Investigation on the overpressure of methane-air mixture gas explosions in straight large-scale tunnels

Journal Pre-proof Investigation on the overpressure of methane-air mixture gas explosions in straight large-scale tunnels Yunfei Zhu, Deming Wang, Zhenlu Shao, Xiaolong Zhu, Chaohang Xu, Yutao Zhang

PII:

S0957-5820(19)31741-0

DOI:

https://doi.org/10.1016/j.psep.2019.12.022

Reference:

PSEP 2039

To appear in:

Process Safety and Environmental Protection

Received Date:

4 September 2019

Revised Date:

11 December 2019

Accepted Date:

18 December 2019

Please cite this article as: Zhu Y, Wang D, Shao Z, Zhu X, Xu C, Zhang Y, Investigation on the overpressure of methane-air mixture gas explosions in straight large-scale tunnels, Process Safety and Environmental Protection (2019), doi: https://doi.org/10.1016/j.psep.2019.12.022

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Investigation on the overpressure of methane-air mixture gas explosions in straight large-scale tunnels Yunfei Zhua, b, Deming Wanga, b, Zhenlu Shaoa, b, Xiaolong Zhua, b, Chaohang Xua, b, Yutao Zhangc

a. China University of Mining and Technology, Xuzhou, Jiangsu Province, 221116, China b. Key Laboratory of Gas and Fire Control for Coal Mines (China University of Mining & Technology), Ministry of Education, Xuzhou, 221116, China

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c. Xi’an University of Science and Technology, Xi’an, Shaanxi Province, 710054, China

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Corresponding authors:

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Deming Wang, [email protected], +86 17312840851 Zhenlu Shao, [email protected], +86 15896422058

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Province, 221116, China

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Address: China University of Mining & Technology, No. 1 of Daxue Road, Xuzhou, Jiangsu

Abstract: To investigate the overpressure of methane-air explosions in straight large-scale tunnels, the computational fluid dynamics (CFD) code of Flame Accelerator Simulator (FLACS) was used and validated against experiments conducted at three different scales, and the effects of the volume concentration of methane in air, the blockage ratio (BR), the tunnel length, and the cross-section were studied. When analysed using the GaussAmp mathematical model, the maximum peak overpressure

appears at a volume concentration of 10.30% of methane in air. Blockage ratios (BR) of 0.15 and 0.3 resulted in the combustion of methane-air mixtures with the volume concentration of 6.5% and 14.0% of methane in air, producing a fatal overpressure of 21 kPa. When the BR increases up to 0.75, both the lean and rich mixtures cause a peak overpressure of over 60 kPa. Combustion of the same methaneair mixture produces the same overpressure, which decays approximately linearly at the same slope owing to a smooth wall roughness before travelling near the outlet, independent of the specific tunnel

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length. A method to characterise the cross-sections was proposed, and the maximum peak overpressure of different lengths of methane-air mixtures in different cross-sectional tunnels was found, presenting various regimes from a hump shape to a wave-like uplift and bowl shape. The cross-section parameters

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determine the degree of confinement and further control the maximum peak overpressure in the

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modelled tunnels. An exponential asymptotic model can be used to conveniently obtain the maximum peak overpressure. These phenomena indicate that approximately square-shaped cross-sections should

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be selected to avoid an extremely high overpressure in large-scale tunnels with the potentially

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significant accumulation of methane-air mixtures.

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Keywords: methane-air explosion; overpressure; cross-section; safety; CFD

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1. Introduction

Coal is an important primary energy source and raw industrial material. As the world’s biggest

coal producer, China extracted 3523.2 million tons of coal, accounting for 45.6% of the global production in 2017 (British Petroleum, 2018). Gas explosions are the most destructive accidents occurring in a coal mine, and 232 gas explosions with over ten deaths each occurred from 2000 to

2016, accounting for 47.5% of such accidents in China (Zhu et al., 2019). In addition, an increasing number of large-scale underground constructions have appeared in modern cities and industrial areas, therefore, the promotion of studies on explosions occurring in large-scale tunnels is needed for improving health and safety. Methane-air gas mixtures are widely used in process industries and can be found distributed throughout underground coal mines. Lewis and Elbe (1987) studied the flammability range of

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methane-air mixtures experimentally, and concluded the lower flammability limit (LFL) is approximately 5% of methane in air, and the upper flammability limit (UFL) reaches approximately 16% of methane in air, under ambient temperature and pressure. Robinson and Smith (1984) and Crowl

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(2003) reported the autoignition temperature, known as the lowest spontaneous combustion

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temperature, was up to be more than 600 °C at a volume concentration of 7% of methane in air. Eckhoff (2013) and Lewis and Elbe (1987) summarised the ignition energy of electric sparks or discharge at

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different concentrations, and found that the minimum ignition energy (MIE) is as low as 0.3 mJ, which

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indicates that methane-air explosions can be easily triggered. In addition, Gamezo et al. (2012) and Ajrash et al. (2017) experimentally studied the deniability of natural gas-air mixture and concluded

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high ignition energy accelerates the flame to produce a higher overpressure and cause more severe outcomes. Kundu et al. (2016) reviewed the experiments investigating the flame speed and

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overpressure of methane-air explosions in small-scale vessels and tubes, and found obstructions in the fuel zone stimulate the flame speed, sustain deflagration, and promote the process of deflagration to detonation, producing more destructive blast waves. Chan et al. (1983) and Moen et al. (1980) investigated the flame acceleration due to turbulence produced by obstacles, and found the dramatic influence of obstacles is interpreted in terms of the positive feedback coupling between the flame itself

and the turbulence and flow field distortions produced by the obstacles. Kundu et al. (2017) explored the explosion characteristics of methane-air mixtures in a spherical vessel connected with a duct, and found explosion severity was very high in the turbulent field of the methane-air mixture and in the presence of strong ignition energies. In addition, Shchelkin and Troshin (1964) used Shchelkin spirals, Lee et al. (1985) tested orifice plates, Baker et al. (2012) and Liberman (2013) changed degrees of wall roughness to study the turbulence effect caused by numerous types of obstructions and drew

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similar conclusions. These studies in small-scale tubes and vessels have provided a basic understanding of methane-air explosions, moreover, they helped to improve the safety of process industries (Crowl and Louvar, 1990). However, a methane-air explosion is a high-speed, chemical,

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thermal, and aerodynamic process (Lewis and Elbe, 1987), and the quantitative features regarding the

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flame propagation and pressure evolution derived from small-scale experiments cannot represent large-scale situations, such as vapor cloud explosions in a chemical plant, an oil platform on the ground,

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or methane-air mixture gas explosions occurring in underground coal mines and construction areas.

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The fatalities and limitations of small-scale experiments caused by an explosion have driven a few scholars to study the methane-air explosions that occur in large-scale tunnels. During the 1960s

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and 1970s, Cybulski conducted many experiments in the Maja mine and the Barbara experimental mine, using 70-1,000 m3 methane-air mixtures and measured the peak overpressure of 0.2-1.5 MP at

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a flame speed of approximately 1,200 m/s (Zipf et al., 2007). In 1968, by conducting a full-scale experiment in an underground coal mine, Genthe found that the peak overpressure of the shock wave was less than 1.0 MPa with a flame speed of lower than 330 m/s, and that the peak overpressure reached 1.8 MPa when the flame was accelerated to 1,200 m/s (Zipf et al., 2007). During the 1980s and 1990s, Sapko et al. (2000) and Zipf et al. (2007) conducted numerous experiments on methane-air and coal

dust explosions in the Lake Lynn Experimental Mine (LLEM) of the National Institute for Occupational Safety and Health (NIOSH) in the USA to investigate the overpressure and flame propagation in real underground coal mines. These studies were extremely valuable in providing an overall perception of large-scale methane-air explosions. Nevertheless, the scholars simply employed a small-volume methane-air mixture and a few transducers in large-scale tunnels with a simple structure owing to the high level of danger and cost. These experiments were therefore insufficient to

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fully reflect the characteristics of a flame and shock wave propagation in large-scale tunnels or draw practical conclusions on improving the evacuation and rescue after a coal mine explosion.

For large-scale flammable gas explosions, empirical models, such as the TNT equivalency

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method, TNO method, multi-energy concept, and Baker–Strehlow method, are used to predict the blast

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wave for a safety assessment. However, these models are too oversimplified to represent the real explosion process (Lea and Ledin, 2002). As modelling techniques are changing from empirical to

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advanced CFD approaches, models are more based on fundamental physics and able to represent the

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real explosion process accurately. Li et al. (2017) and Li and Hao (2019) applied CFD method to investigate the internal and external pressure and impulse from vented gas explosions in large-

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cylindrical tanks. Jiang et al. (2016b, 2016a, 2013), Sun et al. (2015) and Tang et al. (2016) used the AutoReaGas code to simulate the explosions in small-scale tubes shaped like a longwall coalface.

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Zhang et al. (2011a, 2011b), Zhang and Ma, (2015) investigated the overpressure decay in a 3 m × 3 m tunnel and the scale effect of methane-air explosions in tunnels using the AutoReaGas code and concluded that the pressure decay follows the power law. Pang et al. (2014, 2012) numerically studied the effects of laneway supports on methane-air explosions based on the AutoReaGas code and found that the support spacing results in obvious differences in the overpressure and impulse. Hansen et al.

(2010) and Hansen and Johnson (2015) validated the FLACS code using many full-scale explosive experiments in a coal mining and processing industry. In addition, Davis et al. (2015) analysed the reverse flow of a blast wave to describe the bidirectional deformed pans in the UBB gas explosion causing 29 casualties using the FLACS code. Zipf et al. (2007) validated the FLACS code for designing explosion-proof pressure criteria of the seals used in coal mines in the USA. Advanced CFD modelling codes are powerful tools for investigating the process and destructive effect of methane-air explosions

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in large-scale tunnels.

In this paper, the CFD code of FLACS for a flammable gas explosion simulation is used and validated based on the results of three different scale experiments. The peak overpressure and its

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influencing factors of the methane-air mixture explosion in straight large-scale tunnels are studied.

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2. Introduction and validation of FLACS code

2.1.1 Main computational code

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2.1 Introduction of FLACS code

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FLACS is a CFD code used to solve the compressible conservation equations on a 3D Cartesian grid using a finite volume method. The conservation equations for the mass, momentum, enthalpy, and

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mass fraction of species, enclosed by the ideal gas law, are included. These equations are shown as follow with the example in the j direction.

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Conservation of mass:

𝜕 𝜕 𝑚̇ (𝛽𝑣 𝜌) + (𝛽𝑗 𝜌𝑢𝑗 ) = 𝜕𝑡 𝜕𝑥𝑗 𝑉

Where t is time, s; βv is volume porosity, a dimensionless parameter; ρ is density, kgm-3; xj is the length coordinate in the j direction, m; βj is the area porosity in the j direction; uj is the mean velocity in the j direction, ms-1; ṁ is mass rate, kgs-1; V is volume, m3.

Momentum equation: 𝜕 𝜕 𝜕𝑝 𝜕 (𝛽𝑣 𝜌𝑢𝑖 ) + (𝛽𝑗 𝜌𝑢𝑖 𝑢𝑗 ) = −𝛽𝑣 + (𝛽 𝜎 ) + 𝐹𝑜,𝑖 + 𝛽𝑣 𝐹𝑤,𝑖 + 𝛽𝑣 (𝜌 − 𝜌0 )𝑔𝑖, 𝜕𝑡 𝜕𝑥𝑗 𝜕𝑥𝑗 𝜕𝑥𝑗 𝑗 𝑖𝑗 Where ui is the mean velocity in the i direction, ms-1; p is absolute pressure, Pa; σij is stress tensor, Nm2

; Fw,i is flow resistance due to walls and Fo,i is flow resistance due to sub-grid obstructions in the i

direction, Pa; ρ0 is initial density, kgm-3; gi, is gravitational acceleration in the i direction, ms-2. Transport equation for enthalpy:

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𝜇𝑒𝑓𝑓 𝜕ℎ 𝐷𝑝 𝑄̇ 𝜕 𝜕 𝜕 (𝛽𝑣 𝜌ℎ) + (𝛽𝑗 𝜌𝑢𝑗 ℎ) = (𝛽𝑗 ) + 𝛽𝑣 + 𝜕𝑡 𝜕𝑥𝑗 𝜕𝑥𝑗 𝜎ℎ 𝜕𝑥𝑗 𝐷𝑡 𝑉

Where h is specific enthalpy, Jkg-1; μeff is effective viscosity, Pa∙s; σh is Prandtl-Schmidt number of

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enthalpy; Dp is diffusion coefficient of particle; Dt is diffusion coefficient of turbulence; 𝑄̇ is heat

Transport equation for fuel mass fraction:

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rate, Js-1.

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𝑢𝑒𝑓𝑓 𝜕𝑌𝑓𝑢𝑒𝑙 𝜕 𝜕 𝜕 (𝛽𝑣 𝜌Υ𝑓𝑢𝑒𝑙 ) + (𝛽𝑗 𝜌𝑢𝑗 Υ𝑓𝑢𝑒𝑙 ) = (𝛽𝑗 ) + 𝑅𝑓𝑢𝑒𝑙 𝜕𝑡 𝜕𝑥𝑗 𝜕𝑥𝑗 𝜎𝑓𝑢𝑒𝑙 𝜕𝑥𝑗

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Where Yfuel is mass fraction of fuel; σfuel is Prandtl-Schmidt number of fuel; Rfuel is reaction rate for fuel, kgm-3s-1.

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For turbulence, FLACS uses a Reynold-averaged Navier–Stokes (RANS) approach based on the

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standard k–ε model to close the equations. It is an eddy viscosity model that solves two additional transport equations, one for turbulent kinetic energy and one for dissipation of turbulent kinetic energy. Transport equation for turbulent kinetic energy: 𝑢𝑒𝑓𝑓 𝜕𝑘 𝜕 𝜕 𝜕 (𝛽𝑣 𝜌𝑘) + (𝛽𝑗 𝜌𝑢𝑖 𝑘) = (𝛽𝑗 ) + 𝛽𝑣 (𝑃𝑘 − 𝜌𝜀) 𝜕𝑡 𝜕𝑥𝑗 𝜕𝑥𝑗 𝜎𝑘 𝜕𝑥𝑗 Where k is turbulent kinetic energy, m2s−2; σk is Prandtl-Schmidt number of turbulent kinetic energy;

Pk is the production of turbulent kinetic energy, kgm-1s-3; ε is the dissipation of turbulent kinetic energy, m2s-3. Transport equation for the dissipation rate of turbulent kinetic energy: 𝑢𝑒𝑓𝑓 𝜕𝜀 𝜕 𝜕 𝜕 𝜀2 (𝛽 𝜌𝜀) + (𝛽 𝜌𝑢 𝜀) = (𝛽 ) + 𝛽𝑣 (𝑃𝜀 − 𝐶2 𝜌 ) 𝜕𝑡 𝑣 𝜕𝑥𝑗 𝑗 𝑗 𝜕𝑥𝑗 𝑗 𝜎𝜀 𝜕𝑥𝑗 𝑘 Where σε is Prandtl-Schmidt number of dissipation of turbulent kinetic energy; Pε is the production of

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dissipation of turbulent kinetic energy; C2 is constant in the k–ε equation, typically C2 = 1.92. In FLACS, the flame zone is thickened by increasing the diffusion with a factor of β and reducing the reaction rate with a factor of 1/β. Hence, the flame model in FLACS is called the β-model (Arntzen,

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1998), which is applied correlations for both the laminar and turbulent burning velocities that originate from the experiments. The SIMPLE pressure correction algorithm is used and extended to handle

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compressible flows with additional source terms for compression in the enthalpy equation (Patankar,

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1980). In addition, FLACS uses the porosity concept and applies the area and volume porosity to represent obstacles with small details to balance the computational time (Spalding and Launder, 1974).

(Gexcon, 2009).

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More specific and detailed information regarding the FLACS code can be found in the FLACS manual

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2.1.2 Simplifying assumptions

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Simplifying assumptions are mainly related to the mathematical models of the computational code, constructions of physical models and setting of initial conditions. In the mathematical models of computational code, the RANS approach decomposes the flow field variables into its time-averaged and fluctuating quantities and deals with the buoyancy-driven flow with Boussinesq approximation, which ignores density differences except where they appear in terms multiplied by the acceleration due to gravity. The standard k–ε model ignored the intermolecular viscosity, assumed the turbulent

viscosity is isotropic, and the ratio between Reynolds stress and mean rate of deformations is the same in all directions. As for modelling flame, β flame model applied correlations for both the laminar and turbulent burning velocities that originate from the experiment. In the SIMPLE pressure correction algorithm, reasonability of initial guessed pressure field plays a major role in SIMPLE. The β flame model and the SIMPLE pressure correction algorithm are resolved under the supports of RANS approach and standard k–ε model, which means the simplifying assumptions of RANS approach and

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standard k–ε model limit the applications of β flame model and SIMPLE pressure correction algorithm. During the constructions of physical models, the constructed models ignore the cross-sectional area change existing in tunnels, and wall roughness cannot be specified in FLACS code. About the initial

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conditions, FLACS code recommends using the default values for explosion scenarios as shown in

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Table 4. This causes the initial values of gravity constant, characteristic velocity, relative turbulence intensity, turbulence length scale, temperature, ambient pressure and composition of air cannot

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represent all real scenarios. In addition, ignition source was deemed to be an instantaneous point

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ignition source in all simulations, since its duration time and dimension is set to zero, ignoring the overdriven effect caused by a strong ignition source.

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2.2 Experimental verification of FLACS code Many validation studies have contributed to the wide acceptance of FLACS code as a reliable

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tool for the prediction of fuel-air explosions occurring in real processing areas (Davis et al., 2015; Gavelli et al., 2011; Hansen et al., 2010; Hansen and Johnson, 2015; Zipf et al., 2007). To verify the capability of FLACS code to predict methane-air explosions under scenarios with different scales, validations applied in different scale tubes and tunnels are described in this section. 2.2.1 Experiment verification in a small-scale tube

In an experiment conducted by Jiang et al. (2013), a 5 m long tube with a 0.08 m × 0.08 m crosssection was filled with premixed methane-air with a volume concentration of 10% of methane in air, and both ends of the tube were closed. The volume concentration of methane in air can be calculated as follow 𝐶𝑚 =

𝑉𝑚 𝑉𝑎 + 𝑉𝑚

Where Cm is the volume concentration of methane, %; Vm is the volume of methane in the methane-air

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mixture, m3; and Va is the volume of air in the methane-air mixture, m3. Nine pressure sensors were placed along the pipe at 0.5 m intervals. A 2 J ignition source was placed at one end of the tube. Based on the experiment, simulations were conducted to verify the capability of FLACS code to predict

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small-scale methane-air explosions. According to the instructions for FLACS code, the maximum

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control volume (CV) should be less than 10% of the gas cloud diameter. Therefore, the maximum size of the CV can be simply obtained as less than approximately 3 cm. Four different CV sizes were

results under different CV sizes.

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applied to analyse simulation sensitivity. Table 1 shows a comparison of the experiment and numerical

Position (m)

Experiment

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Table 1 Comparison of experiment and numerical results for different CV sizes Numerical pressure (MPa) and relative error (RE) 2 cm

RE

1 cm

RE

0.67 cm

RE

0.5 cm

RE

0.5

1.081

0.957

-11.48%

0.938

-13.22%

1.046

-3.20%

0.980

-9.31%

1

1.022

0.926

-9.44%

0.909

-11.02%

0.993

-2.88%

0.951

-6.91%

1.5

0.995

0.896

-9.94%

0.893

-10.23%

0.970

-2.48%

0.942

-5.26%

2

1.044

0.870

-16.66%

0.878

-15.96%

0.951

-8.92%

0.923

-11.64%

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pressure (MPa)

2.5

0.997

0.852

-14.62%

0.863

-13.44%

0.931

-6.63%

0.899

-9.84%

3

0.961

0.871

-9.33%

0.876

-8.84%

0.960

-0.06%

0.921

-4.17%

3.5

1.054

0.896

-14.96%

0.891

-15.47%

0.988

-6.22%

0.935

-11.28%

4

1.081

0.928

-14.15%

0.906

-16.22%

1.010

-6.56%

0.940

-13.07%

4.5

1.113

0.970

-12.81%

0.930

-16.47%

1.067

-4.15%

0.975

-12.37%

5

1.149

1.005

-12.53%

0.973

-15.35%

1.253

8.99%

1.033

-10.10%

As indicated in Table 1, the simulation results between four CV sizes showed little difference, and when the CV sizes were reduced to 0.67 and 0.5 cm, the simulation results became extremely close.

In addition, the relative errors versus the experiment results decrease to less than 10% at a CV size of 0.67 cm. Fig. 1 shows the experimental and numerical results at a CV size of 0.67 cm, verifying the capability of FLACS code to predict small–scale methane-air explosions. 1.5

Experiment Simulation of CV size = 0.67 cm

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1.2 1.1 1.0 0.9 0.8 0.7 0.6 0.5 0

1

4

3

2

5

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Position (m)

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Peak overpressure (MPa)

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Fig. 1 Experiment and numerical results at a CV size of 0.67 cm

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2.2.2 Experiment verification in a tunnel with a cross-sectional area of 7.2 m2 Qu (2010) and Xu et al. (2011) conducted several methane-air explosion experiments in a 900 m

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long square tunnel with a cross-sectional area of 7.2 m2 and one end closed. An approximately 100 m3

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methane-air mixture gas with a volume concentration of 10% filled a tunnel approximately 14 m long, and a chemical ignition source was placed at the closed end. According to the requirements of FLACS

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code, the maximum CV size was calculated as approximately 1.86 m. CV sizes of 1 and 0.5 m were applied to simulate the experiment. The results of the experiment and simulations are listed in Table 2.

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Table 2 Results and relative errors of the experiment and simulations

Experimental

Numerical pressure (kPa) and relative error (RE)

pressure (kPa)

1m

RE

0.5 m

RE

20

145.33

147.06

1.19%

147.12

1.23%

60

165.67

157.22

-5.10%

156.59

-5.48%

80

169.67

158.20

-6.76%

158.81

-6.40%

120

154.00

153.72

-0.18%

154.33

0.22%

200

147.33

145.53

-1.22%

146.13

-0.82%

280

137.67

137.78

0.08%

138.23

0.41%

340

136.00

128.36

-5.62%

128.67

-5.39%

Position (m)

380

129.00

120.72

-6.42%

120.97

-6.22%

420

121.67

113.51

-6.71%

113.72

-6.53%

460

107.00

107.22

0.20%

107.41

0.39%

500

100.33

102.03

1.70%

102.21

1.87%

780

85.33

84.94

-0.46%

85.08

-0.30%

820

81.67

82.70

1.26%

82.83

1.42%

As listed in Table 2, the simulation results of two different CV sizes are similar, proving the independence of the CV size during the simulations. Furthermore, the simulation results show very little difference compared with the experiment results, as indicated in Table 2 and Fig. 2, validating

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the ability of FLACS code to predict large-scale methane-air explosions. 180

Simulation of CV=0.5 m Experiment

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140

120

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Peak overpressure (kPa)

160

100

0

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80 80

160

240

320

400

Position (m)

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Fig. 2 Experiment and numerical results at CV size = 0.5 m

2.2.3 Experiment verification in underground mine tunnels

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Numerical simulations of gas explosions using FLACS code have been conducted as part of the NIOSH investigations of the Sago mine accident (Gates et al., 2007). Furthermore, to establish the

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explosion pressure design criteria of new seals in US coal mines, FLACS code was validated against the full-scale methane-air explosions in the LLEM and used to extend scenarios that are difficult to study experimentally or theoretically (Zipf et al., 2007). The tunnels in LLEM are approximately 480 m long, and its cross-sectional area is approximately 12.5 m2. Researchers conducted six validation experiments using methane-air mixture gases with a volume concentration of 10% filling 3.66-18.30

m of the tunnel space near the closed end, where an ignition source existed. Table 3 summarises the experiment and numerical results. As listed in Table 3, the experiment and numerical results are close. Table 3 Comparison of the experiment and numerical results

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469

470

484

485

Experiment pressure (kPa)

Numerical pressure (kPa)

Relative error

1

21.53

18.70

-13.14%

10

18.59

17.60

-5.33%

1

58.87

57.50

-2.33%

10

48.13

51.90

7.83%

1

75.03

76.30

1.69%

10

74.80

71.00

-5.08%

10

83.89

71.20

-15.13%

526

29.43

28.66

10

101.30

97.64

526

34.14

42.34

10

48.12

57.06

526

24.88

23.10

-2.62% -3.61%

24.02% 18.58% -7.15%

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486

Gauge Number

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Test Number

Although the results of simulations and experiments are close, the deviations need be interpreted.

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In Fig. 2, a big variation of peak overpressure with the experimental data appeared, and the deviation

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between simulation and experiment reached up to 24.02% as shown in Table 3. This is caused by the simplifying assumptions as stated in 2.1.2. All approaches and algorithms just describe turbulent fluid

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flow field, flame and pressure approximately. Meshing simulation domain into numerous CVs will cause continuous space and explosion process discrete. Besides, the experimental tunnels are likely to

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be not perfectly straight and have various wall roughness and its cross-sectional area may change. The

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ignition sources applied in experiments are not instantaneous point ignition sources, mostly are chemical ignition sources or electrical sparks, whose energy will produce an obvious overdriven effect on explosions. Methane-air explosions are very sensitive with these factors exactly because of the positive feedback between flame acceleration and turbulence resulted by these factors. These approximations and simplification explained the deviation between simulations and experiments. By validating the ability of FLACS under different scaled experiments, the numerical deviations

meet the requirements of the engineering applications and studies, as also concluded by Lea and Ledin (2002). The accurate prediction ability of FLACS makes it a reliable tool to study methane-air explosions in large-scale scenarios such as chemical plants on the surface and in underground tunnels.

3. Physical models and simulation settings 3.1 Physical models Four sets of physical models were built for exploring the effects of the volume concentration of

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methane in air, the blockage ratio, the tunnel length, and the cross-section on the overpressure of methane-air mixture gas explosions in straight large-scale tunnels. At this section, a generic model was presented in Fig. 3 to help to describe models at the following sections. The generic model is a long

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straight tunnel with one end close, whose wall thickness is 1 m in all simulations. The instantaneous

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point ignition source is at the centre of the dead end, 1 cm from the wall. Tunnel’s length, width and height are denoted as l, w and h. The green part denotes methane-air mixture gas, whose length is

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denoted as lm. By defining lm, the volume of methane-air mixture gas can be calculated conveniently

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as the product of w, h, and lm. Fig. 3 (c) presents a 3D geometry of generic tunnel in FLACS code, in all physical models, the tunnel length is drawn on the x-axis, width is on the y-axis and height is on

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the z-axis. The specific parameters will be introduced at the corresponding section below for better

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integrity of the content.

(a) Profile of the side view

(b) Look inside from the outlet

(c) 3D geometry of generic tunnels in FLACS code Fig. 3 Generic physical model of the simulation

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3.2 Simulation domain and meshing method The simulation domain is enclosed within the boundary of the wall, including the wall thickness, extending 5 m at the outlet. FLACS code meshes the simulation domain by dividing the domain evenly

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along directions of x, y, and z to form CVs as basic computing units. Hence, every CV is cuboid, and CV size refers to the side length of the cube. FLACS code recommends cubic CVs for a more precise

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prediction of flame propagation, and requirements the maximum CV size should be approximately 10%

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of gas cloud diameter. Due to the minimum volume of the methane-air mixture is 120 m3 in the flowing simulation, the maximum CV size is calculated as 0.49 m. In addition, according to the verification at

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a 900 m-long tunnel with a cross-sectional area of 7.2 m2, it is found the CV size of 0.5 m supports an acceptable accuracy in the simulation of explosions in large-scale tunnels. Due to the sizes of

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constructed physical models are roughly in the same scale as the size of the verification work, the CV

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sizes were set to 0.5 m to mesh the simulation domains for a better prediction of those explosion simulations.

3.3 Boundary condition The physical models of tunnels were all constructed with one end closed. Therefore, five sides of the simulation domain are solid walls with integer CVs in the wall to ensure the flame and blast will not leak out the wall, according to the requirement of FLACS code. At the outlet, the simulation

boundary was set at a distance of 5 m away from the outlet. FLACS code provides two boundary conditions of EULER and PLAN_WAVE for explosion simulation. As the instruction, EULER boundary condition can and should be used for most explosion simulations, PLAN_WAVE is recommended for the explosion in low confinement. Tunnels are high confinement scenarios, therefore, EULER boundary condition was set out of the outlet. 3.4 Initial condition

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Initial conditions set values for turbulence fields, temperature and pressure at the beginning of the simulation. Information about the gravity conditions, parameters for the atmospheric boundary layer and the composition of the air is also set here. FLACS code recommends using the default values

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for explosion scenarios. Table 4 summarizes the default value of the initial conditions provided by the

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Manual of FLACS code (Gexcon, 2009).

Table 4 Default value of the initial conditions Value

Gravity constant Characteristic velocity Relative turbulence intensity Turbulence length scale Temperature Ambient pressure Air Ignition time Dimension of ignition

9.8 m/s2 0 m/s 0 0m 20 ℃ 100 kPa 20.95% oxygen and 71.05% nitrogen in mole fractions 0s 0 m3

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Items

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4. Results and discussions

4.1 Effects of volume concentration of methane in air on peak overpressure The volume concentration of methane in air is a key factor affecting the flame propagation and

overpressure of methane-air explosions. The complete reaction of methane with oxygen can be expressed as follow: 𝐶𝐻4 + 2𝑂2 = 𝐶𝑂2 + 2𝐻2 𝑂

According to the chemical equation, the ratio of the amount of substance of methane to oxygen is 1:2. The normal volume concentration of oxygen in air is about 21%, therefore, when the volume concentration of methane in air is 9.5%, the stoichiometric ratio will be theoretically realised. However, many small-scale experiments indicate that the maximum explosive intensity can be obtained at a volume concentration of methane in air slightly higher than the stoichiometric ratio (Crowl, 2003). To investigate the explosive peak overpressure at different volume concentrations of methane in

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air in large-scale tunnels, a 200 m long tunnel a with a cross-sectional area of 4 m × 4 m was modelled with one end closed, where an instantaneous point ignition source occurred. A methane-air mixture filled an 8 m long tunnel at the closed end. A CV size of 0.5 m was applied to the simulation. Fig. 4

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shows that the maximum peak overpressure was also obtained at a volume concentration of methane

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in air of slightly higher than 9.5% in the modelled large-scale scenario, which was consistent with the experiments conducted in small-scale closed vessels (Kundu et al., 2016).

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70

50 40

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Peak overpressure (kPa)

60

30 20

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10

0

6

8

10

12

14

Volume concentration of methane in air (%)

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Fig. 4 Explosive peak overpressure at different volume concentrations of methane in air in the modelled tunnel

The GaussAmp mathematical model was used to fit the scatter plot to find the specific

concentration at which the maximum peak overpressure occurred. The fitting result is shown in the following: 𝑃 = −2.33 + 70.55𝑒 −

(𝐶𝑚 −10.30)2 2.91

where P is the peak overpressure, kPa; and Cm is the volume concentration of methane in air, %. The

determination coefficient of R2 was reported to be about 0.99, which suggests that the scatter plot was well fitted by the model. From the fitting result, the maximum peak overpressure obtained at the volume concentration of methane in air reached approximately 10.30% in the modelled scenarios. Because the length of the tunnels in modern coal mines is generally between 2 to 6 m, the results derived from the modelled tunnel of 4 m × 4 m has a benefit for understanding the methane-air mixture gas explosions occurring in large-scale tunnels. In a real gas explosion, the combustion of a methane-

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air mixture gas incurs a heat loss, and the mixtures are not homogeneous. Furthermore, the high-speed turbulence will challenge the reaction of the methane and oxygen at a stoichiometric ratio. Therefore, a reasonably excessive amount of methane is needed to react with oxygen in a lean region of the

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maximum peak overpressure in a real gas explosion.

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mixtures and supplement the heat loss, thereby consuming all of the oxygen and producing the

Table 5 shows the effects of an explosion overpressure on humans (Ji et al., 2017). The

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overpressure is static pressure and causes damage through compression. As indicated in Table 5, people

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may be killed by an overpressure of higher than 21 kPa, and the peak overpressure of very lean or rich mixtures is too low to cause damage or injury. Obstructions will lead to turbulence and an accelerated

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flame, producing a higher overpressure, and thus the effect of an obstruction on the peak overpressure from the combustion of extremely rich or lean mixtures needs to be further studied.

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Table 5 Effects of explosion peak overpressure on humans

Peak overpressure/kPa

Effect

17–21 25–35 50–100

Maximum survivable blast 1% fatalities 50% fatalities

4.2 Effects of blockage ratio on combustion of lean and rich methane-air mixtures A 200 m long tunnel with a cross-section of 4 m × 4 m with one end closed is modelled. Methaneair mixture clouds of 6.5% and 14% fill an 8 m long tunnel at the closed end, where an instantaneous

point ignition source was placed. Square columns of different cross-sections were constructed in a 40 m long tunnel at the closed end to achieve different BRs. The ratio of the longitudinal cross-sectional area of the column to the cross-sectional area of the tunnel is taken as the average BR. Scenarios with BRs of 0.15, 0.30, 0.45, 0.60, and 0.75 are modelled and simulated. As shown in Fig. 5, a BR of 0.15 resulted in the combustion of a 6.5% methane-air mixture producing an overpressure of 21 kPa, reaching the fatal criteria shown in Table 5. For a methane-air mixture of 14.0%, a BR of 0.3 caused

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the peak overpressure to increase to 21 kPa. From Fig. 5, the peak overpressure of the lean methaneair mixture increases slowly before reaching a BR of 0.45, but is higher than the peak overpressure produced by a rich mixture because more oxygen exists in lean mixtures, and methane is easy to

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completely consume, producing carbon dioxide and a higher overpressure. By contrast, rich mixtures

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have less oxygen, which is insufficient for the reaction to produce carbon dioxide or a higher overpressure. After a BR of 0.45, the lean mixture is more sensitive to the BR than a rich mixture.

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When the BR increases to 0.75, the combustions of both lean and rich mixtures produce a peak

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overpressure of approximately 60 kPa. A high BR will cause high turbulence, accelerating the flame and promoting the reaction of methane and oxygen. Therefore, turbulence causes methane to react

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completely with oxygen and produce a high overpressure. 70

Concentration=6.5% Concentration=14.0%

Peak overpressure (kPa)

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60

50

40

30

20

10 0.1

0.2

0.3

0.4

0.5

BR

0.6

0.7

0.8

Fig. 5 Peak overpressure of 6.5% and 14% methane-air mixtures under different BR scenarios

4.3 Effects of tunnel length on the combustion of same methane-air mixture To investigate the effects of the tunnel length on the combustion of a methane-air mixture, tunnels with a cross-section of 4 m × 4 m and a length of 100–800 m were constructed. Methane-air mixtures with a concentration of 10.5% filled 8 m of the tunnels at the closed end, where an instantaneous point ignition source was placed. Fig. 6 shows a peak overpressure of combustion for the same methane-air

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mixtures in tunnels of different lengths. Although the tunnel lengths differ, the peak overpressure at the closed end and outlet is similar. In the 100 m tunnel, the peak overpressure at the closed end is slightly smaller than that in the other tunnels because the methane-air mixture and its combustion flame

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are near the outlet, which is an explosive venting that reduces the degree of confinement of the 100 m tunnel. Therefore, its maximum peak overpressure at the closed end is smaller than that in the other

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tunnels. In tunnels with a length of 200-800 m, the venting outlet is far from the flame zone, the

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confinement of the space far away from the outlet is the same, and only when the blast wave travels near the outlet, the overpressure will dramatically decay. From Fig. 6, the peak overpressure decays

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approximately linearly along with a consistent slope before the outlet in tunnels with a length of 200– 800 m. Wall friction influences the blast wave movement, it is closely related to the effect of boundary

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modelled by:

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layers, which are regions in the flow field close to walls. In FLACS code, the wall friction force is

𝐹𝑤 = −𝛽𝑣 𝜏𝑤

𝐴𝑤 𝑉

(1)

where Fw is the wall friction force, N; v is the volume porosity, a dimensionless parameter; Aw is the area of the wall in the CV, m2; and V is the volume of the CV, m3; w is the wall shear stress, N·m-2. In engineering fluid mechanics, λ is the frictional resistance coefficient expressing the resistance of a homogeneous tunnel/tube (Franzini and Joseph, 1977) to a specific fluid. The mean frictional

resistance of the fluid flowing in a homogeneous tunnel can be calculated as follows: 𝐹𝑤 = −λ𝑃𝑛 𝐷𝑣

(2)

where Fw is the resistance force, N; λ is the frictional resistance coefficient, which is a dimensionless parameter; Pn is the normal pressure against the wall, which should be the overpressure in this modelled situation, Pa; D is the perimeter of the tunnel/tube, m; and v is the mean velocity of the fluid, m/s. Choosing an element of area, the Eq. (4) can be transformed into: 𝐹𝑤 = −λ𝑃𝑛 ∆𝑠

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(3)

Where s is the area of the chosen element area closed to a wall, m2. Eq. (1) and (3) both describe the wall friction force in a microelement, set them equal to each other, and the frictional resistance

λ=

𝛽𝑣 𝜏𝑤 𝐴𝑤 𝑃𝑛 𝑉∆𝑠

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coefficient in a microelement can be expressed as: (4)

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Eq. (4) passes the dimensional analysis test and its unit, a non-dimensional parameter is correct. That

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means if the frictional resistance coefficient of  were chosen to express the wall friction characteristic in FLACS, its accurate calculation can be given by Eq. (4), and it is controlled by the static pressure,

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shearer stress, and the volume and porosity of the CV closed to a wall as shown in Eq. (1). According to the porosity algorithm of the CV closed to a wall in FLACS code, v can be calculated as the mean

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porosity of two adjacent CVs inside and outside the wall, it is 0.5 in the simulation. Due to the CV is

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0.5 m × 0.5 m in the simulation, so the Aw is 0.25 m2, V is 0.125m3, and s is 0.25m2. Hence, the accurate  in this simulation can be calculated by: λ=

4𝜏𝑤 𝑃𝑛

(5)

Eq. (5) indicates the value of  varies with the wall shear stress and overpressure in explosion simulations because of the viscosity of mixtures is changed. Although the tunnels’ lengths are different, the wall roughness, shape and area of the cross-section are the same, resulting in the confinement of

these tunnels is the same to produce turbulence. Therefore, under the same turbulence degree, the combustions of the same methane-air mixtures in these tunnels produce the same initial explosion sources, including the flame and initial blast wave. Furthermore, the same initial blast wave has the same static pressure and shearer stress, based on Eq. (5), the wall friction resistance at the same time is the same. In addition, for large-scale tunnels, the proportion of wall roughness and boundary layer close to a wall in the total cross-sectional area is too small to influence the flame acceleration and air

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movement in the explosion of large-scale tunnels obviously. Thus, these features indicate that the combustion of the same methane-air mixture will produce the same peak, which decays in the same pattern before travelling near the outlet, which is independent of the specific tunnel length.

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70

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60 55 50 45

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Peak overpressure (kPa)

65

40 35

200

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0

400

100m 200m 300m 400m 500m 600m 700m 800m

600

800

Position (m)

Fig. 6 Peak overpressure of combustion of same methane-air mixtures in tunnels with different lengths

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4.4 Effects of cross-sectional shape and mixture volume on peak overpressure 4.4.1 Characterisation of the cross-sectional shape

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There are many tunnels with various shaped cross-sections in underground constructions. To

investigate the effect of a cross-sectional shape on the peak overpressure, a cross-sectional area of 24 m2 was selected to represent the most typical condition found in modern underground constructions; in addition, 400 m long tunnels with different cross-sectional aspect ratios were constructed, namely, 2 m × 12 m, 3 m × 8 m, 4 m × 6 m, 6 m × 4 m, 8 m × 3 m, and 12 m × 2 m. When the width and height

are equal, the aspect ratio is 1. A method was proposed to characterise the degree of deviation of different shapes with respect to the case in which the aspect ratio is 1. This is expressed as follows: 𝐷𝑣 = 1 +

𝑤−ℎ 2

2 √𝑆

where Dv is the degree of deviation, a dimensionless parameter; w and h are the width and height, m; and S is the cross-sectional area of the tunnel, m2. This method ensures that shapes with an interchangeable width and height have the same deviation degree with respect to an aspect ratio of 1.

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Table 6 summarises the Dv of the cross-section of constructed tunnels. Dv

2 × 12 3×8 4×6 6×4 8×3 12 × 2

0.49 0.74 0.90 1.10 1.26 1.51

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Shape / w × h (m)

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Table 6 Dv of the cross-section of constructed tunnels

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4.4.2 Peak overpressure in tunnels with different shaped cross-sections The combustion of different lengths of methane-air mixtures in tunnels with different shaped

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cross-sections will produce a maximum peak overpressure presenting various regimes. Mixtures with lengths of 5, 10, 11, 12, 13, 14, 15, 20, and 30 m were applied to fill in the closed end of the tunnels.

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Fig. 7 shows the maximum peak overpressures in tunnels with different shaped cross-sections produced by the combustion at a mixture length of 5–30 m. When the mixture length is as short as 5–

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10 m, the explosions are weak. The closer Dv is to 1, the greater the peak overpressure that occurs. The curves change in a hump shape. When the mixture length becomes longer at 11–13 m, the explosions increase in strength. A Dv of greater than 1 will obtain a higher peak overpressure. As the mixture length increases from 14 to 30 m, the explosions become extremely powerful. A Dv of closer to 1 generates a lower peak overpressure, whereas when Dv deviates too much, a higher overpressure is

produced. The curves change in a bowl shape. 48

22 20 18 16 14

Maximum peak overpressure (kPa)

48

Maximum peak overpressure (kPa)

Maximum peak overpressure (kPa)

24 46 44 42 40 38 36

46

44

42

40

12 0.4

0.6

0.8

1.0

1.2

1.4

34 0.4

1.6

0.6

0.8

Mixture length = 5 m

1.2

38 0.4

1.6

0.6

0.8

1.0

1.2

1.4

1.6

Dv

Mixture length = 11 m

62

65

50 48 46 44

58 56 54 52 50 48 46 44

0.6

0.8

1.0

1.2

1.4

42 0.4

1.6

0.8

1.0

1.2

1.4

1.6

Dv

50

0.4

Mixture length = 13 m

70

50

45

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55

80 75 70 65 60

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Maximum peak overpressure (kPa)

85

60

0.8

1.0

1.2

1.4

1.6

Mixture length = 14 m

110

90

65

0.6

Dv

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Dv

Mixture length = 12 m

55

45

0.6

55 50

Maximum peak overpressure (kPa)

0.4

60

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Maximum peak overpressure (kPa)

52

Maximum peak overpressure (kPa)

60

54

42

Maximum peak overpressure (kPa)

1.4

Mixture length = 10 m

56

Maximum peak overpressure (kPa)

1.0

Dv

Dv

100

90 80 70 60 50

45

0.4

0.6

0.8

1.0

1.2

1.4

1.6

Dv

0.4

0.6

0.8

1.0

Dv

1.2

1.4

1.6

40 0.4

0.6

0.8

1.0

1.2

1.4

1.6

Dv

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Mixture length = 15 m Mixture length = 20 m Mixture length = 30 m Fig. 7 Regimes of peak overpressure in tunnels with different shaped cross-sections

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Flame propagation is a reaction process of oxygen and methane, which produces massive heat and expands the burnt and unburnt gas. The heated gas tends to move and expand in the vertical

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direction driven by thermal buoyancy. When the mixture length is short at 5–10 m, the flame propagates a limited distance at a low velocity with a slow reaction rate. These factors allow the burnt and unburnt gas to have sufficient time to expand into free space, not only toward the outlet, but also along with the excessive space provided by the cross-sectional shape for Dv far from 1. By contrast, the cross-sectional shape for Dv closer to 1 has a short width and height, forming a relatively confined

space, and therefore the weak combustion of the methane-air mixtures in these tunnels produces a higher peak overpressure, as shown in Fig. 7, at a mixture length of less than 10 m. As the mixture length increases, a flame will accelerate to a fast speed with a high reaction rate, and there is insufficient time for the burnt and unburnt gas to expand toward the excessive space provided by the cross-sectional shape of Dv far from 1, and a narrow space in width or height becomes a high confinement area, producing a higher peak overpressure. In addition, comparing the cross-

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sections with a large and small Dv, the shape with a large Dv has a low height. This will confine the movement and expansion of the burnt and unburnt gas under the effect of thermal buoyancy. Hence, the tunnels with a cross-section with a large Dv have a higher maximum peak overpressure, as shown

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in all plots of Fig. 7. Under strong combustion with a long mixture length, both the confinement from

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the cross-section and the wall roughness play an important role in producing a higher peak overpressure. The wall frictional resistance, Fw, is the main factor generating turbulence to accelerate

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flame and produce high overpressure. Here, Fw is positively correlated with the perimeter and airflow

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velocity, as stated in Eq. (2). For strong explosions, the airflow velocity is extremely fast. As for the perimeter, in the modelled cross-section, the correlation can be expressed as follows:

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𝐹𝑤 ∝ 2(𝑤 + ℎ)

Where Fw is the wall frictional resistance coefficient, N; and w and h are the width and height, m. It is

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known that, in a rectangle, the larger the difference in length and width, the larger the perimeter, and correspondingly, the greater the value of Fw. This relationship can also be expressed as follow: 𝐹𝑤 ∝ |𝐷𝑣 − 1|

This indicates that an approximately square cross-section should be selected to avoid an extremely high overpressure in the potential tunnels under a significant accumulation of methane-air mixture.

4.4.3 Maximum peak overpressure in tunnels with different shaped cross-sections The correlation of the peak overpressure and mixture length in differently shaped cross-section tunnels is presented in Fig. 8, which shows that the maximum peak overpressure exists in the methaneair explosions in long, straight, relatively smooth, non-obstructed tunnels with one end open. It is necessary to discuss why the combustion flame does not produce a higher overpressure along with an increase in the methane-air mixture length. In the modelled tunnels, the length of the methane-air gas

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and its combustion flame is limited, therefore, the pressure wave produced by the flame acceleration is strictly restrained by the total energy from the combustion and the confinement of the tunnels. However, when considering a case in which the mixture length and modelled tunnel are sufficiently

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long, the deflagration flame will not accelerate continuously and detonation will not occur. In essence,

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the deflagration-to-detonation transition (DDT) is a transient phenomenon resulting from the acceleration of a deflagration flame into a detonation from turbulence and compressive heat effects

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(Grosse, 2002). Existing methane-air detonation experiments were conducted in double-ended tubes

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filled with a methane-air mixture and numerous turbulence generators (Kundu et al., 2016) because the natural acceleration of a methane-air flame is much weaker than that of other heavy hydrocarbons

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(Lewis and Elbe, 1987), and the arrangement of turbulence generators and double-ended settings fulfil the requirements of high turbulence and a compressive heat effect for a DDT. In these modelled tunnels

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with one end open, the natural acceleration and combustion-generated turbulent flow of a methane-air combustion flame are restricted by the low local turbulence intensity, such as non-obstructed pathways and straight and relatively smooth walls. Furthermore, the unburnt airflow will be expanded and vented out, rather than compressed and heated to produce a higher burning velocity, and therefore, the absence of a compressive heat effect caused the flame will not continuously accelerate and produce a higher

overpressure. 90

50

50

80

60 50 40 30

Peak overpressure (kPa)

70

Peak overpressure (kPa)

Peak overpressure (kPa)

45 40 35 30 25

20

20

10

15

45

40

35

30

25

20 5

10

15

20

25

5

30

10

15

20

25

30

5

10

Mixture length (m)

Mixture length (m)

Dv = 0.49

15

20

25

30

Mixture length (m)

Dv = 0.74

Dv = 0.90

80 50

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100

40

35

30

Peak ovrepressure (kPa)

45

Peak overpressure (kPa)

60 50 40 30

80

60

40

20

25

20

20

10 5

10

15

20

25

30

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Peak overpressure (kPa)

70

0

5

Mixture length (m)

10

15

20

Mixture length (m)

Dv = 1.26

30

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Dv = 1.10

25

5

10

15

20

25

30

Mixture length (m)

Dv = 1.51

Fig. 8 Maximum peak overpressure in tunnels with different shaped cross-sections

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Fig. 6 shows that the peak overpressure is independent of the tunnel length and is determined by the methane-air mixture length and cross-sectional parameters. As shown in Fig. 8, when the mixture

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length is greater than a specific value, the maximum peak overpressure will be independent of the mixture length. That is, the cross-section parameters control the maximum peak overpressure in the

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modelled scenarios. In experiments conducted on most fuel-air explosions in completely enclosed

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vessels and tubes, the final pressure is within the range of 8-12-times the initial absolute pressure (Eckhoff, 2013; Zipf et al., 2007). These experiments indicate that the confinement degree determines the final pressure. In fact, for large-scale tunnels with one end open, the radial direction is relatively free, and the width and height of the cross-section confine the combustion and airflow movement, and further cause a pressure increase. In the modelled smooth tunnels without obstructions, confinement of the cross-section becomes the main source of turbulence, and hence determines the maximum peak

overpressure, as shown in Fig. 8. When the methane-air mixture length is short, its combustion is limited to reach the maximum peak overpressure under the turbulence of the confinement degree determined by the cross-section. In turn, when the mixture length is too long, the limited confinement degree is too low to produce sufficient turbulence to reach a higher peak overpressure. The phenomena presented in Fig. 7 and 8 indicate that reducing the confinement degree of space is helpful for mitigating the peak overpressure, not only by setting vents, reducing wall roughness, clearing

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obstructions, and cancelling bends and corners, but also choosing a large area cross-section with Dv close to 1.

The exponential asymptotic model was used to fit the scatter plot of Fig. 8, and the fitting results

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are recorded in Table 7. The determination coefficient R2 of all fitting results was reported to be over

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0.99, which suggests the correlation of the peak overpressure and mixture length in the simulated scenarios is well presented by the model. The exponential asymptotic model is y = 𝑎 − 𝑏𝑐 𝑥 . In the

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exponential asymptotic model and the fitting process, numerically, y equals the peak overpressure, P,

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kPa; a equals the maximum peak overpressure, Pm, kPa; b and c are undetermined parameters, and x equals the mixture length, l, m. Hence, the correlation of the peak overpressure and mixture length in

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the simulated scenarios can be expressed as follows: 𝑃 = 𝑃𝑚 − 𝑏𝑐 𝑙

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Based on the mathematical model, the maximum peak overpressure can be predicted through several experiments or simulations on the modelled scenarios, rather than to conduct many times until finding the maximum value of peak overpressure. Table 7 Fitting records and comparison with simulation results Dv

a

b

c

R2

0.49 0.74

101.75 48.22

129.21 102.2

0.93 0.79

0.992 0.993

0.9 1.1 1.26 1.51

48.5 47.94 81.03 124.53

424.82 358.68 104.39 155.69

0.57 0.58 0.91 0.94

0.997 0.998 0.991 0.993

5. Conclusions For methane-air explosions in large-scale tunnels, because a reasonably excessive amount of methane is needed to supplement the heat loss and react with oxygen sufficiently in lean regions of the mixtures, as analysed using the GaussAmp mathematical model, the maximum peak overpressure was

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found at a volume concentration of 10.30%, which is slightly richer than the stoichiometric ratio of 9.5%. This is consistent with the results of the small-scale experiments.

Very rich or lean methane-air mixtures in large-scale tunnels will also produce a fatal overpressure

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under the excitation of blockage. BRs of 0.15 and 0.3 resulted in combustions of methane-air mixtures

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of 6.5% and 14.0%, producing a fatal overpressure of 21 kPa. The peak overpressure of a lean methane-

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air mixture increases slowly before a BR of 0.45, but is more sensitive to a high BR. When the BR increases up to 0.75, the combustions of both lean and rich mixtures produce a peak overpressure of

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as high as over 60 kPa.

In the modelled tunnels, the venting outlet is far from the flame zone, the confinement of the

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space far from the outlet is the same, and only when the blast wave travels near the outlet, the overpressure will dramatically decay. Therefore, the combustion of the same methane-air mixture will

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produce the same peak pressure at the closed end, and the overpressure decays at the same pattern before travelling near the outlet, which is independent with the specific tunnel length. A parameter, Dv, was proposed to characterise the different cross-sectional shapes, and its effects were further analysed. The maximum peak overpressure of different lengths of methane-air mixtures presents various regimes in different cross-sectional tunnels. For weak explosions with a mixture

length of 5-10 m, the closer Dv is to 1, the greater the peak overpressure, which presents a hump shape. When the mixture length increases to 11-13 m, a Dv of greater than 1 will obtain a higher peak overpressure, which is a wave-like uplift. As the mixture length increases from 14 to 30 m, a Dv of far from 1 generates a higher overpressure, and the curves change into a bowl shape. This indicates an approximately square cross-section should be selected to avoid an extremely high overpressure in the potential tunnels with significant accumulation of methane-air mixture. In addition, owing to the low

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local turbulence intensity and the absence of a compressive heat effect of the modelled tunnels, it was found the cross-sectional parameters determine the confinement degree and further control the maximum peak overpressure in the modelled scenarios. The exponential asymptotic model fits this

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phenomenon well, and through the mathematic model, the maximum peak overpressure can be

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predicted using several experiments or simulations of the modelled scenarios. These phenomena indicate that reducing the confinement degree of space is helpful to mitigate the peak overpressure,

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not only by setting vents, reducing the wall roughness, clearing obstructions, and cancelling the bends

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and corners, but also by choosing a large-area cross-section with Dv close to 1.

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Conflict of interest

There is no interest conflict in this manuscript. The FLACS code has been authorized to our co-

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authors, the author list and the content of the manuscript and its revisions has got the approval from all authors.

Acknowledgements This study was supported by the Outstanding Innovation Scholarship for Doctoral Candidate of

CUMT (grant number: 2019YCBS046).

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