Two-Dimensional Mathematical Models to Simulate Deciduous Tree Ignition

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Mathematical Statement

A freestanding deciduous tree (referred to as angiosperms) is considered. At a fixed point in time, a cloud-to-ground lightning discharge strikes the tree trunk. An electrical current from a cloud-to-ground lightning discharge flows inside and along the surface of the tree trunk. It is assumed that the current parameters are the same in different sections of the trunk. It is assumed that 1) the vessels are located in groups, that is, they can be combined into large bundles (vessels); 2) the vessels are flattened along the line of contact; 3) vessels are connected between separate growth rings. The combination of the above assumptions is the basis for the approximation of large vessels. As a result of the release of Joule heat, the wood is heated. When critical values ​​of heat fluxes from large vessels of the core or from the surface layer of water to the ignition surface and its temperature are reached, the deciduous tree ignites. The problem is solved in polar coordinates in a flat setting. A tree trunk cut along the z-axis is modeled.

The solution area diagram is shown in Fig. 1.a, 1.c, where 1 - core, 2 - bark, 3 - large vessels, 4 - surface layer of water; R_w is the boundary of the surface water layer, R_s is the outer radius of the trunk, R₁ is the boundary between the core and the bark. Гi, Гj, Гj.k - boundaries of the regions, which are shown in Fig. 1.b, 1.d.

Figure 1.

Geometry of the solution region (a, c) and boundaries (b, d)

Mathematically, the process of heating a tree by a cloud-to-ground lightning discharge until the moment of ignition is described by a system of non-stationary differential equations of thermal conductivity:

,(1)

,(2)

,(3)

,(4)

Boundary conditions for equations (1) - (4):

Г_0.1,Г_0.2,(5) Г_i, T₁=T₂,(6) Г_j, T₁=T₃,(7), T₁=T₃,(8)

Key Terms in this Chapter

Forest Fire: Uncontrolled aerothermochemical phenomenon characterized by step-by-step mechanism which includes following stages: inert heating, moisture evaporation, high temperature terpens evaporation, dry organic matter pyrolysis, flammable combustion and smoldering.

Prediction: Under the prediction of forest fires is the calculation of the parameters of forest fire danger with a certain projection in advance in order to have enough time to anticipate an emergency. The calculation in this case is carried out in a mode ahead of the real time of the development of the catastrophe - the occurrence of a forest fire.

Mathematical Simulation: The production of a computer model of forest fire conditions and prerequisites, especially for the purpose of study.

Lightning Activity: An atmospheric phenomenon characterized by discharges of the cloud-to-cloud and cloud-to-cloud-to-ground class.

Cloud-to-Ground Lightning Discharge: An electrical discharge during a thunderstorm that occurs between a cloud and the earth’s surface. It is a natural source of forest fires.

Monitoring: Monitoring refers to the periodic calculation of the parameters of forest fire danger with a portion of information available in real time.

Forest Fuel: It can be considered like dead and live forest fuel. Main types of forest fuel which can be involved in combustion during forest fire: cloud-to-ground forest fuel (needles, leaves and dry grass, small branches) and crown forest fuel (needles, small branches).

Meteorological Parameters: Physical characteristics of local weather conditions in the forested area under consideration. Key parameters include ambient temperature, soil temperature, precipitation, wind speed, solar radiation, cloud cover, dew point temperature. These parameters are used for mathematical modeling of the drying of a layer of forest fuel.

Ignition Delay: Time before flame flash after forest fuel heating.

Ignition: Inflammation of forest fuel caused by definite source of high temperature or energy.