The shape of the cohesive arch in hoppers and silos — Some theoretical considerations

Models of the cohesive arch have been developed in order to predict arch shape and test the circular arc hypothesis first proposed by Enstad [‘On the theory of arching in mass flow hoppers’, Chem. Eng. Sci., 1975, 30, 10, 1273–1283].A 2-dimensional arch was modelled. Vertical and horizontal force balances described the system, with 2 unknown: arch stress, σarc, and vertical arch co-ordinate y. A rotationally symmetrical system was also modelled and included the azimuthal stress σaz. σaz was related to σarc by a Mohr–Coulomb yield type of equation. These equations were solved numerically by an Euler method.A least squares fit was used to predict the equivalent circular arc radius R and circular arc co-ordinate ycirc. The square of deviation from the circle Σ(y − ycirc)2 was used as a statistical measure of the goodness of fit to the circular arc.The arch shape was generally an excellent approximation to the circular arc. However, the arch diverged from the circular as arch span increased.A dimensionless group was defined: the Stress–Radius Number, NSR, which incorporates the ratio of equivalent circular arc radius to the arch stress at the apex. NSR was constant for a given set of conditions and was ideally equal to 1 for a 2-dimensional arch and 2 for a rotationally symmetrical arch with no overpressure.Arch thickness models had little effect upon arch shape but had a great influence upon stress. This affected the critical outlet dimension for flow.Rotationally symmetrical arches were very sensitive to the relation between azimuthal and arch stresses. This affected both arch shape and stresses.A critical outlet dimension was calculated and showed great variation dependent upon assumptions made. Jenike's approach of an arch of constant thickness with no overpressure yielded a conservative value.

Cohesive arch shape was predicted for wedge and conical hoppers, using fundamental force balance equations. This was compared to the circular arch. The arch shape was found to be a good approximation to the circular over a wide range of conditions, with significant deviations at larger arch spans.Figure optionsDownload full-size imageDownload as PowerPoint slide