The quasi-steady rate of spread for cell (i+1) is calculated as described by 

 Rothermel (1972): 



^i.i = ^^R^i.i^i.iCt'^s^^i.i/^^g^i.r 



The transition rate of spread as the fire moves from cell i into cell (i+1) is 

 calculated from the heat source terms of cell i and the absorption terms of 

 cell (i+1): 



This can also be written as; 



Rj, = CR.^^ (8) 



where 



C = (I,,)-g-((l) ,<i> )/(Ir,)- -,g- , (* ,(t> ). 



^ R-^i^i^^w'^s-^' ^ R-^ 1+1^1 + 1 ^^w'^s-^ 



Substituting equations (7) and (8) into equations (2) and (3) and evaluating 

 the integrals over the intervals illustrated in figure 8, the distance that the 

 fire spreads into a cell in a given time is as follows: 



Case I : (x ) • < (t ) ■ , 

 ^ r 1 — r 1+1 



CR(t - t^/2(Tp.) + R(t^'2(Tp.^^), for < t < (x^. (9a) 



s = 



CR(x^)./2 + R(t2/2(xp.^^), for (x^. < t < (x^.^^ (9b) 



D' + (t - Cxp.,^)R, for (xp.^^ < t < t^ (9c) 



Case II: (x ) . > (x ) . , 

 r 1 r 1+1 



CR(t - t /2(xp.^J + R(tV2(xp.^^),for < t < (x^.^^ (10a) 



- ^\h.i)^' ^Vi+1 ^ -^^d ^1°^^ 



where R = R. , 

 1 + 1 



The transition distance, D' , can be obtained by solving equations (9b) and 

 (10a) at t = Ctp.^^. 



Case I : (x ) . < (x ) . , 

 ^ r 1 — r 1+1 



D' = R(C(xp. + (xp,^J/2 



(11) 



26 



