Design method for centripetal turbine volute and nozzle ring of micro gas turbine

Through near-zero vortex design method and three-dimensional modeling optimization, the problems of flow loss and wheel circumference efficiency in the centripetal turbine design of traditional micro-fueling engines are solved, and more efficient aerodynamic performance and static strength are achieved.

CN119558005APending Publication Date: 2025-03-04SHENYANG LIMING AERO-ENGINE GROUP CORPORATION
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Patent Information

Application Number
CN202411635738.5
Authority / Receiving Office
CN · China
Patent Type
Applications(China)
Current Assignee / Owner
Filing Date
2024-11-15
Publication Date
2025-03-04

AI Technical Summary

Technical Problem

The design of traditional micro-gas engine centripetal turbines cannot achieve the minimum flow loss and the highest circumferential efficiency at the same time. At the same time, the energy conversion rate of the nozzle ring guide casing is low and the static strength of the volute is insufficient.

Method used

The near-zero vortex design method is used to design the one-dimensional design of the volute shell and nozzle ring. By adjusting the speed ratio, the absolute airflow angle of the impeller outlet is greater than 90°, and the three-dimensional geometric modeling of the nozzle ring and volute shell is constructed in three-dimensional modeling, and the structural parameters are optimized to reduce airflow loss and improve aerodynamic performance.

Benefits of technology

Effectively reduce airflow flow loss, improve the wheel circumferential efficiency and aerodynamic performance of the turbine, enhance the static strength of the volute, and meet the overall design requirements of the micro-fuel engine.

✦ Generated by Eureka AI based on patent content.

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Abstract

The invention relates to the technical field of design of centripetal turbines of micro gas turbines, in particular to a design method for a volute and a nozzle ring of a centripetal turbine of a micro gas turbine. One-dimensional design of a volute and a nozzle ring is carried out by adopting a near-zero vortex design method, after a three-dimensional model is established, three-dimensional flow analysis and structure optimization are carried out, and the blade profile design of the nozzle ring adopts an integer operation mode in the design of an axial flow plane cascade gas turbine under the framework of the near-zero vortex design. The required annular blade grid is obtained through shape-preserving transformation, the blade profile can achieve a better pneumatic effect, and the energy conversion rate is increased; ribs are additionally arranged at an outlet in the volute design, and the problem that static strength of the volute is insufficient possibly is solved. Due to the design of the nozzle ring and the volute, the overall design requirement of the centripetal turbine of the micro gas turbine is met, the aerodynamic performance of the turbine is further improved, and the flowing loss of airflow is reduced.
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Description

Technical Field

[0001] The invention relates to the technical field of micro-gas engine centripetal turbine design, in particular to a design method for a micro-gas engine turbine volute and a nozzle ring. Background Art

[0002] Micro gas turbines are widely used in small distributed power generation devices, meeting the development needs of diversified energy supply. Advanced micro gas turbines have a series of advanced technical features such as multiple integrated expansion, multi-fuel, low fuel consumption rate, low noise, low emissions, low vibration, low maintenance rate, remote control and diagnosis.

[0003] The turbine volute and nozzle ring are key parts for the flow of gas in micro gas turbines. They can guide the gas drawn from the compressor in the gas turbine through a certain flow path and enter the turbine impeller, making it more efficient in working.

[0004] The zero vortex design method is commonly used in the traditional design method of the centrifugal turbine, but this design method cannot achieve the minimum flow loss and the highest circumferential efficiency. At the same time, the traditional nozzle ring guide vane adopts island blades, thin plate blades, wedge-shaped symmetrical blades and some aerodynamic blades, and the energy conversion rate of these traditional blades is relatively low.

[0005] The place where the volute outlet is connected to the nozzle ring for ventilation is usually designed as a cavity structure in traditional cases. This design will reduce the strength reserve of the volute to a certain extent. Summary of the invention

[0006] In summary, the purpose of the present invention is to provide a design method for a micro gas turbine centripetal turbine volute and nozzle ring, so as to meet the overall performance parameters of the gas turbine, such as inlet temperature, intake pressure, turbocharger ratio, exhaust temperature, turbine speed, and wheel circumferential efficiency, and to further improve the aerodynamic performance of the turbine and reduce the flow loss of the airflow.

[0007] The technical solution adopted by the present invention is: a design method for a micro-gas turbine centripetal turbine volute and a nozzle ring, comprising the following contents:

[0008] (1) One-dimensional design of volute and nozzle ring using near-zero vortex design method;

[0009] The near-zero vortex design method is based on the zero vortex design method, and the speed ratio u1 in the zero vortex design method is appropriately adjusted so that the absolute airflow angle α2 at the impeller outlet is greater than 90°;

[0010] Among them, the one-dimensional design of the volute and nozzle ring using the near-zero vortex design method is as follows:

[0011] Through the entropy increase law and enthalpy reduction principle in thermodynamics, the aerodynamic parameters and structural parameters before the impeller inlet are used to obtain the structural parameters and aerodynamic parameters before and after the nozzle ring guide vane. The structural parameters and aerodynamic parameters before the guide vane obtained in the one-dimensional design of the nozzle ring are used to obtain the one-dimensional geometric structural parameters of the volute. The cross-sectional shape of the volute is a trapezoidal cross-section.

[0012] (2) Use 3D modeling software to construct the 3D geometric modeling of the nozzle ring and volute;

[0013] When modeling the nozzle ring, we must first determine the mid-arc of the guide vane, then select the blade shape of the guide vane, and finally build the three-dimensional geometric modeling of the nozzle ring in the three-dimensional modeling software based on the one-dimensional structural parameters of the nozzle ring;

[0014] Among them, when modeling the nozzle ring, a single arc is used as the middle arc line of the guide vane cascade, and the nozzle outlet angle needs to be reduced when determining the middle arc line of the guide vane cascade;

[0015] Wherein, a bladeless radial gap is provided between the guide vane cascade and the impeller;

[0016] The blade profile of the guide cascade is obtained by conformal transformation using integer calculation method in the design of axial-flow plane cascade gas turbine;

[0017] (3) Three-dimensional flow analysis and structural optimization;

[0018] The numerical simulation of three-dimensional flow analysis is carried out using the flow field analysis software STAR-CD. During the simulation, the dynamic and static parts of the centripetal turbine are calculated simultaneously, and the implicit method of multi-rotating reference frames in STAR-CD is used to deal with the relationship between the dynamic and static parts. The turbulence model selected is the standard k-ε two-equation turbulence model, that is, the high Reynolds number k-ε two-equation turbulence model. The fluid can be regarded as an ideal gas, satisfying the ideal gas state equation, and is a compressible fluid (Ma>0.3), so when selecting physical parameters, ρ=f(T,p), Ma-Mach number, ρ-density, T-temperature, p-pressure; specific heat is a constant, with a size of 1148.86 J / kg·K; the boundary condition inlet is a stagnation point, given the total pressure, total temperature and incoming flow direction, the total pressure P0 * =34200 (Pa), total temperature T0 * =1173(K); the outlet condition is given exhaust pressure P2=109960(Pa), exhaust temperature T2=903(K);

[0019] According to the overall design requirements of the micro gas turbine, the structures of the volute and nozzle ring are optimized, including:

[0020] When the impeller inlet flow rate does not meet the design requirements, the impeller inlet flow rate can be increased by increasing the blade height of the nozzle ring guide vane;

[0021] When the absolute flow velocity at the impeller inlet is significantly lower than the one-dimensional design inlet flow velocity, and the total temperature at the impeller outlet is also higher than the one-dimensional design value, the outlet blade angle of the nozzle ring guide vane and the outlet blade angle of the moving blade can be reduced to increase the absolute flow velocity at the impeller inlet.

[0022] Further, a plurality of ribs are provided at the outlet of the volute;

[0023] Furthermore, a radial gap section is provided between the nozzle ring inlet and the rib outlet.

[0024] The beneficial effects of the present invention are:

[0025] 1. The present invention adopts a design method of a near-zero swirl centripetal turbine, which can effectively reduce airflow loss and improve the circumferential efficiency of the rotor.

[0026] 2. Nozzle ring blade design: In the framework of near-zero vortex design, the required annular blade is obtained by conformal transformation using the integer operation method in the design of axial plane blade gas turbines. This blade type will achieve better aerodynamic effects and improve energy conversion efficiency.

[0027] 3. Ribs are added to the outlet of the volute to avoid the problem of insufficient static strength of the volute.

[0028] 4. The design of the nozzle ring and volute not only meets the overall design requirements of the micro-turbine centripetal turbine, but also further improves the aerodynamic performance of the turbine and reduces the flow loss of the airflow. BRIEF DESCRIPTION OF THE DRAWINGS

[0029] Figure 1 This is a three-dimensional modeling diagram of the centripetal turbine volute and nozzle ring;

[0030] FIG2 is a three-dimensional modeling diagram of the volute;

[0031] FIG3 is a three-dimensional modeling diagram of the nozzle ring;

[0032] Figure 4 It is a schematic diagram of the structural parameters of the trapezoidal interface of the nozzle ring;

[0033] Figure 5 It is a schematic diagram of the arc line in the nozzle ring;

[0034] Figure 6 The 3D modeling diagram of the guide blade:

[0035] Figure 7 This is a 3D modeling diagram of the ribs on the volute:

[0036] Figure 8 Schematic diagram of the gap between the nozzle ring inlet and the rib outlet. DETAILED DESCRIPTION

[0037] To make the purpose, technical solution and advantages of the present invention clearer, the technical solution of the present invention will be described in detail below. Obviously, the described embodiments are only part of the embodiments of the present invention, rather than all the embodiments. Based on the embodiments of the present invention, all other implementation methods obtained by ordinary technicians in this field without creative work belong to the scope of protection of the present invention.

[0038] The present invention provides a design method for a micro gas turbine centripetal turbine volute and a nozzle ring, and the specific steps are as follows:

[0039] (1) One-dimensional design of volute and nozzle ring;

[0040] like Figure 1 -3, in this embodiment, the design of the volute and the nozzle ring adopts a near-zero vortex design method based on a zero vortex design method;

[0041] Among them, the design principle of the zero vortex design method is that when the centrifugal turbine is exhausted axially, the residual speed loss of the centrifugal turbine is minimized. Therefore, the design of the centrifugal turbine is usually based on the so-called zero tail vortex principle, that is, the absolute airflow angle α2 = 90° at the impeller outlet.

[0042] However, the centrifugal turbine does not have the minimum flow loss and the highest circumferential efficiency at the same time under the design principle of zero tail vortex. In fact, the minimum flow loss of the centrifugal turbine appears when the absolute flow angle α2 at the impeller outlet is greater than 90°.

[0043] Therefore, the near-zero vortex design method used in this embodiment is based on the zero vortex design method by adjusting the speed ratio in the zero vortex design method. The ratio of the input speed to the output speed of the gas in the centripetal turbine is appropriately adjusted to make the absolute airflow angle α2 at the impeller outlet slightly larger than 90°. In addition, the near-zero vortex design method can also make the circumferential efficiency η u Get some improvement.

[0044] Specifically, according to the overall performance parameter requirements of the micro gas turbine, after preliminary estimation, the circumferential efficiency η of the centrifugal turbine is u Cannot be less than 85%. Under this condition, according to the near-zero vortex design method, the aerodynamic parameters and structural parameters of the impeller can be obtained. Through the law of entropy increase and the principle of enthalpy reduction in thermodynamics, the aerodynamic parameters and structural parameters of the impeller before the impeller inlet, such as the absolute airflow angle of the impeller inlet, the speed coefficient of the guide device, the total pressure, total temperature, flow rate, airflow rate, turbine efficiency, etc., the structural parameters and aerodynamic parameters before and after the guide vane are obtained to complete the one-dimensional design of the nozzle ring.

[0045] In this embodiment, the one-dimensional design parameters of the nozzle ring can be obtained by calculation, as shown in Table 1:

[0046] Table 1: One-dimensional design parameters of nozzle ring

[0047]

[0048] The structural parameters and aerodynamic parameters before the guide vane obtained in the one-dimensional design of the nozzle ring are used as the parameters of the volute outlet and the one-dimensional geometric structural parameters of the volute are obtained. The cross-sectional shape of the volute is a trapezoidal cross-section to further reduce the airflow loss. The one-dimensional geometric structural parameters of the volute and the cross-sectional dimensions of each cross-sectional angle θ are shown in Table 2. The specific meaning of each parameter can be referred to Figure 4 :

[0049] Table 2: One-dimensional geometric structure parameters of the volute

[0050]

[0051] (2) Use 3D modeling software to construct the 3D geometric modeling of the nozzle ring and volute;

[0052] Specifically, when modeling the nozzle ring, the middle arc line of the guide leaf must be determined first;

[0053] Generally, the middle arc line has single arc, double arc and parabola. In this embodiment, a single arc is used as the middle arc line. According to the diameter D of the nozzle ring outlet in the one-dimensional design, N , the nozzle ring inlet cross-sectional diameter D0, the blade installation angle, i.e. the difference between the nozzle inlet angle α0 and the nozzle outlet angle α1, and the nozzle outlet angle α1 can determine the center arc line of the guide vane.

[0054] Furthermore, since the outlet airflow angle of the annular blade cascade changes continuously along the circumferential direction, the nozzle outlet angle α1 determined in the one-dimensional design is actually an average value, so the nozzle outlet angle α1 needs to be reduced when determining the middle arc of the guide blade cascade. In this embodiment, the nozzle outlet angle α1 is reduced to 17°, as shown in FIG. Figure 5-7 shown.

[0055] Furthermore, a bladeless radial gap is provided between the guide vane cascade and the impeller to prevent the flow field at the outlet of the guide vane cascade from being uneven, which would cause some adverse effects on the operation of the impeller.

[0056] When the annular radial gap is large enough, for example, greater than 8 mm, the flow field of the airflow at the radial gap outlet tends to be uniform through the energy exchange between particles. In addition, the radial gap is also necessary to reduce noise and erosion of the guide vanes and prevent the impeller from rubbing.

[0057] Secondly, select the blade type of the guide cascade;

[0058] In the past, the guide vanes of the centrifugal turbine mostly adopted island blades, thin plate blades, wedge-shaped symmetrical blades and some aerodynamic blades, such as the TC series blades of Russia. Since the flow loss of the first three blades is relatively large and the aerodynamic effect is poor, these blades are not adopted in the present invention. Although the TC series blade has better aerodynamic performance, there is no volute in front of this guide vane, so it is not adopted.

[0059] Since the combustion chamber of the micro-turbine adopts an annular combustion chamber, the guide device of the turbine part is required to consist of two major components, namely the guide vane and the volute. In this way, a new blade shape was decided when selecting the blade shape.

[0060] The present invention utilizes the relatively mature integer operation method in the design of axial flow plane blade gas turbines to obtain the required annular blade through conformal transformation. This blade shape not only has good aerodynamic effect, but also can meet the structural requirements of adding a volute before it.

[0061] Then, the blades are thickened according to the blade thickness distribution law and the determined mid-arc line in the integer operation method in the gas turbine design;

[0062] Finally, after a series of coordinate transformations, the required blade shape is obtained. The blade shape data of the guide vane obtained is input into the PRO-E 3D modeling software to complete the 3D modeling of the nozzle ring.

[0063] The modeling of the volute is also carried out in PRO-E 3D modeling software in this embodiment, and the drawing sequence is described as follows:

[0064] The volute consists of two parts; the air intake section and the volute;

[0065] The modeling method for the cochlear tube is as follows:

[0066] ①Use sweep blend method;

[0067] ②The scanning trajectory is a 358.5-degree arc;

[0068] ③ Each scanned section is a trapezoid with rounded corners;

[0069] ④ According to the needs, use a plane to cut off a part at both ends of the volute;

[0070] The modeling method for the intake section is as follows:

[0071] Using the sweep blend method, the scanned trajectory line a is a spline curve, and the trajectory line b is a curve composed of a small straight line segment and a spline curve.

[0072] One end of the inlet section is used to connect with the volute, and the other end is round.

[0073] (3) Three-dimensional flow analysis and structural optimization;

[0074] The numerical simulation of three-dimensional flow analysis is carried out using the flow field analysis software STAR-CD. During the simulation, the dynamic and static parts of the centripetal turbine are calculated simultaneously, and the implicit method of multi-rotating reference frames in STAR-CD is used to deal with the relationship between the dynamic and static parts. The turbulence model selected is the standard k-ε two-equation turbulence model, that is, the high Reynolds number k-ε two-equation turbulence model. The fluid can be regarded as an ideal gas, satisfying the ideal gas state equation, and is a compressible fluid (Ma>0.3), so when selecting physical parameters, ρ=f(T,p), Ma-Mach number, ρ-density, T-temperature, p-pressure; specific heat is a constant, with a size of 1148.86 J / kg·K; the boundary condition inlet is a stagnation point, given the total pressure, total temperature and incoming flow direction, the total pressure P0 * =34200 (Pa), total temperature T0 * =1173(K); the outlet condition is the given exhaust pressure P2=109960(Pa), the exhaust temperature T2=903(K).

[0075] According to the overall design requirements of the micro-turbine, the structures of the volute and nozzle ring are optimized;

[0076] Among them: when the impeller inlet flow rate does not meet the design requirements, the impeller inlet flow rate can be increased by increasing the blade height of the nozzle ring guide vane;

[0077] Specifically, in this embodiment, the flow rate can be effectively increased by increasing the height of the guide vane grid, but the blade height at the impeller inlet is usually slightly greater than the height of the nozzle ring guide vane grid. Therefore, the guide vane grid height and the impeller blade height need to be increased at the same time to reduce the vortex in the impeller to a very small range or even eliminate it.

[0078] After adjusting the guide vane cascade and impeller blade height appropriately, the flow analysis of the entire centripetal turbine was conducted, and the flow rate was indeed greatly improved, from 0.82 kg / s obtained in the first simulation to 1.006 kg / s. Since the design flow rate is required to reach 1.03 kg / s, the guide vane cascade blade height is increased to 17.5 mm. When the guide vane cascade height is increased to 17.5 mm, the flow rate reaches 1.029 kg / s, so the flow rate meets the overall design requirements of the micro gas turbine.

[0079] When the absolute flow velocity at the impeller inlet is significantly lower than the one-dimensional design inlet flow velocity, and the total temperature at the impeller outlet is also higher than the one-dimensional design value, the outlet blade angle of the nozzle ring guide vane and the outlet blade angle of the moving blade can be reduced to increase the absolute flow velocity at the impeller inlet.

[0080] Specifically, in this embodiment, it is found that the absolute flow velocity at the impeller inlet is significantly lower than the inlet flow velocity during the one-dimensional design, and the total temperature at the impeller outlet is also higher than the one-dimensional design value, so the outlet blade angle of the nozzle ring guide vane and the outlet blade angle of the moving blade are reduced. In order to keep the flow rate at 1.03 kg / s unchanged, the blade height of the nozzle ring guide vane is further increased to 18 mm. After the flow analysis, it still cannot meet the design requirements, so the nozzle ring outlet angle is further reduced to 14°. The flow analysis results show that the impeller inlet flow velocity increases significantly.

[0081] In addition, in order to improve the strength of the volute, the present invention provides a plurality of ribs at the volute outlet, such as Figure 7 As shown, four ribs are added to the volute outlet;

[0082] After adding ribs and changing the meridian flow channel accordingly, the model was remodeled and flow analysis was performed. The analysis results showed that the flow rate failed to meet the design requirements 100%. When checking the flow field distribution of the internal flow field, it was found that the airflow behind each rib had a certain separation, so a radial gap section was also added between the nozzle ring inlet and the rib outlet, such as Figure 8 As shown, this can improve the flow of airflow in the guide vane and make the flow field as uniform as possible.

[0083] After optimized design, the micro gas turbine turbine inlet volute and nozzle ring designed in the present invention can meet the design requirements of aerodynamic flow of the volute and nozzle ring. The volute is composed of an inlet pipe and a volute, and the nozzle ring has 24 blades to form a cascade channel.

[0084] After the above design and optimization, the micro-turbine turbine volute and nozzle ring of the present invention can meet the design index inlet temperature T3 * =1173(K), intake pressure P3 * =34200 (Pa), turbocharger ratio Π T =3.11, exhaust temperature T2=903(K), turbine speed is 60000(rpm), and wheel efficiency is 88.826%.

[0085] The micro-turbine turbine air intake volute and nozzle ring shape can meet the requirements of turbine impeller inlet airflow and greatly reduce airflow loss. Due to the unique volute and nozzle ring shape design, the parts designed by the present invention greatly improve the turbine efficiency, save energy, and create good economic and social benefits. The design method of the turbine air intake volute and nozzle ring can also be applied to other internal combustion engine superchargers, small and medium-sized gas turbine devices, and turbine air intake volutes and nozzle rings used in turbo expanders in chemical and refrigeration equipment.

[0086] The above description is only a specific implementation mode of the present invention, but the protection scope of the present invention is not limited thereto. Any technician familiar with the technical field can easily think of changes or substitutions within the technical scope disclosed by the present invention, which should be covered by the protection scope of the present invention.

Claims

1. A design method for a micro gas turbine centripetal turbine volute and nozzle ring, characterized in that: Includes the following: (1) One-dimensional design of volute and nozzle ring using near-zero vortex design method; The near-zero vortex design method is based on the zero vortex design method by adjusting the speed ratio in the zero vortex design method. Make appropriate adjustments so that the absolute airflow angle α2 at the impeller outlet is greater than 90°; (2) Use 3D modeling software to construct the 3D geometric modeling of the nozzle ring and volute; When modeling the nozzle ring, first determine the mid-arc line of the guide vane, then select the blade shape of the guide vane cascade, and finally build the three-dimensional geometric modeling of the nozzle ring in the three-dimensional modeling software in combination with the one-dimensional structural parameters of the nozzle ring; (3) Three-dimensional flow analysis and structural optimization; The numerical simulation of three-dimensional flow analysis is carried out using the flow field analysis software STAR-CD. During the simulation, the dynamic and static parts of the centripetal turbine are calculated simultaneously. When dealing with the relationship between the dynamic and static parts, the implicit method of the multi-rotating reference system in STAR-CD is used; the turbulence model selected is the standard k-ε two-equation turbulence model, that is, the high Reynolds number k-ε two-equation turbulence model; the fluid can be regarded as an ideal gas, satisfying the ideal gas state equation, and is a compressible fluid (Ma>0.3). When selecting physical parameters, ρ=f(T,p), Ma-Mach number, ρ-density, T-temperature, p-pressure; the specific heat is a constant, and its size is 1148.86 J / kg·K; the boundary condition inlet is a stagnation point, and the total pressure, total temperature and incoming flow direction are given, and the total pressure P0 * =34200 (Pa), total temperature T0 * =1173(K); the outlet condition is, given exhaust pressure P2=109960(Pa), exhaust temperature T2=903(K); According to the overall design requirements of the micro gas turbine, the structures of the volute and nozzle ring are optimized, including: When the impeller inlet flow rate does not meet the design requirements, increase the blade height of the nozzle ring guide cascade to increase the impeller inlet flow rate; When the absolute flow velocity at the impeller inlet is less than the inlet flow velocity during one-dimensional design and the total temperature at the impeller outlet is higher than the one-dimensional design value, the outlet blade angle of the nozzle ring guide vane and the outlet blade angle of the moving blade are reduced to increase the absolute flow velocity at the impeller inlet.

2. The design method of the micro-gas turbine centripetal turbine volute and nozzle ring according to claim 1 is characterized in that: The one-dimensional design of the volute and nozzle ring using the near-zero vortex design method is as follows: Through the law of entropy increase and the principle of enthalpy reduction in thermodynamics, the structural parameters and aerodynamic parameters before and after the nozzle ring guide vane are obtained using the aerodynamic parameters and structural parameters before the impeller inlet. The one-dimensional geometric structural parameters of the volute are obtained using the structural parameters and aerodynamic parameters before the guide vane obtained in the one-dimensional design of the nozzle ring. The cross-sectional shape of the volute is a trapezoidal cross-section.

3. The design method of the micro-gas turbine centripetal turbine volute and nozzle ring according to claim 1 is characterized in that: A plurality of ribs are arranged at the outlet of the volute.

4. The design method of the micro-gas turbine centripetal turbine volute and nozzle ring according to claim 3 is characterized in that: A radial gap section is provided between the nozzle ring inlet and the rib outlet.

5. The design method of the micro-gas turbine centripetal turbine volute and nozzle ring according to claim 1 is characterized in that: When modeling the nozzle ring, a single circular arc is used as the center arc line of the guide vane cascade. When determining the center arc line of the guide vane cascade, the nozzle outlet angle needs to be reduced.

6. The design method of the micro-gas turbine centripetal turbine volute and nozzle ring according to claim 1 is characterized in that: There is a bladeless radial gap between the guide vane grid and the impeller.

7. The design method of the micro-gas turbine centripetal turbine volute and nozzle ring according to claim 1 is characterized in that: The blade profile of the guide vane cascade is obtained through conformal transformation using integer operations in the design of axial-flow plane cascade gas turbines.