# A Mathematical Model for Conical Hopper Mass Efficiency
## 3 Results  
The research was conducted using a full-factorial designed experiment. The experimental methodology was based on [35], taking into account the significance of each factor and decoding the terms of regression dependence into the coefficients of the equation for the natural values of the factors. The response criterion was the parameter of the discharge performance of bulk material from the circular conical hopper, \(Q_{\text{K}}\) [kg/s], the factors being the diameter of the hopper's discharge orifice, D, m \(\rightarrow\) x\({}_{1}\), and the cone angle of the hopper, \(\alpha\), degrees \(\rightarrow\) x\({}_{2}\). The values of the factors and the coding coefficients, calculated according to the theory of experimental design, are presented in Table 2.  
To ensure that the design matrix has the property of orthogonality, a column in Table 2 with corrected-level values of \(x\) was introduced. These values were calculated using Formula (33):  
\[\left(x_{i}^{\prime}\right)^{2}=x_{i}^{2}-\frac{\sum x_{i}^{2}}{N}\;. \tag{33}\]  
The matrix for calculating the coefficients of the equation is presented in Table 2, where columns 2-6 constitute an orthogonal design matrix, and columns 7 and 8 contain the response values of the experiment. According to the data from Table 3, the coefficients of the regression equation were calculated. The values of the regression coefficients characterize the contribution of each factor to the value of the response function and are calculated by the following formulas:  
\[b_{1}=\frac{\sum(x_{1}\cdot y)}{6},\;b_{2}=\frac{\sum(x_{2}\cdot y)}{6}, \tag{34}\]  
\[b_{22}=\frac{\sum\Bigl{(}(x_{2}^{\prime})^{2}\cdot y\Bigr{)}}{2},\;b_{11}= \frac{\sum\Bigl{(}\left(x_{1}^{\prime}\right)^{2}\cdot y\Bigr{)}}{2},\;b_{12}= \frac{\sum(x_{1}\cdot x_{2}\cdot y)}{12}, \tag{35}\]  
\[b_{0}=\frac{\sum y}{9}-0.67\cdot b_{11}-0.67\cdot b_{22}\;. \tag{36}\]  
The results of the calculation of the coefficients of the regression equation are given in Tables 4 and 5. To use the natural values of factors in the regression equation, the linear terms of the equation were converted from coded values into natural ones. They were determined by the formula given in [36]. \begin{table}
\begin{tabular}{c c c c c c c} \hline \hline \multirow{2}{*}{**Factors**} & \multirow{2}{*}{**Factors**} & \multirow{2}{*}{**Factors**} & \multirow{2}{*}{**Lenses**} & \multicolumn{4}{c}{**Levels of Factors**} \\ \cline{3-6}  & & & **Upper** & **Null** & **Lower** & **Variation** \\ \cline{3-6}  & & & & **Code Values** & & **Interval** \\ \cline{3-6}  & & & **+1** & **0** & **\(-\)1** & \\ \hline Hopper orifice diameter, D & x\({}_{1}\) & m & 1.0 & 0.075 & 0.050 & 0.025 \\ Hopper cone generatrix angle, \(\alpha\) & x\({}_{2}\) & deg. & 10 & 8 & 6 & 2 \\ \hline \hline \end{tabular}
\end{table} 
Table 2: Levels of variation of factors and their code values in the planned experiment. With the decoded values of the coefficients of the regression equation, the model of the productivity of the flow of bulk material from a cone hopper of circular section will accordingly take the following form for physical variable factors. For crushed coal:  
\[Q_{k2}=3.557+1039.99\cdot D-6.0\cdot\alpha+9066.72\cdot D^{2}+0.667\cdot\alpha^{2 }-140\cdot D\cdot\alpha. \tag{37}\]  
For buckwheat:  
\[Q_{k1}=-0.123+338.676\cdot D-1.692\cdot\alpha+3253.28\cdot D^{2}+0.196\cdot \alpha^{2}-44\cdot D\cdot\alpha. \tag{38}\]  
The regression equations for the dependence of the discharge performance of bulk material from a conical hopper with a circular cross-section on the factors are graphically represented in Figures 5 and 6, corresponding to regression dependencies (37) and (38), respectively.