A centrifugal fan impeller reinforcing ring structure and a parameterized design method thereof
Patent Information
- Application Number
- CN202611154558.4
- Authority / Receiving Office
- CN · China
- Patent Type
- Applications(China)
- Current Assignee / Owner
- Filing Date
- 2026-07-31
- Publication Date
- 2026-11-03
AI Technical Summary
The existing centrifugal fan impeller reinforcing ring design, while ensuring structural rigidity, leads to a decrease in aerodynamic performance, especially problems such as flow blockage, separation, and outlet flow field distortion, which affect the fan efficiency.
By optimizing parameters such as the starting position, tilt angle, and extension length of the reinforcing ring, a negative tilt angle guide surface is designed. Combined with CFD simulation and multi-objective optimization methods, a guide path is formed to reduce flow loss and improve aerodynamic efficiency.
While ensuring structural rigidity, the static pressure efficiency of the fan is significantly improved, the impeller deformation is reduced, and the high-efficiency operating range is expanded, without changing the existing manufacturing process.
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Abstract
Description
Technical Field
[0001] This invention belongs to the field of centrifugal fan technology, and in particular relates to a centrifugal fan impeller reinforcing ring structure and its parametric design method. Background Technology
[0002] A backward-curved centrifugal fan impeller typically consists of a front plate, a rear plate, and several backward-curved plate blades. For larger impellers, to improve the impeller's structural rigidity and operational reliability, it is usually necessary to weld an annular reinforcing ring extending in the circumferential direction onto the blades to enhance blade rigidity and suppress blade vibration.
[0003] Existing reinforcing ring designs typically follow this approach: the reinforcing ring is welded to the middle section of the blade or at the exit, with its surface extending horizontally radially, i.e., a flat plate ring. The width and position of the ring are mainly determined based on structural mechanics requirements, without fully considering its impact on the aerodynamic flow field. A few designs, in order to further reduce aerodynamic losses, tilt the reinforcing ring at a certain angle towards the inside of the impeller to act as a guide, but the selection of the tilt angle lacks systematic research and optimization methods based on flow field analysis, resulting in limited optimization effects.
[0004] The flow inside a centrifugal fan impeller is a complex three-dimensional viscous flow. As an additional structural component within the impeller flow channel, the reinforcing ring inevitably alters the local flow channel geometry, typically leading to the following negative effects: First, flow obstruction; the reinforcing ring occupies flow channel space, reducing the effective flow area and causing local acceleration and pressure loss. Second, intensified separation; as airflow passes through the reinforcing ring, flow separation zones or vortex zones may form on its leeward side and at its ends, increasing flow losses. Third, distortion of the outlet flow field; the reinforcing ring at the outlet obstructs or disturbs the outlet airflow, disrupting the uniformity of the outlet flow and reducing fan efficiency.
[0005] Therefore, in engineering practice, reinforcing rings are often considered a "necessary evil"—structurally essential, but inevitably leading to aerodynamic performance degradation. Conventionally designed flat rings can reduce the static pressure efficiency of a fan by 1 to 5 percentage points, with the specific reduction depending on the ring's size and location. How to minimize or even eliminate this performance loss while ensuring structural rigidity has been a long-standing technical challenge in centrifugal fan impeller design. Summary of the Invention
[0006] The summary section of this application is intended to provide a brief overview of the concepts, which will be described in detail in the detailed description section below. This summary section is not intended to identify key or essential features of the claimed technical solutions, nor is it intended to limit the scope of the claimed technical solutions.
[0007] To address the technical problems mentioned in the background section, some embodiments of this application provide a centrifugal fan impeller reinforcing ring structure, including an impeller front disc, an impeller rear disc, and blades distributed along the circumferential direction. A reinforcing ring extending along the circumferential direction is provided on the blades. The configuration of the reinforcing ring is defined by three parameters: the ring starting position zs, the ring tilt angle α, and the ring extension length L. The meridional coordinate system uses the inner wall of the impeller rear disc as the axial origin and the direction towards the impeller front disc as the positive axial direction. zs is the axial coordinate of the starting point of the reinforcing ring welding. The ring tilt angle α is the angle between the centerline of the reinforcing ring (the geometric centerline of the reinforcing ring cross-sectional thickness) and the radial ray of the impeller. α < 0 indicates that the reinforcing ring is tilted towards the impeller rear disc; α > 0 indicates that the reinforcing ring is tilted towards the impeller front disc; α = 0 indicates that the reinforcing ring extends horizontally along the radial direction. The ring starting position zs is the starting point position where the reinforcing ring is welded to the blade surface, represented by its axial coordinate zs (mm) in the meridional coordinate system. This parameter determines the specific position of the reinforcing ring along the flow channel between the leading and trailing edges of the blade. An increase in zs means the ring's starting point moves closer to the front disc. The starting position of the inlet ring is restricted to the leading edge of the blade; the starting position of the outlet ring is restricted to the trailing edge of the blade; the ring extension length L is the length (mm) of the reinforcing ring centerline from the welding start point to the free end. The extension length is decomposed radially and axially, with Lr representing the radial length and Lz representing the axial length. These two parameters determine the radial / axial extension degree of the reinforcing ring within the flow channel. By optimizing these three parameters, the reinforcing ring forms a guiding surface within the impeller flow channel, ensuring that the fan's static pressure efficiency is not lower than the reference impeller static pressure efficiency without the reinforcing ring.
[0008] Preferably, the reinforcing ring is an inlet reinforcing ring disposed at the blade inlet. For the inlet reinforcing ring, the axial coordinate zs of the ring's starting position is based on the midpoint of the meridional projection of the blade inlet extension line. The ratio of the ring's starting position zs to the impeller's outer diameter is 0.1433 to 0.2267. Based on an impeller diameter of 1200 mm, the value of zs ranges from 172 to 272 mm, preferably 240 to 272 mm. The ring tilt angle α ranges from -75° to -45°, preferably -65° to -55°. The ratio of the ring extension length L to the impeller's outer diameter is 0.029 to 0.054. Based on an impeller diameter of 1200 mm, the value of L ranges from 35 to 65 mm, preferably 45 to 55 mm. The thickness of the reinforcing ring is 3 to 10 mm.
[0009] Preferably, the reinforcing ring is an outlet reinforcing ring located at the blade outlet and the outer diameter of the impeller. The outlet reinforcing ring has a rectangular cross-section and a three-dimensional conical structure. The inclination angle α ranges from -20° to -10°. The ratio of the radial component Lr of the ring extension length to the outer diameter of the impeller is 0.042 to 0.125. Based on an impeller diameter of 1200 mm, the radial component Lr of the ring extension length is 50 to 150 mm, preferably 60 to 75 mm. The thickness of the reinforcing ring is 3 to 10 mm.
[0010] Preferably, the reinforcing ring includes an inlet reinforcing ring simultaneously disposed at the blade inlet and an outlet reinforcing ring simultaneously disposed at the blade outlet. The inclination angle α of both the inlet reinforcing ring and the outlet reinforcing ring is negative, and both are inclined towards the impeller back disk. The initial axial height of the outlet ring is 10 to 135 mm, forming a continuous "U"-shaped guide path in the meridional plane.
[0011] On the other hand, the present invention provides a parametric design method for a centrifugal fan impeller reinforcing ring, comprising the following steps: Step S1: Determine the aerodynamic performance parameters of the reference impeller without a reinforcing ring, including static pressure efficiency, static pressure rise, and shaft power under multiple flow conditions; Step S2: Establish a parametric geometric model of the reinforcing ring, with the ring starting position zs, ring inclination angle α, and ring extension length L as design variables; Step S3: In the parameter space composed of three parameters, sample points are selected using the experimental design method, and the aerodynamic performance of the fan corresponding to each sample point is obtained through computational fluid dynamics simulation. Step S4: Based on CFD simulation data, construct a proxy model with three parameters and flow conditions as inputs and aerodynamic performance as output; Step S5: Based on the surrogate model, determine the optimal parameter combination that takes into account multiple traffic conditions through a multi-objective optimization method; Step S6: Apply the optimal parameter combination to the engineering design of the reinforcing ring.
[0012] Preferably, in step S3, the experimental design method adopts central composite design (CCD) or Latin hypercube sampling (LHS).
[0013] Preferably, in step S4: the surrogate model is a Gaussian process regression (GPR) model or a Kriging regression model, the kernel function is a Matern-2.5 or radial basis function (RBF), and the model prediction accuracy is evaluated by leave-one-out cross-validation.
[0014] Preferably, in step S5, the multi-objective optimization is defined as follows: Optimization objectives: To maximize the aerodynamic performance of the fan: maximize the static pressure efficiency η_sd under the design flow rate; maximize the static pressure efficiency η_sl under the low flow rate (preferably 80%~90% of the design flow rate); maximize the static pressure efficiency η_sh under the high flow rate (preferably 110%~120% of the design flow rate).
[0015] Constraints: Structural strength constraints must be met: the maximum equivalent stress of the impeller shall not exceed the allowable value of the safety factor of the material yield strength (safety factor ≥ 1.5); the maximum total deformation of the impeller shall not exceed 2.4 mm (0.2% of the maximum outer diameter).
[0016] Optimal solution selection: For the Pareto front solution set, the knee point method is used to select the parameter combination that achieves the best balance among multiple objectives as the final design point.
[0017] Compared with the prior art, the present invention has the following beneficial effects: 1. Overcoming technical biases and achieving aerodynamic efficiency enhancement. This invention, through systematic parameter optimization, discovered a region in the parameter space of the reinforcing ring where aerodynamic efficiency surpasses that of the ringless reference, overturning the traditional perception that "reinforcing rings inevitably reduce performance." Taking a backward centrifugal fan with a diameter of 1200mm and a rotation speed of 1480rpm as an example, the static pressure efficiency at the inlet ring design point is 0.86 percentage points higher than that of the ringless reference, and 0.48 percentage points higher than that of the outlet ring. The efficiency improvement of the combined inlet and outlet scheme can reach 1.3~1.5 percentage points.
[0018] 2. Significant structural enhancement effect. The addition of reinforcing rings effectively reduces the maximum deformation of the impeller. The inlet ring can reduce the maximum deformation from 2.81mm to 2.05mm, and the combined inlet and outlet design can reduce it to 1.21mm, significantly improving the impeller's structural stiffness and operational reliability.
[0019] 3. Strong adaptability to all operating conditions. The inlet reinforcing ring improves efficiency more significantly under low flow conditions, and the Pareto optimal solution can simultaneously take into account both low flow and design flow conditions; the negative tilt angle guiding effect of the outlet reinforcing ring is valid under all flow conditions, with strong robustness, which can effectively expand the high-efficiency operating range of the wind turbine.
[0020] 4. Good process compatibility. This invention only changes the welding angle, position, and length of the reinforcing ring, without adding extra parts or changing the welding process. It can be implemented under the existing manufacturing system without additional modification costs.
[0021] 5. High applicability. The three-parameter geometric description system and systematic design method provided by this invention can be directly applied to the design of reinforcing rings for centrifugal fan impellers of different diameters and rotational speeds, requiring only proportional scaling of the corresponding parameter ranges. Attached Figure Description
[0022] The accompanying drawings, which form part of this application, are used to provide a further understanding of the application and to make other features, objects, and advantages of the application more apparent. The illustrative embodiments and descriptions of this application are used to explain the application and do not constitute an undue limitation of the application.
[0023] Furthermore, throughout the accompanying drawings, the same or similar reference numerals denote the same or similar elements. It should be understood that the drawings are schematic, and the elements are not necessarily drawn to scale.
[0024] In the attached diagram: Figure 1 is a three-dimensional structural diagram of the impeller of an embodiment of the inlet and outlet combined reinforcing ring of the present invention; Figure 2 is a three-dimensional structural diagram of the blade and welded reinforcing ring in an embodiment of the inlet and outlet joint reinforcing ring of the present invention; Figure 3 is a schematic diagram of the structure of the inlet reinforcing ring and the outlet reinforcing ring of the present invention on the impeller meridional plane; Figure 4 shows the main effect response surface diagram of impeller inlet ring tilt angle and efficiency; Figure 5 shows the main effect response surface diagram of efficiency at the initial position of the impeller inlet ring; Figure 6 shows the RSM response of the interaction effect of the inlet ring tilt angle and the initial position on efficiency. Figure 7 shows the Pareto front diagram for the entire inlet loop under all operating conditions; Figure 8 shows the main effect response surface diagram of impeller outlet ring tilt angle and efficiency; Figure 9 The response surface plot is the radial width of the impeller outlet ring versus the main effect of efficiency. Detailed Implementation
[0025] Embodiments of this disclosure will now be described in more detail with reference to the accompanying drawings. While some embodiments of this disclosure are shown in the drawings, it should be understood that this disclosure can be implemented in various forms and should not be construed as limited to the embodiments set forth herein. Rather, these embodiments are provided to provide a more thorough and complete understanding of this disclosure. It should be understood that the accompanying drawings and embodiments of this disclosure are for illustrative purposes only and are not intended to limit the scope of protection of this disclosure.
[0026] It should also be noted that, for ease of description, only the parts relevant to the invention are shown in the accompanying drawings. Unless otherwise specified, the embodiments and features described in this disclosure can be combined with each other.
[0027] It should be noted that the concepts of "first" and "second" mentioned in this disclosure are used only to distinguish different devices, modules or units, and are not used to limit the order of functions performed by these devices, modules or units or their interdependencies.
[0028] It should be noted that the terms "a" and "a plurality of" used in this disclosure are illustrative rather than restrictive, and those skilled in the art should understand that, unless otherwise expressly indicated in the context, they should be understood as "one or more".
[0029] This disclosure will now be described in detail with reference to the accompanying drawings and embodiments.
[0030] To make the objectives, technical solutions, and advantages of this invention clearer, the invention will be further described in detail below with reference to the accompanying drawings and specific embodiments. The following embodiments use the same backward centrifugal fan as a verification platform, and its main reference parameters are as follows:
[0031] The performance of the ringless reference (i.e., the original impeller without any welded reinforcing rings) at the design flow point is used as a comparison reference for the following embodiments.
[0032] Example 1
[0033] like Figure 1-3 As shown, a centrifugal fan impeller reinforcing ring structure includes an impeller front disc 1, an impeller rear disc 2, and blades 3 distributed along the circumferential direction. Reinforcing rings extending along the circumferential direction are provided on the blades. The reinforcing rings are inlet reinforcing rings 4, welded to the blade inlet, with a straight line segment as their centerline and a thickness of 6 mm. The reference starting position of the ring in the meridional plane coordinates is (r=312.92mm, z=222.00mm).
[0034] The variable design parameters in this embodiment are: The initial axial position zs of the ring is 172 / 197 / 222 / 247 / 272 mm, with a total of 5 levels. As zs increases, the starting point moves along the outer extension line of the blade inlet towards the impeller front disk, and the radial coordinate r_s changes accordingly.
[0035] Ring tilt angle α: -10° / -15° / -20° / -25° / -30° / -36.87° / -45° / -50° / -53.13° / -55° / -60° / -65° / -70° / -75°, a total of 14 levels.
[0036] This embodiment employs a design of batch experiments (DOE), completing a total of 31 independent belt loop designs (32 cases including the baseline). Each case operates under three flow conditions (24.47 / 27.35 / 29.78 kg / s): Under the conditions of fixed zs=222mm and L=50mm, efficiency response surface analysis of the inclination angle α changing from -10° (close to radial) to -75° (significantly tilted towards the impeller rear plate) shows that there is a clear optimal inclination angle range for the inlet ring [-65°, -53°], with the center around -60°. An inclination angle that is too flat (close to 0°) cannot form effective flow guidance; an inclination angle that is too steep (above -65°) increases the deviation between the guide surface and the incoming flow, thus introducing additional impact losses.
[0037] Further proxy models (GPR, kernel function Matern-2.5, LOO-R) 2 Analysis (=0.964) shows that the optimal tilt angle is stable at around -60° across all zs levels (172~272mm), indicating that the optimal value of this parameter has strong robustness and only changes by about 8° under the influence of zs variations.
[0038] Comparisons with fixed zs=222mm, α=-60°, and L=35 / 50 / 65mm show that: L=50mm is the optimal length (efficiency exceeds the benchmark); at L=35mm, the guide surface is too short, and the efficiency drops below the benchmark; at L=65mm, the blockage effect increases, and the efficiency is slightly lower than at L=50mm but still higher than the benchmark. A value for L in the range of 45~55 mm is recommended (L / d2 = 0.038~0.046).
[0039] The recommended parameters for the inlet reinforcement ring in this embodiment are shown in the table below:
[0040] Enhancement Mechanism: The inlet ring adopts a negative tilt angle design, making the centerline of the ring surface approximately aligned with the inflow direction at the blade inlet. This is equivalent to adding a guide vane at the inlet, effectively suppressing secondary flow at the inlet and reducing flow separation. The benefits of guiding the flow outweigh the blockage loss of the ring itself, ultimately achieving a net efficiency improvement. The closer the starting position is to the leading edge of the blade, the earlier the guiding effect intervenes, and the stronger the control effect on the flow throughout the entire channel.
[0041] Example 2
[0042] The reinforcing ring is the outlet reinforcing ring 5, which is welded to the impeller outer diameter (Φ1200 mm) at the blade outlet, with its centerline being a straight segment. The fixed parameters are: ring thickness 6 mm, ring initial axial height (distance from the rear plate) 135 mm, and ring outer diameter (welding line) 1200 mm.
[0043] This embodiment completed 32 independent designs (including the baseline), covering two phases of research. The tilt angle-efficiency response surface analysis under the condition of fixed Lr=60mm shows that the effect of the outlet ring tilt angle on efficiency exhibits a monotonic trend—the more tilted towards the impeller back plate (the more negative α), the higher the efficiency. This tilt angle effect remains consistent across all five test flow conditions, proving that the negative tilt angle guiding effect of the outlet ring is robust under all operating conditions.
[0044] Based on a quadratic response surface model (R) with 22 negative tilt angle design points 2 Based on the analysis (=0.861), the recommended parameters for the outlet reinforcing ring are as follows:
[0045] Enhancement Mechanism: The outlet ring adopts a negative tilt angle design, and its conical surface guides the outlet airflow towards the impeller rear disc, forming two tapering cross-sections within the flow channel. This helps maintain the uniformity of the outlet flow field and reduce mixing losses. If it were tilted towards the front disc, it would create an asymmetrical flow channel with one expanding and one contracting, exacerbating the instability of the outlet diffuser flow and increasing separation losses. There is an optimal radial width: if it is too narrow, the guiding coverage is insufficient; if it is too wide, the blocking effect outweighs the guiding benefit.
[0046] Example 3
[0047] In this embodiment, parameterized reinforcing rings are set at both the blade inlet and outlet, forming a combined inlet and outlet reinforcing ring scheme.
[0048] Based on the conclusions of independent optimization of the inlet and outlet loops, and considering the synergistic effect of the inlet and outlet loops, the starting position of the outlet loop is adjusted. The recommended design parameters for the joint scheme are as follows:
[0049] The key feature of this design is that both the inlet ring and the outlet ring have negative inclination angles, meaning they are both inclined towards the impeller rear plate. In the meridional view, the inlet ring guides the airflow into the flow channel along the rear plate side at the front section, while the outlet ring maintains the airflow out along the rear plate side at the rear section, forming a continuous guiding structure from front to back.
[0050] After optimization, the efficiency improvement brought by the combined inlet and outlet scheme is +1.3 to +1.5 percentage points, which is better than the combined gain of independent inlet ring and independent outlet ring; the maximum deformation of the impeller is reduced from 2.81 mm to 1.21 mm, and the structural enhancement effect is significant.
[0051] Parameter scaling rules The above embodiments are based on an impeller outer diameter d2 = 1200 mm, and the specific parameter values are given. When the impeller diameter is different, the values are scaled according to the following proportional relationship: The ring's initial position zs is scaled proportionally to the impeller's outer diameter d2; The ring tilt angle α does not scale with the impeller diameter; the angle value is used directly. The ring extension length L is scaled proportionally to the impeller outer diameter d2; The ring thickness is recommended to be selected within the range of 3~10mm according to the impeller load, and is not strictly scaled proportionally.
[0052] The above description is merely a selection of preferred embodiments of this disclosure and an explanation of the technical principles employed. Those skilled in the art should understand that the scope of the invention involved in the embodiments of this disclosure is not limited to technical solutions formed by specific combinations of the above-described technical features, but should also cover other technical solutions formed by arbitrary combinations of the above-described technical features or their equivalents without departing from the above-described inventive concept. For example, technical solutions formed by substituting the above-described features with (but not limited to) technical features with similar functions disclosed in the embodiments of this disclosure.
Claims
1. A centrifugal fan impeller reinforcing ring structure, comprising an impeller front disc, an impeller rear disc, and blades distributed along a circumferential direction, wherein the blades are provided with reinforcing rings extending along a circumferential direction, characterized in that: The configuration of the reinforcing ring is defined by three parameters: the ring starting position zs, the ring tilt angle α, and the ring extension length L. The meridional coordinate system uses the inner wall of the impeller rear disk as the axial origin and the direction towards the front disk as the positive axial direction. zs is the axial coordinate of the reinforcing ring welding start point. The ring tilt angle α is the angle between the centerline of the reinforcing ring and the radial ray of the impeller. α < 0 indicates that the reinforcing ring is tilted towards the impeller rear disk, α > 0 indicates that the reinforcing ring is tilted towards the impeller front disk, and α = 0 indicates that the reinforcing ring extends horizontally radially. The ring extension length L is the length of the centerline of the reinforcing ring from the welding start point to the free end. By optimizing these three parameters, the reinforcing ring forms a guide surface within the impeller flow channel, ensuring that the static pressure efficiency of the fan is not lower than the static pressure efficiency of the reference impeller without the reinforcing ring.
2. The centrifugal fan impeller reinforcing ring structure according to claim 1, characterized in that: The reinforcing ring is an inlet reinforcing ring installed at the blade inlet.
3. The centrifugal fan impeller reinforcing ring structure according to claim 2, characterized in that: The ratio of the ring's initial position zs to the impeller's outer diameter is 0.1433 to 0.2267, the ring's tilt angle α ranges from -75° to -45°, the ratio of the ring's extension length L to the impeller's outer diameter is 0.029 to 0.054, and the thickness of the reinforcing ring is 3 to 10 mm.
4. The centrifugal fan impeller reinforcing ring structure according to claim 1, characterized in that: The reinforcing ring is an outlet reinforcing ring located at the blade outlet and the outer diameter of the impeller. The cross-section of the outlet reinforcing ring is rectangular, and its three-dimensional configuration is a conical ring.
5. The centrifugal fan impeller reinforcing ring structure according to claim 4, characterized in that: The tilt angle α ranges from -20° to -10°, the ratio of the radial component Lr of the ring extension length to the impeller outer diameter is 0.042 to 0.125, and the thickness of the reinforcing ring is 3 to 10 mm.
6. The centrifugal fan impeller reinforcing ring structure according to claim 1, characterized in that: The reinforcing ring includes an inlet reinforcing ring simultaneously disposed at the blade inlet and an outlet reinforcing ring simultaneously disposed at the blade outlet. The inclination angle α of both the inlet reinforcing ring and the outlet reinforcing ring is negative, and both are inclined towards the impeller back plate. The initial axial height of the outlet ring is 10-135 mm, forming a continuous "U"-shaped guide path in the meridional plane.
7. A parametric design method for a centrifugal fan impeller reinforcing ring, characterized in that, Includes the following steps: Step Step S1: Determine the aerodynamic performance parameters of the reference impeller without the reinforcing ring, including static pressure efficiency, static pressure rise, and shaft power under multiple flow conditions; Step S2: Establish a parameterized geometric model of the reinforcing ring, using the ring initial position zs, ring tilt angle α, and ring extension length L as design variables; Step S3: In the parameter space composed of the three parameters, select sample points using the design of experiments method, and obtain the aerodynamic performance of the fan corresponding to each sample point through computational fluid dynamics simulation; Step S4: Based on CFD simulation data, construct a surrogate model with the three parameters and flow conditions as inputs and aerodynamic performance as output; Step S5: Based on the surrogate model, determine the optimal parameter combination that takes into account multiple flow conditions through a multi-objective optimization method; Step S6: Apply the optimal parameter combination to the engineering design of the reinforcing ring.
8. The parametric design method for the centrifugal fan impeller reinforcing ring according to claim 7, characterized in that: In step S3, the experimental design method employs central composite design (CCD) or Latin hypercube sampling (LHS).
9. The parametric design method for the centrifugal fan impeller reinforcing ring according to claim 7, characterized in that: In step S4: the surrogate model is selected as a Gaussian process regression (GPR) model or a Kriging regression model, the kernel function is selected as Matern-2.5 or radial basis function (RBF), and the model prediction accuracy is evaluated by leave-one-out cross-validation.
10. A centrifugal fan, characterized in that: The centrifugal fan impeller reinforcing ring structure includes any one of claims 1 to 6.
