## 2 Materials and Methods
### Object and Heat Exchange Calculation Methods  
This paper considers the heat exchange accounting issue for a low-flow stage with the help of two main methods, i.e., coupled heat exchange and a simplified method. The influence of heat exchange on the calculated stage gas-dynamic characteristics is considered. Modeling is carried out with the Numeca Fine/Turbo software package. The presented studies are a part of a multilateral detailed analysis of the CC low-flow stage modeling problem and serve as the basis for creating a calculation model of the flow parts low-speed stage, taking into account its main features. The objective of this research was to model the SVD-1 stage of a low-flow CC with a conditional flow coefficient \(\Phi\) = 0.008 and a theoretical head coefficient \(\psi_{\text{t}}\) = 0.48. The conditional flow coefficient is determined by the following formula:  
\[\Phi=4\overline{\text{m}}/\big{(}\varrho^{*}_{\text{in}}\pi\text{D}^{2}{}_{2 \text{u}2}\big{)}, \tag{1}\]  
where \(\overline{\text{m}}\) is mass flow, \(\varrho^{*}_{\text{in}}\) is total density in stage inlet, \(\text{D}_{2}\) is impeller outlet diameter, and \(\text{u}_{2}\) is impeller circumferential speed. The stage was developed and tested at the Department of \"Compressor, Vacuum and Refrigeration equipment\" of the Peter the Great St. Petersburg Polytechnic University. The sketch of the level is shown in Figure 1a. There are several approaches to heat exchange modeling; the two main approaches are as follows:  
1. For conjugate heat exchange, in addition to calculating the gas flow, the heat exchange of the flow with washed solids is modeled. For a centrifugal compressor, these solid washable bodies are the disks of the impeller, diaphragms, and the housing. The task statement implies the setting of boundary conditions on the external surfaces of solids that do not border the gas-dynamic path of the stage. One of the factors limiting correct conjugate heat exchange modeling is the unknown boundary conditions on the external solid surfaces. In order to include the heat transfer from the stator and housing elements to the environment, it is necessary to have either the temperature distribution on the external surfaces or the heat transfer coefficient. In most cases of modeling,  
Figure 1: **(a) Sketch of the model stage SVD-1 flow part and leakage diagram (red arrows); (b) The SVD-1 stage mesh model for conjugate heat transfer modeling. **these parameters are unknown and can be approximately calculated by thermal analysis of the flow part construction. However, for an experimental stage consideration, in order to measure and calculate the total pressure more accurately, a high level of thermal insulation of the housing elements must be ensured. Accordingly, it is accepted as an assumption that, in the system, \"flow parts-housing elements\" thermal equilibrium is established with temperature distribution close to the flow temperatures. 2. Simplified accounting of heat exchange involves setting one of several options applied to the flow part walls, i.e., temperature distribution, heat flux, or heat transfer coefficient. In the task of modeling the working process of the centrifugal compressor, simplified accounting of heat exchange seems to be simpler and more convenient, since there is no need to calculate the temperature distribution in solids. This greatly simplifies the mathematical and grid models. With a justified approach, simplified heat transfer calculation can provide acceptable engineering accuracy. An option, considered in this paper, is setting the indicative temperatures of the flow part walls according to the results of the conjugate heat exchange calculation. One of the main tasks is to ensure the correct temperature distribution in the near-field areas, which refers to the prevention of incorrect leaks' overestimation, which affects the gas-dynamic characteristics of the stage.