# Effect of Impeller Trimming on the Energy Efficiency of the Counter-Rotating Pumping Stage
## 1 Introduction  
Impeller trimming is widely used when adjusting centrifugal pumps and expanding their application area [1]. Trimming the impeller along the outer diameter by 10-15% is permitted so as to change parameters and sequential operation of a couple of pumps. One of the most affordable ways is to trim the impeller along the outer diameter [2]. Many studies have been devoted to this issue, the essence of which is to obtain experimental coefficients for calculating the pressure head, hydraulic losses, and energy efficiency depending on the trimming value. However, each type of pump requires different experiments to be carried out. When using a counter-rotating stage, the main pressure head is created due to the negative swirl of the flow, which is itself created when the liquid leaves the impeller and enters the vane disk [3]. In this case, the primary characteristic is not the speed of the liquid flow, but the angle of its inflow to the entrance to the vane disk. Impeller trimming changes the impeller's overall dimensions and the liquid outlet's angle. This fact affects the pressure head, power consumption, and energy characteristics of the counter-rotating stage. Impeller trimming has been thoroughly investigated for standard impellers. However, the effect of impeller trimming on the counter-rotating stage is unknown. This is because transferring energy to the liquid in a counter-rotating stage is highly complicated. Various researchers worldwide have highlighted the significance of the stated problem in ensuring the energy efficiency of centrifugal pumps. [PERSON] [4] studied the impeller trimming of an industrial centrifugal viscous oil pump, and the effect of increasing efficiency was revealed when a trimmed impeller delivered highly viscous oils. [PERSON] et al. [5] proposed an algorithm for the prediction of the effect of impeller trimming on the hydraulic performance of low specific-speed centrifugal pumps. The proposed approach overestimated the reduction in the pressure head after trimming. [PERSON] and [PERSON] [6] evaluated the flow characteristics of a centrifugal pump with different impeller trimming methods. They showed that the efficiency of a polygon-shaped trimmed impeller was less than the best efficiency point. [PERSON] et al. [7] studied the effect of impeller trimming on performance in a double-suction centrifugal pump. The losses in the solute diffuser channel were substantiated by the vortices generated within flow separation and backflow. Radial force variation caused by impeller trimming in the double-suction centrifugal pump was also investigated in [8]. Moreover, many research tasks concerning impeller trimming are also essential for compressors, turbines, and fans. [PERSON] and [PERSON] [9] studied the performance impact of impeller blade trimming on centrifugal compressors. They proved that blade trimming could be applied to reduce the pressure ratio while maintaining flow range. [PERSON] et al. [10] also investigated the effect of impeller trimming on the performance of a modified compressor using a numerical simulation approach. [PERSON] et al. [11] conducted experimental research on a single stage regarding the effects of impeller trimming for a turbine. The influence of parallel impeller inlet trimming on a mixed flow pump as a turbine was studied in [12]. [PERSON] et al. [13] experimentally and numerically investigated the effect of blade trimming on squirrel cage fan aerodynamic performance. [PERSON] et al. [14] numerically studied the impeller trimming effect on the performance of an axial flow fan, and [PERSON] et al. [15] analyzed the performance of the axial fan after blade trimming. Moreover, [PERSON] et al. [16] provided an experimental and numerical analysis of combined hydrogen as a combustion enhancer applied to the Wankel engine. The design of this pump is based on a dynamic type of energy transfer. Simultaneously, the Wankel engine is based on the rotary-piston principle of energy transfer, or positive displacement engines.