1 Introduction

Recently, many high-altitude tunnels have emerged the world, such as the Yankou mountain tunnel in China and the Eisenhower-Johnson Memorial tunnel in the United States. Smoke is the most fatal factor for the tunnel construction and the trapped people [1]. Vertical shafts as a natural smoke extraction mode are widely used in tunnels due to their economy and environmental protection [2]. For example, the high-altitude Kunming metro system using vertical shafts has achieved great operational and economic benefits. Meanwhile reduced pressure which alters fire smoke transportation brings a major challenge to the naturally ventilated vertical shaft design of high-altitude tunnels [3].

Considerable research has been conducted on tunnel fires with naturally ventilated vertical shafts, and mainly focuses on the fire smoke characteristics (Li et al.[4],Tang et al. [5], and Ren et al. [6], etc.), smoke exhaust efficiency and optimization (Kashef et al. [7], Yi et al. [8], and Ji et al. [9], etc.), tunnel fire evacuation (Cano-Moreno et. al [10], Chen et al. [11], Krol et. al [12]), smoke back-layering length (Li et al. [13], Zhang et al. [14], Zhang et al. [15], etc.), and influence factors for smoke layer plug-holing (Zhao et al. [16], Liu et al. [17], Liu et al. [18], etc.). In the aspect of smoke layer plug-holing behavior in tunnel shafts which havs a significant impact on the smoke extraction efficiency, Ji et al. [19] firstly revealed two special phenomena (plug-holing and boundary layer separation) and proposed a critical plug-holing judging criterion (i.e., a new Richardson number Ri’). Then, some researchers [20, 21] verified the applicability of the critical plug-holing criterion under different conditions. For example, Jiang et al. [20] conducted a series of tunnel fire experiments on smoke exhaust under the vertical shaft and longitudinal ventilation, which showed that the critical plug-holing criterion Ri’ = 1.4 was still applicable. The smoke exhaust characteristics of shafts arranged in the middle and above the sides of the tunnel were also studied by Jiang et al. [21], who concluded that Ri’ = 1.4 is still applicable to explain the plug-holing and boundary layer separation phenomena associated with smoke boundary layers. Zhang et al. [22] further studied the influence of the shaft size on the plug-holing height which indicates the degree of smoke layer plug-holing, and built a model for predicting the critical shaft height through a series of numerical tunnel fi re simulations. However, Zhao et al. [23] gave a new dimensionless Ri* by finding that Ji’s plug-holing criterion [19] could not be applied to large cross-section shafts, and then proposing a prediction model for the plug-holing height based on tunnel fire simulated data. As can be seen, Ri’ or Ri* can describe the smoke flow pattern within tunnel shafts, and its critical value can predict the occurrence of the smoke layer plug-holing. However, the above studies are carried out at standard atmospheric pressure (101 kPa), and then the critical Ri’ criterion and the plug-holing height model may not be applicable to the tunnel fires at reduced pressure owing to the various smoke movement patterns at reduced pressure.

Ji et al. [24] carried out a series of simulations in full-scale highway tunnels to explore the effect of ambient pressure on the temperature distribution in tunnel fires, and the results showed that the longitudinal temperature distribution increased with the decrease of ambient pressure. Tang et al. [25] found that longitudinal smoke temperature decays faster at reduced pressure through numerical simulations. Feng et al. [26] and Liu et al. [27] found that the CO concentration and smoke movement speed in high altitude tunnel fires were different from those standard atmospheric pressure, and the smoke movement speed increased with the increase in altitude. Wang et al. [28,29,30,31,32,33,34] investigated the influence of reduced pressure on the ceiling temperature profile, CO concentration profile and flame height by full-scale aircraft cargo compartment fire tests to derive prediction models of these fire plume characteristics applicable at reduced pressure. Obviously, the decrease of ambient pressure can change the tunnel fire smoke transportation characteristics, which have a significant impact on the smoke movement pattern during natural smoke extraction from the shaft and the crucial plug-holing criteria values. For this aspect, Yan et al. [35] calculated the critical plug-holing criterion Ri’ at 60 kPa, 80 kPa, and 100 kPa and found that Ri’ decreased at reduced pressure. However, the mathematical relationship between the critical plug-holing criterion and reduced pressure is not clarified. Moreover, the plug-holing height reflecting the degree of smoke layer plug-holing at reduced pressure has rarely been addressed in previous studies.

Therefore, this work conducted theoretical analysis and numerical simulations to clarify the reduced pressure effects on smoke temperature and velocity in a natural ventilated tunnel with a vertical shaft to present the critical plug-holing criteria and plug-holing height. A new plug-holing criterion and a new prediction model for plug-holing height of tunnel fires with a natural ventilated vertical shaft are proposed to extend the application range of the criterion and prediction model including the standard atmospheric pressure and reduced pressure.

2 Theoretical Analysis

2.1 Plug-Holing Criterion for Vertical Shafts at Reduced Pressure

Ji et al. [19], based on the analysis of the main driving dynamics in the natural smoke exhaust process of the shaft, proposed the dimensionless Ri’ number to determine the flow field structure in the shaft, which is defined as the ratio of the vertical thermal buoyancy (Fv) to the horizontal inertial force (Fh) of the smoke, i.e.

$$Ri^{\prime} = \frac{{F_{v} }}{{F_{h} }} = \frac{{\Delta \rho ghA_{shaft} }}{{\rho_{s0} v^{2} dw_{s} }}$$
(1)

where Δρ is density difference between smoke and ambient air without smoke exhaust, g is the gravitational acceleration, h is the shaft height, Ashaft is the vertical shaft cross-sectional area, ρs0 is the density of smoke without smoke exhaust, v is the horizontal flow rate of the smoke below the shaft without smoke exhaust, d is the smoke layer thickness below the shaft without smoke exhaust, and ws is the shaft width.

If the ambient pressure changes, Δρ and ρs0 will change as well. According to the ideal gas law:

$$PV = nRT$$
(2)

where P is the pressure of ideal gas, V is the volume of ideal gas, n is the amount of gas substance, R is the ideal gas constants, and T is the thermodynamic temperature of ideal gas. It can be obtained from the ideal gas equation:

$$p \propto \rho T$$
(3)

where ρ is the smoke density. The density difference between smoke and ambient air can be estimated by [36]:

$$\Delta \rho = \frac{{\Delta T_{s} }}{{T_{s} }}\rho_{air} = \frac{{\Delta T_{s} }}{{T_{a} + \Delta T_{s} }}\rho_{air}$$
(4)

where Ts and ΔTs are the smoke temperature and the smoke temperature rising below the shaft, Ta is the ambient temperature, ρair is the air density.

He [37] proposed an empirical equation for predicting the velocity of smoke below the ceiling based on experimental results and theoretical analysis of narrow and long space fires, ie.

$$v = 0.8\left( {\frac{{gQ_{c} T_{s} }}{{2C_{p} \rho_{air} T_{a}^{2} w}}} \right)^{1/3}$$
(5)

where Qc is the convective heat release rate, Cp is the specific heat of air, and w is the width of the tunnel.

The heat release rate can be expressed as [38]:

$$Q_{c}\,=\,\dot{m}h_{c}$$
(6)

where \(\dot{m}\) is burning rates and hc is the heat of combustion.

Wieser et al. [39] obtained the average burning rate of n-heptane fire is approximately proportional to the 1.3 power of atmospheric pressure:

$$ \dot{m} \propto p^{1.3} $$
(7)

where p is the ambient pressure.

Combining Eqs. (6) and (7), the relationship between the heat release rate and pressure can be expressed as:

$$Q_{c} \propto p^{1.3}$$
(8)

Then, the velocity of smoke under the same heat release rate and reduced pressure can be expressed as (ignoring the influence of pressure on the gravitational acceleration):

$$v \propto \left( {\frac{{Q_{c} T_{s} }}{{\rho_{air} }}} \right)^{\frac{1}{3}}$$
(9)

The smoke temperature Ts can be expressed as:

$$T_{s} = T_{a} + \Delta T_{s}$$
(10)

Yin et al. [40] obtained the relationship between plume temperature and pressure through high-altitude combustion chamber experiments, which can be expressed as:

$$\frac{{\Delta T_{s} }}{{T_{0} }} \sim (\frac{p}{{Q_{c} }})^{ - 2/3}$$
(11)

Gong et al. [41] pointed out that the longitudinal temperature distribution of the tunnel can be expressed as:

$$\frac{{\Delta T_{s} }}{{T_{a} }} = A_{1} \exp ( - \frac{x/w}{{A_{3} }}) + A_{2} \exp ( - \frac{x/w}{{A_{4} }})$$
(12)

where A1,A2,A3 and A4 are just fitting coefficients, and x is the longitudinal position from the fire source.

Finally, Yan et al. [35] have pointed out that the influence of pressure on the smoke layer thickness is very small, so ignoring the influence of pressure on the smoke layer thickness. Combining the Eqs. (1),  (3), (4), (5), and (9), the expression of Ri’ under the same heat release rate at reduced pressure can be expressed as:

$$Ri_{x} ^{\prime} = \left[ {k_{1} \left( {\frac{{p_{x} }}{{p_{std} }}} \right)^{-0.3} + k_{2} \left( {\frac{{p_{x} }}{{p_{std} }}} \right)^{-0.1} } \right]^{{ - \frac{2}{3}}} Ri^{\prime}$$
(13)

where k1, k2 are just fitting coefficients, pstd and px are the standard atmospheric pressure and reduced pressure of x kPa.

2.2 Plug-Holing Height at Reduced Pressure

According to previous studies [23], the plug-holing height is related to Ri’ and shaft shape. In order to reveal the quantitative relationship between ambient pressure and plug-holing height, the concepts of pressure coefficient and dimensionless shaft height are introduced here. The pressure coefficient is defined as the ratio of ambient pressure to standard atmospheric pressure:

$$\alpha = \frac{{p_{x} }}{{p_{std} }}$$
(14)

The plug-holing height is more sensitive to the shaft opening length [22], and the dimensionless shaft height (h*) is defined as the ratio of the shaft length to the shaft height as follows.

$$h^{*} = \frac{{l_{shaft} }}{{h_{shaft} }}$$
(15)

where lshaft is the shaft length and hshaft is the shaft height.

The dimensionless heat release rate can be calculated as follows [42].

$$Q^{*} { = }\frac{{Q_{{\text{c}}} }}{{\rho_{air} T_{a} C_{p} g^{1/2} H^{5/2} }}$$
(16)

where H is the hydraulic tunnel height defined as the ratio of 4 times the tunnel cross- sectional area to the tunnel perimeter.

Therefore, the dimensionless plug-holing height can be expressed as:

$$h_{p}^{*} = \frac{{h_{p} }}{{h_{shaft} }} = f(Ri^{\prime},h^{*} ,Q^{*} ,\alpha ){ = }k_{3} Ri^{{\prime}{k_{4} }} h^{{*k_{5} }} Q^{{*k_{6} }} \alpha^{{k_{7} }}$$
(17)

where hp is the plug-holing height. k3, k4, k5, k6 and k7 are the fitting coefficients.α is the pressure coefficient. h* is the dimensionless shaft height.

3 Numerical Modeling and Setting

3.1 Fire Scene

A natural ventilated tunnel of 120 m (length) × 9 m (width) × 6 m (height) with a vertical shaft model is established by FDS 6.6, as shown in Fig. 1. The shaft is located at the longitudinal centerline of the tunnel and 57.5 m from the left tunnel opening. The shaft cross-sectional size is 9 m2, 3 m length (l) and 3 m width (w), and the shaft height is adjustable in the range of 0–5 m. For comparison, a fire scenario with no shaft is set up to evaluate the original horizontal inertia force of the smoke. In order to obtain the best simulation result, additional extension areas denoted by yellow are added to the model in Fig. 1.

Figure 1
figure 1

Schematic diagram of the FDS model tunnel

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The ambient pressures are set to 60 kPa, 70 kPa, 80 kPa, 90 kPa, and 101 kPa, which is sufficient to cover the ambient pressure variation characteristics from 0 to 4500 m in altitude [43]. N-Heptane fire sources of 2 m (length) × 2 m (width) are located at the longitudinal centerline of the tunnel floor, 37.5 m away from the left tunnel opening. The heat release rates are 3 MW and 10 MW to represent the most common car fires (3 MW) and truck fires (10 MW) in the tunnel. The initial ambient temperature of the tunnel is 20 °C and the duration of the numerical simulation is 1200 s to form the steady state of the smoke movement in the tunnel and the shaft. The fire simulation conditions are summarized in Table 1.

Table 1 Fire Condition Setting
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Temperature and velocity vector slices are located at the longitudinal centerline of the shaft to visualize the smoke temperature distribution and flow field. In addition, a series of vertical temperature and velocity measurement points are set up below the shaft opening to get the vertical thermal buoyancy and horizontal inertia forces of the smoke layer below the shaft. The detailed arrangement of measurement points in the model is shown in Fig. 2.

Figure 2
figure 2

Setting of measurement points in tunnels and shafts

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3.2 Mesh Analysis

The grid size has an important influence on the accuracy of simulation and calculation time. According to the FDS User’s Guide, when the grid size (dg) follows: dg = 1/16D* ~ 1/4D*, the FDS model has high simulation accuracy [44]. The characteristic diameter D* can be expressed as follows:

$$D^{*} = \left( {\frac{{Q_{c} }}{{\rho_{air} C_{p} T_{a} \sqrt g }}} \right)^{{{2 \mathord{\left/ {\vphantom {2 5}} \right. \kern-0pt} 5}}}$$
(18)

According to the ideal gas equation, ρair = PM/RTa, then Eq. (18) can be changed to:

$$D^{*} = \left( {\frac{{Q_{c} R}}{{PMC_{p} \sqrt g }}} \right)^{{{2 \mathord{\left/ {\vphantom {2 5}} \right. \kern-0pt} 5}}}$$
(19)

where M is molar mass of air.

It can be seen from Eq. (19) that the characteristic diameter increases with the increase of HRR and decreases with the increase of pressure. Therefore, the grid independence study is conducted under the conditions of minimum HRR (3 MW) and maximum ambient pressure(101 kPa). Figure 3 shows the vertical distribution of temperature and velocity in the tunnel 12 m to the right of the fire source under different grid sizes. Obviously, when the grid size is less than 0.15 m, there is no obvious difference in the temperature curve. Therefore, considering the calculation accuracy and simulation time, this paper selects the grid size of 0.15 m for FDS simulation. The simulation with different fire source and pressure takes the same grid size (0.15 m) in order to ensure the consistency of the simulation accuracy for each group of working conditions.

Figure 3
figure 3

Velocity and temperature distribution under different grid sizes

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3.3 Model Validation

Considering the numerical simulation accuracy and calculation time, the FDS grid size was selected as 0.15 m × 0.15 m × 0.15 m. Since most of the simulated conditions in the work are carried out under pressure boundary conditions, the numerical simulation results by FDS at reduced pressure are compared and verified before carrying out the simulations. Yan et al. [45] conducted six full-scaled fire experiments in the Baimang Snow Mountain No. 1 Tunnel, Yunnan Province, where the ambient pressure was approximately 62.63 kPa, the air density was approximately 0.8353 kg/m3, and the ambient temperature was approximately 13.5 °C in the experimental section of the tunnel. The tunnel ambient characteristics are replicated using FDS, and a typical experimental situation with a fire source power of 0.72 MW is simulated. In comparison to the experimental longitudinal temperature distribution, the results of the numerical simulation of the smoke temperature at 0.12 m below the tunnel ceiling are displayed in Fig. 4a. The smoke temperature is selected from the experimental data at the fully developed stage of the fire. The results show that the simulated temperature is in good agreement with the experimental data.

Figure 4
figure 4

Comparison between simulated and experimental results

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At the same time, the FDS simulation results under standard atmospheric pressure are also verified. Cong et al. [46] conducted a series of experiments in a 1:15 small-scale tunnel. The full-scale tunnel corresponding to the experimental model is almost the same size of the model tunnel constructed in this paper. Enlarge the small-scale model to a full-scale model scale, and compare the numerical simulation results with the experimental results of the small-scale model, as shown in Fig. 4 (b). It can be seen that under the standard atmospheric pressure, the temperature simulation results below the tunnel ceiling are not significantly different from the experimental data.

In order to verify the accuracy of the FDS simulation of vertical shaft smoke exhaust, the simulation results were compared with the smoke movement pattern in the 1:6 reduced-size model tunnel by Ji et al. [19]. Ji et al. [19] presented detailed experimental conditions in their study, and the boundary conditions can be well reproduced by FDS. Two representative operating conditions with HRR of 20.21 kW are reproduced in FDS. Figure 5 shows the centerline temperature distribution in the shaft area (left) compared with the experimental results of Ji et al. [19] (right). It can be seen that when the shaft height is 0.2 m, no plug-holing occurs, and when the shaft height is 0.4 m, significant plug-holing has occurred. The two smoke movement patterns are in good agreement when the modeling findings and experimental phenomena are compared side by side. In summary, numerical simulations can satisfy the research needs.

Figure 5
figure 5

Vertical shaft plug-holing: simulated by FDS (left) and experimental results by Ji et al. [19] (right)

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To verify the accuracy of numerical simulation results under other heat release rates, the smoke flow velocity below the tunnel ceiling was compared with previous studies [47]. The comparison between the FDS simulation results and the smoke velocity prediction model Eq. (5) proposed by He [37].

Figure 6 shows the comparison between the smoke flow velocity at a distance of 12 m to the left of the fire source and the predicted results of the Eq. (5). It can be seen that the numerical simulation results are in good agreement with the predicted values of the empirical equation.

Figure 6
figure 6

Comparison between simulation results of smoke velocity and predicted values using empirical equation

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In summary, the accuracy of the FDS numerical model is reliable, and it is feasible to simulate fires in high-altitude tunnels under different environmental pressure.

4 Results and Discussion

4.1 Smoke Temperature Profile and Velocity Field in the Vertical Shaft

Figure 7 shows smoke temperature distribution at the longitudinal central plane of vertical shafts with different heights at 60 kPa and 101 kPa for 3 MW tunnel fires. When the shaft height is 2 m, the smoke layer below the shaft can still maintain a stable layered structure. However, a low-temperature depression area below the shaft has occurred, where smoke with temperature lower than 40 °C starts to invade the smoke layer at the bottom of the shaft. When the shaft height approaches 2.5 m, smoke with a temperature below 25 °C has obviously descended into the shaft. Since more cold air is sucked into the shaft directly due to the stronger stack effect of high vertical shaft, the smoke layer thickness beneath the bottom of the shaft decreases to zero, and then smoke layer plug-holing occurs.

Figure 7
figure 7

Smoke temperature distribution for different shaft heights at 60 kPa and 101 kPa (HRR = 3 MW)

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With the increase in shaft height, the highest point of the recessed region surpasses the bottom of the shaft and then an obvious cold air cone appears in the shaft. In accordance with the Ri’ criteria put forward by Ji et al. [19], smoke layer plug-holing appears in the shaft when the vertical thermal buoyancy causing the stack effect of the smoke is greater than its horizontal inertia force. Thus, for tunnel fires at same pressure, the degree of plug-holing is enhanced by the increasing shaft height, just like the elevating smoke depression point of higher shaft in Fig. 7. Comparing smoke depressed areas for the same shaft height at different pressures, the smoke depression point at 60 kPa is lower than it at 101 kPa, that is, the degree of plug-holing is weakened by the reduced pressure. The quantitative analysis of plug-holing degree and its induced mechanism at reduced pressure is expressed in Sect. 4.3.

The smoke flow velocity field in the shaft can be used to determine the boundary layer separation which means that a vortex area appears on the upstream sidewall of the shaft [48]. Figure 8 shows the smoke flow velocity field in the shaft with different shaft heights at 60 kPa and 101 kPa for 3 MW tunnel fires. When the shaft height is 2 m, a vortex area near the upstream sidewall of the shaft is observed, indicating that boundary layer separation has occurred. And the horizontal inertia force of the smoke is the main driving factor for smoke flow in the shaft, since the stack effect is insufficient to overcome the adverse resistance to smoke exhaust caused by the inverse pressure gradient. The boundary layer separation phenomenon exists when the shaft height is less than 2 m. With the shaft height increasing to 2.5 m, a vortex area near the sidewalls of the shaft disappears and vertical flow vectors appear at the shaft center. Smoke layer plug-holing replaces boundary layer separation and continues to occur for higher shafts, while the stack effect of the shaft is the main driving factor for smoke flow. Clearly, the reduced pressure lengthens the existence of boundary layer separation phenomenon and postpone the occurrence of plug-holing.

Figure 8
figure 8

Smoke flow velocity field for different shaft heights at 60 kPa and 101 kPa (HRR = 3 MW)

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According to the smoke temperature and flow velocity field in the shaft, the smoke boundary layer separation and plug-holing phenomena for different shaft heights at different pressures are summarized in Table 2.

Table 2 Smoke Boundary Layer Separation and Plug-Holing Phenomena for Different Shaft Heights at Different Pressures
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4.2 Plug-Holing Criterion at Reduced Pressure

According to the numerical data of smoke temperature rise, smoke layer thickness and smoke velocity listed in Table 3, the critical Ri’ at different pressures is calculated using Eq. (1) as shown in Fig. 9. The critical Ri’ at standard atmospheric pressure (1.38) agrees with the conclusion reached by Ji et al. [19], and the critical Ri’s at 60 kPa and 80 kPa (1.08 and 1.34 respectively) correspond to Yan’s study [35] which further verifies the accuracy of our simulation results further. Obviously, reduced pressure leads to a reduction in the critical Ri’ for smoke layer plug-holing. Because the faster smoke flow velocity at reduced pressure caused by a higher smoke temperature as presented in Sect. 4.1, enhances the horizontal inertial force of the smoke layer below the shaft. Figure 10 shows the relationship between Ri’ and ambient pressure, i.e. the ratio of Rix’ at reduced pressure to Ri’ at standard atmospheric pressure has a non-linear relationship to the pressure coefficient, which corresponds to the theoretical analysis (Eq. 13).

Table 3 Smoke Layer Parameters Below the Shaft Opening
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Figure 9
figure 9

Ri’ values of smoke plug-holing tunnel shaft at different pressures

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Figure 10
figure 10

Correlation between Rix’/Ri’ and ambient pressure coefficient

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Therefore, the judging criterion of smoke layer plug-holing at reduced pressure Rix’ can be expressed as Fig. 10.

4.3 Reduced Pressure Effect on Plug-Holing Height

The concept of plug-holing height is proposed to describe the degree of plug-holing, which is defined as the vertical height difference between the bottom opening of the shaft and the highest point of the low-temperature depression zone at 5 °C temperature rise in the shaft, denoted as hp [23]. Figure 11 gives the plug-holing height at different pressures for 3 MW tunnel fires with 5 m shaft height through smoke temperature fields in the shaft. The plug-holing height decreases by 66% when the pressure decreases from standard atmospheric pressure to 60 kPa. Reduced pressure has a non-negligible influence on the plug-holing height.

Figure 11
figure 11

Plug-holing height of the shaft at different pressures (HRR = 3 MW, h = 5 m)

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Figure 12 presents all plug-holing heights for tunnel fires of different heat release rates with various shaft heights at different pressures, showing that the plug-holing height decreases with the decrease of the ambient pressure for tunnel fires with the same heat release rate, and the heat release rate decreases the plug-holing height for the same pressure and shaft height condition. In addition, at the same pressure, the plug-holing height shows an overall upward trend with the increase in shaft height, which supports Zhao’s [23] study at standard atmospheric pressure.

Figure 12
figure 12

Plug-holing height at different pressures (60, 70, 80, 90, 101 kPa) and shaft heights

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Through the theoretical analysis in Sect. 2.2, the dimensionless plug-holing height can be expressed by Eq. (17). Based on the critical Ri’ values from the numerical results, Fig. 13 gives the correlation of dimensionless plug-holing height by introducing pressure coefficient α, and the dimensionless shaft height h*, expressed as:

$$h_{p}^{*} = 0.09\alpha^{0.83} Ri^{\prime 1.77} h^{*1.22} Q^{* - 0.17}$$
(20)
Figure 13
figure 13

Dimensionless plug-holing height correlation at different pressures

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5 Conclusion

The plug-holing behavior of smoke layer of tunnel fires with natural ventilated vertical shafts at reduced pressure is studied through theoretical analysis and numerical simulations. The impact of reduced pressure on smoke temperature and flow velocity in the vertical shaft is clarified. And the plug-holing criterion for the smoke layer at reduced pressure, together with the critical judging values, are presented. Then the plug-holing heights at reduced pressure are analyzed. The main conclusions include:

  1. (1)

    The plug-holing criterion Ri’ for tunnel fires with natural ventilated vertical shafts at reduced pressure is given through theoretical analysis, which has a non-linear relationship to the ambient pressure. The critical judging values of Ri’ at 60 kPa, 70 kPa, 80 kPa, 90 kPa are calculated as 1.08, 1.18, 1.25, 1.34 respectively to verify that the critical plug-holing criterion decreases with the reduction of the ambient pressure.

  2. (2)

    The plug-holing height decreases with the decreasing pressure, that is, the plug-holing degree of the smoke layer in tunnel fires with a vertical shaft is weakened by the reduced pressure. Because the horizontal inertia force of fire smoke is enhanced by the increasing smoke flow velocity and temperature at reduced pressure.

  3. (3)

    A global prediction model (Eq. (20)) for the plug-holing height of tunnel fires with a vertical shaft is proposed by introducing a pressure coefficient and extending the engineering application range. The plug-holing height is proportional to the 1.77 power of Ri’, 1.22 power of h*, the − 0.17 power of the Q* and the 0.83 power of the pressure coefficient.