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ABSTRACT
Thermo acoustic refrigeration is a new technology that provides cooling
without the need for refrigerants such as chlorofluorocarbons. There are different
mechanisms used to produce the refrigerating effect. Here the system is based on the
Stirling cycle. The basic mechanism is very simple and efficient. A loudspeaker creates
sound in a hollow tube which is filled with an ordinary gas. These sound waves in turn
create the cooling effect by creating a thermo acoustic phenomenon. The project
developed a working model of an acoustic refrigerator, which develops a cooling of
around few degrees of temperature when using air and being driven with moderate
acoustic power. This project also studies the effects of the design and location of the
thermo acoustic stack and the relative efficacy of different gases. Eliminating the need to
recycle CFC gases in refrigeration units would also eliminate any incidental releases of
coolant gases. CFC’s have been linked to ozone destruction.
Using inexpensive materials, a thermo acoustic refrigerator model can be built.
This model can be used to show that cooling by using sound waves is possible. The
refrigerator is also easy to build even though the theoretical concept looks quite
complicated. The temperature drop of this model is not even close to temperatures
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obtained by other more expensive models but also gives hope to the thought that acoustic
refrigeration may be an inexpensive alternative to other refrigeration technologies. But
the performance of these refrigerators is less than those of their conventional
counterparts. There are still many problems with the inexpensive models, but for all
practical purposes, they can show an area of physics that was almost unknown only a
decade or so ago.
CHAPTER 1 INTRODUCTION
1.1 Thermoacoustics
Thermoacoustics is the study of how heat and sound interplay. In the standard
university physics courses, sound behavior in a resonator is simply described as
depending solely on where one is along the length of the resonator. This neglects an
important reality of gases – namely that they are viscous. We also typically neglect the
thermal effects included when you compress or rarefy a gas. When these effects are
included, though, there is a slight, but important change in the gas behavior. Near the
wall (close enough for a thermal effect to be seen, but far enough away that viscosity
doesn’t dominate), the oscillating gas trades heat back and forth with the wall. Outside of
this region (or about the remaining 99% of the system), the gas pretty much behaves as
described in the university physics sequence. Thermoacoustics takes a small region in the
resonator, and replaces the empty resonator with a porous material that is effectively all
wall, so that the thermoacoustic exchange is effectively utilized. This material is referred
to as a “stack.” Applying a temperature gradient across this stack creates a prime mover
class of heat engine, resulting in the thermoacoustic generation of sound. Gas moves
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toward the warmer end of the stack, expanding and jostling neighboring gas parcels. Gas
moving toward the cooler end contracts, and also jostles the neighboring parcels. As this
is in a resonator, the random motion is amplified at the resonating frequency of the
system, and the gas rapidly transitions from low-amplitude, random oscillation to a large
amplitude resonant oscillation. It’s a neat party trick.
The importance of Thermoacoustics arises when one considers the reverse of a
prime mover – a refrigerator. Consider an acoustic system with a stack in the resonator.
The acoustics forces the gas back and forth, oscillating the temperature as it compresses
and rarefies. These temperature fluctuations induce a thermal exchange with the wall, and
the net effect is similar to a bucket brigade model, where the gas picks up heat from one
end of the wall, transports it, and drops it of on the other end. This results in one end of
the stack cooling off and the other end warming up. Using a thermoacoustic prime mover
to drive a thermoacoustic refrigerator, one can create a refrigerator that operates with no
moving parts, without environmentally hazardous gases.
1.2 Motivation
The motivation for this study is the requirement of alternative sources of
refrigeration as the age-old systems are dying because of the various protocols emerging
out to control the green house effect. This compels one to go for the other sources of
refrigeration’s. Moreover this thermoacoustic refrigeration is the simplified system
compared with the other systems. It also doesn’t create any environmental problems. The
refrigerants used in this system are inert gasses. They are abundantly available and also
cheap, so to make use of them this is one of the most appropriate systems available. To
study such systems, which are in their latest stages of developments, helps one to have
the total hand on the concepts and to develop them. To study such concepts helps to
diversify the field of study that makes the inventions possible in very rapid pace.
The reason that thermo acoustic technology has progressed so rapidly during the
past decade is that there has been an excellent theoretical understanding of the thermo
acoustic heat pumping process which was developed by N. Rott in the late 1960's and
early 1970's, and by J. Wheatley and G. Swift in the 1980's and G. Swift in the 1990's.
Unfortunately, that understanding has been limited to a fairly small portion of the
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available "parameter space." In particular, existing models have been limited to fairly low
acoustic Mach Numbers (Mac < 3% or p1/pm < 5%), due to the one-dimensional nature of
the equations, the limitations of linear acoustics, the absence of mean flow, and the
assumption of a stable laminar boundary layer.
Since the power density of thermo acoustic devices depends upon (p1/pm) 2
, there is quite
a strong motivation to understand thermo acoustics at higher amplitudes. Progress in this
direction will require the construction of thermo acoustic refrigerators which can achieve
higher acoustic Mach Numbers and theoretical advances which could require a solution
to the full non-linear thermo-hydrodynamic equations in two- or three-dimensions. It
would also be useful to study new structures for components such as stacks, resonators,
heat exchangers and electro acoustic driver mechanisms. At the present time, there are no
models for the stack/heat exchanger interface. There are no models for heat transport
between the thermo acoustically oscillating gas and the heat exchanger surfaces which
could be used to suggest what geometries would optimize the useful transfer of heat on
and off of the stack. All electrically-driven thermo acoustic refrigerators to date have
employed electrodynamic drive mechanisms (moving coil or moving magnet). There are
several "solid-state" materials, such as piezoelectric and magnetostrictive compounds,
which have high energy densities and low losses, but which have not been adapted to
thermo acoustic loads. Most of the world's machines are powered by rotary motors. What
is the best way to incorporate such rotary drive mechanisms in thermo acoustic devices?
The above is only a small subset of the possibilities, which could lead to a more
complete understanding and better devices. With an increase in the number of working
devices and motivated investigators, the rate at which thermo acoustics will progress
should increase steadily for many more years.
1.3 Literature Survey
As this is relatively new area the research done is very less. The content on which
they previously worked remains mostly unnoticed because the publications for general
usage are very less. Only available sources are the thesis and patents available in the
World Wide Web (internet). This makes the study a tedious task. The number of
institutes or universities or organizations working on this area only a few to name.
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Most of the research in this Thermoacoustic area is done by a few organizations.
They are Los Alamos National Laboratory, Penn State University, Purdue University,
Rockwell Scientific Company, University of Colorado, Naval Postgraduate School,
Kettering University are some of the organizations taking up serious research in this area.
These all institutes are providing details about their research in their institutes Web pages.
Their research is also being funded by some of the organizations who want this
technology for their future applications. The major organizations in this league are Ford
Motor Corporation, IBM to make their applications ahead of the others by adopting the
latest technology at the earliest.
The theory is taken from the publications of the persons working on this particular
field. They made many publications to name them specifically here. So to quote their
names here is more appropriate and with their names one can locate all the publications
in World Wide Web. Hofler invented a standing-wave thermoacoustic refrigerator. The
most of the work is associated with Gregory W. Swift, Los Alamos National Laboratory.
Because he is the person foremost in this field. With him his colleagues Konstantin I.
Matveev, Scott Backhaus. The model developed in this project is taken from the work of
Daniel A. Russell and Pontus Weibulla, Science and Mathematics Department, Kettering
University, Flint, Michigan. Thermal-Relaxation Dissipation in Thermoacoustic Systems
is the work of M.E.H. Tijani S. Spoelstra P.W. Bach. The heat exchanger analysis in this
project is the main work from Insu Paek, James E. Braun, and Luc Mongeau from Purdue
University. The optimized design conditions are given by Hofler (University of
California). The first law analysis of the thermoacoustic refrigerator is given by Martin
Wetzel and Cila Herman (The Johns Hopkins University). The Development of Miniature
Thermoacoustic Refrigerators is associated with Reh-Lin Chen, Ya-Chi Chen, Chung-
Lung Chen Chialun Tsai, and Jeff DeNatale Rockwell Scientific Company. G.W.Swift
and Wollan are associated with the development of Thermoacoustic Natural gas liquefier.
This is briefly about the sources of information from where the details can be had about
Thermoacoustic Refrigerator.
1.4 Objectives
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The objective of this project is to develop the model for demonstrating the
phenomenon of thermoacoustic refrigeration. By developing such model the project
demonstrates that the drop in temperature is possible by using acoustic waves. There are
no moving parts in this model which makes frictionless operation possible there by
eliminating unnecessary vibrations which makes the system more stable. In addition to
that comparison is made between the conventional refrigerators and the thermoacoustic
refrigerator and is shown that provided the proper equipment, correct analysis of the
process makes this system to reach the performance of the conventional systems.
CHAPTER 2 THERMOACOUSTIC FUNDAMENTALS
2.1 Thermoacoustic Phenomenon
In the 19th century Lord Rayleigh proposed a criterion for heat-driven acoustic
oscillation:
“If heat be given to the air at the moment of greatest condensation, or be taken from it at
the moment of greatest rarefaction, the vibration is encouraged. On the other hand, if
heat be given at the moment of greatest rarefaction, or abstracted at the moment of
greatest condensation, the vibration is discouraged.”
This rule can be conveniently expressed via transformation of thermal energy to
acoustic energy. Considering conservation equations, Culick derived an expression for
the energy addition to the acoustic mode
where p´ is the pressure perturbation; Q´ is the fluctuation of the heat release rate; γ is the
gas constant; po is the mean ambient pressure; V is the chamber volume; and T is the
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cycle period. Equation is an explicit interpretation of Rayleigh’s criterion, showing that
instability is encouraged when the heat release fluctuates in phase with the pressure
perturbation. If the energy input exceeds acoustic losses, and then the system becomes
unstable.
2.1.1Acoustic Waves in a Tube
Consider an acoustic energy source (e.g., a tuning fork) placed near one of the
ends of an open-open tube as shown in Figure2.1.A. A part of the acoustic energy
produced by such a source enters into the tube in the form of a ‘traveling acoustic wave’.
As this wave travels through the tube, it loses some of its energy due to friction. When it
reaches the other end of the tube, a part of the remaining energy reflects back into the
tube, again in the form of a traveling acoustic wave. The rest of the energy transmits
through the open tube boundary and comes out of the tube. So, as shown in Figure 3.1.A,
in the presence of an acoustic energy source, the part of the acoustic wave that reflects
from the open end interacts with the oncoming traveling wave to produce what is termed
as a standing wave or a stationary wave. Figure2.1.B shows the waveform which results
when two traveling waves moving in opposite directions interact to form a stationary
wave. The resultant waveform can be seen to have a magnitude that changes along the
length of the tube. The positions where the magnitude is zero are termed as nodes
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(labeled‘N’inthefigure2.1),
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Figure 2.1 Behavior of the acoustic waves (standing)
figure).One must also note that the oscillation at every point along the tube is in phase.
Thus, unlike traveling waves where the waveform moves ahead (‘travels’) in time, for a
standing wave, the waveform appears to be stationary or standing.
Figure2.1.C shows the variation of acoustic pressure p´ (p´ is the sum of mean
pressure p and acoustic pressure p) and acoustic velocity v (v =v¯+v´) due to the
stationary wave at different times in the cycle for the fundamental mode in a Rijke tube.
The fundamental mode is the one with the lowest possible frequency and the largest
wavelength that satisfies the boundary conditions. In this case, the largest wavelength is
clearly λ=2L, where L is the length of the tube. For the fundamental mode, the acoustic
pressure has one peak at the middle of the tube while the ends of the tube always have
zero acoustic pressure. The acoustic velocity node and antinodes are exactly the reverse
of those for the pressure. It is usually the fundamental mode that is heard in Rijke tube
experiments, so we confine our discussion of acoustic waves in tubes to the fundamental
mode.
2.2 Thermoacoustic Applications
Within thermoacoustics a distinction is made between a thermoacoustic engine or
prime mover (TA-engine) and a thermoacoustic heat pump or refrigerator (TA-heat
pump). The first relates to a device creating an acoustic wave by a temperature difference
while in the second an acoustic wave is used to create a temperature difference.
2.2.1 Thermoacoustic Heat Engine
When a temperature gradient is imposed across a regenerator by for example a
cold and a hot heat exchanger, the following happens with a parcel of gas when an
acoustic wave passes by from the cold side.
The gas is being compressed by the passing pressure wave (compression).
Successively the gas parcel is moved to a hotter part of the regenerator. Since the
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temperature over there is higher than the gas parcel, the gas is heated (heating). Then the
pressure wave that first compressed the gas parcel is now expanding it (expansion).
Finally, the gas parcel is moved back to its original position. The parcel of gas is still
hotter than the structure (regenerator) resulting in heat transfer from the gas to the
structure (Cooling).
During this cycle the gas is being compressed at low temperature, while
expansion takes place at high temperature. This means that work is performed on the gas.
The effect of this work is that the pressure amplitude of the sound wave is increased.
The thermodynamic cycle just described resembles the well-known Stirling cycle.
The acoustic wave has the function of both pistons normally present in a Stirling engine.
In this way it is possible to create and amplify a sound wave by a temperature difference.
The thermal energy is converted into acoustic energy that can be regarded as a kind of
mechanical energy.
2.2.3 Thermoacoustic Refrigerator
Thermoacoustic refrigeration is the process by which acoustical energy is
harnessed to pump heat out of a relatively cold section of air into a warmer one and thus
refrigerate. The pursuit of research on thermo acoustic refrigerators is very promising
because of the countless benefits and applications they offer. For example, since thermo
acoustic refrigerators and engines use no moving parts, and thus cause less friction and
loss of energy, they have the potential of reaching a level of efficiency very close to that
of the Carnot engine and have a much lower probability of mechanical malfunction and
therefore may be used efficiently and reliably in isolated locations such as outer space.
2.3History of the Development of Thermoacoustics Field
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A sound wave in a gas is usually regarded as consisting of coupled pressure and
motion oscillations, but temperature oscillations are always present, too. When the sound
travels in small channels, oscillating heat also flows to and from the channel walls. The
combination of all such oscillations produces a rich variety of “thermoacoustic” effects.
Research in thermoacoustics began with simple curiosity about the oscillating
heat transfer between gas sound waves and solid boundaries. These interactions are too
small to be obvious in the sound in air with which we communicate every day. However,
in intense sound waves in pressurized gases, thermoacoustics can be harnessed to
produce powerful engines, pulsating combustion, heat pumps, refrigerators, and mixture
separators. Hence, much current thermoacoustics research is motivated by the desire to
create new technology for the energy industry that is as simple and reliable as sound
waves themselves.
The rich history of thermoacoustics has many roots, branches, and trunks
intricately interwoven, supporting and cross-fertilizing each other. It is a complicated
history because in some cases invention and technology development, outside of the
discipline of acoustics, have preceded fundamental understanding; at other times
fundamental science has come first.
Rott took the meaning of the word “thermoacoustics” to be self-evident–a
combination of thermal (heat) effects and sound. He developed the mathematics
describing acoustic oscillations in a gas in a channel with an axial temperature gradient,
with lateral channel dimensions of the order of the gas thermal penetration depth δk
(typically of the order of 1 mm), this being much shorter than the wavelength (typically
of the order of 1 m). The problem had been investigated by Rayleigh and by Kirchhoff,
but without quantitative success. In Rott’s time, motivation to understand the problem
arose largely from the cryogenic phenomenon known as Taconis oscillations–when a gas-
filled tube reaches from ambient temperature to a cryogenic temperature, the gas
sometimes oscillates spontaneously, with large heat transport from ambient to the
cryogenic environment. Yazaki demonstrated most convincingly that Rott’s analysis of
the Taconis oscillation was quantitatively accurate.
A century earlier, Rayleigh understood the qualitative features of such heat-driven
oscillations: “If heat be given to the air at the moment of greatest condensation [i.e.,
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greatest density] or be taken from it at the moment of greatest rarefaction, the vibration is
encouraged.” He had investigated Sondhauss oscillations, the glassblowers’ precursor to
Taconis oscillations. Rayleigh’s criterion was also understood to apply to Rijke
oscillations. Similar oscillations can also occur when combustion takes place in a cavity.
The oscillations occur spontaneously if the combustion progresses more rapidly or
efficiently during the compression phase of the pressure oscillation than during the
rarefaction phase. Such oscillations must be suppressed in rockets to prevent catastrophic
damage, but they are deliberately encouraged in some gas-fired residential furnaces and
hot-water heaters to improve their efficiency.
Applying Rott’s mathematics to a situation where the temperature gradient along
the channel was too weak to satisfy Rayleigh’s criterion, Hofler invented a standing-wave
thermoacoustic refrigerator, and demonstrated again that Rott’s approach to acoustics in
small channels was quantitatively accurate. In this type of refrigerator, the coupled
oscillations of gas motion, temperature, and heat transfer in the sound wave are phased in
time so that heat is absorbed from a load at low temperature and waste heat is rejected to
a sink at higher temperature. The offspring of Hofler’s refrigerator are still under study
today.
Meanwhile, completely independently, pulse-tube refrigeration was becoming the
most actively investigated area of cryogenic refrigeration. This development began with
Gifford’s accidental discovery and subsequent investigation of the cooling associated
with square-wave pulses of pressure applied to one end of a pipe that was closed at the
other end. Although the relationship was not recognized at the time, this phenomenon
shared much physics with Hofler’s refrigerator (but in boundary-layer approximation).
Mikulin’s attempt at modest improvement in heat transfer in one part of this “basic”
pulse-tube refrigerator led unexpectedly to a dramatic improvement of performance, and
Radebaugh realized that the resulting “orifice” pulse-tube refrigerator was in fact a
variant of the Stirling cryocooler. Orifice pulse-tube refrigerators and Stirling
refrigerators are available today from several companies, and are used for cooling
infrared sensors on satellites as well as on Earth.
Development of Stirling engines and refrigerators started in the 19th century, the
engines at first as an alternative to steam engines. Crankshafts, multiple pistons, and other
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moving parts seemed at first to be essential. An important modern chapter in their
development began in the 1970s with the invention of “free-piston” Stirling engines and
refrigerators, in which each piston’s motion is determined by interactions between the
piston’s dynamics and the gas’s dynamics rather than by a crankshaft and connecting rod.
Analysis of
Such complex, coupled phenomenon is complicated, because the oscillating motion
causes oscillating pressure differences while simultaneously the oscillating pressure
differences cause oscillating motion. Urieli analyzed these by assuming sinusoidal time
oscillations of all important variables and using complex numbers to account for
amplitudes and time phases. Ceperley added an additional acoustic perspective to Stirling
engines and refrigerators when he realized that the time phasing between pressure and
motion oscillations in the heart of their regenerators is that of a traveling acoustic wave.
Many years later, acoustic versions of such engines were demonstrated by Yazaki,
deBlok, and Backhaus, the latter achieving a heat-to-acoustic energy efficiency
comparable to that of other mature energy conversion technologies. Stirling and
thermoacoustic-Stirling engines are under development today for applications including
spacecraft power and combined-heat-and-power systems on Earth.
Typically, the interface between one component and another is accompanied by a
dramatic change in geometry or boundary conditions, which enables a desired
macroscopic phenomenon such as refrigeration. For example, the regenerators of Stirling
engines and refrigerators have pore sizes much smaller than the thermal penetration
depthδk, and stacks of standing-wave engines and refrigerators have pore sizes
comparable to δk.
The so-called “pulse tubes” in pulse-tube refrigerators and other open spaces in other
systems are much wider than δk, and these components are insulated from their
surroundings while the heat exchangers abutting them are tied to external thermal
reservoirs. Wheatley highlighted the importance of the abrupt changes in the gas’s
environment at such interfaces between components by using the phrase “broken
thermodynamic symmetry.” In one important new development based on the
thermoacoustic approach, Olson extended Rott’s analysis of Rayleigh streaming in a tube
with an axial temperature gradient to include arbitrary p—u time phasing, and showed
13

how slightly tapering the tube can suppress Rayleigh streaming in it. This work
effectively eliminates a harmful source of heat leak in some thermoacoustic devices,
especially pulse-tube refrigerators. Another new development is based on the discovery
of thermoacoustic mixture separation by Spoor, in which radial oscillating thermal
diffusion and axial oscillating viscous motion in a gas mixture in a tube create time-
averaged separation of the components of the gas mixture along the length of the tube.
Geller has used this method in a 2.5-m long tube to separate a 50—50 helium—argon
mixture into 30% helium and 70% argon at one end and 70% helium and 30% argon at
the other end. Neon, a mixture of 9% Ne and 91% Ne, was separated to create 1%
isotope-fraction differences from end to end. The separation occurs because the sound
wave’s oscillating pressure causes radial oscillating temperature gradients in the tube,
which in turn cause opposite oscillating radial thermal diffusion of the light and heavy
components of the mixture. Thus, the two components of the gas take turns being
partially immobilized in the viscous boundary layer, so that the wave’s axial oscillating
motion carries light-enriched gas toward one end of the tube and heavy-enriched gas
toward the other end. This summary highlights only some of the interesting inventions,
discoveries, insights, and fundamental demonstrations of thermoacoustics in the past half-
century. Hoping that this review won’t already look silly next year (and mindful of
previous “experts” predicting, e.g., that no market for personal computers would “ever”
develop, or that household robots would “soon” be commonplace).
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CHAPTER 3 THEORY OF THERMOACOUSTIC
REFRIGERATION
3.1 Principle of Thermoacoustic Refrigeration
Simply put, thermo acoustic effect is the conversion of heat energy to sound
energy or vice versa. Utilizing the Thermo acoustic effect, refrigerators can be developed
that use heat as an energy source and have no moving parts!
To explain the thermo acoustic effect, consider a high amplitude sound wave in a
tube. As the sound wave travels back and forth in the tube, the gas compresses and
expands (that's what a sound wave is). When the gas compresses it heats up and when it
expands it cools off. The gas also moves back and forth, stopping to reverse direction at
the time when the gas is maximally compressed (hot) or expanded (cool).
Figure 3.1 Thermoacoustic Refrigerator Principle.
Now, put a plate of material in the tube at the same temperature as the gas before
the sound wave is started. The sound wave compresses and heats the gas. As the gas
slows to turn around and expand, the gas close to the plate gives up heat to the plate. The
gas cools slightly and the plate below the hot gas warms slightly. The gas then moves,
expands, and cools off, becoming colder than the plate. As the gas slows to turn around
15

and expand, the cool gas takes heat from the plate, heating slightly and leaving the plate
below the gas cooler than it was.
So, what has happened is one part of the plate gets cooler, and one part gets
hotter. If we stack up many plates atop each other (making sure to leave space for the
sound to go through), place the plates of an optimal length in the optimal area of the tube
and attach heat exchangers to get heat in and out of the ends of the plates, we have
created a useful refrigerator.
Even more spectacular is the fact that it can work in reverse. If we have a stack of
plates and force one end to be hot and the other cold and put that in a tube, we can create
a very loud sound. Thus by using waste heat (say from a fire) we could create sound in a
tube and use that sound to cool off another part of the tube (say where a water bottle is
sitting). We have now created a refrigerator that can cool water bottle at one end by
putting the other end in the campfire. A device that creates sound from heat is called a
thermo acoustic heat engine.
Figure 3. 2. Thermoacoustic Refrigeration cycle
3.2 Analysis of the Behavior of the Gas (refrigerant)
The process resembles the operation of a typical Stirling cycle. The analysis
given here is not the cycle’s analysis but the analysis of the behavior of the gas. Analysis
of change in pressure, temperature, velocity etc;
The temperature drop is shown in the graph is with respect to location. The drop
produced is in the region of stack only. This shows an idealized graph. But actually after
16

reaching the low temperature the temperature starts slowly rising towards the source.
This is due to the improper insulation and also due to the heavy pressure perturbations in
that region.
Figure 3.3 Pressure, Velocity and Temperature distribution in Thermoacoustic
Refrigerator.
The pressure variations and velocity changes are given briefly here. The graph in
the figure 3.3 depicts their behavior. When the wave front reaches the closed end, the
pressure reaches maximum value and to the minimum value of the sources end. Here as
the wave will reverse its direction, the velocity will be maximum. This is the same case
when wave front reaches the source’s end but the pressure magnitudes changes from
maximum to minimum and vice versa. When the pressure throughout the system is of
equal magnitude then the velocity will reach its maximum point.
These are the variations obtained for throughout the system. Now a closer look
makes the study more clear. Consider the area in stack a small portion. Shown in the
Figure 3.4. The air packet or blob, the interest of study is observed more closely. When
17

this air blob waves back and forth the temperature, pressure and volume of the blob are
plotted with respect to its location. This allows to study closely on the behavior of
refrigerant in stack. So the modifications can be made appropriately.
Figure 3.4. Behavior of the Air Packet (Blob) in the Stack.
Now the closed dead end of the stack is of main concentration. Concentrated area is
shown in the Figure 3.3 in the portion selected in the ellipse. The change in pressure and
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Figure 3.5 Temperature, pressure variation as a function of thermal penetration
depth.
Temperatures with respect to the thermal penetration depth ratio (y/dk) is plotted. Its
behavior is shown here in the Figure3.5. The loop area gives the work done by the piston
(wave front). Thus work is a function of the change in position. The derivative of work
with respect to position is displayed. This shows the dissipation per unit volume as a
function of position. It is maximum at one thermal penetration depth from the wall,
which is where the gas is farthest from both of the two ideal, reversible cases; adiabatic
and isothermal oscillations. Also note the interesting fact that the gas roughly four
penetration depths from the surface does work on its surrounding, rather than dissipating
work.
19

(wave front). Thus work is a function of the change in position. The derivative of work
with respect to position is displayed. This shows the dissipation per unit volume as a
function of position. It is maximum at one thermal penetration depth from the wall,
which is where the gas is farthest from both of the two ideal, reversible cases; adiabatic
and isothermal oscillations. Also note the interesting fact that the gas roughly four
penetration depths from the surface does work on its surrounding, rather than dissipating
work.
CHAPTER 4 MODEL CONSTRUCTION AND RESULTS
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4.1 Model Construction and Operation
The thermoacoustic refrigerator demonstration described in this note is of the
standing wave variety, and consists of a quarter wavelength resonator an open-closed
tube driven by a loudspeaker. While this is the easiest resonator shape to build, it is the
least efficient of the standing-wave type refrigerators. Since the primary purpose of this
apparatus is to demonstrate the action of an acoustic refrigerator, efficiency was not a
primary concern. A schematic drawing of the refrigerator is shown in Figure 4.1. The
resonator for this refrigerator was a 23 cm length of acrylic tubing with an inner diameter
of 2.5 cm. The length defines the resonance frequency of the system, which was 385 Hz
for our apparatus. A hole was cut in the center of an Acrylic cover sheet and the tube was
glued to the cover sheet, which was then placed over the speaker. The speaker was a
4-inch boxed speaker capable of handling 40 W, and a 4-inch diameter O-ring was used
to provide a seal around the edge of the speaker. An aluminum plug was milled to fit
snugly into the end of the tube, forming the closed end. The most important part of an
acoustic refrigerator is the stack, which consists of a large number of closely spaced
surfaces aligned parallel to the length of the resonator tube. The stack for this apparatus
was constructed, as suggested Hofler by winding a roll of 35-mm photographic film
around a central spindle so that adjacent layers of the spirally wound film provide the
stack surfaces. Lengths of 15-lb nylon fishing line separates adjacent layers of the spirally
wound film stack so that air could move between the layers along the length of the stack
parallel to the length of the resonator tube. Figure 4.3.b shows a cross section of the
rolled-film stack, with layers separated by fishing line. The primary constraint in
designing the stack is the fact that stack layers need to be a few thermal penetration
depths apart, with four thermal penetration depths being the optimum layer separation.
The thermal penetration depth, δk, is defined as the distance that heat can diffuse through
a gas during the time t=1/πf , where f is the frequency of the standing wave. It depends
on the thermal conductivity, k, and density, r, of the gas and the isobaric specific heat per
unit mass, cp according to δk=√(k/πfρcp) if stack layers are too far apart the gas cannot
effectively transfer heat to and from the stack walls. If the layers are too close together
viscous effects hamper the motion of the gas particles. For a frequency of 385 Hz in air
21

one thermal penetration depth is 1.33x 10-4
m. The 15-lb nylon fishing line has a diameter
of 3.40x10-4
m; the stack layers in this apparatus were therefore separated by about 2.5
thermal penetration depths.
Figure 4.1.
a. Schematic diagram of the demonstration thermoacoustic
refrigerator;
b. Cross section of the stack showing how the film layers were
separated by fishing line.
To construct the stack, a roll of 35-mm film was unrolled. Lengths of fishing line
were glued across the width of the film at equal intervals using a spray adhesive. To keep
lines straight the line was first wound onto a ‘‘loom,’’ a cardboard frame with slits cut
every 5 mm. After spraying the glue onto the lines, the frame was placed over the film
and a Teflon weight was placed on top, to press the lines against the film. Once the glue
was set, the fishing line was cut flush with the edges of the film. This process was
repeated for approximately 1 meter of film. The film was then rolled around a small
diameter acrylic rod and layers were gradually peeled off until the film roll fit snugly into
the tube. The stack was positioned in the tube approximately 4 cm from the closed end so
as to be close to the pressure maximum, but away from the particle displacement
22

minimum. Two thermocouples were made by soldering copper and constantan wires
together. One thermocouple was inserted through the outermost winding of the stack to
detect the temperature below the stack, while the other was allowed to dangle just above
the stack. Digital displays were used to display the temperature above and below the
stack. The loudspeaker was driven by a sine wave generator through a 100 W audio
amplifier. The pressure amplitude inside the resonator tube was not measured, but the
power to the speaker was increased until a second harmonic became barely audible,
indicating that the system was becoming nonlinear.
4.2 Data Acquisition Summary of Experimental Results
The apparatus is made to run for about 10 minutes at a high sound level. The
temperature below the stack; it started at 66 °F and dropped to 29 °F. Temperature above
the stack; it started at 66 °F and increased to 75 °F. A temperature difference of 46 °F
~25.6 °C was obtained across the stack after just 10 minutes with air as the ‘‘coolant,’’
and with the loudspeaker cone being the only moving mechanical part. Figure4. 2 shows
the typical results for the temperatures above the stack (Thot) and below the stack (Tcold) as
a function of time. The starting temperatures were normalized to zero, so the plot shows
the changes in temperature as measured by each thermocouple. The plot shows that the
temperature below the stack (Tcold) begins decreasing immediately after the sound is
turned on, dropping 4 °C in the first 15 seconds, with the rate of temperature change
decreasing with time. After 4 minutes of operation the temperature below the stack has
dropped by 10.5 °C and is still decreasing. The temperature above the stack (Thot)
increases, also more rapidly at first, as the heat is being pumped through the stack. After
approximately 2 minutes the temperature above the stack has increased by 5 °C. After
that it stops increasing as the rate at which heat is moved through the stack equals the rate
at which heat is conducted through the aluminum cap into the surrounding room. After 4
minutes of operation, the temperature difference between the top and bottom of the stack
is about 15.5 °C, a difference large enough to be detected by touching a finger along the
outside of the acrylic tube. The trends in Figure 4.2 are similar to those found in the
literature.
23

Figure 4.2. Temperature variation above (Thot) and below (Tcold) the stack as a
function of time.
CHAPTER 5 ANALYSIS OF THE REFRIGERATOR
24

5.1 Objectives and Assumptions
5.1.1 Objectives
The objective of this project is to develop the model for demonstrating the
phenomenon of thermoacoustic refrigeration. By developing such model the project
demonstrates that the drop in temperature is possible by using acoustic waves. There are
no moving parts in this model which makes frictionless operation possible there by
eliminating unnecessary vibrations which makes the system more stable. In addition to
that comparison is made between the conventional refrigerators and the thermoacoustic
refrigerator and is shown that provided the proper equipment, correct analysis of the
process makes this system to reach the performance of the conventional systems.
5.1.2 Assumptions
The word “thermo acoustics” represents one unifying analytical and conceptual
approach to all of these devices and phenomena. The thermoacoustic approach begins
with the assumptions that the oscillations of pressure p, temperature T, density ρ, velocity
u, and entropy s can be thought of as “small” and that they are adequately represented as
sinusoidal functions of time. Results of engineering interest are obtained as time-
averaged products of the oscillating variables: heat fluxes are proportional to the product
of T and u, work to the product of p and u, mass fluxes to the product of ρ and u, etc.
Surprisingly, despite the assumption that the oscillations must be small and mono
frequency, the results of the thermoacoustic approach are usefully accurate even for large
oscillations with substantial harmonic content. The spatial dependences of the amplitudes
and time phases of the oscillating variables can be very complex, varying smoothly
within components and abruptly at the interfaces between components.
In this refrigerator the heat exchangers to transfer heat from and to the system are
not present. The insulation provided is also of low quality. So, while analyzing the
system all these parameters should be kept in view so that an appropriate analysis of the
system is made.
5.2 Analysis of the Model
25

In thermoacoustic refrigerator the external work is supplied by the standing sound
wave in the resonator. The longitudinal standing sound wave causes the gas particles to
oscillate back and forth parallel to the walls of the stack. The alternating compression and
rarefaction of the gas causes the local temperature of the gas to oscillate due to the
adiabatic nature of sound waves. If the local temperature of the gas becomes higher than
that of the nearby stack wall, heat is transferred from the gas to the stack wall. If the local
temperature of the gas drops below that of the stack wall, heat is transferred from the wall
to the gas. The second most important factor in the performance of a thermoacoustic
refrigerator is the critical longitudinal temperature gradient▼Tcrit=p/ξρcp, where p and ξ
are the acoustic pressure and displacement amplitudes, respectively. No heat is
transferred when the peak-to-peak temperature variation caused by adiabatic compression
of the gas, 2p/ρcp exactly matches the variation in the local wall temperature, 2ξ▼Tcrit,
between the extremes of the gas particle motion. Only when the sound wave induced
temperature variation in the gas is greater than the temperature gradient between the cold
and hot ends of the stack will heat be moved from lower temperature to higher
temperature causing refrigeration. This requires a rather intense sound wave inside the
resonator. A boxed loudspeaker with as tight a seal as possible between the speaker and
resonator helps to reduce the sound level in the room to tolerable levels. The
thermoacoustic refrigeration cycle is illustrated in Figure 5.1. As the motion of the sound
wave causes a gas parcel in the stack to move left towards the closed end of the tube.
The pressure increases and the gas is compressed. The compressed gas parcel is now
hotter than the nearby stack wall so it dumps heat to the cooler stack, thus shrinking in
volume. As the standing wave continues through its cycle the parcel is pulled back to the
right where the pressure is lower. The rarefied parcel is now cooler than the nearby stack
wall so it absorbs heat from the warmer stack wall and expands. The cycle repeats with
the net effect of a small amount of heat being moved a short distance along the stack
from the colder towards the hotter end. A ‘‘bucket brigade’’of particles can move a
significant amount of heat from one end of the stack to the other.
26

Figure 5.1. P–V diagram showing the four stages in the thermoacoustic refrigerator
cycle. The left end of the stack wall is towards the closed end of the
resonator tube.
5.3 First Law Analysis
In this section the first law analysis of the thermoacoustic refrigerator is given. As
a result of the analysis it is shown that for optimization purposes the thermoacoustic
refrigerator should be divided into four main modules.
(i) Thermoacoustic Core,
(ii) Resonance Tube,
(iii) Heat Exchangers and
(iv) Acoustic Driver.
This modular description is suitable for design purposes as it allows the designer
to optimize each module separately, and obtain a global performance maximum of the
thermo acoustic refrigerator as a result.
To describe the overall thermodynamic performance of refrigerators, the
coefficient of performance, COP, defined as
COP = heat extracted at lower temperature Tr
Work done on the machine
= Qload (1)
Eel
27

Is typically used. For a thermoacoustic refrigerator, the overall COP can be defined as the
ratio of the cooling load Qload, introduced in to the system through the cold heat exchanger
from the cold temperature reservoir, to the electric power input, Eel, in to the acoustic
driver. The conservations of these two energies with in the components of the
thermoacoustic refrigerator system are illustrated in the energy flux diagram in figure 5.2.
The electric power input Eel, introduced in to the system, is converted in to the
acoustic power Wtot by the acoustic driver. For this energy conversion the electroacoustic
efficiency ηelac can be defined as follows
ηelac= Wtot = Wtc+Wdis = Wtc+Wres+Wex (2)
Eel Eel Eel
The electro acoustic efficiency ηelac is of order of 3% for commercial loudspeakers. For
acoustic drivers dedicated for thermoacoustic refrigerators an efficiency of 50% has been
demonstrated and values up to 90% should be possible.
Equation (2) shows that not all of the converted acoustic power Wtot can be
exploited to pump heat through the thermoacoustic core. The reason for this is that the
thermoacoustic effect occurs not only within the thermoacoustic core, where it is
responsible for the pumping of heat, described by the component Wtc in the equation(2).
Simultaneously, it occurs on surfaces of the resonance tube and the heat exchangers,
where its contribution is dissipative, Wdis, as indicated in the equation(2). These two
components, Wtc and Wdis, comprising the total acoustic power Wtot, are shown in figure
5.2. The acoustic power losses with in the hot portion of the resonance tube and at the hot
heat exchanger (included in terms Wres and Wex in equation (2)and shown as Whres and
Whex in figure 5.2, respectively), acts as additional heating loads on the hot heat
exchangers. The acoustic power losses in the cold portion of the resonance tube and at the
cold heat exchanger (shown as Wcres and Wcex in figure 5.2, respectively),acts as
additional cooling loads on the cold heat exchanger. Considering figure 5.2 and
remembering the fact that heat pumping takes place within the thermoacoustic core, we
con clued that the upper limit of the thermoacoustic refrigerator’s performance is
determined by the performance of the thermoacoustic core, defined as
COPtc = heat extracted at lower temperature Tc
Work done on the thermoacoustic core
= Qc (3)
Wtc
28

Figure 5.2 Energy Fluxes in Thermoacoustic Refrigerator.
The COPtc can be estimated applying a simplified linear model describing
thermoacoustic processes, the short stack boundary layer approximation.
The least understood and analyzed components of the thermoacoustic
refrigerators are the two heat exchangers. In previously built thermoacoustic refrigerators,
they were designed with out any optimization, and in some cases they can cause a
substantial flow blockage with attendant large pressure drop penalty. This pressure drop
is another mechanism that contributes to the dissipation of acoustic power with in the
heat exchangers. The difficulty of modeling thermoacoustic heat transfer within the heat
exchangers can be attributed to the fact that we are deal in with oscillatory flow with zero
mean velocity. In such situation, standard heat exchanger design methods, such as
29

effectiveness-NTU(number of transfer units) method or the LMTD(logarithmic mean
temperature difference) method cannot be applied directly. Nevertheless, we can define
an effectiveness Є for the cold heat exchanger, as done in conventional heat exchanger
analysis, as
Є == heat transferred = Qload
Maximum heat transferable Qc
=1 - Qcdis = 1 - Qcres + Qcex (4)
Qc Qc
Such a definition can be introduced, because the heat transfer through the cold heat
exchanger is limited by the amount of heat the thermoacoustic core is capable of
absorbing at the cold temperature Tc. Therefore the maximum heat transferable
corresponds to Qc. How ever, through dissipation of acoustic power, as discussed above,
the transferred heat is limited to the cooling load Qload.
Substituting equations (2), (3) and (4) into equation (1) we obtain the following
expression for the overall COP of a thermoacoustic refrigerator:
COP = ηelacЄ Wtc Qc = ηelacЄηacCOPtc (5)
Wtot Wtc
We should note the presence of an additional factor, the acoustic power efficiency
ηac=Wtc/Wtot in equation (5). This factor accounts for the fact that not all of the converted
acoustic power Wtot can be used to pump heat through the thermoacoustic core. In other
words, the acoustic power efficiency ηac achieves its maximum value of one, when there
is no acoustic power dissipation, Wdis=0, in the heat exchangers and in the resonance
tube. By definition, the maximum possible values for the electroacoustic efficiency ηelac
and the effectiveness Є of the cold heat exchanger are also one. Thus, the overall COP of
the thermoacoustic refrigeration system reaches its maximum value, that corresponds to
the COPtc of the thermoacoustic core, when the two efficiencies, ηelac and ηac, as well as
the effectiveness Є reach their maximum value of one, which is the case of optimization
algorithm described in our paper. For the optimization of the thermoacoustic refrigerator
system this result suggests the option to the design and optimize the four modules:
(i) Thermoacoustic Core, (ii) Resonance Tube, (iii) Heat Exchangers and (iv) Acoustic
Driver separately. Reducing the heat losses in the acoustic driver improves the
electroacoustic efficiency ηelac. Coming up with an optimum design of the cold heat
exchanger by decreasing the acoustic power losses Wcex, improves the effectiveness Є as
30

well as the acoustic power efficiency ηac. And finally, decreasing the acoustic power
losses in the resonance tube Wres as well as in the hot heat exchangers Whex improves the
acoustic power losses Wcres in the cold portion of the resonance tube, the effectiveness Є
of the cold heat exchanger increases as well.
The above statement implies that at this stage of the analysis, it is assumed that
the four elements of equation (5), ηelac, Є, ηac and COPtc are independent. This assumption
certainly holds for ηelac. Since the COPtc for some design solutions may depend on the
design of the resonance tube and the heat exchangers, interdependence of COPtc, Є and
ηac will have to be considered in the future. Clearly, this would require an accurate model
of heat exchangers, one that is not yet available. Because of this an effectiveness Є for
the hot heat exchanger is not included, since it is not known what the limiting
mechanisms for its performance evaluation are.
5.4 Results and Discussion
This simple and inexpensive thermoacoustic refrigerator effectively demonstrates
the basic physical principles behind its operation. As shown, however, it is rather
inefficient as a heat transfer device. If both ends of the stack were connected to heat
exchangers, thus coupling the stack to a heat source or heat sink, the transfer of heat
would be more efficient. Other improvements could be made by modifying the shape of
the resonator4 or increasing the stack layer separation to an optimal four thermal
penetration depths. One could also study the performance as a function of sound level
inside the resonator. Such studies might make for an interesting senior research project.
5.5 Comparison with conventional systems
The efficiency of the best thermo acoustic devices is well below an optimized
conventional design, but about the same as the cheapest commercial designs. Given that
thermo acoustics is less efficient than conventional designs (and perhaps always will be)
why would someone be interested in thermo acoustics? There are many reasons but most
prominent are environmental ecology and reliability.
31

At the present time, the efficiency of thermo acoustic refrigerators is 20-
30% lower than their vapor compression counterparts. Part of that lower efficiency is due
to the intrinsic irreversibility’s of the thermo acoustic heat transport process. These
intrinsic irreversibility’s are also the favorable aspects of the cycle, since they make for
mechanical simplicity, with few or no moving parts. A greater part of the inefficiency of
current thermo acoustic refrigerators is simply due to technical immaturity. With time,
improvements in heat exchangers and other sub-systems should narrow the gap. It is also
likely that the efficiency in many applications will improve due only to the fact that
thermo acoustic refrigerators are well suited to proportional control. One can easily and
continuously control the cooling capacity of a thermo acoustic refrigerator so that its
output can be adjusted accurately for varying load conditions. This could lead to higher
efficiencies than conventional vapor compression chillers which have constant
displacement compressors and are therefore only capable of binary (on/off) control.
Proportional control avoids losses due to start-up surges in conventional compressors and
reduces the inefficiencies in the heat exchangers, since the proportional systems can
operate over smaller temperature gaps between the coolant fluid and the heat load.
The second law of thermodynamics sets an absolute limit on the performance
("efficiency") of a refrigerator of any design. The larger the temperature difference which
a refrigerator must produce, the less efficient it can be, even if it is perfectly designed and
built. One feature of thermo acoustic devices which may allow them to overcome some
of the inefficiency of the cycle is that they can use proportional control. Proportional
control means that the output of the device may be turned up or down gradually
depending on conditions. A dimmer switch on a lamp is an example of this kind of
control. In contrast, an ordinary light switch is an example of binary control-it is either on
or off, with no in-between. A vapor compression refrigerator uses binary control: it
comes on for a while, and then it goes off. If the conditions require more output, the unit
comes on more frequently, but it is never partially on. A thermo acoustic cooler, on the
other hand, can be partially on. The advantage to this is that the less hard a refrigerator is
working, the more efficient it becomes. When producing maximum output, a vapor
compression refrigerator is more efficient than a thermo acoustic fridge of the same
capacity, but when less output is needed (which is most of the time), the thermo acoustic
32

device increases in efficiency, but the vapor compression fridge does not. There are other
advantages to proportional control. You can imagine that it would be nicer if your home
air conditioner would keep the house at a constant cool temperature rather than cycling
between somewhat too hot and somewhat too cold. Similarly, the performance and
lifetime of some types of electronics could be increase by the steadier temperatures
available through proportional control. Proportional control also eliminates the
electronics-damaging "power surges" that occur throughout the electrical system when
the compressor in a conventional chiller turns on or off.
In the mid 1990's, the production of CFC refrigerants (mainly Freon) was banned
by the Montreal Protocol. CFCs are a major player in the depletion of the ozone layer. In
addition, CFC's are nasty greenhouse gasses. While the recent HFC and HCFC
replacements are less harmful, they will still be major contributors to the greenhouse
effect and there is concern over possible health hazards for many of the newer chemicals.
In addition, since both HFCs and HCFCs are expected to be banned early in the 21st
century, an entirely new refrigeration technology needs to be developed. Thermo acoustic
refrigerators utilize no environmentally hazardous gasses. They use inert gasses which
are both readily available, inexhaustible, and completely environmentally benign! And,
because they can utilize waste heat as an energy source, they are extremely
environmentally friendly.
In addition, since thermo acoustic refrigerators do not use the compressors,
lubricants, sliding seals, and other gizmos present in vapor compression refrigerators, the
thermo acoustic refrigerator should be more reliable. In fact, a heat driven refrigerator
does not need any moving parts in the refrigeration cycle!
For all thermodynamic devices, there will always be a trade-off between
efficiency and power density. For the small power devices built thus far (less than 1,400
Btu/hr = 400 W thermal) and the larger devices currently under construction (36,000
Btu/hr = 10 kW thermal), the size and weight are similar to their vapor compression
equivalents. The cooling capacity of vapor compression units depends upon operating
pressure and the amount of phase-change fluid. The size of a thermo acoustic device is
determined (roughly) by its operating frequency. If small size is important, higher
frequency operation may be required.
33

Finally, since thermo acoustic systems tend to work at higher frequencies and at
single frequencies, noise and vibration control is actually easier than for a conventional
refrigerator - so thermo acoustic refrigerators should be even quieter than the refrigerators
of today.
The new thermo acoustic-Stirling cycle heat engine developed is a fantastic
breakthrough. It provides high efficiency basis for the conversion of heat to mechanical
power without moving parts. Whether it is used to drive a refrigerator or an electric
generator, the new heat engine looks to be the beginning of a revolution.
34

CHAPTER 6 THERMAL-RELAXATION DISSIPATION IN
THERMOACOUSTIC SYSTEMS
6.1. Introduction
Pressure oscillations in a sound wave are accompanied by temperature
oscillations. In the presence of a solid boundary, the heat transfer from the oscillating gas
to the solid boundary causes dissipation of the acoustic energy. This results in the
attenuation of the sound wave. This thermal-relaxation dissipation process has a negative
effect on the performance of thermoacoustic heat pumps and engines. A simple
analytical model describing the interaction between an acoustic wave and a solid
boundary is presented. The effect of the solid material and gas type on thermal-relaxation
dissipation is analyzed. The main result of this model is that the choice of a solid material
with the smallest possible heat capacity per unit area in combination with a gas with the
largest possible heat capacity per unit area minimizes the thermal-relaxation dissipation.
From the different combinations solid-gas used in the calculations, the combination cork-
helium leads to the lowest thermal attenuation of the sound wave. In this case, the heat
transfer from the gas to the wall less damps the temperature oscillations. However,
because of the porosity of cork that may cause some problems, it is suggested that the
combination polyester-helium can be used in practice to minimize the thermal-relaxation
losses. Thermoacoustic heat pumps are devices that use acoustic power to transfer heat
from a low temperature to a high temperature source. Reversibly, thermoacoustic engines
are systems that use a temperature difference to produce sound. Typically, standing-wave
thermoacoustic devices consist mainly of an acoustic resonator filled with a gas. In the
resonator, a stack consisting of a number of parallel plates, and two heat exchangers, are
appropriately installed. In the case of an engine (Figure 6.1), the heat exchangers are used
to maintain a temperature gradient over the stack. The hot heat exchanger supplies heat
Qh to the hot end of the stack and the cold heat exchanger extracts the heat Qc from the
cold end of the stack. The stack is the heart of the engine where the thermoacoustic cycle
is generated. A detailed description of these systems and the way they work can be found
in the literature. One of the ways in which sound waves are affected near solid boundaries
35

is by viscosity and thermal-relaxation. This has a negative effect on the performance of
thermoacoustic systems.
The viscous losses are due to the viscosity that dissipates acoustic energy by
viscous shear in the viscous boundary. To reduce the effect of viscosity, and hence to
improve the performance of thermoacoustic systems, gas-mixtures with a low Prandtl
number can be used. This has resulted in a 70 % improvement of the performance of a
thermoacoustic cooler. The thermal relaxation loss is due to dissipation of the acoustic
energy in the thermal boundary layer. The magnitude of this effect depends on the
temperature difference between the core of the gas and the solid boundary. Although an
analytical expression for the thermo-viscous dissipation was derived, a detailed analysis
of the thermal-relaxation dissipation and possible solutions to reduce its negative effect
on thermoacoustic systems is still lacking.
The aim is to present a simple analytical model describing the interaction between
the periodic temperature changes in a gas with a solid boundary. The effect of the solid
material and gas type on this interaction is analyzed. The calculation results of the model
will be presented and commented and some conclusions related to the minimization of
thermal relaxation dissipation in thermoacoustic systems will be drawn. To the
knowledge of the authors it is the first time that such model is used to analyze this effect.
Figure 6.1 A simple illustration of a thermoacoustic engine. A stack of parallel plates
and two heat exchangers are placed in a gas filled resonator. Heat Qh is supplied to the
engine at temperature Th, and the waste heat Qc is extracted at temperature Tc, so that
sound is produced.
36

The remaining of this paper is organized as follows: Section 2 is devoted to the analytical
model describing the heat transfer from the sound wave to the solid wall. Section 3 is
devoted to the outline of the model results. In section 4, a discussion and interpretation of
the model results is given. In the last section some conclusions related to the
minimization of the thermal-relaxation dissipation in thermoacoustic systems are
summarized.
6.2. Formulation of the problem
The thermal-relaxation attenuation of an acoustic wave in contact with a solid
surface can be dealt with by considering the interaction between the periodic temperature
changes of a gas and a solid boundary. This problem is analogue to a problem that was of
interest in mechanical engineering. In an internal combustion engine, for instance, the gas
temperature undergoes cyclic changes, and it was important to learn how far the cylinder
wall follows these changes (and thus periodic thermal stress). The high temperature used
in combustion engines does not damage the cylinder wall, only because of the fact that
the temperature changes are greatly damped by the transfer of heat.
The pressure oscillations in a sound are accompanied by temperature oscillations.
Heat transfer at boundaries usually damps these periodic temperature changes. This
dissipation of the adiabatic compressive energy results in the attenuation of the sound
wave. The physical origin for this dissipation process can be understood by considering
an acoustic wave in a tube (Figure 6.1). In response to the acoustic wave, the gas in the
tube oscillates and is compressed and expanded. As a consequence of this, oscillating
temperature gradients in the direction normal to the tube wall are produced. During the
forward part of the oscillation, the sound wave compresses the gas and heat flow takes
place from the hotter gas within the core to the colder gas in the thermal boundary layer
and to the colder wall tube. During this thermal-relaxation step, the gas experiences
thermal contraction at high pressure. During the reverse phase of oscillation, the sound
wave expands the gas that becomes colder and heat diffuses from the gas in the boundary
layer and wall into the gas core. During this step, the gas experiences thermal expansion
at low pressure. Since the gas contracts at high pressure and expands at low pressure, a
net work is done on it by the sound wave. The work lost from the sound wave is
37

dissipated and appears as heat near the average temperature Tm. Because the loss of
acoustic energy has a negative effect on the efficiency of thermoacoustic systems, it is
important to minimize this dissipation process. The thermal dissipation of acoustic power
takes place over the whole internal tube-, stack- and heat exchangers surfaces (Figure
6.1). The geometry used in the analysis and discussion of the thermal-relaxation
attenuation is illustrated in Figure 6.2. In this geometry a portion of the tube wall (or a
stack plate) of Figure 6.1 is shown. The x-axis is along the direction of acoustic vibration
and the y-axis perpendicular to the wall of the tube, with y=0 at the gas-wall interface.
Figure 6.2 Illustration used in the analysis and discussion of the thermal-relaxation
dissipation. The thermal boundary layer of the gas, δk, and that of the solid, δs, are also
illustrated. The interface gas-wall is at y=0
We assume that in the gas a one-dimensional acoustic wave with frequency . exists and
that the gas is confined in an acoustic resonator (tube). The temperature oscillations in the
gas have the same frequency, .. We suppose that the temperature of the gas is given by:
(1)
Where Tm is the mean temperature of the gas and Ta is the amplitude of the temperature
oscillation in the gas. Re( ) signifies the real part. Furthermore, we consider only
unidirectional heat flow in y-direction, so that the temperature in the wall will be given
by the partial differential equation:
38

(2)
where ks is the thermal diffusivity of the wall material. We seek a solution of type:
(3)
Where Tm is the mean temperature of the wall and the second term on the right of
equation (3) represents the temperature excess of a point of the wall over the mean
temperature. Tb(y) is complex to account for both the amplitude and the time phasing.
The boundary condition for equation (2) is given by:
(4)
where Ks is the thermal conductivity of the wall. We notice that ks=Ks/ρscs , where ρs and
cs are the density and specific heat per unit mass of the wall, respectively. The parameter
h is the heat transfer coefficient between the gas and the wall. Substituting (3) in (2) it
follows that Tb(y) must satisfy
(5)
The solution of (5), which is finite as y→∞ is:
(6)
where A is an integration constant and δs is given by:
(7)
The thermal boundary layer of the wall, δs, is the distance the temperature wave travels in
a time interval 1/ω. The solution of Eq. (2) that satisfies the boundary condition (4) is:
39

(8)
Where b = h/Ks, and m= δs
−1
. Evaluation of the real part of expression (8) yields (9):
(9)
Substituting Tw from Eq. (9), performing the differentiation, and finally substituting y=0
yield:
(13)
Eq. (13) represents the heat flow entering the wall per unit area.
6.3 Results
Expression (9) represents the excess temperature of a point in the wall over the
mean temperature Tm. The temperature distribution from Eq. (9) oscillates with the same
40

frequency . as the temperature in the gas. Eq. (9) describes a progressive temperature
wave of wave number m and wavelength . given by:
(14)
Some features of the temperature oscillations in the wall can be recognized from Eq. (9):
• The frequency,ω , and the sinusoidal form of the temperature wave in the wall do
not change with the depth y; only its amplitude decreases like:
(15)
and thus falls off more rapidly for large ω. At a distance of one thermal boundary layer
the amplitude is reduced by a factor e-1
, so the temperature waves are strongly attenuated.
• With increasing y, the amplitude of oscillation is delayed; it occurs at the time:
(16)
• Because the temperature maximum (minimum) always needs a time interval,
ω/2π, to proceed by a distance λ, its velocity is:
(17)
Since the aim of this study is to learn how to minimize the thermal-relaxation losses, we
are thus more interested in the temperature excess at the surface (y=0) of the wall which
is given by:
(18)
Expression (18) shows that only at the instant t=φ/ω does the temperature at the surface
have its first maximum. The ratio φ/ω is the time lag of the amplitude of oscillation at
y=0, counted from the previous amplitude of the temperature excess of the gas. Since
B=1, the amplitude at the surface is smaller than in the gas, and B is the fraction to which
the amplitude is reduced by the heat transfer to the wall (thermal-relaxation).
41

Equations (10) and (11) show that the time lag along with the amplitude of the
temperature at the surface is both a function of the ratio m/b. This ratio is in its turn
controlled by the thermo physical parameters of the gas and the wall material. The
temperature at the surface of the wall would have the same temperature amplitude and in
phase with the temperature in the gas whenever the ratio m/b → 0. In this extreme case
only there would be no thermal-relaxation losses. On the other hand if m/b →∞, then
Tw-Tm → 0 which would form a boundary condition on the gas temperature. The heat
transfer coefficient is a function of many variables that are specific for the flow and heat
conduction of a gas. Our objective is to determine the ratio m/b as function of the thermo
physical parameters of the gas and wall material so that men can learn how to minimize
this ratio. The next step is to determine the heat transfer coefficient. The sound wave in
the tube causes the gas to oscillate along the surface of the tube while compressing and
expanding. From the adiabatic gas law pvγ
=constant, the change in the gas core
temperature Ta, due to acoustic pressure change, p, can be expressed in terms of the mean
temperature and pressure, Tm and pm, as:
(19)
where γ is the ratio of the specific heat at constant pressure to the specific heat at constant
volume. Since in an acoustic wave, the dynamic pressure p is a function of location x, the
temperature Ta is also a function of location. Therefore, a local convective heat transfer
from an acoustically oscillating flow has to be considered. Since the radius of the tube is
large compared to the thermal boundary layer, the curvature of the tube wall can be
neglected and we can consider the problem of a laminar flow over a flat plate. The
correlation function for this case is given by:
(20)
42

Where Nu is the average Nusselt number, L is a characteristic length, Pr is the Prandtl
number, and Re is the Reynolds number defined as [10]:
(21)
Where <u> is the time average of the acoustic velocity over one half cycle. In the case of
a standing wave the amplitude of the acoustic velocity in the tube is given by:
(22)
Where ka is the acoustic wave number (ka=ω /a), a is the sound velocity, and x is the
distance from the nearest velocity node. The use of expression (20) means that although
the acoustic character of the flow, the corresponding heat-transfer rate equals the rate in
steady flow with the same instantaneous speed. Because the heat transfer is independent
of the flow direction, the net heat-transfer is obtained by using the time average of the
acoustic velocity over one half cycle. In the stack where the spacing between the plates is
very small, the correlation function (20) holds indeed. In the resonator where the flow
may be turbulent and because of the curvature of the wall tube, expression (20) may be
less appropriate. However, this has no consequences for our analysis and expression (20)
is also used for the resonator to keep our model as simple as possible. The use of a
correlation function of a turbulent fully developed flow in a tube leads to the same
analysis results as in Figure 3.1. Thus expression (20) will also be used for the tube to
keep our model as simple as possible. The harmonic character of the flow suggests also
that the length L can be given by the peak displacement of the gas particles, L=u/ω. From
Eqs.(20), (21), and (22) it follows then
(23)
In the derivation of Eq.(23) the expression of the thermal conductivity for a pure
gas Kg=15µ gR/4M and cp=5R/2M have been used. Where R is the universal gas
43

constant, ρg is the gas density, M is the molecular weight, and cp is the isobaric specific
heat per unit mass. Using the definition of m and b with expression (23) leads to the
expression:
(24)
Where the constant has a value of order one for gases. Expression (24) shows that
the ratio m/b is proportional to the square root of the ratio of the heat capacities per unit
area of the solid and the gas. Notice that expression (24) can also be written as a function
of the ratio of the solid's thermal boundary layer and that of the gas. The numerical values
of the thickness of the solid’s boundary layer and that of the gas are given in Table 3.1.
Since the ideal ratio m/b → 0 can not be realized in practice, one can only try to
minimize the ratio by choosing a wall material with the lowest possible combination Ks ρs
cs and a gas with the largest possible Kg ρg cp. Expression (24) is closely related to the
reciprocal of the parameter εs:
(25)
This parameter was introduced in Thermoacoustics to characterize the fact that if Ks ρs cs
is not large compared to Kg ρg cp, then Ta at the boundary is non zero. The temperature
change experienced by the thermal boundary layer of the gas as it relaxes to the boundary
layer of the wall is thereby reduced; consequently, the hydrodynamic heat flux is
reduced. Since in the stack the hydrodynamic flux has to be made maximal
(thermoacoustic effect), the combination Ks ρs cs has to be large compared to Kg ρg cp
(m/b large). As a consequence, the stack material has to be chosen with a large ratio m/b,
to improve the hydrodynamic heat transfer between the gas and the stack. On the other
hand to minimize the thermal-relaxation at the resonator surface, the resonator material
has to be chosen with a small ratio m/b.
44

Table 6.1 Thermo-physical parameters of some gases and wall materials. The
boundary layer thickness in gas and solid are also given
In Figure 6.3 is the temperature distribution from Eq.(9) plotted as function of the
dimensionless depth y/δs for six different combinations gas-wall material, at three
different instants t1=φ/ω, t2=τ/4, and t3==τ/2. Where τ =ω/2π is the time needed for one
full cycle of the wave. As discussed in section 3, t1 is the instant at which the temperature
at the surface has its first maximum (Eq.16). The thermo-physical properties of the gases
and wall materials used in the calculations are given in Table 6.1. The numerical values
of the ratio m/b are given in Table 6.2. In the calculations a mean pressure pm=10 bar, a
mean temperature Tm =300 K, a frequency f=50 Hz (f=ω/2π), and a practical value of the
gas temperature amplitude Ta=10 K are used.
Table 6.2 Numerical values of the ratio m/b for six different combinations solid-gas
Figure 6.3 shows that the temperature distribution in the wall at subsequent instants of
time (t1<t2< t3) has the character of a wave moving into the wall. The full line represents
the temperature distribution curve at t1= φ/ω, the dashed line the distribution at t2=τ/4,
and the dotted line the distribution at t3==τ/2. The dash-dot lines represent the envelope
functions ±Be-y/δs
(Figure 6.3a). For all calculation results for the six combinations wall-
gas shown in Figure 6.3, the temperature wave decreases with an exponential function of
the dimensionless depth y/δs Eq.(9). Furthermore, the temperature wave completely
disappears at a depth of approximately 4δs. This implies that a minimum thickness for the
45

stack plate of about 8δs is necessary, so that the thermoacoustic interaction gas-stack
would be optimal.
Figure 6.3 Temperature distribution in the wall (y>0) as function of the dimensionless
depth y/ δs, at three different instants t1=φ/ω, t2= τ/4, and t3= τ/2, and for six
different
combinations wall-gas. The period τ = ω/2 π is the time needed for one full cycle of
the wave. The wall is in contact with a gas (y<0) where a sound wave exists with an
oscillating temperature of amplitude Ta=10K. The full line represents the temperature
distribution curve at t1, the dashed line the distribution at t2, and the dotted line the
distribution at t3 (t1< t2< t3). The dash-dot lines represent the envelope functions
46

±Be-y/ δs
6.4 Discussion
In Figure 6.3, the region y>0 corresponds to the wall and y<0 to the gas where a
periodic temperature oscillation of amplitude Ta=10 k is present. Figure 6.3 shows how
the temperature oscillations are damped by heat transfer to the wall surface (y=0). From
the different combinations gas-wall used in the calculations, the combination stainless
steel-argon leads to the largest damping. The temperature amplitude at the wall surface is
about 10-3
of that in the gas (Figure 6.3a). The parameter m/b=1096 is large for this
combination (c.f. Table 3.2). Figure 6.3f shows that the combination cork-helium leads to
the lowest damping, as the temperature oscillations at the surface of the cork wall follows
better the oscillations in helium gas. This combination has the lowest ratio m/b=5.9. All
other combinations gas-wall lead to intermediate results between the results of cork-
helium and stainless steel-argon (Figure 6.3). Since cork is not an appropriate material
that can be used in the construction of an acoustic resonator at high pressure, one can use
a thin layer of cork (4δs =0.17 mm) inside a metallic resonator. The inside surface of cork
layer has to be very smooth, so that viscous losses can be minimised.
Although, the cork-helium combination leads to the lowest thermal losses, it is
clear from Figure 6.3f that the heat transfer still has an appreciable effect. The
temperature at the surface of the wall is only about 10 % of that in the gas. In the
boundary-layer approximation, in which all dimensions of the resonator are much larger
than the boundary layers, an expression for thermal-relaxation dissipation, E, was
derived, and it is given by:
(26)
Where the thermal boundary layer of the gas, δk, is given by:
47

(27)
Expression (26) represents the acoustic energy dissipated by thermal-relaxation per unit
of surface area of the tube, within the thermal boundary layer δk near the wall (c.f. Figure
6.2). Expression (26) shows that increasing εs (decreasing m/b) will result in a decrease of
the thermal relaxation dissipation. This result is equivalent to the previous discussion
concerning m/b. Expression (26) shows also that the energy dissipated by thermal-
relaxation is proportional to the root square of the mean pressure pm (because δkpm~ pm
1/2
)
at constant drive ratio. The drive ratio is defined as the ratio of the acoustic pressure
amplitude p to the mean pressure of the gas pm. This suggests that using a high mean
pressure, will result in an increase of the losses at the same drive ratio. Furthermore,
thermal relaxation dissipation is proportional to the square of the drive ratio. As
expression (26) represents the energy dissipated by thermal-relaxation per unit area, this
suggests that the shape of the resonator can be optimized to obtain minimal loss. In
general, a minimal surface has to be used in regions of high dynamic pressure (pressure
Antinode). We note that, contrary to the expression of the acoustic energy dissipated by
thermal-relaxation, the expression of the energy dissipated by viscous shear does not
depend on the parameters of the wall material. This means that walls of different
materials but with surfaces with the same smoothness will cause the same viscous
dissipation for the same flow conditions.
An example of the numerical values of the acoustic energy dissipated by thermal
relaxation in the thermal boundary layer, using three different wall materials and helium
as working gas at pm =10 and 30 bar, are given in Table 6.3. The data used in the
calculations are given in the caption of Table 6.3. As can be seen the combination
helium-cork leads to the lowest dissipation. The dissipation for helium-stainless steel is
about 15 % higher than for helium-cork at pm =10 bar. The calculations show that this
difference reaches about 25 % at pm=30 bar for the same drive ratio (Table 6.3).
However, because of some problems which may be associated with the porosity of cork,
48

it is suggest to use a thin layer of polyester (4δs =76 µm) inside a metallic tube to
minimize the thermal-relaxation dissipation. This will lead to about 20% less thermal-
relaxation losses than stainless steel at pm=30 bar (Table 6.3). Because the thermal-
relaxation losses are typically about 30% of the total thermo-viscous losses in
thermoacoustic systems this will hence improve the performance of this systems.
Table 6.3 Acoustic energy dissipated per unit of surface area of a tube due to the
thermal relaxation (Eq.26). Helium is used as working gas at a mean pressure of 10
and 30 bar, the frequency is 50 Hz, and the drive ratio is 8.3%
6.5 Remarks
A simple analytical model describing the interaction between a sound wave and a
solid surface is presented. The thermal-relaxation dissipation at the gas-resonator
interface is analyzed. This is minimal whenever the temperature oscillations in the wall
can better follow the temperature oscillations in the gas. The main result of the model is
that thermal-relaxation losses can be minimized by using a tube material with the
smallest possible combination Ks ρs cs and a gas with the largest possible combination Kg
ρg cp. For this case a minimal ratio m/b is realized. This result follows also from the
expression of the acoustic energy dissipated by thermal relaxation Eq. (26). From the
different combinations gas-solid analyzed in the study, cork-helium leads to the lowest
thermal relaxation dissipation. However, it is concluded that because of some problems
which can be associated with the porosity of cork, a thin film (4δs=76 µm) of polyester
may be used inside a metallic tube to minimize the thermal-relaxation dissipation. This
will lead to about 20% less thermal relaxation losses than stainless steel at pm=30 bar and
hence
49

improving the performance of thermoacoustic systems. Using the expression of the
acoustic energy dissipated by thermal-relaxation per unit of surface area of the tube, it is
shown that the losses can be furthermore minimized by optimizing the shape of the
resonator in the regions of high acoustic pressure. In contrast, for the stack where the
thermoacoustic effect takes place and the hydrodynamic heat flow has to be maximized, a
stack material with a large Ks ρs cs in comparison to Kg ρg cp is suitable. It is concluded
that a plate thickness of about 8δs forms a minimum. This completes the set of
parameters needed for the design of stacks besides the spacing between the plates and
length.
50

CHAPTER 7 HEAT EXCHANGERS ANALYSIS
(Theoretical Study)
7.1 Introduction
The performance of a thermoacoustic cooler is very sensitive to heat exchanger
design and performance. The volume available for heat exchange is limited since
convective heat transfer between the heat exchangers and a stack is a result of oscillating
particles and the particle displacements are only a few millimeters. It is necessary to
estimate heat transfer coefficients between the heat exchangers and stack ends in order to
predict the overall cooler performance. Very little work has been done to characterize
heat transfer within oscillating flows for heat exchanger geometries encountered within
thermoacoustic coolers. In the work described in this paper, the gas-side heat transfer for
heat exchangers installed within a one-half wavelength standing wave thermoacoustic
cooler was investigated. Dimensionless heat transfer coefficients, Colburn-j factors,
were determined from the measurements and compared with different models from the
literature. A new method was presented for calculating heat transfer coefficients for
oscillating flows using steady-flow heat transfer correlations.
Most heat exchanger models described in the literature assume steady flow
conditions for both working fluids. Garrett et al. (1994) showed that an acoustic heat
transfer coefficient derived from a conduction heat transfer model between the gas
particle and the heat exchanger could be used for the preliminary design of heat
exchangers within thermoacoustic coolers. He also suggested that the root mean square
value of the heat transfer coefficient obtained from the conduction model could predict
the heat transfer in an oscillating acoustic flow. Mozurkewich (1995) measured the heat
transfer from heated wires located at a velocity antinode in a standing acoustic wave. The
51

Nusselt number was accurately predicted using a steady flow, forced-convection
correlation at high Reynolds numbers and by a natural convection model at low Reynolds
numbers, where acoustic Reynolds numbers were employed. More recently,
Mozurkewich performed experiments within a quarter-wavelength modular
thermoacoustic refrigerator using aluminum heat exchangers with simple geometries. In
this study, a correlation based on Zukauskas’ single-tube steady cross-flow correlation
was developed using a time–average, steady-flow equivalent (TASFE) approximation
(Richardson). The TASFE approximation was found to hold for simple tube heat
exchangers placed near the velocity node. Poese and Garrett assumed that the time
-averaged value of the convective heat transfer coefficient was relevant to characterize
heat transfer in oscillating flows, proposed a modified laminar correlation. The modified
correlation was obtained by averaging a laminar parallel flow correlation for flow over a
flat plate during one half of the period. Swift suggested that the substitution of a root
mean-square RMS) Reynolds number into published steady-flow correlations could be
used to predict the heat transfer coefficient in oscillating flow. There have been other
noteworthy contributions to this field. Wetzel and Herman studied the effect of heat
transfer from a single plate. Brewster et al. studied the effects of heat transfer between
the elements in a thermoacoustic system. Despite these important contributions to
thermoacoustic heat transfer, no one has studied the heat transfer performance of realistic
heat exchanger geometries within a working thermoacoustic cooler. The objective of the
study presented in this paper was to develop methods for calculating heat transfer
coefficients of heat exchangers in oscillating flow, i.e. to provide heat transfer correlation
models to predict the performance of heat exchangers in an oscillating flow environment.
In order to accomplish this goal, both steady- and oscillating-flow experiments were
performed for heat exchangers that were specifically designed for a thermoacoustic
cooler. For oscillating flows, special procedures were established for estimating the heat
transfer coefficients from measured data, and for correlation of the heat transfer
coefficients using dimensionless parameters. It was found that a simple method obtained
by modifying the TASFE model predicted the heat transfer coefficients in oscillating
flow better than other correlation models currently used for the design of thermoacoustic
systems.
52

7.2 Heat Exchanger Analysis
Exchangers with small primary fluid pore spacing and thin (normal to the flow
direction) secondary fluid tubing were used. Figure 7.1 depicts the heat exchangers, and
the inner liquid flow path. The fin spacing was 0.54 mm, and the tube width was 1.9 mm.
The water flow directions are shown using dotted arrows. The dotted line in the center of
the heat exchanger shows the boundary between the two opposite vertical water flows.
Figure 7.1: Sketch of the heat exchanger (dimensions in millimeters).
7.2.1 Steady Flow Measurements
53

A small wind tunnel was used to measure steady-flow, steady-state, gas-side heat
transfer coefficients. The heat exchangers were operated with water flowing through the
tubes. The wind tunnel, shown in Figure 7.2, consisted of a centrifugal fan powered by an
electric motor and a 150 mm diameter PVC pipe. A flow straightener was used to
minimize inflow swirl and transverse flow motion. An inlet bell mouth and a plenum-
settling chamber downstream of the heat exchangers were utilized to improve the flow
velocity profile uniformity across the duct. One heat exchanger was mounted between
two sections of PVC pipe. A pitot tube, in conjunction with a probe traversing
mechanism and an inclined manometer, was used to measure the airflow velocity
distribution across the pipe at one stream wise location. The velocity distribution was
used to determine the air mass flow rate. The water flow rates through the heat exchanger
were measured using an axial paddle wheel turbine type flow meter at the water outlet.
Figure 7.2: Schematic of the wind tunnel used for the steady flow measurements.
A differential temperature transducer was used to measure the water temperature
difference. At the inlet and the outlet of the wind tunnel, arrays of type T thermocouples
were installed, at the temperature measurement locations shown in Fig.7.2 to measure air
temperatures. The rate of heat transfer was calculated from the measured flow rates and
temperatures for both the air-side and the water-side o f the exchanger.
7.2.2 In-Situ Measurements
A thermoacoustic cooler prototype (Mongeau et al.) was used for the heat transfer
rate measurements in oscillating flow. A schematic of the thermoacoustic cooler is shown
in Figure 7.3. It is driven by a 300 W moving magnet linear actuator mounted on metal
leaf springs to provide suspension stiffness. Type T thermocouples were used to perform
detailed temperature measurements. To estimate the average gas temperature of the
54

different locations, four thermocouples were radially distributed with about equal
distance from each other from the center of the heat exchanger, and averaged by
considering their area weightings. The sealed bodies of the thermocouples were carefully
bonded to the heat exchangers, ensuring no contact between the measuring points of the
thermocouples and the heat exchangers.
Figure 7.3: Schematic of a standing wave thermoacoustic cooler (dimensions in
millimeters).
The driver was instrumented with an accelerometer on the driver piston. A
pressure sensor was installed in a port near the piston. The dynamic pressure and velocity
measured at the piston face allowed the estimation of the acoustic particle velocity within
each heat exchanger using a DELTAE (Ward and Swift) model of the thermoacoustic
cooler prototype. The particle velocities on both the hot side and cold side heat
exchangers were used to determine acoustic Reynolds numbers. The device was operated
with two heat exchangers and a stack producing a temperature difference between the
circulating water stream and thermoacoustic working fluid at each heat exchanger. The
stack was composed of a 76 µm thick polyester film and 254 µm thick nylon wire
constructed by using the wire as a spacer and rolling the film into a cylinder. A
differential temperature transducer was used to measure the water temperature difference
between at the inlet and at the exit of the heat exchangers and water flow rates through
55

the heat exchangers were measured using axial paddle wheel turbine type flow meters at
the water outlets.
7.3 Estimation of Heat Transfer Coefficients
7.3.1 Steady Flow
For heat transfer in steady flow, a local heat transfer coefficient is defined as the
ratio of the heat transfer per unit area to the temperature difference between the surface
and “bulk” fluid adjacent to the surface. However, in practice, a global heat transfer
coefficient that is derived from a lumped analysis of the heat exchanger (UA-LMTD or
effectiveness-NTU methods) is utilized. Generally, a global heat transfer coefficient is
estimated from an heat transfer conductance (UA) determined from measurements, using
existing correlations for secondary fluid (water side) heat transfer coefficient, and overall
fin efficiency. The UA of the heat exchanger is estimated from the measurements as the
ratio of heat transfer rate to a log-mean temperature difference (LMTD) between the
primary (gas side), and the secondary (water side) fluids. The overall heat transfer rate is
estimated from measured primary or secondary fluid temperature differences and flow
rates. The equation for the overall heat exchanger conductance model (Incropera and
Dewit) is
(1)
where η, h, and A are overall fin efficiency, the global heat transfer coefficient, and the
surface area, Rtw is the resistance of the tube wall, and the subscripts o and i denote outer
56

(gas side) and inner (water side) surfaces of the micro channel heat exchanger. The
overall conductance of a heat exchanger operating in steady flow and at steady state can
be estimated from experimental data using a log-mean temperature difference model. A
cross-flow heat exchanger is considered in this study with water flowing through the
tubes and air flowing over the finned tubes. For the operating conditions considered in
this study, the capacitance rate (product of mass flow rate and specific heat) of the water
is much greater than that of the air. In this case, the overall heat exchanger conductance is
estimated according to CF LMTD
(2)
where Q is the heat transfer rate determined from measurements and LMTDCF is the log-
mean temperature difference for a cross flow heat exchanger defined as (Incropera and
Dewitt),
(3)
where Thi, Tci, Tho, and Tco are the inlet (i) and outlet (o) temperatures of the hot (h) and
cold (c) streams and F is a correction factor for a conversion from a counter flow LMTD
to a cross flow LMTD. The correction factor, F has been investigated for various cross
flow heat exchangers by others (Bowman et al., Jakob). In order to non-dimensionalize
the data for various gases and to facilitate comparison with theoretical calculations, the
experimental outer heat transfer coefficients are converted into the Colburn-j factor
(Incropera and Dewitt) defined as:
57

(4)
where St, Pr, Nu, and Re are the Stanton, Prandtl, Nusselt, and Reynolds numbers,
respectively.
7.3.2 Oscillating Flow
For cases where the heat exchanger is placed in an acoustically oscillating flow
(within a thermoacoustic system), Eqs. (1), and (2) are also used to estimate the outer
thermoacoustic fluid to surface) heat transfer coefficient from measured data, and the
heat transfer coefficient is cast into a Colburn-j factor using Eq. (4). However, inlet and
outlet fluid temperatures are not defined for the oscillating flow stream and little
temperature change can be attributed to fluid particles oscillating either over the heat
exchanger or between the heat exchanger and the stack. In this case, the log mean
temperature difference was calculated using
(5)
where tac stands for thermoacoustic, Tw,o and Tw,i are the water outlet and inlet
temperatures, and Ts is the spatially averaged, stack-end gas temperature. For the
oscillating flow, the acoustic Reynolds number was obtained using the amplitude of the
acoustic particle velocity u1, from
(6)
Where Dh is the hydraulic diameter, and ν is the kinematic viscosity. An RMS acoustic
Reynolds number was also employed, where an RMS velocity replaces velocity
amplitude in Eq. (6). In the use of Eq. (5), it is assumed that the gas-side temperature
change between the inlet and the exit is negligible. This assumption yields a correction
58

factor for the cross flow LMTDtac equal to unity. Note that the gas-side heat transfer
coefficient was non-dimensionalized using Eq. (4).
7.4 Correlations
To obtain data for a wide range of acoustic Reynolds numbers in oscillating flow,
four different helium and argon mixtures (55%He-45%Ar, 47%He-53%Ar, 34%He-
66%Ar, and 21%He-79%Ar) were employed. The measured data in steady and
oscillating flow were used to calculate the Colburn-j factors. Regress on curves for the
calculated Colburn-j factors were obtained numerically using a Nelder-Mead simplex
method (Nelder and Mead, Mathews and Fink), and they are given by
(7) and (8)
For comparison, predictions from a thermoacoustic steady-flow equivalent model
(TASFE), an RMS Reynolds number model consisting of using the RMS acoustic
Reynolds number directly in a known steady flow correlation model, and a boundary
layer conduction model (Ward and Swift) which is used in DELTAE simulations were
obtained. For consistency, the Colburn-j factors were converted to be based upon an
RMS Reynolds number using the definition in Eq. (4) (even if acoustic Reynolds number
were used in the correlation).
The TASFE model is based on the steady-flow regression model given in Eq. (7).
A sinusoidal dependency of the particle velocity was assumed, and the oscillating
velocity u=uo sinωt was substituted into Eq. (7). The time dependent Colburn-j factors
were numerically averaged over one-half period of the cycle using a recursive adaptive
Simpson quadrature method (Mathews and Fink). The TASFE model yielded
(9)
59

In DELTAE, the heat transfer coefficient used to find the heat exchanger metal
temperature is calculated using a boundary layer conduction model described by,
(10)
Where k is the thermal conductivity of the fluid, rh is the hydraulic radius of the fin
spacing of the heat exchanger, and δκ is the thermal penetration depth which is defined as
(11)
where α is the thermal diffusivity of the fluid, and ω is the angular frequency of the
oscillations. The Colburn-j factor for the boundary layer conduction model can be
calculated using Eq. (4), and it is given by
(12)
where ρ is the density, Cp is the isobaric specific heat, urms is the root mean square value
of the particle velocity, xhx is the heat exchanger length in the direction of the gas
oscillation, and xp is the peak to peak particle displacement of the working fluid in
oscillating flow. In Eq. (12), the heat transfer coefficient was scaled by the ratio of the
peak-to peak particle displacement to the heat exchanger length in the gas flow direction
for consistency because the measured heat transfer coefficients were calculated with the
total heat exchanger surface area.
The Colburn-j factors based upon measurements and different prediction methods
are presented as a function of RMS Reynolds numbers for oscillating flows in Figure 7.4.
The TASFE model prediction is in a good agreement with the measured Colburn-j factors
at low Reynolds numbers. For high Reynolds numbers, the discrepancy between the
TASFE model results and the measured heat transfer coefficients are about 25%. The
steady-flow heat transfer model using RMS acoustic Reynolds numbers overestimated
60

both the measured data and TASFE results. The time averaged Reynolds numbers used in
the TASFE are about 120% of the RMS Reynolds numbers leading to smaller Colburn-j
factors for this model. The Colburn-j factors obtained from the boundary layer
conduction model used in DELTAE are also plotted in Figure 7.4. This simple model
underestimated the heat transfer coefficients at low Reynolds numbers, and overestimated
them at high Reynolds numbers. Relative differences of up to 100% between the laminar
conduction results and experimental data were found. This is consistent with the accuracy
of this model stated in previous studies (Swift).
Figure 7.4: Measured Colburn-j factor vs. RMS acoustic Reynolds number –
comparison with predictions from various models.
7.5 Discussion
Existing heat transfer correlation models failed to accurately predict the
performance of the heat exchangers in oscillating flows. The TASFE model, although it
over predicted heat transfer coefficients in oscillating flow at higher Reynolds numbers,
was found to perform better than other models in this particular experiment. In the
TASFE model, the Reynolds number is assumed to be Re(t)=Re1 sinωt, and it is
substituted into the known steady flow regression model. To find the time-averaged heat
61

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