Temperature gradient → Electrical work

Matrix context

D = disequilibrium row · C = coupling family column. A cell reads the state of the atlas for that pair.

D.01 × C.03 · Temperature gradient × Thermoacousticfamily: ThermoacousticD.01 × C.10 · Temperature gradient × Piezoelectricfamily: Piezoelectric

Typed bounds on this route

Every limit the atlas holds for this route, as constraint records: reached through bounded_by claims on the route's own entities. Hard bounds (an upper bound or a formula) decide the thermodynamic-bound check above; benchmarks, constitutive relations and resource limits are listed and never decisive. No prose limit exists anywhere (pass 42).

• Carnot limit · formula-bound · hard, applies to this route
  η ≤ 1 − T_c/T_h for any heat engine between reservoirs at T_h and T_c
• Electromechanical coupling factor · formula-bound · hard, applies to this route
  converted fraction per cycle ≤ k² = d²/(s·ε)

Steps

1. 01
   Temperature gradient —drives→ Thermoacoustic effectestablishedclaim record

   thermal → acoustic · loss to thermal

   |∇T_crit| = β T_m ω |p₁| / (ρ_m c_p |u₁|); prime-mover onset requires |∇T| > |∇T_crit| · critical-gradient factor β T_m ω/(ρ_m c_p u₁) [K/(Pa·m)]

   p₁ and u₁ are the local first-order acoustic pressure and particle-velocity amplitudes; β is the volumetric thermal expansion coefficient (β T_m ≈ 1 for an ideal gas); this is the linear short-stack / standing-wave criterion of Swift 1988 — above the critical gradient the stack amplifies the wave, below it the same interaction pumps heat — not a closed-form performance law for every travelling-wave regenerator.

   • temperature gradient along the stack steeper than the critical gradient set by the acoustic pressure and displacement amplitudes
   • a resonant gas column

   gas · temperature gradient required

   refs [1]
2. 02
   Thermoacoustic effect —produces→ Acoustic waveestablishedclaim record

   acoustic → acoustic

   • travelling-wave thermoacoustic regime with a non-zero time-averaged acoustic power flux; pressure and particle velocity have the phase relation required for directed acoustic power (a looped tube or a travelling-wave regenerator), unlike the standing wave of a straight resonator

   gas

   refs [2] [3]
3. 03
   Acoustic wave —drives→ Direct piezoelectric effectdemonstratedclaim record

   acoustic → electrical · loss to thermal

   • sound pressure strains a compliant piezoelectric membrane or plate; standing and travelling waves both serve, the response following the pressure oscillation

   solid

   refs [4] [5]
4. 04
   Direct piezoelectric effect —produces→ Separated surface chargeestablishedclaim record

   electrical → electrical

   refs [6]
5. 05
   Separated surface charge —converts into→ Electrical workestablishedclaim record

   electrical → electrical · loss to thermal

   • electrodes on the charged surfaces connected through a load

   refs [7]

Evidence for the complete composition · 0

No source on record for this exact composition. The sources below support the individual steps, not the route as a whole.

Evidence for the constituent relations · 7

1. [1]
   G. W. Swift. Thermoacoustic engines (1988)

   Journal of the Acoustical Society of America 84, 1145–1180 · doi:10.1121/1.396617 · crossref ✓ 2026-09-19 · record
2. [2]
   T. Yazaki et al.. Traveling Wave Thermoacoustic Engine in a Looped Tube (1998)

   Physical Review Letters 81, 3128–3131 · doi:10.1103/PhysRevLett.81.3128 · record
3. [3]
   S. Backhaus, G. W. Swift. A thermoacoustic-Stirling heat engine: Detailed study (2000)

   Journal of the Acoustical Society of America 107, 3148–3166 · doi:10.1121/1.429343 · record
4. [4]
   Y. Zheng et al.. Thermoacoustic energy harvesting using thermally-stabilized polyacrylonitrile nanofibers (2022)

   Nano Energy 95, 106995 · doi:10.1016/j.nanoen.2022.106995 · record
5. [5]
   J. Smoker et al.. Energy harvesting from a standing wave thermoacoustic-piezoelectric resonator (2012)

   Journal of Applied Physics 111, 104901 · doi:10.1063/1.4712630 · record
6. [6]
   J. F. Nye. Physical Properties of Crystals: Their Representation by Tensors and Matrices (1985)

   Oxford University Press · book, no DOI · record
7. [7]
   H. B. Radousky, H. Liang. Energy harvesting: an integrated view of materials, devices and applications (2012)

   Nanotechnology 23, 502001 · doi:10.1088/0957-4484/23/50/502001 · crossref ✓ 2026-09-19 · record