Process Safety and Environmental Protection 1 1 4 ( 2 0 1 8 ) 48–63
Contents lists available at ScienceDirect
Process Safety and Environmental Protection
journal homepage: www.elsevier.com/locate/psep
A simplified statistic-based procedure for gas
dispersion prediction of fixed offshore platform
Jihao Shi a,b , Jingde Li b , Yuan Zhu a,∗ , Hong Hao b , Guoming Chen a , Bin Xie c
a Center for Offshore Engineering and Safety Technology, China University of Petroleum, Qingdao 266580, China
b Centre for Infrastructural Monitoring and Protection, School of Civil and Mechanical Engineering, Curtin University,
WA 6102, Australia
c GexCon China, Shanghai 200135, China
a r t i c l e
i n f o
a b s t r a c t
Article history:
In explosion risk analysis, Frozen Cloud Approach (FCA) and Dimensionless Response Sur-
Received 20 February 2017
face Method (DRSM) are both commonly used to achieve a balance between simulation
Received in revised form 20 October
workloads and accurate results. However, the drawbacks of these two approaches are obvi-
2017
ous. FCA is not reliable for risk study of fuel-dominated regions. Whereas DRSM usually
Accepted 1 December 2017
couples the dimensionless parameters and generates a large numbers of correlations to
Available online 8 December 2017
predict the flammable cloud size, which brings a heavy computation burden for engineers.
Therefore, this paper aims to propose a simplified procedure which can quickly and accu-
Keywords:
rately provide a large number of non-simulation data based on limited CFD simulation data.
Automatically Selected Model
Full Factorial Design of Experiment (FFDOE) based RSM is adopted. Codification is applied
Technology
to couple all the dimensional parameters into a single correlation. Automatically Selected
Computation cost reduction
Model Technology (ASMT) is used to easily determine the suitable structure of correlation.
Explosion risk analysis
Compared to the conventional procedures, the simplified procedure is proven to be more
Frozen Cloud Approach
robust. For subsequent Explosion risk analyses (ERAs) in the fuel-dominated regions, the
Dimensionless respond surface
simplified procedure becomes a superior alternative.
© 2017 Published by Elsevier B.V. on behalf of Institution of Chemical Engineers.
method
Fuel-dominated region
1.
Introduction
different leakage and wind conditions. The representative gas clouds
are subsequently used in gas explosion simulations to obtain DALs.
Due to the fact that performing a large number of dispersion simu-
Explosion risk analyses (ERAs) are widely used to derive the dimensioning accidental loads (DALs) for design of offshore topside facilities.
lations by using CFD tools to derive DALs is inefficient and expensive,
ERAs can predict explosion loads in detail, including overpressures,
differential pressure, and drag loads (FABIG, TN-08). Loads with return-
Frozen Cloud Approach (FCA) and Dimensionless Response Surface
Method (DRSM) have been proposed by others to increase the effi-
ing frequencies of 1 × 10−04 per year are then adopted as DALs and are
incorporated with standards and legislations such as NORSOK Z013
ciency of the risk analysis and to obtain acceptable results based
on a limited number of CFD simulations. DNV initially proposed the
(NORSOK, 2010) and ISO19901-3 (2015) in safety study.
commonly-used Frozen Cloud Approach (FCA) (GexCon, 2015). In this
In a standard ERA, Computational fluid dynamic (CFD) simulation
approach, linear relations amongst gas concentration, leak rate and
is usually employed. Gas dispersion simulation, which plays an important role in ERAs, is performed before gas explosion simulation. The
the wind speed for each leak scenario are firstly assumed and the gas
cloud sizes from the non-numerically-simulated scenarios are then
obtained. However, the FCA can exclusively provides satisfactory pre-
major aim of conducting gas dispersion simulation is to identify credible gas cloud sizes, gas concentrations and cloud locations under
diction results for leakages in ventilation dominated regions while
poor estimations are expected for leakages in fuel dominated regions
(GexCon, 2015).
∗
Corresponding author.
E-mail addresses: shi jihao@163.com (J. Shi), jingde.li@curtin.edu.au (J. Li), zhy3323@163.com (Y. Zhu), Hong.Hao@curtin.edu.au (H.
Hao), offshore@126.com (G. Chen), xie.bin@gexcon.com (B. Xie).
https://doi.org/10.1016/j.psep.2017.12.002
0957-5820/© 2017 Published by Elsevier B.V. on behalf of Institution of Chemical Engineers.
49
Process Safety and Environmental Protection 1 1 4 ( 2 0 1 8 ) 48–63
Afterwards, Cleaver et al. (1999) developed a set of simple correlations to predict the flammable gas cloud size for different gas release
scenarios. Based on these correlations, Huser and Kvernvold (2000)
proposed the concept of DRSM. The DRSM calculates the flammable
gas cloud size by using several response surfaces with a large number
of simulations. Compared to other old approaches, the DRSM indeed
improves the accuracy of gas cloud estimation. Nevertheless, the utilization of these equations incurs heavier computation requirements
for engineers (Ferreira and Vianna, 2014). In order to decrease the
number of those correlations, Qiao and Zhang (2010) adopted a very
conservative way combining the DRSM with the FCA, which however
leads to large safety redundancy and large cost of offshore platform
construction. Later, Ferreira and Vianna developed the DRSM to couple
the most relevant dimensionless parameters into a single mathemati-
S is the energy term, xi is the coordinate in i-direction, ui is
the velocity component in i-direction. The equation consists of
four distinct terms and the first in the left represents transient
term while the second takes into account the convection. The
terms in the right represent diffusion term and the last is the
source term.
It is critical to model the Reynolds stress tensor as follows to
describe the turbulence of dispersion process for the leakage
gas.
Reynolds stress tensor:
−ui uj = ueff
cal model, which though is exclusively suitable for early design phase
(Ferreira and Vianna, 2014).
Precisely, DRSM has a significant drawback even though it improves
the accuracy of gas cloud estimation. As RSM is a collection of mathematical and statistical techniques for analyzing the relationship
between different parameters and response (Bezerra et al., 2008), the
derived correlations estimate data for the ERAs without providing the
physical meaning of gas dispersion. The coupled dimensionless parameters by DRSM result in a variation of two coupled parameters at a fixed
ratio which will not influence the final gas cloud size calculation. In
other words, the gas cloud size will be significantly under/over predicted if the coupled parameters change simultaneously. Alternatively,
if the variations of individual dimensionless parameters are all considered in a correlation, several correlations need to be regenerated, which
may cause extra amount of calculation burden (Ferreira and Vianna,
2014).
Therefore, this study aims to propose a simplified procedure to
quickly generate thousands of non-simulation data with acceptable
interval for further ERAs based on a limited number of simulations.
Compared to conventional procedures, the new simplified proposed
procedure is more robust and user-friendly. It can also reduce the
computation cost. Moreover, for the fuel-dominated region of offshore
platform, the simplified procedure becomes a better alternative than
FCA. In combination with stochastic simulation technique (e.g. Monte
Carlo simulation), the simplified procedure can be eventually used for
further ERAs.
2.
Numerical dispersion model
In this study, FLACS (Flame Acceleration Simulator) is used
to model gas dispersion. FLACS is a 3D CFD software validated over the last 40 years against numerous experiments
for a extensive range of different dispersion scenarios (Middha
et al., 2009, 2010; Hansen et al., 2010; Bleyer et al., 2012). The
software has been widely used in the oil and gas industry to
simulate gas dispersion and vapor cloud explosion in both
offshore and onshore facilities (Vianna and Cant, 2012; Qiao
and Zhang, 2010; Huser and Kvernvold, 2000; Patankar, 1980;
Hansen et al., 2013).
2.1.
FLACS solves the compressible RANs equations on a 3D Cartesian grid using a finite volume method. The conservation
equations for mass, momentum, enthalpy, and mass fraction
of species, closed by the ideal gas law, are included. The conservation equations can be represented in general as:
∂ ueff ∂
∂(ui )
∂( )
=
(
)+S
+
∂t
∂xi
∂xi ∂xi
(1)
In this equation, represents the dependent variable. ueff is
the effective turbulence viscosity, is the gas mixture density,
− 2kıij /3
(2)
where xj is the coordinate in j-direction, uj is the velocity component in j-direction, k is turbulent kinetic energy, ıij is stress
tensor.
Following Boussinesq eddy viscosity assumption, the eddy
viscosity models are used to model Reynolds stress tensor and
solve k–ε model as follows:
Turbulent kinetic energy and one for dissipation of turbulent kinetic energy:
∂
∂(ˇi kui )
∂(ˇv k)
=
+
∂t
∂xi
∂xi
∂(ˇv ε)
∂(ˇi εui )
∂
+
=
∂t
∂xi
∂xi
∂k eff
ˇi
ck ∂xi
∂ε eff
ˇi
cε ∂xi
+ ˇv Pk − ˇv ε
+ ˇv Pε − C2ε ˇv (3)
ε2
k
(4)
where t is time, Pk is the production of turbulent kinetic energy,
Pε is the production of dissipation, C2␧ is constant in FLACS and
equal to 1.92, ˇi is area porosity in the i-direction, ˇv is volume
porosity, ck and cε is Prandtl–Schmidt number.
2.2.
Wind boundary model
Monin–Obukhov length is adopted by FLACS to describe the
properties of the atmospheric boundary layer close to Earth’s
surface and explain the buoyancy effect on the atmospheric
boundary layer. The characteristic length scale L is defined
based on Pasquill classes as follows:
L=
1
z0
log
Ls
zs
(5)
where z0 is an atmospheric roughness length, zs and Ls are
1.0 m and 0 m respectively under Pasquill class D. With the
Pasquill class, the wind velocity profile can also be calculated
as follows:
u(z) =
Turbulent model
∂uj
∂ui
+
∂xj
∂xi
∗ a
u(z) = 0
If
ln
(z − zd ) + z0
z0
− u (z)
If
z0 > 0
(6)
(7)
z0 < 0
where z is the height above the ground, and zd is the canopy
height, a is Von Karman constant, typically equal to 0.41. u0 is
average wind velocity, z0 is an atmospheric roughness length,
* is the friction velocity, which is defined as follows for the
neutral and non Pasquill class:
∗ =
ln
u0 a
(zref −zd )+z0
z0
(8)
− u (zref )
50
Process Safety and Environmental Protection 1 1 4 ( 2 0 1 8 ) 48–63
Fig. 1 – 3-Parameter-2-level FFDOE (three independent parameters A B C and two levels 1, 2 of each parameter result in 8
combinations) (Croarkin et al., 2002).
where zref is the height relative to the ground where velocity equals the wind speed, u equals to 0 for Pasquill class D
adopted in this study.
3.
Response Surface Method based on full
factorial Design of Experiment (FFDOE)
Response Surface Method (RSM) is a collection of mathematical and statistical techniques for problem analysis and
the relationship building between different parameters and
response. By selecting reasonable Design of Experiment (DOE),
performing accurate simulations and applying regression
analysis, a model to describe the relationship may be obtained
and its quantitative form can be represented as below (Box and
Draper, 1987):
y = f (X1 , X2 , X3 , ..., Xn ) ± ε
(9)
where y is the response, f is the response function, ε is the
experimental error, and Xi (i = 1, 2. . . k) are independent parameters.
One of the important steps is to choose the reasonable
DOE to generate the RSM model to describe the complicated
relationship between the flammable cloud volume and its
affecting parameters.
In statistics, FFDOE is an experiment whose design consists of two or more parameters, each with different possible
levels, and whose experimental units take on all possible combinations of these levels across all such parameters (Box et al.,
2005). Fig. 1 shows the 3-parameter-2-level factorial design. As
can be seen, there are 3 parameters named A, B and C and each
parameter has two levels (level 1 and level 2). Thus, there are
8 kinds of combinations considering all the parameters and
levels.
4.
Procedure of FFDOE-based RSM on
dispersion simulation
Fig. 2 presents the procedure of how to apply the FFDOE-based
RSM on dispersion result prediction. The detailed procedure
is described below.
Step 1: Geometry modelling and ventilation simulation
In this step, 3D geometry is imported into FLACS. The
final as-built facility is represented by adding anticipated congestion. Anticipated Congestion Method (ACM) (Davis et al.,
Fig. 2 – Procedure of FFDOE-based RSM on dispersion
simulation.
2011), which is a standard procedure for generating small geometrical details based on process information from detailed
drawings, is used.
After geometry modelling, ventilation simulations for 8 different wind directions using the most frequent wind speed
will be performed according to standard procedures commonly used in practice as defined in GexCon (GexCon, 2015)
and NORSOK (NORSOK, 2010).
Step 2: Codification of the levels of the parameters based
on FFDOE
The following equation are applied to transform a real
value (zi ) into a coded value (xi ) by:
xi =
z0 =
zi − z0
zmax − zmin
z
max − zmin
2
(10)
(11)
where zmax is the real value in the upper limit and zmin is the
lower limit level of a parameter, and z0 is the real value in the
central point.
It is noted that the leak direction is coded as matrix form
in order to obtain the single correlation, which would be illustrated in Section 5.3.
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Process Safety and Environmental Protection 1 1 4 ( 2 0 1 8 ) 48–63
Table 1 – Configuration of FFDOE of 2 levels of leak rate, wind speed, wind direction and 3 levels of leak direction with
codification method.
Std
Leak rate
Wind speed
Wind direction
Leak direction
Std
Leak rate
Wind speed
Wind direction
Leak direction
1
2
3
4
5
6
7
8
9
10
11
12
−1
1
−1
1
−1
1
−1
1
−1
1
−1
1
−1
−1
1
1
−1
−1
1
1
−1
−1
1
1
−1
−1
−1
−1
1
1
1
1
−1
−1
−1
−1
[1 0]
[1 0]
[1 0]
[1 0]
[1 0]
[1 0]
[1 0]
[1 0]
[0 1]
[0 1]
[0 1]
[0 1]
13
14
15
16
17
18
19
20
21
22
23
24
−1
1
−1
1
−1
1
−1
1
−1
1
−1
1
−1
−1
1
1
−1
−1
1
1
−1
−1
1
1
1
1
1
1
−1
−1
−1
−1
1
1
1
1
[0 1]
[0 1]
[0 1]
[0 1]
[–1 –1]
[–1 –1]
[–1 –1]
[–1 –1]
[–1 –1]
[–1 –1]
[–1 –1]
[–1 –1]
After codification, configuration of FFDOE of different levels
of varied affecting parameters can be conducted. Table 1 takes
2 levels of leak rate, wind speed, wind direction and 3 levels
of leak direction as an example to illustrate the configuration
of FFDOE based on codification method. As shown in Table 1,
FFDOE is orthogonal so that each parameter can be evaluated
independently of all the other parameters. Moreover, the interactive effect of varied parameters can be completely analyzed,
which make the correlation derived more accurate and robust
(Box et al., 2005).
Step 3: Mathematical modeling of dispersion simulation
results
The most suitable and robust polynomial order is determined in this step. ASMT, namely F-test, coefficient of multiple
determination R2 , Adjusted-R2 and Predicted-R2 is adapted
to select the polynomial order. F-value, R2 , Adjusted-R2 and
Predicted-R2 can be calculated as below:
2 m m
−
SSreg /(p − 1)
ŷj − y
=
SS/ res (p − 1)
j
F value =
/
j
yj − ŷj
2
The single and interactive effect of different parameters on
dispersion results are then derived from ANOVA results.
Step 4: Further diagnosis analysis of mathematical model
Diagnosis analysis is conducted to further determine the
most suitable transformation of correlation, e.g. Ln transformation, Exponent transformation etc. Internally studentized
residuals and Box-Cox plot are used to diagnosis the outliers from varied transformation and to determine the correct
power law transformation.
Step 5: Dispersion results output: 2D contour plots, 3D surfaces and vapor cloud volume prediction
After deriving the suitable correlation, 3D surfaces and
2-D predicted lines with 95% confident intervals and prediction intervals are obtained. Those surfaces can also be used
for ERAs in combination with stochastic probability analysis.
However, it should be noted that this procedure are useful for
statistical estimation rather than deterministic estimation. In
other words for further ERAs, not only the predicted results
but also the interval’s upper and lower limits should be used
together with stochastic probability method.
(12)
R2 = 1 −
m
m j
j
m − 2
ŷj − y
/
j
yj − ŷi
2
5.
− 2
ŷj − y
Adjusted-R2 = 1 −
(13)
1 − R2 ∗ (m − 1) / (m − p − 1)
To demonstrate the simplified procedure, a case study of an
offshore platform is performed in this paper. Accordingly, this
case study is also used to verify the feasibility and accuracy of
the procedure.
(14)
5.1.
2
Predcited-R = 1 −
m +
j
m
2 m /
j
yj − ŷj
2
− 2
ŷj − y
j
ˆ
yj − yi,−i
(15)
where m is total number of simulation results in the design; p
is number of parameter of model; yj is the simulation results
for the level j; ŷj is estimated value by the model for the level
−
Case study of an offshore platform
+
j; y is the average value of m simulation results; SSreg is the
sum of the square due to regression and SSres is the sum of
the square due to residuals. It is worth noting that Predictedˆ )
R2 is calculated by systematically removing each output (yi,−i
from the data set, which can be used to estimate the regression equation, and determine how well the model predicts the
removed output.
Numerical modeling
The geometry of this case study is a typical fixed and naturally ventilated offshore platform. CAD models are imported
to FLACS. A 3D view of the central part of the offshore platform
is shown in Fig. 3.
The overall length, width and height of the offshore platform are 25 m (X), 60 m (Y), 40 m (Z), respectively. The platform
consists of process module and accommodation module. The
process module (central part), which is the focus in this study,
has three deck levels, i.e. the lower deck level, the middle deck
level and the upper deck level. Most of the gas equipment
locate on the lower and middle deck level. The lower deck
level of the process module is occupied by accommodation
modules on +Y side and the middle deck level is occupied by
the steel plates and accommodation module on −X and +Y
side, respectively. The length, width and height of the process
module are 25 m, 24 m, and 10 m, respectively.
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Process Safety and Environmental Protection 1 1 4 ( 2 0 1 8 ) 48–63
Fig. 3 – Geometric model of a typical fixed and naturally ventilated offshore platform.
5.2.
Ventilation simulations
In order to generate a ventilation distribution map by means
of rate, direction, and probability, ventilation simulations are
performed in the beginning. Based on the ventilation information, wind conditions for FFDOE of the dispersion simulations
are determined subsequently.
According to NORSOK Z-013 standard (NORSOK, 2010),
the wind speed of 5 m/s is used for 8 cases with different
wind directions in all ventilation simulations. The simulation dimensions are 90 × 110 × 65 m3 in x, y and z directions.
The core simulation region is 96 m–123 m in x direction,
88 m–122 m in y direction and 26 m–36 m in z direction. The
grid size used in the ventilation simulation is 1 m at core
domain and the stretching factor 1.2 is used from the core
domain to simulation boundary.
Boundary condition of WIND is used for all inflow and parallel boundaries and NOZZLE is used for the outflow and sea
surface boundaries. Pasquill class D is used, the atmospheric
roughness length z0 and the height relative to the ground zref
are set as 0.0002 and 10 m, respectively. CFLC = 20 and CFLV = 2
are adopted.
The Air Changes per Hour (ACH) is calculated by the volume
flux per unit time entering the core region, which is divided
by the total volume including blocked areas inside the core
region. Fig. 4 presents the ventilation conditions in the whole
module. It is shown that the ventilation conditions are generally good when wind speed is 5 m/s in this module. However,
the ventilation rates are lower for wind coming from the North
(0◦ ) as compared to other wind directions. The main reason is
that the presence of accommodation module blocks wind in
North.
Fig. 5 shows the ventilation conditions in both deck levels
when wind speed is 5 m/s and wind direction is 270◦ , respectively. The upper figure presents the steady wind vector in
XY plane at Z = 29 m in the middle deck level of the module.
It shows that the maximum wind speed in this deck level is
about 4 m/s and distribution area of wind speed above 2 m/s
is small.
The lower figure shows the steady wind vector in XY plane
at Z = 26.5 m in the lower deck level. As can be seen, the maximum wind speed approaches 6.8 m/s whose distribution area
Fig. 4 – Air changes per hour with 5 m/s wind speed of the
process module.
is comparatively large and most of the part in this level is
occupied by wind speeds above 2 m/s. Therefore, the results
show that the ventilation condition in the middle deck level
is worse than that in the lower deck level, which is attributed
to the high confinement in the middle deck level, i.e. the middle deck level is enclosed by steel plate in −X direction and
accommodation module in +Y direction. Whereas the lower
deck level is only confined by accommodation module in +Y
direction.
5.3.
Dispersion simulations
5.3.1.
Codification of parameters
For dispersion simulations, 5 main parameters, namely, leak
position, leak rate, leak direction, wind speed, and wind direction, are considered. To identify the leak information, the first
step is to choose the representative segment of the module.
Amongst all parameters, the ones have potential to lead to the
worst-case consequences are taken into account to find the
representative segment. According to the standard (NORSOK,
Process Safety and Environmental Protection 1 1 4 ( 2 0 1 8 ) 48–63
53
Fig. 5 – Ventilation conditions inside the facility with 5 m/s wind speed and wind direction is 270◦ . The upper figure shows
the wind vector at XY plane at Z = 1.8 m (the middle deck level). The lower figure shows the wind vector at XZ plane at
Z = 2.8 m (the lower deck level). The wind vectors are shown as m/s.
a) The middle deck level. b) The lower deck level.
2010), the leak point (115,110,29.5) as an example, is set at the
center of the middle deck level. A typical composition of natural gas is adopted for gas dispersion simulation. Table 2 shows
the composition of natural gas. Table 3 shows the leak property
of critical segment.
Leak rates used in this study cover the range of all critical
hole sizes. 4 levels of leak rates are considered. The leak rate
can be calculated by (Spouge, 1999):
Q0 = Cd Av P0
M
RT0
−1
2 + 1
−1 P >P ( 2 ) a +1
0
+1
(16)
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Process Safety and Environmental Protection 1 1 4 ( 2 0 1 8 ) 48–63
5.3.2.
Table 2 – Natural gas composition for FLACS analysis.
Component
Concentration (%)
Carbon dioxide
Methane
Ethane
Propane
Butane
Pentane
Hexane
Heptane
n-Octane
n-Nonane
n-Decane
0.84
57.18
5.53
3.85
2.60
1.62
1.16
1.68
1.81
1.33
22.38
Table 3 – Leak property of critical segment.
Item
Leak medium
Pressure (bar)
Temperature (K)
Molecular weight (kg/kmol)
Ratio of specific heat
Natural gas
20
300
16.04
1.328
Table 4 – Codification of leak rate, wind speed and wind
direction.
A:Leakrate (kg/s)
B:Wind speed (m/s)
C:Wind direction
Coded value
Real value
Coded value
Real value
Coded value
Real value
−1
5
−1
1.5
−1
0
−0.6
20
−0.333
5
−0.333
90
0.467
60
0.048
7
0.333
180
1
80
1
12
1
270
where Q0 is mass leak rate, Cd is the coefficient, Av is the leakage area, P0 is the internal pressure, Pa is ambient pressure, M
is the molecular weight, T0 is the initial temperature, ␥ is the
ratio of specific heat.
There are 6 leak directions same as the 6 axial directions.
Wind conditions are selected based on the results of ventilation simulations. 4 levels of wind conditions are considered.
All the coded levels of 3 parameters are summarized in
Table 4. Table 5 shows the codes of different leak directions.
The matrix is used to present 6 leak directions aligned along
the 6 axial directions. It is noted that −Z direction is coded
as (−1,−1,−1,−1,−1,−1) in the Table 5 in order to enable the
FFDOE to meet the orthogonal characteristics.
After coding of parameters, different dispersion simulation
scenarios are defined. The domain of dispersion simulation
boundary is as same as that of ventilation simulations. Grid
size 1 m is used for core domain and a stretching parameter 1.2
is adopted to smooth the grid from core domain to boundaries.
Grid sensitivity analysis is conducted. For different leak rates,
grid refinements around the leak areas are performed to keep
moderate calculation times while getting acceptable results
based on the grid refinement guidance (GexCon, 2015). The
boundary conditions used for dispersion simulations are as
same as those in ventilation simulations.
Dispersion simulation results analysis
Based on the FFDOE, a total number of 384 dispersion simulations (4 leak rates × 4 wind speeds × 4 wind directions × 6
leak directions) are performed. Fig. 6 shows the 3D plot of
flammable cloud distribution under varied leak directions,
leak rate is 20 kg/s, wind speed is 5 m/s and wind direction is
270◦ . The refined grid size near the leakage area (0.005 m2 ) is
0.1 m. The flammable range of leakage natural gas is between
the equivalence ratio of 0.015 and 0.1.
Fig. 6a) and b) shows the results when leak direction is −X
at 5 s and 38 s after leakage. It is seen in Fig. 6a) that the leakage
natural gas firstly disperse along the leak direction (−X) while
ventilation condition is not dominated. Afterwards, as the gas
enters into the more confined process module on both −X and
−Y side, the leakage natural gas accumulates until a portion
of natural gas arrives beyond the upper limit of flammable
range. Eventually, most of flammable gas dominate the module process. Fig. 6c) shows the flammable cloud distribution
when leak direction is +X. As can be seen, the leakage natural
gas directly disperses out the module, thereby resulting in a
small flammable cloud.
Fig. 7 shows the equivalent stoichiometric cloud volumes
(Q9) history profiles of varied leak rates under different wind
speeds when leak direction is −X and wind direction is 270◦ .
Fig. 7a)–d) shows the results of varied leak rates under wind
speed 1.5 m/s, 5 m/s, 7 m/s and 12 m/s, respectively. As can be
seen, only one peak of Q9 occurs for leak rate 5 kg/s under
varied wind speeds while at two obvious peaks are seen for
larger leak rates, i.e. 10 kg/s–80 kg/s. It should be noted that
wind dilution strongly affects the latter peaks. To be precise,
the increasing wind speed after the steady states of the first
peaks eventually induce the wind dilution that significantly
increase the second peaks. All maximum peaks of Q9 are then
summarized in e) Q9 versus different leak rates under different wind speed). Fig. 7e) also indicate the data of different
leak rates under different wind speeds when leak direction is
−X and wind direction is 270◦ . Generally, as seen in Fig. 7e),
the maximum Q9 peaks initially increase and then decrease
with the increase of wind speed for small leak rates, i.e. 5 kg/s
and 10 kg/s. However, for the medium and large leak rates, i.e.
20 kg/s to 80 kg/s, continuous increases of these peaks along
with the increase of wind speed are seen.
5.3.3.
Dispersion mathematical modeling
Based on the gas dispersion simulation results, ASMT is used
to quickly determine the suitable polynomial order of the
mathematical model.
Two criteria are applied to determine the correlation order.
The first criterion is the sensitivity of added terms on the accuracy of the correlation. Table 6 shows results of F-test, which
can indicate the corresponding sensitivity on results. As can
be seen, with more additional terms added into the correlation, F-value becomes smaller. The small F-value indicate that
the added terms become less sensitive. As shown in Table 6 the
F-value of 4.49 corresponding to the Cubic and Quartic orders,
is small enough to become the order determination threshold.
Table 5 – Codification of leak direction.
D:leak direction
Coded value
Real value
Coded value
Real value
{1,0,0,0,0,0}
+X
{0,0,0,1,0,0}
−Y
{0,1,0,0,0,0}
−X
{0,0,0,0,1,0}
+Z
{0,0,1,0,0,0}
+Y
{−1,−1,−1,−1,−1,−1}
−Z
55
Process Safety and Environmental Protection 1 1 4 ( 2 0 1 8 ) 48–63
Fig. 6 – 3D plot of flammable cloud distribution when leak rate is 20 kg/s, wind speed is 5 m/s and leak directions are −X &
+X, wind direction is 270◦ .
a) 3D plot of flammable cloud distribution when leak rate is 20 kg/s, wind speed is 5 m/s and leak direction is –X, wind
direction is 270 at time 4 s after leakage. b) 3D plot of flammable cloud distribution when leak rate is 20 kg/s, wind speed is
5 m/s and leak direction is –X, wind direction is 270 at time about 70 s after. c) 3D plot of flammable cloud distribution when
leak rate is 20 kg/s, wind speed is 5 m/s and leak direction is +X, wind direction is 270 at the time about 70 s after leakage.
Table 6 – Results of F-test.
Source
SSres
p
MSres
Mean vs total
Linear vs mean
2FI vs linear
Quadratic vs 2FI
Cubic vs quadratic
Quartic vs cubic
Residual
Total
17,895.15
403.13
29.04
10.20
16.07
9.96
17.70
18,381.25
1
8
18
3
40
62
247
379
17,895.15
50.39
1.61
3.40
0.40
0.16
0.072
48.50
The other criterion is combination results of R2 , Adjusted R2
and Predicted R2 . Fig. 8 shows the coefficients of determination
results. As shown in Fig. 8, R2 , Adjusted R2 and continuously
increase by increasing the polynomial order. However, at a certain point of Cubic order, the Predict R2 starts to decrease. It is
seen that the Predict R2 at Quartic order is smaller than that at
Cubic order, which means the Quartic correlation is not robust.
Therefore, in combination with the criterion 1, the Cubic order
is chosen for the mathematical model.
F-value
P-value prob > F
Remark
224.74
10.53
27.14
4.49
2.24
<0.0001
<0.0001
<0.0001
<0.0001
<0.0001
Suggested
Aliased
Parametric sensitivity study on simulation results of Q9 is
also conducted based on F-test. The smaller P-values indicate
the higher significance of the corresponding parameter. As
shown in Table 7, the terms of leak rate A, wind speed B, leak
direction D, leak rate and wind speed AB, leak rate and leak
direction AD, wind speed and leak direction BD play vital roles
in affecting Q9 since the corresponding P-values are less than
0.05.
Based on the results derived by ASMT, Cubic model is chosen and Ln is used to transform the response values of Q9. In
56
Process Safety and Environmental Protection 1 1 4 ( 2 0 1 8 ) 48–63
Fig. 7 – Equivalent stoichiometric cloud volumes versus time of varied leak rates under different wind speeds when leak
direction is −X and wind direction is 270◦ .
a) Simulation results of varied leak rates under 1.5 wind speed. b) Simulation results of varied leak rates under 5 wind
speed. c) Simulation results of varied leak rates under 7 wind speed. d) Simulation results of varied leak rates under 12
wind speed. e) Q9 versus different leak rates under different wind speed.
Table 7 – Parametric sensitivity on simulation results
(Q9).
Fig. 8 – Results of coefficient of determination.
Source
SSres
p
MSres
F-value
P-value prob > F
A
B
C
D
AB
AC
AD
BC
BD
CD
Residual
Cor total
33.92
6.21
0.033
360.15
3.40
0.60
21.04
0.016
2.45
1.37
53.93
486.09
1
1
1
5
1
1
5
1
5
5
352
378
33.92
6.21
0.033
72.03
3.40
0.60
4.21
0.016
0.49
0.27
0.15
108.50
221.41
40.55
0.21
470.18
22.22
3.89
27.47
0.11
3.19
<0.0001
<0.0001
<0.0001
0.6437
<0.0001
<0.0001
0.0494
<0.0001
0.7443
0.0078
Process Safety and Environmental Protection 1 1 4 ( 2 0 1 8 ) 48–63
57
Fig. 9 – Comparison of internally studentized residuals between transformed correlation and original correlation.
a) Normal plot of residuals from the original. b) Normal plot of residuals from the transformed. c) Residuals versus the
experimental run order of the original. d) Residuals versus the experimental run order from the transformed.
order to simplify the correlation, stepwise multiple regression
method (Draper and Smith, 2014) is used to delete the terms
that have less effect on the prediction ability. Final equation
of actual factors in terms of leak direction −X is determined
as below:
Ln(Q9)=+7.52986+0.023671 × A − 0.063698 × B − 6.48637E −
003 × C+3.44187E − 003 × A × B+4.54538E − 005 × A × C −
4.21516E − 004 × B × C − 9.41053E − 004 × A
−005 × C
2 − 3.34552E − 005 × A
−007 × A × C
−006 ∗ A
5.3.4.
2 × B − 1.38770E
2+1.96656E − 006 × B × C
3 − 1.28618E − 007 × C
2+5.75984E
2+7.94896E
3
(17)
Diagnosis analysis on correlation transformation
Diagnosis analysis is to check if the determined correlation
satisfies the normal assumption initially made. However, the
variances of the raw residuals at different input samples may
differ, which leads to the raw residuals belong to different normal populations or normal distributions even if the variances
of the errors are equal at these different samples. Therefore,
internally studentized residuals should be used to map all the
different normal distributions into a single standard normal
distribution.
As correlation transformation is conducted in above section, the part is to check the feasibility of transformed
correlation. Meanwhile the transformed one (Ln(cubic)) is compared with the original one (cubic). Fig. 9 shows comparison
results of diagnosis analysis of internally studentized residuals between original and transformed correlation. As can be
seen, both the residuals from the transformed and original
one fall on a straight line, which means both residuals are normally distributed. However, Ln(cubic) indicates better residuals
normal distribution as less outliers occur compared with the
cubic. Fig. 9c) and d) presents residuals versus the experimental run order from both correlations. The random pattern of
residuals indicates the suitability of both correlations. As can
be seen, Ln(cubic) is relatively more robust as less experimental
points are above the 95% confident intervals.
In order to effectively determine the correct power law
transformation, i.e. b of Ln(cubic + b), Box-Cox plot for power
transform is used. As can be seen in Fig. 10, pink color line
indicates the upper (0.17) and the lower (0) limits of the 95%
confident intervals for the Box-Cox lambda value, the blue
(overlapping with the pink color line) and green lines represent
the current (0) and the best (0.09) lambda values, respectively.
The data in Fig. 10 indicates that b = 0 is suitable for the transformed correlation as green line locates between the upper
and lower limits of 95% confident intervals.
Fig. 11 shows comparison results between Ln(cubic) and
(cubic) in terms of predicted results versus simulation results.
As can be seen, the majority of predicted results by Ln(cubic)
are within 50% intervals, which further verify the suitability
of transformation compared with (cubic), as lots of predicted
results by (cubic) locate beyond the acceptable range.
The fitness and accuracy of the Ln-Cubic mathematical
model are verified in the above diagnosis analysis. Therefore,
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Process Safety and Environmental Protection 1 1 4 ( 2 0 1 8 ) 48–63
5.4.2.
Fig. 10 – Box-Cox plot for power transform.
the mathematical model is used to predict dispersion results
in next step. Extra simulations are also performed to verify the
generalization of the procedure in Section 5.5.
5.4.
Response surfaces supporting Q9 with acceptable
interval for ERAs
The visual 3D surfaces and 2D lines, which consists of different
Q9s with acceptable interval, are generated in the simplified
procedure.
5.4.1.
Surface of leak rate and wind speed on Q9
3D surface of leak rate and wind speed data vs. Q9 is shown
in Fig. 12. The leak direction is aligned to −X and wind
direction is 0◦ (coming from the North). As can be seen, the
flammable cloud volume Q9 firstly increase and then decrease
with increasing leak rates under a specific wind speed. In addition, the Q9 decreases with increasing wind speeds when leak
rate is 5 kg/s while Q9 increases when leak rate is 80 m/s. The
above observations are corresponding to the simulation data
shown in Fig. 7, where the wind direction is 270◦ .
Fig. 13 shows 2D plot presenting the predicted lines with
95% confident intervals and 95% prediction intervals under
varied leak rates and wind speeds. The leak direction is aligned
to −X and wind direction is 270◦ . From Fig. 13, one can see
most of simulation results locate between the upper limits and
lower limits of 95% confident intervals and all the simulation
results are within 95% prediction intervals.
Surface of leak direction on Q9
In Table 7, the most different parameter (i.e. leak direction)
has been determined by using the AVOVA. Therefore, the Least
Significant Difference (LSD) Bars are further used to determine
the specific leak direction which is significantly different from
others under same condition. The width of the bars is determined by the sub-sample size and the confidence level (i.e.
95% in this case). If the LSD bars for two directions are not
overlapping, then these two directions are significantly different.
The single effect of leak direction on Q9 is demonstrated in
Fig. 14. The leak rate is 80 kg/s, wind speed is 12 m/s and wind
direction is from South to North (270◦ ). On one hand, the lower
and upper values of LSD bar for direction +X are 74 and 213
as shown in Fig. 14, these values are significantly lower than
other Q9 values. The LSD bar does not overlap with others,
which means that factor of +X direction is significantly different from others. However, little attention should be paid for
scenarios with +X leak direction, since the flammable clouds
from this direction are much smaller (i.e. 100 m3 of Q9) than
other clouds from other directions, as seen in Fig. 13. The small
gas cloud size of 100 m3 would not lead to meaningful gas
explosion in the end. On the other hand, it is worth noting
that more attention should paid for −Z direction since the
flammable cloud in −Z direction is considerably larger than
these from other directions. The LSD bar does not overlap
with that of +Z direction. Overall, after the ANOVA and LSD
bar calculation, −Z is determined to be the most different factor. Therefore, for safety engineers, the most attention should
be paid in −Z leak direction.
5.5.
Comparison between FFDOE-based RSM and
Frozen Cloud Approach (FCA)
Overall, 864 simulation results are derived by using FLACS for
varied leak rates, i.e. 5 kg/s, 10 kg/s, 20 kg/s, 40 kg/s, 60 kg/s,
80 kg/s, varied wind speed, i.e. 1.5 m/s, 3 m/s, 5 m/s, 6 m/s,
7 m/s, 12 m/s, 4 wind directions and 6 leak directions. Except
for the results developing the RSM, the rest are used as a
benchmark to compare both FFDOE-based RSM and FCA.
FCA is proposed by DNV and it assumes that the fuel volume fraction is approximately proportional to the ratio (e.g.
leak rate/ventilation rate) for given dispersion or ventilation
scenarios. With this assumption, if a Q91 at leak rate A1 and
ventilation rate C1 is estimated by using FLACS, to determine
an unknown Q92 at a different leak rate A2 or ventilation rate
Fig. 11 – Comparison between the transformed and original correlation in terms of predicted results versus simulation
results.
a) Is the original one (cubic). b) Is the transformed one Ln (cubic).
Process Safety and Environmental Protection 1 1 4 ( 2 0 1 8 ) 48–63
59
Fig. 12 – 3D results of predicted flammable cloud size of leak rate and wind speed when the leak direction is aligned to −X
and wind direction is 270◦ (red dots present the simulation results, contour are curves of constant response drawn in the
leak rate and wind speed plane keeping all other parameters fixed. Each contour corresponds to a particular value of Q9).
(For interpretation of the references to color in this figure legend, the reader is referred to the web version of this article.)
Fig. 13 – Predicted lines with corresponding lines of 95% confident intervals and 95% predicted intervals under varied leak
rates and wind speeds when the leak direction is aligned to −X and wind direction is 270◦ .
a) 1.5 m/s wind speed. b) 5 m/s wind speed. c) 7 m/s wind speed. d) 12 m/s wind speed.
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Process Safety and Environmental Protection 1 1 4 ( 2 0 1 8 ) 48–63
Fig. 14 – Single effect of leak direction when leak rate is
80 kg/s, wind speed is 12 m/s and wind direction is from
South to North (270◦ ).
C2 , the Q92 is calculated by multiplying the Q91 by a factor of
A2 C1 /A1 C2 (GexCon, 2015).
Extra non-simulation dispersion results are predicted by
both of the simplified procedure and FCA, where the gas dispersion conditions are the same. Fig. 15 shows the comparison
results between the simulation results and predicted results
from these two methods when wind speed is 3 m/s, wind
direction is 270◦ and leak direction is −X. The results from RSM
are predicted by using a single correlation. For cases with wind
speed of 3 m/s and varied leak rates of 5 kg/s, 20 kg/s, 60 kg/s
and 80 kg/s, the FCA results are predicted by using given dispersion scenarios’ data when wind speed is 1.5 m/s (multiplied
by a factor of 0.5 as the ventilation rate is 0.013 s−1 ) and wind
speed is 5 m/s (multiplied by a factor of 1.6 as the ventilation
rate is 0.027 s−1 ). As can be seen Fig. 15a), there is a significant difference between the simulation results and that of
FCA when wind speed is 1.5 m/s for small leak rate. Fig. 15b)
also shows the comparison results under leak rates 20 kg/s and
60 kg/s when wind speed is 3 m/s, wind direction is 270◦ and
leak direction is −X. For FCA, the result under leak rate 20 kg/s
is predicted by using given scenario data when leak rate is
10 kg/s (a factor of 2 is used for the multiplication). While the
result under leak rate 60 kg/s is calculated by multiplying given
scenario (i.e. leak rate is 40 kg/s) data by a factor of 1.5. From
it, one can see the results predicted by the single correlation
are closer to the simulation results compared to those of FCA.
Fig. 16 shows comparison results among the simplified procedure, FCA and CFD of 864 simulation scenarios. As can be
seen in Fig. 16a), the results predicted by the simplified procedure agree well with those of CFD. The majority of these
results are within +50% confident intervals. Whereas more
results derived by FCA in Fig. 16b) are smaller than those estimated by using CFD. Fig. 16c) shows the exceedance frequency
curves for all simulation results. CFD simulation data is used
as a benchmark.
The leak frequency used for those curves is calculated by
using DNV LEAK software based on the HSE database. These
databases provide leak frequency depending on the size or
type of equipment and piping elements. The wind frequency
is derived based on the statistics of wind conditions. Fig. 16c)
indicates that the curve derived from FFDOE-based RSM is
closer to the CFD curve than that of FCA. In other words, the
simplified procedure is more reliable than FCA in ERA when
wind condition is not dominant.
5.6.
Comparison between FFDOE-based RSM and
DRSM
The disadvantages of conventional DRSM are shown in the
comparison of the predicted results between FFDOE based
RSM and conventional DRSM.
Fig. 15 – Comparison results between FFDOE-based RSM, FCA and simulation results when wind speed is 3 m/s, wind
direction is 270◦ and leak direction is −X.
a) Results for varied leak rates. b) Results when leak rates are 20 kg/s and 60 kg/s. c) Simulation results of 3 m/s by CFD. d)
Predicted simulation result of 3 m/s by FCA when wind speed is 1.5 m/s.
Process Safety and Environmental Protection 1 1 4 ( 2 0 1 8 ) 48–63
61
Fig. 16 – Comparison results between FFDOE-based RSM, FCA and CFD under 846 simulation scenarios. Horizontal axis
shows all the CFD results.
a) Predicted results by this procedure versus CFD results. b) Predicted results by FCA versus CFD results. c) Exceedance
frequency curve versus equivalent stoichiometric cloud volume.
Fig. 17 – Dimensionless data predicted by FFDOE-based RSM (the dots present the simulation dimensionless data, the lines
present the dimensionless results generated by FFDOE-based RSM, Vf presents the maximum flammable cloud volume and
V presents the volume of the process module, R is the ratio of the volume flow of flammable gas Qg and the volume of air Qa .
a) Results of wind speed 1.5 m/s. b) Results of wind speed 5 m/s. c) Results of wind speed 3 m/s and this dimensional
simulation results are shown in Fig. 15c).
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Process Safety and Environmental Protection 1 1 4 ( 2 0 1 8 ) 48–63
Generally, conventional DRSM considers dimensionless
parameters to generate several curves or surfaces. There are
two traditional ways to predict gas cloud volume, namely,
non-conservative DRSM and conservative DRSM (Huser and
Kvernvold, 2000; Qiao and Zhang, 2010). For non-conservative
DRSM, if dimensionless data when wind speeds are 1.5 m/s
and 5 m/s are used, two correlations should be firstly developed. These two correlations then will be used to predict
non-simulated data for varied leak rates under these two wind
speeds. However, the gas cloud volume under wind speed
of 3 m/s can not be predicted by using the two correlations
derived above (i.e. when wind speeds are 1.5 m/s and 5 m/s).
Alternatively, new simulations under wind speed of 3 m/s
should be conducted to derive a new correlation accordingly,
which may bring extra workload for engineers. In term of the
conservative way, the correlations in the conventional RSM
do not distinguish the differences between parameters if the
ratio of R, which is equal to the volume flow of flammable
gas Qg divided by the volume of air Qa , is constant in all different scenarios. Taking Fig. 17 as an example, for the same
ratio of R = 0.2, the predicted results by this procedure are
0.34, 0.37, 0.40 with 95% confident intervals, respectively, while
the conservative DRSM will only provide a same result as the
ratio of R is constant. Furthermore, the curve of wind speed
3 m/s in Fig. 17 is composed of non-simulated data by using
FFDOE-based RSM. It is seen that the curve with 95% confident
intervals agrees well with simulated results at wind speed
of 3 m/s. Whereas in order to derive a corresponding curve
by DRSM, a large number of CFD simulations are required,
thereby increasing the computation burden.
6.
Discussion
Although the advantages of the FFDOE-based RSM (i.e. the
simplified procedure) have been discussed in the comparisons
above, there are two concerns of this procedure.
First of all, there are still a large number of simulations
conducting in this study. 384 simulations are used to develop
the correlation and 500 simulations are used for validation.
It is true that 384 simulations bring lots of computation
burden. However, only 16 simulations for each set of gas
dispersion including different leak directions and wind directions, etc. are performed. In other words, the total number of
simulation cases to develop a RSM correlation would significantly decrease if the number of interactive parameters (e.g.
leak direction and wind direction, etc.) in one gas dispersion
set decrease. Assuming that a combination of 3 interactive
parameters are used for one gas dispersion simulation, the
total simulation numbers are actually 48, which is on a moderate simulation amount level. Even though only 3 interactive
parameters are considered, the FFDOE-based procedure can
predict more accurate results than FCA.
The second concern is about the decision of the ignition
model for future ERAs after the gas dispersion risk analysis
in study. So far, there are two widely-used ignition models,
i.e. UKOOA and TDIIM models. Due to the fact that only peak
values are used in this paper, UKOOA is more suitable and convenient for future gas explosion simulations. It is noted that
the FFDOE-based procedure couples the time factor into the
correlation to consider the transient variation of Q9, Q6 and
FLAM, complicated correlation is not surprisingly seen. Future
research work should be performed to build the transient relationship between the Q9, Q6 and FLAM.
7.
Conclusion
This paper proposes a simplified and more robust procedure
to predict the flammable cloud size with acceptable intervals.
Compared with two widely-used methods, namely DRSM and
FCA, the proposed procedure has several advantages.
Firstly, compared with DRSM, the procedure is computationally efficient as it uses dimensional parameters to derive
the single correlation rather than dimensionless ones.
Secondly, due to the fact that the ASMT (i.e. the combination of F-test and coefficient determination R2 calculation) is
used to quickly determine the suitable polynomial order and
structure, the procedure is more robust and user-friendly than
DRSM.
Thirdly, compared to the FCA, the simplified procedure can
predict more accurate results for fuel-dominated region of
offshore platform. Because the interactive effect of different
parameters are taken into account by using the FFDOE.
Finally yet importantly, the procedure can be used to generate visual 3D surfaces and 2D lines with 95% confident
intervals. The 3D surfaces and 2D lines can be further used
for ERAs with stochastic probability analysis, which can be
provided as a safety design guidance along with parametric
sensitivity analysis. However, it is noted that steady releasing leak rate is conservatively adopted in this study. Transient
models such as leak rate, ESD and blow down system, are not
considered in gas dispersion simulations.
Acknowledgments
This study was supported by the Fundamental Research
Funds for Innovation Program of Seventh-generation Ultra
Deepwater Drilling Platform [grant numbers 2016[24]], the
Fundamental Research Funds for the Central Universities [grant numbers 15CX05018A], and the Fundamental
Research Funds for the Central Universities [grant numbers
16CX06019A] and the Graduate Student Innovation Projects of
China University of Petroleum [grant numbers YCXJ2016056].
The authors also greatly appreciate the guidance from Mr.
Olav Roald Hansen at Lloyd’s Register Consulting and the technology support about dispersion simulation from Gexcon.
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