Reynolds number–dependent mass flow rate calculation for pneumatic pipes

In fluid power applications, pipes may have important effects on the system performances. In hydraulics, simple pipe models could give a good approximation of the pressure drop, but in pneumatics, accurate approximations are more complex because of the fluid compressibility. Therefore, simple but accurate models of air mass flow rate in pipes are still needed for system design and control purposes. The main contribution of this article is to propose a novel analytical formulation of the mass flow rate in pneumatic pipes by taking into account the pipe friction factor and more specifically its dependency on the flow Reynolds number. Accordingly, it makes it applicable to any circuit conditions (variable pressure at upstream or downstream pipe sides). The proposition here is a physically based adaptation of the flow approximations as defined in the first edition of the ISO 6358 standard and only uses the two conventional flow rate parameters; the novelty is to introduce a specific relationship between these flow parameters and the pipe friction factor. The proposed formulation provides a simple analytical way to predict accurately pipe behaviour in pneumatic circuits for sonic and subsonic flows. Based on this formulation, the calculated mass flow rates show a good agreement with measurements obtained with pipes of different diameters and lengths, even for variable upstream pressure conditions. Finally, limitations of existing formulations are discussed and advantages of the proposed approach highlighted.