## 1 Introduction  
Centrifugal pumps are widely used in various sectors, such as water supply and wastewater for drinking water, energy for transferring heat carriers in heat exchangers, industry for transporting industrial liquids and chemical solutions, particularly in oil fields [1; 2; 3], as well as in agriculture for field irrigation, etc. In the context of today's pursuit of increased competitiveness, compliance with regulatory requirements [4], and reduction in operational costs [5; 6; 7], the task of enhancing the energy efficiency of centrifugal pumps becomes particularly relevant. The impeller of a centrifugal pump generates centrifugal forces that facilitate the movement of fluid from the center to the periphery, converting the mechanical energy of the rotating blades into the kinetic energy of the flow. However, various energy losses occur during fluid flow through the pump. Frictional forces between the fluid and the blade walls, the shape of the diffuser, and turbulent flow contribute to the formation of vortices, which lead to energy loss [8; 9; 10]. Additionally, cavitation phenomena, which occur when the local pressure drops below the saturation vapor pressure, negatively impact the pump's energy efficiency and reliability [11; 12; 13]. To enhance the efficiency of centrifugal pumps, a comprehensive analysis of these factors is required, along with the exploration of ways to minimize them using high-precision methods. The centrifugal pump system comprises several key components: the inlet pipe, impeller, volute, and outlet extension pipe. A significant portion of energy losses occurs at the impeller, making its geometric parameters particularly crucial for optimizing the pump's performance. While hydraulic calculations can provide optimal parameters, the application of detailed CFD modeling enables a more precise analysis and enhancement of the flow characteristics, ultimately contributing to an increase in the pump's hydraulic efficiency. One of the main geometric parameters of the impeller is the angles of the leading and trailing edges of the blades, as well as the wrap angles. In the study by [PERSON] et al. [14], the impact of the wrap angle and the trailing edge angle on the pump's efficiency and head was investigated. The analysis of four parameters, such as the change in static pressure, relative velocity, streamlines, and kinetic energy, showed that at a wrap angle of 126\({}^{\circ}\) and a trailing edge angle of 24\({}^{\circ}\), the efficiency value was significantly higher compared to the original parameters. In their studies, [PERSON] et al. [15] and [PERSON] et al. [16] examined the influence of the leading-edge shape (elliptical, arcuate and circular) on the hydraulic losses in the impeller. They found that blunter and sharper edge shapes cause a flow compression effect, which at high flow rates create a negative velocity gradient at the inlet, leading to energy losses and a localized low-pressure region. The shape of the impeller's trailing edge has also been studied as an optimization parameter, particularly regarding its influence on the backflow and energy losses. Research showed that the shape of the trailing edge has a minimal impact on the impeller's leading section. However, optimizing the trailing edge, specifically by thinning it, significantly reduces the vortices and energy losses, resulting in an improved flow and a 0.59% increase in overall pump efficiency. Conversely, thickening the trailing edge leads to a decrease in the efficiency by approximately 6% [17]. The study [18] examined the impact of modifying the impeller's trailing edge on the efficiency and head of a centrifugal pump at various specific speeds, based on CFD simulations and experiments. The research in [19; 20; 21] demonstrated that optimizing the blade thickness in centrifugal pumps significantly affects their performance, efficiency, and durability, highlighting the importance of precise blade design to enhance the operational characteristics of pumps.