Effect of thermal expansion on the linear stability of planar premixed flames for a simple chain-branching model: The high activation energy asymptotic limit

Abstract

Published paperThe linear stability of freely propagating, adiabatic, planar premixed flames is investigated in the context of a simple chain-branching chemistry model consisting of a chain-branching reaction step and a completion reaction step. The role of chain-branching is governed by a crossover temperature. Hydrodynamic effects, induced by thermal expansion, are taken into account and the results compared and contrasted with those from a previous purely thermal-diffusive constant density linear stability study. It is shown that when thermal expansion is properly accounted for, a region of stable flames predicted by the constant density model disappears, and instead the flame is unstable to a long-wavelength cellular instability. For a pulsating mode, however, thermal expansion is shown to have only a weak effect on the critical fuel Lewis number required for instability. These effects of thermal expansion on the two-step chain-branching flame are shown to be qualitatively similar to those on the standard one-step reaction model. Indeed, as found by constant density studies, in the limit that the chain-branching crossover temperature tends to the adiabatic flame temperature, the two-step model can be described to leading order by the one-step model with a suitably defined effective activation energy.

Thermal-Diffusive Instability of Premixed Flames for a Simple Chain-Branching Chemistry Model with Finite Activation Energy

“…This shows that as T c → 1+Q, the flame speed becomes increasingly sensitive to the activation energy, in that the flame speed initially increases more rapidly with θ −1 ; i.e., increasingly large activation energies are required for the HAEA value to be quantitatively predictive as T c is increased. This is due to the fact that, in the HAEA solution, the length scale of the intermediate diffusion zone, in which completion reactions and heat release occur, rapidly becomes shorter as T c → 1+Q [11,15]. While in the asymptotic limit the chain-branching zone remains infinitesimally thin compared to this heat release zone, for the finite activation energy solutions θ must be sufficiently high for the width of the finite chain-branching zone to be rendered thin compared to the heat release length scales.…”

“…Appropriate values of Lewis numbers are between 0.3 and 1.8, with the lower bound corresponding to lean hydrogen [11] and the upper bound to propane [21]. However, since the flame stability is found to be not very sensitive to the Lewis number of the intermediates [11,15], in this paper Le Y is set to unity throughout, and we concentrate on the effect of varying Le F .…”

“…As usual [11,15], we assume that the completion rate is state independent (zero activation energy), and we also assume that the chain-branching step is thermally neutral, so that all the heat, Q, is released in the completion step. Here Λ is the dimensionless completion rate constant and θ is the dimensionless activation energy of the chain-branching step.…”

“…Sharpe [15] then extended the two-step HAEA results by considering the variable density model, hence taking into account hydrodynamics effects. As for the one-step model, he found that the only qualitative difference from the CDM results is that the flame remains unstable to a long-wavelength hydrodynamic instability, even for Lewis numbers of the fuel above the critical value for purely thermal-diffusive induced instability.…”

“…where Λ ∞ is the value of the flame eigenvalue in the HAEA limit, determined in [11,15]. Figure 3.2(a) shows S as a function of the inverse activation energy, θ −1 ,f o r Q = T c =5andLe F =0.3 (corresponding to the profiles shown in Figure 3.1), which reveals that the flame speed dependence on the activation energy is not monotonic: as θ −1 is increased from zero, the flame speed initially increases above the asymptotic value and reaches a maximum value (of S =1 .15 at θ = 110 in this case) before decreasing for lower activation energies.…”