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This paper studies the phenomenon of sloshing in the field of automatic machines for packaging liquid products, with specific reference to containers with planar motions. After introducing two equivalent discrete models based on a mass-spring-damper system borrowed from the literature (one linear and one non-linear), a novel method is proposed to evaluate the sloshing height of the liquid, namely the deviation of its free surface at the wall of the container from the equilibrium condition. The merits of this method are that it is easy to use, requiring no experimental evaluation of the system parameters or computationally expensive fluid dynamic simulations, and it gives good results also for highly dynamical motions. Moreover, though this paper focuses on cylindrical containers performing rectilinear movements, the presented technique can be extended to containers of arbitrary shape and generic planar motions. The method is validated by experimental tests using cylindrical containers of different dimensions and many rectilinear motion laws with maximum accelerations up to <inline-formula> <tex-math notation="LaTeX">$12~m/s^{2}$ </tex-math></inline-formula>. The results are compared with those obtained by using other methods of equal complexity available in the literature, showing the effectiveness of the proposed technique.

A Simple Model-Based Method for Sloshing Estimation in Liquid Transfer in Automatic Machines