## 2 Methodology  
### Characteristics of the Centrifugal Pump CHP 150-315/1500  
This study examines a typical centrifugal pump model CHP 150-315/1500 (Karlskrona LC AB (Shymkent, Kazakhstan)) (Figure 1a), produced at the Karlskrona factory, which  
width of the impeller. Vortex flows are observed in the inter-blade channels and the volute, leading to the emergence of local hydraulic resistances that negatively affect the overall efficiency of the system. Due to the complex nature of the flow, an analytical solution to the problem is unavailable. Therefore, in practice, calculation methods are applied based on various simplifications and assumptions, where a detailed study of the flow at every point is not required, only at the control sections. As a flow parameter, the actual value is taken as its average integral value over the area of the flow cross section. For such a significantly simplified problem, the theory of one-dimensional homogeneous flows is applicable, where the problem is treated as one-dimensional and depends only on one coordinate, namely the radial vector. The theory based on such assumptions is called the theory of hydraulic machines. To describe the kinematics of the fluid in the theory of hydraulic machines, a velocity triangle or velocity plan is used (Figure 2), where the absolute flow velocity \(\overset{\rightarrow}{c}\) is defined as the sum of the circumferential flow velocity \(\overset{\rightarrow}{u}\) and the relative flow velocity \(\overset{\rightarrow}{w}\)([PERSON], 2018), as expressed in Equation (1). This visual representation of velocities allows for a more accurate analysis and optimization of the pump's flow characteristics and its interaction with the blades of the impeller. Calculation of the Main Geometric Parameters of the Impeller  
To calculate the optimal parameters of a centrifugal pump, specifically the impeller, the optimal velocities that ensure the maximum efficiency are first determined. Only after that are the optimal dimensions of the impeller calculated. These key geometric parameters are determined at three different sections of the impeller (Figure 3a): at the impeller entry point (0-0), at the leading edge (LE) of the blade (1-1) and at the trailing edge (TE) of the blade (2-2). The required volumetric flow rate \(Q\), head \(H\), and rotational speed \(n\) are specified as input data for the calculation. The specific speed coefficient plays an important role in pump classification, determining the blade shape, blade inclination angles, and the pump's performance characteristics ([PERSON], 2018; [PERSON], 2018; [PERSON], 2018; [PERSON], 2018), and is defined by Equation (2):  
\[n_{s}=0.65n\frac{Q^{\frac{1}{2}}}{H^{\frac{3}{4}}}. \tag{2}\]  
The efficiency coefficient \(\eta\) of a centrifugal pump is defined as the ratio of useful power to the power received from the motor. As mentioned earlier, due to losses (leakage,  
Figure 2: Kinematics of fluid flow in the impeller. Velocity triangles. friction, geometry), the pump efficiency decreases. To minimize such losses, three types of efficiency are considered in the hydraulic calculation: volumetric efficiency \(\eta_{v}\), which takes into account the leakage and bypasses from seals, as well as the effect of the cavity between the impeller and the volute casing; hydraulic efficiency \(\eta_{h}\), which includes the losses related to cavitation, local obstacles, hydraulic friction, vortices, and flow separation from the blades; and mechanical efficiency \(\eta_{m}\), which reflects the losses associated with the bearing friction and fluid friction in the cavity between the impeller and the volute. Each efficiency component is determined using the following empirical equations [28].