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Modelling of miniature proton exchange membrane fuel cells for portable applications

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Modelling of miniature proton exchange membrane fuel cells for portable applications. J.O. Schumacher 1 , E. Fontes 3 , D. Gerteisen 1 , F. Goldsmith 1 , R. Klöfkorn 2 , A. Hakenjos 1 , K. Kühn 1 , M. Ohlberger 2 , A.Schmitz 1 , K. Tüber 1 , C. Ziegler 1

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Modelling of miniature proton exchange membrane fuel cells for portable applications

  • J.O. Schumacher1, E. Fontes3, D. Gerteisen1, F. Goldsmith1, R. Klöfkorn2, A. Hakenjos1, K. Kühn1, M. Ohlberger2, A.Schmitz1, K. Tüber1, C. Ziegler1
    • 1. Fraunhofer Institute for Solar Energy Systems, Heidenhofstr. 2, 79110 Freiburg,, Germany
    • 2. Institute of Applied Mathematics, University of Freiburg, Herrmann-Herder-Str. 10, 79104 Freiburg, Germany
    • 3. COMSOL AB, Tegnergatan 23, SE-111 40 Stockholm, Sweden


  • Examples of portable fuel cell systems
  • Model based analysis of impedance spectra
  • Modelling of self-breathing fuel cells
  • Characterisation of an along-the-channel fuel cell
  • Dynamic simulation of two-phase flow
  • Conclusion and outlook

Fuel cell system for a professional broadcast camera

  • Completely integrated system with 4 fuel cell stacks
  • 40 W average system power
  • 2 Metal Hydride Storages (100 Nl H2 or 150 Whel)
  • Integrated DC/DC- Converter
  • Miniature fans for air supply

Mobile power box

  • Portable power supply
  • Power: max. 100 W average 50 W
  • Metal Hydride Storage
  • Control based on micro processor
  • 12 V voltage supply with DC/DC- Converter

Electrode agglomerate model

  • Electrode is assumed to be made of porous spherical catalyst grains
  • Oxygen is dissolved at the outer surface of the agglomerate
  • Diffusion of dissolved oxygen in the grain and the film in radial direction
  • Local current density is given by the Tafel-equation

Graph: Jaouen et al., 2002


Cathode agglomerate model

Mass balance

Charge balance

Oxygen flux in agglomerate


Cathode agglomerate model

Charge balance

Ohm`s law


Comparision of measured and simulated polarisation curves

Small current density: change of Tafel-slope

Influence of surface-to-volume ratio L of agglomerates

L = 6 105 m-1

L = 9 104 m-1

cell potential / [V]

cell potential / [V]

current density / [A/m2]

current density / [A/m2]


Simulation of impedance spectra

  • Perturbation of solution variables of PDEs
  • Small perturbations: linearise and Laplace-transform PDEs
  • Calculate impedance:

Resistance [Wm2]

Resistance [Wm2]


Comparision of measured and simulated impedance spectra

  • Minimum value of the radius of the impedance arc is reached at a current density of 260mA/cm2.
  • Mass transport limitation is observed for higher current density: increase of radius of impedance arc.


density [A/m2]

meas sim


Influence of double layer capacitance on impedance spectra

Double layer capacitance CDL = 3 107 F m-3

Small double layer capacitance:

Two seperate semicircles appear



density [A/m2]


density [A/m2]


of electrode

planar and self breathing fuel cells based on printed circuit board technology
Planar and self-breathing fuel cells based on printed circuit board technology
  • Benefits of technology:
  • Small cell thickness
  • High mechanical strength
  • Low cost components
  • Well known printed circuit board production technology
  • Integration of electronic circuits
modelling domain and assumptions
Modelling domain and assumptions
  • Two dimensional model
  • Plug flow conditions in anodic gas channel
  • Convective flux of species through membrane and on cathode side neglected
  • No phase transition accounted for

Discretisation mesh and governing equations

  • Multicomponent diffusion of gas species: Stefan-Maxwell equation
  • Electronic and protonic potential: Poisson equation
  • Transport of water across membrane: modified Stefan-Maxwell equation
  • Temperature distribution: heat equation
  • l

Hydrogen and oxygen distribution

H2 molar fraction

O2 molar fraction



Arrows: total flux of hydrogen and oxygen.

Vcell = 0.4 V


Water distribution and flux

H2O molar fraction x 10-3

H2O molar fraction



Arrows: total flux of water.

Vcell = 0.4 V


Heat flux and temperature

T [K]



  • Arrows: total flux of heat.
  • Cooling effect of ribs.
  • Vcell = 0.4 V
electronic and protonic potential current direction
Electronic and protonic potential, current direction

Protonic potential

Electronic potential

fe [V]

fp [V]

Arrows indicate the technical current direction.

comparison of experiment and simulation
Comparison of Experiment and Simulation



  • Opening ratio = cathode opening width / current collector rib width.
  • Limiting current is determined by oxygen supply through cathode opening.

Current distribution in cathode gas diffusion layer



cut line (e)





Normalised x-coordinate

Normalised y-coordinate


PEM fuel cell model based on FLUENT CFD-software

  • Submodels:
  • The electrochemical submodel predicts the local current-to-voltage relation in the MEA.
  • The electrical submodel accounts for electron flow and ohmic heat generation.
  • The MEA submodel describes transport of water and ions through a Nafion membrane.

Segmented fuel cell

‚Along - the - Channel‘

  • Flow-field geometry:
  • Parallel channels
  • Determination of
  • spatially resolved
  • current density
  • Measured values:
  • temperature,
  • gas flow-rates,
  • relative humidity

Current distribution along the channel

  • Comparison of measurement (dots) and simulation (lines)
  • Variation of air flow rate on the cathode side
  • All model parameters are kept constant except air flow and average current

gas flow direction:



Relative humidity of air in the channel

Temperature of air in the channel

Relative humidity of air at MEA

Membrane protonic resistivity


Profiles of flow velocity and temperature including inlet region

velocity profile

temperature profile


Dynamic simulation of two phase flow

Modelling concept by Mario Ohlberger (Institute for Applied Mathematics, Freiburg).

Solution of the PDEs for:

  • Two phase flow in porous media
  • Species transport in the gas phase
  • Energy balance in the porous media
  • Potential flow of electrons and protons

Adaptive grid generation in space / time

Colours: pressure distribution for counter-flow case.

Problem: Determination of material parameters


Two-phase flow in porous gas diffusion layer and electrodes


Mass balance


Water and gas saturation

Capillary pressure


Simulation examples





Mass fraction of gas components and saturation of liquid water


Red: 1, Blue: 0


Agglomerate model

  • The agglomerate model reproduces both, measured polarisation curves and impedance spectra.
  • Change of active agglomerate surface-to-volume ratio depending on the operation point?

Planar fuel cells

  • Our two-dimensional one-phase model includes all relevant processes of planar fuel cells: gas transport, heat transport, electrochemical reaction.
  • The model serves as a design tool for self-breathing planar fuel cells.

Current distribution

  • We validated the CDF model with locally distributed current measurements.
  • The CFD model agrees to measurement results if the cell is operated in the one-phase regime.
  • We are working on a dynamic two-phase flow model taking into account liquid water transport in porous media.
  • The model is extended to 3D. Parallel computing and adaptive grid generation is utilised.

Two-phase flow