D. A. Schult

Forced forward smolder combustion

Abstract We consider porous cylindrical samples closed to the surrounding environment except at the ends, with gas forced into the sample through one of the ends. A smolder wave is initiated at that end and propagates in the same direction as the flow of the gas. We employ asymptotic methods to find smolder wave solutions with two different structures. Each structure has two interior layers, i.e., regions of relatively rapid variation in temperature separated by longer regions in which the temperature is essentially constant. One layer is that of the combustion reaction, while the other is due to heat transfer between the solid and the gas. The layers propagate with constant, though not necessarily the same, velocity, and are separated by a region of constant high temperature. A so-called reaction leading wave structure occurs when the velocity of the combustion layer exceeds that of the heat transfer layer, while a so-called reaction trailing wave structure is obtained when the combustion layer is slower than the heat transfer layer. The former (latter) occurs when the incoming oxygen concentration is sufficiently high (low). Reaction trailing structures allow for the possibility of quenching if the gas mass influx is large enough; that is, incomplete conversion can occur due to cooling of the reaction by the incoming gas. For each wave structure there exist stoichiometric, and kinetically controlled solutions in which the smolder velocity is determined, respectively, by the rate of oxygen supply to the reaction site and by the rate of consumption in the reaction, i.e., by the kinetic rate. Stoichiometric (kinetically controlled) solutions occur when the incoming gas flux is sufficiently low (high). For each of the four solution types, we determine analytical expressions for the propagation velocities of the two layers, the burning temperature, and the final degree of solid conversion. We also determine analytical expressions for the spatial profiles of temperature, gas flux, and oxygen concentration. Gravitational forces are considered and are shown to have a minimal effect provided the ambient pressure is large compared to the hydrostatic pressure drop. The solutions obtained provide qualitative theoretical descriptions of various experimental observations of forward smolder. In particular, the reaction trailing stoichiometric solution corresponds to the experimental observations of Ohlemiller and Lucca, while the reaction leading stoichiometric solution corresponds to the experimental observations of Torero et al.

Propagation and extinction of forced opposed flow smolder waves

Abstract Smoldering is a slow combustion process in a porous medium in which heat is released by oxidation of the solid. If the material is sufficiently porous to allow the oxidizer to easily filter through the pores, a smolder wave can propagate through the interior of the solid. We consider samples closed to the surrounding environment except at the ends, with gas forced into the sample through one of the ends. A smolder wave is initiated at the other end and propagates in a direction opposite to the flow of the oxidizer. Previous experimental results show that for opposed flow smolder, decomposition of the solid fuel into char is the chemical process which drives the smolder process. We model this decomposition as a one step reaction. The model suggests that extinction occurs when decomposition is complete. We employ large activation energy asymptotic methods to find uniformly propagating, planar smolder wave solutions. We determine their propagation velocity, burning temperature, final degree of fuel decomposition, and extinction limits. We also determine spatial profiles of gas flux, oxidizer concentration, temperature, and degree of decomposition of the solid. Comparison is made with previous experimental results.

Rewiring networks for synchronization.

We study the synchronization of identical oscillators diffusively coupled through a network and examine how adding, removing, and moving single edges affects the ability of the network to synchronize. We present algorithms which use methods based on node degrees and based on spectral properties of the network Laplacian for choosing edges that most impact synchronization. We show that rewiring based on the network Laplacian eigenvectors is more effective at enabling synchronization than methods based on node degree for many standard network models. We find an algebraic relationship between the eigenstructure before and after adding an edge and describe an efficient algorithm for computing Laplacian eigenvalues and eigenvectors that uses the network or its complement depending on which is more sparse.

Traveling waves in natural counterflow filtration combustion and their stability

We consider two-dimensional filtration combustion in a porous medium in which an exothermic reaction takes place between the solid and a pure gaseous oxidant which is delivered to the reaction zone by filtration through the pores of the medium. As a result of the reaction, oxidant is consumed and a solid product is formed. The consumption of gas in the reaction causes a pressure gradient which drives filtration. Since no external forcing is required, this arrangement is termed natural filtration combustion. The samples are assumed to be open to gas permeation at one end with ignition at the other end so that gas flow is opposite to the direction of reaction propagation. This configuration is termed counterflow, so we study natural counterflow filtration combustion. This reaction scheme and configuration describe conditions of self-propagating high-temperature synthesis (SHS), in which combustion waves are employed to synthesize advanced materials. Asymptotic solutions describing traveling waves are determ...

Downward buoyant filtration combustion

We consider heterogeneous combustion in a porous medium subject to gravity-induced buoyant forces. A vertical sample, open to flow at the top and bottom, is ignited at the top. Buoyancy causes the hot gases to leave the sample through the top, thus drawing in fresh cool gas, containing both oxidizer and inert gases, through the bottom. The incoming gas supplies the reaction with oxidizer. In contrast to forced filtration, in which the flux of gas into the sample is prescribed, here the incoming flux is determined by the combustion process itself. This configuration describes smoldering, self-propagating high-temperature synthesis (SHS) of advanced materials, and a host of other applications. Combustion waves are described for both adiabatic and nonadiabatic conditions in which heat is lost to the external environment through the sides of the sample. The heat loss causes the temperature in the product to decay from the combustion temperature to the temperature of the external environment in a region termed the cooling region. Two nonadiabatic cases are considered, distinguished by whether or not the sample is sufficiently long and the heat losses sufficiently large that the cooling region behind the reaction site is completely contained within the sample. When it is totally contained in the sample, we describe traveling wave solutions whose shape does not change in time, provided there is no net production of gas in the reaction. When it is not completely contained within the sample, we describe quasi-steady combustion waves which change slowly in time due to the increasing buoyant flux as the combustion wave penetrates farther and farther into the sample. Solutions are categorized as gas deficient when the oxidizer is completely consumed, solid deficient when the solid fuel is completely consumed, or stoichiometric when both oxidizer and solid fuel are completely consumed. Extinction is found to occur for solid deficient and stoichiometric solutions when the buoyant flux is sufficiently large. We show that in order to ignite a self-sustained combustion wave, sufficient heat must be supplied, so that in addition to providing a sufficiently high temperature to form a preheat layer, there is sufficient buoyant flux of oxidizer to allow the combustion wave to become self-sustained. In particular, combustion will not occur in microgravity environments unless there are other mechanisms of oxidizer transport. Some samples will not support a combustion wave because the flux required for ignition is larger than the critical level for extinction. Under nonadiabatic conditions, another extinction mechanism exists in which heat loss lowers the temperature below a critical level. The results of our analysis compare favorably with experimental observations of downward buoyant combustion in the region away from the ends of the sample.

Matched asymptotic expansions and the closure problem for combustion wave

The closure problem for combustion waves arises when applying the method of matched asymptotic expansions for large activation energy to many nonsteady combustion problems. The exponential nature of the dependence of the reaction rate on temperature and the large coefficient (activation energy) in the exponent lead to the first order correction for temperature appearing in the equations at leading order. Equations describing the first order correction involve the second order correction, and so on. These terms can be scaled away for steady solutions, but when considering nonsteady propagation, they remain for any constant scaling of temperature. The closure problem refers to the fact that the equations must be solved at all orders before the leading order solution can be determined. One traditional approach to alleviate the problem is to truncate the series. While sacrificing the distinction between scales of temperature variation ahead of and behind the flame, these methods allow the replacement of the d...

Buoyancy Driven Filtration Combustion

Abstract A theoretical study of combustion in porous media driven by a gravity induced gas flux is conducted. Filtration of the oxidizer carrying gas arises in response to heating of the gas due to exothermic conversion of the solid fuel. Specifically, we consider a reaction front propagating through a porous matrix consisting of reactive (fuel) and inert components. Gas, consisting of both oxidizer and inert components, filters through the matrix and reacts with the solid fuel. The hot gases in the medium rise due to buoyancy, thus drawing in fresh gas from below. We employ approximate analytical methods and numerical simulations to analyze all the basic combustion phenomena, including self ignition, external ignition, both upward and downward as well as adiabatic and nonadiabatic propagating combustion waves. Our simulations also describe the dynamics of buoyancy driven combustion waves. In conventional combustion systems the combustion waves are traveling waves, whose wave characteristics, e.g., propag...

Dynamics of smoulder waves near extinction

Smoulder combustion involves a two phase solid/gas reaction inside a porous media that restricts the flow of air. We study opposed flow smoulder, that is, conditions where the smoulder front burns towards an incoming gas that contains oxygen. Thus gas and solid reactants enter the reaction site from the same direction. Extinction occurs when the mass flux of the forced gas is sufficiently high. Asymptotic methods for large activation energy provide us with descriptions of uniformly propagating smoulder waves and their existence (extinction) limit for relevant parameter values. Linear stability analysis shows that the planar uniformly burning smoulder wave can become unstable before extinction. One-dimensional numerical simulations justify these asymptotic results and elucidate the resulting dynamics of the smoulder wave as the incoming gas flux increases. For reasonable parameter values, the system oscillates and then proceeds through a period doubling cascade of bifurcations to chaotic behaviour before extinction occurs. This cascade does not seem to interact with the extinction of dynamic waves as parameter values can be chosen so that extinction occurs anywhere within the cascade. dschult@colgate.edu

Dopant distribution in isothermal frontal polymerization

We develop and study a mathematical model which describes how an inert species contained in the initial monomer/initiator mixture is redistributed by a propagating polymerization front. This study is relevant to the production of polymers for optical applications.

Synchronization dynamics on the picosecond timescale in coupled Josephson junction neurons

Conventional digital computation is rapidly approaching physical limits for speed and energy dissipation. Here we fabricate and test a simple neuromorphic circuit that models neuronal somas, axons, and synapses with superconducting Josephson junctions. The circuit models two mutually coupled excitatory neurons. In some regions of parameter space the neurons are desynchronized. In others, the Josephson neurons synchronize in one of two states, in-phase or antiphase. An experimental alteration of the delay and strength of the connecting synapses can toggle the system back and forth in a phase-flip bifurcation. Firing synchronization states are calculated >70 000 times faster than conventional digital approaches. With their speed and low energy dissipation (10^{-17}J/spike), this set of proof-of-concept experiments establishes Josephson junction neurons as a viable approach for improvements in neuronal computation as well as applications in neuromorphic computing.