dimensional situations, speedup by order-of-magnitude can be achieved over the traditional brute-force finite-difference approach.
In this work, we focus on extending the TLA methodology introduced in our previous work [12] to enable estimating the local sensitivity of the ignition delay time to variations in the thermodynamic properties of individual species. Note that regardless of the parameters being perturbed, the TLA methodology leads to a system of dynamical equations that governs the state-vector perturbations [9], [10], [11], [12], [13], [14], [15], comprising a linear stretching term involving the Jacobian of the source term, and a forcing term that corresponds to the derivative of the source term with respect to the parameters being perturbed. Our effort in Almohammadi et al. [12] focused exclusively on the impact of rate parameters, and consequently involved deriving expressions for the derivative of the source term with respect to these parameters. In Section 2, a similar analysis is applied, focusing on enthalpy and entropy perturbations, assuming both to be temperature independent. As in Almohammadi et al. [12], we rely on a linearized functional approximation to relate the sensitivity of the ignition delay time to the sensitivity of the state vector. In Section 3, a unified framework is presented that enables simultaneous characterization of the impact of reaction rate, enthalpy, and entropy perturbations. Section 4 provides a brief description of the simulation approaches for modeling the ignition problem and simulating the evolution of local sensitivities. The application of the TLA framework presently developed is illustrated in Section 5 for simulations of the ignition of hydrogen/air, iso-octane/air, and n-butanol/air mixtures. The main conclusions of the work are discussed in Section 6.
Section snippets
Governing equations
In this section, we start by summarizing the governing equation for homogeneous combustion of a reacting gas mixture under adiabatic, constant-volume conditions; see Refs. [16], [17] and Appendix A. We denote , the species mass fractions, the number of reacting species, the temperature, and the time. For a detailed chemical mechanism involving elementary reactions, the evolution of the species mass fractions and temperature follows the coupled system,
Rate and thermodynamic parameter sensitivities
The development above enables us to provide a combined treatment of local sensitivities to thermodynamic and rate parameters, in the case where the former are expressed as in Section 2 above, and the latter follow the form given in Almohammadi et al. [12]. Specifically, when the pre-exponents of reaction is parametrized according towhere is the uncertainty factor associated with reaction , the ’s are canonical independent random variables uniformly distributed
Numerical schemes
The numerical solution of (1) is computed using TChem, which relies on an adaptive-step, error-controlled, stiff integration methodology to determine the time profile of state vector, . The output frequency is controlled by specifying the maximum time step, , as well as the maximum temperature difference, , between consecutive records. In all cases presented below, we set K. This methodology results in discrete time profile of the state vector, , , where the
Results
In this section, we present results obtained by applying TLA schemes to three different settings. First, we consider the oxidation of hydrogen in air using a small mechanism [19]. Then, we scale up the problem considering uncertainty in a larger number of species, as they feature in the iso-octane mechanism from [20], [21], [22]. Finally, we apply TLA to estimate local sensitivities of the ignition delay time on rate and thermodynamic uncertainties belonging to rate rules and thermodynamic
Conclusions
The tangent linear approximation presented in Almohammadi et al. [12] was extended to estimate the local sensitivity of the ignition delay time with respect to uncertainty in thermodynamic parameters, namely, species enthalpies and entropies. Attention is focused on a gas mixture reacting adiabatically at constant volume. The variability in the enthalpies and entropies of species is introduced by perturbing selected coefficients in the NASA polynomials so that the corresponding uncertainty
Declaration of Competing Interest
The authors declare that they have no known competing financial interests or personal relationships that could have appeared to influence the work reported in this paper.
Acknowledgments
The research reported in this publication was supported by King Abdullah University of Science and Technology (KAUST).
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