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Acoustic power flow measurement in a thermoacoustic resonator by means of laser Doppler anemometry (L.D.A.) and microphonic measurement
Applied Acoustics 60 (2000) 1±11
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Acoustic power ¯ow measurement in a thermoacoustic resonator by means of laser Doppler anemometry (L.D.A.) and microphonic measurement H. Bailliet a,*, P. Lotton a, M. Bruneau a, V. Gusev b, J.C. ValieÁre a, B. Gazengel a a
Laboratoire d'Acoustique de l'Universite du Maine, UMR-CNRS 6613, IAM, Univ. du Maine, av. O. Messiaen, 72085 Le Mans Cedex 9, France b Laboratoire de I'Etat Condense de la Faculte des Sciences, UPRESA-CNRS 6087, av. O. Messiaen, 72085 Le Mans Cedex 9, France Received 6 April 1999; received in revised form 30 June 1999; accepted 17 August 1999
Abstract Acoustic power ¯ow measurements in the resonator of a thermoacoustic refrigerator are described. The technique of measurement is based on particle velocity measurement by laser Doppler anemometry (L.D.A.) together with microphonic acoustic pressure measurement. The calibration procedure is explained and results of measurements are compared with analytical results. The L.D.A. technique permits the measurement of acoustic power ¯ow at almost any position and for almost any working frequency in the resonator of thermoacoustic devices. # 2000 Elsevier Science Ltd. All rights reserved. Keywords: Acoustic power ¯ow; laser Doppler anemometry; thermoacoustics; sound measurement
1. Introduction The thermoacoustic process is based on the eects that occur in the thermal boundary layers associated with an acoustically oscillating ¯uid close to a rigid wall. These eects are used to convert acoustic work ¯ow into heat ¯ow (thermoacoustic refrigerator) or the contrary (thermoacoustic prime mover) (e.g. [1]). In the case of thermoacoustic refrigerators, this process involves a high amplitude resonant * Corresponding author. Tel.: +33-2-4383-3270; fax: +33-2-4383-3520. E-mail address: [email protected] (P. Lotton). 0003-682X/00/$ - see front matter # 2000 Elsevier Science Ltd. All rights reserved. PII: S0003-682X(99)00046-8
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H. Bailliet et al. / Applied Acoustics 60 (2000) 1±11
acoustic ®eld inside a ¯uid-®lled resonant cavity, that is due to the coupling between an acoustic source and this cavity (Fig. 1). It must be emphasized that the value of the particle velocity amplitude at the face of the loudspeaker, schematically shown to be zero in Fig. 1, actually depends on the coupling between the loudspeaker and the thermoacoustic cavity [2]. A short stack of plates is located in the cavity between a pressure antinode and a velocity antinode. The thickness of the ¯uid layers between two plates of the stack has the same order of magnitude as the thermal boundary layer thickness. In the stack region, the cyclic oscillations of the ¯uid particles induce a heat ¯ux from one end of the stack to the other and a temperature gradient is induced. Heat exchangers are set at each end of the stack so that heat can be removed from (and provided to) the thermoacoustic resonator. In such a process, the acoustic power ¯ow in the resonator strongly in¯uences the heat ¯ow along the stack. According to the energy conservation law expanded to second order in the acoustic amplitude, the thermoacoustic heat ¯ow is equal to the sum of the acoustic
Fig. 1. Schematic representation of a thermoacoustic refrigerator.
H. Bailliet et al. / Applied Acoustics 60 (2000) 1±11
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power ¯ow and of the thermal heat ¯ux via conduction. Acoustic power ¯ow measurement is thus very important for determining the intrinsic eciency of a thermoacoustic refrigerator. The literature on acoustic power and acoustic impedance measurement is extensive. The techniques used prior to 1948 are reviewed by Beranek [3]. The method that involves the use of two closely spaced pressure microphones has received considerable attention since the forties [4]. It has been re®ned so that attenuation in ducts can be taken into account [5,6] arid still motivates research work, especially associated both with progresses in signal processing [7], and with discussions of several sources of experimental errors [8]. This method is based on the idea that acoustic intensity can be determined from the signals provided by two adjacent pressure sensors, using the average of the two signals to obtain the pressure and the dierence between them to obtain the velocity. The high standing wave ratios encountered in thermoacoustic resonators make this method dicult to use because of the extreme phase accuracy and stability required for measurements on resonating sound ®elds [9,10]. For such experimental conditions, the range of possible measurements, in relation to frequency and position, is limited. Acoustic power ¯ow can also be measured by means of the method that involves the use of nearly coincident pressure and pressure gradient microphones [11] which yields a simultaneous measurement of acoustic pressure and particle velocity. However, this method is complicated in terms of transducer fabrication and calibration and the transducers can disturb the acoustic ®eld. Ho¯er [12] designed and calibrated a driver apparatus for the accurate measurement of acoustic power delivered by the driver to the thermoacoustic resonator. The driver was a modi®ed loudspeaker whose voice coil was attached to a rigid moving piston. A small quartz-crystal dynamic pressure transducer was positioned on the face of the driver apparatus housing. A miniature accelerometer was attached to the back side of the piston; the velocity was obtained from the time-integrated accelerometer signal. The calibrated dynamic pressure signal, the calibrated volume velocity and the phase between them enabled the measurement of acoustic power delivered to the resonator. Recently, Yazaki and Tominaga [13] experimentally studied the spontaneous gas oscillation in a resonator from the standpoint of heat engines and presented the ®rst measurements of the work ¯ow emitted by the stack of a thermoacoustic driver. They used small pressure transducers ¯ush mounted on the resonator walls to obtain a measurement of acoustic pressure and laser Doppler anemometry to measure the velocity along the axis. The pressure and velocity signals were digitized and their power and phase spectra were calculated via a fast-Fourier-transform algorithm. The phase shift between particle velocity and acoustic pressure, was obtained from the phase spectra; it was further adjusted by taking into account delays caused by electrical circuits and by the measurement of pressure. For a given working frequency, axial distributions of pressure, core velocity and work ¯ow were extracted from these measurements, for the fundamental and for the second harmonic of the wave generated by the thermoacoustic prime mover. In respect of the fundamental frequency, experimental results agree with numerical simulations based on
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H. Bailliet et al. / Applied Acoustics 60 (2000) 1±11
thermoacoustic theory [14,15]. The thermoacoustic engine has been loaded by a second stack to get a thermoacoustically driven thermoacoustic refrigerator. In this situation, the authors note that the presence of this second stack increases the work ¯ow along the resonator by approximately a factor of three. To sum up, extensive research in dierent ®elds has been devoted to acoustic power measurement, but the procedure is still complicated, especially in the case of a thermoacoustic resonator. This measurement involves precise measurement of the phase dierence ' between acoustic pressure and particle velocity, which is dicult to determine in a thermoacoustic resonator where the standing wave ratio is very high. This is due to the fact that working frequencies are resonance frequencies, around which cos ' varies rapidly, because ' is close to =2. In the following, a technique of calibration and measurement of particle velocity by laser Doppler anemometry together with classical acoustic pressure measurement is proposed, that yields a measurement of the acoustic power ¯ow at almost any position and for almost any working frequency in the resonator. Because the acoustic power measurement requirements in thermoacoustics do not restrict the general utility of developing such a measurement technique, it is expected that this discussion will be of general interest and useful wherever the measurement of the amount of acoustic power is needed; in particular, it can also be applied to the measurement of acoustic impedance. In Section 2, the acoustic power ¯ow is expressed as a function of both the acoustic pressure and the particle velocity component along the axis. The experimental set up, the calibration of measuring chains and results of measurements are presented in Section 3. The measured acoustic pressure, particle velocity and acoustic power ¯ow are compared with analytically calculated corresponding quantities. 2. Acoustic power ¯ow as a function of the acoustic pressure and the particle velocity along the axis As stated before, the important quantities that allow the characterization of a thermoacoustic refrigerator in terms of its eciency are the heat ¯ow extracted to the cold thermal source and the power ¯ow used to extract this heat ¯ow. The acoustic power ¯ow in the resonator is expressed as a function of the acoustic pressure p and the x-component of particle velocity ux , according to P Shg pu x i where S is the cross-section of the resonator (tilde and angular brackets indicate time and radial averages respectively). In the resonant thermoacoustic cavity, the usual thermoacoustic assumptions are used: quasi-plane waves, linear acoustic approximation, laminar ¯ow, no steady ¯ow, no heat source, no acoustic streaming. Then, the pressure can be assumed to be constant across the resonator section giving hpux i phux i. Therefore, the r-component of the particle velocity ur , can be neglected as compared to its x-component ux
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because ur / @r p and ux / @x p (where @i @i@ for example). Moreover, the spatial variation of particle velocity in the x-direction can be neglected!as compared! to its ! ! spatial variation in r-direction, that is @x u @r u , because @@xu / 1l and @@ru / 1 , the viscous boundary layer thickness , being much smaller than the acoustic wavelength l l. In the case of simple harmonic sound at angular frequency !, the particle velocity in a cylindrical tube of radius R, which is the solution of the NavierStokes equation, is thus simply related to the acoustic pressure according to (e.g. [16]): i J0 k r 1ÿ @x p ; ux x; r !0 J0 k R where 0 is the mean density, J0 is a Bessel function of the ®rst kind and k 1ÿi . In particular, the particle velocity along the axis is given by i 1 1ÿ @x p ; ux x; 0 !0 J0 k R and the spatial average over the resonator section of the particle velocity takes the simple form hux xi
i 1 ÿ f @x p ; !0
R where f k2R JJ10 k k R. Finally, the total acoustic power ¯owing down the resonator is given by 3 1 0 2
P
S S B 6 R phux i RB p6ux x; 0 2 2 @ 4
C 1 ÿ f 7 7 C; 1 5 A 1ÿ J0 k R
1
where R denotes the real part and * denotes complex conjugation. Eq. (1) shows that the total acoustic power ¯ow can be obtained from a microphonic measurement of the acoustic pressure p and a measurement of the particle velocity along the axis ux x; 0. 3. Acoustic power ¯ow measurement The experimental apparatus used for acoustic power ¯ow measurement is illustrated schematically in Fig. 2. It consists of a very simpli®ed demonstration thermoacoustic refrigerator, composed of an electrodynamic loudspeaker loaded by a transparent cylindrical resonator ®lled with air at atmospheric pressure. The length of the resonator is l=49.7 cm and its radius is R=2.2 cm. A 6.5 cm length stack is set in this resonator. The center position of the stack is at xs=14 cm. This stack was
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Fig. 2. Schematic representation of the experimental set up used for acoustic power ¯ow measurement.
fabricated from a long sheet of paper that was spirally wound [17]. Mono®lament 0.13 mm radius strings were glued to the sheet to provide spacing between adjacent layers of the spiral. In order to obtain both the pressure and the velocity measurements at the same position in the resonator, an adapted pressure transducer is used. It consists of a 2 mm diameter 3 cm long probe ®tted to a microphone (B&K 1/400 ) ¯ush mounted on the resonator wall. The sensitivity of the microphone is ®rst determined using a gauge source. Then, the eect of the probe is evaluated by comparison with a calibrated microphone using a coupler (B&K). When considering the probe tube as a transmission line and assuming that, in the frequency range of interest, the volume ¯ow at the front of the microphone is equal to zero, the pressure at the entrance of the coupler is related only to the pressure at the front of the microphone. It follows that the ratio of the pressures at the entrance and at the exit of the probe does not depend on the impedance presented to the probe by the environment external to the probe tube, so that the eect of the probe can be estimated by using a coupler. The particle velocity along the axis of the resonator is measured using laser Doppler anemometry. Two coherent laser beams are crossed and focused at the center of the tube to generate an ellipsoidal probe volume composed of equidistant dark and bright interference fringes. Seeding particles are dispersed in the ¯uid. The seeding we used is generated by a fog generator based on water condensation with aerosol; the mean diameter of seeding particles is about 1 mm. When a seeding particle passes through the measuring volume, light is scattered from its surface. The backward scattered light is detected by a photo-multiplier; the electronic signal obtained, called a burst, has a modulation frequency (Doppler frequency) proportional to the velocity of the seeding particle. In order to distinguish the velocity sign, the wavelength of one beam is shifted using a Bragg cell. Consequently, the Doppler frequency of the received signal is shifted up or down around the Bragg frequency according to the velocity direction [18]. Further signal processing is achieved by means of a Burst Spectrum Analyzer (BSA Dantec) [19] based on FFT analysis with interpolation. The fast-Fourier-transform of the signal is calculated and a peak detector permits estimation of the Doppler frequency, and then the particle velocity. Fig. 3 (upper
H. Bailliet et al. / Applied Acoustics 60 (2000) 1±11
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Fig. 3. Example of result of measurement of particle velocity versus frequency. On the upper curve, each point (ti ; uxij ) corresponds to a particle crossing the probe volume at time instant ti (shifted over an acoustic period) with a velocity uxij . The lower curve is the average value uxi of velocity at time ti .
curve) shows an example of such results. Each time ti (shifted over an acoustic period) Pn corresponds to several associated velocities uxij . The average velocity uxi j1 uxij =n for each time ti is calculated, yielding the lower curve in Fig. 3, that is the velocity signal non uniformly sampled over an acoustic period T. The R.M.S. value r 2 u u xi x i1 ti1 ÿ ti and its phase relative to a reference of the velocity ux T1 i 2 signal, chosen to be the source signal applied to the loudspeaker, are ®nally calculated. Because we need a precise measurement of the phase dierence between acoustic pressure and particle velocity, a phase calibration of the two measuring chains has been carried out. It consists in a comparison between experimental and theoretical phase dierence in a simple resonating tube (Fig. 4). First, at a distance (l ÿ xm ) from the rigidly closed end of the tube, and for varying frequency, the acoustic pressure is measured by means of the probe microphone. The associated particle velocity is simultaneously measured by means of L.D.A.; this gives a measurement of the experimental reference phase dierence 'e xm ; f . Secondly, the corresponding theoretical phase dierence can be precisely calculated by using the well-known theory of acoustic propagation in simple lossy tubes [20]. For this purpose, one can use, for example, the expression of acoustic impedance Z xm ; f as a function of the ! speci®c admittance 1i 2 c0 ÿ 1h at x l:
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H. Bailliet et al. / Applied Acoustics 60 (2000) 1±11
Fig. 4. Schematic representation of the experimental set up for phase shift calibration.
Z xm ; f j
p i'th aZ l b je ; ux cZ l a
where
Z Y h sin k l ÿ xm ; c sin k l ÿ xm ; a cos k l ÿ xm ; b k k
and Z l
0 c0 :
Here c0 is the adiabatic sound speed, is the speci®c heat ratio, h is the thermal boundary layer thickness, k is the complex wave number de®ned as r h i 1 ÿ1fh i!0 h R ! 1ÿi i!S , fh kh2R JJ10 k k c0 1ÿf kh R, kh h , Z S 1ÿf and Y h c2 1 ÿ1f . 0 0
h
The comparison between the theoretical 'th xm ; f and experimental 'e xm ; f phase dierences gives the calibration coecient for the set up: C' 'th xm ; f ÿ 'e xm ; f ÿ3; 36 7; 5:10ÿ3 f : Using this calibration coecient and Eq. (1), we can extract the acoustic power ¯ow P from the measurement of both the particle velocity along the axis and the acoustic pressure in the thermoacoustic resonator. Such results can moreover be compared with results of analytical calculation of the same quantities as presented in
H. Bailliet et al. / Applied Acoustics 60 (2000) 1±11
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[2]. This analytical calculation gives the acoustic pressure, particle velocity and acoustic power ¯ow P in the thermoacoustic resonator for any working frequency (in the ®eld of classical thermoacoustic refrigeration) and at any position as functions of
Fig. 5. Results of measurement () and analytical results (solid line) of the particle velocity (modulus of the velocity averaged across a section in m/s), acoustic pressure (modulus in Pa), phase dierence ' between them (in rad), and acoustic power ¯ow (in W) versus frequency at x=4 cm in a thermoacoustic resonator.
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H. Bailliet et al. / Applied Acoustics 60 (2000) 1±11
the voltage applied to the sound source (loudspeaker) only. As an example, Fig. 5 shows the calculated and measured particle velocity modulus (averaged over the section), acoustic pressure modulus, phase dierence between them, and acoustic power ¯ow versus working frequency at x=4 cm (see Fig. 2). The discrepancies between the calculated acoustic power ¯ow and the results of measurement depend on the precision of both the amplitudes of the acoustic quantities (pressure and particle velocity) and the phase dierence between them. The two maxima for P correspond to possible working frequencies for thermoacoustics. The ®rst maximum corresponds to the quarter-wavelength mode and the second maximum to the halfwavelength one. The agreement between analytical and experimental results is good, even at resonance frequencies around which cos ' varies very rapidly. This kind of measurement is the ®rst step in designing a thermoacoustic refrigerator because it allows the experimental determination of the working frequencies (which are frequencies of the maxima for P). The next step consists in measuring the acoustic power ¯ow along the resonator at the chosen working frequency. Yazaki and Tominaga [13] recently reported such measurements but in the case of thermoacoustic prime mover and using a calibration procedure dierent from the one proposed here. Measurements of the acoustic power ¯ow along the resonator can be achieved following the calibration and experimental procedures presented above. They are in progress at present and we hope to obtain the acoustic power ¯ow along the resonator with a very good spatial resolution. 4. Conclusion Calibration and measurements of particle velocity by laser Doppler anemometry together with classical acoustic pressure measurement have been presented; they yield a measurement of the acoustic power ¯ow in a thermoacoustic resonator. Experimental results appear to be in good agreement with results of analytical calculation. They permit the experimental determination of the possible working frequencies of thermoacoustic refrigerator, which are the frequencies of the maximum of acoustic power ¯ow at the entrance of the resonator. With this technique it is also possible to measure acoustic power ¯ow along the resonator with a very good spatial resolution. The technique presented here is expected to be useful for other applications, wherever the measurement of the amount of acoustic power is needed. Nevertheless, it still needs re®nements before it becomes extensively useful for all kind of eligible applications. In particular, it is necessary to evaluate the accuracy of particle velocity measurement in order to evaluate the accuracy of acoustic power measurement. For our experimental set up, an error smaller than 8% is guaranteed down to particle displacement amplitudes of 0.04 mm [21]. Studies are being carried out at present to estimate all the possible sources of experimental errors when using L.D.A. technique, and signal processing tools are being developed to access larger dynamic and frequency range. Progress in these matters should help in developing a precise and widely useful technique of acoustic power ¯ow measurement.
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References [1] Swift GW. Thermoacoustic engines and refrigerators. Physics Today 1995;48(7):22±8. [2] Bailliet H, Lotton P, Bruneau M, Gusev V. Coupling between loudspeakers and thermoacoustic cavities. Acta acoustica, submitted 1998. [3] Beranek LL. Acoustic measurements. New York: Wiley, 1949. [4] Clapp CW, Firestone FA. An acoustic wattmeter, an instrument for measuring energy ¯ow. J Acoust Soc Am 1941;28(2):124±36. [5] Chu WT. Extension of the two-microphone transfer function method for impedance tube measurements. J Acoust Soc Am 1986;80:347. [6] Fusco AM, Ward WC, Swift GW. Two-sensor power measurements in lossy ducts. J Acoust Soc Am 1992;91:2229±35. [7] Fahy FJ. Measurement of acoustic intensity using the cross-spectral density of two microphone signals. J Acoust Soc Am 1977;62:1057. [8] Seybert AF, Soenarko B. Error analysis of spectra estimates with application to the measurement of acoustic parameters using random sound ®elds in ducts. J Acoust Soc Am 1981;69:1190. [9] Pascal JC, Carles C. Systematic measurement errors with two microphone sound intensity meters. J Sound Vib 1982;83:53±65. [10] Fahy FJ. Sound intensity. 2nd edition. London: Elsevier Applied Science, 1995. [11] van Zyland BG, Anderson F. Evaluation of the intensity method of sound power determination. J Acoust Soc Am 1975;57:682±6. [12] Ho¯er TJ. Accurate acoustic power measurements with a high-intensity driver. J Acoust Soc Am 1988;83(2):777±86. [13] Yazaki T, Tominaga A. Measurement of sound generation in thermoacoustic oscillations. Proc R Soc Lond A 1998;454:2113±22. [14] Tominanga A. Thermodynamic aspects of thermoacoustic theory. Cryogenics 1995;35:427±40. [15] Swift GW. Thermoacoustic engines. J Acoust Soc Am 1988;84(4):1145±80. [16] Arnott WP, Bass HE, Raspet R. General formulation of thermoacoustics for stacks having arbitrarily shaped pore cross sections. J Acoust Soc Am 1991;90(6):3228±37. [17] Ho¯er TJ. Thermoacoutic refrigerator design and performance. Ph.D. thesis. San Diego: University of California, 1986. [18] ValieÁre JC, Herzog P, Valeau V, Tournois G. Acoustic velocity measurements in the air by means of laser Doppler velocimetry: dynamic and frequency range limitations and signal processing improvements. J Sound Vib, in press. [19] Scholten J. Developments in laser Doppler anemometry applied to acoustics. User's guide 57n10/ 57n14/57n25 Burst Spectrum Analyser, report of Post-Doctoral research, Laboratoire d'Acoustique de l'Universite du Maine, DeÂcembre 1997. [20] Bruneau M. Manuel d'Acoustique fondamentale. Paris: HermeÂs, 1998. [21] Valeau V. Mesure de la vitesse particulaire acoustique par aneÂmomeÂtrie laser Doppler: estimation de freÂquence instantaneÂe aÁ variation sinusoõÈ dale, validation de la mesure. Ph.D. thesis, Universite du Maine, 1999.
www.elsevier.com/locate/apacoust
Acoustic power ¯ow measurement in a thermoacoustic resonator by means of laser Doppler anemometry (L.D.A.) and microphonic measurement H. Bailliet a,*, P. Lotton a, M. Bruneau a, V. Gusev b, J.C. ValieÁre a, B. Gazengel a a
Laboratoire d'Acoustique de l'Universite du Maine, UMR-CNRS 6613, IAM, Univ. du Maine, av. O. Messiaen, 72085 Le Mans Cedex 9, France b Laboratoire de I'Etat Condense de la Faculte des Sciences, UPRESA-CNRS 6087, av. O. Messiaen, 72085 Le Mans Cedex 9, France Received 6 April 1999; received in revised form 30 June 1999; accepted 17 August 1999
Abstract Acoustic power ¯ow measurements in the resonator of a thermoacoustic refrigerator are described. The technique of measurement is based on particle velocity measurement by laser Doppler anemometry (L.D.A.) together with microphonic acoustic pressure measurement. The calibration procedure is explained and results of measurements are compared with analytical results. The L.D.A. technique permits the measurement of acoustic power ¯ow at almost any position and for almost any working frequency in the resonator of thermoacoustic devices. # 2000 Elsevier Science Ltd. All rights reserved. Keywords: Acoustic power ¯ow; laser Doppler anemometry; thermoacoustics; sound measurement
1. Introduction The thermoacoustic process is based on the eects that occur in the thermal boundary layers associated with an acoustically oscillating ¯uid close to a rigid wall. These eects are used to convert acoustic work ¯ow into heat ¯ow (thermoacoustic refrigerator) or the contrary (thermoacoustic prime mover) (e.g. [1]). In the case of thermoacoustic refrigerators, this process involves a high amplitude resonant * Corresponding author. Tel.: +33-2-4383-3270; fax: +33-2-4383-3520. E-mail address: [email protected] (P. Lotton). 0003-682X/00/$ - see front matter # 2000 Elsevier Science Ltd. All rights reserved. PII: S0003-682X(99)00046-8
2
H. Bailliet et al. / Applied Acoustics 60 (2000) 1±11
acoustic ®eld inside a ¯uid-®lled resonant cavity, that is due to the coupling between an acoustic source and this cavity (Fig. 1). It must be emphasized that the value of the particle velocity amplitude at the face of the loudspeaker, schematically shown to be zero in Fig. 1, actually depends on the coupling between the loudspeaker and the thermoacoustic cavity [2]. A short stack of plates is located in the cavity between a pressure antinode and a velocity antinode. The thickness of the ¯uid layers between two plates of the stack has the same order of magnitude as the thermal boundary layer thickness. In the stack region, the cyclic oscillations of the ¯uid particles induce a heat ¯ux from one end of the stack to the other and a temperature gradient is induced. Heat exchangers are set at each end of the stack so that heat can be removed from (and provided to) the thermoacoustic resonator. In such a process, the acoustic power ¯ow in the resonator strongly in¯uences the heat ¯ow along the stack. According to the energy conservation law expanded to second order in the acoustic amplitude, the thermoacoustic heat ¯ow is equal to the sum of the acoustic
Fig. 1. Schematic representation of a thermoacoustic refrigerator.
H. Bailliet et al. / Applied Acoustics 60 (2000) 1±11
3
power ¯ow and of the thermal heat ¯ux via conduction. Acoustic power ¯ow measurement is thus very important for determining the intrinsic eciency of a thermoacoustic refrigerator. The literature on acoustic power and acoustic impedance measurement is extensive. The techniques used prior to 1948 are reviewed by Beranek [3]. The method that involves the use of two closely spaced pressure microphones has received considerable attention since the forties [4]. It has been re®ned so that attenuation in ducts can be taken into account [5,6] arid still motivates research work, especially associated both with progresses in signal processing [7], and with discussions of several sources of experimental errors [8]. This method is based on the idea that acoustic intensity can be determined from the signals provided by two adjacent pressure sensors, using the average of the two signals to obtain the pressure and the dierence between them to obtain the velocity. The high standing wave ratios encountered in thermoacoustic resonators make this method dicult to use because of the extreme phase accuracy and stability required for measurements on resonating sound ®elds [9,10]. For such experimental conditions, the range of possible measurements, in relation to frequency and position, is limited. Acoustic power ¯ow can also be measured by means of the method that involves the use of nearly coincident pressure and pressure gradient microphones [11] which yields a simultaneous measurement of acoustic pressure and particle velocity. However, this method is complicated in terms of transducer fabrication and calibration and the transducers can disturb the acoustic ®eld. Ho¯er [12] designed and calibrated a driver apparatus for the accurate measurement of acoustic power delivered by the driver to the thermoacoustic resonator. The driver was a modi®ed loudspeaker whose voice coil was attached to a rigid moving piston. A small quartz-crystal dynamic pressure transducer was positioned on the face of the driver apparatus housing. A miniature accelerometer was attached to the back side of the piston; the velocity was obtained from the time-integrated accelerometer signal. The calibrated dynamic pressure signal, the calibrated volume velocity and the phase between them enabled the measurement of acoustic power delivered to the resonator. Recently, Yazaki and Tominaga [13] experimentally studied the spontaneous gas oscillation in a resonator from the standpoint of heat engines and presented the ®rst measurements of the work ¯ow emitted by the stack of a thermoacoustic driver. They used small pressure transducers ¯ush mounted on the resonator walls to obtain a measurement of acoustic pressure and laser Doppler anemometry to measure the velocity along the axis. The pressure and velocity signals were digitized and their power and phase spectra were calculated via a fast-Fourier-transform algorithm. The phase shift between particle velocity and acoustic pressure, was obtained from the phase spectra; it was further adjusted by taking into account delays caused by electrical circuits and by the measurement of pressure. For a given working frequency, axial distributions of pressure, core velocity and work ¯ow were extracted from these measurements, for the fundamental and for the second harmonic of the wave generated by the thermoacoustic prime mover. In respect of the fundamental frequency, experimental results agree with numerical simulations based on
4
H. Bailliet et al. / Applied Acoustics 60 (2000) 1±11
thermoacoustic theory [14,15]. The thermoacoustic engine has been loaded by a second stack to get a thermoacoustically driven thermoacoustic refrigerator. In this situation, the authors note that the presence of this second stack increases the work ¯ow along the resonator by approximately a factor of three. To sum up, extensive research in dierent ®elds has been devoted to acoustic power measurement, but the procedure is still complicated, especially in the case of a thermoacoustic resonator. This measurement involves precise measurement of the phase dierence ' between acoustic pressure and particle velocity, which is dicult to determine in a thermoacoustic resonator where the standing wave ratio is very high. This is due to the fact that working frequencies are resonance frequencies, around which cos ' varies rapidly, because ' is close to =2. In the following, a technique of calibration and measurement of particle velocity by laser Doppler anemometry together with classical acoustic pressure measurement is proposed, that yields a measurement of the acoustic power ¯ow at almost any position and for almost any working frequency in the resonator. Because the acoustic power measurement requirements in thermoacoustics do not restrict the general utility of developing such a measurement technique, it is expected that this discussion will be of general interest and useful wherever the measurement of the amount of acoustic power is needed; in particular, it can also be applied to the measurement of acoustic impedance. In Section 2, the acoustic power ¯ow is expressed as a function of both the acoustic pressure and the particle velocity component along the axis. The experimental set up, the calibration of measuring chains and results of measurements are presented in Section 3. The measured acoustic pressure, particle velocity and acoustic power ¯ow are compared with analytically calculated corresponding quantities. 2. Acoustic power ¯ow as a function of the acoustic pressure and the particle velocity along the axis As stated before, the important quantities that allow the characterization of a thermoacoustic refrigerator in terms of its eciency are the heat ¯ow extracted to the cold thermal source and the power ¯ow used to extract this heat ¯ow. The acoustic power ¯ow in the resonator is expressed as a function of the acoustic pressure p and the x-component of particle velocity ux , according to P Shg pu x i where S is the cross-section of the resonator (tilde and angular brackets indicate time and radial averages respectively). In the resonant thermoacoustic cavity, the usual thermoacoustic assumptions are used: quasi-plane waves, linear acoustic approximation, laminar ¯ow, no steady ¯ow, no heat source, no acoustic streaming. Then, the pressure can be assumed to be constant across the resonator section giving hpux i phux i. Therefore, the r-component of the particle velocity ur , can be neglected as compared to its x-component ux
H. Bailliet et al. / Applied Acoustics 60 (2000) 1±11
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because ur / @r p and ux / @x p (where @i @i@ for example). Moreover, the spatial variation of particle velocity in the x-direction can be neglected!as compared! to its ! ! spatial variation in r-direction, that is @x u @r u , because @@xu / 1l and @@ru / 1 , the viscous boundary layer thickness , being much smaller than the acoustic wavelength l l. In the case of simple harmonic sound at angular frequency !, the particle velocity in a cylindrical tube of radius R, which is the solution of the NavierStokes equation, is thus simply related to the acoustic pressure according to (e.g. [16]): i J0 k r 1ÿ @x p ; ux x; r !0 J0 k R where 0 is the mean density, J0 is a Bessel function of the ®rst kind and k 1ÿi . In particular, the particle velocity along the axis is given by i 1 1ÿ @x p ; ux x; 0 !0 J0 k R and the spatial average over the resonator section of the particle velocity takes the simple form hux xi
i 1 ÿ f @x p ; !0
R where f k2R JJ10 k k R. Finally, the total acoustic power ¯owing down the resonator is given by 3 1 0 2
P
S S B 6 R phux i RB p6ux x; 0 2 2 @ 4
C 1 ÿ f 7 7 C; 1 5 A 1ÿ J0 k R
1
where R denotes the real part and * denotes complex conjugation. Eq. (1) shows that the total acoustic power ¯ow can be obtained from a microphonic measurement of the acoustic pressure p and a measurement of the particle velocity along the axis ux x; 0. 3. Acoustic power ¯ow measurement The experimental apparatus used for acoustic power ¯ow measurement is illustrated schematically in Fig. 2. It consists of a very simpli®ed demonstration thermoacoustic refrigerator, composed of an electrodynamic loudspeaker loaded by a transparent cylindrical resonator ®lled with air at atmospheric pressure. The length of the resonator is l=49.7 cm and its radius is R=2.2 cm. A 6.5 cm length stack is set in this resonator. The center position of the stack is at xs=14 cm. This stack was
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H. Bailliet et al. / Applied Acoustics 60 (2000) 1±11
Fig. 2. Schematic representation of the experimental set up used for acoustic power ¯ow measurement.
fabricated from a long sheet of paper that was spirally wound [17]. Mono®lament 0.13 mm radius strings were glued to the sheet to provide spacing between adjacent layers of the spiral. In order to obtain both the pressure and the velocity measurements at the same position in the resonator, an adapted pressure transducer is used. It consists of a 2 mm diameter 3 cm long probe ®tted to a microphone (B&K 1/400 ) ¯ush mounted on the resonator wall. The sensitivity of the microphone is ®rst determined using a gauge source. Then, the eect of the probe is evaluated by comparison with a calibrated microphone using a coupler (B&K). When considering the probe tube as a transmission line and assuming that, in the frequency range of interest, the volume ¯ow at the front of the microphone is equal to zero, the pressure at the entrance of the coupler is related only to the pressure at the front of the microphone. It follows that the ratio of the pressures at the entrance and at the exit of the probe does not depend on the impedance presented to the probe by the environment external to the probe tube, so that the eect of the probe can be estimated by using a coupler. The particle velocity along the axis of the resonator is measured using laser Doppler anemometry. Two coherent laser beams are crossed and focused at the center of the tube to generate an ellipsoidal probe volume composed of equidistant dark and bright interference fringes. Seeding particles are dispersed in the ¯uid. The seeding we used is generated by a fog generator based on water condensation with aerosol; the mean diameter of seeding particles is about 1 mm. When a seeding particle passes through the measuring volume, light is scattered from its surface. The backward scattered light is detected by a photo-multiplier; the electronic signal obtained, called a burst, has a modulation frequency (Doppler frequency) proportional to the velocity of the seeding particle. In order to distinguish the velocity sign, the wavelength of one beam is shifted using a Bragg cell. Consequently, the Doppler frequency of the received signal is shifted up or down around the Bragg frequency according to the velocity direction [18]. Further signal processing is achieved by means of a Burst Spectrum Analyzer (BSA Dantec) [19] based on FFT analysis with interpolation. The fast-Fourier-transform of the signal is calculated and a peak detector permits estimation of the Doppler frequency, and then the particle velocity. Fig. 3 (upper
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Fig. 3. Example of result of measurement of particle velocity versus frequency. On the upper curve, each point (ti ; uxij ) corresponds to a particle crossing the probe volume at time instant ti (shifted over an acoustic period) with a velocity uxij . The lower curve is the average value uxi of velocity at time ti .
curve) shows an example of such results. Each time ti (shifted over an acoustic period) Pn corresponds to several associated velocities uxij . The average velocity uxi j1 uxij =n for each time ti is calculated, yielding the lower curve in Fig. 3, that is the velocity signal non uniformly sampled over an acoustic period T. The R.M.S. value r 2 u u xi x i1 ti1 ÿ ti and its phase relative to a reference of the velocity ux T1 i 2 signal, chosen to be the source signal applied to the loudspeaker, are ®nally calculated. Because we need a precise measurement of the phase dierence between acoustic pressure and particle velocity, a phase calibration of the two measuring chains has been carried out. It consists in a comparison between experimental and theoretical phase dierence in a simple resonating tube (Fig. 4). First, at a distance (l ÿ xm ) from the rigidly closed end of the tube, and for varying frequency, the acoustic pressure is measured by means of the probe microphone. The associated particle velocity is simultaneously measured by means of L.D.A.; this gives a measurement of the experimental reference phase dierence 'e xm ; f . Secondly, the corresponding theoretical phase dierence can be precisely calculated by using the well-known theory of acoustic propagation in simple lossy tubes [20]. For this purpose, one can use, for example, the expression of acoustic impedance Z xm ; f as a function of the ! speci®c admittance 1i 2 c0 ÿ 1h at x l:
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Fig. 4. Schematic representation of the experimental set up for phase shift calibration.
Z xm ; f j
p i'th aZ l b je ; ux cZ l a
where
Z Y h sin k l ÿ xm ; c sin k l ÿ xm ; a cos k l ÿ xm ; b k k
and Z l
0 c0 :
Here c0 is the adiabatic sound speed, is the speci®c heat ratio, h is the thermal boundary layer thickness, k is the complex wave number de®ned as r h i 1 ÿ1fh i!0 h R ! 1ÿi i!S , fh kh2R JJ10 k k c0 1ÿf kh R, kh h , Z S 1ÿf and Y h c2 1 ÿ1f . 0 0
h
The comparison between the theoretical 'th xm ; f and experimental 'e xm ; f phase dierences gives the calibration coecient for the set up: C' 'th xm ; f ÿ 'e xm ; f ÿ3; 36 7; 5:10ÿ3 f : Using this calibration coecient and Eq. (1), we can extract the acoustic power ¯ow P from the measurement of both the particle velocity along the axis and the acoustic pressure in the thermoacoustic resonator. Such results can moreover be compared with results of analytical calculation of the same quantities as presented in
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[2]. This analytical calculation gives the acoustic pressure, particle velocity and acoustic power ¯ow P in the thermoacoustic resonator for any working frequency (in the ®eld of classical thermoacoustic refrigeration) and at any position as functions of
Fig. 5. Results of measurement () and analytical results (solid line) of the particle velocity (modulus of the velocity averaged across a section in m/s), acoustic pressure (modulus in Pa), phase dierence ' between them (in rad), and acoustic power ¯ow (in W) versus frequency at x=4 cm in a thermoacoustic resonator.
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the voltage applied to the sound source (loudspeaker) only. As an example, Fig. 5 shows the calculated and measured particle velocity modulus (averaged over the section), acoustic pressure modulus, phase dierence between them, and acoustic power ¯ow versus working frequency at x=4 cm (see Fig. 2). The discrepancies between the calculated acoustic power ¯ow and the results of measurement depend on the precision of both the amplitudes of the acoustic quantities (pressure and particle velocity) and the phase dierence between them. The two maxima for P correspond to possible working frequencies for thermoacoustics. The ®rst maximum corresponds to the quarter-wavelength mode and the second maximum to the halfwavelength one. The agreement between analytical and experimental results is good, even at resonance frequencies around which cos ' varies very rapidly. This kind of measurement is the ®rst step in designing a thermoacoustic refrigerator because it allows the experimental determination of the working frequencies (which are frequencies of the maxima for P). The next step consists in measuring the acoustic power ¯ow along the resonator at the chosen working frequency. Yazaki and Tominaga [13] recently reported such measurements but in the case of thermoacoustic prime mover and using a calibration procedure dierent from the one proposed here. Measurements of the acoustic power ¯ow along the resonator can be achieved following the calibration and experimental procedures presented above. They are in progress at present and we hope to obtain the acoustic power ¯ow along the resonator with a very good spatial resolution. 4. Conclusion Calibration and measurements of particle velocity by laser Doppler anemometry together with classical acoustic pressure measurement have been presented; they yield a measurement of the acoustic power ¯ow in a thermoacoustic resonator. Experimental results appear to be in good agreement with results of analytical calculation. They permit the experimental determination of the possible working frequencies of thermoacoustic refrigerator, which are the frequencies of the maximum of acoustic power ¯ow at the entrance of the resonator. With this technique it is also possible to measure acoustic power ¯ow along the resonator with a very good spatial resolution. The technique presented here is expected to be useful for other applications, wherever the measurement of the amount of acoustic power is needed. Nevertheless, it still needs re®nements before it becomes extensively useful for all kind of eligible applications. In particular, it is necessary to evaluate the accuracy of particle velocity measurement in order to evaluate the accuracy of acoustic power measurement. For our experimental set up, an error smaller than 8% is guaranteed down to particle displacement amplitudes of 0.04 mm [21]. Studies are being carried out at present to estimate all the possible sources of experimental errors when using L.D.A. technique, and signal processing tools are being developed to access larger dynamic and frequency range. Progress in these matters should help in developing a precise and widely useful technique of acoustic power ¯ow measurement.
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References [1] Swift GW. Thermoacoustic engines and refrigerators. Physics Today 1995;48(7):22±8. [2] Bailliet H, Lotton P, Bruneau M, Gusev V. Coupling between loudspeakers and thermoacoustic cavities. Acta acoustica, submitted 1998. [3] Beranek LL. Acoustic measurements. New York: Wiley, 1949. [4] Clapp CW, Firestone FA. An acoustic wattmeter, an instrument for measuring energy ¯ow. J Acoust Soc Am 1941;28(2):124±36. [5] Chu WT. Extension of the two-microphone transfer function method for impedance tube measurements. J Acoust Soc Am 1986;80:347. [6] Fusco AM, Ward WC, Swift GW. Two-sensor power measurements in lossy ducts. J Acoust Soc Am 1992;91:2229±35. [7] Fahy FJ. Measurement of acoustic intensity using the cross-spectral density of two microphone signals. J Acoust Soc Am 1977;62:1057. [8] Seybert AF, Soenarko B. Error analysis of spectra estimates with application to the measurement of acoustic parameters using random sound ®elds in ducts. J Acoust Soc Am 1981;69:1190. [9] Pascal JC, Carles C. Systematic measurement errors with two microphone sound intensity meters. J Sound Vib 1982;83:53±65. [10] Fahy FJ. Sound intensity. 2nd edition. London: Elsevier Applied Science, 1995. [11] van Zyland BG, Anderson F. Evaluation of the intensity method of sound power determination. J Acoust Soc Am 1975;57:682±6. [12] Ho¯er TJ. Accurate acoustic power measurements with a high-intensity driver. J Acoust Soc Am 1988;83(2):777±86. [13] Yazaki T, Tominaga A. Measurement of sound generation in thermoacoustic oscillations. Proc R Soc Lond A 1998;454:2113±22. [14] Tominanga A. Thermodynamic aspects of thermoacoustic theory. Cryogenics 1995;35:427±40. [15] Swift GW. Thermoacoustic engines. J Acoust Soc Am 1988;84(4):1145±80. [16] Arnott WP, Bass HE, Raspet R. General formulation of thermoacoustics for stacks having arbitrarily shaped pore cross sections. J Acoust Soc Am 1991;90(6):3228±37. [17] Ho¯er TJ. Thermoacoutic refrigerator design and performance. Ph.D. thesis. San Diego: University of California, 1986. [18] ValieÁre JC, Herzog P, Valeau V, Tournois G. Acoustic velocity measurements in the air by means of laser Doppler velocimetry: dynamic and frequency range limitations and signal processing improvements. J Sound Vib, in press. [19] Scholten J. Developments in laser Doppler anemometry applied to acoustics. User's guide 57n10/ 57n14/57n25 Burst Spectrum Analyser, report of Post-Doctoral research, Laboratoire d'Acoustique de l'Universite du Maine, DeÂcembre 1997. [20] Bruneau M. Manuel d'Acoustique fondamentale. Paris: HermeÂs, 1998. [21] Valeau V. Mesure de la vitesse particulaire acoustique par aneÂmomeÂtrie laser Doppler: estimation de freÂquence instantaneÂe aÁ variation sinusoõÈ dale, validation de la mesure. Ph.D. thesis, Universite du Maine, 1999.