# Effect of Impeller Trimming on the Energy Efficiency of the Counter-Rotating Pumping Stage
## 3 Results
### CFD Modeling  
Initially, impellers with different outer diameters \(D_{2}\) were designed (Figure 3a). For each impeller, two different vane disks were created using the Solid Works software. Figure 2: Counter-rotating stages: (**a**) Reference stage; (**b**) The elongated counter-rotating stage (\(D_{4}\) = constant); (**c**) The replaced counter-rotating stage (\(h\) = constant); (**d**) The diameter \(D_{4}\) of the counter-rotating stage equals to the diameter \(D_{2}\) of the impeller for the reference stage. Using the ANSYS CFX software [25], the 3D models were meshed by the calculation grid (Figure 3b). Grid independence allowed obtaining high-quality numerical simulation by determining the optimal cell size of the grid for further calculations. During the grid independence test, it was determined that the global mesh size of the cells equaled 7 mm. The number of prismatic layers should be at least 11. Moreover, the first layer should be 0.04 mm. Overall, on average, the number of cells in the grid was \(1.5\times 10^{6}\). The next stage for each model was the \"inlet\" and \"outlet\" conditions, setting of the pumped medium and specific boundary conditions. After that, the numerical calculations of counter-rotating stages were carried out. Initially, counter-rotating stages with different trimming values and bladed disks were calculated. The first study concerned impeller trimming by 5 mm (Figure 4). Two counter-rotating systems were designed and simulated for this case study (and subsequent case studies). The first system considered the elongated counter-rotating stage (Figure 4a) with \(D_{4}=\) constant. The second system considered the replaced counter-rotating stage (Figure 4b) with \(h=\) constant. The characteristics of these degrees were as follows: the first option (\(D_{4}=\) constant)--pressure head \(H=411\) m, power consumption \(N=400\) kW, and energy efficiency \(\eta=50\%\); the second option (\(h=\) constant)--\(H=396\) m, \(N=391\) kW, and \(\eta=49\%\). In the designed stages, vortex formation occurs due to the sizeable inter-blade channel. Traditionally, such vortices negatively impact the pressure and energy characteristics. However, in the considered case study, vortex formation did not have an effect because of the similar parameters for all the stages (number of blades, their shape and thickness, and blade angle). Figure 3: 3D model (**a**) and CFD grid (**b**) of the flow part. The next step was the calculation of counter-rotating stages with various impeller trimmings of 10 mm (Figure 5). The characteristics of these degrees were as follows: the first option (\(D_{4}\) = constant)--\(H\) = 431 m, \(N\) = 408 kW, and \(\eta\) = 52%; the second option (\(h\) = constant)--\(H\) = 392 m, \(N\) = 382 kW, \(\eta\) = 50%. Next, counter-rotating stages with different impeller trimmings of 15 mm were calculated (Figure 6). The characteristics of these degrees were as follows: the first option (\(D_{4}\) = constant)--\(H\) = 407 m, \(N\) = 344 kW, and \(\eta\) = 58%; the second option (\(h\) = constant)--\(H\) = 364 m, \(N\) = 313 kW, and \(\eta\) = 54%. Figure 4: Stages with trimming of 5 mm: (**a**) \(D_{4}\) = constant; (**b**) \(h\) = constant. The last step was the calculation of counter-rotating stages with various trimmings of 20 mm (Figure 7). The characteristics of these degrees were as follows: the first option (\(D_{4}\) = constant)--\(H\) = 348 m, \(N\) = 282 kW, and \(\eta\) = 60%; the second option (\(h\) = constant)--\(H\) = 276 m, \(N\) = 241 kW, and \(\eta\) = 65%. Figure 5: Stages with trimming of 10 mm: (**a**) \(D_{4}\) = constant; (**b**) \(h\) = constant. To compare the pressure and energy characteristics, the reference counter-rotating stage was also calculated (Figure 8a). Additionally, the influence of the maximally trimmed impeller of the counter-rotating step with overall dimensions less than for a reference impeller of the pump CPS-180/1900 was studied (Figure 8b). Figure 6: Stages with trimming of 15 mm: (**a**) \(D_{4}\) = constant; (**b**) \(h\) = constant.