Table 5. — Comparison of Fons'^ diagrams with fire size and stiape matfiematical 

 model outputs using a forward spread distance of 5,000 ft 





WIndspeed 



Graphic 



Computed 



Graphic 



Computed 



Version 



Mi/h 



acres 



acres 



miles 



miles 



Pons 



2 



556.4 





3.38 





Model 





512.8 



513.5 



3.28 



3.31 



Pons 



4 



369.5 





2.88 





Model 







376.0 





2.92 



Pons 



6 



265.1 





2.59 





Model 





281.5 



282.0 



2,60 



2.63 



Pons 



8 



196.0 





2.37 





Model 







214.0 





2.42 



Pons 



10 



157.5 





2.27 





Model 







165.0 





2.26 



Pons 



12 



132.4 





2.17 





Model 





127.9 



128.0 



2.12 



2.15 



^See footnote 1 in text. 



Comparison of Fire Size Properties 



Hornby (1936), using 146 fire records and 102 hypothetical 

 fires, showed that for a constant area reference, the most prob- 

 able perimeter was 1.5 times that of a circle with equal enclosed 

 area. In addition, he found that 92 percent of the fire shapes 

 investigated would have perimeters less than 2 times that of a 

 circle of equal area. These are equivalent to length to width 

 ratiosof5:l and 9.7:1, respectively. Mitchell (1937) related the 

 head fire rate of spread to the rate of perimeter increase using 

 the relationship for circles of the perimeter to the diameter. He 

 suggested the simplest approach was to multiply the head fire 

 rate of spread by 3 for an estimate of the rate of perimeter in- 

 crease. Hanson's (1941) analysis of 140 fires in the Forest Serv- 

 ice's Region 4 relates the length of line in chains to the final 

 acreage of the fire. Brown's (1941) analysis of 65 class "C" 

 fires in Region 2 produced a graph of minimum, average, and 

 maximum control line lengths in chains for fires up to \,000 

 acres (404.7 ha) at control. He found that w times the long 

 axis of the fire agreed closely to the perimeter. For fires under 

 20 acres (8. 1 ha) he found the most probable perimeter to be 

 1 .67 times the perimeter of a circle of equal area. 



Length to width ratios did not appear to receive much atten- 

 tion until McArthur published his Australian research on 

 grassland fires (1966). He recognized that the elliptical shape 

 provides a good approximation of fire shape. With the state- 

 ment on his "grassland fire danger meter" that perimeter in- 

 crease can be taken as 2.5 times the forward spread, fire sizes 

 can be analyzed in terms of perimeter, area, and length to 

 width ratio. Assuming an elliptical shape, we have a unique 

 solution for any fire where two of these dimensions are known. 



Using the concept of equal area shapes as Hornby (1936) in- 

 troduced, we can establish the minimum perimeter an elliptical 

 fire can have unless control action has truncated the fire shape. 

 Working with the simple ellipse we need to use the equations 

 for area and perimeter: 



Area = Trab = A, units^ 



Perimeter = (a + b) kTr = P, units 



where: 

 a = semimajor axis, 

 b = semiminor axis, 

 k = equation (9). 



(11) 



(12) 



By using the area equation (1 1) to define b we can substitute 

 into equation (12) and reduce it to a quadratic equation: 



7rka2 - aP -f- kA = (13) 



The two axes, a and b, can be calculated from: 



^p2 - 47rk2A 



a = P -I- 



27rk 



and 



b = P 



^p2 - Airk^A 



2irk 



(14) 



(15) 



and by successive approximations determine the unique com- 

 bination of area, perimeter, k, and length to width ratio that fit 

 a given set of conditions. 



Using this technique, a series of areas and perimeters were 

 evaluated and length to width ratios, i/w, were determined 

 (fig. 5). Note that area will be the square of the unit of 

 measure used for perimeter — ft^ for area when perimeter is in 

 feet, for example. Perimeter may be any unit of length that is 

 suitable. 



1,000,000 - 



100,000 - 



10,000 - 



10, 000 



5000 



2500 



1000 tz 



1000 



18 20 



length; WIDTH RATIO 



Figure 5.— Tlie relationship of area, perimeter, 

 length to width ratio for elliptical-shaped fires 

 and comparison of several appraisals of fire 

 size and shape relationships. 



6 



