Abstract
The results of the numerical implementation for the spatial formulation of the problem of the deciduous tree ignition by an electric current of a cloud-to-ground lightning discharge are presented. The problem was solved in a three-dimensional formulation in a cylindrical coordinate system. A symmetric formulation of the problem is considered. The localization of reactive wood, structural heterogeneity of the bark, volt-ampere characteristics of a cloud-to-ground lightning discharge, and chemical reaction in the gas phase are taken into account. The conditions of deciduous tree ignition in the typical range of the discharge effect are revealed.
TopMathematical Model
According to (Esau, 1980), the electric current of a cloud-to-ground lightning discharge passes through the core of a deciduous tree trunk. The main assumptions and suggestions: 1) reactive wood is formed in the upper part of the branches (Esau, 1980); 2) the approximation of an “ideal” crack in the bark is used; 3) in the calculations, only a part of the branch growing from the trunk is considered. The rest of the continuation is not considered, since it was previously established that during the exposure to an electric current, the outer part of the branch does not have time to warm up; 4) the main product of pyrolysis is carbon monoxide (Grishin, 1997); 5) the leading chemical reaction is the oxidation of carbon monoxide to carbon dioxide (Grishin and Shipulina, 2002); 6) the tree is considered as a conductor of the resistor type, for which Ohm's and Joule-Lenz's laws are valid (Sivukhin, 1977); 7) the approximation of large vessels is used; 8) moisture evaporation is described by the Knudsen-Langmuir equation (Pankratov et al., 1975). To describe the process under study, the following physical model is adopted. A freestanding deciduous tree is considered. At a fixed moment in time, a lightning discharge of a certain polarity and duration strikes the tree trunk. It is believed that the current-voltage characteristics of the discharge are the same for different sections of the tree trunk. As a result of the flow of electric current in a large vessel, the wood heats up due to the release of Joule heat. In the process of further heating, thermal decomposition of wood occurs with the formation of gaseous pyrolysis products, which instantly enter the gas phase region and mix with the oxidizing agent. At certain temperatures and concentrations of reagents, a chemical reaction of carbon monoxide oxidation occurs. It is considered that ignition occurs if critical values of the following parameters are reached: 1) the heat flux from the chemical reaction zone exceeds the heat flux from the subcrustal zone of the tree; 2) the temperature of the gas mixture reaches a critical value. The solution area is shown in Fig. 1.
The process of ignition of a deciduous tree by a cloud-to-ground lightning discharge is described by a system of three-dimensional non-stationary nonlinear equations of heat conduction and diffusion with corresponding initial and boundary conditions. For numerical implementation, a finite-difference method was used (Samarskiy and Vabishchevich, 2001). Two-dimensional and three-dimensional difference equations were solved by a locally one-dimensional method (Samarskiy and Vabishchevich, 2001). Difference analogues of the one-dimensional equations of heat conduction and diffusion were solved by the marching method in combination with the simple iteration method (Samarskiy, 1983).
Mathematical model:
,(1),(2),(3),(4),(5),(6),(7),(8),(9),(10),(11).(12)
Kinetic equations:
,
(13),
(14),
(15),
(16),
(17),
(18),
(19),
(20),
(21).
(22)
[377],
(23).
(24) Equation of state:
,
(25).
(26) Initial conditions:
t=0,
,
i=1,2,3,4,5,6,7
,g (27)
,
i=8,9,10,11
(28)
,
i=12,13,14
(29) Boundary conditions:
Key Terms in this Chapter
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-cloud-to-ground class.
Monitoring: Monitoring refers to the periodic calculation of the parameters of forest fire danger with a portion of information available in real time.
Ignition: Inflammation of forest fuel caused by definite source of high temperature or energy.
Ignition Delay: Time before flame flash after forest fuel heating.
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.
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-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.
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.
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.