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Reactive multilayered foils in the form of thin films have gained interest in various applications such as joining, welding, and ignition. Typically, thin film multilayers support self-propagating reaction fronts with speeds ranging from 1 to 20 m/s. In some applications, however, reaction fronts with much smaller velocities are required. This recently motivated Fritz et al. (2011) to fabricate compacts of regular sized/shaped multilayered particles and demonstrate self-sustained reaction fronts having much smaller velocities than thin films with similar layering. In this work, we develop a simplified numerical model to simulate the self-propagation of reactive fronts in an idealized compact, comprising identical Ni/Al multilayered particles in thermal contact. The evolution of the reaction in the compact is simulated using a two-dimensional transient model, based on a reduced description of mixing, heat release, and thermal transport. Computed results reveal that an advancing reaction front can be substantially delayed as it crosses from one particle to a neighboring particle, which results in a reduced mean propagation velocity. A quantitative analysis is thus conducted on the dependence of these phenomena on the contact area between the particles, the thermal contact resistance, and the arrangement of the multilayered particles.

Reactive multilayered materials have recently gained increasing interest in various applications, including joining, brazing, sealing, and ignition of secondary reactions [

Schematic illustration of an idealized compact of five multilayered particles. Note that each particle contains multiple bilayers, but only one is illustrated. The individual layers are assumed to be separated by a premixed region, labeled NiAl, of thickness,

One of the advantages of multilayered foils concerns their controlled microstructure, which enables a reaction front with a well-defined velocity, which in many cases can be made insensitive to environmental conditions or boundary conditions imposed by the application [

The present study aims at ultimately developing computational models that can predict the behavior of reaction fronts in compacts of multilayered particles and characterize their dependence on the particle distribution and on the layering within individual particles. To this end, in this paper we consider an idealized two-dimensional model of a compact comprising rectangular multilayered particles in thermal contact. The model is used to examine the evolution of self-propagating reactions, particularly as the front traverses neighboring particles. The computations are then used to analyze the dependence of the front velocity on the contact area, the thermal contact resistance, and the number of particles within the idealized compact.

We will focus primarily on a simplified two-dimensional compact comprising rectangularly shaped, multilayered particles that are in thermal contact. A schematic of a typical configuration is shown in Figure

The evolution of the reaction within the particle compact is analyzed using the reduced model developed in [

Within individual particles, the reaction is described in terms of a coarse-grained continuum model, coupling the conservation of energy equation to a mixture evolution equation. Dividing the particle into a finite number of regions or cells, conservation of energy within each cell is expressed in terms of the region-averaged enthalpy equation:

The evolution of the mean concentration and its second moment, ^{2}/s is the preexponent,

In the computations below, we assume the local heat flux

The coupled system, (

Simulations were conducted to characterize the evolution of reactive fronts within the idealized compacts and to examine their dependence on the properties of the compact. To this end, the particle length is held fixed,

The self-propagating reactions are initiated using a thermal spark of temperature

Figure

Flame front position versus time for contact area

To further investigate the mechanism of front crossover, we plot in Figure ^{3} at^{3} at ^{3}. At ^{3}) once again reach values associated with a steadily propagating front.

Instantaneous distributions of temperature in K (left) and local heat release rate in MW/m^{3} (right) at selected times ((a)–(d)). The contact area

Evolution of the maximum temperature within the compact for different values of

It appears that, for the present configuration, the flame crossover from one particle to the next can be characterized by a front propagation phase that essentially leads to the consumption of the reactants in the first particle, followed by an extended heating phase, and followed by a rapid ignition and the formation of a self-propagating front in the neighboring particle. This succession of regimes is further illustrated in Figure

Instantaneous (a) local heat release rate and (b) temperature profiles at selected time instants for

The results obtained for the present configuration indicate that the motion of self-propagating fronts in particle compacts can involve a complex succession of phenomena, and large variations in the flame temperature and in heat release rate. They also point to the possibility of substantial reduction in the mean front propagation velocity due to heating delay. Below, we focus on quantifying this effect and characterizing its dependence on properties of the compact.

The average speed of reaction fronts propagating in Ni/Al laminates of uniform thickness has been analyzed in numerous published works [

In the present work, we extend the concept used for uniform thickness foils to the case of particle compacts as follows. Regardless of the details of the configuration, we form an estimate of

As shown in the previous section, the delay in front crossover may vary appreciably from one contact region to another. To ensure an appropriate estimate of the average velocity, we analyzed the dependence of

Simulations were conducted by systematically varying

Figure

Profiles of

To further illustrate the impact of the heating time,

Flame front position versus time for different values of

The analysis above was repeated for a particle thickness

In this section, we briefly examine the impact of the particle thickness,

Schematic illustration of particles compact with periodic arrangement. Though the individual particles comprise multiple bilayers, only one is shown in the schematic. Periodicity is simulated by restricting the domain to the region lying between the dotted lines, and imposing periodic boundary conditions on these lines. Other boundary and initial conditions are similar to those depicted in Figure

Figure

Flame front position versus time for the single-row configuration, with particle thickness in the range

Also shown on Figure

The thermal contact resistance in a particle compact may depend on a variety of factors, including geometry, applied pressure, as well as surface cleanliness, roughness, and flatness [^{2} K/W. In this section, we briefly investigate the effect of thermal contact resistance on the average velocity of the front, namely, by repeating the analysis for a compact of 10 ^{2} K/W and as before vary

The predicted values of ^{2} K/W, the

Average flame speed versus contact area. Curves are generated for the case of vanishing thermal contact resistance, and for ^{2} K/W. Also shown for comparison are the experimental average flame speed measurements of Fritz et al. [

To investigate the origin of this trend, we plot in Figure ^{2} K/W become close to each other. Thus, in this range, the impact of the thermal contact resistance becomes substantially weaker.

Flame front position versus time for ^{2} K/W. Curves are generated for different values of

We finally compare our present predictions with experimental measurements of Fritz et al. [^{2} K/W. Although differences between the computational and experimental results can arise due to a variety of factors that are not accounted for in the model, such as heat losses, particle geometry, and compact packing density, the agreement between reported measurements with predictions obtained with high

In this work, we developed a simple numerical model for the simulation of self-propagation of reactive fronts in an idealized compact. The compacts comprise identical Ni/Al multilayered particles in thermal contact. The evolution of the reaction front in the compact was simulated using a two-dimensional transient reduced reaction model. The model was used to investigate the propagation of the reaction front within the compact and to analyze the dependence of its average velocity on the contact area between neighboring particles, the particle thickness, and the thermal contact resistance. Analysis of the computations revealed that

The average velocity of the front can be substantially smaller than that of a continuous multilayered foil of the same microstructure. Consistent with the analysis of [

For the case of perfect thermal contact, the average flame velocity decreases as

The average flame velocity decreases slightly as the particle thickness increases. With the present adiabatic computations, the dependence on particle diameter appears insignificant compared to variations induced by

The thermal contact resistance can have a substantial impact on the average front velocity, especially for low values of

The comparison of computed predictions with experimental measurements of Fritz et al. reveals a favorable agreement with predictions obtained using high

Work is currently underway to extend the present model to account for realistic, three-dimensional microstructures, as well as the effect of radiative and convective heat losses.

This work was supported by the Office of Naval Research through Award N00014-07-1-0740 and by the Defense Threat Reduction Agency, Basic Research Award no. HDTRA1-11-1-0063, to Johns Hopkins University.

_{9}Ni

_{2}phase and analysis of its formation