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]. mensional 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. t 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. ch shape was generally an excellent approximation to the circular arc. However, the arch diverged from the circular as arch span increased. nsionless 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. hickness models had little effect upon arch shape but had a great influence upon stress. This affected the critical outlet dimension for flow. onally symmetrical arches were very sensitive to the relation between azimuthal and arch stresses. This affected both arch shape and stresses. ical outlet dimension was calculated and showed great variation dependent upon assumptions made. Jenikeʹs approach of an arch of constant thickness with no overpressure yielded a conservative value.