Semi-Empirical Model of Ceiling Jet in Tranquil Flow Region in Arched Ceiling Tunnel
Article information
Abstract
The present study focuses on the temperature and velocity of the smoke layer in the tranquil flow region of a naturally ventilated tunnel with an arched ceiling. Fire experiments were conducted using a laboratory-scale model tunnel with a height of 0.275 m, a floor width of 0.41 m and a length of 10 m. To predict the temperature at arbitrary positions within the tranquil flow region, the temperature attenuation equation, expressed as a weighted average of two exponential functions, is presented. A semi-empirical prediction formula is proposed to estimate the velocity attenuation of the smoke layer by combining approximate analytical solutions from a simplified theoretical model with laboratory-scale model tunnel experiments. Sub-models for estimating the variables included in the velocity attenuation prediction formula - such as the starting position of the tranquil flow region, the temperature rise at this position, and the smoke layer thickness - are also presented. These sub-models form a closed system of equations for the velocity attenuation equations. The vertical distribution of temperature in the smoke layer propagating in the tranquil flow region has a similar distribution shape regardless of the distance from the fire source. The velocity distribution also has the same characteristics. These distributions are described by applying the profile method. The proposed formulae are expected to be useful tools for smoke flow prediction, evacuation planning and disaster prevention planning in the event of a tunnel fire.
1. Introduction
A tunnel is a man-made civil structure and is an underground space that is axially elongated relative to the height or width of its cross-section. Tunnels are built to improve the efficiency of traffic and transportation, and their cross-sectional shapes vary according to their purpose and the construction environment.
Pioneering research by Delichatsios[1] is carried out on the flow behaviour of the smoke layer in the tranquil flow region. Understanding the properties of the smoke layer propagating in the tranquil flow region is important for smoke control and emergency evacuation, as the thickness and thermal properties of the smoke layer, such as temperature and velocity, affect both the individual evacuation behaviour in a fire and the determination of the capacity of smoke evacuation equipment during the design phase. For this reason, the movement of smoke along a tunnel axis has been subject of extensive research in the field of fire safety engineering.
Most laboratory-scale tunnel fire experiments are carried out in tunnels with rectangular cross-sections, providing valuable data on smoke layer properties such as thickness[2,3,4] and temperature attenuation along the tunnel axis[5,6]. In contrast, detailed experimental results on the flow properties of the smoke layer along the longitudinal axis of tunnels with arched ceilings are limited due to their peculiar ceiling geometry[7,8]. As a result, there has been little research into practical models for predicting temperature attenuation that take into account the starting position of the tranquil flow region and the temperature at that position. There are also few studies that focus on the relationship between temperature and velocity in the smoke layer. The semi-empirical model reported by Oka and Oka[9], which focuses on the relationship between temperature and velocity, is limited in its application because it is derived from the results of experiments carried out in a laboratory-scale tunnel with a rectangular cross-section and a specific aspect ratio, but the smoke layer thickness included in the prediction equation has an experimental value and is therefore of limited applicability.
The purpose of this study is to elucidate the properties of smoke layers propagating in a tunnel with an arched ceiling. Specifically, focusing on the smoke layer in the tranquil flow region, a prediction formula for the temperature attenuation of the smoke layer, expressed as a weighted average of two exponential functions, taking into account the starting position of the tranquil flow region and the temperature at that point under natural ventilation, is introduced. In addition, an easy-to-use model for the velocity attenuation of the smoke layer considering the temperature attenuation and the thickness of the smoke layer in the tranquil flow region, is presented.
2. Theoretical Approach
2.1 Simple method for temperature attenuation of smoke layer
Ingason et al.[10] has been shown that the temperature attenuation of the smoke layer obtained from both full- scale tunnel fire experiments, including the Runehamar Tunnel Test[11-14], and laboratory scale tunnel fire experiments correlates well with a weighted average of two exponential functions. This method is considered a useful engineering tool to show the temperature attenuation of the smoke layer along the tunnel axis and has been used by many researchers[10,15-18]. We also use the weighted average of two exponential functions expressed in Eq. (1) to predict the temperature attenuation of the smoke layer in the tranquil flow region in the tunnel with arched ceiling. The coefficients A1, A2, k1, and k2 included in Eq. (1) will be determined later from experimental results.
where
Oka and Oka[6] proposed analytical solutions for the variables xref and △
where
2.2 Semi-empirical model for predicting velocity attenuation of smoke layer
The CFAST two-zone fire model used the following correlation to calculate the velocity of the smoke layer.
where
The value of Cd = 0.7 recommended by Bailey et al.[19] can be considered representative of the shooting flow region, where inertial forces dominate over buoyancy, and is therefore considered inapplicable to the tranquil flow region. Figure 1 shows the variation of Cd with the distance from the fire source by substituting the mean temperature rise and mean velocity measured in the experimental data obtained in this test into Eq. (5). The result indicates that the Cd in the tranquil flow region is not a constant value but varies as a function of the distance from the fire source. Therefore, a semi-empirical model to estimate the mean velocity was introduced to treat Cd as a function of distance from the fire source.
Variation of flow coefficient Cd with the distance from the fire source.
To determine the functional form of Cd, Eq. (4) is substituted into the following momentum conservation equation.
where f is the friction coefficient which, assuming Reynolds analogy, is related to the Stanton number. Assuming that the Boussinesq approximation can be applied to the smoke layer propagating in the tranquil flow region, Eq. (5) can be transformed into the following ordinary differential equation.
Solving the simultaneous equation as a special solution of Eq. (6) gives F=C(x)exp(-ax). Substituting this relation into Eq. (6) and considering the temperature rise, △
By integrating Eq. (7), the integral constant C(x) can be given in Eq. (8).
By substituting the function C(x) into the variable F, the flow coefficient Cd can be expressed as follows:
The integral coefficient C0 is determined based on the fact that the flow coefficient at x = xref is Cd,ref and that △
Substituting Eq. (10) into Eq. (4) and considering that
3. Experimental Procedure
The laboratory-scale tunnel fire tests used an arched tunnel with a height of 0.275 m, a width at floor level of 0.41 m, and a length of 10 m, as shown in Figures 2(a) and 2(b). The arch section of the tunnel is made of 40 mm thick cylindrical calcium silicate and the floor of 50 mm thick calcium silicate slab.
Outline of laboratory-scale model tunnel; (a) side view and overview of the temperatrue and velocity measurents, (b) sectional view, (c) overview of the setup for the PIV measurement.
The temperature of the smoke layer along the longitudinal axis of the tunnel during the quasi-steady burning period was measured in detail using a temperature measuring device. The temperature measuring device was positioned at a distance of 2.0 m from the opening at one end and was equipped with K-type thermocouples with a strand diameter of 0.2 mm, each thermocouple being installed vertically from the tunnel ceiling at the following positions 0.001, 0.05, 0.01, 0.015, 0.02, 0.03, 0.04, 0.05, 0.065, 0.08, 0.095, 0.115, 0.135, 0.16, 0.185, 0.215, 0.245, and 0.275 m. The recorded temperature is influenced by the heat given off by the flames and by the heated tunnel walls and ceiling. To adjust the distance from the fire source to the measurement position, the position of the fire source was moved 10 times while the temperature measurement position remained fixed. The number in Figure 2(a) represents the position of the fire source. The distance from the fire source to the measurement position was listed in Table 1.
Measurement Positions for Vertical Profiles of Temperature and Velocity
Velocities were measured using a two-dimensional particle image velocimetry (2D-PIV) system as shown in Figure 2(c). Similar to the temperature measurement, the distance from the fire source to the velocity measurement position varied in 9 times by moving the fire source position as listed in Table 1. Three to four velocity measurements were repeated at each distance. Velocity data were obtained at approximately 3 mm intervals using the Koncerto software (Seika Digital Image Corp.). Dioctyl Sebacate (DOS) oil droplets (average particle size of 1 μm) were used as tracer particles.
The fire source was a 0.1 m x 0.1 m square diffusion burner fuelled with liquefied petroleum gas (LPG). The heat release rate varied in three steps of 1.58, 2.11, 2.74 kW. The flame height at these heat release rates corresponds to half to full height of the tunnel.
4. Results and Discussion
4.1 Smoke layer thickness
In fire tests with hydrocarbon fuels, the smoke layer is visible and the thickness of the smoke layer propagating along the tunnel ceiling can be estimated. The average thickness is determined from the visual evidence of soot accumulation on the wall measured at three points at x/H = 4, 6, and 8 and was found to be 0.108 m. Because of the gradation of soot adhesion at the interface between the smoke layer and the fresh air, there is some arbitrariness in determining the height of the lower end of the smoke layer by observation. In this study, the height of the lower end of the smoke layer is therefore determined from the vertical distributions of temperature and velocity. Five methods are used, such as the N-percentage rule (N = 37 and 50)[20], the 2nd order central differences method[21], the buoyancy frequency method[22] and the inflection point method[3], which have been used to determine the height of the lower end of the smoke layer propagating under the ceiling formed by a fire occurring in a tunnel with a rectangular cross-section from the vertical temperature distribution[3,23].
The thickness of the smoke layer is defined as the distance from the height of the lower end of the smoke layer to the tunnel ceiling.
Figure 3 shows the comparisons of the smoke layer thickness obtained by applying the above five methods to the vertical distributions of temperature and velocity with the visual observation. It can be seen from Figure 3 that the buoyancy frequency method is the best for determining the height of the lower end of the smoke layer from the vertical temperature distribution, while the N-percent method (N = 37) is the best for determining the height of the lower end of the smoke layer from the vertical velocity distribution, which is closest to the visual result.
Comparison of the smoke layer thickness obtained from each method for estimating the height of the lower end of the smoke layer with that obtained from visual observation; (a) Smoke layer thickness derived from temperature distribution, (b) Smoke layer thickness derived from velocity distribution.
As shown in Figure 4, the smoke layer thicknesses determined by each method, namely the buoyancy frequency method for the temperature distribution and the N-percent method (N = 37) for the velocity distribution, are consistent regardless of the data such as temperature or velocity. The results of the smoke layer thickness determined by both definition methods showed that the smoke layer occupied approximately 30% of the maximum tunnel height below the ceiling. Figure 4 also shows the variations in smoke layer thickness derived from the measured vertical distributions of temperature and velocity in the range 3.8<x/
Variations in smoke layer thickness derived from temperature and velocity distributions with distance from fire source.
Figure 5 shows the variation of the coefficient b with the heat release rate. The value of the coefficient b can be obtained analytically using Eq. (12)[6] based on the smoke layer thickness, hT/
Variation of coefficient b with heat release rate.
4.2 Estimation method of mean temperature rise and mean velocity
Figure 6 shows the schematic diagram to explain how to calculate the mean value from the vertical distribution measured at the position of 2.25 m from the fire source. Figure 6(a) shows the vertical distribution of temperature rise at the position of 2.25 m from the fire source under the condition of Q = 2.11 kW. The height of the lower end of the smoke layer was determined by applying the buoyancy frequency method to the vertical temperature distribution. The thickness of the smoke layer is defined as the distance from this height to the tunnel ceiling. In this case, the thickness of the smoke layer is 0.095 m. The mean temperature rise at the position of x, △
Schematic diagram to explain how to calculate the mean temperature
rise △
To calculate the mean velocity from the vertical distribution of velocity, as with the temperature, instead of using the maximum measured velocity value, the estimated maximum velocity is employed by applying a quadratic function to the measured data of three data consisting of the maximum velocity and the two adjacent points. The height of the lower end of the smoke layer was taken as the height where the velocity becomes 37% of the estimated maximum velocity. The mean velocity was obtained by dividing the integral value of the velocity from the height of the lower end of the smoke layer derived from the velocity distribution to the height of the tunnel ceiling by the thickness of the smoke layer derived from the velocity distribution.
Figure 7 shows the relationship between the mean value calculated using the method described above and the estimated maximum value, defined as the peak value of a quadratic function fitted to three points including the measured maximum value. It can be seen that, for both temperature and velocity, a consistent relationship is maintained between the mean value and the estimated maximum value, even when the measurement position and the heat release rate change. According to the measured results, the estimated maximum value for temperature can be obtained by multiplying the mean value by 1.27 and for velocity by multiplying the mean value by 1.40.
Relationship between estimated maximumvalues and mean values; (a) temperature, (b) velcoity.
4.3 Estimation of starting position of tranquil-flow region and its dependence on heat release rate
As a representative example, Figure 8 shows the relationship between the mean temperature rise at each temperature measurement position and the distance from the fire source for Q = 2.11 kW. The mean temperature rise was obtained using the method described above. Using the data in the range 0.75 m < x < 5.75 m, the temperature attenuation characteristic with distance from the fire source was approximated by an exponential function. The position of xref was determined as the position where the mean temperature rise calculated by Eq. (3) appears on the curve shown by the solid line in Figure 8.
Mean temperature attenuation as a function of distance from the fire source.
The dependence of the starting position of the tranquil-flow region on the heat release rate is shown in Figure 9. By applying power law approximation, the values of the intercept and power in Eq. (2) were determined to be β = 12.3 and γ = 0.223.
Relationship between starting position of the tranquil-flow region and heat release rate.
4.4 Relationship between Stanton number and heat release rate
The Stanton number at each position was estimated using the correlation given in St = a/(
Figure 10 shows the variation of the Stanton number with the distance from the fire source. It can be seen that the Stanton number tends to increase gradually with distance from the fire source for each heat release rate. However, as the scientific focus of this study is the effect of the heat release rate on the Stanton number, it is assumed that the variation of the Stanton number with distance from the fire source is small, and the average value was taken as representative for each heat release rate.
Variation of Stanton number with distance from fire source.
The heat transfer coefficient can be expressed as a linear function of velocity according to the Jürges' formula and can be transformed into Eq. (13).
where the coefficients A and B are constants. By transforming Eq. (12), which expresses the thickness of the smoke layer, to
There is a relationship
The result of this experiment applied to Eq. (15) is shown in Figure 11, where the values of the coefficients in Eq. (15) are C = 0.00640 and D = 0.0408, respectively, from the application of the least squares method.
Relationship between Stanton number and variable composed of aspect ratio of tunnel cross-section and heat release rate.
4.5 Temperature attenuation in tranquil flow region
Figure 12(a) shows the variation of the mean temperature rise with distance from the fire source by varying the heat release rate. It appears that the similar temperature attenuation property with distance holds regardless of the heat release rate. Figure 12(b) shows the relationship between the dimensionless temperature rise and the dimensionless distance. △
Temperature attenuation of smoke layer along the tunnel axis in the tranquil flow region; (a) measured data, (b) Approximation using the weighted average of two exponential functions given by Eq. (1).
In order to express the temperature attenuation characteristic in Eq. (1), the least squares method was applied to a data set consisting of dimensionless temperature rise and dimensionless distance, and the values of the coefficients incorporated in Eq. (1) were determined under the constraint of A1 + A2 = 1. The determined values of the coefficients are given in Table 2. The values of A1 and A2 differ from those reported by Ingason et al.[14] due to the different reference points for considering the temperature attenuation in the smoke layer. However, a comparison of the magnitude of each coefficient shows that the relationships between A1 and k1, and A2 and k2, are consistent. Furthermore, the values of k1 and k2, which represent the attenuation gradient, are almost identical.
4.6 Velocity attenuation in tranquil flow region
Figure 13(a) shows the variation of the mean velocity of the smoke layer with distance from the fire source by varying the heat release rate. The velocity of the smoke layer decreases much more slowly than the temperature attenuation with the distance from the fire source. It appears that the similar velocity attenuation characteristic with distance holds regardless of the heat release rate. Figure 13(b) shows the relationship between a dimensionless velocity and a dimensionless distance. The y-axis is the dimensionless velocity
Velocity attenuation of smoke layer along the tunnel axis in the tranquil flow region; (a) measured data, (b) Normalised data.
The results of comparing the predicted values obtained by substituting the values of the coefficients A1, A2, k1 and k2 listed in Table 2 and the variable α in Eq. (11) with the measured values are shown in Figure 14. The value of the St number included in the variable α was calculated using Eq. (15). The results of the relative error defined by Eq. (16) are shown in Table 3. From this set of results, it can be confirmed that Eq. (11) reproduces the velocity attenuation characteristics of a smoke layer flowing in the tranquil flow region along the axis of the tunnel with arched ceiling, regardless of the heat release rate.
Comparison of predicted using Eq. (11) and measured values of the smoke layer velocity propagating in the axial direction of the tunnel in the tranquil flow region; (a) Q = 1.58 kW, (b) Q = 2.11 kW, (c) Q = 2.74 kW.
Relative Error
4.7 Application of profile method to vertical distribution of temperature and velocity within a smoke layer
As a representative example, the measurement results of the vertical distribution of the temperature rise at Q = 2.11 kW are shown in Figure 15(a). The temperature distribution shows a convex shape, with a maximum temperature rise at each measurement point at the peak. It can be seen that the shape of the vertical temperature distribution gradually changes from a sharp to a gently decreasing shape as the distance from the fire source increases. The results of plotting the dimensionless temperature rise, △
Vertical distributions of temperature and velocity at Q = 2.11 kW. (a) Temperature distribution, (b) Nomarised temperature distribution, 8c) Velocity distribution (d) Normalised velocity distribution.
The vertical distribution of the velocity along the longitudinal axis of the tunnel at Q = 2.11 kW is shown in Figure 15(c). It can be seen that the velocity decreases much more slowly than the temperature with distance from the fire source. The results of plotting the dimensionless velocity, VxVx_max', which is the velocity divided by the estimated maximum value at the measurement point, against the dimensionless distance, z/hV, which is the distance from the tunnel ceiling divided by the velocity-derived smoke layer thickness determined by the N% method (N = 37), are shown in Figure 15(d). As with the temperature, the measured data at different positions from the fire source converged on a single curve and the vertical velocity distribution showed a similar distribution shape regardless of the distance from the fire source and the heat release rate.
The results of applying the profile method described by Eq. (17) to a set of temperature and velocity derived vertical distribution data obtained at each heat release rate are shown in Figures 16(a) and 16(b). It can be confirmed that the temperature and velocity distributions can be reproduced in the range from the tunnel ceiling to the lower end of the smoke layer. The values of each coefficient using the profile method are given in Table 4. Furthermore, the results of comparing the distribution shapes of the temperature and velocity distributions reproduced by the profile method are shown in Figure 16(c). The distribution shapes are similar to each other, and when comparing the temperature and velocity thicknesses at the positions where the value of △T/△Tmax or V/Vmax is 1/e, the temperature-derived thickness is 1.07 times thicker than the velocity-derived thickness. This shows that temperature diffusion is greater than momentum diffusion, which is consistent with the relationship between turbulent viscosity and turbulent thermal diffusivity[24].
Comparison of measured vertical distributions in the smoke layer propagating in the tranquil flow region with Eq. (17); (a) Temperature distribution, (b) Velocity distribution, (c) Comparison of vertical distributions of temperature and velocity based on Eq. (17).
5. Conclusions
A series of fire tests were carried out using a laboratory scale tunnel with an arched ceiling. The main conclusions of this study are as follows.
•The thickness of the smoke layer in the tranquil flow region was determined from the temperature and velocity distributions within the smoke layer. Under experimental conditions, the heat release rate has little effect on the smoke layer thickness and the smoke layer thickness is almost constant. A method for predicting the smoke layer thickness has been proposed.
•An empirical formula, expressed as the weighted average of two exponential functions, has been proposed to estimate the temperature attenuation of a smoke layer propagating along the longitudinal axis of the tunnel considering the starting position of the tranquil flow region and the temperature rise at that position.
•From the results of temperature and velocity measurements, it was found that the flow coefficient, which is required to calculate velocity from temperature rise, is not a constant in the tranquil flow region, but varies with distance from the fire source. A semi-empirical formula has been proposed to represent the velocity attenuation, treating the flow coefficient as a function of distance from the fire source and taking into account the temperature attenuation characteristic in the tranquil flow region.
•A semi-empirical formula has been proposed to express the Stanton number, which is the key variable governing the temperature attenuation characteristics of the smoke layer propagating along the longitudinal axis of the tunnel, as a function of the heat release rate and the aspect ratio of the tunnel cross-section.
•A polynomial equation is proposed to describe the vertical distribution shapes of temperature and velocity in the smoke layer. The distribution shapes of temperature and velocity were similar to each other. Comparing the temperature and velocity thicknesses of the smoke layer at positions where the value of △T/△Tmax or V/Vmax is 1/e, the thickness derived from the temperature is 1.07 times thicker than that derived from the velocity.
Notes
Author Contributions
Conceptualization and methodology, Y. Oka; theoretical methodology, H. Oka and Y. Sakurai; Experiment and data processing, A. Tanno, C. Iwamoto, and Y. Oka; writing-original draft preparation, Y. Oka; writing-review and editing, H. Oka and A. Tanno. All authors have read and agreed to the published version of the manuscript.
Conflicts of Interest
The authors declare no conflict of interest.
Acknowledgments
The authors would like to thank Professor Ken Matsuyama of the Tokyo University of Science for kindly allowing them to use the experimental facility and the particle image velocimetry device to carry out the laboratory-scale model tunnel experiments. This work was partially supported by a Grant-in-Aid for Scientific Research (B) (No. 18H01593 and 24K01117).