An Improved Method for Determining The Interface Height of Smoke Layer in A Tunnel Fire

. Smoke layer interface height is an important parameter in fire safety science. In this paper, two methods are proposed on the basis of integral ratio method and the least squares method. a series of experiments were conducted in a 1:10 scale model tunnel for determining the smoke layer interface height in tunnel fire. The interface height of the smoke layer determined by the new methods have been compared with other existing approaches, including integral ratio method, least squares method, N-percentage rule. The comparison results show that the results of the least squares method based on the transition layer can be relatively reasonable and agree well with the visual observations for all experimental condition. In addition, the usage scenarios and considerations of the N-percentage rule were also discussed.


Introduction
The interface height of the smoke layer is an important parameter in the fire safety science. It will directly affect the distribution of temperature, visibility and CO concentration, which is of great significance for the safe evacuation of personnel in tunnel fire. The determination of smoke layer interface in case of a fire is a subject of great interest in the field of fire safety science. Cooper et al. proposed a N-percentage rule to determine the smoke layer interface height based on the experiments of multi-room fires. However, different researchers used discrepant N values for various fire scenarios and the selection of N value is subjective and empirical [1]. In Quintiere's research, a method for calculating the interface height of smoke layer based on the mathematical relationship between the average temperature of upper and lower layers was proposed [2]. Emmons proposed the maximum gradient method to determine the height of the smoke layer by calculating the position of the sharp change in vertical temperature [3]. He et al. put forward an integral ratio method and a least squares method to determine the smoke layer interface height based on the analysis of variance or integral ratio to divide the vertical temperature into two relatively uniform regions. This method does not depend on subjective factors and other environmental factors [4]. Gao et al. introduced the buoyancy frequency as the index to judge the stratification position, and proposed the buoyancy frequency method [5]. The accuracy of this method has also been verified by other scholars [6].
In summary, scholars have proposed a number of calculation methods for the height of the smoke layer, such as the N-percentage rule, the integral ratio method and the least square method. Among these methods, the * Corresponding author: mw@cqu.edu.cn integral ratio method and the least square method were considered to be reliable, which excluded the influence of human interference factors and external parameters, and were widely used by scholars. However, these two methods were proposed for the obvious stratification of the smoke in the confined space. Unlike confined spaces, the tunnel is a long semi-constrained space, accompanied by longitudinal winds. The shear effect of the longitudinal velocity on the smoke layer will greatly affect the stratification stability, the interface of the smoke layer will become unclear [7,8]. It is difficult to accurately calculate the interface height of smoke layer in tunnel with longitudinal wind by using the existing methods. Therefore, a calculation method of the smoke layer based on the transition layer is proposed to determine the position of the interface of the smoke layer.

Calculation methods based on transition layer theory
Integral ratio method and least squares method are based on the theory of two-zone model, which divides the temperature distribution of cross-section into two zones, as shown in Fig.1 (a). However, when the smoke layer is mixed with the lower air flow due to the influence of the longitudinal wind, the section temperature gradient becomes smaller, the theory of the two-zone model will be difficult to distinguish the boundary between the two zones, thus obtaining a higher calculation result, which is also the problem of integral ratio method and least square method.
In order to reflect the phenomenon of smoke subsidence in tunnels and obtain relatively accurate interface height of smoke layer, the transition layer theory [9,10]  height of transition layer and air layer. The variance or the integral ratio is used to divide the temperature field of the section into three divisions: lower air layer, middle transition layer and upper strong stratification zone, not just the upper part of the upper smoke layer and the lower air layer, as shown in Fig.1 (b).

Integral ratio method based on transition layer theory
The principle of the integral ratio method based on the transition layer is to divide the vertical temperature field into three regions by using the integral ratio method [4]. The detailed calculation steps are as follows: The integration ratio of the lower layer (including the transition layer and the air layer) is written as: The integral ratio of upper smoke layer (strong stratified zone) is written as: The sum of the two integral ratios is: Where ‫ݎ‬ ଵ and ‫ݎ‬ ଶ are the integral ratios of the temperature data of the lower layer and upper layer, respectively. ‫ܪ‬ is the assumed smoke layer height, ‫ݖ‬ is the height of data points; ‫ܪ‬ is the height of the ceiling; ‫)ݖ(ܶ‬ is the vertical temperature distribution function. ‫ܪ(ݎ‬ ᇱ ) = min(‫ݎ‬ ) (4) ‫′ܪ‬ is the height of the smoke layer obtained by the integral ratio method, and can also be considered as the interface height of the strong stratification zone and the transition layer. Then, the integral ratio method is used again to divide the temperature field below ‫′ܪ‬ into two parts: the transition layer and the air layer.
The integration ratio of the transition layer is: The integral ratio of the lowest air layer is: The sum of the integral ratios of the transition layer and the lowest air layer is: Where ‫ݎ‬ ଷ and ‫ݎ‬ ସ are the integral ratios of the transition layer and the lowest air layer, respectively. ‫ܪ‬ is a new assumed smoke layer height.
‫ܪ(ܴ‬ ᇱᇱ ) = min൫‫ݎ‬ ൯ (8) When ‫ݎ‬ reaches the minimum value, the temperature distribution of transition layer and air layer below ‫′ܪ‬ height reaches a relatively uniform stratified state. ‫′′ܪ‬ is the height of interface between transition layer and air layer, that is, the height of smoke layer determined by integral ratio method of transition layer. The functional relationship between the two integral ratios and height is shown in Fig. 2.

Fig. 2.
Relationship between the two integral ratios.

Least-square method based on transition layer theory
The least square method based on the transition layer is similar to the integral ratio method based on the transition layer. The difference is that the variance is used instead of the integral ratio [4]. The specific calculation method is as follows: Two constant values ܶ ௩ଵ and ܶ ௩ଶ are the average values of the temperatures T(z) in the lower layer [0, ‫ܪ‬ ] and the upper layer region ‫ܪ[‬ , ‫]ܪ‬ , respectively, as follows.
The temperature averages ܶ ௩ଵ and ܶ ௩ଶ of the lower and upper layers are a function of ‫ܪ‬ . The variance of the upper and lower layers is calculated by the following formula: The total intra-layer variance can indicate the degree of temperature difference within the layers, which is defined as follows: Where ߪ ଵ ଶ and ߪ ଶ ଶ are the variance of the temperature data of the upper layer and lower layer, respectively. ‫ܪ‬ is the assumed smoke layer height, ‫ݖ‬ is the height of data points; ‫ܪ‬ is the height of the ceiling; ‫)ݖ(ܶ‬ is the vertical temperature distribution function.
‫ܪ‬ ′ is the height of the smoke layer obtained by the least square method, and can also be considered as the interface height ‫′ܪ‬ of the strong stratification zone and the transition layer. Then, the least square method is used again to divide the temperature field below ‫′ܪ‬ into two parts: the transition layer and the air layer.
Similarly, two constant values ܶ ௩ଷ and ܶ ௩ସ are introduced, which are the average values of temperature The variance of the lowest air layer and the transition layer is calculated by the following formula: The total intra-layer variance is written as follows: Where ߪ ଷ ଶ and ߪ ସ ଶ are the variance of the temperature data of the transition layer and the air layer, respectively: When the variance value ߪ ଶ reaches the minimum value, the temperature distribution of transition layer and air layer below ‫′ܪ‬ height reaches a relatively uniform stratified state. ‫′′ܪ‬ is the height of interface between transition layer and air layer, that is, the height of smoke layer obtained by least square method of transition layer. The functional relationship between the two variances and height is shown in Fig.3.   Fig. 3. Relationship between the two variances.
The numerical integration and differentiation involved in the above two methods are solved by MATLAB.

Small-scale experiments
According to a typical urban road tunnels and Froude similarity model rate. the small-scale model test bench presented here was built in scale 1:10, as shown in Fig.4. The model tunnel cross-section was made of 11 pieces of 1m long tempered glass, with width of 0.5m, height of 1.0m and longitudinal length of 11m. The frame of the tunnel model is welded with stainless steel plates. The roof, floor and side wall of the tunnel were made up of 5 mm thick fire-proof partitions to reduce heat loss. The other side wall is made of 3 mm thick fire-proof glass, which is used to observe the experimental phenomena. The ambient temperature is approximately 28~29 in the experiments. One opening of model tunnel was attached with an axial flow fan to provide longitudinal ventilation. Ethanol (95%) was used as the fuel of the pool fire, and the heat release rate was set as 2.72MW. A 15 cm×15 cm oil pan was used in the test. The mass loss rate of ethanol during combustion in oil pan was measured by electronic scale, and the combustion efficiency of ethanol and the calorific value of ethanol combustion were inquired to obtain the heat release rate of methanol during combustion. Two typical longitudinal velocity in tunnels were discussed. Similarly, two tests were carried out and discussed. The details of the tests are shown in Table 1.
The K-type sheathed thermocouples connected to a data logging system were used to measure the temperature. A thermocouple tree is installed 4.5m from the left opening and 1.5m from the fire source to record the vertical temperature distribution of the observation points. The thermocouple tree is equipped with 10 measuring points, and the vertical adjacent spacing is 0.05m. The highest point is 0.02m below the top of the tunnel, as shown in Fig. 4. To verify the reliability and repeatability of the experiment, the results of two repeated experiments under the same conditions were compared, and the uncertainty of the maximum measurement temperature was determined to be less than 6%. During the experiments, the smoke cake was used as the tracer to observe the spread of smoke in the tunnel. In order to enhance the visualization of the smoke layer, the semiconductor laser was used as the light source to form any piece of light plane in the vertical direction. A camera was located at the front of the tunnel to record the changing of the interface height of smoke layer. The interface height of smoke layer read by visual observation can directly reflect the visibility stratification of fire scene, so the visual reading results are set as the benchmark to measure the accuracy of other adopted methods. In this work, the visual reading results were recorded every 1s, and the average of 3 independent visual reading results was taken.  5 shows the changing of the smoke layer in the small-scale experiment. It can be seen that the smoke layer is not stable in the initial period after ignition, showing a fluctuation pattern, as shown in Fig. 6 of 50s. About 150s after ignition, the interface of smoke layer is much clearer than that of the initial stage. In this stage, the fluctuation of the smoke layer becomes smaller, and the smoke layer becomes easier to observe. However, there is still a mixing region between the smoke flow and the lower air flow, which can be regarded as a transition layer. This fluctuation is caused by the shear effect of the longitudinal wind on the smoke flow. With the development of fire into the attenuation stage, the buoyancy of smoke flow is weakened, and the smoke fluctuation is intensified under the effect of longitudinal wind. It is more difficult to observe the smoke layer. Fig.  6 shows the smoke stratification structure in the stable stage of fire in Test 2. It was shown that the interface between the smoke flow and the lower air flow under the entrainment of longitudinal wind is not smooth, and there is a transition area where the smoke and air mix with each other. Compared with the upper smoke, the smoke in this transition area is not continuous, showing the law of fluctuation. The height of this transition area is between 0.22m and 0.28m.  The vertical temperature distribution in the stable stage of fire in Test 2 and the height of the smoke layer calculated by the integral ratio method, the least square method and the two methods based on the transition layer, and the height of the upper and lower boundary of the transition region is shown in Fig. 7. It can be found that the overall temperature above the transition layer is relatively higher, and the temperature inside the transition region varies greatly, while the temperature below the transition layer changes relatively little, close to the ambient temperature. It can also be seen from Fig.  7 that the results calculated by the integral ratio method and the least square method are closer to the upper boundary of the transition layer, and the two improved methods based on the transition layer are closer to the lower boundary of the transition layer, which can be considered closer to the real height of the smoke layer.
The vertical temperature distribution in the stable stage of fire in Test 2 and the height of the smoke layer calculated by the integral ratio method, the least square method and the two methods based on the transition layer are shown in Fig.7, and the height of the upper and lower boundary of the transition region is also shown in Fig. 7. It can be found that the overall temperature above the transition layer is relatively higher, and the temperature inside the transition region varies greatly, while the temperature below the transition layer changes relatively little, close to the ambient temperature. It can also be seen from Fig. 7 that the results calculated by the integral ratio method and the least square method are closer to the upper boundary of the transition layer, and the two improved methods based on the transition layer are closer to the lower boundary of the transition layer, which can be considered closer to the real height of the smoke layer.   Fig. 9 shows the height of smoke layer varying with time in Test 1 and Test 2, respectively. The integral ratio method, least square method, integral ratio method based on transition layer, least square method based on transition layer and N-percentage rule were used to calculate the height of smoke layer, and the reference standard of visual observation value was also introduced.
The smoke layer interface height exhibits continuous fluctuations with time, as shown in Fig. 9. In the initial stage, the smoke layer fluctuates sharply. After 80s of ignition, the fluctuation amplitude decreases slowly. At this stage, the smoke layer interface height in Test 1 is stable near 0.27 m, while that in Test 2 is stable near 0.23 m, which indicates that the higher longitudinal velocity makes the smoke layer interface height lower. Comparing the results of different methods, it can be found that the results calculated by the integral ratio method and the least square method are obviously higher, which is consistent with the previous conclusions. The difference between the results of the method based on the transition layer and the visual observation value is relatively small, especially the result of the least square method based on the transition layer, which is closest to the visual observation value in the relative stable stage, as shown in Fig. 9. This shows that the least squares method based on transition layer can be used to determine relatively accurate results of interface height of smoke layer in relatively stable stage, even under longitudinal ventilation conditions. However, in the fire decay stage of Test 1, there were some differences between the calculated results and the visual values of the various methods.
Comparing the heights of the smoke layer interface calculated by different methods, it can be found that the results of the N-percentage rule of different N values are quite different, which indicates that the N value has a significant influence on the result of the N-percentage rule. When the longitudinal velocity is large, the temperature at the low point is significantly higher than the ambient temperature.

Conclusions
This work provides a comparative study of different methods for determining the height of the smoke interface in a tunnel fire scenario, a parameter of great interest to fire safety science. The small-scale model tests were carried out to investigate the distribution and variation of the smoke layer interface height of different longitudinal velocity. According to the research results, the following conclusions were drawn: (1) Based on the existing integral ratio method and least square method, the idea of transition layer is introduced, and the integral ratio method based on transition layer and the least square method based on transition layer were proposed. The smoke layer interface height determined by the new method is compared with other existing methods, including visual observation method, integral ratio method and N percentage method.
(2) The calculation results of the least square method based on the transition layer are in good agreement with the visual observation results, which shows that it is more suitable for tunnel fire with longitudinal wind than the existing least square method.
(3) N-percentage rule is not suitable for the situation of relatively poor smoke stratification with longitudinal wind. For the condition of poor temperature stratification, the selected N value should not be too small.