An oscillatory route to extinction for solid fuel combustion waves due to heat losses

Abstract

A numerical method is used to show that heat loss increases can lead to a perioddoubling route to the cessation of propagation of solid fuel combustion. Oscillatory combustion waves are found in certain regions of the parameter space. The behaviour of these oscillatory waves becomes more complex as the heat loss is increased until extinction of the combustion reaction occurs. Large excursions in temperature, above the adiabatic temperature, are possible in the non-adiabatic case close to this extinction point.

Non-adiabatic combustion waves for general Lewis numbers: wave speed and extinction conditions

“…Of particular note is that as reported by Mercer et al [9] for solid fuels, the numerical results clearly show that the oscillatory instability is much more readily observed as the heat loss is increased. Illustrative calculations of maximum and minimum wave speeds have been shown for a typical case with and without heat loss.…”

“…With t regarded as o(l), the same result for the small drop in temperature at the flame and its connection to heat loss is found from (4.2): Figure 5 shows the multiple wave speeds that pertain to the formula (4.6) particularly at moderate /3 values. Marked on this figure are the numerical results as well, and locations where known oscillatory instabilities occur [9] (the maximum and minimum wave speeds in the oscillatory region are shown). The important difference in the case of solid fuels is that this instability is much more readily obtained and on the figure below corresponds to the slow branch of the wave speed plot.…”

“…The paper by Weber et al [12] was able to confirm this with an accurate value determined for the minimum wave speed. It is known from classical large activation energy analysis that heat losses have a great effect on the stability of combustion fronts, and with the new scheme for the grouping of the parameters, Mercer et al [9] have shown numerically that oscillatory instabilities occur for non-adiabatic systems with again a period-doubling route to chaos. This present paper concentrates on the influence of global heat loss on the propagation speeds of combustion fronts for general Lewis numbers.…”

Stabilization of combustion wave through the competitive endothermic reaction

“…Oscillations of flame speed may result in flashback or blow-off phenomena while pulsations in temperature may lead to the emergence of superadiabatic temperatures and possible local overheating. It is demonstrated in [12] that as the conditions for the onset of pulsating regimes are approached the sequential stages of relatively slow, cool combustion and short bursts of a hot, fast-moving reaction front are observed. These bursts can be considered to be practically 'unsafe' regime of combustion even though such behaviour can be transient [13].…”

“…The approximation of zero ambient temperature is used here. This is one of the standard ways to avoid the mathematical issue known as the 'cold boundary problem' [12]. Models with the one-step Arrhenius kinetics have non-zero rate of the reaction at finite initial temperature and the boundary conditions at the cold end cannot be strictly formulated.…”