MMRC — cathodic metal electrowinning

Metal recovery, energy per gram deposited and a deposit-morphology score for Cu, Zn, Ni, Cd, Pb, Cr.

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

1. $$j_{lim} = \frac{n F D C}{\delta}$$

   Source: Textbook limiting-current relation; mmrc.ts cites no source for it.

   Symbol	Meaning	Unit
   $j_{lim}$	diffusion-limited current density	A/m²
   $n$	electrons per metal ion (Cu 2, Cr 3)	1
   $D$	aqueous diffusion coefficient (Cu 7.14×10⁻¹⁰)	m²/s
   $C$	influent metal concentration (default 500 mg/L)	mol/m³
   $\delta$	diffusion-layer thickness (default 30 µm)	m

2. $$j_{op} = \min\left(j_{an}\,(1 + V_{app}),\ 1.2\, j_{lim}\right), \qquad s = \min\left(1,\ \max\left(0,\ 1.2 - \frac{j_{op}}{j_{lim}}\right)\right)$$

   Source: Linear voltage bump and a linear morphology ramp; mmrc.ts cites no source for it.

   The voltage term is a linear approximation of a Tafel response, and the score is a ramp in j/j_lim; Sand's transition time is described in the header but not computed.

   Symbol	Meaning	Unit
   $j_{an}$	anode current density from predict.ts	A/m²
   $V_{app}$	applied voltage (default 0.6)	V
   $s$	deposit morphology score: 1 smooth, 0 dendritic	1

3. $$FE = \max\left(0.1,\ FE_{max} - 0.3\max(0,\ \mathrm{pH} - pK_h) - 0.1\,\lvert E^\circ\rvert\,[E^\circ < 0]\right)$$

   Source: Per-metal values from Pourbaix data and Modin et al. 2012, J. Hazard. Mater. 235-236:291 (as stated in mmrc.ts).

   A linear penalty model of HER competition, not a kinetic branch ratio.

   Symbol	Meaning	Unit
   $FE_{max}$	intrinsic ceiling (Cu 0.95, Zn 0.70)	1
   $pK_h$	hydrolysis pK_a; above pK_h + 1 the metal precipitates as hydroxide and FE = 0	1
   $E^\circ$	standard reduction potential vs SHE	V

4. $$m = \frac{j_{op}\, A\, t\, FE\, M}{n F}, \qquad \mathrm{recovery} = \min\left(99,\ 100\,\frac{m}{C\,V}\right)$$

   Source: Faraday's law of electrolysis.

   Symbol	Meaning	Unit
   $m$	metal mass deposited	g
   $A$	cathode area, fixed at 0.003	m²
   $t$	run duration (default 24 h)	s
   $M$	molar mass (Cu 63.55)	g/mol
   $C\,V$	influent metal mass in the chamber (default 100 mL)	g

   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 · mmrc.metalRecoveryPct = 38.92 %, within ±0.1 %

   By hand · j = 0.15 A/m² from predict.ts at 25 °C, pH 7, 1000 mg/L COD, no anode named (see the MFC worked example). Copper, V_app = 0.6 V: j_op = 0.15 × 1.6 = 0.24 A/m² (j_lim = 36.1 A/m² does not bind). FE = 0.95 at pH 4. m = 0.24 × 0.003 × 86 400 × 0.95 × 63.55 / (2 × 96 485) = 0.019 46 g of 0.050 g, so 38.92 %.

   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 "mmrc.metalRecoveryPct"

• Anode current from the shared Butler-Volmer biofilm kinetics in predict.ts (Geobacter on acetate), with the same Arrhenius, substrate, pH and ohmic corrections as the MFC predictor.
• Simplified Pourbaix speciation: cation below the hydrolysis pK_a, hydroxide above.
• Diffusion-layer thickness δ = 30 µm (stirred cell) by default.
• Catholyte pH 4 and a 500 mg/L copper feed by default — unsourced engineering defaults.
• The applied voltage is read from the legacy `appliedVoltage` field only.

• Morphology score 0–1 from j / j_lim (0 = dendritic, 1 = smooth).

• Multi-metal selectivity in a mixed feed — single-metal feed assumed.
• Sulfide co-precipitation in acid-mine-drainage treatment.
• Anode dissolution / membrane fouling over time.
• Sand's transition time (described in the header, not computed).

• Modin et al. 2012, J Hazard Mater 235-236:291
• Gregory & Lovley 2005, ES&T 39:8943

libs/shared/electrochemistry/src/predictors/mmrc.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.

No benchmark cell reaches the minimum n for these systems — intervals are withheld.

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

No v2 prior row for these systems.

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.