On the validity of the Hagen and Beverloo formulas for grains discharge through thin sidewalls of bins

Abstract

In this work we show through experiments, by using cohesionless granular materials, that a simple formula proposed to estimate the granular outflow from orifices in thick sidewalls is also valid for thin sidewalls. The use of such a formula entails the approximate validity of the classical Hagen and Beverloo formulas for granular samples composed of small and large grains, respectively. Also, an estimation of the involved error when the wall thickness is left aside also is given.

Granular flow through vertical slots: analysis of a general formula

“…As mentioned earlier, the mass flow rate for circular orifices in very thin sidewalls of bins obeys approximately the same formula used in flat-bottomed silos. This was proved experimentally [22,23] by performing the discharge of grains through vertical sidewalls of different thin thicknesses.…”

“…where α is the internal angle of the orifice defined as α = arctan(D/w) and θ r is the angle of repose of the granular material; formula (4) predicts that if the granular material attains the angle of repose along of the orifice thickness the granular flow will be arrested, even if the silo is brimful. Conversely, in the limit of very small thickness, it is satisfied that (α − θ r ) → (π/2 − θ r ) ∼ 1 for different materials [23]. Thus, formula ( 4) is accurately approximated by…”

“…which was put to the test by inspecting how closely it fitted experimental data from silo emptying experiments [23].…”

“…H is a useful phenomenological relationship for quantifying the flow rate across rectangular orifices of negligible thickness, in vertical sidewalls since it accurately describes both extreme cases of vertically or horizontally elongated orifices. It should be noted that if the orifice thickness is not small, then, more complete correlations must take into account the considerable influence of the wall thickness [22,23].…”

“…Formulas (1) and ( 2) also can be used as approximately valid for circular orifices in vertical sidewalls of silos, whenever the orifice has a small thickness [22,23], which will be achieved, for instance, by bevelling the rim of the orifice in thick sidewalls or if the sidewall is itself very thin.…”