Modeling transport of charged species in pore networks: solution of the Nernst-Planck equations coupled with fluid flow and charge conservation equations

A pore network modeling (PNM) framework for the simulation of transport of charged species, such as ions, in porous media is presented. It includes the Nernst-Planck (NP) equations for each charged species in the electrolytic solution in addition to a charge conservation equation which relates the species concentration to each other. Moreover, momentum and mass conservation equations are adopted and there solution allows for the calculation of the advective contribution to the transport in the NP equations. The proposed framework is developed by first deriving the numerical model equations (NMEs) corresponding to the partial differential equations (PDEs) based on several different time and space discretization schemes, which are compared to assess solutions accuracy. The derivation also considers various charge conservation scenarios, which also have pros and cons in terms of speed and accuracy. Ion transport problems in arbitrary pore networks were considered and solved using both PNM and finite element method (FEM) solvers. Comparisons showed an average deviation, in terms of ions concentration, between PNM and FEM below $5\%$ with the PNM simulations being over ${10}^{4}$ times faster than the FEM ones for a medium including about ${10}^{4}$ pores. The improved accuracy is achieved by utilizing more accurate discretization schemes for both the advective and migrative terms, adopted from the CFD literature. The NMEs were implemented within the open-source package OpenPNM based on the iterative Gummel algorithm with relaxation. This work presents a comprehensive approach to modeling charged species transport suitable for a wide range of applications from electrochemical devices to nanoparticle movement in the subsurface

Modeling the Effect of Low Pt loading Cathode Catalyst Layer in Polymer Electrolyte Fuel Cells. Part I: Model Formulation and Validation

A model for the cathode catalyst layer (CL) is presented, which is validated with previous experimental data in terms of both performance and oxygen transport resistance. The model includes a 1D macroscopic description of proton, electron and oxygen transport across the CL thickness, which is locally coupled to a 1D microscopic model that describes oxygen transport toward Pt sites. Oxygen transport from the channel to the CL and ionic transport across the membrane are incorporated through integral boundary conditions. The model is complemented with data of effective transport and electrochemical properties extracted from multiple experimental works. The results show that the contribution of the thin ionomer film and Pt/ionomer interface increases with the inverse of the roughness factor. Whereas the contribution of the water film and the water/ionomer interface increases with the ratio between the geometric area and the surface area of active ionomer. Moreover, it is found that CLs diluted with bare carbon provide lower performance than non-diluted samples due to their lower electrochemical surface area and larger local oxygen transport resistance. Optimized design of non-diluted samples with a good distribution of the overall oxygen flux among Pt sites is critical to reduce mass transport losses at low Pt loading.This work was supported by the projects PID2019-106740RB-I00 and EIN2020-112247 (Spanish Agencia Estatal de Investigación) and the project PEM4ENERGY-CM-UC3M funded by the call "Programa de apoyo a la realización de proyectos interdisciplinares de I + D para jóvenes investigadores de la Universidad Carlos III de Madrid 2019–2020" under the frame of the "Convenio Plurianual Comunidad de Madrid-Universidad Carlos III de Madrid".Publicad

Transport characteristics of saturated gas diffusion layers treated with hydrophobic coatings

Water flooding of cathodic gas diffusion layers (GDLs) in proton-exchange membrane fuel cells at high current densities or low temperatures limits efficient operation due to disturbed transport of reactants to the catalytic sites or products away into the channels. We utilize tomography-based direct pore-level simulations to provide quantitative insights into the transport characteristics of partially saturated GDLs with and without hydrophobic surface treatment to eventually guide the design of better GDLs. High-resolution (voxel size of 1.3 μm) computed tomography images of two different types commercial Toray TGP-H-120 GDLs, one of them untreated and the other treated by hydrophobic coatings, at different water saturation levels were taken. These images were then digitally processed to precisely segment the gas, water, and fiber phases. The digitalized phase information was used in direct pore-level numerical flow simulations to determine effective relative diffusivity of the gas phase, relative permeability of the gas and liquid water phase, and tortuosity in the gas phase. Mathematical morphology operations are used to calculate size distributions of the liquid water phase and the gas phase at different saturation levels for a better understanding of the pore occupation by water at different capillary pressure. Percolation simulations are used to provide information on the connectivity of the gas and liquid phases. The results were validated with available experimental data and semi-empirical correlations. Power law expressions provide a good level of accuracy for curve fitting. The hydrophobic coating does not affect the effective permeability and effective diffusivity of air; however, it improves the water permeability significantly. The quantitative results presented provide insights and guidance for designing GDLs with better suited transport behavior

Dual network extraction algorithm to investigate multiple transport processes in porous materials: Image-based modeling of pore and grain scale processes

The final publication is available at Elsevier via https://doi.org/10.1016/j.compchemeng.2018.12.025 © 2018 This manuscript version is made available under the CC-BY-NC-ND 4.0 license http://creativecommons.org/licenses/by-nc-nd/4.0/Image processing of 3D tomographic images to extract structural information of porous materials has become extremely important in porous media research with the commoditization of x-ray tomography equipment to the lab scale. Extracted pore networks from images using image analysis techniques enable transport properties calculation for bigger domains at a low computational cost, allowing pore-scale investigation of porous media over meaningful macroscopic length scales. The present study reports a pore network extraction algorithm to simultaneously extract void and solid networks from tomographic images of porous materials using simple image analysis techniques. Crucially, it includes connectivity and geometrical information of both void and solid phases as well as the interlinking of these phases with each other. Validation was obtained on networks extracted from simple cubic and random sphere packings over a range of porosities. The effective diffusivity in the void phase and thermal conductivity in the solid phase was then calculated and found to agree well with direct numerical simulation results on the images, as well as a range of experimental data. One important outcome of this work was a novel and accurate means of calculating interfacial areas between grains and voids directly from digital images, which is critical to many phenomena where phase interactions occur. The efficient ‘dual network’ algorithm is written in PYTHON using open source tools and provides a new way to study critical processes that depend on transport in both void and solid phase such as catalytic reactors and electrochemical systems.University of Engineering and Technology Lahore, PakistanNatural Sciences and Engineering Research Council of Canad

Analysis of representative elementary volume and through-plane regional characteristics of carbon-fiber papers: diffusivity, permeability and electrical/thermal conductivity

Understanding the transport processes that occur in carbon-fiber papers (CFPs) used in fuel cells, electrolyzers, and metal-air/redox flow batteries is necessary to help predict cell performance and durability, optimize materials and diagnose problems. The most common technique used to model these thin, heterogeneous, anisotropic porous media is the volume-averaged approximation based on the existence of a representative elementary volume (REV). However, the applicability of the continuum hypothesis to these materials has been questioned many times, and the error incurred in the predictions is yet to be quantified. In this work, the existence of a REV in CFPs is assessed in terms of dry effective transport properties: mass diffusivity, permeability and electrical/thermal conductivity. Multiple sub-samples with different widths and thicknesses are examined by combining the lattice Boltzmann method with X-ray tomography images of four uncompressed CFPs. The results show that a meaningful length scale can be defined in the material plane in the order of 1–2 mm, which is comparable to the rib/channel width used in the aforementioned devices. As for the through-plane direction, no distinctive length scale smaller than the thickness can be identified due to the lack of a well-defined separation between pore and volume-averaged scales in these inherently thin heterogeneous materials. The results also show that the highly porous surface region (amounting up to 20% of the thickness) significantly reduces the through-plane electrical/thermal conductivity. Overall, good agreement is found with previous experimental data of virtually uncompressed CFPs when approximately the full thickness is considered.The authors thank the support team of Calcul Quebec and Compute Canada for their help during the simulation campaign, as well as Dr. Dula Parkinson and Dr. Alastair MacDowell at the Advanced Light Source (ALS) for help in obtaining the tomographic images. This work was funded under the Fuel Cell Performance and Durability Consortium (FC-PAD), by the Fuel Cell Technologies Office (FCTO), Office of Energy Efficiency and Renewable Energy (EERE), of the U.S. Department of Energy under contract number DE-AC02-05CH11231, Project ENE2015-68703-C2-1-R (MINECO/FEDER, UE) and the research grant 'Ayudas a la Investigation en Energia y Medio Ambiente' awarded to the first author by the Spanish lberdrola Foundation. I.V. Zenyuk and A.D. Shum would like to acknowledge support from the National Science Foundation under CBET Award 1605159. X-ray tomography experiments were performed on beamline 8.3.2 at the ALS (Lawrence Berkeley National Laboratory), which is a national user facility funded by the Department of Energy, Office of Basic Energy Sciences under contract DE-ACO2-05CH11231. Numerical calculations were performed on the supercomputing clusters Briaree, Colosse, Guillimin and Mp2, managed by Calcul Quebec and Compute Canada. The operation of these supercomputers is funded by the Canada Foundation for Innovation (CFI), Ministere de l'Economie, de l'Innovation et des Exportations du Quebec (MEIE), RMGA and the Fonds de recherche du Quebec -Nature et technologies (FRQ-NT)

Implications of inherent inhomogeneities in thin carbon fiber-based gas diffusion layers: A comparative modeling study

The final publication is available at Elsevier via http://dx.doi.org/10.1016/j.electacta.2018.09.089. © 2019. This manuscript version is made available under the CC-BY-NC-ND 4.0 license http://creativecommons.org/licenses/by-nc-nd/4.0/version is made available under the CC-BY-NC-ND 4.0 license http://creativecommons.org/licenses/by-nc-nd/4.0/Thin porous media are present in multiple electrochemical energy devices, where they provide key transport and structural functions. The prototypical example is gas diffusion layers (GDLs) in polymer-electrolyte fuel cells (PEFCs). While modeling has traditionally been used to explore PEFC operation, this is often accomplished using volume-averaged (VA) formulations, where the intrinsic inhomogeneities of the GDL are smoothed out and the lack of defining a representative elementary volume is an ever-present issue. In this work, the predictions of a single-phase VA PEFC model are compared to those of a pore-scale PEFC model using GDL tomograms as a part of the meshed domain to delineate important aspects that VA models cannot address. The results demonstrate that while VA models equipped with suitable effective properties can provide a good average estimate for overall performance, the lack of accounting for real structures limits their predictive power, especially for durability and degradation behavior where large deviations are found in the spatial distributions. Furthermore, interfacial effects between the GDL and the microporous layer are explored with the pore-scale model to understand the implications of the layered geometry. It is shown that the actual microstructure of the GDL/MPL transition region can significantly affect the fluxes across the sandwich, something that VA models cannot easily consider. Interfacial design is recognized as a key quality control parameter for large-scale MEA manufacturing and assembly.Fuel Cell Technologies Office, Office of Energy Efficiency and Renewable Energy, U.S. Department of Energy [contract DE-AC02-05CH11231]MINECO/FEDER [Project ENE2015-68703-C2-1-R]Spanish Iberdrola Foundation [grant ‘Ayudas a la Investigación en Energía y Medio Ambiente’]National Science Foundation [CBET Award 1605159

Probing the Structure-Performance Relationship of Lithium-Ion Battery Cathodes Using Pore-Networks Extracted from Three-Phase Tomograms

Pore-scale simulations of Li-ion battery electrodes were conducted using both pore-network modeling and direct numerical simulation. Ternary tomographic images of NMC811 cathodes were obtained and used to create the pore-scale computational domains. A novel network extraction method was developed to manage the extraction of N-phase networks which was used to extract all three phases of NMC-811 electrode along with their interconnections Pore network results compared favorably with direct numerical simulations (DNS) in terms of effective transport properties of each phase but were obtained in significantly less time. Simulations were then conducted with combined diffusion-reaction to simulate the limiting current behavior. It was found that when considering only ion and electron transport, the electrode structure could support current densities about 300 times higher than experimentally observed values. Additional case studies were conducted to illustrate the necessity of ternary images which allow separate consideration of carbon binder domain and active material. The results showed a 24.4% decrease in current density when the carbon binder was treated as a separate phase compared to lumping the CBD and active material into a single phase. The impact of nanoporosity in the carbon binder phase was also explored and found to enhance the reaction rate by 16.8% compared to solid binder. In addition, the developed technique used 58 times larger domain volume than DNS which opens up the possibility of modelling much larger tomographic data sets, enabling representative areas of typically inhomogeneous battery electrodes to be modelled accurately, and proposes a solution to the conflicting needs of high-resolution imaging and large volumes for image-based modelling. For the first time, three-phase pore network modelling of battery electrodes has been demonstrated and evaluated, opening the path towards a new modelling framework for lithium ion batteries.The described here was financially supported by the University of Engineering and Technology Lahore, Pakistan as well as the Natural Science and Engineering Research Council (NSERC) of Canada and in the UK by the Faraday Institution (EP/R042012/1 and EP/R042063/1). Pablo A. García-Salaberri thanks the support from the STFC Early Career Award (ST/R006873/1) during his stay at the Electrochemical Innovation La

Mass transfer in fibrous media with varying anisotropy for flow battery electrodes: Direct numerical simulations with 3D X-ray computed tomography

The final publication is available at Elsevier via https://doi.org/10.1016/j.ces.2018.10.049. © 2018. This manuscript version is made available under the CC-BY-NC-ND 4.0 license http://creativecommons.org/licenses/by-nc-nd/4.0/A numerical method for calculating the mass transfer coefficient in fibrous media is presented. First, pressure driven flow was modelled using the Lattice Boltzmann Method. The advection-diffusion equation was solved for convective-reacting porous media flow, and the method is contrasted with experimental methods such as the limiting current diffusion technique, for its ability to determine and simulate mass transfer systems that are operating at low Reynolds number flows. A series of simulations were performed on three materials; specifically, commercially available carbon felts, electrospun carbon fibers and electrospun carbon fibers with anisotropy introduced to the microstructure. Simulations were performed in each principal direction (x,y,z) for each material in order to determine the effects of anisotropy on the mass transfer coefficient. In addition, the simulations spanned multiple Reynolds and Péclet numbers, to fully represent highly advective and highly diffusive systems. The resulting mass transfer coefficients were compared with values predicted by common correlations and a good agreement was found at high Reynolds numbers, but less so at lower Reynolds number typical of cell operation, reinforcing the utility of the numerical approach. Dimensionless mass transfer correlations were determined for each material and each direction in terms of the Sherwood number. These correlations were analyzed with respect to each materials’ permeability tensor. It was found that as the permeability of the system increases, the expected mass transfer coefficient decreases. Two general mass transfer correlations are presented, one correlation for isotropic fibrous media and the other for through-plane flow in planar fibrous materials such as electrospun media and carbon paper. The correlations are Sh = 0.879 Re0.402 Sc0.390 and Sh = 0.906 Re0.432 Sc0.432 respectively.The authors acknowledge support from the EPSRC under grants EP/L014289/1 and EP/N032888/1, as well as the STFC Extended Network in Batteries and Electrochemical Energy Devices (ST/N002385/1) for funding of travel for Rhodri Jervis to Canada. Paul R Shearing acknowledges the support of the Royal Academy of Engineering. This work was supported by the Natural Science and Engineering Research Council (NSERC) of Canada. MDR Kok is grateful to the Eugenie Ulmer Lamothe Endowment as well as the Vadasz Family Doctoral Fellowship for funding his work, as well the McGill University’s Graduate Mobility Award for funding his travel to the UK

Simulation of a Full Fuel Cell Membrane Electrode Assembly Using Pore Network Modeling

A pore network model has been applied to a both sides of a fuel cell membrane electrode assembly. The model includes gas transport in the gas diffusion layers and catalyst layers, proton transport in the catalyst layers and membrane, and percolation of liquid water. This paper presents an iterative algorithm to simulate a steady state isothermal cell with a 3D pore network model for constant voltage boundary condition. The proposed algorithm provides a simple method to couple the results of the anode and the cathode sides by iteratively solving the uncoupled equations of the transport processes. It was found that local water blockages at the GDL/CL interface not only affect concentration polarization, but also might change ohmic polarization of the cell. Depending on the liquid water configuration in the porous electrodes, the protons generated in the anode need to travel longer paths to reach the active sites of the cathode; consequently, the IR loss will be increased in the presence of liquid water. This finding highlights the strength of pore network models which resolve discrete water blockages in the electrode