Thermoacoustic Engine

What is modelled

In a half-wave resonator closed at both ends the gas pressure oscillates in a standing wave with antinodes at the ends and a velocity antinode in the middle. A porous stack with a temperature gradient exchanges heat with the oscillating gas parcels at the right phase to add acoustic energy. Once the gradient exceeds a threshold the oscillation grows exponentially until nonlinear losses saturate it.

Equations

• Resonance f = c / (2 L), c = √(γ R Tmean).
• Onset gradient ΔTon = 155 · losses / |sin(2π xs)| (reduced-order fit), hot-side temperature Th = Tc + P / 0.66 W/K in steady state.
• Envelope: dA/dt = (g - 0.85 (A/1800)²) A, with growth rate g = 0.9 (ΔT/ΔTon - 1) √(4 bar / p), saturating at A = 1800 √(g/0.85).
• Mode shapes p ∝ cos(π x/L), u ∝ sin(π x/L).

Checks run in the page

• The heater power at which the growth rate changes sign against 0.66 W/K times the onset gradient.
• The simulated saturation amplitude against its fixed point.
• Helium against argon resonance ratio and the closed-form frequency.

Honest simplifications

This is a modal, reduced-order model, not a Rott-equation solver: the onset gradient and the saturation law are fitted constants, there is no stack geometry, no streaming and no heat-exchanger design. The drawn parcel displacement is exaggerated for visibility.

• Choose Below onset: the pressure decays. Choose Just above onset: it grows slowly, then saturates.
• Move the stack towards the end of the tube and read the growth-rate graph shift: the coupling |sin 2 pi x| is largest at one quarter of the length.
• Switch to argon: the same tube sings about three times lower.