A Simple Fire-Growth Model

Abstract

Forest fire control planning sometimes requires a mathematicaL model of how a forest fire grows with time, Such a formula is especially necessary in problems concerning the economics of fire detection and control. AlthOugh the model presented here is not completely original in concept, it is given in a simple but flexible algebraic form not previously used as far as the author is aware, Assume that, after an initial short period of ad­ justment, the fire's linear rate of spread at each point on the perimeter remains constant. This rate will vary continuously from a maximum at the head to a minimum at the rear, For simplicity, select values of this linear rate of spread for the head, flanks, and rear of the fire, and assume a uniform fuel. Next, assume that the fire's head burns a fan­ shaped area that widens as the head advances' flank spread then proceeds from the sides of the fan: Furthermore, assume that the width of the fan is such that the fire's shape remains elliptical for any combination of head and flank rates, Refer to Figure 1, and let the following symbols apply: .-\ fire's area, an ellipse along semiaxis of ellipse b short semiaxis of ellipse \-linear rate of spread at head u 1 linear rate of spread at flanks \\. linear rate of spread at rear t time since ignitioil Th~n according to the formula for the area of an dlip~,e, .\ =- ;-;ab But ;1 = (v + w) l/2 and b = 2 ut/2 = ut Therefore .-\ = ~ (v + w)ut 2 (1) This expression can be used if all required rates are known, or simplified if necessary. For example, if the fire advances at rate u at all points on its peri­ meter, then expression (1) reduces to ~\ =- ;-; u 2 t 2 (2)