Experimental investigation of the effects of ...

EXPERIMENTAL INVESTIGATION OF THE EFFECTS OF MECHANICAL STROKE ON THE ACOUSTIC IMPEDANCE OF LINEAR ALTERNATORS UNDER THERMOACOUSTICPOWER-CONVERSION CONDITIONS A.Y. Abdelwahed and A.H. Ibrahim1 The American University in Cairo, School of Sciences & Engineering, 11835 New Cairo, Egypt 1 On leave from Mechanical Power Department, Faculty of Engineering, Cairo University, Giza, Egypt

Ehab Abdel-Rahman Professor of Physics, Department of Physics, The American University in Cairo, 11835 New Cairo, Egypt e-mail:[email protected] Linear alternators convert acoustic power produced by thermoacoustic engines into electricity. The conditions required for best acoustic-to-electric power conversion efficiency include that they operate under mechanical and electrical resonances simultaneously. When this occurs, the acoustic impedance of the linear alternator equivalent circuit becomes purely real. This work experimentally demonstrates that the effective inductance of the compact moving-magnet linear alternators typically used in thermoacoustic power converters increases with the mechanical stroke amplitude, which requires different tuning capacitors at different mechanical stroke amplitudes to achieve electric resonance and to keep the acoustic impedance real. The work explains the reasons for this behaviour and presents experimental data and a simple linear model to characterize the effective inductance as a function of the mechanical stroke amplitude. Then, the real and imaginary parts of the acoustic impedance of linear alternators are calculated with and without accounting for this effect. The effects of applying this suggested correction on the performance indices of linear alternators under thermoacoustic-power-conversion conditions are presented and discussed. By accounting for this effect, an increase in the output electric power by 9.8% is achieved, together with an increase in the mechanical stroke by 9.7% and an increase in the resonator dynamic pressure by 4.2%.

1.

Introduction

Thermoacoustic power converters consist of thermoacoustic engines and linear alternators. The former convert thermal energy into acoustic energy. The latter convert the generated acoustic energy into electric energy. Thermoacoustic engines convert heat into high-intensity acoustic power based on the time-average thermoacoustic interactions between the working gas and the solid in the stack/regenerator. The thermodynamic cycles employed are either Brayton cycle in standing-wave thermoacoustic engines or Stirling cycles in travelling-wave thermoacoustic engines [1-2]. Compared to traditional heat engines, thermoacoustic engines consist of few pipes, stack/regenerator and heat exchangers. They have no mechanical moving parts since the oscillations in the working gas replace the motion of the piston/cylinder and thus eliminate the need to moving seals and lubrication encountered in conventional engines. This grants them the merits of simple structure, low manufacturing and maintenance costs and high reliability. Additionally, they utilize environmentallyfriendly gases like helium and argon and their mixtures which do not cause global warming nor ozone 1

The 23rd International Congress on Sound and Vibration

depletion problems. Moreover, because the heat addition to the working gas mixture occurs in a heat exchanger external to the engine, these engines can easily be driven by a variety of heat-energy sources including conventional fuels, solar energy and waste heat, making them ideal for renewable energy applications. Recent investigators demonstrated their large efficiency potential. For example, a first-law thermal-to-acoustic conversion efficiency of 32% and a second-law conversion efficiency of 49% were reported for a travelling-wave thermoacoustic engine [3]. Linear alternators convert the generated acoustic energy into electrical energy. The acoustic power applied to the alternator piston causes the piston to oscillate. This oscillatory motion creates an oscillating magnetic field between the permanent magnet moving with the alternator piston and the stationary copper coils. The oscillating magnetic flux induces voltage in the copper coils causing an electric current to flow into the load and electric power to be delivered. Power factor correction generally is required to balance the effective inductance of the alternator copper coils. The exact capacitance value depends on the effective inductance of the copper coils. Recent moving-magnet linear alternators have been developed specifically for thermoacoustic applications. They utilize a new technology based on flexure bearing and clearance seal which allows a non-contact reciprocating motion of the piston allowing operation without sliding seals or lubrication. This grants these devices the merits of durability and reliability. Some linear alternators have been in continuous operation for more than eight years [4]. Combination of thermoacoustic engines with linear alternators adds their merits together to realize a complete thermoacoustic power converter that is efficient, environmentally-friendly, easily integrated into solar energy or waste-heat applications and that is also reliable and durable thanks to operation without sliding seals or lubrication. Because of these merits, several thermoacoustic power converters have been developed recently [5-7]. An equivalent circuit for linear alternators based on electrical–mechanical-acoustical analogy was presented by [8]. They calculated the acoustic impedance of the linear alternator, Z, as: Z=(

1 R e Bl2 X e Bl2 ) [(R + + j − ) (X )], m m A2 𝑅𝑒2 + 𝑋𝑒2 𝑅𝑒2 + 𝑋𝑒2

(1)

where A is the piston’s area, Rm is the damping coefficient, Re is the stator’s resistance, Bl is the transduction coefficient (force factor), Xe is the electric reactance and Xm is the mechanical reactance. This acoustic impedance affects both the acoustic coupling between the linear alternator and the thermoacoustic engine as well as the coupling between the linear alternator and the electric load, and thus affects the electro-acoustic conversion efficiency and the electric power delivered to the load. Both the electric and mechanical reactances affect the power delivered to the load, WL, and the acoustic-to-electric efficiency, LA, as shown in Eq. (2) and Eq. (3), respectively [8]: (ΔP)2 A2 Bl2 R L 1 WL = [ 2 ], 2 (Bl − X m Xe + R m R e )2 + (X m R e + R m Xe )2

where RL is the load resistance and 𝜂𝐿𝐴 =

Bl2 R L , (Bl2 R e + R m (R2e + X e2 ))

(2)

(3)

where P is the pressure difference between the dynamic pressure in front of linear alternator piston (hereinafter referred to as resonator pressure) and the dynamic pressure at the back of piston (hereinafter referred to as enclosure pressure). Eq. (1) shows that both Xm and Xe affect the acoustic impedance of the linear alternator and that this impedance cannot be pure real unless both terms are zero. Under this condition, Eq. (2) and Eq. (3) predict that both WL and LA will increase. The mechanical reactance, Xm, is zero when the linear alternator operates at its mechanical resonance point. This occurs when the frequency of the acoustic power supplied to the linear alternator matches its mechanical resonance frequency. The later depends on the linear alternator’s moving mass, intrinsic stiffness, K, as well as the gas spring stiffness, Kg, that arises from the motion of the 2

ICSV23, Athens (Greece), 10-14 July 2016


piston in its enclosure volume under a large mean gas pressure. This mechanical resonance condition is achieved when the mechanical reactance Xm satisfies the following condition [9]: Xm = M-Ke/ = 0,

(4)

Kg = PA2/V,

(5)

Xe = Le-1/C = 0,

(6)

where  is the angular frequency, M is the moving mass, Ke is the effective stiffness, which is the sum of the intrinsic stiffness and the gas spring stiffness where the latter is given by [9]: where  is the specific-heat ratio, P is the mean gas pressure in the enclosure volume, A is the piston’s area, and V is the volume of the enclosure holding the linear alternator. The electric reactance, Xe, is zero when the inductance of the alternator’s coil is balanced by the added capacitance of an added tuning capacitor in series with the inductance of the alternator’s coil. This electric resonance condition is achieved when the electric reactance Xe satisfies the following condition [8]: where Le is the effective coil’s inductance and C is the capacitance of the tuning capacitor used for power-factor correction. The main objectives of this work are to 1- examine the effects of the mechanical stroke on the effective inductance of the linear alternator’s coil, which affects its electric reactance and in turns the required capacitance to balance it in order to achieve electric resonance, which is a necessary condition to obtain a pure-real acoustic impedance and to achieve best acoustic-to-electric conversion, 2to provide experimental data to quantify how the mechanical stroke increases the effective inductance, 3- to explain the reasons for this experimental observation and to provide a simple linear model for it, and 4- to examine the effects of this suggested correction on the performance indices of the linear alternator. All the experimental conditions used in this work are limited by the thermoacousticpower-conversion conditions, as illustrated in [5-7]. The first three objectives of this work are achieved using the experimental setup described in Section 2 below. The fourth objective is achieved using the linear-alternator test rig, previously presented in [10] and utilized in [11, 12 and 13]. Both setups utilize the same linear alternator.

2.

Experimental setup

First, the technical specifications of the linear alternator used are presented in Table 1. Then, the setup used to realize the first three objectives of this work is presented. Table 1: Technical specifications of the linear alternator used in this work.

Model number and supplier Stator’s resistance, Re (Ω) Measured coil’s inductance at rest, Lo (mH) Transduction coefficient (Force factor or BL product) (N/A) Moving mass, M (Kg) Intrinsic stiffness, K (kN/m) Damping coefficient, Rm(N.S/m) Free-decay frequency, Hz Piston’s diameter, mm Nominal mechanical stroke amplitude, mm Moving magnet relative permeability (neodymium magnet-FeNdB or equivalent)

1S102D (QDrive – Chart Industries) 6.72 68 48.01 0.478 30.94 4.55 40.48 50.8 5.0 1.05

This section describes the setup built to estimate the effective linear alternator coil’s inductance at different mechanical stroke amplitudes. This task can not be realized by a simple use of an LCR meter, which can measure the effective inductance only when the linear alternator is not energized


3


and thus provides data only at zero mechanical stroke. The measured coil’s inductance at zero mechanical stroke is 68 mH, as measured by an LCR meter (model 380193, supplied by Extech). In order to provide experimental data on the effective coil’s inductance at non-zero mechanical strokes, the setup illustrated in Fig. 1 is used. The linear alternator is operated as an acoustic driver driven by a voltage source. The voltage source used for this purpose does not have built-in capacitance or inductance and thus the only reactive component arises from the inductance of the alternator’s coil itself. For this purpose, a function generator (Model AFG3021B, maximum output peak-peak voltage of 10 V, supplied by Tektronix) is used to drive the linear alternator under test and different mechanical strokes are obtained by controlling the input voltage supplied to the function generator. The setup uses variable tuning capacitor in series with the alternator’s inductance and provides simultaneous measurements of the current and voltage signals across the combination made of the alternator’s coil and the added tuning capacitor. The capacitance of the tuning capacitor is varied until the measured current and voltage signals are in phase indicating that the tuning capacitor value has balanced the effective coil’s inductance at the mechanical stroke tested. The tuning capacitors used are of the run-type, of sufficient voltage rating (450 V) and are connected in series to the linear alternator, in order to balance the coil’s inductance. This observation is made experimentally by monitoring the voltage drop signal across the linear alternator’s coil and the added tuning capacitor. The current signal across the same elements is examined by monitoring the voltage drop signal across a current-measuring resistance of sufficient power-rating. These two voltage signals are captured simultaneously on a digital-storage oscilloscope (Model TDS2024 B, four channels, sampling rate up to 2.0 GS/s, supplied by Tektronix). Under this condition, the coil’s effective inductance can be inferred from the capacitor value using: Le = 1/2C. (7)

Figure 1: Experimental setup used to measure the effective linear alternator coil’s inductance for a wide range of mechanical stroke values.

3.

Figure 2: Sample of the measured in-phase current and voltage signals at the linear alternator’s electric resonance. Data acquired for an input voltage of 40.5 VRMS at 51 Hz, a resulting mechanical stroke amplitude of 2.95 mm, a tuning capacitor of 72 F corresponding to an effective coil’s impedance of 135 mH.

Results

The results are divided into two main sections: the first section presents and discusses the effects of mechanical stroke on the effective coil’s inductance and thus on the capacitance value required to balance it in order to achieve electric resonance. The second section quantifies the linear alternator performance indices with and without the suggested correction. All experiments reported in this subsection are carried-out at the mechanical resonance frequency of the linear alternator to satisfy the mechanical-resonance condition. A sample of this measurement showing in-phase current and voltage signals is shown in Fig. 2 for a mechanical stroke amplitude of 2.95 mm (59 % of the rated mechanical stroke). In this experiment, the tuning capacitance needed was 72 F. At the operating frequency of 4



51 Hz, the effective inductance using Eq. (7) as 135 mH, which is larger than the inductance value measured at zero mechanical stroke (68 mH) and thus required a smaller capacitance value to balance it. This observation indicates that the effective coil’s inductance increases with the mechanical stroke amplitude. 3.1 Dependence of the effective coil’s inductance on the mechanical stroke The measured effective coil inductances at different mechanical stroke amplitudes are plotted as symbols in Fig. 3. The results show an increase in the measured effective coil’s inductance as the mechanical stroke increases. At mechanical strokes slightly larger than zero the circular symbol), the non-uniformity of the magnetic field fringes when the alternator’s piston is first introduced affects the results, causing a deviation from the linear behaviour experienced by the rest of the points (shown as square symbols). The reason for this observed behaviour can be explained in view of Fig. 4: at large values of mechanical stroke amplitudes, the penetration of the iron rod holding the moving magnet increases. the increased exposure of the iron rod causes an increase in the effective core permeability, which in turns increases the effective coil’s inductance [14], since the effective coil’s inductance is proportional to the effective core permeability.

Figure 3: The dependence of the effective coil’s inductance on the mechanical stroke amplitude. The symbols denote measured values. The solid line shows the relationship presented in Eq. (8) assuming Lo of 68 mH and relative permeability  of 2.6.

Figure 4: An illustration showing the moving magnet oscillating inside the stationary copper coils. During this oscillatory motion, one part of the coil feels the existence of the magnet and its iron core while the other part does not.

Accordingly, the effective inductance can be modelled as a sum of two terms, as shown in Eq. (8): the first term corresponds to the part of the electric coil feeling the existence of the iron rod holding the permanent magnet, which experiences larger core permeability and increased effective inductance. The second term corresponds to the part of the electric coil that does not experience this effect, and thus has the nominal core permeability and the nominal coil inductance, S 𝐿𝑜 (𝑆𝑅 − 𝑆) Le = 𝐿𝑜 ( ) + , 𝑆𝑅 𝑆𝑅

(8)

where Le is the effective coil’s inductance, Lois the linear alternator’s inductance at rest (68 mH), S is the linear alternator’s mechanical stroke, SR is the rated mechanical stroke amplitude (5 mm) and µ is the relative permeability of the system made of the electric copper coil, the moving permanent magnet and its iron holder and the air between the copper coil and the magnet under reciprocating conditions. The measured effective coil inductances at different mechanical stroke amplitudes are compared to the values estimated using Eq. (8) in Fig. 3: the straight line in Fig. 3 uses the measured Lo value and a relative permeability of 2.6. For comparison, the relative permeability of neodymium magnet, typically used a moving permanent magnet in linear alternators is 1.05 [15]. This value is close to the measured value of 2.6and the difference may be due to the existence of other materials, like the iron ICSV23, Athens (Greece), 10-14 July 2016

5


rod holding the magnet and the gas between the magnet and the copper coil as well as due to the reciprocating motion. It should be noted that the model presented in Eq. (8) does not account for nonlinear effects, like the non-uniformity of the magnetic field when the alternator’s piston is first introduced into the magnetic field, causing the measurements made at near-zero mechanical strokes (circular symbol in Fig. 3) to deviate from this simple linear model. 3.2 Effects of the suggested correction on the linear alternator performance indices This subsection examines the effects of applying this suggested correction on the performance indices of linear alternators, which is the fourth objective of this work. Data presented when applying this correction utilizes different tuning capacitor to balance the effective inductance at each mechanical stroke amplitude. Data presented without application of this suggested correction utilizes a constant tuning capacitor of 143 F over the entire mechanical stroke range, to balance the inductance measured using an LCR meter at zero mechanical stroke. The performance indices examined are the mechanical stroke, the output electric power, the acoustic-to-electric conversion efficiency, the resonator dynamic pressure, the mechanical-motion loss, the Ohmic loss and the fluid-seal loss. These indices are either measured directly or estimated from the measured signals using the equations presented in [11]. The alternator main signals, namely the mechanical stroke, resonator pressure and enclosure pressure, the output current and the output voltage are acquired for each experiment using a data acquisition card (Model NI 6225, 40 differential-input analogue channels, 16-bit resolution, a maximum sampling rate of 250,000 Samples/S, supplied by National Instruments). The sampling parameters are selected to ensure sampling of an integer large number of cycles with fine time and spectral resolutions without aliasing and without significant amplitude leakage. The sampling parameters used in this work are selected to sample 400 samples/cycle for 400 complete cycles with a total number of samples of 160,000 samples. When operating at 50 Hz, this yields a sampling rate of 20,000 samples/s and a total sampling time of 8.0 seconds with a quantization resolution of 0.3 mV, a time resolution of 50 s and a spectral resolution of 0.15 Hz.

Figure 5: Time domain of the linear alternator signals acquired with the suggested tuning capacitor. The signals are the mechanical stroke (mm), resonator pressure (kPa), enclosure pressure (kPa), output voltage (V) and current (A) and are all simultaneously-acquired. This data is for a gas mixture made of 60% helium 40% argon, a mean gas pressure of 20 bar, an operating frequency of 51 Hz and an electric load made an electric resistance 52.4  in series with a series tuning capacitor that applies the suggested correction (85-µF). Note that the angle between the measured volt and current signals shown in this figure shows their angle at the load (made of the resistance and capacitor) and do not include the coil’s inductance.

The experiments reported in this section are carried-out using a working gas mixture of 60% helium and 40% argon at a mean gas pressure of 20 bar, an electric load that consists of a simple resistance (52.4 ) (in series with a tuning capacitor) at an input pressure ratio to the alternator’s piston face of 0.75%. This ratio is defined as the ratio of the dynamic pressure at the face of the alternator’s piston to the mean gas pressure. These operating conditions are selected to match the operating conditions typically used in thermoacoustic power converters [5-7]. Because the experiments have to be carried-out under both mechanical and electrical resonances, the mechanical resonance frequency at the mean pressure used is determined first. This is achieved by operating the linear alternator as an acoustic driver and then disconnecting the input electric power 6



suddenly by sending a control signal to a solid-state relay. The alternator’s piston oscillates at its mechanical resonance frequency during its decay. The value of the mechanical resonance frequency at this mean gas pressure was found to be 51 Hz at 20-bar mean gas pressure. A sample of this measurement is shown in [11]. The effects of applying this suggested correction on the imaginary and real parts of the linear alternator acoustic impedances are shown in Fig.6 and 7, respectively. With the correction, the imaginary part is always zero over the entire mechanical stroke range (as seen in Eq. 1). The real part is constant over the entire range of mechanical strokes and is equal to (Rm + Bl2/Re)/A2. The effects on the performance indices are presented in Table 2.

Figure 6: Imaginary part of the linear alternator acoustic impedance. With (dotted line) and without (solid curve) the suggested correction.

Figure 7: Real part of the linear alternator acoustic impedance. With (dotted line) and without (solid curve) the suggested correction.

In Table 2, three cases are presented: The first case is without the suggested correction, using a tuning capacitor (143F) estimated based on the inductance measured via an LCR meter (corresponding to a zero mechanical stroke, which is 68 mH in this work). The second case applies the suggested correction, in which case a tuning capacitor (85 F) is used based on the effective inductance (114.5 mH) at the actual mechanical stroke amplitude (1.69 mm). For comparison, the case that uses no tuning capacitors is also presented. Table 2: The linear alternator performance indices with and without the suggested correction. Experimentsare carried-out at the same conditions reported in Fig. 5, except for the values of the tuning capacitor.

Performance index Mechanical stroke amplitude, mm Input electric power, W Output electric power, W Acoustic-to-electric conversion efficiency, % Resonator dynamic pressure, kPa (RMS) Mechanical-motion loss, W Ohmic loss, W Fluid-seal loss, W

Without any tuning capacitors 1.48

Without the suggested correction (143-F) 1.54

With the suggested correction (85-F) 1.69

11.5 4.1

13.2 5.1

14.4 5.6

51.6

61.6

66.3

9.91

10.73

11.18

0.52

0.56

0.63

0.53 0.007

0.67 0.012

0.7 0.017


7


4.

Summary and conclusions

Best operation of linear alternators requires operation at conditions that the keep the acoustic impedance pure real. The presented work experimentally shows that the effective inductance of the alternator’s coil increases with the mechanical stroke. This requires the use of different tuning capacitors to achieve electric resonance to satisfy the required condition on the acoustic impedance. The work quantifies this dependence using an operating frequency corresponding to mechanical resonance condition with all experimental conditions directly related to thermoacoustic power conversion. Then, a potential reason for this observed behaviour is given and a simple linear model is proposed. The effects of applying this suggested correction on the linear alternator acoustic impedance and performance indices are presented. The output electric power was found to increase by 9.8% with an increase in the mechanical stroke by 9.7% and an increase in the resonator dynamic pressure by 4.2%.

5.

Acknowledgements

This publication has been produced with the financial assistance of the European Union. The contents of this document are the sole responsibility of the authors and can under no circumstances be regarded as reflecting the position of the European Union. The authors are thankful to Eng. Amr Sayed Taha for the helpful discussions and his help on selecting and installing the appropriate series tuning capacitors used in this work.

REFERENCES 1 Swift, G., Thermoacoustics: A Unifying perspective for some engines and refrigerators, Melville, (2002). 2 Ueda, Y., Biwa, T., Mizutani, U. and Yazaki, T. Thermodynamic cycles executed in a looped-tube thermoacoustic engine (L). The Journal of Acoustical Society of America, 117(6), 3369-3372, (2005). 3 Tijani, M. E. H. and Spoelstra, S. A high performance thermoacoustic engine. Journal of Applied Physics, 110(9), (2011). 4 Reid, G. (2014). Chart Industries. [Email communication]. 5 Wang, K., Sun, D., Zhang, J., Xu, Y., Zou, J., Wu, K., Qiu, L. and Huang, Z. Operating characteristics and performance improvements of a 500-W traveling-wave thermoacoustic electric generator, Journal of Applied Energy, 160, 853-862, (2015). 6 W, Z., Yu, G., Zhang, L., Dai, W. and Luo, E. Development of a 3 kW double-acting thermoacoustic Stirling electric generator, Journal of Applied Energy, 136, 866-872, (2014). 7 Wu, Z., Dai, W., Man, M. and Luo, E. A solar-powered traveling-wave thermoacoustic electricity generator, Journal of Solar Energy, 86(9), 2376-2382, (2012). 8 Sun, D. M., Wang, K., Zhang, X. J., Guo, Y. N., Xu, Y. and Qiu, L. M. A traveling-wave thermoacoustic electric generator with a variable electric R-C load, Journal of Applied Energy, 106, 377-382, (2013). 9 Tijani, M. E. H., Loudspeaker-driven thermo-acoustic refrigeration. Ph.D Thesis, Technische University, Eindhoven, (2001). 10 Abdou, M., Abdelwahed, A.Y., Ibrahim, A. H. and Abdel-Rahman, E.An experimental setup to test linear alternators: design, operation and preliminary results, Proceedings of the 22nd International Congress on Sound and Vibration, Florence, Italy, 12–16 July, (2015). 11 Ibrahim, A. H. , Abdelwahed, A. Y., Abdou, M. and Abdel-Rahman, E. Real-time measurements of linear alternator performance indices under thermoacoustic-power-conversion conditions, Proceedings of the 23rd International Congress on Sound and Vibration, Athens, Greece, 10–14 July, (2016). 12 Ibrahim, A. H., Abdelwahed, A. Y. and Abdel-Rahman, E. Sensitivity analysis of the linear alternator performance indices under thermoacoustic-power-conversion conditions, Proceedings of the 23rd International Congress on Sound and Vibration, Athens, Greece, 10–14 July, (2016). 13 Abdelwahed, A. Y., Ibrahim, A. H. and Abdel-Rahman, E. Performance of linear alternators using passive linear and non-linear loads under thermoacoustic-power-conversion conditions, Proceedings of the 23rd International Congress on Sound and Vibration, Athens, Greece, 10–14 July, (2016). 14 Purcell, E. M. and Morin, D. M., Handbook of electricity and magnetism, Cambridge University Press, (2013). 15 Juha, P., Tapani, J. and Valeria, H., Design of rotating electrical machines, John Wiley and Sons, (2009).

8
