Systematic Derivation of the Weakly Non-Linear Theory of Thermoacoustic Devices

Thermoacoustics is the field concerned with transformations between thermal and acoustic energy. This paper teaches the fundamentals of two kinds of thermoacoustic devices: the ther-moacoustic prime mover and the thermoacoustic heat pump or refrigerator. Two technologies, involving standing wave and traveling wave modes, are considered. We will investigate the case of a porous medium and two heat exchangers placed in a gas-filled resonator, in which either a standing or traveling wave is maintained. The central problem is the interaction between the porous medium and the sound field in the tube. The conventional thermoacoustic theory is reexamined and a systematic and consistent weakly non-linear theory is constructed based on dimensional analysis and small parameter asymptotics. The difference with conventional thermoacoustic theory lies in the dimensional analysis. This is a powerful tool in understanding physical effects which are coupled to several dimensionless parameters that appear in the equations, such as the Mach number, the Prandtl number, the Laucret number and several geometrical quantities. By carefully analyzing limiting situations in which these parameters differ in orders of magnitude, we can study the behavior of the system as a function of parameters connected to geometry, heat transport and viscous effects. 1