Modeling Fire Spread in a Real-Time Coupled Atmospheric- Vegetation Fire: An Analytical Approach

The ability to analyse the rate of fire spread outbreak in a real-time coupled Atmospheric-vegetation fire has become increasingly vital as forest fire fighters are building diverse kinds of models to combat the dangers/effects of fire spread across a given fire vicinity. This paper theoretically examines the analysis of fire spread in a real fire environment. A partial differential equations (PDE) governing the phenomenon is presented. The analytical solution of the model is obtained via direct integration and eigenfunction expansion technique, which displays the influence of the parameters involved in the system. The effect of change in parameters such as Frank-Kamenetskii number, Radiation number, Peclet energy number and Activation energy number are presented graphically and discussed. The results obtained show that FrankKamenetskii number, Radiation number, Peclet energy number, and Activation energy number all reduced transient state temperature.


Introduction
Interest in mathematical models of vegetation fire is caused by a large number of scientific difficulties. Goldammer and Ronde (2004) noted that forest fire models have been developed since 1940 to the present, but a lot of chemical and thermodynamic questions related to fire behaviour are still to be resolved. Forest fires are divided into underground (peatbog) fires, surface fires, active crown fires, running crown fires (also called independent crown fires), and mass fires. A number of researchers notice, that running crown fires possess the greatest speed of propagation. They are extremely dangerous and very difficult to fight, thus mathematical modelling is represented as an important problem (Pastor et al., 2003).
Fire models can be classified by the type of fire in consideration. There are four major types of fire, and each one has different behaviour. Therefore, physical systems and equations vary from one to another. Pastor et al. (2003) analysed an extensive list of forest fire models and classified them according to their typology. Basically, the four types of fire models are: surface fire, crown fire, spotting fire and ground fire.
Surface fire models deals with the fire that burns the vegetation closed to the surface, such as brush, small trees, or herbaceous plants. Crown fire models are somewhat complementary to surface fire models and studied how the fire spreads over the canopy of trees in a given forest. Spotting fire models provides the equations to analyse those new fire caused by incandescent pieces of the main fire transported out of the main fire perimeter. Finally, ground fire models focused their attention on those physical processes that occur in the substrate of the soil when a fire takes place (Carlos, 2014) Heat, an important aspect of the fire triangle, is the energy transferred between an object of greater temperature to an object of lower temperature. It is this heat energy that is crucial in beginning the evaporative or preheating phase of combustion (Johnson and Miyanishi, 2001). Temperature determines the ease of combustion of wildland fuels. Therefore, higher temperatures heat forest fuels and predispose them to ignition provided that an adequate ignition source becomes readily available (lightning or some anthropogenic source).
Vegetation fire and other natural hazards are cases widely studied by scientific community due to its huge environmental, social and economic impact that they produce virtually every year around the world. Considering forest fire, several preventive actions can be applied to reduce the forest fire risk/spread, but unfortunately, they do not wholly eliminate/eradicate the diverse factors unleashed by a fire. Due to this, our effort is targeted on mitigating the escalation of vegetation fire by showcasing the effects of some parameters such as Frank-Kamenetskii number, Radiation number, Peclet energy number, and Activation energy number on the transient state temperature at a fire scene. This will be achieved via direct integration and eigenfunction expansion technique.

Model Formulations
Following Perminov (2018), a wildfire model is formulated based on balance equations for energy and fuel, where the fuel loss due to burning corresponds to the fuel reaction rate. The respective equations governing forest fires propagation are: Volume fraction of dry organic substance ).

METHOD OF SOLUTION Non-dimensionalisation
Here equation (1) - (6) are non-dimensionalize using the following dimensionless variables: and we obtain;

Solution of the Model
Using perturbation method, direct integration and eigenfunction expansion technique, the analytical solution of equations (8)          0.1 (Blue).

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We observed from the discussions that increment in all the four parameters reduced the temperature, as shown or demonstrated in Figures  1, 2, 3 and 4. Therefore, it is important for fire safety precaution. Wildland fire could be suppress taking into account the effects of these parameters as it relates to temperature.

Conclusion
For a high activation energy situation (i.e. as 0  ), we have solved the equations governing the fire spread model using direct integration and eigenfunction expansion technique. From the results obtained, we can conclude that, Frank-Kamenetskii number, Radiation number, Peclet energy number and Activation energy number all decreased the temperature.
This results obtained are not only expected to guide fire fighters to combat or suppress fire outbreak but to also manage the danger associated with forest fire spread.