Mass flow rate of granular material flowing from tilted bins

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

“…Related to the lateral outflow of granular solids from vertical walls of silos, a few years ago we proposed a phenomenological formula that simultaneously spans cases of different sidewall thickness, w, and orifice diameter, D, but it was only validated experimentally when D ∼ w, i.e., for thick sidewalls [24][25][26]. This formula is unique because it encompasses, simultaneously, parameters (up to now) not related in formulas of the mass flow through bottom holes, as the wall thickness w, the angle of repose θ r and the orifice diameter, D, all of them necessary to determine the minimum opening and thickness through which a stable outflow of non cohesive granular material is ensured.…”

“…By the way, only a few studies [19][20][21][22][23][24][25][26][27] have been conducted to explicitly test the validity of the Hagen or the Beverloo correlations for the discharge of grains through orifices on the vertical sidewalls of bins, which we denote bẏ m. To the best of our knowledge, Franklin and Johanson [19] were the first researchers who experimentally studied the problem of the gravity driven lateral outflow of granular material from boxes. In their studies, they used galvanized metal sheet 0.238 cm thickness, and they found that the flow rate, for orifices at the vertical sidewall, scales aṡ m = cρg 1/2 D 2.93 , where c is a factor dependent upon the angle of internal friction and the diameter of the particle.…”

“…Nevertheless, none of the previously referred works analyzed the effect of the wall thickness onṁ systematically. In the case of wall thickness (w) comparable with the outlet diameter (D) we have reported in several studies that for single circular orifices at the sidewall, the relationship valid for the flow rate behaves asṁ =ṁ 0 (α − θ r ) whereṁ 0 = ρg 1/2 D 5/2 when D d [24][25][26]…”

“…5/2 for cases with D ∼ 10d [27]; θ r is the angle of repose of the granular material and α is the angle of wall which has been defined as α = arctan(D/w). As will be shown later, the formula forṁ predicts that when the wall thickness is very small (w D), its dependence on w is weak but finite.…”

“…First, Sec. 2 we will revisit our phenomenological formulaṁ = cṁ 0 (α − θ r ) [24][25][26], to estimate the mass flow rate from orifices with thin lateral walls and we analyze the twofold limits w D and w = 0. Then, in Sec.…”