Moving magnet linear compressor: Operating characteristics under resonance and off-resonance frequencies

2.1 Moving magnet linear compressor

This experiment was carried out on a moving magnet linear compressor designed by Liang et al. (2016). Different from the other moving magnet linear compressors, its flex spring system consists of two sets of axially separated flex springs, which are mainly used to support the moving piston and limit its displacement along the cylinder axis (Liang et al., 2016). Furthermore, high radial stiffness of flexible spring effectively limits the radial displacement of the piston, prevents wear on the piston and cylinder, and reduces gas residues caused by radial gas leakage (Liang et al., 2016). However, the gas residues are still unavoidable. The residual gas affects equivalent gas stiffness and thus the resonant frequency of the compressor, so it is necessary to model and experimentally study this compressor. Table 1 shows the linear compressor parameters of the designed experiment.

2.2 Test rig

Fig. 1 shows a schematic diagram of the experimental system and Fig. 2 shows the complete test rig. The test rig is equipped with a moving magnet linear compressor, a condenser, an evaporator, four pressure transducers, six thermocouples, two mass flow meters, two current transducers, and two displacement transducers. The pressure transducers are used to measure separately the pressure at the compressor discharge, the evaporator inlet, the compressor suction and the compressor body. The thermocouples are used to measure the temperature in various parts of the system, including compressor body, compressor discharge, condenser outlet, evaporator inlet, evaporator wall, evaporator outlet, and compressor suction. The mass flow meters are used to measure the mass flow in the main and leakage circuits. The current transducers are used to measure the change in current in the two compressor internal coils, and the displacement transducers are used for the measurement of the compressor piston position. The instruments model and accuracy for experimental system can be found in the Table 2.

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Fig. 1

Schematic diagram of an experimental system (Chen et al., 2019).

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Fig. 2

The complete test rig for the moving magnet linear compressor system.

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2.3 Date acquisition system

The linear compressor data acquisition system includes a low-speed data acquisition system (LDAQ) and a high-speed data acquisition system (HDAQ). Two NIUSB-6251 data acquisition cards are used in both LDAQ and HDAQ. NIUSB-6251 has 16 analogue inputs (16 bits) and 2 digital outputs with a maximum sampling rate of 1.25 MHz for a single channel. The LDAQ is used to collect parameters such as pressure (discharge, suction, compressor body, and evaporator inlet), temperature (discharge, suction, condenser and evaporator inlet and outlet, compressor body), and mass flow (main flow and leakage flow); the HDAQ is used to collect data on pressure, voltage, current and displacement, the HDAQ has a sampling rate of 5000 Hz. More details can be found in the previous study (Bao et al., 2021).

2.4 Test conditions

The test condition of this study was set as in Table 3. When the measured pressure and temperature have reached steady-state and there is no vapor in the condenser and no liquid in the evaporator, the data acquisition system starts recording and saving the data.

2.5 Uncertainty analysis

Uncertainty analysis is carried out according to the combined uncertainty equations for two independent uncertainties which can be expressed as:

Where $s_{\overline{x}}$ and $w_{\overline{x}}$ are the Type A and B uncertainty.

Based on Eq. (1) and Eq. (2) (Shen et al., 2022), power consumption, volumetric efficiency, cooling capacity, and COP have a relative uncertainty of 0.95 %, 0.24 %, 2 %, and 2.2 %.

2.6 Mathematical model and experimental verification of resonant frequencies

2.6.1

2.6.1 Linear equivalent model

Linear compressor resonance frequency $f_{\mathit{res}}$ can be expressed using Eq. (3), where $m$ is moving mass, $k_s$ is mechanical spring stiffness, and $k_g$ is equivalent gas spring stiffness of a linear compressor (Liang, 2018). The presence of a clearance volume at the front end of the compressor cylinder causes residual gas always exists inside the compression chamber during the operation of the compressor. The energy stored in the residual gas is acting as an energy storage spring, and it is therefore modelled as an equivalent gas spring.

The value of $k_g$ was estimated based on Eq. (4), where $P_d$ and $P_s$ are linear compressor discharge and suction pressures, $A_p$ is piston surface area, and $S$ is piston stroke. The slope for Eq. (4) was acquired based on the slope (green dotted line) between the starting point of the compression stroke (1) and expansion stroke (3) in Fig. 3 (Kim et al., 2011).

(3)

$$f_{\mathit{res}}=\frac{1}{2\pi }\sqrt{\frac{k_g+k_s}{m}}$$

(4)

$$k_g=\frac{P_d-P_s}{S}{A}_p$$

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Fig. 3

Pressure-displacement diagram for linear compressor operated under 38 Hz frequency, pressure ratio 2.5, and piston stroke 11 mm.

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3.1 Estimation of linear compressor resonant frequency

Fig. 4 compares the estimated linear compressor resonant frequency with that obtained experimentally. Compressor pressure ratios 2.0 and 2.5 are considered in this study with compressor stroke varying from 10 ∼ 13.2 mm. Overall, the resonance frequency of the compressor declines with the increase of compressor operating stroke. Nonetheless, the resonance frequency rises with the elevation of the compressor pressure ratio. Both models exhibit an estimation error of less than 10 %, but the Linear Equivalent model is notably more accurate than the Fourier Transform model in approximating the resonant frequency.

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Fig. 4

Comparison of experimental values and calculated values from two models.

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The discrepancies between theoretical models and experimental data can be attributed to the existing theoretical models considering only the residual gas due to the clearance volume at the front end of the compressor cylinder and in between piston and cylinder. The presence of a clearance gap between piston and cylinder causes a fraction of the intake gas to be not compressed thus undesirably leading to an energy loss. In practice, however, the residual gas also penetrates the vent and escapes through valves during the gas exchange process (Hwang and Lee, 2019). The gas leaking through valves was not considered in the present model, thus resulting in a less accurate estimation of the equivalent gas spring stiffness and resonant frequency.

Furthermore, it can be observed also that the Fourier Transform model exhibits a considerably lower accuracy than the Linear Equivalent model in approximating the compressor resonant frequency. The equivalent gas spring stiffness exhibits highly non-linear characteristics at a high-pressure ratio operation (Liang, 2018), but at the present experimental pressure ratios (2.0 and 2.5) where the non-linear characteristics of the equivalent gas spring are less pronounced, the Linear Equivalent model turns out being a more reliable tool in estimating the equivalent gas spring stiffness and linear compressor resonant frequency. At a higher pressure ratio where non-linear characteristics of the equivalent gas spring are magnified, the Fourier Transform model is expected to give a more accurate estimation of resonant frequency (Bijanzad et al., 2020). The present study shows that the Linear Equivalent model is a more precise tool for estimating the linear compressor resonance frequency at pressure ratios 2.0 and 2.5, a regime where non-linear characteristics of the equivalent gas spring are less prominent.

3.2 Resonance characteristics of the linear compressor

Fig. 5 shows the variation of shaft force, spring force, inertia force and damping force over time for a linear compressor under resonance frequency 38 Hz. The shaft force varies continuously with armature, piston position and instantaneous excitation current, so it is difficult to measure the shaft force of the motor directly. The shaft force can be calculated from the Force-Current-Displacement calibration map (Zhu et al., 2021). The shaft force varies sinusoidally over a cycle and peaks at 56.8 N at 0.0128 s. The spring force is related to the mechanical spring stiffness and piston displacement. The mechanical spring stiffness of the linear compressor is 16284.85 N/m. The real-time displacement in the cylinder can be determined from the piston position measured by a displacement transducer. The piston is closest to the top of the cylinder at 0.0054 s. and the mechanical spring force reaches a maximum of 90.1 N. The damping force is dependent on the damping coefficient and piston velocity, and the inertia force is determined by the moving mass and piston acceleration. Viscous damping is the most important form of damping for linear compressor pistons. In the calculation of the damping coefficient, the flow in the radial clearance is assumed to be laminar and the radial clearance is a fixed value (Liang et al., 2014a). Linear compressor damping coefficient is about 0.0475 N*s/m and moving mass is 0.66 kg.

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Fig. 5

The variation of shaft force, spring force, inertia force and damping force over time for a linear compressor under resonance frequency.

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3.3 Off-resonance characteristics of the linear compressor

Linear compressor off-resonance characteristics were also examined in the range of 36–40 Hz band using a linear compressor with an 11 mm stroke and a 2.5 pressure ratio. Fig. 6 illustrates compressor input power and power factor as a function of operating frequency. Compressor power input ${\dot{W}}_{\mathit{in}}$ was calculated based on the measured voltage and current ${\dot{W}}_{\mathit{in}}=\frac{1}{t}{\int }_0^{t}UIdt$ , where $U$ is voltage, $I$ is current, and $t$ is operating period (Sun et al., 2021). The linear motor power factor reflects the utilization ratio of the power input. It can be expressed as $\mathit{PF}=\frac{{\dot{W}}_{\mathit{in}}}{U_{\mathit{rms}}{I}_{\mathit{rms}}}$ (Sun et al., 2021). The power input is lowest at operating frequency of 38 Hz (91.96 W), and the power factor is maximum (0.998). As the operating frequency deviates from 38 Hz, power input rises noticeably. The input power rises to 102.3 W at 36 Hz and 99.09 W at 40 Hz.

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Fig. 6

Compressor input power and power factor against operating frequency.

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One of the possible reasons that contribute to the lowest input power needed at 38 Hz operating frequency in Fig. 6 is the copper and seal losses are found to be minimal at this specific operating point as demonstrated in Fig. 7. The copper loss is the energy loss due to resistive heating. The copper loss ${\dot{W}}_{\mathit{copper}}$ was calculated using ${\dot{W}}_{\mathit{copper}}={I_{\mathit{rms}}}^{2}R$ (Chen et al., 2020a). Where $R$ is the total coil resistance and $I_{\mathit{rms}}$ is root mean square (RMS) value of current supplied to the compressor. The seal leakage power loss ${\dot{W}}_{\mathit{seal}}$ is expressed by Eq. (12) (Liang et al., 2013), where $c$ is radial clearance, $\mu$ is viscosity constant, $L$ is clearance length, and $P_{1a}$ is amplitude.

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Fig. 7

Copper loss and seal loss against operating frequency.

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With reference to Fig. 7, the copper loss is minimum at the operating frequency of 38 Hz (∼10 W). At lower (36 Hz) and higher (40 Hz) operating frequencies, the copper loss elevates by ∼ 5.1 W and ∼ 2 W, respectively. The seal loss does not vary significantly with operating frequency, fluctuating at about 3.08 W, as seen in Fig. 7. This signifies that the seal loss is weakly dependent on the operating frequency.

Fig. 8 shows the motor efficiency and isentropic efficiency against operating frequency. The motor efficiency ${\eta }_m$ and isentropic efficiency ${\eta }_{\mathit{isen}}$ can be expressed as: ${\eta }_m=\frac{{\dot{W}}_{\mathit{in}}-{\dot{W}}_{\mathit{copper}}}{{\dot{W}}_{\mathit{in}}}$ and ${\eta }_{\mathit{isen}}=\frac{\dot{m}\left(h_2-h_1\right)}{{\dot{W}}_{\mathit{in}}}$ (Sun et al., 2021). Where $h_2$ is the isentropic discharge enthalpy of the compressor and $h_1$ is the suction enthalpy of the compressor. Owing to the compressor losses being minimum at 38 Hz, the maximum motor efficiency and isentropic efficiency were acquired at 38 Hz operating frequency, with a motor efficiency of 81.98 % and an isentropic efficiency of 51.31 %. The motor efficiency and isentropic efficiency both fell by 3.32 % and 4.47 %, respectively, when operating at 36 Hz. When the compressor stroke is fixed, volumetric efficiency does not change considerably with the compressor working frequency. The largest change in volumetric efficiency while the compressor runs at 36–40 Hz is about 3.2 %. As a result, the compressor that operates at off-resonance consumes approximately 2.8 % more energy than that at the resonance frequency.

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Fig. 8

Motor efficiency, isentropic efficiency, and volumetric efficiency against operating frequency.

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Cooling capacity and COP are important indicators for evaluating the performance of refrigeration systems. In contrast to conventional compressors with start-stop regulation and frequency regulation, linear compressors can regulate cooling capacity by varying the drive frequency and piston stroke. Cooling capacity $Q_c$ can be calculated using $Q_c=\dot{m}(h_1-h_5)$ (Chen et al., 2020b). Where, $h_5$ is refrigerant enthalpy at compressor inlet. COP was calculated using $\mathit{COP}=\frac{Q_c}{{\dot{W}}_{\mathit{in}}}$ (Chen et al., 2020b). Fig. 9a shows the COP and cooling capacity at a variable drive frequency for a fixed stroke of 11 mm. The cooling capacity does not change substantially with frequency when the compressor is operating near the resonance frequency. The COP, on the other hand, has a nearly symmetrical tendency over the frequency range. The highest COP is 3.39 and the cooling capacity is 312.17 W when the operating frequency is at the resonance frequency. This is because the input power is low at the resonance frequency. Nonetheless, as the operating frequency is changed from the resonance frequency, the input power increases and the COP declines. The COP reduces to 3.01 at 40 Hz.

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Fig. 9

(a) Cooling capacity and COP at a variable drive frequency for a fixed stroke of 11 mm (b) Cooling capacity and COP at a variable stroke for a fixed frequency of 38 Hz.

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The COP and cooling capacity at the resonant frequency with varying compressor strokes are shown in Fig. 9b. With an increasing stroke, cooling capacity and COP rise in a roughly linear fashion. The maximum cooling capacity is 457.96 W and the maximum COP is 3.42 at a stroke of 13 mm. Sun et al. (2021) also found that when piston stroke increases, compressor input power operating at a given frequency with changing strokes also rises. The cooling capacity and input power both increase linearly as the compressor stroke extends, however, the rate of increase in input power is notably lower than the cooling capacity.