Keywords

1 Introduction

A fire in a tunnel is among the most terrifying transportation mishaps. Because tunnels are long, narrow, closed and have few escape routes, a fire in tunnel could pose a significant risk to the lives of the trapped individuals. At the same time, the high temperature will also damage the structure of the tunnel, leading to serious property damage. Therefore, in order to minimize casualties and economic losses, early monitoring and alarming of tunnel fires and effective control of smoke are very necessary.

Temperature is an extremely important factor in studying tunnel fires. It can be reference for the design of automatic fire alarm system and fire extinguishing system in tunnels. Currently, numerous scholars have conducted research on the highest temperature in tunnels. However, most of these researches have been confined to scenarios involving a single fire source. The impact of factors such as the heat release rate of the fire source, the velocity of longitudinal ventilation, and the parameters of the fire source on the highest temperature have been examined. In 1972, Alpert [1] investigated experimentally the variation of the highest ceiling temperature when there is no accumulation of smoke in the tunnel and established a prediction model. In 2003, Kurioka et al. [2] designed fire experiments in five model tunnels to study the highest temperature. A model for the highest temperature was established using both theoretical and experimental data analysis. Hu et al. [3] established Kurioka’s maximum temperature correction model considering the tunnel slope factor through small-size experiments. Li et al. [4] found that Kurioka's model is no longer applicable when wind speeds are low. Therefore, Li et al. designed two tunnel model test platforms to investigate the variation of highest smoke temperature at different longitudinal ventilation rates. Jiang et al. [5] investigated the variation rule of the maximum temperature rise through full-size tunnel fire experiments, and verified the existing highest temperature rise prediction model. Wang [6] designed fire experiments in tunnels with different slopes, and studied the change of the maximum temperature by changing the tunnel slope.

The above studies consider single fire scenarios in tunnels and do not consider multiple fires scenarios. According to the survey results of tunnel fires [7], the proportion of fires caused by vehicle collisions in tunnels is as high as 81%. Vehicle collisions can easily lead to multiple fires accidents, which increases the scale of the fire as well as the hazards significantly. For example, in 2014, in Shanxi Yanhou tunnel 2 trucks collided and led to a fire in the tunnel. It spread rapidly, igniting vehicles around it, and resulting in 40 fatalities and 12 injuries [8]. It can be seen that the hazards caused by dual fires in tunnels are more serious. For dual source fire scenarios, scholars have also carried out corresponding researches. Xu [9] studied the impact of separation distance between the double fires on the combustion characteristics of dual fires through numerical simulation and model experiments, and established a prediction model for the maximum temperature of the smoke below the ceiling. Xu [10] conducted deep research to investigate the correlation between the maximum ceiling temperature and the separation distance between the two fires during longitudinal ventilation, and based on earlier studies, they added the influence factor of the separation distance between the two fires to create a new maximum temperature prediction model. Zhou et al. [11] investigated the effects of fire source spacing as well as the distance between the wall and two fire sources on the maximum smoke temperature by means of small-size experiments and established a maximum temperature prediction model by introducing the above two factors on the basis of previous studies. Chen [12] investigated the impact of the separation distance between the two fires on the maximum temperature in the case of longitudinal ventilation by means of small-size experiments, and a model for the highest temperature below the tunnel ceiling was established by theoretical analysis. Wan et al. [13] investigated the variation of the highest temperature of the smoke below the ceiling for double fire sources with different sizes under natural ventilation through small-size experiments, and developed a prediction model for the highest temperature below the ceiling.

The interactions between two fires makes the combustion of two fires much more complicated than that of one fire. The current researches [9,10,11,12,13] on the highest temperature rise of smoke from double fire sources mainly take into account the impact of separation distance between the two fires, lateral position of the double fires, and longitudinal ventilation speed on the highest temperature of the smoke. However, previous studies on the maximum temperature of tunnel ceilings lacked consideration of the size of the fire source. Therefore, this article combines the factors influencing the maximum temperature of tunnel ceilings in previous studies and introduces the size parameter of the fire source to establish a highest temperature model.

2 Numerical Simulation

2.1 Physical Modeling

In this paper, numerical simulations are carried out using the fire dynamics software FDS (Fire Dynamics Simulator), whose effectiveness in the study of tunnel fires has been widely verified [14]. FDS is developed from a CFD analysis program specifically for simulating fire smoke dispersion. This software employs numerical methods to solve the Navier–Stokes equations for low-velocity, thermally-driven flows, with a focus on smoke motion and heat transfer processes due to fire.

As seen in Fig. 1, referring to specifications for design of highway tunnels [15], a full-size model tunnel of 80 m in length, 10 m in width, and 5.5 m in height is constructed using FDS. The model tunnel has a rectangular cross-section and is configured with two open ends to replicate a fire scenario with natural ventilation. Considering that the temperature is highest in the area directly above the fire source during combustion, thermocouples are set up according to the location of the fire source and the content of the study. A set of thermocouples with a 0.125 m longitudinal spacing is positioned in the longitudinal center and 0.1 m below the tunnel ceiling. The tunnel wall material is set as “concrete”. 20 ℃ and 101 kPa of pressure are the beginning conditions inside the tunnel.

Fig. 1
A schematic of a highway tunnel along the X, Y, and Z axes. It consists of two fire source locations labeled Fire A and B. The dotted horizontal line denotes the thermocouples. The total length of the tunnel is 80 meters while the height is 5.5 meters.

Schematic diagram of tunnel model

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Double fire sources are installed along the tunnel longitudinal centerline, with the heat rate of a single fire source set to be 1.0, 1.25, and 1.5 MW. The fire source has a square surface with four side lengths of 0.8 m, 1.2 m, 1.6 m, and 2.0 m (refer to Fig. 2) and a height of 0.5 m. This study carried out a total of 84 tests by changing the heat release rate of the sources, the size of the fires, and separation distance between the two fires, as shown in Table 1. The simulation time for each test was set to be 200 s, and at this time the fire smoke could reach a stable state.

Fig. 2
A set of four schematics of four different sizes of burner surfaces. a. 0.8 meters. b. 1.2 meters. c. 1.6 meters. d. 2.0 meters.

Burner surface in four different sizes (Top view)

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Table 1 Fire simulation table
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2.2 Grid System

The precision of the numerical simulation results will depend on the grid size. In this paper, we refer to the user manual of FDS to determine the grid size depending on the calculation outcomes of the characteristic diameter of the fire source [16] according to Eq. (1).

$$D^{*} = \left( {\frac{{ \, \dot{Q}}}{{\rho_{0} c_{{\text{p}}} T_{0} \sqrt g }}} \right)^{2/5}$$
(1)

where \(D^{*}\) is the characteristic diameter of the fire source, m; \(\dot{Q}\) is the heat release rate, kW; \(\rho_{0}\) is the density of air, kg/m3, as 1.204 kg/m3; \(c_{{\text{p}}}\) is the constant pressure specific heat capacity of air, kJ/(kg·K), generally 1.005 kJ/(kg·K); \(T_{0}\) is the ambient temperature, K, in this paper as 293 K; \(g\) is the acceleration of gravity, m/s2, as 9.81 m/s2.

For the fire scenario in this paper when the minimum heat release rate of the fire source is 1.0 MW, the characteristic diameter of the fire source \(D^{*}\) is calculated by Eq. (1) to be 0.96 m. According to the grid size formula proposed by Mcgrattan et al. [16], the model can accurately restore the fire scenario if the ratio of the characteristic diameter of the fire source to the grid size is between 4 and 16. Therefore, the numerical simulation results are more reliable when the grid size is between 0.06 and 0.24 m. For grid sensitivity analysis, four different grid systems were selected, as shown in Table 2.

Table 2 Selection of grid size
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Figure 3 gives the temperature comparison between the four grid systems when the fire source size is 1.2 m, the separation distance between the two fires is 0 m, and the heat release rate of a single fire source is 1.0 MW. As the point of origin, we choose the center between the two fire sources. The comparison reveals that the difference in longitudinal temperatures is small for grid system 1, 2 and 3. When the grid size is increased to 0.5 m, the temperature difference between grid system 4 and grid system 1, 2 and 3 is larger. Grid system 1 has been selected for this study in order to obtain precise results.

Fig. 3
A graph depicts the temperature versus X. It includes four plots of four grid systems with various grid sizes in near-fire regions and grid sizes in other regions. The second grid system with 0.15 by 0.15 by 0.15 meters ranges high at 270 degrees and gradually falls to 80 degrees Celsius between 0 and 10 meters.

Distribution of the ceiling smoke longitudinal temperature using various grid sizes

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We consult earlier studies [17] to guarantee the accuracy of the experimental data. The sensitivity test is conducted using four alternative grid systems, as indicated in Fig. 3. In the study conducted by Huang et al. [17], a tunnel model with a width of 10 m and a height of 5 m was used, exhibiting great similarity with the model used in this paper. To assure the accuracy of the results of the numerical simulation, a grid system that is similar to that used in their study is used in this paper.

3 Simulation Results and Discussion

Figure 4 depicts the change of the smoke maximum temperature with respect to the distance between fires for different fire sizes under conditions of natural ventilation when the heat release rate of a single fire source is 1 MW, 1.25 MW, and 1.5 MW, respectively. It can be seen from Fig. 4.

Fig. 4
Three multi-line graphs depict the variation in temperature versus S for four different D values of 0.8, 1.2, 1.6, and 2.0 meters. a. H R R = 1.5 plus 1.5 megawatts. b. H R R = 1.25 megawatts. c. H R R = 1.0 plus 1.0 megawatts.

Variation of maximum temperature rise with fire spacing for fire sources of different sizes

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  1. (1)

    With increasing fire separation and a constant fire size, the maximum smoke temperature drops. The primary reason for this is that when the two fires are farther apart, there is less contact between the two fire sources and more entrainment of the fire plume, which lowers the maximum temperature.

  2. (2)

    The maximum temperature of the smoke decreases with the increase in the fire source size under fixed fire source spacing. The primary reason for this is that as the total heat release rate of the fire source remains constant, the heat release rate per unit area of the fire source decreases as the fire source area increases, and the flame height decreases accordingly, which leads to a decrease in the temperature of the fire plume ceiling jet.

Considering the effect of fire source size on the maximum temperature of the smoke, we try to introduce the fire source size as well as the fire spacing parameter to predict the maximum temperature. According to the research by Alpert [1] on the maximum temperature of smoke under an open ceiling, the maximum temperature of the smoke induced by a single fire ceiling jet was proportional to the 2/3 power of the heat release rate of the fire source and inversely proportional to the 5/3 power of the distance from the surface of the fire source to the ceiling, as shown in Eq. (2).

$$\Delta T_{\max } \propto \frac{{\dot{Q}^{2/3} }}{{H_{ef}^{5/3} }}$$
(2)

The maximum smoke temperature for double tunnel fires drops with increasing fire separation and gradually drops with increasing fire source size. This article investigates the highest smoke temperature under weak plume conditions based on Alpert's research on the maximum ceiling temperature. The maximum smoke temperature for fire sources of four different sizes is correlated using Eq. (2), and the results of the fitting are displayed in Fig. 5.

Fig. 5
Four graphs depict the variation in temperature with fire spacing and fire sizes. They all have three H R R values of 1.0, 1.25, and 1.5 megawatts. a. D = 0.8 meters. b. D = 1.2 meters. c. D = 1.6 meters. d. D = 2.0 meters.

Correlation of maximum temperature rise with fire spacing and fire size

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The expressions related to the maximum temperature of the smoke for four different fire sizes are given in Fig. 5, and based on the equations in Fig. 5, an expression for the maximum temperature for different fire sizes was established in the following Eq. (3).

$$\Delta T_{\max } = k\frac{{\dot{Q}^{2/3} }}{{H_{ef}^{5/3} }}$$
(3)

As can be seen in Fig. 5, the coefficient \(k\) is related to the fire source spacing and fire source size. The expression for the relationship between \(k\) and fire source size and fire source spacing is established from the fitting results as shown below:

$$k\, = \,\alpha e^{{\left( { - 0.88S} \right)}} + 9.21$$
(4)

Combined with Fig. 5 and the expression for the coefficient \(k\), the value of the coefficient \(\alpha\) decreases as the fire source size increases. Figure 6 illustrates the correlation between the coefficient \(\alpha\) and the size of the fire source, as seen by Eq. (5) below.

$$\alpha = - 10.29D + 25.31$$
(5)
Fig. 6
A graph depicts the alpha versus D. It includes a curve of Y = negative 10.29 X plus 25.31. The plot begins at (0.9, 17) and gradually falls to (2.0, 5) approximately. The value of R square = 0.98.

Correlation of the coefficient \(\alpha\) for fire sources of different sizes

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Substituting Eqs. (4) and (5) into Eq. (3), the maximum temperature for fires of different sizes at different spacings can be expressed by Eq. (6):

$$\frac{{\Delta T_{\max } }}{{\dot{Q}^{2/3} /H_{ef}^{5/3} }} = \left( {25.31 - 10.29D} \right)e^{ - 0.88S} + 9.21$$
(6)

Therefore, the maximum temperature for fires in different sizes and at different fire spacings can be expressed as follows:

$$\Delta T_{\max } = \left[ {\left( {25.31 - 10.29D} \right)e^{ - 0.88S} + 9.21} \right]\dot{Q}^{2/3} /H_{ef}^{5/3}$$
(7)

As illustrated in Fig. 7, the numerical simulation results of the maximum temperature are compared to the highest temperature predicted by Eq. (7). According to the research results, the prediction model created in this article is able of accurately predicting the highest temperature for two fires with various sizes.

Fig. 7
A scatterplot depicts the change in temperature for four D values of 0.8, 1.2, 1.6, and 2.0 meters. They all have approximately similar ranges from 100 to 400 degrees. The values are approximate.

Comparison between model predictions and results from numerical simulation (Eq. (7))

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This article assumes a fire scenario when two vehicles collide at the tunnel longitudinal centerline. The model may not be applicable when the lateral position of the fire source changes. In addition, only the two-source fire scenario is considered in this paper, and further expansion of the fire will consider a multiple-source fire scenario. In future work, the effect of the number and location of fire sources on the maximum ceiling temperature could be considered.

4 Conclusions

Studying the highest temperature of double fires is crucial for the safety and design of tunnels. This study uses numerical simulations to investigate the impact of fire source sizes on the highest smoke temperature for double tunnel fires. The study takes into account various fire source heat release rates, different fire source sizes, and different fire source separation distances. These are the primary conclusions:

  1. (1)

    The maximum temperature gradually drops with increasing fire source size and decreases with increasing fire source separation while the fire source heat release rates are fixed.

  2. (2)

    Combined with previous research on maximum ceiling temperatures, a maximum temperature rise prediction model considering the fire source heat release rates, fire source sizes, and fire source separation distances was established. Furthermore, when compared with the prediction model, the study's results revealed a strong correlation. This study is of great significance for the parameter setting of temperature alarms in tunnels and the evacuation of personnel.