Case II: (x } ■ > (x ) -^i 

 ^ r 1 r 1+1 



D' = RCt^.^^CC + l)/2. (12) 



Equations (9) and (10) can be solved for t to obtain the time that it takes 

 the fire to spread a distance, s, in the cell. 



Case 1: (x^. < (x^.^^ 



[-C - + 2b(i - C(Tp.^^/(xp.)]/[l/(Tp.^^ - C/(T^).], (13a) 



for (t ) . < (t ) . , or C ^ 1 

 ^ r 1 ^ r 1+1 



and < s < R(t ) . [c + (t ) . / (t ). J/2 

 — 1 L r'^ 1^ r 1+1-1 



t = 



s/R, for (xp. = (x^).^^, C = 1 (13b) 



and < s < R(t ) . 



— r 1 



V2s(xp.^^/R - C(xp.(xp.^^, (13c) 



for R(xp.[c + (TPi/(xp.^J/2 < s £D' 

 (s - D')/R + (t^.^^, (13d) 



for D' < s < D 



Case II : (x ) ■ > Cx ) ■ . 



^ r-^i ^ r-^i+l 



t = 



+ 2B(1 - C)]/'[(l - C)(xp.^J, for < s <_ D' 



!1_-C + \C + 2B 

 (s - D')/R + (x 



where 



(14a) 



) . , , for D' < s < D (14b) 



r'^i+l' — 



R = R. , 

 1 + 1 



B = s/R(x ) . , . 

 ^ r 1+1 



If the fire reaches its quasi-steady state in a cell, equation (13c) or (14b) 

 for s = D is equivalent to finding the delay time as described in equation (1). 



As stated earlier, the cell size should be large enough for the fire to 

 achieve a quasi-steady state in every cell. Otherwise, the influence of cell i 

 would extend beyond cell (i+1) . At present the algorithm does not handle this 

 situation. Increasing the cell size excessively may result in averaging out the 

 nonunif ormities we originally intended to examine. It is essential to design the 

 array so that a high percentage of the hexagonal cells achieve a quasi-steady 

 state rate of spread and accept the small error resulting from the few exceptions. 

 If the calculated transition distance is larger than the cell size (D' > D) , the 

 delay time is calculated as the time that it takes the fire to travel the distance D. 



27 



