## 2 Numerical Simulation
### Material Model  
#### 2.2.1 Equivalent TNT Model, Jones-Wilkins-Lee Equation of State Model  
In the simulation of hydrogen gas explosions, prior research has been conducted to indirectly predict the explosion pressure using the equivalent trinitrotoluene (_TNT_) model. The following empirical equation was employed in these studies [28; 29; 30]:  
\[M_{TNT}=\frac{E}{Q_{TNT}}=\eta M_{f}\frac{Q_{H_{2}}}{Q_{TNT}} \tag{4}\]  
where \(M_{TNT}\) is the equivalent mass of _TNT_ (kg), \(M_{f}\) is the mass of stored hydrogen (kg), \(E\) is the explosion energy (kJ), \(Q_{TNT}\) is the calorific value of _TNT_ (4.6 MJ/kg), and \(Q_{H2}\) is the calorific value of hydrogen (119.93 MJ/kg). \(\eta\) refers to the explosion yield contributing to the explosion in the gas cloud, and [PERSON] and [PERSON] [31] estimated the explosion yield (yield of the vapor cloud explosions) at 0.04 when predicting the explosion pressure of hydrogen using the equivalent TNT model through indoor experiments. The magnitude of the explosion pressure (\(p\)) for the equivalent _TNT_ explosive can be calculated using the Jones-Wilkins-Lee equation of state (JWL-EOS) model [25]:  
\[p=A\bigg{(}1-\frac{\omega}{R_{1}\overline{p}}\bigg{)}e^{-R_{1}\overline{p}}+ B\bigg{(}1-\frac{\omega}{R_{2}\overline{p}}\bigg{)}e^{-R_{2}\overline{p}}+ \omega pe_{\text{int}} \tag{5}\]  
where \(A\) (Pa), \(B\) (Pa), \(R_{1}\), \(R_{2}\), and \(\omega\) are constants related to the equivalent _TNT_ materials. \(\overline{p}\) is the ratio of the explosive density in the solid state and the current density, \(e_{\text{int}}\) is the specific internal energy at atmospheric pressure (J/kg), and \(\rho\) is the current density (kg/m\({}^{3}\)) [32]. In Equation (5), the first two terms represent the magnitude of the high pressure generated during the explosion, and the last term represents the magnitude of the low pressure associated with the high volume loss owing to the explosion. The material properties of the _TNT_ explosive charges are listed in Table 1.  
The numerical analysis models using equivalent _TNT_ and JWL-EOS material models were validated through a comparison with hydrogen explosion test data obtained by [PERSON] et al. [33]. The experiments were conducted in an open space in a sealed vinyl tent, as shown in Figure 3. \begin{table}
\begin{tabular}{c c c c c c c} \hline
**Mass Density (kg/m\({}^{3}\))** & **Detonation** & **A (GPa)** & **B (GPa)** & \(\Omega\) & \(R_{1}\) & \(R_{2}\) & **Detonation Energy Density (kJ/kg)** \\ \hline
1630 & 6930 & 373.8 & 3.747 & 0.35 & 4.15 & 0.9 & 3689 \\ \hline \end{tabular}
\end{table} 
Table 1: Input properties of _TNT_ explosive for the JWL-EOS model [32]. Hydrogen gas was injected into the tent to achieve 30% hydrogen and 70% air. The setup was designed to initiate a hydrogen explosion through ignition sources, such as electric sparks or explosive devices (small quantities of C4 or _TNT_). A reinforced concrete wall structure with a height, width, and thickness of 2 m, 10 m, and 0.15 m, respectively, was placed at a straight line distance of 4 m from the explosion source. Data on the explosion pressure and displacement characteristics acting on the concrete structure were measured using attached sensors. To validate the reliability of the modeling techniques, the study developed a 3D numerical analysis model capable of directly simulating the experiments conducted by [PERSON] et al. [33]. Figure 4 illustrates the propagation process of explosion pressure resulting from a hydrogen gas explosion in a 3D numerical analysis simulation. The measurement data for the pressure and horizontal displacement at specific locations (P2, P4, and D2) on the nearby reinforced concrete wall after the explosion were compared with the simulation results, as shown in Figure 5. The predicted pressure and horizontal displacement data obtained through the numerical simulation exhibited overall similarity to the measurements obtained from the experiments.