Numerical study for accidental gas releases from high pressure pipelines

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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: nicola.novembre@enitecnologie.eni.it

fabrizio.podenzani@enitecnologie.eni.it †_{Politecnico di Milano}

piazza Leonardo da Vinci 32, 20133 Milan, Italy

e-mail: emanuela.colombo@polimi.it

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 highly-compressible-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.

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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.

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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).

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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: D z K U p p D z K Q Q Q e exit exit a exit exit a e exit entr exit * * 2 1 2 2 1 1 _⎟⎟ = ⋅ ⎠ ⎞ ⎜⎜ ⎝ ⎛ ⋅ − + ⋅ ⎟⎟ ⎠ ⎞ ⎜⎜ ⎝ ⎛ ⋅ ⋅ = + ρ ρ ρ (1) where: 2 1 2 2 1 * * ₁ ⎟⎟ ⎠ ⎞ ⎜⎜ ⎝ ⎛ ⋅ − + ⋅ ⎟⎟ ⎠ ⎞ ⎜⎜ ⎝ ⎛ ⋅ = exit exit a exit exit a U p p z z ρ ρ ρ (2) exit

Q is the mass flow rate through the orifice (rupture);

entr

Q is the mass flow rate of fluid entrained;

e

K is a constant;

z is the distance from orifice; D is the orifice diameter;

a

ρ is the density at atmospheric pressure;

exit

ρ is the density at the orifice;

a

p is the atmospheric pressure;

exit

p is the pressure at the orifice;

exit

U 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

D

z**

: Ke profile seems to be not affected by total pressure in the reservoir (it depends on the coordinate

D

z** _{only). This circumstance suggested to compute a}

mean Ke value for each

D

z**

(among the different pressure values) so that it can be used to evaluate air entrainment over the cross-section chosen (say

D

z** _{). Following this approach, it}

is necessary to determine the

D

z**

corresponding to the atmospheric-pressure cross-section downstream of the nozzle exit.

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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)

D z** _=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

D

z** _{=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, turbulence-related quantities (turbulence intensity: 5%, turbulent viscosity ratio: 1000).

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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:

exit Mach D P P X = ⋅ ⋅ ∞ 0 645 . 0 (3) Mach

X is the distance from nozzle exit of the Mach disc;

o

P is the (total) absolute pressure in the reservoir;

∞

P is the pressure of medium into which fluid discharges (generally air at rest);

exit

D is the nozzle exit diameter.

This relation allows a “local” check of our numerical results in the highly-compressible-fluid 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

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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. κ-ε 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).

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velocity values are even negative). Downstream of the normal shock, velocity values from κ-ω 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 initial velocity

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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) P∞ P₀ Dexit (m) Experimental Xm (m) Computed Xm (m) Pecentage deviation 10 11 1.17 1.17 0 50 51 2.51 2.62 4.4 125 126 3.95 4.13 4.6 250 251 0.546 5.58 5.64 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

D

z** _{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 D z** =5 Ke for D z** =10 Ke for 10⋅XMach 10 barg 0.316 0.231 0.219 (z**_/D=15.3) 50 barg 0.309 0.220 0.204 (z**_/D=15.7) 125 barg 0.308 0.219 0.203 (z**_/D=15.8) 250 barg 0.321 0.237 0.223 (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

D

z** _{), so the total mass flow}

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D z** =5 D z** =10 10⋅XMach Mean Ke From compressible-fluid simulation Mean Ke From compressible-fluid simulation Mean Ke From compressible-fluid simulation 10 barg 610 614 882 898 1263 1304 50 barg 2837 2802 4104 3988 6031 5816 125 barg 7024 6896 10160 9806 15027 14392 250 barg 14010 14347 20267 21132 30164 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

D

z**

=10 criterion provide the best results. Results from ten-times-Mach-disc criterion or

D

z**₌₅

criterion are also satisfactory. Entrainment profile (fig. 28) shows that the ten-times-Mach-disc 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;

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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.

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Figure 1: scheme of the phenomenon and of the approach used 0.2 0.22 0.24 0.26 0.28 0.3 0.32 0.34 0.36 0 10 20 30 40 50 60 70 80 z**/D Ke

Figure 2: __ 10 barg __ 50 barg __ 125 barg __ 250 barg

cross-section where p ≈patmospheric

Domain for compressible-fluid analysis

Domain for incompressible-fluid analysis

pipe end

highly-compressible-fluid region

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11.7 7.6 3.8 0 0.5 1 1.5 2 2.5 3 3.5 4 0 5 10 15 20 25 30 35 40

distance from rupture (m)

Ma c h num ber 11.7 7.6 3.8 0 100000 200000 300000 400000 500000 600000 700000 0 2 4 6 8 10 12 14 16

a b solut e pr es sur e ( P a)

Figure 3: __ CFD solution (P0=10 barg) __ 10*XMach __ z**/D=10 __ z**/D=5

Figure 4: 23700 cells grid, velocity fields (m/s) κ-ε

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0 100 200 300 400 500 600 700 0 2 4 6 8 10 12 14 16 18 20

distance from nozzle exit (mm)

ax ial v e loc ity (m /s )

Figure 5: 23700 cells grid, velocity profile along the jet axis

__ experimental data __ κ-ε solution __ κ-ε RNG solution __ κ-ω SST solution

150 200 250 300 350 400 450 500 550 600 650 0 0.5 1 1.5 2

distance from jet axis (mm)

ax ial v elocity (m /s) 100 150 200 250 300 350 400 450 500 550 0 0.5 1 1.5 2

ax ial vel oc it y (m /s )

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

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Figure 8: 41000 cells grid, velocity fields (m/s) -100 0 100 200 300 400 500 600 700 0 2 4 6 8 10 12 14 16 18 20

a x ia l vel oc ity ( m /s ) 50 100 150 200 250 300 350 400 450 500 550 0 0.5 1 1.5 2

axi a l vel o c it y (m /s)

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

κ-ε

κ-ε RNG

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-100 0 100 200 300 400 500 600 0 10 20 30 40 50 60 70 80

a xia l v eloc ity (m/ s)

Figure 11. Top: 41000 cells grid, velocity profile along the jet axis

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0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1 0 20 40 60 80 100

CH4 mass f ract ion 0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1 0 20 40 60 80 100 120

CH

4 mass

fracti

o

n

Methane concentration along jet axis Figure 12 (left): P0=10 barg Figure 13 (right): P0=50 barg

0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1 0 20 40 60 80 100 120

C H 4 m a ss f rac ti o n 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1 0 20 40 60 80 100 120

C H 4 m a s s fr ac tio n

Methane concentration along jet axis Figure 14 (left): P0=125 barg Figure 15 (right): P0=250 barg

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0 200 400 600 800 1000 0 20 40 60 80 100 120

a x ial v e locity along jet axis ( m /s ) 0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1 0 20 40 60 80 100 120

CH 4 m a s s f rac ti on

Figure 16 (left): P0=10 barg, axial velocity along jet axis Figure 17 (right): P0=10 barg, methane concentration along jet axis

0 20 40 60 80 100 120 140 160 0 1 2 3 4 5 6 7

distance from jet axis (m)

a x ia l v e loc ity (m /s) 0 0.02 0.04 0.06 0.08 0.1 0.12 0.14 0.16 0.18 0 1 2 3 4 5 6 7

CH4 m a ss fra c tio n

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

0 10 20 30 40 50 60 0 2 4 6 8 10 12 14 16 18

a xi a l vel oci ty (m /s ) 0 0.01 0.02 0.03 0.04 0.05 0.06 0.07 0 2 4 6 8 10 12 14 16 18

CH4 mas

s fr

action

__ compressible-fluid solution __ incompressible-fluid solution 10*XMach __ incompressible-fluid solution

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0 200 400 600 800 1000 1200 0 50 100 150 200 250 300

axi a l vel o c it y (m /s ) 0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1 0 50 100 150 200 250 300

C H 4 mass fr action

Figure 22 (left): P0=125 barg, axial velocity along jet axis Figure 23 (right): P0=125 barg, methane concentration along jet axis

0 50 100 150 200 250 300 0 2 4 6 8 10 12 14

a x ia l v e loc ity (m /s) 0 0.05 0.1 0.15 0.2 0.25 0.3 0 2 4 6 8 10 12 14

CH4

m

ass

f

raction

-10 10 30 50 70 90 110 130 150 0 5 10 15 20 25

ax ia l ve locity ( m /s ) 0 0.02 0.04 0.06 0.08 0.1 0.12 0.14 0 5 10 15 20 25

CH4 mass f

ract

io

n

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2000 3000 4000 5000 6000 7000 8000 9000 10000 11000 12000 15 20 25 30 35 40 45 50 55 60 65

tota l m a s s flow r a te (k g/s )

Figure 28: P0=10 barg, entrainment

10000 20000 30000 40000 50000 60000 70000 35 45 55 65 75 85 95 105 115

to tal mass fl o w rate (kg /s)

Figure 29: P0=125 barg, entrainment