European Conference on Computational Fluid Dynamics
ECCOMAS CFD 2006
P. Wesseling, E. Oñate, J. Périaux (Eds)
© TU Delft, The Netherlands, 2006
NUMERICAL STUDY FOR ACCIDENTAL GAS RELEASES
FROM HIGH PRESSURE PIPELINES
Nicola Novembre*, Fabrizio Podenzani*, Emanuela Colombo†
* EniTecnologie S.p.A., dpt. SVIL-IPM
via F. Maritano 26, 20097 San Donato milanese, Milan, Italy
e-mail: [email protected]
[email protected]
†
Politecnico di Milano
piazza Leonardo da Vinci 32, 20133 Milan, Italy
e-mail: [email protected]
Key words: free jet, methane release, high pressure, pipeline, accidental event, gas dispersion
Abstract. This work concerns the analysis of consequences of gas releases from high
pressure pipelines due to accidental events. This analysis has been performed using the CFD
code Fluent. The first step of the work intended to evaluate the capability of Fluent to
adequately simulate a supersonic underexpanded free jet. This validation has been based on
small scale experimental data from literature. Methane jets from high pressure (from 10 to
250 barg), large diameter (0.5 m) pipelines have then been simulated. As second step of our
analysis, a simplified model (based on works by Birch et al.) to handle the highlycompressible-fluid region of such kind of jets has been checked. Birch model resulted reliable
once completed with two relations, one to evaluate the air entrainment and the other to
compute the distance from the release point at which fluid catches up the atmospheric
pressure.
1 INTRODUCTION
Pipelines are very important means to transport fluids. Moreover they are relatively safe.
Nevertheless, if an accident occurs, consequences may be dramatic. Recognising that great
care should be taken to people, environment and property, industry is going toward a
“responsible care” approach1. It means that all possible solutions to prevent accidents and to
mitigate consequences should be activated at any levels. In particular, industry and public
authorities are called to set up strong co-operations. The former should improve safety at any
levels (design, construction, operation, maintenance), while the latter has the responsibility to
state appropriate regulatory approaches and standards.
Several approaches and methodologies have been developed and are still under analysis in
this field in order to address the above mentioned topic. Analysis of consequences focuses on
predicting accident scenarios and consequences in order to understand and quantify the
involved risks. Results will support both industry and all the actors involved in the
development of appropriate codes and standards.
Nicola Novembre, Fabrizio Podenzani and Emanuela Colombo.
Due to different causes (typically third part interferences), pipelines can undergo loss of
integrity which may lead to failure. In the worst case, a failure results in a burst that breaks
the line and, for buried pipelines, removes soil producing a crater2. Consequently, gas is
released. According to rupture configuration, gas can interact with some obstacle and then
spreads out in the ambient. Furthermore, an ignition source can light up the cloud of the
flammable fluid and then a combustion process can occur.
Performing an analysis of consequences means evaluating adverse effects involved in the
phenomenon in order to estimate the key safety-related parameters. In the above described
scenario, the main adverse effect is associated with thermal load at ground level due to
radiation while the associated safety-related parameter is the width of the dangerous area
required to protect people, environment and other assets3,4.
Modelling gas releases is compulsory in order to evaluate jet characteristics and then to
verify whether combustion may take place if ignition occurs.
Gas release process is affected by the depressurization mode of the pipe section (which
depends on the pipe system) and by the mode and the final configuration of rupture. In this
study, a steady-state blow out and a full bore rupture (guillotine rupture) have been assumed.
Fluid releases from high pressure pipeline must be treated as compressible flows; they are
characterized by interaction between expansion waves and compression waves and by a
typical structure made by oblique shocks and one or more normal shocks (“diamond”
structure).
Within safety analyses, interest is focused on the gas dispersion in the far field rather than
on the details of the flow structure near the release point where fluid experiences high changes
in flow quantities. For this reason, some simplified semi-empiric models are available for
calculating jet conditions in the cross-section where the jet has reached the atmospheric
pressure. The given values can be used as inlet boundary conditions for simulations in which
the fluid can be treated as incompressible. Such calculations are computationally by far much
less expensive than those with compressible fluids and therefore are suitable to investigate jet
spreading in very large domains. Such models provide the expanded jet dimension, the mean
velocity, the mean temperature and the mean density over the cross-section. Only few models
take into account also the air entrainment.
This work presents results of the application of such a model using CFD analyses,
performing computations with the commercial code Fluent. The first step intended to evaluate
the capability of Fluent in simulating a supersonic underexpanded free jet. This check has
been based on small scale experimental data taken from literature5. Methane jets from high
pressure (from 10 to 250 barg), large diameter (0.5 m) pipelines have then been simulated.
Results have been compared with some experimental data6,7. In all these simulations, fluids
are regarded as compressible. As second step of our analysis, a simplified model (based on
Birch et al. work8) for handling the highly-compressible-fluid region of such kind of jets has
been used coupling it with CFD incompressible-fluid analyses to investigate the far field
region. Results have been compared with the above mentioned compressible-fluid simulations
(used as reference, fig. 1). Birch model resulted reliable once completed with two relations
which make it a fully predictive model.
Nicola Novembre, Fabrizio Podenzani and Emanuela Colombo.
2 THEORETICAL BACKGROUND
2.1 Turbulence modeling
When a high pressure gas discharges through an orifice into an ambient where pressure is
much lower than the exit pressure, “choked” conditions occur at the outlet and a so called
“underexpanded” free jet expands to the ambient pressure through an interaction between
expansion waves and compression waves, creating a typical structure with oblique shocks and
one or more normal shocks, known as Mach discs.
In the normal shock region (close to the release point), turbulence is negligible; this
explains why such high gradients can exist (sometimes this phenomenon is treated as not
viscous by using Euler equations). Further downstream, as the jet looses its momentum,
turbulence becomes predominant and governs the mixing between released fluid and air. Gas
dispersion in the far field is greatly affected by turbulence. Hence, turbulence plays a key role
in the overall phenomenon.
In the present work, the Reynolds-averaged approach has been employed for the governing
Navier-Stokes equations (RANS approach). Among the two equations models, κ-ε standard,
κ-ε RNG and κ-ω SST have been tested9.
2.2 Simplified models for supersonic underexpanded free jets
As previously said, from an engineering point of view it is more important the overall gas
dispersion in the far field than the highly-compressible-fluid region close to the release point.
This region can therefore be viewed as a “black box” characterized by an appropriate model.
Such a model should provide jet conditions at the atmospheric pressure (once the fluid has
expanded) as output data using the stagnation quantities in the reservoir (pipeline) as input
data.
The 1987 Birch et al. work proposes a method to evaluate the conditions of a supercritical
jet once it has reached the ambient pressure. This method is based on conservation of mass
and momentum through the free expansion region (region between the jet nozzle exit and the
atmospheric cross-section) and assumes isentropic expansion from stagnation (in the
reservoir) to chocked (at the nozzle exit) conditions. It neglects viscous forces over the free
expansion surface and the entrainment of ambient fluid. Temperature at the atmospheric
cross-section is equal to the stagnation temperature. Gas is treated as ideal. Once the
stagnation conditions (pressure and temperature) are known, it is possible to calculate the jet
conditions after expansion (diameter, velocity and density) over the so called “pseudo source”
of the jet.
Clearly, this method has two limitations: it is a zero-dimension mathematical method (it
does not provide the distance from the nozzle exit of the pseudo source) and does not take into
account the air entrainment.
A first improvement consists in including conservation of energy to calculate the
temperature at the atmospheric cross-section (like10,11).
A further improvement was made by ComputIT11 to take into account the effects of air
entrainment on outflow conditions, but the actual amount of the entrained air is an input in the
Nicola Novembre, Fabrizio Podenzani and Emanuela Colombo.
model. A way for evaluating air entrainment is presented here below.
To make the model fully predictive, an additional relation to calculate the distance from
the nozzle exit of the atmospheric-pressure cross-section is also proposed in the following.
Air entrainment. We started from a work by Spalding12 and carried on by Hess13. Hess
proposed the following relation (based on the momentum equation) to describe jet
entrainment:
Qexit + Qentr
z ⎛ ρ
= K e ⋅ ⋅ ⎜⎜ a
Qexit
D ⎝ ρ exit
1
⎞2
⎟⎟
⎠
1
⎛
p −p
⋅ ⎜⎜1 + exit 2 a
ρ exit ⋅ U exit
⎝
⎞2
z **
⎟⎟ = K e ⋅
D
⎠
(1)
where:
⎛ ρ
z ** = z ⋅ ⎜⎜ a
⎝ ρ exit
1
⎞2
⎟⎟
⎠
⎛
p −p
⋅ ⎜⎜1 + exit 2 a
ρ exit ⋅ U exit
⎝
1
⎞2
⎟⎟
⎠
(2)
Qexit is the mass flow rate through the orifice (rupture);
Qentr is the mass flow rate of fluid entrained;
K e is a constant;
z is the distance from orifice;
D is the orifice diameter;
ρ a is the density at atmospheric pressure;
ρ exit is the density at the orifice;
p a is the atmospheric pressure;
p exit is the pressure at the orifice;
U exit is the velocity at the orifice.
Once obtained the total mass flow rates over different cross-sections from our
compressible-fluid simulations, it was possible to evaluate Ke through (1). In fig. 2 Ke is
plotted as a function of
z **
D
: Ke profile seems to be not affected by total pressure in the
reservoir (it depends on the coordinate
mean Ke value for each
z **
D
z **
D
only). This circumstance suggested to compute a
(among the different pressure values) so that it can be used to
evaluate air entrainment over the cross-section chosen (say
is necessary to determine the
z **
D
z **
D
). Following this approach, it
corresponding to the atmospheric-pressure cross-section
downstream of the nozzle exit.
Distance from nozzle exit to atmospheric-pressure cross-section. Two criteria have
Nicola Novembre, Fabrizio Podenzani and Emanuela Colombo.
been adopted to evaluate such distance, suggested by analyses of our compressible-fluid
simulations:
1) ten times the distance XMach computed through experimental relation (3);
2)
z **
D
=10.
Since the goal is to get the minimum distance from which to run a incompressible-fluid
simulation (provided that solution agrees with the compressible-fluid one), also the criterion
z **
D
=5 has been applied. These three distances are shown in fig. 3 (where Mach number and
pressure along the jet axis are plotted versus the distance from pipe release section) for P0=10
barg.
3 NUMERICAL MODELS
This chapter and the following are divided into three sections which corresponds to our
analysis road map. In section 3.1, the activity performed to evaluate Fluent capability in
simulating a supersonic underexpanded free jet is presented. In section 3.2, real cases
simulations (high pressure, large diameter jets) are presented. In section 3.3, the application of
Birch et al. simplified model is presented together with our enhancements.
3.1 Small scale air jets
These simulations are based on the experimental work by Eggins et al.5 which provides
velocity measurements in an under-expanded supersonic free air jet. The jet was produced by
a converging nozzle with an exit diameter of 2.7 mm; the angle is not provided. The nozzle
was rigidly connected to an air reservoir maintained at a (total) pressure P0 of 5.7 bar above
atmospheric and at a (total) temperature T0 of 293 K. This work gives velocity profiles in
some cross-sections downstream of the nozzle exit and the velocity profile along the jet axis
(as far as 80 mm downstream of the nozzle).
A coupled implicit solver has been used to solve the governing equations for mass,
momentum and energy in steady state. Standard κ-ε, κ-ε RNG and κ-ω SST turbulence
models have been employed to get closure. The computational domain is axisymmetric and
comprises an air reservoir (supposed infinite and represented by an appropriate boundary
condition), a 20 mm long converging nozzle (whose axis is the domain axis of symmetry) and
free atmospheric air at rest. Domain is 300 mm long with a radius of 100 mm. Boundary
conditions have been set as follow:
a) converging nozzle inlet: total pressure and temperature equal to those of experiment,
turbulence-related quantities (turbulence intensity: 5%, hydraulic diameter: 4 mm);
b) domain boundaries: total atmospheric pressure, total temperature: 293 K, turbulencerelated quantities (turbulence intensity: 5%, turbulent viscosity ratio: 1000).
Air has been considered an ideal gas. Computational grid has 23700 quadrangular cells,
then increased to verify that solution is grind independent.
Nicola Novembre, Fabrizio Podenzani and Emanuela Colombo.
3.2 High pressure methane jets
Experimental data for high pressure methane jets from large diameter pipes are not
available. Nevertheless, some important conclusions are in6:
a) Mach disc location is insensitive to k (fluid nature);
b) Mach disc location is given by the following equation:
X Mach = 0.645 ⋅
P0
⋅ Dexit
P∞
(3)
X Mach is the distance from nozzle exit of the Mach disc;
Po is the (total) absolute pressure in the reservoir;
P∞ is the pressure of medium into which fluid discharges (generally air at rest);
Dexit is the nozzle exit diameter.
This relation allows a “local” check of our numerical results in the highly-compressiblefluid region.
To evaluate the results in the far field region, experimental results from7 have been
employed.
Jets of total pressure values 10, 50, 125 and 250 bar above atmospheric issuing from a
0.546 m (24”) diameter pipe have been considered.
A coupled implicit solver has been used to solve the governing equations for mass,
momentum and energy with an implicit unsteady scheme toward the steady state.
An additional transport equation for methane has been included.
Standard κ-ω model have been employed to get closure, since this model is suited to
represent free flow (as it is in the far field region) and since in the far field no significant
differences exist among the models tested.
Like the small scale simulations, the computational domain is axisymmetric and comprises
a methane reservoir, a 10 m long straight pipe and free atmospheric air at rest. Domain is 120
m long with a radius of 40 m. Boundary conditions have been set as follow:
a) pipe inlet: total pressure, total temperature (equals 353 K) and turbulence-related
quantities (turbulence intensity: 10%, hydraulic diameter: 0.546 m);
b) downstream domain boundary: static atmospheric pressure and backflow quantities
(total temperature: 300 K, turbulence intensity: 10%, turbulent viscosity ratio: 1000);
c) upstream and lateral boundaries: total atmospheric pressure, total temperature: 300 K,
turbulence-related quantities (turbulence intensity: 10%, turbulent viscosity ratio: 1000).
Methane and air has been regarded as ideal gases. Computational grid has 40000 cells.
Once provided that results from this set of simulations are acceptable, these have been used
both to investigate air entrainment (through the approach presented in section 2.3) and as
reference for incompressible-fluid simulations based on the Birch model.
3.3 Treating the far field
The far field analysis has been performed using the simplified Birch et al. model coupled
Nicola Novembre, Fabrizio Podenzani and Emanuela Colombo.
with a CFD analysis with incompressible fluids. This analysis has involved the same methane
jets presented in the previous section (10, 50, 125, 250 bar above atmospheric in the reservoir,
0.546 m pipe diameter).
A segregated (implicit) solver has been used to solve the governing equations for mass,
momentum and energy in steady state. An additional transport equation for methane has been
included. Standard κ-ω turbulence model have been employed to get closure.
The computational domain is axisymmetric with a inlet surface representing Birch’s
pseudo-source of the jet. Flow conditions (mean velocity, mean temperature and composition)
at ambient pressure provided by the Birch et al. model have been imposed as fixed boundary
conditions at the inlet surface. Inlet surface diameter is also provided by Birch model.
Boundary conditions have been set as follow:
a) Birch’s conditions and turbulence-related quantities at the inlet surface;
b) total atmospheric pressure, total temperature: 300 K and turbulence-related quantities at
the domain boundaries (turbulence intensity: 15%, turbulence viscosity ratio: 104).
Methane and air have been considered incompressible (density is a function of temperature
only) ideal gases. Computational grid has 33600 quadrangular cells.
As previously said, solutions from these (incompressible-fluid) simulations have then been
compared with results from compressible-fluid simulations.
4 RESULTS AND DISCUSSION
4.1 Small scale air jets
Qualitatively, flow structure agrees with visualizations of jets at similar conditions from
literature (fig. 4 where the shadowgraph is taken from14 and refers to a P0=20 bar, Dexit=25.4
mm jet). κ-ε RNG turbulence model permits to better resolve the details of flow structure
(note the normal shock, the reflected oblique shocks and the slip line), while standard κ-ε and
κ-ω SST models do not capture the normal shock close to the nozzle exit.
Comparison of the numerical results with the experimental data is shown in figg. 5 to 7. κε RNG solution captures both the normal shock position and the magnitude of velocity drop
through it. Downstream of it, velocity values are lower than those from experiment.
Standard κ-ε and κ-ω SST models provide the same results: velocity drop is not correctly
reproduced, while velocity values further downstream are of the same order as the
experimental even if shifted in the streamwise direction.
According to the first transverse profile (0.2 mm upstream of Mach disc, fig. 6) all the
three turbulence models provide good results, while to the second one (0.2 mm downstream
of Mach disc, fig. 7) only the κ-ε RNG solution agrees with experimental data.
To investigate results sensitivity to grid refinement, cells number has been increased to
41000 elements through an adapting velocity-gradient-based algorithm.
Standard κ-ε solution is not affected by the refining, while κ-ω SST solution is now similar
to κ-ε RNG solution but with a smaller Mach disc diameter (fig. 8).
From comparison with experimental data (fig. 9), we note that κ-ω SST solution has a even
larger velocity drop through the Mach disc compared to κ-ε RNG solution (the lowest
Nicola Novembre, Fabrizio Podenzani and Emanuela Colombo.
velocity values are even negative). Downstream of the normal shock, velocity values from κω SST solution are closer to experimental data (although not in phase) than those from κ-ε
RNG solution.
From data comparison in the cross-sectional profiles downstream the Mach disc (fig. 10),
we note that κ-ω SST predictions are now comparable with those from κ-ε RNG solution.
Because κ-ω SST was greatly affected by grid refining, a simulation with a further
increased cell number (60700 cells, by the same adapting velocity-gradient-based algorithm)
has been performed. The results have not shown significant changes.
All the velocity profiles (both the experimental one and those from computations) are
qualitatively similar. They are made by four regions, as shown in fig. 11:
1) first region (A-B), acceleration up to a velocity maximum Vmax (supersonic
expansion);
2) second region (B-C), abrupt velocity drop down to a minimum Vmin (normal
shock);
3) third region (C-D), oscillatory zone in which the velocity seems to be the sum of
two components: a basic bell-shaped component and a oscillatory damped
component whose frequency is almost constant and whose amplitude diminishes
while reaching the end of the region;
4) fourth region (D-E), where velocity diminishes asymptotically down to zero
(dispersion).
Such a schematization allows a synthetic evaluation of the three turbulence models
employed as follows:
1) supersonic expansion A-B: all the models provide results that agree with
experimental data;
2) normal shock B-C: κ-ε RNG solution is the closest to experimental data;
3) oscillatory zone C-D: standard κ-ε and κ-ω SST solutions seem to be closer to
experimental data;
4) dispersion D-E: the three models give almost similar results even quite far from
experimental data.
Regarding dispersion region D-E, we point out that a) the experimental curve has only
three points; b) a direct comparison between experimental and computed data is not very
meaningful because each solution is in some way affected by previous zones; c) from a
qualitative comparison of the curves, based for example on the “half distance” of initial
velocity value (see table 1), results are quite close to experimental data.
series
experimental
κ-ε standard
κ-ε RNG
κ-ω SST
initial velocity
(m/s)
~260
~400
~400
~400
Table 1
half distance
(mm)
~50
~40
~40
~40
Nicola Novembre, Fabrizio Podenzani and Emanuela Colombo.
4.2 High pressure methane jets
Qualitatively, flow structure in the near field is quite close to the shape expected. Mach
disc axial location and diameter increase with reservoir total pressure.
The following table reports the numerical results compared with those predicted through
relation (3). Agreement is satisfactory.
Reservoir
total pressure
(barg)
10
50
125
250
P0
P∞
11
51
126
251
Dexit (m)
0.546
Experimental Xm
(m)
Computed Xm
(m)
Pecentage
deviation
1.17
2.51
3.95
5.58
1.17
2.62
4.13
5.64
0
4.4
4.6
1.07
Table 2
In fig. 12 to 15 computed methane concentrations and those predicted by7 are compared.
Agreement is satisfactory. Methane concentration profiles on the jet axis present two zones:
close to the release point only methane exists (air has not yet reached the jet core), while in
the subsequent region methane fraction decays with x-1 law.
From these results it possible to investigate air entrainment according to the approach
presented in section 2.3. In the following table Ke values for the three cross-sections
corresponding to the three distances proposed are shown. Differences are little. Note that
cross-sections taken through the XMach-based criterion have a corresponding
z **
D
very close
each others even if the localization approaches are different; this shows that the proposed
criteria agree with the physics of the phenomenon.
Ke for
10 barg
50 barg
125 barg
250 barg
z **
=5
D
0.316
0.309
0.308
0.321
Ke for
z **
=10
D
0.231
0.220
0.219
0.237
Ke for 10 ⋅ X Mach
0.219
0.204
0.203
0.223
(z**/D=15.3)
(z**/D=15.7)
(z**/D=15.8)
(z**/D=15.9)
Table 3
The mean value for Ke among the four cases for each criterion have been computed. The
inverse process has then been applied: Ke is now known (for each
z **
D
), so the total mass flow
rates for each cross-section corresponding to the three criteria have been computed and
compared directly with those computed from compressible-fluid simulations (see table 4).
Differences are acceptable.
Nicola Novembre, Fabrizio Podenzani and Emanuela Colombo.
z **
=5
D
Mean Ke
10 barg
50 barg
125 barg
250 barg
610
2837
7024
14010
From
compressiblefluid
simulation
614
2802
6896
14347
z **
=10
D
Mean Ke
882
4104
10160
20267
From
compressiblefluid
simulation
898
3988
9806
21132
10 ⋅ X Mach
Mean Ke
1263
6031
15027
30164
From
compressiblefluid
simulation
1304
5816
14392
31612
Table 4
In this way, air entrainmet has been evaluated for the four pressure values in our
interest.
4.3 Methane jets far field
Incompressible-fluid Birch-model-based simulation results are very close to those from
compressible fluid analysis. Axial velocity profiles and methane mass fraction profiles both
along jet axis and over different cross-sections have been compared. Entrainment profiles
have also been compared.
In figg. 16 to 21 results for P0=10 barg are shown. The simulations based on
criterion provide the best results. Results from ten-times-Mach-disc criterion or
z **
=10
D
z **
=5
D
criterion are also satisfactory. Entrainment profile (fig. 28) shows that the ten-times-Machdisc criterion provides very good results.
In fig. 22 to 27 results for P0=125 barg are shown. Differences are more marked. Again,
entrainment profile shows that the ten-times-Mach-disc criterion results are closer to
compressible-fluid results.
5 CONCLUSIONS
Purpose of the work is to investigate the dispersion into the ambient of methane releases
from high pressure pipelines.
Because interest is not focused on the highly-compressible-fluid region close to the release
point, the following approach has been used:
preliminary CFD compressible-fluid analyses over the entire domain have been
performed to validate Fluent code; different turbulence models have been employed;
a simplified model to get flow conditions at ambient pressure of high pressure jets
has been checked, completed and applied;
two relations have been added to the model to evaluate a) air entrainment; b) distance
from rupture of the ambient pressure cross-section;
CFD incompressible-fluid analyses based on Birch model outputs have been
Nicola Novembre, Fabrizio Podenzani and Emanuela Colombo.
performed to investigate gas dispersion.
The main conclusions are as follows:
no turbulence model gives satisfactory results over the entire domain; nevertheless
each model is able to resolve a particular feature of the phenomenon which is
characterized by a complex multi-regimes fluid dynamics.
results from CFD incompressible-fluid analyses are very close to those from direct
CFD compressible-fluid analyses over the entire domain;
This study allows to compute the flow field of a high pressure methane jet far away from
the release point in a fast way (CFD incompressible-fluid analysis). Flow fields will be used
to study combustion processes.
REFERENCES
[1] Report of the OECD Workshop on Pipelines (Prevention of, Preparedness for, and
Response to Releases of Hazardous Substances), Oslo, 3rd-6th June 1996.
[2] Fifteen die in Belgium gas blast, BBC NEWS http://news.bbc.co.uk/2/hi/europe/3939087.stm
[3] A model for sizing high consequence areas associated with natural gas pipelines, Gas
Research Institute, GRI-00/0189, October 2000.
[4] Line rupture and the Spacing of Parallel Lines, PRCI, catalog no. L51861, April 2, 2002.
[5] P. L. Eggins, D. A. Jackson, Laser-Doppler velocity measurements in an under-expanded
free jet, J. Phys, D: Appl. Phys., Vol. 7, 1974.
[6] Crist, P. M. Sherman, D. R. Glass, Study of the Highly Underexpanded Sonic Jet, AIAA
JOURNAL, VOL. 4, NO. 1, 1966.
[7] A. D. Birch, D. R. Brown, M. G. Dodson, F. Swaffield, The Structure and Concentration
Dacay of High Pressure Jets of Natural Gas, Combustion Science and Technology, 1984,
Vol. 36, pp. 249-261.
[8] A. D. Birch, D. J. Hughes, F. Swaffield, Velocity Dacay of High Pressure Jets, ibidem,
1987, Vol. 52, pp. 161-171.
[9] D. C. Wilcox, Turbulence Modeling for CFD, DCW Industries, Inc. La Canada,
California, Second Edition.
[10] I. O. Sand, K. Sjon, J. R. Bakke, Modelling of release of gas from high pressure
pipelines, Int. Journal for Numerical Methods in Fluids, 23: 953-983, 1996.
[11] Kameleon FireEx 2000 Theory Manual, Report no.: R0123, 2001.
[12] F. P. Ricou, D. B. Spalding, Measurements of entrainment by axisymmetrical turbulent
jets, Journal of Fluid Mechanics, Vol. 11, 1961.
[13] K. Hess, W. Leuckel, A. Stoeckel, Ausbildung von explosiblen Gaswolken bei
Uberdachentspannung und Massnahmen zu deren Vermeidung, Chemie-Ing.-Techn. 45:
1973/Nr.5.
[14] B. C. R. Ewan, K. Moodie, Structure and Velocity Measurements in Underexpanded
Jets, Combust. Sci. and Tech., 1986 Vol. 45, pp. 275-288.
Nicola Novembre, Fabrizio Podenzani and Emanuela Colombo.
expanding jet
pipe end
cross-section where
p ≈ patmospheric
highly-compressible-fluid region
incompressible-fluid region
Domain for compressiblefluid analysis
Domain for incompressiblefluid analysis
Figure 1: scheme of the phenomenon and of the approach used
0.36
0.34
0.32
Ke
0.3
0.28
0.26
0.24
0.22
0.2
0
10
20
30
40
50
60
z**/D
Figure 2: __ 10 barg __ 50 barg __ 125 barg __ 250 barg
70
80
Nicola Novembre, Fabrizio Podenzani and Emanuela Colombo.
700000
4
absolute pressure (Pa)
600000
Mach number
3.5
3
2.5
2
1.5
1
0.5
3.8
7.6
500000
400000
300000
200000
100000
11.7
3.8
0
7.6
11.7
0
0
5
10
15
20
25
distance from rupture (m)
30
35
40
0
2
4
6
8
10
distance from rupture (m)
Figure 3: __ CFD solution (P0=10 barg) __ 10*XMach __ z**/D=10 __ z**/D=5
κ-ε
κ-ε RNG
κ-ω SST
Figure 4: 23700 cells grid, velocity fields (m/s)
12
14
16
Nicola Novembre, Fabrizio Podenzani and Emanuela Colombo.
700
600
axial velocity (m/s)
500
400
300
200
100
0
0
2
4
6
8
10
12
14
16
18
20
distance from nozzle exit (mm)
Figure 5: 23700 cells grid, velocity profile along the jet axis
__ experimental data __ κ-ε solution __ κ-ε RNG solution __ κ-ω SST solution
600
550
550
500
500
450
axial velocity (m/s)
axial velocity (m/s)
650
450
400
350
300
250
400
350
300
250
200
200
150
150
100
0
0.5
1
1.5
distance from jet axis (mm)
2
0
0.5
1
1.5
distance from jet axis (mm)
Figure 6 (left): 23700 cells grid, velocity profile over the cross-section upstream of Mach disc
Figure 7 (right): 23700 cells grid, velocity profile over the cross-section downstream of Mach disc
__ experimental data __ κ-ε solution __ κ-ε RNG solution __ κ-ω SST solution
2
Nicola Novembre, Fabrizio Podenzani and Emanuela Colombo.
κ-ε
κ-ε RNG
κ-ω SST
Figure 8: 41000 cells grid, velocity fields (m/s)
700
550
600
500
450
axial velocity (m/s)
axial velocity (m/s)
500
400
300
200
100
400
350
300
250
200
150
100
0
0
2
4
6
8
10
12
14
-100
16
18
20
50
0
distance from nozzle exit (mm)
0.5
1
1.5
distance from jet axis (mm)
Figure 9 (left): 41000 cells grid, velocity profile along the jet axis
Figure 10 (right): 41000 cells grid, velocity profile over the cross-section downstream of Mach disc
__ experimental data __ κ-ε solution __ κ-ε RNG solution __ κ-ω SST solution
2
Nicola Novembre, Fabrizio Podenzani and Emanuela Colombo.
600
axial velocity (m/s)
500
400
300
200
100
0
0
10
20
30
40
50
60
70
-100
distance from nozzle exit (mm)
Figure 11. Top: 41000 cells grid, velocity profile along the jet axis
__ experimental data __ κ-ε solution __ κ-ε RNG solution __ κ-ω SST solution
Bottom: schematisation
80
Nicola Novembre, Fabrizio Podenzani and Emanuela Colombo.
1
0.9
0.8
0.8
0.7
CH4 mass fraction
CH4 mass fraction
1
0.9
0.6
0.5
0.4
0.3
0.7
0.6
0.5
0.4
0.3
0.2
0.2
0.1
0.1
0
0
0
20
40
60
80
100
0
20
40
distance from rupture (m)
60
80
100
120
100
120
distance from rupture (m)
Methane concentration along jet axis
Figure 12 (left): P0=10 barg
Figure 13 (right): P0=50 barg
1
1
0.9
0.9
0.8
CH4 mass fraction
CH4 mass fraction
0.8
0.7
0.6
0.5
0.4
0.7
0.6
0.5
0.4
0.3
0.3
0.2
0.2
0.1
0.1
0
20
40
60
80
100
120
0
20
distance from rupture (m)
Methane concentration along jet axis
Figure 14 (left): P0=125 barg
Figure 15 (right): P0=250 barg
__ experimental data __ CFD solution
40
60
80
distance from rupture (m)
Nicola Novembre, Fabrizio Podenzani and Emanuela Colombo.
0.9
0.8
800
CH4 mass fraction
axial velocity along jet axis (m/s)
1
1000
600
400
0.7
0.6
0.5
0.4
0.3
0.2
200
0.1
0
0
0
20
40
60
80
100
0
120
20
40
60
distance from rupture (m)
80
100
120
distance from rupture (m)
Figure 16 (left): P0=10 barg, axial velocity along jet axis
Figure 17 (right): P0=10 barg, methane concentration along jet axis
0.18
140
0.16
120
0.14
CH4 mass fraction
axial velocity (m/s)
160
100
80
60
40
0.12
0.1
0.08
0.06
0.04
20
0.02
0
0
0
1
2
3
4
5
6
7
0
1
2
distance from jet axis (m)
3
4
5
6
7
distance from jet axis (m)
60
0.07
50
0.06
CH4 mass fraction
axial velocity (m/s)
Figure 18 (left): P0=10 barg, axial velocity over cross-section z=30m from rupture
Figure 19 (right): P0=10 barg, methane concentration over cross-section z=30m from rupture
40
30
20
10
0.05
0.04
0.03
0.02
0.01
0
0
0
2
4
6
8
10
12
distance from jet axis (m)
14
16
18
0
2
4
6
8
10
12
14
16
18
distance from jet axis (m)
Figure 20 (left): P0=10 barg, axial velocity over cross-section z=60m from rupture
Figure 21 (right): P0=10 barg, methane concentration over cross-section z=60m from rupture
__ compressible-fluid solution __ incompressible-fluid solution 10*XMach __ incompressible-fluid solution
z**/D=10 __ incompressible-fluid solution z**/D=5
Nicola Novembre, Fabrizio Podenzani and Emanuela Colombo.
1200
1
0.9
0.8
800
CH4 mass fraction
axial velocity (m/s)
1000
600
400
0.7
0.6
0.5
0.4
0.3
0.2
200
0.1
0
0
0
50
100
150
200
250
300
0
50
100
150
200
250
300
distance from rupture (m)
distance from rupture (m)
300
0.3
250
0.25
CH4 mass fraction
axial velocity (m/s)
Figure 22 (left): P0=125 barg, axial velocity along jet axis
Figure 23 (right): P0=125 barg, methane concentration along jet axis
200
150
100
0.2
0.15
0.1
0.05
50
0
0
0
2
4
6
8
10
12
14
0
2
4
distance from jet axis (m)
6
8
10
12
14
distance from jet axis (m)
Figure 24 (left): P0=125 barg, axial velocity over cross-section z=70m from rupture
Figure 25 (right): P0=125 barg, methane concentration over cross-section z=70m from rupture
150
0.14
130
0.12
CH4 mass fraction
axial velocity (m/s)
110
90
70
50
0.1
0.08
0.06
0.04
30
0.02
10
0
-10 0
5
10
15
20
distance from jet axis (m)
25
0
5
10
15
20
25
distance from jet axis (m)
Figure 26 (left): P0=125 barg, axial velocity over cross-section z=110m from rupture
Figure 27 (right): P0=125 barg, methane concentration over cross-section z=110m from rupture
__ compressible-fluid solution __ incompressible-fluid solution 10*XMach __ incompressible-fluid solution
z**/D=10 __ incompressible-fluid solution z**/D=5
Nicola Novembre, Fabrizio Podenzani and Emanuela Colombo.
12000
total mass flow rate (kg/s)
11000
10000
9000
8000
7000
6000
5000
4000
3000
2000
15
20
25
30
35
40
45
50
55
60
65
distance from rupture (m)
Figure 28: P0=10 barg, entrainment
total mass flow rate (kg/s)
70000
60000
50000
40000
30000
20000
10000
35
45
55
65
75
85
95
105
115
distance from rupture (m)
Figure 29: P0=125 barg, entrainment
__ compressible-fluid solution __ incompressible-fluid solution 10*XMach __ incompressible-fluid solution
z**/D=10 __ incompressible-fluid solution z**/D=5