1. Swiss Fuel Cell Symposium, Yverdon-les-bains, 19./20. May 2003 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

2. Fuel Cell Research Symposium, Modeling and exp. Validation, 18./19. March 2004 Outline PEM Modeling Issues Modeling Goals 2½D Modeling Approach 1D Interaction Model Implementation Preliminary Results Conclusions

3. Fuel Cell Research Symposium, Modeling and exp. Validation, 18./19. March 2004 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

4. Fuel Cell Research Symposium, Modeling and exp. Validation, 18./19. March 2004 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?

5. Fuel Cell Research Symposium, Modeling and exp. Validation, 18./19. March 2004 2½D Modeling Approach 1 Basic IdeaHEXIS Stack Modeling: Calculating 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

6. Fuel Cell Research Symposium, Modeling and exp. Validation, 18./19. March 2004  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 2½D Modeling Approach 2

7. Fuel Cell Research Symposium, Modeling and exp. Validation, 18./19. March 2004  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). 2½D Modeling Approach 3

8. Fuel Cell Research Symposium, Modeling and exp. Validation, 18./19. March 2004 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)

9. Fuel Cell Research Symposium, Modeling and exp. Validation, 18./19. March 2004 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 H 2, H 2 O, O 2, N 2, temperature, electric potential, pressure and the respective flow quantities).  The strong non-linearity asks for efficient and robust methods numerical methods for its discretisation.

10. Fuel Cell Research Symposium, Modeling and exp. Validation, 18./19. March 2004 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 interactions non-locality is handled by domain decomposition methods in order to obtain an efficient, fast algorithm.

11. Fuel Cell Research Symposium, Modeling and exp. Validation, 18./19. March 2004 Implementation 2  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) Map tr(x,y)

12. Fuel Cell Research Symposium, Modeling and exp. Validation, 18./19. March 2004 Preliminary Results 1  Simple Test Model (Co or Counter flow) H 2 O molar fractioncurrent densityH 2 O transfer rate

13. Fuel Cell Research Symposium, Modeling and exp. Validation, 18./19. March 2004 Preliminary Results 2  Inspection of internal states λ value in Membrane Mass fractions in GDL

14. Fuel Cell Research Symposium, Modeling and exp. Validation, 18./19. March 2004 Preliminary Results 3  Test Cell Model Velocity Cathode Anode Pressure

15. Fuel Cell Research Symposium, Modeling and exp. Validation, 18./19. March 2004 Preliminary Results 4  Species molar fractions H2OH2O sheet resistance of Membrane O2O2 H2H2

16. Fuel Cell Research Symposium, Modeling and exp. Validation, 18./19. March 2004  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 Conclusions

17. Fuel Cell Research Symposium, Modeling and exp. Validation, 18./19. March 2004 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?