Numerical studies of integral equation and rod models of solid fuel combustion

We consider modes of solid flame propagation associated with the SHS (Self Propagating High Temperature Synthesis) process of materials synthesis. In this process reactants are ground into a powder, cold pressed into a solid sample, typically a cylinder, and ignited at one end. Synthesis ensues as a high temperature self-sustaining combustion wave propagates through the sample converting reactants to products. When gas plays no significant role in the process the resulting gasless combustion wave is referred to as a "solid flame". First we consider nonadiabatic gasless solid fuel combustion employing a reaction sheet model. We derive an integrodifferential equation for the location of the interface separating the fresh fuel from the burned products. For all values of the scaled activation energy, Z, and heat loss parameter, Gamma, the model admits a uniformly propagating combustion wave. This solution is subject to a pulsating instability for Z sufficiently large. We find that for Z slightly below the adiabatic stability limit, the effect of heat loss is to promote a period doubling cascade leading to chaotic behavior prior to extinction. We also find an interval of laminar behavior within the chaotic window, corresponding to a secondary period doubling sequence.;We next consider an array of interacting rods, each of which supports propagating waves. Thus, we employ an array of interacting 1D rods connected via heat transfer. The heat transfer terms correspond to a discretization of the transverse Laplacian. We consider first a rod model consisting of an outer ring of 3 rods equally spaced along the ring, together with an axial rod. We find a multitude of solutions including spinning, radial, and quasiperiodic solutions. We next consider an 8/8/1 rod model in which 8 equally spaced rods are located at the surface of the cylinder, another 8 equally spaced rods are on a concentric circle located halfway inside the cylinder, and an axial rod is located on the axis. Neighboring rods are connected to one another via heat transfer. We find numerous spin, radial, periodic, and quasiperiodic solutions and compare these solutions with possible 3D analogues.