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Thesis and Dissertation Collection
1989-12
Investigation of a heat driven thermoacoustic
prime mover.
Lin, Hsiao-Tseng
Monterey, California. Naval Postgraduate School
http://hdl.handle.net/10945/26020
»
NAVAL POSTGRADUATE SCHOOL
Monterey
,
California
THESIS
Investigation of a Heat Driven
The rmoacoustic
Prime Mover
by
Lin, Hsiao-Tseng
* *
December 1989
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Co-ad visor
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T. J. Hofler
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INVESTIGATION OF A HEAT DRIVEN THERMOACOUST1C PRIME
MOVER
1
Personal Author(s)
2
13a Type of Report
Master's Thesis
Lin, Hsiao-Tseng
13b Time Covered
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To
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1
December 1989
75
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Supplementary Notation The views expressed in this thesis are those of the author and
policy or position of the Department of Defense or the U.S. Government.
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Acoustics, Thermoacoustics, Thermoacoustic Heat Transport
Subgroup
Abstract (continue on reverse
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work output of a heat driven thermoacoustic prime mover. The
experimental approach was to measure the frequency response of both a simple resonant tube and a prime mover
for a variety of values of mean gas pressure and applied temperature difference across the prime mover stack. A
least squares fit to the frequency response yields the quality factor which can be compared to predictions based on
a short stack, boundary layer approximation theory by Swift [J. Acoust. Soc. Am. 84, 1145-1180 (1988)]. The
results are reported of measurements made on the lowest three modes of the prime mover in helium for mean gas
pressures between approximately 170 kPa and 500 kPa and the applied temperature differences between zero and
onset. The signal waveforms of the sound generated by the prime mover above onset at a mean gas pressure of
308 kPa are also reported. Results for the resonant tube have at most 3% difference with theory. For the prime
mover, the measurements generally agree with predictions for the fundamental mode except close to onset. This
agreement between measured and predicted results worsens with decreasing mean gas pressure. Agreement is
poor for the second and third modes for all pressures used. Finally, the sound generated by the prime mover
above onset is highly distorted, and the distortion becomes more serve as the temperature difference increases.
The peak positive pressure amplitude of this signal at temperature difference of 325 °C, 368 °C and 453 °C are
1.1%, 4.4% and 7.9% of mean gas pressure, respectively.
The goal of
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Investigation of a Heat Driven Thermoacoustic
Prime Mover
by
Lin, Hsiao-Tseng
Captain, Taiwan
B.S.,
Chung Cheng
Submitted
Institute of
Army
Technology
in
Taiwan, 1984
of the requirements
for the degree of
in partial fulfillment
MASTER OF SCIENCE
IN
ENGINEERING ACOUSTICS
from the
NAVAL POSTGRADUATE SCHOOL
December 1989
ABSTRACT
The goal of
this thesis is to investigate the
thermoacoustic prime mover.
work output of
The experimental approach was
a heat driven
to
measure the
frequency response of both a simple resonant tube and a prime mover for a
variety of values of
the prime
mover
quality factor
mean gas
stack.
A
pressure and applied temperature difference across
least squares
which can be compared
to the
fit
to predictions
boundary layer approximation theory by Swift
1180 (1988)].
three
modes of
The
results are reported of
the prime
mover
frequency response yields the
[J.
based on a short stack,
Acoust. Soc.
Am.
84, 1145-
measurements made on the lowest
helium for mean gas pressures between
in
approximately 170 kPa and 500 kPa and the applied temperature differences
between zero and onset.
prime mover above onset
The
at a
signal
waveforms of the sound generated by
mean gas pressure of 308 kPa
Results for the resonant tube have
at
most
3%
the
are also reported.
For the
difference with theory.
prime mover, the measurements generally agree with predictions for the
fundamental
mode
except close to onset. This agreement between measured and
predicted results worsens with decreasing
for the
second and third modes for
mean
all
generated by the prime mover above onset
becomes more serve
gas pressure.
pressures used.
is
Agreement
is
poor
Finally, the sound
highly distorted, and the distortion
as the temperature difference increases.
The peak
positive
pressure amplitude of this signal at temperature difference of 325 °C, 368 °C and
453 °C are 1.1%, 4.4% and 7.9% of mean gas pressure, respectively.
in
CI
TABLE OF CONTENTS
I.
INTRODUCTION
1
II.
THEORY
4
III.
EXPERIMENT APPARATUS AND PROCEDURE
A.
B.
C.
D.
IV.
V.
THE RESONANT TUBE
13
13
1.
The Tube
13
2.
Driver and Microphone
15
THE HEAT DRIVEN PRIME MOVER
15
1.
The Heater Section and Hot Heat Exchanger
16
2.
The Ambient Heat Exchanger
16
3.
The Prime Mover Stack
16
TEMPERATURE CONTROL EQUIPMENT
INSTRUMENTATION AND PROCEDURE
19
21
1.
The Empty Resonant Tube
21
2.
The Prime Mover below Onset
23
3.
The Prime Mover above Onset
23
RESULTS AND DISCUSSIONS
A. THE EMPTY RESONANT TUBE
B. THE PRIME MOVER BELOW ONSET
C. THE PRIME MOVER ABOVE ONSET
27
SUMMARY, CONCLUSIONS AND RECOMMENDATIONS
58
27
28
31
A.
SUMMARY
58
B.
CONCLUSIONS
59
C.
RECOMMENDATIONS
59
IV
APPENDIX A. PARTIAL LISTING OF THE PHYSICAL
PROPERTIES OF HELIUM
60
APPENDIX B. LISTING OF THE PHYSICAL PROPERTIES
OF MATERIALS FOR THE PRIME MOVER STACK AND
HEAT EXCHANGERS
APPENDIX
C.
61
LISTING OF THE SPECIFICATIONS OF THE
COMPOMENTS AND GEOMETRICAL PARAMETERS OF
THE PRIME MOVER
LIST OF REFERENCES
INITIAL
DISTRIBUTION
62
63
LIST
64
LIST
OF SYMBOLS
c
sound speed
Cp
isobaric heat capacity per unit
E
dissipated acoustic
power
e
dissipated acoustic
power per
f
frequency
fo
measured resonance frequency
k
complex wave number, k = k
L
resonator length
/
plate half-thickness
Pa
peak acoustic pressure amplitude
P
complex acoustic pressure
Q
quality factor
Re
real part of
S
cross section area
Sd
surface area of driver
T
temperature
UD
volume velocity of driver
Vd
volume displaced by driver
w
generated acoustic power
X
position along tube axis
mass
unit area
a
- i
VI
LIST OF
SYMBOLS (CONTINUED)
y
plate half-gap
Z mo
complex mechanical impedance,
a
absorption coefficient
P
thermal expansion coefficient
Ax
plate length
5S
thermal penetration depth in the solid plate
8K
thermal penetration depth in the gas
&U
viscous penetration depth in the gas
£s
plate heat capacity ratio
r
normalized temperature gradient
y
ratio of isobaric to isochoric specific heats
n
perimeter
pm
mean gas
ps
density of solid
a
Prandtl
£d
displacement of driver
Z mo = R mo + Zm
density
number
vu
i
ACKNOWLEDGMENTS
First
and foremost,
I
would
like to thank the
so much, not just about acoustics, but
Lord who allowed
me
more importantly, about how
to learn
to believe
and count on Him.
I
want
to
express
my
sincere appreciation to
my
advisors, Dr.
Anthony
Atchley and Dr. Thomas Hofler for their encouragement, words of wisdom and
infinite patience.
Next
I
thank George Jaksha and Steven Blankschein of the physics machine
shop for their help
David Pierce and
to help
me
in
my
in writing
Finally,
I
conducting
my
experiment.
sponsor Michael Weisskopf
and typing
I
who
also thank
classmate
spent a great deal of time
skills.
reserve special thanks for the support from
though they are far away overseas.
May
my
the
Lord bless them
vni
all.
my
family, even
INTRODUCTION
I.
There are two basic classes of thermoacoustic engines: heat pumps and prime
movers
[Refs. 1-3].
standing
wave
A
to transport (or
pump)
uses a high amplitude acoustic
heat along, for example, the boundary of a
This acoustically generated heat flow results
plate situated in the standing wave.
in a
pump
thermoacoustic heat
thermal gradient being established across the plate. In other words, acoustic
energy
is
converted into stored thermal energy, which in turn can be used in a
number of
practical applications.
Thermoacoustic prime movers, on the other
hand, convert stored thermal energy into useful work in the form of sound.
Such a device
is
the subject of this thesis.
Referring to Figure
1,
a typical
stack of plates, called the prime
prime mover configuration consists of
mover
stack (or, simply, the stack),
which
a
is in
thermal contact with two heat exchangers. In this thesis, the two heat exchangers
will be called the hot
mover
held
at
and ambient heat exchangers, because one end of the prime
stack will be held at elevated temperatures while the other end will be
ambient (room) temperature.
However, the important quantity
temperature difference across the prime mover stack, not the
temperatures of either end.
Hence,
if
(what
held at ambient temperature, and (what
we
we
called) the hot end
is
the
absolute
were
to
be
called) the ambient end held at
temperatures below ambient, the process described in this thesis would
fundamentally be the same.
An example
of an ambient/cold prime mover
so called, "Hofler tube", described elsewhere [Refs. 3-4].
Figure
1,
the prime
mover
As
is the,
also indicated in
stack and heat exchangers are housed within an
The prime mover
acoustic resonator.
stack/heat exchanger/resonator assembly
is
called a (thermoacoustic) prime mover.
Thermal energy
stored in the prime
is
mover by imposing
a temperature
difference across the stack, the two quantities being proportional.
the prime
mover
produce net positive work,
to
amount of stored energy converted
to
i.e.
to
In order for
produce audible sound, the
sound must exceed the amount of acoustic
energy dissipated by losses in the prime mover. The dominant loss mechanism
and frequencies of interest here are thermal and viscous
for the type of gases
exchanger surfaces.
losses at the resonator walls and the stack and heat
prime mover
is
said to have reached "onset"
across the stack
is
sufficient for the
detectable levels of sound.
imply
that
The use of
when
the temperature difference
prime mover
the
to
generate and sustain
word "detectable"
that onset is a subtle question of detection thresholds.
when
onset
is
is
not meant to
Our experience
reached, the observer (and everyone else in the room)
it.
The Ambient Heat Exchanger
The Hot Heat Exchanger
The Ambient End
The Prime Mover Stack
Figure
1
-
A
typical
The
The Hot End
prime mover configuration
is
knows
The
investigation of a prime
below and above onset.
approach
will be
to
mover
is
conveniently divided into two parts:
The main goal of
this
paper
onset of a thermoacoustic prime mover.
made of
above onset.
the behavior
number of added complications above
As
investigate the
However,
brief mention
will be discussed, there are a
onset.
work output of
In order to investigate the
is to
made
a prime mover, use will be
of the fact that the temperature difference across the stack affects the net
absorption in the prime mover.
zero to
its
value
at onset, the
in turn,
the temperature difference increases
net absorption decreases to zero.
coefficient can be determined
mover, which,
As
by measuring the quality factor
The absorption
Q
of the prime
can be related to the acoustic power dissipated
The power can be calculated using equations derived by Swift
from
in the tube.
[Ref. 1],
which
express the acoustic power output of a thermoacoustic engine as a function of the
applied temperature difference across the stack.
Swift presents a detailed theoretical development in his review article.
However,
as he shows, the theory results in relatively simple expressions if the
short stack and
boundary layer approximation are made.
The
theoretical
predictions presented in this thesis are based on this approximate theory, in order
to test the
ranges of
its
validity.
THEORY
II.
The measurements performed
in the
is
The dependence of
power
the net acoustic
The relationship between these two
its
Q
as a function of
However, the quantity most conveniently
the applied temperature difference.
computed from theory
prime mover yield
quantities
dissipated by the prime mover.
derived in this chapter.
is
the acoustic pressure with distance for a plane
L and
closed, rigid, cylindrical tube of length
cross section area S
is
wave
in a
given by
[Ref. 5 Equation (9.26)]
cos[k(L-x)]
P(x,t)-P A
where k = k
at
x =
is
-
ia.
cosRL
,
The input mechanical impedance presented
(1)
to a driver located
therefore given by
Z mo =-iP m
In steady state, the
driver,
icot
e
which
is
power dissipated by
CO
ScotkL
-k
the tube
is
-
(2)
equal to that delivered by the
given by
2
.
E=
PA S
2
R m0
2
2Z
'
(3)
where
R mo
= Re{Z mo } and
rigid tube, sin
kL =
0,
Zm0 = Z m0
I
pi
i
E=y
mode
is
coefficient
of the tube by
for a
Equation 3 reduces to
•
The absorption
Under conditions of resonance
I.
a=
a
SaL
.
(4)
can be related to the quality factor
Qn
of the n th
(Dn/2Q n c. (The subscript n will be dropped unless there
potential for confusion).
Substituting this relationship into Equation 4 and
solving for 1/Q gives
2
,
1
v
As
indicated in Figure
1,
4p
c
PlcoLS
a prime
mover
hot end, the hot heat exchanger, the prime
exchanger, and the ambient end.
mover can be separated
comprised of five
mover
stack, the
parts: the
ambient heat
Accordingly, the power dissipated by the prime
into five parts as
E = Eamb+ Eamb + E stack + E
end
where
is
htex
the five terms represent the
hot
htex
+ Ehot
end
power dissipated by
,,
'
^
'
the ambient end, the
ambient heat exchanger, the prime mover stack, the hot heat exchanger, and the
hot end, respectively.
—
Using
results
for the five terms
from
Swift's review article [Ref. 1], expressions can be found
on the right-hand side of Equation
We
6.
begin with Swift's
expression for the acoustic power generated by a short thermoacoustic engine
[Ref.
1
Equation 80]
W = | Itf
2
Axco cos (kx)
—
'
Pm C
(7)
>
G-0
1+
(
\
o+//y y
-1
Ol
5K
(lW^)(l-8 v /y 0+ 5*/2y*
where e s = p m
cp
8K
/
p s cp 8 S
,
J
(l-5 v /y
T = VT / VT cril VT =AT Ax =
/
,
tan (kx)
+5'/2y'
(T hol
-
T amb )/Ax,
and
T{3
„_ = —
VT cnt
- coc
crit
cot
.
assumed
the radian
5v,
»6
y
this
and
/
(8)
.
in deriving this expression that the stack is short
wavelength (Ax
K
kx
l+//y
cP
It is
.
»5
S)
is
«
compared
X/2k), the boundary layer approximation (y
valid and that
AT
equation differs from Swift's in that
«
T.
It
to
>>
should be pointed out that
we assume
the pressure to vary as
cos(kx), rather than sin(kx).
Two
before
it
modifications, involving conventions, must be
can be applied to our problem.
W
in
Equation 7
is
made
to
positive
Equation 7
when power
is
generated.
power
is
However, the convention
dissipated. Hence,
E = - W.
in
Equation 5
E
is that
is
Finally, the sign convention of
reversed relative to Swift's convention because
we
this
AT must
be
measure x from the
will
ambient end, whereas Swift measures x from the hot end.
accommodate
when
positive
order to
In
change, a minus sign will be introduced in front of I\ With
we
these conventions in mind,
arrive at an expression for the acoustic
power
dissipated by a short thermoacoustic engine
1
-n5 K Axco
E=
4
2
——cos
p
(kx)
Pn/
(9)
2
A
Cr-0,
(
1
+
Ol
+
,
(lW^)(l-6 v /y 0+ 5-/2y')
sv
5
I
Again following Swift, the contribution
*
:
to
* (l
+
//y o)
^
The
justification for letting
T=
these parts of the prime mover.
//y
)
The
term accounts for the increase
If there is
no stack then
assumes the role of y
mean
is that
this
in the
term
0,
/
=
that the
0,
power
the temperature
Equation 9 by the
is
absence of the stack.
«y
-
uniform throughout
/
=
is that
volume velocity upon entering
in unnecessary.
dissipated
and assuming that 5 V
justification for setting
in
(kx)
Equation 6 from the ambient
per unit area e of the resonator can be found by dividing
r=
2
- S v /y +5*/2y*)
and hot ends of the resonator can be found by realizing
surface area of the engine IlAx, setting
^tan
tan
the (1
+
the stack.
Finally, the tube radius
R
For the frequencies, gases,
pressures, and tube radius of interest, 6\-/R in on the order of 10" 2 to 10" 3
.
These modifications
to
Equation 9 yield, for the power dissipated per unit area
of the resonator
€
=
— 6^.0)
VJ[
cos (kx)
2
4
(
1
^ + ^(l + // yoJ
+e
sJ
(10)
tan'(kx)
K
Integration of this expression over the surface area of the ambient and hot ends
of the resonator yields expressions for E*jJ and
Eamb =
P A 0)LS8 K (y-l)
2
end
4p m c
x amb
L
R (1+C,)L
Pa
coLSS.
+
f
x amb
Ejjjj
in ( 2kx amb)
kL
(11)
6k
sin(2kx amb )
1)
L (l+ej
kL
4p m cR
(T-
and
Ehot =
end
P A coLS8 K (y-i)
2
(l+e s H
4p m c R
+
L-x hol
P A coLS5 v L - x hot
2
4P m ^
kL
(12)
sin(2kx hol )
kL
R
The thermophysical
+
5 k (Y-l)
L (1+eJ
properties are to be evaluated for either the ambient
or hot end, depending on which term
The
sin(2kx hot )
is to
be computed.
contributions to Equation 6 from the ambient and hot heat exchangers
found by setting
T=
in
Equation 9 (because the temperature
is
assumed
to
is
be
uniform across the heat exchangers) and substituting the result into Equation
The
5.
result is
Eamb/hot=
P A con5 K Ax
(Y-l)
2
COS (kx)
htex
'
(l
4 Pm C
+
O
+
8v
(i
+
'/y„)'
tan (kx)
5
(13)
<(l-8 v /y 0+ V2^
where the geometrical properties of the exchangers and the thermophysical
properties are to be evaluated for either the ambient or hot heat exchangers,
depending on which terms are
The
to
stack's contribution to
be computed.
Equation 6
is
found by substituting Equation 9
directly into Equation 5 to give
P A con5 K Ax
2
cos (kx)
'stack
'
4 Pm C
X
(14)
(y-0
1+
(
hl
OI( W^)(l-8
1
Because, as discussed
measuring the
Q
Vjv
8k
v /yo+ 5jy2y^)
;
later the initial
O+'fro)'
l-5 v /y 0+ V2yJ
phases of
this
experiment consist of
of an empty, rigidly terminated, cylindrical resonator,
an expression for the
Q
of such a system.
The
tan (kx)
losses in the
we
give
empty resonator can
be found by integrating Equation 10 over the surface area of the entire
resonator.
Substitution of the result into Equation 5 yields
1
Qempty
5 K (Y-l)
SV
RfJ+e)
R
where
identical to Swift's result [Ref.
is
In order to obtain the
in the
s J
v
tube
Q
25 K (Y-D
L (l + e)'
s'
^^
v
1
Equation 91].
of the prime mover,
will be driven at frequencies
it
neighborhood of a resonance with a high impedance source and measure
the amplitude of the output of a high
impedance receiver. Both the source and
receiver are located at the rigid end of the ambient section of the prime mover,
i.e. at
x
=
0.
The
fitting
function
is
found by considering the acoustic impedance
The driver used
seen by the driver.
measurements
in these
transducer, which acts as a constant displacement source,
constant.
The volume
velocity of the driver
Ud
is
if
is
an electret
the driving voltage
given by
U D =^ D S D co=V D co,
(16)
where ^d and Sd are the displacement and surface area of the
respectively.
is
Vd
is,
therefore, the
volume displaced by
equal to the product of the acoustic impedance
acoustic impedance
is
Z and
at
x
=
the driver.
the
volume
driver,
The pressure
velocity.
The
impedance by Z = Z m /S 2
related to the mechanical
Therefore, the complex pressure amplitude
is
is
-
given by
2
p(0)
=
^E^-cotkL,
where use has been made of Equation
2.
10
(17)
The magnitude of
=
the pressure at x
pm
p = |p(0)| =
Using the relation k=
to co
and
Q
co/c,
thus
is
LV D
2
kL
kL
cot
-^V-^
koL= n and Q=C0o/2ac,
(18)
the factor
kL
can be related
by
kL=7C ,i^__L
co
By
further defining
A
= pm
Vd L
of the three parameters A, Q, and
co 2 / S,
co
A
Equation 19 can be expressed in terms
as
cot
p=
(19)
2QJ
J- «U"
CO
(20)
CO,
71
I"
«o
A
three parameter, least squares
to obtain
Q
and
co
1
,
of this function to the measured data
arise as to the validity of using the
is
used
assumptions associated
in particular, that the losses, represented
of the complex propagation constant (k = k
along the length of the tube.
The answer
2/
for the analysis discussed below.
The question may
with Equation
fit
'I
2<
Such
lies in the fact that the
is
-
by the imaginary part
ia), are distributed uniformly
certainly not the case in the prime mover.
quantity of interest
11
is
the frequency response
of the prime mover, not the variation of the acoustic pressure with distance, or
even the absolute acoustic pressure
in the
prime mover.
Over
a small band of
frequencies, the pressure amplitude at a particular point should depend only on
the total attenuation in the tube, not on
of the relation
to
a=
co/2Qc,
a
its
spatial distribution.
we
use
is
made
represents the effective absorption coefficient due
uniformly distributed losses that would give the same
measured. As long as
When
Q
as the
one actually
limit ourselves to finding the frequency response, the
formalism associated with Equation
1
is valid.
12
EXPERIMENT APPARATUS AND PROCEDURE
III.
The primary goals
mover below onset
are to
measure the quality factor of the heat driven prime
as a function of the
mean gas pressure and
the applied
temperature difference, and the waveform of the sound generated above onset.
In order to gain confidence in our ability to
measured the
Figure
2.
Q
factor,
of a simple configuration called the resonant tube,
shown
we
in
This resonant tube was then modified by adding the ambient heat
exchanger and changing the upper portion
stack and the hot heat exchanger.
driven prime mover.
above onset.
A
to
one consisting of the prime mover
This configuration (Figure 3)
final configuration
This configuration
mover, except
transducer.
measure the quality
is
was
built to
resonant tube.
(B)
is
called the heat
observe the behavior
virtually identical to the heat driven
that the electret driver is replaced
This chapter
is
prime
by a high intensity pressure
divided into four sections as follows: (A) The
The heat driven prime mover. (C) Temperature
control
equipment. (D) Instrumentation and procedure.
A.
THE RESONANT TUBE
1.
The Tube
The tube
is
made from two 3.82 cm ID copper
brass ambient heat exchanger container.
11.4
cm
and 88.1
cm
long.
The two
The other end of
the
tubes, separated by a
sections of copper tube are
two sections are
fitted
with
flanges which allow them to be soldered to the ambient heat exchanger container.
The
total length
of the resonator, including the heat exchanger container,
13
is
L=
1.02
m
cm
ID = 3.82
Gosed Cap
It*- haaa
p
Tnhp
iudc
Microphone ^"PPcr
Figure 2
-
'^
<
1
Ambient Heat
Exchanger Container
Copper Tube
The resonant tube configuration
L=
1.00
m
ID = 3.82
cm
"I
Prime Mover Stack
Ambient Heat Exchanger
feh
:
*$
7
Thermocouple
Hot Heat Exchanger
Figure 3
-
The Heat Driven Prime Mover
14
1.02m
(internal dimensions).
section,
forming a closed,
However,
The
driver and
is
this
is
soldered to the end of the shorter
There
also a copper end cap for
is
cap houses an electret driver and a tiny
is
epoxied
connected to a gas handling system, through a valve
evacuated the resonator before
In order to sense the
OMEGA
gauge and an
to the
2.
fill
A
with helium.
filling
mean pressure
The
in the
ambient
A vacuum pump
system vent
is
also
inside the tube, a dial pressure
Model PX304-150AV pressure transducer
are connected
line.
Driver and Microphone
It is
desired that the impedance of the end cap containing the driver and
microphone be large compared
to the highest acoustic
impedance of the standing
wave. This requirement can be accomplished by using electret designs.
cm
in the
to the section of tube.
heat exchanger container, to allow pressurization with helium.
provided.
electret
microphone are flush mounted and sealed
The end cap
end cap with epoxy.
resonator
copper cap
rigid, termination.
the longer section.
microphone.
A
electret transducer
signal is provided
was constructed
for use as the acoustic driver.
An
The
input
by a Hewlett Packard Model 8904A Multifunction Synthesizer
and amplified by a Techron Model 7520 power amplifier.
A
tiny,
0.594
diameter, Panasonic electret microphone was used to sense the response.
output signal
B.
is
1.9
cm
The
amplified by a signal amplifier with an open loop gain of 100.
THE HEAT DRIVEN PRIME MOVER
As explained
in the
previous chapter, the heat driven prime
of hot and ambient heat exchangers and a prime
described in this section.
15
mover
stack.
mover
consists
These devices
are
The Heater Section and Hot Heat Exchanger
1.
The purpose of
mover
As shown
stack.
a nickel heater section
3.82
to
cm
ID, 5.00
accommodate
cm
supply heat to the prime
Figure 4, the hot end of the prime mover consists of
in
The heater
and a heat exchanger.
long, nickel tube.
One end
section consists of a
of the tube
is
capped and
drilled
other end of the heater section contains the heat exchanger
consisting of 25, 0.051
cm
thick, 0.762
each pair of adjacent plates
cm
long nickel plates. The gap between
Between each
0.102 cm.
is
stainless steel spacers with a length of 0.102
cm
304
plates there is a
and a diameter of 0.031 cm.
The Ambient Heat Exchanger
2.
impose
In order to
stack, an
ambient heat exchanger
is
employed
to
maintain one end of the stack
The construction of
similar to the hot heat exchanger, except
it
stacks separated by a 0.15
cm
gap as shown
this heat
exchanger
has a length of 1.02
The ambient heat exchanger
copper plates.
prime mover
a temperature gradient across the
a constant ambient temperature.
25
is to
a thermocouple probe, used to sense the hot heat exchanger
The
temperature.
the hot heat exchanger
in
cm
very
and contains
actually consists of
Figure
is
two such
5.
The Prime Mover Stack
3.
This stack
is
the heart of the heat driven
prime mover.
A
temperature
gradient will be established across this stack to supply the required heat flux.
shown
at
in Figure 6, the
prime mover stack consists of a cylindrical stainless
shell containing 35, 0.25
by 0.079
cm
A
cm
thick
304
stainless steel plates 3.42
cm
As
steel
long spaced
.
complete
prime mover stack
is
listing
of the specifications of the heat exchangers and the
provided
in
Appendix C.
16
The Hot Heat Exchanger
nAx
Figure 4
-
= 0.0113
m2
The Nickel Heater Section
17
0.15
T
1
cm
T
2.19
cm
U
3.82
cm
1.02
cm
FIAx = 0.017 m'
Figure 5
-
The Ambient Heat Exchanger
B3
E=
T
3.42
i
cm
i
I~LAx
Figure 6
-
= 0.072 m'
Prime Mover Stack Container
18
C.
TEMPERATURE CONTROL EQUIPMENT
Control of the temperature gradient across the prime mover stack
OMEGA
by an
Model 9151 miniature microprocessor temperature
is
achieved
controller, a
HBA
Model 202040 heater and a Neslab Model RET-110 constant temperature
bath.
The heater was mounted
in
Figure
7,
to
surround the nickle heater section. As shown
output of the temperature controller was fed to a voltage divider and
then connected
to the
Amplitude Modulation input of the function generator.
Output from the function generator
power
amplifier.
through
Water
water jacket
a
is
then connected to the heater through a
circulated by the
is
temperature
constant
which surrounds the ambient heat exchanger.
bath
The
water pipe was also wrapped around the long section of the prime mover
maintain a uniform temperature.
A
type
K
thermocouple
is
soldered to the hot
heat exchanger to sense the temperature of the hot end.
thermocouples were glued
the prime
mover
to the top,
to sense the
the resistance of a thermistor
difficult for
Three type
is
The reference
found by using a 4-wire method
to
measure
mounted on an isothermal aluminum block.
It
is
an operator to monitor temperature using these thermocouples
because their outputs give voltages rather than actual temperature.
order to monitor actual temperature directly, two type
thermometers were glued on
bottom of the prime mover.
maximum
E
middle, and bottom of the long section of
temperature along that section.
temperature for the whole system
to
to the
With
Therefore
in
E thermocouples
ambient heat exchanger container and the
this
temperature control equipment, the
deviation of the applied temperature difference
19
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°C.
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20
D.
INSTRUMENTATION AND PROCEDURE
As discussed
at
the beginning of this chapter, the primary goal
the quality factor of the prime mover.
was
prime mover
to drive the
frequency near resonance and measure the
at a
steady state frequency response was obtained.
fit
measure
The measurement technique employed
steady state amplitude of the microphone output signal.
squares
is to
The
Q
is
In this manner, the
determined by a
least
of the ideal response to this data.
The measurements consisted of
measuring the
Q
three phases.
of the empty resonant tube.
Q
of the prime
difference below onset.
waveform of
The
mover
third
and
first
phase consisted of
The purpose of
give confidence in the measurement technique.
measuring the
The
this
phase was to
The second phase consisted of
as a function of applied temperature
final
the acoustic signal produced
phase consisted of measuring
by the prime mover above
onset.
the
The
instrumentation and procedure was for each of these three phases are discussed in
this section.
1.
The Empty Resonant Tube
The experimental setup
performed by an
PC AT
is
shown
through a
HP 3457A
GPIB
the controlling program, a source signal
the electret driver.
custom-made
3457A
Figure
compatible computer.
with the SR-530 lock-in amplifier, the
multifunction synthesizer
in
is
8.
The data
acquisition
The computer communicated
multimeter and the
interface.
Through
HP 8904A
the execution of
supplied by the function generator to
The output voltage from the microphone was amplified by
signal amplifier.
All data signals of interest were fed to a
multimeter. The output of the
HP 3457A was
21
is
used as the data input
a
HP
to the
A KIKUSUI COS6100A
computer.
oscilloscope
signal to the electret and the output signal
Before the data acquisition
opened and the system
is
was used
from the signal amplifier.
the valve connected to the tube
is started,
pumped down and
refilled
When
is
ASM
An
Model 110
then used to ensure no leakage exists.
the automatic data acquisition begins, the
voltages from the
is
with helium to the desired
pressure three times in order to purge undesired gases.
turbo helium leak detector
monitor the input
to
program
PX-304 pressure transducer and converts
initially takes
to actual pressure.
Next, the approximate resonance frequency and the half power bandwidth are
entered into the computer, which then determines the start and stop frequencies
and the frequency increment. After the valve
turned on and the data acquisition
is
is
started.
closed the function generator
The program
frequency and measures temperatures, frequency,
imaginary part of the signal amplitude).
It
and repeats the process. The time required
X
and
sets
Y
is
the driving
(the real
and
also calculates frequency increment
to
measure the frequency response
is
approximately 5 minutes.
As discussed
frequency
is
earlier, the
resonance behavior of the tube as a function of
given by
2
p=
where P
is
cot (kL)
Mh
the acoustic pressure of the tube,
kL=7i[co/(Ou+i/(2Q)].
A
(21)
kL
least squares fit is
A
performed
22
is
a scaling constant, and
to obtain fo,
Q, and A.
2.
The Prime Mover below Onset
The setup
shown
is
shown
in
Figure 7 (the same portion with Figure 8
The measurement was performed
here).
measurement performed
follows:
earlier with the simple resonant tube
After that measurement, the heater
and then encased
as
was mounted around
in insulating material.
is
First,
not
the
was repeated.
the nickel heater section
The temperature
controller and the
constant temperature bath were turned on and set with a specific temperature.
After thermal equilibrium
magnitude of the input
is
achieved, the data acquisition
started.
The
power was determined by taking the product of
electrical
the voltage and current.
is
The current was measured through
a Tektronix
Model
P6021 current probe.
Since the quality factor increases with temperature, the
synthesizer was replaced by a
frequency resolution.
This
HP 3325A
HP
8904
function generator to get better
measurement was repeated with increasing
temperature until sound was generated.
3.
The Prime Mover above Onset
This configuration (Figure 9) was built to observe the behavior of the
sound generated above onset. The long section of the prime mover was replaced
by a different section of identical length which has an
5 piezoresistive pressure transducer
The transducer
is
A
back volume
to eliminate
Model 8510B-
screwed through the center of the end cap.
housed within another cap which
brass flange.
ENDEVCO
high impedance leak
is
is
bolted to the end cap with
provided between the resonator and the
dc pressure difference with
pressure differences.
23
little
effect
on acoustic
In this phase of the experiment,
state electrical
(1) the
power input
HP 3314A
time.
,
set to
900 °C
to
After reaching heat equilibrium
the data acquisition
HP
Tektronix Model
to
measure the steady
This task was accomplished as follows:
is
started.
ensure the heater was turned on
i.e.
the temperature of hot
The output
pressure transducer was amplified by an
connected to a
was desired
function generator was set up with a small constant voltage.
was
(2) the controller
to the heater.
it
3457 A Multimeter and a
2445A
oscilloscope
AM
signal
502
to
constant
differential amplifier
and
signal analyzer.
A
monitor the output signal and
the dc voltage supplied to the differential amplifier.
24
is
from the piezoresistive
HP 3561 Dynamic
was used
end
the
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26
RESULTS AND DISCUSSIONS
IV.
The
results of the three phases of the
The
discussed in this chapter.
presented
first,
results for the
experiment will be presented and
results for the
empty resonator tube
will be
followed by those for the prime mover below onset. Finally, the
prime mover above onset will be presented.
Investigation of the
empty resonator tube and
the prime
mover below onset
both involve determining the resonance frequency and the quality factor from
the
measured frequency response.
shown
in
Figure 10, which
the electret
microphone
particular data set
across the stack,
The
frequency and
from
Although
this
as a function of drive frequency.
mover with
°C temperature difference
a 198
representative of the data obtained under
in the figure are the
Q
is
a graph of the amplitude of the voltage output
for the prime
it is
of the frequency response data
solid curve represents the ideal response based
Also indicated
A.
is
is
An example
on the
fit
all
to
conditions.
Equation 21.
estimated errors of the fitted resonance
values.
THE EMPTY RESONANT TUBE
The Q's of
mean gas
modes of
the first three
pressure ranging from approximately 170 to 500 kPa.
plotted in Figure 11,
gas pressure.
which
is
a graph of
The square root of
the
Q
represent the theoretical
is
Q
as
Q
<*1 /
6V
<*
P m 1/2 The
27
is
.
16.
for
results are
versus the square root of the
computed from Equation
quite good, although there
The
mean gas pressure was chosen
abscissa because, as seen from Equation 16,
agreement
empty resonator were measured
the
mean
as the
solid lines
In general the
a tendency to slightly over predict the
Q, indicating the presence of unaccounted losses.
each pressure. In
at
The estimated
Two
measurements were made
but a few cases the two results
all
error of the
fit
is
fall
on top of one another.
no larger than the size of the symbols
representing data points.
An
indication of
terminated tube
the
measured
0.01 m.
As
c
empty resonator behaves
all
%
this
to
an ideal, rigidly
graph
the ratio of
is
resonance frequency to the ideal frequency
= 1008 ms* 1
seen,
approximately 0.4
closely the
given in Figure 12. The ordinate of
(fitted)
computed using
±
is
how
(the
sound speed of helium
of the values are within 2 to
3%
at
f
20 °C) and
of 1.00.
An
= nc/2L,
L=
error of
can be attributed to the fact that the actual temperature of
An
resonator during the measurements was closer to 18 °C than 20 °C.
additional source of error
was
1.02
may be due
to the fact that the total length of
1
.02
m
arrived at by adding the individual lengths of the ambient end, the ambient
heat exchanger container, and the hot end measured separately before the
resonator was assembly by soldering the three parts together.
did not
fit
together exactly, the actual length of the resonator
may have been
unassembled length.
slightly longer than the
B.
If the three parts
THE PRIME MOVER BELOW ONSET
The
results of the
mode below
measurements of the
onset for five different
13 through 17, which
across the stack.
The
show 1/Q
mean
Q
of the prime mover's fundamental
gas pressures are presented in Figures
as a function of the temperature difference
solid line represents the theoretical prediction using
Equation 6 and 11 through 14, which are based on the boundary layer
The calculation of
assumption and the short stack approximation.
contributions
from the ambient and hot end
28
is
straightforward.
the
The
thermophysical properties of the gas are computed using either the ambient or
hot end temperature, whichever
In order to calculate the
appropriate.
is
contributions from the heat exchangers, x in Equation 13
is
set equal to the
location of the center of the heat exchanger, and again the thermophysical
properties of the gas are
temperature, whichever
from the
stack.
stack, x in
was
computed using
In order to
appropriate.
Equation 14
is set
compute the contribution
the stack is accounted for in the
Other than the propagation constant, the only temperature
dependent quantities
depths.
end
equal to the location of the entrance to the
The temperature gradient across
following manner.
either the ambient or hot
in
Equation 14 are the thermal and viscous penetration
In order to account for their temperature
converted into a differential by replacing
Ax by
dependency, Equation 14
dx.
By assuming
is
a linear
temperature gradient across the stack, the temperature dependence of the
penetration depths can be converted into an x dependence.
Q
computed by numerically
is
integrating the differential
The
reciprocal of the
form of Equation
with x-dependent penetration depths, along the length of the stack.
14,
The
interpretation of this computation is that, in the short stack limit, the acoustic
parameters in the stack are determined by their values
only modification necessary
the stack
is
at the stack entrance.
presence of
to adjust the particle velocity for the
and the viscous boundary
layer.
Before the integration of the penetration depth can be accomplished,
necessary to determine
performing a
from an
its
temperature dependence. This dependence
least squares fit to viscosity
NBS
table [Ref. 6].
penetration depths in helium
The
is that
The
5
T
29
is
found by
and thermal conductivity data obtained
result for both the thermal
«=
is
it
-
85 .
and viscous
There are several important features evident
overall agreement
decreasing
is
mean gas
the losses at
in Figures 13 to
17.
The
good, through the quality of the agreement diminished with
pressure.
In each case, there is a tendency to over predict
low temperature difference.
pressure, the data tend to
tail
Finally, at the lower values of gas
off near onset, a feature not predicted by the
theory.
In order to understand the discrepancies discussed above, the validity of the
approximations must be investigated. For the fundamental mode, the short stack
approximation
is
only about 10
well satisfied because the length of the prime
%
As
of the radian wavelength.
mover
boundary layer approximation
high
mean
is valid.
and
starts to fail
when
the
If
is
plotted as
y /5 K ^1.5 the
For the viscous penetration depth,
gas pressure the boundary layer assumption
assumption
18.
mean gas
is
is
at
well satisfied, but this
pressure decreases.
layer assumption for the thermal penetration depth
is
boundary layer
for the
assumption, the penetration depth for the average stack temperature
a function of the temperature difference in Figures 17
stack
The boundary
not satisfied very well
at
any pressure.
In order to determine if there
is
any connection between the failure of the
boundary layer approximation and the
tailing off of the data at
pressures, the 1/Q data are replotted in Figures 20 through 24.
these figures represent a linear least squares
data,
where
the viscous
the 500, 376, and
fit
to the
The
low gas
solid lines in
low temperature difference
boundary layer approximation
is valid.
These
lines
fit
308 kPa data well over the entire range of temperature
differences, as might be expected from the fact that the boundary layer
approximation
is
valid in these cases at almost
30
all
temperature differences below
onset.
at
However,
the
239 and 170 kPa data begin
to depart
from the
straight lines
an approximate temperature difference of 200 and 100 °C, respectively.
Referring to Figure 18, these are the same points where the approximation
begins to
The
fail.
results for the
second and third modes of the prime mover are given
in
Figure 25 through 28 with two different pressures of 308 kPa and 170 kPa. The
agreement
is
not very good, probably because the short stack approximation
violated at high modes.
%
20 and 30
In the second and third
of the radian wavelength.
modes
the prime
mover
stack
Although the quantitative agreement
poor, the theory does predict the correct dependence of 1/Q at
AT =
AT
and
is
is
is
at
onset.
THE PRIME MOVER ABOVE ONSET
C.
This final series of data contains the results above onset for the prime mover.
Figures 29 and 30 show the waveform and the spectrum of the sound generated
by the prime mover above onset.
The mean gas pressure
temperature difference across the stack
is
325 °C which
is
is
The
307 kPa.
slightly
above onset.
The
signal exhibits slight distortion, particularly in the positive cycle (Figure
29).
Figure 30 shows that the difference in spectrum level between the
modes
is
larger than 15 dB.
difference of 368 °C.
The
Figures 31 and 32
signal
is
show
first
results for a temperature
distorted even further and there
energy stored
in
(Figure 33)
distorted sharply in both positive and negative half-cycles.
34
is
high modes.
illustrates the
between for the
At a temperature difference of 453 °C,
spectrum for the same signal. The difference
first
few modes has decreased further
energy has been spread
to
higher modes.
31
few
in
to less than
is
more
the signal
Figure
spectrum levels
6
dB and more
The
signals
shown
in these figures
have very high amplitude.
The peak
positive pressure in Figures 29, 31, and 33 are approximately 3.3, 13.5, and 24.2
kPa, respectively.
7.9% of mean gas
These amplitudes correspond
pressure.
32
to
approximately
1.1, 4.4
and
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33
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Q
T
160
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140
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120
-
100
-
80
-
60
-
•
•
Fundamental
Second mode
Third
mode
Q
40
400
P
Figure 11
700
600
500
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2
m
(
Pa2
)
Resonant tube with helium
34
800
l.UU
o
r"
—I
'
1
1
Fundamental
0.99
o
•
Second mode
D
0.98
D
Third
mode
•
D
•
D
•
D
•
D
8
•
0.97
0.96
n
qc;
400
i
i
1
1
P^
Figure 12
-
i
700
600
500
1
l
(Pa
J)
Resonant tube with helium
35
800
0.03
0.02 -
1
^
Q
0.01
-
0.00 -
-0.01
200
100
300
DeliaT (°C)
Figure 13 - Prime mover with helium at
500 kPa (the fundamental mode)
36
400
0.03
0.02 -
0.01
-
Q
0.00 -
-0.01
200
100
300
DeltaT(°C)
Figure 14 - Prime mover with helium at
376 kPa (the fundamental mode)
37
400
0.04
0.03 -
0.02 -
Q
0.01
-
0.00 -
-0.01
200
100
400
300
DeltaT (°C)
Prime mover with helium
308 kPa (the fundamental mode)
Figure 15
-
38
at
0.04
i
L
»
i
'
-
0.03
0.02
_
0.01
-
~
<fr\
Q
•
0.00 -
-0.01
xj^
«
-0.02
i
200
100
i
i
i
300
400
DeltaT (°C)
Prime mover with helium
238 kPa (the fundamental mode)
Figure 16
-
39
at
0.05
Q
-0.01
-
-0.02
200
100
400
300
DeliaT (°C)
Figure 17 - Prime mover with helium
170 kPa (the fundamental mode)
40
at
I
3.0
2.0
X
D
O
A
h°v
L'
Do
oo,
D
*X
A A,
1.0
'Oo o
to.
'MB,
.X K
5..
kPa
kPa
kPa
kPa
500 kPa
170
238
308
376
xy *x
°oo 0(
ooo
" Aaaa
>w«r*^aa
XX
XXXxx
xxxxx^xx EDQD ^
*xxxx x
-
0.0
200
100
Delta
300
400
T (°C)
Figure 18 - Ratio of the plate half-gap to the viscous
penetration depth Vs. temperature difference
41
X
A
o
oo
yo
XX
8„
1
170 kPa
238 kPa
308 kPa
376 kPa
500 kPa
Oo o
°D D
Xx xxx
oooi
AAAa
-
XXXxx xxxx
200
100
Delta
300
BDrt
xxxjc
400
T (°C)
Figure 19 - Ratio of the plate half-gap to the thermal
penetration depth Vs. temperature difference
42
0.03
0.02
-
0.01
-
Q
0.00 -
-0.01
400
Figure 20 - Prime mover with helium at
500 kPa (the fundamental mode)
43
7"
0.03
1
'
1
1
l
8s.
0.02
•
-
^vO
•
-
h
01h
CL^
0.00 -
0.01
i
1
200
100
Delta
I
300
T (°C)
Figure 21 - Prime mover with helium at
376 kPa (the fundamental mode)
44
400
0.04
0.03
0.02 -
Q
0.01
-
0.00
-
0.01
200
100
Delta
300
T(°C)
Figure 22 - Prime mover with helium at
308 kPa (the fundamental mode)
45
400
0.04
0.03 -
Q
0.02
-
0.01
-
0.00 -
0.01
-
0.02
200
100
Delta
T(°C)
Figure 23 - Prime mover with helium
238 kPa (the fundamental mode)
46
400
300
at
0.06
-0.02
200
100
400
300
Delta T(°C)
Figure 24 - Prime mover with helium
170 kPa (the fundamental mode)
47
at
0.03
T
i
i
o
0.02 -
i
o
^***^»„^
"
8
o
Q
0.01
-
0.00 -
-0.01
1,1,
1
200
100
400
300
DeltaT (°C)
Figure 25 - Prime mover with helium
308 kPa (the second mode)
48
at
0.03
0.02
-
0.01
-
Q
0.00 -
0.01
400
Figure 26 - Prime mover with helium
308 kPa (the third mode)
49
at
0.04
1
1
1
-
I
'
1
0.03
sO
0.02 -
I"
V
1
fc
Q
0.01
1
o
-
o
•
j ,
1
-0.01
1
1
200
100
400
300
DeltaT (°C)
Figure 27 - Prime mover with helium
170 kPa (the second mode)
50
at
-»-
0.04
1
1
">-T
'~^f—,
I
o
o
0.03
o
0.02
Q
0.01
-
"\°
°
-
X;
»
0.00 -
0.01
'
1
1
1
200
100
1
300
DeltaT (°C)
Figure 28 - Prime mover with helium at
170 kPa (the third mode)
51
400
O
UD
-p
in
r-l
O
>
>
E
E
Figure 29
-
Waveform
>
n h
\
of the sound generated by the prime
temperature difference of 325 °C
52
O d
<
OJ
in
x
CD
II-..
mover
at a
m x
Figure 30
-
Spectrum of
the
sound generated by the prime mover
temperature difference of 325 °C
53
at a
i
o
E
ID
<e-l
>
(J
E
o
c\j
o
o
4-J
>
i—
»—
i
C\J
>
e
Figure 31
O ro c
<
n
e
-
Waveform
of the sound generated by the prime
temperature difference of 368 °C
54
mover
at
—
N
O Z
..
LT
o i- ro n ^
<
hCD X
K-t
.
i
Figure 32
-
ovei
Spectrum of the sound generated by the prime mover
•
temperature difference of 368 °C
55
at a
.
01
Figure 33
-
Waveform
of the sound generated by the prime mover
temperature difference of 453 °C
56
X
at a
C/D
Figure 34
-
Spectrum of the sound generated by the prime mover
temperature difference of 453 °C
57
at a
X
SUMMARY, CONCLUSIONS AND RECOMMENDATIONS
V.
A.
SUMMARY
The purpose of
this thesis is to investigate the
work output of
The experimental approach was
thermoacoustic prime mover.
a heat driven
to
measure the
frequency response of both a simple resonant tube and a prime mover for a
variety of values for
across the prime
mean
mover
yield the quality factor
stack,
least squares
which can be compared
results are reported of
the prime
to
A
stack.
fit
to a the
frequency response
to predictions
based on a short
boundary layer approximation theory.
The
kPa
gas pressure and applied temperature difference
mover
in
measurements made on the lowest three modes of
helium for mean gas pressures between approximately 170
500 kPa and the applied temperatures between zero and
waveform of
the sound generated
onset.
by the prime mover above onset
The
at a
signal
mean
gas
pressure of 307 kPa are also reported.
The
overall results can be
tube have
at
most
3%
summarized
difference
Results of the resonant
as follows.
with theory.
measurements generally agree with predictions
For the prime mover, the
for the
fundamental mode except
close to onset where the boundary layer approximation
is
not satisfied very well.
This agreement between measured and predicted results worsens with decreasing
mean gas
pressure.
Agreement
is
poor for the second and
pressures used, the source of the discrepancy
short stack assumption at higher modes.
58
may
arise
Finally, the
third
modes
for all
from the violation of
the
sound generated by the
prime mover above onset has been noticeably distorted. The distortion becomes
more severe
as the temperature difference increases.
CONCLUSIONS
B.
Several conclusions can be drawn from our results.
that the
method used
to
Q
determine the
The
first
conclusion
is
works well judging by the close
agreement between theory and measurement for the simple resonant tube. The
second conclusion
is
boundary layer theory describes the
that short stack,
fundamental mode of the prime mover below onset
fairly well.
There
is
generally good agreement with theory for low temperature differences, although
the agreement worsens
worsen
as
mean gas
somewhat
as onset is approached.
The agreement
pressure decreases, probably due to the break
viscous boundary layer approximation.
A
down
also
of the
third conclusion is that the short stack
theory does not adequately describe higher modes, although the theory shows the
same
qualitative features as measurement.
The
final
conclusion
is
that the
sound
generated by the prime mover above onset exhibits a great deal of nonlinear
distortion.
Reasons for
this are not fully
understood.
RECOMMENDATIONS
C.
In order to gain better understanding of the thermoacoustic process
and
to
determine the source of discrepancy, particularly for higher modes and the
region close to onset, several recommendations are proposed as follows:
•
Use
•
Determine whether the nonlinear distortion seen above onset is primarily
due to the presence of high amplitude waves in the resonator, independent
of the presence of the stack, or if the stack is the dominant source of the
theory not
approximations.
a
limited
by
short
distortion, or if both play equal roles.
59
stack
and
boundary
layer
APPENDIX
PARTIAL LISTING OF THE PHYSICAL
A.
PROPERTIES OF HELIUM
TABLE
p
T
MPA
K
0.170
0.238
0.308
0.376
0.507
293
293
293
293
293
1:
PHYSICAL PREPERTIES OF HELIUM
DEN
KG/M 3
Cv
0.280
0.392
0.504
0.616
0.833
3123
3123
3123
3123
3123
Cp
C
M/S
J/KG-K
5197
5197
5197
5197
5197
60
1008
1008
1009
1009
1009
[Ref. 6]
vise
PA-S*E+6
19.6
19.6
19.6
19.6
19.6
COND
MW/M-K
152.4
152.4
152.5
152.5
152.6
APPENDIX
B.
LISTING OF THE PHYSICAL PROPERTIES
OF MATERIALS FOR THE PRIME MOVER STACK AND
HEAT EXCHANGERS
TABLE 2: PHYSICAL PROPERTIES OF MATERIALS
MATERIAL
AISI304
DEn(|^)
SPECIFIC HEAt(——- )
3
[Ref. 7-8]
COND (^^)
8027
451.9
16.3
8890
8900
443.8
384.0
350.0
STAINLESS STEEL
NICKEL
COPPER
61
89.9
APPENDIX
C.
LISTING OF THE SPECIFICATIONS OF THE
COMPOMENTS AND GEOMETRICAL PARAMETERS OF THE
PRIME MOVER
TABLE 3: SPECIFICATIONS OF THE COMPONENTS AND GEOMETRICAL
PARAMETERS OF THE PRIME MOVER
•PRIME MOVER STACK
n=
n
205.8
cm
Ax = 3.50 cm
= 148.3 cm
TI= 83.7
= 0.950
x
hol
x
stack
L=
cm
m
= 0.925m
1.0
cm
•HOT HEAT EXCHANGER
= 0.0255 cm
Ax = 0.762 cm
•AMBIENT HEAT EXCHANGER
= 0.0254 cm
Ax = 2.032 cm
•PARAMETERS OF PRIME MOVER
/
= 0.0125
m
62
= 0.0395 cm
/
y = 0.051
/
y
m
x aSb = 0.894 m
x amb = 0.883
y
x
hot
R=
cm
= 0.051 cm
= 0.946
0.0191
m
m
LIST OF REFERENCES
1.
2.
G. W. Swift, "Thermoacoustic engines,"
1145-1180(1988).
heat engines,"
3.
Acoust. Soc.
Am
Vol. 84,
.
W.
Swift and A. Migliori, "Understanding
simple phenomena in thermoacoustics with applications to acoustical
John Wheatley, T. Hofler, G.
some
J.
Am.
J.
Phvs 53, 147-162 (1985).
.
John Wheatly, T. Hofler, G.
W.
Swift and A. Migliori, "An instrinsically
Vol 74,
J. Acoust. Soc.
Am
irreversible thermoacoustic heat engine,"
.
153-170(1983).
4.
John Wheatley, G. W. Swift and A. Migliori,
Los Alamos Science, Fall 1986.
5.
L.
"
"
The Natural Heat Engine ."
A. R. Frey, A. B. Coppens, and J. V. Sanders,
Fundamental of Acoustics ." Third Edition, John Wiley & Sons, Inc.,
E.
Kinsler,
1982.
6.
Helium-4 from 2-1500 K
with Pressure to 1000 Atmospheres Washington D. C., NBS Technical
Note 631, National Bureau of Standards, 1972.
McCarty R.
D.. Thermodvnamical Properties of
.
7.
Allegheny Ludlum Steel corporation, Stainless Steel Handbook Allegheny
.
Ludlum
8.
Steel Corporation, 1956.
Weast, Robert C, ed., CRC Handbook of Chemistry and Physics 61 st
Boca Raton, Flordia: CRC Press Inc., 1980.
.
63
ed.,
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