MEC — Volmer-Heyrovsky hydrogen evolution

Volumetric H₂ production rate, cathodic hydrogen recovery and applied-voltage energy efficiency for a driven MEC.

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

1. $$V_{app} - \lvert E_{eq}\rvert = \eta_a(j) + \eta_c(j) + j\,R_{int}, \qquad \lvert E_{eq}\rvert = 0.13\ \mathrm{V}, \quad R_{int} = 0.04\, d$$

   Source: Voltage budget with R_int = 0.04 Ω·m² per cm; mec.ts cites no source for it.

   Symbol	Meaning	Unit
   $V_{app}$	applied cell voltage (default 0.6)	V
   $E_{eq}$	acetate-anode / H₂-cathode equilibrium voltage, −0.13	V
   $\eta_a, \eta_c$	anode and cathode overpotentials at current j	V
   $j$	kinetic current density	A/m²
   $R_{int}$	internal resistance	Ω·m²
   $d$	electrode spacing, floor 0.1	cm

2. $$\eta_a = \max\!\left(0,\ 0.045\,\log_{10}\frac{j}{j_{0,a}}\right), \qquad j_{0,a} = 1.0\ \mathrm{A/m^2} \cdot e^{\frac{E_a}{R}\left(\frac{1}{298.15}-\frac{1}{T}\right)} \cdot \frac{S}{K_s+S}$$

   Source: j₀ = 1.0 A/m² for a Geobacter biofilm on graphite brush, attributed in mec.ts to Logan 2008 Table 4.1 and not re-checked against the book (2026-10 citation audit).

   The MEC anode is a Tafel term, not the Butler-Volmer solve the MFC predictor uses; both share the Arrhenius and Monod factors from anode-kinetics.ts.

   Symbol	Meaning	Unit
   $\eta_a$	anode overpotential	V
   $0.045$	Geobacter anode Tafel slope	V/dec
   $j_{0,a}$	anode exchange current density after temperature and substrate factors	A/m²
   $E_a$	activation energy, 25 000	J/mol
   $S, K_s$	influent COD and its half-saturation constant	mg/L

3. $$\eta_c = b\,\log_{10}\frac{j}{j_{0,c}}\quad (j > j_{0,c}), \qquad b = 0.03\ \mathrm{V/dec} \times m$$

   Source: Approximate HER exchange-current table and Tafel slopes; mec.ts cites no source for it.

   The j₀ table is labelled approximate in the code; the spread is wide.

   Symbol	Meaning	Unit
   $\eta_c$	cathode hydrogen-evolution overpotential	V
   $b$	Tafel slope	V/dec
   $m$	slope multiplier: 1 for a Pt cathode, 4 for carbon, 2 otherwise	1
   $j_{0,c}$	HER exchange current density: 10 on Pt/C, 0.02 on stainless steel, 0.005 on plain carbon, 0.05 default	A/m²

4. $$j = \min\!\left(j_{kin},\ \frac{S \cdot \frac{24}{\mathrm{HRT}} \cdot V \cdot F}{8 \cdot 86\,400 \cdot A}\right)$$

   Source: COD stoichiometry, 1 mol O₂ = 32 g COD = 4 mol e⁻ (electron-supply.ts).

   An absolute ceiling: it assumes 100 % COD removal and 100 % coulombic efficiency, so it errs toward permitting too much current.

   Symbol	Meaning	Unit
   $j_{kin}$	current the voltage budget can drive	A/m²
   $S$	influent COD (g/m³ equals mg/L)	g/m³
   $\mathrm{HRT}$	hydraulic retention time	h
   $V$	chamber volume (default 28 mL)	m³
   $A$	anode area (default 0.024)	m²
   $8$	grams COD per mole of electrons (32 g O₂ per 4 e⁻)	g/mol

5. $$Q_{H_2} = \frac{j\,A}{2F} \cdot 86\,400 \cdot V_m \cdot 10^{-3} \cdot \frac{r_{H_2}}{V}$$

   Source: Faraday's law; magnitude anchor Cheng & Logan 2007, PNAS 104:18871 (1.1 m³ H₂/m³·day at 0.6 V). · doi:10.1073/pnas.0706379104

   Symbol	Meaning	Unit
   $Q_{H_2}$	volumetric hydrogen production rate at STP	m³ H₂/m³·day
   $j$	operating current density	A/m²
   $A$	anode area	m²
   $V_m$	molar gas volume at 0 °C and 1 atm, 22.414	L/mol
   $r_{H_2}$	cathodic hydrogen recovery (default 0.85)	1
   $V$	chamber volume	m³

6. $$\eta_E = \frac{n_{H_2}\,\Delta H_{H_2}}{j\,A\,V_{app}\,t} = \frac{r_{H_2}\,\Delta H_{H_2}}{2F\,V_{app}}$$

   Source: Logan et al. 2008, Environ. Sci. Technol. 42:8630. · doi:10.1021/es801553z

   With coulombic efficiency taken as 100 %, current cancels and η_E depends on V_app alone. The code flags it `appliedVoltageEfficiencyIsIdentity`: a thermodynamic ceiling, not a modelled result.

   Symbol	Meaning	Unit
   $\eta_E$	electrical energy efficiency; above 100 % is expected because the substrate supplies the balance	1
   $n_{H_2}$	hydrogen recovered over time t	mol
   $\Delta H_{H_2}$	higher heating value of H₂, 285 800	J/mol
   $r_{H_2}$	cathodic hydrogen recovery	1
   $V_{app}$	applied 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
   cellVoltageV	0.6	V

   Output · mec.appliedVoltageEfficiency = 209.8 %, within ±0.1 %

   By hand · r_H₂ = 0.85 (default). η_E = 0.85 × 285 800 / (2 × 96 485.33 × 0.6) = 242 930 / 115 782.4 = 2.098, i.e. 209.8 %. The current (0.163 A/m², substrate-limited) cancels.

   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 "mec.appliedVoltageEfficiency"

• Anode: Tafel term with a 45 mV/dec slope and j₀ = 1 A/m² at 25 °C and saturating substrate, scaled by the shared Arrhenius and Monod factors (anode-kinetics.ts).
• b_HER = 30 mV/dec on Pt, ×4 on carbon, ×2 otherwise; j₀_HER = 10 A/m² on Pt/C, 0.02 on stainless steel, 0.005 on plain carbon, 0.05 by default.
• Coulombic-to-hydrogen recovery r_H₂ defaults to 0.85 (range 0.6–0.95; lower with active methanogens).
• Defaults when not supplied: V_app = 0.6 V, chamber 28 mL, anode area 0.024 m².
• H₂ volume reported at STP (0 °C, 1 atm).

• Substrate-limited current cap from the COD supply (`substrateLimited`).

• COD removal and coulombic efficiency: both need a substrate mass balance the snapshot solver does not have, so they are returned as not computed.

• Liu, Grot & Logan 2005, Environ. Sci. Technol. 39:4317
• Cheng & Logan 2007, PNAS 104:18871 · doi:10.1073/pnas.0706379104
• Call & Logan 2008, Environ. Sci. Technol. 42:3401
• Logan et al. 2008, Environ. Sci. Technol. 42:8630 · doi:10.1021/es801553z
• Logan & Rabaey 2012

libs/shared/electrochemistry/src/predictors/mec.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.