# Improving Energy Efficiency of Grain Cleaning Technology
## 4 Results and Discussion  
The mathematical model [1]\(\widetilde{y}=f(x_{1},x_{2},x_{3})\) for ZAV-40 in the form of a second-order regression equation is as follows:  
\[\widetilde{y}=1.012-0.313x_{1}+0.283x_{2}+0.228x_{3}-0.102x_{1}x_{2}-0.082x_{ 1}x_{3}+0.07x_{2}x_{3}-0.102x_{1}^{2}+0.084x_{2}^{2}+0.084x_{3}^{2} \tag{1}\]  
The analysis of Equation (1) shows that the function of specific electricity consumption has an extremum. In the physically acceptable range of operating modes of the production line, the function of specific electricity consumption is continuous and has no other extreme points, which means that at the estimated points of minimum, the function reaches the lowest value [4]. The tasks of static optimization of objects are solved by search methods. In addition, some mathematical transformations allow a graphical and analytical interpretation of the optimum domain. For this purpose, canonical transformation of the mathematical model and the method of two-dimensional cross-sections of the response surface were used. According to Equation (1), the specific electricity consumption for the ZAV-40 unit is calculated depending on the performance, capacity, and load factor of the electrical equipment, which results in obtaining corresponding theoretical response surfaces. To analyse Equation (1) by method of two-dimensional cross-sections, the obtained second-order regression equation was differentiated for each factor and set to zero: \begin{table}
\begin{tabular}{c c c c c} \hline \hline \multirow{2}{*}{**Factor Variability Levels**} & \multirow{2}{*}{**Factors in Nominal Units**} & \multicolumn{4}{c}{**Factors in Physical Units**} \\ \cline{3-5}  & & \(Q\), \(t\)/h & \(P\), \(k\)W & \(K_{1}\), \(x\).u. \\  & & \(x_{1}\) & \(x_{2}\) & \(x_{3}\) \\ \hline Upper \(X_{i,u}\) & \(x_{i,u}=+1\) & 40.0 & 45.0 & 0.8 \\ Lower \(X_{i,l}\) & \(X_{x,l}=-1\) & 20.0 & 25.0 & 0.5 \\ Base \(X_{i,0}\) & \(x_{i,0}=0\) & 30.0 & 35.0 & 0.65 \\ Variability intervals \(\Delta X_{i}\) & \(\Delta x_{i}=\pm 1\) & 10.0 & 10.0 & 0.15 \\ Star arm size & & & & \\ \(+\alpha\) & \(+1.215\) & 42.85 & 47.15 & 0.832 \\ \(-\alpha\) & \(-1.215\) & 17.85 & 22.85 & 0.468 \\ \hline \hline \end{tabular}
\end{table} 
Table 1: Factor levels and variability intervals for ZAV-40. \[ \begin{array}{l}\frac{\partial\widetilde{v}}{\partial x_{1}}=-0.313-0.102x_{2}-0.08 2x_{3}-0.204x_{1}=0;\\ \frac{\partial\widetilde{v}}{\partial x_{2}}=0.283-0.102x_{1}+0.07x_{3}+0.168x_{ 2}=0;\\ \frac{\partial\widetilde{v}}{\partial x_{3}}=0.228-0.082x_{1}+0.07x_{2}+0.168x _{3}=0.\end{array}  \tag{2}\]  
After solving the system of equations, coordinates of the centre were obtained in coded units:  
\[x_{1S}=-0.396;\ x_{2S}=-1.548;\ x_{3S}=-0.906;\ y_{S}=0.894,\]  
which correspond to the following values of the factors and the target function in physical units:  
\[Q=40\ \mathrm{t/h};\ P=25\ \mathrm{kW};\ K_{l}=0.583.\]  
The optimum value of the target function is equal to \(W_{sp. }=0.894\ \mathrm{kW}\cdot\mathrm{h/t}\). The possible two-dimensional sections with the most practical significance for the specific electricity consumption were considered. 1. Two-dimensional cross-section of the response surface characterizing the connected capacity and the load coefficient of electric motors (Figure 2) at \(x_{1}=0\):  
\[ \begin{array}{l}\frac{\partial\widetilde{v}}{\partial x_{2}}=0.283+0.07x_{ 3}+0.168x_{2}=0;\\ \frac{\partial\widetilde{v}}{\partial x_{3}}=0.228+0.07x_{2}+0.168x_{3}=0;\\ x_{2S}=-1.35;\ x_{3S}=-0.79;\ y_{S}=0.73,\end{array}  \tag{3}\]  
the values of factors in physical units is as follows:  
\[P=21.5\ \mathrm{kW};\ K_{l}=0.5\ \mathrm{and}\ W_{sp.}=0.73\ \mathrm{kW}\cdot \mathrm{h/t}.\]  
For canonical transformation of the equation, the following system was solved:  
\[f(\mathrm{B})=\left| \begin{array}{cc}0.168-\mathrm{B}&0.5\cdot 0.07\\ 0.5\cdot 0.07&0.168-\mathrm{B}\end{array}