## 3 Results and Validation of the Developed Models
### Results of CFD Modeling  
To ensure the validity of the hydrodynamic model, numerical simulations of the flow in a single-stage centrifugal pump with single suction were performed under various flow rate conditions, corresponding to the experimental test data. The convergence of the solution was achieved in approximately 600 to 1000 iterations, depending on the flow rate. The comparison between the numerical results and the experimental data for pressure head and efficiency showed strong correlation, as illustrated in Figure 5. The maximum relative error in the pressure head calculations did not exceed 3.9%, while the largest deviation in efficiency was less than 5.4%. These findings demonstrate the reliability of the developed algorithm and its capability to accurately reproduce the real performance characteristics of the pump under the tested conditions.  
Figures 6 and 7 show the pressure and velocity distributions inside the impeller and solute casing of a centrifugal pump, obtained as a result of numerical simulation using computational fluid dynamics (CFD) methods, at a flow rate of \(Q\) = 300 m\({}^{3}\)/h. The pressure distribution reflects the typical dynamics for a centrifugal pump (Figure 6). The inlet pressure is about 0.3 \(\times\) 10\({}^{5}\) Pa, which corresponds to the specified value of 0.3 atm. Minimum pressure values are observed at the beginning of the impeller blades, reaching \(-\)2.75 \(\times\) 10\({}^{5}\) Pa. In this zone, the liquid begins to interact with the blades, where the centrifugal forces significantly accelerate the flow, which causes a sharp drop in pressure. As the liquid moves from the inlet to the outlet, the pressure in the impeller increases, which is associated with the conversion of the kinetic energy of the flow into potential energy due to the action of the centrifugal forces. In the volute casing, where the flow slows down due to the expanding section, the pressure continues to increase. At the pump outlet, it reaches values of about 3.3 \(\times\) 10\({}^{5}\) Pa, which indicates the successful operation of the pump in converting the energy of liquid movement into hydraulic pressure.  
Figure 7 shows the velocity distribution fields inside the impeller and volute casing of a centrifugal pump, obtained as a result of numerical simulation using the CFD methods. At the center of the pump, where the liquid enters, the flow velocity is low, which is typical for the initial conditions when the liquid has not yet been affected by the blades. As the liquid enters the impeller, the velocity begins to increase under the action of the centrifugal forces and reaches its maximum at the periphery of the blades. In the volute, the velocity gradually decreases, facilitating the conversion of the kinetic energy of the flow into static pressure. Also observed are zones with reduced velocity on the up  
Figure 5: Comparison of the hydraulic performances of the centrifugal pump. (**a**) Head (\(H\))—flow rate (\(Q\)); (**b**) efficiency (\(\eta\))—flow rate (\(Q\)).  
which may indicate localized areas of turbulence or uneven flow. These features illustrate the complex interaction of the flow with the impeller blades and volute casing, ensuring the efficient operation of the pump.