Abstract
A model for the description of the transient regime leading to steady-state sound in a quarter-wavelength thermoacoustic prime mover is proposed, which is based on the description of the unsteady heat transfer in the system, coupled with an ordinary differential equation describing wave amplitude growth/attenuation. The equations are derived by considering a cross-sectional averaged temperature distribution along the resonator, and by assuming that both the characteristic time associated with heat diffusion through the stack and that associated with the thermoacoustic amplification are much larger than the acoustic period. Attention is here focused on the only mechanism of saturation due to heat transport by sound within the stack. The numerical solving of the governing equations leads to the prediction of the transient regime, which is compared with experimental results for several values of the heat power supplied to the system and for several positions of the stack in the resonator. The model reproduces the experiments quite well, notably showing that a small diminution of the temperature in the vicinity of the hot end of the stack is associated to an overshoot of wave amplitude growth, while heat diffusion through the whole stack impacts the subsequent evolution of wave amplitude leading to steady state. Additional experimental results exhibiting complicated regimes of wave amplitude evolution are provided, which are not reproduced by the present model.
Highlights
Several works have been devoted to the description of the transient regime of wave amplitude growth and its saturation due to nonlinear processes in thermoacoustic systems
Other studies have described the transient regime under an assigned heat input in different kinds of engines: in these models, sound saturation occurs via the diminution of the temperature gradient due to heat transport by the thermoacoustic effect along the stack, while the propagation of acoustic waves is assumed linear, and it is described either by lumped elements16 or by two-ports
It is worth mentioning that a few papers dealt with the use of commercial computational fluid dynamics simulation tools to compute the transient regime in thermoacoustic devices of complicated geometry,18,19 leading to results that are still quite different from experiments
Summary
Several works have been devoted to the description of the transient regime of wave amplitude growth and its saturation due to nonlinear processes in thermoacoustic systems. It is worth mentioning that a few papers dealt with the use of commercial computational fluid dynamics simulation tools to compute the transient regime in thermoacoustic devices of complicated geometry, leading to results that are still quite different from experiments. Time consuming, these latter approaches might be, appropriate to describe thermoacoustic engines accurately, but it is still of interest to pursue the investigation of the transient regime by means of a simplified modeling in order to get a deeper insight about the dominant mechanisms controlling the saturation of the acoustic wave. Note that the walls of the resonator are assumed to rest at constant temperature T1 1⁄4 300 K, as well as both ends of the device
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