Enclosure Fire Dynamics
Enclosure Fire Dynamics:
A collection of equations,
tables and figures
Pool fire .............................................................................................................................2
Fire Plumes and Flame Heights .........................................................................................2
The Zukoski Plume ....................................................................................................2
The Heskestad Plume ................................................................................................2
The Thomas Plume ....................................................................................................2
Line Source Plume.....................................................................................................2
Ceiling Jet ..................................................................................................................3
Flame Extensions Under Ceilings .............................................................................3
Pressure Profiles and Vent Flows, Well-Ventilated Enclosures ........................................3
The Well-Mixed Case................................................................................................3
Taking into Account the Mass Produced in the Room ..............................................4
The Stratified Case ....................................................................................................4
Mass Flow out through a Ceiling Vent ......................................................................4
Gas Temperatures in Ventilated Enclosure Fires ..............................................................4
Predicting time to Flashover ......................................................................................4
The Energy and Mass Balance...................................................................................5
Conservation Equations and Smoke Filling ......................................................................5
The Conservation of Energy ......................................................................................5
Smoke filling of an Enclosure with Leaks .................................................................5
Estimating Gas Temperature for the Floor Leak Case ..............................................6
Smoke Control in Large Places .................................................................................6
Natural Ventilation from Upper Layer ......................................................................6
Lower layer Pressurization by Mechanical Ventilation .............................................7
Combustion Products.........................................................................................................7
BK,061015 / HA, 071023
-1-
Equations, tables and figures
Pool fire
 = A f m"
Energy release rate: Q
  H c

-kD
Mass loss rate: m
   m
   1  e
(3.5)

(3.6)
Fire Plumes and Flame Heights
 2 / 5  1.02D
Flame height: L  0.235Q
Continuity equation:
(4.3)
d 2
(b u ) 2 u b
dz

(4.10)
.

d 2 2
Qg
b u 
Momentum-boyancy equation:
dz
 u c p T  
(4.13)
The Zukoski Plume
 2 g  . 1 / 3
5 /3
Mass flow equation: m p  0.21   Q z
c
T
 p  
1/ 3
.
(4.21)
The Heskestad Plume
 2 / 5  1.02D
Virtual origin: z 0  0.083Q
(4.23)
Plume radius: b 0.12 T0 / T 
(4.24)
1/ 2
z  z0 
 T  . 2 / 3
5 / 3
Centerline temperature: T0  9.1 2 2  Q c z  z 0 
gc p  
1/ 3
 g 
Centerline velocity: u 0  3.4

 c p T   
(4.25)
1/ 3
 1 / 3 z  z 1 / 3
Q
c
0
(4.26)
Plume mass flow rates:
For z > L:
m p 0.071Q c z  z 0 
For z < L:
m p  0.0056Qc
.
.
.
1/ 3
.
5/3
z
L
.
 1.85 10 3  Q c
(4.27)
(4.28)
The Thomas Plume
 p  0.188  P  z 3 / 2
Mass flow rate: m
(4.31)
Line Source Plume

Q
Flame height: L  0.035  
B
2/3
(4.34)
-2-
Enclosure Fire Dynamics
 2/3
Q
 p  0.21    z
Mass flow rate: m
B
(4.35)
Ceiling Jet
For r/H < 0.18:
Tmax  T 
 2/3
16.9  Q
H5/ 3
 
Tmax
 r
5.38  Q
 T 
H
For r/H < 0.15:
u max
 1 / 3
Q
 0.96   
H
For r/H > 0.15:
u max 
For r/H > 0.18:
(4.36)
2/3
(4.37)
(4.38)
 1 / 3  H1 / 2
0.195  Q
r5/ 6
(4.39)
Flame Extensions Under Ceilings
0.96
r
L  H
Radial flame extension: f  0.5
D
D 
(4.40)
Pressure Profiles and Vent Flows, Well-Ventilated Enclosures
The Bernoulli equation: P1 
T
1 2
1
v1 1  h1 1 g  P2  v22 2  h 2 2 g
2
2
353
353
and  

T
(5.2)
(5.9)
 g  Cd A u g
Mass flow through thin top vent: m
 a  Cd A l a
Mass flow through thin lower vent: m
2h u  a   g g
g
2h l  a   g g
a
(5.11)
(5.12)
The Well-Mixed Case
  Cd  vdA
Non-constant velocity: m
A
g 
Mass flow rate through upper vent: m
2 a   g g 3 / 2
2
C d W g
hu
3
g
(5.18)
a 
Mass flow rate through lower vent: m
2 a   g g 3 / 2
2
C d W a
hl
3
a
(5.19)
 a  0.5 A H o
Mass flow into tall opening: m
-3-
(5.24)
Equations, tables and figures
Taking into Account the Mass Produced in the Room
a 
Mass flow rate of ambient air: m
The Stratified Case
g 
Mass flow rate out: m
a 
Mass flow rate in: m
2.1  A H o
1  1.6  1  m
b
 a 2 / 3
m

3/ 2
2 a   g g
2
H o  H N 3 / 2
C d W g
3
g
(5.29)
(5.31)
2 a   g g
2
H N  H D 1 / 2  H N  1 H D  (5.36)
Cd  W  a 
3
a
2


Mass Flow out through a Ceiling Vent
c 
Mass flow: m
C d A c  a 2gH  H D Tg  Ta Ta

Tg Tg  A c2 Ta A l2
(5.44)

Gas Temperatures in Ventilated Enclosure Fires
1/ 3
.2




Q
Temperature difference: T 6.85

 Ao Ho h k AT 


(6.11)

4
2
Thermal penetration time: t p 
(6.14)
kc
t
For t < tp
hk 
For t  tp
hk 
k

For t < tp
hk 
A W ,C
kcW,C
AT
t
For t  tp
hk 
AW, C k W,C
For composite layers: h k 
(6.15)
(6.16)
AT  W,C


AF
AT
kcF
t
AF k F
AT  F
(6.17)
(6.18)
1
(6.19)
n
1

i 1 h k ,i
Predicting time to Flashover
.

Necessary energy release rate: Q FO 610 h K A T A o H o
-4-

1/ 2
(6.20)
Enclosure Fire Dynamics

 Q
0.63
Mechanically ventilated area:
m
Ta
  c p Ta
Tg




0.72
 h K AT

 m
  cp




0.36
(6.21)
The Energy and Mass Balance
 q q q q
Total energy balance: Q
L
W
R
B
(6.26)
 0.09A H H
Maximum energy release rate: Q
o
o
eff , wood
(6.27)
 g c d (Tg Ta ) 0.5A o H o c p (Tg Ta )
Hot gases leaving: q L m
(6.28)
Through walls: q W  A t  A o 
1


kc
 Tg  Ta 
t
(6.29)

(6.30)
Radiation through opening: q R Ao f  Tg4 Ta4

Conservation Equations and Smoke Filling
Gauss’ formula:
d
dt
  dV    v
n
CV
dS  0
(8.3)
CS
Universal gas law: P =  R T
(8.12)
The Conservation of Energy
d
 W

With work:
  u  dV     u  v n  dS  Q
dt 
CV
CS
(8.14)
With enthalpy:
d

udV   h v n dS  Q
dt 
CV
CS
(8.21)
.
P - Pa
Qt
=
Sealed pressure rise:
Pa
V a c v Ta
.
.
c V dP
Leaky pressure rise: v
+ m e c p Te  Q
R
dt
.
.
When leakage is stabilized: m e c p Te  Q
(8.24)
(8.30)
(8.31)
.

1 
Q
Leaky pressure difference: P 


2e c p Te Ae Cd 
2
Smoke filling of an Enclosure with Leaks
z
Dimensionless height: y =
H
-5-
(8.32)
(8.36)
Equations, tables and figures
.
Q
*=
Dimensionless heat release rate: Q
(8.37)
 a c p Ta g H 5 / 2
g H2
Dimensionless time:  = t
H S
Dimensionless equation:
(8.38)
dy  *
 * )1/3 y 5/3 = 0
 Q + 0.21 (Q
d
 
 2  0.21  *
Q
Solution for y: y  1 
3

1/ 3



(8.39)
3 / 2
(8.41)
Estimating Gas Temperature for the Floor Leak Case
.
Energy room size relation: Qt = SH(1 - y)c p Ta a g 
(8.45)
 *  (1 - y) 1 - g 
Dimensionless equation: Q
  
a 

(8.46)
Smoke Control in Large Places
0.21   a2 g
Control constant: k =
 g  c p Ta
1/ 3




(8.50)
3 / 2
 1 / 3 2t 1 n / 3 
1 
 2 / 3 
Interface layer height: z  k
 S
n3
H 
(8.52)


 t n 1

Upper layer density:  g   a 1 
 n  1H  z Sc 353 
p


(8.55)
Heat transfer: h 
kc
t
(8.61)
Gas temperature: Tg  Ta 

Q
 e  hA w
cp m
(8.62)
Natural Ventilation from Upper Layer
Pressure difference: Pl =
2
m
2
2 a C d A D 
(8.64)
 e = Cd A E
Upper opening mass flow rate: m
-6-


2  g - Pl +  a -  g  g H E - z 
(8.67)
Enclosure Fire Dynamics
Lower layer Pressurization by Mechanical Ventilation
Pressure difference: Pl =
m
0
 p 2
m
(8.71)
2 a C d A D 
2
 e = Cd A E
Upper opening mass flow rate: m


2  g Pl +  a -  g  g H E - z 
(8.72)
Combustion Products
Yield of species: y i 
mi
mf
(9.1)
Mass fraction of species: i =
Equivalence ratio:  =
mi
m
(9.2)
m f m ox
r
Fuel mixture fraction: f =
(9.5)
f
m

m
Fuel mixture fraction based on equivalence ratio: f =
Control volume formulation: m

+
1
r
N
dYi
 j  Yi , j - Yi  = y i m
 f -m
 i, loss
+ m
dt j=1
(9.8)
(9.20)
net out
Steady state mass fraction: Yi ( t )  y i
f
m
 yi  f
 f m
p
m
-7-
(9.25)
Equations, tables and figures
Q (MW)
20
4.8 m
15
3.0 m
10
2.4 m
5
1.7 m
0.9 m
0
0
300
600
900
Time (s)
Figure 3.1 Energy release rate measured when burning 1.2 m by 1.2 m wood pallets,
stacked to different heights.
Q (kW)
4000
3000
2000
1000
0
0
300
600
900
Time (s)
Figure 3.6 Typical Energy Release Rate from a wood pallet stack.
Figure 3.7 Dependence of pallet stack height on peak energy release rate (from
Babrauskas [3.1]).
-8-
Enclosure Fire Dynamics
Table 3.2 Burning rate per unit area and complete heat of combustion for various materials (from Tewarson [3.4]).
m
 " (kg/m2s)
Hc (MJ/kg)
Polyethylene
0.026
43.6
Polypropylene
0.024
43.4
Heavy fuel oil (2.6 - 23 m)
0.036
Kerosene (30 - 80 m)
0.065
Crude oil (6.5 - 31 m)
0.056
n-Dodecane (0.94 m)
0.036
Gasoline (1.5 - 223 m)
0.062
JP-4 (1 - 5.3 m)
0.067
JP-5 (0.6 - 1.7 m)
0.055
n-Heptane (1.2 - 10 m)
0.075
44.6
n-Hexane (0.75 - 10 m)
0.077
44.8
Material (values in brackets indicate pool diameters tested)
Aliphatic Carbon-Hydrogen Atoms:
Transformer fluids (2.37 m)
44.1
44.2
0.025 - 0.030
Aromatic Carbon-Hydrogen Atoms:
Polystyrene (0.93 m)
0.034
39.2
Xylene (1.22 m)
0.067
39.4
Benzene (0.75 - 6.0 m)
0.081
40.1
Polyoxymethylene
0.016
15.4
Polymethylmethacrylate, PMMA (2.37 m)
0.030
25.2
Methanol (1.2 - 2.4 m)
0.025
20
Acetone (1.52 m)
0.038
29.7
Flexible polyurethane foams
0.021 - 0.027
23.2 - 27.2
Rigid polyurethane foams
0.022 - 0.025
25.0 - 28.0
Polyvinylchloride
0.016
16.4
Tefzel (ETFE)
0.014
12.6
Teflon (FEP)
0.007
4.8
Aliphatic Carbon-Hydrogen-Oxygen Atoms:
Aliphatic Carbon-Hydrogen-Oxygen-Nitrogen Atoms:
Aliphatic Carbon-Hydrogen-Halogen Atoms:
-9-
Equations, tables and figures
Table 3.3: Data for large pool (D > 0.2 m) burning rate estimates (from Babrauskas [3.1])
Material
Density(kg/m3)
"
m
  (kg/m2s) Hc (MJ/kg)
k (m-1)
Cryogenics:
Liquid H2
70
0.017
120.0
6.1
LNG (mostly CH4)
415
0.078
50.0
1.1
LPG (mostly C3H8)
585
0.099
46.0
1.4
methanol (CH3OH)
796
0.017
20.0
*
ethanol (C2H5OH)
794
0.015
26.8
*
butane (C4H10)
573
0.078
45.7
2.7
benzene (C6H6)
874
0.085
40.1
2.7
hexane (C6H14)
650
0.074
44.7
1.9
heptane (C7H16)
675
0.101
44.6
1.1
xylene (C8H10)
870
0.09
40.8
1.4
acetone (C3H6O)
791
0.041
25.8
1.9
dioxane (C4H8O2)
1035
0.018’’
26.2
5.4**
diethyl ether (C4H10O)
714
0.085
34.2
0.7
benzine
740
0.048
44.7
3.6
gasoline
740
0.055
43.7
2.1
kerosine
820
0.039
43.2
3.5
JP-4
760
0.051
43.5
3.6
JP-5
810
0.054
43.0
1.6
transformer oil, hydrocarbon
760
0.039**
46.4
0.7**
fuel oil, heavy
940 - 1000
0.035
39.7
1.7
crude oil
830 - 880
0.022-0.045
42.5 - 42.7
2.8
polymethylmethacrylate (C5H8O2)n
1184
0.020
24.9
3.3
polypropylene (C3H6)n
905
0.018
43.2
1050
0.034
39.7
Alcohols:
Simple organic fuels:
Petroleum products:
Solids:
polystyrene (C8H8)n
*
Value independent of diameter in turbulent regime.
**
Estimate uncertain since only two points available
- 10 -
Enclosure Fire Dynamics
Figure 3.8 Typical upholstered furniture energy release rates (from Babrauskas [3.1]).
1200
Hospital bed
HHR (kW)
1000
Improved bed
800
600
400
200
0
0
300
600
Time (s)
900
1200
Figure 3.9 Typical mattress energy release rate (from Särdqvist [3.2]).
400
4.1 kg
300
HRR (kW)
3.51 kg
2.34 kg
200
1.17 kg
100
0
0
120
240
360
480
600
Time (s)
Figure 3.10 Energy release rates for trash bags (adopted from Babrauskas [3.1]).
- 11 -
Equations, tables and figures
HRR (kW)
300
200
100
0
0
300
600
900
1200
Time (s)
Figure 3.11 Energy release rates from two experiments with television sets (adopted
from Babrauskas [3.1])
800
Test 16
HRR (kW)
600
Test 17
400
Test 18
200
0
0
120
240
360
Time (s)
480
600
Figure 3.12 Energy release rates from three experiments with Christmas trees (adopted
from Babrauskas [3.1]).
Table 3.4: Conversion to Equivalent Fire Load Density and Equivalent Opening Factor
- 12 -
Enclosure Fire Dynamics
5000
HRR (kW)
4000
3000
ultra fast
2000
fast
1000
medium
slow
0
0
300
600
900
Time (s)
Figure 3.14 Energy release rates for different growth rates.
Table 3.5 Values of  for different growth rates according to NFPA 204M [3.6].
Growth rate
 kW/s2
Time (s) to reach 1055 kW
ultra fast
0.19
75
fast
0.047
150
medium
0.012
300
slow
0.003
600
Table 3.7 Typical growth rates recommended for various types of occupancies.
Type of occupancy
Growth rate 
Dwellings, etc.
medium
Hotels, Nursing homes, etc.
fast
Shopping centers, entertainment centers
ultra fast
Schools, offices
fast
Hazardous industries
Not specified
- 13 -
Equations, tables and figures
Table 3.6 Energy release rate data from Nelson [3.7]
Description
Growth rate
kW/m2 of
floor area
fire retarded treated mattress (including normal bedding)
S
17
light weight type C upholstered furniture**
M
170*
moderate weight type C upholstered furniture**
S
400*
mail bags (full) stored 5 feet high
F
400
cotton/polyester innerspring mattress (including bedding)
M
565*
light weight type B upholstered furniture**
M
680*
medium weight type C upholstered furniture**
S
680*
methyl alcohol pool fire
UF
740
heavy weight type C upholstered furniture**
S
795*
polyurethane innerspring mattress (including bedding)
F
910*
moderate weight type B upholstered furniture**
M
1020*
wooden pallets 1 - 1/2 feet high
M
1420
medium weight type B upholstered furniture**
M
1645*
light weight type A upholstered furniture**
F
1700*
empty cartons 15 feet high
F
1700
diesel oil pool fire (>about 3 Ft. dia.)
F
1985
cartons containing polyethylene bottles 15 feet high
UF
1985
moderate weight type A upholstered furniture**
F
2500*
particle board wardrobe/chest of drawers
F
2550*
gasoline pool fire ( >about 3 Ft. dia.)
UF
3290
thin plywood wardrobe with fire retardant paint on all surfaces
UF
3855*
wooden pallets 5 feet high
F
3970
medium weight type A upholstered furniture**
F
4080*
heavy weight type A upholstered furniture**
F
5100*
thin plywood wardrobe (50in. X 24in. X 72in. high)
UF
6800*
wooden pallets 10 foot high
F
6800
wooden pallets 16 foot high
F
10200
Notes:
* Peak rates of energy release were of short duration. These fuels typically showed a rapid rise to the peak and a
corresponding rapid decline. In each case the fuel package tested consisted of a single item.
** The classification system used to describe upholstered furniture is as follows:
LIGHT WEIGHT = Less than about 5 lbs/ft2 of floor area. A typical 6-foot long couch would weigh under 75 lbs.
MODERATE WEIGHT = About 5-10 lbs/ft2 of floor area. A 6-foot long couch would weigh between 75 and 150 lbs.
MEDIUM WEIGHT = About 10-15 lbs/ft2 of floor area. A 6-foot long couch would weigh between 150 and 300 lbs.
HEAVY WEIGHT = More than about 15 lbs/ft2 of floor area. A typical 6-foot long couch would weigh over 300 lbs.
Type A = Furniture with untreated or lightly treated foam plastic padding and nylon or other melting fabric.
Type B = Furniture with lightly- or un-treated foam plastic padding or nylon or other melting fabric but not both.
Type C = Furniture with cotton or treated foam plastic padding, having cotton or other fabric that resists melting.
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Enclosure Fire Dynamics
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Equations, tables and figures
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Enclosure Fire Dynamics
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Equations, tables and figures
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