# A Mathematical Model for Conical Hopper Mass Efficiency
## 2 Materials and Methods
### Mathematical Model of Granular Material Discharge from the Conical Hopper--Assessments
Then, Formula (8), which is used for finding the pressure in an arbitrary section of a vertical cylindrical hopper, will take the form:  
\[p=\frac{\rho\cdot(g-a)\cdot D}{8\cdot f\cdot\tan(\beta)}\cdot\left(1-\mathrm{ e}^{-\frac{8\cdot f\cdot\tan(\beta)}{D}\cdot x}\right)\,. \tag{9}\]  
Under conditions where a \(\approx 0\) and at a sufficient depth (x \(\gg\) D) in the hopper, the pressure is practically constant and is determined by Formula (10):  
\[G=\frac{\pi\cdot D^{2}}{4}\cdotp\cdotp\cdotp\cdotp \tag{2}\]0  
For a hopper with a variable cross-sectional diameter of \(dm\), the elementary volume of mass can be represented by Formula (11):  
\[G=\frac{\pi\cdot D^{2}}{4}\cdotp\cdotp\cdotp\cdotp \tag{2}\]1  
where:  
\(dx\)--height of the elemental volume (EV) of the granular material [m];  
\(S(x)\)--cross-sectional area, varying with the displacement of the EV [m\({}^{2}\)]. The change in the diameter of the silo with a circular cross-section along the height is calculated using Formula (12):  
\[ \begin{array}{l}d(x)=D_{0}-2\cdot x\cdot\tan(\alpha),\ 0\leq x\leq H_{1}\,\\ \tan(\alpha)=(D_{0}-D)/(2\cdot H_{1})\end{array}  \tag{12}\]where:  
\(D_{0}\)--diameter at the upper part of the hopper [m];  
\(D\)--diameter at the lower part of the hopper [m];  
\(H_{1}\)--hopper height [m];  
\(\alpha\)--inclination angle of the cone generatrix [\({}^{\circ}\)]. Thus, the cross-sectional area of the round hopper will be equal to:  
\[S(x)=\frac{\pi\cdot(D_{0}-2\cdot x\cdot\tan(\alpha))^{2}}{4}\]  
To model the velocity of the dispersed material movement in the hopper, the following differential equation of motion for the elementary volume was developed:  
\[ \begin{array}{l}\rho\cdot S(x)\cdot dx\frac{dx}{dt}=p\cdot S(x)-(p+dp)\cdot (S(x)+ds)++dG-2\cdot p\cdot\tan\left(\beta+\psi\right)\cdot H(x)\cdot dx\\ H(x)=\tan\left(\beta+\varphi\right)\cdot\pi\cdot dx\end{array}  \tag{13}\]  
where:  
\(\frac{dp}{dt}\)--acceleration of the material in elemental volume [\(\frac{m}{s^{2}}\)];  
\(ds\)--increase in the cross-sectional area of the elementary volume [m\({}^{2}\)];  
\(dG\)--weight of the elementary volume of the material [N];  
\(\psi\)--angle of repose for the considered material [\({}^{\circ}\)];  
\(\varphi\)--angle of the given material's external friction (material-to-vessel) [\({}^{\circ}\)]. Dividing the equation of motion for the elementary volume (Formula (13)) by S(x)\(\cdot\)dx, a first-order differential equation with variable coefficients is obtained:  
\[\rho\cdot\frac{dv}{dt}=-\frac{dp}{dx}-K(x)\cdot p+\rho\cdot g \tag{14}\]  
where:  
\(K(x)\)--the simplifying coefficient [\(\frac{1}{m}\)]. \[K(x)=\frac{1}{S(x)}\cdot\frac{dS}{dx}+2\cdot\frac{h(x)}{S(x)}\cdot\tan(\beta+\psi) \tag{15}\]  
The efficiency of the hopper outlet can be expressed through the velocity \(v\) of the movement of the elementary volume of material, using Formula (16):  
\[Q_{bn}=\rho\cdot S\cdot v \tag{16}\]  
After rearranging Formula (16) and taking the derivative with respect to time:  
\[\frac{dv}{dt}=\frac{1}{\rho\cdot S}\cdot\frac{dQ_{bn}}{dt}-\frac{Q_{bn}}{ \rho\cdot S^{2}}\cdot\frac{dS}{dx}\cdot\frac{dx}{dt}. \tag{17}\]  
Since \(\frac{dx}{dt}=v\), Formula (17) can take an alternative form:  
\[\frac{dv}{dt}=\frac{1}{\rho\cdot S}\cdot\frac{dQ_{bn}}{dt}-\frac{Q_{bn}}{ \rho^{2}\cdot S^{3}}\cdot\frac{dS}{dx} \tag{18}\]  
Hereafter, the authors assumed that the process is stationary, with Qbn = const; therefore, \(\frac{dQ_{bn}}{dt}=0\). Substituting Formula (18) into Formula (14), a first-order differential equation with variable coefficients with respect to pressure \(p(x)\) is obtained.