1 Introduction

Due to the limitation of ground space, various forms of one-end blocking tunnels have gradually appeared in the planning, design and construction of mountainous cities, which is different from the common open tunnels at both ends. Due to the existence of the blocking end, the flue gas is easier to gather and is not easy to discharge out of the tunnel, and the flue gas flow law of such one-end blocking tunnels in case of fire is less involved. Putting forward a suitable ventilation and smoke exhaust design scheme for one end blocked tunnel has become an important solution to the difficulty of smoke control.

The research shows that compared with the top opening natural ventilation and smoke exhaust mode, the shaft natural ventilation has stronger chimney effect, which can produce greater pressure difference and obtain better smoke exhaust effect. Its smoke exhaust effect has been verified by a large number of experimental data and examples [1,2,3,4]. At the same time, the influence of shaft length on flue gas flow under natural ventilation is carried out, and the theoretical model of critical shaft length based on two open tunnels under different conditions is obtained [5,6,7,8,9]. However, the above research is mainly based on the open tunnel with shaft at both ends. Whether the research conclusion is applicable to the tunnel blocked at one end needs further research.

FDS numerical simulation method is used to study the critical shaft length of one end blocking tunnel when the shaft width is constant, the shaft in the tunnel is rectangular, and the shaft width is equal to the tunnel width. The smoke spread length with different parameters such as fire source location, fire source heat release rate and shaft height is discussed, and then the critical shaft length is analyzed.

2 Critical Shaft Length

The concept of critical shaft length was proposed in the study of the influence of open tunnel shafts on flue gas flow. The so-called critical shaft length usually refers to the minimum shaft length to ensure that the flue gas generated by the fire source and spread to the shaft is completely discharged from the shaft and will not spread to the rear of the shaft. In the blocked tunnel at one end, when the section of the shaft is too small or the height of the shaft is too low, the buoyancy generated by the pressure difference inside and outside the shaft is small, and the chimney effect is limited. A large amount of flue gas will spread through the shaft and continue to gather in the tunnel area behind the shaft, so the flue gas cannot be discharged through the shaft, which is not conducive to personnel evacuation. In order to make the shaft discharge flue gas effectively and reduce the spread distance of flue gas to the greatest extent, the critical shaft length of one end blocking tunnel is studied.

Fig. 1.
figure 1

Schematic diagram of the smoke exhaust effect and the length of the shaft in the tunnel with one closed portal (a) Xf < Xs (The fire source is located between the plugging end and the shaft); (b) Xf > Xs (The fire source is located between the shaft and the open end).

Assuming that only one shaft is set in the blocked tunnel at one end and close to the blocked end, \({L}_{c}\) is the critical shaft length. The relationship between smoke exhaust effect of tunnel shaft blocked at one end and shaft length is shown in Fig. 1. It can be seen from Fig. 1 (a) that when the fire source is located between the blocking end and the shaft, in order to meet the conditions of full exhaust of flue gas in the critical shaft length, the shaft needs to continuously discharge all flue gas generated by the fire source to ensure that the tunnel area behind the shaft is a smoke-free area. It can be seen from Fig. 1 (b) that when the fire source is located between the open end and the shaft, the appropriate critical shaft length can discharge all the flue gas spreading to the blocking end, avoiding the accumulation of flue gas at the blocking end, so that the flue gas in the tunnel will not settle for a long time and ensure the safety of personnel in the tunnel.

3 FDS Numerical Simulation

3.1 Validation of FDS Numerical Simulation

Before using FDS for numerical simulation, the experimental model is established according to the small-scale experimental conditions, and the accuracy of FDS is demonstrated by comparing the small-scale experimental results with the FDS numerical simulation results. Taking a blocked tunnel at one end of a rail transit in Chongqing as the prototype, a 1:15 small-scale model test-bed was built with Froude similarity model rate [10]. The small-scale test-bed is a tunnel blocked at one end, in which the tunnel is 5.0 m long, 0.32 m wide and 0.48 m high. The model uses K-type armored thermocouples to measure the tunnel temperature. 38 thermocouples are arranged on the longitudinal centerline in the tunnel. The spacing above and near the fire source is 0.1m, and the spacing of the far fire source is 0.2 m. Using ethanol as fuel, oil pans of different sizes are designed and calculated according to the fuel loss rate. The heat release rate of four ethanol oil pools was designed, which was 8 × 8,10 × 10,12 × 12,15 × 15. According to the above experimental conditions, the tunnel geometry, fire source fuel and tunnel material settings established by FDS are consistent with the experiment. The comparison between small-scale test and FDS simulation results is shown in Fig. 2.

Fig. 2.
figure 2

Comparison of small-scale experimental data and FDS numerical simulation result (a) Temperature distribution in longitudinal position of tunnel roof; (b) Temperature distribution of thermocouple tree 0.5 m away from tunnel opening.

As can be seen from Fig. 2 (a), for the longitudinal temperature distribution of the tunnel roof, the FDS numerical simulation results are slightly smaller than the experimental results near the fire source, while there is little difference between the experimental results and FDS numerical simulation results in the far fire source area. There are individual experimental data measurement points that fluctuate. In general, FDS numerical simulation can better reflect the experimental results.

3.2 Grid Independence Test

Before FDS numerical simulation, grid independence and reliability verification shall be carried out. The FDS user manual [11] provides a grid division criterion for users. The grid size can be determined by the calculated value of \({D}^{*}/\delta x\), \(\delta x\) is the grid size and \({D}^{*}\) is the characteristic diameter of fire source. The calculation formula is as follows:

$$ D^* = \left( {\frac{Q}{{\rho_a c_p T_a g^{{\raise0.7ex\hbox{$1$} \!\mathord{\left/ {\vphantom {1 2}}\right.\kern-\nulldelimiterspace}\!\lower0.7ex\hbox{$2$}}} }}} \right)^\frac{2}{3} $$
(1)

At present, it is generally believed that when the calculated value of \({D}^{*}/\delta x\) is between 4–16, the numerical simulation will have a better result. The fire source power used in this paper is 2.5 MW, 5 MW and 7.5 mw. Through calculation, it can be obtained that the more reasonable grid size should be between 0.114 m–0.458 m. Four grids with different scales of 0.100 m, 0.125 m, 0.160 m and 0.200 m are selected to calculate the same fire condition. Under different grid sizes, the longitudinal temperature distribution diagram of tunnel ceiling blocked at one end is shown in Fig. 3.

It can be seen from Fig. 3 that the calculation results of the four grid sizes have little difference. The maximum temperature at the fire source location of the grid of 0.2 m is low, while the grid of 0.160 m is close to the maximum temperature of 0.100 m and 0.125 m, and the temperature in the far fire source area of the four grids has little difference. In this simulation, 0.160 m is uniformly used as the simulated grid size.

Fig. 3.
figure 3

Grid independence analysis

3.3 FDS Simulation Scheme

The heat release rate of fire source is 2.5 MW, 5.0 mw and 7.5 mw respectively. Four fire source positions are selected. The shaft center is fixed at \({X}_{s}=10\) m. The shaft length is taken according to the shaft height (1.2 m–2.2 m, with an interval of 0.2 m). The specific simulation arrangement is shown in Table 1.

Table 1. Simulation scheme setting

When the height of the shaft is 5 m, the heat release rate of the fire source is 5 MW and the fire source is located at \({X}_{f}=5\) m, the temperature distribution of flue gas in the roof of one end blocked tunnel under different shaft lengths is shown in Fig. 5. When the difference between the tunnel ceiling temperature and the ambient temperature is less than 5 \(\mathrm{^\circ{\rm C} }\), it can be approximately considered that this area is a smoke-free area. Since the tunnel ceiling temperature measuring points are all at the ambient temperature of 20 \(\mathrm{^\circ{\rm C} }\) after 50 m, in order to better compare the smoke propagation length under different shaft lengths, this figure only shows that the tunnel length is 50 m. It can be seen from Fig. 5 that under the working conditions of different shaft lengths, the temperature of each tunnel has a sudden drop at x = 10 m, but the cooling range is different. Obviously, the longer the shaft length, the greater the cooling range. When the temperature of flue gas directly drops to the ambient temperature after crossing the shaft, the shaft length at this time can be considered as the critical shaft length. The change of shaft length will not have a great impact on the tunnel ceiling flue gas temperature between the blocking end and the shaft. From the shaft to the open end, the flue gas propagation length is significantly shortened with the increase of shaft length. When the shaft length L = 1.6 m is increased to 1.8 m and 2.0 m, the smoke spread length is shortened from 25 m to 18 m and 12 m. When the shaft length L = 2.2 m, the smoke spread length is shortened to 6 m, which is approximately equal to the distance from the shaft to the fire source. There is no smoke spread in the tunnel area behind the shaft, and the tunnel behind the shaft is a smoke-free area. Therefore, when the shaft height \({H}_{s}=5\) m, the fire source heat release rate is 5 MW and the fire source is located at\( {X}_{f}=0.5\) m, the critical shaft length \({L}_{c}=2.2\) m.

According to the calculation method shown in Fig. 4, the critical shaft length under different fire source heat release rates and different shaft height conditions is counted, as shown in Table 2.

Fig. 4.
figure 4

The tunnel ceiling’s smoke temperature distribution under different shaft lengths.

Table 2. The critical shaft length at the fire source location X = 5 m.

4 Analysis of Simulation Results

4.1 Influence of Fire Source Location

The distribution diagram of flue gas temperature at one end of the blocked tunnel roof at four different fire source locations is shown in Fig. 6, in which the shaft height \({\mathrm{H}}_{\mathrm{s}}=10\mathrm{ m}\), the shaft length L = 1.2 m, and the heat release rate HRR of the fire source is 5 MW. It can be seen from Fig. 6 that fire source position 1 is located between the blocking end and the shaft, and the maximum temperature of flue gas in the tunnel ceiling blocked at one end is significantly higher than that in the other three fire source positions. The conclusion shows that when the fire source is close to the plugging end and the fire source is located between the plugging end and the shaft, the flue gas generated by the fire source will continue to gather at the plugging end, resulting in a significantly higher maximum temperature. When the fire source is located between the shaft and the open end, the fire source will deflect violently to the plugging end. The continuous consumption of air at the plugging end and the continuous discharge of flue gas from the shaft, while the continuous input of air at the open end leads to the low side pressure at the plugging end of the fire source, resulting in the imbalance of pressure difference at both ends of the fire source and severe deflection. When the fire source is located between the plugging end and the shaft, in order to achieve the effect of full exhaust of shaft flue gas at the critical shaft length, the shaft needs to discharge all flue gas generated by the combustion of the fire source. When the fire source is located between the shaft and the open end, the value of the critical shaft length only needs to discharge about half of the flue gas generated by the combustion of the fire source,then \({l}_{{x}_{f}=5}\ge {l}_{Other fire source locations}\). It can be seen from Fig. 5 that the shaft length L = 1.2 m, the temperature of the flue gas generated by the combustion of the three fire sources at fire source locations 2, 3 and 4 drops sharply below 25 \(\mathrm{^\circ{\rm C} }\) after crossing the shaft, while the flue gas generated by the combustion of fire source location 1 continues to spread to x = 15 m after crossing the shaft, which also proves that the critical shaft length of fire source locations 2, 3 and 4 is less than or equal to 1.2 m. The critical shaft length of fire source position 1 is the largest. Among the four fire source positions, the next section will study the change law of critical shaft length when the heat release rate of fire source changes for fire source position 1 with \(\mathrm{x}=5\mathrm{ m}\).

Fig. 5.
figure 5

The tunnel ceiling’s smoke temperature distribution under different fire locations

4.2 Effect of Fire Source on Heat Release Rate

The flue gas temperature distribution of one end blocked tunnel ceiling under different fire source heat release rates is shown in Fig. 6. When the parameters of the shaft are the same, there is a large difference in the temperature of flue gas produced by fire sources with different fire source heat release rates before crossing the shaft. The greater the fire source heat release rate, the higher the flue gas temperature of the tunnel ceiling; However, after the flue gas passes through the shaft, the flue gas temperature of the heat release rate of the three fire sources decreases sharply, but the flue gas propagation length of the three fire sources is almost the same. Comparing the three shaft lengths, it is found that with the increase of shaft length, the smoke spread length of the heat release rate of the three fire sources decreases to the same extent, and when the shaft length L = 2.2M, the smoke spread length of the three fire sources decreases to 6 m, which is approximately equal to the distance between the fire source and the shaft. Therefore, it can be considered that under the conditions of FDS simulation in this section, the critical shaft length of the three fire source heat release rates is equal, and the critical shaft length is almost independent of the fire source heat release rate, which is the same as the previous research results [12]. This shows that Heskestad and Yuan’s theory is also applicable to blocking the tunnel at one end. Since the fire plume and the smoke exhaust mass flow of the shaft are directly proportional to the 1/3 power of the fire source heat release rate, the value of the critical shaft length has nothing to do with the fire source heat release rate.

4.3 Theoretical Model of Critical Shaft Length

According to the above analysis, the fire source location at \({X}_{f}=5\) m. \({X}_{f}\) is a relatively unfavorable situation in the study, and its critical shaft length is the largest, which can meet the conditions for the complete discharge of flue gas from other fire source locations to the shaft along the shaft. The following will mainly analyze and summarize the critical shaft length model at \({X}_{f}=5\) m fire source location. In case of fire at the fire source, the flue gas generated by the fire source will continue to spread to both sides of the tunnel after reaching the tunnel ceiling, and part of the flue gas will be discharged along the shaft (\(\mathrm{L}<{\mathrm{L}}_{\mathrm{c}}\)) and take away the heat in the tunnel. The longer the shaft length, the more flue gas will be discharged along the shaft. At this time, the less flue gas will spread to the tunnel area behind the shaft, the lower the tunnel ceiling temperature, and the shorter the flue gas spread distance, which will provide a safer evacuation environment for the trapped people until the shaft length increases to completely discharge the flue gas generated by fire source combustion (Critical shaft length \(\mathrm{L}={\mathrm{L}}_{\mathrm{c}}\)). If the length of the shaft is \(\mathrm{L}>{\mathrm{L}}_{\mathrm{c}}\), there will be a section of the shaft that is completely smoke-free, which will not only increase the investment cost, but also cause a serious suction through phenomenon in the shaft. Therefore, we should try our best to avoid the situation of \(\mathrm{L}>{\mathrm{L}}_{\mathrm{c}}\).

Fig. 6.
figure 6

The tunnel ceiling’s smoke temperature distribution under different fire source heat release rates (a) L  =  1.8 m; (b) L  =  2.0 m; (c) L  =  2.2 m.

According to heskestad’s theory [13], the plume mass flow (M0) generated by fire source combustion can be expressed by Eq. (2);

$$ m_0 \propto \rho_0 g^{{\raise0.7ex\hbox{$1$} \!\mathord{\left/ {\vphantom {1 2}}\right.\kern-\nulldelimiterspace}\!\lower0.7ex\hbox{$2$}}} Q^{*{\raise0.7ex\hbox{$1$} \!\mathord{\left/ {\vphantom {1 3}}\right.\kern-\nulldelimiterspace}\!\lower0.7ex\hbox{$3$}}} H^{{\raise0.7ex\hbox{$5$} \!\mathord{\left/ {\vphantom {5 2}}\right.\kern-\nulldelimiterspace}\!\lower0.7ex\hbox{$2$}}} $$
(2)

Where: Q is the heat release rate of fire source, HRR, kW; \({Q}^{*}\) is the heat release rate of dimensionless fire source; H is the height of the tunnel, m.

According to Yuan’s theory [7], the mass flow at the boundary of the control body (\({m}_{1}\)), the mass flow of shaft smoke exhaust (\({m}_{s}\)), and the temperature rise of flue gas \(\Delta {T}_{1}/{T}_{0}\) meet the following laws:

$$ m_1 = m_{ref}^* \rho_0 g^{{\raise0.7ex\hbox{$1$} \!\mathord{\left/ {\vphantom {1 2}}\right.\kern-\nulldelimiterspace}\!\lower0.7ex\hbox{$2$}}} Q^{*{\raise0.7ex\hbox{$1$} \!\mathord{\left/ {\vphantom {1 3}}\right.\kern-\nulldelimiterspace}\!\lower0.7ex\hbox{$3$}}} H^{{\raise0.7ex\hbox{$5$} \!\mathord{\left/ {\vphantom {5 2}}\right.\kern-\nulldelimiterspace}\!\lower0.7ex\hbox{$2$}}} $$
(3)
$$ m_s = \left( {2m_1 l^2 hw^2 \rho^2 g\varepsilon^{ - 1} \frac{\Delta T_1 }{{T_0 }}} \right)^{{\raise0.7ex\hbox{$1$} \!\mathord{\left/ {\vphantom {1 3}}\right.\kern-\nulldelimiterspace}\!\lower0.7ex\hbox{$3$}}} $$
(4)
$$ \Delta T_1 /T_0 = \Delta T_{ref}^{*} Q^{{*}{\raise0.7ex\hbox{$2$} \!\mathord{\left/ {\vphantom {2 3}}\right.\kern-\nulldelimiterspace}\!\lower0.7ex\hbox{$3$}}} exp\left[ { - k\left( {s - X_{ref} } \right)} \right] $$
(5)

Where: \(l\) is the shaft length, \(h\) is the shaft height, \(w\) is the shaft width, \(\varepsilon \) is a fixed constant, \(ref\) is the reference point position of the fire source section, \({X}_{ref}\) is the reference point position coordinate, s is the coordinate of the shaft center, and k is the ceiling temperature attenuation coefficient of the fire source section.

For the tunnel blocked at one end, when the flue gas is completely discharged along the shaft, the relationship between \({m}_{0}\), \( {m}_{1}\) and \({m}_{s}\) can be expressed as the following formula (6):

$$ m_0 = m_1 = m_s $$
(6)

Combining the above formula, formula (7) can be obtained:

$$ h \propto \beta l^{ - 2} \omega^{ - 2} H^5 e^{ks} $$
(7)

Where:\(\beta =\frac{2}{2}o{({m}_{ref}^{*})}^{2}\Delta {T}_{ref}^{*-1}{e}^{-kxref}\) and \({m}_{ref}^{*}\)(dimensionless flue gas mass flow at the reference position) and \({({m}_{ref}^{*})}^{2}\Delta {T}_{ref}^{*}\) (dimensionless temperature rise at the reference position) are constants.

According to the simulation results, the value of k is approximately equal to 0.372, and then fit the results according to formula (7). The results are shown in Fig. 7. The experimental results show that for fire source position 1, the value of \(h/{\omega }^{-2}{H}^{5}{e}^{ks}\) is approximately linear with the value of \({l}^{-2}\), and the following expression is obtained:

$$ \frac{h}{{\omega^{ - 2} H^5 e^{ks} }} = 0.1678l^{ - 2} - 0.0144,{ }0.1m^{ - 2} < l^{ - 2} < 0.7m^{ - 2} $$
(8)

Equation 7 can also be rewritten as:

$$ h = 0.1678l^{ - 2} \omega^{ - 2} H^5 e^{ks} - 0.0144\omega^{ - 2} H^5 e^{ks} ,{ }1m < l < 3m $$
(9)
Fig. 7.
figure 7

Fitting graph of the experimental results

5 Conclusion

By comparing the small-scale experiment and FDS simulation, the reliability of FDS for fire smoke simulation of one end blocked tunnel is verified. Taking the planned one end blocked tunnel train inspection depot project as an example, a full-scale FDS numerical simulation calculation model is established. After grid independence verification, the relationship between critical shaft length and fire source location, fire source heat release rate and shaft height is analyzed. According to the numerical simulation results, the following results are obtained:

  • Setting appropriate shaft size can effectively discharge all flue gas inside the tunnel and ensure the safety of tunnel structure and personnel evacuation.

  • When the fire source is located between the plugging end and the shaft, the natural ventilation and smoke exhaust volume of the shaft is significantly greater than that of other fire sources, and the critical shaft length is greater. At the same time, the critical shaft length does not change with the change of fire source heat release rate, but decreases with the increase of shaft height.

  • When the fire source is located between the blocking end and the shaft, a prediction model of the critical shaft length of the tunnel blocked at one end is proposed.