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, [email protected], 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

Overview for portable applications

  • 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 50 W for portable applicationsmax laptop

Fuel cell system for a professional broadcast camera for portable applications

  • 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 for portable applications

  • 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 for portable applications

  • 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 for portable applications

Mass balance

Charge balance

Oxygen flux in agglomerate

Cathode agglomerate model for portable applications

Charge balance

Ohm`s law

Comparision of measured and simulated polarisation curves for portable applications

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 for portable applications

  • 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 for portable applications

  • 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 for portable applications

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 circuit board technology

  • 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 circuit board technology

  • 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 circuit board technology

H2 molar fraction

O2 molar fraction



Arrows: total flux of hydrogen and oxygen.

Vcell = 0.4 V

Water distribution and flux circuit board technology

H2O molar fraction x 10-3

H2O molar fraction



Arrows: total flux of water.

Vcell = 0.4 V

Heat flux and temperature circuit board technology

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 circuit board technology

Protonic potential

Electronic potential

fe [V]

fp [V]

Arrows indicate the technical current direction.

Comparison of experiment and simulation
Comparison of Experiment and Simulation circuit board technology



  • 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 circuit board technology



cut line (e)





Normalised x-coordinate

Normalised y-coordinate

PEM fuel cell model based on FLUENT CFD-software circuit board technology

  • 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 circuit board technology

‚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 circuit board technology

  • 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:

Analysis circuit board technology

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 region

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 region


Mass balance


Water and gas saturation

Capillary pressure

Simulation examples region





Mass fraction of gas components and saturation of liquid water


Red: 1, Blue: 0

Conclusion region

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.

Conclusion region

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