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Prandtl number and thermoacoustic refrigerators
M. E. H. Tijani, J. C. H. Zeegers, and A. T. A. M. de Waele
Department of Applied Physics, Eindhoven University of Technology, P.O. Box 513, 5600 MB Eindhoven,
The Netherlands
͑Received 28 November 2001; revised 25 April 2002; accepted 4 May 2002͒
From kinetic gas theory, it is known that the Prandtl number for hard-sphere monatomic gases is 2/3.
Lower values can be realized using gas mixtures of heavy and light monatomic gases.

Transcript

Prandtl number and thermoacoustic refrigerators
M. E. H. Tijani, J. C. H. Zeegers, and A. T. A. M. de Waele
Department of Applied Physics, Eindhoven University of Technology, P.O. Box 513, 5600 MB Eindhoven,The Netherlands
Received 28 November 2001; revised 25 April 2002; accepted 4 May 2002
From kinetic gas theory, it is known that the Prandtl number for hard-sphere monatomic gases is 2/3.Lower values can be realized using gas mixtures of heavy and light monatomic gases. Prandtlnumbers varying between 0.2 and 0.67 are obtained by using gas mixtures of helium–argon,helium–krypton, and helium–xenon. This paper presents the results of an experimentalinvestigation into the effect of Prandtl number on the performance of a thermoacoustic refrigeratorusing gas mixtures. The measurements show that the performance of the refrigerator improves as thePrandtl number decreases. The lowest Prandtl number of 0.2, obtained with a mixture containing30% xenon, leads to a coefﬁcient of performance relative to Carnot which is 70% higher than withpure helium. ©
2002 Acoustical Society of America.
DOI: 10.1121/1.1489451
PACS numbers: 43.35.Ud, 43.35.Ty
RR
I. INTRODUCTION
The basic understanding of the physical principles un-derlying the thermoacoustic effect is well established and hasbeen discussed in many papers.
1,2
However, a quantitativeexperimental investigation of the effect of some importantparameters on the behavior of the thermoacoustic devices isstill lacking. One of these parameters is the Prandtl number,a dimensionless parameter characterizing the ratio of kine-matic viscosity to thermal diffusivity. Viscous friction has anegative effect on the performance of thermoacousticsystems.
2,3
Decreasing the Prandtl number generally in-creases the performance of thermoacoustic devices. Kineticgas theory has shown that the Prandtl number for hard-sphere monatomic gases is 2/3. Lower Prandtl numbers canbe realized using mixtures of heavy and light monatomicgases, for example, binary gas mixtures of helium and othernoble gases.
3–5
The calculations of the Prandtl number shown in thispaper use more rigorous expressions for the transport coefﬁ-cients than those used previously.
4,5
For all binary gas mix-tures of helium with other noble gases, calculations showthat the coefﬁcient of performance of the refrigerator ismaximized when the Prandtl number is near its minimumvalue.This paper presents the results of an experimental inves-tigation into the effect of Prandtl number on the performanceof a thermoacoustic refrigerator using gas mixtures of heliumwith argon, krypton, and xenon. These combinations pro-vided gas mixtures with Prandtl numbers varying between0.2 and 0.67. These results are discussed below.The thermoacoustic refrigerator used for the measure-ments is shown in Fig. 1. The acoustic resonator is ﬁlled withan inert gas at a pressure of 10 bar. A channeled stack isstrategically located in the resonator to facilitate heat trans-fer. At both ends of the stack heat exchangers are installed.The temperature of the hot heat exchanger is ﬁxed at roomtemperature by circulating water. At the cold heat exchangercooling power is generated. A loudspeaker generates a stand-ing wave in the resonance tube, causing the gas to oscillatewhile compressing and expanding. The interaction of themoving gas in the stack with the stack surface generates heattransport.
2
A detailed description of the refrigerator can befound in the literature.
3,6
II. KINETIC THEORY OF BINARY GAS MIXTURES
The Prandtl number is given by
c
p
K
,
1
where
is the dynamic viscosity,
K
is thethermal conductiv-ity, and
c
p
is the isobaric speciﬁc heat. The Prandtl numbercan also be written in terms of the thermal and viscous pen-etration depths
k
and
as
k
2
,
2
where
k
and
are given by
k
2
K
c
p
,
3
and
2
.
4
Here,
is the angular frequency of the sound wave and
isthe density.The viscosity has a negative effect on the performanceof thermoacoustic devices, so a reduction of the effect of viscosity means an increase in efﬁciency. This can be accom-plished by lowering the Prandtl number. In the Appendix asurvey of the kinematic theory for pure and binary gas mix-tures is given. The text of Hirschfelder
et al.
7
forms the prin-cipal source for this survey. By substituting the expressionsof viscosity
mix
, thermal conductivity
K
mix
, and the iso-baric speciﬁc heat
c
p
for binary gas mixtures into Eq.
1
, thePrandtl number of binary gas mixtures can be calculated. The
134 J. Acoust. Soc. Am.
112
(1), July 2002 0001-4966/2002/112(1)/134/10/$19.00 © 2002 Acoustical Society of America
resultant expression is too long to be given explicitly here.This was incorporated in a computer program to accomplishthe calculations.
III. CALCULATION OF GAS MIXTURES PROPERTIES
In Fig. 2, some calculated properties of binary gas mix-tures, consisting of He–Ne, He–Ar, He–Kr, and He–Xe, areplotted as functions of mole fraction,
x
, of the heavy compo-nent. The temperature and pressure used in the calculationsare 250 K and 10 bar, respectively. These values apply to theexperimental situation. Figure 2
a
shows that, for all mix-tures, the density
increases linearly as a function of
x
, inaccordance with Eq.
A14
. From Figs. 2
b
and
c
, it can beseen that the kinematic viscosity
/
and thermal diffu-sivity
k
K
/
c
p
decrease as a function of
x
. This behavior isto be expected, since both properties are approximately in-versely proportional to the square of the apparent mass of themixture. But, in the range 0
x
0.4 the rate of decrease for
is larger than for
k
. This results in a decrease of the ratio of these two quantities which is the Prandtl number
. For
x
0.4,
k
still decreases while
remains nearly constant; thisresults in an increase of
until the value for a pure noble gasis reached. From this behavior of the Prandtl number de-creasing and then increasing as function of the increase of the mole fraction of the heavier component, a minimumvalue is to be expected as shown in Fig. 2
f
. Similar calcu-lation results are obtained by Giacobbe
4
and Belcher
et al.
5
using approximative expressions for the viscosity and ther-mal conductivity.The behavior of the sound velocity
a
as a function of
x
isillustrated in Fig. 2
d
. It is also a decreasing function of
x
, inaccordance with Eq.
A16
, since the apparent mass of themixture increases. The behavior of
c
p
is shown in Fig. 2
e
.As can be seen from Fig. 2
f
, the Prandtl number has aminimum at
x
0.38 for all binary gas mixtures. The valueof
at the minimum is a function of the molecular weight of the added heavier component. The lowest value of 0.2 isreached with the heaviest noble gas xenon. The minimumPrandtl number is plotted in Fig. 3 as a function of the molarmass of the heavy component. Extrapolation of the data forradon shows that using a helium–radon mixture, a Prandtlnumber of about 0.1 can be reached which can be consideredas the lowest Prandtl number for helium–noble gases mix-tures. The Prandtl number is also calculated for different gasmixtures, at different temperatures, as shown in Fig. 4. Theinﬂuence of the temperature on the Prandtl number is small.The effect of temperature is most pronounced for He–Xemixture. At the helium mole fraction corresponding to theminimum Prandtl number, a change in temperature of 100 Kresults in a relative change in the value of the Prandtl of only6.5% for a helium–xenon mixture.Figures 4
a
and
b
show the thermal and viscous pen-etration depths
k
and
, respectively. The behavior of
as a function of
x
can be illustrated by means of Eq.
4
.Since
varies approximately as 1/
M
, it follows that
willvary as 1/
4
M
, thus a decreasing behavior as shown in Fig.4
b
. A similar analysis can be done to explain the behaviorof
k
. It is interesting to note that
k
has a maximum for abinary mixture at about
x
0.1, as shown in Fig. 4
a
. Wenote that for the calculations of
k
and
, the wavelength iskept constant (
2.35 m) and the frequency is allowed tovary as the sound velocity varies with the composition.The product
a
is calculated for a temperature of 250 Kand at a pressure of 10 bar, and it is plotted in Fig. 4
c
. Thebehavior can be understood as follows: the density increaseslinearly and
a
decreases as 1/
M
; as a result the product
a
increases as
x
increases, as
M
. The product
a
will beneeded later for discussions concerning the cooling power of the thermoacoustic refrigerator.
IV. PERFORMANCE CALCULATIONS
The purpose of a refrigerator is to remove a heat (
Q˙
C
) ata low temperature (
T
C
) and to reject heat (
Q˙
H
) to the sur-roundings at a high temperature (
T
H
). To accomplish this,work
W
is required. The coefﬁcient of performance
COP
is deﬁned asCOP
Q˙
C
W
.
5
The quantityCOPC
T
C
T
H
T
C
,
6
FIG. 1. Schematic diagram of the thermoacoustic refrigerator, showing thedifferent parts.
135J. Acoust. Soc. Am., Vol. 112, No. 1, July 2002 Tijani
et al.
: Prandtl number and thermoacoustic refrigerators
is called the Carnot coefﬁcient of performance which deﬁnesthe optimal performance for all refrigerators. The coefﬁcientof performance relative to Carnot’s coefﬁcient of perfor-mance is deﬁned asCOPR
COPCOPC.
7
The performance measurements for the refrigerator are pre-sented in plots of COP, COPR, and
T
given by
T
T
C
T
H
,
8
as functions of the total heat load, which is the sum of theheat load applied by the heater and the heat leak. To under-stand the behavior of the cooler as a function of the variedparameters, the measured
T
data will be ﬁt with the equa-tion
T
T
0
Q˙
,
9
where
Q˙
is the total heat load.
3
During a given performancemeasurement, the drive ratio, deﬁned as the ratio of the dy-namic pressure to the average pressure, is kept constant
FIG. 2. Calculated properties of binary gas mixtures ata temperature of 250 K and a pressure of 10 bar.
a
Density.
b
Kinematic viscosity
.
c
Thermal diffu-sivity
k
.
d
Speed of sound
a
.
e
The isobaric speciﬁcheat
c
p
.
f
Prandtl number
.FIG. 3. Minimal Prandtl number, at a helium mole fraction of 0.62, as afunction of the molecular weight of the heavy component. The ﬁrst pointcorresponds to the Prandtl number of pure helium. The last point corre-sponds to the minimal Prandtl number of a helium–radon binary gas mix-ture obtained by extrapolation.
136 J. Acoust. Soc. Am., Vol. 112, No. 1, July 2002 Tijani
et al.
: Prandtl number and thermoacoustic refrigerators
while stepwise the heat load is applied to the cold heat ex-changer and the temperature is allowed to stabilize. The heatload is applied by an electric heater placed at the cold heatexchanger. Two thermometers are used to monitor the tem-peratures at the hot heat exchanger and at the cold heat ex-changer. Incorporating Eqs.
A5
and
A10
into the expres-sions for the energy ﬂow and work ﬂow in the stack,
8
theCOP has been calculated as function of
x
. Figure 5 shows thebehavior of COP for helium–xenon, helium–krypton, andhelium–argon mixtures. For all binary gas mixtures, as thePrandtl number decreases the performance improves. Amaximum is reached nearly at the point where the Prandtlnumber is a minimum
cf. Fig. 2
.
V. MIXTURES PREPARATION
A container is used to prepare the gas mixtures. The lowmole fraction component is ﬁrst ﬁlled in the container untilthe pressure fraction needed is reached. Then, the secondcomponent is ﬁlled up to 33 bar. The mixture is allowed toreach equilibrium prior to ﬁlling the refrigerator to a pressureof 10 bar. The composition of the mixture was checked bymeasuring the resonance frequency of the refrigerator. Thisfrequency is then related to the resonance frequency of thesystem for pure helium by the expression
f
He
f
mix
xM
2
1
x
M
He
M
He
,
10
where
f
He
, and
f
mix
are the resonance frequencies for purehelium and mixture, respectively.
M
2
and
x
are the molarmass and mole fraction of the heavy component, and
M
He
isthe molar mass of helium.
VI. MEASUREMENTS
With our thermoacoustic refrigerator
3,6
three differentmixtures have been investigated: He–Ar, He–Kr, and He–Xe. The performance measurements were all made using a
FIG. 4. The thermal and viscous penetration depths,
k
and
, the product
a
, and Prandtl number
forhelium–other noble gas binary mixtures. We note that,for the calculations
k
and
, the wavelength is keptconstant (
2.35 m) and the frequency is allowed tovary as the sound velocity varies with
x
. The tempera-ture and pressure used in the calculations are alsoshown in the graphs. The Prandtl number has been cal-culated at two different temperatures.FIG. 5. Calculated COP as function of
x
.
137J. Acoust. Soc. Am., Vol. 112, No. 1, July 2002 Tijani
et al.
: Prandtl number and thermoacoustic refrigerators

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