Thermoacoustics and its applications

The energy carried by a sound wave is mechanical energy. It is partly kinetic (due to the oscillatory motion of the air) and partly potential (due to the local compression of the air). Like every form of energy, it can be dissipated as heat by friction forces, but the study of thermoacoustics shows that it can also be put to good use, and that machines can be designed that use sound energy in several interesting ways.

Thermoacoustics studies heat exchange in the presence of sound, and is based on a few simple acoustic and thermodynamic principles.

Thermoacoustic devices are divided into engines and refrigerators^[1]

1. Thermoacoustic engines turn the heat flow between a hot body and a cold one into mechanical energy in the form of a sound wave.
2. Refrigerators, instead, use the mechanical energy of the sound wave to produce a heat flow from a cold body to a hot one.

Some advantages of thermoacoustic technology. The devices

• are simple (with no moving parts);
• are reliable: having no parts moving against each other, they are free from wear and lubrication problems
• use low-cost materials and production technologies;
• can deliver an adjustable power by acting on the amplitude of the oscillation.

The simplest thermoacoustic devices use air as the working fluid; more efficient thermoacoustic engines and refrigerators use inert gases such as helium or argon. In no case are toxic or environmentally hazardous mixtures used.

How to extract heat or work from sound waves

So can a sound wave locally heat and cool the air it passes through?

Certainly, but the temperature changes induced by a sound wave are of the order of 0.0001 C. It is not surprising, then, that thermoacoustic effects are not evident unless some special measures are taken.

How can we design thermoacoustic machines if no heat flows in the air?

In reality a small amount of heat can be exchanged between the air and the wall of the tube, but we must consider the question quantitatively, and put it in these terms:

Over what distance can heat actually be exchanged between the air and the wall in a time comparable to the period of a sound wave?

For times shorter than one period of oscillation, some regions of the air are slightly warmer than others, but after a period the situation is reversed, and any heat flows are reversed too. Where there was compression, rarefaction follows, and vice versa. Heat exchange is possible only over the short distance heat can travel in the brief span of a single acoustic oscillation. This important parameter is called the thermal penetration depth and, just to give an idea, for air carrying a 1000 Hz sound it is about 0.1 mm. In other words, when a standing wave is set up in a sound tube, all the useful heat exchanges between the air and the walls of the tube are due to the very thin layer of air lying closer to the walls than the thermal penetration depth.

The working principle of thermoacoustic machines is based on this very thin layer.

But these heat exchanges are tiny. How can a visible effect be obtained from them?

For example by using a porous material, that is, by dividing a section of the sound tube into a great many small tubes, each with a diameter comparable to the thermal penetration depth, and therefore able to take an active part in the heat exchange. A porous element can add up a great many small heat exchanges until a macroscopic effect appears.

Explaining the quantitative details of these simple principles is beyond the scope of this page, but below we briefly examine two examples of the main types of machine that can be built on them.

How to turn a car radio into a refrigerator: standing-wave thermoacoustic refrigerators

A thermoacoustic refrigerator consists of three parts:

1. A sound source (usually a loudspeaker of the woofer type) able to produce a sound of adjustable frequency. The loudspeaker injects sound energy into the system.
2. A resonator tube, in which the standing wave forms when the sound produced by the loudspeaker has the right resonance frequency. One end of the tube is closed by an aluminium cap able to dissipate heat. The other end rests on the woofer.
3. A stack: a small block of porous material (like the ceramic elements found in car catalytic converters). It is placed inside the resonator tube. The exchange between acoustic energy and thermal energy takes place inside it.

The refrigerator works through the following exchanges of energy and heat:

1. an energy source (the woofer) produces the sound wave, that is, a series of compressions and rarefactions of the air
2. the air exchanges heat with the walls of the stack, pumping heat from the colder end to the warmer one.

Let us examine the dynamics of the system in detail:

• The sound coming from the woofer, suitably tuned to the resonance frequency of the tube, sets the air in the tube resonating and sets up a standing wave at the frequency of the fundamental harmonic of the duct.
• The air molecules start oscillating back and forth and, given the stationary pressure field, they oscillate between two points at different pressures. Locally there are small temperature oscillations, consistent with the fact that sound propagates essentially in an adiabatic regime.
• For convenience, we can divide into four phases the thermodynamic cycle that the small volume of air goes through in each period of oscillation. This cycle repeats at the frequency of the sound in the tube (for the refrigerator shown at the start of the section the frequency is 286 Hz).
• Remember that all heat exchanges can occur only for the air lying within one thermal penetration depth of the walls. The animation and drawings therefore show a single micro-channel of the stack, not the whole tube. The stack acts both as a heat sink and as a heat source at the different ends of the cycles of the individual particles. It is therefore essential for the stack to be made of a material with low thermal conductivity: this keeps the maximum and minimum temperatures separate, both within a single cycle and overall.

compression		During the first half of the oscillation the small volume of air is pushed by the sound wave towards a high-pressure region; it contracts quickly and therefore its temperature rises, but it has no time to exchange heat with its surroundings because of the poor thermal contact.
heat release + compression		The walls of the stack in which the gas moves are initially at room temperature, that is, at a temperature lower than that reached by the small volume at the end of its stroke, so heat is released by the small volume to the walls. Correspondingly, the small volume continues to contract where it is.
rarefaction		Pulled back by the second half-oscillation of the sound wave, the small volume returns towards the lower-pressure region; the rarefaction causes an adiabatic drop in its temperature.
heat absorption + rarefaction		Now the small volume is at a lower temperature than the walls of the channel, so heat is absorbed from the walls by the air, with a further small increase in volume. The cycle can start again.

Remarks:

• At each oscillation, each small volume of air in the channel can pump a little heat from one end of the stack to the other, from the lower-temperature region to the higher-temperature one. The combined action of the various small volumes of air is like that of a chain of firefighters passing buckets of water: each firefighter handles only one bucket at a time without moving from their place, but the effect of the whole chain is actually to pump water from one end to the other.

• Note that the machine absorbs energy from the sound wave and turns it into heat. Indeed, the expansion of the gas (when the small volume in the drawing is all the way to the left) takes place at a lower pressure than the compression (which happens when the small volume is all the way to the right), and consequently the work done in the cycle is negative

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• the tube behaves like a half-open pipe. The open end is the one resting on the woofer, where the air molecules have their maximum velocity, while the closed end corresponds to the aluminium cap, where the air molecules are ideally motionless.
• The stack is the heart of the thermoacoustic refrigerator. It is a small block of porous material, that is, made of a series of small straight, parallel channels. The conversion between heat and sound takes place inside these channels. The stack must be much smaller than the wavelength of the standing wave, and must itself be a poor conductor of heat.
• A simple refrigerator does not require an advanced research laboratory to be built: for example, a ceramic element is not necessary for the stack. An alternative technique is to wind photographic film into a spiral, using nylon threads to space one turn from the next. The image shows a refrigerator built for the Engineering degree thesis of Giulio Allesina, able to obtain a temperature difference of 16.8 C using a small car-radio woofer.

ceramic stack obtained from a catalytic converter component	stack assembled from photographic film and nylon threads	temperature at the ends of the stack in the experimental refrigerator built by Giulio Allesina

Applications

Some implementations of the thermoacoustic refrigerator principle:

• The non-standing-wave refrigerator built for Ben&Jerry's (the US ice-cream giant) in a collaboration between Penn State University and the American company. The efficiency (more correctly called COP) of this machine is 19% of that of the corresponding ideal Carnot cycle.
• The device installed on the destroyer U.S.S. Deyo to cool radar equipment. The maximum refrigeration capacity measured in sea trials is 400 W. The lowest temperature reachable allows operation without the risk of freezing the instruments, working at around 4 C. This refrigerator operates at 26% of the efficiency of the corresponding Carnot cycle. The overall efficiency, however, is lower because of the low efficiency of the heat exchangers.

Acoustic lasers, or standing-wave thermoacoustic engines

In the thermoacoustic engine we want the opposite energy flow to that of the refrigerator

1. an external energy source produces a temperature difference between the ends of the stack
2. the stack gives heat to the air, favouring its compression and rarefaction so as to trigger a standing sound wave in the tube. For this to happen, heat must be given and taken away "at the right moment", according to the founding principle stated by Rayleigh:

This principle is nothing but the well-known principle of the swing, which applies in general to all oscillating systems: if the swing is pushed in phase with its oscillation, the oscillation is encouraged, while if it is pushed in opposition of phase, it is damped.

Let us then examine the different phases of the thermodynamic cycle for the thermoacoustic engine: note that, while compression and rarefaction are essentially the same phases as in the refrigerator, the direction of the heat flows is opposite in the other two phases. The reversal is due to the different condition of the stack which, while in the refrigerator it is initially at a uniform room temperature, in the engine must be prepared by imposing a suitable temperature gradient on it.

heat absorption + rarefaction		The air absorbs heat from the hot end of the stack and, consequently, expands rapidly.
rarefaction		During the rarefaction the air moves towards a part of the stack where the walls are colder. Because air conducts heat poorly, this expansion is essentially adiabatic. The small volume keeps its heat and, consequently, cools slightly.
heat release + compression		At the point of greatest rarefaction the air is at rest, and has time to give heat to the stack if the stack is at a lower temperature. Consequently the small volume contracts slightly. Immediately afterwards, by virtue of its elasticity, the small volume is pulled back towards the point from which it started, and continues its compression.
compression		The air returns towards maximum compression and is again in contact with a warmer portion of the stack. It is therefore ready to absorb its heat, and the cycle can start again

Remarks:

• for the engine to work, it is essential that the temperature change along the stack be greater than the one produced by adiabatic compression of the air; otherwise, at the ends of its motion, the small volume could not absorb heat from the hot part of the stack and release it to the cold part. In other words, there is a critical temperature gradient below which the device cannot trigger the oscillations, and therefore does not work.
• If the cycle is triggered, the engine can actually produce work (although not very efficiently), because the phase in which the volume of the gas increases takes place at a higher pressure than the one in which it decreases. Over the whole cycle, therefore, the mechanical work is positive

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• The intensity of the sounds induced by thermoacoustic oscillations can be extremely high, which is why the engines must be acoustically sealed. Moreover, note that a coherent oscillation of all the air molecules in the tube is obtained starting from a heat source, in which the agitation of the molecules is random. For these reasons standing-wave thermoacoustic engines are sometimes called "acoustic lasers".

Applications

• A first application, developed at the University of Utah, consists of micro-devices able to convert the heat coming from electrical equipment (CPUs, radars, etc.) back into electricity. The device becomes more efficient as the temperature rises, so the hotter the processor, the more sense it makes to use this technology.
• Researchers at Los Alamos National Laboratory, with Northrop Grumman Space Technology, have developed a compact generator that converts heat into electricity with an efficiency of 18%. The generator is fairly simple to build, with excellent durability and reliability. These features make it particularly suitable for spacecraft applications. ^[3]