Explosion of Flammable Propane Refrigerants Leaked in an MiC Unit

Abstract

Modular Integrated Construction (MiC) has been strongly promoted in many dense urban areas, including the Greater Bay Area. There might be an explosion risk if leaked flammable clean refrigerants accumulate in a confined unit. Experimental and modeling studies on the explosion of flammable refrigerant propane in an MiC unit were carried out with a rectangular unit model to explore well-covered or partially covered conditions, representing the scenario of an MiC unit with its door open or closed. The experimental results were used in developing an analytical model to predict the flame surface and pressure change, with acceptable results. This study could be used as a reference for estimating pressure changes and designing ventilation systems to prevent deflagration in MiC units.

1. Introduction

A Modular Integrated Construction (MiC) [1] or prefabricated unit employs the technique of having freestanding volumetric modules manufactured off-site and then transported to the site for assembly together to form a complete building. Taken as a green and high-productivity construction technology, many dense urban areas are promoting the adoption of MiC [2] in the construction industry. In the Hong Kong Special Administrative Region (HKSAR), it started in 2017, driving the adoption of MiC through pilot applications in capital works projects [3,4,5,6]. Fire protection for MiC focused on system approaches was discussed in a seminar in 2018 [7]. The Hong Kong Fire Service Department (FSD) launched a FSD Circular Letter No. 3/2019 in 2019 as guidance notes for the fire services installations and equipment in MiC building projects [8,9]. The Architectural Services Department (ArchSD) circulated an updated General Specification for Building Services in 2022, with MiC introduced [10]. However, they mention general fire service requirements, but explosions were ignored.

The HKSAR government has made a plan to adopt a standardized simple design and Modular Integrated Construction (MiC) approach to build about 30,000 Light Public Housing (LPH) units in the first five-year period (i.e., from 2023–2024 to 2027–2028). Those are not intended to provide an air conditioner and an exhaust fan in the cooking space [11]. Consequently, the users have to provide their own home appliances.

Some clean refrigerants (CRs) used in air conditioners are flammable, but this is impossible to control because CRs are too expensive and have a low coefficient of performance (CoP). It is not easy for the Customs Department to control the use of fake products. Most of the users are not well-off and would select cheap air conditioners, which will increase the risk induced by the low quality of air conditioners and freezers. The floor area of the LPH units is 13 m², 25 m², and 31 m², respectively, for three typical units [4]. There will be a higher fire/explosion risk if there is leakage of refrigerants from the air conditioner or freezer installed for the small MiC units. It is difficult to deal with the flashover [12] which might be induced by a severe fire and explosion. However, there is limited information available on the fire safety of prefabricated units and the component materials. It is necessary to study the fire and explosion hazards in developing big, dense areas such as the Guangdong–Hong Kong–Macao Greater Bay Area [13].

Many clean refrigerants [14,15,16,17,18] replacing chlorofluorocarbons (CFCs) used in air-conditioning and refrigeration systems are based on hydrocarbons (for example, propane in blend R290) [16,19,20]. Although this type of clean refrigerant has zero Ozone Depletion Potential, low Global Warming Potential (GWP of 3), and good thermodynamic and heat transfer properties, it is flammable and poses fire and explosion hazards. Propane refrigerant (R-290) is characterized by its colorless and odorless nature, with a low boiling point of −42 °C, high latent heat, and high flammability (the flammability limit ranges from 2.1% to 9.5% by volume). Its molecular weight is 41.1 g/mol, critical temperature is 97 °C, and critical pressure is 4.3 MPa.

Flammable refrigerants may cause potential fire hazards when malfunctioning of the systems occurs. For example, an explosion will occur when propane refrigerants leak from the air-conditioning unit, mix with room air in a ratio higher than the lower flammability limit (LFL), and are ignited by a small amount of energy [18]. Flammable propane refrigerant explodes when its concentration is over the lower explosion limit, say 2.1% for propane R290 [18,20]. Fires resulting from explosions of leaked refrigerant formulated with propane in air-conditioning units have been reported all over the world in densely populated urban areas [14,21].

There have been at least three explosions [22,23] involving flammable refrigerant propane in air-conditioning systems in Hong Kong since the phasing out of CFCs in early 2000. Further, freezers are put in cabinets of open kitchens in small apartments or hotels. A hazard scenario was identified in the explosion of a leakage of flammable refrigerant from a freezer in a small room [24]. More accidents involving explosions of flammable refrigerants are expected, as the systems are starting to deteriorate.

The allowable amount of leaked propane in a room should give a well-mixed gas concentration lower than the LFL. However, the mechanism and extent of mixing with air depends on the locations of the air inlets and outlets. As propane is heavier than air, higher propane concentrations above the LFL can be found at lower levels in a small flat when the fresh air intake is at a high level. Explosions might occur when there are small ignition sources, as already experienced. Further, freezers might be put inside cabinets in kitchens or small hotel rooms. The interior of a freezer is a closed or half-closed MiC model. Flammable refrigerants leaked into the freezer would create an environment that might possibly cause a confined or partly confined explosion. There is an urgent need to better understand [25] the explosion hazards of flammable refrigerants leaking from malfunctioning refrigeration systems.

Bradley and Mitcheson [26,27] studied the venting of gaseous explosions in spherical vessels. Ogunfuye et al. [28] investigated the dynamics of explosions in cylindrical vented enclosures. Yang et al. [29] studied the effect of room layout on natural gas explosions in kitchens. The change of overpressure in a cuboid space with different pressure relief ports was studied by Dobashi [30] and Jo and Kim [31], with the flame propagation pictures provided. Chen et al. [32], Ibrahim and Masri [33], and Oh et al. [34] comprehensively studied the influence of obstacle size, blocking rate, and relief pressure on the overpressure in the propagation process of premixed flame. The effects of ignition location, vent size, and obstacles on vented explosion overpressure in propane–air mixtures were studied [35,36]. Ogle [37] determined the explosion limit and explosion pressure of different gases under different sealing conditions. Through these studies, the factors that affect flame propagation speed and overpressure in confined or partially confined space have become relatively clear.

There are many studies on flammable refrigerants, which are usually heavier than air and tend to stay at a lower level [19,38,39]. Therefore, the distribution of leaked refrigerants and their mixing with air are not uniform in space. The deflagration propagation of non-premixed gases is quite different from that of premixed gases. Friedrich et al. [40] carried out a hydrogen deflagration experiment with a pressure outlet. It was found that the whole combustion process lasted for about 4 s. The non-premixed gasoline–air mixture deflagration experiment by Du et al. [41] lasted for seconds. The overpressure value was also lower. These are very different from the millisecond duration of premixed gas deflagration.

Although measures are taken to avoid fire during the use of flammable refrigerants [42,43], the speed, pressure changes, and the maximum pressure range of deflagration due to leakage of refrigerants containing propane still need to be studied thoroughly in case an ignition source is present for some reason [44].

An example case with a rectangular MiC unit model was constructed to study experimentally [24,45,46] the ignition of leaked propane (contained in flammable propane refrigerant) to justify the identified scenario. This MiC model was used to study two key scenarios:

• the well-covered condition when the freezer door is closed;
• the partially covered condition when the freezer door is open.

The variation of pressure under the well-covered condition and the partially covered condition was measured. The transient propagation speed of the flame surface and the correlation between the propagation time and location of the flame surface in the MiC model were obtained. As the flame resulting from explosion was too bright, the deflagration intensity could not be observed by photographs taken in the test. However, measuring the transient pressure changes would provide useful information.

A model based on experimental observation with flame spread rate and area was used to predict pressure changes without considering the chemical reaction and temperature change. A basic equation for the pressure change in an explosion in a small MiC model due to leaked gas deflagration is derived and experimentally validated. Maximum pressure and flame propagation are the main objectives of the analytical study. The pressure will directly damage the freezer door and other vulnerable structures.

2. Materials and Methods

2.1. Experimental Studies

Experiments on leakage of flammable propane refrigerant in an MiC model unit were carried out to investigate the gas concentration distributions, burning temperature, and transient pressure. The experimental data obtained were used in validating the mathematical model built in this paper.

The volume of a freezer is usually about 300~800 L, the internal cavity is mostly cuboid, and the corresponding refrigerant mass is generally 20~60 g. In order to simulate the refrigerant leakage inside a freezer, a rectangular MiC model of length 1 m, width 0.6 m, and height 1 m (the size of a typical domestic freezer) was constructed as shown in Figure 1a. Such a design would give a simple geometry for analysis.

The MiC model floor area was 0.6 m², and the volume was 0.6 m³. The floor and four walls of the MiC model were made of 2 mm thick stainless-steel plate. Then, 31.5 g (the refrigerant content in general small freezers) of propane refrigerant was injected into the MiC model through a gas injection pipe, giving an average concentration of propane of 3% and a temperature of 300 K. The duration of the gas injection was 720 s, to ensure the propane was evenly distributed in the model. The oxygen content in the MiC model was sufficient for complete combustion of the injected propane. The fuel concentration distribution has been described in previous articles [24].

The experiment simulated two situations. The first was the worst case in which the freezer was damaged, resulting in internal and external communication, and all the refrigerant leaked into the freezer. The second was that only internal leakage occurred, and all the refrigerant remained in the freezer.

Two covering conditions were employed, respectively, in studying the consequences of igniting propane with reference to the common interior design of freezer with a closed or half-closed door:

• The partially covered scenario represented the first situation.
  A pressure relief port at the top surface was provided. The pressure relief port was 4 cm × 4 cm, representing the partially covered space conditions.
  The leakage of propane indicated that the freezer casing might be damaged. Air inside may be connected to the outside. The assumption of this experiment referred to the worst case of air-tightness failure. Although a small area of the freezer casing was damaged, all the propane leaked inside the freezer and stayed at the bottom.
• The well-covered scenario represented the second situation.
  The constructed MiC model was well-covered during the injection and ignition of the gas. The top surface is made of a 1 mm iron sheet. In order to simulate the actual situation of the freezer, the same sealing strip as the freezer is used for sealing. Note that overpressure might damage the MiC model and lead to leakage at the top after the deflagration occurs. The MiC model was not a pressure-resistant enclosure (as freezers in common use are not a pressure-resistant enclosures).

For the partially covered scenario with pressure relief, the top surface of the MiC model was made of 5 mm thick plexiglass plate for observation and taking video of the flame propagation process. For the well-covered scenario, the top surface was made of 2 mm steel plate and fixed on the MiC model, and so there was no visual record for this condition. The pressure sensor and observation positions are shown in Figure 1b.

The ignition was achieved using an electrical spark-ignition device located at 0.2 m above the center of the floor level of the model. The concentration of the combustible gas at this height was appropriate for the flame to spread rapidly. The combustible gas in the MiC model was selected to be propane, with a mass fraction up to 17%. Experiments on the well-covered scenario and the partially covered condition with pressure relief were carried out.

There were three pressure measuring points (P1–P3), located, respectively, in front and on the left and right of the ignition source near the wall, as shown in Figure 1b. These positions were at a level of 0.1 m above the ignition position. The pressure of the MiC model was measured three times under each condition to ensure repeatability.

The frequency of the sensors was 1000 Hz. The data collection rate maximum of this instrument was 1000 Hz, distributed to each sensor at 333 Hz, as the collection rate of each sensor was 333 Hz. However, there were far too many data collected, and so these were treated carefully to plot figures. All the raw data were inspected carefully before presentation.

After starting the data acquisition unit for 2 s to confirm it worked properly, the propane was ignited. The data acquisition system would not be turned off until the pressure became stable for more than 30 s.

Digital cameras were set up to take images of the flame propagation above the ignition position. All the results were recorded by a data storage system.

2.2. Experimental Results

After the leaked propane was ignited, the explosion occurred almost immediately, and the flame surface was observed in the experiment. The flame surface extended outward from 0 s to 0.467 s after ignition for the scenario with pressure relief, as shown in Figure 2. It was observed that the flame propagation surface is spherical, so the instantaneous surface area of the flame can be easily obtained in the subsequent theoretical analysis.

Each experiment was repeated three times, and the results were similar. As explained above, there were many raw data on transient pressure changes collected by the three pressure sensors. The average measured transient pressure changes are shown in Figure 3.

With the pressure relief scenario (Figure 3a), the deflagration lasted for a longer time, with the explosion pressure developing slowly. The maximum deflagration pressure was only about 400 Pa because gas was able to flow out of the MiC model. Negative pressure resulted at the moment when burning stopped upon using up the fuel while gas was still rushing out of the MiC model due to momentum. The pressure rose quickly to the ambient atmospheric pressure subsequently.

For the well-covered scenario (Figure 3b) the top surface was quickly opened after ignition; pressure rose up to 3 kPa quickly, with the explosion lasting for a very short time. P1 and P3 were symmetrically distributed and close to the ignition point; P2 was further away. When deflagration occurred, the overpressure first increased and then decreased with the increase in propagation distance. P2 was smaller than P1 and P3, which indicated that the overpressure had reached the maximum before arriving P2, but whether the overpressure value had reached the maximum before reaching P1 and P3 cannot be given according to this study. Because the sealing strip of the MiC model was easily opened by the pressure produced by deflagration, the maximum pressure was not very high. However, this pressure was large enough to lift the freezer cover, causing damage, and the flame would spread from the freezer, which may cause fire. The MiC model was slightly damaged, so there was a negative pressure at P2 from 5 s to 6 s.

2.3. Flame Development Model

When burning flammable gas in an MiC model without strong convective flow, the flame spreads out spherically upon ignition. Putting in an obstacle would block the flame propagation, and the spherical flame surface cannot be kept. The flame propagation speed was slow in this experiment. The wall had little effect on the flame surface before striking the wall.

As observed in the experiments, the flame propagation distance increases as shown in Figure 4. The flame surface changes from the complete spherical to the hemispherical. Eventually, it becomes a curved surface with the area increasing first and then decreasing. The variation pattern and the surface area during the combustion process would affect the accuracy of the pressure calculation under the pressure relief scenario.

Figure 4 on flame development is discussed based on the following assumptions:

• A spherical flame surface before touching the wall.
• The flame stopped moving when touching the wall and was partially spherical. The curvature of the flame surface reduced with the smaller surface area.

The flame surface development is shown to be spherical under ideal conditions.

However, consistent pressure distribution at the three typical positions of measurement cannot be achieved even for an ignition position under symmetrical conditions. Due to the difference in propane concentration distribution and uncertainty of ignition position, the maximum pressure at P1 and P3 has a delay of less than 0.1 s.

Figure 5 and Figure 6 show the flame surface area and volume as a function of the distance from the ignition position. The average growth rate of the flame radius could be taken as constant under the ideal state. Some part of the flame surface (say surface S1) would reach a wall before the other parts (say S2). However, the flame surface S1 did not affect the propagation of the flame surface S2.

The flame with surface area $A$ and the entire flame region volume $V_b$ can be estimated from experimental observations. This is similar to cutting the flame ball with a growing radius by a cuboid surface. The flame propagation pattern could be described by four stages when the flame touched different walls. $A$ and $V_b$were calculated at several flame surface positions during each stage. The curves of $A$ and $V_b$ in different stages can be obtained by fitting to get the curves in Figure 5 and Figure 6.

The change of the flame surface area is related to the pressure change rate. The above observation is useful in developing an analytical model for the temperature of the flame surface before and after the explosion. Calculating the flame surface positions would give a better estimation for the temperature of the gas volume after burning.

Figure 5 and Figure 6 show the flame surface area and volume change of the flaming zone after burning under ideal conditions. In the actual measurements, it can be seen from Figure 2 and Figure 3 that there are differences in time delay and maximum pressure distribution.

2.4. Modeling of Partially Covered Scenario with Pressure Relief

This MiC model, as shown in the below Figure 7, was formulated under the same conditions as the common flame propagation velocity, under the assumption that there was no heat input or output on the boundary in the system. The temperature of unburned reactants in the system is assumed to be the initial temperature. The temperature of the burned reactants is the same as the temperature of the final combustion products. For deflagration, the rise of pressure and the gas temperature of the unburned zone and burned zone will change. As the deflagration time is very short, there is no time for heat exchange. The change of temperature and pressure is approximately regarded as an adiabatic process. In the adiabatic deflagration model, both the equation of state and the conservation equation of mass and energy are applied.

In the isothermal model for the MiC unit, the partially covered scenario with pressure relief satisfies the following gas state equations for pressure $P$, total volume $V$ and gas temperature $T$:

$$P_{0}V=n_{0}R{T}_u$$

(1)

$$P(V-V_b)=n_{u}R{T}_u$$

(2)

$$P{V}_b=n_{b}R{T}_b$$

(3)

Law of mass conservation [47] can be expressed by mole numbers:

$$n_0=n_u+n_b+n_e$$

(4)

The subscript “$0$” indicates the initial state before ignition, the subscript “$u$” indicates the region where the deflagration has not passed, subscript “$b$” in the above equations indicates the region where the deflagration passes, and the subscript “$e$” indicates the exhaust.

$n_e$ (in mol) is zero when no gas is released. The fuel consumed rate during the fast combustion can be expressed by the radical expansion velocity $K_r$ and surface area $A$ of the burnt volume as

$${\displaystyle \frac{\mathrm{d}{n}_b}{\mathrm{d}t}}=-{\displaystyle \frac{\mathrm{d}{n}_u}{\mathrm{d}t}}={\displaystyle \frac{K_{r}AP}{R{T}_u}}$$

(5)

Equation (5) can be further modified by considering the turbulence factor $\alpha$ for the turbulent fuel–air mixture:

$${\displaystyle \frac{\mathrm{d}{n}_b}{\mathrm{d}t}}={\displaystyle \frac{\alpha K_{r}AP}{R{T}_u}}$$

(6)

The outflow rate [47] on gas amounts $n_e$ is

$${\displaystyle \frac{\mathrm{d}{n}_e}{\mathrm{d}t}}=K_V{\displaystyle \frac{A_V{(P-P_0)}^{1/2}}{{T_u}^{1/2}}}$$

where the pressure relief coefficient of subsonic flow $K_V$ is $918{ \mathrm{molK}}^{1/2}/{\mathrm{m}}^2\cdot {\mathrm{Pa}}^{1/2}\cdot \mathrm{s}$. $K_V$ is a constant. Many works were reported on theoretical study [48,49,50,51,52].

Differentiating both sides of Equation (4) and rearranging:

$${\displaystyle \frac{\mathrm{d}{n}_u}{\mathrm{d}t}}=-{\displaystyle \frac{\mathrm{d}{n}_b}{\mathrm{d}t}}-{\displaystyle \frac{\mathrm{d}{n}_e}{\mathrm{d}t}}=-{\displaystyle \frac{\alpha K_{r}AP}{R{T}_u}}-{\displaystyle \frac{K_V{A}_V{(P-P_0)}^{1/2}}{{T_u}^{1/2}}}$$

(7)

Eliminating $V_b$ from Equations (1) and (2) and substituting $P$ for $n_u$ gives

$${\displaystyle \frac{\mathrm{d}P}{\mathrm{d}t}}={\displaystyle \frac{\alpha K_{r}AP(P_m-P_0)}{V{P}_0}}-{\displaystyle \frac{R{T_u}^{1/2}{A}_V{K}_V{(P-P_0)}^{1/2}}{V}}$$

(8)

As Equation (8) is a first-order nonlinear nonhomogeneous ordinary differential equation, a fourth-order Runge–Kutta method was used to solve it numerically.

The above formula gives the pressure change rate before the flame surface reaches the pressure relief port. When the flame surface reaches the pressure relief port, the released gas temperature is $T_b$. The pressure change rate at this time is

$${\displaystyle \frac{\mathrm{d}P}{\mathrm{d}t}}={\displaystyle \frac{\alpha K_{r}AP(P_m-P_0)}{V{P}_0}}-{\displaystyle \frac{R{T}_u{A}_V{K}_V{(P-P_0)}^{1/2}}{V{T_b}^{1/2}}}$$

(9)

As observed in the experiment, there were three stages of explosion. The calculation process on pressure change is then divided into three stages, namely, (1) the deflagration stage, (2) the pressure relief stage, and (3) the recovery to atmospheric pressure stage.

The different stages were determined from the flame surfaces touching the wall. For the well-covered scenario, the first stage was spherical flame development before touching the wall. The second stage was based on experimental observation. The first stage stopped at 3 s, and the model then has a limitation. The third stage stopped at differential pressure at 0 Pa for predicting overpressure.

• Stage 1 (Deflagration Stage): from deflagration to flame off.

The duration was about 0.8 s, as observed in the experiment shown in Figure 3a. The flame had not reached the relief port at this stage. The pressure change rate is calculated according to Equation (8). As mentioned above, with the spread of the flame surface, the law of change of flame surface area is different. According to the law of flame propagation, two processes are involved in this stage: the spherical change process and the one-dimensional diffusion process.

The spherical flame surface begins from the start of deflagration and ends with the flame surface reaching the MiC model bottom (in about 0.2 s in the experiment). The flame surface $A$ is spherical and propagates with steady velocity during this process, and the formula is

$$A=4\pi r^2\cong 4\pi t^2$$

(10)

The one-dimensional diffusion process starts from the end of stage 1 and ends when the flame surface has reached the relief port (taking about 0.6 s in this experiment). According to the flame surface area changes with time, the weighted average from Figure 5 is used to get the average area at this stage. The values of the related parameters [44] are shown in

Table 1. Values of the parameters.

Parameters	Values
Turbulence and burning rate influence coefficient ${\alpha K}_r$	0.48 m/s
Maximum pressure in the closed state $P_m$	103,000 Pa
Initial pressure $P_0$	100,000 Pa
$\mathrm{MiC}\mathrm{model} \mathrm{volume} V$	0.6 m3
$\mathrm{Gas} \mathrm{constant} R$	287 J/kg·K
$\mathrm{Initial} \mathrm{temperature} T_u$	300 K
Pressure relief area $A_V$	6 × 10−6 m2
Pressure relief coefficient $K_V$	918 molK1/2/m2Pa1/2 s

.

In the whole deflagration process, Equation (8) is a first-order nonlinear nonhomogeneous ordinary differential equation, the analytical solutions of which are difficult to obtain. The equations are then solved numerically by the fourth-order Runge–Kutta method.

In solving the differential equation of pressure (denoted as u in the following treatment) and time using the Runge–Kutta method, the following equation is assumed:

$$\left\{\begin{array}{l}{\displaystyle \frac{du}{dt}}=f\left(t,u\right),t_0<t<T\\ u\left(t_0\right)=a\end{array}\right.$$

(11)

For the initial problem, the approximation values ${\left\{u_m\right\}}_{m=1}^N$ on $N$ discrete points ${\left\{{\mathrm{t}}_m\right\}}_{m=1}^N$ ($t_1,t_2,\dots ,t_N\le T$ are called nodes) satisfy the relation:

$$t_{m+1}=t_m+h_{m+1},m=0,1,\dots ,N-1$$

(12)

${\left\{{\mathrm{h}}_m\right\}}_{m=1}^N$ is a set of positive numbers known as step length. In this way, a continuous domain is divided into discrete nodes. In order to reduce the trouble of determining the step length, the adaptive format of the Runge–Kutta method [53] is used in the following equations:

$$u_{m+1}=u_m+h\left({\displaystyle \frac{25}{216}}{k}_1+{\displaystyle \frac{1408}{2565}}{k}_3+{\displaystyle \frac{2197}{4104}}{k}_4-{\displaystyle \frac{1}{5}}{k}_5\right), k_1=f\left(t_m,u_m\right)$$

(13)

$$k_2=f\left(t_m+{\displaystyle \frac{h}{4}},u_m+{\displaystyle \frac{h}{4}}{k}_1\right), k_3=f(t_m+{\displaystyle \frac{3h}{8}},u_m+{\displaystyle \frac{3h}{32}}{k}_1+{\displaystyle \frac{9h}{32}}{k}_2)$$

(14)

$$k_4=f\left(t_m+{\displaystyle \frac{12h}{13}}, u_m+{\displaystyle \frac{1932h}{2197}}{k}_1-{\displaystyle \frac{7200h}{2197}}{k}_2+{\displaystyle \frac{7296h}{2197}}{k}_3\right)$$

(15)

$$k_5=f\left(t_m+h, u_m+{\displaystyle \frac{439h}{216}}{k}_1-8h{k}_2+{\displaystyle \frac{3680h}{513}}{k}_3-{\displaystyle \frac{845h}{4104}}{k}_4\right)$$

(16)

$$u_{m+1}=u_m+h\left({\displaystyle \frac{16}{135}}{k}_1+{\displaystyle \frac{6656}{12825}}{k}_3+{\displaystyle \frac{28561}{56430}}{k}_4-{\displaystyle \frac{9}{50}}{k}_5+{\displaystyle \frac{2}{55}}{k}_6\right)$$

(17)

$$k_5=f\left(t_m+{\displaystyle \frac{h}{2}}, u_m-{\displaystyle \frac{8h}{27}}{k}_1+2h{k}_2-{\displaystyle \frac{3544h}{2565}}{k}_3+{\displaystyle \frac{1859h}{4104}}{k}_4-{\displaystyle \frac{11h}{40}}{k}_5\right)$$

(18)

By comparing the difference between the calculated values at the same time step, one can judge whether the given precision is satisfied. If it is not satisfied, the step size is reduced. The relative deviation precision given in this paper is 10⁻⁴. A lower value of 0.005 s is adopted for the step length in the prediction calculations.

The transient pressure curve at stage 1 is shown in Figure 8.

This is mainly determined experimentally by when the flame surface touched the wall, by trial and error.

Time intervals of 3 s in Figure 8 were determined from flame spread rates and the MiC model size. The flame spread regions were divided into different regions, and then the data were analyzed by appropriate methods. At the opening with conflagration, 0.8 s was taken as the sustainable time, with consideration on the flame surface spread to the wall.

Deflagration started at 2.2 s in the experiment, so the same time was used for the calculation. The pressure rose rapidly for 0.8 s after ignition and reached the maximum at 3.0 s, with a relative pressure of 474 Pa.

• Stage 2 (Pressure Relief Stage): Deflagration is completed and the expanding gas has been released through the relief port.

The duration was about 4 s in the experiment. At the beginning of the pressure relief process, the flame surface reaches the relief port. The pressure formula is the Equation (9). As deflagration is completed, there is no gas generation. The pressure change rate at this stage becomes

$${\displaystyle \frac{\mathrm{d}P}{\mathrm{d}t}}=-{\displaystyle \frac{R{T}_u{A}_V{K}_V{(P-P_0)}^{1/2}}{V{T_b}^{1/2}}}$$

(19)

Since the initial pressure of the relief phase is the maximum pressure at the end of the deflagration phase, that is $P_0=100474$ Pa, the pressure in the process drops. Therefore, the above formula ${(P-P_0)}^{1/2}$ should be changed to ${(P_0-P)}^{1/2}$.

Assume that $Q={(P_0-P)}^{1/2}$; then, the above equation becomes

$${\displaystyle \frac{2\mathrm{Qd}Q}{\mathrm{d}t}}=-{\displaystyle \frac{R{T}_u{A}_V{K}_{V}Q}{V{T_b}^{1/2}}}$$

(20)

Integrating the equation gives

$$Q=-{\displaystyle \frac{R{T}_u{A}_V{K}_V}{2V{T_b}^{1/2}}}t$$

(21)

$$P=P_0-Q^2=P_0-{({\displaystyle \frac{R{T}_u{A}_V{K}_V}{2V{T_b}^{1/2}}})}^2{t}^2$$

(22)

The time t in the above equation is the time of the relief phase. As seen in Figure 3a, the absolute time in the whole process should be $(t+3)s$. In addition, the top cover steel is lifted a little by the expanding gas, leading to a certain interspace at the top of the MiC model to achieve unconstrained pressure relief. There is a certain connection between the area of the pressure relief and the internal expansion. In the pressure relief process, the pressure is gradually reduced. The pressure relief area should be reduced compared to the explosion stage:

$$A_V=2.0\times {10}^{-6}{\mathrm{m}}^2$$

The pressure change at this stage as given by Equation (22) is shown in Figure 9.

At the end of deflagration, gas is discharged through the pressure relief port, resulting in a decrease in the pressure inside the MiC model. Since the expansion and discharge of the gas cannot be stopped suddenly due to the momentum effect, negative pressure appears inside the MiC model.

• Stage 3 (Recovery Stage): Internal pressure of the MiC model starts to rise for a certain period.

The duration was about 3 s (around 2 s to 5 s in Figure 3a) in the experiment, with linear pressure rise.

The theoretical transient pressure curve of the whole deflagration process is compared with experimental results for the pressure relief scenario in Figure 10. The two curves roughly match, indicating that the model and the theoretical results are reasonable.

However, except that the maximum and the minimum of the pressure change had small relative deviation predictions, the results did not agree so well with the experimental values. This is because the actual change of temperature during the deflagration process is ignored in the calculation while using the isothermal model. However, $T_u$ and $P_0$ change in reality. Based on this assumption, the final state is affected more by the assumption made at the initial state. The effect of the initial state conditions on the final state is amplified.

As $T_u$ and $P_0$ are assumed to be unchanged, the change rate $dP/dt$ is different from the real scenario with variations of $T_u$ and $P_0$ of the unburned gas. The pressure drop in the experiment is due to the reduction in flame intensity upon extinction. The relatively flat section in the experimental curve from 6 s to 7 s results from the flame instability during extinction. The isothermal model is not able to predict the pressure changes well when the flame is extinct.

2.5. Modeling of Well-Covered Scenario

Another scenario, the well-covered scenario in the MiC unit, is studied by omitting the mass outflow term in Equation (8) for the pressure relief scenario. The pressure change rate in the well-covered scenario is then given by

$${\displaystyle \frac{\mathrm{d}P}{\mathrm{d}t}}={\displaystyle \frac{\alpha K_{r}AP(P_m-P_0)}{V{P}_0}}$$

(23a)

In the above equation, $A$ is the surface area of the flame, which can be obtained using the transfer equation of the three-dimensional changes of the flame inside the MiC model.

Rearranging Equation (23a) gives

$${\displaystyle \frac{\mathrm{d}P}{P}}={\displaystyle \frac{\alpha K_r(P_m-P_0)}{V{P}_0}}A\mathrm{d}t$$

(23b)

The term $P$ on the left side of the above formula is a variable, and the $A$ on the right side depends on $t$, which can be further expressed as a polynomial of $t$. The equation can be solved through integration by assuming $I(t)={\displaystyle \int Adt}$:

$$\ln P={\displaystyle \frac{\alpha K_r(P_m-P_0)}{V{P}_0}}I(t)+C$$

(24)

$$P=C\exp ({\displaystyle \frac{\alpha K_r(P_m-P_0)}{V{P}_0}}I(t))$$

(25)

In the above equation, $C$ is the integration constant, which can be determined from the initial conditions. By obtaining the expression for $I\left(t\right)$, $P$ can be calculated.

The following empirical formula was reported [54]:

$$Kr=f_3{S}_L{({\displaystyle \frac{2\pi r_f}{{\lambda }_c}})}^{(n_i-1)/n_i}=2m/s$$

(26)

The laminar flame speed ${\mathrm{S}}_L$ [55] was 0.4 m/s, the flame radius $r$ was 1 m, the cut-off wavelength ${\mathsf{\lambda }}_c$ was 0.005 m [56], ${\mathrm{n}}_i$was 1.65, and $f$ was 0.3.

The flame propagation process can be divided into two stages based on experimental observation in this study:

• The first stage starts from the ignition and ends with the flame surface contacting the bottom surface of the container (0 m < $r$ < 0.2 m). This part is calculated according to the formula for a spherical flame surface.
• The second stage begins at the end of the first stage and ends with the flame surface reaching the exit (0.2 < $r$ < 0.8 m), since the spherical formula is no longer applicable. The change in flame surface area for the whole process is shown in Figure 10.

The first stage (0 m < $r$ < 0.2 m, 0 < $t$ < 0.1 s):

$$P=P_0+(P_m-P_0){({\displaystyle \frac{P_m}{P_0}})}^2{\displaystyle \frac{{\alpha }^3{K_r}^3{t}^3}{a^3}}$$

(27)

In the above equation, $P_0$ = 100,000 Pa, $K_r$ = 2 m/s, and $a$ = 0.2 m, according to the maximum pressure in the first stage of the preceding paragraph, taking $P_m$ = 100,011 Pa.

The second stage (0.2 m < $r$ < 0.8 m, 0.1 s < $t$ < 0.4 s):

$${\displaystyle \frac{\mathrm{d}P}{\mathrm{d}t}}={\displaystyle \frac{\alpha K_{r}AP(P_m-P_0)}{V{P}_0}}$$

(28)

In the above equation, where $V$ is the volume of the MiC model (which is constant), considering the change of $A$ alone is enough. The propagation of the flame surface area $A$ is plotted in Figure 10.

In the second stage, the area is obtained based on the linear feature and the weighted method to get A = 0.9 m². Integrating Equation (28) would give pressure as

$$P=P_1\exp [{\displaystyle \frac{A\alpha K_r(P_m-P_1)}{V{P}_1}}t]$$

(29)

In the above equation, t is the time of the second stage. If the first stage is considered, then the absolute time here should be ($t$ + 0.1), $P_1$ = 100,016 Pa, which is the final state pressure of the first stage and the initial pressure of the second stage.

3. Results

3.1. Results for Partially Covered Scenario with Pressure Relief

Prediction results for the partially covered scenario given by Equations (9), (19) and (22), as described in Section 2.4, are shown in Figure 11.

The difference between the predicted and experimental results of the maximum pressure (time = 3 s) and minimum pressure (time = 4 s) in Figure 9 is acceptably small. The accuracy of the predictions depends on assumptions made on spherical flame change.

3.2. Results for Well-Covered Scenario

Prediction results for the well-covered scenario given by Equation (29) are shown in Figure 12. The complexity of the calculation can be reduced without substantially affecting the accuracy. The difference between the two-stage method and the four-stage method is only in the equations of the flame surface area. The two-stage method is a simplification of the four-stage method.

The peak pressure is 3012 Pa, compared with the experimental value of 3011 Pa, giving a very low relative deviation of 0.03%.

3.3. Comparison of Pressure Change Rate

To facilitate calculation of the pressure change rate $dP/dt$ using Equations (23a) or (23b), the following conversion is made.

In the process of flame propagation, there is a proportional relationship between flame surface area and volume of the flame. Assuming that

$$A=k_1{x}^2,V_b=k_2{x}^3$$

(30)

where $x$ is a characteristic length related to the surface area of the flame.

Similarly, for the MiC model, there is also a proportional relationship between its surface area $S$ and volume $V$.

$$S=k_1{X}^2,V=k_2{X}^3$$

(31)

where $X$ is a characteristic length related to the MiC model.

The flame surface area $A$ can be expressed in terms of $S$ and $V$as

$$A=S{({\displaystyle \frac{V_b}{V}})}^{2/3}$$

(32)

Under the isothermal condition:

$${\displaystyle \frac{V_b}{V}}={\displaystyle \frac{1-P_0/P}{1-P_0/P_m}}$$

(33)

Then

$${\displaystyle \frac{\mathrm{d}P}{\mathrm{d}t}}={\displaystyle \frac{\alpha K_{r}S{P_m}^{2/3}}{V{P}_0}}{(P_m-P_0)}^{1/3}{(1-{\displaystyle \frac{P_0}{P}})}^{2/3}P$$

(34)

Under the adiabatic condition, $T_b$and $T_m$are not constants but will change with the pressure in the MiC model. Since the duration of flame front propagation is short, the process can be regarded as approximately an adiabatic process. The temperature of the unburned gas is increased by adiabatic compression in the partially covered scenario, and is controlled by pressure relief, namely:

$$T_u=T_0{({\displaystyle \frac{P}{P_0}})}^{1-{\displaystyle \frac{1}{\gamma }}}$$

(35)

$$T_b=T_m{({\displaystyle \frac{P}{P_m}})}^{1-{\displaystyle \frac{1}{\gamma }}}$$

(36)

Then

$${\displaystyle \frac{V_b}{V}}={\displaystyle \frac{1-(P_0/P{)}^{\frac{1}{\gamma }}}{1-(P_0/P_m{)}^{\frac{1}{\gamma }}}}$$

(37)

The pressure change rate is expressed as

$${\displaystyle \frac{\mathrm{d}P}{\mathrm{d}t}}={\displaystyle \frac{\alpha \gamma K_{r}S{P_r}^{\beta }{P_m}^{2/3\gamma }}{V{P_0}^{2-\frac{1}{\gamma }}}}{({P_m}^{{\displaystyle \frac{1}{\gamma }}}-{P_0}^{{\displaystyle \frac{1}{\gamma }}})}^{1/3}{(1-{({\displaystyle \frac{P_0}{P}})}^{{\displaystyle \frac{1}{\gamma }}})}^{2/3}{P}^{3-{\displaystyle \frac{2}{\gamma }}-\beta }$$

(38)

In the above equation, $P_0$ = 100,000 Pa is the reference pressure, and $\beta =0.2$ is the burning rate coefficient affecting pressure.

The variation of pressure change rate during the rise of pressure is shown as Figure 13, in which the non-dimensional pressure $p={\displaystyle \frac{P-P_0}{P_m-P_0}}$ is used for the x-axis.

Obviously, as the deflagration pressure increases, the pressure change rate $dP/dt$increases due to the increasing intensity of the chemical reactions. At the beginning of the explosion, the pressure in the MiC model is small, the chemical reaction is slow, and the pressure increases slowly. As the deflagration is gradually accelerated, the pressure increases, and the reaction is faster than that at the initial stage. Both the isothermal and adiabatic results illustrate this trend. Compared with the adiabatic condition, the pressure increase in the isothermal condition is slower. This is because the pressure increase is only due to the compression of the gas in the isothermal condition with a slower process.

4. Discussion and Conclusions

An identified explosion hazard scenario was studied experimentally. Leaked propane in a small MiC unit model under well-covered and partially covered scenarios were considered. The same propane concentration was used in the modeling and experiment. The pressure and flame patterns generated by the explosion were recorded. The flame propagation speed was estimated. The results, achievements, and concluding remarks of the present work are summarized as follows:

• The development of flame in the explosion process was studied. The relationships between flame surface area, flame volume, and the distance of flame surface to ignition point were obtained.
• The burning of leaked gas in the partially covered scenario can be divided into three stages, with differential equations derived. The numerical solutions of these equations using the Runge–Kutta method agree well with the experimental results.
• Combustion in the well-covered scenario can be simplified using two-stage equations. The equation for the pressure change rate in the well-covered scenario is derived with analytical solutions.

The model is formulated based on isothermal conditions. As the flame propagates, the gas temperature inside the model is not high except at the flame surface. Fast flame propagation would transfer heat to the surrounding slowly. Therefore, this model depends mainly on the flame propagation speed, which is affected by the heat of combustion, whether the model is covered or open, and independent of the opening size and position.

The prediction results of the developed model are acceptable compared with the experimental data. This study could be used as a reference for estimating the pressure changes and designing ventilation systems to dilute explosive gases to prevent deflagration in an MiC unit. However, the current study is based on a small, simple model. There might be higher deviation for many fuels in a complex geometric structure, which should consider potential impacts due to uncertain factors.