The gas turbine is a rotary heat engine with double conversion of energy. In guide vanes (nozzles), the potential energy of steam is converted into kinetic energy, which is then converted into mechanical work by rotating the turbine shaft (rotor). The turbine rotor drives the rotor of the consumer machine, like in alternators, compressors, pumps, etc.  To increase the efficiency of turbine installations one must increase the thermal efficiency of the cycle, as well as the efficiency of individual elements of the installation’s thermal scheme. A familiar method of increasing the efficiency of the thermal cycle is by increasing the temperature of the working fluid in front of the turbine. However, this not only requires using high-temperature materials, but also requires cooling the blade apparatus. As a result, installation costs increase, and the efficiency of the turbine stages decreases .
One way to mitigate the cost of the installation while maintaining relatively high efficiency is by introducing highly loaded stages with low volumetric flow rates of the working fluid. With these stages, we can reduce both the need for expensive alloys and steels, and the number of stages in the turbine altogether.
In the supersonic stage of turbines, one of the main elements affecting the overall efficiency is the first nozzle apparatus. For this reason, considerable attention in the literature has been given to the study of fluid flow processes in supersonic nozzles [4, 5] and methods of their profiling [6–8]. In fact, the impact is so substantial that the nozzle velocity coefficient has a greater effect on turbine stage performance than the rotor velocity coefficient. An increase in the velocity coefficient of the nozzle by 1% corresponds to an increase in the internal efficiency by 1–2.5% [3, 9, 10]. For air microturbines , the influence of the rotor velocity coefficient ψ on the stage efficiency is 3–4 times weaker than the influence of the nozzle velocity coefficient φ.
Depending on the parameters at which a turbine will operate, various types of nozzles can be used. In particular, at lower Mach numbers (1.4 or less) (Figure 2 (1)) it is possible to use nozzle arrays with converging channels. At higher Mach numbers, it is necessary to use channels formed by Laval type nozzles (Figure 2 (2)), the shape of which matches the geometry of the nozzle blades.There are various nozzle profiling methods. The most developed and widespread is the Method of Characteristics . A nozzle designed via this method dampens the rarefaction waves that occur during expansion in front of the exit section. This happens when a rarefaction wave meets a wall the direction of which coincides with the direction of the flow velocity vector when passing through this wave. When all waves are attenuated, the flow becomes uniform. An infinite number of rarefaction waves of infinitely small intensity is replaced by a finite number of waves of finite intensity (deflecting the flow at a certain angle).
For the design of short nozzles, we can use a method based on the properties of supersonic flow around obtuse angles. The characteristic vectors come from this method, and the wall break angle will be equal to half the deflection angle. The use of such nozzles is preferred due to design limitations related to the axial width of the section area. But in practice, a simplified method for profiling short supersonic nozzles is quite common . Despite the high efficiency of nozzle profiling by other methods, the simplified method allows us to design nozzle contours using straight lines and circular arcs.
Simplified profiling methods compare favorably to others due to the ease of setting the cascade contours on the drawing and the precision of nozzle production. In addition, nozzles profiled by these methods are less sensitive to mode changes than nozzles profiled by the Method of Characteristics.
Supersonic nozzles can be either milled, where the cross-sections are rectangular (Figure 3), or axisymmetric, where the cross-sections are circular (Figure 4). Axisymmetrical nozzles enjoy some advantages over milled nozzles; making the former more widespread.
However, axisymmetric nozzle cascades are not without drawbacks. When it is necessary to reduce the design angle of flow exit from the nozzle cascade (especially at large enthalpy differences and low flow rates of the working fluid), and at the same time ensure uniform flow in the impeller, turbines with milled nozzles may be more efficient.
The level of losses in drilled nozzles depends on the choice of various parameters, like the angle of drilling, the throat length, the fillet radius, and others. If nozzles of this type are located in a cascade, then an interaction occurs between the supersonic flows exiting each nozzle. This phenomenon can significantly affect losses. Therefore, when developing methods for calculating losses in nozzle cascades, it is necessary to take into account the location of the nozzles (Figure 5).
To date, there is not enough experimental data available to develop loss models taking into account all the various factors that affect the efficiency of the supersonic turbine cascade. Luckily, CFD programs help when it comes to performing gas-dynamic computational studies of complex three-dimensional flows in the flow parts of turbomachines. This significantly reduces the time and material costs for obtaining the relevant result.
However, to generalize the results of computational studies and obtain data for loss systems, very many calculations are typically required. To overcome this, various techniques are used by way of reducing the number of calculations needed. One such technique involves obtaining a response surface. This reduces the number of calculations when solving the optimization problem by replacing the objective function with its approximation or interpolation dependence, i.e., with a formal macromodel (FMM).
Using a novel version of this approach, a method for calculating losses in cascades with conical drilled nozzles was developed and implemented in AxSTREAM. With this method, it is possible to account for the influence of convergent-divergent nozzle arrangements on aerodynamic losses in turbine cascades.
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