# Small-scale BLEVE: Near-field aerial shock overpressure and impulse

E.M. Laamarti, A.M. Birk, Frederic Heymes

► To cite this version:

E.M. Laamarti, A.M. Birk, Frederic Heymes. Small-scale BLEVE: Near-field aerial shock overpressure and impulse. Journal of Loss Prevention in the Process Industries, 2024, 92, pp.105463. 10.1016/j.jlp.2024.105463. hal-04753091

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# Small-scale BLEVE: Near-field aerial shock overpressure and impulse

E.M. Laamarti<sup>a,b</sup>, A.M. Birk<sup>a</sup>, F. Heymes<sup>b,*</sup>

<figure><img src="image_3.png" /></figure>

## ARTICLE INFO

**Keywords:**

Blast overpressure

Shock wave

Impulse

Propane

Boiling liquid expanding vapor explosion

BLEVE

Vessel failure

Liquid full

Near-field

## ABSTRACT

This paper presents an investigation into one of the most damaging hazards associated with Boiling Liquid Expanding Vapor Explosion (BLEVE), specifically focusing on the near-field overpressure and impulse effects. Experiments were conducted at a small-scale to study the overpressure and impulse using aluminum tubes with a diameter of 50 mm and a length of 300 mm. The tubes were filled with pure propane liquid and vapor. The controlled variables, in this work included the failure pressure, the liquid fill level, and the weakened length along the tube top. These variables control the strength of the overpressure characterized by the peak over-pressure amplitude, duration of this overpressure event, and the resultant impulse. Notably, these experiments at a small scale included experiments with a 100 % liquid fill level. This further confirmed that the vapor space is the main driver of the lead overpressure hazard. High-speed cameras and blast gauges effectively illustrated the progressive formation of the shock wave in both temporal and spatial dimensions. Furthermore, various predictive models available in the literature are discussed in this paper and new correlations were developed to quantify the overpressure duration and impulse. The current analysis aims to predict the potential consequences of overpressure events during a BLEVE.

## 1. Introduction

The boiling liquid expanding vapor explosion (BLEVE) constitutes a distinctive form of a thermomechanical physical explosion involving a pressure liquefied gas (PLG). It can happen when the pressure vessel containing the PLG is severely weakened by some process such as fire heating. Not all explosions observed fall under the BLEVE classification as it requires rapid total loss of containment (i.e., full opening of the vessel) as proposed by Birk et al. (2007). The BLEVE happens during the sudden explosive release of the vapor and liquid energy upon the rapid opening of the vessel. The BLEVE results in rapidly expanding vapor and flashing liquid that will generate hazards including projectiles, fireballs if the commodity is flammable, toxic or flammable vapor clouds, ground loading, and blast overpressures. The intensity of these hazards depends on the operating conditions and the specific vessel geometry of the failure, as illustrated by Laamarti et al. (2024b). These hazards close to urban areas, can cause significant consequences in terms of material losses and human casualties.

Numerous researchers have devised models aimed at facilitating hazard mitigation and prediction. Casal and Salla (2006); Planas-Cuchi et al. (2004) developed models based on expansion energy using a flash

fraction of the liquid to quantify the lead shock overpressure. Birk et al. (1996) in response to risk and prevention, developed a model to predict the fireball diameter and duration. Moreover, Baum et al. (2005) established correlations to predict the projectile's velocity and radius of action based on the isentropic expansion energy. As for the ground load, Laamarti et al. (2024a) formulated correlations for estimating peak ground load variables associated with small-scale BLEVE events. The present study focuses on the near-field overpressure and impulse. We define the near-field as being between three to ten times the vessel's diameter.

Overpressures are produced by the expanding vapor and the flashing liquid. They manifest as waves that propagate over finite distances, yet they possess sufficient strength to induce structural damage to buildings, fracture surrounding glass, and even initiate a cascading chain reaction. Birk et al. (2007) identified three overpressures from a BLEVE. The first two are produced primarily by the vapor space release and the third by both the vapor phase and flashing liquid. The first overpressure is a steep shockwave, followed by a negative phase rarefaction wave caused by the overexpansion of the vapor space. The overexpansion leads to a second weaker shock wave. The third wave is produced by the flashing liquid and is not a shock wave. The prediction of this specific hazard has been

<footnote><sup>a</sup> Queen's University Department of Mechanical and Materials Engineering, Kingston, ON, K7L 3N6, Canada</footnote>

<footnote><sup>b</sup> Laboratoire des Sciences des Risques, IMT Mines Ales, Ales, France</footnote>

<footnote>* Corresponding author.</footnote>

<footnote>E-mail address: frederic.heymes@mines-ales.fr (F. Heymes).</footnote>
<figure><img src="image_4.png" /><figcaption>Fig. 1. BLEVE 2022 Experimental apparatus.</figcaption></figure>

<figure><img src="image_5.png" /><figcaption>Fig. 2. Summary Laamarti BLEVE experiments 2022</figcaption></figure>

extensively explored experimentally and numerically by several authors.

The exact nature of whether the shock wave overpressure results exclusively from vapor and liquid remains a subject to controversy. Giesbrecht et al. (1981) conducted experiments involving rapid catastrophic failure of fully propylene-filled glass spheres, showing the generation of an overpressure wave. However, it is not clear if this overpressure was a shock wave (supersonic). Consequently, conclusions on the role of the liquid phase in shockwave formation remain elusive.

Van den Berg et al. (2006) formulated a one-dimensional expansion-controlled model assuming that the flash fraction of the liquid contributes to shock formation. While this assumption leans towards a conservative approach, its accuracy remains uncertain. This perspective contrasted with Baker et al. (1983) suggesting that BLEVE shock formation is only a result of vapor expansion as the liquid flashing is too slow to produce a shock wave. Presently, advancements in data acquisition technologies are leaning toward the understanding that the primary shockwave from BLEVE, is an outcome of the vapor expansion alone, with no significant contribution from the liquid phase. The liquid produces a slower and more powerful contribution in the form of ground loading, projectiles, and dynamic pressure (drag) effects. Across different scales, Birk et al. (2007), Eyssette et al. (2021), and current experiments provided experimental evidence on the contributions of vapor and liquid to BLEVE hazards.

Currently, to estimate the overpressure theoretically, a predominant approach has been based on isentropic expansion energy derived from

thermodynamic first principles. Researchers like Brode et al. (1959), and Prugh et al. (1991) adopted this method. In contrast, some other models integrated irreversibly to account for heat losses as the one developed by Planas-Cuchi et al. (2004) and Casal et al. (2006). While both approaches yielded conclusive results when compared to experiments, these models, while informative, fail to account for the complexity of opening dynamics in real-world scenarios and do consider the flash fraction of the liquid in shock formation. Nevertheless, a new approach, far from energy models, was presented by Birk et al. (2018), Birk et al. (2020) with the shock start model using the 1D shock tube equation and spherical shock wave theory.

This paper presents a set of new overpressure data from the 2022 experimental campaign done by Laamarti et al. (2024a,b) complemented by data from Eyssette (2018). The collected data include a broader range of controlled variables, including failure pressure, liquid fill level, and weakened length, thereby extending the scope of the existing data. Notably, analysis of unaddressed aspects of overpressure was carried out using all the experimental results from a small-scale apparatus. This study further investigates the use of the Friedman-Whitham approach for spherical shock waves combined with the shock tube model. Correlations are introduced to address the interplay of dependent overpressure variables, specifically the lead overpressure peak, its duration, and impulse.

## 2. Small scale propane BLEVE experiments

### 2.1. *Experimental apparatus*

The experiments were conducted with small-scale tubes of 300 mm in length and 50 mm in diameter. A total of 36 experiments were conducted in 2022. The tubes were made from aluminum type 6061 T6 annealed to T0 temper to reduce the material's yield and ultimate strength. The tubes were machined along the top to reduce the wall thickness along a defined weakened length. To reduce the effects of age hardening, the tubes were stored in a freezer at -40 °C before the testing. The tube preparation facilitated tube rupture at desired failure pressures, thereby delineating both the location and size of the burst opening. To initiate the BLEVE, the tubes were filled with propane and subjected to an external heater, along the tube bottom (Fig. 1. [Left]). The heater primarily heated the liquid to produce a saturated mixture of liquid and vapor. The heating caused the pressure to rise in the vessel until it failed. The failure occurred in the weakened length when the von Mises stress reached the yield strength of the annealed aluminum. High and low-speed instruments were used to capture the hazards produced by the small-scale BLEVE. This included thirteen pencil blast gauges positioned at varied locations, diverse angles, and elevations [vertical, horizontal, 45° angle] around the subjected tubes. The vertical gauges PCB 137B28 labelled TOP 1,2,3 and 4, are positioned respectively at 15,20,30 and 40 cm above the tube. These gauges boast a range of 3.45
<figure><img src="image_6.png" /><figcaption>Fig. 3. Summary Eyssette (2018) BLEVE experiments.</figcaption></figure>

<figure><img src="image_7.png" /><figcaption>Fig. 4. Overpressure measured at 20 m from the side of a 2000 L propane tank, Birk et al. (2007).</figcaption></figure>

bar, a response time of 1 µs, a sensitivity of 14.5 mV/kPa, and an uncertainty of under 2%. The remaining gauges, PCB 137A23, are placed at different angles and sampled at 200 kHz. They also have a range of 3.45

bar, a response time of 4 µs, a sensitivity of 14.5 mV/kPa, and an uncertainty of 0.2 %.

Three high-speed cameras were strategically positioned at the front, side, and window views to capture the details of the explosion and observe the overpressure details. Operating at a rate of approximately 24,000 fps, these cameras featured an exposure duration of 40 µs. A detailed description of the apparatus is presented in Laamarti et al. (2024a,b).

## 2.2. Experimental apparatus

The small-scale BLEVE experiments carried out in 2022 [Fig. 2] for various operating conditions has been complemented by data from Eyssette (2018) [Fig. 3]. The controlled variables that were selected to attain the desired burst conditions included.

- Failure pressure $P_f$ [Barg or Bara]: g and a stand for gauge and absolute pressure

- Liquid fill (volume fraction) $\varphi$ [%]

- Weakened length $L_c$ [m]

These variations in operating conditions help understand the impact of controlled variables and whether similar conditions can yield to same outcomes.

New data for 100 % liquid fill level was achieved to discern the specific contribution of the liquid phase in the formation of shockwaves. The experimental variables were systematically modified progressively, thereby facilitating an investigation into the individual influence exerted by each variable on the resultant blast wave. Among the experiments, a total of nine experiments resulted in partial opening of the vessel, thus showing the effect of opening size on the strength of the hazards produced by the BLEVE.

## 2.3. BLEVE overpressures

Variations in the operating conditions generate various data on the dependent near-field overpressure variables. Conventionally, a BLEVE produces a sequence of overpressure events, consisting of an initial first lead shock overpressure from the vapor phase followed by an under-pressure wave, a second wave from the vapor phase, and a third wave overpressure from both the vapor phase and liquid flashing. Fig. 4 shows typical blast data measured at 20 m from a 2000 L propane tank. Three successive wave overpressures generated during a BLEVE are visible on the graph. The magnitude, duration, and impulse of the lead shock overpressure will be explored in this study.

<figure><img src="image_8.png" /><figcaption>Fig. 5. Shock wave for [A] unsteady and [B] steady cases.</figcaption></figure>
<figure><img src="image_9.png" /><figcaption>Fig. 6. Lead overpressure TOP 1 vertical gauge (15 cm) [a] Eyssette (2018) [b] Laamarti Experiments 2022</figcaption></figure>

<figure><img src="image_10.png" /><figcaption>Fig. 7. [a]. Comparative study of the first lead peak overpressure from TOP 1 blast gauge.</figcaption></figure>

The objective is to formulate a physics-based analysis of the dependent variables, specifically extracted from vertical blast gauges, because they measured the strongest overpressure for the top opening failure. The variables were.

* Peak overpressure of the first wave [Bar]

* Duration of the first wave overpressure [s]

* Impulse per unit area of the first wave overpressure [Bar.s]

The lead shock decays with distance and is assumed to follow the spherical shock theory. The data suggests that vessels filled to 100 % capacity with liquid generate solely an overpressure with no shock-waves. Thus, the shock tube equations and the Friedman-Whitham spherical shock theory cannot be applied.

### 3. Numerical methods

#### 3.1. Shock tube theory

During a BLEVE, the pressure vessel fails, and the vapor leaves the vessel, creating pressure on the surrounding air. This rapid pushing on the atmosphere results in a series of compression waves that pile up and form a supersonic shock wave to meet the flow constraints (Fig. 5).

Commonly referred to as a “surface of discontinuity”, the shock wave refers to a sharp pressure rise that then tails off with distance. The shock strength and shape depend on many factors, such as.

* The operating conditions: Failure pressure, weakened length, liquid fill.

* The type of opening: whether partial or full opening of the pressure vessel.

* The rate of opening due to the uneven thickness of the weakened length

The BLEVE resulting from a cylindrical vessel will generate a cylindrical shock that will transition to a spherical shock with distance as it propagates through an unconfined environment at a transient state. The initial overpressure can be calculated using the shock tube equations, especially if the opening of the vessel is extremely rapid. During an explosion, Fig. 5 [A], a shock wave propagates at supersonic speed S, accompanied by a blast wind $u_b$ behind it. In the unaffected region, the surrounding air remains unaware of the impending shock. In this unsteady state, traditional normal shock equations cannot be applied since they pertain to steady-state conditions. Thus, we need to adopt a reference frame that moves with the shock. To transition to a steady state, in Fig. 5 [B], we consider the ambient air approaching the explosion side at supersonic velocity S, resulting in a pressure jump at the shock interface (dashed line). Behind the shock, the flow transitions to a subsonic velocity, allowing for the application of shock tables or equations. However, to determine stagnation pressures, we must go back to the stationary reference frame.

The 1D normal shock tube equations developed by Rankine-Hugoniot (1870) can then be applied to address the stationary problem. These equations satisfy the continuity, momentum and energy equations
<figure><img src="image_11.png" /><figcaption>Fig. 7. [b]. High-speed camera Experiment 20; Frame 1 (time = 0 ms): no failure, Frame 2 (time = 0.4 ms): Lead shock reached TOP 1 (15 cm), small white cloud (vapor) starting. Frame 3 (time = 0.8-1.2 ms): Flashing process. Horned beast expansion of contents. Frame 4–5 (time = 1.6-2 ms): Internal vessel pressure drops progressively to atmospheric pressure and the second wave reaches TOP 1 gauge. Frame 6 (time = 3.1 = 5.2 ms): Cloud dispersion.</figcaption></figure>

<figure><img src="image_12.png" /><figcaption>Fig. 8. Lead peak overpressure Laamarti Experiments 2022</figcaption></figure>
<figure><img src="image_13.png" /><figcaption>Fig. 9. Lead peak overpressure Eyssette (2018).</figcaption></figure>

<figure><img src="image_14.png" /><figcaption>Fig. 10. Lead peak overpressure from TOP blast gauges.</figcaption></figure>

which result in:

$$ \frac{P_4}{P_1} = \frac{P_2}{P_1} \left[ 1 - \frac{(\gamma_4 - 1) \frac{a_1}{a_4} \left( \frac{P_2}{P_1} - 1 \right)}{\sqrt{2\gamma_1 \left( 2\gamma_1 + (\gamma_1 + 1) \left( \frac{P_2}{P_1} - 1 \right) \right)}} \right]^{\frac{-2\gamma_4}{\gamma_4 - 1}} \quad (1) $$

The static pressures obtained by transitioning to steady state are: (White, 1998):

$$ \frac{P_2}{P_1} = \frac{2\gamma_1 M^2}{(\gamma_1 + 1)} - \frac{(\gamma_1 - 1)}{(\gamma_1 + 1)} \quad (2) $$

$$ \Delta p_{st} = (P_2 - P_1) = > \text{overpressure} \quad (3) $$

➤ State 2: Explosion side (before shock = downstream part)

➤ State 1: Unaffected side (after shock = upstream part) at ambient conditions

<figure><img src="image_15.png" /><figcaption>Fig. 11. Lead peak overpressure from all gauges.</figcaption></figure>

h₁ and h₂: Enthalpy at state 1 and 2 S: Supersonic shock speed

T₁ and T₂: Static temperature at state 1 and 2 P₄: Static failure pressure

P₁ and P₂: Static pressure at state 1 and 2 γ₁: Air specific heat ratio (1.4)

S₁ and S₂: Entropy at state 1 and 2 a₄: Speed of sound in propane

γ₄: Propane specific heat ratio (1.3) M: Mach number

uₕ: Blast wind speed (speed of air behind the shock) a₁: Speed of sound in air

All terms are expressed in SI units: meters (m), seconds (s), pascals (Pa), Joule (J), Kelvin (K).

## 3.2. Friedman Whitham approach

Under the assumption of spherical expansion, the variation of shock strength with distance can be predicted using the Friedman Whitham relation (Friedman, 1960) as follows:
<figure><img src="image_16.png" /><figcaption>**Experiment 2: 99% | 150 mm | 24 Barg**</figcaption></figure>

<figure><img src="image_17.png" /><figcaption>**Experiment 6: 50% | 150 mm | 23.1 Barg**</figcaption></figure>

<caption>**Fig. 12.** Comparative study of the prediction of the overpressure from blast vertical top gauges.</caption>

**Experiment 2:** 99% | 150 mm | 24 Barg **Experiment 6:** 50% | 150 mm | 23.1 Barg.

<figure><img src="image_18.png" /><figcaption>**Fig. 13.** Comparative study of the prediction of the overpressure from blast TOP gauges.</figcaption></figure>

$$ \left[ \frac{R}{R_0} \right]^2 = e^{(2\gamma_1 - 2) \frac{1}{2}} \sin^{-1} \left[ \frac{2Y^2 - (\gamma_1 - 1)Z^2}{(\gamma_1 + 1)^2 M^2} \right] \quad (4) $$

Where

$$ Y = (2\gamma_1 M^2 - \gamma_1 + 1)^{\frac{1}{2}} \quad (5) $$

$$ Z = ((\gamma_1 - 1)M^2 + 2)^{\frac{1}{2}} \quad (6) $$

$$ W = \frac{(Y - Z)^2}{M} \left[ Y(\gamma_1 - 1)^{\frac{1}{2}} + Z(2\gamma_1)^{\frac{1}{2}} \right] \left( \frac{2\gamma_1}{\gamma_1 - 1} \right)^{\frac{1}{2}} Y^{\frac{2}{\gamma_1}} \quad (7) $$

R: Radius of the hemispherical shock

R₀: Radius of the hemispherical sphere surface of the vapor phase where the shock started

$$ R_0 = \left[ \frac{3(1 - \varphi)}{4\pi} \frac{V_{\text{tube}}}{2} \right]^{\frac{1}{3}} \quad (8) $$

$V_{\text{tube}}$: Volume of the tube

$$ \left[ \frac{\Delta p_{\text{exp}}}{\Delta p_{\text{st}}} \right] = \left[ \frac{R}{R_0} \right]^n \approx \left[ \frac{\Delta p_{\text{FW}}}{\Delta p_{\text{st}}} \right] = \left[ \frac{R}{R_0} \right]^n \quad (9) $$

$\Delta p_{\text{FW}}$: Overpressure from Friedman Whitham

$\Delta p_{\text{st}}$: Overpressure from shock tube equations

$\Delta p_{\text{exp}}$: Experimental overpressure

n: Index.

### 3.3. Duration of lead overpressure

The duration of the lead overpressure is an important dependent variable that helps evaluate the strength of the shock. The longer the duration of lead overpressure prolongs the exposure of structures to the intense force exerted by the blast wave potentially leading to great damage. Moreover, prolonged overpressure can result in secondary effects such as structural fatigue and domino effects. The experimental value is found from the base of the triangular-shaped peak overpressure. The duration of lead overpressure, being solely an outcome of the vapor phase, can be rendered non-dimensional by dividing it by the duration of vapor release.

$$ t_{\text{ov}}^* = \frac{t_{\text{ov}}}{t_{\text{vap}}} \quad \text{with} \quad t_{\text{vap}} = \frac{D}{a} \quad (10) $$

t<sub>ov</sub>: Experimental duration of overpressure [s] D: Tube diameter [m]

t<sub>ov</sub>*: Dimensionless duration of overpressure t<sub>vap</sub>: Duration of the vapor phase [s]

a: Speed of sound in vapor propane [m/s].

### 3.4. Impulse per unit area

The impulse per unit area is determined by integrating the first peak overpressure over its duration. The wave form is nearly an isosceles triangle, and the integral is approximately equal to:

$$ I_{\text{ov}} = \frac{\Delta p_{\text{exp}} t_{\text{ov}}}{2} \quad (11) $$

I<sub>ov</sub>: Experimental Impulse per unit area [bar. s]

The formula corresponds to the impulse per unit area experienced by an object exposed to a blast. The condition is that the object is small enough for the impulse per unit area to remain constant across its surface.
<figure><img src="image_19.png" /><figcaption>Fig. 14. [a]. Duration of TOP 1 gauge [15 cm] lead overpressure from BLEVE 2022.</figcaption></figure>

<figure><img src="image_20.png" /><figcaption>Fig. 14. [b]. Duration of TOP 1 gauge [15 cm] lead overpressure from Eyssette(2018).</figcaption></figure>

### 3.5. Correlation equation

The duration and the impulse from the first overpressure have been correlated to the independent variables to formulate a predictive equation equal to:

$$t_{ov}^{*} = \frac{t_{ov}}{t_{vap}} \frac{t_{ov\_cor}}{t_{vap}} = t_{ov\_cor}^{*} = K_{1} \left( \frac{P_{f}}{P_{atm}} \right)^{a} \left( \varphi \right)^{b} \left( \frac{L_{c}}{L_{v}} \right)^{c}$$

$$t_{ov} \approx t_{ov\_cor} = K_{1} \left( \frac{P_{f}}{P_{atm}} \right)^{a} \left( \varphi \right)^{b} \left( \frac{L_{c}}{L_{v}} \right)^{c} t_{vap} \quad (12)$$

t<sub>ov\_cor</sub>: Correlated duration of lead overpressure [s]

a, b, c, K<sub>1</sub>, are constants

For the impulse per unit area, and the corresponding equation is:

$$I_{ov\_cor} = \frac{\Delta p_{fw} t_{ov\_cor}}{2} \quad (13)$$

## 4. Results and discussion

### 4.1. Opening dynamics

The opening dynamics of the vessel are believed to strongly influence the resultant overpressure. Experiment 30 from Laamarti BLEVE experiments 2022 can be compared with Experiment 26 by Eyssette (2018) under similar initial conditions of failure pressure, liquid fill, and weakened length. Fig. 6 [Left] shows the lead overpressures and Fig. 6 [Right] shows the tube opening. As it can be seen, the overpressure vs
<figure><img src="image_21.png" /><figcaption>Fig. 15. [a]. Dimensionless duration of lead overpressure at TOP 1 gauge (15 cm vertically).</figcaption></figure>

<figure><img src="image_22.png" /><figcaption>Fig. 15. [b]. Experimental vs correlated duration of lead overpressure from TOP 1 gauge (15 cm vertically).</figcaption></figure>

time is quite similar except the Eyssette (2018) data shows a rising pressure after the initial overpressure. This could be attributed to the larger size of the laboratory used in the Eyssette (2018) experimental campaign and the difference of 5–10 % liquid fill, which couldn’t be precisely controlled due to leaking issues.

## 4.2. Influence of failure conditions on top vertical blast overpressure

The controlled failure conditions influence the lead overpressure

strength and impulse. While prior studies on near-field overpressure predominantly focused on the blast wave’s peak overpressure, this investigation extends its scope to include the study of the impulse. To discern the impact of the controlled variables, experiments were selected with similar conditions and with only one parameter changing at a time (Fig. 7).

The sequence of events remains consistent across the experiments; however, the duration of these events varies depending on the operating conditions and the opening dynamics involved.
<figure><img src="image_23.png" /><figcaption>Fig. 16. [a]. Experimental Lead peak overpressure vs overpressure duration.</figcaption></figure>

<figure><img src="image_24.png" /><figcaption>Fig. 16. [b]. Experimental Impulse per unit area vs Correlated Impulse per unit area.</figcaption></figure>

Using experiment 20 as a reference point, experiments 18, 25, and 29 respectively represent an increase in liquid fill, a decrease in failure pressure, and a decrease in weakened length. Fig. 7[1] shows the overpressures measured by the blast sensor TOP 1 positioned 15 cm above the top of the tube. Below are summaries of the experiments.

**Experiment 20 \ Experiment 18:** Increase the liquid fill level from 20 % to 90 %

* reduced maximum peak overpressure.

* increased duration of the peak wave.

* reduced impulse.

* formed two clear pressure peaks.

* negative phase amplitude is similar for both experiments.

High liquid fill level does imply a lower amount of vapor phase energy and thus a more rapid decay of overpressure with distance.

**Experiment 20 \ Experiment 25:** Decrease the failure pressure from 23 to 14 Barg.

* reduced maximum peak overpressure.

* increased duration of the peak wave

* reduced impulse.

* reduced the negative phase amplitude.

* ratio of failure pressures is equal to the ratio of peak overpressures (0.67)

The failure pressure describes the pressure difference between the atmospheric pressure and the tube pressure. As expected, lower failure pressure produces a weaker pressure wave.

**Experiment 20 \ Experiment 29:** Decrease the weakened length from 100 to 75 mm.

* smaller opening

* reduced maximum peak overpressure.

* no change in duration of peak wave

* slightly reduced the negative phase amplitude.

The contents exiting the pressure vessel through a smaller opening area had an impact on the maximum peak overpressure, as the reduced opening of the vessel walls interfered with the formation of the shock.

Overall, it was observed that all controlled operating conditions resulted in changes to the overpressure characteristics measured at 15 cm above the tube. An important dependence was noted for failure pressure and liquid fill level, while a partial dependency was identified for weakened length. This underscores the necessity to develop correlations that link lead overpressure characteristics to the controlled variables. While the crack velocity, which describes the way the pressure vessel opens, is believed to exert a certain impact, control over this parameter has not been achieved to date.

### 4.3. Estimation of the lead peak overpressure

The lead peak overpressure from the vertical blast gauge at 15 cm above the tube is presented in Figs. 8 and 9 for both the Laamarti BLEVE 2022 campaign and Eyssette (2018) as a function of the failure pressure, liquid fill, and weakened length. The graphical representations reveal a discernible pattern: the magnitude of the experimental overpressure consistently increases with an increase in failure pressure and weakened length, while it decreases with a rise in liquid fill. This phenomenon can be attributed to the vapor phase; a larger volume of the vapor phase correlates to a slower pressure decay over distance, resulting in a higher magnitude of overpressure at a specified distance. Moreover, the graph shows that at a similar failure pressure, the 100 % liquid fill cases failed partially and gave the lowest peak overpressure.

The experimental peak overpressures were normalized by the calculated peak overpressures from the shock tube equation under identical operating conditions. These normalized values are represented in Fig. 10 with respect to dimensionless distance.

Results from both campaigns exhibit similarities, overlaying each other and revealing the general behavior of the shock wave. As distance increases, the overpressure shows a 1/R rate of decay, indicative of the weakening shock because of energy dissipation in the surrounding atmosphere and an expanding shock surface area. Experimental findings demonstrate that the vertical overpressures are more powerful compared to those measured by the 45° angle gauges and the horizontal gauges, as depicted in Fig. 11. This can be primarily attributed to the dynamics of opening, particularly the energy required to flatten the vessel walls.

The experimental findings presented above can then be contrasted with the prediction from the Friedman-Whitham theory for each experiment, for instance, experiments 2 and 6 are illustrated in Fig. 12. The overpressures were normalized by the overpressure from the shock tube equations.

The comparison between experiments and the Friedman Whitham approach for experiment 2 indicates a good approximation of the
solution with a minor discrepancy observed in the exponent value. This deviation in results can be attributed to small differences in the opening of the vessel. For lower liquid fill levels, the overpressures are faster and stronger and the Friedman-Whitham model tends to overpredict the actual experimental data. Furthermore, the shock fully forms within the 3–4 times the tube diameter range, thus it may fully form at TOP 2, 20 cm above the tube, rather than TOP 1 (15 cm). This is visible for experiment 6 where the second experimental overpressure data point at 20 cm surpasses the first one at 15 cm.

The Friedman-Whitham prediction is compared with the experimental overpressure data obtained for all experiments at various heights above the tube as illustrated in Fig. 13.

The graph distinctly illustrates that the Friedman-Whitham relation tends to slightly overestimate the experimental overpressure. However, it is noteworthy that dividing the Friedman-Whitham formula by approximately half underestimates the weakest overpressures.

In the overall experiments, the exponent for vertical blast gauges was obtained between −0.8 and −1.02 for the Friedman-Whitham approach and between −0.87 and −1.25 for the experiments such as:

$$ \left[ \frac{\Delta p_{exp}}{\Delta p_{st}} \right] = \left[ \frac{R}{R_0} \right]^{-0.87 \text{ to } -1.25} \approx \left[ \frac{\Delta p_{Fw}}{\Delta p_{st}} \right] = \left[ \frac{R}{R_0} \right]^{-0.8 \text{ to } -1.02} $$

### 4.4. Estimation of the duration of lead overpressure

The power of a shock aligns with the energy of the overpressure over time – indicating that a longer duration of overpressure corresponds to a stronger shock. Thus, this investigation targets the duration of overpressure quantified by measuring the integral under the maximum peak overpressure, evenly distributed along the isosceles triangle. Fig. 14 illustrates the overpressure duration under various operating conditions derived from 2022 and Eyssette (2018) experimental campaigns. In Fig. 14, it is observed that the experimental duration of lead overpressure increases with higher liquid fill. Conversely, an increase in failure pressure leads to a decrease in experimental duration. The influence of weakened length is less evident. Importantly, cases with 100 % liquid fill showed the longest duration of lead overpressure.

The dimensionless duration of lead overpressure [Equation (10)] has been calculated for both Laamarti BLEVE 2022 experiments and Eyssette (2018). A correlation [Equation (12)] with specific exponents has been numerically derived to match the dimensionless duration from 2022 experimental data based on operating conditions. Subsequently, this correlation has been then applied to Eyssette (2018) experiments, showing a very consistent fit with the results (y = x<sup>0.99</sup>). Fig. 15 [a] shows the dimensionless duration of overpressure with respect to the correlation for both campaigns.

The graph reveals five data points with a distinct behavior from the rest. These points are 100 % liquid fill scenarios, and they have a prolonged duration of lead overpressure compared to other experiments because of the different opening dynamics involved. Consequently, the 100 % liquid fill cases are not included in the formulation of the duration of lead overpressure. The average duration of lead overpressure found numerically is expressed as follows:

$$ t_{ov\_cor} = 9.05 \left( \frac{P_f}{P_{atm}} \right)^{-0.66} \left( \varphi \right)^{-0.20} \left( \frac{L_c}{L_v} \right)^{0.02} t_{vap} $$

Fig. 15[b] shows the experimental vs correlated duration of lead overpressure. From the graph, the highest duration covering the upper bound of the data cloud (100 % liquid fill excluded) is

$$ t_{ov\_cor\_max} = kt_{ov\_cor} \quad \text{with } k = 1.64 $$

This correlation is applicable within 3–4 times the diameter (3-4D) range, where the maximum peak overpressure was registered, and has undergone validation exclusively at small scale (D = 50 mm L/D = 6) for both Laamarti Experiments 2022 and Eyssette (2018) data. They are not applicable for 100 % liquid fill, and they require validating at a larger scale.

### 4.5. Estimation of the impulse per unit area

The experimental peak overpressure is shown as a function of the lead overpressure duration in Fig. 16 [a]. The plot shows that the highest peak overpressures have shorter durations. The duration will depend on the weakened length, the failure pressure, the liquid fill, and the opening dynamics. Fig. 16 [a] shows again the five points, each with 100 % liquid fill, as observed in Fig. 15. These points exhibit a combination of low overpressure magnitude and long duration, despite having similar failure pressures as the other liquid fill cases. This suggests the distinctive characteristics of BLEVEs with 100 % liquid fill. Further investigation is required for this specific BLEVE scenario. Fig. 16 [b] shows that most of the experimental results lie beneath the average predicted impulse line, aligning with the intended goal of overpredicting the results. The upper and lower bounds of the experimental impulse are determined by twice and one-fourth the average impulse.

By employing Equation (13), the correlation for impulse per unit area can be written as:

$$ I_{ov\_cor} = \frac{\Delta p_{Fw} 9.05 \left( \frac{P_f}{P_{atm}} \right)^{-0.66} \left( \varphi \right)^{-0.20} \left( \frac{L_c}{L_v} \right)^{0.02} t_{vap}}{2} $$

$$ I_{ov\_cor} = \frac{\Delta p_{st} \left[ \frac{R}{R_0} \right]^{-0.8 \text{ to } -1.02} 9.05 \left( \frac{P_f}{P_{atm}} \right)^{-0.66} \left( \varphi \right)^{-0.20} \left( \frac{L_c}{L_v} \right)^{0.02} t_{vap}}{2} $$

## 5. Conclusion

The investigation into near-field overpressure from small-scale propane BLEVE experiments (D = 50 mm, L = 300 mm, volume = 589 cm<sup>3</sup>) provides new insights into the calculation of impulse per unit area from the lead blast overpressure. The study focused on vertical blast gauges positioned above the tube, precisely where the maximum peak overpressure is registered.

Analysis of the Laamarti BLEVE experiments 2022 and Eyssette (2018) experimental campaign revealed that experiments with high failure pressures, low liquid fill levels, and long weakened lengths produced the highest peak overpressures. Notably, experiment 16 from BLEVE experiments 2022 exhibited the greatest peak overpressure, aligning with the highest failure pressure (35 Barg), the longest weakened length (150 mm), and the lowest liquid fill level (15 %). This observation suggests that the vapor phase contributes to high overpressure magnitude and is mainly responsible for shock formation.

The duration of lead overpressure was found to increase with high liquid fill levels and low failure pressures. However, when the liquid fill level reached 100 % (compressed liquid), the duration of lead overpressure was relatively longer. Consequently, the established correlation is not applicable to scenarios involving 100 % liquid fill.

Finally, a formula for the impulse per unit area was derived based on the Friedman-Whitham spherical approach and the developed correlation for the duration of lead overpressure. Presently, this impulse formula is validated only at a small scale and still requires validation at a larger scale. Moreover, it is important to reiterate that the correlation does not apply to liquid full fill cases where the vessel fails under hydrostatic conditions. Hence, additional investigation is necessary for scenarios involving 100 % liquid fill, as they represent a specific case of BLEVE.

### CRediT authorship contribution statement

E.M. Laamarti: Writing – review & editing, Writing – original draft, Visualization, Validation, Software, Resources, Project administration, Methodology, Investigation, Formal analysis, Data curation,
**Conceptualization.** **A.M. Birk:** Writing – review & editing, Validation, Supervision, Project administration, Investigation, Funding acquisition, Conceptualization. **F. Heymes:** Writing – review & editing, Validation, Supervision, Resources, Project administration, Funding acquisition, Conceptualization.

Declaration of competing interest

The authors declare the following financial interests/personal re-
lationships which may be considered as potential competing interests:El
Mehdi Laamarti reports financial support was provided by Queen's
University. El Mehdi Laamarti reports financial support was provided by
IMT mines Ales. If there are other authors, they declare that they have no
known competing financial interests or personal relationships that could
have appeared to influence the work reported in this paper.

Acknowledgements

The authors gratefully acknowledge the financial contribution from
Queen's University, Kingston, Canada and IMT mines Ales, Ales, France.

Data availability

Data will be made available on request.

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