Numerical Modeling of PEM Fuel Cells in 2

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Numerical Modeling of PEM Fuel Cells in 2½D
M. Roos, P. Held ZHW-University of Applied
Sciences Winterthur Switzerland F. Büchi, St.
Freunberger PSI-Paul Scherrer Institut Switzerland

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Outline

• PEM Modeling Issues
• Modeling Goals
• 2½D Modeling Approach
• 1D Interaction Model
• Implementation
• Preliminary Results
• Conclusions

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PEM Modeling Issues

• Large variation of important length scales
• Electrochemistry at nm
• Porous Flow at um
• Flow Fields at mm
• Balance of Plant at m
• Multi-domain physical modeling mandatory for many
  technical important applications
• Electrochemistry and Water transport
• Flow and heat generation and transport
• Charge transport, diffusion and reaction
• Complexity of geometrical structure
• Layered structures
• Repeated sub systems

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Modeling Goals

• Understanding local processes in detail
• reaction mechanisms in electrochemistry, e.g., by
  state space models
• Investigation coupled part processes
• interaction of mass transport and electrochemical
  reaction in PEM flow field structures
• Understanding cell / stack behavior
• water management in PEM systems strong
  dependence on operation conditions. Influence of
  processes in distant parts of a cell / stack.
• Dynamic behavior of full stacks including
  auxiliary systems
• Setting up control systems
• Performance optimization

• Flexibility with respect to Interactions
• Demanding Geometries
• Fast Calculation

How to realize?
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2½D Modeling Approach 1
Basic Idea
HEXIS Stack ModelingCalculating effective
transport parameters and deploying them in
rotational symmetric 2D models ? large reduction
in computational effort,? justified method
Application to PEM
Problem there is no dimension to map form 3D to
2D (if full stacks or cells are in the focus) ?
Resort to two or more interacting 2D models
describing the cathode and anode flow
fields ? Volume Averaging Method works in
this case
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2½D Modeling Approach 2

• Single (repeated) fuel cell decomposed into 3
  parts
• Anode Flow Field
• Membrane Electrode Assembly (MEA)
• Cathode Flow Field
• Modeling of flow fields by two 2D domains,
  Discretisation of Transport Equations with FEM
  and Volume Averaging Method (VAM)
• El. chem. Reactions treated as non-local
  interaction of the 2D part models

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2½D Modeling Approach 3

• The models of the 2D part domains can be tailored
  to describe transport phenomena
• Fluid Flow
• Diffusion (Channels)
• Heat Transport
• Method of choice for the determination of
  effective material parameters is the volume
  averaging method, preferably its numerical
  variant.
• Mass transport in flow fields, expressed by
  anisotropic Darcy tensor. ? Different pressure
  drops for flow parallel or perpendicular to the
  channels (underneath the rims).

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1D Interaction Model 1

• The electrochemical interaction within the MEA is
  1D
• low in-plane electrochemical interaction and
  associated transport processes
• coupling of different positions in the MEA plane
  is effected by the gas composition of the 2D
  domains and by the (per bipolar plate) constant
  electric potential.
• PEM transport properties (water concentration
  dependence) does not allow for analytic
  expressions of the electrical current density in
  terms of partial pressures, potentials and
  temperatures of the corresponding points in flow
  fields.

• Modeling the 1D interaction by a set of ordinary
  differential equations (ODE)

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1D Interaction Model 2

• The 1D interaction model accounts for
• Darcy flow in the gas diffusion layer
• Diffusion (Maxwell-Stefan for multi species)
• Heat transport
• Charge Transport
• Electrochemical reactions at the anode and
  cathode (including the kinetics)
• Water transport in the PEM (drag and diffusion)
• Interactions temperature dependent material and
  gas properties, source rates for temperature
  field due to reversible and irreversible heat
  release processes.
• Due to the 1D domain, the mathematical form of
  these equations is a system of nonlinear, coupled
  first order ordinary differential equations for
  14 physical quantities (the potentials molar
  concentration for H2, H2O, O2, N2, temperature,
  electric potential, pressure and the respective
  flow quantities).
• The strong non-linearity asks for efficient and
  robust methods numerical methods for its
  discretisation.

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Implementation 1

• The 2D domains are implemented as NM SESES models
  for anode and cathode, respectively with help of
  the finite element method.
• The independent degrees of freedom are the gas
  species, the (averaged) temperature field, the
  pressure distribution and the velocity field for
  each compartment.
• The 1D electrochemical interaction model turns
  out to be a source rate for the chemical species
  from the point of view of the 2D part models,
  i.e., it is cast into an effective, homogeneous
  chemical reaction.
• Electrochemical reaction couples fields of the
  anode to the cathode domain? interaction is
  non-local for FE model our model set up.
• The non-locality of the interaction is,
  unfortunately, not a standard type of interaction
  in FEM (not physical for 3D or true 2D
  situations).
• NM SESES offers user friendly and efficient
  interfaces for user-defined interactionsnon-local
  ity is handled by domain decomposition methods in
  order to obtain an efficient, fast algorithm.

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Implementation 2
Map tr(x,y)

• Implementation of the 1D interaction
• Shooting Method for non-linear ODE system
• Realization as C-program
• Setting up mapping transformation
• Formulation of interactions in terms of functions
  of the form
• Defining iteration algorithms
• Using domain decomposition techniques to
  decompose the degrees of freedom
• Adopting convergence acceleration methods (e.g.
  SOR)

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Preliminary Results 1

• Simple Test Model (Co or Counter flow)

H2O molar fraction
current density
H2O transfer rate
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Preliminary Results 2

• Inspection of internal states

Mass fractions in GDL
? value in Membrane
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Preliminary Results 3

• Test Cell Model

Pressure
Cathode Anode
Velocity
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Preliminary Results 4

• Species molar fractions

H2O
H2
sheet resistance of Membrane
O2
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Conclusions

• 2½D Modeling is an interesting approach to bridge
  the gap between complex geometries / interactions
  and the need for PEM cell / stack models
• NM SESES is easily equipped with user defined
  interactions that model electrochemistry, water
  transport, etc. in PEM membranes
• Efficient numerical iteration schemes allow for
  fast solutions of these numerical models.
• Full modeling for cells of technical relevance is
  in reach
• There is a lot of work to do extension of the
  water transport in GDL and Membrane (e.g., two
  phase description)
• Extensive validation of the models is a demanding
  work in itself, but mandatory for save
  application in technical developments
• Further Steps Extensions to dynamic models to
  support the control system development

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Proposal

• Many Different Modeling approaches are (and will
  be) discussed. Each variant with its specific
  pros and cons.
• Models, which are in vertical relation to each
  other, can provide parameter input to the lower
  level models (e.g., similar to the VAM)
• Models which are on the same level can serve as
  test bench to improve the quality, accuracy and
  efficiency

Benchmarking Choosing a test cell of technical
relevance including all important data regarding
geometry, material properties, typical operation
conditions, etc. (provided, e.g., by the PSI
group) Defining operation states with extensive
experimental data (already present?). Benchmarking
the approaches Model Accuracy, using detailed
models to evaluate parameters for coarser models,
etc. Setting up an Internet site with the
contributions (including a news group?) Financing?