Journal of Applied Electrochemistry

, Volume 40, Issue 5, pp 921–932 | Cite as

Mathematical modeling of high-pressure PEM water electrolysis

  • S. A. GrigorievEmail author
  • A. A. Kalinnikov
  • P. Millet
  • V. I. Porembsky
  • V. N. Fateev
Original Paper


This paper is devoted to the modeling and numerical optimization of proton-exchange membrane (PEM) water electrolysers for operation at elevated pressures (up to 130 bars). The model takes into account different geometrical parameters of the PEM cell, the kinetics of the hydrogen and oxygen evolution reactions, the electro-osmotic drag of water molecules, the permselectivity of the solid polymer electrolyte and associated gas cross-over phenomena. The role of various operating parameters (such as pressure, temperature, current density, flow rate of water) on cell efficiency, faradaic yield and heat produced during water electrolysis is evaluated and discussed. The model is also used for the purpose of optimizing the performances of PEM cells. In particular, optimal values of some critical operating parameters (current density, rate of water supplied to the anodes) are recommended.


Proton-exchange membrane (PEM) Water electrolysis High pressure Mathematical modeling 

List of symbols

Cg, Cl, CH, CO (mol m−3)

Molar density of gas, liquid, hydrogen, oxygen.

υg, υl (m s−1)

Velocity of gas and liquid.

υ0 (m s−1)

Velocity of cooling water

Pg, Pl, Pc (Pa)

Pressure of gas, liquid and capillary pressure

Kp (m2)

Permeability of porous current collector

Rp (m)

Average pore radius of porous current collector

krl, krg

Relative permeabilities of liquid and water

s (adim.)

Saturation of porous current collector

sfc (adim.)

Saturation at the current collector–flow channel interface

scl (adim.)

Saturation at the current collector–catalytic layer interface

θc (deg.)

Wetting angle of porous current collector

ε (adim.)

Porosity of gas diffusion layer

Rbc (m)

Critical bubble radius

Ng, Nl, N (mol m−2 s−1)

Flux of gas, liquid, total flux

Nga, Nla (mol m−2 s−1)

Gas and liquid fluxes at current collector–anodic layer interface

Ngc, Nlc (mol m−2 s−1)

Gas and liquid fluxes at current collector–cathodic layer interface

CH (mol m−3)

Hydrogen concentration in the gas phase

Rp (m)

Effective pore radius

iΣ, i0, ip (A m−2)

Total, useful, and parasitic current densities

np (adim.)

Electro-osmotic drag coefficient

Ps (Pa)

Pressure of saturated water vapor

hm (m)

Membrane thickness

h (m)

Gas diffusion layer thickness

hfc (m)

Flow channel thickness

ΔT (°C)

Temperature difference along the membrane-electrode assembly (MEA)

SMEA (m2)

Active area of MEA

iex (A m−2)

Exchange current density

ρm (Ohm−1 m−1)

Specific ionic conductivity of solid polymer electrolyte

Physicochemical parameters

Cpw = 4,217 (J kg−1 K−1)

Specific heat of liquid water at T = 100 °C

μl = 2.822 × 10−2 (Pa s)

Dynamic viscosity of liquid water at T = 100 °C and P = 13 MPa

μH = 1.09 × 10−5 (Pa s)

Dynamic viscosity of hydrogen at T = 100 °C and P = 13 MPa

μO = 2.47 × 10−5 (Pa s)

Dynamic viscosity of oxygen at T = 100 °C and P = 13 MPa

σ = 0.05884 (Pa m)

Surface tension of water at T = 100 °C and P = 13 MPa

ГH = 1.67 × 10−3 (mol mol−1)

Water solubility of hydrogen at T = 100 °C and P = 13 MPa

ГO = 1.85 × 10−3 (mol/mol)

Water solubility of oxygen at T = 100 °C and P = 13 MPa

Eeq = 1.29 (V)

Equilibrium cell voltage at T = 100 °C and P = 13.0 MPa

ΔH° = −285.0 (kJ/mol)

Enthalpy change for reaction \( {\text{H}}_{2} + {\frac{1}{2}}{\text{O}}_{2} \to {\text{H}}_{2} {\text{O}}\left( l \right) \) at T = 100 °C

Fitted equations

np = 2.3 + 0.0212 · (T − 80)

Number of solvated water molecules associated with each proton (drag factor) in solid polymer electrolyte at full humidification versus temperature T in  °C (approximated from data in [1]).

\( D_{\text{OM}} = 2.44\, \times \,10^{ - 8} \exp \left( { - {\frac{2,100}{T}}} \right)\, \)(m2 s−1)

Diffusion coefficient of oxygen in solid polymer electrolyte at full humidification (approximated from data in [2]).

\( D_{\text{HM}} = 5.65\, \times \,10^{ - 8} \exp \left( { - {\frac{2,100}{T}}} \right)\,\, \)(m2 s−1)

Diffusion coefficient of hydrogen in solid polymer electrolyte at full humidification (approximated from data in [2]).

\( F_{\text{c}} (s) = \left( {\begin{array}{*{20}c} {1.417(1 - s) - 2.12(1 - s)^{2} + 1.263(1 - s)^{3} \quad } & {{\text{if}}\,\theta_{\text{c}} < 90^{^\circ } } \\ {1.417s - 2.12s^{2} + 1.263s^{3} } & {{\text{if}}\,\theta_{\text{c}} > 90^{^\circ } } \\ \end{array} } \right. \)

Leverett function which is used to determine capillary pressure in porous media as a function of the wetting angle θ c (taken from [3]).

\( E_{\text{r}} = E^{(0)} - 8.5\, \times \,10^{ - 4} \left( {T - 298.15} \right) + 4.308\, \times \,10^{ - 5} T\ln \left( {P_{\text{H}} P_{\text{O}}^{0.5} } \right) \) (V)

Reversible voltage of electrolysis cell as a function of temperature T and gas pressures.

\( E_{\text{tn}} = - {\frac{{\Updelta H^{(0)} }}{2F}} - 8.5\, \times \,10^{ - 4} \left( {T - 298.15} \right) + 4.308\, \times \,10^{ - 5} T\ln \left( {P_{\text{H}} P_{\text{O}}^{0.5} } \right) \) (V)

Thermoneutral voltage of electrolysis cell as a function of temperature T and gas pressures.



This work has been supported by the Federal Agency for Science and Innovations of the Russian Federation within the framework of the Federal Principal Scientific-Technical Programme “Researches and development on priority directions in development of scientific technological complex of Russia for 2007–2012” and by the Commission of the European Communities (6th Framework Programme, STREP project GenHyPEM no 019802). Financial support from the Global Energy International Prize Non-Profit Foundation (Grant no MG-2008/04/3) is also gratefully acknowledged.


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Copyright information

© Springer Science+Business Media B.V. 2009

Authors and Affiliations

  • S. A. Grigoriev
    • 1
    • 2
    Email author
  • A. A. Kalinnikov
    • 1
  • P. Millet
    • 3
  • V. I. Porembsky
    • 1
  • V. N. Fateev
    • 1
    • 2
  1. 1.Hydrogen Energy and Plasma Technology Institute of Russian Research Center “Kurchatov Institute”MoscowRussia
  2. 2.Moscow Power Engineering Institute (Technical University)MoscowRussia
  3. 3.Institut de Chimie Moléculaire et des Matériaux, UMR CNRS, n° 8182Orsay CedexFrance

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