Thermoacoustic engines and heat pumps use the heat exchange between an oscillating gas and a porous solid stack to sustain self-oscillation or pump heat, and their miniaturization targets electronics cooling. At micro-scale stack spacings the gas becomes rarefied: the Knudsen number KnKn (molecular mean free path over plate spacing) enters the slip-flow range 0.01≤Kn≤0.10.01\le Kn\le 0.1, where Offner and Ramon (2016) showed, within a linear standing-wave framework with velocity-slip and temperature-jump boundary conditions and constant properties, that slip lowers the onset temperature difference of engines but reduces the temperature difference a heat pump can sustain. The open problem is a quantitative theory of thermoacoustic conversion in the transition-flow regime Kn≳0.1Kn\gtrsim 0.1, where continuum equations with slip corrections cease to be reliable and the Boltzmann kinetic equation governs transport: in the source's own words, that regime is "governed by the Boltzmann equation, with some reported solutions for specific cases in thermoacoustics, but a complete theory for this regime has not yet been developed."

Background

A thermoacoustic engine is a gas-filled resonator in which a porous solid matrix (the stack, typically parallel plates or pores) is held across a temperature gradient: when the gradient exceeds a critical value the acoustic wave does net work on the gas and self-sustained oscillations begin, while below it the same physics, driven by a loudspeaker, pumps heat from cold to hot as a refrigerator. The classical linear theory is due to Rott, who reduced the problem to thermoviscous acoustic waves in narrow channels with heat exchange at the walls; Swift and co-workers developed it into practical engines and refrigerators, and Atchley analyzed the below-onset standing-wave regime. Interest in miniaturization came from electronics cooling: Symko's group built high-frequency millimeter-scale heat-driven engines for chip thermal management, and Benavides derived a lower bound on the reachable device size. As the stack spacing y0y_0 shrinks toward the molecular mean free path λ\lambda, the Knudsen number Kn=λ/y0Kn=\lambda/y_0 grows and the continuum no-slip description fails. In the slip-flow range 0.01≤Kn≤0.10.01\le Kn\le 0.1 one still solves continuum (Navier–Stokes/Fourier) equations but with velocity-slip and temperature-jump boundary conditions at the solid walls; beyond Kn≳0.1Kn\gtrsim 0.1, in the transition-flow regime, the gas must instead be described by the Boltzmann equation, the kinetic equation for the molecular velocity distribution function, with gas–surface scattering rules replacing continuum boundary conditions. Offner and Ramon (2016) built the linear, standing-wave theory of micro-scale thermoacoustic engines and heat pumps in the slip regime, taking gas and solid properties constant over the stack: their stability (onset) calculations show that slip lowers the critical temperature difference for engine onset at the high resonance frequencies characteristic of micro devices, while for heat pumps slip is detrimental, reducing the maximum sustainable temperature difference across the stack. They were explicit that the analysis was restricted to this regime and these assumptions. Kinetic solutions relevant to oscillatory rarefied flows existed only for specific simpler cases, notably Kalempa and Sharipov's rarefied-gas calculations of thermoacoustic waves between coaxial cylinders (boundary-temperature-oscillation driven waves) and of sound propagation in binary gas mixtures at arbitrary frequency. Since 2016, kinetic theory of acoustics has progressed on adjacent problems — continuum breakdown in the sound of a pulsating sphere (Ben-Ami and Manela), sound propagation in rarefied molecular gases (Li, Su and Zhang), and oscillatory Couette flow of rarefied gas mixtures (Zhang, Wang and Guo) — but none of these treats the stack geometry, the imposed axial temperature gradient, or the onset and heat-pumping questions of thermoacoustic conversion.

Problem Statement

Develop a quantitative theory of thermoacoustic energy conversion in micro-scale stack geometries that remains valid when gas rarefaction goes beyond the slip-flow regime — that is, for Knudsen number Kn≳0.1Kn\gtrsim 0.1, where the continuum equations with velocity-slip and temperature-jump boundary conditions used by Offner and Ramon (2016) cease to be reliable and transport is governed by the Boltzmann kinetic equation; in the source's own words, "a complete theory for this regime has not yet been developed." The theory should replace the velocity-slip and temperature-jump boundary conditions of that framework with kinetic gas–surface scattering, keeping the linear standing-wave setting and the source's constant-property assumption, and deliver, for rarefied stack flow, the predictive counterparts of that framework's outputs: (i) the onset condition of a thermoacoustic engine — the critical temperature difference for self-oscillation as a function of rarefaction and of the resonance frequency — and (ii) the performance of a thermoacoustic heat pump — the maximum sustainable temperature difference across the stack. The source model used the standing-wave approximation "for the sake of simplicity" and took gas and material properties (viscosity, thermal conductivity, sound speed, heat capacity) as constants over the stack "to retain simple, analytic expressions"; these are acknowledged simplifications of the slip-flow analysis rather than separately posed open problems, and the effect of rarefaction on onset and heat pumping beyond the slip-flow regime is currently unknown. A treatment confined to the slip regime 0.01≤Kn≤0.10.01\le Kn\le 0.1 would not address the problem.

References

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