Mathematical aspects of thermoacoustics

Abstract

This thesis addresses the mathematical aspects of thermoacoustics, a subfield within physical acoustics that comprises all effects in which heat conduction and entropy variations of the gaseous medium play a role. We focus specifically on the theoretical basis of two kinds of devices: the thermoacoustic prime mover, that uses heat to produce sound, and the thermoacoustic heat pump or refrigerator, that use sound to produce heating or cooling. Two kinds of geometry are considered. The first one is the so-called standing-wave geometry that consists of a closed straight tube (the resonator) with a porous medium (the stack) placed inside. The second one is the so-called traveling-wave geometry that consists of a resonator attached to a looped tube with a porous medium (regenerator) placed inside. The stack and the regenerator differ in the sense that the regenerator uses thinner pores to ensure perfect thermal contact. The stack or regenerator can in principle have any arbitrary shape, but are modeled as a collecting of long narrow arbitrarily shaped pores. If the purpose of the device is to generate cooling or heating, then usually a speaker is attached to the regenerator to generate the necessary sound. By means of a systematic approach based on small-parameter asymptotics and dimensional analysis, we have derived a general theory for the thermal and acoustic behavior in a pore. First a linear theory is derived, predicting the thermoacoustic behavior between two closely placed parallel plates. Then the theory is extended by considering arbitrarily shaped pores with the only restriction that the pore cross-sections vary slowly in longitudinal direction. Finally, the theory is completed by the inclusion of nonlinear second-order effects such as streaming, higher harmonics, and shock-waves. It is shown how the presence of any of these nonlinear phenomena (negatively) affects the performance of the device. The final step in the analysis is the linking of the sound field in the stack or regenerator to that of the main tube. For the standing-wave device this is rather straightforward, but for the traveling-wave device all sorts of complications arise due to the complicated geometry. A numerical optimization routine has been developed that chooses the right geometry to ensure that all variables match continuously across every interface and the right flow behavior is attained at the position of the regenerator. Doing so, we can predict the flow behavior throughout the device and validate it against experimental data. The numerical routine can be a valuable aid in the design of traveling-wave devices; by variation of the relevant problem parameters one can look for the optimal travelingwave geometry in terms of power output or efficiency.