**2. Simulation Objects**

#### *2.1. Experimental Apparatus and Geometric Model*

The special-designed experimental device and its size and structure are displayed in Figure 1. This experimental system consisted of two different spherical vessels, where one is big and the other is small comparatively, and they connected each other with pipes having a varied length [13].

(**a**)

(**b**) 

**Figure 1.** Picture and schematic diagram of the experimental device for the gas explosion. (**a**) Photo of the experimental device; (**b**) scheme structural diagram of linked vessels.

Two spherical vessels were fabricated from steel with a design pressure of 20 MPa and their internal diameters were 0.6 and 0.35 m (0.113 m<sup>3</sup> and 0.022 m<sup>3</sup> in volume, respectively). Seamless steel pipes, with an inner diameter and a thickness of 0.06 and 0.015 m, respectively, were connected using flanges and bolts. The pipes were divided into three parts (each part was 2 m in length), so that they could be combined into pipes with varied lengths of 2.45, 4.45, and 6.45 m between the big and small spherical vessels. Suitable monitor points were placed as dictated by the experiment. The entire explosion reactor with three different structures and the mesh are given in Figure 2.

Although only pressure data could be obtained from the experiment, the data regarding the temperature and explosion products were determined through numerical simulation. It provided further analysis and prediction dealing with unidentified explosion processes in linked vessels.

Based on the physical experiments, this study conducted natural gas explosions in three systems, which were a single spherical vessel, a single spherical vessel connected with a pipe, and a big spherical vessel connected to a small spherical vessel, individually. Meanwhile, multiple influence factors, including different structures, sizes of the vessels, lengths of the connection pipes, and ignition positions were investigated in detail.

**Figure 2.** Grid division on the model of the three structures. (**a**) Grid division on the model of the spherical vessel; (**b**) grid division on the model of the single sphere with a pipe; (**c**) grid division on the model of the big vessel connected with the small vessel.

#### *2.2. Boundary Conditions and Initial Conditions*

According to the experimental conditions and the FLACS User's Manual [28], the initial conditions were set as standard pressure and temperature and the boundary conditions were set as Euler equations (inviscid flow equations), which is suitable for most explosion simulations. In FLACS, Euler equations are discretized for a boundary element. Namely, the momentum and continuity equations are solved on the boundary in the case of outflow. The concentration of methane used in the experiment was 10 vol%, and the vessel was filled entirely with the gas mixture. The parameter for the relative fuel–oxygen concentration employed in FLACS is denoted by equivalence ratio (ER), which is defined in Equation (1):

$$\text{ER} = \frac{\left(\text{m}\_{\text{fuel}} / \text{m}\_{\text{oxygen}}\right)\_{\text{actual}}}{\left(\text{m}\_{\text{fuel}} / \text{m}\_{\text{oxygen}}\right)\_{\text{stoichiometric}}} = \frac{\left(\text{V}\_{\text{fuel}} / \text{V}\_{\text{oxygen}}\right)\_{\text{actual}}}{\left(\text{V}\_{\text{fuel}} / \text{V}\_{\text{oxygen}}\right)\_{\text{stoichiometric}}} \tag{1}$$

where m is the gas mass and V is the gas volume. Accordingly, the ER of methane–oxygen was 1.058. In addition, the default choice of heat switch is close in FLACS, as large-scale explosions are not much influenced. Activating radiation model will let walls and objects have background temperature, and gas heat loss from radiation will calculate as well as radiation from hot objects around. This is useful for small-scale confined explosions with important heat e ffects and is suitable to apply in this study. Furthermore, based on the experiment results in small closed vessels, Luo et al. [31] confirmed that adding a radiation model is an e ffective way to improve the precision of calculated results. They found that the simulation results without adding the radiation model had a large error with an average error of 10.05%, while the simulation results with adding the radiation model had a small error with an average error of 1.88%. Therefore, adding a radiation model to the simulation of gas explosions was able to produce a sound agreemen<sup>t</sup> with experimental results. As a result, to examine a realistic situation, this study included a radiation model to improve the accuracy of the results.

#### **3. Simulation Results with Di** ff**erent Structures**

#### *3.1. Simulation Results in the Single Spherical Vessel*

#### 3.1.1. Influence of Vessel Size on the Explosion Pressure

The center of the large sphere was at (0.35, 0.35, 0.35); the sensor mounted on the sidewall was at (0.075, 0.35, 0.35); the center of small sphere was at (0.19, 0.19, 0.19); and the sensor mounted on the sidewall was at (0.025, 0.19, 0.19). The simulation was performed following the methane explosion experiment at 10 vol% in two single vessels. The results of the simulation are plotted in Figure 3.

**Figure 3.** Natural gas explosion pressure varying with time in two single vessels.

The results revealed that the pressure plots at the center of the spheres and the sensor on the sidewall could be superposed. As shown in Figure 3, a nearly consistent maximum explosion pressure can be observed in the big/small vessel, which were 7.78 and 7.79 barg, respectively. However, the maximum explosion pressure in two vessels peaked at 102.9 and 53.6 ms, respectively, and there was a considerable di fference of 49.3 ms, where the amplitude can be attained to 48%. Therefore, the rate of pressure rise in the big vessel was lower than that of the small vessel. As a result, without changing the initial conditions of the scenario, the peak pressure was same, but by contrast, the rate of

the pressure rise was only relevant to the vessel size. Additionally, this finding was consistent with the "explosion cubic law" [32].

3.1.2. Simulation Results of the Explosion Parameters in the Single Vessel

The distributions on the X-Y section of explosion parameters containing pressure, temperature, and explosion products at the 10 vol% CH4 mixture in the big vessel are presented in Figure 4.

**Figure 4.** Contours of the natural gas explosion parameters containing pressure, temperature, and concentration of products in the larger vessel. (**a**) Distribution of pressure; (**b**) distribution of temperature; (**c**) distribution of the concentration of products.

As evidenced in Figure 4a, the pressure in the larger vessel was practically uniform at any specific time. The explosion wave expanded into a spherical wave and the temperature was high near the igniter but gradually decreased until it approached the leading edge of the combustion wave in Figure 4b. Here, the combustion wave reached the sidewall at 125 ms. The temperature in the big vessel began to decrease when the explosion ended. As indicated in Figure 4c, when more natural gas participated in the reaction, the distribution of the explosion products spread further until the gas was consumed. The laws of the distribution of pressure, temperature, and explosion products in the small vessel were the same as those in the big vessel.

The changing temperatures and concentrations of the explosion products in the vessels are plotted in Figure 5. When the explosion occurred, the temperature at the center of the vessel enhanced rapidly, reaching 2161 K at 6 ms. Then the rate of the temperature rise dropped, and the temperature at the center of the small vessel reached 2580 K at 53 ms, whereas in the big vessel it reached 2581 K at 101 ms. The center and sidewall reached the highest temperature simultaneously, but the energy was lost when the combustion wave contacted the sidewall. The highest temperature near the sidewall was lower than that at the center of the sphere. After it was ignited at the center of the sphere, the concentration of products at the center escalated promptly, reaching its highest value (27.7%) at 12 ms in Figure 5b. The theoretical concentration of products reached 28.35% based on the methane combustion chemical equation. Accordingly, the simulation results fitted the theoretical values.

**Figure 5.** Profiles of the natural gas explosion parameters in the single vessel. (**a**) Profiles of explosion temperatures against time in the vessels; (**b**) concentration profiles against time for the explosion products in the vessels.

#### *3.2. Simulation Results in for Single Vessel with Pipes Connected to IT*

#### 3.2.1. Simulation Results of the Explosion Parameters in the Single Vessel with Pipes Connected to It

The big sphere with the 4.25-m pipe was adopted as an example; the igniter was located at the center of the sphere. The distributions of the pressure, temperature, and concentration of products of the methane explosion are shown in Figure 6. The plots of the explosion temperature and the concentration of products are delineated in Figure 7.

Figure 6a shows that the peak pressure in the single vessel with a pipe connected to it was lower than that of the single vessel. As provided by Figure 6b, because of the oscillation sparked by the blast, the gas in the pipe was carried to the vessel, and a jet flow formed. Moreover, from Figure 6c, when the ignition source was located at the center of the vessel, the combustion wave spread 2.3 m in the pipe when it spread from the center to the sidewall in the vessel. Therefore, compared with the vessels alone, the pipes significantly accelerated the spread of the explosion.

In virtue of the oscillation of the explosion, the plots of temperature displayed a fluctuation at the center of the vessel (in Figure 7a). Moreover, the energy was lost when the combustion wave contacted the sidewall. The highest temperature near the sidewall was low and the fluctuations were small comparatively. The plots in Figure 7b reveal that as the explosion wave spread, the entire vessel was filled with combustion products, which reflected that the variation of products was not be affected by the oscillation.

**Figure 6.** Contours of the natural gas explosion parameters (pressure, temperature, and concentration of products) in the single vessel with a pipe connected to it; (**a**) distribution of pressure. (**b**) distribution of temperature; (**c**) distribution of the concentration of products.

**Figure 7.** Profiles of the natural gas explosion parameters (temperature and concentration of products) in the single vessel with a pipe connected to it.

3.2.2. Influence of Changing the Ignition Position

The big vessel with the 4.25-m pipe was adopted as an example, the gas was ignited at the sidewall of the big vessel, the center of the sphere, location 1, and location 2 (in Figure 2). The plots of the explosion pressure as it changed with time are illustrated in Figure 8.

**Figure 8.** Natural gas explosion pressure versus time by changing the ignition positions.

As revealed in Figure 8, changing the ignition position substantially influenced the explosion pressure in the big vessel. The peak pressure locations in the big vessel with different ignition positions could be sorted from highest to lowest as follows: location 4, location 2, the center, location 1, and the sidewall, orderly. By contrast, the maximum rate of pressure rise could be sorted as follows: location 4, location 2, location 1, the center, and the sidewall, orderly. The combustion wave contacted the sidewall during the explosion when the ignition source was located at the sidewall. Due to the wall effect, the abundant energy was consumed, which resulted in the explosion intensity to be weaker than at the other ignition positions. Accordingly, the explosion pressure and the rate of pressure rise were diminished. Locations 1, 2, and 4 were located within the pipe, where the explosion occurred with a tighter constraint and the pipes accelerated the explosion wave. Location 4 was at the end of the pipe, at which the acceleration distance was the longest. The explosion intensity was most vigorous when the combustion wave spread to the big vessel. Therefore, the explosion pressure and the rate of pressure rise were the largest when the ignition source was at location 4.

#### 3.2.3. Influence of the Length of the Connection Pipes

The ignition source was located at the center of the vessel. The explosion pressure in different lengths of pipe is shown in Figure 9.

**Figure 9.** Natural gas explosion pressure versus time by changing the length of the pipe connected to the single vessel.

Figure 9 shows that the size effect [33] of each pipe length had a noticeable influence on the explosion intensity in the vessel. The peak explosion pressure lessened as increasing length of the connection pipe, with the influence being the greatest in the small vessel. In any single vessel with a connected pipe, the energy from the explosion was consumed by the wall. More energy was lost when the lengths of the connected pipes increased on account of the greater area of the inner wall. Consequently, the peak explosion pressure decreased with the increase in the lengths of the connected pipes.

#### *3.3. Simulation Results of the Big Vessel Connected to the Small Vessel*

#### 3.3.1. Simulation Results of the Explosion Parameters in the Big Vessel Connected to the Small Vessel

Regarding the simulation of the big vessel with the 4.45-m pipe and the small vessel, the ignition source was located at the center of the big vessel. Changes in the distribution of the pressure, temperature, and concentration of products are presented in Figure 10. The plots of the temperature and the concentration profiles of the explosion products are depicted in Figure 11.

**Figure 10.** *Cont*.

**Figure 10.** Contours of the natural gas explosion parameters in the big vessel connected to the small vessel. (**a**) Distribution of pressure; (**b**) distribution of temperature; (**c**) distribution of the concentration of products.

**Figure 11.** Profiles of the natural gas explosion parameters (temperature and concentration of products versus time) in the big vessel connected to the small vessel. (**a**) Profiles of explosion temperatures against time in the vessels; (**b**) concentration profiles against time for the explosion products in the vessels.

As illustrated in Figure 10a, the peak pressure was attained in the small vessel at 75 ms and was considerably higher than that in the big vessel. The temperature in the pipe was lower than that in the sphere because energy was absorbed by the wall in the long pipe in Figure 12b. Gas at a lower temperature was brought into the sphere as a result of the oscillation originated from the explosion wave. Here, the low-temperature jet cuts the temperature distribution in the sphere to a symmetrical structure.

**Figure 12.** Natural gas explosion pressure versus time in the large and small vessels at different ignition positions.

In Figure 11a, the explosion wave reached its highest temperature (2700 K) at 70 ms in the small vessel. By contrast, the temperature was 2400 K in the big vessel at the same time. Because of the oscillation of the combustion wave, the temperature at the center of the big vessel plummeted to 1300 K at 74 ms after the combustion wave spread into the small vessel, which then formed a second peak.

Figure 11b shows that the concentration of products enhances rapidly at 62 ms. Namely, the combustion wave spread into the small vessel through the connected pipe at that moment. Furthermore, the combustion wave spread to the sidewall of the big vessel at 65 ms. This result again demonstrated the fact that the pipes accelerated the combustion wave.

#### 3.3.2. How Changing the Ignition Position Influenced the Gas Explosion

Considering the big vessel with the 4.45-m pipe and small vessel as an example, the ignition positions were the sidewall of the big vessel, the center of the big vessel, location 1, location 2, location 4, location 6, the center of the small vessel, and the sidewall of the small vessel, as depicted in Figure 2. A methane mixture with a concentration of 10 vol% was intentionally selected in the simulations. The plots of the explosion pressure in the large and small vessels are shown in Figure 12.

As exhibited in Figure 12a, the peak pressure was attained in the big vessel when the ignition was positioned at the sidewall of the small vessel. The minimum pressure value was obtained when the ignition was positioned at the sidewall of the big vessel. Figure 12b shows that the peak pressure decreased when the ignition was moved from the sidewall in the big vessel to location 2, then the peak pressure increased slightly when the ignition was moved from location 2 to the sidewall in the small vessel. Therefore, the pressure piling [21,22] originated from the oscillation was the weakest at location 2.

#### 3.3.3. How Changing the Length of the Connection Pipe Influenced the Gas Explosion

In the experiment, the vessel that contained the ignition source was defined as the initiating vessel, and the other was defined as the secondary vessel. The explosion results were determined variably with the change in the length of the pipe, as drawn in Figure 13.

Figure 13a,b exhibited that the length of the pipe affected the pressure in the initiating vessel. The peak pressure in the initiating vessel was reached when the length of the connecting pipe was 4.45 m. As the length increased, the peak pressure in the initiating vessel increased, ye<sup>t</sup> decreased when the pipe was longer than 4.45 m. The acceleration from the pipes on the combustion wave reinforced

the explosion intensity, but as the length of the pipe increased, more energy was dissipated by the cause of the wall. When the length of the connecting pipe was excessive, a more momentous influence on the acceleration from the wall than that from the pipes could be demonstrated. Comparing Figure 13a together with Figure 13c,b together with Figure 13d revealed that the peak pressure in the initiating vessel was lower than that in the secondary vessel. Moreover, when the initiating vessel was the big vessel, the peak pressure in the second vessel was considerably higher.

**Figure 13.** Natural gas explosion pressure-time curves in the vessels with changing the length of the pipe.
