Thermoacoustics

Thermoacoustics is the interaction between temperature, density and pressure variations of acoustic waves. Thermoacoustic heat engines can readily be driven using solar energy or waste heat and they can be controlled using proportional control. They can use heat available at low temperatures which makes it ideal for heat recovery and low power applications. The components included in thermoacoustic engines are usually very simple compared to conventional engines. The device can easily be controlled and maintained. Thermoacoustic effects can be observed when partly molten glass tubes are connected to glass vessels. Sometimes a loud and monotone sound is spontaneously produced. A similar effect is observed if one side of a stainless steel tube is at room temperature (293 K) and the other side is in contact with liquid helium at 4.2 K. In this case, spontaneous oscillations are observed which are named "Taconis oscillations". The mathematical foundation of thermoacoustics is by Nikolaus Rott. Later, the field was inspired by the work of John Wheatley and Swift and his co-workers. Technologically thermoacoustic devices have the advantage that they have no moving parts, which makes them attractive for applications where reliability is of key importance.

Historical review of thermoacoustics Thermoacoustic-induced oscillations have been observed for centuries. Glass blowers produced heat-generated sound when blowing a hot bulb at the end of a cold narrow tube. This phenomenon also has been observed in cryogenic storage vessels, where oscillations are induced by the insertion of a hollow tube open at the bottom end in liquid helium, called Taconis oscillations, but the lack of heat removal system causes the temperature gradient to diminish and acoustic wave to weaken and then to stop completely. Byron Higgins made the first scientific observation of heat energy conversion into acoustical oscillations. He investigated the "singing flame" phenomena in a portion of a hydrogen flame in a tube with both ends open. Physicist Pieter Rijke introduced this phenomenon into a greater scale by using a heated wire screen to induce strong oscillations in his Rijke tube. Feldman mentioned in his related review that a convective air current through the pipe is the main inducer of this phenomenon. The oscillations are strongest when the screen is at one fourth of the tube length. Research performed by Sondhauss in 1850 is known to be the first to approximate the modern concept of thermoacoustic oscillation. Sondhauss experimentally investigated the oscillations related to glass blowers. Sondhauss observed that sound frequency and intensity depends on the length and volume of the bulb. Lord Rayleigh gave a qualitative explanation of the Sondhauss thermoacoustic oscillations phenomena, where he stated that producing any type of thermoacoustic oscillations needs to meet a criterion: "If heat be given to the air at the moment of greatest condensation or taken from it at the moment of greatest rarefaction, the vibration is encouraged". This shows that he related thermoacoustics to the interplay of density variations and heat injection. The formal theoretical study of thermoacoustics started by Kramers in 1949 when he generalized the Kirchhoff theory of the attenuation of sound waves at constant temperature to the case of attenuation in the presence of a temperature gradient. Rott made a breakthrough in the study and modeling of thermodynamic phenomena by developing a successful linear theory. After that, the acoustical part of thermoacoustics was linked in a broad thermodynamic framework by Swift.

Sound Usually sound is understood in terms of pressure variations accompanied by an oscillating motion of a medium (gas, liquid or solid). Thermoacoustic machines rely more on the temperature-position variations than the usual pressure-velocity variations. The sound intensity of ordinary speech is 65 dB. The pressure variations are about 0.05 Pa, the displacements 0.2 μm, and the temperature variations about 40 μK. So, the thermal effects of sound cannot be observed in daily life. However, at sound levels of 180 dB, which are normal in thermoacoustic systems, the pressure variations are 30 kPa, the displacements more than 10 cm, and the temperature variations 24 K. A full theory of thermoacoustics should account for the propagation of heat in the fluid as it makes compression cycles during the propagation of the sound wave. Good insights can however be gained by making the usual assumption of adiabatic compression. Even if no heat is exchanged during adiabatic compression, the temperature of the fluid does change and will tell the correct direction of heat flow. Under the adiabatic approximation, the one-dimensional wave equation for sound reads

c 2 ∂ 2 v ∂ x 2 − ∂ 2 v ∂ t 2 = 0 {\displaystyle c^{2}{\frac {\partial ^{2}v}{\partial x^{2}}}-{\frac {\partial ^{2}v}{\partial t^{2}}}=0}

with t time, v the gas velocity, x the position, and c the sound velocity given by c2=γp0/ρ0. For an ideal gas, c2=γRT0/M with M the molar mass. In these expressions, p0, T0, and ρ0 are the average pressure, temperature, and density respectively. In monochromatic plane waves, with angular frequency ω and with ω=kc, the solution is

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