A Mathematical Model For Predicting Fire Spread in Wildland Fuels
A Mathematical Model For Predicting Fire Spread in Wildland Fuels
A Mathematical Model For Predicting Fire Spread in Wildland Fuels
A MATHEMATICAL MODEL
FOR PREDICTING
FIRE SPREAD
IN WILDLAND FUELS
Richard C. Rothermel
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 t o 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 t o 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 t o these broader aspects of vegetative manipulation activities, but also t o 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 alternatives t o 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
t o 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 t o 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 t o
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 t o 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. I t 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 t o 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 t o predict conditions during which severe fire phenomena develop, even though
the model does not include mass fire effects.
Concurrently, studies designed t o 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 W e a p ~ nAircraft
s
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 t o 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.
I t 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 t o 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
-
..............................4
Propagating Flux . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4
Reaction Intensity . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6
Effect of Wind and Slope . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6
Approximate Rate of Spread Equation . . . . . . . . . . . . . . . . . . . . . . 6
EVALUATION O F PARAMETERS. NO.WIND. OR SLOPE . . . . . . . . . .7
HeatSink . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 7
Heat Required for Ignition
Heat Source . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 9
Experimental Design . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 14
Experimental Results . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 16
EVALUATION O F WIND AND SLOPE COEFFICIENTS . . . . . . . . . . . 2 1
Wind Coefficient . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2 1
Slope Coefficient . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 24
SUMMARY O F FIRE SPREAD EQUATIONS . . . . . . . . . . . . . . . . . . 25
THE FIRE SPREAD MODEL . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 28
. . . . . . . . . . . . . . . . . . . . . . . . 30
APPLICATION TO THE FIELD . . . . . . . . . . . . . . . . . . . . . . . . . . . 34
Fuel Models and Applications . . . . . . . . . . . . . . . . . . . . . . . . . . . 35
LITERATURE CITED . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 40
Formulation of Fire Spread Model
Heading fire -
5 m.p.h.
Tg=Temperature of air
, minutes
Figurs 1.--&el temperature history prior t o i g n i t i o n for heading, no-wind, and backing
fires.
Considering f i r e as a s e r i e s of i g n i t i o n s h e l p s i n breaking down t h e problem f o r
a n a l y s i s . Heat i s s u p p l i e d from t h e f i r e t o t h e p o t e n t i a l f u e l , t h e s u r f a c e i s
dehydrated, and f u r t h e r h e a t i n g r a i s e s t h e s u r f a c e temperature u n t i l t h e f u e l begins t o
pyrolyze and r e l e a s e combustible gases. When t h e gas evolution r a t e from t h e p o t e n t i a l
f u e l i s s u f f i c i e n t t o support combustion, t h e gas i s i g n i t e d by t h e flame and t h e f i r e
advances t o a new p o s i t i o n . F i n a l l y , a constant r a t e of spread i s achieved; t h i s i s
c a l l e d t h e " quasi-steady s t a t e " wherein t h e f i r e advances a t a r a t e t h a t i s t h e average
of a l l t h e elemental r a t e s .
This process i s i l l u s t r a t e d i n f i g u r e 1, which i s based on a laboratory t e s t i n
which we monitored t h e s u r f a c e temperature of a f i n e f u e l element and t h e a i r adjacent
t o i t ahead of an advancing f i r e . I n t h e no-wind f i r e and backing f i r e s , t h e f u e l
temperature r o s e slowly u n t i l t h e f i r e was w i t h i n 1 o r 2 inches of t h e f u e l element
where i t suddenly r o s e t o i g n i t i o n . During t h e preheating phase, t h e f u e l temperature
exceeded t h e a i r temperature; t h i s i n d i c a t e s t h a t convective h e a t i n g o r d i r e c t flame
contact does not occur u n t i l t h e f i r e f r o n t reaches t h e p a r t i c l e . Consequently, r a d i a t i o n must have accounted f o r t h e energy imparted t o t h e f u e l elements on t h e upper s u r face while simultaneously t h e p a r t i c l e was being cooled by convective i n d r a f t s . This
does not occur i n t h e heading, o r wind-driven f i r e , i n which t h e temperature of t h e
f u e l r o s e s t e e p l y even when t h e f i r e was 2 f e e t from t h e thermocouple t h a t had been
i n s e r t e d i n t h e p a r t i c l e . During t h e r i s e t o i g n i t i o n , t h e a i r temperature was h i g h e r
than t h e f u e l s u r f a c e temperature; t h i s shows t h a t convective h e a t i n g can be p r e s e n t i n
a d d i t i o n t o r a d i a t i o n . Such temperature h i s t o r i e s i n d i c a t e t h a t b a s i c d i f f e r e n c e s e x i s t
i n the mechanisms t h a t b r i n g f u e l s t o i g n i t i o n . These b a s i c d i f f e r e n c e s provided
us with a method f o r c h a r a c t e r i z i n g f i r e s and developing s i m i l a r methods f o r
mathematical modeling.
CONCEPTION O
W
MATHEMATICAL MODEL
The model was d e v e l o p e d from a s t r o n g t h e o r e t i c a l b a s e t o make i t s a p p l i c a t i o n a s
wide a s p o s s i b l e . T h i s b a s e was s u p p l i e d b y F r a n d s e n (1971) who a p p l i e d t h e c o n s e r v a t i o n o f e n e r g y p r i n c i p l e t o a u n i t volume o f f u e l a h e a d o f a n a d v a n c i n g f i r e i n a
homogerieous f u e l b e d . H i s a n a l y s i s l e d t o t h e f o l l o w i n g :
'be
Qig
where :
I?
I . = h o r i z o r l t a l h e a t f l u x a b s o r b e d b y a u n i t volume o f f u e l a t t h e t i m e o f i g n i t i o n ,
x1g
~.t.u./ft.~-mill.
= e f f e c t i v e b u l k d e n s i t y ( t h e amount o f f u e l p e r u n i t volume o f t h e f u e l bed
r a i s e d t o i g n i t i o n a h e a d o f t h e a d v a ~ l c i n gf i r e ) , l b . / f t . 3
Q.
= h e a t of p r e i g n i t i o l l ( t h e h e a t r e q u i r e d t o b r i n g a u n i t weight of f u e l t o
ignition), B.t.u./lb.
lg
(2')
d e p t h , z c , o f t h e f u e l b e d , B . t . u . / f t . 3-min.
C
where :
M
= r a t i o o f f u e l m o i s t u r e t o ovendry weight
= i g n i t i o n temperature.
ig
The amount o f f u e l i n v o l v e d i n t h e i g n i t i o n p r o c e s s i s t h e e f f e c t i v e b u l k d e n s i t y , 'be.
To a i d i n t e r p r e t a t i o n and a n a l y s i s , an e f f e c t i v e h e a t i n g number i s d e f i n e d a s t h e r a t l o
o f t h e e f f e c t i v e bulk d e n s i t y t o t h e a c t u a l b u l k d e n s i t y .
E E -'be
'
'b
The e f f e c t i v e h e a t i n g number i s a d i m e n s i o n l e s s number t h a t w i l l be n e a r u n i t y f o r f i n e
f u e l s and d e c r e a s e toward z e r o a s f u e l s i z e i n c r e a s e s . T h e r e f o r e ,
'be
= f (bulk d e n s i t y , f u e l s i z e )
C41
Propagating Flux
The p r o p a g a t i n g f l u x i s t h e numerator o f t h e RHS ( r i g h t - h a n d s i d e ) o f e q u a t i o n
(1) and h a s t h e u n i t s o f h e a t p e r u n i t a r e a , p e r u n i t t i m e . The p r o p a g a t i n g f l u x i s
r e p r e s e n t e d by I .
P'
Figure 2 . --Schematic of
no-wind f i r e .
Wind
Figure 3. --Schematic o f
wind- driven f i r e .
Figure 4 . --Schematic o f
ups lope f i r e .
E q u a t i o n (6) p e r m i t s ( I p ) o t o b e e v a l u a t e d from e x p e r i m e n t s w i t h s p r e a d i n g f i r e s i n
t h e no-wind c o n d i t i o n b y m e a s u r i n g Ro o v e r a wid? r a n g e o f f u e l c o n d i t i o n s . Note t h a t
t h e propagating f l u x occurs a t t h e f r o n t of t h e f i r e ; t h e r e f o r e (Ip)o i s expected t o
be c l o s e l y r e l a t e d t o t h e f i r e i n t e n s i t y of t h e f r o n t :
Reaction Irrtensity
The e n e r g y r e l e a s e r a t e o f t h e f i r e f r o n t i s p r o d u c e d b y b u r n i n g g a s e s r e l e a s e d
from t h e o r g a n i c m a t t e r i n t h e f u e l s . T h e r e f o r e , t h e r a t e of change o f t h i s o r g a n i c
m a t t e r from a s o l i d t o a g a s i s a good a p p r o x i m a t i o n o f t h e s u b s e q u e n t h e a t r e l e a s e r a t e
of t h e f i r e .
The h e a t r e l e a s e r a t e p e r u n i t a r e a o f t h e f r o n t i s c a l l e d t h e r e a c t i o n
i n t e n s i t y and i s d e f i n e d a s :
where :
dW - mass l o s s r a t e p e r u n i t a r e a i n t h e f i r e f r o n t ,
dt
l b . / f t . 2-min.
'The r e a c t i o n i n t e n s i t y i s a f u n c t i o n o f s u c h f u e l p a r a m e t e r s a s t h e p a r t i c l e s i z e , b u l k
d e n s i t y , m o i s t u r e , and c h e m i c a l c o m p o s i t i o n .
An
The r e a c t i o n i n t e n s i t y i s t h e s o u r c e o f t h e no-wind p r o p a g a t i n g f l u x , (Ip),.
i m p o r t a n t c o n c e p t upon which t h e model i s b a s e d t h a t ( I ) o and I R c a n b e e v a l u a t e d i n d e p e n d e n t l y and c o r r e l a t e d .
Knowing t h e c o r r e l a t i o n , (?p)o c a n b e d e t e r m i n e d from t h e
r e a c t i o ~ ii n t e n s i t y , which i s i n t u r n d e p e n d e n t on f u e l p a r a m e t e r s o b t a i n e d from t h e
f u e l bed complex.
I f t h i s c o n c e p t i s k e p t i n mind, i t w i l l a i d i n u n d e r s t a n d i n g t h e development o f t h e
model.
EVALUATION O F PARAMETERS,
NO -WIND, OR SLOPE
The c o n c e i v e d f u n c t i o n a l r e l a t i o n s h i p s n e c e s s a r y f o r e v a l u a t i n g e q u a t i o n (1) a r e
d i v i d e d and c o n s i d e r e d f i r s t as t h o s e f o r m i n g a h e a t s i n k , and s e c o n d a s t h o s e s e r v i n g
as a h e a t s o u r c e .
Heat Sink
Heat of Preignition
The h e a t o f p r e i g n i t i o n and t h e e f f e c t i v e b u l k d e n s i t y a r e t h e two t e r m s t h a t had
t o be e v a l u a t e d b e f o r e t h e p r o p a g a t i n g f l u x c o u l d be computed. Qig was e v a l u a t e d
a n a l y t i c a l l y f o r c e l l u l o s i c f u e l s by c o n s i d e r i r l g t h e change i n s p e c i f i c h e d t from a m b i e n t
t o i g n i t i o n t e m p e r a t u r e and t h e l a t e n t h e a t o f v a p o r i z a t i o r l o f t h e m o i s t u r e .
where :
= s p e c i f i c h e a t o f d r y wodd
pd
AT. = t e m p e r a t u r e r a n g e t o i g r l i t i o r l
18
bI = f u e l m o i s t u r e , l b . w a t e r / l b . -Try wco
f
Pw
ATB
= s p e c i f i c heat of water
=
t e m p e r a t u r e rarlge t o b o i l i n g
V = l a t e n t heat o f vaporizatiorl.
D e t a i l s o f t h e c a l c u l a t i o ~ la r e
assumed t o r a n g e from 20" t o 320'
e q u a t i o n (11) becomes :
C.
by F r a r ~ d s e r l . ~The t e m p e r a t u r e t o i g n i t i o n i s
and b o i l i n g t e m p e r a t u r e t o be a t 100' C . , t h e n
2 ~ H. . F r a n d s e n .
The e f f e c t i v e h e a t i n g o f f u e l p a r t i c l e s a h e a d o f a s p r e a d i n g
f i r e . USDA F o r e s t S e r v . , I n t e r m o u n t a i n F o r e s t and Range Exp. S t a . , Ogden, Utah ( i n
preparation).
E f f e c t i v e Bulk Density
To e v a l u a t e t h e e f f e c t i v e bulk d e n s i t y (pbe), we needed t o d e t e r m i n e t h e e f f i c i e n c y
of h e a t i n g a s a f u n c t i o i l o f p a r t i c l e s i z e . T h i s was e v a l u a t e d by p l a c i n g thermocouples
w i t h i n s e c t i o n s o f two s t i c k s t h a t were l o c a t e d on t h e upper s u r f a c e 3 f e e t from one
end of s t a n d a r d wood c r i b s .
The i n s t r u m e n t e d s e c t i o n s were o r i e n t e d i n b o t h t h e l o n g i t u d i n a l and l a t e r a l d i r e c t i o n s ( f i g . 5 ) . The t e m p e r a t u r e d i s t r i b u t i o n w i t h i n t h e
s t i c k s was analyzed t o determine t h e amount o f h e a t absorbed by t h e s t i c k s up t o t h e
time of i g n i t i o n .
R e s u l t s of t h e a n a l y s i s a r e show11 i n f i g u r e 6 .
An e x p o n e n t i a l f i t t o t h e d a t a i s :
where :
Figure 6 . - - E f f e c t i v e heating
number versus Z/a. a i s
the s urface-area- to-volume
r a t i o o f the p a r t i c l e ; E
i s a measure of t h e fraction
o f the p o t e n t i a l fuel t h a t
must be raised t o i g n i t i o n .
Measured values
Heat Source
Reaction I n t e n s i t y
The most complex f u n c t i o n e v a l u a t e d was r e a c t i o r i i n t e n s i t y u s i n g a new c o n c e p t
The e v a l u a t i o n
t h a t e v o l v e d by d e r i v i n g f i r e i n t e n s i t y from t h e w e i g h t l o s s d a t a . 4
was made from a s e r i e s o f experimerlts u t i l i z i n g an i ~ i s t r u m e r l t s y s t e m t h a t r e c o r d e d t h e
w e i g h t o f a p o r t i o n o f t h e f u e l bed d u r i n g f i r e s p r e a d .
E q u a t i o n (7) c a n b e r e a r r a n g e d t o e x p r e s s r e a c t i o n i r l t e ~ i s i t yi n t h e f o l l o w i n g malnier:
where :
dx = R , t h e q u a s i - s t e a d y r a t e o f s p r e a d .
dt
Therefore,
I d x = -Rh dw.
R
(16)
This gives
I D = Rh(w -W ) .
R
n r
where :
D = r e a c t i o n zone depth ( f r o n t t o r e a r ) , f t .
w
w
= net i n i t i a l f u e l loading, ~ b . / f t . ~
=
r e s i d u e l o a d i n g immediately a f t e r p a s s a g e of t h e
r e a c t i o n zone, l b . / f t .
S u b s t i t u t i n g t h e r e a c t i o n time i n t o e q u a t i o n (18) g i v e s
The r e a c t i o n zone e f f i c i e n c y i s t h e n d e f i n e d a s
IR
(wn-wr)
Q
- 6
=I
W
Rmax
I1
where :
w
S
= ovendry f u e l l o a d i n g , ~ b . / f t . ~
=
f u e l m i n e r a l c o n t e n t , l b . rnilierals
l b . dry f u e l
'
Reaction Velocity
The r e a c t i o n v e l o c i t y i s a dynamic v a r i a b l e t h a t i n d i c a t e s t h e completeness and
r a t e o f f u e l consumption, T h e r e f o r e , i t r e p r e s e n t s t h e dynamic c h a r a c t e r of t h e f i r e
and i s t h e key t o s u c c e s s f u l development o f t h e model.
The r e a c t i o n v e l o c i t y i s d e f i n e d a s t h e r a t i o o f t h e r e a c t i o n zone e f f i c i e n c y t o
t h e r e a c t i o n time,
I'
"
T
r'
Then :
p o t e n t i a l r e a c t i o n v e l o c i t y , min.-l
rlM =
ns
i- =
r'
74"s
(26)
Figure 7 . --Determination of
moisture damping c o e f f i c i e n t from ponderosa
pine needle fuel beds.
(I)
.-
.6
(I)
Mf/Mx
Mineral Damping C o e f f i c i e n t
The m i n e r a l damping c o e f f i c i e n t was e v a l u a t e d from thermogravimetric a n a l y s i s (TGA)
d a t a of n a t u r a l f u e l s by P h i l p o t (1968). I n t h i s s t u d y , i t was assumed t h a t t h e r a t i o
of t h e normalized decomposition r a t e would be t h e same a s t h e normalized r e a c t i o n
i n t e n s i t y . The maximum decomposition r a t e used f o r n o r m a l i z a t i o n was a t a m i n e r a l
c o n t e n t of 0.0001, a value t h a t was assumed t o be t h e lowest f r a c t i o n a l mineral c o n t e n t
f o r n a t u r a l f u e l s . P h i l p o t found t h a t s i l i c a d i d n o t a f f e c t t h e decomposition r a t e .
T h e r e f o r e , t h e s i l i c a - f r e e ash c o n t e n t was t a k e n a s t h e independent parameter. The
d a t a a r e shown i n f i g u r e 8. The e q u a t i o n f o r t h e curve , i n f i g u r e 8 is
where:
Se = e f f e c t i v e mineral c o n t e n t ( s i l i c a f r e e ) .
5 ~ avoid
o
confusion i n e q u a t i o n development, t h e m o i s t u r e and mineral v a l u e s a r e
expressed as a r a t i o r a t h e r than a percent.
where :
B = packing r a t i o , dimensionless
pb = f u e l a r r a y bulk d e n s i t y , lb . / f t .
p
The s u r f a c e - a r e a - t o - v o l u m e
r a t i o i s used t o q u a n t i f y t h e f u e l p a r t i c l e s i z e .
Let a = t h e f u e l p a r t i c l e surface-area-to-volume
with respect t o t h e i r t h i c k n e s s ,
ratio.
For f u e l s t h a t a r e l o n g
4
a = , f t .- 1
d
where :
d = d i a m e t e r of c i r c u l a r p a r t i c l e s o r edge length o f square p a r t i c l e s , f t .
The p a c k i n g r a t i o o f t h e f u e l a r r a y , 8, and t h e s u r f a c e - a r e a - t o - v o l u m e r a t i o o f
t h e f u e l p a r t i c l e , a , a r e t h e primary independent v a r i a b l e s used throughout t h e
remainder of t h e paper f o r e v a l u a t i n g c o r r e l a t i o n e q u a t i o n s .
Experimental Design
'To e v a l u a t e t h e r e a c t i o n v e l o c i t y , a w e i g h i n g p l a t f o r m was c o n s t r u c t e d as p a r t of
t h e f u e l s u p p o r t s u r f a c e f o r t h e e x p e r i m e n t a l f u e l b e d s . T h i s w e i g h i n g p l a t f o r m , which
was 18 i n c h e s s q u a r e , was s u p p o r t e d b y f o u r l o a d c e l l s , which were p r o t e c t e d from t h e
h e a t b y a s e r i e s o f b a f f l e s and c e r a m i c c y l i n d e r s . A l l f o u r s i g n a l s from t h e s e l o a d
c e l l s were e l e c t r o n i c a l l y summed, a m p l i f i e d , and s p l i t i n t o two e q u i v a l e n t s i g n a l s .
One s i g n a l was r e c o r d e d d i r e c t l y ; t h e s e c o n d w a s e l e c t r o n i c a l l y d i f f e r e n t i a t e d b e f o r e
b e i n g r e c o r d e d . T h i s d u a l a r r a n g e m e n t gave c o n t i n u o u s r e c o r d s o f t h e w e i g h t o f t h e f u e l
or1 t h e p l a t f o r m a s w e l l a s t h e t i m e r a t e o f change o f t h e w e i g h t .
The e x c e l s i o r f u e l b e d s were 3 f e e t w i d e , 8 f e e t l o n g , and 4-1/2 i n c h e s d e e p . The
f r o n t o f t h e w e i g h i n g p l a t f o r m was p l a c e d 4 f e e t from t h e f r o n t o f t h e f u e l b e d and
c e n t e r e d l a t e r a l l y . T h i s arrangement p e r m i t t e d t h e f i r e t o reach a q u a s i - s t e a d y r a t e
of spread b e f o r e b u r n i n g onto t h e p l a t f o r m . I n c o n s i s t e n c i e s i n b u r n i n g r a t e n e a r t h e
e d g e s were minimized b y a l l o w i n g 9 i n c h e s o f f u e l on e i t h e r s i d e o f t h e p l a t f o r m . Fuel
c r i b s were c o n s t r u c t e d u s i n g 1 / 4 - i n c h and 1 / 2 - i n c h s t i c k s ; c r i b s were a p p r o x i m a t e l y 5
f e e t l o n g , 3 f e e t w i d e , and 5 t o 6 i n c h e s d e e p . The same s i z e w e i g h i n g p l a t f o r m was
u s e d f o r b o t h t h e s t i c k c r i b s and t h e e x c e l s i o r b e d s ,
The c o n c e p t o f a r e a c t i o n zone and a r e a c t i o n t i m e can b e v i s u a l i z e d by c o n s i d e r i n g
t h e f u e l - r e a c t i o n zone i n t e r f a c e a s moving t h r o u g h t h e f u e l on t h e w e i g h i n g p l a t f o r m
( f i g . 9 ) . When t h i s i n t e r f a c e r e a c h e d t h e f u e l b e i n g w e i g h e d , t h e s t r i p c h a r t r e c o r d e r
i n d i c a t e d t h e t i m e o f a r r i v a l by t h e s t a r t o f w e i g h t l o s s . As t h e f i r e i n t e r f a c e p r o c e e d e d i n t o t h e ~ i e i g h e df u e l , t h e w e i g h t l o s s r a t e c o n t i n u e d t o i n c r e a s e . The l e n g t h
o f t h e w e i g h i n g p l a t f o r m was l o n g e r t h a n t h e d e p t h o f t h e r e a c t i o n z o n e ; h e n c e , t h e
r a t e o f w e i g h t l o s s s t a b i l i z e d when t h e f i r e advanced o n t o t h e p l a t f o r m a d i s t a n c e
c c l u i v a l e n t t o t h e d e p t h o f t h e r e a c t i o n z o n e . The l a p s e d t i m e from i n i t i a l w e i g h t l o s s
t o t h e onset of s t a b i l i z a t i o n i s t h e r e a c t i o n time, TR.
Reaction time determination i s
g r e a t l ; ~e n h a n c e d b y d i f f e r e n t i a t i n g t h e w e i g h t l o s s s i g n a l . The m a j o r c o n v e r s i o n o f
woody f u e l s t o c o m b u s t i b l e g a s e s o c c u r s w i t h i n t h i s t i m e .
I n f i g u r e 1 0 , t h e r e a c t i o n t i m e , T R , i s d e f i n e d on t h e d e r i v a t i v e c u r v e a s t h e t i m e
from i n i t i a l mass l o s s u n t i l t h e l o s s s t a b i l i z e s a t a s t e a d y r a t e . The o b s e r v a t i o n o f a
~ z n e a rmass l o s s r a t e d u r i n g t h e r e a c t i o n t i m e was a s u r p r i s i n g b u t c o n s i s t e ~ tf e a t u r e
o f o u r m e a s u r e m e n t s . The d u r a t i o n o f constant mass l o s s r a t e was d e p e n d e n t on t h e l e n g t h
of t h e w e i g h i n g p l a t f o r m ; i t h a d no b e a r i n g on t h e d u r a t i o n o f t h e r e a c t i o n t i m e .
7
Note a l s o t h a t t h e r e a c t i o n t i m e c o u l d b e t a k e n a s t h e f i r e b u r n e d o f f t h e w e i g h i n g
p l a t f o r m . ?'he c o n c e p t o f r e a c t i o r l t i m e , as a s s o c i a t e d w i t h w e i g h t l o s s , was f i r s t n o t e d
i n t h i s manner. Iloi\.ever, d a t a t a k e n a s t h e f i r e b u r n e d o f f t h e w e i g h i n g p l a t f o r m w e r e
n o t as c o n s i s t e n t as t h e y were when i t b u r n e d o n t o t h e p l a t f o r m .
/ 'I
Figure 9 . --Fire-fue Z i n t e r f a c e
moving through weighed f u e l .
/Fire
fuel interface
arrives at weighed fuel
s
[Weight loss begins and
continues to increase
Depth of
III
where I.V e q u a l s t h e w i d t h o f t h e w e i g h i n g p l a t f o r m .
f i r e s can now b e e x p r e s s e d a s
The e f f i c i e n c y o f t h e e x p e r i m e n t a l
Combining t h e e f f i c i e n c y w i t h t h e r e a c t i o n t i m e , TR ( a s i n d i c a t e d by e q u a t i o n ( 2 5 ) and
t a k e r from t h e w e i g h t l o s s d a t a , f i g u r e l o ) , g i v e s t h e e x p e r i m e n t a l l y d e t e r m i n e d r e a c t i o n
velocity,
The p o t e n t i a l r e a c t i o n v e l o c i t y i s c a l c u l a t e d u s i n g e q u a t i o n (26) t o d i s a ~ s o c i a t e
t h e e x p e r i m e n t a l l y measured r e a c t i o n v e l o c i t y , r , from t h e e f f e c t s o f t h e m o i s t u r e and
minerals o f t h e f u e l s t h a t were used i n t h e experiments.
Figure 10 . - - I l l u s t r a t i o n
of mass l o s s and i t s
derivative.
dm
-
dt
+-Ed
Experimental Results
Reaction Velocity
The r e s u l t s of t h e experiments u t i l i z i n g t h e d e r i v a t i v e o f t h e weighing system t o
determine r e a c t i o n v e l o c i t y a r e shown i n f i g u r e 11. As e x p e c t e d , t h e r e was an optimum
packing r a t i o f o r each o f t h e 1/4-inch and 1/2-inch f u e l s . I t was n o t p o s s i b l e t o
i d e n t i f y a drop i n r e a c t i o n v e l o c i t y a t very low packing r a t i o s with t h e e x c e l s i o r
because o f ( a ) t h e d i f f i c u l t y i n c o n s t r u c t i n g a f u e l bed h a v i n g only a few s t r a n d s o f
e x c e l s i o r p e r s q u a r e f o o t , and ( b ) t h e lack of s e n s i t i v i t y on t h e weighing system a t
extremely l i g h t f u e l l o a d i n g s . tiowever, i t i s e v i d e n t t h a t t h e r e a c t i o n v e l o c i t y must
drop t o z e r o i f t h e r e i s no f u e l t o s u p p o r t combustion, j u s t a s i t does f o r t h e l a r g e r
fuels .
Legend:
Excelsior
Y4"
.02
.04
.06
Packing ratio
Cribs
%" Cribs
.08
.10
-0
.12
Figure 2 2 . --Determination o f
corre l a t i ons for maximum
reaction v e l o c i t y and
optimum packing r a t i o .
Fuel
surface area
to volume ratio
-U,f t ~ '
fl
Figure 14. - - D e t e d n a t i o n of
propagating flux r a t i o , 5.
Z
I
i
Legend:
Excelsior
%" Cribs
'/a' Cribs
.02
.04
.06
.08
Packing ratio
.10
.12
-6
Legend:
Excelsior
'h"Cribs
'hwCribs
..
*C
**
~ - l l l \ ~ l l
+\,
r"
\.,
-
Figure 15.--Confirmation of
propagating f l u x equation
with original data.
.08 .10
Packing ratio -
/3
.12
Legend:
Figure 1 6 . - -Confirnation o f
rate of spread equation
w i t h o r i g i n a l data.
Excelsior
l/qu
Cribs
A '/a" Cribs
Packing ratio
-0
(36)
where :
The e q u a t i o n s t h a t r e l a t e r e a c t i o n v e l o c i t y , r e a c t i o n i n t e n s i t y , p r o p a g a t i n g f l u x ,
and r a t e o f s p r e a d were developed as a s e t t o f i t n o t only t h e dependent v a r i a b l e b u t
a l s o t h e d a t a shown i n f i g u r e s 11 through 16. Note a l s o t h a t e q u a t i o n s (36) , (37) , (38) ,
and (39) w i l l p r e d i c t r e a c t i o n v e l o c i t y f o r any combination o f f u e l p a r t i c l e s i z e , a ,
and any packing r a t i o , 8 . The form o f t h e e q u a t i o n s h a s been chosen t o p r e d i c t r e a s o n a b l e
v a l u e s when i n p u t parameters a r e e x t r a p o l a t e d beyond t h o s e t e s t e d ; i .e . , curves do n o t
go n e g a t i v e o r t o i n f i n i t y when t h e y obviously s h o u l d n o t .
Reaction I n t e n s i t y
The r e a c t i o n i n t e n s i t i e s a r e c a l c u l a t e d from e q u a t i o n (23) and t h e d a t a o b t a i n e d
from t h e weight l o s s experiments a r e shown i n f i g u r e 13.
The c o r r e l a t i o n e q u a t i o n s t h a t p r e d i c t t h e r e a c t i o n v e l o c i t y - - ( 3 6 ) , ( 3 7 ) , ( 3 8 ) , and
( 3 9 ) - - a r e combined w i t h e q u a t i o n (27) t o p r e d i c t r e a c t i o n i n t e n s i t y f o r t h e t h r e e f u e l
s i z e s used i n t h e e x p e r i m e n t s . The curves from t h e s e e q u a t i o n s a r e a l s o p l o t t e d i n
f i g u r e 1 3 , where t h e f i t can be compared t o t h e o r i g i n a l d a t a .
D i r e c t comparison o f r e a c t i o n i n t e n s i t y between t h e f u e l s used i n t h e experiments
i s n o t i n t e n d e d , n o r can i t b e made because f u e l l o a d i n g was n o t h e l d c o n s t a n t . The
s t u d y d a t a were only i n t e n d e d t o a i d i n t h e development of e q u a t i o n s t h a t could b e used
t o p r e d i c t r e a c t i o n i n t e n s i t y and, s u b s e q u e n t l y , r a t e o f s p r e a d o v e r a wide range o f
f u e l and environmental combinations
Propagating FLUX
The no-wind p r o p a g a t i n g f l u x i s c a l c u l a t e d from e q u a t i o n (6),
A ratio,
5 , i s now computed; i t r e l a t e s t h e p r o p a g a t i n g f l u x t o t h e r e a c t i o n i n t e n s i t y :
The v a l u e s computed f o r 5 a r e p l o t t e d i n f i g u r e 14 as a f u n c t i o n o f 6 f o r t h e t h r e e f u e l
s i z e s . The f o l l o w i n g c o r r e l a t i o n e q u a t i o n was found f o r 5 as a f u n c t i o n of 0 and a :
R a t e of Spread
Combining t h e h e a t s o u r c e and h e a t s i n k terms produces t h e f i n a l no-wind r a t e o f
spread equation :
EVALUATION OF W I N D
A N D SLOPE COEFFICIENTS
4,
I f t h e f u e l p a r a m e t e r s i n e q u a t i o n (6) a r e assumed c o n s t a n t , t h e p r o p a g a t i n g f l u x i s
p r o p o r t i o n a l t o t h e r a t e of s p r e a d and e q u a t i o n (44) becomes
where :
R
r a t e of s p r e a d i n t h e p r e s e n c e of a h e a d i n g wind
Similarly ,
where :
R
= r a t e of s p r e a d up a s l o p e .
W i n d Coefficient
Rate of s p r e a d measurements i n t h e p r e s e n c e of wind o r on s l o p e s i l l f u e l a r r a y s
amenable t o t h e no-wind model a r e needed t o e v a l u a t e e q u a t i o n s (45) and ( 4 6 ) .
Figure 17.--Double
t r i p o d Sue Z bed used
i n wind tunne Z
experiments.
at the center.
d e s i r e d packing
i s f a r superior
sticks collapse
Field Data
McArthurls (1969) d a t a on r a t e of s p r e a d f o r h e a d i n g g r a s s l a n d f i r e s i n A u s t r a l i a
a r e show11 i n f i g u r e 1 9 . However, no d a t a a r e a v a i l a b l e on t h e p a r t i c l e s i z e , d e p t h , o r
l o a d i n g of t h e v a r i o u s a r e a s burned; t h e r e f o r e , i t was assumed t h a t t h e s e v a l u e s were
s i m i l a r t o t h o s e of a t y p i c a l a r i d g r a s s a r e a i n t h e Western United S t a t e s .
3,500 f t . - '
*re It'. - - R ~ r n i n g i m b Ze
r"po2
f ~ Ze bad i~ c
2rge c i n i tl*nneZ.
, Wo =
0.75 t o n / a c r e ,
and depth = 1 . 0 f t .
10.0
8.0
Source of data
x Geelong Fires
E
2
E'
17-1-65
7-2-67
6.0
(5,
20
--z
4.0
-a
0
x0
2.0
S
%4
**
Ana Zysis
Before a c o r r e l a t i o n c o u l d be found between wind v e l o c i t y and t h e m u l t i p l i c a t i o n
f a c t o r f o r wind, it was n e c e s s a r y t o f i n d an i n t e r r e l a t i o n s h i p between 4, and t h e
To do t h i s , t h e e x c e l s i o r and 1 / 4 - i n c h s t i c k d a t a from t h e
f u e l p a r a m e t e r , a and 6/Bo
wind t u n n e l were p l o t t e d aPong with McArthurts f i e l d d a t a . H a l f - i n c h s t i c k d a t a d i d
n o t c o r r e l a t e and had t o be d i s c a r d e d . Apparently t h e e f f e c t i v e bulk d e n s i t y i s a l t e r e d
by t h e r a p i d h e a t i n g caused by a heading f i r e ; t h u s t h e assumption of corlstant f u e l prope r t i e s needed f o r o b t a i n i n g e q u a t i o n (45) i s n o t v a l i d f o r f u e l s a s l a r g e a s o n e - h a l f
inch.
where:
= 7.47 e x p ( - 0 . 1 3 3 ~ * ~ ~ )
B = 0.02526~*~~
E = 0.715 exp(-3.59 x 1 0 - ~ a ) .
Legend
80
Gross
Excelslor
60 -
C]
'14"
sticks
40 -
20
Slope Coefficient
The e f f e c t of s l o p e was determined f o r f i n e f u e l s by burrling e x c e l s i o r f u e l beds
on s l o p e s of 25, 50, and 75 p e r c e n t . The experimerits were conducted i n a l a r g e
combustion l a b o r a t o r y under t h e same e n v i r o n m e r ~ t a l c o n d i t i o n s used f o r t h e no-wind and
wind t u n n e l f i r e s . Fuel was e x c e l s i o r c o n s t r u c t e d a t f o u r packing r a t i o s : 0 . 0 0 5 , 0 . 0 1 ,
0 . 0 2 , and 0 . 0 4 . A c o r r e l a t i o r ~of t h e d a t a 1 s shown i n f l g u r e 22. The equation f o r t h e
line i s
where t a n
tion i s
i s t h e s l o p e of t h e f u e l bed.
@//3l 5
SUMMARY OF
FIRE SPREAD EQUATIONS
The complete s e t o f p a r a m e t r i c e q u a t i o n s developed i n t h i s work i s give11 on page
2 6 . The r e q u i r e d i n p u t p a r a m e t e r s a r e given on page 27. These e q u a t i o n s a r e e a s y t o
program f o r computer c o m p u t a t i o n s . S t u d e n t s of f i r e b e h a v i o r can g a i n a p e r s p e c t i v e
u n d e r s t a n d i n g o f t h e e f f e c t s o f v a r i o u s i n p u t p a r a m e t e r s by computing and c r o s s p l o t t i n g
curve f a m i l i e s f o r r e a c t i o n v e l o c i t y , r e a c t i o n i n t e n s i t y , and o t h e r i n t e r n a l v a r i a b l e s
t h a t g o v e n ~ f i r e s p r e a d . The e q u a t i o n s might a l s o be used f o r a n a l y z i n g e x p e c t e d
b e h a v i o r o f p l a n n e d l a b o r a t o r y e x p e r i m e n t s . A word of c a u t i o n - - t h e f u e l bed w i d t h must
be s u f f i c i e n t t o s i m u l a t e a l i n e f i r e (Anderson 1 9 6 8 ) ; and t h e f u e l beds must be c a r e f u l l y c o n s t r u c t e d t o i n s u r e a uniform d i s t r i b u t i o n of t h e f u e l e l e m e n t s . ?'he e q u a t i o n s
i n t h i s form have l i m i t e d u s e i n t h e f i e l d b e c a u s e few f u e l t y p e s a r e composed o f f u e l s
t h a t a r e homogeneous i n s i z e . The remainder o f t h i s p a p e r i s d e v o t e d t o a d a p t i o n of t h e
p a r a m e t r i c eo,uations i n t o a m a t h e m a t i c a l model s u i t a b l e f o r f i e l d a p p l i c a t i o n .
R =
pbEQig
r 'wllh rlM'ls
IR =
Rate of s p r e a d , f t . / m i l l .
(52)
Reaction i n t e n s i t y , B. t . u . /
f t . min.
(27)
where :
r'
= ~ * m a x ( ~ / ~ o p ) A e ~mp o[ ~
p )( ]l - Optimum r e a c t i o n v e l o c i t y ,
min. - l
'max
Bop
= 01.5(495
+ .05940~'~)-~
Maximum r e a c t i o n v e l o c i t y , ( 3 6 )
min.-l
3.3480--'~'~
A = 1/(4.774a.l
1' M
1 - 2.59
Optimum p a c k i n g r a t i o
- 7.27)
P.1f
Mx
hlf
+
5.11
Mf
(%)
- 3.52
(%)
+w =
(192 + 0 . 2 5 9 5 a ) - ~ e x ~ [ ( 0 . 7 9 2+ 0 . 6 8 1 0 . ~ )(B + 0 . 1 ) ]
Propagating f l u x r a t i o
cuB
(29)
h l i n e r a l damping c o e f f i c i e i l t (30 1
= 0.174 Se-.I9
, =
(37)
(39)
!>loisture da-nping c o e f f i c i e n t
IIS
(38)
(*)
-L
Wind c o e f f i c i e n t
(42)
(47)
C = 7 . 4 7 exp ( - 0 . 1 3 3 0 . ~ ~ )
(48)
B = 0 . 0 2 5 2 6 0 . 54
(49)
W,,
$S
0.715 exp ( - 3 . 5 9
w0
5.275
tan
pb = w0/6
= exp(-138/0)
B = %
P
$j2
10-~0)
(50)
Net f u e l l o a d i n g , 1 b . / f t 2
(24)
Slope f a c t o r
(51)
Overldry b u l k d e n s i t y ,
lb./ft. 3
(401
E f f e c t i v e h e a t i n g number
(14)
Heat o f p r e i g n i t i o n ,
B.t.u.
lb .
(12)
Packing r a t i o
( 3 1-1
0'
ovendry f u e l l o a d i n g , I b . / f t .
8, f u e l d e p t h , f t .
a , f u e l p a r t i c l e surface-area-to-volume r a t i o , l / f t .
h , f u e l p a r t i c l e low h e a t c o n t e n t , B . t . u . / l b .
ovendry p a r t i c l e d e n s i t y , l b . / f t . 3
P'
Mf, f u e l p a r t i c l e moisture content, l b . moisture
l b . ovendry wood
p
ST, f u e l p a r t i c l e t o t a l m i n e r a l c o n t e n t , l b . min,erals
l b . bvendry wood
S e , f u e l p a r t i c l e e f f e c t i v e m i n e r a l c o n t e n t , l b . s i l i c a - fi-ee m i n e r a l s
l b . ovendry wood
U , wind v e l o c i t y a t midflame h e i g h t , f t ./min.
tan 4
Mx,
To a i d i n t h e understanding of f u e l d i s t r i b u t i o n . , we i n t r o d u c e d t h e concept of a
u n i t f u e l c e l l . A u n i t f u e l c e l l i s t h e s m a l l e s t volume of f u e l w i t h i n a s t r a t u m of mean
depth t h a t h a s s u f f i c i e n t f u e l t o b e s t a t i s t i c a l l y r e p r e s e n t a t i v e of t h e f u e l i n t h e
e n t i r e f u e l complex. T h i s concept p e r m i t s t h e mathematical r e p r e s e n t a t i o n o f t h e f u e l
d i s t r i b u t i o n t o b e r e f e r e n c e d t o a u n i t f u e l c e l l r a t h e r t h a n t o t h e e n t i r e complex.
( w o ) i j = ovendry l o a d i n g , l b . / f t . 2
.
( l b . m i n e r a l s / l b . wood)
j = s u r f a c e - area-to-volume
);(
(F ) . .
r a t i o , ( f t 2 / f t . 3,
= mineral content,
T 11
( l b . minerals - l b , s i l i c a )
(Te) j = e f f e c t i v e m i n e r a l c o n t e n t
l b . wood
( h ) i j = low h e a t v a l u e , B . t . u . / l b .
(Mf)i j = m o i s t u r e c o n t e n t , ( l b
(Fp)i
. m o i s t u r e ) / ( l b .wood)
= ovendry p a r t i c l e d e n s i t y ,
(lb . / f t . 3 ) .
(M,)
. moisture) / ( l b . ovendry
Mean f u e l a r r a y p r o p e r t i e s :
tan 9
= depth of f u e l , ( f t . )
=
= wind v e l o c i t y a t midflame h e i g h t , ( f t . / m i n .)
t o t a l number of c a t e g o r i e s
wood)
mean t o t a l s u r f a c e a r e a o f f u e l p e r u n i t f u e l c e l l .
Ai
mean t o t a l s u r f a c e a r e a o f f u e l o f ith c a t e g o r y p e r u n i t f u e l c e l l .
*.
I
'
(3
ij
A.. =
Go)
i j ,
fij
ij
= T
Ai
A.
f.
1
-
th
Ratio of surface a r e a of j
size class t o t o t a l surface area
of i t h category p e r u n i t f u e l c e l l
th
Ratio of s u r f a c e a r e a of i
category t o t o t a l surface area
per unit fuel c e l l
Using t h e w e i g h t i n g p a r a m e t e r s , t h e b a s i c f i r e s p r e a d e q u a t i o n s a r e modified a s
follows :
Reaction i n t e n s i t y becomes :
(wnIij =
Net l o a d i n g of j t h c l a s s w i t h i n
i t h category
j =n
fii
f..F..
= C
j =O
(Qi
( 5e ) i.
11 11
0. 174(Se)i--19
Low h e a t c o n t e n t v a l u e of i
category
(60)
th
(61)
Mineral damping c o e f f i c i e n t
of i t h category
(62)
Characteristic effective
th
m i n e r a l c o e f f i c i e n t of i
category
(63)
M o i s t u r e damping c o e f f i c i e n t
of i t h c a t e g o r y
(64)
j =n
fij(Se)ij
= C
j =1
Moisture r a t i o of it h
category
j =n
(Mf)i = C
f . . (Mf)ij
j = 1 11
Moisture c o n t e n t of
ith category
(66)
To complete t h e c a l c u l a t i o n of t h e r e a c t i o n i n t e n s i t y , t h e p o t e n t i a l r e a c t i o n
v e l o c i t y , I-', must b e c a l c u l a t e d . A s i n g l e v a l u e of r e a c t i o n v e l o c i t y i s c a l c u l a t e d
f o r t h e f u e l complex.
r'
F/BOP 1 I
where :
C h a r a c t e r i s t i c surface-areato-volume r a t i o o f t h e f u e l
complex
(71)
C h a r a c t e r i s t i c surface-areato-volume r a t i o of ith f u e l
category
(72)
Mean packing r a t i o
Mean bulk d e n s i t y
T h i s completes t h e computations n e c e s s a r y f o r c a l c u l a t i n g r e a c t i o n i n t e n s i t y .
The p a r a m e t e r s w i t h i n t h e b a s i c r a t e of s p r e a d e q u a t i o n
where :
(
lg
1 . . = 250
1J
1,116 ( E f ) i j
The h e a t o f p r e i g n i t i o n
f o r jth s i z e c l a s s within
t h e ith c a t e g o r y
and :
where :
U = mean w i n d s p e e d a t midflame h e i g h t , ( f t . / m i n . )
(81)
I R ~
where :
q = f r e e s t r e a m dynamic p r e s s u r e l b . / f t .
J = 778 f t . l b . / B . t . u .
- mechanical equivalent o f h e a t
E v a l u a t i n g t h i s r a t i o a t t h e l i m i t i n g v a l u e o f s p r e a d r a t e f o u n d by McArthur (1969)
( f i g . 20) g i v e s :
T h i s l i m i t i s t a k e n f o r ($ )
w max
If
U > 0 . 9 , t h e n $w
-
IR
= $w a t U = 0 . 9 1
R'
eleva-
Heat Content
Mineral Content
P a r t i c l e Density
2 . Fuel Array Arrangement
Wind V e l o c i t y
Fuel Moisture Content
Slope
Ratio
The f u e l p a r t i c l e p r o p e r t i e s a r e n o t e x p e c t e d t o vary g r e a t l y w i t h i n v e g e t a t i o n
t y p e s . Such v a l u e s can be r e a d i l y determined i n t h e l a b o r a t o r y and assembled i n a mann e r t h a t s h o u l d have wide a p p l i c a b i l i t y .
Fuel a r r a y arrangement p a t t e r n s must b e determined i n t h e f i e l d . These i n v e n t o r y
t a s k s w i l l b e more d i f f i c u l t than measuring f u e l p a r t i c l e p r o p e r t i e s . However, i t i s
e x p e c t e d t h a t p a t t e r n s w i l l b e found t h a t a r e r e p e a t a b l e w i t h i n t h e l i m i t s n e c e s s a r y f o r
c a l c u l a t i n g p o t e n t i a l f i r e h a z a r d u s i n g t h e model. The f u e l t y p e , age of s t a n d , expos u r e , s o i l s , r a i n f a l l p a t t e r n s , and f i r e h i s t o r y may b e used as indexes f o r c a t a l o g i n g
f u e l arrangement p a t t e r n s . Broader c l a s s i f i c a t i o n by e c o t y p e o r h a b i t a t w i l l a l s o p r o v e
v a l u a b l e f o r s o r t i n g out f u e l p a r a m e t e r s .
The e n v i r o n m e n t a l r e l a t e d p a r a m e t e r s can b e i n s e r t e d t o i n v e s t i g a t e t h e e f f e c t of
t h e range o f wind, m o i s t u r e , o r s l o p e t h a t might b e e x p e c t e d t o b e imposed upon t h e f u e l s
b e i n g modeled.
~ 2 . 91 - a
[ l - ? 10
(M)
f dead
] -O.226,withalowerlimitof0.30,
(88)
where: a = r a t i o o f m a s s - o f - f i n e - l i v e - f u e l t o m a s s - o f - t o t a l - f i n e - f u e l ; f i n e f u e l i s
t a k e n a s f u e l Q1/4-inch d i a m e t e r . (Mf) dead = m o i s t u r e c o n t e n t ( f r a c t i o n , n o t p e r c e n t )
o f f i n e dead f u e l .
;t:
P.+
I
1
I
I
. .
Ln
0
0
Ln*
0
0
Ln*
0
0
Ln
0
M
0
0
L
I
0
M
0
M
,+
0
M
l
I
I
I
I
I
0
M
0
M
0
M
Figure 23. - - P o t e n t i a l r e a c t i o n
ve Zocity o f t y p i c a 2 wildland
fue 2s. The two Zines repres- e n t t h e extreme values o f
a, one f o r s h o r t grass, t h e
o t h e r f o r heavy logging
slash.
Black O a k litter
litter
13 , Pocking ratio
Legend:
--..-
..-..-0-0
-.--
Slash (light
4 0 T./A.)
Slosh (medium 120 T./A.)
Slash (heavy 2 0 0 T./A.)
A-.
.-m
0-0
Fuel.
Dm = % r "
Depth=l
0'
Loaded for
05
13 ,
10
15
20
25
30
Fuel moisture , MI
7
s
.- 0 4
h
&
v,
Q,
.c&
o
~2
.&
o)
P:
LITERATURE CITED
Anderson, H . E .
1968. F i r e s p r e a d and flame s h a p e .
F i r e T w h n o l . 4(1) :51-58.
Anderson, H. E .
1969. Heat t r a n s f e r and f i r e s p r e a d . USDA F o r e s t S e r v . Res. Pap. INT-69, 20 p . ,
illus.
B e r l a d , A. L.
1970. F i r e s p r e a d i n s o l i d f u e l a r r a y s . Combust. and Flame 14:123-236.
Brown, J . K .
1972. F i e l d t e s t o f a r a t e - o f - f i r e - s p r e a d model i n s l a s h f u e l s . USDA F o r e s t
Serv. Res. Pap. INT-116, 24 p . , i l l u s .
Countryman, C . M . , M. A. Fosberg, R. C . Rothermel, and M. J . S c h r o e d e r
1968. F i r e weather and f i r e b e h a v i o r i n t h e 1966 loop f i r e . F i r e Technol. 4(2) :
126- 141, i l l u s
Analysis of f i r e spread i n l i g h t f o r e s t f u e l s .
illus.
Frandsen , W H .
1971. F i r e s p r e a d through porous f u e l s from t h e c o n s e r v a t i o n of energy. Combust.
and Flame 16:9-16, i l l u s .
McArthur, A. G.
1969. The Tasmanian b u s h f i r e s of 7th February, 1967, and a s s o c i a t e d f i r e behavi o u r c h a r a c t e r i s t i c s . I n The T e c h n i c a l Co-operation Programme. Mass F i r e
Symposium (Canberra, ~ u x r a l i a1969) v . I . 23 p . Maribyrnong, V i c t o r i a :
Defence S t a n d a r d s L a b o r a t o r i e s .
P h i l p o t , C . W.
1968. Mineral c o n t e n t and p y r o l y s i s of s e l e c t e d p l a n t m a t e r i a l s .
S e r v . Res Note INT-84, 4 p .
USDA F o r e s t
P h i l p o t , C h a r l e s W . , and R . W . Mutch
1971. The s e a s o n a l t r e n d s i n m o i s t u r e c o n t e n t , e t h e r e x t r a c t i v e s , and energy of
ponderosa p i n e and D o u g l a s - f i r n e e d l e s . USDA F o r e s t S e r v . Res. Pap.
INT-102, 21 p . , i l l u s .
Rothermel, R . C . , and H . E. Anderson
1966. F i r e s p r e a d c h a r a c t e r i s t i c s determined i n t h e l a b o r a t o r y .
Res. Pap. INT-30, 34 p . , i l l u s .
U. S . F o r e s t S e r v .
T a r i f a , C. S . , and A. M. T o r r a l b o
1967. Flame p r o p a g a t i o n a l o n g t h e i n t e r f a c e between a gas and a r e a c t i n g medium.
P. 533-544, i l l u s . , I n : Eleventh Symposium ( I n t e r n a t i o n a l ) on Combustion
( B e r k e l e y , C a l i f o r n i a 1967). P i t t s b u r g h : The Combust. I n s t .
A mathematical model f o r predicting fire spread i n wildland fuels, USDA Forest S e w . Res. Pap. INT-115, 40 p . ,
illus.
1972.
A mathematical model for predicting f i r e spread i n wildland fuels, USDA Forest Serv. Res. Pap. INT-115, 40 R ,
illus.
A mathematical fire model for predicting r a t e of spread and
intensity t h a t i s applicable to a wide range of wildland fcels and environment i s presented. Methods of incorporating mixtures of fuel
s i z e s a r e introduced by weighting input p a r a m e t e r s by surface a r e a .
The input parameters do not require a p r i o r knowledge of the burning characteristics of the fuel.
1972.
ROTHERMEL, RICHARD C.
A mathematical model for predicting f i r e spread in wildland fuels, USDA Forest Serv. Res. Pap. INT-115, 40 p.,
illus.
ROTHERME L, RICHARD C.
1972.
A mathematical model f o r predicting f i r e spread in wildland fuels, USDA Forest Serv. Res. Pap. INT-115, 40 p.,
illus.
ROTHERMEL, RICHARD C.
1972.
ROTHERMEL, RICHARD C.