Research progress on multi-overpressure peak structures of vented gas explosions in confined spaces

Journal Pre-proof Research progress on multi-overpressure peak structures of vented gas explosions in confined spaces Kai Yang, Qianran Hu, Siheng Sun, Pengfei Lv, Lei Pang PII:

S0950-4230(19)30569-8

DOI:

https://doi.org/10.1016/j.jlp.2019.103969

Reference:

JLPP 103969

To appear in:

Journal of Loss Prevention in the Process Industries

Received Date: 10 July 2019 Revised Date:

6 September 2019

Accepted Date: 25 September 2019

Please cite this article as: Yang, K., Hu, Q., Sun, S., Lv, P., Pang, L., Research progress on multioverpressure peak structures of vented gas explosions in confined spaces, Journal of Loss Prevention in the Process Industries (2019), doi: https://doi.org/10.1016/j.jlp.2019.103969. This is a PDF file of an article that has undergone enhancements after acceptance, such as the addition of a cover page and metadata, and formatting for readability, but it is not yet the definitive version of record. This version will undergo additional copyediting, typesetting and review before it is published in its final form, but we are providing this version to give early visibility of the article. Please note that, during the production process, errors may be discovered which could affect the content, and all legal disclaimers that apply to the journal pertain. © 2019 Elsevier Ltd. All rights reserved.

Research progress on multi-overpressure peak structures of vented gas explosions in confined spaces Kai Yang a,b, Qianran Hu a, Siheng Sun a,b, Pengfei Lv a,b, Lei Pang a,b,* a

School of Safety Engineering, Beijing Institute of Petrochemical Technology, Beijing 102617, China

b

Beijing Academy of Safety Engineering and Technology, Beijing 102617, China

Abstract: Explosion venting of flammable gases within confined spaces, which may be an explosion hazard or a hazard mitigation measure, has gained widespread attention from researchers in various fields. A substantial number of scientific studies have found that complex initial and boundary conditions lead to complex and diverse peak structures in vented gas explosion pressure curves. This poses major challenges for the accurate prediction and scientific research on vented gas explosion overpressures as well as incident prevention and control. By reviewing the existing literature, we analysed the distribution characteristics of multi-overpressure peak structures and the formation mechanisms of typical peak structures in vented gas explosion pressure curves. On this basis, the influence of different factors on various peak structures are described in detail, and a quantitative method for evaluating the correlation between influencing factors and peak structures is proposed. Our analysis revealed that peaks Pb, Pext, Pmfa, and Pac had the highest occurrence frequencies in the vented explosion pressure curves and were the dominant peaks during a vented gas explosion process. Influencing factors with the highest degrees of correlation with Pb, Pext, Pmfa and Pac include vent opening pressure, ignition location, obstacles, and gas concentration. In particular, gas concentration was identified as a key condition that influenced all typical pressure peak structures. Our conclusions may serve as a reliable theoretical basis for future research on safety relief theories and prevention of gas explosion incidents. Key words: Confined space; gas explosion; vented explosion; overpressure; peak structure

1. Introduction The unregulated use of various flammable gases within buildings has led to frequent explosion incidents with serious casualties. When such incidents occur, the combined influence of building walls, doors, and windows with different dimensions, opening pressures, interior accessories, and furniture typically results in the induction of complex vented gas explosion processes in confined spaces as a result of the aforementioned building elements (Tomlin et al., 2015). However, among the various control measures for gas explosion hazards discussed by researchers, vented gas explosions also serve as an effective, low-cost technical measure and have been applied in the protection of a wide variety of confined spaces, such as production equipment and devices. When a gas explosion occurs within a confined space, high-temperature, high-pressure combustion products can be swiftly released through explosion vents (Cao et al., 2016), thus causing the internal pressure to rapidly decrease below the pressure at which structural damage occurs. Thus, the degree of damage to buildings and equipment and the number of casualties should be reduced as much as possible (Bauwens and Dorofeev, 2014; Fakandu et al., 2014). Therefore, research on vented gas explosions is of great significance regardless *

Corresponding author. School of Safety Engineering, Beijing Institute of Petrochemical Technology, Beijing 102617, China.

E-mail addresses: [email protected] (L Pang).

of whether the process results in a hazard or serves as a hazard mitigation measure. Previous research has shown that the degree of severity of confined space explosion hazards is dependent on the magnitude of the vented explosion overpressure and pressure duration (Bao et al., 2016). The peak structures in pressure curves are primarily generated by the pressure balancing process that occurs during vented gas explosions in confined spaces. Gas venting in a confined space is extremely complex because it involves different physical processes, including combustion, thermodynamic and fluid dynamic processes, and is highly prone to influences by vent characteristic parameters, such as the opening pressure and vent size as well as factors, such as the room size, gas concentration, ignition location, and obstacles (Chen et al., 2006). This ultimately leads to the formation of overpressure peak structures as a result of different physical mechanisms (Bauwens et al., 2012). Cubbage and Simmons (1955) experimentally confirmed the existence of a double-peak structure in vented explosion overpressure curves for the first time. Subsequently, Yao (1974) performed reasonable theoretical analyses and predictions of the double-peak structure in overpressure curves (Pasmen et al., 1974; Bradley and Mitcheson, 1978a, 1978b; Molkov and Nekrasov, 1981). In a study by Chen et al. (2006) on the vented explosion of a methane-air mixture in a cylindrical vessel with a length: diameter ratio (L/D) of 5:3, a double-peak structure was also observed in the pressure curve, and it was concluded that the second peak pressure increased as the vent opening pressure increased. Chow et al. (2000) performed vented explosion experiments in a cylindrical vessel with a length: diameter ratio (L/D) of 3:1 using methane/air, propane/air, and ethylene/air mixtures. The results indicated a triple-peak structure in the pressure curves, with the first peak gradually forming the main characteristic of the overpressure-time plane with an increase in the vent opening pressure. Bauwens et al. (2008) and Rocourt et al. (2014) also observed triple-peak structures in overpressure curves during their vented explosion experiments. The results of these experiments indicate that each pressure peak structure may be influenced by different physical mechanisms and may become the dominant pressure peak under different initial experimental conditions. Cooper et al. (1986) conducted a vented explosion experiment in a cubic vessel with a side length of 0.91 m and observed a four-peak structure in the overpressure curve, with the peak induced by acoustic oscillations being the dominant peak. A pressure curve with four peaks was also obtained by Guo et al. (2015) during a vented hydrogen explosion experiment in a cylindrical vessel with a length and internal diameter of 0.25 m; however, the formation mechanisms of the second and fourth peaks were different from those of the corresponding peaks previously reported by Cooper et al. (1986). In another study, Fakandu et al. (2016) observed a six-peak structure in the pressure curve in an experiment conducted in a 0.1-m3 cylindrical vessel, and additional pressure peaks were formed by the venting of unburned gases and the continued combustion of backflow gases. Based on the findings of the aforementioned studies, it can be deduced that flame propagation and the flow of combustion products and unburned gases during confined space gas explosions are highly complex processes. Relevant research has also found that the maximum pressure achieved during a vented explosion may be controlled by any peak belonging to a certain number of specific pressure peaks (Chow et al., 2000). These pressure peaks may correspond to different physical phenomena, such as vent opening, external explosion, obstacle-induced increases in the flame surface area, and development of flame acoustic oscillations (Bauwens and Dorofeev, 2014). Different boundary and initial conditions also influence the evolution of explosion pressures, which leads to significant differences among the various peak structures. Although the multi-peak structures in the overpressure curves of vented explosions in confined spaces have been expounded in detail by many researchers, there are many factors that influence the vented explosion process, and the degree of

influence on different peak structures differs among the various factors. This has resulted in a lack of effective statistical analyses of the occurrence frequencies and dominance of peak pressures as well as in-depth summarizations and quantitative analyses of the trends and degrees of the influence of various factors. On this basis, herein we aimed to achieve the following: (1) perform a statistical analysis of the peak characteristics based on the results of selected literature on the multi-overpressure peak structures of vented explosions in confined spaces; (2) provide a summary of the formation mechanisms of typical peak structures; (3) elucidate the evolution patterns of various peak structures under different influencing conditions; (4) propose a novel quantitative method for evaluating the degrees of influence of different influencing factors on the peak structures. The results of this paper can serve as an important reference for the prevention, mitigation, and control of confined space gas explosions and the investigation of such incidents.

2. Evolution patterns of multi-overpressure peak structures 2.1 Distribution characteristics of multi-overpressure peak structures When a vented gas explosion occurs in a confined space, it usually experiences a series of gas-dynamic processes that result in the formation of different pressure peak structures (Gao et al., 2018). After a gas explosion occurs, the internal pressure continuously increases. When the static opening pressure of the vent is reached, the vent opens, and the indoor unburned gas and air mixture starts to release under the internal and external pressure difference. In the early stage of the explosion, because the rate of pressure rise in the confined space is low, the pressure as a whole shows a downward trend, forming a peak Pb. After the internal and external pressure difference is balanced, the indoor gas gradually stops venting. Under the action of the turbulence induced by the previous venting, the indoor combustion rate increases, and the rate of pressure rise increases, and further a new internal and external pressure difference is formed, and the unburned gas and air mixture is vented again. When the discharge rate is gradually greater than the indoor gas generation rate, the pressure begins to decrease and a peak Pfv is formed. When the burnt gases begin to flow out of the vent, the rate of pressure increase is drastically reduced, leading to the formation of peak Pcv. As the flames propagate out from the confined space, the external gas cloud is ignited, causing a second explosion and resulting in a significant characteristic pressure peak, Pext, owing to changes in internal pressure. During vent opening, Helmholtz oscillations may be initiated, which leads to the formation of the oscillation peak, Phel (Bauwens et al., 2011; Guo et al., 2018; Rui et al., 2018); The Helmholtz oscillations ultimately decay as the flames continue to spread inside the confined space. When the flames of the explosion propagate to the walls of the room, the rate of pressure increase reaches a maximum, thereby forming peak Pmfa (Alexiou et al., 1996). During the late stage of deflagration venting, a certain amount of fresh air and flames are drawn into the room owing to the negative pressure, and the remaining unburned gases are combusted, leading to an increase in pressure and the formation of peak Prev (Guo et al., 2017). Under certain specific circumstances, interactions between the room structure and acoustic waves may induce flame oscillations, thereby forming the pressure oscillation peak, Pac. According to the results of existing literature, the aforementioned pressure peak structures were not formed under the same experimental conditions because the types of peaks formed were dependent on the initial and boundary conditions of the gas explosion. Therefore, to accurately determine the distribution characteristics of the overpressure peak structures of vented gas explosions and to investigate the occurrence frequencies and variations of various peak structures under different experimental conditions, the research results of 31 articles related to constrained vented flammable gas

explosions in confined spaces are summarised in this paper, as shown in Table 1.

Table 1 Distribution characteristics of multi-overpressure peak structures. Literatures

Experimental conditions

Peak structures

Dominant peaks

Chamber size

2/3

Kv (V /Av)

Ignition location

Fuel

11.11/ 5.56

Front/ centre/

Propane

Pb

Pfv

Pcv

Pext

Phel

Pmfa

Prev

Pac

(length× width× height, m) Solberg et al.

2.5 × 3.5× 4

(1981) Wingerden

P1

P2

P2

back 2× 2× 1.2

2.86

Centre

Propane

P1

P2

3× 3× 3

20

Centre/ back

Methane

Pv

0.91× 0.91× 0.91

0.89

Centre

Methane

P1

P2

4× 3.7× 2.6

3.70/ 2.78/ 2.17

Centre

Methane

P1

P2

L= 0.864, D= 0.288

4

Centre/ back

Ethylene/ propane/

P1

P3

P3

Pac

Pv

P4

P4

P3

P2

and Zeeuwen (1983) Mccann et al.

Pac

(1985) Cooper et al.

P3

(1986) Mercx et al.

P3

(1993) Chow et al. (2000) Kumar

Pb

P2

Pb

methane 10× 4 × 3

50/ 25 /11.11

Centre

Hydrogen

4.6× 4.6 × 3

5.88

Centre

Propane

4.6× 4.6× 3

5.88

Front/ centre/

P1

P2

P1

P1

P3

P2

P1

P2

Methane

P1

P3

P2

P2

P3

Hydrogen/

P1

P1

P3

(2009) Bauwens et al. (2010) Chao et al. (2011) Bauwens et al.

back 4.6× 4.6 × 3

5.88/ 11.11

Front/ centre/

P2

P3

P3

(2011) Bauwens et al.

4.6× 4.6× 3

5.88/ 11.11

(2012) Wen et al.

back

methane/ propane

Front/ centre/

Hydrogen

Pext

Pvib

Pext

Pvib

back 0.15× 0.15× 0.5

0.15

Back

Methane

L= 0.46, D= 0.162

3.57/ 5.26/ 11.11

Back

Ethylene

4.6× 4.6× 3

11.11

Centre/ back

Hydrogen

0.15× 0.15× 0.15

0.42

Centre

Hydrogen

1× 0.96× 0.98

3.85

Back

L= D= 0.25

25

L= D= 0.25

25

Pv

Pmax

Pmax

(2013) Fakandu et al.

Pfv

Pext

Pfv

Pext

(2014) Bauwens and

Pext

Pvib

Pext

Pv

P1

P2

P1

Hydrogen

P1

P2

P3

P2

Centre

Hydrogen

P1

Front/ centre/

Hydrogen

P1

P2

P3

P2

P2

Pext

Pmfa

Pext

Pmfa

∆P2

∆P3

∆P4

∆P1

∆P4

P4

P2

Dorofeev (2014) Rocourt et al. (2014) Kuznetsov et al. (2015) Guo et al.

P2

P3

P4

P1

P3

(2015a) Guo et al. (2015b) Hisken et al.

P3

P1

back 1.5× 0.3× 0.3

2.94

Back

Propane

P1

9× 4.5× 4.5

1.59

Back

Methane

Pv

2× 2× 3

8.33

Centre

Methane

∆P1

L= D= 0.25

25

Front/ centre/

Hydrogen

P1

(2015) Tomlin et al.

Pb

(2015) Bao et al. (2016) Cao et al. (2016)

back

P2

P3

P4

Fakandu et al.

linked vessel L= 0.46, D= 0.162

0.54

Back

Methane

Pburst

(2016)

and L= D= 0.5

Guo et al.

Pfv

Pext

Pmfa

L= D= 0.25

25

Centre

Methane

P1

L= D= 0.25

25

Centre

Hydrogen

P1

P2

5× 4× 1.5/ 4.75× 4× 3/10× 4× 3

50

Centre

Hydrogen

P1

P2

2× 1.2× 0.6

1.79

Back

Ethylene

1× 0.55× 1.8

5

Centre

Hydrogen

Pv

4.6× 4.6× 3

5.88/ 11.11

Centre/ back

Methane

P1

1× 0.55× 1.8

3.57

Front

Hydrogen

P1

1× 0.55× 1.8

3.57

Centre

Hydrogen

P1

Prev

Pac

P2

Pext

P1

(2016) Guo et al.

P3

P1

(2017) Liang

P3

P1

P3

(2017) Sun et al.

P1

P2

P2

(2018) Wang et al.

P1

P2

P1

P3

P4

P3

P4

Phel

Pacc

P1

Pacc

P2

P3

P1

P3

(2018) Bauwens et al.

P2

(2018) Rui et al. (2018) Zhang et al. (2019)

Pext

As shown in the above table, the overpressure curves obtained by various researchers under different experimental conditions typically consist of combinations of several peak structures, and each peak structure may possibly form the dominant peak in the vented explosion pressure-time curve. This poses certain challenges for the investigation of the distribution patterns of constrained vented explosion peak structures under different conditions. Therefore, analysis of the correlations between complex, diverse pressure peak structures and the maximum internal pressure must be performed to facilitate correlation of the maximum explosion pressure with certain peak structures and the provision of a key reference for the investigation and analysis of explosion incidents. To quantitatively determine the distribution characteristics of multi-overpressure peak structures in vented explosion experiments and to analyse the correlations between various peak structures and the dominant pressure peaks in confined spaces, for each peak structure reported by the selected articles in Table 1, the occurrence frequencies ratio and occurrence frequencies ratio that each peak formed the dominant peak in a pressure curve were statistically analysed, as shown in Fig. 1. The results indicated that peaks Pb, Pext, Pmfa and Pac had the highest occurrence frequencies in vented explosion experiments reported in the literature with each peak occurring in more than 48% of all pressure curves. In addition, these four peaks were the dominant peaks in more than 29% of the pressure curves obtained by the selected studies. Evidently, these four peaks were most commonly observed during the vented gas explosion process, and there is a high likelihood that any of these peaks represent the dominant pressure within the confined space. Therefore, accurate determination of the evolution patterns of these four typical peaks during the vented gas explosion process in confined spaces will contribute to the theoretical basis for reasonable pressure relief designs for buildings, production equipment, and devices as well as incident investigations and analyses.

Fig. 1. Distribution characteristics of multi-overpressure peak structures.

2.2 Formation process and mechanisms of typical multi-overpressure peaks Solberg et al. (1981) performed a vented propane explosion experiment in a 2.5 m × 3.5 m × 4 m vessel and reported the detailed formation process for a double-overpressure peak structure, as shown in Fig. 2. The process began with back wall ignition, which led to the forward propagation of the flame front. When the high-temperature combustion gases reached the vent, the first peak, Pcv, was formed. Subsequently, as the flame propagation continued, Rayleigh-Taylor instability was generated at the boundary layer between the low-flow-rate unburned gases and high-flow-rate burned gases, leading to

an increase in the flame area. The second and larger peak, Pmfa, was formed when the flames propagated to the other walls of the vessel.

Fig. 2. Double-peak structure observed by Solberg et al. (1981).

Cooper et al. (1986) performed experiment of a weakly constrained vented methane explosion in a steel enclosure with a volume of 0.76 m3 and obtained an overpressure curve with four typical peaks during the vented gas explosion process (Sun et al., 2018), as shown in Fig. 3. According to the conclusions in Section 2.1, the four peak structures observed by Cooper have been considered to be the highest occurrence frequencies peaks during the vented gas explosion process, and each pressure peak structure corresponds to a specific physical phenomenon. Table 2 shows the detailed formation mechanisms of the peaks.

(a) Flame propagation and vent opening

(b) Development of external explosion

(c) Maximum flame area and acoustic oscillation process Fig. 3. Schematic of the gas-dynamic development process of multi-peak structures (Cooper et al., 1986). Table 2 Formation mechanisms of typical peak structures Peaks

Physical events

Peak characteristic

Mechanism

P1

Vent failure

Significant peak under

The unburned gas and air mixture begins to vent after

(Fig. 3a (A-B))

higher opening pressures

vent opening, and the internal pressure drops to form a peak (Cooper et al., 1986).

P2

External explosion

Directly related to the

The explosion mechanism can be explained in the

(Fig. 3b (C-D))

intensity of the external

following ways:

explosion and vent area

(1) The external explosion hinders the gas venting process (Kuznetsov et al., 2015); (2). The external explosion induces a fluid instability within the confined space, which increases the generation rate of the combustion products (Bauwens et al., 2009; Proust and Leprette., 2010); (3). The pressure waves of the external explosion propagate back into the confined space, causing an increase or decrease in the internal overpressure (Harrison and Eyre, 1987; Cao et al., 2017).

P3

Maximum flame area

Significantly increased in

The flames expand, propagate forward under the

(Fig. 3c (E))

the presence of

effects

turbulence factors, such

Rayleigh-Taylor instability, and ultimately contact the

as obstacles

walls of the confined space, which leads to a

of

venting-induced

turbulence

and

maximum flame area and an increase in the rate of pressure increase. P4

Acoustic-vibration

Occurs under specific

Instability in gas combustion induces acoustic pressure

coupling

circumstances

oscillations. The acoustic waves undergo multiple

(Fig. 3c (F))

(controlled by the vessel

reflections by smooth wall surfaces and subsequently

shape and physical

couple with acoustic patterns of the room, leading to

responses)

flame folding and the intensification of unstable burning. Such a mechanism results in high-frequency pressure oscillations (Rocourt et al., 2014).

According to previous studies, the difficulty in modelling gas explosions lies in the fact that the maximum pressure achieved during the explosion process may be controlled by any peak belonging to a certain number of specific pressure peak structures. In addition, these peak structures correspond to different physical phenomena. Bauwens et al. (2009, 2011) observed that numerical models were unable to capture the pressure peaks related to room acoustics. This is because the room walls were set as rigid walls during the modelling process, which led to the inability of the models to predict the structural responses of the room, thereby resulting in the lack of interaction effects between acoustic vibrations and the room structure and the disappearance of the Pac peak. In addition, during the numerical modelling process, the deformation and fracture of the room structure under vented explosion conditions is typically neglected. Therefore, the calculated pressure peak values are usually excessively high because the fluid-structure interactions between the explosion flow field and the room structure are neglected. In experimental studies, vent opening typically occurs after a certain process; however, during numerical modelling, it is usually assumed that complete shattering of the vent occurs immediately when the dynamic opening pressure is reached. This ultimately leads to significant differences in the model-predicted and actual pressure peak values. In view of these issues, there is a need to examine and refine the idealistic assumptions made during the numerical modelling process and to develop numerical calculation methods with higher accuracy and reliability during future numerical studies.

3. Influencing factors of multi-overpressure peak structures Overpressure peaks are a dominant factor that determines the severity of damage caused by gas explosions. The distribution characteristics and peak magnitudes of multi-overpressure peak structures are related to the characteristic parameters of the confined space vent, such as the opening pressure (Pv), opening time (T0), inertia (ω), shape (Vs), area (Av) or vent coefficient (Kv), and are also influenced by the room conditions, such as the room shape (Rs) and volume (V), gas concentration (φ), ignition location (Ignloc), obstacles (Obs), and initial turbulence (Initur). Therefore, the influences of different initial and boundary conditions on a number of major peak structures are explained in this section.

3.1 Peak Pb Table 3 summarises the variation patterns of Pb under the influence of different factors. Upon comparison, it was found that characteristic parameters such as the vent inertia and opening pressure influenced the vent opening process, thereby exerting a key influence on the pressure peak, Pb. Cubbage et al. (1955) proposed a mathematical relationship between Pb and vent inertia, as shown in Equation (1). Although this relationship has been experimentally validated by Zhang et al. (2019), it is only applicable to vent opening pressures of < 2 kPa, which considerably limits its applicability to real-life situations.

Pb =

Su ⋅ ( 0.43 ⋅ K ⋅ ω + 2.78 V

1

(1)

3

Where Su is the burning velocity in m/s; K is the vent area coefficient (ratio of area of enclosure cross section to area of relief); ω is the surface density, i.e., inertia, of the vent cover in kg/m2; V is volume of the enclosure in m3. In 2004, by taking the influence of vent opening-induced turbulence effects on the combustion rate into consideration, Molkov et al. (2004) proposed a modified version of the Pb equation based on experimental data reported by Cooper et al. (1986):

Pb = Pv + 0.00043 ⋅ ( Su ⋅

χ 2 Acs ω ⋅ ) ⋅ µ F V 13

(2)

Where Acs is the area of enclosure cross section which is parallel to a wall with relief panel in m2; F is the area of vent cover in m2; χ is deflagration-outflow interaction (DOI) number. µ

However, when gas combustion within a confined space is excessively intense, a vent opening will not lead to a rapid reduction in the internal pressure. For instance, when the initial conditions, such as a high gas concentration or strong turbulence, exist within the room, these factors can contribute to the maintenance of a higher combustion rate. Consequently, the generation rate of combustion products will exceed the venting rate of gases, leading to a continued increase in the internal pressure and ultimately resulting in an indistinct peak for Pb (Kumar, 2009; Liang, 2017). Table 3 Evolution patterns of Pb under different conditions. Relevant factor

Variable

Influence

Activation of the vent

Pv

Positively correlated with Pv (Guo et al., 2015a; Cao et al., 2016), T0

Opening process of the vent

T0

(Pang et al., 2019) and ω (Zhang et al., 2019)

ω Burning state

Pressure relief

Av

No significant changes in Pb

Overpressure rise

φ

May result in the disappearance of Pb (Chen et al., 2019)

Obs

No significant changes in Pb

Initur

May result in the disappearance of Pb (Kumar, 2009)

3.2 Peak Pext Table 4 shows the various factors governing peak Pext from two aspects, viz. the flame burning state and external explosion. Upon comparison, it was found that Pext is mainly controlled by the intensity of the secondary explosion occurring outside the room; back-wall ignition can also lead to a higher Pext. Increased vent inertia may influence the propagation behaviour of the vent explosion flame and the formation of external clouds of unburned gas, leading to a reduction in external explosion intensity and a lower Pext. Although the gas concentration considerably influences the external explosion, a clear linear relationship has not been identified between Pext and φ. In addition, the influence of the external explosion on the internal pressure is constrained by Kv (V2/3/Av). In an experiment conducted by Rocourt et al. (2014), it was found that the peak Pext disappeared when Kv ≥ 9, despite an intense secondary explosion occurring outside the enclosure. One reasonable explanation for this phenomenon is that the centre of the external explosion is located further from the vent as Kv increases, and a smaller vent opening limits the propagation of pressure waves from the external explosion back into the enclosure, thereby reducing the influence of the external explosion on the internal pressure. Conversely, within a certain range, as Kv decreases, Pext is influenced by the external explosion to a greater extent and gradually forms the dominant peak in the pressure-time curve (Tomlin et al., 2015). Therefore, for vented explosions within structures that have a large volume and low strength, secondary damage in the structures caused by Pext should be an important consideration, and the vent area should not be increased recklessly just to enhance the pressure relief effects. Bauwens et al. (2012) proposed a single-peak structure calculation method based on the principles of previously reported simplified analysis models (Bradiey and Mitcheson et al., 1978a, 1978b;

Molkov et al., 1999; Tamanini, 2001), as shown in Equation (3). In this equation, the external explosion pressure (Pe), flame area (Af), and burning speed (Su) can control and determine the magnitudes of Pext, Pmfa, and Pac, respectively. During the comparison of this model with experimental data and the design of a reasonable pressure relief system, key parameters, such as external explosion pressure and flame area, must still be estimated. In addition, owing to the complexity of the physical mechanisms that induce pressure peak structures, a comprehensive consideration of fluid dynamic factors may be difficult to achieve. Therefore, a high degree of uncertainty exists in the use of this method to calculate pressure values (Chao et al., 2011).  S u A f (σ − 1)  P Pe   = 1 − G  P0 P0   acd Av  

2

−1

    = Pe 1 − G  2 *  P0  A  v   

−1

(3)

Where P is different peak structure in the confined space; Pe is the outside explosion pressure; Af is the flame area; Av* is the vent parameter; σ is the expansion ratio; G = ((γ+1) / 2) γ/(γ-1), acd = (RTv γ (γ+1) / (2Mv))1/2, where γ , cd, R, Tv, and Mv are the specific heat ratio, a discharge coefficient, the universal gas constant, and the temperature and molar mass of the vented gas, respectively. Table 4 Evolution patterns of Pext under different conditions. Relevant factor Burning state

Overpressure rise

Variable

Influence

Pv

Positively correlated with Pv (Cooper rt al., 1986), T0 (Pang et al.,

T0

2019).

φ

Within a certain range, the laminar burning velocity is positively correlated with φ (Chen et al., 2019).

Flame propagation

ω

Influences flame propagation behaviour in vented explosions (Sun et al., 2019).

Vs

Square vents result in lower external flame velocities compared to round vents.

Ignloc

Back-wall ignition produces a significantly higher Pext compared with other ignition locations.

Pressure relief

Kv

Within a certain range, Pext is positively correlated with Kv.

External

External explosion

Pv

Pext is positively correlated with Pv, T0.

explosion

intensity

T0 ω

Pext is negatively correlated with ω.

Vs

Square vents result in a lower Pext compared to round vents (Nagy et al., 1983; Andrews et al., 1994; Fakandu et al., 2014).

φ

No clear linear relationship between Pext and the gas concentration (Guo et al., 2015b).

Ignloc

Back-wall ignition results in more unburned gases outside the room and a maximum Pext (Mercx et al., 1993; Rocourt et al., 2014; Guo et al., 2015b).

Effect on the internal overpressure

Kv

The degree of influence of the external explosion on the internal pressure is negatively correlated with Kv (Harrion and Eyre, 1987).

3.3 Peak Pmfa

Table 5 summarises the evolution patterns of Pmfa under different influencing conditions. Upon comparison, it was found that an increase in the opening pressure and a decrease in the vent area can lead to increases in the volume of unburned gas remaining in the room and the combustion reaction time, thereby resulting in increases in the flame area and propagation rate as well as a significantly higher Pmfa. The gas concentration has similar influences on Pmfa and Pext (Chen et al., 2019). Based on previous studies, the presence of obstacles and initial turbulence within the room and central ignition usually lead to the formation of distinct peak structures, as shown in Fig. 4. First, the presence of turbulence will lead to significant increases in the flame propagation speed, flame area, and combustion rate (Ibrahim et al., 2001; Ciccarelli et al., 2010), causing Pmfa to become the maximum internal pressure. Second, with central ignition, the flames can simultaneously propagate towards opposite directions, which facilitate the development of flame instability and delay the contact of the flames with the walls of the room, thereby resulting in a smaller heat loss and larger flame area (Cao et al., 2016). Therefore, in experimental studies on vented explosions with central ignition, variations in the peak pressures caused by the maximum flame area should not be underestimated. Table 5 Evolution patterns of Pmfa under different conditions. Relevant factor Burning state

Overpressure rise

Variable

Influence

Pv

Pmfa is positively correlated with Pv (Cao et al., 2016)

φ

No clear linear relationship between Pmfa and φ (Guo et al., 2015b)

Obs

The presence of turbulence factors leads to a significant increase

Initur

in Pmfa (Bauwens et al., 2010)

Pressure relief

Av

Pmfa is negatively correlated with Av (Kumar, 2009)

Flame propagation

Ignloc

Central ignition leads to a maximum flame area and a significant increase in Pmfa

(a) Obstacles (Bauwens et al., 2010)

(b) Initial turbulence (Liang, 2017)

(c) Ignition location (Solberg et al., 1981) Fig. 4. Patterns of influence of obstacles, initial turbulence, and ignition location on Pmfa.

3.4 Peak Pac Pac represents a unique pressure behaviour generated by combustion instabilities during a large-scale vented gas explosion process (Baker et al., 1983) and has a significant influence on the entire pressure–time curve. Table 6 provides a summary of the evolution patterns of Pac under different initial and boundary conditions. Upon comparison, it was found that the front-wall ignition can lead to a significant peak structure for Pac (Chao et al., 2011). The presence of obstacles and initial turbulence within the room induces severe flame folding during the vented explosion, which hinders the acoustic-vibration coupling process and reduces Pac. In small rooms, the duration of acoustic wave-flame coupling is shorter, which increases flame instability (Wingerden and Zeeuwen, 1983); This has been experimentally validated by Liang (2017), who conducted a vented hydrogen gas explosion experiment in explosion chambers at volumes of 25 m3 (5 m × 4 m × 1.5 m), 57 m3 (4.75 m × 4 m × 3 m), and 120 m3 (10 m × 4 m × 3 m), and the obtained experimental data was subjected to 50 Hz low-pass filtering, the smoothing experimental data is shown in Fig. 5. the following conclusions can be made: (1) relatively higher Pac values may be achieved in small chambers; (2) the emergence of Pac is not strongly correlated with the chamber volume; (3) Pac appeared earlier in the cuboid chambers (25 m3, 120 m3) than in the near-cubic chamber (57 m3), which demonstrates that chamber shape influences the acoustic-vibration coupling process. Additionally, experimental studies conducted by Chow et al. (2000), Wingerden and Zeeuwen (1983) revealed that asymmetry in room shape and glass wool-covered room walls effectively curbed the interactions between the acoustic waves and flames, leading to a significant reduction or disappearance of Pac. Table 6 Evolution patterns of Pac under different conditions. Relevant factor Burning state

Overpressure rise

Variable

Influence

Pv

Changes in Pv and φ may result in the disappearance

φ

of Pac; however, there is no clear linear relationship between these two variables (Bao et al., 2016; Liang, 2017)

Pressure relief

Av

Pac is negatively correlated with Av (Rocourt et al., 2014; Liang, 2017)

Flame propagation

Ignloc

Front-wall ignition may lead to a higher Pac (Bauwens et al., 2010; Chao et al., 2011)

Acoustic coupling

Effect on the acoustic coupling

V

Pac is increased in smaller rooms

Break the acoustic coupling

Rs

Asymmetry in room shape leads to a decrease in Pac

Obs

Under conditions of turbulence, severe flame folding occurs, and Pac decreases or disappears (Kumar, 2009; Bauwens et al., 2010; Liang, 2017)

Fig. 5. Influence of different explosion chamber volumes on Pac (Liang, 2017).

3.5 Other peak structures During vented explosions in confined spaces, pressure oscillations induced by gas-dynamic processes, such as Helmholtz oscillations and Rayleigh-Taylor instability, are typically observed. In particular, because the pressure peak induced by Helmholtz oscillations, Phel, may reach the same order of magnitude as the maximum internal pressure (Bauwens et al., 2011) and influence the distribution of the internal pressure (Cooper et al., 1986; Bauwens et al., 2010; Rocourt et al., 2014), some researchers have conducted in-depth studies on Helmholtz oscillations. Mccann et al. (1985) proposed a theoretical formula for calculating the period (T) of Helmholtz oscillations, as shown in Equation (4), and reported that significant Helmholtz oscillations could only be observed with relatively large vent areas. Rui et al. (2018) performed a vented hydrogen gas explosion experiment in a moderately-sized explosion chamber with dimensions of 1 m × 0.55 m × 1.8 m and observed that Phel consistently appeared after vent opening; therefore, the researchers concluded that Phel increases with opening pressure.

2π  L + α  T= V C  A 

1

2

(4)

Where: L is the neck length; V is the explosion chamber volume; A is the vent area; α is the end correction, α is 0.51 r according to Alster (1972), r is the vent radius. The peak Pfv doesn’t appear frequently during the vented gas explosion process and hardly becomes the dominant peak of the vented explosion pressure-time curve. Fakandu et al. (2016) conducted a methane explosion experiment using a linked vessel with L = 0.46 m, D = 0.162 m and L = D = 0.5 m and founded that the peak Pfv is formed significantly under the low vent burst pressure, but the peak pressure is only about 4 kPa. Chow et al. (2000) utilised a cylindrical vessel with a length/diameter ratio of 3:1 to investigate

the influence of different constraints on Pb and Pcv. The experimental results, which are shown in Fig. 6, indicate the following: (1) under a high vent opening pressure (Pv), peak Pcv increased slightly and Pb replaced Pcv as the maximum peak pressure during the entire explosion process; (2) as K (the ratio of the cross-sectional area of the cylindrical vessel to the vent area) increased, peak Pcv increased significantly; (3) Pb values obtained for vented explosions of ethylene, propane, and methane did not vary significantly, but Pcv exhibited a decreasing trend with the gas burning speeds for different activities; (4) with central ignition, combustion gas mixtures were rapidly vented, and unburned gases were rarely vented after vent opening. Therefore, Pb increased, while Pcv gradually decreased and ultimately disappeared.

(a) Opening pressure (Pv)

(c) Fuel

(b) Vent area coefficient (K)

(d) Ignition location

Fig. 6. Patterns of influence of different constraints on Pb and Pcv (Chow et al. 2000).

Prev, which is commonly known as the result of residual combustion, typically appears towards the end of the vented explosion pressure curve (Molkov et al., 2006; Guo et al., 2015a). Because most of the flammable gases are consumed and vented during the early phases of a vented explosion, there is an insufficient amount of flammable gases available for explosion reactions with fresh air that back flows into the room. This results in a lower pressure peak for Prev compared to other peaks in the pressure curve. In experiments conducted by Cao et al. (2016) and Guo et al. (2017), it was observed that Prev was only maintained at a relatively low level of several kPa; therefore, Prev was not sufficiently high to form the maximum peak pressure in the confined spaces.

4. Correlation analysis between typical pressure peak structures and influencing factors

4.1 Correlation analysis methods Sustek and Janovsky (2013) proposed a quantitative method to evaluate the accuracy of equations for calculating the maximum overpressure in an explosion chamber based on relevant studies (Razus and Krause, 2001; Sustek et al., 2009). First, the maximum calculated overpressure values from several experimental setups were calculated using various overpressure calculation equations. A dimensionless pressure ratio was obtained by dividing the calculated value by the experimentally measured value, and a score was assigned based on the average, maximum, and minimum values of this ratio (representing the average calculation accuracy, largest degree of underestimation, and largest degree of overestimation, respectively), with the range of assigned scores being 0-5 (six levels). The sum of the scores corresponding to the three ratio values was representative of the overall accuracy of the equation, with higher scores indicating a higher proximity of values calculated by the equation to the experimental values. By applying the research ideas and the pressure ratio scoring method described above to the content described in this paper, we propose a quantitative evaluation method to characterise the correlations between various overpressure peak structures and influencing conditions to provide a scientific basis for the accurate determination of the influence of various conditions on different peak structures. The specific steps of this method are as follows: (1) Peak structures and influencing factors to be included in the correlation analysis were determined; (2) From the literature of experimental studies that investigated the selected peak structures and influencing factors, the minimum and maximum values of a specific peak structure when a certain influencing factor was set as the experimental variable were identified; (3) By dividing the maximum value of the selected peak structure by the minimum value, the dimensionless ratios of the peak value and corresponding score were obtained (If the peak disappears within a range of influence factors in the literature, the minimum value of the peak was considered to be 0, and a value of >10 was assigned to the ratio). On the basis of the study by Sustek, the obtained ratios were divided into six levels corresponding to scores of 0-5, as shown in Table 7. Higher ratios were assigned higher scores, which indicated a higher degree of correlation between the peak structure and the influencing factor; (4) Correlation analysis: a certain number of similar articles were selected, and the calculations described in Step (3) were performed. The scores calculated by the peak structure in each literature were summed and averaged to obtain the comprehensive correlation score of the influencing factors, and the degrees of correlation were divided into four intervals (low, medium, high, significant), as shown in Table 8. A higher degree of correlation indicated a greater influence of a certain factor on a certain peak structure. Taking the correlation evaluation of the peak Pb and the opening pressure (Pv) as an example. Guo et al. (2015a) studied the effect of the vent opening pressure on the hydrogen explosions in a cylindrical vessel. The opening pressure (Pv) in the literature varies from 23 to 275 kPa, and the corresponding peak Pb range is 23 - 275 kPa. First, the maximum value (275kPa) of peak Pb was divided by the minimum value (23 kPa) to obtain the dimensionless pressure ratio (11.96) for the influence of the opening pressure on peak Pb, and the score was given with reference to Table 7 (5 points); Then this method was taken to evaluate several other related literatures on the effect of opening pressure on peak Pb, and average the scores of several similar literatures; Finally, the correlation between the peak Pb and the opening pressure (Pv) was evaluated with reference to Table 8. This method was used to

evaluate the correlation between several other influencing factors and the peak Pb, so as to obtain the influence degree of each influencing factor on the peak Pb. Table 7 Method for assigning scores to pressure ratios. Scores

0

1

2

3

4

5

Pressure ratio

1.0-1.3

1.3-1.5

1.5-3.0

3.0-6.0

6.0-10

>10

Table 8 Evaluation method for degrees of correlation. Scores range

0-1.25

1.25-2.5

2.5-3.75

3.75-5

Correlation level

Low

Medium

High

Significant

4.2 Analysis of results Because experimental studies on influencing conditions, such as vent inertia, room shape, and room size, are relatively scarce, we investigated the correlations between the typical peaks (Pb, Pext, Pmfa, and Pac) and the vent opening pressure (Pv), vent area (Av), gas concentration (φ), ignition location (Ignloc), and obstacles (Obs) based on the some experimental studies from the literatures. The selected studies mainly involved vented explosion experiments conducted with four types of gases (hydrogen, ethylene, propane, and methane) in cuboid, cubic, or cylindrical explosion chambers/vessels. Tables 9-12 show the results of the correlation analysis. For the different peak structures, the various factors ranked in descending order of degree of correlation are as follows: (1) Peak Pb: Pv > Av > φ > Ignloc > Obs; (2) Peak Pext: Ignloc = φ > Av > Pv = Obs; (3) Peak Pmfa: Obs > Pv > φ > Av = Ignloc; (4) Peak Pac: φ > Av > Obs > Ignloc > Pv. Based on the comparisons in this paper, the influencing factors with the highest degrees of correlation with peaks Pb, Pext, Pmfa, and Pac were Pv, Ignloc, Obs, and φ, which basically corroborate the patterns of influence discussed in Section 3. In addition, it was found that the gas concentration was highly correlated with all four typical peak structures, indicating that the significant influence of gas concentration on multi-peak structures of vented explosions should not be overlooked. The aforementioned analysis method enables a relatively accurate reflection of the degrees of correlation between the major influencing conditions and the four typical peak structures, and it provides a scientific basis for the quantitative determination of the relationships between the factors and peak structures. However, there are certain issues and limitations with this method. For instance, when investigating the correlation between a certain influencing condition and the peak structures, standardization of experimental conditions was not comprehensively considered. Therefore, differences in the experimental conditions, such as room size and gas type, may lead to different levels of correlation for the same influencing factor among different peak structures. Furthermore, the different variation ranges of influencing factors from the literature may also lead to differences between the minimum and maximum values of the peak structures. Therefore, average values were adopted in this study to reduce the influence of such differences, and further improvements will be made to the proposed evaluation method in our future research to enhance the scientific basis and accuracy of the evaluation results. Table 9 Correlation analysis of Pb. Av

Pv Pressure

Scores

Pressure

Ignloc

φ Scores

Pressure

Scores

Pressure

Obs Scores

Pressure

Scores

ratio

ratio

ratio

ratio

ratio

>10

5

>10

5

>10

5

>10

5

1.89

2

11.96

5

>10

5

8

4

6

3

1.36

1

>10

5

1.7

2

>10

5

1

0

1.05

0

>10

5

>10

5

>10

5

--

--

--

--

4.17

3

>10

5

1.63

2

--

--

--

--

Average

4.6

4.4

4.2

2.7

1

scores Correlation

Significant

Significant

Significant

High

Low

level The corresponding experimental data of factors Pv comes from Cooper et al. (1986), Guo et al. (2015a), Bao et al. (2016), Guo et al. (2016), Wang et al. (2019); The corresponding experimental data of factors Av comes from Cooper et al. (1986), Tomlin et al. (2015), Qi et al. (2016), Liang et al. (2018), Chen et al. (2019); The corresponding experimental data of factors φ comes from Guo et al. (2015b), Bao et al. (2016), Liang. (2017), Wang et al. (2018), Wang et al. (2019); The corresponding experimental data of factors Ignloc comes from Cao et al., (2016), Hooker et al. (2017), Chen et al. (2019); The corresponding experimental data of factors Obs comes from Tomin et al. (2015), Wen et al. (2015), Wang et al. (2019). Table 10 Correlation analysis of Pext. Pv Pressure

Av Scores

ratio

Pressure

Ignloc

φ Scores

ratio

Pressure

Scores

ratio

Pressure

Obs Scores

ratio

Pressure

Scores

ratio

>10

5

4.17

3

3

3

18.67

5

1.32

1

1.25

0

20

5

6

4

3.67

3

1.84

2

--

--

2.27

2

8.7

4

>10

5

1.53

2

--

--

5.56

3

12.7

5

3.82

3

46.9

5

--

--

>10

5

48

5

>10

5

--

--

Average

2.5

3.6

4.2

4.2

2.5

scores Correlation

Medium

High

Significant

Significant

Medium

level The corresponding experimental data of factors Pv comes from Cooper et al. (1986), Fakandu et al. (2015); The corresponding experimental data of factors Av comes from Cooper et al. (1986), Chow et al. (2000), Fakandu et al. (2014), Rocourt et al. (2014), Tomlin et al. (2015); The corresponding experimental data of factors φ comes from Wingerden and Zeeuwen (1983), Cooper et al. (1986), Bauwens and Dorofeev (2014), Wang et al. (2018), Chen et al. (2019); The corresponding experimental data of factors Ignloc comes from Bauwens et al. (2010), Chao et al. (2011), Bauwens et al. (2012), Bauwens and Dorofeev, (2014), Rocourt et al. (2014); The corresponding experimental data of factors Obs comes from Bauwens et al. (2010), Chao et al. (2011), Bauwens et al. (2012), Tomlin et al. (2015). Table 11 Correlation analysis of Pmfa. Pv

Av

φ

Ignloc

Obs

Pressure

Scores

ratio

Pressure

Scores

ratio

Pressure

Scores

ratio

Pressure

Scores

ratio

Pressure

Scores

ratio

>10

5

>10

5

>10

5

2.33

2

>10

5

>10

5

5.56

3

5.8

4

3

2

>10

5

3.25

3

5.78

3

>10

5

>10

5

59

5

>10

5

>10

5

2.5

3

6.7

4

>10

5

--

--

2.85

2

41

5

>10

5

>10

5

Average

4.5

3.6

4.4

3.6

5

scores Correlation

Significant

High

Significant

High

Significant

level The corresponding experimental data of factors Pv comes from Cooper et al. (1986), Guo et al. (2015a), Bao et al. (2016), Chen et al. (2019); The corresponding experimental data of factors Av comes from Cooper et al. (1986), Kumar (2009), Bauwens et al. (2010), Bauwens et al. (2011), Tomlin et al. (2015); The corresponding experimental data of factors φ comes from Wingerden and Zeeuwen, (1983), Kumar (2009), Guo et al. (2017), Yang et al. (2018), Wang et al. (2019); The corresponding experimental data of factors Ignloc comes from Solberg et al. (1981), Bauwens et al. (2010), Bauwens et al. (2011), Schiavetti et al. (2014), Chen et al. (2019); The corresponding experimental data of factors Obs comes from Bauwens et al. (2010), Chao et al. (2011), Tomlin et al. (2015), Wan et al. (2018), Wang et al. (2019). Table 12 Correlation analysis of Pac. Pv Pressure

Av Scores

ratio

Pressure

Ignloc

φ Scores

ratio

Pressure

Scores

ratio

Pressure

Obs Scores

ratio

Pressure

Scores

ratio

>10

5

>10

5

3

3

4.6

3

4.3

3

>10

5

12.22

5

>10

5

1.83

2

7.13

4

>10

5

4.09

3

8.4

4

>10

5

>10

5

1.23

1

5.57

3

>10

5

>10

5

--

--

1.33

1

111

5

11.6

5

4.7

3

--

--

Average

3.4

4.2

4.4

3.6

4

scores Correlation

High

Significant

Significant

High

Significant

level The corresponding experimental data of factors Pv comes from Mccann et al. (1985), Cooper et al. (1986), Cao et al. (2016), Hooker et al. (2017), Rui et al. (2018); The corresponding experimental data of factors Av comes from Cooper et al. (1986), Bauwens et al. (2010), Chao et al. (2011), Bauwens et al. (2012), Bauwens and Dorofeev (2014); The corresponding experimental data of factors φ comes from Mccann et al. (1985), Kumar (2009), Bauwens et al. (2012), Guo et al. (2015b), Liang (2017); The corresponding experimental data of factors Ignloc comes from Bauwens et al. (2010), Chao et al. (2011), Bauwens and Dorofeev (2014), Cao et al. (2016), Hooker et al. (2017); The corresponding experimental data of factors Obs comes from Bauwens et al. (2010), Chao et al. (2011), Bauwens et al. (2012).

5. Conclusions and future prospects In this study, we analysed the distribution characteristics of multi-overpressure peak structures and formation mechanisms of typical peak structures in vented flammable gas explosion pressure curves. On this basis, the patterns of influence of different factors on various peak structures were described in detail, and a quantitative method for evaluating the correlations between the influencing factors and peak structures was proposed. Through our results and analysis, the following conclusions were drawn: (1) A vented gas explosion process in a confined space can lead to the formation of eight different types of pressure peak structures, which exhibit significant differences owing to different formation mechanisms, initial conditions, and boundary conditions. In particular, peaks Pb, Pext, Pmfa and Pac induced by vent opening, external explosion, maximum flame area, and flame-acoustic wave coupling had the highest occurrence frequencies and were the dominant peaks in the vented explosion pressure curves in previous experimental studies. Because these pressure peaks may result in severe damage to building structures, there is a need for an in-depth investigation of the evolution patterns of these typical peaks in explosion flow fields. (2) A preliminary understanding of the patterns of influence of initial and boundary conditions on the overpressure peak structure has been achieved. We found that the characteristic parameters of a vent had a higher influence on the typical pressure peak structures, and the influence on Pb was highly significant. However, excessively intense combustion of gases within a room may lead to the disappearance of Pb. Under conditions of back-wall, central, and front-wall ignition, Pext, Pmfa, and Pac, respectively, may form the maximum vent pressure within the room. External explosion has a profound influence on Pext; however, the interdependence between Pext and the vent area must be taken into consideration because a smaller vent area may reduce the influence of the external explosion on the internal pressure. Asymmetry in room shape and the presence of obstacles and initial turbulence may significantly inhibit the formation of Pac. The influence of changes in gas concentration were consistent across the four typical peak structures, and the presence or absence of each peak could be controlled by the gas concentration. Although constrained vented gas explosions have been widely studied, further exploration is required for other aspects, such as experimental studies on the influence of constraints, such as the vent opening time and the presence of a series of large obstacles in the room on the explosion characteristics. Through the comprehensive determination of the mechanisms by which peak structures change under the influence of different factors, losses caused by explosion incidents can be effectively reduced. (3) The accuracy of gas explosion numerical models based on fluid dynamic calculations requires further improvement. Studies on gas explosion numerical models should be focused on the vent opening process, the structural responses of rooms, and the influence of interactions between the room, acoustic waves and flames, to increase the accuracy and completeness of calculations performed using numerical analysis methods. (4) Experience-based equations for the prediction of overpressures in constrained vented gas explosions lack robustness. Pressure peak structures during vented gas explosions in confined spaces are highly complex and are influenced by different initial and boundary conditions. In addition, each pressure peak structure may become the dominant peak in the pressure curve. Therefore, future research should be focused on the development of experience-based prediction equations that account for the influence of multiple factors on each specific peak structure, particularly for peaks with high occurrence frequencies, to enhance the accuracy and timeliness of the predicted explosion results. (5) Through the use of a novel quantitative evaluation method to comparatively analyse the

degrees of correlation between initial/boundary conditions and typical multi-overpressure peak structures, it was found that influencing factors with the highest degrees of correlation with Pb, Pext, Pmfa, and Pac included the vent opening pressure, ignition location, obstacles, and gas concentration, and the gas concentration was identified as a key condition that influenced all typical pressure peak structures. The accurate determination of the degrees of correlation between pressure peak structures and constraints in confined spaces will be of great significance for future research on vented gas explosions.

Acknowledgements The authors appreciate the financial support from the National Natural Science Foundation of China Project (No. 51404029, 51604031), Beijing Natural Science Foundation-Municipal Education Committee Joint Funding Project (No. KZ201910017020), the Beijing Science and Technology Nova Program (No. Z181100006218092), and the Training Funded Project of the Beijing Youth Top-Notch Talents of China (No. 2016000026833ZK05).

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Highlights


The causes, evolution patterns of multi-overpressure peaks and their correlations with influence factors are clarified.




Peaks Pb, Pext, Pmfa, and Pac are most commonly observed during the vented gas explosion process.




A quantitative evaluation method to characterize correlations between pressure peaks and various conditions is proposed.




Pb, Pext, Pmfa, and Pac have highest correlation with venting pressure, ignition, obstacles and

concentration, respectively.