Low-grade waste heat to electricity: refuted magnet mechanisms and a thermomagnetic engine

Below 60 °C the heat engines of CA.5 stall, tempting heat-plus-magnet-plus-copper electricity claims. Three are tested and refuted, a static magnet does no work (Lorentz force ⊥ v), a heat pump and an engine are inverses (COP·η_Carnot = 1), and black copper only absorbs, replaced by a Carnot-bounded thermomagnetic engine reaching η ≈ 2.6–4.4%.

Chapter CA.5 left heat-to-electricity to a thermoelectric or Stirling engine and marked the "rotation-amplitude → electric field" generator as unproven. This chapter takes the case down to low-grade waste heat (40–60 °C: air-conditioner condensers, data centres), where those engines stall and a family of "heat + magnet + copper → electricity" claims becomes tempting. Three such mechanisms are tested and refuted — a static magnet does no work (Lorentz force ⊥ v), a heat pump and an engine are inverses (COP·η_Carnot = 1), and black copper is an absorber, not a converter — and replaced by a thermomagnetic engine in which heat switches a Curie-point ferromagnet's magnetisation across Tc. It is Carnot-bounded (1−T_c/T_h = 7.7% at ΔT = 25 K) yet operates on low-grade heat; with regeneration it reaches η ≈ 2.6–4.4%.

CA (chemistry applications), CM (conduction/magnetism, Chapter 3). Status grades: [F] forced (zero free parameters) · [F?] forced-candidate · [CAL] calibration input (measured constant) · [V] simulation-measured · [H] hypothesis · [O] open / refuted. Every numeric claim is reproduced by a deterministic, standard-library module (2× sha256 identical); module and sha256 anchors are given inline and collected in CA.13.

This chapter continues the discipline of CA.0 and platinum (Chapter CT): where a speculative mechanism is proposed, it is tested by direct simulation; what does not survive is reported as refuted and re-grounded on validated physics; what survives is kept and labelled. Here three device claims fail and one is kept — and the kept device is a heat engine, not a violation of one. Its active element is a Curie-point ferromagnet, not copper; copper stays non-magnetic (3d¹⁰, Chapter CM) throughout.

CA.8 The question: where the heat engines of CA.5 stall

CA.5 bounded heat-to-electricity by Carnot and used a thermoelectric (Seebeck, ZT = 1 → ~5.5%) or a Stirling engine (~½ Carnot) at a hot store. Those numbers assume a hot source. The waste heat that is actually abundant is low-grade: an air-conditioner condenser at ~50 °C, a data-centre coolant loop at ~40 °C. Two facts govern this regime. First, the absolute ceiling is the exergy of the source: η_max = 1 − T₀/T = 4.8% for 40 °C heat against a 25 °C ambient, 7.8% for 50 °C [F]. No collection scheme exceeds it — "collecting better" raises the quantity of heat, never the efficiency, which temperature fixes. Second, a real thermoelectric or Stirling engine reaches only a fraction of that ceiling and, mechanically, a Stirling barely runs on a 15–25 K span. This is the gap a tempting family of claims tries to fill: take the abundant low-grade heat, add a magnet and the free-electron sea of copper (Chapter CM), and "make electricity." The claims are tested below.

CA.9 Three refuted mechanisms [O]

CA.9a A static magnet aligns copper's conduction electrons into a current [O]

Module vp_magnet_lorentz.py integrates the conduction electrons in a static field with the Boris pusher (energy-exact for a magnetic force). The Lorentz force F = q(E + v×B) with a static B = B ẑ is perpendicular to the velocity, so it does no work: in the simulation the mean kinetic energy is constant to machine precision while each electron traces a closed cyclotron circle ω_c = qB/m. The electrons do keep oscillating — but the motion is circular, so the ensemble drift ⟨v_x⟩ ≈ 0 and the net current is zero. Adding an electric field instead produces a steady drift (a current) and rising kinetic energy. The conclusion is forced: a current is driven by an E-field — from a changing flux (Faraday), a junction's built-in field, or a thermal gradient (Seebeck) — and never by a static magnet. Copper compounds the failure twice: it is non-magnetic (3d¹⁰ closed shell, Chapter CM, so a magnet does not magnetise it) and its Seebeck coefficient ≈ 1.8 µV/K is negligible. A magnet has N/S poles, not the ± that an E-field uses to push charge. Module sha256 7c8246ab… (2× identical) [V].

CA.9b A heat pump "concentrates" the waste heat, then an engine generates from it [O]

Module vp_heatpump_exergy.py evaluates the loop. A heat pump and a heat engine are thermodynamic inverses: for the same reservoir pair, COP·η_Carnot = 1. Chaining them — pump the heat up, then run an engine down — therefore recovers, in the reversible limit, exactly the waste heat's exergy (4.8% at 40 °C) and returns the pumping work with no net gain; with a real pump (COP ≈ 2) and a real low-temperature engine (~10%) the loop is strongly negative (≈ −65% of the processed heat). The heat pump's celebrated "300–400% efficiency" is real but is heat moved, not energy created (4 units of heat = 1 of electricity + 3 of moved ambient heat), so it cannot be converted back to more electricity than was spent. The honest use of the condenser is the opposite of generation: lowering its temperature raises the air-conditioner's own COP, and the electricity saved exceeds anything generated. Module sha256 64e26ee6… (2× identical) [V].

CA.9c Black copper "generates" electricity from the collected heat [O]

This is the CA.5/CA.6 verdict carried one step further. Black copper is an absorber — α ≈ 0.96 by graded-index impedance matching and intrinsic CuO absorption (CA.6), pure optics — and converts radiation into heat. It is not a converter of heat into electricity: it has no thermoelectric, photovoltaic, or engine mechanism, its Seebeck is negligible, and it melts at 1085 °C, so it cannot even serve as the high-temperature emitter of a thermophotovoltaic cell (whose converter is the semiconductor cell, not the copper). The "rotation- amplitude → electric field" generator of CA.5 remains [H]; the absorber-as-generator reading is refuted outright. Module vp_blackcu_absorber_not_generator.py, sha256 0067cbe1… (2× identical) [V].

CA.10 The thermomagnetic engine, kept and quantified [V]

The valid "heat + magnet → electricity" device replaces the static field with a changing one, produced by heat itself. A soft-magnetic element with Curie temperature Tc tuned into the waste-heat band (Gd has Tc = 293 K; La(Fe,Si,Mn)₁₃ or rare-earth-free Mn-Fe-P-Si tune Tc to 40–60 °C with a sharp magnetisation step) sits on a spring between a hot surface near a permanent magnet and a cold sink. Below Tc the element is magnetised and the magnet pulls it onto the hot surface; there it heats above Tc, its magnetisation M collapses, the spring returns it to the cold side, it cools below Tc, M recovers, and the magnet pulls it in again. The thermal lag makes the magnetic force stronger on the inward (cold, high-M) stroke than the outward (hot, low-M) stroke, so it does net positive work each cycle — a self-sustained relaxation oscillation — and the moving magnetised element induces an EMF in a copper pick-up coil. Module vp_thermomagnetic_oscillator.py exhibits the limit cycle and a positive cycle-averaged harvested power (sha256 d6d8a9ab…, 2× identical) [V]. The decisive property is that the relevant temperature scale is the Curie point, not a large ΔT: the device operates on a 15–25 K span around Tc, exactly where a Stirling engine stalls. The active element is the Curie-point ferromagnet; copper is only the coil winding.

The bare-device efficiency is set by the magnetic work against the sensible-plus-latent heat shuttled each cycle: η ≈ ΔM·ΔB ⁄ [ρ(c_pΔT_sw + L)]. With a sharp-transition alloy (ΔM ≈ 0.7×10⁶ A/m, ΔB ≈ 0.35 T from an NdFeB N42), this is ≈ 0.4% at ΔT = 25 K — small, and bounded by Carnot 1 − T_c/T_h = 7.7%. The sensible/latent heat, not the magnetic work, is the loss. Module vp_thermomagnetic_design.py, sha256 0a3af6ec… (2× identical) [V].

CA.11 Regeneration and the practical ceiling [V]/[F?]

Because the loss is the heat shuttled to change the element's temperature, recovering it raises efficiency. Run the device as an active magnetic regenerator (AMR) — the mechanism of magnetic refrigeration, in reverse: a stack of thin magnetocaloric films with an oscillating water flow through a counterflow microchannel matrix that passes the released heat from a cooling film to a heating one. The regenerative efficiency is η = Carnot⁄[1 + (1−ε)R], with R ≈ 18 the ratio of shuttled to isothermal-magnetic heat: ε = 0.90 → 2.7%, ε = 0.95 → 4.0%, approaching Carnot as ε → 1 [V]. Thin films (f ∝ 1/thickness) give ~10× the power density and make a high-ε regenerator physically realisable. The microchannel is governed by two forced relations: effectiveness ε = K/(K+f) with K = Nu·k_f ⁄ (2 d_c² ρ_f c_f) (smaller channels run faster at a given ε), and pumping loss P_pump/P_gen = 48 μ f L² ⁄ (d_c t_p w), which a short flow length (L ≈ 20 mm) holds to 2–5% — so pumping is not the limiter. The practical window is η ≈ 2.6–4.4% (34–58% of Carnot) at f ≈ 1.5–4 Hz, the choice set by material cost versus output. Modules vp_thermomagnetic_regenerative.py (sha256 aa45e145…) and vp_regenerator_microchannel.py (sha256 aa296c82…), each 2× identical [V]. The Carnot ceiling itself (7.7% at ΔT = 25 K) is the wall no design crosses [F].

CA.12 Application scale and the honest order [H]/[CAL]

For a 10 kW air-conditioner rejecting ~13 kW at ~50 °C, the efficiency-optimised harvester yields ~290 W — about 9% of the unit's cooling electricity, and peak-coincident (most output on the hottest hours). Across India's ~110 million room air-conditioners this is of order 30 TWh/yr [CAL]. But the discipline of CA.9b applies: efficiency first. Better heat rejection and a higher-COP unit save ~30% of the same electricity — three to four times the harvester — so the harvester is an add-on, not the lever, for cooling in general. Its economic case is the always-on large load with no heat-reuse demand: a 1 MW data centre in a hot region (where the waste heat cannot be sold as district heating because nothing needs heating) gives ~23 kW continuous, ~205 MWh/yr, with a payback of ~4–6 years [H]. Module vp_datacenter_economics.py, sha256 13d054c5…; fleet module vp_waste_heat_scale.py, sha256 b59d22a5… (each 2× identical). The honest split mirrors CA.5: a Carnot-bounded conversion, the wrong mechanisms refuted, the validated one kept and sized — and the larger saving left where it actually is, in not spending the electricity in the first place.

CA.13 Master ledger — waste heat to electricity

[F] forced (zero free parameters): exergy ceiling 1 − T₀/T (4.8% at 40 °C, 7.8% at 50 °C); Lorentz force ⊥ v does no work; heat-pump/engine inversion COP·η_Carnot = 1; Carnot ceiling 1 − T_c/T_h = 7.7% at ΔT = 25 K; microchannel ε = K/(K+f) and pumping 48 μ f L²/(d_c t_p w). [V] simulation-measured (2× sha256): cyclotron drift ≈ 0 and constant kinetic energy (7c8246ab…); heat-pump loop −65% real (64e26ee6…); black-copper absorber/non-converter (0067cbe1…); self-sustained thermomagnetic oscillation (d6d8a9ab…); bare η ≈ 0.4% (0a3af6ec…); regenerative η 2.7–4.0% (aa45e145…); microchannel window η 2.6–4.4% at 1.5–4 Hz (aa296c82…). [CAL] calibration inputs: Gd Tc = 293 K; copper Seebeck 1.8 µV/K; NdFeB N42 ΔB ≈ 0.35 T; water properties; India fleet ~110 M units. [H] hypothesis: device-level performance and economics projections (per-unit ~290 W; data-centre ~23 kW, payback ~4–6 yr). [O] refuted: static-magnet current; heat-pump-upgrade generation; black-copper-as-generator. Anchors: CA.9–CA.12; repro folder repro/chemistry/07-low-grade-waste-heat-electricity/.