Weakly nonlinear thermoacoustics for stacks with slowly varying pore cross-sections

Abstract

A general theory of thermoacoustics is derived for arbitrary stack pores. Previous theoretical treatments of porous media are extended by considering arbitrarily shaped pores with the only restriction that the pore cross-sections vary slowly in the longitudinal direction. No boundary-layer approximation is necessary. Furthermore, the model allows temperature variations in the pore wall. By means of a systematic approach based on dimensional analysis and small parameter asymptotics, we derive a set of ordinary differential equations for the mean temperature and the acoustic pressure and velocity, where the equation for the mean temperature follows as a consistency condition of the assumed asymptotic expansion. The problem of determining the transverse variation is reduced to finding a Green's function for a modified Helmholtz equation and solving two additional integral equations. Similarly the derivation of streaming is reduced to finding a single Green's function for the Poisson equation on the desired geometry.

Linear and nonlinear modelling of a theoretical travelling-wave thermoacoustic heat engine

“…This resulted in the development of the prediction software package DeltaE (Ward & Swift 1994) (replaced now by DeltaEC), which, together with similar modeling tools such as SAGE and REGEN, is still actively used in the academic literature as well as in industry. Other examples of advanced low-order modeling relying on Rott's theory and systematic asymptotic approximations (Bauwens 1996;In 'T Panhuis et al 2009;Hireche et al 2010) suffer from similar shortcomings. While the prediction of global quantities of interest such as acoustic amplitude, efficiency, and frequency of operation can be made accurate in low-pressure amplitude cases (not without some heuristics required on the user's end), it is not possible with such an approach to directly account, for example, for the interaction of high-amplitude acoustic wave with complex geometries, the effects of transitional turbulence and higher-order harmonics on thermoacoustic transport (Olson & Swift 1997) and acoustic energy dissipation (Ward & Swift 1994).…”

Generalization of the thermoacoustic form functions for evaluating the transverse temperature gradient effect

“…Froude number as defined in literature [7] F 0 Viscous form function δ ν Scaled viscous penetration depth defined as in literature [7,21], according the length scale d * ɛ…”

“…In this study, the fluid conservation laws of mass, momentum and heat equation are used in the dimensionless form according to the work of Panhius et al [7] in order to analyze the magnitude of the thermoacoustic phenomena and limiting the number of parameters. Indeed scale parameters are taken following the last literature for space and time that give in consequence the aspect ratio of the channel ɛ and the Helmholtz number h n .…”

“…However, the last models [12,17] use the mathematical tools of the standard linear theory named the form functions [2,7,21] which are established under the hypothesis of the uniform transverse time-average temperature. Hence, the cited models are hybrid between standard linear theory and computational fluid dynamic method and allow a simpler analysis of the heat conversion.…”

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Section average of the viscous form function F 0 for perfect fluid g i (0) Section average of the ith thermal form function G i for perfect fluid G i Thermal form functions g i Section average of the viscous form function G i h n Helmholtz number express the ratio l * / * r between the length of channel and the scale of the acoustic wave length * r = 2π ω * r C * r using the scales C * r and ω * r for speed of sound and for angular frequency respectively MaMach number expressing the ratio formed by scales of fluid speed u * r and speed of sound C * r q i Dimensionless heat flux density respectively following the axial and transversal direction i = 1 or 2 q y0 0th order heat flux density following the transverse directionDimensionless coordinate of y * = d * y using length scale d * Y| 0 , Y| 1 Scaled variable of fluid at the interface Γ 1 and Γ 2where

or 1 is the order of the variable Y according to Mach number

Greek letters δ κScaled thermal penetration depth defined as in literature [7,21], according the length scale d *

AbstractThe main purpose of this paper is to study analytically the effect of transverse gradient of time averaged temperature on the oscillating flow between parallel plates that occurs in thermoacoustic device such as resonator, stack and heat exchanger. In fact, this transverse gradient of temperature is not taken into account in standard linear theory and can have considerable consequences on the thermoacoustic machines operating as the onset parameters.

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