Energy Technology Innovation Initiative

Pore Network Modelling of PEMFC Materialswith OpenPNM

Tom Tranter

Types of PNMTopology and Geometry are key to realistic simulations.

e.g. Simple Cubic Network with “Stick-and-ball” geometryRandom Delaunay Network with Voronoi geometry

Notes

Pore Network Models represent a series of 1-D transport calculations mapped onto a 2 or 3 dimensional network or graph. Parameters are evaluated for each pore and throat based on initial boundary conditions and resistance to flow through the network which is determined by the geometry and topology i.e. pore and throat sizes and coordination. For a cubic network the distribution of sizes is imposed and data is fitted to match experiments, whereas for the random network geometry and topology are intrinsically linked and more realistic. When using the Voronoi geometry which is a compliment to the Delaunay network, the location of pores and the proximity to nearest neighbours directly controls the local pore and throat sizes. A higher pore density leads to a lower average pore size.

Transport CalculationAs the pore space is idealised the solution of rigorous transport equations, such as those employed by finite element or finite volume approaches, is replaced by simpler nodal balances.

This has both advantages and disadvantages, multiphase calculations become far simpler with the percolation processes following rule based algorithms and the speed of the calculations can be many orders of magnitude greater. However, details of the local variation of properties within a single pore are lost such as velocity streamlines and linear approximations to transport must be used.

Convection – Darcy’s Law: Mass balance

Diffusion – Fick’s Law: Species balance

The key to accurate representation is the pore and throat size distributions and coordination or connectivity of the pore space.

Pore Size DistributionThe distribution of pore sizes and the coordination number (number of connections per pore)are the key parameters dictating the flow through the network.Cubic:

Pore Size DistributionThe distribution of pore sizes and the coordination number (number of connections per pore)are the key parameters dictating the flow through the network.Random:

Random PNM

Notes

The probability that a point will be placed in a certain region can be adjusted according to a probability density function applied to each co-ordinate. This will create regions with higher pore density and as a consequence lower average pore volume. Anisotropy and heterogeneity can be introduced using this technique as well as scaling the co-ordinates along a particular axis to stretch or compress the domain.

Random PNM

Random PNM

Notes

A Delaunay triangulation for a set P of points in a plane is a triangulation DT(P) such that no point in P is inside the circumcircle of any triangle. It is a simple and efficient way of connecting nearest neighbours and it’s compliment or dual is the Voronoi diagram. Connecting the centers of the circumcircles produces the Voronoi Diagram.

In mathematics, a Voronoi diagram is a partitioning of a plane into regions based on 'closeness' to points or seeds. For each seed there is a corresponding region consisting of all points closer to that seed than to any other. These regions are called Voronoi cells. The edges of the cells are perpendicular to the connections formed by the Delaunay triangulation and are used in OpenPNM to represent the fibres in the porous media.

Random PNM

Gostick, J. ECS, 2013

OpenPNM

Notes

In 3d, the region around a pore is termed a Voronoi cage and has a series of planar facets defined by vertices which are shared by neighbouring pores. The vertices form the outer points of each throat connection in the network and porosity and dimension is given to the throats by eroding it’s binary image by the fibre radius. Throat areas, centroids and maximum inscribed spheres can be calculated easily from the offset vertices using python’s image analysis tools

Random PNM

OpenPNM StructureThe structure of the code is object orientated and methods are added sequentially in a simple script.Written in Python which is easy to learn

Network

Geometry

Fluids

Physics

Algorithm

Simple Cubic

Ball & Stick

Air & Water

WashburnPc

Ordinary Percolation

import OpenPNM" Create Network Object "pn = OpenPNM.Network.Cubic(shape=[5,6,7],spacing=0.0001,name='net')pn.add_boundaries()

" Create Geometry Object and invoke "geom = OpenPNM.Geometry.Toray090(network=pn,pores=Ps,throats=Ts)

" Create Fluid Objects and invoke "air = OpenPNM.Phases.Air(network=pn,name='air')water = OpenPNM.Phases.Water(network=pn,name='water')air['pore.Dac'] = 1e-7 # Add custom properties directly

" Define Pore Scale Physics "phys_a = OpenPNM.Physics.Standard(network=pn,phase=air,pores=Ps,throats=Ts)phys_a.add_model(model=OpenPNM.Physics.models.diffusive_conductance.bulk_diffusion, propname='throat.gdiff_ac', pore_diffusivity='pore.Dac')phys_w = OpenPNM.Physics.Standard(network=pn,phase=water,pores=Ps,throats=Ts)

" Run a drainage simulation "" Create and algorithm object, define injection sites and run "OP_1 = OpenPNM.Algorithms.OrdinaryPercolation(network=pn,invading_phase=water, defending_phase=air)Ps = pn.pores(labels=['bottom_boundary'])OP_1.run(inlets=Ps)

IP Cubic PNM

Notes

Invasion Percolation on a cubic network of 1000 pores using a weibull distribution for pore sizes. Some connections between pores have been removed to introduce variation of coordination, most pores have six neighbours.

Saturation of the pore space by the invading liquid water is quite high at breakthrough (i.e. water creates a spanning cluster and reaches the top of the domain).

IP algorithm with same rules on random network with same number of pores but higher coordination and wider distribution leads to different results. Saturation at breakthrough is lower.

IP Random PNM

Pc – S CurvesAn important relationship for characterising porous fuel cell media is the Capillary Pressure – Saturation (Pc-S) curve.

These are produced by starting with a dry sample then placing it in contact with a water reservoir and increasing the pressure in the reservoir.

As liquid water is non-wetting for typical GDL the process of displacing air is called drainage and the reverse process of air displacing liquid is called imbibition.

Hysteresis can clearly be seen meaning a range of saturations are possible for a given Pc depending on the history of the situation.

Dullien, Porous Media – Fluid Transport and Pore Structure, 1992

NotesScanning curves are produced by reversing the pressure gradient at intermediary saturations of the sample. The primary drainage curve (R0) can be seen to go from 100 saturation of the wetting phase to the irreducible saturation (Swi) which for fuel cells with high porosity and connectivity is typically zero. The secondary imbibition curve (A) shows displacement of the non-wetting phase from the Swi to the residual saturation (Snwr) which for a fuel cell would be trapped pockets of water that become disconnected from the bulk (this can represent 10 –20% of the total saturation). It is clear that hysteresis occurs along the curves R-A

Hysteresis Explained

The scanning curve hysteresis can be explained by considering the structure of theconstrictions that each phase must pass through and the wetting characteristics. The throats ofa fibrous media have been considered as a toroid by Purcell (1950) and discussed in detail byMason & Morrow (1994).

As the interface passes through the constriction the curvature depends on both the contact angle (θ) and the filling angle (β) as well as the dimensions of the fibre (R) and fibre spacing (r).

https://www.onepetro.org/journal-paper/SPE-950369-Ghttp://www.sciencedirect.com/science/article/pii/S0021979784714020

Hysteresis ExplainedNormalized Cmax

Filling Angle (β)

(θ)

At some point during the transition from one side to the other the interface will always inflect. This means that irrespective of whether wetting or non-wetting phase displaces the other there will always positive pressure required to force the transition. This does not occur for straight capillary tubes where a simple dependence on cos(θ) determines the sign of the capillary pressure.

ResultsUsing the Purcell toroid model for capillary pressure and matching the through plane porosity profile given by Fishman & Bazylak good agreement between simple OpenPNM simulation and experimental data collected by Gostick et al. is achieved

http://jes.ecsdl.org/cgi/doi/10.1149/1.3481443http://linkinghub.elsevier.com/retrieve/pii/S0013468608010931

ResultsMore than one algorithm can be run during a simulation. For example a Fickian Diffusion algorithm can be run at each invasion step to calculate relative effective diffusivity.

Air only occupies a fraction of the network and conductance is adjusted to account for the presence of water by multiplying by a small number or setting to zero, effectively removing water invaded pores and throats from the network

Results

Results

Notes

Relative diffusivity for air in the through-plane direction is significantly affected by the presence of liquid water. This result may be under-predicted as the air phase, which is non-wetting will likely retain some connectivity in the corner of pores and throats even at high capillary pressure and high saturation.

Air permeability is more significantly affected than diffusivity by the presence of liquid water. The saturation in the network is determined by Ordinary Percolation and so represents the steady state saturation for a given capillary pressure. Access limitations are considered so that only pores with throats having entry pressures low enough for invasion and connected to the invading liquid front will be invaded. The selection of inlets for liquid water is therefore an additional determining factor on the saturation profile and the resulting relative transport results. A large number of inlets will produce a higher saturation for a given capillary pressure resulting in more liquid blocking air –flow. This effect is also accentuated by the surface pores having on average larger volumes. Liquid is thus likely to invade across the surface blocking gas flow through the network perpendicular to the surface plane. In this case the chosen inlet is the bottom surface and so through-plane transport is hindered most. Reducing porosity at the surfaces should therefore improve the surface flooding both by increasing the capillary pressure required to invade and also increasing lateral resistance to liquid flow.

ObservationsThe throats with normal aligned through-plane suffer less of a reduction in area than those perpendicular (in-plane).

ObservationsCompression of ordered networks with small perturbation factors has a significant impact on the through-plane relative permeability of liquid water. The effect is less pronounced on random networks. A more linear relationship between liquid phase relative permeability and saturation indicates straighter percolation which would result in faster liquid removal and better air distribution.

ObservationsDecreasing the z-spacing, i.e. decreasing the throat areas perpendicular to z, creates a more linear through-plane relative permeability.

Conclusions

• Pore Network Modelling is demonstrated as a powerful and efficient predictive tool for analysing transport in porous media. Representative size distribution and connectivity is key.

• Hysteresis in the saturation vs capillary pressure curves is reproduced with a few simple rules and physical considerations – Purcell toroid model for Pc

• Important relations, normally imposed by other modelling techniques, such as relative permeability and diffusivity can be calculated with OpenPNM’s modular framework.

• Relations are found to depend on compression… at the moment the effect on performance is not established. We are working on adding electrochemistry to next release.

Acknowledgements

• Professor Jeff Gostick, Harold Day and Mahmoudreza Aghighi of the PMEAL group at McGill

• James Hinebaugh and other members of the Bazylak Group at UofT• Michael Hoh of Aachen U and Julich• Andreas Putz of AFCC

Q&A

Thank You For Your Attention

Any Questions Please

---------------------------------------------------------------------------------------------------------------Further Information---------------------------------------------------------------------------------------------------------------Website: www.openpnm.orgInstallation: cmd type pip install openpnm for latest release

or source code at https://github.com/PMEAL/OpenPNMEmail: pmtgt@leeds.ac.uk

http://www.openpnm.org/https://github.com/PMEAL/OpenPNM