# Effect of Impeller Trimming on the Energy Efficiency of the Counter-Rotating Pumping Stage
## 3 Results
### Overall Evaluation Results  
All the obtained data after the CFD simulations are summarized in Table 1.  
The following data were evaluated as a result of applying the mathematical model (4)-(8). Firstly, for the elongated counter-rotating stage (\(D_{4}\) = const), the matrix of trimming  
\begin{table}
\begin{tabular}{c c c c c c} \hline \hline \multirow{2}{*}{**Trimming Value,**} & \multicolumn{3}{c}{\(D_{4}\) = Const**} & \multicolumn{3}{c}{\(h\) = Const**} \\ \cline{2-6}  & **Pressure Head** & **Power Consumption \(N\),** & **Energy Efficiency,** & **Pressure Head** & **Power Consumption \(N\),** & **Energy Efficiency,** \\  & \(H\)**, \(m\)** & **kW** & **kW** & \(H\)**, \(m\)** & **kW** & **kW** \\ \hline \(0^{*}\) & 387 & 331 & 57 & 387 & 331 &  
coefficients \([T]\) (6), the Moore-Penrose pseudoinverse matrix \([\Theta]\) (8), and the compliance matrix \([K]\) (7) were as follows:  
\[[T]=\begin{bmatrix}1&0&0\\ 1&5&25\\ 1&10&100\\ 1&15&225\\ 1&20&400\end{bmatrix};\ [\Theta]=\begin{bmatrix}0.886&0.257&-0.086&-0.143&0.086\\ -0.154&0.037&0.114&0.077&-0.074\\ 0.006&-0.003&-0.006&-0.003&0.006\end{bmatrix};\]  
\[[K]=\begin{bmatrix}383.2&10.36&-0.6\\ 336.086&16.006&-0.954\\ 55.743&-0.977&0.063\end{bmatrix}.\]  
However, for the replaced counter-rotating stage (\(h=\text{const}\)), these matrices were as follows:  
\[[T]=\begin{bmatrix}1&0&0\\ 1&5&25\\ 1&10&100\\ 1&15&225\\ 1&20&400\\ 1&25&625\end{bmatrix};\ [\Theta]=\begin{bmatrix}0.821&0.321&0&-0.143&-0.107&0.107\\ -0.118&0.001&0.066&0.077&0.035&-0.061\\ 0.004&-0.001&-0.003&-0.003&-0.001&0.004\end{bmatrix};\]  
\[[K]=\begin{bmatrix}385.893&5.612&-0.549\\ 341.929&10.513&-0.759\\ 54.857&-0.903&0.057\end{bmatrix}.\]  
After comparing these matrices, it could be concluded that parameters \(K_{12}\) and \(K_{22}\) were much higher for the elongated counter-rotating stage (\(D_{4}=\text{const}\)). This fact substantiated a decrease in pressure head and power consumption.  
The approximating curves evaluated by Formula (4) are presented in Figure 9.  
Figure 9: Pressure and energy characteristics for various trimming values: 1—pressure head, m; 2—power consumption, kW; 3—energy efficiency, %; a—the elongated counter-rotating stage (\(D_{4}=\text{const}\)); b—the replaced counter-rotating stage (\(h=\text{const}\)).  
The solid blue line 1a represents the approximating curve for the pressure head for the elongated counter-rotating stage (\(D_{4}\) = const), shown by filled round dots. The dashed blue line 1b is the approximating curve for the pressure head for the case of the replaced counter-rotating stage (\(h\) = const), represented by unfilled round dots. Red curves 2a and 2b represent the power consumption (marked by square dots) for the abovementioned cases. Curves 3a and 3b, marked by the swamp color, approximate the power consumption marked by triangular dots.  
Since \(K_{12}\) and \(K_{22}\) were positive for both cases, the curves of \(H(t)\) and \(N(t)\) ascended. However, negative values of \(K_{32}\) led to a descending angle of energy efficiency \(\eta(t)\) for low trimming values. Since \(K_{13}\) and \(K_{23}\) were negative for both cases, the curves of \(H(t)\) and \(N(t)\) were convex. However, positive values of \(K_{33}\) led to a concave curve of energy efficiency \(\eta(t)\).  
According to Formulas (10) and (11), the minimum value of energy efficiency \(\eta_{min}\) = 52% corresponded to the trimming of 7.8 mm for the case of the replaced counter-rotating stage (\(h\) = const). However, the maximum values of pressure head \(H_{max}\) = 428 m and power consumption \(N_{max}\) = 403 kW were reached for the trimmings of about 8.5 mm for the case of the elongated counter-rotating stage (\(D_{4}\) = const).