FEATURE ARTICLE
The Power of Sound
Sound waves in "thermoacoustic" engines and refrigerators can replace the pistons and cranks that are typically built into such machinery
Steven Garrett, Scott Backhaus
Speech and Hot Air
The interaction of heat and sound has interested acousticians since 1816, when Laplace corrected Newton's earlier calculation of the speed of sound in air. Newton had assumed that the expansions and compressions of a sound wave in a gas happen without affecting the temperature. Laplace accounted for the slight variations in temperature that in fact take place, and by doing so he derived the correct speed of sound in air, a value that is 18 percent faster than Newton's estimate.
Such thermal effects also explain why 19th-century glassblowers occasionally heard their heated vessels emit pure tones—a hint that thermoacoustics might have some interesting practical consequences. Yet it took more than a century for anyone to recognize the opposite effect: Just as a temperature difference could create sound, sound could produce a temperature difference—hot to one side, cool to the other. How acoustic cooling can arise is, in retrospect, rather easy to understand.
Suppose an acoustic wave excites a gas that was initially at some average temperature and pressure. At any one spot, the temperature will go up as the pressure increases, assuming the rise happens rapidly enough that heat has no time to flow away. The change in temperature that accompanies the acoustic compressions depends on the magnitude of the pressure fluctuations. For ordinary speech, the relative pressure changes are on the order of only one part per million (equivalent to 74 decibels, or dB, in sound pressure levels), and the associated variation in temperature is a mere ten-thousandth of a degree Celsius. Even for sounds at the auditory threshold of pain (120 dB), temperature oscillates up and down by only about 0.02 degree.
Most refrigerators and air conditioners must pump heat over considerably greater temperature ranges, usually 20 degrees or more. So the temperature swings that typical sound waves bring about are too small to be useful. To handle larger temperature spans, the gas must be put in contact with a solid material. Solids have much higher heat capacities per unit volume than gases, so they can exchange a considerable amount of heat without changing in temperature by very much. If a gas carrying a sound wave is placed near a solid surface, the solid will tend to absorb the heat of compression, keeping the temperature stable. The opposite is also true: The solid releases heat when the gas expands, preventing it from cooling down as much as it otherwise would.
The distance over which the diffusion of heat to or from an adjacent solid can take place is called the thermal penetration depth. Its value depends on the frequency of the passing sound wave and the properties of the gas. In typical thermoacoustic devices, and for sound waves in air at audio frequencies, the thermal penetration depth is typically on the order of one-tenth of a millimeter. So to optimize the exchange of heat, the design of a thermoacoustic engine or refrigerator must include a solid with gaps that are about twice this dimension in width, through which a high-amplitude sound wave propagates. The porous solid (frequently a jelly-roll of plastic for refrigerators or of stainless steel for engines), is called a "stack," because it contains many layers and thus resembles a stack of plates.
When an acoustically driven gas moves through the stack, pressure, temperature and position all oscillate with time. If the gas is enclosed within a tube, sound bounces back and forth creating an acoustic standing wave. In that case, pressure will be in phase with displacement—that is, the pressure reaches its maximum or minimum value when the gas is at an extreme of its oscillatory motion.
Consider how this simple relation can be put to use in a thermoacoustic refrigerator, which in its most rudimentary form amounts to a closed tube, a porous stack and a source of acoustic energy. As a parcel of gas moves to one side, say to the left, it heats up as the pressure rises and then comes momentarily to rest before reversing direction. Near the end of its motion, the hot gas deposits heat into the stack, which is somewhat cooler. During the next half-cycle, the parcel of gas moves to the right and expands. When it reaches its rightmost extreme, it will be colder than the adjacent portion of the stack and will extract heat from it. The result is that the parcel pumps heat from right to left and can do so even when the left side of the stack is hotter than the right.
The span of movement for an individual parcel is quite small, but the net effect is that of a bucket brigade: Each parcel of oscillating gas takes heat from the one behind and hands this heat off to the next one ahead. The heat, plus the work done to move it thermoacoustically, exits one end of the stack through a hot heat exchanger (similar to a car radiator). A cold heat exchanger, located at the other end of the stack, provides useful cooling to some external heat load.
One can easily reverse this process of refrigeration to make a thermoacoustic engine. Just apply heat at the hot end of the stack and remove it at the cold end, creating a steep temperature gradient. Now when a parcel of gas moves to the left, its pressure and temperature rise as before, but the stack at that point is hotter still. So heat flows from the stack into the gas, causing it to expand thermally just as pressure reaches a maximum. Conversely, when the parcel shifts to the right, it expands and cools, but the stack there is cooler still. So heat flows into the solid from the gas, causing thermal contraction just as pressure reaches a minimum. In this way, the temperature variation imposed on the stack drives heat into and out of the gas, forcing it to do work on its surroundings and amplifying the acoustic oscillations. Maintenance of the steep thermal gradient requires an external source of power, such as an electric heater, concentrated sunlight or a flame—which explains why glassblowers sometimes observe the spontaneous generation of sound when they heat the walls of a glass tube (serving as a stack) in such a way as to create a strong temperature gradient, a phenomenon first documented in a scholarly journal in 1850.
Indeed, this "singing tube" effect arises easily enough that Reh-lin Chen, a student in the Graduate Program in Acoustics at the Pennsylvania State University, was able to build a thermoacoustic engine with only three parts. The stack consists of a plug of porous ceramic (material that is normally used for automotive catalytic converters). Electrical current passing through a heater wire attached at one end of the plug imposes a temperature gradient. A Pyrex test tube acts as a miniature organ pipe and sets up a standing acoustic wave. Because the cold end of the stack faces the mouth of the test tube, no cold heat exchanger is needed: Air streaming in and out of the open end of the tube provides sufficient cooling. Despite its simplicity, Chen's engine is capable of producing sound at uncomfortable levels.
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