a mathematical model for predicting fire spread in wildland fuels

8/18/2019 A MATHEMATICAL MODEL FOR PREDICTING FIRE SPREAD IN WILDLAND FUELS

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USDResearc

MATHEMATICALMOD

FOR PREDICTING

FIRE SPREAD

IN WILDLAND FUEL

RichardC

Rothermel


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PREF

Forest managers as well as those engaged in research involvinests, brush fields, and grasslands need a consistent method for pspread and intensity in these fuels. The availability of the mmodel of fire spread presented in this paper offers for the method for making quantitative evaluations of both rate of sprintensity in fuels that qualify for the assumptions made on the

and weather parameters measurable in the field are featured as model. It is recognized that this model of the steady -state fireonly a beginning in modeling wildland fires, but the initial apthe National Fire-Danger Rating System and to fuel appraisal

wide applicability.The introduction of this model will permit the use of syst

techniques to be applied to land management problems. As a rdimension is offered to land managers for appraising the cons proposed programs. Questions can be answered such as: What is

fuel hazard when thinning is done in overstocked areas? Can ltices be modified to reduce the potential fire hazard of the fuduce? How much slash should be left on the ground to produc

site treatment for the next crop of trees? How long after cutticessful bum still be achieved? What is the hazard buildup in chafields of the Los Angeles Basin in years subsequent to the last bu

Systems analysis can be applied not only to these broader asptative manipulation activities, but also to traditional activities,suppression planning and prescribed burning. As we learn mo


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burning conditions, if sufficient heat is generated, the fire can grow into the treetops causing a crown fire to develop. The nature andisms of heat transfer in a crown fire are considerably different than

a ground fire. Therefore, the model developed in this paper is not ato crown fires. An exception can be made for brush fields. Brushchamise, is characterized by many stems and foliage that are reason

tiguous to the ground, making it suitable for modeling as a ground fiContributions to the spread of the fire by firebrands have not

cluded. At first this may seem to be a serious limitation to the mcause everyone who has been on a large fire (most investigators gofires, the fires not presently being modeled) knows the importanceting. However, seeing firebrands in the air and landing ahead of the

does not mean that they are effective in advancing the fire. Berlahas shown that not all firebrands have a significant effect in spreadi

To be significant, firebrands must release sufficient heat when the

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

overrun the descent point as a result of conventional heat transfer meFurthermore, the model has been designed to simulate a firestabilized into a quasi-steady spread condition. Most fires begin fromsource and spread outward, growing in size and assuming an elliptiwith the major axis in the direction most favorable to spread. Wheis large enough so that the spread of any port ion is independent of icaused by the opposite side, it can be assumed to have stabilized ifire. A line fire behaves like a reaction wave with progress that is ste

time in uniform fuels.

All input parameters can be determined from knowledge of the cistics of fuels in the field. This does no t imply that all the parameter

and environment are readily available or can easily be measuredhowever, delineate what parameters should be cataloged and eliminathat are not needed. A convenient method of cataloging input para

through the concept of fuel models tailored to the vegetation pattein the field. The companion fuel models are thus a set of input pthat describe the inherited characteristics that have been found i

fuel types in the past. The environmental parameters of wind, sloppected moisture changes may be superimposed on the fuel models. model concept has already been incorporated into the National Fir

Rating System (Deeming and others 1972).The mathematical model produces quantitative values of sprea

tensity that should be regarded as appraised or mean values for the


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THE

Richard C. Rothermel is a Research Engineer stationed atNorthern Forest Fire Laboratory in Missoula, Montana. H

the Research Project Leader for the Fire Physics research wunit. Rothermel received his B.S. degree in Aeronautical Eneering at the University of Washington in 1953. He servethe U. S. Air Force as a Special

eap~ns

Aircraft DevelopmOfficer from 1953-1955. Upon his discharge, be was emplo

at Douglas Aircraft Company as a troubleshooter in the Arment Group. From 1957 to 196 1 Rothermel was employedthe General Electric Company in their Aircraft Nuclear Pro

sion Department at the National Reactor Testing StationIdaho. In 1961, Rothermel joined the staff at the Northern est Fire Laboratory, where he has been engaged in research

the mechanisms of fire spread. He received his master's dein Mechanical Engineering at the University of Colorado in FCollins in 1971.


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ABSTRA

The development of a mathematical model for predicting rate

spread and intensity applicable to a wide range of wildland fuels is pfrom the conceptual stage through evaluation and demonstration oto hypothetical fuel models. The model was developed for and is noused as a basis for appraising fire spread and intensity in the Natio

Danger Rating System. The initial work was done using fuel arra

posed of uniform size particles. Three fuel sizes were tested overange of bulk densities. These were 0.026-inch-square cut excelsior,

sticks, and 112-inch sticks. The problem of mixed fuel sizes was

solved by weighting the various particle sizes that compose actual fuby either surface area or loading, depending upon the feature ofbeing predicted.

The model is complete in the sense that no prior knowledge ofburning characteristics is required. All tha t is necessary are inputs d

the physical and chemical makeup of the fuel and the environmenttions in which i t is expected to burn. Inputs include fuel loading, fufuel particle surface-area-to-volume ratio, fuel particle heat cont

particle moisture and mineral content, and the moisture content extinction can be expected. Environmental inputs are mean wind and slope of terrain. For heterogeneous mixtures, the fuel prope

entered for each particle size. The model as originally conceived wasfuels in a uniform stratum contiguous to the ground, such as l itterIt has been found to be useful, however, for fuels ranging from pinlitter to heavy logging slash and for California brush fields


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CONTE

HOW FIRE SPREADS

CONCEPTION OF MATHEMATICAL MODEL

Heat Required for Ignition

Propagating FluxReaction Intensity

Effect of Wind and Slope

Approximate Rate of Spread Equation

EVALUATION OF PARAMETERS.NO WIND

OR SLOPE

HeatSink

Heat SourceExperimental Design

Experimental Results

EVALUATION OF WIND AND SLOPE COEFFICIENTS

Wind Coefficient

Slope Coefficient

SUMMARY OF FIRE SPREAD EQUATIONSTHE FIRE SPREAD MODEL

Formulation of Fire Spread Model

APPLICATION TO THE FIELD

Fuel Models and Applications


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HOW FIRE SPR

Early work i n f i r e spre ad rese arch conducted by th e USDA Fores t Sepr i mari ly a imed a t developing relat io ns hi ps between burning condit i ons var i able s t h at would ai d fo re st managers t o cope with f i r e problems. Sf ue l moi s tu re , f ue l l oad ing , wind ve loc i ty , r e l a t i ve humidi ty, s l ope , a

were a l l r ecognized as producing important e f fe c t s on f i r e . These e f feand corr el at ed t o some form of f i r e behavio r. The work was prim ari ly dsome very good results were obtained, consider ing the complexit ies of th

the va r i ab i l i t y o f wea the r , pa r t i cu l a r ly w ind. Much of t h e pre se nt dayr a t i n g , f u e l c l a s s i f i c a t i o n , and o t h e r u s es o f f i r e r e se a rc h a r e b as ed

ing work.

W. R. Fons (1946) was th e f i r s t t o a t tempt t o descr ibe f i r e spreadmat ica l model. Fons focused h i s a t te nt io n on the head of th e f i r e whe

car ry the f i re and where there i s ample oxygen t o sup port combustion. t h a t s u f f i c i e n t h e a t i s n eed ed t o b r i n g t h e a d j o in i n g f u e l t o i g n i t i o n the f i r e f r on t . T he re f o r e, Fons r easoned tha t f i r e sp r ead i n a f u e l be

i zed a s p r oceed ing by a se r i e s o f success ive i gn i t i ons and tha t i t s r a tp r imar i ly by the i gn i t i o n t ime and the d i s t ance between p a r t i c l e s .

Fon s ea r l y ide as have been conf irmed by rec ent work i n f lame spreT a r i f a and T or ra lbo ( 1967) s t a t e t h a t :

Heat ing of the fuel ahead of the f lame as i t pr ogr e sse s i s t hand most e s s en t i al proces s of th e flame propaga tion mechanismf o r e , it i s very imp orta nt t o know t h e flame pro paga tion mechfrom f lame t o f ue l and t o s tudy t he t ime consumed f or th e heapr ocess s ince i t may con tro l propa gati on speed i n many cas es.t h e l e s s , t h e r e i s l i t t l e i n f o r ma tion on the se p roblems.


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Heading fire- 5 m p h

-g=Temperature of air

T s =

Temperature f surface

Distances from fire

I I I I I I I

Time since beginning of fire , minutes

Figurs 1.-- el temperature hi stor y prior t o ig ni ti on for heading, n

f ires.

Considering f i r e as a ser ie s of ig ni ti on s helps i n breaking dowana lysi s . Heat i s suppl ied from the f i r e t o the poten t ia l fue l , thedehydrated, and fur t her hea t ing ra is es th e surface temperature un t i lpyrolyze and re lea se combustible gases. When th e gas evolution r a t efuel i s su ff ic ie nt t o support combustion, the gas i s ig nited by the fadvances t o a new pos it i on. Finall y, a constant r at e of spread i s aca l led the "quasi-steady state" where in the f i r e advances a t a ra te

of a l l the e lemental r a te s .

T hi s pr oc es s i s i l l u s t r a t e d i n f i gu re 1, which i s based on a lawhich we monitored t h e su rf ac e temper ature of a f i ne f ue l element ant o i t ahead of an advancing f i r e . In t he no-wind f i re and backing temperature ros e slowly un t i l t he f i r e was within 1 o r 2 inches of thwhere i t suddenly rose t o igni t ion . During the preheating phase, th


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CONCEPTI

MATHEMATICAL M

The model was developed 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 tw id e a s p o s s i b l e . Th i s b a s e was s u p p l i ed b y Frandsen (1971) who ap pl i e

t i o n o f e n er gy p r i n c i p l e t o a u n i t v ol um e o f f u e l ah ead o f an ad van c i n

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 ll o w in g :

R =

be ig

where :

I = q u a s i - s t e a d y r a t e o f s p re a d, f t . / m i n .

I . = h o r i z o r l t a l h e a t f l u x ab s or b ed b y a u n i t v olume o f f u e l a t t h ex1g

~ . t . u . / f t . ~ - m i l l .

= e f f e c t i v e b ul k d e n s i ty ( t h e amount o f f u e l p e r u n i t vo lume o fr 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 .

Q. = h e a t o f 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 w ei gh tl

i g n i t i o n ) ,B . t . u . / l b .

= t h e g r a d i e n t o f t h e v e r t i c a l i r l t e n s i t y e v a l ua t e d a t a p l a n e a

(2 ) d e p t h ,zc ,

o f t h e f u e l b e d ,B . t . u . / f t . 3 - m i n .

zC

T h e h o r i z o n t a l and v e r t i c a l c o o r d in a t e s a r e x and z , r e s p e c t i v e l y .


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Heat Required for Ignition

The he a t r e qu i re d f o r i gn i t i oni s

dependent upon (a) ig ni t i on temois ture content of th e fue l , and (c ) amount of f ue l involved i n the

The energy per uni t mass requi r ed f o r ign i t io n i s t h e h e a t o f p r

where:

M = r a t i o o f f u e l m oi st ur e t o ovendry weightf

= i g n i t i on t e mpe ra ture .g

The amount of fu e l involved i n th e ign i t io n process i s t h e e f f e c t i v e To a id i n t e r pr e ta t io n and ana ly s is , an e ff ec t i ve hea t ing number i s deo f t h e e f f e c t i v e bu lk d e n s i t y t o t h e a c t u a l b u lk d e n s i t y .

be

E E -

b

The effect ive heat ing number i s a dimensionless number t h a t w i l l b e nfu e l s and decrease toward ze ro a s fu e l s i z e inc reas es . There fore ,

be

= f ( bu lk d e n s i t y , f u e l s i z e )

Propagating Flux

The propagat ing flux i s th e numera tor of t heRHS ( r ight-ha nd s id

(1) and ha s t he un i t s o f he a t pe r un i t a r e a , pe r un i t t ime . The proprepresented by I .

P

The propagat ing flux i s composed of two terms, th e hor i z o n ta l f lo f t h e v e r t i c a l f l u x i n t e g r a t e d from minus i n f i n i t y t o t h e f i r e f r o n tcan be char acte rize d a s shown i n fi gu res 2, 3 , and 4. The f i g u r e s i n


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Figure 2 . --Schematic of

no-wind fire.

Wind

Figure 3. --Schematic ofwind -driven f ire .

Solid mass tra


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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 fr om e xp e ri m e nt s w i t h s p r et h e no-wind co nd i t i on by measur ing o o v e r a wid? r an ge o f f u e l c o n d i

t h e p r op ag at i ng f l u x oc c ur s a t t h e f r o n t o f t h e f i r e ; t h e r e fo r e I p ) 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 rrtensity

The e ne rg 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 od u ced b y b u r n i n gf r o m t h e o r g a n i c m a t te 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 o f ch an g e m a t t e r from a s o l i d t o a g as i s a good approximat ion o f t h e su bs e qu en

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

i n t e n s i t y

and i s d e f i n e d a s :

where :

- mass l o s s r a t e p er u n i t a r e a i n t h e f i r e f r o n t , l b . / f t . 2 -md t

h = h e a t c o n t e n t o f f u e l , B . t . u . / l b .

'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 uc h f u e l p a ra m et e rs a s t h e pd en s i t y , m o i s t u r e , an d ch em i ca l co m p o s i t i o n .

T h e 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-wi nd p r o p ag a t i n g imp or ta nt concept upon which th e model i s b as ed t h a t ( I ) o and I R c a n

d ep en d en t l y and co r r e l a t e d . Knowing t h e co r r e l a t i o n , ?p)o can be de

r e a c t i o ~ i n t e n s i t y ,

which i s i n t u r n d e pe nd en t on f u e l p a r a m e t er s o b tfuel bed complex.

I f t h i s c on ce pt i s k ep t i n m in d, i t w i l l a i d i n u n d e r s t a nd i n g t h e d ev

model .

Effect of Wind and Slope

Wind a nd s l o p e c h an ge t h e p r o p a g a t i n g h e a t f l u x b y e x p os i n g t h e

a d d i t i o n a l c o n ve c t i v e and r a d i a n t h e a t ( f i g s . 3 and 4 ) .

Let and@ s

r e p r es e n t t h e a d d i t i o n a l p r o p a g a t i n g f l ux p roduced

T h ey a r ed i m e ~ ~ s i o r l l e s so e f f i c i e r l t s

t h a t a r e f u n c t i o n s of w in d, s l o p e

m e t e r s . T hey m us t b e ev a l u a t ed f ro m ex p e r i m en t a l d a t a . The t o t a l p

r e p r e s e n t e d b y t h e e x p re s s i o n,


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EVALUATION OF PARAM

NO -W I ND OR

The c o nc e iv 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 ed i v i d e d a nd c on s i d er e d f i r s t as t h o s e f o rm i ng a h e a t s i n k and s eco nd

a s 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 ul k d e n s i t y a r e t h e tw

t o be ev a l u a t e d b e f o r e t h e p r o p ag a t i n g f l u x co u l d b e com pu ted. Qig wa

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 b y cons ider i r l g t he change i n s p e c i f i

t o i g n i t i o n t e m p e r a t u r e a n d t h e l a t e n t h e a t o f vapo r i za t i o r l o f t h e m o

where :

C = s p e c i f i c h e a t o f d r y w o d dpd

AT. = t em p e r a t u r e r ange t o igr l i t ior l18

b = f u e l m o i st u r e , l b . wa t e r / l b . Try wcof

C = s p e c i f i c h e a t o f w a te rPw

ATB = t e m p e r a t u r e rarlge t o b o i l i n g

h f


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Figure 5. Instrwnentation fordetermining the ef fect ivebulk densi ty.

Moisture i s the primary independent v a r ia b l e i n t h e e v a l u a t i o n of

i s r ecognized tha t o ther parameters should eventual ly be inc luded i n ther i t i~ lg

r a t e ,i n o r g a n i c

impur i t i e s , and nonpyro ly t i c vo la t i l e s .

Ef fect ive Bulk Dens ity

To evaluate th e ef fe c t iv e bulkdens i ty pbe) ,

weneeded

t o dete rmi

ofh e a t i n g

a s afunct io i l

of pa rt ic le s i ze . This was evaluated by plac

with in s e c t i o n s of two s t i ck s th at were locat ed on the upper surfa ce 3

end

of st an da rd wood c r i b s . Thei n s t r u men ted s ec t i o n s

were or ien ted

g i t u d i n a l

and l a te r a l d i r e c t io ns ( f ig . 5 ) . The t empera ture d i s t r i bu t io

st ic ks was analyzed t o determine th e amount of heat absorbed by the s t

t ime of igni t ion .

Resul ts of the anal ys is ar e show11 i n f i g u r e 6. An exponent ia l f i


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Figure 6 . - - E f f ec t ive heatingnumber versus Z/a a i sthe s urface-area- to-volumera t i o o f the par t i c l e ;

E

i s a measure of the fractionof the potent ial fuel thatust be rai sed t o ign i t io n .

Heat Source

Reac tion In tens i t y

The most complex f u n c t i o n eva l ua te d was r ea c ti or i i n t e n s i t y using

t ha t e vo l ve d byd e r i v i n g

f i r ei n t e n s i t y

f rom th e we ight los sd 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~ is t rumer l ts ys t e m tw ei gh t o f a p o r t i o n o f t h e f u e l b ed d u ri n g f i r e s p r ea d .

Equat ion (7) can b e r e a r ra n g e d t o e x p r e ss r e a c t i o n i r l t e ~ i s i t y n t

where :

dx = R t h e q u a si - s t e ad y r a t e o f s p r e a d .d t

T he r e f o r e ,I dx = -Rh dw.R


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This gives

I D Rh w W ) .R n r

where:

D = r eac t ion zone dep th ( f r on t t o r ea r ) , f t .

w = n e t i n i t i a l f u e l l o ad in g, ~ b . / f t . ~n

w = res idue loading immediately a f t e r passage of th er

rea ct io n zone, l b./ f t

The ne t i n i t i a l fue l loading was cor rec ted fo r the presence of noncomband minerals.

The t ime t aken f o r t he f i r e f r on t t o t r ave l a d i s t ance equ iva len t one reaction zone i s t he r eac t ion t ime

R

Subs t i t u t ing the r ea c t ion t ime in to equa t ion (18) g ive s

We now def i ne a maximum rea ct io n in te ns i t y where th ere i s no loading rea f t e r t he r eac t ion zone i s passed and where the r e a c t i o n time remains umaximum react ion intensi ty i s represen ted by

The reac t ion zone e f f ic iency i s then def ined as

IR wn-wr)Q =I-

Rmax

WI1

Repl acin g (w -wr) i n equa t ion 201, we : ~ v c i n terms of measurable fI1

parameters.t

Th t f l l di f t i (23) b bt i d f


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Figure 7 Determination of

moisture damping coeffi

c i e n t

from ponderosa

pine needle fuel beds.

I)

.

6

I)

Fuel moisture

t may vary between 0 .10 and 0 . 4 0 . ~

Kecent f i e l d experiments i n loggi

1972) ind ic a te tha t x may be between 0.10 and 0.15 for logging slash,

p or ou s t h an l i t t e r .

The e q ua t io n f o r t h e c u rv e i n f i g u r e 7 i s

The mois ture damping coe f f ic ie nt accounts fo r th e decrease i n in t en s i t

combust iorl o f f ue l s th a t i n i t i a l l y c on ta ined m ois tu re . The exa c t e f f e

tu r e has no t bee n a dequa te ly exp laine d i n t er m s o f r e a c t ion k i ne t i c s .

Q i g s i nc luded im pl i c i t ly i n th e developmen t o f nM I f f u r t h e r

r e v e a l

~t

t o be non l in e a r , t hen t he c u rve form o f nM

w i l l

change.

Mineral Damping Coefficient

The miner al damping coe ff ic ie nt was eval uat ed from thermogravime

d a t a o f n a t u r a l f u e l s b y P h i l p o t 1 96 8) . I n t h i s s t u d y, t was assum

of t he normal ized decomposi t ion r a t e would be t he same as th e normaliz


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Figure 8 . --Mineral . damping

c o e f f i c i e n t o f n a t u r a l

f u e l s , d e r i v e d from t h e

w ork o f P h i l p o t

1 9 6 8 ) .

Effective mineral con

Phys ica l Fue l Parameters

Two v a r i ab l e s r emai n t h a t h ad t o be co n s i de red i n ev a l u a t i n g t h e

in te ns i ty - - fu e l bed compactness and fue l p a r t i c l e s i z e . Both a r e kno

can t e f f ec t s upon co m b u s t i b i l i t y , b u t t o d a t e , i n t eg ra t ed r e s ea rch h a

ed t o s ep a r a t e and q u an t i fy t h e e f f e c t s o f t h e s e v a r i ab l e s o n t h e dyn

f i r e .

I t s h y po t hes i zed t h a t low v a l ues o f f i r e i n t en s i t y and r a t e o f

th e two ex t remes o f compactness loose and dense) . In dense beds , t h

t o low a i r - t o - f u e l r a t i o and t o p oo r p en e t r a t i o n o f t h e h ea t beyond t

o f th e fue l a r ra y . In loose beds a t th e o th er ex t reme], low in te ns i

a r e a t t r i b u t e d t o h ea t t r a n s f e r l o s s e s b et ween p a r t i c l e s and t o l a ck

th es e two extremes, th er ef or e, th er e must be an optimum arrangement o

p ro du ce t h e b e s t b a l ance o f a i r , f u e l , and h ea t t r a n s f e r f o r b o th max

and reac t io n ve lo c i ty . I t i s no t expec ted th a t t he optimum ar rangeme

f o r d i f fe r e n t s i z e f u e l p a r t i c l e s .

The compact ness o f t h e fu e l b ed i s q u an t i f i ed by t h e p ack i ng r a t

d e f i n ed a s t h e f r ac t i o n o f t h e fu e l a r r ay v olume t h a t

s

occupied by

r a t i o c an b e e a s i l y c a l c u l a t e d by e v a l u a t i n g t h e r a t i o o f t h e f u e l a r

t o t h e f u el p a r t i c l e d e ns i ty ,


21/48

T he s u r f a ce - a r e a- t o - vo l ume r a t i o i s u se d t o q u a nt i fy t h e

f u el p a r t i

L e t a = t h e f u e l p a r t i c l e s ur fa ce -a re a- to - vo l um e r a t i o . F or f u e l s

w it h r e s pe c t t o t h e i r t h i c k n e s s ,

a

= f t 1

d

where

d

=

d i a me t e r o f c i r c u l a r p a r t i c l e s o r e dg e le n gt h o f s qu a re p a r t i c l

Th e p a ck i ng 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

t h e f u e l p a r t i c l e ,

a ,

a r e t h e p r i m ar y i n d ep e n de n t v a r i a b l e s u s ed t h r o ug h

r em ai n de r of t h e p a p e r 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 .

Exper imenta l e s ign

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 ei g hi n g p l a t f o r m was c o n s t r u

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 ex p e ri m e nt a l f u e l b ed s . T h i s w ei g hi n g

was 18 i nc h e s s q ua r e , was s uppo r t e d by f ou r l oa d c e l l s , wh ic h w er e p r o t e

h e a t by a s e r i e s o f b a f f l e s a nd c er am 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 fro

c e l l s we re e l e c t r o n i c a l l y summed, a m p l i f i ed , and s p l i t i n t o tw o e q u i v a l e

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 ec o nd w s e l e c t r o n i c a l l y d i f f e r e n

b e i n g r e c or d e d . T h i s d u a l a r ra n g em e nt ga ve c o n t i n u o u s r e c o r d s o f t h e w

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 c ha ng e o f t h e w e i g h t .

T he e x c e l s i o r f u e l b e d s w er e f e e t w i d e, 8 f e e t l ong , a nd 4 - 1/ 2 i n

f r o n t o f t h e we i gh i ng p l a t f o r m was p l a c e d f e e t f rom t h e f r o n t o f t h e

c e n te r ed l a t e r a l l y . T h i s a rr an ge me nt p e r m i t te d t h e f i r e t o re a ch a q u a

o f s p r e a d b e f o r e b u r n i n g o n to t h e p l a t f o r m . I n co n s is t en c i es i n b ur ni n g

edges were minimized by a l lowing 9 i n c he s o f f u e l on e i t h e r s i d e of t h e

c r i b s we re c o n s t r u c t e d u s i n g 1 / 4 - i nc h an d 1 /2 - in c h s t i c k s

;

c r i b s w er e a

f e e t l o ng , f e e t w i d e, and

5

t o 6 i nc he s de e p . T he same s i z e w e i gh i n

u se d f o r bo 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 ,

Th e c o n ce p t o f a r e a c t i o n z o ne an d 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

t h e f u e l - r e a c t i o n z on e i n t e r f a c e a s m ovi ng t h r ou g h t h e f u e l on t h e w e ig

( f i g .

9 ) .

When t h i s i n t e r f a c e r e a c h ed t h e f u e l be i n g w ei gh ed , t h e s t r i

i n d i c a t e d t h e ti m e o f a r r i v a l by t h e s t a r t o f w ei gh t l o s s .

As

t h e f i r e

c ee de d i n t o t h e ~ i e i g h e d u e l , t h e we ig ht l o s s r a t e c on t in u ed t o i n c r e a s

o f t h e we i gh i ng p l a t f o r m was l o n g e r th a n t h e d e pt h o f t h e r e a c t i o n z o ne

r a t e o f w e ig h t l o s s s t a b i l i z e d when t h e f i r e a dv an ce d o n t o t h e p l a t f o r m

c cl ui va le nt t o t h e de p th o f t h e r e a c t i o n z o ne . The l a p s e d t i m e from i n i

t o t h e o ns et o f 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 t i m e,

T R .

R e a c t i on t i m e


22/48

Figure 9 Fire fue interf ace

moving through weighed fu e l .

I

Fire fuel

arrives a

I

Fire interface approachin

[We igh t lo

continues

Fire burning into weighed

Depth of

W

b

Steady weight loss rate a

T he mass l o s s r a t e ,

h

o b t a i n e d fr om t h e w ei g ht l o s s d a t a , was r e

lowi rl g phy s i c a l f e a t u r e s :

where I V e q u a l s t h e w id t h o f t h e w ei g hi n g p l a t f o r m . The e f f i c i e n c y o f

f i r e s can now be expresse d as

C ombi ni ng 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 , T a s i n d i c a t e d by e

t a k e r fro m 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

v e l o c i t y ,


23/48

Figure 1 . I l l u s t ra t i o n

of mass lo ss and i t s

der i va t i ve .

I

I

Mass los

m

dm

- E d

xperim ental R esults

Reaction Velocity

The r e s u l t s o f t h e e xp er im e nt s 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 w e

determine r e ac t i on ve lo c i ty ar e shown i n f i g u r e 1 1 . s ex pe ct ed , t h e r

p ac k in g r a t i o f o r e ac h o f t h e 1 / 4- in ch and 1 /2 -i nc h f u e l s . I t was n o t

i d e n t i f y a dr op i n r e a c t i o n v e l o c i t y a t v e ry low pa c ki ng r a t i o s w i th t

b ec a us e 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 be d h a v i n g o nl y a

e x c e l s i o r p e r sq u a r e f o o t , and

b)

t h e l ack o f s e n s i t i v i t y on t h e w e ig

ex t remely l i g h t fue l load in gs . t iowever, t i s e v id e nt t h a t t h e r e a c t i

d ro p t o z er o i f t h e r e

s

no fue l to suppor t combus tion, j us t

a s

t

doe

f u e l s

L


24/48

Figure 2 2 --Determination of

corre la t ions for maximum

reaction velocity and

optimum packing ratio.

Fuel surface are a

to volume ratio

-U

t

The r e a c t i on v e l o c i t y f o r f i ne f c e l s e xc e l s i o r ) i s much g r e a t e r n

p a ck in g r a t i o t h a n t i s f o r t h e l a r ge r 1 / 4 -i nch and 1 / 2 -i nch s t i c k s

e x p ec t ed , t h e optimum p a ck in g r a t i o i s n o t t h e same f o r a l l f u e l s and s

r i g h t as t h e fu e l s i n c re a se i n t h i c k ~ ~ e s s .

ote a l s o t h a t t i g h t l y p ack e

a c t ua l l y have l ow er r e a c t io r l ve l o c i t i e s t han do l a r ge r f ue l s a t t he sam

T he l o s s o f r e a c t i on ve l oc i t y o f f i ne f ue l call be s ee r] i n t h e f i e l d by

t h e d i f f e r e nc e i n f l am i ng v i go r bet we en p i ne ne e d l e s - on a b roke n t r e e t o

above t h e g round and com pacted p i n e ne e d l e l i t t e r ; t he l a t t e r bu rns w i t

The d a t a p o i n t s i n f i g u r e s

11 th rough 16 a r e th e average of th re e

t i o n s i n t h e e x c e l s i o r , a nd two o r more i n t h e s t i c k c r i b s .


25/48

Figure 1 4 - -D e t edna t i on o f

propagating flux ratio 5

-

-

7

-

i

C

Le

-

0 .02 .04 .06 .08

Packing ratio

Legend

Excelsior

- 'h Cribs

A 'hw

ribs

l l l \ ~ l

C

**

r

-


26/48

Figure

16 -

C o n f i r n a t i o n o f

r a t e o f s p r e a d e q u a t i o n

w i t h

o r i g i n a l d a t a .

Packing ratio

The ma the m at i ca l f i t t o the da ta o f f ig u r e 11 was assumed to bk a

a P oi sson d i s t r ib u t io n . To de te r mine th e ge ne r a l e qua t io r ~as a fun cti

a , t h e equa tio ns f o r the maximum valu e of r and th e optimum be ta ,

s i ze were found as a func t io n of a ( f ig . 12) .

max ~ . ~ / ( 4 9 5 0 . 0 5 9 4 a ~ - ~ )

These were then combined with an a rb it ra ry va r i ab le ,

A

t o g iv e:

where

:

The e q ua t io n s t h a t r e l a t e r e a ct i o n v e l o c i t y , r e a ct i o n i n t e n s i t y ,

and r a t e o f sp r e a d wer e de ve lope d a s a s e t t o f i t no t only the depende

a l so th e da ta shown in f ig ure s 11 through 16 . Note a ls o t h a t equa t ion

and (39) w 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 c omb in at io n o f f u e l p a

and any packing ra t io , 8

The form of t he eq ua t ion s has been chosen t


27/48

The c o r r e l a t i o n eq u a t i on 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 9 ) - -a r e combined w i th e q u at i on 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

s i z e s u se d i n t h e e x pe r im e nt s . The c ur ve s from t h e s e e q u a t io n s a r e a l

f i g u r e 1 3 , where t h e f i t can b e compared t o t h e o r i g i n a l d a t a .

D i r e c t c ompar ison o f r e a c t ion in t e n s i ty be twee n t h e f ue l s u se d i n

i s no t in t e nde d , no r c an

i t

be made because fu el loading was not hel d

s tudy da ta we re on ly in t e nde d t o a id i n t h e developme nt o f e qua t ions t

to p r e d ic t r e a c t ion in t e ns i t y a nd , subse que n t ly , r a t e o f sp r e a d ove r a

fu el and environmental combinations

P r o p a g a t i n g LUX

The no-wind propag a t ing f lu x i s ca lc ul a te d from equa t ion 6),

A

r a t i o ,

5

i s now computed;

i t

r e l a t e s t h e p ro pa ga ti ng f l u x t o t h e r e

The valu es 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 nc t io n o f

s i z e s . The f o l lowing c o r r e l a t io n e qua t ion was f ound f o r 5 as a fun cti

The f i t o f t h e d a t a t o t h i s e qu a t i o n can b e se en i n f i g u r e 1 4.

The f i t o f t h i s e q ua t io n t o t h e o r i g i n a l v a l ue s o f p r op a ga t in g f l

from equa t ion 6) can be seen in f ig ur e 15 . The da t a show th a t Ip) ,

i n c r e a s i n g

0

b u t a t a d e c re a s i ng r a t e . E x t r ap o l a ti o n o f e q u a ti o n 4 1

i n d i ca t e s t h a t

i t

would ac tu al ly reach a maximum and th en decr eas e. T

r e a sona b le p r e d ic t ion , c ons ide r ing the

f a c t t h a t t h e f ue l a r r a y i s bec

t h a t t h e i n t e n s i t y h as a l s o d ec re as ed f i g . 1 3) .

R a t e

o

S p r e a d

Combining t h e hea t source and hea t s in k terms produces t h e f i n a l

sp r e a d e qua t ion


28/48

EV LU TIO N O F

N D S LO P E C O E F FIC

To in t rod uce wind and s lop e i n t o th e model , we must eva lua te th e

4 and $s . Rearranging equat ion (9) with $s 0 :

I f t h e f u e l p a ra m e te r s i n e q ua t i o n ( 6) a r e a ssumed c o n s t a n t , t h e p r o p a

prop or t ion al t o th e r a t e o f sp rea d and equat ion (44) becomes

where :

R r a t e o f s p r ead i n t h e p r e s ence o f a h ead in g w in d

S i m i l a r l y

where

R r a t e o f s p r ead up a s l o p e .

s

For exped ie~lcy t was assumed t ha t no in t er ac t i on ex is te d between wind

W i n d C o e f f i c i e n t

Rate of spread measurements

i n

th e p resence o f wind or on s l ope s

amenable t o th e no-wind model a re needed t o eval ua te equ at i ons (45) a

Wind Tunne xperiments


29/48

Figure 17.--Double

tr ip od Sue bed used

i n wind tunne

experiments.

a t t h e cen t e r . T hese d o ub l e t r i p o d s were a rr an ged a t v a r i o u s s p ac i n

d e s i r e d

p ack i ng r a t i o ( f i g s . 1 7 and 1 8 ) . For v e ry low p ack i n g r a t i o

i s f a r s u p e r i o r t o t h e t r a d i t i o n a l c r i b c o n s t r uc t i o n b ec au se c r i b s w

s t ic ks co l l ap se when t h e c r oss members bum o u t .

E x ce l s i o r f u e l b ed s must be ca re f u l l y co n s t ru c t ed t o achi ev e t h

o r t h e b u lk d e n s i ty w l l be a l t e r e d w it h d r a s t i c e f f e c t s on t h e r a t e

Field Data

M cA rt hu rl s ( 1969) d a t a on r a t e o f s p r e a d f o r he a d in g g r a s sl a n d

a re show11 i n f i g u re 1 9 . However, n o d a t a a re a v a i l ab l e on t h e p a r t i

l o ad i ng of t h e v a r i o u s a r e a s b u rn ed ; t h e r e f o r e ,

t

was assumed th a t

s i m i l a r t o t h o s e o f 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 W estern U ni te d

a

3 , 5 0 0 f t . -

W o

0 . 75 t o n / ac re , and d ept h 1 . 0


30/48

Figure 19. --Reproduction of

McArthur s (1969) ra te o f

spread data fo r grass .

Average wind velocity at

33

in the open , m.p.h

Ana Zysis

10.0

Before a co r r e l a t io n could be found between wind ve lo c i ty and th e

f a c t o r f o r wi nd ,

t

was ne c e ssa r y to f in d an in t e r r e l a t io ns h i p betwe en

fue l pa ramete r , a and 6/Bo To do t h i s , th e exc e ls io r and 1 /4- inch s

wind tunne l were p l o t te d aPong wi th McArthur ts f i e l d da ta . Ha lf - inch s

n o t c o r r e l a t e a nd had t o b e d i s c ar d e d . A pp ar en tl y t h e e f f e c t i v e b ul k

by th e r a p id he a t ing c aused by a he ad ing f i r e ;

thus the assumpt ion of

e r t i e s need ed f o r o b t a i n i n g e q ua t io 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

inc h .

-

Kon

Another p l o t o f t he f u e l par a m ete r s and m ul t ip l i c a t io r l f a c t o r v s .

p roduce d th e f i na l c o r r e l a t i on g ive n by e qua t ion 47 ). F igu re 20 show

pa r am e ter s u s ing the o r i g i na l da ta .

8.0

-

x Gee

4

Long

4

o

Tasm

2

6 . 0

E

5,

g

4.0

-

3

z

-

-

2.0

-

-

S

a

4

4

where:

c =

7.47 e x p - 0 . 1 3 3 ~ * ~ ~ )

= 0 . 0 2 5 2 6 ~ * ~ ~

E = 0.715 exp -3.59 x 1 0 - ~ a ) .

Legend

8 0 - Gross


31/48

Figure 21. --I nfl uen ce of packing

r a t i o

and

part ic l e s i z e on wind

coe f f i c i en t a t 12 m p h . I n t h e

absence of wind th e fuel

condit ion s for b es t burning

would occur at B B o p =

I

I n

th e presence of wind f i re s

spread fas t er i n Zess dense

fue l e . ~ a ~ ~Zess than

I The wind co ef f i c i en t

increases markedly

s

surface-

area-to-voZwne r at i o incr ease s.

The shape of t h e cu rves i s

n

good agreement with th e concept s

Rothermel and Anderson 19 66 ). t t h a t t i m e , i t was s p e c ul a t ed t h a t

f u e l , t h e s h a r pe r t h e r e s u l t i n g i n c r e a s e i n s p r e ad r a t e wi th w ind v e

ex p ec t ed , fu e l s t h a t were t o o s p a r s e t o bu rn we l l i n t h e abs en ce o f

a r ap i d f i r e s p read when wind i s ap p l i ed . I n e f f e c t , t h e optimum p a

t ow ar d more l i g h t l y l oa de d f u e l s a s wind i n c r e a s e s . T h is e f f e c t i s

f i gu re 21 and can be seen in th e f i e l d where spar se fue l s - -such as p

cheat grass burr]

poor ly without wind but become a f l ash y f ue l when w

Slope oefficient

The e f f e c t o f s l o p e was d e te rmi n ed f o r f i n e f u e l s by b ur rl in g ex

on slo pe s of 25, 50, and 75 pe rc en t. The experimerits were conducted

combust ion lab or ato ry under th e same envi ronm er~ talcond i t io ns used f

win d t u n n e l f i r e s . Fue l was ex c e l s i o r co n s t ru c t e d a t fo u r p ack in g r

0 .02 , and 0 . 0 4 . c o r r e l a t i o r ~o f t h e d a t a

1 s

shown i n f lgur e 22 . T

l in e i s

whe re t an i s t h e s l o p e of t h e fu e l b ed. The f i n a l form o f t h e r a t

t io n i s


32/48

SUM M

FIR E SP R E D EQU


33/48

Swmnary o f as ic Fir e Spread Equ at ions

I ~ c ( 1 $w

R

=

R at e o f s p r e a d , f t

/ m i l l .

pbEQig

I

R

= r 'wllh rlM'ls

R e a c t i o n i n t e n s i t y ,

B .

t . u

f t . min.

where :

r

=

~ * m a x ( ~ / ~ o p ) A e ~ p [ ~ ( l -o p ) ] Optimum r e a c t i o n v e l o c

min.

l

'max

= 01.5(495 . 0 5 9 4 0 ~ ' ~ ) - ~

aximum r ea ct io n ve lo c

min . - l

Bop = 3 . 3 4 8 0 - - ' ~ ' ~ Optimum p a c ki n g r a t i o

A

= 1 / ( 4 . 7 7 4 a . l - 7.27)

P.1

f

h l f 2

'1M

=

1 - 2.59

Mf

M

x 5 .11 ( )

-

3.52 ( )

>loi s tu re da-np ing co ef f i c i

I I S = 0.174 Se- . I9 h l ineral damping coef f i c i

€ = (192 0 . 2 5 9 5 a ) - ~ e x ~ [ ( 0 . 7 9 2 0 . 6 8 1 0 . ~ ) B 0 . 1 ) ]

P r op a g at i ng f l u x r a t i o

w = cuB

*)

L

Wind coef f i c i en t

C = 7 .4 7 exp ( - 0 . 1 3 3 0 . ~ ~ )

B =

0.025260.

54

E

= 0.715 exp ( -3 .59

x

1 0 - ~ 0 )

W =

w

Net f u e l l o a d i n g , 1 b . / f t

1

+

S = 5.275 tan $ j 2 S l op e f a c t o r


34/48

Input Parameters for asic Equations

w

ovendry f ue l l oad ing , Ib . / f t .

0

8

f u e l d ep th , f t .

a ,

f u e l p a r t i c l e su rf ac e- ar ea -t o- vo lu me r a t i o , l / f t .

h , f u e l p a r t i c l e low h e at c o n t e nt , 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

M f

fu e l pa r t i c l e m oi s tu re con t en t , l b . m o is ture

lb. ovendry wood

ST, fu e l pa r t i c l e t o t a l m inera l con t en t , l b . m in ,e ra ls

lb. 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 mi ne ra l c o n t e n t , l b . s i l i c a - fi-ee

lb . ovendry woo

U

wind ve l oc i t y a t m id flame he ig h t , f t /min.

t a n 4 , s l o p e , v e r t i c a l r i s e / h o r i z o n t a l d i s t a n c e

Mx

mois ture conten t o f ex t in ct i on . This t erm needs exper im

de t e rmina t i on . We ar e p rese n t l y u s ing 0.30, t h e f i b e r s

poi n t of many dead fu el s . For a er ia l fuel s B


35/48

T H E F I R E S P R E D M ODEL

Rate o f sp rea d and in t en s i ty p re d ic ted by th e model a re based on

and 27)

These equa t ions had t o be modified,however, t o acc ept

f u e l s

composed o f h e te ro gen eou s m i x t ure s o f f u e l t y p es and p a r t i c l e s i ze s .

p i n e n e e dl e l i t t e r , g r a s s , b r u s h , a nd lo gg i ng s l a s h a r e t h e e a s i e s t t

f u e ls - - ac c u mu l a ti o ns o f b r ok en b r a nc h e s, t r e e t o p s , s n a g s , f o l i a g e l i t

o t h e r l e s s e r v eg e t a t i o n a r e more d i f f i c u l t t o model b ecau s e of t h e d i

p a t t e r n s i n which t h ey a re fou nd . For t h e m od el , ho weve r, t h e s e v a r i o


36/48

To a id i n the understanding of fu e l d i s t r ib ut ion . , we in t roduced t h

u n i t f u e l c e l l . u n i t f u e l c e l l i s t h e sma lle st volume of fu el within

dep th t h a t h as 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 se n t at i v e o f t h

en t i re f ue l complex. This concept pe rmi ts the mathematica l rep resen ta t i

d i s t r i b u t i o n t o b e r e fe re n ce d t o a u n i t f u e l c e l l r a t h e r t h an t o t h e e n

Primari ly t h i s concept a ids i n mathemat ica l ly weighting input para

sequent ly , i t i s n o t n ec es s ar y t o s p ec i f y t h e s i z e o f t h e u n i t f u e l c e l

under s tudy ; r a t he r t o p rov ide mean va lue s pe r u n i t fue l c e l l which the n

fuel complex that i s being modeled. Representat ive inpu ts can be pre sel

fue l models t ha t a re t a i l o re d fo r a na lys i s by t h e f i r e sp rea d model.

The model

i s

based on the concept th a t a s in gul a r ch ar ac te r i s t ic p

be

found by proper ly weight ing the var ia t ion s in th e paramete r i n t he h

mixture . To implement t h i s concept , we had t o consi der how each fu el p

model exerts i t s e f f e c t on t h e t h r e e c h a r a c t e r i s t i c f e a t u r e s o f a sp r e a

The energy source; 2 ) t h e e ne rgy s ink ; 3) the f low of

air

o r o f h e a t

The processes tha t

contr ol combustion ra te--evaporat ion of moisture

t ran . s fe r of hea t in to th e fue l , and evolut ion of combust ib le gases by t h

t hr ou gh t h e s u r f ac e o f t h e f u e l p a r t i c l e .

The fue ls having the h ig hes t

volume ra t i o f i ne fu e l s )

w l l

r es po nd t h e f a s t e s t ; t h e r e f o r e , t h e s e

w

i n t h e l ea di n g p o r t io n s of a f i r e . I t i s no r e ve la ti on . t o f i r e f i g h t e r s

s c i e n t i s t s t h a t f i n e f u e l s m ig ht b e e xp ec te d t o r e a c t t h e f a s t e s t . H w

r e a l i s t i c t o a r b i t r a r i l y s t a t e t ha t 90 percent of pa r t ic le s 1 /8 inch in .

under a r e consumed and th a t some f ix ed r a t i o of t h e o the r s i z e c lass es

Weighting by surf ace a rea e l imina tes t he problem of making a rb i t ra ry de

which fue l s i ze s t o inc lude and which not t o inc lude .

Mathematical ire Spread Mode Inputs

Mean va lue s wi th in

it

category and

jth

s iz e c las s of fu e l complex

wo)i j = ove ndry l oa d ing , l b . / f t . 2

j = su r fa c e - a re a- to -volume ra t i o , f t / f t . 3

F )

=

mineral con ten t , lb miner als/ l b wood)

T

l b . m in er al s l b , s i l i c a )

Te) j = e f f e c t i ve minera l c on ten t

l b . woodh) i j

=

low hea t va lue , B. t .u . / lb .

Mf)i

j

= mois ture conte nt , lb mois ture ) / lb .wood)

Fp)i

=

ov end ry p a r t i c l e d e n s i t y , l b . / f t . 3 ) .


37/48

ormulations f ire Sp read M ode

The model i s now formula ted f rom the ba s i c equa t i ons o f f i r e spr

weight ing concept . The ba s i c equa t ions a re ava i la bl e on page 26. The

weighting concept must be developed.

Weigh ting pa r a me te rs ba se d on th e su r f a c e a r e a o f t he f ue l w i th i

and cat eg ory a r e developed from in pu t pa rame ters a s shown on page 29 .

Le t :

-

= mean t o t a l s u r f a c e a re a o f f u e l p e r u n i t f u e l c e l l .

-

i

=

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 te g or y p e r u n i t f

. = mean to t a l su r f a c e a r e a o f f ue l o f j th c la s s a nd

ith

c a te g

I u n i t f u e l c e l l .

The mean t o t a l s u r f a c e a r e a p e r u n i t f u e l c e l l o f e ac h s i z e c l a s

c a te gor y

i s

de termined f rom th e mean loading of t ha t s i z e c l as s and t

volume r a t i o and pa r t i c ke de n s i ty .

-

3

G o )

i

A..

=

i j

The mean to t a l s u r f a c e a r e a o f t he ith c a te gor y pe r un i t f u e l c e

t o t a l s u r f a c e a r e a p er u n i t f u e l c e l l a r e t h en o b t a i n ed by s ummation o

each ca tegory and wi th in th e fu e l ce l l wi th equa t ion s 54) and 55) .

Two weight ing paramete rs a re now c a lc ul a t ed t ha t a r e used throug

of th e model

-

H

i

R a t i o o f s u r f a c e a r e a o f j

t

h

f i j T

s i z e c l a s s t o t o t a l s u r f ac e a re a


38/48

React ion in te ns i t y becomes

:

where t h e ch a r ac t e r i s t i c p a ramet e rs w e ig ht ed by s u r f ace a r e a a r e

:

Net loading of

it

category

(wnIij

=

Net loading of j th cl a ss wi t h in

1 i t h ca t eg o ry

j =n

fii

C

f . . F . . Low hea t con ten t va lue o f

i

h

j O

ca tegory

(Qi = 0. 174(Se)i--19

Mineral damping coeff icient

o f i t h ca t eg o ry

j =n

5 ) . = C

f i j ( S e ) ij C h a r a c t e ri s t i c e f f e c t i v e

e i

j =1 mi n era l co e f f i c i e n t o f

i

h

category

Mois ture damping co ef f i ci en t

of

i t

category

Mo is tu r e r a t i o o f

i

h

ca tegory

j =n

-

(Mf)i

=

C

f .

.

(Mf)ij Moisture con ten t of

j = 1

i t

category

To compl et e t h e ca l c u l a t i o n of t h e r ea c t i o n i n t e n s i t y , t h e p o t en

v e l o c i t y ,

I- ,

must be calculated . 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

fo r the fue l complex .


39/48

where

C h a r a c t e r i s t i c s u rf a c e- a r e

to-vo lume r a t i o of t h e fue

complex

C h a r a c t e r i s t i c s u rf a c e- a r e

to-volume r a t i o of

it

u e

category

Mean packing r a t i o

Mean bulk density

Th i s com plet es t h e com pu ta t ions neces sa ry fo r ca l cu l a t i ng r e ac t

The parameters wi th in the ba s i c ra te of spread equat ion

a r e t r e a t e d s i m i l a r l y .

The no-wind propaga t ing f lu x r a t i o , 5 ,

s

a fu nct io n of th e mea

and ch a r ac t e r i s t i c su rface-a rea - to -volume ra t i o .

In t h e hea t s i nk t e rm s , t h e bu lk dens i t y i s dependent upon bulk


40/48

The model

i s

c om pl et ed b y i n c l u s i o n o f t h e w ind a n d s l o p e m u l t i p l

f a c t o r s

and

where

U = mean w i nd s pe e d a t m id fl am e h e i g h t , ( f t . / m i n . )

I f 17 5, t h e w ei gh te d f u e l s i z e

i s

t o o l a r g e f o r t h e wind f a c t o r .

f u e l s i z e i n c r e a s e s . ) We h av e n o t fou nd t h i s l i m i t a t i o n t o b e r e s t r i c

t h e f u e l models t e s t e d t o d a t e . The r e a s o n, o f c o u r s e , i s t h a t w i l dl a

composed p r i m a r i l y o f f i n e f u e l s w i t h c o n s e q ue n t l a r g e v a l u e s o f

a

An upp er

l i m i t

i s p l a c e d on t h e w in d m u l t i p l i c a t i o n f a c t o r . R ot h

Ander son (1966) found th a t t h e ang l e o f f l ame

t i l t

c o ul d b e c o r r e l a t e d

t h e e n e rg y of t h e wind a nd t h e e n e r g y o f t h e f i r e :

q

u

I R ~

where

q = f r e e s t r e a m dy na mi c p r e s s u r e l b . / f t .

J = 778 f t . l b ./ B . t . u .

-

m e ch a ni c al e q u i v a l e n t o f h e a t

E v al u at i ng t h i s r a t i o a t t h e l i m i t i n g v al ue o f s p r e a d r a t e f ou nd by

Mc

( f i g . 20) g i v e s :


41/48

FFLIC TION TO

TM

FIELD

The mathematipal mpdel has ap pl ic at io n to management problems i n

1 The h ~ o t h e t i c a l i r e s i t u a t i o n i n which o p e r a t i on s ' r e se a r c

u t i l i z e d f o r E ir e p l an n in g , f i r e t r a i n i n g , and f u e l a p p r a i s a l.

2

The poss ib le f i re s i tu a t io n fo r which advance p lann ing

i s

ne

f i r e -d an g e r r a t i n g and p re s u p p re s s i o n p l an n i ng . P red i c t i o n s o f p o t en

fo r pos s ib le management p ra c t i c es ( i . e . methods o f th in n in g , s l a sh t r

p r e s c r i b ed b u rn i n g) h av e t h e p o t e n t i a l f o r be i ng t h e most v a l u ab l e co

f i r e model ing can perform.

A t h i r d s j t u a t i o n - - f o r e c a s ti n g t h e b e ha v io r o f e x i s t i n g w i l d f i r e s

a g r ea t e r degree o f so ph i s t i c a t io n than t h i s model and our knowledge o

m i t a t t h e p r e s en t t i me . V a r i a t i on s i n f u e l and w ea th e r cau s e d ep a r t u

s p re a d and i n t e n s i t y t h a t p o se r i s k s u n ac c ep ta bl e i n f i r e s up p re s si on

method f o r fo r ecas t i n g t h e b ehav io r o f a s p ec i f i c f i r e ev en t u a ll y

w i l

most

l i k e l y ,

t w i l l

b e p a t t e r n e d on a p r o b a b i l i t y b a s i s s i m i l a r t o t h

cas t in g weather . To accompl ish th i s , a t echn ique must be developed fo

f u e l i n v e n t o r i e s on t h e t h r e a t e ne d s i t e .

Choos ing inpu t parameters f o r th e model f rom th e i n f i n i t e var i e t y

enviro nment al arran geme nts and combinations seems almost overwhelming

i n t h e g rowth of v e ge t a ti o n e x i s t t h a t can b e u t i l i z e d t o g r e a t l y s i m

p ro ces s . I t a l s o p ro ves h e l p fu l t o g roup t h e i n p u t s i n t h e fo l l o w i n g

1

F uel P a r t i c l e P r o p e r t i e s

Heat Content

Mineral Content

Pa r t i c l e D en s i t y


42/48

The fu e l p a r t i c l e p ro p e r t i e s a r e n o t exp ec ted t o v ary g r ea t l y w i t h

ty pe s . Such values can be re ad i l y determined in th e labor atory and ass

n e r t h a t s h o u l d h av e wide ap p l i c ab i l i t y .

Fue l a r r ay a r rangement pa t t e r ns must be dete rmined i n th e f i e ld .

t a s k s w l l be more d i f f i c u l t t han measur ing fu e l pa r t i c l e p rop er t i es .

e x p e c t e d t h a t p a t t e r n s

w l l

b e fou nd t h a t a r e r ep ea t ab l e w i t h in t h e

l m

ca l cu l a t i n g p o t e n t i a l f i r e h azard u s i n g t h e model. The fu e l t y p e , ag e

s 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 u se d a s in de xe s

fu e l a r rang emen t p a t t e r n s . B ro ad er c l a s s i f i c a t i o n by eco ty p e o r h a b i t a

v a l u abl e f o r s o r t i n g o ut f u e l p a r ame t e r s .

The en vi ro nmen ta l r e l a t ed p a rame te r s can b e i n s e r t e d t o i n v es t i g a t

th e range o f wind, mois t u re , o r s lop e t ha t might be expec ted t o be impo

being modeled.

Fuel Models and pplication

On-the-spot sampl ing of a l l inpu t parameters i s co st ly , t ime consw

Ca t al o gi n g fu e l p ro p e r t i e s and r e l a t i n g them t o o b se rvab l e s i t e ch a r ac

e l i mi n a t e t h e fu e l s amp li ng p ro ces s , b u t

t w l l

permit a wide appl i cat

r e s u l t s . T hese r e s u l t s can b e fu r t h e r r e f i n ed fo r u s e i n t h e ma th ema t

assembling them into

fuel models

t h a t r e p re s e nt t y p i c a l f i e l d s i t u a t i o n

models con ta in a comple te s e t o f inp u t s f o r the mathematical f i r e sp rea

Land managers can be t r ai ne d t o choose the f u e l model, th at i s mos

t o t h e f u e l s and c l im a t e f o r t h e i r a r e as o f i n t e r e s t . I f fu r t h e r r e f i n

i n t e r n a l p r o p e r t i e s e . g . , f u e l l oa di ng i n l o gg in g s l a s h , t h e r a t i o of

fue l i n brush , and t h e amount and typ e of unders t ory i n t imber ) of each

b e t a i l o r ed t o p ermi t t h e model t o more

c l o s e l y match s p ec i f i c fu e l s .

Work has al re ad y begun on f u e l models f o r t h e Na ti on al Fire-Danger

Eleven f u e l models

t ab l e 1) h av e b een a ss embl ed t h a t r ep re s en t a l a rg e

fo re s t s , b rush f i e ld s , and g rass l ands found i n th e t empera te c l imates o

The var i a t ion s in sp read and in te ns i t y be tween f ue l types

as pr ed i

may re ad il y be seen from r e s u l t s o bta ine d from computations with t h e 11

i n p u t s .

To a s s i s t i n u n de rs ta nd in g t h e s e n s i t i v i t y o f t h e i n p u t s , t h e

p ro per t i e s have been he ld cons tan t and th e var i a t i ons among fue l types

t o t h e l o ad i n g by s i z e c l a s s and f u e l d ep th s shown i n t a b l e I . The f

l iv ing fuel

category

i s

a l l t h a t

i s

assumed t o e n t e r i n t o t h e r e a c t i o n .

r e a so n a bl e v a l u e s of r e a c t i o n i n t e n s i t y f o r f u e l models t h a t c o nt a in l i

mo i st u re o f e x t i n c t i o n v a l u e f o r t h e l i v i n g fu e l must b e ad j u s t ed t o a

th a t used fo r th e dead fu e l s . Very l i t t l e research has been done on th

l i v i ng fue l s . Ph i lpo t and Mutch 1971) sugges t tha t c rowning po te n t i a l

pin e and Dougla s-fir f o r e s t s of Montana may be dependent upon t h e hi gh e


43/48

;t:

O L n O O Ln N

O N O M

N N d O d N

P.

1 I Ln Ln Ln

I

L n l I I

I I I

I I I

,+

I 1

I

I I M M M M M M M


44/48

Figure

23

- -Poten t ia l reac t ion

ve Zocity o f ty pi ca wildland

fue 2s. The two Zines re pre -

sen t the ex treme va lues o f

-

a,

one fo r sho rt grass the

oth er fo r heavy logging

s la sh .

Closed timbe

Black

3 Pocking ra

The e f f e c t o f t h e c ha ng e i n f u e l a rr an ge me nt on p o t e n t i a l r e a c t i o n

shown i n f i g u r e 23 . The f l a s h y fu e l s (g ra s s and b ru s h) h ave t h e h i g h es

c l o s e d t im be r l i t t e r h a s t h e l o we st . Note t h a t t h e g r a s s and b ru sh l i e

t h e optimum packing r a t i o under no-wind co ndi t i ons . Inasmuch

as

one o f

wind

i s

t o s h i f t t h e optimum p ac ki ng r a t i o t o t h e l e f t , t h e s e f u e l s w i l

we1 under windy co ndi t i ons .

The p r e d i c t i o n o f r e a c t i o n i n t e n s i t y f o r t h e f u e l mode ls

i s

sho

f o r f u e l m o i s tu r e r a ng i ng f r om 0 t o e x t i n c t i o n .

l l

f u e l models e x t i n g

which i s t h e v a l u e s e t by x f o r t h e d ea d f u e l . The h i g h e r o r d e r v a r i a

fu e l m odels a r e cau s ed by t h e l i v i n g f u e l com po nent s i n ab i l i t y t o b u rn

fu e l mois tu r e becomes h igh . Th is

i s

a t t r i b u t a b l e t o F os be rg a nd S c h r o

Legend


45/48

Figure 24. --Reac tion

i n t e n s i t y of t y p i c a l

w i dland fue 2s

computed with

heterogeneous

formulations for

the model from

data i n t ab l e 1.

..

.. .. S

0 0 S

-.--

S

0 0

H

. B

m

bead fuel moisture content (percent)

Heavy logging s l as h has by f a r the h ighes t r eac t ion i n t en s i ty bu

of sp read ; chapar r a l has bo th a h igh reac t ion in te ns i t y and a h igh ra

i s gr at i f y i ng th at the model pred ict ion s a re h igh i n both values beca

d esi g ned t o r ep r e s en t t h e b ru sh f i e l d s o f t h e So ut hw es t.

These brush

sev er e f i r e haz ard Countryman, Fosberg, Rothermel, and Schroed er 196


46/48

F i gu r e 2 6. E f f e c t o f m o i s t u r e

and m i nera l s

on

r a t e o f s p r e a d

i n a h y p o t h e t i c a l f u el m od el .

Effective min

0 0 5 10 15

Fuel moisture

The pos i t i ons o f th e cu r ve s i n both f igu r e s

4

and

25

would b e r e f

p a r t i c l e p r o p e r t i e s h , p Se ,

hlx

o f t h e a c t u a l f u e l s w e re s u b s t i t u t e

va lues th a t we re use d i n P i e f ue l m odel s.

To i l l u s t r a t e t h e r e l a t i v e e f f e c t o f m in e r a ls and m o i s tu r e on r a t e

t y p i ca l homogenous fu el w s chosen and ra t e of sp read versus mois ture

w

26) . No a t te m pt o the r tha n t o d i sc oun t s i l i c a w s made t o d is t i ng ui sh

e f f e c t s o f d i f f e r e n t m in e ra l s.

F igu r e 27 i l l u s t r a t e s th e use f u lne ss o f t he m odel t o a ppr a i se f u e l

de c i s io ns . Th i s f ig u r e shows th e change i n sp r e a d and in t e n s i ty th a t c

i n lo gg in g s l a s h i f t were burned under no-wind and 10 pe rce nt moist ur

v a ri o us s t a g e s i n d e c om po si ti on . The a b i l i t y o f t h e model t o p r e d i c t f

r e f le c t ed i n f i g u r e

7

shou ld o f f e r new op por tun i t i e s f o r r e sour c e ma n

f ue l m anagement i n t o r e sour c e p la nn ing a c t iv i t i e s .


47/48

LITER TURE

CITED

Anderson, H. E .

1968. Fi re spread and f lame shape. Fi re Tw hn ol . 4 1) :51-58.

Anderson,

H.

E .

1969. Heat t r a n s f e r and f i r e sp rea d. USDA Fo res t Ser v. Res.

i l l u s .

Ber lad ,

A.

L.

1970. F i re spre ad i n s o l i d fu e l arr ay s. Combust . and Flame 1

Brown,

J K .

1972.

F i e ld t e s t of a ra te -of - f i re - spread model i n s l as h fu e l

Serv. Res. Pap. INT-116, 24 p . , i l l u s .

Countryman,

C .

M . M.

A.

Fosberg,

R. C .

Rothermel, and M .

J

Schro

1968. F i r e we ather and f i r e be ha v io r i n the 1966 loop f i r e .

126- 141, l u s

Deeming,

J

E .

J W .

La nc a s te r , M. A . Fosberg, R.

W .

Furman, and M

1972. The Na ti on al F i r e Danger Rocky Mountain Ra ti ng System

Se rv . Res. Pap. RM-84, i l l u s .

Fons,

W .

1946. A na ly si s of f i r e s p re a d i n l i g h t f o r e s t f u e l s .

J

Agr.

i l l u s .

Fosberg, Michael A. and Mark

J

Schroeder

1971. Fine herbaceous fu el s i n f i re- da ng er ra t i ng . USDA For

RM-185, 7 p .

Frandsen

W H

1971. Fi r e spr ead through porous fue ls f rom th e con serv atio n

and Flame 16:9-16, i l l u s .

McArthur, A. G .

1969. The Tasmanian bu sh fi re s of 7th February, 1967, and as so

i o u r ch ar ac te r i st ic s . In The Technic al Co-operation P

Symposium C an be rr a, ~ u x r a l i a 969) v . I . 23 p . Mar iby

Defence Standards Laborator ies.

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1968. Minera l conten t and py ro ly s i s of se le c t ed p l an t mat e r i

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a mathematical model for predicting fire spread in wildland fuels

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