Comparative analysis of the prediction accuracy of hydrogen cloud explosion overpressure peak based on three kinds of theoretical models

Abstract

Hydrogen energy is pivotal in the energy transition due to its high efficiency and zero-emission characteristics. However, the potential for explosions constrains its broader application. Gaining insights into the dynamics of overpressure in hydrogen explosions is vital for the safe design of explosion-proof facilities and the determination of equipment spacing. This study investigates hydrogen explosions in open spaces of 1 and 27 m³ volumes, analyzing flame propagation and overpressure distribution. It also evaluates the accuracy of three theoretical models in predicting peak overpressure. The results reveal that the spherical flame from a hydrogen cloud explosion transforms into an ellipsoidal shape upon contact with the ground. The average flame propagation velocity across different equivalent ratio is ordered as follows: V_a (φ = 1.0) > V_a (φ = 1.5) > V_a (φ = 2.5) > V_a (φ = 0.5). At equivalent distances, the peak overpressure of hydrogen cloud explosions is comparable across both scales. The traditional trinitrotoluene model overestimates the peak overpressure of hydrogen cloud explosions at both scales. The optimized trinitrotoluene model achieves over 90% accuracy in predicting hydrogen cloud explosions in 1 m³ volumes but shows decreased accuracy in 27 m³ explosions. At source intensity level 3, the Nederlandse Organisatie voor Toegepast Natuurwetenschappelijk Onderzoek multi-energy model exhibits a prediction accuracy of over 70% for peak overpressure in hydrogen cloud explosions, with consistent performance across different scales, rendering it a more reliable model for such predictions. This research enhances hydrogen safety assessment technologies by providing a more precise method for evaluating large-scale hydrogen cloud explosion risks.

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1. Introduction

Hydrogen, as a clean energy source, holds significant application value and potential [1, 2]. However, its low ignition energy and wide explosion limit make it prone to explosive accidents under specific conditions [3–5], impeding its widespread adoption. Understanding the overpressure behavior of hydrogen cloud explosions in open spaces is essential for designing blast-resistant pressure-bearing components, establishing safe distances between process devices, and planning explosion-proof facilities. Due to the scarcity of experimental data on large-scale hydrogen clouds, developing accurate overpressure prediction models is critical [6, 7]. By comparing the explosive power of traditional solid explosives, one can gain a profound understanding of explosion intensity at the macroscopic level. This understanding provides a theoretical foundation for the development of large-scale hydrogen explosion models in numerical simulation software [8, 9].

Common theoretical models for predicting explosion overpressure include the traditional trinitrotoluene (TNT) model [10–12] and the Nederlandse Organisatie voor Toegepast Natuurwetenschappelijk Onderzoek (TNO) multi-energy model [13]. The traditional TNT model, an empirical or phenomenological approach, converts the destructive effect of a gas cloud explosion into that of a TNT explosion, translating the amount of combustible cloud into TNT equivalent [14, 15]. The TNO multi-energy model, which primarily considers the acceleration effect of obstacles on flames, can better evaluate the explosion load generated by industrial explosions [16–18]. The TNT model is capable of assessing the response of structures to blast shocks under conditions of low blockage density and highly reactive fuels [19, 20]. However, it cannot accurately reflect the actual situation in cases of deflagration due to slow flame propagation and combustion response [21]. The TNT model does not fully consider key factors such as secondary combustion, gas expansion, volume change, and the impact of obstacles on overpressure, limiting its predictive capacity [22].

The TNO multi-energy model, suitable for vapor cloud explosions in environments with medium and low blockage density, can predict the injury threshold of explosion overpressure to humans based on blast wave parameters [23, 24]. It provides more accurate predictions of overpressure, pressure wave morphology, and duration [25]. In experimental studies of propane and methane vapor cloud explosions, the TNO multi-energy model’s accuracy in predicting deflagration-to-detonation transition (DDT) was stable at 33% and was not affected by changes in the pressure change standard [26]. However, the TNO multi-energy model has certain limitations, as it considers the degree of congestion and fuel activity in the explosion scene; the choice of obstacles is subjective, and the volumetric filling rate does not fully reflect the impact of obstacles on the explosion effect in practical applications [27].

In the practical application of the TNT and TNO models for predicting the positive impulse of open-space gas explosions, it has been observed that the TNT model generally underestimates while the TNO multi-energy model overestimates the predictions [28]. Mueschke and Joyce’s simplified TNT model can predict the order of magnitude of methane–hydrogen–air explosion overpressure in a 6.13 m³ steel explosion chamber (open to the atmosphere), but its accuracy for specific spaces is relatively low [29]. This discrepancy arises because the TNT model only considers the instantaneous release of explosion energy and neglects the duration of the explosion load, whereas the TNO model overemphasizes the impact of confined spaces on combustible gas explosions [30–32].

Previous studies have also conducted overpressure predictions for open-space hydrogen explosions. Sato et al. performed deflagration tests with homogeneous hydrogen concentrations in volumes of 5.2, 37, and 300 m³, finding that the Sachs scale is applicable to small and medium scales [33]. Otsuka et al. conducted a 1400 L spherical latex balloon hydrogen explosion experiment, observing overpressure values lower than those estimated using Strehlow’s theory and measured maximum velocity [34]. Kim tested hydrogen cloud explosions of different volumes (9.4, 75, and 200 m³) and found that the observed overpressure of hydrogen–air mixture explosions was significantly lower than theoretical expectations based on TNT equivalence, although the explosion pulse scale was consistent with theoretical predictions [35]. Mukhim et al.’s comparative study of multiple papers’ hydrogen explosion overpressure data revealed that only 75% of the data points fell within the scaled overpressure versus scaled distance curve defined by the TNT model, indicating that the TNT model grossly overestimates hydrogen explosion overpressure [36]. Li et al. conducted a 1 m³ scale experiment of hydrogen cloud explosion with internal obstacles and found that the maximum explosion overpressure predicted by both the TNT and TNO models decreased with increased distance from the ignition source, but theoretical predictions were generally higher than experimental values at all test distances [37].

In this paper, through the innovative design of 1 and 27 m³ multi-scale experiments, the influence of scale effect on the explosion characteristics of hydrogen cloud is deeply explored, especially in the aspects of flame propagation velocity and overpressure distribution. The study comprehensively analyzes flame propagation and overpressure distribution, offering a detailed perspective on key parameters in hydrogen cloud explosions. Additionally, we compare and optimize the traditional TNT model, the optimized TNT model, and the TNO multi-energy model, enhancing the accuracy of overpressure predictions for hydrogen cloud explosions. Furthermore, by integrating experimental data with theoretical models, we validate these models through experimentation and use them to elucidate experimental phenomena, thereby improving the reliability and applicability of the research findings. Ultimately, this study provides theoretical support for hydrogen cloud explosion safety assessment, particularly in determining safety distances and designing explosion-proof measures, and offers new perspectives and tools for the safe use and risk management of hydrogen energy.

2. Experimental systems and methods

The experimental platform consists of 1 and 27 m³ cubic steel frames, a gas distribution system, a concentration monitoring system, a high-pressure ignition system, a data acquisition system, and a synchronous control system, as depicted in Fig. 1.

The bottom of the cube steel frame is sealed with a stainless steel plate, and during the experiment, the device is enclosed with polyethylene film to contain the premixed gas. The hydrogen–air distribution system comprises a manifold row, 2 diverters, and 10 inlet lines. The concentration monitoring system includes a circulation pipeline, a hydrogen vacuum pump, and an optical wave interference gas concentration detector. The high-voltage ignition system consists of a high-voltage igniter and ignition electrodes. The ignition electrodes are positioned at the center of the bottom of the device, with ignition heights of 0.2 m for the 1 m³ hydrogen cloud and 0.6 m for the 27 m³ hydrogen cloud. The high-voltage pulse generator is amplified through a 1:50 transformer and releases energy by controlling the internal capacitor, reaching a voltage of 30 kV during the experiment. The ignition energy, calculated using Equation (3), is determined to be 36 J [38, 39].

The capacitance C is 200 μF, the voltage U is 600 V, and the energy E of the electric spark is calculated to be 36 J.

The data acquisition system comprises pressure sensors, a data acquisition instrument, and high-speed cameras. The data acquisition instrument operates the PCB 106B pressure sensor (PCB Piezotronics, USA) to measure external overpressure. The sensor has a sensitivity of 43.5 mV/kPa, a resolution of 0.00069 kPa, and a sampling period of 10 μs. According to the similarity criterion, the equivalent distance is calculated using Equation (4).

Here, D represents the equivalent distance. The variable d denotes the distance from the pressure measurement point to the ignition center, measured in meters. L is the edge length of the membrane-enclosed cubic space (L_{1 m³} = 1 m, L_{27 m³} = 3 m).

When the pressure measurement points for a 1 m³ hydrogen cloud are located at 2, 3, 4, 5, and 6 m from the ignition center, and for a 27 m³ hydrogen cloud at 6, 9, 12, 15, and 18 m, they correspond to the same equivalent distances, specifically 2, 3, 4, 5, and 6 m. High-speed cameras, operating at a recording rate of 3000 frames per second, capture the changes in flame morphology. The synchronization control system sequentially triggers the high-voltage igniter, data acquisition instrument, and high-speed camera.

The experimental procedure is as follows:

1. Verify the pipeline’s gas tightness to ensure the normal operation of each subsystem.

2. Encase the cube device with a polyethylene film.

3. Use the gas circulation system and hydrogen concentration detector to monitor the hydrogen concentration at the remote end of the circulation pipeline.

4. When the hydrogen concentration reaches the preset experimental value, turn off the air intake system and proceed with pressurization and ignition.

5. After completing each set of experiments, promptly record the data and clean the site in preparation for the next set.

3. Results and analysis

3.1 Flame propagation process and overpressure distribution of hydrogen cloud explosion

In the experiment, the hydrogen–air mixture equivalence ratios were φ = 0.5, 1.0, 1.5, and 2.5, with corresponding volume concentrations of 17.4%, 29.6%, 37.8%, and 51.2%, respectively. These ratios encompass lean combustion, stoichiometric ratio, and rich combustion. A high-speed camera recorded the flame propagation images during the hydrogen cloud explosion process, and the development of flame morphology was analyzed.

3.1.1 Flame propagation process of hydrogen cloud explosion

Figure 2 illustrates the evolution of flame morphology in hydrogen cloud explosions at two scales: 1 and 27 m³, with different equivalencies.

The analysis of Fig. 2 reveals that prior to the rupture of the polyethylene film under pressure (referred to as the broken film), the flame area of hydrogen cloud explosions with different equivalences varies at the same time, with the flame area at an equivalence of 1.0 being the largest and its development process the fastest. When the spherical flame areas of 1 and 27 m³ exceed the critical values of 0.503 and 4.524 m², respectively, the spherical flame contacts the bottom of the device and propagates vertically downward. Thermal diffusion instability (irregular changes in the flame’s surface structure due to uneven heat and mass diffusion) and hydrodynamic instability (uneven flame propagation due to variations in fluid flow characteristics such as velocity, pressure, and density) exacerbate the flame’s tensile deformation, gradually changing its morphology to a flattened ellipsoidal shape. From the moment the flame reaches the bottom to just before the film breaks, the non-combustible gas is continuously compressed by explosion pressure, causing more non-combustible gas to participate in the combustion. This results in the explosion pressure causing the film to bulge outward and rupture. Subsequently, the unburned hydrogen comes into contact with outside air, mixing to form a more violent combustion. At this point, flame propagation is no longer constrained, and the form becomes completely unstable. It is evident that the spherical flame of a hydrogen cloud explosion is influenced by factors such as ignition height, frame limitation, and buoyancy, and the overall shape tends to become ellipsoidal when the flame area exceeds the critical value.

To thoroughly analyze the propagation characteristics of flames, it is essential to consider the flame propagation radius in various directions: the vertical upward radius (R_u), the vertical downward radius (R_d), and the horizontal right radius (R_r) (note that the horizontal flame propagation is influenced by buoyancy). By binarizing the propagation images of spherical flames at different times, and obtaining the pixel values corresponding to the flame edges of spherical flames at different times, the radius of spherical flames at different times can be converted according to the number of pixels in the frame in the experimental apparatus. The laminar flame radius and thermal expansion ratio are determined using Equations (3) and (4).

In this context, r denotes the laminar spherical flame radius (m), σ represents the thermal expansion ratio, which is determined by the ratio of unburned density ρ_u to burned density ρ_b, and S_L represents the laminar combustion rate (m/s). t denotes the flame propagation time (s).

The values of σ and S_L for hydrogen cloud combustion at different equivalence ratios are presented in Table 1. The flame radii of hydrogen cloud explosions at scales of 1 and 27 m³ in different directions are illustrated in Figs 3 and 4.

Comparing the real flame propagation radius with the theoretical laminar flame radius, Figs 3 and 4 reveal the following patterns under identical scale conditions:

1. The order of the laminar flame radius at the same time for hydrogen cloud explosions with different equivalence ratios is as follows: r (φ = 1.5) > r (φ = 1.0) > r (φ = 2.5) > r (φ = 0.5).

2. For φ = 0.5, the hydrogen cloud concentration is low, leading to significant thermal diffusion and hydrodynamic instability, and the flame radius in all three directions exceeds that of the laminar flame (see Figs 3a and 4a). For φ = 1.0, the hydrogen cloud burns more intensely. Except for the flame radius of the 1 m³ hydrogen cloud explosion, which is very close to r, the flame radius of 1 and 27 m³ in three directions is greater than that of the laminar flame (see Figs 3b and 4b). For φ = 1.5, the buoyancy effect becomes more pronounced due to the increased hydrogen volume, and the flame radius of the 1 m³ hydrogen cloud explosion in three directions gradually approaches the laminar flame radius over time (see Fig. 3c). The flame radius of the 27 m³ hydrogen cloud explosion differs from the laminar flame radius in three directions, transitioning from close to gradual (see Fig. 4c). For φ = 2.5, the hydrogen cloud is in a fuel-rich combustion state, slowing the flame combustion rate. The flame radius in all three directions is smaller than the laminar flame radius before the flame touches the bottom of the device. After the downward flame propagation is limited, the flame radius in the other two directions exceeds the laminar flame radius (see Figs 3d and 4d).

3. The flame radius of the hydrogen cloud explosion is influenced by the volume, equivalence ratio, and buoyancy of the hydrogen cloud. Compared with the flame radius in the three directions, R_u is always larger than R_d and R_r, meaning the vertical upward flame radius is greater than that in the other two directions at the same time, due to the buoyancy effect on the high-temperature gas combustion.

The flame radius with the smallest thermal diffusion instability and hydrodynamic instability was selected, and the flame propagation velocity of hydrogen with different equivalent ratios was calculated by the flame radius, and the variation law of its average flame velocity (V_a) was analyzed. The variation of flame propagation velocity of hydrogen cloud explosion is shown in Fig. 5.

Figure 5 illustrates the oscillatory behavior of the flame propagation velocity following the explosion of the hydrogen cloud over time. This phenomenon can be explained from the point of view of fluid dynamics and heat transfer. In the initial phase of a hydrogen cloud explosion, the flame front comes into contact with an unreacted mixture of hydrogen and air, and the flame propagation velocity increases rapidly due to the high reactivity of hydrogen. As the flame propagated, combustion products such as water vapor and hot gases began to accumulate, which had an inhibitory effect on flame propagation, resulting in a temporary decrease in the flame velocity. In addition, turbulence and vortex during flame propagation can also cause fluctuations in flame velocity. This oscillatory behavior is the result of dynamic equilibrium during combustion, involving complex interactions among combustion chemistry, fluid dynamics, and heat transfer.

The average flame propagation velocity at different equivalent ratios is ranked as follows: V_a (φ = 1.0) > V_a (φ = 1.5) > V_a (φ = 2.5) > V_a (φ = 0.5). This trend can be explained by chemical reaction kinetics and combustion theory. When the equivalent ratio is φ = 1.0, the mixing ratio of hydrogen and oxygen reaches the stoichiometric ratio, and the combustion is the fullest, and the energy released is the largest, so the flame propagation speed is the fastest. When the equivalent ratio deviates from the stoichiometric ratio, the adequacy of combustion decreases whether it is rich in hydrogen (φ < 1.0) or poor in hydrogen (φ > 1.0), resulting in a slowdown in flame propagation. In particular, when the equivalent is relatively low (φ = 0.5), the flame propagation speed is the slowest due to the lack of hydrogen and the limited combustion reaction.

At the same equivalent ratio, the effect of scale on the average flame propagation velocity V_a is as follows: for φ = 1.0 or 1.5, V_a (27 m³) > V_a (1 m³) and for φ = 0.5 or 2.5, V_a (27 m³) < V_a (1 m³). This phenomenon can be explained by the scale effect and the geometrical properties of flame propagation. In large-scale (27 m³) hydrogen cloud explosions, the flame has a larger contact area with the outside world, which may lead to increased heat loss, which affects the flame propagation velocity. However, in the vicinity of stoichiometric ratios (φ = 1.0 and 1.5), the flame propagation of the 27 m³ hydrogen cloud explosion is less affected by the heat loss due to the higher combustion intensity, so the flame propagation is faster. Conversely, at a relatively long distance from the stoichiometric (φ = 0.5 and 2.5), the combustion intensity is lower and the flame propagation is more susceptible to heat loss, resulting in a slower flame propagation velocity of the 27 m³ hydrogen cloud explosion.

3.1.2 Overpressure distribution of hydrogen cloud explosion

The explosion overpressure in this study refers to the overpressure outside the frame. The overpressure of hydrogen cloud explosions at equivalent distances for volumes of 1 and 27 m³ is compared, as depicted in Fig. 6.

After the rupture of the film, the rapid expansion of high-temperature and high-pressure hydrogen combustion products allows the external pressure sensor on the frame to keenly detect pressure changes. Analysis of Fig. 6 reveals the following:

1. The overpressure curve of the hydrogen cloud explosion exhibits a “peak-trough-small amplitude oscillation” pattern over time.

2. Under the same scale conditions, the explosion overpressure of the hydrogen cloud initially increases and then decreases with the increase in the hydrogen cloud equivalent ratio, reaching its maximum peak at an equivalent ratio of 1.0.

3. Under the same scale and equivalent ratio, the peak overpressure of the hydrogen cloud explosion decreases as the equivalent distance increases, with the corresponding peak overpressure time being delayed. This indicates that the explosion pressure wave gradually decays with increasing propagation distance.

4. Due to the scale of the gas cloud and the buoyancy effect, the explosion overpressure rise rate of the 27 m³ hydrogen cloud at the same equivalent distance is slower compared to that of the 1 m³ hydrogen cloud.

The peak overpressure (P_max) of hydrogen cloud explosions at equivalent distances for volumes of 1 and 27 m³ is compared, as shown in Fig. 7. The absolute value (Ab-Value) of the peak change amplitude of overpressure at equivalent distances for the two volumes is presented in Table 2.

Based on the analysis in Fig. 7 and Table 2, it is observed that the overpressure from hydrogen cloud explosions of 1 and 27 m³ exhibits similar trends at equivalent distances. However, at an equivalence ratio of 1.0, the overpressure of the 27 m³ hydrogen cloud is slightly greater than that of the 1 m³ at equivalent distances 4 and 5. A similar phenomenon is also observed at an equivalence ratio of 1.5 at equivalent distance 4, while at other equivalent distances, the overpressure of the 1 m³ hydrogen cloud explosion is greater than that of the 27 m³.

The reasons for the higher overpressure in the 27 m³ hydrogen cloud compared to the 1 m³ may lie in the following factors:

1. Scale effect: Due to the dynamic characteristics of flame propagation and pressure wave dynamics differ in larger volumes of hydrogen cloud explosions, interactions between the flame and the surroundings in larger volumes may lead to reflections and superposition of pressure waves, resulting in higher overpressure at certain locations.

2. Complexity of flame propagation: Turbulence and eddies during the flame propagation process can lead to non-uniform pressure distribution. In larger scales, the complexity of flame propagation in hydrogen cloud explosions increases, potentially leading to intensified pressure waves in local areas, and thus higher overpressure at certain equivalent distances.

3. Combustion efficiency and heat release rate: At an equivalence ratio of 1.0, the reaction between hydrogen and oxygen is most complete, and the energy released from combustion is at its maximum. In larger scales, this efficient combustion may lead to more intense pressure waves, resulting in higher overpressure at certain locations. These factors collectively contribute to the observed phenomenon of higher overpressure in the 27 m³ hydrogen cloud compared to the 1 m³ under specific conditions.

The absolute value of the peak overpressure change amplitude at the same equivalent distance in Table 2 (Avg Ab-value) was used to obtain the average change amplitude of the peak overpressure. When the equivalence ratios are 0.5, 1.0, 1.5, and 2.5, the average changes in peak overpressure for the 27 m³ hydrogen cloud compared to the 1 m³ hydrogen cloud are 27.52%, 3.66%, 4.56%, and 23.44%, respectively. This indicates that the scale effect significantly impacts hydrogen cloud explosions with concentrations far from the stoichiometric ratio, while it has less influence on explosions with concentrations close to the stoichiometric ratio. This conclusion can be extended to the approximate assessment [40] of the overpressure peak of hydrogen cloud explosions on a larger scale.

3.2 Overpressure prediction of hydrogen cloud explosion by traditional TNT model

The traditional TNT model is a straightforward method for equating combustible gas to an equivalent mass of TNT. This model is utilized to preliminarily predict explosion overpressure at various locations surrounding the explosion source. The equivalent mass of TNT can be calculated using Equation (5), and the proportional distance can be determined from the TNT equivalent and the r-value of the distance from the source, as shown in Equation (6).

In this context, η represents the explosion efficiency, m denotes the mass of the combustible gas, and ΔH_c refers to the explosive energy or heat of combustion of the combustible gas (with hydrogen’s heat of combustion being 142 351 kJ/kg). E_TNT stands for the explosive energy of TNT, typically 4686 kJ/kg. Z is the proportional distance, and r is the distance between the measurement point and the explosion center.

Additionally, according to accident statistics, the explosion efficiency of the combustible cloud (η) ranges from 1% to 10%, and the proportional overpressure (p_s) is calculated using Equation (7). For a TNT explosion on flat ground, the relationship between p_s and Z is depicted in Fig. 8.

Among these variables, p_s represents the proportional overpressure, which is dimensionless; p₀ denotes the peak lateral overpressure; and p_a refers to the ambient pressure.

Due to the influence of unobstructed objects and confined spaces surrounding the explosion in this study, the explosion efficiency η is taken as the minimum value of 1% based on available statistics. Utilizing this theory in conjunction with the traditional TNT model and as illustrated in Fig. 8, the predicted overpressure data for the hydrogen cloud explosion is presented in Table 3.

Table 3 demonstrates that, compared to the explosion of solid gunpowder with an equivalent TNT yield, the explosion overpressure measured at the two scales in this experiment is lower. The overpressure predicted by the traditional TNT model for hydrogen cloud explosions significantly exceeds the experimentally measured values, with deviations ranging from 10 to 35 times. Consequently, the explosion efficiency derived from accident statistics using the TNT equivalence method, denoted as η, introduces substantial discrepancies in theoretical studies. Therefore, the traditional TNT model has limitations in accurately predicting the overpressure of hydrogen cloud explosions, primarily for the following reasons:

1. The explosion process is characterized by complex and variable conditions that are influenced by both the explosion source and the surrounding structures, which can exacerbate the explosion in intricate scenarios. Additionally, some explosions may generate raised dust, potentially leading to an overestimation of η.

2. Accident statistics are derived from large-scale incident reports, and explosion efficiency may be affected by factors such as the size of the explosion source.

3.3 Overpressure prediction of hydrogen cloud explosion by O-TNT model

Based on the analysis presented in the previous section, it is evident that the η derived from the accident statistics in the TNT model introduces significant bias when applied to theoretical research. Consequently, the η in the theoretical framework is adjusted according to the experimental conditions. When the explosion efficiency and hydrogen equivalent ratio exhibit the relationship described in Equation (8), the theoretical calculation results align well with the experimental data from the open-space hydrogen cloud explosion.

Where φ is the hydrogen equivalent ratio.

Considering factors such as the scale of the hydrogen cloud, the location of the overpressure measurement point, and the equivalent distance, the O-TNT model is established to predict the peak overpressure of a hydrogen cloud explosion. The calculation formula is presented in Equation (9):

Where V represents the volume of the hydrogen cloud, r denotes the distance from the pressure measurement point to the ignition center, and D indicates the equivalent distance.

The peak overpressure of the hydrogen cloud explosion, based on the O-TNT model, was calculated using Equations (5), (6), (8), and (9), as presented in Table 4. The O-TNT model also predicts the deviation (E_r) of the overpressure from the experimental values, as illustrated in Fig. 9.

The analysis presented in Fig. 9 indicates that the O-TNT model effectively predicts the peak explosion overpressure of a 1 m³ hydrogen cloud, characterized by the following observations:

1. At φ = 0.5, the predicted peak explosion overpressure at close ranges (D = 2, 3) shows a deviation from the experimental values of less than 10%, resulting in a prediction accuracy greater than 90%. As the equivalent distance increases, the prediction accuracy decreases, with deviations exceeding 10%.

2. For φ values of 1.0 and 1.5, the average deviations of the predicted peak explosion overpressure at various equivalent distances are 3.69% and 6.35%, respectively, yielding a prediction accuracy exceeding 93%.

3. At φ = 2.5, the average deviation across different equivalent distances is the highest, reaching 14.97%. Additionally, the prediction deviation at longer distances (D = 4, 5, 6) exceeds 15%, resulting in the lowest prediction accuracy of peak overpressure for hydrogen cloud explosions among the four equivalence ratios considered.

However, the predictive characteristics of the O-TNT model for the peak overpressure of the 27 m³ hydrogen cloud explosion vary as follows:

1. For φ = 0.5, the average deviation of the predicted overpressure at different equivalent distances is 36.03%, resulting in a prediction accuracy of only 63.97%.

2. For φ = 1.0 and 1.5, the average deviations of the peak predicted explosion overpressure at different equivalent distances are 15.14% and 15.08%, respectively, with prediction accuracy exceeding 84%.

3. For φ = 2.5, the prediction accuracy significantly improves, with an average deviation of only 2.13% at different equivalent distances; however, this does not fully represent the overall prediction accuracy of the O-TNT model method.

Based on the analysis above, it is evident that the prediction accuracy of the O-TNT model for hydrogen clouds at φ = 0.5 and 2.5 varies significantly between the two scales. At the 1.0 and 1.5 equivalents, where the hydrogen cloud explosion intensity is high, the O-TNT model achieves an overall accuracy exceeding 84% in predicting hydrogen cloud explosion overpressure. This finding holds considerable reference value for predicting hydrogen cloud explosion overpressure.

3.4 Overpressure prediction of hydrogen cloud explosion by TNO multi-energy model

The TNO multi-energy model takes into account constraints when evaluating the explosion power of gas clouds. Initially, the model calculates the constrained volume of the gas cloud, followed by an estimation of the combustion energy based on the average combustion energy density of hydrocarbons (E₀ = 3.5 × 10⁶ J/m³) to analyze the explosion power. Figure 10 illustrates the relationship between the peak overpressure of the gas cloud explosion and the comparison distance in the TNO multi-energy model [1]. Once the energy and initial explosion intensity of the equivalent fuel-air mixture have been estimated, the lateral overpressure of the simulated explosion at various comparison distances is presented in Fig. 10. The intensity of the detonation source is represented by a scale from 1 to 10, where 1 indicates extremely weak detonation source intensity and 10 denotes a strong detonation, suitable for conservative estimates in the near field. A value of 6 signifies a centered explosion intensity, appropriate for far-field estimation, while a source intensity level of 7 may provide a more accurate reflection of the actual explosion.

As illustrated in Fig. 10, there is no significant difference in source strength levels between 7 and 10 for explosions with lateral overpressure below 0.5 bar. For explosions originating from an unrestricted and unobstructed partial gas cloud, a minimum intensity of 1 is assumed. In contrast, for most components that are not stationary but are in a state of low-intensity turbulent motion (e.g. due to the momentum released by the fuel), an intensity of 3 can be assumed.

The calculation processes for the analogous distance (⁠$R¯$⁠) and the analogous overpressure (⁠$Δp¯s$⁠) are presented in Equations (9) and (10), respectively, with the results summarized in Table 5.

Where $R¯$ is the dimensionless parameter representing the approximate distance from the center of the explosion. R denotes the distance from the center of the explosion in meters (m). E represents the combustion energy of the fuel in joules (J). P_a indicates the ambient atmospheric pressure in pascals (Pa). p₀ refers to the lateral explosion overpressure in pascals (Pa). $Δp¯s$ is analogous to the lateral explosion overpressure.

Based on the $Δp¯s$ values presented in Table 5, the corresponding explosion intensity classes are illustrated in Fig. 10, while Fig. 11 depicts the hydrogen cloud explosion intensity levels across two scales and different equivalents.

Figure 11 illustrates that (1) when the equivalent ratio is 1.0, the source intensity reaches its peak, while at φ = 0.5, the source intensity is at its lowest. (2) For φ values of 1.0 and 1.5, the intensities of the hydrogen cloud explosion sources with volumes of 1 and 27 m³ range between 3 and 4. In contrast, for φ values of 0.5 and 2.5, the intensities of the hydrogen cloud blast sources of 1 and 27 m³ vary between 2 and 3. Notably, when $R¯$ <1, the intensity of the 27 m³ hydrogen cloud explosion source at an equivalent ratio of 0.5 is below 2, whereas when $R¯$ >1.5, the intensity of the 1 m³ hydrogen cloud explosion source at an equivalent ratio of 2.5 exceeds 3.

The TNO multi-energy model predicts the explosion overpressure of hydrogen clouds, requiring a subjective selection of the explosion source intensity. This intensity is calculated from experimental data, revealing that the explosion source intensity levels within the hydrogen equivalent ratio range of 0.5–2.5 in this experiment fall between 2 and 4. Given that an equivalent ratio of 1.0 yields the highest overpressure for hydrogen cloud explosions, it is more practical to predict the explosion overpressure at varying distances when φ = 1.0. Assuming explosion source intensity levels of 3 and 4, the comparison of the lateral explosion overpressure p₀ calculated using Equation (13) against the experimental overpressure values is presented in Fig. 12.

Figure 12 shows that when the intensity of the hydrogen cloud explosion source is assumed to be 3, the lateral explosion overpressure p₀ predicted by the TNO multi-energy model is relatively low. Furthermore, the prediction accuracy decreases as the equivalent distance increases, yet the overall prediction accuracy remains above 70%. Conversely, when the hydrogen cloud explosion source intensity is set to 4, the predicted lateral explosion overpressure p₀ increases significantly, with the predicted value being approximately 1.5 times greater than the experimental value. Thus, it can be concluded that the TNO multi-energy model effectively predicts the peak explosion overpressure of the open-space hydrogen cloud examined in this study. To achieve a more accurate prediction, it is recommended that the explosion source intensity level be selected as 3.

3.5 Comparison of overpressure prediction accuracy of different models

Due to the significant deviation of the traditional TNT model in predicting the peak overpressure of hydrogen cloud explosions, this section focuses solely on the predictive accuracy of the O-TNT model and the TNO multi-energy model regarding the overpressure of hydrogen cloud explosions. Additionally, it evaluates the applicability of both models. The peak overpressure of hydrogen cloud explosions at two scales with φ = 1.0 is compared to the predictions made by the O-TNT model and the TNO multi-energy model, as illustrated in Fig. 13.

Figure 13 demonstrates that the accuracy of the O-TNT model exceeds 90% for predicting the peak overpressure of 1 m³ hydrogen cloud explosions. However, the accuracy of the peak overpressure prediction for 27 m³ hydrogen cloud explosions decreases, indicating a significant influence of scale effects on the O-TNT model’s predictive performance. To enhance the prediction accuracy for the peak overpressure of large-scale hydrogen cloud explosions, the O-TNT model requires further verification and optimization through additional experimental data. In contrast, the prediction accuracy of the TNO multi-energy model remains unaffected by the scale of the hydrogen cloud, showcasing broader adaptability. Its robustness renders it more reliable for practical applications, making it suitable for hydrogen cloud explosion scenarios across various environmental conditions. Among the three models discussed in this paper, the TNO multi-energy model is recommended for predicting the peak overpressure of hydrogen cloud explosions, as it balances prediction accuracy and stability, offering significant advantages for practical engineering applications.

4. Conclusions

In this study, an experimental platform was established to investigate the flame morphology and overpressure evolution of hydrogen cloud explosions in open space, specifically for two sizes of 1 and 27 m³. Utilizing the experimental data, we examined the prediction accuracy and application range of three models: the traditional TNT model, the optimized O-TNT model, and the TNO multi-energy model, in relation to the peak overpressure of large-scale hydrogen cloud explosions. The main conclusions are as follows:

1. The flame propagation distance, velocity, and explosion overpressure of the hydrogen cloud explosion are influenced by the size of the hydrogen cloud, the equivalent ratio, and buoyancy factors as described in the experiment. In the initial phase of the hydrogen cloud explosion, the flame developed spherically; however, thermal diffusion instability and hydrodynamic instability caused the spherical flame to elongate horizontally, leading to instability in the flame morphology after the membrane rupture. The average flame propagation velocities at different equivalence ratios were as follows: V_a (φ = 1.0) > V_a (φ = 1.5) > V_a (φ = 2.5) > V_a (φ = 0.5). The scale effect significantly impacts the peak explosion overpressure of hydrogen clouds in both fuel-rich and fuel-lean scenarios, with the peak overpressures at the same equivalent distance being comparable across both scales.

2. The predicted peak overpressure for open-space hydrogen cloud explosions using the traditional TNT model is 10 to 35 times greater than the experimental values. Additionally, the explosion efficiency derived from accident statistics in this model exhibits substantial deviation when applied to theoretical predictions. The optimized O-TNT model demonstrates over 90% accuracy in predicting the peak overpressure of a 1 m³ hydrogen cloud explosion, but its accuracy diminishes for the 27 m³ hydrogen cloud explosion. When assuming the intensity level of the hydrogen cloud explosion source to be level 3, the overpressure predicted by the TNO multi-energy model is relatively low, and its prediction accuracy decreases with increasing equivalent distance; nevertheless, the overall prediction accuracy of this model for the peak overpressure of hydrogen cloud explosions exceeds 70%.

3. A comparison of the prediction accuracy and applicability of the three theoretical models reveals that the TNO multi-energy model’s prediction accuracy is independent of the hydrogen cloud scale, making it suitable for predicting peak overpressure in larger-scale hydrogen cloud explosions. Considering practical engineering applications, the TNO multi-energy model emerges as the optimal choice among the three models, balancing prediction accuracy with reliability.

Acknowledgements

The authors would like to express sincere appreciation for the financial support rendered by the National Key Research and Development Program of China (2021YFB4000900) and the National Natural Science Foundation of China (52374233, 52174208).

Author contributions

Bin Peng (Conceptualization [lead], Data curation [lead], Methodology [lead], Visualization [lead], Writing—original draft [lead]), Qiuhong Wang (Conceptualization [lead], Formal analysis [lead], Investigation [lead], Methodology [lead], Validation [lead], Visualization [lead], Writing—original draft [lead], Writing—review & editing [lead]), Wei Gao (Conceptualization [equal], Data curation [equal], Visualization [equal]), Huahua Xiao (Conceptualization [equal], Data curation [equal], Supervision [equal]), Zhenmin Luo (Conceptualization [equal], Data curation [equal], Methodology [equal]), Mingshu Bi (Methodology [lead], Writing—original draft [lead], Writing—review & editing [lead], Conceptualization [equal], Data curation [equal], Supervision [equal]), Yifei Liu (Visualization [equal], Writing—original draft [equal]), He Zhu (Visualization [equal], Writing—original draft [equal]), and Jianxiong Liu (Conceptualization [equal], Data curation [equal])

Conflict of interest statement

The authors declare that they have no known competing financial interests or personal relationships that could have appeared to influence the work reported in this paper.

Data availability

The data underlying this article will be shared on reasonable request to the corresponding author.

References