The axial and free surfaces are geometric features of tumbling mill flows. Their shape is a characteristic of the particle interactions within the charge body. This means that any modelling scheme capable of predicting these surfaces automatically satisfies the underlying physics of the system and thus can be used as a global continuum signature for breakage and mixing. Elbasher et al (2021) presented a model that successfully captured the shape of the axial free surface by comparing their results to axial free surface predictions from Chou et al (2009). Due to the encouraging results from that work, we leverage off the geometric symmetry between the axial free surface and sigmoid-shaped (S- shaped) free surfaces and present an analytical solution, free from any empirical fitting parameters, to predict S-shaped free surface. These predictions are then statistically compared to those derived from previously conducted Discrete Element Method (DEM) simulations and positron Emission Particle Tracking (PEPT) experiments. This model provides an alternate to complicated numerical routines that fit S-shaped free surfaces to experimental data; these are often prone to regions of numerical instability depending on the nature of the data-set.
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