Linear and Nonlinear Modeling of a Traveling-Wave Thermoacoustic Heat Engine

Abstract

We have carried out three-dimensional Navier-Stokes simulations, from quiescent conditions to the limit cycle, of a traveling-wave thermoacoustic heat engine (TAE) composed of a long variable-area resonator shrouding a smaller annular tube, which encloses the hot (HHX) and ambient (AHX) heat-exchangers, and the regenerator (REG). Simulations are wall-resolved, with no-slip and adiabatic conditions enforced at all boundaries, while the heat transfer and drag due to the REG and HXs are modeled. HHX temperatures have been investigated in the range 440K - 500K with AHX temperature fixed at 300K. The initial exponential growth of acoustic energy is due to a network of traveling waves amplified by looping around the REG/HX unit in the direction of the imposed temperature gradient. A simple analytical model demonstrates that such thermoacoustic instability is a Lagrangian thermodynamic process resembling a Stirling cycle. A system-wide linear stability model based on Rott's theory is able to accurately predict the frequency and the growth rate, exhibiting properties consistent with a supercritical Hopf bifurcation. The limit cycle is governed by acoustic streaming -- a rectified flow resulting from high-amplitude acoustics. Its key features are explained with an axially symmetric incompressible model driven by the wave-induced stresses extracted from the compressible calculations. These features include Gedeon streaming and mean recirculations due to flow separation. The first drives the mean advection of hot fluid from the HHX to a secondary heat exchanger (AHX2), in the thermal buffer tube (TBT), necessary to achieve saturation of the acoustic energy growth. The direct evaluation of the energy fluxes reveals that the efficiency of the device deteriorates with the drive ratio and that the acoustic power in the TBT is balanced primarily by the mean advection and thermoacoustic heat transport.