# A Mathematical Model for Conical Hopper Mass Efficiency
## 2 Materials and Methods
### Mathematical Model of Granular Material Discharge from the Conical Hopper--Simulation  
In order to evaluate the developed model, the authors decided to test it using two different bulk materials (buckwheat grains), assuming six different hopper generatrix angles and five different hopper outlet diameters. The simulation parameters for simulation regarding the material, along with parameters referring to the hopper setup, are listed in Table 1.  
Figures 3 and 4 show the modeling results. The cone angle of the hopper, starting from \(4^{\circ}\), affects the discharge performance. At an angle of \(\alpha=2^{\circ}\), the bulk material discharges  
\begin{table}
\begin{tabular}{c c c} \hline \hline
**Feature** & **Buckwheat** & **Coal** \\ \hline Density, \(\rho\) [\(\frac{\mathrm{kg}}{\mathrm{m}^{3}}\)] & 560 & 900 \\ Particle packing angle, \(\beta\) [\({}^{\circ}\)] & 56 & 35 \\ External friction angle, \(\varphi\) [\({}^{\circ}\)] & 15 & 25 \\ Internal friction angle, \(\psi\) [\({}^{\circ}\)] & 17 & 34 \\ Material-wall friction & 0.47 & 0.47 \\ coefficient [\(\cdot\)] & 0.47 & 0.47 \\ Cone angle, \(\alpha\) [\({}^{\circ}\)], step: \(2^{\circ}\) & 4–12 & 2–12 \\ Hopper height, [m] & 0.5 & 0.5 \\ Hopper orifice diameter, [m], & 0.025–0.125 & 0.025–0.125 \\ step: 0.025 m & & \\ \hline \hline \end{tabular}
\end{table}
Table 1: Selected characteristics of the tested materials and the hopper.  
under the influence of gravity, and the friction angles do not impact the discharge process significantly. At an angle of \(\alpha=4^{\circ}\), the discharge performance of the bulk material is 1.5-2 times lower than at \(\alpha=2^{\circ}\).  
The modeling results shown in Figures 3 and 4 indicate that it is better to select a cone angle for the hopper within the range of \(\alpha\in(4^{\circ},10^{\circ})\) and a discharge orifice diameter of \(D\in(0.05\,\,\mathrm{m},\,0.1\,\mathrm{m})\).  
For buckwheat (Figure 4), the discharge performance increases with the diameter of the discharge orifice but decreases along with the increase in the generatrix angle of the hopper. When the orifice diameter is D = 0.025 m, the discharge efficiency is minimal. Increasing the diameter of the orifice to 0.05 m results in an average increase in discharge performance by 4.5 to 5.0 times. Therefore, using a discharge opening diameter of 0.025 m could be impractical in the design of a round conical hopper.  
Figure 4: Dependence of the discharge performance (Qbn) of crushed coal from a hopper of diameter \(D\) of the discharge orifice and the hopper cone angle \(\alpha\). Please note the Qbn scale when comparing the results with Figure 3.  
Figure 3: Dependence of the discharge performance (Qbn) of buckwheat from a hopper of diameter \(D\) of the discharge orifice and the hopper cone angle of \(\alpha\). Please note the Qbn scale when comparing the results with Figure 4.  
This pattern of bulk material discharge, as shown by the modeling results, applies to both buckwheat and coal. In order to confirm the results from the simulation, the authors decided to perform a full-factorial-designed experiment. The purpose of conducting the full-factorial-designed experiment was to verify the correspondence between the analytical modeling of the discharge performance of bulk materials from a circular conical hopper and the data from experimental studies.