Mathematical Model
for Fire Spread
Prof. Nita H. Shah
Department of Mathematics
Gujarat University
Ahmedabad
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What is Fire?
• Heat is supplied from the fire to the
potential fuel, the surface is dehydrated
and further heating raises the surface
temperature until the fuel begins to
pyrolyze and release combustible
gases.
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When the gas evolution rate from the
potential fuel is sufficient to support
combustion, the gas is ignited by the
flame and the fire advances to a new
position.
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• Gradually, a constant rate of spread is
attained which is called “quasi-steady
state” wherein the fire advances at a
rate that is the average of all the
elemental rates.
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A mathematical model to determine rate
of fire spread and its intensity is
formulated. The model is developed by
R.C. Rothermal (1972) and it is
considered as the basis in the National
Fire Danger Rating System.
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Objective
The model considers physical and
chemical properties of the fuel and the
environmental conditions in which it is
expected to tingle.
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• The physical properties incorporated are
1. fuel loading
2. fuel depth
3. fuel particle surface area to volume ratio
4. fuel particle moisture, and
5. the moisture content at which extinction is
expected.
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• Environmental inputs are
1. mean wind velocity
2. slope of terrain.
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Mathematical Model
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quasi-steady rate of spread, ft/min
Horizontal heat flux absorbed by a unit volume of the fuel
at the time of ignition B.t.u./ft2 min
Effective bulk density (the amount of fuel per unit volume of
the fuel bed raised to ignition ahead of the advancing fire),
lb/ft3
Heat of pre-ignition (the heat required to bring a unit weight
of fuel to ignition), B.t.u./lb
The gradient of the vertical intensity evaluated at a plane at
a constant depth, 𝑧𝑐, of the fuel bed, B.t.u./ft3 min
Horizontal coordinate
𝑧 Vertical coordinate
R
XigI
be
igQ
c
z
z
I
z
x
• Using conservation of energy principle
to a unit volume of fuel ahead of an
advancing fire in a homogeneous fuel
bed (by Fransen (1971))
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0heat flux received
from the source(1)
heat required for
ignition by the potential fuel
c
zXig
z
be ig
II dx
zR
Q
• The fuel-reaction zone interface is fixed
and the unit volume is moving at a
constant depth, zc, from x = - infinity
towards the interface at x = 0. The unit
volume ignites at the interface.
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Analysis
• The heat required for ignition depends upon
(a)Ignition temperature
(b) moisture content of the fuel, and
(c)Amount of fuel involved in the ignition
process.
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• The energy per unit mass required for
ignition is the heat of pre-ignition, .
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igQ
( , ), . . . / (2)
where
: ratio of fuel moisture to ovendry weight
: ignition temp.
ig f ig
f
ig
Q f M T B t u lb
M
T
• The amount of fuel involved in the
ignition process is the effective bulk
density, .
• An effective heating number is ratio of
the effective bulk density to the actual
bulk density.
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be
(3)
be
b
• The effective heating number is
dimensionless which will be nearly
unity for fine fuels and decrease to zero
as fuel size increases. Therefore,
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bulk density, fuel size (4) be f
• Propagating Flux
It is given by
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0
gradient of the vertical flux integrated from
-infinity to the fire fron
horizontal
flu
5
x
t
c
z
p Xig
z
II I dx ( )
z
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• The figures indicate that the vertical
flux is more significant during wind-
driven and upslope fires because the
flame tilts over the potential fuel,
thereby, increasing radiation, causing
direct flame contact and convective
heat transfer to the potential fuel.
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• Assume that the vertical flux is small
for no-wind fires and let . This is
the basic heat flux component to which
all additional effects of wind and slope
are related.
• With (3) – (5) in (1) and and R=R0
for the no-wind case, we get
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0
p pI I
0
p pI I
00
(6) p b igI R Q
• The energy release rate of the fire front
is produced by burning gases released
from the organic matters in the fuels.
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Reaction Intensity
• The rate of change of this organic
matter from a solid to a gas is a good
approximation of the subsequent heat
release rate of the fire.
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• The heat release rate per unit area of
the front is called the reaction intensity
and is defined as
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Heat content of fmass loss
rate per unit area
(7
in
)
the fire front
l ue
R
dwI h
dt
• The reaction intensity is a function of
fuel parameters such as particle size,
bulk density, moisture, and chemical
composition.
• The reaction intensity is the source of
the no-wind propagating flux
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0
(8)p RI f I
• Wind and slope change the
propagating heat flux by exposing the
potential fuel to additional convective
and radiant heat.
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Effect of Wind & Slope
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• Let and 𝜙𝑠 represent the additional
propagating flux produced by wind and
slope. The total propagating flux is
• Approximate rate of spread (eq.(1)) becomes
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w
0
1 (9) p p w sI I
0
1(10)
p w s
b ig
IR
Q
Specific heat of dry wood
Temperature range to ignition
Fuel moisture lb. water / lb. dry wood
Specific heat of water
Temperature range to boiling
Latent heat of vaporization
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Heat of Pre-ignition
pdC
igT
fM
pwC
bT
V
• for cellulosic fuels is determined by
considering the change in specific heat
from ambient to ignition temperature
and the latent heat of vaporization of
the moisture as
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igQ
(11)ig pd ig f pw bQ C T M C T V
• Taking temperature to ignition in the
range of 200c to 3200c and boiling
temperature to be @ 1000c, eq. (11)
becomes
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250 1116 (12) ig fQ M ,B.t.u. / lb
• To determine the effective bulk density,
we need to compute the efficiency of
heating as a function of particle size.
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Effective Bulk Density
• An exponential fit is given by
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1
138(13)
where is particle surface area to volume ratio, ft
exp
• Rearrange eq.(7) as
• To solve (16), integrate over the
reaction zone depth D, and over the
limits of loading in the reaction zone.
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(15)
where the quassy-steady rate of spread
(16)
R
R
dw dxI h
dx dt
dxR,
dt
I dx Rhdw
x
w
where
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0
( ) (18) r
n
wD
R R n r
w
I dx Rh dw I D Rh w w
D Reaction zone depth (front to rear)
Net initial fuel loading, lb/ft2
𝑤𝑟 Residue loading immediately after passage of the
reaction zone, lb/ft2
nw
• The time taken for the fire front to travel
a distance equivalent to the depth of
one reaction zone is the reaction time,
Then eq. (18) becomes
(20)
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(19)R
D
R
n r
R
R
h w w I =
• Next, we define a maximum reaction
intensity where there is no loading
residue left after the reaction zone is
passed and where the reaction time
remains unchanged.
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max (21) nR
R
hwI
• The reaction zone efficiency is
In (23), the net fuel loading is given by
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max
(22)
(23)
n rR
R n
nR
R
w wI
I w
w hI
0
2
0
(24)1
where
ovendry fuel loading, lb/ft
fuel mineral content, lb minerals/lb dry fuel
n
T
T
ww
S
w
S
• The reaction velocity denotes the
completeness and rate of fuel
consumption. It is defined as the ratio
of the reaction zone efficiency to the
reaction time.
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Reaction Velocity
*reaction velocity/min. (25)R
1. Moisture content
2. Mineral content
3. Particle size, and
4. Fuel bed bulk density
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Effective Fuel Parameters for
Reaction Velocity
Then
One can evaluate reaction velocity and the moisture
and mineral damping coeffiecients by experiments.
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Potential reaction velocity/min
Moisture damping coefficient having values
ranging from 1 to 0
Mineral damping coefficient having value
ranging from 1 to 0
M
s
(26)
(27)
then the reaction intensity is
M s
R n M sI w h
• It is defined as
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Moisture Damping coefficient
max 0
(28)
f
RM
R M
I
I
In adjacent fig., is fuel
moisture and 𝑴𝒙 is the
moisture content of the
fuel at which the fire will
not spread.
Moisture damping
coefficient accounts for
the decrease in intensity
caused by the
combination of fuel that
initially contained
moisture.
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fM
• It was evaluated from thermo gravimetric
analysis (TGA) data of natural fuels by Philot
(1968). It is assumed that the ration of the
normalized decomposition rate will be same
as the normalized reaction intensity.
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Mineral Damping coefficient
where 𝑺𝒆 is effective
mineral content.
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0.190.174 (30)s eS
• Next, we need to consider two
parameters:
1. The reaction intensity – fuel bed
compactness, and
2. Fuel particle size
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Physical Fuel Parameters
Assume that low values of fire intensity
and rate of spread occur at the two
extremes of compactness (loose and
dense).
In dense beds, this can be attributed to
low air-to-fuel ratio and to poor
penetration of the heat beyond the upper
surface of the fuel array. April 2, 2015 IWM2015 Delhi Uni. 45
• In loose beds, low intensity and poor
spread are attributed to heat transfer
looses between particles and to lack of
fuels.
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• Between these two extremes, there
must be an optimum best equilibrium
of air, fuel and heat transfer for both
maximum fire intensity and reaction
velocity.
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• The compactness of the fuel bed is
quantified by the packing ration, which
is the ratio of the fuel array to fuel
particle density.
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3
3
packing ratio, (31)
where : fuel array bulk density, lb/ft
: fuel particle density, lb/ft
b
p
b
p
• The surface area-to-volume ratio, is
used to quantify the particle size.
• For fuels that are long w.r.t. the
thickness
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4(32)
here denotes diameter of circular particles
or edge length of square particles.
d
d
The reaction time,
is defined on the
derivative curve as the
time from initial mass
loss until the loss
stabilizes at a steady
rate.
During the reaction
time mass loss rate is
linear. Also the
duration of the
constant mass rate is
dependent on the
length of platform. April 2, 2015 IWM2015 Delhi Uni. 50
The reaction time
could be thought of as
the fire burned off.
R
The mass loss rate is
The efficiency of fire is
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(33)
W : width of the platform.
n r
dmw w RW
dt
1(34)
n
dm
w RW dt
• With efficiency and the reaction time,
the reaction velocity is
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1(35)
n R
dm
w RW dt
The potential reaction
velocity to disassociate
the reaction velocity
from the effects of the
moisture and minerals
of the fuel is .
The reaction velocity must
drop to zero if there is no
fuel to support combustion.
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M s
Tightly packed fine fuels
have lower reaction
velocity than do longer
fuels at the same packing
ratio.
The loss of reaction
intensity of the fine
fuel can be seen as
the difference in
flaming vigor.
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To determine the
general equation as a
function of and 𝜎 is
given by
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1.5
1.5
0.8189
(36)495 0.0594
3.348 (37)op
Combining these two eqs. gives
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max
0.1
exp 1 (38)
1with (39)
4.77 7.27
A
op op
A
A
These will predict reaction velocity for
any combination of fuel particle size and
any packing ratio. The eqs. will predict
reasonable values when input
parameters are extrapolated. This will
help us to predict reaction intensity and,
subsequently, rate of spread over a wide
range of fuel and environmental
combinations. April 2, 2015 IWM2015 Delhi Uni. 57
The no-wind propagating flux is
A ratio 𝝃 relating the propagating flux to
the reaction intensity is
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Propagating Flux
00( we know from (6))p b igI R Q
(41)
p o
R
I
I
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A respective fit is
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1exp (0.792 0.681 )( 0.1) (42)
192 0.259
It is seen that
increases with
increase in , but at a
decreasing rate.
will attain a
maximum and then
decreases.
This is reasonable,
considering the fact
that the fuel array is
becoming so compact
that the intensity has
decreased.
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0pI
0pI
Combining the heat source and heat sink
terms results into the no-wind rate of
spread equation as
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Rate of Spread
0 (43)R
b ig
IR
Q
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Fire Flames
Reference
1. Richard C. Rothermal (1972):
Intermountain Forest and Range
experiment station Forest service, Res.
Paper 115.
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