The Effect of Slope on Smoke Characteristics of Natural Ventilation Tunnel with Shafts

Abstract

Tunnels with natural ventilation and extraction have become the focus of ventilation research in recent years. It is significant to study the characteristics of smoke in tunnel fires to ensure the safety of people and the tunnel structure. Previous research has mainly focused on natural ventilation in horizontal tunnels, and there are few studies on sloped tunnels. In this paper, we studied the smoke characteristics of natural ventilation extraction in slope tunnel fires both experimentally and theoretically. The small-scale experimental results showed that the position of the fire source, heat release rate (HRR), and the size of the shaft had little effect on the deflection angle of the fire plume. The deflection angle of fire plume was only related to the tunnel slope and increased with the tunnel slope. The slope had no effect on the smoke temperature distribution on the downside of the tunnel, while the smoke temperature on the upside decreased with the increase in the slope. The calculation models of the maximum smoke temperature rise and the smoke temperature distribution were obtained based on the experimental results and theoretical analysis. Compared with the experimental data, the developed semi-empirical models could provide a reliable prediction of smoke temperature.

1. Introduction

Due to the narrow and long structure of tunnels, an effective ventilation and smoke extraction design is crucial. Taking the subway tunnel as an example, when the subway vehicle is operating normally in the tunnel, the subway traction system, the condenser, and the friction of the subway track will generate a lot of heat in the tunnel. A ventilation system is essential to dissipate the heat and provide enough fresh air for the passengers. Once a fire occurs, the high temperature and toxic smoke generated by combustion threaten the people in the tunnel and endanger the tunnel structure. Thus, the ventilation system is required to control the smoke flow as well as maintain the temperature and visibility in the tunnel within a safe range to ensure the safe evacuation of people [1].

At present, there are two main ways of ventilation and smoke extraction in subway tunnels. One is longitudinal ventilation and smoke extraction, which controls the smoke flow through fans, but it is not conducive to the escape of people from the downstream, and the operating cost of the fans is relatively high (Figure 1a). Another is natural ventilation and smoke extraction. In daily operation, the heat generated by moving trains raises the air temperature in the tunnel, forming a chimney effect in the shaft. Combined with the piston wind generated by the moving train, the ventilation between the tunnel and the ground environment is guaranteed, as shown in Figure 1b. When a fire breaks out, the high-temperature smoke generated by combustion will enhance the chimney effect, and the high-temperature smoke will be discharged out of the tunnel through the nearest shaft under thermal pressure, as shown in Figure 1c, which can ensure safe evacuation from the tunnel. Compared with the longitudinal ventilation and smoke extraction, natural ventilation and smoke extraction reduces the fan operation costs and has unique advantages for evacuation [2].

Some scholars have studied the effectiveness of natural ventilation. Mao et al. [3] conducted model tests on the Nanjing Chengdong Tunnel, analyzing the smoke temperature distribution and CO₂ concentration distribution. The results indicated that the safety of the evacuees could be guaranteed. Tong et al. [4] analyzed the effect of smoke extraction with shafts through full-scale experiments, which could effectively control the flow of smoke. The natural ventilation system in subway tunnels with shafts has been widely used [5,6,7]. Therefore, it is necessary to analyze the characteristics of smoke with natural ventilation tunnels, which is important for the design of tunnel ventilation systems and smoke management.

Many scholars have carried out research on the smoke characteristics in natural ventilation tunnels. Through model tests and numerical simulations, they studied the law of smoke diffusion and temperature distribution beneath the tunnel ceiling with varying shaft sizes, proposing mathematical models to predict the range of smoke diffusion and temperature distribution [6,7,8,9]. Takeuchi et al. [10] conducted experimental studies on the natural smoke exhaust effect in an underground highway tunnel with six shafts, and established a simplified model for smoke temperature distribution under the tunnel ceiling based on a theoretical analysis. Additional experiments focused on smoke diffusion in a natural ventilation model with four shafts were also conducted [11]. Several researchers have conducted experiments based on reduced-scale model tests with a scale ratio of 1/15 and conducted a systematic study on the smoke characteristics of natural ventilated tunnels with shafts including the extraction temperature, the smoke exhaust volume, and the smoke temperature below the tunnel ceiling [1,12,13]. In terms of the full-scale test, Wang et al. [14] and Wang et al. [15] measured the velocity and temperature of smoke gas and established a prediction model to calculate the smoke diffusion distance. The model’s accuracy was verified through numerical simulation.

For smoke control, scholars have proposed the concept of critical shaft size [16,17,18,19]. Zhang et al. [18] studied the road tunnel fires with natural ventilation through numerical simulations. They defined the minimum shaft height at which plug-holing occurs as the critical shaft height; the factors affecting the critical shaft height were studied, and a dimensionless critical shaft height prediction model was established. Zhao et al. [19] defined the critical shaft height as the lowest shaft height to ensure complete smoke extraction in the tunnel. Through experiments, they identified the most unfavorable fire source position in naturally ventilated tunnels and derived a predictive model for critical shaft height in the event of a fire. In addition, other scholars have carried out important investigations on natural smoke extraction with shafts [20,21].

Most of the above research has been carried out on horizontal tunnels, with limited studies on the natural ventilation of sloped tunnels. Yang et al. [22] investigated the length of smoke backlayering in sloped tunnels with natural ventilation using saltwater experiments. Wan et al. [2] analyzed the smoke flow characteristics of a tunnel fire with a slope, examining the impact of the slope on smoke temperature distribution within the tunnel and shaft. Moreover, they also studied the smoke backlayering length and the wind velocity at the tunnel entrance and analyzed the plug-holing effect of the shaft under different slopes. Based on the equations of continuity, energy, and pressure, researchers have developed simplified models and obtained the theoretical values of airflow velocity and temperature in single-slope tunnel and double-slope tunnel through simulation calculations [23]. Yi et al. [24] conducted full-scale numerical simulations to analyze the smoke distribution and inlet airflow velocity of the smoke gas under different tunnel shapes and slopes. They proposed a simple method for estimating the smoke temperature and velocity in inclined tunnel fires.

For horizontal tunnels, extensive research has been conducted on natural smoke extraction, but studies on slope tunnels with natural smoke exhaust are lacking. In practical engineering, most tunnels have a slope. In sloped tunnels, the height difference along the tunnel creates a chimney effect on the high-temperature smoke gas generated during a fire. Thus, the diffusion of smoke will be affected by the thermal pressure. Compared to horizontal tunnels, sloped tunnels experience changes in airflow velocity, smoke temperature distribution, the direction of the smoke diffusion, the distance of the diffusion, and the entrainment of the fire plume [2,24,25,26,27]. Therefore, the natural ventilation of slope tunnel fires needs to be further researched.

In this paper, a reduced-scale tunnel model with natural smoke extraction was constructed to investigate the characteristics of the smoke in slope tunnel fires including the shape of the fire plume, the temperature distribution of the smoke gas under the top wall, and the smoke exhaust characteristics of the shaft. This can provide a certain basis and reference for the smoke exhaust design of natural ventilation in slope tunnels.

2. Experimental Settings

2.1. Reduced-Scale Tunnel Model

According to Froude modeling, a 1:20 reduced-scale model was built for the experiments, and the relation between the reduced-scale model and full-scale model can be expressed as follows [28]:

$$\frac{Q_m}{Q_f}={\alpha }^{5/2}$$

(1)

$$T_m=T_f$$

(2)

$$\frac{V_m}{V_f}={\alpha }^{5/2}$$

(3)

The model system consists of a tunnel bracket, the main body of the tunnel, and the shafts. The tunnel bracket was made square, and the tunnel slope could be adjusted between 5% and 0. The model tunnel was 8 m long, 0.25 m wide, and 0.25 m high, and the full-scale tunnel was 160 m long, 5 m wide, and 5 m high. The inner side distance of the shaft was 3 m. The tunnel model was made of two kinds of materials: one of the side walls was made of 5 mm thick fireproof glass to observe the fire plume and smoke in the tunnel, and the rest of the walls were made of 10 mm fireproof plate, which could prevent the tunnel ceiling from burning down due to excessive temperature. To simplify the descriptions in the following sections, the zone between two shafts was defined as the fire section, and the other zone of the model tunnel was defined as the non-fire section. The model is shown in Figure 2.

In the experiment, the mass loss rate of the fuel during the combustion process was measured by an electronic balance. The placement of the electronic balance is shown in Figure 3. The distance between the bottom of the fuel pan and the bottom of the tunnel was 2 cm.

2.2. Temperature System

In order to measure the smoke temperature beneath the tunnel ceiling, thermocouples were arranged along the longitudinal centerline of the ceiling between two shafts with an interval of 10 cm and 1 cm below the top wall. At the same time, six thermocouples were arranged behind the upstream shaft, and nine thermocouples were arranged behind the downstream shaft of the non-fire source sections. The schematic diagram of the thermocouple layout is shown in Figure 4. K-type thermocouples used in the experiment had a diameter of 1 mm and a temperature measurement range of 0–1300 °C; two thermocouples were arranged in the environment on both sides of the tunnel to measure the ambient temperature. The temperature data were collected every 5 s, and the final data used for the analysis were the average value of the steady stage.

2.3. Heat Release Rate

Methanol was used as the fuel in the experiment. Methanol can well simulate the combustion state of combustibles in real tunnels. The HRR can be calculated by the following equation:

$$Q=\chi m_f\Delta H$$

(4)

where χ is the combustion efficiency, m_f is the mass loss rate, and ΔH is the heat of combustion.

According to Tewarson [29], the combustion efficiency χ of methanol can be considered as 1, and ΔH is 19.93 kJ/g. The mass loss rate m_f was measured by the electrical balance, and the average mass loss rate at the steady period was adopted to calculate the HRR. Figure 5 shows the relationship between the fuel mass and time in a 5.5 cm fuel pan when the tunnel slope was 4% with the fire source located at the center, and the length of the shaft was 0.1 m and the height was 0.1 m. It can be seen from Figure 5 that the slope of the curve was stable between 400 s and 600 s; this period can be considered as the stable period of methanol combustion, and the average in this period was used to calculate the HRR. Smoke incense was placed near the fire, and a laser light was adopted to visualize the smoke flow [30], as shown in Figure 6. In this paper, we used three different sizes of square fuel pans with dimensions of 5.5 cm, 7.5 cm, and 9.5 cm, and the height of the fuel pans was 2 cm. The heat release rate of the three fuel pans was calculated as 0.935, 1.599, 2.733, respectively, by Equation (4).

2.4. Experimental Scenarios

Nine different fire source locations were selected on the central axis of the tunnel in the experiment (Figure 7). Position A is the longitudinal center of the tunnel, position E is just below the inner wall of the shaft, position B is the midpoint between position A and position E, position C is the midpoint between position B and position E, and position D is the position where the outsider of the fire source close to the inner wall of the shaft, which can be adjusted according to the heat release rate of the fire source. As the slope tunnel was different from the horizontal tunnel and had no symmetry, the positions B’, C’, D’, and E’ were selected at the positions where positions B, C, D, and E were symmetrical with respect to position A. All fire sources were located on the central axis of the tunnel. Some tests are shown in

Table 1. Summary of the experimental conditions.

Slope	HRR/kW	Shaft Length/m	Shaft Height/m	Shaft Interval/m	Fire Source Location
5%	0.935	0.1	0.13, 0.16	3	A
5%	0.935, 1.599, 2.733	0.1	0.1	3	A
5%	0.935, 1.599, 2.733	0.1	0.1	3	B
5%	0.935, 1.599, 2.733	0.1	0.1	3	C
5%	0.935, 1.599, 2.733	0.1	0.1	3	D
5%	0.935, 1.599, 2.733	0.1	0.1	3	E
5%	0.935, 1.599, 2.733	0.1	0.1	3	B’
5%	0.935, 1.599, 2.733	0.1	0.1	3	C’
5%	0.935, 1.599, 2.733	0.1	0.1	3	D’
5%	0.935, 1.599, 2.733	0.1	0.1	3	E’
4%	0.935, 1.599, 2.733	0.1	0.1	3	A
4%	0.935, 1.599, 2.733	0.1	0.1	3	B
4%	0.935, 1.599, 2.733	0.1	0.1	3	D
4%	0.935	0.1	0.17, 0.175, 0.18	3	D
4%	0.935	0.06, 0.07, 0.08	0.15	3	B
2%	0.935	0.1, 0.13, 0.16	0.14	3	D
2%	0.935, 1.599, 2.733	0.1	0.1	3	A
2%	0.935, 1.599, 2.733	0.1	0.1	3	B
2%	0.935, 1.599, 2.733	0.1	0.1	3	D
0%	0.935, 1.599, 2.733	0.1	0.1	3	A
0%	0.935, 1.599, 2.733	0.1	0.1	3	B
0%	0.935, 1.599, 2.733	0.1	0.1	3	D

.

3. Results and Discussion

3.1. Diffusion of Fire Smoke in Tunnel

3.1.1. Fire Plume

Figure 8 shows the shapes of the fire plume in the horizontal tunnel and 5% slope tunnel when the heat release rate was 0.935 kW. In the horizontal tunnel, the air entrainment on both sides of the fire was the same, so the fire plume was vertically upward and symmetrically distributed about the center of the fire (Figure 8a).

In the 5% slope tunnel, the fire plume deflected to the upside because of the influence of the chimney effect caused by the height difference along the tunnel, and the position of the smoke plume impacts on the ceiling shifted to the upside (Figure 8b). At the same time, the plume had a longer rising section before hitting the tunnel ceiling and was disturbed more severely, so more fresh air was sucked into the plume.

3.1.2. Deflection Angle of Fire Plume

The definition of the deflection angle of the fire plume is shown in Figure 9a, where the angle between the fire plume and the horizontal plane is considered as θ. The fire plume was recorded by the camera during the combustion process, and the angle was measured when the fire plume shape was stable. Figure 9b shows the result of angle θ when the fire source located at position B and HRR = 1.599 kW with a 5% slope.

Table 2. Results of angle θ.

No.	Slope	Fire Source Location	HRR/kW	Shaft Length/m	Shaft Height/m	Angle θ/°
1	5%	A	0.935	0.1	0.1	59.79
2		A	0.935	0.1	0.13	57.57
3		A	0.935	0.1	0.16	61.28
4		A	1.599	0.1	0.1	58.18
5		A	2.733	0.1	0.1	57.05
6		B	0.935	0.1	0.1	59.63
7		B	1.599	0.1	0.1	57.1
8		B	2.733	0.1	0.1	60.65
9		D	0.935	0.1	0.1	57.07
10		D	1.599	0.1	0.1	58.75
11	4%	A	0.935	0.1	0.1	64.5
12		A	1.599	0.1	0.1	65.97
13		A	2.733	0.1	0.1	64.61
14		D	0.935	0.1	0.1	66.02
15		D	2.733	0.1	0.1	65.65
16		D	0.935	0.1	0.17	66.98
17		D	0.935	0.1	0.18	64.36
18		D	0.935	0.1	0.175	67.35
19		B	0.935	0.08	0.15	67.72
20		B	0.935	0.07	0.15	65.59
21		B	0.935	0.06	0.15	65.98
22	2%	D	0.935	0.13	0.14	74.02
23		D	0.935	0.1	0.14	74.57
24		D	0.935	0.06	0.14	76.14

shows the results of angle θ under different experimental conditions. From the test results, it can be concluded that:

(1)

Cases 1–15 show that the HRR has little effect on the deflection angle of fire plume for the same location of the fire source.

(2)

Cases 4–15 show that the location of the fire source has little effect on the deflection angle of the fire plume, which indicates that different longitudinal fire source positions have little effect on the plume entrainment.

(3)

Cases 1–3 and 16–24 show that the change in the height and length of the shaft has little effect on the entrainment of the fire plume, and the deflection angle of the fire plume hardly changed.

It can be concluded that the deflection angle of the fire plume has nothing to do with the position of the longitudinal fire source, HRR, and the size of the shaft. The main factor affecting the deflection angle of the fire plume is the slope of the tunnel.

3.2. Effect of Slope on Angle θ of Fire Plume

Under the same tunnel slope, the change trend of angle θ was small, and the average value was taken for analysis. The diagram in Figure 10 illustrates tunnel angle θ at a 5% slope. When the slope was 5%, 4%, and 2%, the average values of the angles between the fire plume and the horizontal plane were 59.38°, 65.96°, and 74.58°, respectively.

Figure 11 illustrates the relationship between angle θ and the slope. It can be seen that when the slope was not 0, the angle θ decreased with the increase in the slope, and the deflection angle of the fire plume increased with the slope, which was mainly due to the thermal pressure of the high temperature smoke and the height difference along the tunnel caused by the slope. There was a linear relationship between angle θ and the tunnel slope, which can be expressed as Equation (5). Since the minimum slope in this experiment was 2%, when the slope is smaller, the disturbance to the fire plume is reduced, and whether Equation (5) is applicable in the lower slope tunnel needs further verification.

$$\theta =88.77-5.92\beta$$

(5)

3.3. Maximum Smoke Temperature Beneath the Ceiling

In the following section n, we adopted ΔT to analyze the difference between smoke temperature and ambient temperature. Figure 12 shows the relationship between the maximum temperature rise of smoke (ΔT_max) and the slope under different heat release rates when the shaft was 0.1 m long and 0.1 m high. The ΔT_max increased with the increasing HRR and decreased with the increase in the slope.

The reason for this is that the increasing slope results in a longer length of the fire plume, and more heat will be lost through radiation heat transfer. Moreover, as the slope increases, the fire plume is more disturbed, and more smoke mass flow is formed at a constant heat release rate, so the maximum smoke temperature beneath the tunnel decreases with the increase in the slope. At the same time, with the increase in the heat release rate, the length of the fire plume increases and the heat loss increases. When the slope of the tunnel is larger, the heat loss increases and the maximum smoke temperature decreases. Air entrainment is an important reason for the decrease in smoke temperature [31].

The effect of slope on smoke temperature can be equivalent to the effect of longitudinal ventilation on a horizontal tunnel fire, and the equivalent wind velocity is defined as the induced wind velocity V_slope [27]. In a longitudinal ventilation tunnel, the ΔT_max of a small fire can be expressed by Equation (6) [32].

$$\Delta T_{\max }=\left\{\begin{array}{c}\frac{2.68C_T{\left(1-{\chi }_r\right)}^{2/3}{g}^{1/3}}{{\left({\rho }_0{c}_p{T}_0\right)}^{1/3}}\frac{Q}{{u{b}_f}^{1/3}{H}^{5/3}},V^{*}>0.19\\ 14.1C_T{\left(1-{\chi }_r\right)}^{2/3}{\left(\frac{Q}{H^{5/2}}\right)}^{2/3},V^{*}\le 0.19\end{array}\right.$$

(6)

When the dimensionless ventilation velocity is small (V* ≤ 0.19), the maximum smoke temperature rise has a 2/3 power relationship with the HRR, and the longitudinal ventilation velocity has little disturbance on the fire plume, so the fire plume does not deflect, and the maximum smoke temperature rise is independent of the velocity. When the dimensionless velocity is larger than 0.19, the ΔT_max decreases with the increasing longitudinal ventilation velocity and increases linearly with the increasing HRR.

In our experiments, when the slope β ≥ 2%, the fire plume had a deflection angle, so the induced wind velocity in this experiment can be considered V* > 0.19.

Raj et al. [33] studied the pool fire in the ventilated tunnel and concluded that the relationship between the flame deflection angle θ and the dimensionless longitudinal ventilation velocity V* can be expressed as Equation (7).

$$\sin \theta =\left\{\begin{array}{c}1,V^{*}\le 0.19\\ {\left(5.26V^{*}\right)}^{-1/2},V^{*}>0.19\end{array}\right.$$

(7)

where $V^{*}=\frac{u}{{\omega }^{*}}$, ${\omega }^{*}={\left(\frac{Qg}{b_f{\rho }_0{c}_p{T}_0}\right)}^{1/3}$.

Combing Equations (5)–(7), we finally have:

$$\Delta T_{\max }=14.0968C_T{\left(1-{\chi }_r\right)}^{2/3}\frac{{\left(\sin \left(\frac{84.827-4.959\beta }{180}\pi \right)Q^{1/3}\right)}^2}{H^{5/3}}$$

(8)

where $\sin \left(\frac{84.827-4.959\beta }{180}\pi \right)$ is the sinθ.

Data analysis was performed based on Equation (8), and the results can be seen in Figure 13. It can be seen that the $\Delta T_{\max }$ and ${\left(\sin \theta Q^{1/3}\right)}^2/H^{5/3}$ are linearly related, so the maximum smoke temperature rise under the tunnel ceiling of the sloped tunnel can be expressed as Equation (9).

$$\Delta T_{\max }=\frac{24.94{\left(\sin \left(\frac{84.827-4.959\beta }{180}\pi \right)Q^{1/3}\right)}^2}{H^{5/3}}$$

(9)

3.4. Smoke Temperature Distribution

From previous research, tunnel slope significantly influences the deflection angle of the fire plume due to its impact on the chimney effect. Figure 14 shows the temperature distribution of smoke at various slopes when the fire source is located at the longitudinal center. The experimental results indicate that:

(1)

Near the fire source (within 0.15 m): The tunnel slope significantly impacts the smoke temperature beneath the ceiling. In this range, the smoke temperature is mainly influenced by the fire plume and the larger slopes causing a stronger disturbance of the fire plume, resulting in a decrease in smoke temperature as the slope increases.

(2)

Far from the fire source: As the smoke spreads, the smoke temperature distribution on the upside does not change much with the slope.

(3)

Downside of the tunnel: The smoke temperature increases gradually with the decrease in the tunnel slope. When the tunnel slope is 0, the temperature on the downside and upside is symmetrically distributed on both the downside and upside, with the fire source at the center.

For a sloped tunnel, the temperature decay below the ceiling can be expressed as [34]:

$$\frac{\Delta T}{\Delta T_{ref}}=e^{-kks(x-x_{ref})}$$

(10)

The distance from the fire source can be non-dimensionalized to $\left(x-x_{ref}\right)/H$, so Equation (11) becomes the following equation:

$$\frac{\Delta T}{\Delta T_{ref}}=e^{\frac{-kks(x-x_{ref})}{H}}$$

(11)

The dimensionless temperature rise $\Delta {T_{ref}}^{*}$ at the reference point can be expressed as [1]:

$$\Delta {T_{ref}}^{*}=\frac{\Delta T_{ref}}{Q^{*2/3}{T}_0}$$

(12)

where $Q^{*}$ is given by

$$Q^{*}=\frac{Q}{{\rho }_0{c}_p{T}_0{g}^{1/2}{H}^{5/2}}$$

(13)

It can be seen from Figure 14 that the position near the fifth thermocouple on the upside and the position near the fourth thermocouple on the downside are the positions where the smoke diffusion is completely converted into one-dimensional diffusion. Therefore, the thermocouple (0.45 m from the fire source) was selected as the reference point. Taking the 5% slope tunnel as an example for data analysis, Figure 15 shows that there is a linear relationship between $\Delta T_{ref}$ and $Q^{*2/3}/T_0$. We can conclude that the dimensionless smoke temperature on the upside is 3.059 ($\Delta {T_{ref}}^{*}$ = 3.059) and 2.271 on the downside.

Combining Equations (11)–(13), we obtain:

$$\Delta T=\Delta {T_{ref}}^{*}{Q}^{*2/3}{T}_0{e}^{-k{k}_s\frac{\left(x-x_{ref}\right)}{H}}$$

(14)

According to Equation (14), the temperature distribution of smoke gas beneath the ceiling is shown in Figure 16. The attenuation coefficient on the upside is $k{k}_s=0.123$, and $k{k}_s=0.273$ on the downside. The same method was used for data processing for the 0%, 2%, 4%, and 5% slope tunnels, and the results are shown in

Table 3. Fitting results of the slope tunnel.

Slope/%	$\mathbf{Upside}\Delta {\mathit{T}}_{\mathit{r}\mathit{e}\mathit{f}}^{*}$	$\mathbf{Downside}\Delta {\mathit{T}}_{\mathit{r}\mathit{e}\mathit{f}}^{*}$	$\mathbf{Upside}\mathit{k}{\mathit{k}}_{\mathit{s}}$	$\mathbf{Downside}\mathit{k}{\mathit{k}}_{\mathit{s}}$
5	3.059	2.271	0.123	0.273
4	3.033	2.458	0.129	0.234
2	2.992	2.805	0.141	0.180
0	2.973	2.973	0.131	0.131

.

The attenuation coefficient on the upside remained stable with the change in the tunnel slope, so we adopted the average of all slopes as the attenuation coefficient of the smoke temperature of all slopes, that is, $k{k}_s=0.131$. The reason is that the dimensionless smoke temperature of the reference point on the downside increases as the tunnel slope decreases, so the attenuation coefficient decreases with the decrease in the slope. When the slope decreases to 0, it is equal to that on the upside. The reason is that when the tunnel slope is small, more smoke flows to the downside, and the smoke flowing down the slope side is disturbed and weakened by the flow from the lower layer of the tunnel.

According to the experimental results, the smoke temperature rise beneath the tunnel ceiling of natural smoke exhaust in horizontal tunnels can be expressed as Equation (15).

$$\Delta T=2.973Q^{*2/3}{T}_0{e}^{-0.131\left(\frac{x-x_{ref}}{H}\right)}$$

(15)

Combined with the temperature attenuation coefficient when the tunnel has no slope, the attenuation coefficient k_s of the sloped tunnel can be obtained. Figure 17 shows that the coefficient k_s on the downside of a sloped tunnel is linearly related to the tunnel slope.

The relationship between $\Delta T_{refs}^{*}/\Delta T_{ref0}^{*}$ (which is the ratio of dimensionless smoke temperature at the reference point of slope tunnel to that in horizontal tunnels) and the tunnel slope is shown in Figure 18.

The smoke temperature rise beneath the tunnel ceiling of the fire source section in a slope tunnel fire with natural smoke extraction can be expressed as Equation (16), and the slope is β.

$$\Delta T=\left\{\begin{array}{cc}(0.991+0.008\beta )2.973Q^{*2/3}{T}_0{e}^{-0.131\frac{(x-x_{ref})}{H}}& ,upside\\ (1.068-0.006\beta )2.973Q^{*2/3}{T}_0{e}^{-(0.896+0.232\beta )0.131\frac{(x-x_{ref})}{H}}& ,downside\end{array}\right.$$

(16)

Figure 19 shows the comparison between the calculated results of Equation (16) and the experimental results. It can be seen from Figure 10 that the calculated value was in good agreement with the experimental value, and that the smoke temperature rise below the ceiling of the fire source section in slope tunnels can be predicted by Equation (16).

4. Conclusions

This paper investigated the smoke characteristics of natural ventilation in slope tunnels through reduced-scale experiments. The main conclusions are as follows:

(1)

In slope tunnels with natural smoke extraction, the fire plume deflects to the upside when a fire occurs.

(2)

For slope tunnels with natural smoke extraction, the deflection angle of the fire plume is not affected by the longitudinal fire source position.

The deflection angle of the fire plume is only related to the slope of the tunnel, increases linearly with the tunnel slope, and is unaffected by the size of the shafts or the HRR. The deflection angle of the fire plume can be predicted by Equation (5).

(3)

The maximum smoke temperature rise decreases with the increase in the slope, so $\Delta T_{\max }$ and ${\left(\sin \theta Q^{1/3}\right)}^2/H^{5/3}$ are linearly related. This can be predicted by Equation (10).

(4)

Within 0.15 m of the fire source, the temperature of the smoke decreases with increasing slope. While in the area far from the fire source (the distance from the fire source was greater than 0.15 m), the slope had little effect on the smoke temperature distribution in the upside, but increased with the decrease in the slope on the downside.