thermoelectric-gen

name	thermoelectric-gen
description	Thermoelectric generator (TEG) — Seebeck effect (V = α×ΔT), Peltier and Thomson effects, figure of merit ZT = α²σT/k, TEG power output and efficiency (Carnot × η_max), device geometry (unicouple, module), maximum power and efficiency operating points, thermal resistance network, Bi₂Te₃/PbTe/SiGe materials, matched load resistance, module characterization, and waste heat recovery applications.
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Thermoelectric Generator (TEG) — Complete Skill

Thermoelectric Effects

Seebeck Effect (Power Generation)

Seebeck effect: voltage generated by temperature gradient across dissimilar conductors V_Seebeck = α_AB × ΔT = (α_A − α_B) × (T_H − T_C) α_AB = α_A − α_B = Seebeck coefficient of thermocouple pair [V/K; typical: 100–300 μV/K for semiconductors]

Sign convention: α > 0 (p-type): current flows from hot to cold inside material α < 0 (n-type): current flows from cold to hot inside material

Physical origin: hot-side carriers have higher energy → diffuse toward cold side → charge buildup → electric field

Peltier Effect (Refrigeration/Heating)

Peltier effect (inverse of Seebeck): heat pumped at junction when current flows Q_Peltier = α_AB × T × I [W; T = junction temperature; I = current] Reversible (not I²R dissipation): Q_Peltier sign changes with current direction

Thomson Effect

Thomson effect: heat generated/absorbed in homogeneous conductor with temperature gradient and current: Q̇_Thomson = τ_T × I × dT/dx [τ_T = Thomson coefficient = T × dα/dT] Usually neglected in device analysis (smaller contribution); included in rigorous TEG analysis

Figure of Merit

ZT Definition and Physical Interpretation

Material figure of merit: z = α² × σ / k [S²/m² / (W/m·K)^(-1) = 1/K; σ = electrical conductivity; k = thermal conductivity] ZT = z × T = α² × σ × T / k [dimensionless; T = absolute temperature [K]]

Components: α² × σ = power factor [W/(m·K²)] — maximize → high Seebeck × high conductivity k = thermal conductivity — minimize → reduces heat flow; maintains ΔT

Wiedemann-Franz law conflict: σ and k_electronic both increase with electron concentration → trade-off Phonon engineering: introduce nanostructures, grain boundaries, alloy scattering → reduce k_lattice without affecting σ

ZT values for materials:

Material	Temperature [°C]	ZT	α [μV/K]	Application
Bi₂Te₃ (commercial)	25–230	0.8–1.0	220–260	Room-temperature
PbTe:Na (p-type)	400–500	1.4–2.0	150–280	Mid-temperature
PbTe:I (n-type)	400–600	1.0–1.7	−230	Mid-temperature
GeTe-based	400–600	2.0–2.5	200	Mid-T; R&D
SiGe (NASA)	600–1,000	0.6–1.0	−150	RTG (space)
Half-Heusler	400–700	0.8–1.5	200	Automotive WHR
SnSe crystal	600–800	2.6	160	Record; not practical

Current state of the art: ZT ≈ 1.0 (commercial); ZT ≈ 2.5 (lab nanostructured); Carnot efficiency × η_max scales with ZT

TEG Device Physics

Unicouple Model

Single p-n pair (unicouple): p-leg: α_p > 0 (Bi₂Te₃-p; Sb₂Te₃); n-leg: α_n < 0 (Bi₂Te₃-n; Bi₂Se₃) Total Seebeck coefficient: α = α_p − α_n [sum of absolute values for p-n pair; typical 350–500 μV/K]

Open-circuit voltage: V_OC = α × (T_H − T_C) = α × ΔT

Internal resistance: R_int = ρ_p × L_p / A_p + ρ_n × L_n / A_n [ρ = electrical resistivity; L = leg length; A = cross-section]

Thermal conductance (open-circuit): K_element = k_p × A_p / L_p + k_n × A_n / L_n [W/K]

Power Output and Efficiency

Matched load (R_L = R_int): V_matched = V_OC / 2 = α × ΔT / 2 I_matched = V_OC / (2 × R_int) = α × ΔT / (2 × R_int) P_max = V_OC² / (4 × R_int) = α² × ΔT² / (4 × R_int)

General load: P = V_OC² × R_L / (R_int + R_L)² = α² × ΔT² × R_L / (R_int + R_L)² I = α × ΔT / (R_int + R_L) Maximum at R_L = R_int

TEG efficiency: η_TEG = P_out / Q_H [Q_H = heat absorbed from hot source]

Q_H energy balance: Q_H = α × T_H × I − (1/2) × I² × R_int + K × ΔT [Peltier heating − Joule (half appears on hot side) + conduction]

Maximum efficiency (optimum R_L for maximum efficiency, not maximum power): η_max = (ΔT/T_H) × (√(1 + ZT_mean) − 1) / (√(1 + ZT_mean) + T_C/T_H) ZT_mean = Z × T_mean = α² × T_mean / (R_int × K_element) [T_mean = (T_H + T_C)/2]

Relationship to Carnot: η_max = η_Carnot × (√(1 + ZT_mean) − 1) / (√(1 + ZT_mean) + T_C/T_H) For ZT → ∞: η_max → η_Carnot = ΔT/T_H For ZT = 1: η_max / η_Carnot ≈ 0.40 (40% of Carnot) For ZT = 2: η_max / η_Carnot ≈ 0.55

Optimum load for maximum efficiency: R_L_opt = R_int × √(1 + ZT_mean) [different from R_int for maximum power]

Numerical Example

Bi₂Te₃ module for waste heat recovery: T_H = 200°C = 473 K; T_C = 25°C = 298 K; ΔT = 175°C α = 420 μV/K (pair); R_int = 0.8 Ω; K_element = 0.04 W/K; ZT_mean = 0.85 at T_mean = 385 K

V_OC = 420×10⁻⁶ × 175 = 73.5 mV per couple P_max = (0.0735)² / (4 × 0.8) = 0.00540 / 3.2 = 1.69 mW per couple

Q_H = α × T_H × I + K × ΔT = 420e-6 × 473 × (73.5e-3/(2×0.8)) + 0.04 × 175 = 420e-6 × 473 × 0.0459 + 7.0 = 0.00912 + 7.0 = 7.01 W per couple

η_max = (175/473) × (√1.85 − 1) / (√1.85 + 298/473) = 0.370 × (1.360 − 1) / (1.360 + 0.630) = 0.370 × 0.360 / 1.990 = 0.067 = 6.7%

TEG Module and System

Module Construction

Commercial TEG module: arrays of unicouples (N_couples = 127, 254, 511) electrically in series, thermally in parallel Module V_OC = N × α × ΔT [N = number of couples] Module R_int_total = N × R_int_per_couple Module P_max = N × P_max_per_couple (at matched load)

Typical module (Laird, Ferrotec, Marlow): 62×62 mm; N = 127 couples; α_module = 40–60 mV/K; R_int = 1–4 Ω; K_th = 0.5–2 W/K At ΔT = 100°C: V_OC = 5 V; R_int = 2 Ω; P_max = 6.25 W; η ≈ 4–6%

Thermal Resistance Network

Heat flow path: Q_source → R_contact1 → R_module → R_contact2 → Q_sink

Effective temperature across module: ΔT_module = ΔT_total − Q × (R_h + R_c) [R_h = hot-side thermal resistance; R_c = cold-side; Q = total heat] Iterative solution (Q depends on ΔT_module which depends on Q)

TEG efficiency including heat exchangers: η_system = P_out / Q_total = η_TEG × (ΔT_module / ΔT_total) Fin/heat exchanger optimization: maximize ΔT_module → minimize R_contact

Contact resistance: thermal paste R_contact ≈ 0.001–0.01 K·m²/W; high-conductivity bonding important

Waste Heat Recovery Applications

Automotive (Exhaust TEG)

ORC competitor: TEG produces electricity directly (no moving parts) vs. ORC (turbine/expander) Target: 1 kW electrical from exhaust (500–700°C hot gas; coolant cold side) TEG cost challenge: ZT~1 → ~5% efficiency; 20 kW waste heat × 5% = 1 kW; cost goal: < $1/W ($1,000 total)

Weight/volume: TEG mass for 1 kW ≈ 5–10 kg (current TRL 5–6); bypass valve required for low-load conditions

Industrial Waste Heat

Steel plant, cement kiln: T_exhaust = 300–800°C; large surface area available → MW-scale potential RTG (Radioisotope Thermoelectric Generator): PuO₂ → SiGe → 300 W_e; 25 year life; Voyager, Cassini, Mars Curiosity

Building/Solar TEG

Solar-TEG: concentrated solar → T_H = 200–300°C; combined with PV (multi-junction) Performance: η_TEG ≈ 5–8% at this T_H; less than PV efficiency but complementary spectral use

TEG Characterization Testing

Harman method: direct ZT measurement in situ Apply DC current → Seebeck voltage builds up → steady-state Z = V_Seebeck / (V_Joule) × 1/T I-V curve: sweep R_L from 0 to ∞ at fixed ΔT → extract V_OC, I_SC, R_int, P_max ASTM E2456: terminology for thermoelectric energy conversion (in development)

Standards and References

Standard	Scope
ASTM E2161	Standard terminology for thermoelectric elements
IEC 62492-1	Radiation thermometers (Seebeck background)
NASA SP-4014	RTG design (SiGe TEG for space)
DOE Vehicle Technologies Office	Automotive TEG R&D targets

Output

Provide: application (waste heat source: T_H [°C]; heat flow Q [W]; cold side T_C [°C]; ΔT [°C]; required P_out [W] or maximize efficiency), material selection (T_H range → Bi₂Te₃/PbTe/SiGe; ZT_mean at T_mean; α [μV/K]; ρ_e [Ω·m]; k [W/mK]), unicouple design (α_pair [μV/K]; R_int [Ω]; K_element [W/K]; ZT_mean; leg geometry L [mm]; A [mm²] — optimized for matched ZT condition), module specification (N_couples; V_OC [mV/K × ΔT = V]; R_int_module [Ω]; P_max at matched load [W]; η_max [%]; comparison with Carnot η [%]), thermal resistance analysis (R_hot [K/W]; R_cold [K/W]; effective ΔT_module vs. ΔT_total; efficiency penalty from heat exchangers), system performance (P_system [W]; η_system [%]; number of modules; total hot-side area [cm²]; flow rate/fin design), power electronics (MPPT DC-DC converter; efficiency 90–95%; output voltage/current for load), and applicable reference (NASA SP-4014 for RTG; DOE WHR program for automotive; commercial module datasheet for Bi₂Te₃).