Subsonic and sonic flow

When a pressurized gas is released, the velocity of the flow will heavily depend on the pressure difference between stagnant pressure and downstream pressure. By assuming an isentropic expansion of an ideal gas from its stagnant conditions (P₀ , meaning the velocity of the gas is zero) to downstream conditions (P₁, positioned at the exit plane of the nozzle or orifice), the subsonic flow rate of the source term is given by Ramskill's formulation:^[4]

The flow rate of an ideal gas can be represented by the graphed line. As the pressure ratio decreases and the critical value is approached, the flow changes from non-choked to choked, setting an upper limit to the velocity of the gas to the speed of sound of the medium.

${\displaystyle Q\;=\;C_{D}\;A_{o}\;\rho _{1}\;{\sqrt {\;2\;{\frac {P_{0}}{\rho _{0}\ }}\;\left[{\frac {\gamma \ }{\gamma \ -1}}\right]\left[1-\;\,\left({\frac {\;P_{1}}{P_{0}}}\right)^{\frac {\gamma \ -1}{\gamma \ }}\;\right]}}}$

As the ratio between downstream condition pressure and stagnant condition pressure decreases, the flow rate of the ideal gas will increase. This behavior will continue until a critical value is reached (in air, P₁/P₀ is roughly 0.528,^[5] dependent on the heat capacity ratio, γ), changing the condition of the jet from a non-choked flow to a choked flow. This will lead to the a newly defined expression for the aforementioned pressure ratio and, sub-sequentially, the flow rate equation.

The critical value for the pressure ratio is defined as:

${\displaystyle {\frac {\;P_{1}}{P_{0}}}\;=\;\left[{\frac {2}{\gamma \ +1}}\right]^{\left({\frac {\gamma \ }{\gamma \ -1}}\right)}}$

This newly defined ratio can then be used to determine the flow rate for a sonic choked flow:

${\displaystyle Q\;=\;C_{D}\;A_{o}\;\rho _{1}\;V_{c}}$

The flow rate equation for a choked flow will have a fixed velocity, which is the speed of sound of the medium, where the Mach number is equals to 1:

${\displaystyle V_{c}\;=\;{\sqrt {\;{\frac {T_{1}\;\gamma \ R}{M}}}}}$

It is important to note that if P₁ keeps on decreasing, no flow rate change will occur if the ratio is already below the critical value, unless P₀ also changes (also assuming that the orifice/nozzle exit area and upstream temperature stay the same).

Under-expanded jet structure

An under-expanded jet is one that manifests when the pressure at downstream conditions (at the end of a nozzle or orifice) is greater that the pressure of the environment where the gas is being released in. It is said to be under-expanded since the gas will expand, trying to reach the same pressure of its surroundings. When under-expanded, the jet will have characteristics of a compressible flow, a condition in which pressure variations are significant enough to have a strong effect on the velocity (where it can exceed the speed of sound of the gas), density and temperature.^[6] It is important to note that as the jet expands and incorporates gases from the surrounding medium, the jet will behave more and more like an incompressible fluid, allowing for a general definition of the structure of a jet to be the following:^[1]

The subdivision of a high pressure jet as it expands to the pressure of the surrounding conditions.

• Nearfield zone: this zone is composed of a core layer that is isolated from the surrounding medium, with its behavior being mostly dominated by compressible effects, and an outer layer that is in contact with the surrounding medium fluid. Due to turbulent effects, the outer layer, nominated as mixing layer, permits gas entrainment as it is facilitated, diluting the jet. In this shearing zone, a subsonic and supersonic section may be distinguished, where temperature, density and pressure vary wildly in a few centimeters of distance from the source. This zone has the characteristics of a compressible fluid.

• Transition zone: the beginning of this zone represents the ending of the nearfield zone, where variations (longitudinally and radially to the axis of the jet) are small compared to the previous one. Density and temperature variations are mostly because of mixing with the surrounding fluid.

• Farfield zone: this final zone is one of a fully expanded and incompressible jet. Longitudinal velocity and temperature are now inversely proportional to the distance from the source and radial evolution can be described by a gaussian dispersion model. It is important to note that this zone can be further split into inertial (dominated by initial acceleration), buoyant (dominated by internal buoyancy forces) and turbulent zones (dominated by ambient turbulence).

Under-expanded jet classification

Further classification of the jet can be related to how the nearfield zone develops due to the compressible effects that govern it.^[1] When the jet first exists the orifice or nozzle, it will expand very quickly, resulting in an over-expansion of the flow (which will also reduce the temperature and density of the flow as quickly as it depressurized). Gases that have expanded to a pressure lower than the one of the surrounding fluid will be compressed inwards, causing an increase in the pressure of the flow. If this re-compression leads to the fluid having a higher pressure than the surrounding fluid, another expansion will happen. This process will repeat until the pressure difference between ambient pressure and jet pressure is null (or close to null).^[7] Compression and expansion are accomplished through a series of shock waves, formed as a result of Prandlt-Meyer expansion and compression waves.^[8]

Development of the aforementioned shock waves will be related to the difference in pressure between the stagnant conditions or downstream conditions and the ambient conditions (η₀ = P₀/P_amb and η_e = P₁/P_amb, respectively), as well as the mach number (Ma = V/V_c, where V is the velocity of the flow and V_c is the speed of sound of the medium). With varying pressure ratios, under-expanded jets can be classified as:^[1]

A CFD simulation of a natural gas extremely under-expanded jet. Sections where the Mach Number is much higher than 1 are points in which the velocity of the fluid is much higher than the speed of sound, an effect that manifest due to the sudden depressurization of the gas. P_∞ is ambient pressure of 101325 Pa

• Moderately under-expanded jet: Nearfield with diamond shaped structures (each structure is called a cell). A Prandlt-Meyer expansion generates oblique expansion waves that expand the fluid downstream from the exit orifice. As these waves attain constant pressure from the surrounding fluid, they are deflected back as compression waves, converging in oblique shock waves (called intercepting shock). When they meet on the axis of the jet, reflected shock waves move outwardly until they attain constant pressure from the surrounding fluid, repeating the process, and in turn, recreating the cell structure (this phenomenon occurs in air at a range of 2 ≤ η₀ ≤ 4 or 1.1 ≤ η_e ≤ 3).

• Highly under-expanded jet: Nearfield with barrel shaped structures. As the pressure ratio increases, the intercepting shock waves can't meet on the axis of the jet anymore, which forces the generation of a normal shock wave when the intercepting shock waves go beyond a certain critical angle (The normal shock wave is called Mach Disk). From the interception point of the mach disk and the intercepting shock, a residual slipstream will reflect outwardly, until it reaches constant pressure from the surrounding fluid, repeating the process, recreating the barrel shaped cell structure (this phenomenon occurs in air at a range of 5 ≤ η₀ ≤ 7 or 2 ≤ η_e ≤ 4 ).

• Extremely under-expanded jet: Nearfield with a single cell structure. When the pressure ratio goes beyond a critical value (in air at a range of η₀ ≥ 7 or η_e ≥ 4 ), cell numbers within the nearfield of the jet decrease, until they all coalesce into a single cell with a single mach disk. Due to the increase in velocity and lower pressure zones around the jet, ambient fluid entrainment will increase.

Computational Fluid Dynamics

Experimental data of high pressure jets have to be limited in terms of size and complexity of the scenario due to the inherit dangers and expenses correlated to the experiment itself. Alternative methods to gather data, such as representative models, can be used in order to predict what the maximum extend of the gas cloud at its LFL concentration can reach. Simpler models like a gaussian gas dispersion model (e.g., SCREEN3 - a dispersion model) or integral model (e.g., PHAST- an integral model) can be useful to have a quick and qualitative overview on how the jet may extend. Unfortunately, their inability to properly simulate jet-obstacle interactions make them impossible to use beyond preliminary calculations. This is the reason why Computational Fluid Dynamic (CFD) simulations are generally preferred for more complex scenarios.^[15]

Although there exists several approaches for CFD simulations, a common approach is the use of a finite volume method that discretizes the volume into smaller cells of varying shapes. Every single cell will represent a fluid-filled volume where the scenarios parameters will be applied. Every cell that was modeled solves a set of conservation equations of mass, momentum and energy, along with the continuity equation. Fluid-obstacle interaction is then modeled with varying algorithms based on the closure turbulent model used.^[16] Depending on the number of total cells within the volume, the better the quality of the simulation, the longer the simulation time. Convergence problems can arise within the simulation as large momentum, mass and energy gradients appear in the volume. The points where these problems are expected to appear (like in the nearfield zone of the jet) need to have a higher number of cells to achieve gradual changes between one cell and another. Ideally, through CFD simulations, a simpler model can be derived which, for a specific set of scenarios, allows to have results with an accuracy and precision level similar to the CFD simulation itself.^[17]

Birch's Approach

Through a set of small scale experiments at varying pressures, Birch et al. formulated an equation that allowed the estimation of a virtual surface source, considering the conservation of mass between the exit plane of the orifice and the virtual surface.^[18] This approach allows to simulate a compressible, under-expanded jet as an incompressible, fully-expanded jet. As a consequence, a simpler CFD model can be simulated by using the following diameter (named pseudo-diameter) as the new exit plane:^[19]

${\displaystyle d_{ps}\;=\;d\;{\sqrt {\;C_{D}\;\left({\frac {T_{2}}{T_{0}}}\right)^{0.5}\left({\frac {P_{0}}{P_{2}}}\right)\left({\frac {2}{\gamma \ +1}}\right)^{\left({\frac {1+\gamma \ }{2\gamma \ -2}}\right)}}}}$

Where:
dps	= Birch's pseudo diameter, m
d	= orifice diameter, m
T0	= temperature in stagnant conditions, K
T2	= temperature in ambient conditions, K
P0	= pressure in stagnant conditions, Pa
P2	= pressure in ambient conditions, Pa

Ground and obstacle interaction

In the process industry, there exist a variety of cases where a high pressure jet release incident can occur. LNG storage facilities or NG pipeline systems leakage^[20] can degenerate into a jet fire and, through a domino effect, cause heavy damage to the workforce, equipment and surrounding environment. For different scenarios that may happen, safety protocols have to be engineered that aim to set minimum distances between equipment and the workforce, along with preventive systems that reduce the danger of the potential incidental scenario. The following are some of the most common scenarios that may be encountered in an industrial environment:^[19][21][22]

Jet-tank-ground interaction will have an effect on the maximum extension of the cloud at LFL concentrations. Eddies generated after their interactions will promote gas dissipation. s represents the distance between the center of the two tanks, D is the diameter of the tanks

• Jet-Ground interaction: this is one of the most common scenarios, where the free jet does not interact with any other obstacle apart from the ground. Although the jet can dissipate to concentrations below its LFL after roughly 16 meters with no interactions (interaction with concrete and at stagnant pressure of 65 bar, a common pressure for NG pipelines,^[23] with an orifice size of 2.54 mm), as the jet gets closer to the ground and makes contact with it, a dragging effect leads to a further extension. The jet has the tendency of bending downwards if close enough to the ground because of lower pressure zones below it.

• Jet-horizontal tank interaction: jet-tank interaction will depend of the type of material used. For cylindrical steel tanks, the distance from the exit plane will have an effect with the interaction, along with its angle with respect to the axis of the jet and the height of the release of the jet. Generally, when the jet impinges the horizontal tank along the axis of the tank, the jet is brought downwards, allowing an interaction with the ground. This will often lead to the extension of the jet at its LFL concentration with respect to its free jet (the same scenario is simulated with no other obstacles but the ground). Furthermore, crosswise extension is also expected.

• Jet-vertical tank interaction: for a cylindrical steel tank, the distance from the exit plane will have an effect with the interaction. Generally, when the jet impinges the vertical tank along the axis of the tank, the obstacle will act as a limiting factor for the extension of the jet at its LFL concentration with respect to its free jet. The impact with the tank will generate more eddies downstream the tail of the jet and limit ground interaction (at low enough flow rates and speeds), speeding up dilution of the gas below its lower flammable limit.

• Jet-horizontal tandem tank interaction: by adding a second horizontal tank behind the first, a shortening effect is achieved. The presence of the second obstacle leads to eddy generation after the first, promoting dissipation. Moreover, the second obstacle may have a detachment effect of the jet from the ground, as it will tend to adhere to the round surface of the second tank due to the coanda effect. Distance between the two tanks will have an effect, since, after a certain distance, the second obstacle will no longer have an effect on the cloud at LFL concentration.

• Jet-vertical tandem tank interaction: by adding a second vertical tank behind the first, a shortening effect will generally be achieved. The presence of the second obstacle leads to eddy generation after the first, promoting dissipation. Contrary to the previous scenario, the second obstacle can allow for jet-ground interaction and elongate the jet, a quality that be worsen due to the coanda effect. Distance between the two tanks will have an effect, since, after a certain distance, the second obstacle will no longer have an effect on the cloud at LFL concentration.