Thermionic Energy Converter (2D)

Electrode Temperatures

Emitter temperature T_e 1800 K

Collector temperature T_c 600 K

Materials & Geometry

Emitter work function W_e 1.80 eV

Vacuum gap d 20 µm

Live Output

Current density J

– A/cm²

Barrier voltage V₀

– V

Power density P

– W/cm²

Efficiency η / Carnot

– %

—

How it works

A thermionic converter is two electrodes facing each other across a vacuum gap. The hot emitter boils electrons off its surface; they cross the gap and land on the cooler collector, driving current through an external load. No moving parts — heat goes in, electricity comes out.

Richardson-Dushman:  J = A·T² · exp(−W / (k·T))
Net current:          J = J_e(T_e, W_e) − J_c(T_c, W_c)
Barrier voltage:      V₀ = W_e − W_c   (volts, from eV)
Child-Langmuir limit: J_CL = (4/9)·ε₀·√(2e/mₑ)·V₀^1.5 / d²
Electron cooling:     Q_e = J·(2k·T_e/e + W_e)
Efficiency:           η = J·V₀ / (Q_e + σ·ε·(T_e⁴ − T_c⁴))

T_e, T_c — emitter and collector temperature. A bigger gap raises the theoretical output but also the heat lost to radiation, so efficiency has an interior optimum.
W_e — emitter work function. Lower work function emits far more current (it enters the exponential), but also lowers the barrier voltage per electron — real converters use caesium vapour to hold W_e low without destroying V₀.
Vacuum gap d — the electrons in flight form a negative space-charge cloud. The Child-Langmuir law caps how much current a gap of width d can actually carry regardless of how many electrons the emitter tries to boil off; when this cap is below the Richardson emission, the device is space-charge-limited and the electron stream visibly bunches near the emitter instead of streaming smoothly across.
The collector is modelled with a fixed low work function of 1.0 eV (a caesium-coated collector, as in real converters) so the barrier voltage stays positive across the slider ranges.

Real-world relevance: this exact mechanism (with a Cs vapour reservoir instead of a hard vacuum) powered early space-reactor concepts and is being revisited for waste-heat recovery from industrial furnaces and concentrated solar receivers, where there is a hot surface but no working fluid to spin a turbine.

Emitter (hot) Collector (cool) Electron stream