| Summary: | The critical velocity plays a significant role in the design of the ventilation system in tunnels, from which the ventilation velocity of fresh air for preventing the smoke back-layering in tunnel fires can be determined. In this work, a full-range analytical solution to the equation for the critical velocity was derived. Specifically, we first proved that the unconstrained equation for the critical velocity has and only has one positive real solution, and then we proved that the positive real solution has a unified expression for different cases of tunnel fires. Finally, the analytical solution that satisfies the constraints was obtained, which was also compared with the iterative solution. The result shows that the analytical solution can avoid the negative and complex solutions automatically, which is more stable than the iterative solution. Furthermore, the analytical solution was compared with the simulation and experimental results, which were found to be in excellent agreements with each other. For tilted tunnels, the influence of the Froude number on the analytical solution of the critical velocity was also investigated by comparing the critical velocities obtained based on two different correlations of the Froude number.
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