# Parametric Optimization of a New Gear Pump Casing Based on Weight Using a Finite Element Method
## 4 Discussion
### Optimization of Gear Pump Parts  
#### 4.2.1 Optimization of Gear Pump Back Cover  
As the pump back cover (Figure 18) has a large safety factor, but also a large number of mounting holes, optimization was carried out in terms of diameter.  
Parametric optimization of the pump cover was carried out by diameter because the safety factor in this zone tends to infinity. Figure 18 shows in gray the zone that will be removed as a result of optimization.  
By brute force, four variants of back covers were loaded (pressure 28 MPa), with an optimization step of 2 mm in the direction of the decrease. The results for the back cover are presented in Table 7.  
\begin{table}
\begin{tabular}{c c c c} \hline \hline
**Material** & **Casing** & **Back Cover** & **Front Cover** \\ \hline Aluminum & 19\% & 62\% & 38\% \\ Cast iron & 46\% & 81\% & 68\% \\ Polycarbonate (PC) & 6\% & 30\% & 28\% \\ \hline \hline \end{tabular}
\end{table}
Table 6: Level of resistance to damage to gear pump parts.  
\begin{table}
\begin{tabular}{c c c c} \hline \hline
**Variant** & **Diameter,** & **Maximum Stress,** & **Weight,** \\  & **D, mm** & \(\sigma_{\text{max}}\)**, MPa** & **m, kg** \\ \hline  & \multicolumn{3}{c}{Aluminum} \\
1 (original) & 256 & 73.76 & 1.84 \\
2 & 254 & 77.24 & 1.61 \\
3 & 252 & 84.12 & 1.43 \\
4 & 250 & 91.12 & 1.20 \\  & \multicolumn{3}{c}{Cast iron} \\
1 (original) & 256 & 67.89 & 6.90 \\
2 & 254 & 73.24 & 6.51 \\
3 & 252 & 79.01 & 6.14 \\
4 & 250 & 85.16 & 5.52 \\  & \multicolumn{3}{c}{Polycarbonate (PC)} \\
1 (original) & 256 & 110.47 & 1.21 \\
2 & 254 & 117.53 & 1.14 \\
3 & 252 & 123.58 & 1.06 \\
4 & 250 & 129.31 & 1.01 \\ \hline \hline \end{tabular}
\end{table}
Table 7: Optimization options for the pump back cover.  
Figure 18: Gear pump back cover.  
In all four variants (aluminum, cast iron, and polycarbonate), the standard safety factor is provided. The polycarbonate pump back cover has the minimal weight. The cast iron back cover has the maximal weight.  
The subsequent reduction in the diameter of the cover is impossible because of the presence of the holes for fastening.  
For the back cover of a pump made of three materials, the optimality criterion was calculated using Formula (1). The optimality criterion for the back pump cover is given in Table 8.  
The polycarbonate back pump cover has a minimal weight. However, taking into account the mass ratios and safety factor, aluminum should be selected for the back pump cover. With this construction material, the optimization criterion is reduced to a minimum.  
According to Formulas (2)-(6), a regression analysis was performed to establish a mathematical model of quadratic regression and its adequacy. The results show an inversely proportional relationship between the diameter of the cover (X) and the maximum stresses (Y). The dependence of the stress values on the diameter of the following cover has the form:  
\[\text{Y}=0.1750\text{x}^{2}-91.9290\text{x}+12,132.9120\]  
At the same time, the correlation coefficient is 0.99, the determination coefficient is 0.99, and the average approximation error is 0.23. The obtained correlation coefficient and determination coefficient show that the parameter relationship between the studied covers is very strong. As the approximation error is less than 15%, this equation can be used as a regression equation.  
By changing the number and location of the holes in the back cover, the weight of this part can be further reduced. There is no analysis of the back and front covers in the literature; studies of casing [47; 48; 50; 51; 52] and pump impellers [53] predominate.