03-11.QXD
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Page 280
3–280
Hazard Calculations
H
Radiation
receiving
element
X
(a) Right circular source
H
Radiation
receiving
element
θ
The turbulent flame is approximated by a cylinder.
Under wind-free conditions, the cylinder is vertical (Figure 3-11.19(a)). Under the influence of wind, the cylinder
is assumed to be tilted (Figure 3-11.19(b)). View factors for
horizontal and vertical targets of a vertical cylinder for
no-wind conditions are as follows:
¡
¢
ˆ
‡
† (S > 1)
1
h
h
£
¤
>1
>1
ƒ
tan
>
tan
F12, V C
(S = 1)
9S
9S
S2 > 1
(22a)
ˆ
‡
† (A = 1)(S > 1)
Ah
>1
= ƒ
tan
(A > 1)(S = 1)
9S A2 > 1
ˆ
‡
† (B = 1)(S > 1)
(B > 1/S)
F12, H C ƒ
tan>1
(B > 1)(S = 1)
9 B2 > 1
ˆ
‡
† (A = 1)(S > 1)
(Aƒ > 1/S)
>1
>
tan
(A > 1)(S = 1)
9 A2 > 1
where
β
X
AC
(b) Inclined cylindrical source
Figure 3-11.19. Configuration factor calculation geometries for right and inclined cylinders.
pools in the determination of the flame height. For noncircular pools, the effective diameter will be defined as the
diameter of a circular pool with an area equal to the actual
pool area, given by Equation 13 as
ˆ
‡
† 4A
(13)
DC
9
In the presence of wind, the flame length is given by
the following correlation developed by Thomas40 (Equation 2):
¡
¢0.67
m
g ã!
H
C 55 £ ƒ ¤
(u!)>0.21
(2)
D
:a gD
where u! is the nondimensional wind velocity given by
Equation 2:
uw
(3)
u! C
(gm
g !ãD/:v)1/3
where uw is the wind speed, m/s, and :v is the fuel vapor
density, kg/m3.
In the presence of a significant wind, the flame may
not remain vertical, and a flame tilt angle due to the wind
is relevant to the assessment of radiation. The American
Gas Association (AGA)39 proposed the following correlation to determine the flame tilt (Equation 7):
™
§1
for u! D 1
ƒ
(7)
cos 1 C
›1/ u!
for u! E 1
where u! is the nondimensional wind velocity given by
Equation 20 with the wind velocity measured at a height
of 1.6 m above the ground.
(22b)
SC
2L
D
h2 = S2 = 1
2S
hC
2H
D
BC
1 = S2
2S
and
L C distance between the center of the pool fire and the target
H C height of the pool fire
D C pool fire diameter
If the target is either at ground level or at the flame height,
then a single cylinder can represent the flame. If the target
is above the ground, then two cylinders must be used to
represent the flame. One cylinder represents the flame below the height of the target, and the other represents the
flame above the height of the target. See Figure 3-11.13b
for an illustration of the nomenclature. The overall view
factor is the sum of the two component view factors.
The maximum configuration factor at a point is given
by the vectorial sum of the horizontal and vertical target
configuration factors given by Equation 19:
„
2
2
= F12,
(19)
F12, max C F12,
H
V
The configuration exchange factor for windblown flame
has been given by Mudan,54 who employed a contour integral approach developed by Sparrow55 to determine closedform equations for view factors from a tilted cylinder.
The view factor for a tilted cylindrical flame with a
circular base is as follows:
9FV C
Ÿ1/2
„ ‹
a cos 1 a 2 = (b = 1)2ƒ> 2b(1= sin 1)
b>1
tan>1 A
B b= 1
b > a sin 1
AB
#
2
ab >
1
(b2 > 1) sin
cos 1
ƒ
ƒ 1Ü
ƒ (b >1)sin
= ‚ ? Ÿtan>1
= tan>1 ƒ
C
b2 > 1 C
b2 > 1 C
ˆ
‡
†b > 1
a cos 1
(23a)
>
tan>1
(b > a sin 1)
b= 1
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Fire Hazard Calculations for Large, Open Hydrocarbon Fires
C tan>1
ˆ
‡
†b = 1
b>1
ˆ
‹
Ÿ1/2
‡
†A b>1
a 2 = (b = 1)2 >ƒ2(b = 1 = ab sin 1)
? tan>1
B b= 1
AB
#
2
1/2
ab >
1
(b2 >1)
sin 1
ƒ (b >1)sin
‚ sin 1Ü
‚
= ‚ ? Ÿtan>1
= tan>1
2
C
C
b >1 C
>
(23b)
where
a C H/R
b C L/R
A C a 2 = (b = 1)2 > 2a(b = 1) sin 1
B C a 2 = (b > 1)2 > 2a(b > 1) sin 1
C C 1 = (b2 > 1) cos2 1
0.50
θ = 0°
θ = 30°
θ = 45°
0.20
Maximum view factor
9FH
0.10
0.05
When the angle of tilt is zero, Equations 23a and 23b reduce to Equations 22a and 22b, respectively. The maximum configuration factors at the target location are
determined using Equation 19. Figures 3-11.20 and
3-11.21 show the calculated configuration factor for nowind conditions and underwind conditions for a target at
ground level. For the no-wind condition, Figure 3-11.14
can also be used to determine the view factor for a more
general set of target locations.
θ = –30°
θ = –45°
0.02
0.01
1
5
2
10
20
Dimensionless distance
Figure 3-11.21. Maximum configuration factors for
tilted circular cylinders with targets at ground level. The
dimensionless distance is L/R.
The emissive power, E, of the flame is given by the
following correlation:
Height-to-radius ratio
Maximum view factor at ground level
03-11.QXD
E C Emaxe >sD = Es [1 > e >sD]
0.5
1.0
3.0
6.0
0.1
(24)
where
E C equivalent blackbody emissive power, 140 kW/m2
s C extinction coefficient, 0.12 m–1
D C equivalent pool diameter, m
Es C emissive power of smoke, 20 kW/m2
Atmospheric Absorption
0.01
0.002
1
2
4
6 8 10
20
40
60 80
Nondimensional distance from flame axis
Figure 3-11.20. Maximum configuration factors for a
ground-level object from a right circular cylinder. The
nondimensional distance from the flame is L/R.
The radiation from the fire to surrounding objects
will be partially attenuated by absorption and scattering
along the intervening path. The principal constituents of
the atmosphere that absorb thermal radiation are water
vapor (H2O) and carbon dioxide (CO2). Table 3-11.1 indicates the composition of various gases in the atmosphere.
The CO2 content in the atmosphere is generally constant
at about 330 ppm by volume. The water vapor content
varies strongly with temperature and humidity. Figure
3-11.22 indicates the relationship between atmospheric
temperature, relative humidity, and the amount of precipitable water vapor in a given pathlength.
The principal absorption bands for water vapor are at
1.8, 2.7, and 6.27 5m. Minor absorption bands also exist at
0.94, 1.1, 1.38, and 3.2 5m. Strong absorption by CO2
existing in the 2.7-5m region, the 4.3-5m region, and the