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We examine the problem of sub-micron aluminium drop combustion when both O diffusion and Al diffusion occur. During the unsteady evolution of the combustion field the oxide that is generated has the form of a shell whose inner surface encloses an aluminium bath and whose outer surface is exposed to an air bath. The diffusing atoms can react at these surface baths and also, in principle, between themselves in the interior of the oxide. However, the mass concentrations of the atoms are small so that the interior collisions necessary for this reaction are negligible.With this important simplification we are able to define a model that leads to an analytical discussion. For small Damkohler numbers (of which there are two, one defined by the O atoms the other by the Al atoms) the relative values of the bath reaction rates determine the proportion of oxide accretion on the outer surface to accretion on the inner surface. For large Damkohler numbers the relative values of the diffusion coefficients play the same...

We pose the following question: ‘Within the context of the classical shrinking-core model for sub-micron aluminium combustion, allowing only for the diffusion of atomic oxygen (no aluminium diffusion), but with two non-classical ingredients (large Knudsen number heat losses to the ambient atmosphere and a fractal reacting surface), are the solutions consistent with four experimental facts and data sets: (i) burn times; (ii) particle temperatures; (iii) maximum temperatures independent of particle size for small Damkohler numbers; and (iv) also for small Damkohler numbers a burn law of the form d1 − ν–t (where d is the particle diameter and ν ∼ 0.7 or so)?’ In the analysis we first consider a non-fractal model and scale the lengths with (where is the diffusion coefficient and k is the reaction-rate constant) and time with (where ρ− is the aluminium density and co is a characteristic value of the O density within the alumina). Burn times are calculated which follow a linear d–t law for small values of kL/D(...

We revisit the shrinking-core model of sub-micron aluminum combustion with particular attention to the mass flux balance at the reaction front which necessarily leads to a displacement velocity of the alumina shell surrounding the liquid aluminum. For the planar problem this displacement simply leads to an equal displacement of the entire alumina layer, and therefore a straightforward mathematical framework can be constructed. In this way we are able to construct a single curve which defines the burn time for arbitrary values of the diffusion coefficient of O atoms, the reaction rate, the characteristic length of the combustion field, and the O atom mass concentration within the alumina provided that it is much smaller than the aluminum density. This demonstrates a transition between a ‘d 2–t’ law for fast chemistry and a ‘d–t’ law for slow chemistry. For the spherical geometry, the one of physical interest, the outward displacement velocity creates not a simple displacement, but a stress field which, wh...

Flame ball interactions are numerically investigated in a reaction–diffusion system characterized by single-step Arrhenius kinetics and radiative heat losses. It is found that the interactions of two neighbouring flame balls are characterized by two distinct regimes – a repulsion regime and an attraction regime, depending upon the separation distance. The two regimes join at a critical separation distance, which corresponds to an unstable equilibrium state. For supercritical separation distances, the two flame balls repel and drift apart from each other; whereas for sub-critical separation distances, they move towards each other and eventually merge into a single stationary flame ball. In this connection, flame ball interactions are found to exhibit a qualitatively reverse character in comparison with the well-known van der Waals curve which characterizes intermolecular forces.

We use numerical strategies to examine the stability of reverse smolder waves in the context of a model that can permit both fuel-rich and fuel-lean waves. The steady-state response for such waves, maximum temperature versus blowing rate, is characterized, for increasing blowing rate, by a fuel-rich branch of rising temperature followed by a fuel-lean branch of falling temperature, followed by quenching. The propagation speed at the quenching point is non-zero. For the parameters that we consider, the entire fuel-rich branch is unstable to two-dimensional disturbances, but the dynamic consequences are modest. An interval of the fuel-lean branch whose left boundary is at the point of stoichiometry is stable, but the remainder of the branch, all the way to the quenching point, is unstable. These instabilities are destructive, and the smolder wave becomes fragmented. Tribrachial fragments can emerge, analogous to the tribrachial or triple flames familiar from gaseous edge-flame studies. Their emergence is ch...

A two-dimensional numerical model is developed to simulate combustion and heat transfer in ‘Swiss roll’ combustors (SRC). This model couples heat transfer and chemical reaction in the gas and heat diffusion in the conducting walls. The goal of the model is to gain insight into micro-scale combustion to prevent the occurrence of extinction. In the numerical modelling, the ‘Swiss roll’ geometry is unwrapped into a straight channel, with heat recirculation between the unburnt and burnt gas being appropriately taken into account. The simulation is carried out using a fourth-order scheme to discretize the spatial variables, with time integration to steady state, based on a first-order Runge–Kutta scheme. We investigate the extinction mechanisms and heat transfer characteristics in the combustor, and perform parametric studies to examine their effects on combustion and extinction. Numerical predictions of extinction limits agree well with experimental data.

We examine sandwich propellants constructed from sheets of pure ammonium perchlorate (AP) interleaved with an AP/binder blend, and construct solutions numerically using a code that fully couples gas-phase and solid-phase processes via an unsteady moving interface. This code has been used elsewhere to simulate the burning of random packs of spherical AP particles embedded in binder. We show that for a stoichiometric configuration, variations of the burning rate with α (a measure of the oxygenation of the AP/binder blend) are not monotonic, but display a weak maximum; and variations of the burning rate with sandwich thickness are monotonic for small α, but display a minimum for large α (e.g. α = 0.5). When the equivalence ratio is varied, the burning rate displays a maximum on the fuel-lean side when α is small, on the fuel-rich side when α is large. These results, and the manner in which the sandwich topography varies with the different parameters, suggest that the configuration could be invaluable for val...

We examine twin premixed flames in a plane counterflow and uncover, in the parameter space, a hitherto unknown domain of cellular instability. This leads us to hypothesize that for small Lewis numbers a two-dimensional (2D) steady solution branch bifurcates from the one-dimensional (1D) solution branch at a neutral stability point located near the strain-induced quenching point. Solutions on this 2D branch are constructed indirectly by solving an initial-value problem in the edge-flame context defined by the multiple-valued bistable 1D solution. Three kinds of solution are found: a periodic array of flame-strings, a single isolated flame-string and a pair of interacting flame-strings. These structures can exist for values of strain greater than the 1D quenching value, corresponding to sublimit solutions.