This section presents the discussion of the results of CFD simulations of the experimentally recorded pressure peaking phenomenon in the large-scale enclosure. The comparison of two different approaches to model the hydrogen release source is mainly focused on the pressure dynamics recorded in the enclosure, which is the primary concern of the safety analysis for such scenarios.
4.1. Inflow Boundary at the Notional Nozzle Exit Approach
The first analysis is focused on the reproduction of the experimental pressure dynamics within the enclosure for Test No.14 by employing the inflow boundary conditions at the notional nozzle exit.
Figure 3 shows the comparison between experimentally recorded pressure transient and simulated pressure dynamics obtained with a different resolution of the nozzle, i.e., 1 × 1 CV and 2 × 2 CVs. The simulations with a 2 × 2 CVs resolution at the notional nozzle exit boundary used time steps (Δ
t) 0.17 ms and 0.34 ms, which correspond to CFL numbers at the release area of approximately 25 and 50, respectively. Simulated pressure transients were shifted by 1.4 ms, which corresponds approximately to the time needed by the mass flow rate to reach a constant value at the release nozzle in the experiments. The experimental pressure dynamics in the enclosure is well reproduced in simulations with both CFL numbers 25 and 50. The pressure in the enclosure increases as long as the volumetric flow rate of hydrogen entering the chamber is higher than the volumetric flow rate of gases leaving the chamber. The latter increases in time due to two factors: (a) the decrease of density of the gas mixture in the enclosure by the increase of hydrogen fraction, and (b) the increase of pressure in the enclosure to compensate for the growth of pressure due to hydrogen inflow by the increase of gas mixture outflow rate through the enclosure vent. The maximum pressure in the enclosure is reached when the outflow volumetric rate through the vent is equal to the volumetric flow rate of hydrogen into the chamber. Simulations well predict the experimental peak overpressure of 21.1 kPa with an accuracy of 3% for Δ
t = 0.17 ms (20.5 kPa) and 5% for Δ
t = 0.34 ms (20.1 kPa). Given the negligible difference in the obtained pressure up to 6 s, it is concluded that the time step equal to 0.34 ms can be accepted. Thus, simulation then continued only for Δ
t = 0.34 ms, given the associated significant saving of computational time. At approximately 9 s the hydrogen release is stopped (see
Figure 3) and maximum velocity in the calculation domain starts to decrease, allowing to gradually increase time step. The time step was doubled five times till it reached 5.4 ms, each time letting the simulation run for 20 time steps before the next time step increase. The time step 5.4 ms is maintained for the rest of the simulation.
Figure 3 shows that beyond 9 s, the pressure rapidly decreases reaching negative values. Experimental negative pressure peak is seen to reach approximately −4 kPa, whereas CFD simulation prediction is about −2 kPa. Experimentalists noted that water vapour started condensing at the cold walls of the enclosure. Condensation could affect the pressure dynamics but was not accounted for in the model. This may be a reason for the discrepancy between the simulated and experimental negative pressure transients. Another reason could be the resolution of the vent area.
Figure 4a shows the distribution of hydroxyl (OH) mole fraction in the plane
x = 0 perpendicular to the enclosure walls in the location of vents. The presence of hydroxyl is an indicator of the location of chemical reactions i.e., the combustion zone in the jet fire. A limit to OH mole fraction equal to 0.001 is generally considered to indicate the most reacting zone and estimate a jet fire flame length [
23]. The latter was seen to be consistent with the visible flame length corresponding to the region with temperature in the range 1300–1500 K [
24] in the numerical study by [
25]. It can be seen that the jet fire hits the enclosure ceiling. With time, the jet fire increases in width and the maximum OH mole fraction decreases, as an indication of a decreased reaction rate due to decreased concentration of air in the enclosure.
Figure 4b shows the temperature distribution in the enclosure with time. The high-temperature zone (>1800 K) impinges on the ceiling and the hot combustion products move along the ceiling and then descend along the enclosure walls. By the time 5.4 s, the enclosure starts to be filled up with hot combustion products. This creates potentially harmful conditions for humans by temperature criteria and oxygen depletion. Similar to what observed from the OH mole fraction distribution, the jet fire increases in width (see zone T > 1300 K in
Figure 4b) and the maximum temperature along the jet axis decreases with time.
Four experimental thermocouples were located close to the enclosure walls to assess the thermal load on the structure (see
Figure 1). The thermocouples were coated with protective Inconel alloy, which would affect their response time due to heat transfer through the Inconel layer. On the other hand, numerical simulations provide the instantaneous “non-inertial” temperature of the hot combustion gases, preventing a direct comparison with experimental measurements and requiring the data manipulation to include heat transfer through the sensors’ Inconel layer. This process may be affected by inaccuracies rendering the comparison not reliable and, for this reason, this analysis is omitted.
The calculation time for simulations of 1 s is approximately 45 h with a time step of 0.34 ms on a 60 CPU workstation. Thus, the calculation time for a complete release simulation is approximately two weeks. The numerical grid employing 1 × 1 CV to discretise the inflow boundary allows the use of a larger time step while maintaining the same CFL number due to the larger cell size at the release area. Simulations are performed by using a time step equal to 0.5 ms (CFL = 50) or 0.25 ms (CFL = 25).
Figure 3 shows the resulting overpressure dynamics for both grids and nozzle resolutions. The maximum variation in the simulated pressure peaks was within 0.5%, confirming the independence of the grid resolution at the inflow boundary for the same CFL number. The simulation time was 30 h for 1 s of hydrogen release, which is not yet sufficient to have a time-efficient and accurate calculation strategy.
4.2. Volumetric Source Model Approach
To further reduce the calculation time without affecting the solution accuracy, a volumetric source model is used to simulate hydrogen release. As the first step, the analysis is conducted to find convergence by CFL number. This was changed in the range 0.37–50.0, which corresponds to the time step range 0.12–16 ms.
Figure 5 shows the resulting overpressure dynamics for the CFL number in the range 0.37–3.0. It can be observed that variation in simulated pressure dynamics from experimental measurement decreases with the decrease of CFL. The calculated pressure peaks are 20.37 kPa and 20.86 kPa (approximately 2% relative difference) for CFL numbers 1.5 (Δ
t = 0.5 ms) and 0.75 (Δ
t = 0.25 ms) respectively. A further decrease of CFL to 0.37 (Δ
t = 0.125 ms) results in a pressure peak equal to 21.00 kPa, which is higher by merely 0.7% than the pressure peak for CFL = 0.75. Thus, a CFL = 0.75 is deemed to lead to a converged solution while maintaining an acceptable calculation time of 17 h to simulate 1 s of hydrogen release. With the volumetric source model, the simulations of a test can be completed in approximately 5 days, which is a significant decrease in calculation time compared to the case employing the notional nozzle exit as the inflow boundary (approximately 2 weeks). A CFL = 0.75 is applied for the rest of the simulations. The calculated pressure peak is 20.9 kPa, which agrees well with the experimentally measured peak pressure of 21.1 kPa.
Figure 5 shows that the pressure dynamics is reproduced well, even though a slight difference towards the descending phase of the curve can be noticed. Once the release of hydrogen is stopped, the computational time step size was gradually increased with the same procedure as described for the notional nozzle exit inflow boundary modelling approach. Simulated negative overpressure decreases to approximately −3 kPa, whereas about −4 kPa was measured in the experiment. The effect of the number of iterations per time step is assessed by increasing it from 20 to 40. A variation of about 2% in the simulated pressure peak is observed, thus confirming the good accuracy of a solution with 20 iterations per time step. This value is maintained in the simulations.
The described above CFD model is used then to simulate Tests No.18 and No.19.
Figure 6 shows the comparison between the experimental and simulated pressure dynamics in the enclosure. The simulated pressure peak for Test No.18 is 35.10 kPa, which conservatively predicts the experimentally measured 33.22 kPa with 7% accuracy. On the other hand, simulated overpressure in the enclosure for Test No.19 reaches the maximum value of 42.21 kPa, which underpredicts the experimental pressure peak of 48.1 kPa by 14%. This variation is considered to be acceptable in the engineering calculations of such phenomenon. The dynamics and magnitude of the negative pressure phase are well reproduced for both Tests No.18 and No.19. However, it can be observed that the maximum negative pressure for Test No.19 simulation is achieved 4 s earlier than in the experiment. The last is though due to the absence of water vapour condensation in the CFD model.
4.2.1. Effect of Heat Transfer to the Enclosure Walls
The effect of heat transfer on pressure dynamics is assessed by comparing simulations for adiabatic and thermally conductive enclosure walls, see
Figure 7. In both cases, radiative heat transfer from the jet fire is included through the DO model. Pressure peak slightly increases from 20.86 kPa to 21.05 kPa when heat losses are not included in the model, which is considered to be a negligible effect on the pressure peak and pressure dynamics. This result confirms the conclusion drawn in [
10] that the heat transfer to the walls has a negligible effect on the positive phase of the overpressure dynamics. The negative phase may be affected by condensation of water vapour at the enclosure walls, which will be dependent on the heat transfer.
4.2.2. Time-Efficient Approach of the Uniform Cube-Shaped Control Volumes Grid
The use of a volumetric source model is found, as demonstrated above, to reduce significantly the calculation time by approximately a factor of 3. In this section, a novel and simplified approach is developed to further enhance the time efficiency of calculations while maintaining an acceptable solution accuracy. This is based on the building of a uniform hexahedral grid throughout the calculation domain with the same size as the vent (7 cm square opening). The volumetric source is modelled as a cube with a 7 cm side. This dimension corresponds to approximately eight times the notional nozzle diameter. The CV number in the domain is 317,878. Maintaining the same CFL = 0.75 at the release area for this grid increases time step by three times (Δt = 0.75 ms). Thus, it is expected to significantly decrease the computational time.
Figure 8a shows the pressure dynamics for Test No.14. A larger volumetric source results in a larger simulated overpressure (22.66 kPa) by approximately 8% relative difference, whereas the negative phase is simulated similarly. The hydrogen mass flow rate provided by the volumetric source (VS) is monitored. Maximum variation of released hydrogen between the two VS sizes, respectively 3.8 and 7.0 cm, is calculated to be within 0.7%. However, even though the hydrogen mass flow rate is maintained nearly the same for both cases, the distribution of hydrogen, combustion products and temperature can vary greatly, affecting as a consequence the composition and density of gases exiting the enclosure and the reached overpressure. A lower volumetric flow rate through the vents is observed for the simplified approach with uniform CV size 7 cm. All the above could be a direct consequence of the application of the volumetric source approach beyond the validity range stated in [
18] and the use of a coarser grid. Calculation time is approximately 6 h to simulate 1 s of hydrogen release. With full understanding of the simplified approach limitations, this is seen as a pragmatic way to reduce the calculation time by approximately a factor of 3. Thus, the authors yet consider the simplified approach as valuable for PPP prediction to obtain time efficient calculations. However, it should be highlighted that the reduction of computational time may come at a cost of a lower accuracy of calculations that should be maintained within reasonable for engineering applications precision.
The CFL independence of a solution for the volumetric source (VS) model with uniform “cube grid” of 7.0 cm size was assessed for Test No.14 for CFL number 0.50 and 0.75 (see
Figure 8a). The pressure dynamics is not affected by the CFL number change by 50%. Thus, the CFL = 0.75 was maintained for the simulations using the “cube grid”.
Figure 8b,c show the overpressure dynamics for Test No.18 and Test No.19 respectively. Maximum pressure for Test No.18 demonstrates variation by +5.8%, i.e., the increase from 35.10 kPa for a VS size 3.8 cm to 37.20 kPa for VS size 7.0 cm, and the difference of maximum overpressure simulated for VS = 7.0 cm from the experimentally measured value is +11% (conservative). For Test No.19, the simulated overpressure peak with VS = 3.8 cm underestimates experimental measurement by −14%, whereas the case with VS = 7.0 cm differs from experiments by only −7.4%. Conversely to what is observed for Test No.14, the negative pressure in Tests No.18 and No.19 is affected by the change of VS modelling approach, resulting in approximately half of the maximum negative pressure recorded in experiments when the VS = 7.0 cm approach is employed. Simulation results on the maximum overpressure peaks for the two approaches VS = 3.8 cm and VS = 7.0 are summarized and compared to experiments in
Table 3.
The simplified uniform “cube grid” VS = 7.0 cm approach is seen to reduce significantly the calculation time by approximately 3 times compared to the VS = 3.8 cm approach, and by approximately 7.5 times compared to the previously used notional nozzle exit approach for hydrogen inflow boundary. The accuracy of the maximum simulated overpressure is found to be ±11%. This is well within the acceptable engineering accuracy for such complex phenomenon. It can be concluded that the unified “cube grid” VS = 7.0 cm approach is a valuable hydrogen safety engineering tool for the assessment of the pressure peaking phenomenon.
Both the inflow boundary at the notional nozzle exit and volumetric source approaches are seen to represent well experiments with storage pressure up to 12.5 MPa and hydrogen mass flow rate up to 11.5 g/s. The validation domain of the CFD model is not limited by storage pressure but is defined by the released mass flow rate, being this the key input for PPP. A release of hydrogen from a storage with pressure 12.5 MPa and TPRD diameter of 4 mm would be equivalent to a release from a 70 MPa storage and 0.65 mm TPRD diameter, as calculated through e-Laboratory developed within NET-Tools project [
26] (11.5 g/s for conservative
Cd = 1). Thus, the conclusions of the present study are valid for current onboard storage systems with such TPRD diameter and the CFD model can be applied for the associated safety assessments.
4.3. Effect of Cryogenic Storage Temperature on the Pressure Peaking Phenomenon
Hydrogen may be stored in cryo-compressed conditions, i.e., storage temperature below 120 K as generally considered for cryogenics [
27] and pressure up to 35 MPa [
28]. Cryogenic storage pressure below 20 MPa is considered to provide a better gain in gravimetric and volumetric capacities against the energy required for the compression and cooling down of the hydrogen gas. With the increase of storage pressures above 20 MPa, these benefits are seen to reduce [
29].
This section assesses the effect of hydrogen storage temperature on the PPP dynamics for the same enclosure and the volumetric source model approach with VS size 3.8 cm and CFL = 0.75 (see detailed description in
Section 4.2) is used for the cases listed in
Table 4. Simulations include the effect of heat transfer through the enclosure walls. The selected scenario is that of Test No.14, i.e., storage pressure 11.78 MPa, nozzle diameter 4 mm and constant hydrogen mass flow rate. All three experimental vents of the enclosure are simulated as fully open. The storage temperature,
TS, is varied from ambient 277 K, as per Test No.14, to an intermediate temperature of 200 K and to cryogenic temperature of 100 K. Four cases of hydrogen releases at a storage temperature lower than atmospheric are simulated and details are given in
Table 4 along Test No.14 data. Effect of heat transfer through the release pipe walls is not taken into account to isolate the effect of storage temperature on the pressure peak. Parameters at the notional nozzle exit are calculated as in [
30], i.e., through the under-expanded jet theory implementing the NIST database [
31] using the Helmholtz Free Energy EOS. The calculated notional nozzle parameters are used to define the source terms in the volumetric source approach. It is considered that the potential for hydrogen phase change during expansion of cryo-compressed hydrogen can be neglected for this range of applications and the CFD model.
Table 4 shows that conditions at the notional nozzle exit would be above hydrogen critical point. Furthermore, the volumetric source approach was seen in [
32] to well reproduce experimental temperature distribution for a transient hydrogen unignited jet from a storage with initial pressure and temperature equal to 20 MPa and 80 K respectively [
33]. Cases 2 and 3 maintain the same storage pressure, pipe diameter of 4 mm and discharge coefficient as per Test No.14. The decrease of temperature causes an increase of released hydrogen mass flow rate from 11.37 g/s for
TS = 277 K, to 14.11 g/s for
TS = 200 K and 23.16 g/s for
TS = 100 K. Cases 4 and 5 consider the effect of storage temperature on the PPP dynamics for scenarios of the same storage pressure and the same mass flow rate as per Case 1 (Test No.14). For releases at ambient temperature, the PPP magnitude depends on the released mass flow rate of hydrogen for unvaried enclosure volume and vent area. This analysis aims at quantifying the sole effect of cryogenic hydrogen temperature at the release on the combustion products temperature and enthalpy, and, thus, on the resulting maximum overpressure peak. To match the same mass flow rate as in Test No.14 (Case 1) the discharge coefficient is reduced for the Cases 4 and 5 at cryogenic temperatures. This is equivalent to reducing the pipe diameter, as the mass flow rate is proportional to the product of discharge coefficient and pipe cross section area being proportional to diameter squared. The change in calculated notional nozzle diameters,
, in cases 4 and 5 is due to the application of NIST EOS which provides larger densities for cryogenic hydrogen compared to the Abel Noble EOS employed for case 1 at ambient temperature.
Figure 9 presents the resulting overpressure dynamics for the simulated cases (see
Table 4). As expected, Cases 2 and 3 demonstrate a higher pressure peak than for case 1, due to the higher hydrogen mass flow rate (MFR) for the same storage pressure and discharge coefficient but lower storage temperature. The pressure peak recorded for Case 2 with storage temperature
TS = 200 K and MFR = 14.11 g/s is 26.95 kPa. The pressure peak increases up to 42.82 kPa for the Case 3 (
TS = 100 K, MFR = 23.16 g/s). The higher is the positive pressure peak the higher is the negative pressure peak, which reaches −5.36 kPa.
It is expected that the higher pressure peak for cryogenic releases is associated with a larger difference between the volumetric flow rate entering the enclosure, due to the larger hydrogen MFR and its combustion, and the volumetric flow rate of gases exiting the enclosure through vents. To confirm this generic concept, two limiting cases, i.e., Case 1 (
TS = 277 K) and Case 3 (
TS = 100 K) are compared. The assessment is based on the PPP theory for ignited hydrogen releases [
7]. For the complete combustion of hydrogen in air, the volumetric flow rate “entering” the enclosure is determined by “competition” between the rate of consumption of cold hydrogen and enclosure air during combustion, and the rate of generation of hot combustion products in reaction at adiabatic flame temperature,
. For an arbitrary molar flow rate of hydrogen and assuming an isobaric combustion at atmospheric pressure, the balance of gas volume consumption and generation associated with hydrogen combustion in air can be written as [
7]:
where
is the hydrogen molar mass (2 kg/kmol),
R is the universal gas constant (8314.4 J/kmol/K) and
P is the ambient pressure.
is the initial temperature of the mixture of hydrogen at notional nozzle temperature,
, and air at ambient temperature,
. The conservation of enthalpy is used to calculate
as follows:
Table 5 shows the parameters calculated for the selected Cases 1 and 3. A diffusion flame consumes reactants at a stochiometric composition, meaning that a mole of hydrogen requires 2.38 mole of air. This corresponds to 1 g of hydrogen,
in Equation (3), consuming 34.8 g of air,
in Equation (3). Ambient temperature,
, is 277 K. The specific heat for air,
, is equal to 1010 J/kgK, whereas the specific heat for hydrogen,
, is obtained from NIST Chemistry WebBook [
31]. Thus,
is calculated as 263.8 K and 228.1 K for
equal to 277 K and 100 K, respectively.
A minor effect of hydrogen storage temperature is observed on
, which presents a difference of approximately 20 K. The multiplier
represents the difference between the volumetric flow rate of the unignited release and the ignited release (jet fire) from the same source [
7]. The term
increases from 22.5 for
Ts = 277 K to 26.3 for
Ts = 100 K. The variation in
α and the more significant double increase in mass flow rate for decreasing storage temperature, cause the volumetric flow rate “entering” the enclosure, i.e., the volumetric flow rate resulting from the combustion of the released hydrogen in air, to rise from 2.77 m
3/s to 5.70 m
3/s.
Figure 10 compares the calculated inlet volumetric flow rate resulting from combustion,
, and the simulated volumetric flow rate exiting the enclosure through the three vents
. It can be observed that for
Ts = 277 K, the maximum
almost equalises
. On the other hand, for
Ts = 100 K,
is almost twice the volumetric flow rate going out from the enclosure, and this is the cause for the higher overpressure recorded in the enclosure.
Cases 4 and 5 investigate the effect of only hydrogen storage temperature on the pressure peaking phenomenon while maintaining the same storage pressure and mass of released hydrogen as in Case 1 by reducing the discharge coefficient. Reduction of temperature leads to a slight decrease of pressure peak in the enclosure from 20.95 kPa for
Ts = 277 K to 20.51 kPa for
= 200 K and to 19.96 kPa for
= 100 K if the mass flow rate (11.37 g/s) and the storage pressure (11.78 MPa) are kept the same. This result is expected as a decrease of the hydrogen temperature mixing with air would lead to a decrease of combustion temperature. As a consequence, after stopping the hydrogen release, combustion products cool down faster for lower hydrogen release temperature. This causes a more pronounced negative pressure as shown in
Figure 9.
It can be concluded that for the same storage pressure and hydrogen mass flow rate (requires, in general case, the decrease of piping diameter for lower temperature), the decrease in storage temperature causes the decrease of enthalpy associated to the entering hydrogen, see Equation (2), and, thus, only a slight reduction of the PPP overpressure in the enclosure. On the other hand, for the same storage pressure, and the same piping system (with the same discharge coefficient) resulting in the increased hydrogen mass flow rate for lower temperature, causes, as a consequence, the increase of the pressure peak in the storage enclosure. As an example, it is observed that for a storage pressure of 11.78 MPa, the hydrogen mass flow rate increases from 11.37 g/s for = 277 K to 23.16 g/s for = 100 K. Such a variation in mass flow rate leads to a twofold increase in the pressure peak from 20.95 kPa to 42.82 kPa.