Simulation of fingering behavior in smoldering combustion using a cellular automaton

Smoldering is the slow, low-temperature, flameless burning of porous fuels and the most persistent type of combustion phenomena. It is a complex physical process that is not yet completely understood, but it is known that it is driven by heat transfer, mass transfer, and fuel chemistry. A specific case of high interest and complexity is fingering behavior. Fingering is an instability that occurs when a thin fuel layer burns against an oxygen current. These instabilities appear when conduction rather than convection is the dominant mode of heat transfer to the fuel ahead and the availability of oxygen is limited during the combustion of a thin fuel, such as paper. The pattern of the fingers can be characterized through the distance between them and their width, and can be classified into three different regimes: isolated fingers, tip-splitting fingers, or no fingers forming and a smooth continuous front. In this paper, a multilayer cellular automaton based on three governing principles (heat, oxygen, and fuel) is shown to reproduce all the regimes and the details of finger structures observed in previous experiments. It is shown how when oxygen is not limited, a smooth smoldering front is formed. If the oxygen speed decreases beyond a critical value, fingers appear first as tip-splitting fingers and later as isolated fingers, increasing the distance between them and decreasing their thickness. The oxygen consumed during oxidation influences these critical values with a positive correlation. This cellular automaton provides an alternative approach to simulate smoldering combustion in large systems over long times. That the model is able to reproduce the complex pattern formation seen in a fingering experiment validates the model. In the future, we could apply the model in various other geometries to make predictions on the outcome of smoldering combustion processes.

Figure 1

Schematic of the evolution of the front formed when a fluid is injected at the central point of a circular Hele-Shaw apparatus filled by another fluid. (a) The injected fluid is more viscous than the one in the apparatus. The front is stable. (b) The injected fluid is less viscous than the one in the apparatus. The front is unstable; fingers are formed. (c) Schematic pressure distributions along two radii, one passing through a finger [green (upper)] and the other passing in between two fingers [blue (lower)].

Figure 2

Schematic of the experiments in a Hele-Shaw apparatus that are modeled by the cellular automaton. Top (left) and side (right) views. Notice that oxygen flow is opposite to the propagation of the smoldering front.

Figure 3

Graphical explanation of the terms of the overall heat transfer parameter. The conductive term, ${\gamma }_{cd}$, is directly related with the fuel, and has a constant value. The convection term, ${\gamma }_{cv}$, depends on the speed of the oxygen flow and is assumed to be proportional to the square root of the oxygen speed.

Figure 4

Effective diffusivities (heat transfer parameters) in relation to the direction of the oxygen flow, for opposed propagation. The effective diffusivity on the opposite direction of the reaction front is larger than the one on the same direction.

Figure 5

Evolution of the oxygen flow, with ten cells per time step, corresponding to an oxygen speed of $10/18\approx 0.56$ layers per time step. The first step consists of the addition of ten new oxygen-rich cells in random locations at the oxygen front. The second step consists of lateral diffusion of oxygen-rich cells along the oxygen front. In (c), all new oxygen-rich cells that were color coded off-blue (light gray) in (a,b) are colored blue (dark gray).

Figure 6

Lateral diffusion along the oxygen front. (a) The diffusion occurs to the side with greatest height difference with the site, to the left in this case. (b) If both sides have the same difference, diffusion will occur to each of the sides with probability $\frac{1}{2}$.

Figure 7

Schematics of the formation of an oxygen shadow. Opposed smoldering is represented, with the direction of the smoldering front in the positive $x$ direction and the oxygen flow in the negative $x$ direction. Gray color represents the inner part of the oxygen shadow, where oxygen is not sufficient for oxidation. (a–d) represent a sequence in time of a growing oxygen shadow. Courtesy of Belen Otero, 2017.

Figure 8

Sketch illustrating the effect that oxidation has on the oxygen layer. (a,b) Oxidation takes place at the gray site, consuming the oxygen within the red (darker) square of sides $2\mathrm{\Theta }$. (b,c) Lateral diffusion will smooth the oxygen front. The radius of the oxygen shadow $\mathrm{\Theta }$ is two cells. Lateral diffusion occurs after the oxidation.

Figure 9

Four-state structure obtained with the cellular automaton described, left, and corresponding finger structure, right. Only cells in the ash state are considered as part of the fingers, while cells in any of the other three states (wet fuel, dry fuel, and char) are not.

Figure 10

Figure processing to find fingers and their characteristics. From left to right and from top to bottom: (a) result of the cellular automaton with only two states (Fig. 9, right); (b) enclosed gaps filled; (c) clusters of less than six cells were eliminated; (d) isolated cluster were connected to the main fingers. Structures as obtained in (d) were further analyzed.

Figure 11

Patterns generated as the oxygen speed is varied, for an oxygen shadow radius equal to ten cells, at the time step expressed behind each pattern. As the oxygen speed increases, the width of the fingers increases and the distance between them decreases, until the fingers disappear and a smooth continuous front is observed. The arrows represent the direction of the oxygen flow. Initial ignition takes place along the yellow line at the bottom edge.

Figure 12

Fingers patterns for an oxygen shadow radius of five cells, at the time step expressed behind each pattern. The lowest oxygen speed already produces a tip-splitting pattern, and no appearance of isolated fingers is observed in this case. The arrows represent the direction of the oxygen flow. Initial ignition takes place along the yellow line at the bottom edge.

Figure 13

Evolution of the patterns for two oxygen speeds; the top five correspond to two layers per time step and the bottom five correspond to 12 layers per time step, at the time step expressed behind each pattern. At low oxygen speed, for the five studied oxygen shadow sizes fingers appear. At higher speed, the smallest radius produces a continuous front, while the highest radius produces isolated fingers. The arrows represent the direction of the oxygen flow. Initial ignition takes place along the yellow line at the bottom edge.

Figure 14

Staircase diagrams and histograms of the values of the distance between fingers times the number of fingers for Θ equal to ten cells and oxygen speed equal to five layers per time step. Each one of the pairs of graphs (staircase and histogram) is a repetition of the simulation with these values. The red (gray) line in the staircase diagrams represents the average value for the rows from tenth to 80th.

Figure 15

(a) Distance between fingers times the number of fingers versus oxygen speed for different oxygen shadow sizes; (b) width of the fingers versus oxygen speed for different oxygen shadow sizes. The dots show the average (ten simulations) of the different oxygen speed values used in the simulation, while the shadow represents the standard deviation of the results. The inset schematic shows the modeled process, with the directions of the air flow and the smoldering spread, and a horizontal line showing fingers (blue) and distance between fingers (red).