Marginal conditions of thermoacoustic oscillations in a looped tube based on thick and thin diffusion-layer theories

Abstract

Marginal conditions for the onset of thermoacoustic oscillations of a gas in a looped tube with a “stack” inserted are examined by using two approximate equations for thick and thin thermoviscous diffusion layers in comparison with a span length of a gas passage. The equations are derived from the general thermoacoustic-wave equation valid for any thickness of the layer in the linear framework. Applying those approximate equations, respectively, to the gas in the stack and that in the outside of the stack, a frequency equation is derived by imposing matching conditions at both ends of the stack. Seeking a real solution for the frequency, the marginal conditions are obtained numerically for the temperature ratio at both ends of the stack. The ratio depends not only on the span length of one passage in the stack but also on its porosity. It is revealed that the temperature ratio decreases with increasing the span length and the porosity as well. This is the case when the thick diffusion layer is assumed in the stack. It is also revealed that a traveling wave tends to emerge in the tube outside of the stack in the sense from the hot end to cold end.