Experimental study on the maximum ceiling gas temperature driven by double fires in a tunnel with natural ventilation

The maximum gas temperature below the ceiling is an important parameter for tunnel safety. The present study analyzed the characteristics of the maximum excess ceiling gas temperature driven by double fire sources in a naturally ventilated tunnel. A series of small-scale tunnel fire experiments were carried out with different fire separation distances and heat release rates. Theoretical analysis based on the equivalent virtual origin was also performed. The results showed that there exists only one peak gas temperature when the two fire plumes are merged before reaching the ceiling, while two peak gas temperatures can be observed when the two fire plumes are completely separated. The maximum excess gas temperature below the tunnel ceiling gradually decreases with an increasing fire separation distance in the plume merging region ( S < S cp ). When the fire separation distance increases further ( S > S cp ), the effect of the fire separation distance on the maximum gas temperature below the ceiling is very limited. Furthermore, a model using an equivalent fire source was proposed to predict the maximum excess gas temperature below the ceiling, considering different plume merging states. The present study contributes to the understanding of the maximum excess gas temperature characteristics of the smoke flow driven by double fires with an equal heat release rate in naturally ventilated tunnels.


Introduction
Tunnel fire safety has raised public concern in recent decades, mainly due to the increasing number of tunnel fire accidents (Lönnermark and Ingason, 2006).The maximum gas temperature below the tunnel ceiling is an important parameter to evaluate the tunnel structure (Connolly, 1998;Yao et al., 2019a).An empirical model was developed by Kurioka et al. (Kurioka et al., 2003) to predict the maximum excess gas temperature below the tunnel ceiling in a tunnel under longitudinal ventilation.Hu et al. (Hu et al., 2006) further confirmed this empirical model through a series of full-scale tunnel fire tests and numerical simulations.However, this empirical model is not suitable for tunnels with low velocities or for very large tunnel fires.Based on the axisymmetric fire plume theory, Li et al. (Li et al., 2011;Li and Ingason, 2012) conducted a theoretical analysis of the maximum ceiling gas temperature beneath the tunnel ceiling and developed models for small and large fires that correlated well with both small-scale and full-scale test data under different ventilation conditions.Yao et al. (Yao et al., 2020) further proposed a revised model for the maximum excess gas temperature with a relatively larger longitudinal ventilation velocity.Gao et al. (Gao et al., 2014) studied the effect of sidewall on the maximum gas temperature below the ceiling in a naturally ventilated tunnel and correlated the gas temperature with the flame height.Tang et al. (Tang et al., 2018) proposed a normalized equation for the maximum gas temperature, considering different burner aspect ratios and ceiling smoke extraction rates.
However, the above studies only focused on the smoke flow driven by a single fire source.Multiple vehicles may burn simultaneously due to collisions (Ji et al., 2016), or fire spread in large tunnel fires (Lönnermark and Ingason, 2006).The burning of multiple fire sources indicates more fuel burning in the tunnel and a larger total heat release rate with increased gas temperature.This makes it more difficult for evacuees to reach the safe region and for firefighters to reach the position where they can effectively extinguish a fire (Ingason et al., 2014).Therefore, an increasing number of studies have been carried out to study the multiple fires in the tunnel.Hansen andIngason (Hansen andIngason, 2011, 2012) investigated the total heat release rate in a longitudinally ventilated tunnel with multiple HGVs (heavy goods vehicles).Ji et al. (Ji et al., 2016) analyzed the characteristics of burning rate and flame merging behavior of the double fires.He et al. (He et al., 2019) further disclosed the influence of fire source number on the burning characteristics of multiple fires.Wan et al. (Wan et al., 2022) experimentally investigated the performance of ceiling jet induced by two unequal strong plumes in a naturally ventilated tunnel.Chen et al. (Chen et al., 2020b) experimentally studied the effect of the sealing effects at the tunnel entrances on the fire behaviors with multiple fires in an arched tunnel.Chen et al. (Chen et al., 2022) studied the effects of cross airflow and burner distance on temperature profile and flame morphology of dual tandem pool fires.To optimize the effect of smoke control, Tsai et al. (Tsai et al., 2010) qualitatively studied the influence of double fires on the critical ventilation velocity.Zhang et al. (Zhang et al., 2019) and Tang et al. (Tang et al., 2020) further studied critical ventilation velocity for double fires and developed related revised models.Recently, four full-scale fire experiments were performed by Guo et al (Guo et al., 2023) to investigate the characteristics of double fire sources.The heat release rate, longitudinal ceiling gas temperature, vertical temperature, carbon monoxide and flame deflection angle were analyzed.Besides, the longitudinal ceiling gas temperature attenuation was also studied experimentally and numerically (He et al., 2023;Jia et al., 2021;Meng et al., 2023;Ren et al., 2021;Zhou et al., 2021).For the maximum ceiling gas temperature in case of double fire sources, Zhang et al. (Zhang et al., 2020) and Ma et al. (Ma et al., 2021) proposed related revised empirical models to enable the prediction of the maximum gas temperature in the longitudinally ventilated tunnel.Zhao et al. (Zhao et al., 2022) investigated the fire evolution and ceiling temperature distribution of the double fires in the naturally ventilated tunnel, and proposed an empirical model for maximum ceiling gas temperature.
It can be known from the above review that the related studies on the maximum gas temperature below the tunnel ceiling for the scenario with two fire sources are still limited.For double fires in a naturally ventilated tunnel, the maximum ceiling gas temperature is closely related to the air entrainment of two fire plumes, which is affected by flame interactions between the double fires.However, the effect of flame merging state on the maximum ceiling gas temperature has not been fully considered in the previous studies.A complete theoretical analysis is still needed.Therefore, the present study conducts a series of smallscale experimental tests (1:10) to investigate the maximum gas temperature below the ceiling in a naturally ventilated tunnel.The effect of flame interaction on the maximum gas temperature below the ceiling is carefully addressed by considering different fire separation distances and heat release rates.A theoretical analysis is conducted and a theoretical model is proposed.The present study is of great significance for understanding the maximum temperature characteristics of the smoke flows driven by double fires with an equal heat release rate in naturally ventilated tunnels.

Experimental set-up
The Froude scaling method has been widely used in the tunnel fire safety field (Ingason and Li, 2010;Yao et al., 2019b;Chen et al., 2020a;Fan et al., 2023;).The related scaling correlations are shown in Table 1, where F and M represent the full and model scales, respectively.More details about scaling theories can be found in the reference (Ingason et al., 2014).
The 1:10 scale tunnel was 14 m long, 0.5 m high and 0.4 m wide.The ceiling, floor and back walls of the tunnel were made of 0.02 m fire-proof plates, while the front wall was made of 0.01 m fire-proof glass to enable experimental observation.The small-scale model tunnel was placed in an experiment hall.The doors and windows of the experimental hall were closed to ensure a stable and quiescent environment.During each test, the two exists of the model tunnel were kept open for natural ventilation.K-type thermocouples with a diameter of 1 mm were used to measure the gas temperature at 0.02 m below the tunnel ceiling along the tunnel centerline.The longitudinal interval of the thermocouples varied from 0.05 m to 0.2 m, as shown in Fig. 1.
Two gas burners with square cross-sections were used to simulate the fire source.Each burner's side length and wall thickness were 0.1 m and 3 mm, respectively.The two burners were placed along the longitudinal center line of the tunnel, and its top surface is 0.02 m above the tunnel floor.Propane gas with 99 % purity was used as fuel.An independent digital mass flow controller controlled the propane flow rate of each
The corresponding full-scale values are 1.7 MW, 3.4 MW, 5.1 MW and 6.8 MW, respectively, referring to different stages of a passenger car fire.
As shown in Fig. 1, the distance between two burner sides is defined as the fire separation distance (S) in this paper.A total of 40 experiments were carried out, as shown in Table 2.

Flame behavior and ceiling gas temperature profile
Fig. 2 shows the typical flame images of double fires in the naturally ventilated tunnel with different fire separation distances.The flame behavior of the double fires is affected by fire interactions and therefore different compared to a single fire.When the fire separation distance is relatively large, the two flames tilt inward because the longitudinal induced air flow formed by the air requirement for plume entrainment and combustion leads to unbalanced air entrainment on both sides of a single fire source.With the decrease in the fire separation distance, the two separate flames begin to contact and merge vertically when the lateral deviation of the flames is less than the fire separation distance.It can be also found that the flame height increases when the flame merging occurs due to increasingly restricted air entrainment between the double fires as the continuous decrease of the fire separation distance.For zero fire separation distance, the two flames merge from the base and behave as a single flame with the maximum flame height.
The fire interaction between the two fire sources leads to complex flame behavior, which also results in the different characteristics of the ceiling gas temperature distribution.Fig. 3 shows the longitudinal gas temperature profile below the tunnel ceiling under different fire separation distances.The time-average gas temperature data at the quasisteady stage are used for further analysis.As shown in Fig. 3, there will be only one peak temperature for a specific temperature curve when the fire separation distance is relatively small, such as S < 0.5 m.This indicates the two fire plumes finally merge into one plume in the vertical direction before impinging the tunnel ceiling.However, it should be noted that there is still only one peak gas temperature beneath the ceiling, even the two flames are not merged in a few fire scenarios when S < 0.5 m.For example, the two flames are not merged when S = 0.2-0.3m for Q s = 5.4 kW, as shown in Fig. 2, but only one peak ceiling gas temperature for a specific temperature profile could be observed in Fig. 3 (a).This phenomenon is attributed to the flame inclination, which leads to the inward lateral deviation of the two plumes.When the fire separation distance is smaller than the inward lateral deviation of the two plumes, the two plumes still merge vertically before impinging the tunnel ceiling.However, the inward lateral deviation of the two plumes is restricted by the height limit of the tunnel ceiling.With the continuous increase of fire separation distance, the two fire plumes are completely separated, and the two fire plumes impinge the tunnel ceiling individually.Two peak gas temperatures could be observed for a specific temperature profile when the fire separation distance is large enough, such as for S ≥ 0.5 m.Besides, it can be observed that the distribution of  gas temperature below the ceiling is symmetrical for the double fire sources with equal heat release rates in the naturally ventilated tunnel.Fig. 4 shows the maximum excess gas temperature below the tunnel ceiling under the impact of the fire separation distance.It can be observed that the maximum excess gas temperature below the ceiling gradually decreases first and then approaches a relatively constant value with a longer fire separation distance.The reasons can be explained as follows: for the two fire sources with zero fire separation distance, the flames merge entirely from the base of the burner with the least air entrainment, and the total convective heat release rate is 2Q s,c , which leads to the maximum ceiling gas temperature under a total given heat release rate.With an increased fire separation distance, the flame merging point becomes higher, which increases the air entrainment before impinging the tunnel ceiling.Based on the energy conservation,  the maximum excess gas temperature decreases with the increasing fire separation distance.Note that it is possible that the two flames do not merge but the two plumes merge before reaching the tunnel height in some fire scenarios, e.g., for Q s = 5.4 kW and S = 0.3 m.As the fire separation continues to increase, due to the limitation of the tunnel ceiling, the merging point can approach the maximum height at ceiling level.Therefore, one could expect a critical plume merging separation distance.Beyond this distance, the two fire plumes could not contact and merge in the vertical direction when the fire separation is longer than the critical plume merging separation distance.In this situation, the two fire plumes impinge on the tunnel ceiling individually, and consequently, the effect of the fire separation distance on the maximum gas temperature below the ceiling is minimal.Note that the two flames with a relatively large fire separation distance tilt inward due to the longitudinal induced airflow, as shown in Fig. 2. Based on previous studies (Li and Ingason, 2014;Yao et al., 2020), at a given height, the total air entrainment of the inclined plume increases when the fire plume tilts.As expected from the energy conservation for a given heat release rate, the maximum excess gas temperature below the ceiling is the lowest when the two plumes are completely separated.As shown in Fig. 4, the maximum gas temperature becomes almost constant, insensitive to the heat release rate, when the fire separation distance is greater than 0.5 m.Therefore, it could be expected that the critical fire plume merging distance lies between 0.4 m and 0.5 m (the dashed line), insensitive to the heat release rate.This will be further explored in the following section.

Model for maximum excess gas temperature below the ceiling
Based on the ideal axisymmetric plume theory, Zukoski et al. (Zukoski et al., 1981) established a model to predict the mass flow rate of fire plume at different heights, which is expressed as: For the double fire source scenarios with S = 0, the flame merging from the burner's base could be considered as one fire plume.The smoke mass flow rate at the ceiling level can be calculated as follows: For the double fire source scenarios with S > 0, the induced airflow in the naturally ventilated tunnel leads to the incline of the two flames.When the ventilation velocity is of a relatively great value, the flame deflects and the ventilated flow induces extra air entrainment into the fire plume, so that the plume mass flow rate increases with the ventilation velocity and the ceiling gas temperature decreases with the velocity (Li et al., 2011).Based on Quintiere et al. (Quintiere et al., 1981), Li and Ingason (Li and Ingason, 2014) confirmed that the mass flow rate of fire plume at a given height in a ventilated flow almost equals that in the open by use of the inclined path as the plume height by an experimental data analysis and their equation for mass flow rate of the inclined fire plume can be expressed as follows (Li and Ingason, 2014): As the fire separation distance increases, the merging point gradually becomes higher.Due to the height limit of the tunnel ceiling, the maximum height of the merging point can approach the effective tunnel height.The corresponding fire separation distance can be defined as the critical plume merging distance (S cp ), which can be expressed as S cp = 2H ef tanθ-D according to Fig. 5.For the double fire source scenarios with 0 < S ≤ S cp , due to the two plumes are separated below the merging point, the smoke mass flow rate from each fire source is assumed to be estimated by Eq. ( 8).Therefore, the total mass flow rate of the two plumes at height of the merging point could be calculated as follows: where l traj,m is the trajectory of each fire plume between the center of the fire source and the plume merging point.As shown in Fig. 5, assuming there exists an equivalent virtual fire source, whose heat release rate is equal to 2Q s , producing the equal smoke mass flow rate at the height of the merging point, namely: ṁp,total = 0.071 ( 2 Qs,c where H m,eq is the vertical distance between the equivalent virtual fire source and the plume merging point.Based on Eq. ( 10), the relationship between H em and l traj,m can be obtained as: The trajectory of each fire plume from the centre of the fire source to the plume merging point is related to the flame inclination angle, l traj,m = 0.5(S + D)/sinθ (12) It reaches the maximum value when the merging point is at the ceiling.
As shown in Fig. 5, the height difference between the merging point and the tunnel ceiling could be calculated as follows: where H m is the height difference between the fire source and the plume merging point, which could be expressed as follows: Due to the height limit of the tunnel ceiling, the merging point reaches the highest point at the ceiling, namely H m = H ef , and the fire separation distance is equal to the critical plume merging distance, namely S = S cp .
Therefore, the distance between the equivalent virtual origin and the tunnel ceiling could be expressed as: After the two plumes are separated (S > S cp ), the flame inclination angle for each plume is relatively stable and the trajectory for each plume remains closely constant within a certain fire separation distance, which can be expressed as follows: Therefore, the total smoke mass flow rate at the height of the tunnel ceiling can be expressed as follows: Based on the energy conversation principle, the average excess gas temperature below the tunnel ceiling can be calculated as follows: The maximum excess gas temperature below the tunnel ceiling is proportional to the average excess gas temperature, namely ΔT max ∝ΔT avg .Furthermore, the convective heat release rate could be assumed as Qs,c = (1 − χ r ) Qs .Therefore, combining the Eqs.( 17) and ( 18), the maximum excess gas temperature below the tunnel ceiling can be calculated as follows: with where C T is a non-dimensional coefficient between the average gas temperature and the maximum temperature and χ r is the fraction of radiative heat release rate.The fraction of radiative heat release rate for common typical fuels is within 20 % to 40 %, and the convective heat release rate is often assumed to be 70 % of the total heat release rate (Karlsson and Quintiere, 1999) which is also used in this study.The flame inclination angle of the two fires in a naturally ventilated tunnel could be estimated using the AGA model (Raj et al., 1979): but with a different non-dimensional induced air velocity as proposed by He et al. (He et al., 2021): Note that Eq. ( 21) is similar to the equation proposed by Li and Ingason (Li and Ingason, 2014) for the flame angle defined based on the position of maximum ceiling gas temperature.
Test data are used to determine the parameter C T .However, it should be noted that Eq. ( 21) and ( 22) are proposed for the flame tilt angle for relatively large fire separation distances.When the fire separation distance is small, the flame tilt angle may slightly deviate from the predicted value.However, note that H m could be of a relatively small value for a relatively small fire separation distance.Consequently, Eq. ( 21) and ( 22) could be used in general, which will be verified by experimental data in the following part.Based on Eq.s ( 14), ( 21) and ( 22), the critical fire separation distance, S cp , can be calculated to be 0.48 m, and it is independent of the heat release rate.This is consistent with the results shown in Fig. 4 and the analysis presented there.In addition, Eq.s (7)-( 9) may be not valid when the flames impinge on the tunnel ceiling, especially for the cases with a continuous combustion zone higher than the tunnel height.In such cases, the tunnel ceiling lies in the continuous combustion zone, and the maximum temperature approaches constant, the so-called continuous flame temperature.Therefore, we first obtain C T using the experimental data when the continuous flame is lower than the tunnel ceiling and all the excess temperatures are lower than 600 K.As shown in Fig. 6, experimental data can be correlated well as follows: A correlation coefficient of 0.94 was obtained for the correlation.According to Eq.s ( 19) and ( 23), the value of C T is obtained as 1.76.The C T in the present study is slightly higher than 1.59 in the previous study of Li et al (Li et al., 2011).This may be attributed to the fact that the aspect ratio of the model tunnel (W T /H T = 0.8 in our study) is smaller, probably resulting in less entrainment into the fire plume due to the restriction of walls and thus causing higher maximum gas temperatures.This is in accordance with previous research, e.g., (Tang et al., 2018).
Further, we may assume that Eq.s ( 7)-( 9) are still valid for slightly larger fires.Fig. 7 shows a comparison between the experimental data and the predictions given by Eq. ( 23).In Fig. 7, all the experimental data of maximum excess gas temperatures are plotted, including the experimental data with mean flame impinging on the tunnel ceiling.It can be found that most experimental data points are distributed within 20 % error lines.This indicates that the calculated maximum ceiling excess temperatures predicted by Eq. ( 23) are with reasonable accuracy for larger fires with the temperature lower than 900 • C.
To further verify the accuracy of the model, the experimental data from previous studies with symmetric double fires in the natural ventilation tunnel are also used for comparison.As shown in Fig. 7, the small-scale experimental research data from the study of Zhao et al. (Zhao et al., 2022) can be well predicted by the model proposed in this study.However, the full-scale experimental data from the study by Guo et al. (Guo et al., 2023) are slightly lower than the model predictions.The possible reason is that these experiments (Guo et al., 2023) were conducted in a tilted tunnel with a slope of 5 %.According to a previous study by Hu et al. (Hu et al., 2013), the maximum ceiling gas temperature decreases as the tunnel slope increases under a given heat release rate.Therefore, it should be pointed out that the model proposed in this study is only suitable for symmetric double fires with equal heat release rates in a naturally ventilated horizontal tunnel.

Conclusions
In this study, a series of small-scale tunnel fire experiments was carried out to address the effects of interactions of two fire sources on the maximum excess gas temperature beneath the ceiling in a naturally ventilated tunnel.The fire separation distance and heat release rate were considered.A theoretical analysis of the merged plume using an equivalent virtual fire origin was also carried out.The major conclusions are listed as follows: 1.There is only one peak gas temperature when the two fire plumes are merged before impinging on the tunnel ceiling.But two peak temperatures can be observed when the two fire plumes are completely separated and impinge on the tunnel ceiling independently.The distribution of gas temperature below the ceiling is essentially symmetrical for the double fire sources with equal heat release rates in the naturally ventilated tunnel.2. With an increased fire separation distance in the flame merging and plume merging region (S < S cp ), the plume merging point becomes higher, and the maximum gas temperature below the tunnel ceiling gradually decreases.With a further increased fire separation distance (S > S cp ), the two fire plumes are completely separated in the vertical direction, and the effects of fire separation distance on the maximum temperature below the ceiling are very limited.Moreover, the maximum excess gas temperature below the ceiling is the lowest when the two plumes are completely separated.3. Based on the mass conservation principle, a model using an equivalent fire source is developed to predict the maximum excess gas temperature below the ceiling for the double fire sources with equal heat release rates in the naturally ventilated horizontal tunnel, considering different plume merging states.The good predictions of experimental data by Eq. ( 23) indicate that the maximum excess gas temperature below the ceiling increases with the 2/3 power of the total heat release rate and decreases with the − 5/3 power of the equivalent tunnel height.

Declaration of competing interest
The authors declare that they have no known competing financial interests or personal relationships that could have appeared to influence the work reported in this paper.Comparison between the experimental value and predicted value given by Eq. ( 23).

Fig. 2 .
Fig. 2. Typical pictures at quasi-stable moments of two fire sources in a tunnel.

Fig. 3 .
Fig. 3. Longitudinal gas temperature profile below the tunnel ceiling near the fire source under different fire separation distances.

Fig. 4 .
Fig. 4. The maximum temperature below the tunnel ceiling as a function of separation distance.

Fig. 6 .
Fig. 6.Correlation for maximum gas temperature below the tunnel ceiling with different fire separation distances.Fig. 7.Comparison between the experimental value and predicted value given by Eq. (23).