A non-dimensional criterion and its proof for transient flow caused by fire in ventilation network

Application of 1D Transient Flow Model to a network analysis finds that the fluids in two branches (which are connected in parallel and contain no drive source) may no longer flow in a same direction while the network flow state changes from being steady into being unsteady due to a fire in the ventilation network. This phenomenon can be referred to as non-parallel flows in parallel branches. A non-dimensional criterion ξ is deduced theoretically. It is proved that for any two branches which are connected in parallel and contain no drive source, Non-parallel flows can be expected in the branches only if ξ is not equal to unity. If ξ is equal to unity, the non-dimensional flow velocities in the two branches will always be equal to each other. If ξ is not equal to unity, a prediction can be made on which one of the non-dimensional velocities in the two parallel branches will change more rapidly.