A tangent linear approximation of the ignition delay time. I: Sensitivity to rate parameters

Abstract

A tangent linear approximation is developed to estimate the sensitivity of the ignition delay time with respect to individual rate parameters in a detailed chemical mechanism. Attention is focused on a gas mixture reacting under adiabatic, constant-volume conditions. The uncertainty in the rates of elementary reactions is described in terms of uncertainty factors, and are parametrized using independent canonical random variables. The approach is based on integrating the linearized system of equations governing the evolution of the partial derivatives of the state vector with respect to individual random variables, and a linearized approximation is developed to relate the ignition delay sensitivity to the scaled partial derivatives of temperature. The efficiency of the approach is demonstrated through applications to chemical mechanisms of different sizes. In particular, the computations indicate that for detailed reaction mechanisms the TLA leads to robust local sensitivity predictions at a computational cost that is order-of-magnitude smaller than that incurred by finite-difference approaches based on one-at-a-time rate perturbations.

A tangent linear approximation of the ignition delay time. II: Sensitivity to thermochemical parameters

“…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 O(N ) 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 [9][10][11][12] have particularly focused on estimating ignition delay time sensitivities to rate parameters.…”

“…These include methodologies based on tangent linear approximations [9,[12][13][14], adjoint techniques [10], as well as optimization approaches [11]. Applications in [9][10][11][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 [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.…”

“…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 [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.…”

“…In section 2, a similar analysis is applied, focusing on enthalpy and entropy perturbations, assuming both to be temperature independent. As in [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.…”