## 1 Introduction  
Multistage centrifugal pumps are widely used in many industries, including energy, domestic water supply and agriculture, as part of fluid transport systems where high pressure (head) is required. It is well known that turbomachinery is one of the main consumers of electricity worldwide. According to Ref. [1], the world market volume for centrifugal pumps is in the order of USD 20 billion per year. The increasing demand for energy efficiency necessitates improvements in the components of these machines, which are significant energy consumers worldwide. In-depth research on this problem in Bulgaria has been performed at the University of Ruse. In Ref. [2], the author proposes a large-scale experimental and theoretical study concerning the return device of a multistage centrifugal pump 11 MT32, manufactured by VIPOM, Vidin, Bulgaria. The studies are related to the determination of the impact of the geometry of the return guide vanes (RGVs) on the indicators of the studied stage as well as of the entire pump. The number of RGVs is equal to the number of Radial diffuser vanes downstream of the pump impeller. The vanes are profiled on arcs of circles. The authors in Ref. [3] modeled the flow in two-, three- and four-stage centrifugal pumps using ANSYS CFX, applying the \(k-\varepsilon\) turbulence model. The numerical resultsobtained are validated using data provided by the pump manufacturer. As a result of the study performed, the impact of the number of impeller blades, the number of vanes of the return device and the number of stages of the pump on its head, as well as the coefficient of efficiency and power, was determined. In Ref. [4], the authors studied the possibility of reducing the overall size of a multi-stage centrifugal pump using a circumferential twisted return guide vane. The study is based on applying CFD modeling of the flow through a two-stage pump, using ANSYS FLUENT. A turbulence model of the type RNG k - \(\varepsilon\) is used in this study. As a result, the authors propose replacing the existing six-stage pump with a four-stage pump that ensures the same head. Considering Ref. [5], it is clear that the interaction between the impeller and guide vanes has the most significant impact on the energy loss and pump coefficient of efficiency. The results provided concern a numerical study of a 10-stage centrifugal pump. The method LES (Large Eddy Simulation) is used. In Ref. [6], the authors presented an experimental study on the impact of the vanes of the reciprocating apparatus of a multistage pump on pump performance. It is found that the return vane, whose trailing edge is set at the outer wall radius of the downstream annular channel, which discharges the swirl-less flow, ensures a good pump performance. Additionally, the impeller blade is modified to obtain shock-free conditions at the inlet. The blade length of the modified impeller with the smaller blade inlet angle is slightly longer than that of the standard (original) impeller, and the modified impeller slightly improves the pump coefficient of efficiency. In Ref. [7], a new method for profiling the guide vanes of the guide and return devices of a multistage centrifugal pump is proposed. Four types of two-dimensional and three-dimensional new vanes are presented, which, according to the authors of Ref. [7], lead to an increase in the pump efficiency. However, the study does not provide information on the results obtained after performing research on the operation of pumps with these newly developed vanes. In Ref. [8], a method to optimize the design of a typical multistage centrifugal pump based on the energy loss model and Computational Fluid Dynamics (CFD) is proposed. External characteristic experiments are also performed to benchmark numerical simulations. It is found that reducing volumetric leakage and interstage leakage losses is the most effective technique for increasing the efficiency of standard centrifugal pumps. In Ref.