Hornby (1936) expresses fire size and shape in terms of the 

 length of perimeter for an ellipse to the circumference of a cir- 

 cle of equal area. For the no-wind, no-slope condition, the cir- 

 cle and the ellipse are equal in perimeter and the //w ratio is 

 1:1. He expressed the most probable fire size as when the 

 perimeter of the ellipse is 1.5 times the circumference of a circle 

 of equal area and the //w ratio is 5:1. The fires Hornby 

 analyzed showed that 92 percent of all the perimeters in- 

 vestigated were less than 2.0 times the circumference of a circle 

 with equal area — or having a i/w ratio of 9.7:1. These three 

 descriptions of fire size and shape are identified in figure 5 by 

 the vertical lines at: 



1. Minimum perimeter = circumference = 1:1 i/w ratio, 



2. Most probable perimeters 1.5 circumference = 5:1 //w 

 ratio, 



3. Maximum perimeter - 2.0 circumference = 9.7:1 //w 

 ratio. 



Figure 5 presents the data previously cited by Hanson (1941), 

 Brown (1941), McArthur (1966), Pirsko (1%1), and Banks 

 (1963), and shows how the other interpretations compare to 

 Hornby's. 



Brown (1941) and Hanson (1941) both used an analysis 

 method of fire size and shape that used the standard error to 

 define the expected minimum and maximum values of 

 perimeter for a given fire size. McArthur's (1966) values for 

 area and perimeter, when wind is a factor, agree with the most 

 probable values found by Brown and Hanson. These results for 

 most probable and maximum values are shown in figure 5 as 

 the solid lines through the data points for i/w ratios from 2: 1 

 to 7: 1 and 8: 1 to 17: 1 . These data points can be expressed 

 mathematically in empirically determined equations: 



For //w ratios < 7 



A = 4.74 (i'/w)'*-638, area. 

 For i/w ratios > 7 



A = 1.62 X 10-^(f/w)6-285 area. 



This suggests that a greater range of combinations for area, 

 perimeter, and i/w ratio occurs than the procedure used by 

 Hornby (1936) can accommodate. This is a result of constrain- 

 ing the perimeters to 1.5 and 2 times the circumference of a cir- 

 cle with equal area. The i/w ratios are then fixed at 5 and 9.7 

 as representing the most probable and the expected maximum 

 //w ratio respectively. 



The relationship of the length to width ratio to the average 

 wind on the flame can be expressed with the equations 

 developed from Fons' wind tunnel data. Using the dimensions 

 of figure 1 where d equals 1 for normalizing, we can describe 

 the ratio of total fire length to maximum fire width: 



10 1- 



t/w = {I + c)/2b 



(16) 



Since the backing and flanking dimensions are expressed as 

 fractions of the forward rate of spread distance, the forward 

 distance has a value of unity. Combining equations 2 and 6 as 

 indicated above and clearing the fractional form, we can ex- 

 press the length to width ratio (fig. 6) by: 



t/v/ = 0.936 EXP(0.1147U) + 0.461 EXP( - 0.0692U) (17) 



where U = windspeed at 1.5 ft or midflame miles per hour. 



< 



OH 



W INDSPEED 1. 5 FEET ABOVE FUEL BED (Ml/ H) 



Figure 6.— Relation of the length to width ratio 

 and windspeed 1 .5 ft (45.7 cm) above the fuel 

 tied. The dotted lines at 2 and 12 mi/h (3.2 and 

 19.3 km/h) show the range of experimental data 

 used by Fons.^ 



Fons found a relationship of length to width to wind that is 

 linear in nature over the range of winds examined: 



d -I- c 

 2b 



1.0 + 0.5 U 



(1) 



where U is miles per hour. The solid line in figure 6 represents 

 this equation, and the dashed line presents a similar equation 

 with a coefficient of 0.44 for fires burning in ponderosa pine 

 needle litter beds in the forest. 



These equations have nearly twice the slope of equation 16 

 or 17, primarily because only the downwind distance and the 

 distance to one side of the centerline of the fire shape are used. 

 Equation 16 can be reduced to a similar form by disregarding 

 the backing distance, c, and using the minor axis dimension, b, 

 as the width: 



d/b = 1/b = 1/0.534EXP[- 

 or 1.873 EXP[0.1147U] 



0.1147U] 



(18) 



7 



