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USDA Forest Service 

Research Paper INT-115 

January 1972 

A MATHEMATICAL MODEL 

FOR PREDICTING 

FIRE SPREAD 

IN WILDLAND FUELS 

Richard C. Rothermel 

INTERMOUNTAIN FOREST AND RANGE EXPERIMENT STATION 

Forest Service 

U.S. Department of Agriculture 

Ogden, Utah 84401 

Robert W. Harris, Director 
PREFACE 

Forest managers as well as those engaged in research involving fires in forests, brush fields, and grasslands need a consistent method for predicting fire 

spread and intensity in these fuels. The availability of the mathematical 

model of fire spread presented in this paper offers for the first time a 

method for making quantitative evaluations of both rate of spread and fire 

intensity in fuels that qualify for the assumptions made on the model. Fuel 

and weather parameters measurable in the field are featured as inputs to the 

model. It is recognized that this model of the steady-state fire condition is 

only a beginning in modeling wildland fires, but the initial applications to 

the National Fire-Danger Rating System and to fuel appraisal illustrate its 

wide applicability. 

The introduction of this model will permit the use of systems analysis 

techniques to be applied to land management problems. As a result, a new 

dimension is offered to land managers for appraising the consequences of 

proposed programs. Questions can be answered such as: What is the resultant 

fuel hazard when thinning is done in overstocked areas? Can logging practices be modified to reduce the potential fire hazard of the fuels they produce? How much slash should be left on the ground to produce the desired 

site treatment for the next crop of trees? How long after cutting can a successful bum still be achieved? What is the hazard buildup in chaparral brush 

fields of the Los Angeles Basin in years subsequent to the last burn? 

Systems analysis can be applied not only to these broader aspects of vegetative manipulation activities, but also to traditional activities, such as presuppression planning and prescribed burning. As we learn more about the 

growth and decay patterns of our fuels, the long-range consequences of management policy can be examined and appraised on a quantitative basis. Decisions will be more often in line with the desired outcome when the alterna- ' tives to proposed practices can be compared and evaluated before a stick of 

wood is cut. 

This mathematical model has been developed for predicting rate of spread 

and intensity in a continuous stratum of fuel that is contiguous to the 

ground. The initial growth of a forest fire occurs in the surface fuels (fuels 

that are supported within 6 feet or less of the ground). Under favorable 
burning conditions, if sufficient heat is generated, the fire can grow vertically 

into the treetops causing a crown fire to develop. The nature and mechanisms of heat transfer in a crown fire are considerably different than those for 

a ground fire. Therefore, the model developed in this paper is not applicable 

to crown fires. An exception can be made for brush fields. Brush, such as 

chamise, is characterized by many stems and foliage that are reasonably contiguous to the ground, making it suitable for modeling as a ground fire. 

Contributions to the spread of the fire by firebrands have not been included. At first this may seem to be a serious limitation to the model because everyone who has been on a large fire (most investigators go to large 

fires, the fires not presently being modeled) knows the importance of spotting. However, seeing firebrands in the air and landing ahead of the fire front 

does not mean that they are effective in advancing the fire. Berlad (1970) 

has shown that not all firebrands have a significant effect in spreading a fire. 

To be significant, firebrands must release sufficient heat when they land to 

ignite the adjacent fuels, and they must do so before the fire would have 

overrun the descent point as a result of conventional heat transfer mechanisms. 

Furthermore, the model has been designed to simulate a fire that has 

stabilized into a quasi-steady spread condition. Most fires begin from a single 

source and spread outward, growing in size and assuming an elliptical shape 

with the major axis in the direction most favorable to spread. When the fire 

is large enough so that the spread of any portion is independent of influences 

caused by the opposite side, it can be assumed to have stabilized into a line 

fire. A line fire behaves like a reaction wave with progress that is steady over 

time in uniform fuels. 

All input parameters can be determined from knowledge of the characteristics of fuels in the field. This does not imply that all the parameters of fuels 

and environment are readily available or can easily be measured. It does, 

however, delineate what parameters should be cataloged and eliminates those 

that are not needed. A convenient method of cataloging input parameters is 

through the concept of fuel models tailored to the vegetation patterns found 

in the field. The companion fuel models are thus a set of input parameters 

that describe the inherited characteristics that have been found in certain 

fuel types in the past. The environmental parameters of wind, slope, and expected moisture changes may be superimposed on the fuel models. This fuel 

model concept has already been incorporated into the National Fire-Danger 

Rating System (Deeming and others 1972). 

The mathematical model produces quantitative values of spread and intensity that should be regarded as appraised or mean values for the given fuel 

and environmental conditions. The National Fire-Danger Rating System, 

however, will display the values on a relative scale in the form of indexes. 

The indexes developed from this mathematical model can be designed to predict conditions during which severe fire phenomena develop, even though 

the model does not include mass fire effects. 

Concurrently, studies designed to confirm portions of the model through 

field tests have been conducted and are reported by J. K. Brown (1972). 
THE 

Richard C. Rothermel is a Research Engineer stationed at the 

Northern Forest Fire Laboratory in Missoula, Montana. He is 

the Research Project Leader for the Fire Physics research work 

unit. Rothermel received his B.S. degree in Aeronautical Engineering at the University of Washington in 1953. He served in 

the U. S. Air Force as a Special Weap~ns Aircraft Development 

Officer from 1953-1955. Upon his discharge, be was employed 

at Douglas Aircraft Company as a troubleshooter in the Armament Group. From 1957 to 1961 Rothermel was employed by 

the General Electric Company in their Aircraft Nuclear Propulsion Department at the National Reactor Testing Station in 

Idaho. In 1961, Rothermel joined the staff at the Northern Forest Fire Laboratory, where he has been engaged in research on 

the mechanisms of fire spread. He received his master's degree 

in Mechanical Engineering at the University of Colorado in Fort 

Collins in 1971. 
ABSTRACT 

The development of a mathematical model for predicting rate of fire 

spread and intensity applicable to a wide range of wildland fuels is presented 

from the conceptual stage through evaluation and demonstration of results 

to hypothetical fuel models. The model was developed for and is now being 

used as a basis for appraising fire spread and intensity in the National FireDanger Rating System. The initial work was done using fuel arrays composed of uniform size particles. Three fuel sizes were tested over a wide 

range of bulk densities. These were 0.026-inch-square cut excelsior, 114-inch 

sticks, and 112-inch sticks. The problem of mixed fuel sizes was then resolved by weighting the various particle sizes that compose actual fuel arrays 

by either surface area or loading, depending upon the feature of the fire 

being predicted. 

The model is complete in the sense that no prior knowledge of a fuel's 

burning characteristics is required. All that is necessary are inputs describing 

the physical and chemical makeup of the fuel and the environmental conditions in which it is expected to burn. Inputs include fuel loading, fuel depth, 

fuel particle surface-area-to-volume ratio, fuel particle heat content, fuel 

particle moisture and mineral content, and the moisture content at which 

extinction can be expected. Environmental inputs are mean wind velocity 

and slope of terrain. For heterogeneous mixtures, the fuel properties are 

entered for each particle size. The model as originally conceived was for dead 

fuels in a uniform stratum contiguous to the ground, such as litter or grass. 

It has been found to be useful, however, for fuels ranging from pine needle 

litter to heavy logging slash and for California brush fields. 

The concept of fuel models is introduced, wherein parameters of wildland 

fuels necessary for inputs to the model are categbrized and tabulated. These 

are then used to predict fire spread and intensity; this eliminates the necessity for repeatedly measuring such parameters. The conceptual approach 

recognizes that fuels have inherent characteristics that are repeatable. 
CONTENTS - - - 

HOW FIRE SPREADS ................................... 1 

CONCEPTION OF MATHEMATICAL MODEL .................. 3 

Heat Required for Ignition .............................. 4 

Propagating Flux ..................................... 4 

Reaction Intensity .................................... 6 

Effect of Wind and Slope ............................... 6 

Approximate Rate of Spread Equation ...................... 6 

EVALUATION OF PARAMETERS. NO.WIND. OR SLOPE .......... 7 

HeatSink .......................................... 7 

Heat Source ........................................ 9 

Experimental Design ................................. 14 

Experimental Results ................................. 16 

EVALUATION OF WIND AND SLOPE COEFFICIENTS ........... 21 

Wind Coefficient .................................... 21 

Slope Coefficient .................................... 24 

SUMMARY OF FIRE SPREAD EQUATIONS .................. 25 

THE FIRE SPREAD MODEL ............................. 28 

Formulation of Fire Spread Model ........................ 30 

APPLICATION TO THE FIELD ........................... 34 

Fuel Models and Applications ........................... 35 

LITERATURE CITED .................................. 40 
HOW FIRE SPREADS 

Early work in fire spread research conducted by the USDA Forest Service was 

primarily aimed at developing relationships between burning conditions and obvious 

variables that would aid forest managers to cope with fire problems. Such variables as 

fuel moisture, fuel loading, wind velocity, relative humidity, slope, and solar aspect 

were all recognized as producing important effects on fire. These effects were studied 

and correlated to some form of fire behavior. The work was primarily done outdoors and 

some very good results were obtained, considering the complexities of the problem and 

the variability of weather, particularly wind. Much of the present day fire - danger 

rating, fuel classification, and other uses of fire research are based on this pioneering work. 

W. R. Fons (1946) was the first to attempt to describe fire spread using a mathematical model. Fons focused his attention on the head of the fire where the fine fuels 

carry the fire and where there is ample oxygen to support combustion. He pointed out 

that sufficient heat is needed to bring the adjoining fuel to ignition temperature at 

the fire front. Therefore, Fons reasoned that fire spread in a fuel bed can be visualized as proceeding by a series of successive ignitions and that its rate is controlled 

primarily by the ignition time and the distance between particles. 

Fon's early ideas have been confirmed by recent work in flame spread theory. 

Tarifa and Torralbo (1967) state that: 

Heating of the fuel ahead of the flame as it progresses is the first 

and most essential process of the flame propagation mechanism. Therefore, it is very important to know the flame propagation mechanism 

from flame to fuel and to study the time consumed for the heating 

process since it may control propagation speed in many cases. Nevertheless, there is little information on these problems. 

McAlevy and others l theori zed that : 

The phenomena of flame spreading over an igniting propellant surface 

is viewed herein as one of continuous, diffusive, gas-phase ignition; 

thus, the flame spreading phenomena is linked inextricably to the 

ignition phenomena. 

l~obert F. McAlevy, 111, Richard S. Magee, and John A. Wrubel. Flame spreading 

at elevated pressures over the surface of igniting solid propellants in oxygen/inert 

environments. (Paper presented at spring meeting of Western States Sect. Combust. Inst., 

1967. ) 
Heading fire - 5 m.p.h. 

- Tg=Temperature of air 

Ts= Temperature %f surface 

- 

Distances from fire 

I I I I I I I I I I 

Time since beginning of fire , minutes 

Figurs 1.--&el temperature history prior to ignition for heading, no-wind, and backing 

fires. 

Considering fire as a series of ignitions helps in breaking down the problem for 

analysis. Heat is supplied from the fire to the potential fuel, the surface is 

dehydrated, and further heating raises the surface temperature until the fuel begins to 

pyrolyze and release combustible gases. When the gas evolution rate from the potential 

fuel is sufficient to support combustion, the gas is ignited by the flame and the fire 

advances to a new position. Finally, a constant rate of spread is achieved; this is 

called the "quasi-steady state " wherein the fire advances at a rate that is the average 

of all the elemental rates. 

This process is illustrated in figure 1, which is based on a laboratory test in 

which we monitored the surface temperature of a fine fuel element and the air adjacent 

to it ahead of an advancing fire. In the no-wind fire and backing fires, the fuel 

temperature rose slowly until the fire was within 1 or 2 inches of the fuel element 

where it suddenly rose to ignition. During the preheating phase, the fuel temperature 

exceeded the air temperature; this indicates that convective heating or direct flame 

contact does not occur until the fire front reaches the particle. Consequently, radiation must have accounted for the energy imparted to the fuel elements on the upper surface while simultaneously the particle was being cooled by convective indrafts. This 

does not occur in the heading, or wind-driven fire, in which the temperature of the 

fuel rose steeply even when the fire was 2 feet from the thermocouple that had been 

inserted in the particle. During the rise to ignition, the air temperature was higher 

than the fuel surface temperature; this shows that convective heating can be present in I 

addition to radiation. Such temperature histories indicate that basic differences exist 

in the mechanisms that bring fuels to ignition. These basic differences provided - 

us with a method for characterizing fires and developing similar methods for 

mathematical modeling. 
CONCEPTION OW 

MATHEMATICAL MODEL 

The model was developed from a strong theoretical base to make its application as 

wide as possible. This base was supplied by Frandsen (1971) who applied the conserva - 

tion of energy principle to a unit volume of fuel ahead of an advancing fire in a 

homogerieous fuel bed. His analysis led to the following: 

R = C 

'be Qig 

where : 

I? = quasi-steady rate of spread, ft./min. 

I . = horizorltal heat flux absorbed by a unit volume of fuel at the time of ignition, x1g ~.t.u./ft.~-mill. 

= effective bulk density (the amount of fuel per unit volume of the fuel bed 

raised to ignition ahead of the adva~lcing fire), lb./ft. 3 

Q. = heat of preignitioll (the heat required to bring a unit weight of fuel to 

lg ignition), B.t.u./lb. 

= the gradient of the vertical irltensity evaluated at a plane at a constant (2') depth, zc, of the fuel bed, B.t.u./ft. 3-min. z 

C 

The horizontal and vertical coordinates are x and z, respectively. 

111 Fraridsen's analysis, the fuel- reaction zone interface is fixed and the unit 

volume is moving at a co~lstant depth, zc, from x = -m toward the interface at x = 0. 

The unit volume ignites at the interface. 

111 one sense, equation (1) shows that the rate of spread during the quasi-steady 

state is a ratio between the heat flux received from the source in the numerator and 

the heat required for ignitioii by the potential fuel in the denominator. Equation (1) 

corltairls heat flux terms for which the mechariisms of heat transfer are not known; 

consequently, it could not be solved analytically at this time. To solve equation (I), 

it was necessary to examine each term and determine experimental and analytical 

methods of evaluation. This required the definition of new terms that ultimately 

provided an approximate solution to equation (1). 
Heat Required for Ignition 

The heat required for ignition is dependent upon (a) ignition temperature, (b) 

moisture content of the fuel, and (c) amount of fuel involved in the ignition process. 

The energy per unit mass required for ignition is the heat of preignition, Qig: 

where : 

M = ratio of fuel moisture to ovendry weight f 

T = ignition temperature. i g 

The amount of fuel involved in the ignition process is the effective bulk density, 'be. 

To aid interpretation and analysis, an effective heating number is defined as the ratlo 

of the effective bulk density to the actual bulk density. 

'be EE-' 

'b 

The effective heating number is a dimensionless number that will be near unity for fine 

fuels and decrease toward zero as fuel size increases. Therefore, 

'be = f (bulk density, fuel size) C41 

Propagating Flux 

The propagating flux is the numerator of the RHS (right-hand side) of equation 

(1) and has the units of heat per unit area, per unit time. The propagating flux is 

represented by I . 

P ' 

The propagating flux is composed of two terms, the horizontal flux and the gradient 

of the vertical flux integrated from minus infinity to the fire front. These fluxes 

can be characterized as shown in figures 2, 3, and 4. The figures indicate that the 

vertical flux is more significant during wind-driven and upslope fires because the 

flame tilts over the potential fuel, thereby increasing radiation, but more significantly 

causing direct flame contact and convective heat transfer to the potential fuel. 

We will assume the vertical flux is small for no-wind fires and let Ip = (Ip)o. 

In the model, (Ip)o is the basic heat flux component to which all additional effects of 

wind and slope are related. 

When we substitute equations (3) and (5) into equation (1) and let I = (lp)o and R = R for the no-wind case, then P 

0 
Figure 2. --Schematic of 

no-wind fire. 

Wind 

Figure 3. --Schematic of 

wind-driven fire. 

Solid mass transport 

Figure 4. --Schematic of 

ups lope fire. 
Equation (6) permits (Ip)o to be evaluated from experiments with spreading fires in 

the no-wind condition by measuring Ro over a wid? range of fuel conditions. Note that 

the propagating flux occurs at the front of the fire; therefore (Ip)o is expected to 

be closely related to the fire intensity of the front: 

Reaction Irrtensity 

The energy release rate of the fire front is produced by burning gases released 

from the organic matter in the fuels. Therefore, the rate of change of this organic 

matter from a solid to a gas is a good approximation of the subsequent heat release rate 

of the fire. The heat release rate per unit area of the front is called the reaction 

intensity and is defined as: 

where : 

- dW - mass loss rate per unit area in the fire front, lb./ft. 2-min. 

dt 

h = heat content of fuel, B.t.u./lb. 

'The reaction intensity is a function of such fuel parameters as the particle size, bulk 

density, moisture, and chemical composition. 

The reaction intensity is the source of the no-wind propagating flux, (Ip),. An 

important concept upon which the model is based that (I )o and IR can be evaluated i n - 

dependently and correlated. Knowing the correlation, (?p)o can be determined from the 

reactio~i intensity, which is in turn dependent on fuel parameters obtained from the 

fuel bed complex. 

If this concept is kept in mind, it will aid in understanding the development of the 

model. 

Effect of Wind and Slope 

Wind and slope change the propagating heat flux by exposing the potential fuel to 

additional convective and radiant heat (figs. 3 and 4). 

Let and @s represent the additional propagating flux produced by wind and slope. 

They are dime~~siorlless coefficierlts that are functions of wind, slope, and fuel para - 

meters. They must be evaluated from experimental data. The total propagating flux is 

represented by the expression, 

Approximate Rate of Spread Equation 

Inserting the approximate relationsllips, equation (1) becomes : 
EVALUATION OF PARAMETERS, 

NO -WIND, OR SLOPE 

The conceived functional relationships necessary for evaluating equation (1) are 

divided and considered first as those forming a heat sink, and second as those serving 

as a heat source. 

Heat Sink 

Heat of Preignition 

The heat of preignition and the effective bulk density are the two terms that had 

to be evaluated before the propagating flux could be computed. Qig was evaluated 

analytically for cellulosic fuels by considerirlg the change in specific hedt from ambient 

to ignition temperature and the latent heat of vaporizatiorl of the moisture. 

where : 

C = specific heat of dry wodd 

pd 

AT. = temperature range to igrlitiorl 18 

bI = fuel moisture, lb. water/lb. -Try wco f 

C = specific heat of water 

P w 

ATB = temperature rarlge to boiling 

V = latent heat of vaporizatiorl. 

Details of the calculatio~l are by Frar~dserl.~ The temperature to ignition is 

assumed to range from 20" to 320' C. and boiling temperature to be at 100' C., then 

equation (11) becomes : 

2~. H. Frandsen. The effective heating of fuel particles ahead of a spreading 

fire. USDA Forest Serv. , Intermountain Forest and Range Exp. Sta. , Ogden, Utah (in 

preparation). 
Figure 5. --Instrwnentation for 

determining the effective 

bulk density. 

Moisture is the primary independent variable in the evaluation of Qig; however, it 

is recognized that other parameters should eventually be included in this evaluation: 

heriti~lg rate, inorganic impurities, and nonpyrolytic volatiles. 

Effective Bulk Density 

To evaluate the effective bulk density (pbe), we needed to determine the efficiency 

of heating as a functioil of particle size. This was evaluated by placing thermocouples 

within sections of two sticks that were located on the upper surface 3 feet from one 

end of standard wood cribs. The instrumented sections were oriented in both the longitudinal and lateral directions (fig. 5). The temperature distribution within the 

sticks was analyzed to determine the amount of heat absorbed by the sticks up to the 

time of ignition. 

Results of the analysis are show11 in figure 6. An exponential fit to the data is: 

where : 

a = particle surface- area-to-volume ratio, ft.-] 

If we take unity as being 100-percent heating for a hypothetical zero-thickness 

fuel, figure 6 shows that 22 percent of the 1/2-inch stick and 50 percent of the 

1/4-inch stick must be heated to ignition; it also predicts that 92.8 percent of 

the excelsior is heated. This agreed with our original assumption of 100 percent for 

fine fuels. 

3~randsen, ibid. 
Figure 6. --Effective heating 

number versus Z/a. a is 

the s urface-area- to-volume 

ratio of the particle; E 

is a measure of the fraction 

of the potential fuel that 

must be raised to ignition. 

Measured values 

- 

Heat Source 

Reaction Intensity 

The most complex function evaluated was reactiori intensity using a new concept 

that evolved by deriving fire intensity from the weight loss data.4 The evaluation 

was made from a series of experimerlts utilizing an i~istrumerlt system that recorded the 

weight of a portion of the fuel bed during fire spread. 

Equation (7) can be rearranged to express reaction irlte~isity in the following malnier: 

where : 

- 

dx = R, the quasi-steady rate of spread. 

dt 

Therefore, 

I dx = -Rh dw. R (16) 

To solve equation (16) integrate x over the reactiori zone depth, D, and w over the limits 

of the loading in the reaction zone. 

4\~. H. Frarldsen arid R. C. Rothermel. bleasurirlg the energy release rate of a spreading fire. USDA Forest Serv., Illtermountairl Forest and Range Exp. Station, Ogden, 

Utah (in preparation) . 
This gives 

I D = Rh(w -W ). R n r 

where : 

D = reaction zone depth (front to rear), ft. 

w = net initial fuel loading, ~b./ft.~ n 

w = residue loading immediately after passage of the 

r reaction zone, lb. /ft. 

The net initial fuel loading was corrected for the presence of noncombustibles, water, 

and minerals. 

The time taken for the fire front to travel a distance equivalent to the depth of 

one reaction zone is the reaction time T R' 

Substituting the reaction time into equation (18) gives 

We now define a maximum reaction intensity where there is no loading residue left 

after the reaction zone is passed and where the reaction time remains unchanged. This 

maximum reaction intensity is represented by 

The reaction zone efficiency is then defined as 

Q6=I I R (wn-wr) - 

Rmax W I1 

Replacing (w -wr) in equation (201, we 11:~vc 1 in terms of measurable fuel and fire I1 parameters. It 

The net fuel loading necessary for equation (23) can be obtained from equation (24). 

where : 

w = ovendry fuel loading, ~b./ft.~ 

0 

lb . rnilierals S = fuel mineral content, T lb. dry fuel ' 
Reaction Velocity 

The reaction velocity is a dynamic variable that indicates the completeness and 

rate of fuel consumption, Therefore, it represents the dynamic character of the fire 

and is the key to successful development of the model. 

The reaction velocity is defined as the ratio of the reaction zone efficiency to 

the reaction time, 

" 

I' = reaction velocity, min.-l. T R 

Four fuel parameters are considered to have a major effect on the reaction velocity -- 

moisture content, mineral content, particle size, and fuel bed bulk density. 

Fuel moisture and mineral content are introduced through two damping coefficier~ts 

that operate on the potential reaction velocity; the latter is the reaction velocity 

that would exist if the fuel were free of moisture and contained minerals at the same 

concentration as alpha cellulose. The presence of moisture or minerals reduces the 

reaction velocity below its poterltial value. 

Let: 

r' = potential reaction velocity, min.-l 

rlM = moisture damping coefficierit having values raligir~g from 1 to 0, dimensionless. 

ns = mineral damping coefficient having values ranging from 1 to 0, dimensionless. 

Then : i- = r' 74"s . (26) 

Substitutirlg equations (25) and (26) into equation (23) produces the final expression 

for reaction interlsity. 

The reaction velocity and the moisture and mineral damping coefficients must be evaluated by experimentation. 

Yoisture Damping Coefficient 

The moisture damping coefficient is defined as 

Anderson (1969) tested identical fuel beds of ponderosa pine needles over a wide 

moisture range. The ratio I /I Rmax or nM, as plotted in figure 7, was obtained from 

his data. 

The abscissa in figure 7 is the ratio of Mf, the fuel moisture, to >Ix, the moisture 

of extinction. Mx is the moisture content of the fuel at which the fire will not I spread. For litter fuels of poliderosa pine needles, Mx z 0.30; for other dead fuels, 
Figure 7. --Determination of 

moisture damping coefficient from ponderosa 

pine needle fuel beds. 

- 

C 

(I) .- 

U .6 (I) 

8 

Fuel moisture ratio- Mf/Mx 

it may vary between 0.10 and 0.40.~ Kecent field experiments in logging slash (Brown 

1972) indicate that Mx may be between 0.10 and 0.15 for logging slash, which is more 

porous than litter. 

The equation for the curve in figure 7 is 

The moisture damping coefficient accounts for the decrease in intensity caused by the 

combustiorl of fuels that initially contained moisture. The exact effect of the moisture has not been adequately explained in terms of reaction kinetics. 

Qig is included implicitly in the development of nM. If further studies of Qig 

reveal ~t to be nonlinear, then the curve form of nM will change. 

Mineral Damping Coefficient 

The mineral damping coefficient was evaluated from thermogravimetric analysis (TGA) 

data of natural fuels by Philpot (1968). In this study, it was assumed that the ratio 

of the normalized decomposition rate would be the same as the normalized reaction 

intensity. The maximum decomposition rate used for normalization was at a mineral 

content of 0.0001, a value that was assumed to be the lowest fractional mineral content 

for natural fuels. Philpot found that silica did not affect the decomposition rate. 

Therefore, the silica-free ash content was taken as the independent parameter. The 

data are shown in figure 8. The equation for the curve ,in figure 8 is 

where: Se = effective mineral content (silica free). 

5~o avoid confusion in equation development, the moisture and mineral values are 

expressed as a ratio rather than a percent. 
Figure 8. --Mineral. damping 

coefficient of natural 

fuels, derived from the 

work of Philpot (1968). 

Effective mineral content - Se 

Physical Fuel Parameters 

Two variables remain that had to be considered in evaluating the reaction 

intensity--fuel bed compactness and fuel particle size. Both are known to have significant effects upon combustibility, but to date, integrated research has not been conducted to separate and quantify the effects of these variables on the dynamic character of 

fire. 

It is hypothesized that low values of fire intensity and rate of spread occur at 

the two extremes of compactness (loose and dense). In dense beds, this can be attributed 

to low air-to-fuel ratio and to poor penetration of the heat beyond the upper surface 

of the fuel array. In loose beds (at the other extreme], low intensity and poor spread 

are attributed to heat transfer losses between particles and to lack of fuel. Between 

these two extremes, therefore, there must be an optimum arrangement of fuel that will 

produce the best balance of air, fuel, and heat transfer for both maximum fire intensity 

and reaction velocity. It is not expected that the optimum arrangement will be the same 

for different size fuel particles. 

The compactness of the fuel bed is quantified by the packing ratio, which is 

defined as the fraction of the fuel array volume that is occupied by fuel. The packing 

ratio can be easily calculated by evaluating the ratio of the fuel array bulk density 

to the fuel particle density, 

where : 

B = packing ratio, dimensionless 

pb = fuel array bulk density, lb ./ft. 

p = fuel particle density, lb./ft.3. 

P 
The surface-area-to-volume ratio is used to quantify the fuel particle size. 

Let a = the fuel particle surface-area-to-volume ratio. For fuels that are long 

with respect to their thickness, 

4 

a = - , ft. -1 d 

where : 

d = diameter of circular particles or edge length of square particles, ft. 

The packing ratio of the fuel array, 8, and the surface-area-to-volume ratio of 

the fuel particle, a, are the primary independent variables used throughout the 

remainder of the paper for evaluating correlation equations. 

Experimental Design 

'To evaluate the reaction velocity, a weighing platform was constructed as part of 

the fuel support surface for the experimental fuel beds. This weighing platform, which 

was 18 inches square, was supported by four load cells, which were protected from the 

heat by a series of baffles and ceramic cylinders. All four signals from these load 

cells were electronically summed, amplified, and split into two equivalent signals. 

One signal was recorded directly; the second was electronically differentiated before 

being recorded. This dual arrangement gave continuous records of the weight of the fuel 

or1 the platform as well as the time rate of change of the weight. 

The excelsior fuel beds were 3 feet wide, 8 feet long, and 4-1/2 inches deep. The 

front of the weighing platform was placed 4 feet from the front of the fuel bed and 

centered laterally. This arrangement permitted the fire to reach a quasi-steady rate 

of spread before burning onto the platform. Inconsistencies in burning rate near the 

edges were minimized by allowing 9 inches of fuel on either side of the platform. Fuel 

cribs were constructed using 1/4-inch and 1/2-inch sticks ; cribs were approximately 5 

feet long, 3 feet wide, and 5 to 6 inches deep. The same size weighing platform was 

used for both the stick cribs and the excelsior beds, 

The concept of a reaction zone and a reaction time can be visualized by considering 

the fuel-reaction zone interface as moving through the fuel on the weighing platform 

(fig. 9). When this interface reached the fuel being weighed, the strip chart recorder 

indicated the time of arrival by the start of weight loss. As the fire interface proceeded into the ~ieighed fuel, the weight loss rate continued to increase. The length 

of the weighing platform was longer than the depth of the reaction zone; hence, the 

rate of weight loss stabilized when the fire advanced onto the platform a distance 

ccluivalent to the depth of the reaction zone. The lapsed time from initial weight loss 

to the onset of stabilization is the reaction time, TR. Reaction time determination is 

great l;~ enhanced by differentiating the weight loss signal. The major conversion of 

woody fuels to combustible gases occurs within this time. 

In figure 10, the reaction time, TR, is defined on the derivative curve as the time 

from initial mass loss until the loss stabilizes at a steady rate. The observation of a 7. 

~znear mass loss rate during the reaction time was a surprising but consiste~t feature 

of our measurements. The duration of constant mass loss rate was dependent on the length 

of the weighing platform; it had no bearing on the duration of the reaction time. 

Note also that the reaction time could be taken as the fire burned off the weighing 

platform. ?'he concept of reactiorl time, as associated with weight loss, was first noted 

in this manner. Iloi\.ever, data taken as the fire burned off the weighing platform were 

not as consistent as they were when it burned onto the platform. 
Figure 9. --Fire-fue Z interface 

moving through weighed fuel. / 'I /Fire fuel interface 

arrives at weighed fuel 

I Fire interface approaching weighed fuel 

1 [Weight loss begins and 

s continues to increase 

Fire burning into weighed fuel 

Depth of J Weight loss rate 

becomes constant 

III Steady weight loss rate achieved 

The mass loss rate, h, obtained from the weight loss data, was related to the followirlg physical features : 

where I.V equals the width of the weighing platform. The efficiency of the experimental 

fires can now be expressed as 

Combining the efficiency with the reaction time, TR (as indicated by equation (25) and 

taker from the weight loss data, figure lo), gives the experimentally determined reaction 

velocity, 

The potential reaction velocity is calculated using equation (26) to disa~sociate 

the experimentally measured reaction velocity, r, from the effects of the moisture and 

minerals of the fuels that were used in the experiments. 

The potential reaction velocity may now be correlated with the physical features of the 

fuel array. 
Figure 10 .--Illustration 

of mass loss and its 

derivative. 

I I I 

1 Mass loss trace 

m 

0 

0 

- dm 

d t +-Ed 

Experimental Results 

Reaction Velocity 

The results of the experiments utilizing the derivative of the weighing system to 

determine reaction velocity are shown in figure 11. As expected, there was an optimum 

packing ratio for each of the 1/4-inch and 1/2-inch fuels. It was not possible to 

identify a drop in reaction velocity at very low packing ratios with the excelsior 

because of (a) the difficulty in constructing a fuel bed having only a few strands of 

excelsior per square foot, and (b) the lack of sensitivity on the weighing system at 

extremely light fuel loadings. tiowever, it is evident that the reaction velocity must 

drop to zero if there is no fuel to support combustion, just as it does for the larger 

fuels . 

Figure 11. --Detemination 

of corre lation equation 

for potential reaction 

velocity. 

Legend: 

Excelsior 

Y4" Cribs 

A %" Cribs 

0 .02 .04 .06 .08 .10 .12 

Packing ratio -0 
Figure 22. --Determination of 

corre lati ons for maximum 

reaction velocity and 

optimum packing ratio. 

Fuel surface area 

to volume ratio -U, ft~' 

The reaction velocity for fine fcels (excelsior) is much greater near the optimum 

packing ratio than it is for the larger 1/4-inch and 1/2-inch sticks (fig. 11). As 

expected, the optimum packing ratio is not the same for all fuels and shifts to the 

right as the fuels increase in thick~~ess. Note also that tightly packed fine fuels 

actually have lower reactiorl velocities than do larger fuels at the same packing ratio. 

The loss of reaction velocity of fine fuel call be seer] in the field by observing 

the difference in flaming vigor between pine needles -on a broken treetop supported 

above the ground and compacted pine needle litter; the latter burns with much less vigor. 

The data points in figures 11 through 16 are the average of three or more replications in the excelsior, and two or more in the stick cribs. 

Figure 23. --Confirmation of reaction 

intensity equation with original 

data. Direct comparison of reaction 

intensity between fuels is not intended, 

nor can it be made because loading 

was not held constant. 

Packing ratio - fl 
Figure 14. --Detednation of 

propagating flux ratio, 5. 

Figure 15.--Confirmation of 

propagating flux equation 

with original data. 

- C 

- 7 

- Z 

I 

- i 

C 

Legend: 

- Excelsior 

%" Cribs 

A '/a' Cribs 

0 .02 .04 .06 .08 .10 .12 

Packing ratio -6 

Legend: 

Excelsior - 

- 'h" Cribs 

A 'hw Cribs 

. 

~-lll\~ll - - *C ** \+, 

.. \., r" 

- i - 

I 

0 .02 .04 .06 .08 .10 .12 

Packing ratio - /3 
Figure 16. - -Confirnation of 

rate of spread equation 

with original data. 

Legend: 

Excelsior 

l/qu Cribs 

A '/a" Cribs 

Packing ratio -0 

The mathematical fit to the data of figure 11 was assumed to bk a modification to 

a Poisson distribution. To determine the general equatior~ as a function of both B and 

a, the equations for the maximum value of r' and the optimum beta, , for each fuel 

size were found as a function of a (fig. 12). 

'max = ~'.~/(495 + 0.0594a~-~) (36) 

These were then combined with an arbitrary variable, A, to give: 

where : 

The equations that relate reaction velocity, reaction intensity, propagating flux, 

and rate of spread were developed as a set to fit not only the dependent variable but 

also the data shown in figures 11 through 16. Note also that equations (36) , (37) , (38) , and (39) will predict reaction velocity for any combination of fuel particle size, a, 

and any packing ratio, 8. The form of the equations has been chosen to predict reasonable 

values when input parameters are extrapolated beyond those tested; i. e. , curves do not 

go negative or to infinity when they obviously should not. 

Reaction Intensity 

The reaction intensities are calculated from equation (23) and the data obtained 

from the weight loss experiments are shown in figure 13. 
The correlation equations that predict the reaction velocity-- (36), (37), (38), and 

(39)--are combined with equation (27) to predict reaction intensity for the three fuel 

sizes used in the experiments. The curves from these equations are also plotted in 

figure 13, where the fit can be compared to the original data. 

Direct comparison of reaction intensity between the fuels used in the experiments 

is not intended, nor can it be made because fuel loading was not held constant. The 

study data were only intended to aid in the development of equations that could be used 

to predict reaction intensity and, subsequently, rate of spread over a wide range of 

fuel and environmental combinations . 

Propagating FLUX 

The no-wind propagating flux is calculated from equation (6), 

A ratio, 5, is now computed; it relates the propagating flux to the reaction intensity: 

The values computed for 5 are plotted in figure 14 as a function of 6 for the three fuel 

sizes. The following correlation equation was found for 5 as a function of 0 and a: 

The fit of the data to this equation can be seen in figure 14. 

The fit of this equation to the original values of propagating flux calculated 

from equation (6) can be seen in figure 15. The data show that (Ip), increases with 

increasing 0, but at a decreasing rate. Extrapolation of equation (41) solved for (I ) 

indicates that it would actually reach a maximum and then decrease. This is a P 0 

reasonable prediction, considering the fact that the fuel array is becoming so compact 

that the intensity has also decreased (fig. 13). 

Rate of Spread 

Combining the heat source and heat sink terms produces the final no-wind rate of 

spread equation : 

Predictions from this equation are shown with the original data in figure 16. 

Figure 16 illustrates the difference in spread characteristics between the fine 

fuel, excelsior; and the sticks. A family of curves for any particle size could be 

calculated, using the equations developed in this section. 
EVALUATION OF WIND 

AND SLOPE COEFFICIENTS 

To introduce wind and slope into the model, we must evaluate the koefficients 

4, and $s. Rearranging equation (9) with $s = 0: 

If the fuel parameters in equation (6) are assumed constant, the propagating flux is 

proportional to the rate of spread and equation (44) becomes 

where : 

R = rate of spread in the presence of a heading wind W 

Similarly , 

where : 

R = rate of spread up a slope. s 

For expedie~lcy it was assumed that no interaction existed between wind and slope. 

Wind Coefficient 

Rate of spread measurements in the presence of wind or on slopes ill fuel arrays 

amenable to the no-wind model are needed to evaluate equations (45) and (46). 

Wind Tunne Z Experiments 

Fuel beds were built using three fuel sizes at packing ratios porous enough to 

cause flameout and compact enough to exceed natural conditions. They were bunled in 

the large wind tunnel at the Northern Forest Fire Laboratory. The tunnel temperatures 

were held between 85" and 9oe F.; and the relative humidities between 20 and 25 percent. 

Mean tunnel velocity was set at 2, 4, 6, or 8 m.p.h. The fuel beds were 3 feet wide and 

12 feet long. The excelsior fuel was 4-112 inches deep. The stick fuels were constructed using a new method: three sticks were stapled together near the center and 

spread so they stood or, three legs to form a double tripod, one up and one down, joined 
Figure 17.--Double 

tripod Sue Z bed used 

in wind tunne Z 

experiments. 

at the center. These double tripods were arranged at various spacings to achieve the 

desired packing ratio (figs. 17 and 18). For very low packing ratios, this arrangement 

is far superior to the traditional crib construction because cribs with widely spaced 

sticks collapse when the cross members bum out. 

Excelsior fuel beds must be carefully constructed to achieve the exact fuel depth 

or the bulk density will be altered with drastic effects on the rate of spread. 

Field Data 

McArthurls (1969) data on rate of spread for heading grassland fires in Australia 

are show11 in figure 19. However, no data are available on the particle size, depth, or 

loading of the various areas burned; therefore, it was assumed that these values were 

similar to those of a typical arid grass area in the Western United States. 

a = 3,500 ft.-' , Wo = 0.75 ton/acre, and depth = 1.0 ft. 

*re It'. --R~rning imb Ze 

r"po2 f~e Z bad i~ c 

2rge cini tl*nneZ. 
Figure 19. --Reproduction of 

McArthur's (1969) rate of 

spread data for grass. 

Average wind velocity at 33' in the open , m.p.h 

Ana Zysis 

10.0 

Before a correlation could be found between wind velocity and the multiplication 

factor for wind, it was necessary to find an interrelationship between 4, and the 

fuel parameter, a and 6/Bo . To do this, the excelsior and 1/4-inch stick data from the 

wind tunnel were plotted aPong with McArthurts field data. Half-inch stick data did 

not correlate and had to be discarded. Apparently the effective bulk density is altered 

by the rapid heating caused by a heading fire; thus the assumption of corlstant fuel properties needed for obtaining equation (45) is not valid for fuels as large as one-half 

inch. 

- * Source of data 

Kongorong Fire S. A. 17-1-59 

Another plot of the fuel parameters and multiplicatiorl factor vs. wind velocity 

produced the final correlation given by equation (47). Figure 20 shows the correlation 

parameters using the original data. 

8.0 - x Geelong Fires Vic. 17-1-64 

4 Longwood Fire Vic. 17-1-65 

4 

E o Tasmanian Fires 7-2-67 

2 6.0 

E' 

(5, 

2 

0 

g 4.0 - x 0 

3 0 b 

- z 

- 

0 2.0 - 

- 0 S M a % 4 4 ** 

where: c = 7.47 exp(-0.133~*~~) 

B = 0.02526~*~~ 

E = 0.715 exp(-3.59 x 10-~a). 

Figure 20. --Come Zation 

parameters for determining 

wind coefficient. 

Legend 

80 - Gross 

Excelslor 

60 - C] '14" sticks 

40 - 

20 - 

I I 
Figure 21. --Influence of packing 

ratio and particle size on wind 

coefficient at 12 m.p. h. In the 

absence of wind, the fuel 

conditions for best burning 

would occur at B/Bop = I. In 

the presence of wind, fires 

spread faster in Zess dense 

fuel, i .e., ~/a~~ = Zess than 

I. The wind coefficient 

increases markedly as surfacearea-to-voZwne ratio increases. 

The shape of the curves is in good agreement with the concept suggested by 

Rothermel and Anderson (1966). At that time, it was speculated that the finer the 

fuel, the sharper the resulting increase in spread rate with wind velocity. As 

expected, fuels that were too sparse to burn well in the absence of wind will sustain 

a rapid fire spread when wind is applied. In effect, the optimum packing ratio shifts 

toward more lightly loaded fuels as wind increases. This effect is illustrated in 

figure 21 and can be seen in the field where sparse fuels--such as poor stands of 

cheat grass--burr] poorly without wind but become a flashy fuel when wind is applied. 

Slope Coefficient 

The effect of slope was determined for fine fuels by burrling excelsior fuel beds 

on slopes of 25, 50, and 75 percent. The experimerits were conducted in a large 

combustion laboratory under the same environmer~tal conditions used for the no-wind and 

wind tunnel fires. Fuel was excelsior constructed at four packing ratios : 0.005, 0.01, 

0.02, and 0.04. A correlatior~ of the data 1s shown in flgure 22. The equation for the 

line is 

where tan + is the slope of the fuel bed. The final form of the rate of spread equation is 

Figure 22. --Come Zation parameter 

for s Zope coeffi eient . 

Slope factor, tan @//3 l5 
SUMMARY OF 

FIRE SPREAD EQUATIONS 

The complete set of parametric equations developed in this work is give11 on page 

26. The required input parameters are given on page 27. These equations are easy to 

program for computer computations. Students of fire behavior can gain a perspective 

understanding of the effects of various input parameters by computing and crossplotting 

curve families for reaction velocity, reaction intensity, and other internal variables 

that goven~ fire spread. The equations might also be used for analyzing expected 

behavior of planned laboratory experiments. A word of caution--the fuel bed width must 

be sufficient to simulate a line fire (Anderson 1968); and the fuel beds must be care- I 

fully constructed to insure a uniform distribution of the fuel elements. ?'he equations 

in this form have limited use in the field because few fuel types are composed of fuels - 

that are homogeneous in size. The remainder of this paper is devoted to adaption of the 

parametric eo,uations into a mathematical model suitable for field application. 
Swmnary of Basic Fire Spread Equations 

I~c(1 + $w + 

R = Rate of spread, ft ./mill. (52) 

pbEQig 

I R = r 'wllh rlM'ls Reaction intensity, B. t.u./ 

ft. min. (27) 

where : 

r' = ~*max(~/~op)Ae~p[~(l- mop)] Optimum reaction velocity, 

min. -l (38) 

'max = 01.5(495 + .05940~'~)-~ Maximum reaction velocity ,(36) 

min.-l 

Bop = 3.3480--'~'~ Optimum packing ratio (37) 

A = 1/(4.774a.l - 7.27) (39) 

P.1 f hlf 2 '1M = 1 - 2.59 - Mf 

M x + 5.11 (%) - 3.52 (%) 

!>loisture da-nping coefficient (29) 

IIS = 0.174 Se-.I9 hlineral damping coef ficieilt (30 1 

€, = (192 + 0.2595a)-~ex~[(0.792 + 0.6810.~) (B + 0.1)] 

Propagating flux ratio (42) 

+w = cuB (*) -L Wind coefficient (47) 

C = 7.47 exp (-0.1330.~~) (48) 

B = 0.025260. 54 (49) 

E = 0.715 exp (-3.59 x 10-~0) (50) 

W,, = w 0 Net fuel loading, 1b./ft2 (24) 1 + 

$S = 5.275 tan $j2 Slope factor (51) 

pb = w0/6 Overldry bulk density, 

lb./ft. 3 (401 

E = exp(-138/0) Effective heating number (14) 

Qig - 250 + 1,116 >If - Heat of preignition, 

B.t.u. (12) 

lb . 

B=% Packing ratio (3 1-1 

P 
Input Parameters for Basic Equations 

w ovendry fuel loading, Ib. /ft. 

0' 

8, fuel depth, ft. 

a, fuel particle surface-area-to-volume ratio, l/ft. 

h, fuel particle low heat content, B. t.u./lb. 

p ovendry particle density, lb./ft.3 

P ' 

Mf, fuel particle moisture content, lb. moisture 

lb. ovendry wood 

ST, fuel particle total mineral content, lb. min,erals 

lb. bvendry wood 

Se, fuel particle effective mineral content, lb. silica- fi-ee minerals 

lb. ovendry wood 

U, wind velocity at midflame height, ft ./min. 

tan 4 , slope, vertical rise/horizontal distance 

Mx, moisture content of extinction. This term needs experimental 

determination. We are presently using 0.30, the fiber saturation 

point of many dead fuels. For aerial fuels (B <.02) with low 

wind velocity (4 m.p.h .) Mx - 0.15. 
THE FIRE SPREAD MODEL 

Rate of spread and intensity predicted by the model are based on equations (52) 

and (27) . These equations had to be modified,however, to accept fuels that were 

composed of heterogeneous mixtures of fuel types and particle sizes. Such fuels as 

pine needle litter, grass, brush, and logging slash are the easiest to model. Patchy 

fuels--accumulations of broken branches, treetops, snags, foliage litter, brush, and 

other lesser vegetation are more difficult to model because of the discontinuous 

patterns in which they are found. For the model, however, these various size fuels 

are assumed to be uniformly distributed within the fuel array. This assumption is 

especially critical for the fine fuels (foliage and twigs under 114 inch in diameter). 

It is also assumed that the fuel can be grouped into categories according to 

similar properties. For example, there would be one category for living fuel and a 

second for dead fuel. It is also desirable to have separate categories for foliage and 

branchwood. Grouping by species is not sufficient because foliage and branchwood can 

have significant differences in particle properties. A further breakdown by size class 

is required within these categories if the fuel particles vav greatly in size. The 

size classes used can be arbitrarily established but should include a class for fine 

fuels. Experience will show to what extent size class breakdowns are necessary. Our 

initial work indicates that the larger fuels have a negligible effect on fire spread; 

thus, these can often be eliminated from consideration. 
To aid in the understanding of fuel distribution., we introduced the concept of a 

unit fuel cell. A unit fuel cell is the smallest volume of fuel within a stratum of mean 

depth that has sufficient fuel to be statistically representative of the fuel in the 

entire fuel complex. This concept permits the mathematical representation of the fuel 

distribution to be referenced to a unit fuel cell rather than to the entire complex. 

Primari ly , this concept aids in mathematically weighting input parameters . Consequently, it is not necessary to specify the size of the unit fuel cell within an area 

under study; rather to provide mean values per unit fuel cell which then represent the 

fuel complex that is being modeled. Representative inputs can be preselected to form 

fuel models that are tailored for analysis by the fire spread model. 

The model is based on the concept that a singular characteristic parameter can 

be found by properly weighting the variations in the parameter in the heterogeneous 

mixture. To implement this concept, we had to consider how each fuel parameter in the 

model exerts its effect on the three characteristic features of a spreading fire : (1) 

The energy source; (2) the energy sink; (3) the flow of air or of heat within the array. 

The processes that control combustion rate--evaporation of moisture from the fuel, 

tran.sfer of heat into the fuel, and evolution of combustible gases by the fuel--occur 

through the surface of the fuel particle. The fuels having the highest surface-area-tovolume ratio (fine fuels) will respond the fastest; therefore, these will be involved 

in the leading portions of a fire. It is no revelation. to firefighters or to fire 

scientists that fine fuels might be expected to react the fastest. Hwever, it is not 

realistic to arbitrarily state that 90 percent of particles 1/8 inch in. diameter and 

under are consumed and that some fixed ratio of the other size classes is consumed. 

Weighting by surface area eliminates the problem of making arbitrary decisions as to 

which fuel sizes to include and which not to include. 

Mathematical Fire Spread Mode Z Inputs 

Mean values within ith category and jth size class of fuel complex: 

- 

(wo)ij = ovendry loading, lb./ft.2 

(;) j = surface- area-to-volume ratio, (ft . 2/ft. 3, 

(F ) . . = mineral content, (lb . minerals/lb . wood) T 11 (lb. minerals - lb, silica) (Te) j = effective mineral content - lb. wood 

(h)ij = low heat value, B.t.u./lb. 

(Mf)i j = moisture content, (lb . moisture) / (lb .wood) 

(Fp)i = ovendry particle density, (lb ./ft. 3). 

Mean value within ith category: 

(M,) = moisture content of extinction (lb . moisture) / (lb . ovendry wood) . 

Mean fuel array properties : 

- 

6 = depth of fuel, (ft.) 

tan 9 = slope, (ft. vertical rise/ ft. horizontal). 

U = wind velocity at midflame height, (ft./min .) 

m = total number of categories 

n = number of size classes within. ith category. 
Formulation sf Fire Spread Model 

The model is now formulated from the basic equations of fire spread and the 

weighting concept. The basic equations are available on page 26. The detailed 

weighting concept must be developed. 

Weighting parameters based on the surface area of the fuel within each size class 

and category are developed from input parameters as shown on page 29. 

Let: 

- 

% = mean total surface area of fuel per unit fuel cell. 

- 

Ai = mean total surface area of fuel of ith category per unit fuel cell. 

*. . = mean total surface area of fuel of jth class and ith category per 

I' unit fuel cell. 

The mean total surface area per unit fuel cell of each size class within each 

category is determined from the mean loading of that size class and its surface-area-tovolume ratio and particke density. 

- 

(3 Go) 

i j A.. = ij , 

The mean total surface area of the ith category per unit fuel cell and the mean 

total surface area per unit fuel cell are then obtained by summation of the areas within 

each category and within the fuel cell with equations (54) and (55). 

Two weighting parameters are now calculated that are used throughout the remainder 

of the model : - 

H i j Ratio of surface area of j t h 

fij =T 

Ai size class to total surface area 

of ith category per unit fuel cell 

- 

A. 1 f. = - Ratio of surface area of i th 

1 9 category to total surface area 

per unit fuel cell 

Using the weighting parameters, the basic fire spread equations are modified as 

follows : 
Reaction intensity becomes : 

where the characteristic parameters weighted by surface area are : 

Net loading of ith category 

- 

(wnIij = Net loading of jth class within 

1 + ith category (60) 

j =n 

fii =C f..F.. Low heat content value of i t h 

j =O 11 11 category (61) 

(Qi = 0. 174(Se)i--19 Mineral damping coefficient 

of ith category (62) 

j =n 

(5 ). = C fij(Se)ij Characteristic effective 

e i j = 1 mineral coefficient of i th 

category (63) 

Moisture damping coefficient 

of ith category (64) 

Moisture ratio of i th 

category 

j =n - - (Mf)i = C f. . (Mf)ij Moisture content of 

j=1 11 ith category (66) 

To complete the calculation of the reaction intensity, the potential reaction 

velocity, I-', must be calculated. A single value of reaction velocity is calculated 

for the fuel complex. 

r' is dependent on the packing ratio and fuel particle size. The packing ratio 

regulates the heat and airflow within the fuel array. This regulation of flow is 

dependent upon whether or not the space is occupied or vacant. Therefore the ratio 

should be entered as a mean value of all particle sizes. However, the surface-areato-volume ratio is a parameter that characterizes the particle size of the fuel complex 

that is regulating the combustion processes in the fire front and a must be weighted by 

surface area. 

Applying these concepts, 

r' - 

- 'max (F/F~,) 'exp [A ( 1 - F/B OP 1 I 
where : 

Characteristic surface-areato-volume ratio of the fuel 

complex (71) 

Characteristic surface-areato-volume ratio of ith fuel 

category (72) 

Mean packing ratio 

Mean bulk density 

This completes the computations necessary for calculating reaction intensity. 

The parameters within the basic rate of spread equation 

are treated similarly. 

The no-wind propagating flux ratio, 5, is a function of the mean packing ratio 

and characteristic surface-area-to-volume ratio. 

In the heat sink terms, the bulk density is dependent upon bulk properties of 

the array: The effective heating number, E, and the heat of preignition are dependent 

upon fuel surface. Therefore, the bulk properties must be separated from the particle 

properties when summing and weighting. 

where : 

( 1.. = 250 + 1,116 (Ef)ij The heat of preignition lg 1J for jth size class within 

the ith category 
The model is completed by inclusion of the wind and slope multiplication 

factors : 

and : 

where : 

U = mean windspeed at midflame height, (ft./min.) (81) 

If ; < 175, the weighted fuel size is too large for the wind factor. (a decreases as 

fuel size increases.) We have not found this limitation to be restrictive on any of 

the fuel models tested to date. The reason, of course, is that wildland fuels are 

composed primarily of fine fuels with consequent large values of a . 

An upper limit is placed on the wind multiplication factor. Rothermel and 

Anderson (1966) found that the angle of flame tilt could be correlated to a ratio of 

the energy of the wind and the energy of the fire: 

- q u 

IR~ 

where : 

q = free stream dynamic pressure lb./ft. * 

J = 778 ft. lb./B. t.u. - mechanical equivalent of heat 

Evaluating this ratio at the limiting value of spread rate found by McArthur (1969) 

(fig. 20) gives: 

Assuming air temperature and density for a nominal summer day, T = 80° F., elevation = 3,000 ft.; this reduces to, 

This limit is taken for ($ ) w max 

If - U > 0.9, then $w = $w at U= 0.9 1 

I R R' 
AFFLICATION TO TME FIELD 

The mathematipal mpdel has application to management problems in two situations : 

1. The "h~othetical fire" situation in which operations' research techniques are 

utilized for Eire planning, fire training, and fuel appraisal. 

2. The "possible fire" situation for which advance planning is needed, such as 

fire-danger rating and presuppression planning. Predictions of potential fire severity 

for possible management practices (i .e. , methods of thinning, slash treatment, and 

prescribed burning) have the potential for being the most valuable contribution that 

fire modeling can perform. 

A third s jtuation-- forecasting the behavior of existing wildfires --will require 

a greater degree of sophistication than this model and our knowledge of fuels will permit at the present time. Variations in fuel and weather cause departures from predicted 

spread and intensity that pose risks unacceptable in fire suppression activities. A 

method for forecasting the behavior of a specific fire eventually will be developed; 

most likely, it will be patterned on a probability basis similar to that used for forecasting weather. To accomplish this, a technique must be developed for rapidly updating 

fuel inventories on the threatened site. 

Choosing input parameters for the model from the infinite variety of fuel and 

environmental arrangements and combinations seems almost overwhelming. However, patterns 

in the growth of vegetation exist that can be utilized to greatly simplify the inventory 

process. It also proves helpful to group the inputs in the following manner: 

1. Fuel Particle Properties 

Heat Content 

Mineral Content 

Particle Density 

2. Fuel Array Arrangement 

Loading by Size Class--Living and Dead 

Mew Size Within Each Class--Surface-Area-To-Volume Ratio 

Mean Depth of Fuel 

3. Environmental Related Values 

Wind Velocity 

Fuel Moisture Content 

Slope 
The fuel particle properties are not expected to vary greatly within vegetation 

types. Such values can be readily determined in the laboratory and assembled in a manner that should have wide applicability. 

Fuel array arrangement patterns must be determined in the field. These inventory 

tasks will be more difficult than measuring fuel particle properties. However, it is 

expected that patterns will be found that are repeatable within the limits necessary for 

calculating potential fire hazard using the model. The fuel type, age of stand, exposure, soils, rainfall patterns, and fire history may be used as indexes for cataloging 

fuel arrangement patterns. Broader classification by ecotype or habitat will also prove 

valuable for sorting out fuel parameters. 

The environmental related parameters can be inserted to investigate the effect of 

the range of wind, moisture, or slope that might be expected to be imposed upon the fuels 

being modeled. 

Fuel Models and Application 

On-the-spot sampling of all input parameters is costly, time conswhing, and tedious. 

Cataloging fuel properties and relating them to observable site characteristics does not 

eliminate the fuel sampling process, but it will permit a wide application of sampling 

results. These results can be further refined for use in the mathematical model by 

assembling them into fuel models that represent typical field situations. Such fuel 

models contain a complete set of inputs for the mathematical fire spread model. 

Land managers can be trained to choose the fuel model, that is most applicable 

to the fuels and climate for their areas of interest. If further refinement is desired, 

internal properties (e.g., fuel loading in logging slash, the ratio of dead-to-living 

fuel in brush, and the amount and type of understory in timber) of each fuel model could 

be tailored to permit the model to more closely match specific fuels. 

Work has already begun on fuel models for the National Fire-Danger Rating System. 

Eleven fuel models (table 1) have been assembled that represent a large portion of the 

forests, brush fields, and grasslands found in the temperate climates of North America 

The variations in spread and intensity between fuel types as predicted by the model 

may readily be seen from results obtained from computations with the 11 fuel models as 

inputs. To assist in understanding the sensitivity of the inputs, the fuel particle 

properties have been held constant and the variations among fuel types may be attributed 

to the loading by size class and fuel depth as shown in table I. The fine fuel of the 

living fuel category is all that is assumed to enter into the reaction. To obtain 

reasonable values of reaction intensity for fuel models that contain living fuels, the 

moisture of extinction value for the living fuel must be adjusted to a higher value than 

that used for the dead fuels. Very little research has been done on the burning of 

living fuels. Philpot and Mutch (1971) suggest that crowning potential in ponderosa 

pine and Douglas-fir forests of Montana may be dependent upon the higher content of 

ether extractives (waxes, terpenes , and oi 1s) that do not require pyrolysis for producing the combustible constituents. It also appears that the proportion of dead fuels 

within a fuel complex has an influence on how much of the living fuel bums. Fosberg 

and Schroeder (1971) provide a formulation for predicting the moisture of extinction of 

living fuels based on the ratio of living-to-dead fuels and the moisture content of the 

find dead fuel. 

1 - a 10 

(Mx) living ~2.9 [l-?(M) ] -O.226,withalowerlimitof0.30, (88) f dead 

where: a = ratio of mass-of-fine-live-fuel to mass-of-total-fine-fuel; fine fuel is 

taken as fuel Q1/4-inch diameter. (Mf) dead = moisture content (fraction, not percent) 

of fine dead fuel. 
;t: OLnOO Ln N . . ONOMO 

N N a 0 dOdNM 

P.+ 0 0 0 0 

I 0 0 0 0 

1 I Ln* Ln* Ln I LnlIII 

G I I * I I I I I d d d ,+ 

I1 10 0 0 00000 

E I I IM M M MMMMM 
Figure 23. --Potential reaction 

ve Zocity of typica 2 wildland 

fue 2s. The two Zines represent the extreme values of - 

a, one for short grass, the 

other for heavy logging 

slash. 

Closed timber - understory 

Black Oak litter 

litter 

13 , Pocking ratio 

The effect of the change in fuel arrangement on potential reaction velocity is 

shown in figure 23. The flashy fuels (grass and brush) have the highest values; the 

closed timber litter has the lowest. Note that the grass and brush lie to the left of 

the optimum packing ratio under no-wind conditions. Inasmuch as one of the effects of 

wind is to shift the optimum packing ratio to the left, these fuels will burn extremely 

we1 1 under windy conditions. 

The prediction of reaction intensity for the 11 fuel models is shown in figure 24 

for fuel moisture ranging from 0 to extinction. All fuel models extinguish at M = 0.3, f which is the value set by Mx for the dead fuel. The higher order variations for some 

fuel models are caused by the living fuel component's inability to burn when the dead 

fuel moisture becomes high. This is attributable to Fosberg and Schroeder's formulation 

(equation 88). 

The prediction of rate- of spread is shown in figure 25 at M = 0.04 (4 percent 

moisture content) in the dead fuel over a range of windspeeds from 0 to 12 m.p.h. 

(1,056 ft./min.) at the midflame height. Comparison between figures 24 and 25 reveals 

the sensitivity of the model to changes in fuel arrangement and the apparent agreement 

of the model to what can be expected qualitatively between the fuel models. The 

closed timber litter and the short grass have similar and low reaction intensities. 

However, the rate of spread differs dramatically for the two models in the presence 

of wind; the grass has the highest rate of spread, the litter the slowest. This is 

attributed to the contrast in porosity of the two fuels (6 = 0.001 for grass, and 

= 0.036 for the litter). This example illustrates the common misconception that 

rate of spread and reaction intensity are directly related. 
Figure 24. --Reaction 

intensity of typical 

wi ldland fue 2s 

computed with 

heterogeneous 

formulations for 

the model from 

data in table 1. 

Legend: 

--..- Timber (litter and understory) 

..-..-- Slash (light 40 T./A.) 

0-0 Slosh (medium 120 T./A.) 

-.-- Slash (heavy 200 T./A.) 

0-0 Hardwood , oak liner 

.-A Brush (not chaparral) 

m-. Closed timber liner 

bead fuel moisture content (percent) 

Heavy logging slash has by far the highest reaction intensity but a medium rate 

of spread; chaparral has both a high reaction intensity and a high rate of spread. It 

is gratifying that the model predictions are high in both values because the model was 

designed to represent the brush fields of the Southwest. These brush fields pose a 

severe fire hazard (Countryman, Fosberg, Rothermel, and Schroeder 1968). 

Figure 25. --Rate of 

spread of typical 

wi ldland fue 2s 

computed with 

heterogeneous 

formulations for 

the model from 

data in table 1 at 

Mf = 0.04 (4 percent 

moisture content) 

and windspeed = 

12 m.p.h. Wind velocity 'U' (m. p h.) 
Figure 26. --Effect of moisture 

and minerals on rate of spread 

in a hypothetical fuel model. 

Fuel. 

Dm =%r" 

Depth=l 0' 

Loaded for 13 , 

: Effective mineral content 

0 05 10 15 20 25 30 

Fuel moisture , MI 

The positions of the curves in both figures 24 and 25 would be refined if fuel 

particle properties (h, p Se, hlx) of the actual fuels were substituted for the mean 

values that were used in Pie fuel models. 

To illustrate the relative effect of minerals and moisture on rate of spread, a 

typical homogenous fuel was chosen and rate of spread versus moisture was plotted (fig. 

26). No attempt other than to discount silica was made to distinguish between the 

effects of different minerals. 

Figure 27 illustrates the usefulness of the model to appraise fuels for management 

decisions. This figure shows the change in spread and intensity that could be expected 

in logging slash if it were burned under no-wind and 10 percent moisture conditions at 

various stages in decomposition. The ability of the model to predict fire severity as 

reflected in figure 27 should offer new opportunities for resource managers to integrate 

fuel management into resource planning activities. 

n 

Y 

i 

I 

Figure 27.--Change in spread and intensity 6 

in logging slash after aging. 1. New-- m 

when it had dried but retcrined its = 

needles; 2. Intermediate--after suf- 7 s 

ficient aging to remove 50 percent of & 

h 04 .- 

v, x the needles and the depth of fuel had Q, 

& settled to 75 percent of its original . c - 

value; 3. Old--sufficient aging to ~2 o remove 100 percent of the needles and . - 

& 

U 

the depth of the fuel had settled to o) 

50 percent of its original value. P: 

0 
LITERATURE CITED 

Anderson, H. E. 

1968. Fire spread and flame shape. Fire Twhnol. 4(1) :51-58. 

Anderson, H. E. 

1969. Heat transfer and fire spread. USDA Forest Serv. Res. Pap. INT-69, 20 p., 

illus. 

Berlad, A. L. 

1970. Fire spread in solid fuel arrays. Combust. and Flame 14:123-236. 

Brown, J. K. 

1972. Field test of a rate-of-fire-spread model in slash fuels. USDA Forest 

Serv. Res. Pap. INT-116, 24 p.,illus. 

Countryman, C. M., M. A. Fosberg, R. C. Rothermel, and M. J. Schroeder 

1968. Fire weather and fire behavior in the 1966 loop fire. Fire Technol. 4(2) : 

126- 141, i llus . Deeming, J. E., J. W. Lancaster, M. A. Fosberg, R. W. Furman, and M. J. Schroeder 

1972. The National Fire Danger Rocky Mountain Rating System 1972. USDA Forest 

Serv. Res. Pap. RM-84, illus. 

Fons, W. 

1946. Analysis of fire spread in light forest fuels. J. Agr. Res. 72(3) :93-121, 

illus. 

Fosberg, Michael A. , and Mark J. Schroeder 

1971. Fine herbaceous fuels in fire-danger rating. USDA Forest Serv. Res. Note 

RM-185, 7 p. 

Frandsen , W . H . 1971. Fire spread through porous fuels from the conservation of energy. Combust. 

and Flame 16:9-16, illus. 

McArthur, A. G. 

1969. The Tasmanian bushfires of 7th February, 1967, and associated fire behaviour characteristics. In The Technical Co-operation Programme. Mass Fire 

Symposium (Canberra, ~uxralia 1969) v. I. 23 p. Maribyrnong, Victoria: 

Defence Standards Laboratories. 

Philpot, C. W. 

1968. Mineral content and pyrolysis of selected plant materials. USDA Forest 

Serv. Res . Note INT-84, 4 p. 

Philpot, Charles W., and R. W. Mutch 

1971. The seasonal trends in moisture content, ether extractives, and energy of 

ponderosa pine and Douglas-fir needles. USDA Forest Serv. Res. Pap. 

INT-102, 21 p., illus. 

Rothermel, R. C., and H. E. Anderson 

1966. Fire spread characteristics determined in the laboratory. U. S. Forest Serv. 

Res. Pap. INT-30, 34 p., illus. 

Tarifa, C. S., and A. M. Torralbo I 

1967. Flame propagation along the interface between a gas and a reacting medium. 

P. 533-544, illus., In: Eleventh Symposium (International) on Combustion 

(Berkeley, California 1967). Pittsburgh: The Combust. Inst. 
ROTHERMEL, RICHARD C. 

1972. A mathematical model for predicting fire spread in wildland fuels, USDA Forest Serv. Res. Pap. INT-115, 40 p., 

illus. 

A mathematical fire model for predicting rate of spread and 

intensity that is applicable to a wide range of wildland fuels and environment is presented. Methods of incorporating mixtures of fuel 

sizes are introduced by weighting input parameters by surface area. 

The input parameters do not require a prior knowledge of the burning characteristics of the fuel. 

ROTHERME L, RICHARD C. 

1972. A mathematical model for predicting fire spread in wildland fuels, USDA Forest Sew. Res. Pap. INT-115, 40 p., 

illus. 

A mathematical fire model for predicting rate of spread and 

intensity that is applicable to a wide range of wildland fuels and environment is presented. Methods of incorporating mixtures of fuel 

I sizes are introduced by weighting input parameters by surface area. 

I The input parameters do not require a prior knowledge of the burning characteristics of the fuel. 

ROTHERMEL, RICHARD C. 

1972. A mathematical model 

land fuels, USDA Fore

illus. 

A mathematical fire model 

intensity that is applicable to a wid

ronment is presented. Methods 

sizes are introduced by weighting 

The input parameters do not requ

ing characteristics of the fuel. 

ROTHERMEL, RICHARD C. 

1972. A mathematical model 

land fuels, USDA For

illus. 

A mathematical fire model 

intensity thatis applicable to a wid

ronment is presented. Methods 

sizes are introduced by weighting 

The input parameters do not requ

ing characteristics of the fuel.