MFC (served by the API) — calibrated material table

Areal power density, coulombic efficiency, cell voltage and current density for an MFC as served by /api/ml/predict, the MCP server and the chat assistant. An empirical lookup over anode and cathode materials, not an electrochemical model.

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

1. $$P = \min\!\left(30000,\ \max\!\left(50,\ P_{\mathrm{anode}} \min\!\left(1 + 0.3\,\frac{P_{\mathrm{cathode}}}{P_{\mathrm{anode}}},\ 2\right) s\, f_T\, f_{\mathrm{pH}}\right)\right)$$

   Source: Hand-calibrated material table, cathode-enhancement and synergy multipliers in enhanced-performance-models.ts; the file cites no source (calibration note 2026-07).

   Symbol	Meaning	Unit
   $P$	areal power density, clamped to the MFC range	mW/m²
   $P_{\mathrm{anode}},\ P_{\mathrm{cathode}}$	per-material base power from the hand-calibrated table; the catalog peak power for a material with no entry	mW/m²
   $s$	anode × cathode synergy multiplier from a pair table; 1.0 for a pair not listed	1
   $f_T,\ f_{\mathrm{pH}}$	temperature and pH factors below	1

   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
   anode	carbon-felt	material
   cathode	pt-c	material
   temperatureC	25	°C
   ph	7	pH

   Output · powerOutput = 3,172 mW/m², within ±0.1 %

   By hand · Carbon felt 2600 and Pt/C 3800 mW/m² base power: 2600 × min(1 + 0.3 × 3800/2600, 2) = 2600 + 1140 = 3740. Synergy 1.1 gives 4114. f_T = exp(−|25 − 30| / 20) = exp(−0.25) = 0.7788; pH_opt = (7.2 + 7.0)/2 = 7.1, so f_pH = exp(−0.01) = 0.9900. P = 4114 × 0.7788 × 0.9900 = 3172 mW/m², inside [50, 30000].

   Reproduce · Pinned by libs/shared/ml/src/__tests__/served-mfc-registry.test.ts. Run it from the repository root:

   pnpm exec jest -c libs/shared/electrochemistry/jest.config.js served-mfc-registry.test.ts -t "powerOutput"

2. $$f_T = \exp\!\left(-\frac{\lvert T - T_{\mathrm{opt}}\rvert}{20}\right), \qquad T_{\mathrm{opt}} = 30\,^{\circ}\mathrm{C}$$

   Source: Exponential decay in enhanced-performance-models.ts (calculateTemperatureFactor); no source cited.

   Symbol	Meaning	Unit
   $T$	operating temperature	°C
   $T_{\mathrm{opt}}$	the MFC profile’s optimum temperature	°C

3. $$f_{\mathrm{pH}} = \exp\!\left[-\left(\frac{\mathrm{pH} - \mathrm{pH}_{\mathrm{opt}}}{1.0}\right)^{2}\right]$$

   Source: Gaussian factor in enhanced-performance-models.ts (calculatePHFactor); no source cited.

   Symbol	Meaning	Unit
   $\mathrm{pH}$	operating pH	pH
   $\mathrm{pH}_{\mathrm{opt}}$	mean of the anode and cathode table optima; 7.0 for a material with no entry	pH

4. $$CE = \min\!\left(95,\ \max\!\left(10,\ 65.2\,\min\!\left(1,\ \frac{\sigma_{\mathrm{anode}}}{2000}\right) e^{-\left(\frac{T - 32}{15}\right)^{2}} e^{-\left(\frac{\mathrm{pH} - 7}{1.5}\right)^{2}}\right)\right)$$

   Source: Base CE 65.2 % ("average from experimental data") and Gaussian factors in enhanced-performance-models.ts (predictCoulombicEfficiency); no source cited.

   Symbol	Meaning	Unit
   $CE$	coulombic efficiency, clamped to [10, 95]	%
   $\sigma_{\mathrm{anode}}$	catalog conductivity of the anode; above 2,000 S/m it no longer limits CE	S/m
   $T$	operating temperature	°C
   $\mathrm{pH}$	operating pH	pH

5. $$V = \min\!\left(0.8,\ \max\!\left(0.2,\ 0.45 + 0.1\,\log_{10}\!\left(\frac{P}{1000} + 1\right)\right)\right)$$

   Source: Heuristic in serverless-predictor.ts ("typical MFC: 0.2–0.8 V"); no source cited.

   Symbol	Meaning	Unit
   $V$	cell voltage, a heuristic of power, not a polarisation solve	V
   $P$	areal power density from the first relation	mW/m²

6. $$j = \frac{P}{V}$$

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

   Symbol	Meaning	Unit
   $j$	current density	mA/m²
   $P$	areal power density	mW/m²
   $V$	cell voltage heuristic	V

• P_anode and P_cathode are per-material base powers from a hand-calibrated table. Calibration note (2026-07): they are the TYPICAL areal power a cell on that material reaches, rescaled from record values (one 636 W/m² novel-electrode paper) to the literature-typical 0.1–2 W/m² band; the material ordering and the temperature, pH and synergy multipliers were kept.
• s(anode, cathode) is a synergy multiplier from a small pair table; a pair not listed is 1.0.
• A catalog material with no table entry uses the catalog peak power density as its base power, pH optimum 7.0 and a paper count of 1.
• σ_anode is the catalog conductivity in S/m.
• Power confidence = min(0.95, (papers_anode + papers_cathode) / 30 + 0.5) from the table paper counts; CE confidence is the anode table reliability (0.7 for a material with no entry); voltage and current carry a fixed 0.9.

• No electrode kinetics: power does not depend on overpotential, substrate, COD, HRT or geometry.
• The anode surface area is accepted but unused (area factor 1.0).
• Voltage and current are a heuristic of power, not a polarisation solve.
• The table carries a per-material temperatureCoeff that no calculation reads; temperature acts only through f_T.

libs/shared/ml/src/enhanced-performance-models.ts

none — The predictor ships no band of its own. Served through /api/ml/predict, power, current, coulombic efficiency and COD removal get the benchmark out-of-fold band below, or no interval when no benchmark cell is large enough. Voltage gets no interval. How served intervals are built

The base powers and pH optima the served number is computed from, as the code holds them.

• 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.