A traveling-wave thermoacoustic electric generator with a variable electric R-C load
D.M. Sun, K. Wang, X.J. Zhang*, Y.N. Guo, Y. Xu, L.M. Qiu Institute of Refrigeration and Cryogenics, Zhejiang University, Hangzhou 310027, PR China *Corresponding author. Tel/Fax: +86-571-87952793. E-mail address: [email protected] (K. Wang), [email protected] (X.J. Zhang)
Abstract A traveling-wave thermoacoustic electric generator, which is composed of a traveling wave thermoacoustic engine and linear alternators, is promising in solar power generation and energy recovery due to its high efficiency, high reliability, and capability of utilizing low-grade heat. An equivalent acoustic circuit of a linear alternator is first built and analyzed using electro-mechano-acoustical analogy. It is found that the acoustic coupling of the linear alternators to the traveling-wave thermoacoustic engine is crucial to the performance of the system. A traveling-wave thermoacoustic electric generator with a variable electric R-C load is then constructed and experimentally studied. Both the theoretical analysis and the experimental results show the importance of mechanical and electrical resonances to the overall performance of the system. Furthermore, the thermal-to-electric efficiency and the electric power are found to be proportional to the pressure amplitude and the square of it in front of the piston of the linear alternator, respectively. By optimizing the load impedance, the traveling-wave thermoacoustic electric generator has achieved a maximum electric power of 345.3 W with a thermal-to-electric efficiency of 9.34% and a maximum efficiency of 12.33% with an electric power of 321.8 W at around 65 Hz when helium of 3.0 MPa is used as the working gas.
Keywords: thermoacoustic, electric generator, linear alternator, heat
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1. Introduction The conversion of waste heat to electricity represents an important trend of worldwide efforts to reduce greenhouse gas emissions and fossil fuel dependence because of the severe problems with the supply and use of fossil fuel. Thermoacoustic electric generators, which are composed of thermoacoustic engines and linear alternators, are regarded as a promising technology in harvesting solar energy and waste heat from plants, industrial facilities, and so forth. A thermoacoustic engine converts heat into high-intensity acoustic oscillations based on the time-averaged thermodynamic interaction between working gas and solid. Compared to traditional heat engines, a thermoacoustic engine has no mechanical moving components and simply consists of a few pipes and heat exchangers. Thus, it has the merits of simple structure, high reliability, and low manufacturing and maintenance costs [1-7]. The thermodynamic cycle occurring in the regenerator of a traveling-wave thermoacoustic engine is the same as the Stirling cycle. Thus, it is intrinsically of high thermal efficiency. Up to now, the most efficient traveling-wave thermoacoustic engine has achieved a thermal efficiency of 32% corresponding to 49% of Carnot efficiency [8], which shows its great potential and prospect in the applications of energy conversion. The linear alternators used in the thermoacoustic electric generator convert acoustic power into electricity. Flexure bearing and clearance seal adopted in a linear alternator allow a non-contact reciprocating motion of the piston. As a result, the friction and vibration can be kept at a very low level, and the efficiency can reach up to 90% [9, 10]. A thermoacoustic electric generator combines the advantages of a thermoacoustic engine and a linear alternator to realize the conversion from heat to electricity with high efficiency, high reliability, and low costs. The first traveling-wave thermoacoustic electric generator was constructed by S. Backhaus et al. in 2004 for deep space application [11, 12]. At its most efficient operating point, the system generates an electric power of 39 W with a thermal-to-electric efficiency of 18%. This is also the first successful attempt to replace the resonator of a traveling-wave thermoacoustic engine with the alternators, which makes the whole system more compact and efficient. In 2008, Sunpower, Inc. modified a coaxial Stirling convertor into a coaxial traveling-wave thermoacoustic electric generator by replacing the displacer of the original machine with an acoustic feedback circuit and achieved an electric power of 50 W [13]. Yu et al. have recently built a low-cost thermoacoustic electric generator by placing a loudspeaker in the traveling-wave looped-tube [3, 14]. The achieved electric power output and the overall efficiency are limited due to the low conversion efficiencies of the engine and the loudspeaker. In 2010, Luo et al. constructed a traveling-wave thermoacoustic electric generator that achieved a maximum electric power of 481.0 W with an efficiency of 15.0% [15]. Based on this work, a solar powered thermoacoustic electric generator was then developed [16]. A maximum electric power of about 200 W was obtained in their experiments. Recently, Sun et al. have also started the research on thermoacoustic electric generation. The effects of alternator, acoustic transfer line, and electrical loads on the working characteristics of a traveling-wave thermoacoustic electric generator are discussed [17, 18]. In a traveling-wave thermoacoustic electric generator, the linear alternators act as the acoustic loads to the thermoacoustic engine. Thus, the acoustic coupling between the linear alternators and the thermoacoustic engine is crucial to the overall performance. However, the coupling mechanisms, especially the effects of the mechanical and the electrical resonances on the power generation performance, have rarely been discussed systematically in the previous studies. In this study, the key factors that affect the electric power generation and the efficiency of the linear alternators are first analyzed theoretically. A traveling-wave thermoacoustic electric generator with a vibration-balanced pair of linear alternators is developed and tested with a variable electric R-C load. The approaches to achieve a better performance in the thermoacoustic electric generator are analyzed systematically.
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2. Theoretical analysis of linear alternator A linear alternator realizes the conversion from acoustic energy to electric power through strongly coupled acoustical, mechanical and electrical effects. As a complex electroacoustic transducer, the linear alternator is an intrinsic acoustic system which is coupled with the acoustic field of the thermoacoustic engine. Ignoring the nonlinear behavior and hysteresis losses, a simplified physical model for the moving-magnet linear alternator is shown in Fig. 1. The linear alternator has a moving mass M, a mechanical stiffness K, and a mechanical resistance Rm. The winding resistance and winding inductance are re and Le, respectively. A load resistance Rl and a capacitance Ce are connected to the coil. The force factor is Bl and the back volume is Vb. When an oscillating pressure p1 acts on the piston which has an area A, the piston reciprocates with a velocity v1, and consequently acoustic power is converted into electric power. The current generated in the circuit is I1 and the oscillating pressure in the back volume is denoted as p1,b.
Fig. 1. Simplified physical model of moving-magnet linear alternator.
The electro-mechano-acoustical analogy is used to analyze acoustic impedance of the linear alternator. The equivalent circuit of the simplified physical model of the linear alternator is shown in Fig. 2(a). The acoustical, mechanical, and electrical parts are coupled by two transformers. U1 denotes the volume flow rate in front of the piston, and p1 denotes the complex pressure difference between the front and the back sides of the piston, i.e. p1 p 1 p 1,b . The mechanical and the electrical impedances are defined as the following equations, respectively.
Zm R m jX m (1)
Ze R e jX e (2) where XMKm , Re R l r e , and XLCe e1 e . To simplify the analysis, all the components in Fig. 2(a) are converted to equivalent acoustic impedances, as shown in Fig. 2(b). With the contributions of the mechanical and electrical parts denoted in the figure, the equivalent acoustic impedance of the linear alternator can be obtained by 2 22 11Bl Ree Bl X Bl Za2 Z m 2 R m 2 2 j X m 2 2 . (3) AZARXRXe e e e e It is shown that the acoustic impedance of the linear alternator is determined by the mechanical and the electrical parameters. Therefore, adjusting the parameters of the linear alternator can change the acoustic impedance of the linear alternator and further affects the acoustic coupling between the thermoacoustic engine and the linear alternators. With the expression for the acoustic impedance, the input acoustic power Wa can be calculated by 2 pA2 2 p A2 Bl 2 R R R 2 X 2 R 11% 1 1 e m e e m WaaRe[ p11 U ] Re[ Z ] . (4) 222 Z 222 2 a Bl Xm X e R m R e X m R e X e R m As shown in Fig. 2(b), the acoustic power entering the linear alternator is dissipated by the
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Fig. 2. Equivalent circuits for the physical model of moving-magnet linear alternator. (a) the coupled acoustical, mechanical, and electrical parts of the system; (b) the converted equivalent acoustic impedances of the system. three equivalent acoustic resistances, which are caused by the mechanical resistance Rm, the winding resistance re, and the load resistance Rl, corresponding to the mechanical power loss, the Joule heat of the coil and the output electric power, respectively. The mechanical power loss can be calculated by 2 p2 A2 R 2 X 2 R RRmm2 p1 1 1 e e m WUm 1 . (5) 2AAZ22 2 2 2 2 2 a Bl Xm X e R m R e X m R e X e R m The electric power extracted by the load resistance is 2 22 Rl 1 p1 A Bl Rl WWWl a m . (6) R 2 2 2 2 e Bl Xm X e R m R e X m R e X e R m The acoustic-to-electric efficiency of the linear alternator is given by W Bl2 R ll . (7) ae 2 2 2 Wa Bl Re R m R e X e As is shown in Eq. (7), the efficiency of the linear alternator is affected by the magnitudes of the force factor Bl, the load resistance Rl, the real and imaginary parts of the electrical impedance Ze, and the mechanical resistance Rm. For a specific linear alternator, all the parameters except for the angular frequency , the load resistance Rl, and the capacitance Ce are unadjustable. Thus, decreasing the imaginary part of the electrical impedance Xe by connecting an appropriate capacitance Ce can improve the efficiency of the linear alternator. According to Eq. (4), under the electrical resonant state (i.e. Xe=0), more acoustic power can be absorbed by the linear alternator when the mechanical part is forced to be resonant (i.e. Xm=0). Moreover, the extracted electric power is also increased according to Eq. (6). Thus, the resonance of the electrical and the mechanical parts are the two crucial factors that affect the performance of the linear alternator as well to the whole system. When the mechanical and the electrical resonances are both achieved, the acoustic impedance given by Eq. (3) becomes a pure real value, which shows that an acoustically resonant state of the linear alternator is achieved. Under practical conditions, the pressure amplitude of p1,b in the back volume of the linear 4 alternator is relatively small and can be ignored compared to the pressure amplitude of p1 in front of the piston. According to Eq. (6), the extracted electric power can be simplified as follows when the resonances are achieved. 2 22 1 p1 A Bl Rl Wll 2 F p1 , R . (8) 2 Bl2 R R me As shown by Eq. (8), the electric power is a function of the pressure amplitude of p1 and the load resistance Rl. Particularly, it is roughly proportional to the square of the pressure amplitude in front of the piston. Thus, for given load resistances Rl, the amount of extracted electric power can be effectively enlarged by increasing the pressure amplitude. The variation of the load resistance Rl has a more complex effect and should be optimized to obtain a large electric power.
When the mechanical resonance is achieved in a well-designed alternator, the mechanical resistance Rm and reactance Xm are relatively small and can be eliminated in Eq. (3). Then the peak stroke of the linear alternator can be expressed by
U1 p 1 p 1 A 22 x1 2 Ree X . (9) A A Za Bl From Eq. (9), it is found that the peak stroke |x1| is approximately proportional to the pressure amplitude |p1|. Therefore, the pressure amplitude should be within a range to assure the safe operation of the linear alternators when improving the electric power output by increasing the pressure amplitude.
3. Experimental setup As shown in Fig. 3, the traveling-wave thermoacoustic electric generator is composed of a traveling-wave thermoacoustic engine and a vibration-balanced pair of moving-magnet linear alternators. The parameters of the two linear alternators are given in Table 1. The traveling-wave thermoacoustic engine consists of a main ambient heat exchanger (HX), a regenerator, a hot HX, a thermal buffer tube, a secondary ambient HX, a compliance tube, a feedback tube and a resonator. The dimensions of the thermoacoustic engine can be found elsewhere [19, 20]. It should be pointed out that a cross tube with an axial length of 0.29 m is added here between the torus and the resonator to connect the linear alternators. The total length of the branches connecting the linear alternators of the cross tube is 0.38 m. The regenerator is the key component where thermoacoustic conversion occurs. The resonator provides the required high acoustic impedance in the regenerator to attain a high efficiency. With a large temperature gradient along the regenerator that is exerted by the hot HX and the main ambient HX, the regenerator amplifies the feedback acoustic power by converting heat into acoustic power. A portion of the amplified acoustic power feeds back into the regenerator through the feedback tube and the compliance tube, and the rest enters the resonator. The linear alternators convert a portion of the acoustic power entering the resonator into electricity with the rest dissipated by viscous effect. The coils of the linear alternators are connected in series with a variable electric R-C load to extract electric power. The electric power is measured when the system works steadily with the heating temperature at approximately 650 °C. The voltage across the load resistance is measured by Keithley digital multimeter 2700, and the magnitude of the load resistance is measured by Fluke digital multimeter (model 15B) to determine the electric power. A piezoresistive pressure sensor supplied by Huba Control (model 511.933003142) with an accuracy of ±0.3% is installed in front of the piston of #2 linear alternator to measure the mean pressure and the pressure oscillation, as indicated by p1 in Fig. 3. The pressure oscillation of the back volume of the linear alternator, which is relatively small, is measured by a PCB piezoelectric pressure sensor (model 102B15) to determine the stroke of the piston, as denoted
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Fig. 3. Schematic of the traveling-wave thermoacoustic electric generator.
Table 1. Parameters of the linear alternators. 2 No. Bl (N/A) Le (mH) re (Ω) K (N/m) Rm (Ns/m) M (kg) A (m ) Vb (L) #1 108.69 315.9 3.58 189235 15.0 1.097 1.9635×10-3 1.63 #2 111.72 313.5 3.56 188844 11.2 1.079 1.9635×10-3 1.63
by p1,b in Fig. 3. The peak stroke is then calculated by using x1 p 1,bb V/ p 0 A, where is the ratio of specific heats and p0 is the mean pressure.
4. Results and discussions As analyzed above, the resonances of the mechanical and the electrical parts of the linear alternators are crucial to the performance of the whole system. To achieve the mechanical resonance, i.e. XMKm 0 , the mechanical resonant frequencies of the linear alternators, which are 66.6 Hz and 66.8 Hz respectively, should equal the working frequency of the thermoacoustic engine. With nitrogen as a working gas, the traveling-wave thermoacoustic engine operates at about 25 Hz [19], which is far below the resonant frequencies. Previous experiments with nitrogen as the working gas showed that the maximum extracted electric power is only several watts with a thermal-to-electric efficiency of less than 0.25% [17]. It proved that the performance of the thermoacoustic electric generator is very sensitive to the working frequency. To make the working frequency of the thermoacoustic engine to be around the mechanical resonant frequencies, helium is used as the working gas and the length of the connecting cross tube for the linear alternators is finally chosen to be 0.29 m. Individual operation of the thermoacoustic engine shows that the working frequency is about 66 Hz, which agrees well with the target value. The working frequency of the whole system is about 64.5 Hz and a near resonant state is achieved.
After the mechanical resonance is achieved, the electrical resonance, i.e. XLCe e 10 e , can be realized by adding an appropriate capacitance Ce in the circuit. The measured imaginary impedance of the circuit X e is reduced from 258.2 Ω to 5.3 Ω after an electric capacitance is introduced, which can be considered that an electrical resonant state is roughly achieved. Fig. 4 shows the effects of the electrical resonance to the electric power and the thermal-to-electric efficiency with helium of 2.1 MPa as a working gas. As shown, the electric power and the thermal-to-electric efficiency both increase with the load resistance whatever the electrical 6 resonance is achieved or not. It is worth noting that the performance of the traveling-wave thermoacoustic electric generator is significantly improved when the electrical resonance is achieved. The electric power is increased from 182.5 W to 205.8 W, and the efficiency is improved from 7.09% to 10.24% when the load resistance is 160 Ω. The experimental results agree well with the theoretical analysis in showing that the electric power and the efficiency can be increased when the electrical resonance is achieved.
Fig. 4. Electric power and thermal-to-electric efficiency vs. load resistance. The electrical resonance is achieved by adding a capacitance to make the imaginary electrical impedance X e to be decreased from 258.2 Ω at the non-resonant state to 5.3 Ω.
Fig. 5 shows the peak stroke and pressure amplitude in front of the piston of #2 linear alternator. As shown, when the linear alternators are not at electrical resonant state the peak strokes at different load resistances are all above 5 mm, which is close to the maximum allowable stroke of the linear alternators, i.e. 6.5 mm. The large peak stroke is mainly due to the large imaginary electrical impedance caused by the winding inductance of the coils, as analyzed in Eq. (9). As the imaginary electrical impedance ( Xe =258.2 ) in the non-resonant state is much larger than the load resistance
( Rl 160 ), the peak stroke varies little when the load resistance varies from 20 Ω to 160 Ω.
When the approximate electrical resonance ( Xe 5.3 ) occurs, the peak stroke decreases significantly, which agrees with the theoretical analysis. As shown in Fig. 4 and Fig. 5, the peak stroke decreases from 5.16 mm to 3.25 mm at the most efficient and powerful operating point. The reduced peak stroke provides more space to achieve a better performance. Moreover, the pressure amplitude in the thermoacoustic engine decreases after the resonant state is achieved. Therefore, the viscous dissipation in the resonator is decreased and more power is extracted by the linear alternators, which improves the overall performance.
Fig. 5. Peak stroke of the piston of #2 linear alternator and the pressure amplitude in front of the piston vs. load resistance.
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Furthermore, the theoretical analysis shows that the electric power can be increased by increasing the pressure amplitude and optimizing the load resistance. The thermoacoustic electric generator is tested at three charge pressures, i.e. 2.1 MPa, 2.5 MPa, and 3.0 MPa, with a variable load resistance. Fig. 6 shows the effects of pressure amplitude |p1| in front of #2 linear alternator and load resistance Rl on the electric power under electrical resonance. When the charge pressure is increased from 2.1 MPa to 3.0 MPa, the pressure amplitude |p1| is increased significantly. Consequently, the electric power is increased significantly and rises proportionally to the square of the pressure amplitude |p1| when the load resistance is fixed, which agrees well with the theoretical analysis. For example, when the pressure amplitude at a charge pressure of 3.0 MPa is 1.29 times that at 2.1 MPa with the load resistance of 100 Ω, the ratio of the electric power is 1.69 which is roughly the square of 1.29. Thus, increasing the pressure amplitude by increasing the charge pressure is an effective way to improve the electric power. Furthermore, it is shown that the electric power increases rapidly at small load resistances and then stabilizes at a relatively high level, which indicates that the electric power can be maximized by optimizing the load resistance. When the load resistance is 80 Ω and the charge pressure is 3.0 MPa, the thermoacoustic electric generator generates a maximum electric power of 345.3 W, with a thermal-to-electric efficiency of 9.34% which can be found in Fig. 8.
Fig. 6. Electric power map vs. pressure amplitude and load resistance under electrical resonance.
Fig. 7. Effects of pressure amplitude and load resistance on the peak stroke of #2 linear alternator under electrical resonance.
Fig. 7 shows the effects of pressure amplitude and load resistance on the peak stroke of #2 linear alternator under electrical resonance. As shown, the peak stroke of #2 linear alternator is always proportional to the pressure amplitude |p1| at any load resistance, which agrees well with the theoretical analysis. The peak stroke increases with the load resistance at a constant charge pressure. When the load resistance is 160 Ω and the charge pressure is 3.0 MPa, the peak stroke reaches a 8 maximum value of 4.06 mm. At the most powerful operating point where the maximum electric power is obtained, the peak stroke is 3.12 mm, which is far less than the maximum allowable stroke of 6.5 mm. Therefore, much space is still left for the improvement of the electric power.
Fig. 8 shows the variation of thermal-to-electric efficiency he with peak stroke at different charge pressures and load resistances. It is shown that the thermal-to-electric efficiency is linearly proportional to the peak stroke, which is denoted by the linear function in the figure. So, increasing the peak stroke of the linear alternator is beneficial for the improvement of thermal-to-electric efficiency. As shown in Fig. 7, a higher charge pressure and a larger load resistance both help to achieve a larger peak stroke. When the load resistance is 160 Ω and the charge pressure is 3.0 MPa, the thermal-to-electric efficiency reaches the maximum value of 12.33% with an electric power of 321.8 W.
Fig. 8. Effect of peak stroke of #2 linear alternator on thermal-to-electric efficiency under electrical resonance.
It is worth noting that the acoustic resonance for the thermoacoustic engine is mainly accomplished by the large resonator due to the relatively small swept volumes of the two linear alternators at present. The large resonator tube in the current design introduces much dissipation. For example, when the linear alternators produce an electric power of 345.3 W at the most powerful operating point, the calculated acoustic power dissipated in the resonator is about 470 W, which is evaluated by DeltaEC [21, 22]. If linear alternators are acoustically coupled to the thermoacoustic torus replacing the resonator completely, the performance of the traveling-wave thermoacoustic electric generator will be further improved.
5. Conclusions A traveling-wave thermoacoustic electric generator shows its great potential in thermal energy utilization. An equivalent acoustic circuit of a linear alternator is first built by using electro-mechano-acoustical analogy, which sheds light on the way to improve the overall performance of the acoustic system. A traveling-wave thermoacoustic electric generator consisting of a traveling-wave thermoacoustic engine and two linear alternators was constructed and tested with a variable electric R-C load. Both the theoretical analysis and the experimental results show the importance of the mechanical and the electrical resonances in improving the performance of the thermoacoustic electric generator. Furthermore, the electric power and the thermal-to-electric efficiency can both be increased significantly by increasing the pressure amplitude in the engine. With helium of 3.0 MPa as a working gas, the traveling-wave thermoacoustic electric generator produces an electric power of 345.3 W with a thermal-to-electric efficiency of 9.34% at the most powerful operating point. At the most efficient operating point, the system produces an electric power of 321.8 W with a thermal-to-electric efficiency of 12.33%. This research shows that further improvements can be expected by further increasing the pressure amplitude and reducing the
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Acknowledgements This research is financially supported by a grant from the Major State Basic Research Development Program of China (973 Program) under contract No. 2010CB227303 and the National Natural Science Foundation of China under contract No. 61077035.
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Nomenclature
English letters A area Bl force factor Ce capacitance F force F a function I current j square root of -1 K mechanical stiffness Le winding inductance of coil M moving mass p pressure Qh heating power R resistance re winding resistance of coil U volume flow rate Vb back volume v velocity W work X reactance x piston stroke Z impedance Greek letters difference ratio of isobaric to isochoric specific heats
ae acoustic-to-electric efficiency
he thermal-to-electric efficiency angular frequency Subscripts a acoustic b back volume of linear alernator e electric l electric load m mechanical 0 mean 1 first order Special symbols
Re[ ] real part of | | magnitude of complex number tilde complex conjugate
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