Effect of disorderness in two-dimensional discrete adiabatic combustion waves in a heterogeneous solid medium under stable regime

The effect of disorderness for the propagation of stable combustion waves in a two-dimensional discrete adiabatic system are studied by considering two systems, namely, (i) periodic and (ii) disorder. The discrete periodic system is modeled by regular arrangement of burnt and unburnt point heat sources. In contrast, the discrete disordered system is modeled by the concatenation of regular burnt and randomly distributed unburnt heat sources. Four different kinds of discrete disordered systems (S1, S2, S3, and S4) are modeled by varying the joint frequency count of unburnt heat sources without changing the domain size of the system. Results show that the disordered structure of the discrete system plays a crucial role in the combustion dynamics, shape of the combustion wave, and it affects the propagation of stable combustion waves with the manifestation of sparse and chaotic finger patterns. Quantitative analysis performed for the width and roughness of combustion front show that the disordered structure and ignition temperature (ε) of the discrete system plays a crucial role. The burn rates of discrete system are sensitive to change in structure of the system and ε. It is observed that the burn rates obtained for the discrete disordered system is always less than that of burn rates of the discrete periodic system. The burn rates calculated for the present model are compared with the available experimental data on combustion of thermite systems.

Figure 1

Plot for nondimensional ignition temperature ε and burn rate ω.

Figure 2

Structure of a two-dimensional discrete periodic system. ’$•$’ represents the burnt particles and “o” is for unburnt particles.

Figure 3

Ignition time profiles of a discrete periodic system using Eq. (5) at different ignition temperatures.

Figure 4

Structure of combustion front in a discrete periodic system for ignition temperature (ε) = 0.4.

Figure 5

Structure of the two-dimensional discrete disordered system. ‘$•$’ represents the burnt particles and ‘$\circ$’ is for unburnt particles.

Figure 6

Joint frequency count of 30 × 200 unburnt heat sources for different disordered systems. (a), (b), (c), and (d) in the figure represents distribution of $x$ and $y$ coordinates for different systems S1, S2, S3, and S4, respectively. S1 resembles those actual mixtures where the chemical components are more densely packed in the middle region of combustion chamber. Similarly, S2—densly packed in the initial part, S3—densely packed at the tail end, and S4—densely packed at initial and tail end of the combustion chamber.

Figure 7

Ignition time profile for discrete disordered systems S1, S2, S3, and S4 at different nondimensional ignition temperatures (ε). Circle represents the combustion stop.

Figure 8

Snapshot of instantaneous combustion front for ignition temperature ε = 0.05 with burnt heat sources 300, 900, 2700, 4500 at different ignition times for different disordered systems S1 (first row), S2 (second row), S3 (third row), and S4 (fourth row). S1—structure of combustion front shifts from irregular to regular pattern. S2—regular to irregular. S3—sparse fingers. S4—chaotic fingers.

Figure 9

Snapshot of instantaneous combustion front for ignition temperature ε = 0.4 with burnt heat sources 300, 900, 2700, 4500 at different ignition times for different disordered systems S1 (first row), S2 (second row), S3 (thirrd row), and S4 (fourth row). S1—irregular to regular combustion front. S2—irregular and increased fingers. S3—fingers with tip split. S4—increased chaotic fingers.

Figure 10

Width of the instantaneous combustion front for discrete system S1, S2, S3, and S4 at different ignition times and different ε: (a) ε = 0.05, (b) ε = 0.20, (c) ε = 0.30, (d) ε = 0.40. Dashed circles represent the initiation effect and circles represent the domain size effect.

Figure 11

Roughness of the combustion front for discrete systems S1, S2, S3, and S4. (a) $\varepsilon =0.05$; (b) $\varepsilon =0.20$; (c) $\varepsilon =0.30$; (d) $\varepsilon =0.40$. Symbols ‘$\circ$’ represent S1, ‘$\square$’ represent S2,‘$▵$’ represent S3, and ‘$◊$’ represent S4. Deviation indicates the intensity of fingering patterns.

Figure 12

Comparison plots of burn rates calculated for periodic and disordered systems using Eqs. (5) and (6).

Figure 13

Comparison plots of burn rates for different thermite mixture obtained from previous experimental work [5].

Figure 14

Comparison plots of burn rates calculated from present model (represented by solid line) and experimental work [5] (represented by symbol).

Figure 15

Temperature profile for unburnt cells.

Figure 16

Comparison plot for burn rates of a discrete periodic system. Solid line is obtained using Eq. (7) and symbols are obtained using Eqs. (5) and (6). “o” represents numerical burn rates using MPI and “x” using MPI and slice reconfiguration scheme.

Figure 17

Snapshot of instantaneous combustion front for ignition temperature ε = 0.2 with burnt heat sources 300, 900, 2700, 4500 at different ignition times for different disordered systems S1 (first row), S2 (second row), S3 (third row), and S4 (fourth row). S1—irregular fingers to regular combustion front. S2—sparse fingers. S3—sparse fingers with increased roughness. S4—chaotic fingers with increased roughness.

Figure 18

Snapshot of instantaneous combustion front for ignition temperature ε = 0.3 with burnt heat sources 300, 900, 2700, 4500 at different ignition times for different disordered systems S1 (first row), S2 (second row), S3 (thirrd row), and S4 (fourth row). S1—irregular to regular combustion front. S2—irregular with increased number of fingers. S3—increased fingers and roughness. S4—increased roughness and chaotic fingers.