MFC — Butler-Volmer + Monod single-overpotential snapshot

Areal power and current density, coulombic efficiency and COD removal for an air- or O₂-cathode MFC.

Each relation is transcribed from the implementation. Where the code cites no source, the source line says so.

1. $$j_{BV} = i_0\left[\exp\!\left(\frac{\alpha n F \eta_a}{RT}\right) - \exp\!\left(-\frac{(1-\alpha) n F \eta_a}{RT}\right)\right]$$

   Source: Textbook Butler-Volmer relation (calculateButlerVolmer in scientificConstants.ts); predict.ts cites no source for it.

   Symbol	Meaning	Unit
   $j_{BV}$	anodic current density before substrate and transport limits (code works in A/cm², reports ×10⁴)	A/m²
   $i_0$	exchange current density after the Arrhenius and roughness corrections	A/cm²
   $\alpha$	anodic transfer coefficient (catalog value, else 0.5)	1
   $n$	electrons transferred, fixed at 2	1
   $F$	Faraday constant, 96 485.332 12	C/mol
   $\eta_a$	anode overpotential, a single snapshot at 0.15 V	V
   $R$	gas constant, 8.314 462 618	J/(mol·K)
   $T$	temperature	K

2. $$i_0 = i_{0,\mathrm{cat}}\, \exp\!\left[\frac{E_a}{R}\left(\frac{1}{298.15} - \frac{1}{T}\right)\right] f_R, \qquad f_R = \min\!\left(100,\ \max\!\left(1,\ 2\log_{10} a_{BET} + 1\right)\right)$$

   Source: Arrhenius form with E_a = 25 kJ/mol; roughness form calibrated in-house (Phase B, 2026-04-29); predict.ts cites no source for it.

   The roughness factor is a log-scaled heuristic with a floor of 1, not a measured area ratio.

   Symbol	Meaning	Unit
   $i_{0,\mathrm{cat}}$	catalog exchange current density for the named anode; 5×10⁻⁸ when none is named or the slug is unknown	A/cm²
   $E_a$	activation energy of a mixed-culture biofilm, 25 000	J/mol
   $R$	gas constant, 8.314 (anode-kinetics.ts)	J/(mol·K)
   $f_R$	roughness factor; 1 when the catalog has no BET area	1
   $a_{BET}$	specific surface area of the anode from the catalog	m²/g

3. $$f_S = \frac{S}{K_s + S}$$

   Source: Rittmann & McCarty 2001, Environmental Biotechnology (K_s = 120 mg/L, mixed anaerobic culture); other buckets calibrated against the validation-tuple corpus.

   Symbol	Meaning	Unit
   $f_S$	substrate factor applied to the current	1
   $S$	influent COD	mg/L
   $K_s$	half-saturation constant from the substrate bucket, or a citation-backed override	mg/L

4. $$j = j_{BV} f_S \left[1 - \exp\!\left(-\frac{j_{lim}}{j_{BV} f_S}\right)\right], \qquad j_{lim} = \max\!\left(0.5,\ 50\,\frac{S}{1000}\,\frac{T}{298.15}\right)$$

   Source: Unsourced engineering default in predict.ts (2026-10 citation audit).

   A smooth saturation that bounds the unphysical high edge; it does not model species transport.

   Symbol	Meaning	Unit
   $j$	served current density	A/m²
   $j_{lim}$	substrate-diffusion ceiling	A/m²
   $S$	influent COD	mg/L
   $T$	temperature	K

5. $$V_{cell} = 0.75\, f_d f_{pH} - \min\!\left(\frac{j}{10^4}\left(ASR_a + ASR_c\right),\ 0.7 \cdot 0.75\, f_d f_{pH}\right), \qquad f_{pH} = e^{-(\mathrm{pH}-7)^2/2}$$

   Source: Representative OCV, Gaussian pH factor and linear spacing penalty; predict.ts cites no source for it.

   The IR drop is capped at 70 % of the working voltage so pathological inputs cannot drive it negative.

   Symbol	Meaning	Unit
   $V_{cell}$	operating cell voltage	V
   $0.75$	representative open-circuit voltage of an acetate-fed membraneless MFC	V
   $f_d$	spacing penalty: 1 up to 2 cm, then 0.12 per cm, floor 0.4	1
   $f_{pH}$	Gaussian pH factor, optimum 7, σ = 1	1
   $ASR_a, ASR_c$	electrode area-specific resistances from the catalog, else 5 each	Ω·cm²
   $j$	served current density	A/m²

6. $$P = 10^{3}\, j\, V_{cell}$$

   Source: Definition of electrical power, P = I·V.

   Symbol	Meaning	Unit
   $P$	areal power density	mW/m²
   $j$	served current density	A/m²
   $V_{cell}$	operating cell voltage	V

   Worked example

   These inputs, passed to the predictor, return the output below. A test re-runs the example on every change, so this number is the one the code produces.

   input	value	unit
   temperatureC	25	°C
   ph	7	pH
   electrodeSpacingCm	2	cm
   hrtHours	24	h
   codMgL	1,000	mg/L

   Output · result.powerDensity = 115 mW/m², within ±0.5 %

   By hand · No anode named: i₀ = 5×10⁻⁸ A/cm², f_R = 1, Arrhenius factor = 1 at 25 °C. F·η_a/(R·T) = 96 485.33 × 0.15 / (8.314 46 × 298.15) = 5.838, so j_BV = 5×10⁻⁸ × (e^5.838 − e^−5.838) × 10⁴ = 0.1716 A/m². f_S = 1000/1120 = 0.893, j = 0.1532 A/m² (the transport factor is 1.000). V_cell = 0.75 − 0.1532/10⁴ × 10 = 0.7498 V. P = 0.1532 × 0.7498 × 1000 = 114.9, reported to the nearest whole number as 115 mW/m².

   Reproduce · Pinned by libs/shared/electrochemistry/src/predictors/worked-examples.test.ts. Run it from the repository root:

   pnpm exec jest -c libs/shared/electrochemistry/jest.config.js worked-examples.test.ts -t "result.powerDensity"

7. $$CE = \min\left(80,\ 20\, f_{HRT} f_{pH}\right), \quad \mathrm{COD} = \min\left(95,\ 75\, f_{HRT} f_S f_T\right), \quad f_{HRT} = \min\left(1.2,\ 0.3 + 0.5\log_{10}(\mathrm{HRT}+1)\right)$$

   Source: Empirical saturation forms; predict.ts cites no source for it.

   Heuristic saturation curves, not a substrate mass balance.

   Symbol	Meaning	Unit
   $CE$	coulombic efficiency	%
   $\mathrm{COD}$	COD removal	%
   $f_{HRT}$	retention-time saturation factor	1
   $\mathrm{HRT}$	hydraulic retention time	h
   $f_T$	temperature step: 1 above 20 °C, else 0.8	1

• Single anode overpotential snapshot η_anode = 0.15 V — not a polarisation sweep.
• Mixed-culture Monod constant K_M = 120 mg COD/L unless the substrate lookup gives another.
• Arrhenius activation energy E_a = 25 kJ/mol for a mixed-culture biofilm; i₀ normalised to 298 K.
• pH response is a Gaussian around 7.0 with σ = 1.0.
• An anode the catalog cannot resolve, or none, uses i₀ = 5×10⁻⁸ A/cm².

• Electrode roughness factor on i₀ from the electrode catalog.
• Ohmic penalty from electrode spacing (linear above 2 cm).
• Ohmic IR drop from electrode area-specific resistance (gated by NEXT_PUBLIC_LAB_PHYSICS_OHMIC).
• Mass-transport ceiling on the current (gated by NEXT_PUBLIC_LAB_PHYSICS_MASS_TRANSPORT).
• Biofilm thickness clamped to 0–5 mm.

libs/shared/electrochemistry/src/predict.ts

heuristic_fixed_fraction — A fixed ±25 % placeholder band. It is NOT a calibrated interval; when served through /api/ml/predict it is replaced by the benchmark band below, or withheld. How served intervals are built

• Benchmark out-of-fold residual quantiles: bes-benchmark-v1 · gkf5 · generated 2026-09-09
• Split-conformal calibration: split-conformal · generated 2026-05-21
• Hierarchical-priors v2 trust snapshot: populated · generated 2026-05-16

Each row is a stratum with n ≥ 30 residuals at nominal level 90%. Coverage is the cross-validated share of held-out points inside the band.

The state the lab badge shows for each served parameter and system, from the v2 priors fit.

pnpm verify:science runs these rules over the committed priors and validation artifacts in pre-commit. Known violations are baselined per rule and slug; a new one blocks the commit. Baseline from hierarchical-priors-v2.json@2026-10-26T00:10:47.721880+00:00.