Study of the critical velocity of the tunnels using an analytical approach

Abstract

Fire safety is one of the major design issues of tunnel engineers. Longitudinal ventilation is a common method for exhausting smoke and hot gases in tunnel fires. The minimum ventilation air velocity along the tunnel to prevent smoke back layering is called the critical velocity. This critical velocity is the key element in designing the longitudinal exhaust system. Several researchers have studied the impact of different parameters on critical velocity, including tunnel geometry, tunnel height, and fire magnitude, numerical and experimentally. In this study, an analytical solution was used to solve a third-order non-linear differential equation to determine the critical velocity of the tilted tunnel.

Dimensional analysis can significantly reduce the costs associated with the experimental study in the full-scale tunnels. The Froude number representing the power of the buoyancy force against the inertial force is widely used in the critical velocity studies. This study validated our analytical results with critical values obtained from an experimental and numerical simulation in a scaled model with a ratio of 1:8.

The results showed that the values calculated for the critical velocity employing an analytical solution were lower than the numerical and experimental studies' values.

Data from the latest international standard was used to enhance the precision of the critical velocity calculation. We showed that using a modified Froude number can significantly increase the accuracy of the analytical solution. The critical velocity values obtained using the modified Froude number were then compared with experimental results from full-scale tests.

This study emphasized that the analytical solution for the critical velocity saves a significant amount of time compared to the iterative solutions while keeping the accuracy in a reasonable range.

Introduction

To facilitate travel in large cities around the world, tunnels are necessary. Tunnels are crucial components for developing railway networks and roads under the oceans or in remote high mountainous areas. Tunnels are subject to fire risk due to vehicle collisions or tunnel maintenance equipment incidents, regardless of their size and location. One of the key challenges for tunnel engineers is the construction of an efficient exhaust system for smoke and toxic gases in the event of a fire. According to the National Fire Protection Association (NFPA), 70% of fatalities in tunnel accidents are due to inhaling toxic gases [1]. A further risk factor leading to the number of deaths in tunnel accidents is the decrease in visibility [2,3]. Therefore, an effective exhaust system plays a vital role in the safety of tunnels [1,4].

The longitudinal exhaust system is one of the most common designs for extracting smoke generated by a fire in tunnels. In this design, the fresh air moves the smoke downstream, and the exhaust fans will remove it. An important parameter for calculating the exhaust system's efficiency is called the smoke back-layering length (BL) where the BL is defined as the length of the reversed smoke flow upstream of the fire (Fig. 1). The design objective of the tunnel exhaust system is to keep the BL as low as possible, where the ideal BL value is zero, but this may not be achieved. The longitudinal air velocity to drive the smoke downstream towards the exhaust fans and make the BL zero is called critical velocity.

There have been various theoretical and experimental studies on the factors contributing to the critical velocity in the tunnels [[5], [6], [7]]. These studies showed that the tunnel geometry, heat release rate (HRR) from the fire and the tunnel slope are the most important factors in the critical velocity [[8], [9], [10]]. Thomas [11] first introduced a semi-empirical correlation for the minimum air velocity to prevent the smoke from returning in a 90 cm × 90 cm wind tunnel. The results suggested the critical velocity was proportional to the cube-root of the heat release rate. Danziger [12] introduced an alternative correlation to calculate the critical velocity in a horizontal tunnel equipped with a longitudinal exhaust system. They concluded that in fires with low HRR, (when the flames do not reach the tunnel's ceiling) the critical velocity was proportional to ${\left(HRR\right)}^{1/3}$and the smoke temperature. The equation defined the smoke temperature as a function of the heat release rate [[13], [14], [15]] and therefore was a non-linear equation. Although the smoke temperature was considered in the critical velocity equation, it neglected the effect of the tunnel slope and therefore was only valid for horizontal tunnels.

The slope of the tunnel is a crucial element in managing fire and smoke in the tunnels. Studies have shown that the critical velocity in tunnels tilted along the downstream is greater than the critical velocity in the horizontal tunnels [[16], [17], [18], [19], [20]] and this leads to a different strategy for controlling the back-layering effect in the non-horizontal tunnels. Kennedy [21] accounted for the effect of the tunnel slope on the critical velocity equation by adding the Froude Number (Fr) concept to Danziger's [12] critical velocity correlation. However, the equation remained non-linear. Kennedy's non-linear equation has been widely used in many tunnel ventilation programs, including the Subway Environmental Simulation (SES) Computer Programme [22].

Iterative methods are a common technique for solving non-linear equations and have been used by engineers and researchers to determine the critical velocities of tunnels. Despite the advantages of iterative methods in finding a solution for complicated equations, it suffers from many drawbacks. For example, developing a program for an iterative solution is time-consuming; it requires a good initial guess to initiate the solution influencing the convergence rate, and a residual error is inherent with iterative solutions. With the advancement of high-speed computers, numerical simulations are becoming a standard for predicting the critical velocity and distribution of the hot gases in a tunnel [17,[23], [24], [25]]. While the numerical simulations provide better visualization of the toxic gas behavior and distribution in the tunnels, they require an iterative solutions strategy.

Tarada [26] later introduced an analytical solution for the critical velocity, which avoided an iterative solution. Despite the significant advantages of Tarada's method compared with the iterative solutions (e.g. the need for a reliable initial guess, residual errors, and programming time), it has not been well-received by researchers and engineers so far. In this paper, the authors investigated the accuracy of Tarada's method in predicting the critical velocity in horizontal and tilted tunnels at different HRRs. The critical velocity was obtained from experimental and numerical work by Li et al., [25].

However, due to the limited available full-scale experiments, using the Froude number conservation is a common method to verify the simulation or analytical calculations of the critical velocities. The experimental results from three full-scale tunnels have been used to validate the analytical solution presented in this study [27,28].