MDC — Donnan-exclusion salt flux

Salt removal rate, desalination efficiency and energy per kg salt for a three-chamber CEM/AEM stack.

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

1. $$N_{NaCl} = \frac{CE_{desal}\, j}{F}\, P_{CEM}\, P_{AEM}$$

   Source: Faraday's law; Cao et al. 2009, Environ. Sci. Technol. 43:7148 (about 90 % desalination in the first MDC).

   Symbol	Meaning	Unit
   $N_{NaCl}$	salt removal flux	mol/m²/s
   $CE_{desal}$	coulombic-to-salt efficiency (default 1.0, clamped 0–1.5)	1
   $j$	anode current density from predict.ts	A/m²
   $F$	Faraday constant, 96 485	C/mol
   $P_{CEM}, P_{AEM}$	membrane permselectivities (default 0.95 each)	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
   temperatureC	25	°C
   ph	7	pH
   electrodeSpacingCm	2	cm
   hrtHours	24	h
   codMgL	1,000	mg/L

   Output · mdc.saltRemovalRate = 1.403e-6 mol Na⁺/m²/s, 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). N = 1.0 × 0.15 / 96 485 × 0.95 × 0.95 = 1.403×10⁻⁶ mol/m²/s.

   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 "mdc.saltRemovalRate"

2. $$C(t) = C_{floor} + \left(C_0 - C_{floor}\right) e^{-k t}, \qquad C_{floor} = 0.1\, C_0, \quad k = \frac{N A / V}{C_0 - C_{floor}}$$

   Source: First-order approach to a 10 % floor; Cao et al. 2009 reached about 10 % residual after 24 h (as stated in mdc.ts).

   k is chosen so the initial rate equals the Faradaic salt flux; the floor is a fixed fraction, not a conductivity model.

   Symbol	Meaning	Unit
   $C_0$	initial middle-chamber salt concentration (default 35)	g/L
   $C_{floor}$	residual floor where conductivity collapses	g/L
   $A$	smaller of the CEM and AEM areas (default 0.001)	m²
   $V$	middle-chamber volume (default 50 mL)	m³
   $t$	run duration (default 24 h)	s

3. $$E_{spec} = \frac{\left(j A V_{app} - P_{an} A\right) t}{3.6\times 10^{6}\ m_{salt}}$$

   Source: Energy balance as stated in mdc.ts; Cao et al. 2009 observed power output during desalination.

   Symbol	Meaning	Unit
   $E_{spec}$	net energy per kg salt removed; negative when the anode out-produces the bias	kWh/kg
   $V_{app}$	external bias (default 0)	V
   $P_{an}$	anode power density from predict.ts	W/m²
   $m_{salt}$	salt mass removed	kg

• 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.
• CE_desal defaults to 1.0 (literature 0.85–1.2; Cao et al. 2009 report ~90 % desalination efficiency).
• Membrane permselectivity ≈ 95 % for both CEM and AEM; co-ion leakage is a multiplicative loss.
• Anode area is taken equal to the smaller membrane area.

• Co-ion leakage factors on CEM and AEM.
• Conductivity collapse as the middle chamber depletes.

• Osmotic water flux (significant only above ~35 g/L feeds).
• Biofouling kinetics over weeks of operation.

• Cao et al. 2009, Environ. Sci. Technol. 43:7148 (first MDC, ~90 % desalination efficiency)
• Kim & Logan 2013, Desalination 308:122

libs/shared/electrochemistry/src/predictors/mdc.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.