The theoretical calculation of the flow rate of granular matter from an inclined orifice

A theoretical calculation method for the flow rate of granular matter from an inclined orifice is discussed in this article and for the inclination angles at $\theta \le 90^{\circ }$ , a theoretical relation between the flow rate $Q$ and inclination angle $\theta $ is derived; and for the inclination angles at $\theta >90^{\circ }$ , a semi-theoretical relation is established. From the relations, we found that the ratio of the flow rate from a vertical orifice, $Q_{90}$ , to that from a horizontal orifice, $Q_{0}$ , is equal to the sine of the angle of repose $\theta _{\mathrm{r}}$ , i.e., $Q_{90} /Q_0 =\sin \theta _{\mathrm{r}} $ . The theoretical relations are tested by means of the experimental data and the results indicate that the theoretical calculating values are in good agreement with the experimental data over a wide range of the inclination angles. Therefore, the formula proposed in this article can be used for the theoretical calculation of the flow rate of granular matter from an inclined orifice. The relation $Q_{90} /Q_0 =\sin \theta _{\mathrm{r}}$ may be used as an alternative approach to obtaining $\theta _{\mathrm{r}}$ : measuring $Q_{90}$ and $Q_{0}$ , and then calculating $\theta _{\mathrm{r}} $ by using formula $\theta _{\mathrm{r}} =\arcsin (Q_{90} /Q_0 )$ .