Electrochemical cells: Faraday balance, electricity and heat¶

process/cells.yaml (schema cheme/electrochemical-cells/v1) describes the electrochemical cells of a process: electrolysers for hydrogen, chlor-alkali cells and similar units at steady state. cheme process cell evaluates them, cheme validate reports their findings and cheme maturity gives a flowsheet step a cell is linked to the model basis electrochemical-cell. The code is chemeascode.domains.process.electrochemical_cell (model faraday-balance v1); the decision is ADR 0023.

Specified performance, not predictive electrochemistry¶

The cell voltage and the Faradaic (current) efficiencies are inputs: measured, vendor or literature figures with their provenance and source. What follows from them is exact bookkeeping (Faraday's law, the charge, element and energy balances). No polarisation curve, Nernst equation or Butler-Volmer kinetics predicts a voltage; a cell with performance: predictive is refused (cheme.cell-unsupported). cheme maturity therefore counts a cell as specified: it can support balanced, never predictive.

Measured performance¶

A cell with performance: measured takes its voltage and Faradaic efficiencies from a measured series (electrochemical measurements, ADR 0041) at its current density I / active_area:

  - id: BPM-1
    performance: measured
    reaction: CO2R
    electrons: 2
    current: {value: 0.75, unit: A, provenance: given, source: "..."}
    active_area: {value: 5, unit: cm2, provenance: given, source: "..."}   # required
    side_reactions:
      - {reaction: WS, electrons: 2}       # current_fraction from the measurement
    measurements:
      file: experiments/electrochemistry/rbpm-1m-koh.yaml
      product: CO                          # the measurement's product of the overall reaction
      side_products: [H2]                  # their FE -> the side reaction their entry names
    thermal: {kind: isothermal, temperature: {value: 25, unit: °C, provenance: given}}

The values are the measurement's point means, interpolated linearly in the current density; the result names the file, the bracketing points and the standard deviations. Outside the measurement's validity range of current density or temperature (within 0.5 K), or beyond its measured points, the cell is refused (cheme.cell-operating-range, error): a measured cell is never extrapolated.
voltage, faradaic_efficiency or a side reaction's current_fraction may be left out. A value given anyway that differs from the measured one by more than the replicates' standard deviation is cheme.cell-measured-conflict (warning); the measured value is used.
A side product whose measurement entry names a reaction of side_reactions gives that reaction its current fraction; one that names none counts as unassigned current (heat), with an info finding and a limitation.
The measurement's voltage basis must be a terminal voltage: electrode potentials against a reference are refused; an iR-corrected voltage is used with the limitation that the power and heat leave out the ohmic loss.
The measured Faradaic efficiencies are of one electrode: the product's and the side products'. A product the measurement forms by another reaction than the cell's (or, with no reaction: there, one that is no product of the cell's reaction), or a side product at the other electrode, is refused (cheme.cell-measurement). A side product measured at 0 (below detection) gives its side reaction no extent.
Measured efficiencies that add up to more than 1 at the cell's current density passed the measurement's own fe-sum-high check (a file with errors refuses the cell), so they are scaled to add up to 1 (cheme.cell-measurement, info; a limitation and measurement.normalised_from give the sum). Side fractions stated in cells.yaml on top of them are not scaled: above 1 in all, the cell is refused (cheme.cell-charge).
A missing measurement file, one with error findings, or a product it does not have is cheme.cell-measurement (error, refused). For maturity a measured cell is specified (never predictive) and sourced when the measurement file states its source.

Membrane¶

An optional membrane: block reports the transport across the membrane apart from the Faraday balance, which it never changes:

    membrane:
      kind: bipolar                        # bipolar | cation_exchange | anion_exchange | diaphragm | none
      bias: reverse
      water_dissociation_efficiency: {value: 0.94, provenance: given, source: "..."}   # or from the measurement
      co_ion_crossover: {K+: 0.34, HCO3-: 0.66}   # shares of the current not carried by water dissociation
      source: "..."

Quantity	Relation
co-ion current	`(1 - WDE) N I`
co-ion flux of ion `i` (charge `z_i`)	`f_i (1 - WDE) N I / (abs(z_i) F)`, mol/s and kmol/h
anolyte hydroxide deficit	`(1 - WDE) N I / F`, mol/s and kmol/h

A measured cell takes the water dissociation efficiency from its measurement (with its sd and the measured carbon share of the co-ion current); a stated one that disagrees is cheme.cell-measured-conflict. Shares that do not add up to 1, an unreadable ion or an efficiency outside [0, 1] are cheme.cell-membrane errors; no efficiency at all is an info. The water dissociation efficiency is not a Faradaic efficiency: it does not change the extents, the products or the heat.

Stacks¶

An optional stacks: section (same schema, optional fields only) sizes electrolyser stacks of a cell of this file for a product duty: the cells and stacks, their arrangement on the DC bus, power, output and envelope, and the skid equipment a duty of process/duties.yaml sizes. cheme process stack evaluates it; the cell's Faraday balance does not read it. See electrolyser stacks and ADR 0044.

The contract¶

schema: cheme/electrochemical-cells/v1
kind: electrochemical_cells
cells:
  - id: MEM-1
    reaction: CA                       # overall reaction of process/reactions.yaml
    electrons: 2                       # per extent of it
    electrodes:
      anode: {species: {Cl-: -2, Cl2: 1, e-: 2}}
      cathode: {species: {H2O: -2, e-: -2, H2: 1, OH-: 2}}
    dissociation:                      # dissolved components as ions
      NaCl: {Na+: 1, Cl-: 1}
      NaOH: {Na+: 1, OH-: 1}
    cells: 1                           # in series, each carrying the current
    current: {value: 6, unit: kA, provenance: given, source: "..."}
    active_area: {value: 1, unit: m2, provenance: given, source: "..."}
    faradaic_efficiency: {value: 0.95, provenance: given, source: "..."}
    voltage: {value: 3.0, unit: V, provenance: given, source: "..."}
    side_reactions:                    # optional: Faradaic losses that form something
      - {reaction: OER, electrons: 4, current_fraction: {value: 2, unit: "%", provenance: given, source: "..."}}
    crossover: {H2: 0.01}              # fraction of a formed product leaving at the other electrode
    purge: {H2: 0.005}                 # fraction of a formed product purged
    thermal:
      kind: isothermal                 # adiabatic: not supported in v1
      temperature: {value: 90, unit: °C, provenance: given, source: "..."}
    operating_range: {temperature: [353.15, 363.15], current_density: [1400, 6500], source: "..."}
    flowsheet: {step: ELY, case: design, electricity: E1, heat: Q1}   # optional

Values are the process Quantity (value, unit, provenance, source, uncertainty, origin); the reaction, its stoichiometry and heat_of_reaction are those of process/reactions.yaml, the formulas and molar masses those of process/materials.yaml. Units: current A, kA, MA; voltage V, mV; area m2, cm2; efficiencies as a fraction (no unit, -, 1) or %; temperature K or °C.

What is calculated¶

For N cells in series carrying the current I at the voltage U:

Quantity	Relation
extent of the overall reaction	`xi = FE N I / (n F)`, F = 96485.33212 C/mol (CODATA 2018)
extent of a side reaction	`xi_k = f_k N I / (n_k F)`
unassigned current	`1 - FE - sum f_k`: no net chemical change (recombination, shunt currents), becomes heat
electric power (DC, at the terminals)	`P = N U I`, counted once
thermoneutral voltage	`U_tn = dH / (n F)`
heat to remove at the isothermal boundary	`Q = P - sum xi dH` (negative: heat to supply)
electricity per kg of a product	`P / m_formed` and `P / m_delivered`

Q is never added to the electricity: the energy balance is "electricity in = reaction enthalpy + heat out", and the report shows its residual (zero). The electricity per kg is the whole cell's electricity per kg of each product: for co-products (chlorine, caustic soda, hydrogen) it is not allocated; allocation is the LCA's decision.

Checks and refusals¶

Every finding names process/cells.yaml, the line, a rule id and a hint.

Refused (error, no numbers): no current, no Faradaic efficiency, no voltage, no thermal boundary, a reaction without heat_of_reaction or a molar mass (cheme.cell-input-missing); performance: predictive or an adiabatic boundary (cheme.cell-unsupported); current fractions above 1 (cheme.cell-charge); an unknown or unbalanced reaction. Nothing is assumed in place of a missing input.
Electrodes: each half-reaction balances its atoms and its charge (ions written as formula and charge, OH-, Na+, SO4-2; electrons e-); the anode releases electrons, the cathode takes them; scaled to electrons, the two add up to the overall reaction once the dissolved components are written as their ions (cheme.cell-electrodes). The home electrode of each product (where it, or one of its ions, is formed) decides where crossover sends it.
Operating range (warning): the temperature or current density outside the declared range; (info) no range declared. The data are never extrapolated.
Temperature (info): a heat of reaction taken at another temperature than the cell's; the sensible heat and the temperature dependence of dH are then left out and listed as a limitation.
Measured cells and membranes: see the two sections above (cheme.cell-measurement, cheme.cell-measured-conflict, cheme.cell-operating-range, cheme.cell-membrane).
Flowsheet link: the step exists; the electricity stream flows into it and carries P; the heat stream carries Q (out of the step when positive); one stream named for both is an error and any other energy stream at the step a warning (cheme.cell-double-count).

Limitations¶

Electrolyte properties (activities, heats of dilution, conductivity) are not modelled. The heats of reaction are those of process/reactions.yaml on their standard states; for brine and caustic at process concentrations they differ by the heats of dilution. Electrolyte thermodynamics is a linked dependency.
No adiabatic boundary, temperature rise or stream enthalpies (v1).
One current efficiency for the overall reaction: below 1 it reduces every product alike, and the current lost becomes heat, not chemistry. In a chlor-alkali cell the cathode forms H2 at about 100 %, while hydroxide back-migration forms hypochlorite, chlorate and O2 in the anolyte (the BREF, 1.4.2, gives 1.070–1.128 t NaOH/t Cl2 for this reason). The H2 and NaOH flows at a current efficiency below 1 are therefore approximate. The electricity per t Cl2 is not affected. To include those reactions, declare them as side reactions with their current fractions.
Several cells linked to one step: the step counts as sourced only when every one of them is (cheme.cell-flowsheet warns).
Not a unit of the DWSIM simulation: a linked step is balanced with the cell's specified values.
Rectifier, transformer and busbar losses are outside the cell (the power is DC at the terminals).

Qualification¶

tests/test_electrochemical_cell.py checks the model against data that do not come from the code:

Case	Reference	Result	Tolerance
water electrolysis: `U_tn`	dfH(H2O, l) = -285.830 kJ/mol, CODATA Key Values (Cox, Wagman, Medvedev 1989); ATcT through `chemicals`	1.4812 V (tabulated 1.481 V)	0.0005 V (the data's +-0.04 kJ/mol is 0.0002 V)
PEM cell, 1.9 / 1.8 / 1.6 V, FE 1 (assumed)	US DOE, Technical Targets for PEM Electrolysis (stack): 51 / 48 / 43 kWh/kg H2	50.52 / 47.86 / 42.54 kWh/kg	the published integer, by rounding (see below)
at `U = U_tn`	HHV of H2 141.88 MJ/kg (DOE Hydrogen Analysis Resource Center)	`Q = 0`, 141.78 MJ/kg	0.2 % (molar mass and data source)
FE 0.975 with crossover and purge	the DOE footnote's 2.5 % crossover loss	extents x 0.975, `Q = N I (U - FE U_tn)`	1e-12 relative
chlor-alkali: `U_tn`	dfH(Cl-, aq) -167.080, dfH(OH-, aq) -230.015, dfH(H2O, l) -285.830 kJ/mol (CODATA), infinite dilution	2.3101 V	0.0005 V
products per t Cl2	EU BREF chlor-alkali (Brinkmann et al. 2014, EUR 26844 EN, doi:10.2791/13138), section 2.1: 1.128 t NaOH, about 28 kg H2	1.1283 t NaOH, 28.4 kg H2	0.0005 t; rounding to 1 kg
membrane cell at 3.0 V, 6 kA/m2, CE 0.95 (BREF 4.3.2.3.2)	BREF Table 3.10, bipolar membrane cells 2 279-2 865 AC kWh/t Cl2; rectifiers 94-98 % (3.3.4.3.1)	2 387 DC kWh/t Cl2, 2 436-2 540 as AC	inside the range

The DOE does not state a current efficiency, so the reference cell assumes 1 (provenance: assumed). Its published kWh/kg are U 2F / M(H2) at that efficiency. Footnote e of the same table says they include a 2.5 % H2 crossover loss, but with it they would round to 52 / 49 / 44 kWh/kg. The comparison therefore checks the electricity per kg against the DOE's own arithmetic. It is not evidence for the efficiency.

Charge (residual current), elements (residual per element) and energy (P - sum xi dH - Q) close to round-off in every case. The qualified range is what the table says: 1.6-1.9 V per cell for water electrolysis, 3.0 V at 6 kA/m2 for a chlor-alkali membrane cell, heats of reaction at 298.15 K. The chlor-alkali enthalpy is on the infinite-dilution standard state; a cell at 90 °C with 32 wt-% caustic is reported with the cheme.cell-temperature limitation.

cheme process cell --json records the model id and version, the core version, the assumptions, limitations, the qualified cases and the sha256 of every input file.