#### Tip Clearance Loss Models in Centrifugal Compressors

This blog discusses tip clearance loss models in centrifugal compressor impellers with large relative clearances

In the flow path of turbomachines, there is a clearance between the tip of the rotor blades and the housing parts of the machine. This clearance is necessary in order to prevent the rotor from touching the stator during rotation of the impeller. The tip clearance value depends on the following features:

• Deformation of the rotor under the action of gas, thermal and centrifugal loads
• Housing deformations under the influence of air pressure and uneven heating
• Clearance in bearings
• Design features

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There is a pressure gradient between the suction side and the pressure side, which results in a flow from one side of the blade to the other through the clearance. Studies of the flow in the tip clearance of the blades of turbomachines indicate its complex nature. The flow through the tip clearance affects the flow in the shroud section of the blade and has a significant impact on performance and efficiency. According to the results of the studies, an increase in the relative clearance by 1% reduces efficiency by 2%. Known methods for evaluating the effect of tip clearance on efficiency are most often reduced to a linear dependence of the reduction in efficiency on the relative clearance. This provides acceptable accuracy for engineering calculations with a relative clearance of no more than 3%.

The typical value of the tip clearance for the centrifugal compressor impeller is 0.2-0.5 mm. However, in some cases, the clearance is significantly higher and reaches 1-3 mm. An example would be the impellers of low-pressure compressors, which are made of plastic. Plastic is not a sufficiently rigid material, which requires the designer to significantly increase the tip clearance in order to avoid the impeller touching the housing part of the compressor in operation.

A feature of centrifugal compressors is the low blade height at the outlet of the impeller. Figure 1 shows the impeller of the compressor designed for pressure ratio ptr=2.4 with the diameter and height of the blades at the outlet, respectively, 220 mm and 15.1 mm. For such an impeller, with an absolute clearance of 0.5 mm, the relative clearance will be 3.3%. This means that simple clearance loss estimation methods will have a large margin of error for such an impeller. It should be taken into account that an impeller designed for the same outlet diameter, but at pressure ratio ptr=5, will have approximately half the blade height, respectively, and the relative clearance is twice as large.

Recently, there has been strong interest in small turbomachines. The impeller diameter of such compressors ranges between 50-70mm. The real estimation of clearance losses for this kind of compressor is a problem due to the large relative tip clearance.

The SoftInWay team performed several calculations using 1D and 3D approaches. The purpose of the study was to determine how accurate the predictions of the methods used in 1D for calculating losses in the tip clearance are, especially for relative clearances greater than 3%. The object of study was the compressor shown in Figure 1 and described above. Calculations were carried out for relative clearances in the range of 1-10%. A study for relative clearances greater than 10% has not been carried out, the reasons for this will be described below.

The results of the 3D CFD calculation demonstrate a linear dependence of the Pressure ratio and efficiency on the value of the relative clearance (Figure 2, Figure 3). This is consistent with the general ideas outlined above. However, for this impeller, increasing the relative clearance by 1% reduces the efficiency by approximately 0.8%.

In the range of 1-3%, the 1D approach shows almost complete agreement in terms of efficiency and close values in terms of pressure ratio. The maximum error in pressure ratio is less than 2%, which is an acceptable level of accuracy for engineering calculations.

The 3D results show a more dramatic drop in pressure ratio with increasing relative clearance compared to the 1D approach. This leads to an unacceptable error in determining the pressure ratio for relative clearances of more than 3%. It can also be seen that there is an error in determining efficiency.

Figure 4 shows the meridional velocity distributions for tip clearance of 1%, 5%, and 10%. An increase in the clearance leads to an increase in the zone of low velocities in the shroud section. It is clearly seen that the zone of low speeds creates a blockage, which increases with the clearance. This increases the velocity in the core of the flow, which leads to an increase in unevenness in the direction of the Hub-to-Shroud, and also negatively affects the deviation.

With an increase in the tip clearance of more than 7-10%, the growth of the low-velocity zone at the tip section leads to the separation of flow from the surface of the shroud. This negatively affects the parameters of the impeller, and also significantly complicates convergence or makes it impossible to achieve convergence in 3D simulation. For this reason, studies for clearances greater than 10% have not been carried out.

Loss models for large relative clearances (greater than 3%) should increase the loss more aggressively as the clearance increases and additionally account for the increase in deviation. The SoftInWay team has upgraded the loss model to better describe clearance losses. Figure 5 shows the results of calculating the compressor with the standard loss model (black line) and the upgraded loss model (green line), as well as the results of 3D simulation (red line). The 1D approach with the upgraded clearance loss model better predicts the decrease in pressure ratio with increasing clearance by accounting for the increase in deviation. The efficiency dependence line predicted by the 1D approach also has the same slope as the line obtained by 3D simulation. The efficiency error is less than 0.8% over the entire range.

To successfully use 1D approaches for calculating impellers with large clearances (more than 3%), clearance loss models have to take into account not only losses due to flow from the pressure side to the suction side, but also more complex effects. For example, an increase in blockage and an increase in the velocity gradient in the direction of the hub-to-shroud with an increase in clearance and related effects. However, it is quite possible to calibrate the loss models so that the prediction of 1D approaches is reasonably accurate for large relative clearance. This significantly expands the possibilities of 1D approaches in engineering practice.

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