# A Mathematical Model for Conical Hopper Mass Efficiency
## 2 Materials and Methods  
### Mathematical Model of Granular Material Discharge from the Conical Hopper--Assessments  
The most commonly occurring types of hoppers have either round or rectangular cross-sections. For the sake of the model, the authors assumed an elementary volume with a height of \(dx\) (Figure 2). The granular material's characteristics, such as particle size, shape, stiffness, surface characteristics, water absorption, etc., were indirectly taken into account as the following coefficients:  
\(\beta\)--particle packing angle,  
\(f\)--coefficient of friction of the material mass against the hopper walls,  
\(\rho\)--bulk material density,  
\(\varphi\)--external friction angle (material-to-hopper-walls). The force counteracting the movement of the given elementary volume of granular material inside the hopper in circular cross-section with a vertical load can be described as in Formula (1):  
\[F_{\mathrm{T}}=f\cdot N=f\cdot\pi\cdot D\cdot dx\cdot p_{N} \tag{1}\]  
where:\(F_{T}\)--frictional force [N];  
\(N\)--force normal to the hopper walls [N];  
\(f\)--coefficient of friction of material sliding along the hopper walls [-];  
\(D\)--diameter of the hopper outlet [m];  
\(dx\)--height of the considered elementary volume [m];  
\(p_{N}\)--pressure of the material acting on the hopper walls [Pa]. The gravitational force acting on the elemental volume can be calculated using Formula (2):  
\[G=\frac{\pi\cdot D^{2}}{4}\cdotp\cdotp\cdotp\cdotp \tag{2}\]  
where:  
\(G\)--weight of the elemental volume [N];  
\(\rho\)--density of the granular material [\(\frac{kg}{m}\)];  
\(g\)--acceleration of gravity [\(\frac{m}{g^{2}}\)]. The normal pressure acting on the hopper walls surface can be described by Formula (3):  
\[p_{N}=1\cdotp\cdotp\tan(\beta) \tag{3}\]  
where:  
\(\beta\)--packing angle of the material inside the hopper [\({}^{\circ}\)];  
\(p\)--pressure acting on the upper surface of the elemental volume [Pa]. The force of inertia can be described with Formula (4):  
\[F_{i}=\rho\cdot\frac{\pi\cdot D^{2}}{4}\cdotp\cdotp\cdotp \tag{4}\]  
where:  
Figure 2: Diagram of the hopper and the forces acting on the dispersed material of the elementary volume. \(a\)--downward acceleration of the elemental volume [\(\frac{m}{g^{2}}\)]  
According to the d'Alembert principle, the forces acting on the rheonomic set of material points are, at a given moment, balanced by the force of inertia; thus, \(\sum_{k=1}^{n}F_{kx}=0\). All the projections of the forces were summed on the X axis (Formula (5)), taking into account the force generated by the pressure p and (\(p+dp\)) on the upper and lower faces of the selected elementary volume:  
\[p\cdot\pi\cdot\frac{D}{4}-\left(p+\frac{dp}{dx}\cdot dx\right)\cdot\frac{\pi \cdot D^{2}}{4}+G-F_{i}-F_{T}=0. \tag{5}\]  
Combining all the dependencies from Formulas (1)-(4) into Formula (5), the following equation can be obtained:  
\[p\cdot\pi\cdot\frac{D}{4}-\left(p+\frac{dp}{dx}\cdot dx\right)\cdot\frac{\pi \cdot D^{2}}{4}+G-F_{i}-F_{T}=0. \tag{6}\]  
After division by \(\frac{\pi\cdot D^{2}}{4}\cdot dx\) and the necessary rearrangement, a differential equation is needed to find the pressure (Formula (7)):  
\[\frac{dp}{dx}+\frac{8\cdot f\cdot\tan(\beta)}{D}\cdot p=\rho\cdot(g-a). \tag{7}\]  
When the acceleration \(a\) is considered to be constant, Formula (7) can be integrated into the following form:  
\[p=\frac{\rho\cdot(g-a)\cdot D}{8\cdot f\cdot\tan(\beta)}+C\cdot\mathrm{e}^{- \frac{8\cdot f\cdot\tan(\beta)}{D}\cdot x} \tag{8}\]  
where:  
\(x\)--distance from the origin to the cross-section [m]. The integration constant can be found using the boundary conditions when \(x=0\) and \(p=0\).