# Engineering Acoustics/Thermoacoustics

One ordinarily thinks of a sound wave as consisting only of coupled
pressure and position oscillations. In fact, temperature oscillations
accompany the pressure oscillations and when there are spatial gradients
in the temperature oscillations, oscillating heat flow occurs. The
combination of these oscillations produces a rich variety of
"thermoacoustic" effects. In everyday life, the thermal effects of sound
are too small to be easily noticed; for example, the amplitude of the
temperature oscillation in conversational levels of sound is only about
0.0001 °C. However, in an extremely intense sound wave in a pressurized
gas, these thermoacoustic effects can be harnessed to create powerful
heat engines and refrigerators. Whereas typical engines and
refrigerators rely on crankshaft-coupled pistons or rotating turbines,
thermoacoustic engines and refrigerators have no moving parts (or at
most only flexing parts without the need for sliding seals). This
simplicity, coupled with reliability and relatively low cost, has
highlighted the potential of thermoacoustic devices for practical use.
As a result, thermoacoustics is maturing quickly from a topic of basic
scientific research through the stages of applied research and on to
important practical applications\[1\]. Recently, thermoacoustic
phenomena have been employed in the medical field for imaging of
tissues.

## History

The history of thermoacoustic engines is long but sparsely populated. A
review of Putnam and Dennis\[2\] describes experiments of Byron
Higgins\[3\] in 1777 in which acoustic oscillations in a large pipe were
excited by suitable placement of a hydrogen flame inside. The Rijke
tube\[4\], an early extension of Higgins\' work, is well known to modern
acousticians. Higgins\' research eventually evolved into the modern
science of pulse combustions\[5\] whose applications have included the
German V-I rocket (the \"buzz bomb\") used in World War II and the
residential pulse combustion furnace introduced by Lennox, Inc., in
1982. The Sondhaus tube is the earliest thermoacoustic engine that is a
direct antecedent of the thermoacoustic prime movers. Over 100 years
ago, glass blowers noticed that when a hot glass bulb was attached to a
cool glass tubular stem, the stem tip sometimes emitted sound, and
Sondhauss quantitatively investigated the relation between the pitch of
the sound and the dimensions of the apparatus.

The English physicist, Lord Rayleigh explained the Sondhauss tube
qualitatively in 1896: \"In almost all cases where heat is communicated
to a body, expansion ensues and this expansion may be made to do
mechanical work. If the phases of the forces thus operative be
favorable, a vibration may be maintained. For the sake of simplicity, a
tube, hot at the closed end and getting gradually cooler towards the
open end, may be considered. At a quarter of a period before the phase
of greatest condensation, the air is moving inwards, i.e., towards the
closed end, and therefore is passing from colder to hotter parts of the
tube; but in fact the adjustment of temperature takes time, and thus the
temperature of the air deviates from that of the neighboring parts of
the tube, inclining towards the temperature of that part of the tube
from which the air has just come. From this it follows that at the phase
of greatest condensation heat is received by the air, and at the phase
of greatest rarefaction heat is given up from it, and thus there is a
tendency to maintain the vibrations.\"

The history of imposing acoustic oscillations on a gas to cause heat
pumping and refrigeration effects is even briefer and more recent than
the history of thermoacoustic prime movers. In a device called a
pulse-tube refrigerator, Gifford and Longsworth produced significant
refrigeration by applying a very low-frequency high-amplitude pressure
oscillation to the gas in a tube. As they explained the phenomenon, \"If
any closed chamber is pressurized and depressurized by delivery and
exhaustion of gas from one point on its surface and the flow is
essentially smooth, heat pumping will occur away from the point on its
surface,\" because of the temperature changes that accompany the
pressure changes in the gas, and their time-phasing relative to the
oscillatory gas flow.

## Principle of operation

When a sound wave is sent down a half-wavelength tube with a vibrating
diaphragm or a loudspeaker, the pressure pulsations make the gas inside
slosh back and forth. This forms regions where compression and heating
take place, plus other areas characterized by gas expansion and cooling.
!Working
principle{width="500"}!Working
principle{width="500"}

A thermoacoustic refrigerator is a resonator cavity that contains a
stack of thermal storage elements (connected to hot and cold heat
exchangers) positioned so the back-and-forth gas motion occurs within
the stack. The oscillating gas parcels pick up heat from the stack and
deposit it to the stack at a different location. The device \"acts like
a bucket brigade\" to remove heat from the cold heat exchanger and
deposit it at the hot heat exchanger, thus forming the basis of a
refrigeration unit.
1
The governing mathematical equations of the thermoacoustic phenomenon
are given below.

!Working principle{width="700"}

### Standing-wave systems

The thermoacoustic engine (TAE) is a device that converts heat energy
into work in the form of acoustic energy. A thermoacoustic engine is
operating using the effects that arise from the resonance of a
standing-wave in a gas.  A standing-wave thermoacoustic engine typically
has a thermoacoustic element called the "stack". A stack is a solid
component with pores that allow the operating gas fluid to oscillate
while in contact with the solid walls. The oscillation of the gas is
accompanied with the change of its temperature. Due to the introduction
of solid walls into the oscillating gas, the plate modifies the
original, unperturbed temperature oscillations in both magnitude and
phase for the gas about a thermal penetration depth δ=√(2k/ω) away from
the plate, where k is the thermal diffusivity of the gas and ω=2πf is
the angular frequency of the wave. Thermal penetration depth is defined
as the distance that heat can diffuse though the gas during a time 1/ω.
In air oscillating at 1000 Hz, the thermal penetration depth is about
0.1 mm. Standing-wave TAE must be supplied with the necessary heat to
maintain the temperature gradient on the stack. This is done by two heat
exchangers on both sides of the stack.[^1]

If we put a thin horizontal plate in the sound field the thermal
interaction between the oscillating gas and the plate leads to
thermoacoustic effects. If the thermal conductivity of the plate
material would be zero the temperature in the plate would exactly match
the temperature profiles as in Fig. 1b. Consider the blue line in Fig.
1b as the temperature profile of a plate at that position. The
temperature gradient in the plate would be equal to the so-called
critical temperature gradient. If we would fix the temperature at the
left side of the plate at ambient temperature *T*~a~ (e.g. using a heat
exchanger) then the temperature at the right would be below *T*~a~. In
other words: we have produced a cooler. This is the basis of
thermoacoustic cooling as shown in Fig. 2b which represents a
thermoacoustic refrigerator. It has a loudspeaker at the left. The
system corresponds with the left half of Fig. 1b with the stack in the
position of the blue line. Cooling is produced at temperature *T*~L~.

It is also possible to fix the temperature of the right side of the
plate at *T*~a~ and heat up the left side so that the temperature
gradient in the plate would be larger than the critical temperature
gradient. In that case we have made an engine (prime mover) which can
e.g. produce sound as in Fig. 2a. This is a so-called thermoacoustic
prime mover. Stacks can be made of stainless steel plates but the device
works also very well with loosely packed stainless steel wool or
screens. It is heated at the left, e.g., by a propane flame and heat is
released to ambient temperature by a heat exchanger. If the temperature
at the left side is high enough the system starts to produces a loud
sound.

Thermoacoustic engines still suffer from some limitations, including
that:

-   The device usually has low power to volume ratio.
-   Very high densities of operating fluids are required to obtain high
    power densities
-   The commercially-available linear alternators used to convert
    acoustic energy into electricity currently have low efficiencies
    compared to rotary electric generators
-   Only expensive specially-made alternators can give satisfactory
    performance.
-   TAE uses gases at high pressures to provide reasonable power
    densities which imposes sealing challenges particularly if the
    mixture has light gases like helium.
-   The heat exchanging process in TAE is critical to maintain the power
    conversion process. The hot heat exchanger has to transfer heat to
    the stack and the cold heat exchanger has to sustain the temperature
    gradient across the stack. Yet, the available space for it is
    constrained with the small size and the blockage it adds to the path
    of the wave. The heat exchange process in oscillating media is still
    under extensive research. 
-   The acoustic waves inside a thermoacoustic engines operated at large
    pressure ratios suffer many kinds of non-linearities such as
    turbulence which dissipates energy due to viscous effects,  harmonic
    generation of different frequencies that carries acoustic power in
    frequencies other than the fundamental frequency. 

The performance of thermoacoustic engines usually is characterized
through several indicators as follows:[^2]

-   The first and second law efficiencies.
-   The onset temperature difference, defined as the minimum temperature
    difference across the sides of the stack at which the dynamic
    pressure is generated.
-   The frequency of the resultant pressure wave, since this frequency
    should match the resonance frequency required by the load device,
    either a thermoacoustic refrigerator/heat pump or a linear
    alternator.
-   The degree of harmonic distortion, indicating the ratio of higher
    harmonics to the fundamental mode in the resulting dynamic pressure
    wave.
-   The variation of the resultant wave frequency with the TAE operating
    temperature

### Travelling-wave systems

Figure 3 is a schematic drawing of a travelling-wave thermoacoustic
engine. It consists of a resonator tube and a loop which contains a
regenerator, three heat exchangers, and a bypass loop. A regenerator is
a porous medium with a high heat capacity. As the gas flows back and
forth through the regenerator it periodically stores and takes up heat
from the regenerator material. In contrast to the stack, the pores in
the regenerator are much smaller than the thermal penetration depth, so
the thermal contact between gas and material is very good. Ideally the
energy flow in the regenerator is zero, so the main energy flow in the
loop is from the hot heat exchanger via the pulse tube and the bypass
loop to the heat exchanger at the other side of the regenerator (main
heat exchanger). The energy in the loop is transported via a travelling
wave as in Fig. 1c, hence the name travelling-wave systems. The ratio of
the volume flows at the ends of the regenerator is *T*~H~/*T*~a~, so the
regenerator acts as a volume-flow amplifier. Just like in the case of
the standing-wave system the machine "spontaneously" produces sound if
the temperature *T*~H~ is high enough. The resulting pressure
oscillations can be used in a variety of ways such as in producing
electricity, cooling, and heat pumping.

## Thermoacoustic machines

There are two basic kinds of thermoacoustic machines:

1.  Thermoacoustic prime movers
2.  Thermoacoustic refrigerator

!A standing wave demonstration
refrigerator{width="550"}

## Basic components of a thermoacoustic system

A thermoacoustic machine generally consists of:

1.  Acoustic driver
2.  Stack or regenerator
3.  Heat exchanger
4.  Resonator

## Acoustic driver

Electrodynamic drivers are used in a class of electrically driven
thermoacoustic refrigeration systems. The mechanical and electrical
characteristics of the driver, in conjunction with the acoustic load
impedance at the driver piston, determine the electroacoustic efficiency
of the actuator. The electroacoustic efficiency is, of course, a key
factor in the overall efficiency of the cooling system. For this reason,
it is useful to develop models that allow the efficiency of any such
driver to be predicted for varying operating conditions and loads. A
detailed description of linear models of loudspeakers using equivalent
electrical circuits is readily available.\[2\] !Equivalent electrical
circuit of an electroacoustic
driver
Several methods based on such linear models have been proposed in order
to determine the model parameters experimentally.

## Stack

In the thermoacoustic refrigerator the stack is the main component where
the thermoacoustic phenomenon takes place. Below shown are two stacks of
different materials used in a standing wave thermoacoustic refrigerator.

!A ceramic stack{width="400"} !A
parallel plate
stack{width="400"}

## Heat exchanger

The heat exchangers employed in a thermoacoustic refrigerator influence
the acoustic field created in the resonator. There are many design
constraints such as porosity of the heat exchanger and high heat
transfer coefficient for efficiency. Due to these constraints, special
kind of heat exchangers are used. One typical micro channel aluminum
heat exchanger is shown below.

!Heat exchanger{width="400"}

## Resonator

This the part of refrigerator which is only there for maintaining the
acoustic wave. Because it is a dead volume which causes heat loss and
adds bulk, quarter wavelength resonators are preferred over half
wavelength.

## References

-   Greg Swift *et al.*,\"Thermoacoustics for Liquefaction of Natural
    Gas\",LNG Technology.
-   L. L. Beranek, *Acoustics*, McGraw--Hill, New York.
-   R. W. Wakeland, ''Use of electrodynamic drivers in thermoacoustic
    refrigerators,''J. Acoust. Soc. Am. 107, 827--832,2000.


[^1]: M. Emam, Experimental Investigations on a Standing-Wave
    Thermoacoustic Engine, M.Sc. Thesis, Cairo University, Egypt
    (2013).

[^2]: G.W. Swift, A unifying perspective for someengines and
    refrigerators, Acoustical Society of America, Melville, (2002).
