Theoretical & Applied Mechanics Letters 8 (2018) 109-114
Contents lists available at ScienceDirect
Theoretical & Applied Mechanics Letters
journal homepage: www.elsevier.com/locate/taml
Letter
A new equivalent method to obtain the stoichiometric fuel-air cloud from
the inhomogeneous cloud based on FLACS-dispersion
Yulong Zhang*, Yuxin Cao, Lizhi Ren, Xuesheng Liu
PowerChina railway construction Co., LTD, Beijing,100044,China
H I G H L I G H T S
• The pseudo-component gas and its explanation.
• Deduction of new equivalent method.
A R T I C L E
I N F O
Article history:
Received 17 July 2017
Received in revised form 8 December
2017
Accepted 18 January 2018
Available online 1 February 2018
Keywords:
Explosion
Equivalent approach
Homogenous stoichiometric fuel-air
cloud
TNO multi-energy method
A B S T R A C T
The fuel-air cloud resulting from an accidental discharge event is normally irregular in shape and
varying in concentration. Performance of dispersion simulations using the computational fluid
dynamics (CFD)-based tool FLACS can get an uneven and irregular cloud. For the performance of
gas explosion study with FLACS, the equivalent stoichiometric fuel-air cloud concept is widely
applied to get a representative distribution of explosion loads. The Q9 cloud model that is
employed in FLACS is an equivalent fuel-air cloud representation, in which the laminar burning
velocity with first order SL and volume expansion ratio are taken into consideration. However,
during an explosion in congested areas, the main part of the combustion involves turbulent flame
propagation. Hence, to give a more reasonable equivalent fuel-air size, the turbulent burning
velocity must be taken into consideration. The paper presents a new equivalent cloud method
using the turbulent burning velocity, which is described as a function of SL, deduced from the TNO
multi-energy method.
©2018 The Authors. Published by Elsevier Ltd on behalf of The Chinese Society of Theoretical and
Applied Mechanics. This is an open access article under the CC BY-NC-ND license
(http://creativecommons.org/licenses/by-nc-nd/4.0/).
With the application of quantity risk analysis to the 18th subway line of Chengdu, China, FLACS code, as a computational
fluid dynamics (CFD) tool, plays a key role. It embedded Q4, Q5,
Q8 and Q9 as the models for equivalent fuel-air cloud volume.
The benefit of the equivalent approaches is ease to get a representative distribution of explosion loads with minimum number
of simulations.
In these models, the laminar burning velocity and volume
expansion ratio are taken into account in the Q5 and Q9 methods, only the laminar burning velocity is taken into account in
Q4, the volume expansion ratio is only taken into account in the
Q8 method. Detailed information about them can be found in
* Corresponding author.
E-mail address: jadedragon_zhang@163.com (Y.L. Zhang).
literature[1].
Q9, as the latest version, is widely used to assess the explosion loads as part of a risk or consequence analysis [2], and is described by Eq. (1) in FLACS:
Q9 =
n
P
i=1
Vi (Ve(E R i ) ¡ 1) E R f ac(E R i )
max [(Ve (E R ) ¡ 1) E R f ac (E R ) : E R LF L · E R · E R UF L ]
; (1)
(F=O)
(F/O is the ratio
(F=O)
stoi
th
of fuel and oxygen); Vi is the i control volume of the numerical
grid inside the fuel-air region where the fuel-air is in the range of
lower flammability limit (LFL) and the upper flammability limit
(UFL), that is E R LF L · E R · E R UF L ; Ve(E R i ) is volume
where ER is equivalence ratio, E R =
http://dx.doi.org/10.1016/j.taml.2018.02.006
2095-0349/© 2018 The Authors. Published by Elsevier Ltd on behalf of The Chinese Society of Theoretical and Applied Mechanics. This is an open access article under the
CC BY-NC-ND license (http://creativecommons.org/licenses/by-nc-nd/4.0/).
110
Y.L. Zhang et al. / Theoretical & Applied Mechanics Letters 8 (2018) 109-114
expansion ratio at constant pressure in the ith control volume, its
value depend on the ERi; E R f ac (E R i ) is shown with Eq. (2):
E R f ac (E R i ) =
S L (E R i )
;
max (S L (E R ) : E R LF L · E R · E R UF L )
(2)
where SL is the laminar burning velocity.
In Eqs. (1) and (2), the volume expansion ratio and laminar
burning velocity are the two key factors for representing the inhomogeneous fuel-air cloud as a homogeneous fuel-air cloud in
Q9 model.
The factor, volume expansion ratio, employed in Q9 denotes
that the part of ignitable heated fuel-air is expelled out of the
control volume in explosion process because the Q9 can be got
in the dispersion simulation stage prior to explosion simulation
stage, thus, the donation of the expelled fuel-air to explosion
load is ignored and its effect may be underestimated. However,
in most realistic cases, the computing domain will not be completely filled with an ignitable fuel-air cloud, the deemed expelled fuel-air is still in the computing domain and also plays an
important role in the explosion process, its donation on the explosion load cannot be neglected. It seems inappropriate that
the volume expansion ratio is introduced into the equivalent approaching.
On the other hand, Q9 only employs SL with first order to describe the combustion process. Most of the realistic fuel-air explosion process, the fuel-air flow field turns into turbulent regime and the flame propagation is also in violent and turbulent
status.
To give a more reasonable equivalent fuel-air cloud size, a
turbulent burning velocity ST is proposed. Many models, such as
Zimont correlation, Peters Correlation and Mueller
correlation[3], describe ST as a function of SL and turbulence
quantities. So, it is also reasonable to describe ST as the SL with a
non-one order based on TNO (abbreviation for the Netherlands
Organization) multi-energy method (ME).
In the multi-energy method, an idealized fuel-air explosion
scenario model is put forward, shown in Fig. 1. The explosion is
based on a ground-level hemispherical fuel-air cloud which is
filled with a fuel-air mixture at a stoichiometric concentration.
Figure 1 shows the main features of an idealized explosion
scenario, a ground-level hemispherical fuel-air cloud is ignited
in the center, the flame front will then propagate symmetrically
from the centre. the initial peak overpressure in the hemispherical fuel-air cloud zone is assumed as a constant P0 whereas the
side-on overpressure and dynamic pressure will decay with disPeak overpressure
P0
x
Vapour cloud
Central lgnition
Blast wave
Fig. 1. Idealized fuel-air cloud explosion scenario model.
tance outside of the fuel-air zone.
The peak overpressure calculations in the fuel-air cloud zone
are given with the Eqs. (3) and (4). Equation (3) shows the 2 dimensional (2D) explosion expansion, and Eq. (4) shows 3 dimensional (3D) explosion expansion.
2D expansion:
µ
¶2:75
VB R ¢ L p
(3)
P 0 = 3:38
S L 2:7 D 0:7 ;
D
3D expansion
µ
¶2:75
VB R ¢ L p
P 0 = 0:84
S L 2:7 D 0:7 ;
D
(4)
where P0 is the peak overpressure; VBR is the volume blockage
ratio of the obstructed region; Lp is the maximum flame path
length; D is the typical diameter of the obstacles.
Here LP is calculated as
µ
¶1=3
3Vgr
(5)
LP =
;
2¼
where Vgr is the obstructed cloud volume in an obstructed
region.
Detailed descriptions of the multi-energy method and the
calculation of other variables in Eqs.(3) and (4) are found in the
literatures [4-11].
It is practicable to assess the peak overpressure of a fuel-air
mixture with any concentration by the introduction of stoichiometric concentration. that is, a certain concentration fuelair mixture can be assumed as a kind of new pure flammable gas
with a new SL1 when the flammable gas’s ER value equals 1.0, the
new flammable gas is called pseudo-component gas in literature [6, 12 ]. As Eq.(3) or Eq.(4) indicate the different peak overpressure value between stoichiometric fuel-air cloud and the
other concentration fuel-air cloud depends only on S L 2:7 if the
other variables remain unchanged (that means the same obstructed region, the same ignition location and the same ignition energy), notably, S L 2:7 also indicates that turbulent burning
plays a key role during an explosion.
The new approach is trying to transform an inhomogeneous
fuel-air cloud into a smaller stoichiometric fuel-air cloud where
the explosion can generate similar peak overpressure as the inhomogeneous cloud.
The inhomogeneous fuel-air cloud gets the overpressure P0,
and the stoichiometric fuel-air cloud gets the overpressure P1.
Then, by setting P0=P1 , the following Eq.(6) can be derived from
the Eq. (3) or Eq. (4).
µ
¶2:75 µ
¶2:7
P1
L p1
SL1
(6)
=
¢
= 1;
P0
L p0
SL0
µ
L p1
L p0
¶2:75
L p1
=
L p0
µ
=
SL0
SL1
µ
SL0
SL1
2:7
¶ 2:75
:
¶2:7
;
(7)
(8)
Thus, the following Eq. (9) can be derived by the use of the Eq. (5)
µ
¶1=3 µ
¶ 2:7
L p1
Vgr1
S L 0 2:75
(9)
=
=
;
L p0
Vgr0
SL1
Y.L. Zhang et al. / Theoretical & Applied Mechanics Letters 8 (2018) 109-114
Vgr1
=
Vgr0
"µ
Vgr1 =
Ã
SL0
SL1
#3
2:7
¶ 2:75
2:7
S L 0 2:75
SL1
!3
=
=
µ
µ
SL0
SL1
SL0
SL1
¶2:945
¶2:945
(10)
;
Vstoi =
n
X
111
(13)
CVgr1 i :
i=1
The Eq. (13) is the proposed new equivalent cloud method.
A series of simulations are conducted to validate the new
equivalent cloud method, For simplifying the representation of
the inhomogeneous cloud, 1 m3 methane-air clouds with a certain ER value is employed. The obstacles include 61 m-length
and 0.2 m-diameter pipes for the representation of blockage.
The different ER values and the corresponding laminar burning
velocities of methane-air mixtures are shown in Fig. 2, the data
source is from the literature [1].
The obstacles and computing domain is shown in Fig. 3. As
stated in Fig. 2, the laminar burning velocity curve for methaneair mixture is approximately symmetrical around the axis of ER
value at 1.08, the laminar burning velocities where the ER values
equals 0.7, 0.8, and 0.9 are selected for the validation simulations, the results from these ER values are the representative res-
(11)
¢ Vgr0:
In the view of control volume (CV) from FLACS code, Eq. (12)
can be shown as following:
µ
¶2:945
SL0
(12)
CVgr1 i = CVgr0 ¢
:
SL1 i
The whole computing domain, which is divided into many
control volume using the gridding mechanism, is filled with the
different concentration fuel-air cloud (concentration between
LFL and UFL). Correspondingly, the homogenous stoichiometric fuel-air cloud volume Vstoi is expressed by the Eq. (13).
0.50
0.45
0.43
Laminar buming velocity (m/s)
0.40
0.38
0.45 0.45
04.0
0.35
0.30
0.30
0.31
0.25
0.20
0.20
0.18
0.15
0.10
0.08
0.05
0.08
0
0
0
0.53 0.60 0.70 0.80 0.90 1.00 1.08 1.10 1.20 1.30 1.40 1.50 1.60
ER
Fig. 2. Laminar burning velocities vs. ER.
ults for ER values at 1.37, 1.30 and 1.20 respectively because of
the symmetry. When ER equals 1.08 or 1.10, the equivalent stoichiometric clouds volume is approximate same as the original
gas clouds respectively, so, they are not selected any more.
56
56
50
44 38
20 14
32 26
8
Fuei
2
50
44
38
32
26
20
14
8
n
regio
y
Fig. 3. Obstacles and computing domain.
x
z
For comprehensive analysis of the explosion load, three typical cases are taken into consideration, one is the methane-air
explosion in open space with obstacles, the other one is the
methane-air explosion in open space without obstacles, the last
one is the methane-air cloud explosion is in an enclosed space
with obstacles [13,14]. The simulating results are shown in Fig. 4.
There is a good agreement by the comparison of the explosion overpressure between equivalent clouds and the cloud
which ER equals 0.9 for cases of the cloud explosions in open
space. For other scenarios, it seems that there is a poor agreement, in fact, the low peak overpressures indicate the ignited
non-stoichiometric clouds are not the explosion process but the
combustion process.
For the cloud explosions in the enclosed space, there are
good agreements for each ER value respectively. The cloud
which ER equals 0.7 can generate the peak overpressure up to
6.561 barg, the corresponding equivalent stoichiometric cloud
can generate up to 7.840 barg overpressure, The relative error is
within 20%. The relative errors of the other two sets of scenarios
112
Y.L. Zhang et al. / Theoretical & Applied Mechanics Letters 8 (2018) 109-114
consideration. However, during an explosion in congested areas,
the main part of the combustion involves turbulent flame
propagation. Hence, to give a more reasonable equivalent fuelair size, the turbulent burning velocity must be taken into consideration. The paper presents a new equivalent cloud method
using the turbulent burning velocity, which is described as a
function of SL , deduced from the TNO multi-energy method. To
validate the new equivalent cloud method, a series of simulations was conducted. There are good agreements by the comparison of the overpressures between equivalent clouds and the
cloud with different ER values if explosion happened.
are also within 20%.
In this paper, the fuel-air cloud resulting from an accidental
discharge event is normally irregular in shape and varying in
concentration. Performance of dispersion simulations using
CFD-based tool FLACS can get an uneven and irregular cloud.
For the performance of gas explosion study with FLACS, the
equivalent stoichiometric fuel-air cloud concept is widely applied to get a representative distribution of explosion loads. The
Q9 cloud model that is employed in FLACS is an equivalent fuelair cloud representation, in which the laminar burning velocity
with first order SL and volume expansion ratio are taken into
MP 83
0
20
Run: 000001
P (barg)
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s10: (3.846, 0.004)
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s486: (157.553, 0.001)
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0.006
0.004
0.002
0
−0.002
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Run: 001002
s11: (3.651, 0.014)
0.010
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0
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0.010
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0
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0.010
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Run: 000003
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s10: (3.648, 0.014)
MP 83
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MP 83
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c
P (barg)
0.06
0.04
0.02
0
−0.02
−0.04
s12: (3.822, 0.015)
0.015
0.010
0.005
0
−0.005
−0.010
−0.015
0
Run: 000001
e
s10: (3.646, 0.007)
P (barg)
0.03
0.02
0.01
0
−0.01
−0.02
P (barg)
P (barg)
a
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Y.L. Zhang et al. / Theoretical & Applied Mechanics Letters 8 (2018) 109-114
g
h
7
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2
1
0
s44: (12.632, 7.118)
P (barg)
P (barg)
s202: (56.622, 6.561)
0
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8
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s570: (145.089, 8.029)
s218: (56.494, 7.840)
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2 500
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s109: (29.145, 7.571)
7
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Run: 000003
20
40
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80
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s135: (34.810, 8.028)
8
7
6
5
4
3
2
1
0
0
20
Run: 001003
40
60
80
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100
120
Fig. 4. Explosion of cloud with different ER values and the corresponding equivalent cloud. a ER=0.7 and the corresponding equivalent cloud in
open space and obstacles. b ER=0.8 and the corresponding equivalent cloud in open space and obstacles. c ER=0.9 and the corresponding equivalent cloud in open space and obstacles. d ER=0.7 and the corresponding equivalent cloud in open space without obstacles. e ER=0.8 and the
corresponding equivalent cloud in open space without obstacles. f ER=0.9 and the corresponding equivalent cloud in open space without
obstacles. g ER=0.7 and the corresponding equivalent cloud in enclosed space and obstacles. h ER=0.8 and the corresponding equivalent cloud
in enclosed space and obstacles. i ER=0.9 and the corresponding equivalent cloud in enclosed space and obstacles.
References
Nomenclature
D
typical diameter of the obstacles
m
ER
equivalence ratio
-
Lp
flame path length
m
P0
peak overpressure
bar
SL
laminar burning velocity
m/s
VBR
volume blockage ratio
-
Ve
volume expansion ratio
-
Vgr
obstructed cloud volume
m3
Vi
ith control volume
m3
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