Criteria for Beverloo's scaling law

Beverloo's scaling law can describe the flow rate of grains discharging from hoppers. In this paper, we show that the Beverloo's scaling law is valid for varying material parameters. The flow rates from a hopper with different hopper and orifice sizes (D, D-0) are studied by running large-scale simulations. When the hopper size is fixed, the numerical results show that Beverloo's law is valid even if the orifice diameter is very large and then the criteria for this law are discussed. To eliminate the effect of walls, it is found that the criteria can be suggested as D - D-0 >= 40d or D/D-0 >= 2. Interestingly, it is found that there is still a scaling relation between the flow rate and orifice diameter if D/D-0 is fixed and less than 2. When the orifice diameter is close to the hopper size, the velocity field changes and the vertical velocities of grains above the free fall region are much larger. Then, the free fall arch assumption is invalid and Beverloo's law is inapplicable.