Controlling the Fluid Induced Instability of a Supercritical CO₂ Compressor Supported by Magnetic Bearing

1. Introduction

The sCO₂ power cycles provide highly efficient, highly dense and less corrosive power generation compared with other cycle (Incumbent Steam Rankine or Air Brayton Cycles) over a wide range of applications such as waste heat recovery, concentrating solar power, nuclear, and fossil energy.^1,2 Efficiency and power density in power conversion systems are the key criteria according to increasing energy demands over the world. In addition, corrosiveness is also significant issues for turbines and compressors in most power cycles.

Since the sCO₂ turbomachinery are generally small and operate at high rotational speeds, bearings pose a significant challenge to the design of turbo-compressors/expanders for sCO₂ power cycles.² Although the basic sCO₂ cycle consists of a compressor, turbine, heater and cooler as well as a recuperator, efficient, reliable and compact turbine and compressor for high-pressure environment are most significant components to realize robust operation and high-efficiency.¹

Magnetic bearings present several advantages over pure mechanical bearings such as rolling element bearing, hydrostatic/dynamic bearing, and gas foil bearing although bearing selection is rather complex and depends on many factors such as cost, duty cycle, load, speed, size/weight, efficiency, and dynamic performance.^3-6 For examples, a wide range of operating conditions, controllable bearing dynamics, and measurement of bearing forces are the well-known strong point of magnetic bearings. In addition, magnetic bearing technology has been gaining more and more area in turbomachinery and rotating equipment for couple of decades together with remarkable development and progress of integrated power electronics and semiconductor industries.⁷

Although journal bearing may have instability known as oil whirl and whip, fluid induced instability like oil whirl may also happen even with magnetic bearing under both high pressure and speed conditions of sCO₂ turbomachinery.⁸ Dedicate analysis was required to evaluate instability of journal bearing with sCO₂ since the supercritical fluids are generally 100 to 1,000 times denser than gases and slightly lighter than liquids.⁹ In addition, experimental investigation of rotordynamic performance is very important for designing critical components of turbomachinery such as rotor, bearing, seal and so on.¹⁰ However, there are few studies and practical guidelines on sCO₂ turbomachinery supported by magnetic bearings.

With magnetic bearing, vibration or instability of journal bearing can be reduced as well as the working condition of the rotating machinery can be extended or optimized.^11-13 Adding damping, coupled stiffness or preload with magnetic bearing may improve the rotordynamic stability.^14,15 In addition, the proper control of magnetic bearing enhances the rotordynamic performance.¹⁶

This paper presents experimental investigation on instability of a sCO₂ compressor supported by magnetic bearing. First, we introduce the sCO₂ compressor supported by magnetic bearing. Procedure to guarantee rotordynamic performance of the sCO₂ compressor supported by magnetic bearing is provided. Then, effects of working condition such as pressure and rotating speed on the fluid induced instability are investigated experimentally. Finally, a strategy to resolve the fluid-induced instability with conventional PID control are proposed and experimentally verified.

2. sCO₂ Compressor Supported by Magnetic Bearing

2.1 Specifications

The test rig for a sCO₂ compressor supported by magnetic bearing is shown in Fig. 1. Specifications of the compressor are summarized in Table 1.

Fig. 1

sCO2 Compressor supported by magnetic bearings

Table 1

Specifications of sCO2 compressor supported by magnetic bearings

2.2 Rotor Bearing System

The 60 kW turbo-compressor is composed of one shaft with a built-in motor and two impellers, two radial magnetic bearings and thrust foil bearings. A PMSM (Permanent Magnet Surface Mount) motor with two poles is used to drive the compressor. The rotor bearing system of the compressor is designed to have symmetric shape and to satisfy the separation margin of the API 617. Finite element modeling is used to analyze the lateral vibrations considering the gyro and shear effects, as shown in Fig. 2.¹⁷ The model is composed of 20 beam elements and totally has 84 DOF (Degrees of Freedom). Campbell diagram of the rotor bearing system is shown in Fig. 2(b) and the shaft has enough speed margin from the 1st bending mode (79,091 rpm).

Fig. 2

FE model and critical speed analysis

Radial magnetic bearing is designed considering G2.0 balancing grade and safety factor, and its geometric specifications are summarized in Table 2. The safety factor for load capacity of the radial magnetic bearing is determined considering the effect of fluid dynamic forces. The required static load to support the rotor is 46 N, while the dynamic load due to the rotor unbalance is 35 N. However, the load capacity of the radial magnetic bearing is determined as 186 N considering the safety factor 4. The designed heteropolar radial magnetic bearing is shown in Fig. 3.

Table 2

Specifications of radial magnetic bearing

Fig. 3

Radial magnetic bearing

An inductive displacement sensor is used to control the magnetic bearings and its transducer is integrated into the magnetic bearing controller. Sine wave with 20 kHz drives the sensing coil and the current flowing through the coil is measured to calculate the inductance of the sensing coil. The sensors are mounted differentially on both side of the object and the displacement of the object is obtained from the inductance difference between the two sensors. The measuring range is ±500 μm in radial direction and ±1,000 μm in axial direction. The current control is a simple PI controller, while the displacement control is a PID controller with some filters. In addition, an imbalance controller is used to reduce the synchronous vibration of the rotor. Specifications of the controller are summarized in Table 3.

Table 3

Specification of magnetic bearing controller

3. Fluid Induced Instability

3.1 Response under Atmospheric Pressure

Simple unbalance test is performed under atmospheric pressure and the vibrations near first resonance (15,000 rpm) is shown in Fig. 4. With proper PID gains, the magnitude of the unbalance response is less than 0.035 mm, which is satisfied with ISO14893-2 zone A/B.

Fig. 4

Vibrations at 15,000 rpm under atmospheric pressure

3.2 Responses under the Pressurized sCO₂

The waterfall chart of the rotor vibration from 24,000 to 36,000 rpm under 10 bar of sCO₂ are shown in Fig. 5. Not only dominant synchronous component but also sub-harmonic one appear at 270 Hz. The frequency of the sub-harmonic component is close to half of the synchronous one and coincides with the first resonance frequency in Fig. 4. In addition, not the frequency but the magnitude of the sub-harmonic component varies according to the rotating speed.

Fig. 5

Waterfall chart of the rotor vibration under 10 bar

Under the pressure of 70 bar, another low-frequency subharmonic vibration appears, as shown in Fig. 6. The frequency of this sub-harmonic vibration not only increase with the rotating speed, but also the amplitude increase with the rotating speed, as shown in Fig. 7.

Fig. 6

Frequency spectrum of the rotor vibration under high pressure (70 bar)

Fig. 7

Sub-harmonic vibration according to rotating speed

4. Control of Fluid Induced Instability

This sub-harmonic vibration frequency, near 270 Hz, is caused by cross-couple stiffness of the pressurized sCO₂ whirling. The cross-couple stiffness of the pressurized sCO₂ whirling equivalently reduces the damping of the bearing and causes the subharmonic vibration of the system, which is the average whirling velocity of the fluid between the rotor and the bearing.¹⁸

The damping of the sCO₂ compressor can be increased by tuning the PID control gains. After the gain is re-tuned, the vibration amplitude at 270 Hz can be reduced successfully, as shown in Fig. 8. The proportional gain remains low for the system stability while the derivative gain increases for sub-harmonic vibration. The vibration amplitude at 140 Hz is reduced by about 56%, from 5.9 to 2.6 μm, and the vibration amplitude at 270 Hz is reduced by about 67%, from 6.2 to 2.0 μm. The amplitude of the synchronous vibration was reduced by 45% from 12.4 to 6.8 μm.

Fig. 8

Frequency spectrum of vibration under high pressure (70 bar) after re-tuning the position control gains

5. Conclusion

This paper presents experimental investigation on instability of a sCO₂ compressor supported by magnetic bearings. First, we introduce the sCO₂ compressor supported by magnetic bearings. Then, effects of working condition such as pressure and rotating speed on the fluid-induced instability are investigated experimentally. Finally, a strategy to resolve the fluid-induced instability with conventional PID control are proposed and experimentally verifie.

Acknowledgments

This paper was supported by Korea Institute for Advancement of Technology (KIAT) grant funded by the Korea Government (MOTIE) (No. P0006915, Korea-China Joint R&D Project).

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