Stresses in a bulk solid in a cylindrical silo, including an analysis of ratholes and an interpretation of rathole stability criteria

Abstract

A two-dimensional model for stresses in a cohesive bulk solid in a vertical, cylindrical vessel has been developed, assuming a rotated, circular arc orientation of principal stresses, after Enstad [1995. On the theory of arching in mass flow hoppers. Chemical Engineering Science 30 (10), 1273–1283] and Li [1994. Mechanics of arching in a moving bed standpipe with interstitial gas flow. Powder Technology 78, 179–187]. The model assumes that principal stresses form a spherical dome surface within the vessel. At the wall, the surface makes a constant angle with the wall normal, necessitating a progression of arc centre up the vessel. An incremental element of upper and lower surfaces with vertical height δ x at the wall has variable thickness with arc angle ε measured from the vessel centre. Positions within the vessel are located by height at which the arc cuts the vessel wall x, and arc angle ε . Rotational symmetry is assumed through azimuthal angle θ . This gives 3 principal stresses: σ R ; σ ε ; σ θ . Static vertical and horizontal force balances yielded two partial differential equations, and a third was obtained by assuming a known ratio between effective stresses σ ε ′ and σ R ′ . These equations were integrated numerically to give stress surfaces in x – ε space. The model simulated variations in the state of stress across the arc surface, and in some conditions, this lead to large variations in stress between the wall and vessel centre. The model accurately predicted the minimum outlet radius required for flow of a cohesive bulk material. The presence of a vertical, co-axial rathole was modelled with minor geometrical modifications. The rathole was assumed to be an unconfined plane of principal stress where the radial stress, σ R , acted vertically downwards and its value increased linearly with bed depth, contrary to Jenike's [1964. Storage and flow of solids, Bull. 123, University of Utah, USA] assumption of a limiting stress in deep beds. Conventional stability criteria ( σ R < = f c for stability), suggest that only shallow ratholes can exist. However, a modified yield criterion, allowing for the curvature of the rathole, enabled deep, stable ratholes to exist. Results show that the stability of deep ratholes depends upon the rathole radius and the vessel radius.