A. P. Aldushin

Instabilities, Fingering and the Saffman - Taylor Problem in Filtration Combustion

We consider planar, uniformly propagating combustion waves driven by the filtration of gas containing an oxidizer which reacts with the combustible porous medium through which it moves. We find that these waves are typically unstable with respect to hydrodynamic perturbations. For both forward (cofiow) and reverse (counterflow) filtration combustion (FC), in which the direction of gas flow is the same as or opposite to the direction of propagation of the combustion wave, respectively, the basic mechanism leading to instability is the reduction of the resistance to flow in the region of the combustion products, due to an increase of the porosity in that region. Another destabilizing effect in forward FC is the production of gaseous products in the reaction. In reverse FC this effect is stabilizing. We also describe an alternative mode of propagation, in the form of a finger propagating with constant velocity. The finger region occupied by the combustion products is separated from the unburned region by a f...

Effects of gas-solid nonequilibrium in filtration combustion

To determine the effects of gas-solid nonequilibrium on forced filtration combustion (FC) waves, a two-temperature model is employed, with separate temperature fields for the solid and gas particles. We consider heterogeneous (solid/gas) combustion in a porous sample with a prescribed gas flux at the inlet. If the reaction is initiated at the inlet (outlet) of the sample and the combustion wave travels in the direction of (opposite to ) gas filtration it is referred to as coflow (counterflow) FC. We determine the effects of gas-solid nonequilibrium on various aspects of forced FC waves. First, we consider coflow FC waves, in which case the gas infiltrating through the hot product region significantly enhances the propagation of the combustion wave. For a relatively small gas flux, the infiltrating gas delivers heat from hot product to the combustion layer, thus increasing the combustion temperature and, hence, the combustion rate. Propagation of such waves is controlled by conduction of the heat released in the reaction to the preheat zone. Conductively driven coflow FC waves have been studied extensively using one-temperature models, which assume a very large rate of interphase heat exchange between the solid and the gas, so that thermal equilibrium is attained almost immediately. However, if the gas flux is sufficiently large, the phases do not have sufficient time to equilibrate and, hence, the underlying assumption of one-temperature models is no longer valid. One-temperature models are only appropriate for describing slowly propagating coflow FC waves in which the time of contact between the solid particles and the gas is sufficiently large for rapid thermal equilibrium to occur. However, not all coflow FC waves are slowly propagating. There can also be rapidly propagating coflow FC waves, in which case a two-temperature model, with the solid and the gas attaining distinct temperatures, is more appropriate. For a relatively large gas flux, an alternative mechanism of enhancement occurs in the combustion temperature is increased as a result of increasing the effective initial temperature of the unurned solid ahead of the reaction zone. Propagation of such waves is controlled by convection of heat stored in the product to the preheat zone. Convectively driven coflow FC waves depend on a pronounced temperature difference between the phases and, therefore, cannot be described by a one-temperature model. We employ a two-temperature model to study coflow FC waves. The two-temperature model must also account for the fact that the oxidizer concentration at the surface of the solid particles, where the reaction occurs, may be significantly less than the average oxidizer concentration in the gas stream. Effects of gas-solid nonequilibrium are determined by the interphase heat transfer rate, as well as the rate of oxidizer diffusion to the solid particle surface. In the appropriate limits, the model can describe both conductive and convective coflow FC waves. We determine steady solutions in each of the limits and compare and contrast the structures, the combustion characteristics, and the parameter dependences of the two modes of propagation. Transition between conductive and convective steady-traveling coflow FC waves is discussed. We determine how gas-solid nonequilibrium affects such features as the wave velocity, the extinction limit, net gas production or consumption in the reaction, and the ability of the wave to accumulate energy near the reaction site. The results of our analyses are verified by numerical computations of the time-dependent problem. Next, we consider the effects of gas-solid nonequilibrium on counterflow FC waves, in which case the solid near the reaction site loses heat to the cool infiltrating gas. Counterflow FC waves are necessarly driven by conduction because the convective mechanism relies on the gas to deliver heat to, rather than carry heat away from, the unburned solid fuel. It is known from one-temperature-model analyses that extinction of steady-traveling waves will occur if the gas influx is sufficiently large. However, when the gas flux is sufficiently large to cause extinction, the two phases may not have sufficient time to attain thermal equilibrium, in which case a two-temperature model is required. Though only conductively driven counterflow FC waves are possible, we find that thermal nonequilibrium has a significant effect on the intrinsic extinction limit and the combustion characteristics.

Combustion of Porous Samples with Melting and Flow of Reactants

Abstract We formulate and analyze a model describing the combustion of porous condensed materials in which a reactant melts and spreads through the pores of the sample. Thus there is liquid motion relative to the porous solid matrix. Our model describes the cases when the melt either fills all the pores or when some gas remains in the pores. In each case the melt occupies a prescribed volume fraction of the mixture. We employ both analytical and numerical methods to find uniformly propagating combustion waves, to analyze their stability and to determine behavior in the instability region. The principal physical conclusion which follows from our analysis is that the flow of the melted component can result in nonuniform composition of the product. Unlike models which do not take into account the relative motion of the components, this model exhibits a dependence of the structure of the product on the mode of propagation of the combustion front. Thus, if the initial mixture is uniform, models which do not al...

Rapid Filtration Combustion Waves Driven by Convection

The propagation of filtration combustion (FC) waves in a porous solid which reacts with a gaseous oxidizer flowing through its pores may be significantly enhanced by increasing the infiltrating gas flux through the hot porous product region. For relatively small flux, enhancement occurs by the superadiabatic effect, i.e., by increasing the maximum temperature (Tb ), and thus, the reaction rate W(Wb ) in the combustion front. Though convection of heat from the product region increases Tb , the mechanism of FC wave propagation is controlled by diffusion of the heat released in the reaction. An alternative mechanism of enhancement, which occurs for relatively large gas fluxes, and corresponds lo pronounced temperature nonequilibrium between the gas and solid phases, leads to an increase of the FC wave velocity without increasing the combustion temperature. The propagation of such waves is controlled by the convection of heat stored in the products, rather than by diffusion of the heat released in the reactio...

Interaction of Gasless and Filtration Combustion

Abstract We study the propagation of combustion waves through porous samples in which two reactions occur. The first is a gasless solid-solid reaction between two solid species in the porous solid matrix to form a solid product, while the second is a solid-gas reaction in which gas delivered to the reaction site through the pores of the sample reacts with one of the solid species to form both solid and gaseous products. We consider the case of coflow filtration, in which the direction of gas flow is the same as the direction of propagation. The gas, consisting of both chemically active and inert components, filters to the reaction zone through the product region thus transferring heat from the high-temperature products to the unburned mixture. Using kinetics motivated by large activation energy considerations we determine the structure of a uniformly propagating combustion wave and, in particular, such important characteristics as the propagation velocity of the wave, the burning temperature and the compo...

EXTREMUM PRINCIPLES FOR SELECTION IN THE SAFFMAN-TAYLOR FINGER AND TAYLOR-SAFFMAN BUBBLE PROBLEMS

We consider the Saffman–Taylor problem describing the displacement of one fluid by another having a smaller viscosity, in a porous medium or in a Hele–Shaw configuration, and the Taylor–Saffman problem of a bubble moving in a channel containing moving fluid. Each problem is known to possess a family of traveling wave solutions, the former corresponding to propagating fingers and the latter to propagating bubbles, with each member characterized by its own velocity and each occupying a different fraction of the channel through which it propagates. To select the “correct” member of the family of solutions we propose two related extremum principles. Employing these principles for a symmetric family with zero surface tension (σ=0) selects the solution (finger or bubble, viscous or inviscid) which happens to be the same as that obtained by taking the limit σ→0 of the nonzero isotropic surface tension problem. For other problems, e.g., perturbation by anisotropic surface tension, the fingers selected by the two ...

Stoichiometric flames and their stability

We consider a flame in a stoichiometric combustible mixture of two reactants, A and B, having different diffusivities. We employ a thin reaction zone approximation and assume that the reaction ceases when the concentrations and temperature approach their thermodynamic equilibrium values. Thus, our analysis accounts for the possibility of a reversible stage in the combustion reactions. We find uniform flames and analyze both their cellular and pulsating instabilities. We compare our results with those for a one reactant flame as well as with previously obtained results for a stoichiometric mixture of two reactants. Studies of the latter assumed complete consumption of one of the reactants in the reaction front and that both Lewis numbers, LA and LB, are close to 1. They showed that the cellular stability boundary for bimolecular reactions is determined by an effective Lewis number which is the arithmetic mean of LA and LB, i.e., 12(LA + LB). By considering the limiting case of a negligibly small equilibrium constant, so that the final concentrations of the reactants approach zero, we show that for general Lewis numbers, not limited to being close to 1, the cellular stability boundary is determined by an effective Lewis number which, for equimolecular, e.g., bimolecular reactions, is the harmonic mean of the Lewis numbers of the two reactants, i.e., Leff = 2(LA−1 + LB−1)−1 The leading term of an asymptotic expansion of our effective Lewis number, for both LA and LB close to 1, is the simple arithmetic mean, previously obtained. For significantly different Lewis numbers, a case not covered by previous results, our solution shows that stability is determined by diffusion of the lighter reactant, in accord with experimental observations. Thus, we provide a theoretical basis for the effect of preferential diffusion. We also find that dissociation of the product is stabilizing.

Diffusion Driven Combustion Waves in Porous Media

Filtration of gas containing oxidizer, to the reaction zone in a porous medium, due, e.g., to a buoyancy force or to an external pressure gradient, leads to the propagation of filtration combustion (FC) waves. The exothermic reaction occurs between the fuel component of the solid matrix and the oxidizer. In this paper, we analyze the ability of a reaction wave to propagate in a porous medium without the aid of filtration. We find that one possible mechanism of propagation is that the wave is driven by diffusion of oxidizer from the environment. The solution of the combustion problem describing diffusion driven waves is similar to the solution of the Stefan problem describing the propagation of phase transition waves, in that the temperature on the interface between the burned and unburned regions is constant, the combustion wave is described by a similarity solution which is a function of the similarity variable The difference between the two problems is that in the combustion problem the temperature is not prescribed, but rather, is determined as part of the solution. We will show that the length of samples in which such self-sustained combustion waves can occur, must exceed a critical value which strongly depends on the combustion temperature Tb,. Smaller values of Tb require longer sample lengths for diffusion driven combustion waves to exist. Because of their relatively small velocity, diffusion driven waves are considered to be relevant for the case of low heat losses, which occur for large diameter samples or in microgravity conditions. Another possible mechanism of porous medium combustion describes waves which propagate by consuming the oxidizer initially stored in the pores of the sample. This occurs for abnormally high pressure and gas density. In this case, uniformly propagating planar waves, which are kinetically controlled, can propagate. Diffusion of oxidizer decreases the wave velocity. In addition to the reaction and diffusion layers, the uniformly propagating wave structure includes a layer with a pressure gradient, where the gas motion is induced by the production or consumption of the gas in the reaction as well as by thermal expansion of the gas. The width of this zone determines the scale of the combustion wave in the porous medium.

Solid Flame Waves

We consider the gasless combustion model of the SHS (Self-Propagating High Temperature Synthesis) process in which combustion waves are employed to synthesize desired materials. In this case the combustion phenomenon is referred to as a “solid flame”. Specifically, we consider the combustion of a solid sample in which combustion occurs on the surface of a cylinder of radius R. In addition to uniformly propagating planar waves there are many other types of waves. The study of different waves is important since the nature of the wave determines the structure of the product material. For the fixed value of the Zeldovich number Z which we employ the uniformly propagating planar wave is unstable. We describe stable waves only. We consider solution behavior as R is increased. For R sufficiently small, slowly propagating planar pulsating flames are the only modes observed. As R is increased, transitions to more complex modes of combustion occur, including (i) spin modes in which one or several symmetrically spaced hot spots rotate around the cylinder as the flame propagates along the cylindrical axis, thus following a helical path, (ii) counterpropagating (CP) modes, in which spots propagate in opposite angular directions around the cylinder, executing various types of dynamics. These include spots which pass through each other essentially unchanged, much the same as solitons, and spots which appear to be annihilated, only to be regenerated further along the sample in one of a variety of ways, (iii) alternating spin CP modes (ASCP), where rotation of a spot around the cylinder is interrupted by periodic events in which a new spot is spontaneously created ahead of the rotating spot. The new spot splits into counterpropagating daughter spots, one of which collides with the original spot leading to their mutual annihilation, while the other continues to spin, (iv) modulated spin waves consisting of either one or two symmetrically located rotating spots which exhibit a periodic modulation in speed and temperature as they rotate, (v) asymmetric spin waves in which two spots of unequal strength and not separated by angle π, rotate together as a bound state, (vi) modulated asymmetric spin waves in which the two asymmetric spots oscillate in a periodic manner as they rotate, alternately approaching each other and then moving apart periodically in time, (vii) asymmetric ASCP modes in which a slowly varying bound state of two spots rotates around the cylinder with the leading spot, and subsequently the trailing spot, exhibiting episodes of ASCP behavior, and (viii) 3-headed spins in which three spots rotate around the cylinder in a nonuniform fashion so that each cell alternately approaches one of its neighbors and then the other. In one case the motion is quasiperiodic, with neighboring spots approaching and departing from each other periodically in time as they rotate. In another case the motion is apparently chaotic. Two neighboring spots nearly collide, after which one spot is rapidly propelled away from the other as they rotate. Finally, for a slightly higher value of R, two neighboring spots collide, leading to annihilation of one spot and collapse of the 3-headed spin to a 2-headed spin mode.

Selection in the Saffman-Taylor finger problem and the Taylor-Saffman bubble problem without surface tension

Abstract We consider the Saffman-Taylor problem describing the displacement of one fluid by another having a smaller viscosity, in a porous medium or in a Hele-Shaw configuration, and the Taylor-Saffman problem of a bubble moving in a channel containing moving fluid. Each problem is known to possess a family of solutions, the former corresponding to propagating fingers and the latter to propagating bubbles, with each member characterized by its own velocity and each occupying a different fraction of the porous channel through which it propagates. To select the correct member of the family of solutions, the conventional approach has been to add surface tension σ and then take the limit σ → 0. We propose a selection criterion that does not rely on surface tension arguments.