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					                                   Applied Acoustics 60 (2000) 1±11
                                                                      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. Valiere a, B. Gazengel a
                                 Á
             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 ana-
lytical 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 e€ects that occur in the thermal
boundary layers associated with an acoustically oscillating ¯uid close to a rigid wall.
These e€ects 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: lotton@laum.univ.lemans.fr (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 mea-
surement is thus very important for determining the intrinsic eciency of a thermo-
acoustic refrigerator.
  The literature on acoustic power and acoustic impedance measurement is exten-
sive. 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 con-
siderable 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
di€erence between them to obtain the velocity. The high standing wave ratios
encountered in thermoacoustic resonators make this method dicult to use because
of the extreme phase accuracy and stability required for measurements on resonat-
ing 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. How-
ever, 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 measure-
ment 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 accel-
erometer 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 fre-
quency, 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 funda-
mental 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 di€erent ®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 di€erence 9 between acoustic pressure and particle velocity, which is dicult
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 9 varies rapidly, because 9 is close to %a2.
   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 eciency 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

    € ˆ Shg i
          pu x

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 ˆ phux i. Therefore, the r-compo-
nent 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            5

                                                   d
because ur G dr p and ux G dx p (where di ˆ di for example). Moreover, the spatial
variation of particle velocity in the x-direction can be neglected3as compared to its
                                                3        3                            3
                                                                        u             u
spatial variation in r-direction, that is dx u ( dr u , because ddx G 1 and ddr G 1# ,
                                                                             l
the viscous boundary layer thickness # , being much smaller than the acoustic
wavelength l …# ( l†. In the case of simple harmonic sound at angular frequency 3,
the particle velocity in a cylindrical tube of radius R, which is the solution of the Navier-
Stokes equation, is thus simply related to the acoustic pressure according to (e.g. [16]):
                                      
                  i          J0 …k# r†
     ux …xY r† ˆ       1À               dx p Y
                 3&0        J0 …k# R†

where &0 is the mean density, J0 is a Bessel function of the ®rst kind and k# ˆ 1Ài . In
                                                                                 #
particular, the particle velocity along the axis is given by
                                    
                  i            1
     ux …xY 0† ˆ       1À             dx p Y
                 3&0       J0 …k# R†

and the spatial average over the resonator section of the particle velocity takes the
simple form

                   i
     hux …x†i ˆ       …1 À f# †dx p Y
                  3&0

                J
where f# ˆ k#2R J1 …k# R†. Finally, the total acoustic power ¯owing down the resonator is
                  0 …k# R†
given by
                                 H P                      QÃ I
          S              S f T                      1 À f# U g
     €ˆ     R…phux ià † ˆ RfpTux …xY 0†                        U gY                      …I†
          2              2 d R                           1 S e
                                                  1À
                                                     J0 …k# R†

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 …xY 0†.


3. Acoustic power ¯ow measurement

  The experimental apparatus used for acoustic power ¯ow measurement is illu-
strated schematically in Fig. 2. It consists of a very simpli®ed demonstration ther-
moacoustic 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
6                        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 Â3 cm long probe ®tted to a microphone (B&K 1/4HH ) ¯ush mounted
on the resonator wall. The sensitivity of the microphone is ®rst determined using a
gauge source. Then, the e€ect of the probe is evaluated by comparison with a cali-
brated 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 e€ect of the probe can be estimated by using a coupler.
   The particle velocity along the axis of the resonator is measured using laser Dop-
pler 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 velo-
city 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                        7




Fig. 3. Example of result of measurement of particle velocity versus frequency. On the upper curve, each
point (ti Y 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 per-
iod) corresponds to several associated velocities uxij . The average velocity uxi ˆ
€n
  jˆ1 uxij an 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
                          1         u ‡u
of the velocity ux ˆ T Æi xi 2x…i‡1† …ti‡1 À ti † and its phase relative to a reference
signal, chosen to be the source signal applied to the loudspeaker, are ®nally calcu-
lated.
  Because we need a precise measurement of the phase di€erence 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 di€erence in a simple resonating tube (Fig. 4). First, at a distance (l À 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 di€erence 9e …xm Y f †. Secondly, the correspond-
ing theoretical phase di€erence 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 Y f † as a function of the
                            3
speci®c admittance  ˆ 1‡i c0 … À 1†h at x ˆ l:
                         2
8                          H. Bailliet et al. / Applied Acoustics 60 (2000) 1±11




         Fig. 4. Schematic representation of the experimental set up for phase shift calibration.




                      p i9th aZ…l† ‡ b
      Z…xm Y f† ˆ j      je ˆ           Y
                      ux      cZ…l† ‡ a
where
             È          É      "
                               Z#    È          É      "
                                                       Yh    È          É
      a ˆ ™os k…l À xm † Y b ˆ    sin k…l À xm † Y c ˆ    sin k…l À xm † Y
                               k                       k

and
               & 0 c0
      Z…l† ˆ          X
                 

  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
    3
k ˆ c0    1‡…À1†fh
                             , fh ˆ kh R J1 …kh R†, kh ˆ 1Ài, Z# ˆ S…1Àf# † and Yh ˆ & c2 ‰1‡…À1†f Š.
                                     2 J                      "     i3&0        "           i3S
               1Àf#                       0 …kh R†        h                        0 0         h


  The comparison between the theoretical 9th …xm Y f † and experimental 9e …xm Y f †
phase di€erences gives the calibration coecient for the set up:

      C9 ˆ 9th …xm Y f † À 9e …xm Y f † ˆ À3Y 36 ‡ …7Y 5X10À3 f †X


  Using this calibration coecient and Eq. (1), we can extract the acoustic power
¯ow € 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                            9


[2]. This analytical calculation gives the acoustic pressure, particle velocity and
acoustic power ¯ow € 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 (Â) 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 di€erence 9 between
them (in rad), and acoustic power ¯ow (in W) versus frequency at x=4 cm in a thermoacoustic resonator.
10                    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 di€erence 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 di€erence between them. The two maxima for €
correspond to possible working frequencies for thermoacoustics. The ®rst maximum
corresponds to the quarter-wavelength mode and the second maximum to the half-
wavelength one. The agreement between analytical and experimental results is good,
even at resonance frequencies around which cos 9 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 fre-
quencies of the maxima for €). 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 thermo-
acoustic prime mover and using a calibration procedure di€erent from the one pro-
posed 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 cal-
culation. They permit the experimental determination of the possible working fre-
quencies 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 velo-
city 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.
                           H. Bailliet et al. / Applied Acoustics 60 (2000) 1±11                       11


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