Water transport in polymer electrolyte membrane fuel cells

Progress in Energy and Combustion Science 37 (2011) 221e291 Contents lists available at ScienceDirect Progress in Energy and Combustion Science journal homepage: www.elsevier.com/locate/pecs Review Water transport in polymer electrolyte membrane fuel cells Kui Jiao a, Xianguo Li a, b, * a b 20/20 Laboratory for Fuel Cells and Green Energy RD&D, University of Waterloo, Waterloo, Ontario, Canada State Key Laboratory of Engines, Tianjin University, Tianjin, China a r t i c l e i n f o a b s t r a c t Article history: Received 24 February 2010 Accepted 7 June 2010 Available online 14 July 2010 Polymer electrolyte membrane fuel cell (PEMFC) has been recognized as a promising zero-emission power source for portable, mobile and stationary applications. To simultaneously ensure high membrane proton conductivity and sufficient reactant delivery to reaction sites, water management has become one of the most important issues for PEMFC commercialization, and proper water management requires good understanding of water transport in different components of PEMFC. In this paper, previous researches related to water transport in PEMFC are comprehensively reviewed. The state and transport mechanism of water in different components are elaborated in detail. Based on the literature review, it is found that experimental techniques have been developed to predict distributions of water, gas species, temperature and other parameters in PEMFC. However, difficulties still remain for simultaneous measurements of multiple parameters, and the cell and system design modifications required by measurements need to be minimized. Previous modeling work on water transport in PEMFC involves developing rule-based and first-principle-based models, and first-principle-based models involve multi-scale methods from atomistic to full cell levels. Different models have been adopted for different purposes and they all together can provide a comprehensive view of water transport in PEMFC. With the development of computational power, application of lower length scale methods to higher length scales for more accurate and comprehensive results is feasible in the future. Researches related to cold start (startup from subzero temperatures) and high temperature PEMFC (HT-PEMFC) (operating at the temperatures higher than 100  C) are also reviewed. Ice formation that hinders reactant delivery and damages cell materials is the major issue for PEMFC cold start, and enhancing water absorption by membrane electrolyte and external heating have been identified as the most effective ways to reduce ice formation and accelerate temperature increment. HT-PEMFC that can operate without liquid water formation and membrane hydration greatly simplifies water management strategy, and promising performance of HTPEMFC has been demonstrated. Ó 2010 Elsevier Ltd. All rights reserved. Keywords: Polymer electrolyte membrane fuel cell (PEMFC) Water management Water transport Cold start High temperature PEMFC (HT-PEMFC) Contents 1. 2. Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 224 1.1. Fundamental principles . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 224 1.2. Origin and importance of water management . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 226 1.3. Strategy and impact of water management . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 227 1.4. Scope and objective . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 228 State of water . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 229 2.1. In membrane . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 230 2.2. In gas diffusion layer . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 231 2.3. In catalyst layer . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 232 2.4. In flow channel . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 233 2.5. Summary . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 233 * Corresponding author. 20/20 Laboratory for Fuel Cells and Green Energy RD&D, University of Waterloo, Waterloo, Ontario, Canada. Tel.: þ1 519 888 4567x36843; fax: þ1 519 888 6197. E-mail address: [email protected] (X. Li). 0360-1285/$ e see front matter Ó 2010 Elsevier Ltd. All rights reserved. doi:10.1016/j.pecs.2010.06.002 222 3. 4. 5. 6. 7. 8. 9. 10. 11. 12. K. Jiao, X. Li / Progress in Energy and Combustion Science 37 (2011) 221e291 Mechanism of water transport . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 233 3.1. In membrane . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 233 3.1.1. Proton transport . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 234 3.1.2. Diffusion of water . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 236 3.1.3. Electro-osmotic drag effect . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 237 3.1.4. Hydraulic permeation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 238 3.1.5. Reactant transport . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 238 3.1.6. Membrane expansion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 238 3.2. In gas diffusion layer . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 239 3.2.1. Diffusion and convection . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 239 3.2.2. Surface tension and wall adhesion effects in porous media: capillary effect . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 240 3.2.3. Condensation and evaporation of water . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 242 3.3. In catalyst layer . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 242 3.4. In flow channel . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 244 3.5. Summary . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 244 Experimental observation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 244 4.1. Current distribution measurement . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 244 4.2. High frequency resistance distribution measurement . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 245 4.3. Gas species concentration measurement . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 245 4.4. Temperature distribution measurement . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 246 4.5. Water visualization . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 246 4.6. Summary . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 249 Overview of numerical models . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 249 5.1. Level of scale . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 249 5.2. History of model development . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 251 5.3. Summary . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 252 Multi-dimensional multi-component multiphase model with full cell geometry . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 252 6.1. Computational domain . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 253 6.2. Two-fluid model . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 253 6.3. Mixture model . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 255 6.4. Boundary conditions and numerical implementation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 257 6.5. Two-fluid model vs mixture model . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 258 6.6. Summary . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 260 Modeling water transport in membrane . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 260 7.1. Macroscopic approach . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 260 7.1.1. Diffusive model . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 260 7.1.2. Chemical potential model . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 260 7.1.3. Hydraulic model . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 261 7.1.4. Combinational model . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 261 7.2. Bottom-up approach and physical models . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 262 7.2.1. Modeling ionomer self-organization . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 262 7.2.2. Modeling proton transport . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 262 7.2.3. Physical models with simplified membrane micro-structure . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 262 7.3. Summary . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 263 Modeling water transport in gas diffusion layer . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 264 8.1. Homogeneous approach . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 264 8.2. Structure generation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 264 8.3. Volume-of-fluid model . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 265 8.4. Lattice Boltzmann model . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 266 8.5. Rule-based model . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 267 8.5.1. Full morphology model . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 267 Pore-network model . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 268 8.5.2. 8.6. Summary . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 268 Modeling water transport in catalyst layer . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 268 9.1. Agglomerate model . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 269 9.2. Molecular dynamics and office-lattice pseudo particle simulations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 269 9.3. Summary . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 270 Modeling water transport in flow channel . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .271 10.1. Volume-of-fluid model . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 271 10.2. Other aspects . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 272 10.3. Summary . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 272 Cold start . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .273 11.1. Experimental work . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 274 11.2. Numerical model . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 274 11.3. Cold start characteristics . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 275 11.4. Summary . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 279 High temperature polymer electrolyte membrane fuel cell . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 280 12.1. Experiential work . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 280 12.2. Numerical model . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 281 12.3. Performance . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 282 K. Jiao, X. Li / Progress in Energy and Combustion Science 37 (2011) 221e291 12.4. Summary . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Summary and outlook . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Acknowledgments . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 13. Nomenclature a A Bo c C Ca Cp d D DL e E EW f F g h H I j J j0 k K Kn l L _ m M n ! n nd ! nw p Q_ r R Re RH s S SH t T T0 ! tw ! u U V We x water activity, specific area (m 1) cell geometric area (m2) bond number molar concentration (kmol m 3) electric capacity (F m 2) capillary number specific heat (J kg 1 K 1) diameter (m) mass diffusivity (m2 s 1) doping level velocity (m s 1) electrical potential (V), energy (J mol 1) equivalent weight of ionomer (kg kmol 1) distribution function, pre-exponential factor Faraday’s constant (96,487 C mol 1) Gibbs free energy (J kg 1 or J kmol 1), gravity (m s 2) enthalpy and latent heat (J kg 1 and J kmol 1), surrounding heat transfer coefficient (W m 2 K 1) Henry’s constant (Pa m3 kmol 1) current density (A cm 2) reaction rate (A m 3) mass or molar flux (kg m 2 s 1 or kmol m 2 s 1) volumetric exchange current density (A m 3) thermal conductivity (W m 1 K 1) permeability (m2) Knudsen number mean free path (m) characteristic length (m) mass flow rate (kg s 1) molecular weight (kg kmol 1) number of moles of electron for per mole of hydrogen or oxygen unit vector normal to interface electro-osmotic drag coefficient (H2O per Hþ) unit vector normal to wall pressure (Pa) heat transfer rate (W) pore radius (m) universal gas constant (8.314 J mol 1 K 1), radius (m) Reynolds number relative humidity entropy (J kg 1 K 1 or J kmol 1 K 1), volume fraction source term, mass or heat transfer term, entropy (J kg 1 K 1 or J kmol 1 K 1) analogous to the Sherwood number time (s) temperature (K) volume averaged cell temperature (K) unit vector tangential to wall velocity (m s 1) velocity (m s 1) electrical potential (V) Weber number position or coordinate (m) X Y z . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 283 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 283 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 284 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 284 mole fraction mass fraction charge number or valence Abbreviation AFC alkaline fuel cell BP bipolar plate BGK Bhatnagar–Gross–Krook CCD charge-coupled device CFD computational fluid dynamics CG coarse grained CGMD coarse grained molecular dynamics CL catalyst layer CO carbon monoxide DSC differential scanning calorimetry DMFC direct methanol fuel cell DPD dissipative particle dynamics EIS electrochemical impedance spectroscopy EOD electro-osmotic drag FM full morphology GDL gas diffusion layer GC gas chromatograph HFR high frequency resistance HOR hydrogen oxidation reaction HT-PEMFC high temperature polymer electrolyte membrane fuel cell IR infrared LB lattice Boltzmann LG lattice gas MC Monte Carlo MCFC molten carbonate fuel cell MD molecular dynamics MEA membrane electrode assembly MPL micro porous layer NMR nuclear magnetic resonance ORR oxygen reduction reaction PAFC phosphoric acid fuel cell PN pore network PBI polybenzimidazole PEM polymer electrolyte membrane PEMFC polymer electrolyte membrane fuel cell PFSA perfluorosulfonic acid PTFE polytetrafluoroethylene SOFC solid oxide fuel cell VOF volume-of-fluid Greek letters transfer coefficient, water transport coefficient (kmol2 J 1 m 1 s 1) b convective transport coefficient g water phase change rate (s 1) G uptake coefficient d thickness or average distance (m), time interval (s) P chemical potential (J kmol 1) 3 porosity z water transfer rate (s 1) a 223 224 K. Jiao, X. Li / Progress in Energy and Combustion Science 37 (2011) 221e291 h q w i k l m n x r s 2 s Y f u efficiency, overpotential (V) contact angle ( ) mobility (m2 kmol J 1 s 1) interfacial drag coefficient electrical conductivity (S m 1), surface curvature (m water content in ionomer dynamic viscosity (kg m 1 s 1) kinematic viscosity (m2 s 1) stoichiometry ratio density (kg m 3) surface tension (N m 1) condensation and evaporation rate coefficients tortuosity, relaxation time relative mobility electrical potential (V) volume fraction of ionomer in catalyst layer 1 ) Subscripts and superscripts 0 intrinsic value a anode act activation air air B bulk or binary BP bipolar plate c cathode, capillary cap capillary ce condensation/evaporation cell cell characteristic CL catalyst layer conc concentration cond condensation cs charged site desb desublimation diff diffusive eff effecitive ele electronic EOD electro-osmotic drag equil equilibrium evap evaporation f frozen F Fickian fl fluid phase FPD freezing point depression fmw frozen membrane water g gas 1. Introduction 1.1. Fundamental principles A fuel cell is an energy conversion device that converts the chemical energy stored in fuels and oxidants into electricity through electrochemical reactions. With different kinds of electrolyte, fuel cells can be classified into different types. The most common types of fuel cells are the polymer electrolyte membrane (also called the proton exchange membrane) fuel cell (PEMFC), direct methanol fuel cell (DMFC) (the same as PEMFC but uses methanol instead of hydrogen as the fuel), alkaline fuel cell (AFC), phosphoric acid fuel cell (PAFC), molten carbonate fuel cell (MCFC) and solid oxide fuel cell (SOFC). Different types of fuel cells are suitable for different applications. Some types of fuel cells are most suitable for stationary power generation, such as PAFC, MCFC and SOFC, and some other types of fuel cells are mostly used for GDL H2 H2 O hyd I i-g i, j ice in ion K l-i lattice lq m mem N n-f n-i n-v nf nmw O2 ohm open out pore ref r sat sl sld surf surr t T th u v-i v-l vp vp/lq wall gas diffusion layer hydrogen water hydraulic intro gas phase in species i the ith and jth components ice inlet ionic Knudsen liquid water to ice (vice versa) lattice liquid water mass (for source term) membrane or ionomer normal condition non-frozen membrane water to frozen membrane water (vice versa) non-frozen membrane water to ice non-frozen membrane water to vapour (vice versa) non-frozen non-frozen membrane water oxygen ohmic open circuit outlet pore reference state reversible saturation solid solid surface or interface surroundings time energy (for source term) thermodynamic momentum (for source term) vapour to ice vapour to water liquid (vice versa) water vapour interface between vapour and liquid surrounding wall of the cell vehicular or portable applications, such as PEMFC, DMFC and AFC. Benefiting from the advantages such as low operating temperature, high power density and zero/low emission, PEMFC has increasingly become the most promising candidate as the power source for automotive and backup power applications. PEMFC is also the most popular one under research and development compared with other types of fuel cell. However, further improvements in terms of performance, durability and cost are necessary before commercial application. Fig. 1 shows the schematics of a single PEMFC and a PEMFC stack with three single cells. Typically a single PEMFC (Fig. 1a) consists of an anode and a cathode, and a polymer electrolyte membrane (PEM) in between. At the anode, hydrogen flows into the flow channel through the gas diffusion layer (GDL) to the catalyst layer (CL). In the anode CL, hydrogen splits into protons (hydrogen ions) and electrons. The protons pass through the membrane and travel to the cathode CL, however, the electrons cannot pass through the K. Jiao, X. Li / Progress in Energy and Combustion Science 37 (2011) 221e291 225 Fig. 1. Schematics of (a) a single PEMFC and (b) a PEMFC stack with three single cells. Reversible voltage 1.2 Open circuit voltage is less than reversible voltage due to fuel crossover and internal currents Cell V oltag e, V 1 Voltage initially falls fast mainly due to activation loss Voltage falls more linearly and slowly mainly due to ohmic loss 0.8 0.6 0.4 membrane, but travel through an external circuit to the cathode, thus generating electricity. At the same time, on the cathode side, air or oxygen flows into the flow channel through the GDL to the CL. In the cathode CL, oxygen reacts with the protons and electrons from the anode, producing water and heat. Due to the water concentration and pressure differences between the anode and cathode, and the proton transport across the membrane, water can travel through the membrane in both directions. On the anode side, the reaction that hydrogen splits into protons and electrons is a hydrogen oxidation reaction (HOR): H2 /2Hþ þ 2e On the cathode side, the reaction that oxygen, protons and electrons form water is an oxygen reduction reaction (ORR): Voltage falls faster at high currents mainly due to concentration loss 0.2 0 0 0.2 0.4 0.6 (1) 0.8 1 -2 Current Density, A cm Fig. 2. Sample polarization curve of a single PEMFC. 1.2 1 O þ 2Hþ þ 2e /H2 O 2 2 (2) The overall reaction is simply hydrogen reacting with oxygen producing water, electrical energy and heat: 226 K. Jiao, X. Li / Progress in Energy and Combustion Science 37 (2011) 221e291 1 H2 þ O2 /H2 O þ Electrical Energy þ Heat 2 (3) Single cells are often connected in series to form a PEMFC stack to produce higher voltages. As shown in Fig. 1b, with three single cells connected in series, the supplied gases have to be distributed into the single cells through the inlet manifolds, and the exhaust gases have to be removed through the outlet manifolds. Fig. 2 presents a sample currentevoltage graph of a single PEMFC, which is also called the polarization curve. This figure shows the voltage outputs at different current outputs. The current has been normalized by the area of the fuel cell in terms of a standard unit of current density (A cm 2), because PEMFCs with different sizes obviously produce different currents, and a normalized current unit makes fuel cell performance comparable. Theoretically, a PEMFC would supply any amount of current under the condition of sufficient fuel supply, while maintaining a constant voltage determined by thermodynamics. In practice, however, the actual voltage output of a fuel cell is less than the ideal thermodynamically predicted value (reversible voltage). Furthermore, the more current that is drawn from a real fuel cell, the lower the voltage output of the cell, limiting the total power that can be delivered. The reversible (theoretically maximum possible) cell potential, Er (V), can be calculated as Er ¼ Dg (4) nF where Dg (J mol 1) is the change of Gibbs free energy of the overall reaction per mole of hydrogen, n (¼ 2) the number of moles of electrons transferred per mole of hydrogen, and F (¼96,487 C mol 1) the Faraday’s constant. It should be noticed that the values of Dg are different for different reaction products (e.g. liquid water and vapour), and the product water is typically in liquid form at the PEMFC operating condition [1]. Since the change of Gibbs free energy (Dg, J mol 1) represents the maximum useful work available from the reaction, the thermal (theoretically maximum possible) efficiency, hth, can be defined as hth ¼ Dg ¼ 1 Dh T Ds Dh 1 (5) where Dh (J mol ) is the change of enthalpy of the overall reaction per mole of hydrogen (the total energy change of the reaction), T the temperature (K), and Ds (J mol 1 K 1) the change of entropy of the overall reaction per mole of hydrogen. The total amount of energy transformed into heat at the reversible cell voltage is TDs, which is also called the reversible heat. By using Equation (5), the thermal efficiency can be calculated to be 83% at 25  C with product water in liquid state. Practical fuel cells can have efficiency close to this theoretical maximum efficiency, while practical heat engines are very difficult to achieve even about 50% of the theoretical maximum (or Carnot) efficiency [1]. It is difficult to maintain the cell voltage at a high level under current load. As shown in Fig. 2, the voltage output of a PEMFC in operation is less than the reversible voltage due to the irreversible losses. The total loss increases with the increment of the current density. There are four major types of fuel cell losses [2]: the loss due to fuel crossover and internal currents, activation loss, ohmic loss, and mass transport or concentration loss. The fuel crossover and internal currents are the waste of fuel passing and electron conduction through the electrolyte, respectively. The electrolyte should only transport protons, however very small amount of fuel diffusion and electron flow will always be possible. It does have a marked influence on the open circuit voltage (the cell voltage at zero current), which explains why the open circuit voltage is always smaller than the reversible voltage. However, this type of loss reduces considerably when a meaningful amount of the current is drawn from the cell. The activation loss is caused by the slowness of the reactions taking place on the surface of the electrodes. A proportion of the voltage generated is lost in driving the electrochemical reaction that transfers the electrons and protons to or from the electrode. In Fig. 2, it is represented by the initial sharp drop of the cell voltage. The ohmic loss is the straightforward resistance to the transport of electrons and protons through the materials of the electrodes, membranes and the various interconnections. This voltage drop is essentially proportional to current density, represented by the almost linear fall in the middle of the performance curve in Fig. 2. The concentration or mass transport loss results from the change in concentration of the reactants at the surface of the electrodes as they are consumed along the flow channel from the inlet to the outlet. Concentration affects voltage via the change of differential pressure of reactant. That is why this type of irreversibility is called concentration loss. On the other hand, since the reduction in concentration is the result of a failure to transport sufficient reactant to the electrode surface or catalyst sites, this type of irreversibility is also called mass transport loss. In Fig. 2, such loss can be observed at high current density range as a nonlinear rapid drop, because sufficient reactant supply is the controlling factor to obtain large amount of current. However, even with sufficient reactant supply, the water flooding in the cell caused by improper water removal may also result in concentration or mass transport loss due the blockage of the reaction sites. The operating voltages of a PEMFC at different current densities can be calculated by using the reversible voltage subtracting all the voltage losses due to the irreversibilities, and because the open circuit voltage is equal to the reversible voltage subtracting the voltage loss due to the fuel crossover and internal currents, therefore a relationship between the operating cell voltage and open circuit voltage can be obtained Ecell ¼ Eopen hohm hact hconc (6) where Ecell and Eopen are the operating and open circuit voltages, respectively; and hact, hohm and hconc represent the voltage losses due to the activation loss, ohmic loss and concentration or mass transport loss, respectively. As mentioned before, the voltage loss due to fuel crossover and internal currents reduces considerably when a meaningful amount of current is drawn from the cell (Eopen z Er). The operating voltage is usually lower than 0.8 V when drawing a useful current. Therefore, as mentioned earlier in this section, many single cells have to be connected in series to form a PEMFC stack to produce a higher voltage (as demonstrated in Fig. 1b). 1.2. Origin and importance of water management The water management issue has been going with PEMFCs since the initial development by General Electric Company in 1950s [3e5] and the first practical application for U.S. Gemini space missions from 1962 to 1966 [6]. Until nowadays, tremendous studies in this area are still continuing for achieving better performance. The major cause of the water management issue is the proton conductor e the membrane. The mostly widely used type of membrane as the proton conductor is the perfluorosulfonic acid (PFSA) polymer membrane. The most famous one in the PFSA family in the last thirty years is the Nafion membrane invented by E.I. DuPont de Nemours due to its relatively high durability and proton conductivity. Other PFSA polymer membranes like Aciplex, Dow and Flemion have also been widely tested and used. Even though the chemical structures are different for the different PFSA polymer membranes, the membrane morphologies and the base K. Jiao, X. Li / Progress in Energy and Combustion Science 37 (2011) 221e291 mechanisms of proton transport are similar. The PFSA polymer membranes have acceptable proton conductivity only when they are well hydrated. The electrical resistance increases when the membrane is drying out, resulting in the increment of the ohmic loss and heating. The increased local heating further accelerates the local drying by evaporating more water, resulting in vicious self-accelerated destruction of cell performance. The PFSA polymer membranes are also unstable at high temperatures [1], therefore the local heating also limits the lifetime of the membranes, hence the cell lifetime as well. Since PFSA polymer membranes feature the similar characteristics (water adsorption/desorption, proton conduction etc.), only Nafion membrane is discussed in the major part of this paper. Other types of membranes (such as hydrocarbon membranes) have also attracted attentions, with the main objective of operating at higher temperatures and less relative humidity. The development of such membranes is currently under active research, and one of the most promising membranes, acid doped polybenzimidazole (PBI) membrane that can tolerate higher operating temperatures and has acceptable proton conductivity without humidification, is discussed in Section 12. As mentioned above, membrane hydration is of paramount importance for PEMFC. In order to hydrate the membrane, humidified reactant gases are often supplied. However, due to the water production from the electrochemical reactions, as well as the low operating temperature of PEMFC (typically around 80  C) that leads to the almost unavoidable water condensation, liquid water may be present and flood the electrode pore regions, giving rise to the so-called water flooding phenomena, which severely reduce the rate of reactant supply to the reaction sites and degrade the cell performance considerably. The water flooding is most severe in cathode CL due to the fact that water is produced in this layer and the electro-osmotic drag (EOD) also causes water migrating form anode CL to cathode CL. In fact, the concentration or mass transport loss described previously is mostly likely caused by the water flooding of cathode electrode. On the other hand, the EOD effect also causes drying out of the portions of membrane close to the anode side, resulting unevenly distributed water across the membrane and large ohmic loss close to the anode side. On the stack level, the supplied water (vapour or liquid) may be unevenly distributed to each cell, causing flooding in some cells and drying out in others. Therefore, proper membrane hydration without causing electrode flooding by water, commonly referred to as water management, remains one of the major technical challenges of PEMFC, and maintaining the dynamic balance of water in the cell during operation is essential for better water management and stable performance. 1.3. Strategy and impact of water management The simplest way to avoid the water flooding is to operate PEMFC on non-humidified inlet gases with acceptable performance degradation. Without humidifying the inlet gases, the humidification subsystem can be cast off, and the water and heat removal as well as the mass transport of reactant gases are all improved due to the fact that water is mostly likely in the vapour form. Utilizing the product water for membrane hydration and balancing the water in the cell are the key factors for operating PEMFC on non-humidified gases. However, a meaningful relative humidity of the gas stream in the cathode for membrane hydration can only be obtained if the cell temperature is lower than 60  C, and operating the cell at elevated temperatures (e.g. 80  C) results in severe reduction of the relative humidity of the gas stream and hence reduces the membrane hydration level significantly [2,7]. The idea of balancing the water in the cell is to utilize the water produced in the cathode CL to hydrate the membrane close to the anode side because water 227 can travel through the membrane driven by concentration or pressure gradients. Since the relative humidity of the gas stream in the cathode is often the highest at flow channel exit due to the accumulation of product water, the counter flow arrangement by placing the anode and cathode flows in opposite directions is widely used to enhance the water transport from cathode to anode [2,7]. However, the EOD causing water migrating from anode CL to cathode CL becomes fast at high current densities and leads to severe dehydration of the portions of membrane close to the anode side, and hence further humidification is needed rather than just unitizing the water produced in cathode. For the operating pressure range from 1 to 3 atm, it has been shown that the cell performance when operating on non-humidified inlet gases is about 20e40% lower than operating on fully humidified (the relative humidify is 100%) inlet gases for the operating temperatures lower than 60  C, and the performance degradation is expected to be much more severe at elevated operating temperatures [7]. Since both the electrochemical reaction kinetics and membrane conductivity can be improved at higher temperatures, operating the cell at the highest possible temperature while maintaining the membrane durability is critical to achieve better performance, and therefore the normal operating temperature for PEMFC is usually controlled at about 80  C. At this level of operating temperature, especially when a meaningful current is drawn from the cell, external humidification becomes essential. Rather than utilizing the product water, alternative ways to hydrate the membrane while operating on non-humidified inlet gases were also demonstrated, such as supplying water directly to the membrane through wicks [8] and imbedding catalyst particles in the membrane to produce water from the fuel and oxidants moved into the membrane [9,10]. However, none of these methods have been widely accepted because more complicated cell/system design and operating strategy are needed, which may also cause other problems. As mentioned above, it is very difficult to achieve the “best” performance when operating PEMFC without external humidification, therefore, external humidification of PEMFC by humidifying the reactant gases inside or outside the stack becomes essential. In this scenario, the water flooding mentioned earlier may become a severe problem, and water management is clearly composed of two interrelated issues: complete hydration of the membrane electrolyte and product water removal e a dynamic balance of water needs to be achieved to satisfy the two conditions. External humidification of PEMFC can be achieved by employing water columns, reactant gas recirculation, and direct water injection into either the anode or/and cathode compartment or into the reactant gas streams outside the stack. For the water column technique, process water is stored in a container and reactant gas is introduced at the bottom of the container. The saturated reactant gas stream leaves the container at or near the top of the container. The amount of water that can be picked up by the gas stream and brought into the stack therefore depends on the humidification temperature. Low temperatures result in low water partial pressures, while high temperatures result in low reactant partial pressures. This technique works well for low reactant flow rate, such as pure hydrogen stream. For air at high current density operations, high air flow rate tends to carry small droplets with it and bring them into the cathode electrode, causing water flooding; while air stream itself may be unsaturated due to the short residence time, resulting in membrane drying out locally. The reactant gas-recirculation design utilizes the water taken by the exit gas streams to humidify both the anode and cathode inlet gases. Satisfactory stack operation may be achieved without external process water. However, this method requires an external piece of equipment, often a compressor, to recirculate the gas streams, and the parasitic power consumption may not be neglected for high recirculation 228 K. Jiao, X. Li / Progress in Energy and Combustion Science 37 (2011) 221e291 rates [1]. In the direct water injection design, water can be injected in vapour (steam) or liquid form and injected directly into the anode and cathode compartment, or to the fuel and oxidant streams before their entry to the stack. Direct injection of steam and liquid outside the stack are easy to accomplish, while into the stack compartment directly requires complex designs, and the components needed to achieve this tend to be expensive to construct and difficult to incorporate into a practical stack. For direct liquid water injection, more water can be introduced into the reaction compartment to maintain the membrane hydration than what is possible by just saturating the inlet gases by injecting steam, the evaporation of liquid water also absorb waste heat which facilitates the cell cooling. However, distributing the injected liquid water evenly to each singe cell of a stack is difficult [11,12], resulting in both the flooded and dry cells in a stack. All the external humidification methods described here require extra equipment for the system design and/or more complex stack design, which increase both the cost and parasitic power consumption, and hence need to be evaluated before implementing to practical applications. Even though the membrane may be properly hydrated at the normal operating temperature (around 80  C) with the external humidification methods described above, the excessive water (usually in liquid form) needs to be removed or balanced to avoid the water flooding problem and to better hydrate the membrane. The water removal and balancing are usually accomplished by optimizing the stack/system design and operating condition. Although various water management strategies have been proposed, water is still typically removed by pumping air into cathode flow fields. Therefore, the flow field design is critical for water management. To the geometrical configurations of the flow fields, a variety of different designs are known and the conventional designs typically comprise pin-type [13,14], straight-parallel [15,16], serpentine [17,18], integrated [19,20], interdigitated [21,22], and porous metal/carbon foam flow fields [23,24]. Every single flow field design has its own advantages and disadvantages, and modified flow fields by incorporating the advantages of the different designs are often used. One example is that the straight-parallel flow field design features low pressure drop from the channel inlet to outlet but weak water removal ability, while the serpentine flow field design features high pressure drop but effective water removal, and therefore a so-called serpentine-parallel flow field by using numbers of serpentine flow channels connecting in parallel has been demonstrated for the compromised design [25,26]. Different designs of flow field are also suitable for different water management systems, for example the interdigitated flow field has been demonstrated to be the most suitable design when used with direct liquid water injection [27]. More details about the design considerations for flow field in PEMFC can be found in [28,29]. The GDL and CL properties such as thickness, porosity, permeability, wettability, catalyst loading (for CL) and ionomer fraction (for CL) also have strong impacts on water removal [30e40]. One common way is to add polytetrafluoroethylene (PTFE) to these layers to increase the hydrophobic level to expel water, and inserting a hydrophobic micro porous layer (MPL) between GDL and CL in cathode was also found to be an effective way to push water from cathode CL into membrane [31e34,41e45]. Even though it was found that using MPL could improve the cell performance, the actual function of this layer is still under debate. One explanation is that using MPL between GDL and CL may reduce the electrical and thermal contact resistances. Another explanation is as mentioned above, more water can be retained in CL and membrane to improve the proton conductivity. Therefore, MPL may act as electrical/ thermal transfer smoother and mass transfer resistance, and further investigation is needed to find its impacts on actual transport phenomena. Although the Nafion membrane has been recognized as the “standard” membrane for PEMFC, different thicknesses of the Nafion membrane are available and therefore need to be considered for water removal and balance. Optimizing the operating condition by controlling the operating temperature and pressure is also an effective way to remove and balance the water. Generally, increasing the operating temperature and decreasing the operating pressure may all reduce the amount of liquid water formation, and vice versa [1,2]. However, the promotions for both the electrochemical kinetics and reactant transport by increasing the operating temperature and pressure, the durability of the stack components, as well as the power consumptions for cell cooling and pressurization all need to be considered. The impervious cathode and anode flow distribution plates can also be replaced by porous hydrophilic plates or something similar [13,14,46e48]. With appropriate matching of the pore sizes among the anode electrode, anode porous flow distribution plate, cathode electrode and cathode porous flow distribution plate, a judicially controlled set of pressure differences between the oxidant stream and the adjacent cooling water stream and between the cooling water and the fuel stream transfers the product water in the cathode to the cooling water stream, then to the anode by the pressure differentials between the various streams involved, and from the anode to the cathode via EOD. The water flooding in the cathode can be avoided while the portions of membrane close to the anode side can be further hydrated. Precision monitoring and control of the pressure differentials add the cost of the system and operation and maintenance. It might be also possible to run the anode and cathode at different pressures to accelerate the water permeation from the cathode to anode through the membrane. However, much larger pressure differential is required by comparing to pressurizing the water to the cooling channel, and therefore the mechanical strength of the MEA needs to be carefully evaluated. Many other methods to remove and/or balance the water are also available, such as using a liquid-permeable electricity-conductive layer for storage and transport of water [49], using integrated EOD pumping to remove water [50], and sequentially exhausting each cell of a stack to avoid single cell flooding [51]. Apparently, many water management methods are available and proper implementation is needed. In reality, a number of the water management methods described here can be combined to their best effects (e.g. [13,14,27,47e55]). Understanding of the water transport in PEMFC is therefore critical to achieve the optimal water management strategy, and hence the improved cell performance. Some reviews on water management of PEMFC were published recently [56e61]. Bazylak [56] reviewed the visualization techniques for water distribution in PEMFC. The reviews conducted by Li et al. [57] and Ji and Wei [61] focused on the diagnosis and mitigation of water flooding in PEMFC. Studies on water transport in MEA were reviewed by Dai et al. [58]. Yousfi-Steiner et al. [59] reviewed the voltage degradation issues related to water management, and Schmittinger and Vahidi [60] reviewed the durability issues related to water management. Each of the reviews in [56e61] focused on one or few specific topics of water transport in PEMFC, and the details of state of water and mechanism of water transport in PEMFC were only partially presented in these reviews due to the narrow scopes. Therefore, it is timely to comprehensively summarize the progress on the understanding of water transport in PEMFC. 1.4. Scope and objective The main objective of the present paper is to summarize the current status of understanding on water transport in PEMFC. The various water transport processes in PEMFC are therefore elaborated in different sections by reviewing the experimental and 229 K. Jiao, X. Li / Progress in Energy and Combustion Science 37 (2011) 221e291 CL Gas mixture Gas mixture Carbon support Liquid water Liquid water Catalyst Flow channel Polymer electrolyte Gas mixture SO3− Flexible perfluoracarbon (gas permeable) Solid material Liquid water GDL Liquid water, H+ and H3O+ Hydrophobic backbone Membrane a CL Gas mixture Gas mixture Carbon support Ice Catalyst Flow channel Polymer electrolyte Gas mixture − SO3 Flexible perfluoracarbon (gas permeable) Solid material Ice Ice GDL Liquid water, H+ and H3O+ Hydrophobic backbone Membrane b Fig. 3. Schematics of a single PEMFC with the structure of each cell component illustrated (a: normal operating condition; b: cold start). Normal Operating Condition: In Membrane Non-frozen Membrane Water In CL Non-frozen Membrane Water In GDL and Flow Channel Vapour Liquid numerical studies. This article is organized as follows. The state and transport mechanism of water are explained in Sections 2 and 3, respectively; the water transport related experimental observations are reviewed in Section 4; the water transport related numerical models are reviewed in Sections 5e10 including both the first-principle-based and rule-based models, and both the topdown and bottom-up models are included for the first-principlebased models; the two special cases for starting from subzero temperatures (cold start) and operating above the boiling point of water (high temperature PEMFC (HT-PEMFC)) are introduced in Sections 11 and 12, respectively; and finally the summary and outlook is given in Section 13. Vapour Liquid Cold Start: In Membrane Non-frozen Membrane Water Frozen Membrane Water In CL Non-frozen Membrane Water Liquid In GDL and Flow Channel Vapour Liquid Vapour Ice Ice Fig. 4. Schematics of water phase change in different components of PEMFC for both normal operating condition and cold start. 2. State of water Due to the differences in materials and in local operating conditions among the different components of PEMFC, water can be present in different states with different phase change processes. Many efforts in both the experiment and theory have been paid to understand the state of water in PEMFC. Generally, the 230 K. Jiao, X. Li / Progress in Energy and Combustion Science 37 (2011) 221e291 state and phase change of water are different in membrane, CL, GDL and flow channel. Fig. 3 shows the schematics of a single PEMFC with the structure of each cell component illustrated for both the normal operating condition and cold start, and the schematics of water phase change are given in Fig. 4. As shown in Figs. 3a and 4 for normal operating conditions (cell temperature generally ranges from 60 to 80  C), water exists in forms of vapour and liquid in the flow channel and pore regions of GDL and CL (liquid water is formed if the local water vapour pressure is higher than the local water saturation pressure), the ionomer (polymer electrolyte) in membrane and CL also absorbs certain quantities of water in liquid state or bound to Hþ (e.g. H3Oþ). Due to the fact that PEMFC is often considered to be used for automotive applications, and it is almost unavoidable for vehicles driving below the freezing point of water in winter, PEMFC must be able to successfully start up from subzero temperatures, which is referred to as “cold start”. During a cold start process for PEMFC, the initial cell temperature is usually equal to the surrounding temperature (usually under 0  C in winter), and water will mostly likely freeze in this kind of scenario. As shown in Figs. 3b and 4 for PEMFC cold start, the formation of liquid water can be almost neglected since it freezes to ice (note that ice and liquid water may still co-exist when the local cell temperature increases or decreases to the freezing point of water, resulting in ice melting or liquid freezing). Therefore, water usually exists in forms of vapour and ice in the pore regions of GDL and CL (ice is formed if the local water vapour pressure is higher than the local water saturation pressure) for PEMFC cold start. Since the ice formed can easily stick on the solid materials of CL and GDL and difficult to move, the ice formation in flow channel might be neglected. Figs. 3b and 4 also show that water in the ionomer of membrane and CL may also freeze at subzero temperatures. Apparently, water in PEMFC can create very complex scenarios, and the different states of water need to be classified systematically for better understanding of water transport in PEMFC. In this section, the states of water are classified and described in different components of PEMFC to provide a general view of water to guide the discussions in the following sections. Even though most of the previous studies of PEMFC focused on the normal operating conditions, the subzero temperature conditions (cold start) are also considered in this section to provide a complete view of the state of water. After this section, the normal operating conditions are focused on and the details of the cold start processes are explicitly discussed in Section 11. Note that unless otherwise specified, the water content mentioned starting from Section 3 (except Section 11) represents the non-frozen water content, since no frozen water is present in ionomer in normal operating conditions. The states of water in different components of PEMFC are presented in Table 1, and the detailed descriptions are given in the following subsections. Table 1 State of water in each component of PEMFC. Cell component Specific location/ material State of water Membrane Ionomer GDL Pore region Free, Freezable, Non-freezable (the free and freezable water are all possible to freeze at subzero temperatures) Vapour, Liquid, Icea CL Pore region Ionomer Vapour, Liquid, Icea Free, Freezable, Non-freezable (the free and freezable water are all possible to freeze at subzero temperatures) Flow channel Everywhere Vapour, Liquid, Icea (the ice formation might be neglected) a Only present at subzero temperatures. 2.1. In membrane As shown in Fig. 3, the ionomer of membrane and CL consists of hydrophobic backbones, flexible perfluorocarbons (gas permeable), and hydrophilic clusters with HþSO3 (the region in the middle of the ionomer structure demonstrated in Fig. 3). The SO3 are bound to the material structures and difficult to move, and there is an attraction between the Hþ and SO3 for each HþSO3 . The hydrophilic clusters with HþSO3 can absorb large quantities of water to form hydrated hydrophilic regions. In the hydrated hydrophilic regions, the Hþ are relatively weakly attracted to the SO3 and can move more easily. The hydrated hydrophilic regions can be considered as dilute acids, explaining why the membrane needs to be well hydrated (hydrated regions must be as large as possible) for appreciable proton conductivity, and the SO3 can be considered as the proton exchange sites since the Hþ often move between the SO3 . The water absorption level of ionomer is often represented as the number of water molecules per SO3 referred to as the water content (l). In a well hydrated Nafion membrane, there will be about 20 water molecules for each SO3 , and the proton conductivity can reach higher than 10 S m 1. The thickness of the present Nafion membrane ranges from 25 (Nafion 211) to 175 (Nafion 117) mm, and the size of the hydrophilic region that can contain water is on the level of nanometre. The water concentration ðcH2 O ; kmol m 3 Þ inside the ionomer of membrane and CL can be correlated with the water content (l): l ¼ EW rmem cH2 O (7) where rmem (kg m 3) is the density of dry membrane (ionomer), and EW (kg kmol 1) the equivalent weight represented by the dry mass of the membrane (ionomer) over the number of moles of SO3 [1]: EW ¼ Dry ionomer mass in g 
Mole of proton exchange sites SO3 (8) Generally, with high EW, the mechanical and thermal strengths of membrane are high; and with low EW, the number of proton exchange sites is high, resulting in high proton conductivity. EW is usually equal to 1100 kg kmol 1 (Nafion 112, 115 and 117) or 2100 kg kmol 1 (Nafion 211 and 212) for Nafion membranes. In considering all the operating conditions (normal and cold start) and in the hydrophilic regions, liquid water, water bound to Hþ (e.g. H3Oþ) and ice are all possible to be present. The most suitable classification of the water in ionomer is non-frozen water and frozen water [62,63], which is based on the observations of the freezing behaviour of water by differential scanning calorimetry (DSC) and nuclear magnetic resonance (NMR) [64e68]. DSC has been used to determine the amounts of the different types of water, and the maximum allowed amount of non-frozen water in Nafion membrane at different subzero temperatures has been reported. It has been found that there is a certain amount of water (about 4.8 water content) that does not freeze [64]. The non-frozen water can be further classified into non-freezable, freezable and free water [69e71], based on how tight they are bound to the sulphuric acids (HþSO3 ). The non-freezable water is mostly tightly bound to HþSO3 , and its maximum allowed amount is about 4.8 [64]. The freezable water is loosely bound to HþSO3 and exhibits freezing point depressions, which has been observed in [64e68]. The free water may also appear if the water content is sufficiently high, confirming the observations that the water freezes in ionomer at the temperatures close to the normal freezing temperature of water (0  C) if the water content is high [64e68]. The nanometre confinement of water in the small pores may also lead to freezing point depressions due to the enhanced surface dynamics of water. Therefore the free water may still possess slightly lower freezing K. Jiao, X. Li / Progress in Energy and Combustion Science 37 (2011) 221e291 points than bulk water. An empirical correlation to calculate the maximum allowed amount of non-frozen water content has been given in [62,63] based on the experimental measurement in [64]: lsat 8 > <¼ ¼ > : if T < 223:15 K h4:837 1:304 þ 0:01479T 3:594  10 5T2 if 223:15 K  T < TN if T  TN > lnf i 1 (9) where lsat and lnf are the saturation (maximum allowed nonfrozen water content) and non-frozen water content; and T and TN are the local temperature and the normal freezing temperature of water (273.15 K), respectively. The units of T and TN are all in K. It should be noticed that no further water phase change was detected when the temperature is lower than 50  C [64], and the saturation water content remains at about 4.8. This amount of water content corresponds to the non-freezable water. For the temperature range from 50 to 0  C (223.15e273.15 K), the maximum allowed amount of non-frozen water increases with the increment of temperature. Moreover, water in ionomer does not freeze if the temperature is higher than the normal freezing temperature of water (T  TN), therefore the saturation water content is always higher than the non-frozen water content in the ionomer in this temperature range. The difference between the maximum allowed water content and the local non-frozen water content can be considered as the driving force for the water phase change in ionomer, i.e. if lnf is larger than lsat, water will freeze until the local equilibrium state is reached (lnf ¼ lsat). The proton conductivity of Nafion ionomer is usually calculated based on the correlation reported in [72], and by considering the non-frozen water content rather than the total water content, the correlation becomes
kion ¼ 0:5139lnf    1 0:326 exp 1268 303:15 1 T  (10) where kion (S m 1) is the proton conductivity and T (K) the temperature. It should be noticed that Equation (10) is originally correlated based on experimental measurements between 30 and 80  C, and the original equation overestimates the ion conductivity at subzero temperatures (sharper decrements of the conductivity were observed [64e66]). However, by considering the freezing of membrane water at subzero temperatures, Equation (10) with nonfrozen membrane water content rather than total membrane water Membrane Conductiv ity, S m -1 14 λ = 15 12 10 8 λ = 10 6 Freezing points 4 λ=5 2 0 -50 0 50 100 Temperature, oC Fig. 5. Effects of water freezing on proton conductivity of Nafion ionomer [62]. 231 content provides a more reasonable agreement with the experimental measurements in [64]. Fig. 5 shows the changes of proton conductivity at different total membrane water contents, it can be observed that the proton conductivity starts dropping fast when the membrane water starts freezing. The non-frozen water in ionomer can also be classified into surface and bulk water [73,74], or vapour and liquid [75e77]. The classification of surface and bulk water is related to the strength of the interactions between water and HþSO3 . Surface water strongly interacts with HþSO3 and is mostly likely present close to the SO3 [78]. Bulk water is mainly identified as liquid which are loosely bound to SO3 . The surface and bulk water, and the non-freezable, freezable and free water might be able to be interrelated. The classification into vapour and liquid water in membrane is actually an assumption based on the state of water in the adjacent phase outside the membrane. In this paper, the classification of water into nonfreezable, freezable and free water is used since it comprehensively covers all the operating conditions and is easy to understand. In addition, the state and phase change of water in the different PFSA polymer membranes features the similar characteristics, and only water in Nafion membrane is discussed in this paper. For PEMFC with the other kinds of membranes, the state of water as well as the transport processes could be different, e.g. HT-PEMFC with PBI based membranes is discussed explicitly in Section 12. 2.2. In gas diffusion layer The GDL is usually carbon paper or carbon cloth, and the porosity is around or higher than 0.5. Water can exist in the pore regions of GDL in forms of vapour, liquid and ice. As shown in Figs. 3a and 4 under normal operating conditions, water exists in forms of vapour and liquid, and the condensation and evaporation occur depending on the local operating condition. For cold start as presented in Figs. 3b and 4 and as mentioned earlier, the formation of liquid water can be almost neglected since it freezes to ice, and it should be noticed that ice and liquid water may still co-exist when the local cell temperature increases or decreases to the freezing point of water, resulting in ice melting or liquid freezing. Therefore, water usually exists in forms of vapour and ice in the pore regions of GDL (ice is formed if the local water vapour pressure is higher than the local water saturation pressure) for PEMFC cold start. The pressure inside PEMFCs is usually between 1 to several atm, and the freezing point of water in this pressure range can be treated as constant (0  C). Because the operating pressure of PEMFC is always equal to or greater than the atmospheric pressure, the sublimation process (phase change from ice directly to vapour) can be safely neglected. The difference between the local temperature and the freezing temperature of water (around 0  C in GDL) indicates the phase change direction between liquid and ice: ice will melt to liquid if the local temperature is higher than the freezing point, and vice versa, and liquid and ice may co-exist during such phase change processes. The difference between the water saturation pressure and vapour pressure indicates the phase change direction between vapour and liquid (above the freezing point), and between vapour and ice (below the freezing point). When the local temperature is above the freezing point of water, if the vapour pressure is higher than the saturation pressure, vapour will condense to liquid, otherwise liquid will evaporate to vapour; when the local temperature is below the freezing point of water, if the vapour pressure is higher than the saturation pressure, vapour will desublimate to ice, however, as mentioned earlier that ice will not sublimate to vapour under the operating conditions in PEMFC even when the vapour pressure is lower than the saturation pressure. By checking the experimental data tabulated in [79], it is found that the correlation provided by Springer et al. [72] provides acceptable 232 K. Jiao, X. Li / Progress in Energy and Combustion Science 37 (2011) 221e291 agreement with the experimental data in the temperature range from 50 to 100  C: 2:1794 þ 0:02953ðT Saturation water content 273:15Þ 9:1837  10 5 ðT 273:15Þ2 þ 1:4454  10 7 ðT 273:15Þ3 15 ð11Þ where psat (Pa) is the saturation pressure of water and T (K) the temperature. 2.3. In catalyst layer Water Content log10 ðpsat =101; 325Þ ¼ 20 10 lequil ¼  0:043 þ 17:81a 39:85a2 þ 36:0a3 if 0  a  1 14:0 þ 1:4ða 1Þ if 1 < a  3 Equilibrium water content 0 -30 -25 -20 -15 -10 -5 0 o Temperature, C Fig. 6. Temperature dependence of the equilibrium water content (lequil) for the phase change between the non-frozen water in the ionomer of CL and the vapour in the pores of CL, and the saturation water content (lsat) for the phase change between the nonfrozen water in the ionomer and the ice in the pores of CL or the frozen water in the membrane layer for the temperature range from 30 to 0  C (note that in the calculation of equilibrium water content shown in this figure, the water vapour amount in the pores of CL is assumed to be equivalent to a water vapour activity of 1 at 30  C and liquid water is not considered because it freezes to ice at subzero temperatures; also note that the two lines, equilibrium water content and saturation water content, separate this figure into four regions as marked in the figure, and for the non-frozen water content in the ionomer, lnf, falling in these different regions, the water phase change processes are different, as detailed in Table 2) [63]. between the non-frozen membrane water and vapour occurs when non-humidified inlet gases are supplied [62,63]; only the phase change between the non-frozen membrane water and liquid occurs in cathode CL with fully humidified inlet gases [80]; and only the phase change between the non-frozen membrane water and vapour occurs in anode CL no matter the inlet gases are humidified or not [62,63,80], due to the fact that water is not produced in anode and the EOD effect dries out the anode. Based on the assumptions mentioned above, Fig. 6 shows the temperature dependence of the equilibrium water content (lequil) for the phase change between the non-frozen water in the ionomer of CL and the vapour in the pores of CL, and the saturation water content (lsat) for the phase change between the non-frozen water in the ionomer and the ice in the pores of CL or the frozen water in the membrane layer for the temperature range from 30 to 0  C. Note that in the calculation of the equilibrium water (12) where lequil is the equilibrium water content (the water content in ionomer corresponding to the amount of surrounding water), and a the water activity in the pore regions, defined as Xvp pg a ¼ þ 2slq psat Intersection point 3 4 5 The most complex scenario occurs in CL. As shown in Fig. 3, the carbon and platinum particles together with part of the ionomer (polymer electrolyte) have to mix together to form the reaction sites. The carbon and platinum particles have to be present as the catalyst and for electron transport, and the ionomer has to be present for proton transport. The volume fraction of the ionomer in CL ranges from about 0.2 to 0.4, and the porosity of the CL ranges from about 0.2 to 0.5. The thickness of CL is typically around 0.01 mm. The state of water in the pore regions of CL is the same as in the pore regions of GDL. The water in the ionomer of CL is also the same as in the membrane. It should be mentioned that due to the presence of small pores in CL (on the level of nanometre), the surface dynamics of water is enhanced, and the freezing point of water may be depressed to about 1  C in such small pores [62,63]. In the CL and at the interface between the CL and membrane, both the liquid water and vapour can be desorbed or absorbed by the ionomer to or from the pore regions of CL. The non-frozen membrane water, liquid or vapour can freeze or desublimate to ice, as shown in Fig. 4. The different water phase change processes can occur simultaneously or separately. It should be noticed that the ice in CL in Fig. 4 represents the ice in both the ionomer and pore regions. To simplify the phase change processes, one assumption made in the previous studies [62,63] is that when the water in the ionomer of CL freezes, it leaves the ionomer and only forms ice in the pore regions of CL, based on the experimental observation that the frozen water does not contribute to the proton transport in ionomer [64]. Equation (9) can be used to determine whether the water in the ionomer freezes to ice or not (if the non-frozen water content is higher than the saturation water content, it will freeze). The equation first used in [72] can be used to indicate the mass transfer (phase change) direction between the water in the ionomer and the vapour/liquid in the pore regions: 2 1 (13) where Xvp is the mole fraction of water vapour in the pore regions, pg (Pa) the pressure of the gas mixture in the pore regions, psat (Pa) the water saturation pressure, and slq the liquid water volume fraction in the pore regions. As mentioned in Section 1.3, when non-humidified gases are supplied, the presence of liquid water may be neglected, on the other hand, liquid water will mostly likely be present in cathode CL when fully humidified gases are supplied. Based on this premise, further assumptions to simplify the water phase change processes between the non-frozen membrane water and vapour/liquid have been made [62,63,80]. It is assumed that only the phase change Table 2 Water phase change processes that involve the ionomer in CL for the non-frozen water content in the ionomer (lnf) in the different regions and on the different dividing lines of Fig. 6 [63]. For the non-frozen water content in the different regions of Fig. 6 Phase change between the non-frozen water in the ionomer and the vapour in the pores The non-frozen water in the ionomer freezing to the ice in the pores In Region 1 In Region 2 In Region 3 In Region 4 On the line between Regions 1 and 2 On the line between Regions 2 and 3 On the line between Regions 3 and 4 On the line between Regions 4 and 1 At the intersection point Occur Occur Occur Occur Stop Occur Occur Stop Stop Occur Occur Stop Stop Stop Occur Stop Stop Stop 233 K. Jiao, X. Li / Progress in Energy and Combustion Science 37 (2011) 221e291 6 water and vapour can co-exist, and the condensation and evaporation can happen based on the difference between the vapour and saturation pressures (as described in Section 2.2); for cold start conditions, since water usually freezes in CL and GDL first and stick on the solid materials that is difficult to move, and usually nonhumidified gases are supplied for PEMFC cold start to reduce the amount of ice formation, therefore the chance of ice formation in flow channel is much lower than in CL and GDL, and might be neglected. 4 2.5. Summary 2 The states of water in different components of PEMFC are elaborated in Section 2 and summarized in Table 1. In the ionomer of membrane and CL, water is classified into free, freezable and non-freezable water, based on how tight they are bound to the sulphuric acids. In the pores of GDL and CL, and in flow channel, both vapour and liquid states are possible to be present. At subzero temperatures, water is able to freeze in all the components. The freezing points of water in the pores of GDL and CL and in the ionomer of membrane and CL are depressed due to the enhanced surface dynamics in the small holes. The freezing point of water in the ionomer of membrane and CL is further lowered because water is bound to the sulphuric acids. Even though water is classified into different types, whether phase equilibriums of water exist in different components remain debated. This is mainly due to the presence and arbitrary transport of liquid water, especially in the heterogeneous structures of CL and GDL. Equilibrium Water Content 14 Equation 12 [72] Measurement at 30 oC [81] o Measurement at 80 C [82] 12 10 8 0 0 0.2 0.4 0.6 0.8 1 Water Activity Fig. 7. Equilibrium water content as a function of water vapour activity for Nafion membrane at temperatures of 30  C and 80  C [72,81,82]. content shown in this figure, the water vapour amount in the pores of CL is assumed to be equivalent to a water vapour activity of 1 at 30  C and liquid water is not considered because it freezes to ice at subzero temperatures. In Fig. 6, the equilibrium water content decreases monotonically with temperature, because the water vapour amount in the pores of CL has been assumed to be saturated at 30  C, correspondingly the water vapour becomes more and more unsaturated as temperature is increased, leading to the decrease in equilibrium water content. On the other hand, the saturation water content increases monotonically with temperature since more non-frozen water can be maintained in the ionomer to avoid freezing. The two curves in Fig. 6 divide the figure into four regions, and for the non-frozen water content in the different regions and on the different dividing lines, the two water phase change processes (from the non-frozen membrane water to ice, and between the non-frozen membrane water and vapour) can either occur or stop simultaneously, or one process can also occur when the other stops, as illustrated in Table 2. The principal driving forces for these water phase change processes in the CL are the differences between the non-frozen water content in the ionomer and the equilibrium water content, and between the non-frozen water content and the saturation water content. It should be mentioned that Equation (12) is derived based on the experiential measurement at 30  C. As shown in Fig. 7 [72,81,82] comparing the equilibrium water contents at two different temperatures (30 and 80  C), although the equilibrium water content increases slightly with temperature increment at low water vapour activity, it decreases with increasing temperature at high water activity, especially for the hydration from saturated water vapour. The drying out of the ionomer at elevated temperature suggests that the membrane needs to be better hydrated at higher temperature. In addition, the measurements in [81,82] also revealed that the equilibrium water content is higher when the membrane is immersed to liquid water than that exposed to saturated water vapour, and this is the reason that the water activity is usually calculated to be greater than 1 when liquid water is present for calculating the equilibrium water content (Equations (12) and (13)). The details about the ionomer absorption and desorption processes are given in Section 3.3. 3. Mechanism of water transport With the different materials and local operating conditions in the different components of PEMFC, the mechanisms of water transport are different as well. In this section, the mechanisms of water transport in the different cell components are described one by one to provide a complete view of water transport in PEMFC. Note that starting from this section, only the normal operating conditions are considered, and the cold start processes are explicitly described in Section 11. Therefore, unless otherwise specified, the water content mentioned starting from this section (except Section 11) represents the non-frozen water content, since no frozen water is present in ionomer in normal operating conditions. 3.1. In membrane Effective membrane hydration is of paramount importance for reducing the ohmic loss of PEMFC for all the PFSA polymer Hydrophobic polytetrafluoroethylene (PTFE) backbone F F F F C C C C F F n O Fm m=1 n = 6 ~ 10 F C F F C CF3 Polymer side chain O m F C F F C F 2.4. In flow channel The state of water in flow channel is similar to as in GDL. As shown in Figs. 3 and 4, for normal operating conditions, liquid Sulphuric acid (SO3− H +) O S O − H+ O Fig. 8. Schematic of chemical structure of Nafion. 234 K. Jiao, X. Li / Progress in Energy and Combustion Science 37 (2011) 221e291 At a low water content: At a high water content: SO3− H+ H2O Polymer chain 1 nm Fig. 9. Schematics of micro-structural features of Nafion for low and high water contents (note that the free volume containing water at a low water content is smaller than at a high water content, and such difference is neglected in this figure). membranes. As mentioned earlier, in an operating PEMFC, the membrane close to the anode side may dry out quickly due to the EOD effect, therefore balancing the various water transport processes to evenly hydrate the membrane is critical, and this requires the understanding of the various water transport mechanisms in membrane. In this subsection, the proton transport in membrane is first elaborated (because it largely couples with water transport), followed by the explanations of the various water transport mechanisms in membrane (diffusion, EOD effect and hydraulic permeation). In addition, the reactant transport in membrane and membrane expansion are described as well. 3.1.1. Proton transport The proton transport has significant influence on the water transport in membrane; on the other hand, the proton transport also largely depends on the membrane environment (water SO3− H+ content). Therefore, the proton transport is described first in this subsection to guide the following discussions on the water transport in membrane. Generally, the ionomer (polymer electrolyte) proton conductivity follows the Arrhenius Law [83]. In fact, Equation (10) for calculating the proton conductivity is essentially a modified Arrhenius equation based on experimental measurements, and it shows that the proton conductivity strongly depends on the water content and temperature. Fig. 8 presents the schematic of chemical structure of Nafion. It consists of PTFE backbone providing the mechanical stability with polymer side chains with sulphuric acid for promoting proton conduction. As mentioned earlier, with large amount of SO3 , the proton conductivity is high, but the mechanical and thermal strengths are low; and with water absorbed in the membrane, the Hþ becomes weakly attracted to the SO3 , resulting in easier proton transport. Fig. 9 shows the schematics of micro-structural features of Nafion for both the low and Polymer chain Fig. 10. Schematic of direct proton transport between polymer chains of Nafion. K. Jiao, X. Li / Progress in Energy and Combustion Science 37 (2011) 221e291 high water contents. The solid lines in this figure represent the polymer chains (the hydrophobic backbones and polymer side chains shown in Fig. 8), where those directly connected to the SO3 are the polymer side chains (also shown in Fig. 8). Each polymer side chain attracts one Hþ forming sulphuric acid. The Hþ is relatively weakly bound to the polymer side chains. There are free volumes surrounded by the polymer chains, on the scale of nanometre, and water can be absorbed into these spaces (the hydrophilic regions discussed with Fig. 3 earlier). It should be mentioned that the size of these spaces increases with the increment of water content, and such difference between the spaces at the low and high water contents is neglected in Fig. 9. As shown in Fig. 9 at the low water content, the water in the membrane is mostly likely bound to the slide chains. This kind of water may be classified as the non-freezable water [62e64,69e71] 235 or surface water [73,74], as mentioned in Section 2.1. This part of water is strongly bound to the charged sites (SO3 ). In this scenario, proton has to transfer through the void volume from one charged site to another. Since the Hþ is relatively weakly bound to the SO3 , it is possible for it to jump from site to site directly. The mechanism of proton transport from one charged site to another directly is demonstrated in Fig. 10 (this is the most common type of proton transport in solid conductors). It can be noticed that the polymer side chain can actually vibrate in the free volume, and the movement of the polymer side chain can physically reduce the distance for the Hþ transport. When one extra Hþ is present at the polymer side chain from left hand side (the first picture from the left of Fig. 10), the two polymer side chains may attract each other and therefore vibrate. Again, it should be noticed that increasing the SO3 enhances the proton transport by reducing the distance Fig. 11. Schematics of proton transport in water (a: vehicle mechanism; b: hopping mechanism). K. Jiao, X. Li / Progress in Energy and Combustion Science 37 (2011) 221e291 content and conductivity, it has been observed that the membrane conductivity drops sharply when the water content is lower than 5 [65,66,95,96], and the membrane conductivity starts becoming appreciable when the water content is higher than 2, these values may also be used to explain the percolation threshold for the hopping mechanism for proton transport. If no water is present in a membrane, the proton can only transfer directly between the charged sites, resulting in very low proton conductivity. If water is present in a membrane but the amount is below the percolation threshold, the proton transfer mechanisms will be mainly the direct transfer between the charged sites and vehicle mechanism (with the increment of water content, the water diffusion coefficient is also higher, therefore the vehicle mechanism becomes more dominating). When the amount of water is higher than the percolation threshold, the dominating proton transport mechanisms are the vehicle and hopping mechanisms. It has been reported that for intermediate and low hydration levels, the vehicle mechanism still dominates the proton transport, and for water content higher than 13, the hopping mechanism dominates [97e100]. 3.1.2. Diffusion of water Due to the concentration gradient, water can diffuse through the void space of membrane. Since water is produced in cathode CL, resulting in more water on the cathode side, the diffusion of water is therefore usually from cathode to anode. The diffusional nonfrozen water flux in membrane, Jnmw,diff (kmol m 2 s 1), due to the concentration gradient is a vector quantity and can be written as Jnmw;diff ¼ Dnmw Vcnmw ¼ rmem EW Dnmw Vlnf (14) 20 -2 m s -1 where Dnmw (m2 s 1) is the diffusion coefficient of non-frozen water in membrane, cnmw (kmol m 3) the non-frozen water concentration in membrane; and as mentioned earlier, rmem (kg m 3) is the density of dry membrane, and EW (kg kmol 1) the equivalent weight. The negative sign in Equation (14) presents the fact that the diffusional flux is always in the direction of decreasing the concentration. Note that since only the normal operating condition is considered in this section, the non-frozen water represents all the water in membrane. The diffusion coefficient is sensitively dependent on the membrane hydration. Experimentally, self-diffusion coefficient is relatively easily measured by tracking the tracer in a homogeneously hydrated membrane, due to the random molecular motion. Generally, the water self-diffusion coefficient is similar in the PFSA polymer -10 between the SO3 , but the mechanical and thermal strengths are reduced. The vibration of the polymer side chains enhances the proton transport, explaining why the polymer membranes exhibit higher proton conductivity than most of the other solid proton conductors (without vibration of the polymer side chains) (e.g. ceramics). Proton may also transfer by hitching a ride on water by forming hydrogenewater ions (e.g. H3Oþ, H5Oþ 2 or something similar). The diffusion of the hydrogenewater ions happens from high to low proton concentration regions, therefore facilitating proton transport. This kind of proton transport is called the vehicle mechanism (also called the vehicular diffusion) [84]. However, the diffusion of the hydrogenewater ions may be retarded due to the hydrogen bonding [85] (i.e. H3Oþ may bond with other water molecules and therefore difficult to move). For a well hydrated membrane, more water exists in the membrane, as shown in the second picture in Fig. 9. If the amount of water is high enough to connect the polymer side chains, the proton may also transfer directly from one water molecule to another, and the water molecules in this case are essentially the charged sites. This kind of proton transfer is called the hopping mechanism (also called the Grotthuss mechanism or structure diffusion) [86e90]. These two mechanisms are demonstrated in Fig. 11. The vehicle mechanism can be relatively easily understood as the diffusion of the hydrogenewater ions, as shown in Fig. 11a. The hopping mechanism can be explained with the help of Fig. 11b. The transferring proton can reside on a water molecule forming a hydronium ion (H3Oþ), and an Eigen-ion (H9Oþ 4 ) can be formed since the H3Oþ is bound to three neighbouring water molecules by forming hydrogen bonds, as shown in the top left picture of Fig. 11b. The proton can then transfer to a location symmetrically between two water molecules, and a Zundel-ion (H5Oþ 2 ) can be formed, as shown in the top right and bottom pictures of Fig. 11b. After that, another H9Oþ 4 can be formed but at a different location than the previous H9Oþ 4 . Such transformations þ between the H9Oþ 4 and H5O2 are caused by the hydrogen bond forming and breaking processes, which essentially results in proton transport. It should be noticed that the vehicle mechanism mainly depends on the diffusion coefficient of water (or hydrogenewater ions) in membrane, which depends on the water content and temperature (explained later in Section 3.1.2). The hopping mechanism takes place if enough amount of water is present in membrane to form continuous network between the charged sites. This can be explained by using the percolation theory, by which enough free volume in a solid material has to be ensured for possible fluid flow from one side to another. The minimum amount of volume fraction required is called the percolation threshold [91,92]. The percolation threshold for the hopping mechanism is not determined yet, even though it is believed to be between the water contents of 1 and 7. The water content in Nafion membrane can change from 0 (completely dry) to 22 [93], and for PEMFC operating at 80  C, the maximum membrane water content is considered to be about 16.8 [72,93]. Assuming that all the free volumes in a membrane are occupied by water at its maximum water content (22), based on the percolation threshold for randomly distributed equiaxed ellipsoids (aspect ratio is 1), where the percolation does not occur until the volume fraction of the conducting phase is 28.54% [92], the percolation threshold for the hopping mechanism in membrane is therefore about 6.28 water content, which seems to be high. It was also claimed that the percolation threshold is about 5% of the maximum water content [94], which is only about a water content of 1.1. Another approximation is based on the measured proton conductivity, when the proton conductivity becomes appreciable, the water content ranges from 2 to 5. Even though Equation (10) shows a linear relationship between the membrane water Fickian D iffus ion Coefficient, 10 236 15 10 Equation 17 [105] 5 Equation 16 [104] 0 0 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 Water Content Fig. 12. Fickian diffusion coefficients of water in Nafion membrane at different water contents and at 80  C by using different correlations [104,105]. 237 K. Jiao, X. Li / Progress in Energy and Combustion Science 37 (2011) 221e291 membranes in terms of the magnitude and the trend of variations with the water content and temperature. Due to the small spaces for water diffusion, and the hydrogen bond which retards the water movement, the self-diffusion coefficient in Nafion is merely about  1 2416 > 303 > >   > : 10 10 exp 2416 1 303 > 10 10 exp  ðl  2Þ 

i 1 h 0:87 3 lnf þ 2:95 lnf 2 ; ð2 < l  3Þ T 
i 
1 h 2:95 4 lnf þ 1:642454 lnf 3 ; ð3 < l  4Þ T   1
2 3 2:563 0:33lnf þ 0:0264lnf 0:000671lnf ; ðl > 4Þ T four times lower than the value in bulk liquid water when hydrated by saturated vapour. The self-diffusion coefficient has been measured experimentally at different temperatures and water contents [95,101] using pulsed gradient NMR spectroscopy. The self-diffusion coefficient is measured when the membrane hydration is uniform and homogeneous for the entire membrane often referred to as the intradiffusion coefficient, which is applicable for the completely hydrated membrane in PEMFC operations. In practice, the membrane during the dynamic operation of PEMFC may be partially dried out on the anode side and yet still maintain full hydration on the cathode side. In the presence of such water gradient, the appropriate coefficient describing the water diffusion through such membrane is the interdiffusion (or Fickian diffusion) coefficient, which is related to the self-diffusion coefficient. For systems where the transport number of electrons is zero or unity, the intro- and Fickian diffusion coefficients (Dnmw,I and Dnmw,F) are related through the “Darken factor” [102] Dnmw;F " # vlnðaÞ 
¼ Dnmw;I vln lequil |fflfflfflfflfflfflfflfflfflfflffl{zfflfflfflfflfflfflfflfflfflfflffl} (15) Darken factor where a is the water activity when measuring the intradiffusion coefficient, and lequil the equilibrium membrane water content. The Darken factor can be obtained by taking the reciprocal of the differential of Equation (12). Based on the experimental measurement in [101] at 30  C and the activation energy of water diffusion in Nation membrane [103], and by transforming the intradiffusion coefficient to Fickian diffusion coefficient, the following correlation has been developed [72,104] to calculate the Fickian diffusion coefficient of non-frozen water in membrane (Equation (16)), where the units of Dnmw,F and Dnmw are all m2 s 1, and the unit of T is K. Note that unless otherwise specified, the diffusion coefficient of water in membrane represents the Fickian diffusion coefficient in this paper. Similarly, based on the same experimental data in [101] and similar methods, the following correlation has been developed [105]. Dnmw ¼ Dnmw;F ¼ ( h
 i 1 exp 2346 3:1  10 ; nf exp 0:28lnf T
 0 < lnf < 3 h
 i ð17Þ ; 4:17  10 8 lnf 161exp lnf þ 1 exp 2346 T 
3  lnf < 17 7l where the units of Dnmw,F and Dnmw are all m2 s 1, and the unit of T is K. ð16Þ used for PEMFC modeling. It can also be noticed that the two correlations all show that the maximum diffusion coefficient occurs at the water content of 3, caused by the transformation (Equation (15)) of the diffusion coefficient, and such sharp change of the diffusion coefficient also increases the difficulty for numerical simulations. 3.1.3. Electro-osmotic drag effect The EOD coefficient, nd, depends on the water content of membrane. Water content varies across a Nafion membrane because of several factors. Perhaps most important is the fact that protons traveling through the pores of Nafion generally drag water molecules along with them. Actually, protons travel in the form of hydronium complexes (H3Oþ) or something similar as explained earlier. For simplicity, however, it is straightforward to define the EOD coefficient in terms of the number of water molecules per proton. In other words, EOD coefficient is defined as the ratio of mole-of-water per mole-of-proton transported through the membrane in the absence of concentration and pressure gradient. The water flux (Jnmw,EOD, kmol m 2 s 1) due to the EOD is Jnmw;EOD ¼ nd Iion F (18) where Iion (A m 2) is the ionic current density, and F (9.6487  107 C kmol 1) the Faraday’s constant. For Nafion membranes, nd has been measured to be 2.5 at a water content of 22, and 0.9 at a water content of 11 [81]. Another measurement showed that it is 1.4 for the water contents from 5 to 2.5 Electro-os motic D rag Coefficient Dnmw ¼ Dnmw;F ¼ 8 2:692661843  10 10 ; >   > > 1 > 10 > < 10 exp 2416 303 Fig. 12 compares the diffusion coefficients by using the two correlations at 80  C. Apparently, the difference between the two correlations is not negligible, and there is no conclusion indicating the more accurate one. In fact, the two correlations have all been widely 2 1.5 1 Equation 20 [108] 0.5 Equation 19 [72] 0 0 2 4 6 8 10 12 14 16 18 20 22 Water Content Fig. 13. EOD coefficients of Nafion membrane at different water contents by using different correlations [72,108]. 238 K. Jiao, X. Li / Progress in Energy and Combustion Science 37 (2011) 221e291 14, and gradually decreases to zero for the water contents from 5 to 0 [106]. It was also reported that it is 1 for the water contents from 1.4 to 14 [107]. nd has also been reported for several other PFSA polymer membranes [108]. Correlations have been developed based on the experimental measurements for numerical models, one shows a linear relationship [72]: nd ¼ 2:5lnf 22 (19) and another one shows a stepwise correlation [108]: nd ¼  
lnf  14 0:1875lnf 1:625 ðotherwiseÞ 1 DO2 (20) Fig. 13 compares these two correlations and the difference is not negligible, even though both the correlations have been widely used for PEMFC modeling. 3.1.4. Hydraulic permeation The water flux (Jnmw,hyd, kmol m 2 s 1) associated with the hydraulic permeation of water due to pressure gradient is Jnmw;hyd ¼ cnmw Knmw mnmw Vpnwm ¼ lnf rmem Knmw Vp EW mlq (21) where Knmw (m2) is the permeability of non-frozen water in ionomer, cnmw the non-frozen water concentration in ionomer (kmol m 3), mnmw (kg m 1 s 1) the dynamic viscosity of non-frozen water in ionomer (the property of liquid water is often used instead), and pnmw (Pa) the pressure of non-frozen water in ionomer. It is in the direction of deceasing pressure, represented by the negative sign. The permeability of non-frozen water in membrane is mainly associated with water content because the pore size in membrane increases with the increment of water content, as described in Section 3.1. The following correlation has been developed to calculate Knmw (m2) [109,110]. Knmw ¼ 2:86  10 20 lnf preferred in order to minimize cell performance losses related to mass transfer resistance (or the depletion of the reactants at the reaction sites). This requires optimization of the ionomer fraction in CL to ensure high proton conductivity in CL while maintaining enough amount of reactant transport. Like water in the membrane, the diffusion of reactant gas is much larger than hydraulic permeation, and the diffusion coefficient ðDO2 mem ; m2 s 1 Þ for oxygen in fully hydrated Nafion 117 for oxygen is [111] (22) Apparently, the permeability is very low. In order to provide additional means for the reduction of water in the cathode and hydrate the membrane close to the anode, cells may be differentially pressurized such that the oxidant gas is supplied at a higher pressure than the fuel gas in the anode. Thus, the predominant direction of the diffussional and hydraulic water fluxes can be arranged opposite to that of EOD water flux, balancing the water in the membrane. In addition, the mechanical strength of the membrane needs to be considered when pressurizing the anode and cathode differentially, as well as the parasitic power requirement. 3.1.5. Reactant transport The requirement for the transport of reactant gases through the membrane is self-conflicting: on one hand, low diffusion coefficients, hence low rates of the reactant transfer, through the membrane are mandatory to separate the fuel and oxidant gas from mixing in order to avoid the degradation of cell performance and the occurrence of potential hazards; on the other hand, the electrochemical reaction for electric energy generation in the CLs is heterogeneous, occurring at the surface of the catalyst which is surrounded by the ionomer (polymer electrolyte) (as shown in Fig. 3). The ionomer covering the catalyst surface is essential for the protons to be transported away avoiding reaction product accumulation in the anode CL and to transfer the protons as reactants for the cathode reactions. This creates a significant challenge for the reactant gases (hydrogen and oxygen) to reach the catalyst surface and high values of hydrogen and oxygen diffusion coefficients are mem and for hydrogen ðDH2 is [109] DH2 mem 10 ¼ 2:88  10 ¼ 4:1  10   1 exp 2933 313 mem ; 7 exp m2 s  1Þ 1 T  (23) in fully hydrated Nafion 117  2602 T (24) where T is temperature with the unit of K. It can be noticed that the diffusion coefficients of reactant gases in membrane are very low (about 10 6 of the bulk diffusion coefficients), therefore the transport of reactants through the membrane is usually neglected in PEMFC studies. It should be mentioned that the diffusion of oxygen and hydrogen through the membrane occurs after the oxygen and hydrogen gas have dissolved in the hydrated membrane, following the Henry’s Lay [1]: pi Hi ci ¼ (25) where ci (kmol m 3) is the reactant concentration on the membrane side (i represents hydrogen or oxygen), pi the partial pressure (Pa) of the corresponding reactant on the gas side, and Hi (Pa m3 kmol 1) the corresponding Henry’s constant. For oxygen dissolving in hydrated Nafion 117, the Henry’s constant for oxygen in membrane ðHO2 mem ; Pa m3 kmol 1 Þ is [109] HO2 mem ¼ 101:325exp  666 þ 14:1 T  (26) where T is the temperature with the unit of K. Equation (26) yields about 2.04  107 Pa m3 kmol 1 at the typical PEMFC operating temperature of 80  C. Hydrogen is a weak function of temperature and can be considered as constant: 4.56  107 Pa m3 kmol 1 for the operating temperature range of PEMFC. The Henry’s constants are 1.45  108 and 1.26  108 Pa m3 kmol 1 for hydrogen and oxygen in liquid water, respectively, much higher than in hydrated membrane [112,113]. Apparently, the reactants are more soluble in membrane than in liquid water. In fact, the reactants are more soluble when the membrane is dry, however, the diffusion of the reactant gases is faster when the membrane is better hydrated [114], and again, the slow diffusion of the reactants in membrane may be neglected. 3.1.6. Membrane expansion As mentioned earlier, the pore size in PFSA polymer membranes increases when water is absorbed, in this case the membranes also expand in volume. As a result of the volume change, the concentration of the fixed charged sites also changes, depending on water content. For Nafion membrane, the concentrations of charged sites and water (ccs and cH2 O , kmol m 3) can be expressed as [72] ccs ¼ r 1 1 ; c ¼ lccs ¼ l mem EW 1 þ 0:0126l H2 O EW 1 þ 0:0126l rmem (27) where rmem (kg m 3) is the density of dry membrane, EW (kg kmol 1) the equivalent weight, and l the total water content. 239 K. Jiao, X. Li / Progress in Energy and Combustion Science 37 (2011) 221e291 For this subsection (3.1), and as mentioned previously, only the Nafion membrane is focused on, which is not the ideal membrane but the best or standard so far. More information about the other membranes such as Flemion, Gore and so on is also available in literature [100,115]. 3.2. In gas diffusion layer As the physical support of membrane and CL, GDL is attached to the outside of CL and the whole structure (membrane, CL and GDL) is compressed to form the membrane electrode assembly (MEA). Carbon paper or carbon cloth with thicknesses between 0.2 and 0.4 mm is widely used as the GDL because of the high porosity (usually higher than 0.5) that facilitates the reactant transport towards the catalyst sites. Also the very low electrical resistance of carbon paper or carbon cloth makes the ohmic loss within an acceptable range. PEMFC at high current density operation is of particular interest to vehicular applications for obtaining high power density, as long as with sufficient fuel and oxidant supply, the “concentration or mass transport loss” shown in Fig. 2 comes into play due to the excessive liquid water build up. As mentioned earlier and shown in Fig. 3a, liquid water blocks the porous pathways in CL and GDL thus causing hindered oxygen transport to the reaction sites, such water flooding phenomenon is perceived as the chief mechanism leading to the limiting current behaviour in the cell performance, and this is the reason that the carbon paper or carbon cloth is usually treated to be hydrophobic by coating a layer of PTFE on them to expel water. Therefore, effective water removal in GDL without affecting the membrane hydration is critical, and this requires understanding the water transport mechanism in GDL. In this subsection, the various water transport mechanisms in GDL (diffusion, convection and capillary effect) are explained, as well as the water condensation and evaporation. 3.2.1. Diffusion and convection In GDL and CL, the flow exhibits porous and tortuous structures on the micro- and nanometre length scales, in which the convective forces are resisted. As a result, the flow in GDL can be diffusion dominated, convection dominated, or mixed, depending on the design of the flow channel. Convection refers to the bulk motion of a fluid (under action of a mechanical force), and diffusion refers to the transport of a species due to a concentration gradient. In PEMFC, the convective force that dominates the convective transport is the pressure at the flow channel inlets. High flow rate can ensure good distribution of reactants (and effective water removal) but may require unacceptable high driving pressures or lead to other problems. The concentration gradients that dominate diffusive transport are from species consumption/production in CL: the reactant consumption and water product result in reactant delivery and water removal. Fig. 14 shows the flow characteristics in GDL with different flow channel designs. For parallel flow channel design, only small inlet pressures are needed for reactant flow because the flow distance from flow channel inlet to outlet is short, and the pressures are relatively evenly distributed for each straight channel. Since very small pressure gradient is present in GDL, resulting in slow convective flow, the flow is mostly likely diffusion dominated in GDL. Serpentine flow channel design needs higher inlet pressures than parallel design because the distance from inlet to outlet is longer, resulting in larger pressure gradient and cross flow from channel to channel through GDL directly, therefore the flow can be dominated by both diffusion and convection. Interdigitated design forces all the flow through GDL, resulting in the largest pressure gradients among the three flow channel designs shown in Fig. 14 and the flow becomes convection dominated in GDL. In the porous and tortuous flow structures of GDL and CL, the movement of the gas molecules can be restricted by the pore walls, lowering the diffusional flux. To account for such diffusion resistance, a modified or effective diffusion coefficient based on the porosity and tortuosity can be used [116]. Deff ¼ Di i 3 s (28) where Deff i and Di represent the effective and bulk diffusion coefficients of gas species i, respectively; and 3 and s are the porosity and tortuosity of GDL or CL, respectively. The porosity is defined as the percentage of void volume in the total volume (void and solid volumes), and the tortuosity describes the additional impedance to Parallel design Flow field GDL Diffusion dominated flow Serpentine design Flow field GDL Interdigitated design Flow field GDL Fig. 14. Flow characteristics in GDL with different flow channel designs. Convection dominated flow 240 K. Jiao, X. Li / Progress in Energy and Combustion Science 37 (2011) 221e291 diffusion caused by a tortuous or convoluted flow path. The Bruggeman correlation provides the relationship between porosity and tortuosity as [117] s¼ 3 0:5 (29) therefore Equation (28) can be simplified as Deff ¼ Di 31:5 i (30) Other correlations have also been developed [118e122] to calculate the effective diffusion coefficient in porous medium, and there is no significant difference among the different correlations. Therefore the simplicity of the Bruggeman correlation made it become the most popular one. Considering the blockage of liquid water, it can be assumed that liquid water has the similar effect as the solid materials [62,63], therefore Equation (30) can be modified as
Deff ¼ Di 31:5 1 i slq 1:5 (31) Different values of the exponential for the liquid water volume fraction (slq) term in Equation (31) have also been reported, e.g. a value of 2 was suggested in [121]. The convective flow through GDL (and CL) by pressure gradient is mainly affected by the permeability of the GDL (and CL). The pressure gradient caused due to the porous and tortuous structure can be calculated for gas and liquid phases: Vpg;lq ¼ mg;lq ! Kg;lq u g;lq Klq ¼ K0 s4:0 lq slq 4:0 Dp ¼ s  1 1 þ R1 R2  (35) where the surface curvature is represented by the two radii (R1 and R2, m) of the curved surface (interface) in two orthogonal directions. Considering liquid water transport in PEMFC cathode, the following non-dimensional numbers can be used to evaluate the importance of surface tension effect. Reynolds number : Re ¼ rlq Ulq L mlq (36) Capillary number : Ca ¼ mlq Ulq slq air (37) (32) ! where p (Pa), m (kg m 1 s 1), K (m2) and u (m s 1) are the pressure, dynamic viscosity, permeability and velocity, respectively; and the subscripts g and lq represent gas and liquid phases, respectively. Note that Equation (32) is only valid when the gravitational force can be neglected. For large velocity flow (e.g. with serpentine and interdigitated flow channel), the right hand side of Equation (32) can be added as an extra resistance force to standard momentum conservation equations for the porous media (detailed in Section 6). The intrinsic permeability of GDL is typically on the level of 10 12 m2, and the gas phase and liquid phase permeabilities (Kg and Klq, respectively, m2) depend on the intrinsic permeability of the porous materials (K0, m2) and the local volume fraction of liquid water. The gas phase and liquid phase permeabilities can be calculated as [62,63]
Kg ¼ K0 1 the force acting to minimize the free energy (minimize the forces at the interface) by decreasing the area of the interface, resulting in the droplet squeezing itself together until it has the locally lowest surface area possible. Surface tension can be represented by a fluid property, surface tension coefficient (s, N m 1), representing the tensional force along a line on the interface (therefore having of unit of N m 1). If there is no pressure difference across the interface, the interface remains flat. If the pressure on one side is greater than the other side, the interface is curved to the low pressure side (as for the droplet and air example mentioned before, the shape of water droplet indicates that the interface is curved to the air side). The surface tension coefficient (s, N m 1), pressure difference (Dp, Pa) and surface curvature can be related by the YoungeLaplace equation [126]: (33) Weber number : Bond number : Bo ¼
2L rlq Ulq ¼ Re$Ca slq air (38)  rair gL2 rlq slq (39) air where rlq, Ulq, L, mlq, slq-air, rair and g are the liquid water density, liquid water velocity, characteristic length, liquid water dynamic viscosity, liquid water surface tension coefficient when exposed to air, air density and gravity, respectively; and by using the typical values (at 80  C and 1 atm) for these parameters of 990 kg m 3, 10 5 m s 1, 8  10 5 m (the pore diameter in GDL), 3.5  10 4 kg m 1 s 1, 0.063 N m 1, 1 kg m 3 and 9.81 m s 2, respectively, the Reynolds number (inertia force divided by viscous force) is calculated to be 0.0023, the Capillary number (viscous force divided by surface (34) where slq represents the volume fraction of liquid water in the pores. Other values of the exponentials in Equations (33) and (34) have also been reported in the range from 3.0 to 5.0 [80,123e125]. 3.2.2. Surface tension and wall adhesion effects in porous media: capillary effect Surface tension is a force, acting only at the interface between liquid and liquid, liquid and gas or liquid and vacuum. For example, for a liquid water droplet in air, the inter-molecular forces on the water molecules inside the droplet are balanced, however, at the interface between the droplet and air, the inter-molecular forces (attractive forces) on the droplet side are larger than on the air side. The water molecules at the interface are therefore subject to an inward force of inter-molecular attraction which is only balanced by the liquid water’s resistance to compression. This results in a pressure difference across the interface (the pressure in this case is higher on the droplet side due to the larger inward inter-molecular force and the resistance to compression). In this case, the surface tension can be understood as We ¼ d d Gas Gas R θ θ R θ R Liquid water Liquid water Hydrophilic (θ < 90o) pg > plq Hydrophobic (θ > 90o) pg < plq Fig. 15. Two-phase behaviours in small pores with different surface wettabilities. K. Jiao, X. Li / Progress in Energy and Combustion Science 37 (2011) 221e291 tension) is calculated to be 5.6  10 8, the Weber number (inertia force divided by surface tension) is calculated to be 1.3  10 10, and the Bond number (gravitational force divided by surface tension) is calculated to be 9.9  10 4. Apparently, the surface tension effect plays an important role in liquid water transport in PEMFC, using the typical values for liquid water in CL results in the same conclusion. Note that the liquid water velocity might become much higher when strong convective flow occurs in GDL (as mentioned in Section 3.2.1). Assuming that the interface is identical to a sphere surface, the R1 and R2 in Equation (35) become identical to the radius of the sphere radius (R, m), and Equation (35) can be simplified to 2s Dp ¼ R (40) In GDL (and CL), liquid water transport is strongly affected by the pore walls due to the small pore sizes, and the wall adhesion effect becomes significant in this scenario. The surface wettability of GDL (and CL) therefore plays a significant role in liquid water transport. Fig. 15 shows the two-phase behaviours in small pores with different surface wettabilities (hydrophobic and hydrophilic) of the pore walls. It can be noticed that the angles between liquid water and pore walls are different with different surface wettabilities. The angle q in Fig. 15 represents such angles, and is usually called the contact angle. The contact angle can be considered as a measure of the surface wettability. In the case in Fig. 15, for q less than 90 , the surface is hydrophilic, and for q greater than 90 , it is hydrophobic. A hydrophobic surface is simply more effective in facilitating liquid transport than a hydrophilic surface due to the reduced contact area between liquid and wall. This is why that the GDL and CL are usually treated to be hydrophobic by adding PTFE to them to expel water. With the different surface wettabilities of the pore walls, it can be noticed in Fig. 15 that the interfaces between liquid water and gas are significantly different, and the shape of the interface is determined by the surface wettability. The hydrophilic pore walls result in an interface that is curved to the liquid water side (the pressure on the gas side is therefore higher than on the liquid water side), and vice versa. The pressure difference can drive the flow of liquid water and gas in the small pores, and such movement in small pores is defined as the capillary motion, or simply say that such movement is caused by the capillary effect. The capillary effect is essentially the combined effects of surface Capillary Pressure, kPa 0 Gas diffusion layer -5 -10 Catalyst layer -15 -20 -25 0 0.2 0.4 0.6 0.8 1 241 tension and wall adhesion in small pores. As illustrated in Fig. 15, the sphere radius R (m) at the interface can be calculated by using the pore diameter d (m) and the contact angle q as R ¼ d 2cosq (41) The pressure difference across the interface in this case is also called the capillary pressure (pc, Pa), which can be defined as pc ¼ pg plq ¼ 4slq cosq d (42) where pg (Pa) and plq (Pa) are gas and liquid water pressures, and slq (N m 1) the surface tension coefficient of liquid water (exposed to air or oxygen in cathode, and exposed to hydrogen in anode). Apparently, the capillary pressure (pc, Pa) is an important parameter affecting liquid water transport in PEMFC as a function of liquid water surface tension coefficient (slq, N m 1), contact angle (q), porous structure (represented by the porosity 3 and the intrinsic permeability K0, m2), and liquid water volume fraction (slq), and these parameters can be related based on the Leverett function [127,128]: (
0:5 h

2 1:42 1 slq 2:12 1 slq 3 i
 if q < 90 þ1:26 1 slq
0:5 h i  slq cosq K30 1:42slq 2:12s2lq þ1:26s3lq if q > 90 slq cosq pc ¼ 3 K0 (43) Equation (42) shows that the liquid water pressure can be calculated by using the gas phase and capillary pressures. The liquid ! water velocity ( u lq , m s 1) by neglecting the gravity effect can be calculated based on liquid water pressure by using Equation (32). Fig. 16 shows the changes of capillary pressures with liquid water volume fractions in GDL and CL by using Equation (43). For the calculations in Fig. 16, the surface tension coefficient of liquid water is 0.063 N m 1 (when expose to air); the contact angles in GDL and CL are all 110 ; the porosities of GDL and CL are 0.6 and 0.3, respectively; and the intrinsic permeabilities of GDL and CL are 10 12 and 10 13 m2, respectively. It can be noticed that negative capillary pressures (liquid water pressure is higher than gas pressure) are obtained because the GDL and CL are all hydrophobic. Note that Equation (43) was originally derived based on experimental data of homogeneous soil or a sand bend with uniform wettability, which are different from the GDL and CL structures in PEMFC. Other experimental measurements have been carried out recently trying to assess the real situation in PEMFC, and the other correlations for calculating capillary pressure are available in [80,123,129e133]. However, due to the differences in the measurement approaches, facilities, experimental conditions, and the materials being investigated, the results do not agree with each other very well. Therefore Equation (43) is still widely used for PEMFC studies. In addition, the diffusion coefficient of liquid water (Dlq, m2 s 1) and the relationship between liquid water and gas ! ! velocities ( u lq and u g , m s 1) have also been derived based on the capillary pressures [134] in GDL and CL, which concentrate significantly for modeling water transport in PEMFC: Dlq ¼ Kg dpc mlq dslq (44) Liquid Water Volume Fraction Fig. 16. Capillary pressures in GDL and CL at different liquid water volume fractions calculated by using Equation (43) (the surface tension coefficient of liquid water is 0.063 N m 1 (when expose to air); the contact angles in GDL and CL are all 110 ; the porosities of GDL and CL are 0.6 and 0.3, respectively; and the intrinsic permeabilities of GDL and CL are 10 12 and 10 13 m2, respectively). Klq mg ! ! ! u lq ¼ i u g ¼ ug Kg mlq (45) where i is named as the interfacial drag coefficient, representing the ratio of liquid and gas velocities. i has been assumed to be 1 (gas 242 K. Jiao, X. Li / Progress in Energy and Combustion Science 37 (2011) 221e291 velocity equal to liquid velocity) to solve liquid water transport in flow channel by assuming that liquid water only forms very small droplets, however, the accuracy of this assumption remains debated [80]. 3.2.3. Condensation and evaporation of water The condensation and evaporation of water in PEMFC depend on the local mass and heat transfer conditions. From kinetic theory [135,136], assuming an ideal gas and neglecting interactions between individual molecules, the net mass transfer of the evaporation and condensation can be estimated using the Hertze KnudseneLangmuir equation, as demonstrated in [137]: Sv l 0 rffiffiffiffiffiffiffiffiffiffiffiffi MH2 O B pvp ¼ Avp=lq @2cond pffiffiffiffiffiffiffi 2pR Tvp p 1 lq C 2evap qffiffiffiffiffiffi A Tlq (46) where Sv-l (kg m 3 s 1) is the mass transfer rate of phase change between vapour and liquid water, Avp/lq (m 1) the liquid/vapour specific interfacial area (interfacial area per unit volume) which depends on the volume fraction of liquid water, MH2 O (18 kg kmol 1) the molecular weight of water, R (8314 J kmol 1 K 1) the universal gas constant, 2cond and 2evap the condensation and evaporation rate coefficients, pvp and plq the vapour and liquid water pressures (Pa), and Tvp and Tlq the vapour and liquid water temperatures (K). A comprehensive investigation of the condensation and evaporation process is rather complicated and needs to be performed in the surrounding regions of the liquid/ vapour interface on the molecular level. For the sake of simplicity, for PEMFC modeling on the macroscopic level, it is impractical to incorporate such processes and a revised form of the equation can be used [134,137,138]: Sv l Avp=lq ¼ RT 2ce
pvp psat  (47) where 2ce is the analogous condensation/evaporation rate and it reads 2ce ¼ Gce sffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi RT 2pMH2 O (48) where Gce is an uptake coefficient that accounts for the combined effects of heat and mass transport limitations in the vicinity of the liquid/vapour interface. From the analysis of [121], this coefficient is about 0.006. The specific liquid/vapour interfacial area is calculated as (49) Avp=lq ¼ Gsurf Apore where Apore is the pore surface area per unit volume which varies from 1.3  107 to 3  107 m 1 for different GDL materials [130]. Gsurf is an accommodation coefficient similar to Gce. The study of [139] showed that Gsurf rarely exceeds 20% for spherical droplets with small amount of liquid water. The ranges of Gce and Gsurf were also estimated to be 0.001e0.006 and 1e20% for PEMFC operations [137]. The water condensation/evaporation dynamics are limited by the mass transport in the vicinity of the vapour/liquid interface, and therefore Equation (47) was also modified as [137] Sv l Shce Dvp ¼ Apore d
pvp psat RT  (50) where d (m) is the characteristic length for water diffusion, Dvp (m2 s 1) the mass diffusivity of water vapour, and Shce the dimensionless number accounting for mass transport capability during condensation/evaporation, and it is analogous to the Sherwood number for mass transfer, calculated as [137] Shce ¼ Gce Gsurf sffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi RT d 2pMH2 O Dvp (51) The values of d and Dvp are not important here since they will cancel out in Equation (50). d can be calculated based on the diameter of the solid material of the porous media (e.g. the fibre diameter of the GDL), dsl (m) based on the model in [121]: d ¼ 4dsl (52) It has also been estimated that the range of Shce is from 0.00204 to 0.245 [137]. To differentiate the condensation and evaporation processes, a Langmuir type correction with the porosity and local liquid should be considered [137]: Sv l ¼ 8 <
  pvp psat Shcond Dvp
3 1 slq Apore RT d if pvp > psat ðcondensationÞ
 pvp psat Shevap Dvp 3slq Apore RT d if pvp < psat ðevaporationÞ ð53Þ : Shcond and Shevap are the phase transfer rates of condensation and evaporation ranging from 0.00204 to 0.245 [137]. 3 and slq are the porosity and liquid water volume fraction, respectively. Equation (53) can be further simplified by using constant overall phase change rates gcond and gevap (s 1): Sv l ¼ (
gcond 3 1 gevap 3slq slq ðpvp  ðpvp psat Þ RT psat Þ RT if pvp > psat if pvp < psat ðcondensationÞ ðevaporationÞ (54) Equation (54) is the most widely used equation to calculate the water condensation and evaporation rates in PEMFC modeling on the macroscopic level. However, the values of gcond and gevap from 1:0 to 104 s 1 have all been used in the previous studies [80,140]. Since the water phase change rates are strongly affected by the local conditions such as mass and heat transfer, the accuracy of the mass transfer rate of phase change calculations on the macroscopic level remains debated. 3.3. In catalyst layer On both sides of membrane, CLs usually form in terms of carbon supported platinum powders as the catalyst embedded in part of the membrane ionomer, as shown in Fig. 3. With the presence of both the pore regions and ionomer, the most complex water transport occurs in CL. Since the water transport in ionomer and pores has been described in Sections 3.1 and 3.2, respectively, they are not repeated in this subsection. It should be noticed that the pores in CL are much smaller than in GDL, resulting in lower porosity and permeability. The platinum particles in CL are typically in the range from 1.5 to 2.5 nm, while the carbon support particles are in the size range from a few mm to about 20 mm [1]. With different platinum/carbon and ionomer fractions, the pore diameters range from several nm to about 1 mm [141]. The Knudsen numbers (Kng and Knlq) for both the gas and liquid phases can be calculated by the following equations [142]. mg lg Kng ¼ ¼ d pg sffiffiffiffiffiffiffiffiffiffi pRT 1 2Mg d (55) 243 K. Jiao, X. Li / Progress in Energy and Combustion Science 37 (2011) 221e291 Knlq ¼ dlattice lq (56) d where lg (m) is the mean free path (average distance between collisions) of gas phase, d (m) the pore diameter, mg (kg m 1 s 1) the dynamics viscosity of gas phase, pg (Pa) the pressure of gas phase, R (8314 J kmol 1 K 1) the universal gas constant, T (K) the temperature, the Mg (kg kmol 1) the molecular weight of gas phase, and dlattice lq average distance between liquid water lattices. By using a pore diameter of 10 nm for CL, the properties of air (the dynamic viscosity is 2.08  10 5 kg m 1 s 1 and the molecular weight is 29 kg kmol 1), the operating pressure and temperature of 101,325 Pa and 353.15 K, and the average distance of 0.3 nm between the liquid lattices, the Knudsen numbers for gas and liquid water are calculated to be about 8.2 and 0.03, respectively. Note that the macroscopic top-down approach (by solving the continuity, NaviereStokes and other equations) is only applicable when the Knudsen number is lower than 10 1 with slip boundary conditions on walls and lower than 10 3 without slip boundary conditions on walls, suggesting that the macroscopic top-down approach cannot be used with the real micro-structure of CL (also cannot be used with the nanometre pores in membrane). The main mechanism of gas diffusion is essentially the collision between gas molecules, as occurring in GDL, CL and flow channel. However, in the extremely small pores in CL, as analyzed before by calculating the Knudsen numbers, another mechanism of gas diffusion occurs in CL as well, which is called the Knudsen diffusion due to the collision between gas molecules and walls. The diffusion coefficients related to the mechanism of collision between gas molecules (the binary diffusion coefficient, DBi , m2 s 1) and related to the collision between gas molecules and walls (the Knudsen 2 1 diffusion coefficient, DK i , m s ) are [143,144] DBi ¼ DB;ref i DK i ¼ T Tref !1:5  pref p  (57)   1 8RT 0:5 d 3 pMi (58) where DB,ref (m2 s 1) is the reference binary diffusion coefficient at i the reference temperature (Tref, K) and pressure (pref, Pa); p (Pa) and T (K) are the local pressure and temperature ; R (8314 J kmol 1 K 1), Mi (kg kmol 1) and d (m) are the universal gas constant, molecular weight of gas species i, and the pore diameter; and the subscript i represents different gas species. The binary diffusion occurs in GDL, CL and flow channel, and the Knudsen diffusion is only significant in CL, the combined diffusion coefficient (Di, m2 s 1) can be summarized as Surrounded by water vapour F F F F F F 8 B < Di ðin GDL and flow channelÞ   1 Di ¼ : 1B þ 1K ðin CLÞ D D i As mentioned earlier, the effective diffusion coefficient needs to be further calculated in considering the porous and tortuous flow structures of GDL and CL as well as the liquid water formation (Equations (28)e(31)). The water transfer between the ionomer and pore regions in CL (membrane water absorption/desorption) plays an important role in PEMFC because it determines the hydration/dehydration of membrane. It has been observed that the amounts of water absorption from liquid water and from saturated vapour are not the same (from liquid water is higher than from saturated vapour). Such phenomenon was initially reported by Schroeder in 1903, hence the phenomenon has been called Schroeder’s paradox [145]. Evidence suggests that the PFSA polymer membrane surface is strongly hydrophobic when it is in contact with water vapour (whether saturated or not), and it becomes hydrophilic when in contact with liquid water, as illustrated in Fig. 17. When a liquid water droplet advances on the ionomer surface, wetting more areas of the surface, the hydrophilic sulphuric acid moieties initially inside the ionomer (when in contact with the water vapour on the surface) spring out towards the liquid water now spread on the surface, thus making the surface more hydrophilic and more water absorption occurs when liquid water wets the surface. On the other hand, water absorption from the vapour phase involves water condensation on the hydrophobic surface, leading to less water uptake. As mentioned in Section 2.3, this is the reason that the water activity is usually calculated to be greater than 1 when liquid water is present for calculating the equilibrium water content (Equations (12) and (13)). It should be mentioned that the humidification of the ionomer is a very slow process, especially with water vapour. It has been shown that the time scale for the membrane to reach its absorption equilibrium state in humid air is on the order of 100e1000 s [146,147] or even longer [148]. The following equation has been used to calculate the mass transfer rate of the phase change (water transfer) between the non-frozen water in ionomer and vapour/ liquid in pores (the membrane absorption/desorption rate, Sn-v,n-l, kmol m 3 s 1) [80,149e151]: Sn v;n l ¼ gn Surrounded by liquid water − + SO3− H + SO3 H SO3− H + SO3− H + − + SO3 H SO3− H + rmem
EW lnf lequil  During operation (in case surrounded by both vapour and liquid) F F F SO3− H + SO − H + 3 SO3− H + SO3− H + − Ionomer surface v;n l (60) where gn-v,n-l (s 1) represents the various phase change (water transfer) rates; rmem (kg m 3) is the density of dry membrane, and SO3− H + SO − H + 3 SO3 H + − + SO3− H + SO3 H SO3− H + (59) i SO3− H + SO3− H + Ionomer surface Ionomer surface Fig. 17. Illustration of PFSA membrane surface morphology when it is in contact with vapour and liquid water [1]. K. Jiao, X. Li / Progress in Energy and Combustion Science 37 (2011) 221e291 3.4. In flow channel Flow channel provides pathways for distributing reactant and removing product water in PEMFC. The flow in flow channel is convection dominated, and the driving force is the pressure at the flow channel inlets. Considering a flow velocity of 5 m s 1 in cathode flow channel with a typical cross section of 0.001 m  0.001 m (the characteristic length is therefore 0.001 m), and using the values of 1 kg m 3 and 2.075  10 5 kg m 1 s 1 as the density and dynamic viscosity of air, respectively, the Reynolds number in flow channel is calculated to be about 241 by using Equation (36), indicating that the flow in flow channel is mostly likely laminar. By further considering the values of 990 kg m 3, 3.5  10 4 kg m 1 s 1 and 0.063 N m 1 as the density, dynamics viscosity and surface tension coefficient (when expose to air) of liquid water, respectively, it can be calculated from Equations (37) to (39) that the inertia force, viscous force, surface tension (as well as wall adhesion) and gravitational force all need to be accounted for in flow channel. In fact, to facilitate liquid water removal in flow channel, the channel walls can be modified to be hydrophobic to minimize the wall adhesion effect. Since the various transport mechanisms as well as the flow characteristics related to different flow channel designs have been described in Sections 3.1e3.4, they are not repeated here. 3.5. Summary The mechanisms of water transport in different components of PEMFC are elaborated in Section 3. The transport of water in the ionomer of membrane and CL involves diffusion, EOD and hydraulic permeation, and it is largely coupled with proton transport. In the pores of GDL and CL and in flow channel, the transport of water vapour involves diffusion and convection. The capillary force also plays an important role in liquid water transport in the pores of GDL and CL. The Knudsen diffusion can be neglected in the pores of GDL and in flow channel, but needs to be considered in the much smaller pores of CL. Phase change processes take place until the equilibrium states are achieved. However, as mentioned in Section 2.5, whether phase equilibriums of water exist remain debated because of the presence and arbitrary transport of liquid water, especially in the heterogeneous structures of CL and GDL. 4. Experimental observation Presently available experimental techniques are excellent tools for investigating the transport phenomena in PEMFC. The current and high frequency resistance (HFR) distribution measurements provide important information about the reactant delivery, product removal and water distribution in membrane. The distributions of the different gas species can be obtained from the species concentration measurements at different locations of PEMFC. The temperature distribution measurements and various water visualizations provide valuable information to guide better thermal and water management of PEMFC. Not only help understand the transport phenomena, the experimental observations also provide valuable data to guide more complex and accurate numerical modeling of PEMFC. This section reviews the various experimental work published in literatures, the representing results obtained from the experimental measurements are shown as well. 4.1. Current distribution measurement The printed circuit board technique demonstrated in [152,153] was first used by Cleghorn et al. [154] to the measure the current distribution in a PEMFC. In the measurement of [154], a segmented current collector on the anode side with different flow fields separated was used, the anode GDL and CL were also segmented corresponding to the current collector. This approach is called the partial electrode approach with flow field, GDL and CL all segmented, it allows for mapping of the current distribution on the electrode surface to investigate the reaction kinetics at different locations directly. Stumper et al. [155] demonstrated three methods for current distribution measurement of PEMFC, including the partial electrode approach as described in [154], the subcell approach and the current distribution mapping approach. First, the partial electrode approach involves the segmentation of flow field, GDL and CL, therefore determining the local current density behaviour of the electrode. Second, the subcell technique involves placing small subcells at specific locations in a main cell and isolating them, therefore the performance of the desired location (at a very small scale) can be measured from the subcells. The third is the current distribution mapping approach, in this approach the current distribution is measured from the flow field plate with unmodified MEA, e.g. shunt resistors normal to an unmodified MEA surface were located between the flow field plate and a buss plate, voltage sensors could passively determine the potential drop across each resistor, and via Ohm’s law, current distribution through the flow plate was determined [155]. All the current distribution measurements can be generally categorized into these three approaches, even though the different measurement techniques were used for the same approach. The first and third approaches 1 -2 EW (kg kmol 1) the equivalent weight; and lnf and lequil are the nonfrozen membrane water content and equilibrium water content (Equation (12)), respectively. In Equation (60), the mass transfer rate is assumed to be proportional to the difference between the local ionomer water content and the equilibrium value. Generally, the study of water absorption/desorption is still relatively new and many characteristics for this process remain unclear. A constant value of 1.3 s 1 was suggested for gn-v,n-l [80,149]. The study in [147] showed that the physical mechanism of membrane absorption is different from that of desorption which is mainly limited by the interfacial mass transport. Water absorption process presents a two-step behaviour: the initial 35% of water absorption is described by the same interfacial transport rate coefficient as that of desorption, while for the value above 35%, water absorption is controlled by the dynamics of membrane swelling and relaxation. It is found that the absorption process is 10 times slower than that of desorption in the second stage. Current D ensity, A cm 244 Cell voltage = 0.8 V Cell voltage = 0.65 V Cell voltage = 0.5 V 0.8 0.6 0.4 0.2 0 0 0.2 0.4 0.6 0.8 1 Fractional Distance from Cathode Inlet Fig. 18. Measured current density along fractional distance from cathode flow channel inlet at different cell voltages [168]. 245 K. Jiao, X. Li / Progress in Energy and Combustion Science 37 (2011) 221e291 were all widely adopted, and the second one is relatively less popular, due to the fact that the manufacture of the special MEA and flow field plate is complex and great care is needed to ensure proper alignment [155]. The partial electrode approach has been continuously used and improved [156e160], mainly by increasing the number of the segments to improve the resolution. Natarajan and Nguyen [159,160] segmented the CLs, GDLs and flow fields for both the cathode and anode, and therefore the cell voltage/current density for each segment can be measured and controlled separately. As mentioned earlier, the partial electrode approach generally features the ease of use and the current behaviour on the electrode surface (reaction area) can be directly measured. However, the spatial resolution is also limited by the size of the segmented electrode, and it is also important to utilize a non-segmented electrode to preserve the true operating characteristics. The current distribution mapping approach using unmodified MEA has been widely adopted due to such reasons [155]. Rather than using shunt resistors for the current distribution mapping approach [155], Wieser et al. [161] developed a technique utilizing a magnetic loop array embedded in a current collector plate for the current mapping with non-segmented electrode. Similar to [155], but utilizing small current collector pins connected to the different locations on a flow field plate and to high-resolution resistors for current measurements, the current mapping distribution approach was also used by Noponen et al. [162,163], and similar measurements were also done by Brett et al. [164,165] and Mench and Wang [166,167]. Even though the current mapping distribution approach allows using unmodified MEA, the flow field plate still needs modification, which may still result in unrealistic operating characteristics. Developing new experimental approach and allowing accurate current distribution measurement without affecting cell performance therefore becomes the greatest target for future work. Fig. 18 shows the measured current density along the fractional distance from cathode flow channel inlet at different cell voltages [168] by using the experimental methods developed in [166,167]. For measuring the current density shown in Fig. 18, the cell was operating at 80  C and the inlet relative humidities are 100% and 50% for anode and cathode corresponding to the operating temperature, respectively; and the stoichiometry ratios are 2 and 1.5 for supplying air and hydrogen corresponding to the operating current density, respectively. It can be noticed that when the cell is operating at a high voltage (0.8 V), the variation of current density is insignificant; and for the intermediate and low cell voltages (0.65 and 0.5 V), the current density is higher and the variations are more significant. The variation of current density is caused by the combined effects of reactant consumption, membrane hydration/ dehydration and water production. Therefore investigating the membrane hydration level and water concentration by measuring the HFR and species concentration distribution is needed to interpret the measured current density distribution. 4.2. High frequency resistance distribution measurement By using the experimental approaches for measuring current distribution described in Section 4.1, HFR distribution can be measured simultaneously [154,157,162,163,165,168], mainly by using electrochemical impedance spectroscopy (EIS). The membrane resistance contributes most significantly to the total ohmic resistance, and the other resistances do not vary much during operation. Therefore, the HFR distribution measurement can be used to estimate the membrane hydration level at different locations and to explain the measured current distribution data. Fig. 19 shows the measured HFR along the fractional distance from cathode flow channel inlet at different cell voltages [168]. In fact, the HFR distribution shown in this figure was measured simultaneously with the current distribution measurement in Fig. 18. The two figures together show that at most of the locations with low HFRs (high membrane hydration level), the corresponding current densities are high, and vice versa. Fig. 19 also shows that the locations with the lowest HFRs (highest membrane hydration level) are at the inlet and outlet, due to the humidification of the supplied reactants (hydrating the membrane most significantly at inlet) and the accumulation of product water (hydrating the membrane most significantly at outlet). In addition, the corresponding HFR increment to the current density drop at the outlet in Fig. 18 cannot be observed in Fig. 19, suggesting that the current density drop at the outlet is perhaps due to the concentration or mass transport loss. Therefore, measuring the gas species concentration distribution can help understand the results shown in Figs. 18 and 19 more clearly. 4.3. Gas species concentration measurement Measuring the distribution of reactants provides the information about reactant delivery, and measuring the water vapour distribution helps understand the measured HFR distribution (related to membrane hydration/dehydration). The water amount 0.5 0.14 Water V apour Mole Fraction Hig h Frequency Resistance, Ω cm 2 Cell voltage = 0.8 V Cell voltage = 0.65 V Cell voltage = 0.5 V 0.4 0.3 0.2 Cell voltage = 0.8 V Cell voltage = 0.65 V Cell voltage = 0.5 V 0.1 0 0.12 0.1 0.08 0.06 0 0.2 0.4 0.6 0.8 1 Fractional Distance from Cathode Inlet Fig. 19. Measured HFR along fractional distance from cathode flow channel inlet at different cell voltages [168]. 0 0.2 0.4 0.6 0.8 1 Fractional Distance from Cathode Inlet Fig. 20. Measured water vapour mole fraction in cathode flow channel along fractional distance from inlet at different cell voltages [168]. 246 K. Jiao, X. Li / Progress in Energy and Combustion Science 37 (2011) 221e291 was first measured at the flow channel outlet by collecting liquid water and condensing water vapour [169,170]. Knowing the water supply and removal rates at the inlets and outlets on both sides, the rate of membrane hydration/dehydration and the water transfer rate through membrane can be estimated. Measuring the gas species concentrations at different locations of PEMFC was first carried out by Mench et al. [171]. In the measurement of [171], gas sampling ports were placed at different locations along anode and cathode flow channels, and a micro gas chromatograph (GC) system was used to measure the gas species concentrations corresponding to the gas sampling ports at different locations. More advanced experimental measurements to simultaneously measure current, HFR and gas species concentration distributions have also been developed [168,172,173], and the simultaneously measured results are more valuable and can be more easily understood than a single set of measurement data. Corresponding to the same cell and operating condition for Figs. 18 and 19, Fig. 20 shows the measured water vapour mole fraction in cathode flow channel along the fractional distance from inlet at different cell voltages [168]. It can be noticed that due to the water production and EOD causing water migrating from anode to cathode, the water vapour mole fraction increases along cathode flow channel from inlet. The increasing water vapour mole fraction along cathode flow channel in Fig. 20 and the dramatically changing HFR in Fig. 19 do not agree with each other well, because the water mole fraction in anode perhaps decreases along flow direction, indicating that simultaneous measurements of gas species concentrations on both sides are desirable. 4.4. Temperature distribution measurement Water phase change, membrane hydration/dehydration and electrochemical reaction kinetics are significantly affected by local temperature in PEMFC. Therefore, measuring temperature distribution provides valuable information for thermal and water management. Temperature distribution can be measured by using infrared (IR) cameras [158,174,175] (the thermography technique). Wang et al. [174] designed a PEMFC with an optical window on the anode side allowing IR light. Two-dimensional temperature distribution on the MEA surface was obtained under different operating conditions. Shimoi [175] applied the thermography technique to an operating test cell in a manner similar to [174] as well. The thermography technique was also applied simultaneously with other measurements such as current distribution and liquid 80 Anode Cathode o Temperature, C 75 70 65 60 0 0.2 0.4 0.6 0.8 1 Fractional Location Along Flow Channel Fig. 21. Measured temperature in anode and cathode flow channels along flow direction [181]. water visualization (the optical window allows both the temperature measurement and liquid water visualization) [158]. The main drawback of thermography technique is that it requires major modifications to cell design and component material due to the requirement of optical window. Another way to measure temperature distribution is inserting micro-thermocouples at different locations of PEMFC. Mench et al. [176] measured the temperature distribution at different positions in a MEA by embedding eight micro-thermocouples. Such measurement is not easy to conduct due to the PEMFC configuration, and it is also difficult to prevent the destruction of the thermocouple when clamping the cell. Vie and Kjelstrup [177] measured the temperature profile in the MEA of a PEMFC by using micro-thermocouples. It was shown that the temperature gradient across the MEA surface is not negligible. The measurement in [178] placed micro-thermocouples in the lands (ribs) of bipolar plate (BPs) in direct contact to GDL surface along the flow direction for both the anode and cathode. Similarly, temperature distribution measurements in PEMFC by using micro-thermocouples were also carried out in [179,180]. Recently, Alaefour et al. [181,182] conducted non-destructive temperature distribution measurements for a PEMFC with straightparallel flow channel design. 23 micro-thermocouples were embedded in the arrays of blind holes along the flow channels and lands (ribs). Temperature distributions have been obtained for two principle directions: parallel and normal to the direction of flow channel for both the anode and cathode. The obtained results clearly indicated that the temperature distribution inside PEMFC is very sensitive to operating current density. Almost uniform temperature distribution inside PEMFC was observed at low current densities, and the temperature variations were considerable at high current densities. Fig. 21 shows the measured temperature in anode and cathode flow channels along the flow direction [181] for the cell operating at 0.6 V. It can be noticed that the highest temperature locations are close to the middle along the flow direction, possible explanation is that the membrane is well hydrated due to product water accumulation there and the reactant concentrations are still sufficiently high. Simultaneous measurements of current, HFR, gas species and temperature distributions have not been conducted yet to the best of the authors’ knowledge, which are expected to provide more valuable information for thermal and water management. 4.5. Water visualization Since water management is one of the most important issues for PEMFC, investigation of detailed water behaviours inside PEMFC is therefore important. Experimental methods for investigating water behaviours include: direct imaging on liquid water in transparent PEMFCs [183e189], neutron radiography/tomography [190e198] and X-ray micro-tomography [199,200]. Similar to the temperature distribution measurement using IR cameras mentioned in the previous subsection, optical window is also required for direct imaging on liquid water in PEMFC. ChargeCoupled Device (CCD) camera is often used to capture detailed liquid water movement through optical window. Tuber et al. [183] visualized liquid water transport in cathode flow channels of a transparent PEMFC, and it was found that the air stoichiometry ratio, temperature, inlet air relative humidity and GDL property all have non-negligible influence on liquid water transport. Not only focusing on liquid water transport in cathode flow channel, the water emerging process from cathode GDL surface was visualized by Wang and co-workers [184,185], it was reported that water droplets emerging from the cathode GDL surface only appear at preferential locations, and can grow to a size comparable to the flow channel dimension under the influence of surface adhesion. Rather than only focusing on the liquid water transport in cathode, K. Jiao, X. Li / Progress in Energy and Combustion Science 37 (2011) 221e291 Spernjak et al. [186] and Ge and Wang [187] also investigated liquid water transport in anode flow channel, the visualizations in [186,187] all confirmed that the anode and cathode electrode wettabilities have significant impacts on the presence of liquid water in anode (i.e. increasing the hydrophobicity of cathode electrode and hydrophilicity of anode electrode results in more liquid water in anode due to the enhanced water transfer from cathode to anode through membrane). Such observation explains the reason that using hydrophobic MPL between cathode CL and GDL could result in better cell performance since membrane can be better hydrated. The effects of various operating conditions on liquid water transport in cathode flow channel have also been studied by Hussaini and Wang [188]. In fact, building transparent PEMFC and direct imaging on liquid water is the only available experimental method to capture the detailed liquid water flow behaviours (the neutron and X-ray methods to be discussed later on this subsection cannot distinguish the state of water). However, most of the transparent materials are not electrically conductive: if the BP is purely transparent, the cell may not work because electrons cannot be transferred. Therefore, present transparent PEMFCs still use electrically conductive materials as the land to contact the GDL directly, and the only visible place is inside the flow channel, this is why the experimental investigations reported in [183e188] all showed real-time liquid water behaviours in flow channels but they all neglected the cross flow under the land area. Park and Li [201] conducted both numerical and experimental investigations on the cross flow through the GDL, and they reported that the pressure drop could be reduced by up to 80% due to the cross flow through the GDL under the land area. Therefore, experimental investigations of detailed water transport behaviours caused by such cross flow under the land are necessary for better water management of PEMFCs. Based on this premise, a transparent PEMFC with serpentine flow channel 247 design and with both the optical land and flow channel was build by Jiao et al. [189], promising liquid water visualization in both the flow channel and under the land. The GDL thickness was carefully controlled by inserting metal shims with different thicknesses in parallel with GDL. Fig. 22 shows the visualization of liquid water transport in both the flow channel (Fig. 22a) and under the land (Fig. 22b) [189]. The cell was initially flooded with liquid water, and then the water removal characteristics were investigated. Fig. 22a shows that liquid water films sticking on flow channel walls are hard to be removed, and Fig. 22b shows liquid water flowing under the land. The visualization in Fig. 22b confirms that the cross flow under land plays an important role in liquid water removal. Neutron method relies on the nature of neutron beam: it could detect organic hydrogen-containing substances, and this feature is suitable for PEMFCs since water is the only substance that could be detected. However, it is difficult to use neutron method to distinguish between liquid water and vapour. In addition, for neutron radiography method [190e195], the through-plane location of water (to indicate water in cathode, anode, membrane, CL, GDL, flow channel) is also difficult to be determined since only two-dimensional images for the two in-plane directions can be obtained. The neutron imaging results from Geiger et al. [190] and Pekula et al. [191] both showed significant water concentration in the flow channels, especially at the downstream and at the serpentine corners. Hickner et al. [192] observed that the amount of water accumulation changes dramatically with current density and increasing the reactant flow rate also facilitates water removal. Owejan et al. [193] showed neutron images for an interdigitated flow field, and they reported that water accumulation in the GDL reduced the GDL permeability significantly. Zhang et al. [194] reported that GDL properties such as wettability, porosity etc. affect the characteristics of water removal significantly. With a special design of serpentine flow channels on both sides, the Fig. 22. Visualization of liquid water transport (a) in the flow channel and (b) under the land of a transparent PEMFC with serpentine flow channel design [189]. 248 K. Jiao, X. Li / Progress in Energy and Combustion Science 37 (2011) 221e291 Fig. 23. Neutron images at different current densities with 3 min duration for each load (a: current density is increased up to 0.75 A cm 1 A cm 2) [195]. neutron imaging conducted by Park et al. [195] was able to distinguish the water in anode and cathode flow channels, and the dynamic response of PEMFC together with water distribution was analyzed. Satija et al. [196] used neutron tomography method to reconstruct three-dimensional water distribution for an inactive PEMFC, however, real-time imaging was only obtained in twodimensional by using neutron radiography method. Hickner et al. 2 ; b: current density is increased up to [197] visualized the water profile on a cross section of an operating PEMFC by using neutron radiography method. At different operating current densities, the water amounts in MEA and flow channels were estimated based on the operating conditions. Based on neutron imaging method, Turhan et al. [198] analyzed the throughplane liquid water accumulation, distribution and transport in different components of a PEMFC with different levels of channel K. Jiao, X. Li / Progress in Energy and Combustion Science 37 (2011) 221e291 a 5. Overview of numerical models 1 Increasing current density Decreasing current density Cell V oltag e, V 0.9 0.8 0.7 0.6 0.5 0.4 0 0.2 0.4 0.6 0.8 1 -2 Current Density, A cm b 1 Increasing current density Decreasing current density 0.9 Cell V oltag e, V 249 0.8 0.7 0.6 0.5 0.4 0 0.2 0.4 0.6 0.8 1 Current Density, A cm-2 Fig. 24. Dynamic cell performance corresponding to the test conditions of Fig. 23 (a: current density is increased up to 0.75 A cm 2; b: current density is increased up to 1 A cm 2) [195]. wall hydrophobicity. X-ray method can also be used to detect water [199,200], however, X-ray method is relatively unstable because the X-ray beam is easy to be scattered and absorbed by electrons, and this feature makes X-ray method less popular than Neutron method. Fig. 23 [195] shows the neutron images at different current densities with 3 min duration for each load, it can be noticed from Fig. 23 that more water is accumulated at the serpentine corners, which agrees with the liquid water visualization shown in Fig. 22a [189]. The water distributions at the same current density when increasing and decreasing the load are also different, corresponding to the test conditions of Fig. 23, the cell dynamic performance shown in Fig. 24 indicates different cell voltages at the same current density when increasing and decreasing the load. The different cell voltages are caused by the dynamic liquid water transport blocking the reaction sites, indicating that effective control of liquid water in PEMFC is essential to achieve consistent cell performance. 4.6. Summary Experimental observations on the distributions of current, HFR, gas species, temperature and water are reviewed in Section 4. The distributions of many parameters can be obtained based on the presently available experimental techniques, and simultaneously measuring more parameters with minimum modification of cell and system design is the primary target of future experimental observations. As reviewed in Section 4, many experimental studies have been conducted to observe the various transport phenomena in PEMFC. However, due to the drawbacks such as difficulty to perform the different experimental measurements simultaneously, unrealistic operating conditions due to the modified cell and system designs for experimental measurements, and high cost for materials and testing instruments, numerical modeling of PEMFC is therefore critical for better understanding of transport phenomena in PEMFC. Up to early 2000s, excellent reviews on the modeling work for PEMFC mainly focusing on the macroscopic (top-down) first-principle-based models can be found in [143,202], and on the researches related to the proton conductor of PEMFC (e.g. Nafion membrane) in [100,203]. In this paper, both the first-principlebased (from atomistic to full cell levels) and rule-based models are comprehensively reviewed. The first-principle-based models rely on solving a set of governing partial differential equations including both the top-down (e.g. solving continuity, NaviereStokes and other equations) and bottom-up (e.g. solving Boltzmann equations) approaches, and the rule-based models depend on applying physical rules to simplified or real physical structures. The first-principle-based models are categorized into three levels in this paper based on the characteristic length and numerical methods: microscale, mesoscale and macroscale, as shown in Fig. 25 and detailed in Section 5.1. This paper therefore provides a comprehensive review on most of the PEMFC related models on all the levels of scale. In this section, the numerical models, including both the first-principle-based and rule-based models, are summarized to guide the detailed discussions on the different models in Sections 6e12. A summary of the level of scale for the first-principle-based models is given in Section 5.1, followed by a discussion on the model development history in Section 5.2. 5.1. Level of scale Most of the previously developed PEMFC models are first-principle-based, from atomistic to full cell levels. The number of rulebased model is much less, and such models mainly focused on estimating liquid water transport in PEMFC electrode by applying physical rules to simplified or real electrode micro-structures. Therefore, in this paper, only the first-principle-based models are classified into different levels of scale, which are microscale, mesoscale and macroscale. Fig. 25 illustrates the levels of scale for first-principlebased PEMFC models with representing phenomena and numerical methods. The first-principle-based models depend on either the topdown or bottom-up approach. The top-down approach relies on solving a set of governing partial differential equations essentially based on the continuum assumption, and these partial differential equations include continuity, NaviereStokes and/or other equations for conservations of the macroscopic properties such as mass, momentum, energy and so on. The bottom-up approach includes the molecular dynamics (MD), off-lattice pseudo particle (such as dissipative pseudo particle and Monte Carlo (MC)), lattice gas (LG) and lattice Boltzmann (LB) methods, by solving the partial differential equations for the motions of a set of single molecules (or atoms) or a set of pseudo particles (the pseudo particle represents a group of molecules). The name of “top-down” is obtained because this approach relies on the solutions of the macroscopic properties, and the solutions for the motions of molecules/atoms or pseudo particles pursue “bottom-up” strategies, so-called the bottom-up approach. The top-down approach (the continuum method shown in Fig. 25) can only be applied for small Knudsen number regimes (on the macroscale defined in this paper, as shown in Fig. 25). By neglecting the real micro-structures of GDL, CL and membrane, the 250 K. Jiao, X. Li / Progress in Energy and Combustion Science 37 (2011) 221e291 Fig. 25. Level of scale for PEMFC modeling with representing phenomena and numerical methods (only first-principle-based numerical methods are shown). K. Jiao, X. Li / Progress in Energy and Combustion Science 37 (2011) 221e291 top-down approach can be applied to all the cell components by considering modeled effective transport properties (such as the effective diffusion coefficient in Equation (30)) and/or other assumptions (such as homogeneous material of GDL, CL and membrane, or assuming simplified CL and membrane structures such as the agglomerate models). In fact, most of the previously developed models rely on using the effective transport properties and assuming homogeneous materials, therefore all the cell components can be accounted for simultaneously, allowing full cell modeling. By considering the real micro-structures of GDL, CL and membrane, as analyzed in Section 3.3, the small pores on the level of nanometre in CL and membrane prohibit the use of the topdown approach, while the top-down approach can still be used in GDL with the pores on the level of micrometre. For the bottom-up approaches, the MD and pseudo particle simulations can be used to investigate the transport phenomena and material self-organization in membrane and CL on the microscale level (as defined in this paper and shown in Fig. 25), which provide the fundamental understanding on the transport mechanisms to guide the analysis on the grander scales. The office-lattice pseudo particle, LG and LB methods solve the motions of pseudo particles which include a number of real molecules, therefore the computational time can be saved and the simulations can be performed on a larger scale (e.g. the mesoscale defined in this paper, as shown in Fig. 25). The current computational power also allows the LG and LB simulations performed on the macroscale. In fact, the LG and LB methods allow the micro-structure to be more easily considered than the continuum method (the top-down approach), and the LB method therefore has been used to investigate the liquid water transport in the pores of GDL, CL and membrane and in flow channel. Hence the LG and LB methods can be considered as the scale-bridging method which can be used to capture the transport phenomena on the multi-scales. Detailed discussions on the various numerical models will be given in Sections 6e12 with their applications to different components of PEMFC. 5.2. History of model development The numerical models developed by Springer et al. [72,204] and Bernardi and Verbrugge [109,205] in early 1990s are usually referred to as the pioneering modeling work for PEMFC. These models are essentially one-dimensional models considering the membrane, CL and GDL based on the continuum (top-down) approach by solving the conservation equations by assuming homogeneous materials and using effective transport properties. The water diffusion through the membrane and the effects of water content on the membrane conductivity are accounted for by Springer et al. [72,204], while Bernardi and Verbrugge [109,205] assumed constant water content (fully hydrated) across the membrane. Effective transport properties are used such as the effective diffusion coefficient by using the Bruggeman correlation. In these one-dimensional models [72,109,204,205], the fundamental framework and most of the fundamental formulations for PEMFC modeling based on the continuum (top-down) approach are established and have been widely used in many of the later numerical studies. Following [72,109,204,205], Nguyen and White [206] and Fuller and Newman [207] developed pseudo two-dimensional models by further considering the flow channel with the along-the-channel direction, which considers the effect of inlet water humidity and temperature distributions, providing more detailed water and thermal management capability. After that, more models which are similar to [72,109,204e207] have also been developed [208e210]. As interest grew in fuel cells in late 1990s, more and more numerical models were developed. Yi and Nguyen [211,212] and Gurau et al. [213] all developed two-dimensional models to explore 251 more detailed transport phenomena in PEMFC, these models illustrated the utility of multi-dimensional models in the understanding of the internal conditions of PEMFC, such as the reactant and water distributions. Based on the continuum (top-down) approach, simplified CL structures were proposed by assuming that the ionomer and platinum/carbon particles form large agglomerates on the level of micrometre, so-called the agglomerate models, which had been developed by Gloaguen and Durand [214], Bultel et al. [215e217] and Marr and Li [218]. Investigations on the water and proton transport through the membrane have also been carried out based on the continuum (top-down) approach [219e223]. Not only on the macroscale, physical models with assumed randomly distributed mesoscale pores were studied to relate the water content and membrane conductivity [73], and MD simulations [224,225] were also conducted to study the self-organization of membrane. In 2000s, multi-dimensional models based on the continuum (top-down) approach and solving a complete set of conservation equations (continuity, NaviereStokes, Energy and so on) coupled with electrochemical reactions, were developed by many researchers. Computational fluid dynamics (CFD) code (such as the commercial code Fluent, Star-CD, CFX, CFD-ACEþ through their user coding capability) based on finite volume or finite element methods were modified and used to develop such models, and more complex geometry and transport phenomena were able to be investigated. With three-dimensional geometry considered, the models of Dutta et al. [226,227], Zhou and Liu [228], Berning et al. [229], Mazumder and Cole [230], Lee et al. [231], Um and Wang [232] and Wang and Wang [233] are considered as the pioneering work in this field, these models mainly considered a single flow channel with the major components (e.g. flow channel, GDL, CL, membrane, and some with BP), and some of them accounted for the transient calculation as well, such as in [233]. Large scale simulations considering multi-channel or small stacks were also carried out [234e239]. Note that the multi-dimensional models in [211e213,226e239] neglected liquid water formation by only solving a single equation for vapour and liquid water, therefore super saturated water vapour was observed, and some of the models calculated the amount of liquid water during the post processing (e.g. [226,227,231]). Real two-phase models have also been developed, in which vapour and liquid water move at different velocities, and the other liquid water effects were accounted for, such as the surface tension and water flooding effects. Generally, two major models, two-fluid model and mixture model, were widely used and considered as the real two-phase models. The two-fluid model is mainly attributed to the work of Nguyen and co-workers [80,134,138,240,241], Djilali and coworkers [242e244], Mazumber and Cole [245] and Wu et al. [137,246]; and the mixture model was mainly developed by Wang and co-workers [247e255] and You and Liu [256,257], Note that here “single-phase” and “two-phase” represent the state of water only in pores of GDL and CL and in flow channel, therefore “singlephase” indicates that only water vapour is considered in these regions, and “two-phase” means that both the vapour and liquid water are accounted for in these regions. However, the state of water in the ionomer of the membrane and CL is different (as discussed in Section 2), and a separate equation was used to solve the water transport in ionomer (e.g. [80,137,246], the treatment of water/proton transport in membrane will be detailed in Section 7). Perhaps “multiphase” is more accurate in presenting the state of water. The two-fluid model solves the mass, momentum and species transport conservation equations for the gas mixture, with an extra liquid water transport conversation equation; and the mixture model solves the mass and momentum and species transport conservation equations for the two-phase mixture (mixture of liquid water and gas) mainly based on the mass- 252 K. Jiao, X. Li / Progress in Energy and Combustion Science 37 (2011) 221e291 averaged properties of the two-phase mixture. Both models are able to predict the velocities for both the vapour and liquid water. In consideration with phase change of water, the mixture model essentially accounts for an infinitely large (instantaneous) phase change process, while the phase change process in the two-fluid model can be modeled at different rates (constant phase transfer rates were assumed in previous studies). Detailed discussions on the multiphase multi-dimensional models for PEMFC based on continuum (top-down) approach are given in Section 6. Recently, multiphase multi-dimensional modeling of PEMFC starting from subzero temperatures (cold start) was also carried out, by Wang and co-workers based on the mixture model framework [258e260] and by Jiao and Li based on the two-fluid model framework [62,63,261]. In the work of [258e260], liquid water formation was not considered therefore the simulations were limited at subzero temperatures, and the water freezing in ionomer was not accounted for as well. Both the ice and liquid water formations, as well as water freezing/melting processes in ionomer were all included in the models in [62,63,261], therefore these models provided the capability for simulations in the whole temperature range (from subzero to normal operating temperatures). The two-fluid and mixture models can only be used to estimate the two-phase concentrations (e.g. the volume fractions of liquid and gas phases), and the detailed liquid water transport behaviours cannot be investigated because these models do not allow interface tracking between liquid water and gas. Therefore, in 2000s, rather than focusing on increasing the size of the computational geometry, numerical models focusing on more detailed transport characteristics were also developed. Based on the volume-of-fluid (VOF) model, the liquid water dynamics in a single serpentine flow channel was investigated by Quan et al. [262], and in small PEMFC stacks with straight-parallel and serpentine-parallel flow channels by Jiao et al. [11,12]; and the effects of surface wettability of flow channel on liquid water behaviours were also investigated [263,264]. The water transport behaviours in both the simplified [265e267] and real [268] micro-structures of GDL were investigated by Jiao and co-workers by using the VOF model, in which the effects of surface wettability were investigated as well [266,268]. The development of agglomerate models considering simplified structures of CL was also continued in 2000s [269e271] based on continuum (top-down) approach. Models based on bottom-up approach were also developed in 2000s for investigating liquid water transport in GDL and CL. With micro-structures of GDL and CL, LB method was used to simulate the air flow to obtain the GDL properties (such as permeability) [272,273], and to simulate liquid water transport in GDL and CL [274e277]. Construction of realistic micro-structures of GDL and CL is an essential prerequisite for such simulations, and this can be achieved by either 2D or 3D imaging (such as by using X-ray and magnetic resonance microtomography) or stochastic models (generating micro-structures by computational simulations). Rather than focusing on first-principlebased models, rule-based models were also developed, such as full morphology (FM) models [274,275,278] and pore-network (PN) models [274,279] mainly for investigating liquid water behaviours in GDL (such as liquid water effects on capillary pressure). Models based on bottom-up approach on microscale and mesoscale were also developed in 2000s, mainly focused on the understanding of water and proton transport and material self-organization. MD based on quantum mechanical [78,280,281] and classical [282e290] theories and pseudo particle [291,292] simulations were conducted for Nafion membrane. Quantum mechanical MD and MC simulations have been performed to study the elementary reaction processes on catalyst surfaces [293e298]. Mesoscale simulations to evaluate key factors for CL fabrication by investigating material self-organization (sizes of platinum/carbon/ionomer agglomerates and pores) were also carried out [299]. 5.3. Summary Section 5 provides an overview of the numerical models for PEMFC to guide the detailed discussions in the following sections. The review shows that numerical models based on different principles and physical rules on all the levels of scale have been extensively developed. In the following sections, full cell models based on continuum (top-down) approach are described in Section 6 to provide a complete view for the various transport processes, followed by Sections 7e12 focusing on a specific cell component or operating condition in each section. 6. Multi-dimensional multi-component multiphase model with full cell geometry With the currently available computational power, continuum (top-down) approach by considering homogeneous materials of GDL, CL and membrane with modeled effective transport properties is perhaps the only way to model a full PEMFC (including major or all cell components). By solving a number of conservation equations in a multi-dimensional computational domain, the multi-component multiphase transport with electrochemical reactions and electron/ proton transport has been successfully modeled in the past decade, and as reviewed in Section 5, the two major types of such models are the two-fluid model and mixture model. The names of two-fluid and mixture are obtained based on the modeling approaches of water (vapour and liquid) transport in the pores of GDL and CL and in flow channel. However, different modeling approaches of water transport in ionomer have also been used and can be classified into three groups, namely, the hydraulic (or convective) model [74,205,229], diffusive model [72,300e302], and chemical potential model [75,149,303e307]. The hydraulic model assumes a fully humidified membrane therefore the major water transport mechanism becomes convection (because the pores in membrane is enlarged by water). However, in an operating PEMFC the ionomer close to anode usually dries out quickly and water is produced in cathode, resulting in uneven water distribution, therefore the hydraulic model has only been used for the early days’ modeling work, and is rarely considered nowadays. The diffusive model accounts for diffusive water transport in ionomer by incorporating with experimentally measured diffusion coefficient of water. Therefore unevenly distributed water can be accounted for in the diffusive model, which models a more realistic condition. In the diffusive/hydraulic models, the proton concentration is assumed constant over the ionomer domain and therefore the proton transport can be easily solved by using Ohm’s law (water content only affects proton conductivity of ionomer). In the chemical potential model, however, the concentration of mobile protons is assumed to vary with the water concentration. Utilizing the dusty fluid model [149,303], generalized StefaneMaxwell equations [305e307], or concentrated solution theory [75,304], the water and proton transport are strongly coupled and solved simultaneously. The chemical potential model might be considered as a superclass of diffusive/hydraulic models; the diffusive/hydraulic models are only valid in certain situations (constant proton concentration), while the chemical potential model is a more comprehensive approach which applies to a much larger range. Nevertheless, the present chemical potential models are invariably confined to the membrane region along with many simplifications. Further, several parameters and correlations related to this model class remain unknown, such as the diffusion coefficient of hydrogenewater ions (e.g. hydronium), the interaction properties of water and proton, etc. Hence, the application of the chemical potential model in full cell modeling needs to be explored further. Therefore, most of the two-fluid and mixture models still assume constant proton concentration in membrane, and the diffusive model is most widely used. The EOD causing water migrating from anode to K. Jiao, X. Li / Progress in Energy and Combustion Science 37 (2011) 221e291 cathode, as well as the convective transport of water through membrane (usually can be neglected but might be considered when the cell is pressurized differentially) can also be accounted for by adding extra terms on the water transport equation. The details about modeling water transport in membrane are given in Section 7. For water transport in pores of CL and GDL and in flow channel, three models based on continuum (top-down) approach have been used, which are the two-fluid model, mixture model and VOF model. The two-fluid model solves individual sets of equations for each phase while the interaction among different phases is explicitly taken into account through limited phase transfer terms. The mixture model is a kind of single-fluid model. It solves a single set of conservation equations for the phase mixture assuming phase equilibrium, and the volume fractions of the phases, as well as the relative velocity among different phases are estimated subsequently. In PEMFC modeling, the two-fluid and mixture models are usually simplified by combining the momentum and other equations with the help of Darcy’s law and a capillary pressure function. Power law relations and Leverett J-functions were widely used to calculate capillary pressure (e.g. [247,308]), and recent studies [129e133] also provide alternative correlations. Compared to the mixture model, the advantage of the two-fluid model is that only one extra equation for liquid saturation is added, 253 while allowing for the simulation of non-equilibrium phase transfer processes. Rather than using the two-fluid and mixture models, two-phase water transport in pores of CL and GDL and in flow channel can also be modeled by using the VOF model. The biggest advantage of the VOF model is its ability to trace the trajectory of the liquid water movement. However, due to the nature of the extremely small time-steps and intensive computing time related to VOF methods, its application so far has been restricted to investigating the liquid behaviour in the electrode [265e268] or flow channels [11,12,262e264]. Full geometry PEMFC models that incorporate the VOF approach have also been conducted [309,310], but only with very limited time instances considered due to the computational power limitation. It should be noticed that the transport in real micro-structure of CL cannot be model by any topdown models, due to the presence of nanometre pores. In summary, the “chemical potential þ VOF” approaches may finally evolve to be the main features of the next generation of PEMFC models. At the current stage, however, the “diffusive þ twofluid/mixture” type models still dominate. Therefore, in this section, only the two-fluid and mixture model are described, all with diffusive model for water transport in ionomer. The computational domain required for such full cell models is first introduced. Then the two-fluid model and mixture model are introduced one by one, followed by the specifications of boundary condition and numerical implementation. The comparison between the two models is given as well with representing simulation results. 6.1. Computational domain The requirements of computational domain for the different full cell models are the same. Fig. 26 shows the sample computational domain and mesh including all major cell components (BP, flow channel, GDL, CL and membrane) for a single straight PEMFC. It can be noticed that the micro-structures of GDL, CL and membrane are all neglected and they are considered as homogeneous layers. Therefore, structured mesh can be applied to the whole computational domain, which improves both the computational accuracy and efficiency. Usually 10  10 layers of grid are needed on the cross sections of each layer normal to the flow direction, and the number of grid along the flow direction depends on the flow channel length/ geometry and other conditions (e.g. 100 layers of grid along the flow channel direction is usually considered sufficient for a single straight PEMFC). In fact, the number of grid layers along the through-plane direction (normal to membrane surface) cannot be too high in CL, due to the fact that the CL is very thin (usually around 0.01 mm), which may cause too large aspect ratios of the computational cell and lead to computational instability. The number of grid cell for the single straight PEMFC shown in Fig. 26b is 76,000, such computational domain allows both the steady and unsteady simulations on a single computer. For multi-channel and stack simulations, parallel computing is an effective way to handle large number of grid cell. For example, using 24 processors in parallel is considered sufficient for the number of grid cell on the level of 106. 6.2. Two-fluid model The two-fluid model presented in this section can be concisely summarized by the following conservation equations [80,134,137, 138,240e246]. Mass of gas mixture (solved in flow channel, GDL and CL): Fig. 26. Sample (a) computational domain and (b) mesh including all major cell components for full cell modeling. v

3 1 vt  
 ! slq rg þ V$ rg u g ¼ Sm (61) Momentum of gas mixture (solved in flow channel, GDL and CL): 254 K. Jiao, X. Li / Progress in Energy and Combustion Science 37 (2011) 221e291 Table 3 Source terms in the conservation equations (Equations (61)e(68)) except those for water (for two-fluid model) [137]. Domain Sm Su Si Sele Sion ST BP Flow channel GDL 0 Svp Svp 0 0 0 0 0 0 0 0 0 0 0 kVfele k2 keff ele Spc kVfele k2 keff þ Spc ele ja ja jhact j þ kVfele k2 keff þ kVfion k2 keff ion þ Spc ele mg ! Kg Anode CL Cathode CL Kg ug SH2 ¼ mg ! SO2 þ Svp Membrane ug mg ! SH2 þ Svp Kg 0 ug SO2 ¼ 0 ja M 2F H2 jc M 4F O2 0 ! ! ! rg ! rg ! ug ug ug v
 þ V$
2 vt 3 1 s lq 32 1 slq ! !! !T ! ug ug  þV
 ¼ Vpg þ mg V$ V
3 1 slq 3 1 slq !! ! u 2 m V V$
g  þ Su 3 g 3 1 s (62) Gas species (solved in flow channel, GDL and CL, i represents hydrogen, oxygen or vapour):  

 ! slq rg Yi þ V$ rg u g Yi ¼ V$ rg Deff i VYi þ Si (63) Liquid water (solved in flow channel, GDL, CL):
v 3slq rlq vt  

! þ V$ irlq u g ¼ V$ rlq Dlq Vslq þ Slq (64) Non-frozen membrane water (the total amount of water in membrane for normal operating conditions) (solved in membrane and CL):
rmem v ulnf EW vt  ¼ rmem EW 
V$ Deff nmw Vlnf þ Snmw (65) Energy (solved in whole computational domain): v
vt rCp eff T fl;sl  þ V$
rCp ! u eff T fl 
 ¼ V$ keff fl;sl VT þ ST (66) Electronic potential (solved in CL, GDL, BP): 
0 ¼ V$ keff ele Vfele þ Sele (67) 
0 ¼ V$ keff ion Vfion þ Sion (68) Ionic potential (solved in CL, membrane): The above equations are closely coupled through the right hand side source terms, which either stem from the electrochemical Table 4 Source terms in the conservation equations (Equations (61)e(68)) for water (for two-fluid model) [137]. Domain Sv-l Sv-l Sv l þ Sn Cathode CL Sv þ Sn Membrane Slq Snmw 0 0 v MH2 O Sv-l Sv-l Sv-l v MH2 O Sv-l jc 2F 0 Svp Flow channel GDL Anode CL 0 l 0 Sn v 
n d eff k Vf F ion ion
n  þ V$ d keff Vf F ion ion þ V$ Sn v jc jc 0 lq v

3 1 vt ja 0 jc T D S þ jc jhact j þ kVfele k2 keff þ kVfion k2 keff ion þ Spc ele 2F 2 eff kVfion k kion reactions or from the interfacial mass transfer among different phases. The expressions of these source terms have been summarized in Tables 3 and 4. The related mass transfer functions accounting for the water phase change and membrane absorption/ desorption processes are described in Section 3, and therefore are not repeated in this section. Most of the constitutive and empirical formulas for transport properties, mass transfer rate and other parameters for closing the conservation equations have also been given in Sections 2 and 3, and therefore only those important and not previously mentioned are described in this section. It should be noticed that superficial velocity is used and all the gas species are assumed to be ideal gas for the conservations equations shown above (Equations (61)e(68)). By assuming incompressible flow, the viscous force terms in the momentum equations for gas mixture (Equations (62)) can be further simplified. Strictly speaking, the diffusion terms in the gas species conservation equations (Equation (63)) are only valid for binary diffusion (when only two gas species are present). For PEMFC without impurities (e.g. CO), the only gases in anode are hydrogen and water vapour, which is acceptable; however, oxygen, water vapour and nitrogen are all in gas phase in cathode when air is supplied, which requires StefaneMaxwell formulation for the diffusion terms. However, most of the previous full cell models used simplified binary (Fick’s law) diffusion formulation (the fraction of nitrogen is only estimated from the fractions of oxygen and water vapour because the total fraction is 1). The gas mixture pressure and velocity can be solved through its mass and momentum equations (Equations (61) and (62)), and the liquid water pressure, velocity and diffusion coefficient are derived from the capillary pressure and gas mixture pressure and velocity, as described in Section 3.2 and shown in Equations (42)e(45). Therefore, only one extra equation to solve the liquid water volume fraction is needed to account for liquid water transport. It should be noticed that this approach still needs better correlations to relate the gas and liquid water pressures and velocities to better predict water transport in flow channel, and that is why most of the previous models neglected liquid water transport in flow channel. Previous study has assumed that the gas and liquid water velocities are the same [80] (the interfacial drag coefficient i in Equation (64) is equal to 1), which remains debated especially when large liquid water droplets are present. The effect of liquid water blockage on reaction rate in CL is usually represented by using a linear relationship applied to the ButlereVolmer equation:
ja ¼ 1 jc ¼
1 0 10:5     c 2aa F a H @ 2A h exp slq jref 0;a RT act cref H2  c  O2 slq jref 0;c ref cO2 exp exp    4aa F c hact þ exp RT  2ac F a h RT act  (69) 4ac F c h RT act  (70) 255 K. Jiao, X. Li / Progress in Energy and Combustion Science 37 (2011) 221e291 where j (A m 3) is the reaction rate, slq the liquid water volume fraction, c (kmol m 3) the concentration of gas species, a (0.5 for anode and cathode) the transfer coefficient, F (9.6487  107 C kmol 1) the Faraday’s constant, R (8314 J kmol 1 K 1) the universal gas constant, T (K) the temperature, and hact (V) the overpotential (activational voltage loss). The subscripts/superscripts a and c represent anode and cathode, respectively; and ref and 0 represent reference states. Equations (69) and (70) indicate that when CL is fully flooded by liquid water (slq ¼ 1), the reaction rate becomes 0. The water transport in ionomer is solved in a single conservation equation (Equation (65)). This equation is solved in both membrane and CL, membrane is considered to be full of ionomer (the ionomer volume fraction u is equal to 1) and CL is partially occupied by ionomer (u usually ranges from 0.2 to 0.4). The nonfrozen water diffusion coefficient in ionomer in CL therefore also needs to be corrected in considering the volume fraction of ionomer, for example, the Bruggeman correlation can be used 1:5 Deff Dnmw nmw ¼ u Deff nmw 2 Heat is generated from the electrochemical reactions (activational heat and reversible heat), electron and ion transport (ohmic heat), and water phase change (latent heat), as shown in the sources terms in Table 3. The non-frozen water in ionomer could be assumed to be equivalent to liquid water [137], therefore the latent heat for ionomer absorption/desorption with water vapour could be the same as for water condensation/evaporation, and it is zero for ionomer absorbing/desorbing liquid water. The conservations of electronic and ionic potentials are given in Equations (67) and (68), respectively. Ohm’s law is used for such processes, by assuming that the concentrations of electron and ion are constant in the electron conductive (e.g. BP, platinum and carbon powers) and proton conductive (ionomer) materials. This assumption is usually valid for electron transport but remains debated for proton transport. The interaction between proton and water in ionomer may need to be accounted for, and the details will be given in Section 7. The corresponding source terms (reaction rates) are added to the conservation equations, as shown in Table 3. It should be noticed that the transient terms are neglected in the conservation equations for the electronic potential and ionic potential, the reason can be explained with the help of Table 5. Table 5 shows an analysis of the time constants of the fundamental transient phenomena in PEMFC, which are gas transport, liquid water transport, non-frozen water transport in ionomer, electrochemical double layer charging and discharging, and heat transfer. Different components of PEMFC are used to estimate the typical values of different time constants. GDL is used to calculate the time constants of gas transport and liquid water transport because the diffusion dominated mass transport in GDL is slower than the convection dominated mass transport in flow channel. Water transport in ionomer typically occurs in membrane so that membrane is considered to calculate the time constant. Electrochemical double layer charging and discharging takes place in CL so that CL is considered. Heat transfer is typically slow in membrane due to its low heat conductivity, so that the time constant calculation for heat transfer is conducted for membrane. Based on the calculated values in Table 5, it can be noticed that the time constant of electrochemical double layer charging and discharging is much smaller than the other time constants, explaining why the transient terms can be safely neglected in the conservation equations for the electronic potential and ionic potential. (71) 1 where (m s ) is the effective non-frozen water diffusion coefficient in ionomer, and Dnmw (m2 s 1) the bulk non-frozen water diffusion coefficient in ionomer (as given in Equations (16) and (17)). It can be noticed from Table 4 that the product water is assumed to be the non-frozen water in ionomer, by adding a source term (the term jc/(2F) in Table 4) to the non-frozen membrane water conservation equation, due to the fact that water is produced at the interface of three-phase contact (ionomer, catalyst and reactants). The Vfion Þ in Table 4) water flux term due to EOD (the term V$ððnd =FÞkeff ion is therefore also added as a source term to the non-frozen membrane water conservation equation. In fact, with the present model, different water production (vapour, liquid water and water in ionomer) can be easily implemented by placing the water production term in the corresponding conservation equations. Similarly, the various water phase change processes can also be implemented by simply adding source terms to the water conservation equations, as shown in Table 4. The water phase change and ionomer absorption/ desorption rates can be controlled by adjusting the values of the source terms. Therefore, the non-equilibrium water phase change and ionomer absorption/desorption can be easily modeled. In fact, rather than solving a conservation equation for non-frozen water in ionomer, the water vapour conservation equation can be modified to account for water transport in ionomer, by assuming equilibrium ionomer absorption/desorption. This approach is introduced in Section 6.3 together with mixture model. The energy equation (Equation (66)) is solved in whole computational domain, to fully account for heat generation and transfer processes. Effective heat capacities and thermal conductivities need to be considered to include all the materials (gas mixture, liquid water, ionomer, catalyst layer and other materials). 6.3. Mixture model The mixture model can be concisely presented by the following conservation equations [247e257]. Mass of gas and liquid water mixture (solved in flow channel, GDL and CL): Table 5 Time constants of various transient phenomena in PEMFC. Phenomenon Typical value Gas transport In GDL: dGDL z200 mm; Deff g z10 Time constant Liquid water transport In GDL: dGDL z200 mm; Dlq z10 Non-frozen water transport in ionomer In membrane: dmem z50 mm; rmem z1980 kg m 3 ; Dlnf z10; Iz1 A cm 2 ; EWz1100 kg kmol 1 ; Fz96; 487 C mol Electrochemical double layer charging and discharging In CL: dCL z10 mm; CL specific area ðaÞz105 m Heat transfer In membrane: dmem z50 mm; ðrCp Þeff mem z1650 kJ m 5 m2 s d2GDL 1 Deff g 6 m2 s 1 d2GDL Dlq Electric capacity ðCÞz0:2 F m keff mem z1 W m 1 K 1 2; 1 1; keff z50 S cm 1 ; keff ion z0:1 S cm ele 3 K 1 ; 1 z0:004 s z0:04 s 2F dmem Dlnf rmem z17 s I  EW 
1 1 d2CL aC eff þ eff z0:2 ms; Too small, can be ignored. kele kion d2mem ðrCp Þeff mem keff mem z0:004 s 256 K. Jiao, X. Li / Progress in Energy and Combustion Science 37 (2011) 221e291 Table 6 Source terms in the conservation equations (Equations 66-68 and 72-74) (for mixture model) [251, 252] Domain Su Si Sele Sion ST BP Flow Channel 0 0 0 0 0 0 0 0 kVfele k2 keff ele Spc SH2 ¼ V$ðYH2 Jcap;lq Þ 0 0 kVfele k2 kele þ Spc SH2 O ¼ V$ððYvp Ylq ÞJcap;lq Þ SO2 ¼ V$ðYO2 Jcap;lq Þ 0 0 kVfele k2 kele þ Spc ja ja jhact j þ kVfele k2 keff ele m! Anode GDL K0 u m! Cathode GDL K0 u SH2 O ¼ V$ððYvp m! Anode CL K0 u SH2 ¼ eff eff Ylq ÞJcap;lq Þ ja ja 2F MH2 eff þkVfion k2 kion þ Spc þV$ðYH2 Jcap;lq Þ SH2 O ¼ K0 Ylq ÞJcap;lq Þ jc 4F MO2 þ þV$ððYvp m! Cathode CL eff V$ðnFd kion Vfion ÞMH2 O u SO2 ¼ jc jc jc T DS 2F SH2 O ¼ þ jc jhact j eff þkVfele k2 kele V$ðYO2 Jcap;lq Þ jc 2F MH2 O þkVfion k2 keff ion þ Spc þV$ðnFd keff Vfion ÞMH2 O ion Membrane þV$ððYvp 0 0 Ylq ÞJcap;lq Þ v ! ð3rÞ þ V$ðr u Þ ¼ 0 vt 0 (72) Momentum of gas and liquid water mixture (solved in flow channel, GDL and CL):  !  !! ru u v ru þ V$ ¼ vt 3 32 !! ! !T u u Vp þ mV$ V þV 3 3  ! 2 u mV V$ þ Su 3 3 (73) Species (solved in flow channel, GDL, CL and membrane, i represents hydrogen, oxygen or water (vapour and liquid water are accounted for together)): 
v
eff 3 rYi þ V$ðbr! u Yi Þ ¼ V$ rg Deff i g VYi vt i g  þ Si (74) Energy (solved in whole computational domain): can be represented by Equation (66) Electronic potential (solved in CL, GDL, BP): can be represented by Equation (67) Ionic potential (solved in CL, membrane): can be represented by Equation (68) The source terms for the above conservation equations are given in Table 6. It can be noticed that rather than solving the mass and momentum equations for gas mixture as in the two-fluid model, the mass and momentum conservations for the gas and liquid water mixture is solved in the mixture model. Therefore, the source term for the mass conservation equation is 0 (Equation (72)). For incompressible flow, the transient term for the mass conservation equation (Equation (72)) can be neglected, and the viscous stress terms in the momentum conservation equations (Equation (73)) can be simplified. The density (r, kg m 3) and dynamics viscosity (m, kg m 1 s 1) of gas and liquid water mixture are calculated based on the properties and volume fractions of gas mixture and liquid water as
r ¼ rlq slq þ rg 1 m ¼ h
slq
rlq slq þ rg 1   (75) slq i. Klq =nlq þ Kg =ng where rg (kg m 3) and rlq (kg m 3) are the densities of gas mixture and liquid water, respectively; ng (m2 s 1) and nlq (m2 s 1) are the kinematic viscosities of gas mixture and liquid water, respectively; and Kg (m2), Klq (m2) and K0 (m2) are the gas, liquid water and intrinsic permeabilities, respectively, as described in Section 3.2 and Equations (33) and (34). Rather than solving a conservation equation for liquid water transport as in the two-fluid model, the liquid water volume fraction (slq) in the mixture model is calculated as slq ¼
cH2 O rlq =MH2 O c sat (77) csat where cH2 O (kmol m 3) and csat (kmol m 3) are the vapour/liquid water mixture and saturation water (can be obtained by using saturation pressure of water by ideal gas law) concentrations, respectively; and MH2 O (18 kg kmol 1) is the molecular weight of water. As shown in the source terms of gas species in Table 6, the capillary effect on gas species transport is also accounted for, and by neglecting the effect of gravity, the capillary liquid water flux (Jcap,lq, kg m 2 s 1) can be calculated as Jcap;lq ¼ Yg Ylq m K0 (76) (78) rK0 Vpc ! and liquid water velocity ( u lq , m s 1 ) can be obtained as ! rlq ! u lq ¼ Jcap;lq þ Ylq r u (79) where pc (Pa) is the capillary pressure, as described in Section 3.2 ! and Equations (42) and (43). u (m s 1) is the two-phase mixture velocity. Yg and Ylq are the relative mobilities of gas mixture and liquid water, given as Ylq ¼
Klq =nlq  Klq =nlq þ Kg =ng and Yg ¼ 1 Ylq (80) In flow channel, it might be assumed that gas and liquid water velocities are the same (as mentioned earlier with two-fluid model), and better correlations need to be further explored. The relationship ! between the two-phase mixture velocity ( u , m s 1), gas mixture ! ! velocity ( u g , m s 1) and liquid water velocity ( u lq , m s 1) is ! ! r! u ¼ rg u g þ rlq u lq  eff kVfion k2 kion 0 (81) Rather than solving a separate conservation equation for water transport in ionomer as in the two-fluid model, the water transport in ionomer is obtained from the water (vapour and liquid mixture) 257 K. Jiao, X. Li / Progress in Energy and Combustion Science 37 (2011) 221e291 transport equation by modifying the transport properties. The eff eff effective porosities ð3eff H2 O ; 3H2 ; 3O2 Þ and effective diffusion coefficient of water vapour (Deff H2 O g, m2 s 1, the subscript g indicates that only vapour phase water is considered) are modified by considering both pores and ionomer as 3eff H2 O ¼ 3 þ u rmem RT dlequil EW psat
eff 3eff H2 ¼ 3O2 ¼ 3 1 Deff H2 O g slq (82) da 
1:5 1 ¼ Deff vp ¼ 3 þ u1:5 (83) slq 1:5 Dvp rmem RT dlequil EW psat da Dnmw ð84Þ where 3 and u are the volume fractions of pores and ionomer, respectively; rmem (kg m 3) and EW (kg kmol 1) are the dry density and equivalent weight of membrane; R (8314 J kmol 1 K 1), T (K) and psat (Pa) are the universal gas constant, temperature and water saturation pressure, respectively; lequil and a are the equilibrium water content in ionomer and water activity in pores, and dlequil/da can be obtained by differentiating Equation (12); and Dvp (m2 s 1) and Dnmw (m2 s 1) are the bulk diffusion coefficients of water vapour and the diffusion coefficient of non-frozen water in ionomer. By neglecting the hydrogen and oxygen transport in ionomer, the last term on the right hand side of Equation (84) can be neglected to calculate the effective diffusion coefficients for hydrogen and oxygen. This method can also be applied to the two-fluid model described in Section 6.2. In the mixture model presented this subsection, both the liquid water volume fraction in pores and water content in ionomer can be estimated by proper correlations. Without solving separate conservation equations of liquid water and absorbed water in ionomer, the mass transfer rates of water phase change and ionomer absorption/desorption cannot be implemented (as they are implemented in the two-fluid model described in Section 6.2 by adding source terms to the different conservation equations). The estimated amounts of liquid water and water content are based on the equilibrium condition without considering the non-equilibrium mass transfer. Therefore, the mixture model presented in this subsection can be regarded as the two-fluid model described in Section 6.2 with infinitely large water phase change and ionomer absorption/ desorption rates (to ensure equilibrium states). The two-fluid model features a more straightforward concept, and provides the capability to model the water phase change and ionomer absorption/desorption processes at different rates; and the mixture model is more computational efficient with less amount of conservation equations. It should be noticed that the equilibrium approach described through Equations (82)e(84) can also be implement to the two-fluid model by modifying the corresponding effective porosities and diffusion coefficients in the gas species conservation equations. 6.4. Boundary conditions and numerical implementation The boundary conditions for the two models can be specified in similar ways. At inlets of flow channels, mass flow rates (or velocity or mass flux etc.), volume/mass fractions of the different gas species and liquid water, and temperatures are often specified, and pressures are usually defined at outlets of flow channel. On the outer surfaces of computational domain, either temperatures or heat flux conditions can be specified for heat transfer into or out of computational domain. For the boundary conditions of electronic and ionic potentials, it is worthwhile to be mentioned that two methods have been used, as shown in Fig. 27. In method 1 in Fig. 27, the electronic potential at the anode BP end surface is set at fele ¼ 0, and it is set to be the cell voltage at the cathode BP end surface, fele ¼ Ecell. The overpotential (activational voltage loss) in anode CL is simply the difference between the electronic and ionic potentials, haact ¼ fele fion, while the overpotential in cathode CL is calculated as hcact ¼ fele fion Er, where Er (V) is the theoretical reversible cell voltage (as described in Section 1 and Equation (4)). It can be calculated from the modified form of the Nernst equation by assuming that the overall cell reaction is at thermodynamic equilibrium: Vr ¼ Fig. 27. Two different methods in specifying boundary conditions for electronic and ionic potentials [137]. Dgref 2F þ Dsref
2F T Tref  ! !0:5 # " pO2 p H2 RT þ ln 2F pref pref (85) Here Dgref (J mol 1) and Dsref (J mol 1 K 1) are the changes of Gibbs free energy and entropy for the overall reaction per mole of hydrogen at reference temperature (Tref, K) and pressure (pref, Pa). F (96,487 C mol 1) is the Faraday’s constant, R (8314 J kmol 1 K 1) the universal gas constant, and T (K) the temperature. pH2 (Pa) and pO2 (Pa) are the partial pressures of hydrogen and oxygen, respectively, and the values at flow channel inlets or the averaged values in CL have all been used for calculating the reversible voltage. In method 2 in Fig. 27, a zero electronic potential is set at cathode BP end surface, fele ¼ 0. While at anode BP end surface, the total cell potential loss is imposed, fele ¼ htotal ¼ Er Ecell. The corresponding overpotentials in anode and cathode CLs are all hact¼fele fion. No observable differences in results between these methods in Fig. 27 were found [137]. The sample distributions of potential/overpotential corresponding to the two methods are shown in Fig. 28. Generally, the potential distribution from method 1 is physically more meaningful since it demonstrates the real potential distributions within the cell. In contrast, the potential distribution from method 2 is more intuitive since it reveals in a straightforward manner of the 258 K. Jiao, X. Li / Progress in Energy and Combustion Science 37 (2011) 221e291 φele φion 0.6 0.5 Po tential, V 0.4 0.3 ηc = φele − φ ion − Er 0.2 0.1 0 -2 a 0.7 Current D ens ity, A cm a 0.94 Liquid production Vapour production 0.92 0.9 Equivalent to single phase model 0.88 Equivalent to mixture model 0.86 ηa = φele − φ ion 0.84 -0.1 10 -3 -0.2 CL Membrane CL -1 10 10 0 1 2 10 10 0.6 3 1.2 Production in ionomer Liquid production Current D ensity, A cm -2 φele φion 0.5 10 -1 Anode b Overpotential, V -2 Water Phase Change Rate, s Cathode b 10 ηa = φele − φion 0.4 0.3 1.1 1 Equivalent to equilibrium aproach 0.9 ηc = φele − φion 0.2 0.8 0.1 10 -2 10 -1 10 0 10 1 Water Phase Change Rate, s 0 Cathode CL Membrane CL Anode Fig. 28. Sample distributions of potential/overpotential corresponding to boundary conditions in Fig. 27 (a: corresponding to method 1 in Fig. 27; b: corresponding to method 2 in Fig. 27) [137]. potential loss from each component of the cell. It was also found that method 2 is more computational efficient than method 1 [137]. The model equations described for the two-fluid and mixture models are often implemented into CFD codes (such as the commercial code Fluent, Star-CD, CFX, CFD-ACEþ) based on finite volume or finite element methods, through their user coding capability. Therefore, the multi-dimensional geometry can be easily handled, and the numerical iterations can be properly stabilized. 6.5. Two-fluid model vs mixture model As described in Sections 6.1e6.4, the main difference between the two-fluid and mixture models is that a separate conservation equation of liquid water is solved in the two-fluid model, and the liquid water distribution is estimated from the two-phase mixture properties in the mixture model. The phase equilibrium is always maintained in the mixture model (instantaneous water evaporation and condensation), and the non-equilibrium water phase change can be accounted for by the two-fluid model by setting different source terms in the conservation equations. Even though the non-equilibrium ionomer absorption/desorption approach is described with the two-fluid model, and the equilibrium ionomer absorption/desorption approach is presented with the mixture 10 2 -1 Fig. 29. Effects of water phase change rate (a) between vapour and liquid water and (b) between water in pores and water in ionomer [137]. model, it is important to be noticed that these two approaches can all be applied to the two models. By using different values of the water phase change rate, g (s 1), in equations (54) and (60) for the two-fluid model, the effects of mass transfer rate for water phase change and ionomer absorption/ desorption processes can be examined. Fig. 29 compares the effects of water phase change rate between vapour and liquid water and between water in pores and water in ionomer on the performance of a single straight PEMFC running at a constant voltage. In Fig. 29a, the product water is assumed to be vapour or liquid water corresponding to the two lines in this figure. When water is assumed to be produced in vapour state, the current density decreases significantly with the increment of water phase change rate, due to the fact that the liquid water formation can be neglected at very low water phase rate, therefore the water vapour production condition together with very low water phase change rate can be considered to be equivalent to a single-phase model (formation of liquid water is neglected). On the other hand, when water is assumed to be produced in liquid state, the current density is not very sensitive to the water phase change rate. When the water phase change rate is high enough (on the level of 103 s 1), the current densities corresponding to the two water production mechanisms become similar, due to the fact that the phase equilibrium state can be achieved with large water phase rate. Therefore, the two-fluid model with large water phase change rate between vapour and liquid water can be regarded as equivalent to the mixture model (assumes phase equilibrium between vapour and liquid water). Water is assumed to be produced in liquid state in K. Jiao, X. Li / Progress in Energy and Combustion Science 37 (2011) 221e291 259 Fig. 30. Evolution of liquid water volume fraction upon the change of anode and cathode inlet relative humidities from 100% to 50% [252]. Fig. 31. Multi-channel simulation results (a: water content at membrane/CL interface in cathode; b: current density distribution in membrane; c: temperature distribution in membrane) [238]. 260 K. Jiao, X. Li / Progress in Energy and Combustion Science 37 (2011) 221e291 pores or absorbed in ionomer in Fig. 29b, it can be noticed that the production water in ionomer mechanism is more sensitive to the water phase rate for ionomer absorption/desorption than the liquid water production mechanism. With large enough water phase change rate (on the level of 102 s 1), both water production mechanism result in similar results, and they can be considered to be equivalent to the equilibrium ionomer absorption/desorption approach described in Section 6.3. With the two-fluid and mixture models, transient simulations to investigate liquid water evolutions upon change of operating condition can be conducted. Fig. 30 shows the evolution of liquid water volume fraction upon the change of anode and cathode inlet relative humidities from 100% to 50% based on the mixture model with equilibrium ionomer absorption/desorption [253]. By reducing the inlet relative humidities, it can be observed that the amount of liquid water decreases, and at the time instance of 1.75 s after the relative humidity change, almost all the liquid in electrodes under flow channel is removed, but it is still present under the lands. Large scale simulations considering multi-channels or stacks can also be carried out based on such continuum (top-down) approach by assuming homogeneous materials and effective transport properties. Fig. 31 shows the multi-channel simulation results with inlet gas humidities all at 50%. (liquid water formation was neglected in the simulations), which are the water content at membrane/CL interface in cathode and current density and temperature distributions in membrane. It can be observed from Fig. 31a that the water content at membrane/CL interface in cathode increases along the flow direction mainly due to the product water accumulation. Correspondingly, the current density distribution in Fig. 31b matches well with Fig. 31a: the current density is higher at the locations with higher water content (less ohmic resistance). With higher current density, the reaction rate is also higher, resulting in faster heat generation, therefore the temperature distribution in Fig. 31c matches with the water content and current density distributions in Fig. 31a and b. 6.6. Summary The continuum (top-down) full cell models for PEMFC by considering homogeneous materials of GDL, CL and membrane with modeled effective transport properties are reviewed in Section 6. Such models can be generally classified into two-fluid model and mixture model. The phase equilibrium is always maintained in the mixture model, and the non-equilibrium water phase change can be accounted for by the two-fluid model by setting different phase change rates. Such multi-dimensional multicomponent multiphase full cell models are powerful tools to investigate the transport phenomena simultaneously in different cell components and to evaluate various cell designs and material properties for the cell sizes from single straight channel to multichannel or stacks based on the available computational power. Therefore, such models have attracted many attentions. However, modeling a specific cell component with more realistic microstructures and more accurate formulations provides deeper insights for various transport phenomena and material properties. Therefore, in the following Sections (Sections 7e10), various models applied to specific cell components are introduced. 7. Modeling water transport in membrane Both the top-down and bottom-up models have been extensively carried out for modeling water and proton transport in membrane. As mentioned previously, the top-down approaches mainly include diffusive, chemical potential and hydraulic models. The bottom-up models were used to learn water and proton transport mechanisms and effects of water on ionomer self-organization. Physical models were also developed, and simplified structures of hydrophilic pores in ionomer at different hydration levels were proposed in such models to provide theoretical basis for explaining the water transport mechanism in ionomer. 7.1. Macroscopic approach 7.1.1. Diffusive model The diffusive model can be explained with dilute solution theory [311], by considering ionomer as solvent and water and proton as solute species. The dilute solution theory assumes that the solute species are dilute enough that the interaction between them can be neglected, and only the interaction between solute and solvent is accounted for. The flux of solute species (water and proton) in solvent (ionomer) can be expressed by using the NernstePlanck equation [311]: Ji ¼ zi wi Fi Vfion ! Di Vci þ ci u mem (86) Here the subscript i represents different solute species (water and proton). Ji (kmol m 3 s 1) is the superficial flux, zi the charge number or valence, wi (m2 kmol J 1 s 1) the mobility, F (9.6487  107 C kmol 1) the Faraday’s constant, fion (V) the ionic potential, Di (m2 s 1) the diffusion coefficient, ci (kmol m 3) the ! concentration, and u mem (m s 1) the superficial velocity of solvent (the ionomer). The relationship between wi (m2 kmol J 1 s 1) and Di (m2 s 1) is given by the NernsteEinstein equation [311e313]: (87) Di ¼ RTwi 1 1 where R (8314 J kmol K ) is the universal gas constant, and T (K) the temperature. For water transport, the first term on the right hand side of Equation (86) becomes zero because water has no valence. For proton transport, as mentioned earlier, the diffusive model assumes constant concentration of proton in ionomer, therefore the second term on the right hand side of Equation (86) becomes 0. Further, due the fact that ! the ionomer does not move ð u mem ¼ 0Þ, therefore the last term on the right hand side of Equation (86) becomes 0 for both water and proton transport. As a result, the simplified NernstePlanck equation (Equation (86)) for proton transport becomes Ohm’s law: Iion ¼ (88) kion Vfion 2 1 where Iion (A m ) and kion (S m ) are the ionic current density and conductivity, respectively. The simplified NernstePlanck equation (Equation (86)) for water transport becomes Fick’s law, and by further considering the EOD effect (Equation (18)), The water flux (Jnmw, kmol m 2 s 1) becomes Jnmw ¼ Dnmw Vcnmw þ nd Iion F (89) Here the subscript, nmw, represents the non-frozen water in ionomer (the only water in ionomer in normal operating condition). nd is the EOD coefficient, and F (9.6487  107 C kmol 1) the Faraday’s constant. It can be noticed that the diffusive model described in this subsection is actually used in the two-fluid and mixture models described in Section 6. In fact, the diffusive model has become the most successful top-down model for water transport in ionomer for PEMFC modeling since its initial application by Springer et al. [72,204]. 7.1.2. Chemical potential model In the chemical potential model, the interactions between water, proton and ionomer are all accounted for. This can be achieved by utilizing the concentrated solution theory [75,314], generalized 261 K. Jiao, X. Li / Progress in Energy and Combustion Science 37 (2011) 221e291 StefaneMaxwell equations [305e307,315], dusty fluid model [149,303,316], or the irreversible thermodynamics [317,318]. With the consideration of multi-component (water, proton and ionomer) interactions, these approaches give the fluxes in the following forms (or similar expressions): Iion ¼ Jnmw ¼ nd kion VPion F I anmw VPnmw þ nd ion F kion Vfion due to EOD effect. The model in [307] sheds light on full cell modeling incorporating the chemical potential model for water transport in ionomer, and with further exploration of the transport parameters, the chemical potential model is expected to be the dominating top-down model replacing the diffusive model for water transport in ionomer. (90) (91) where Pion (J kmol 1), anmw (kmol2 J 1 m 1 s 1) and Pnmw (J kmol 1) are the chemical potential of hydrogen ion (proton), water transport coefficient, and chemical potential of non-frozen water, respectively. The major difficulty for the chemical potential model is the lack of known transport parameters, such as Pion, anmw and Pnmw. It can be noticed that by replacing the chemical potential (P) by concentration (c), and by replacing the water transfer coefficient (a) by diffusion coefficient (D), Equations (89) and (91) become identical, and one more term considering the multi-component interaction in Equation (90) presents than in Equation (88). Such replacement has been used as one way to estimate the unknown transport parameters, (e.g. [204,319e321]), and another way is to estimate the transport parameters based on available experimental data (e.g. [304,305,307,322]). The difficulty in obtaining the transport parameters makes the chemical potential model less popular than the diffusive model, and the chemical potential model was also relatively rarely applied for full cell modeling. It is worthwhile to be mentioned that Baschuk and Li [307] have recently applied the chemical potential model to full cell modeling based on StefaneMaxwell equations. In this model, the membrane, CL, GDL and flow channel were all included, and protons were assumed be bound to water molecules to form hydroniums, therefore the transport of hydronium is accounted for rather than proton, (similar approach can also be found in [149,305]). The multi-component transport is systematically accounted for in the whole computational domain in [307]. Fig. 32 shows the simulated water activity in different cell components with fully humidified inlet gases on both sides [307], the water activity in membrane and CL can be transferred to water content by using Equation (12). It can be observed from Fig. 32 that water accumulates at the downstream of cathode, and the anode dries out along the flow direction. The highest water activity is observed in cathode CL close to cathode outlet due to water production and accumulation, and the lowest water activity is observed at the membrane/CL interface in anode near flow outlet 7.1.3. Hydraulic model In the diffusive and chemical potential models described in Sections 7.1.1 and 7.1.2, the convective transport caused by pressure gradient across membrane is not accounted for. As described in Section 3, water enlarges the pores in ionomer to allow larger amount of convective transport. The hydraulic model originates from the models developed by Bernardi and Verbrugge [109,205]. In [109,205], the membrane was assumed to be fully hydrated allowing the maximum possible convective transport. The dilute solution theory was utilized based on the NernstePlanck equation, and the Schlogl’s equation [323,324] has been used to calculate the water flux due to pressure gradient and EOD effect. The models in [109,205] also assumed constant gas volume fractions in membrane, allowing convective gas transport as well. After that, more models utilizing the approach in [109,205] have been developed [74,213,229,325e327]. Generally, the hydraulic model neglects the diffusive transport and accounts for the convective transport water in ionomer, and the water flux can be represented by the follow equation Jnmw ¼ Knmw mnmw Vpnwm þ nd Iion F (92) where Knmw (m2) is the permeability of non-frozen water in ionomer, mnmw (kg m 1 s 1) the dynamic viscosity of non-frozen water in ionomer (the property of liquid water is often used instead), and pnmw (Pa) the pressure of non-frozen water in ionomer. It should be noticed that the membrane close to anode side often dries out quickly due to EOD effect during PEMFC operation. Therefore, the fully hydrated membrane assumption remains questionable. In fact, the water flux due to pressure gradient is only considerable when the cell is differentially pressurized or most of the membrane is well hydrated, and most models only accounted for the diffusive and EOD caused water transport. Therefore, the hydraulic model neglecting the diffusive water transport essentially reflects unrealistic conditions. 7.1.4. Combinational model When a PEMFC is differentially pressurized for anode and cathode, both the diffusive and convective water transport may need to be accounted for, and the most straightforward way is to add all the terms (diffusive, convective and EOD) together, forming the combinational diffusive/hydraulic model [211,255,303,328e332] Jnmw ¼ Fig. 32. Simulated water activity in different cell components with fully humidified inlet gases on both sides [307]. cnmw Dnmw Vcnmw cnmw Knmw mnmw Vpnwm þ nd Iion F (93) It should be noticed that the permeability of non-frozen water in ionomer, Knmw, increases with the increment of water content, as shown in Equation (22) and other experimental measurements (such as [333]). Another approach to account for both the diffusive and convective water transport mainly focused on modifying the chemical potential model with the pressure effect [304,322], and the effects of convective water transport on proton transport can be accounted for in the modified chemical potential model. However, due to the fact that some transport parameters remain unclear, further exploration is needed for this approach. 262 K. Jiao, X. Li / Progress in Energy and Combustion Science 37 (2011) 221e291 Fig. 33. Snapshots of final micro-structures in hydrated Nafion membrane at different water contents (water, hydronium and side chain domains are show in green, and hydrophobic domains are shown in red) [346]. (For interpretation of the references to colour in this figure legend, the reader is referred to the web version of this article.) 7.2. Bottom-up approach and physical models 7.2.1. Modeling ionomer self-organization As mentioned in Sections 2 and 3, with the changes of amount and state of water in PFSA polymer membranes, the ionomer morphology changes as well, resulting in different water and proton transport characteristics. Even though different simplified micro-structures of ionomer were proposed (detailed in Section 7.3), bottom-up molecular simulations are the only computational methods to investigate the details of chemical structure. Therefore, quantum mechanical [78,224,280,281,334] and classical [282e290,335] MD and pseudo particle [291,292,336e340] simulations have been extensively carried out for this purpose. The MD simulations in [224,334] showed the formation of percolating channels with water in ionomer, which agrees with the simplified micro-structure of Nafion ionomer proposed in [341,342]. However, the percolating water channel was observed breaking very fast after formation in the MD simulations in [282]. Both the water being strongly and loosely bound to sulphuric sites were observed in the MD simulations in [224,335], and frequent exchange between the two kinds of water at high water contents was also presented in [335]. Similarly, other MD simulations (e.g. [285,287e290]) were also performed to study the ionomer morphology and state of water. The MD simulations on atomistic level mentioned in the previous paragraph are useful tools for investigating the water phase segregation in ionomer. The current computational power does not allow such simulations to represent the chemical structures in a sufficiently large length scale and long time scale. Pseudo particle simulations are therefore needed by representing a group of molecules as a single molecule with appropriate approximations to save computational time. Coarse grained (CG) MD (CGMD) simulations were conducted in [291,294] by presenting a group of atoms as a CG segment, and it was reported that the shape of hydrophilic regions changes from spherical to elliptical when the water content increases from 6 to 8. MC simulations by using a reference interaction site model [343] were carried out in [336]. In [336], each CF2 and CF3 were presented by a pseudo atom. Threes layers were observed for Nafion ionomer, which are a central water-rich region and two outer layers of side groups strongly associated with water molecules. A linear relationship between ionomer swelling and water content was also reported in [336]. Dissipative particle dynamics (DPD) was also used to study the chemical structure of Nafion ionomer [337e339] by considering the motion of pseudo particles governed by Newton’s equations [344,345]. It was found that the hydrophilic domain size increases linearly with water content [339]. LB simulations based on the morphologies generated in [339] were also performed to simulate water transport in ionomer, and it was confirmed that the permeability of water in ionomer increases with water content [340]. Fig. 33 shows the snapshots of the final micro-structures in hydrated Nafion membrane at different water contents obtained from CGMD simulations [346]. It can be observed that the hydrophilic domains (water, hydronium and polymer side chain) form clusters embedded in the hydrophobic domains (hydrophobic backbone). For the simulations in Fig. 33, it was also reported that the typical water filled channel sizes are 1, 2 and 4 nm at the water contents of 4, 9 and 15, respectively. 7.2.2. Modeling proton transport ab initio quantum mechanical MD simulations are usually needed to study the mechanism of proton transport, however, the current computational power does not allow to perform full MD simulations of proton/water transport in the real structure of Nafion membrane [346]. The structural complexity needs to be simplified, and the rarity of proton transfer events (need large time scale) needs to be efficiently sampled [346e351]. MD simulations to investigate the proton transfer between water and charged sites were conducted in [99,281,352] by considering several polymer side chains attached to a single hydrophobic backbone, therefore the detailed polymerewater interaction could be studied. The MD simulations in [353] found that the flexibility of polymer side chain could be important for proton transport at low hydration level. Similarly, Fig. 34 shows the elementary interfacial proton transport and configuration energy along the reaction path obtained from MD simulations at low hydration level [346], and it can be noticed that tilting and rotation of the acceptor/donor sites facilitate the elementary interfacial proton transport. 7.2.3. Physical models with simplified membrane micro-structure Based on experimental observations (such as X-ray, NMR and IR) and percolation theory, Hsu and Gierke [341,342] postulated a simplified membrane micro-structure, as shown in Fig. 35, a principal spherical and minor cylinder-like hydrophilic micro-structure was assumed. Even though the proposed micro-structure remains debated, it has become the most enduring Nafion morphology. This micro-structure in three-dimension suggested a percolation threshold of 15% (about 3.3 water content) for the hopping proton transport descried in Section 3.1 (the minimum water content to form a continuous water pathway in ionomer), which reasonably agrees with experimental measurement (proton conductivity becomes appreciable when water content ranges from 2 to 5). Dissimilar to the spherical-cylinder micro-structure proposed in [341,342], irregular three-phase [354], lamella [355], sandwiched, [356] and channel-like [357] micro-structures were also proposed. K. Jiao, X. Li / Progress in Energy and Combustion Science 37 (2011) 221e291 263 Fig. 34. Elementary interfacial proton transfer and configuration energy along the reaction path (the nearest neighbour distance of C atoms, dCC, is about 0.7 nm, and the activation barrier is about 0.5 eV) [346]. Similarly, all these models provided low percolation thresholds as the critical feature of proton conductivity in ionomer. Based on the simplified membrane micro-structure in [341,342], a PN model was also developed [73]. This model assumed that the hydrophilic clusters are spherical pores connected by cylindrical channels. The numbers and sizes of the pores and channels are estimated based on water content, and therefore the relationship between proton conductivity and water content was obtained, and a percolation threshold of 20% was suggested (about 4.4 water content). difficulty in obtaining the transport parameters makes the chemical potential model less popular than the diffusive model, and the chemical potential model was also relatively rarely applied for full cell modeling. The modified diffusive model by adding an EOD term is still the most popular model so far. The bottom-up models have been developed to study water and proton transport mechanisms and effects of water on ionomer self-organization. Simplified structures of hydrophilic pores in ionomer at different hydration levels have also been proposed to provide theoretical basis for explaining the water transport mechanism in ionomer. 7.3. Summary The different models for water transport in membrane are reviewed in Section 7. The review shows that both the top-down and bottom-up models have been extensively developed. The topdown models can be classified into diffusive, chemical potential and hydraulic models. Even though the chemical potential model solves the water transport in the most comprehensive way, the 5 nm SO3 − SO3 − SO3− SO3− SO3− SO3− SO3− SO3− SO3− SO3− 4 nm SO3 1 nm SO3− − SO3− SO3− SO3− SO3− SO3− SO3− SO3− SO3− Fig. 35. Simplified Nafion micro-structure [341,342]. Fig. 36. A magnified picture of GDL (carbon paper) [268]. 264 K. Jiao, X. Li / Progress in Energy and Combustion Science 37 (2011) 221e291 8. Modeling water transport in gas diffusion layer Many different models have been developed to simulate transport phenomena in GDL. Generally, the different models can be categorized into two groups, one assumes homogeneous material of GDL, and another utilizes real or simplified micro-structure of GDL for simulations. When GDL is assumed to be homogenous, the micro-structure is neglected and the computational domain becomes pure void. Effective transport properties are used to reflect the effects of the neglected micro-structure. Such homogeneous approach greatly simplifies the geometry generation process and results in more stable numerical iterations. The full cell models described in Section 6 all use this approach. The digital microstructure of GDL needs to be generated first for the second approach. The small pores result in extremely large number of computational grid, therefore usually only a small sample was accounted for. This approach therefore was mostly adopted to simulate liquid water behaviours or to numerically measure the transport properties. In this section, the different models considering homogeneous or real/simplified GDL are introduced, the homogeneous approach is only briefly discussed because it is described in the previous sections, and this section mainly focuses on the models considering real or simplified micro-structures of GDL. In addition, both the first-principle-based and rule-based models are included in this section. 8.1. Homogeneous approach The homogeneous approach is described with the full cell models in Section 6, and the various transport mechanisms are described Sections 2 and 3, therefore the details of this approach are not repeated here. It should be noticed that the StefaneMaxwell formulation for gas diffusion is not used in the full cell models in Section 6. Such approach is only valid when two gas species are present. For PEMFC without impurities (e.g. CO), the only gases in anode are hydrogen and water vapour, which is acceptable; however, oxygen, water vapour and nitrogen are all present in gas phase in cathode when air is supplied, which requires StefaneMaxwell formulation, as shown in the following equation. VYi ¼ X Yi Jj jsi Yj Ji rg Deff i;j used. The micro-structure can be obtained either by a threedimensional volume imaging technique or by constructing a digital micro-structure based on stochastic models, description about such models can be found in [358,359]. Constructing the three-dimensional image using the volume imaging technique requires the use of non-invasive experimental techniques, such as X-ray and NMR. With this technique, the porous material is repeatedly sectioned and imaged automatically. Then the two-dimensional cross-sectional images are combined to construct the three-dimensional image of the micro-structure. This technique can be very expensive and time consuming. It requires the use of a three-dimensional imaging software along with a scanning electron microscope and very accurate sectioning of the GDL. The reconstruction of the image using digital stochastic models requires the knowledge about the pore distribution and pore size of the micro-structure, which can be obtained easily with a porosimeter. This technique is not as expensive and is faster than the experimental imaging technique, and has been used and proved to be successful (e.g. [274,275,278,360]). Assumptions are often needed for the stochastic method, for example, for constructing the micro-structure of carbon paper, the following assumptions are often made [360]: 1. The fibres are considered to be cylindrical, with a constant radius and are infinitely long. 2. The fibres are allowed to overlap. 3. According to the fabrication process of carbon fibre, the fibre system is isotropic in the material plane. Fig. 37 shows a three-dimensional computational domain for GDL (carbon paper) with a 90% porosity obtained by using the stochastic method [360], which is very comparable with the image shown in Fig. 36. Note that simplified micro-structures of the GDL were also used, such as the study in [121] assumed that the GDL is made up of stacked two-dimensional random carbon fibre mats as shown in Fig. 38. It was assumed that the fibres are infinitely long on the plane parallel to GDL surface, and are allowed to overlap. The solid (94) Here the subscripts i and j represent different gas species. Y and J (kg m 2 s 1) are the mass fraction of gas species in gas mixture and diffusional flux. rg (kg m 3) is the density of gas mixture, and Di,jeff (m2 s 1) the effective binary diffusion coefficient of species i in species j. Apparently, the StefaneMaxwell formulation is much more complex than the Fickian diffusion formulation in Section 6. Therefore, many of the previous full cell models neglected it to only consider the Fickian diffusion formulation. As reviewed in Section 5, the VOF and LB are the only two firstprinciple-based models used that can track the interface between liquid and gas phases to investigate the detailed liquid water behaviour in PEMFC. Even though these models can all be used for the homogeneous approach with effective transport properties, the detailed liquid water behaviour can only be investigated when the micro-structure of GDL is used as the computational domain. 8.2. Structure generation A magnified picture of GDL (carbon paper) is shown in Fig. 36 [268]. It can be noticed that the GDL structure is highly anomalous, and therefore special methods are needed to generate the digital micro-structure of GDL. There are two common methods that can be Fig. 37. Three-dimensional computational domain for GDL (carbon paper) with a 90% porosity obtained by using stochastic method [360]. 265 K. Jiao, X. Li / Progress in Energy and Combustion Science 37 (2011) 221e291 Fig. 38. Simplified micro-structure of GDL (a: image of Toray carbon paper; b: single screen made of overlapping fibres with square pore spaces; c: stack of fibre screens; d: pore spaces of stacks with an arbitrary screen position shifting) [121]. structure was modeled as stacks of continuously overlapping fibre screens. PN models often utilizes simplified micro-structures which consist of pores, throats and solids, as shown in Fig. 39 [361]. The flow resistances are modeled to be different in pores and throats in PN models with such simplified micro-structures (detailed in Section 8.5.2). The simplified micro-structures shown in Figs. 38 and 39 are much easier to be constructed than the more realistic structure in Fig. 37, and therefore were also widely adopted. 8.3. Volume-of-fluid model The VOF model is perhaps the only top-down model that has been used to investigate detailed liquid water behaviour in PEMFC [11,12,262e268]. Most of the work focused on liquid water transport in flow channel or in GDL with simplified micro-structure. With the micrometre pores in GDL, it is still possible to use the top-down VOF model with real GDL micro-structures. To the best of the authors’ knowledge, only Park et al. [268] recently adopted the VOF method to simulate water transport in a real GDL. By neglecting the phase change, electrochemical reaction, gravity and heat transfer, the VOF model with two phases, gas and liquid water, can be presented by the following conservation equations. Mass of two-phase mixture: v ! ðrÞ þ V$ðr u Þ ¼ 0 vt (95) Fig. 39. Two-dimensional schematic of pore-network construction (a: relationship among pores, throats and solid; b: structure in terms of void and solid space) [361]. 266 K. Jiao, X. Li / Progress in Energy and Combustion Science 37 (2011) 221e291 Momentum of two-phase mixture: v ! !! ðr u Þ þ V$ðr u u Þ ¼ vt Liquid water:

T  ! ! Vp þ mV$ Vð u Þ þ V u 2 mVðV$ð! u ÞÞ þ Su 3 (96) 
 v
rslq þ V$ rslq ! u ¼ 0 vt (97) r ¼ sg rg þ slq rlq (98) where r (kg m 3) and m (kg m 1 s 1) are the density and dynamic viscosity of the two-phase mixture, respectively, and they are the volume averaged values, such as for density: Here the subscripts g and lq represent the gas and liquid phases, and s the volume fraction. It can be noticed that the two phases rely ! on the same momentum equations, and therefore the velocity ( u , 1 m s ) and pressure (p, Pa) are shared by the two phases in each computational cell as well. The liquid water conservation is solved (Equation (97)), and the gas phase volume fraction can be simply calculated since the total volume fraction is unity. sg þ slq ¼ 1 (99) 2 2 The source term (Su, kg m s ) in the momentum equation (Equation (96)) accounts for the surface tension effect, and it can be calculated as Su ¼ s rkVsg  0:5 rg þ rlq (100)
where s (N m 1) is the surface tension coefficient between the two phases, and k (m 1) the surface curvature defined as ! ! k ¼ V$! n ¼ V$ð n w cosðqÞ þ t w sinðqÞÞ (101) ! where n is the unit vector normal to the interface between the two ! ! phases; near the wall, n w and t w are the unit vectors normal and tangential to the wall surface; and q is the contact angle. It can be noticed that the shape of the interface between the two phases at the wall depends on the wettability of the wall. In addition, it can be noticed from Equations (95)e(97) that the porous media properties such as porosity are not taken into account (as described in Section 6 for the two-fluid and mixture models), this is valid in flow channel (pure void space) and when the micro-structure of GDL (or CL) is represented in the computational domain. By using the VOF model in a GDL micro-structure, Fig. 40 shows the transient liquid water discharging from GDL with a pressure gradient of 6.5  105 Pa m 1 and a contact angle of 135 [268]. It can be observed that when a pressure gradient is exposed, liquid water are being removed, and finally there is a small amount of liquid water left which is difficult to be removed. Fig. 41 presents the transient liquid water volume fractions in GDL with different pressure gradients and contact angles [268]. It can be noticed that the water removal time is significantly affected by the amount of pressure gradient, and the GDL hydrophobicity has to be high enough to initiate liquid water removal. Not only investigating liquid water dynamics, the simulations in [268] also indicated that the pressure gradient and contact angle are the two parameters which determine the initiation of liquid water removal in GDL. Fig. 40. Transient liquid water discharging from GDL with a pressure gradient of 6.5  105 Pa m 1 and a contact angle of 135 obtained from volume-of-fluid model (the time step between the successive sequence plots is 500 ms) [268]. adopted to find the GDL and CL properties (e.g. permeability) [272,273], as well as to investigate liquid water transport in GDL and CL [274e277]. The LB method is based on a finite number of identical particles (it is intrinsically a pseudo particle method) that go through collision and propagation successively on prefixed paths in space. For multiphase flow simulations, the evolution of distribution function for each phase is represented by an evolution equation [276]. For the kth phase, it is 8.4. Lattice Boltzmann model Based on the numerously previous development [362e373], the LB model, originated from its predecessor LG model, has been fik ðx þ ei dt ; t þ dt Þ fik ðx; tÞ ¼ fik ðx; tÞ kðeqÞ fi sk ðx; tÞ (102) K. Jiao, X. Li / Progress in Energy and Combustion Science 37 (2011) 221e291 Liquid Water V o lume Fraction a 1 6.5×103 Pa m-1 4 -1 6.5×10 Pa m 6.5×105 Pa m-1 6.5×106 Pa m-1 0.8 0.6 0.4 0.2 0 0 1 2 3 4 -3 Time, 10 s Liquid Water V o lume Fraction b 1 124 o 125 o 127.5 o 135 o 0.8 0.6 0.4 0.2 0 1 2 3 4 -3 Time, 10 s Fig. 41. Transient liquid water volume fractions in GDL (a: for different pressure gradients when the contact angle is 135 ; b: for different contact angles when the pressure gradient is 6.5  105 Pa m 1) [268]. where fik(x,t) is the number density distribution function for the kth phase in the ith velocity direction at position x and time t, and dt is the time step. sk is the relaxation time of the kth phase, and fik(eq)(x,t) represents the equilibrium distribution function. The (2) (5) (0) (3) (7) right hand side of Equation (102) is the BhatnagareGrosseKrook (BGK) collision term [362]. The LB model is usually represented by the numbers of dimension and velocity directions. Fig. 42 shows the lattice structure of a D2Q9 lattice Boltzmann model [276]. Here D2 represents two-dimensional, and Q9 means 9 directions of velocity at each node. Other configurations such as D3Q15 and D3Q19 were also commonly used. The interactions between fluids and between fluids and wall (e.g. surface tension and wall adhesion) can be taken into account through the modified equilibrium distribution functions. The macroscopic fluid phase properties can be obtained through appropriate averaging of the particle distribution function. It should be noticed that the density ratio between air and liquid water is about 1000 in cathode, while it is much higher in anode (hydrogen and liquid water). However, most of the previous LB simulations assumed that the density ratio is 1 (large density ratio results in numerical instability), which is only valid when the capillary effect dominates. As analyzed in Section 3.2.2, this assumption is valid when the cross flow through GDL is not significant (e.g. with parallel flow channel design small pressure gradient is present in GDL), and it may be failed for large cross flow (e.g. with long serpentine or interdigitated flow channel design large pressure gradient is present in GDL). Even though a few recently developed LB models [370,373] are available for high density ratio up to 1000, their stability is not yet proven in a complicated geometry. Fig. 43 shows the liquid water transport through GDL obtained from the LB simulation in [276], it can be noticed that part of water is trapped in GDL and difficult to move, and this observation is consistent with the water transport shown in Fig. 40 with VOF model. It was also concluded in [276] that the pressure gradient and GDL wettability are the dominating factors for liquid transport in GDL, which agrees with the conclusion in [268] using VOF model. 8.5. Rule-based model 0 (6) 267 (4) (1) (8) Fig. 42. Lattice structure of a two-dimensional (D2Q9) LB model [276]. The VOF and LB models introduced in the previous subsections are first-principle-based models. Rule-based models were also used to investigate liquid water transport in PEMFC. The most representing rule-based models are the FM [274,275,278] and PN [274,279,361] models. Rather than solving a set of governing equations, the rule-based models depend on applying physical rules to simplified or real physical structures, and therefore are more computationally efficient than the first-principle-based models but can only be applied to certain flow conditions. 8.5.1. Full morphology model Based on the previous development in [374,375], the FM model has been adopted to investigate liquid water transport in GDL in [274,275,278]. With a constructed GDL micro-structure (as described in Section 8.2 and shown in Fig. 37), the FM model was used to estimate the steady state liquid water distribution in GDL by assuming that the capillary effect dominates the two-phase transport in GDL [274,275,278]. As analyzed in Section 3.2.2, this assumption is valid when the cross flow through GDL is not significant (e.g. with parallel flow channel design small pressure gradient is present in GDL), and it may be unacceptable with strong cross flow (e.g. with long serpentine or interdigitated flow channel design large pressure gradient is present in GDL). Based on this assumption, if liquid water enters a dry GDL from one side, the pressure difference between the liquid water inlet and the opposite GDL surface (the outlet) can be considered as the capillary pressure. By further assuming that liquid water forms sphere droplets, the capillary pressure, liquid water droplet diameter and contact angle can be related by Equation (42). For different capillary pressures and contact angles, different liquid water droplet diameters can be 268 K. Jiao, X. Li / Progress in Energy and Combustion Science 37 (2011) 221e291 or generalized Poiseulle law [376] or something similar. To investigate liquid water transport through GDL, the PN model in [274,279] applied constant liquid water injection rate at one side of a simplified network structure, constant pressure is applied to the opposite surface (outlet), and the other boundaries are set as walls. Liquid water can only invades to the next pore or throat when the pressure difference between liquid water and gas is higher than the capillary pressure. The pores and throats are invaded one by one from liquid water inlet until a steady state liquid water distribution is reached. Therefore the liquid water invasion process can be investigated by using the PN model, and the liquid water distribution in GDL obtained from the PN model when a steady state is reached is shown in Fig. 45 [279]. The PN model can account for different flow resistances with proper calculation of pressure drop (such as using the HagenePoiseulle law [144] or generalized Poiseulle law [376]), and therefore can be used for different flow conditions. The transient simulation and capability for various flow conditions are the advantages of the PN model by comparing with the FM model. However, the PN model in [274,279] assumed that the pores and throats are completely filled with liquid water after invasion. This assumption remains debated because gas films may still form at the corners between pores and throats [377,378]. The simplified GDL micro-structure may also lead to unrealistic results. These features make the accuracy of the PN model remain debated. 8.6. Summary Fig. 43. Liquid water transport through GDL obtained from LB model (the Reynolds and Capillary numbers are 0.1 and 2.3  10 5, respectively) [276]. obtained. The liquid water droplets with calculated diameters are fitted in a generated GDL micro-structure (such as the one shown in Fig. 37), and those droplets intersecting the solid material and not directly connecting to the liquid water inlet are removed. The rest of the liquid water are then dilated to fill the void space between the connected droplets, and the volume fraction of the dilated liquid water in the void space of the micro-structure is obtained as the liquid water volume fraction corresponding to the capillary pressure and contact angle. By using the FM model, the relationship between the capillary pressure and liquid water volume fraction was studied by using different GDL samples, and reasonable results were obtained [274,275,278]. Fig. 44 shows the visualization of liquid water distribution in GDL obtained from FM model [278]. Fig. 44a presents the micro-structure of GDL without any liquid water, and Fig. 44bef shows that the liquid water volume fraction increases with the pressure difference between the two phases. Generally, the FM model presented here is more computationally efficient than the first-principlebased models. However, it can only be applied to certain flow conditions, such as the capillary dominated flow discussed in this subsection. Since only the steady state liquid water distribution can be obtained from the FM model, the liquid water dynamics cannot be investigated. 8.5.2. Pore-network model In the PN models [274,279,361] developed for GDL, the GDL micro-structures were assumed to consist of three components, which are pores, throats and solid (such as the one shown in Fig. 39). The shapes of the pores and throats are assumed to be simple so that the flow resistance through the pores and throats can be simply calculated. The pressure drop across the pores and throats can be calculated based on the HagenePoiseulle law [144] The different models for water transport in GDL are reviewed in Section 8. This review shows that such models can be classified into two groups, one assumes homogeneous material of GDL, and another utilizes real or simplified micro-structure of GDL for simulations. The homogenous approach is often adopted in full cell modeling (Section 6). The second approach requires digital microstructure of GDL as the computational domain. Such computational domains consist of small pores therefore result in much more computational grids than the homogeneous approach. Therefore, usually only small samples of GDL are considered in this approach. The second approach involves both the first-principle-based and rule-based models. The VOF and LB models are the two most popular first-principle-based models for simulating two-phase flow in GDL, and the FM and PN models are the two mostly widely used rule-based models. 9. Modeling water transport in catalyst layer Since the detailed formulations for the top-down models considering homogeneous materials are described in Section 6, and the various models with porous micro-structures are presented in Section 8, they are not repeated in this section. It should be noticed that the pores in CL are much smaller than in GDL (refer to the analysis in Section 3.3), therefore the VOF model presented in Section 8.3 cannot be applied to CL micro-structures, and the bottom-up approach such as the LB model is needed to simulate liquid water transport in CL pores. Because the calculations for the rule-based models (the FM and PN models) are based on top-down formulation, they may not be able to be directly applied to CL micro-structures as well. Rather than just accounting for the StefaneMaxwell diffusion in the pores of GDL, the much smaller pores in CL also results in Knudsen diffusion (as described in Section 3.3), therefore the gas diffusion function given in Equation (94) for GDL needs to be modified as VYi ¼ X Yi Jj jsi Yj Ji rg Deff i;j Ji rg Deff K;i (103) K. Jiao, X. Li / Progress in Energy and Combustion Science 37 (2011) 221e291 269 Fig. 44. Visualization of liquid water distribution in GDL obtained from FM model (a: GDL micro-structure without any liquid water; bef: steady state liquid water distribution in GDL corresponding to different capillary pressures) [278]. 2 1 where Deff K,i (m s ) is the effective Knudsen diffusion coefficient of gas species i. In this section, only the models not previously included are described. The agglomerate model, with simplified CL micro-structure and based on top-down formulation, is presented first, followed by the MD and off-lattice pseudo particle simulations for investigating the reaction mechanisms and material microstructures. 9.1. Agglomerate model As mentioned previously, the top-down approach cannot be applied to the small pores of CL. However, by assuming that the catalyst, ionomer and part of void space are homogeneously mixed and form micrometre agglomerates, the top-down approach can still be used. Due to the fact that the sluggish electrochemical reaction in cathode CL is the limiting factor of cell performance, such agglomerate models usually focused on cathode CL. The agglomerate model has been extensively studied in the past decade [214e218,269e271,379e381], and different simplified microstructures were proposed such as spherical agglomerate [271], cylindrical agglomerate [379], ordered CL [380], and non-uniform CL [269]. Most of the previous agglomerate models are one- or twodimensional, and a three-dimensional agglomerate model was developed recently to fully account for the effect of agglomerate arrangement on cell performance [271]. By assuming spherical agglomerates with a diameter of 5 mm, Fig. 46 shows the schematics of agglomerate arrangements and the corresponding computational domains in cathode CL for the agglomerate model developed in [271]. The in-line, uni-directional staggered and bi-directional staggered arrangements were considered. Since the top-down formulation is described in Section 6, the top-down formulation of the agglomerate model is not repeated here. Corresponding to the three agglomerate arrangements, Fig. 47 shows the reaction rates at a current density of 0.6 A cm 2 in cathode CL [271]. To achieve the same current density, it can be observed that the reaction rate is the lowest with in-line agglomerate arrangement, and it is the highest with unidirectional agglomerate arrangement. High reaction rate indicates that the activation loss is high. With such significant differences between the three arrangements, it was concluded that the in-line arrangement features much better cell performance than the other two arrangements. Generally, the agglomerate models are useful in optimizing the CL micro-structures and catalyst distribution. However, for more detailed investigation on the reaction mechanisms and material micro-structures in CL, MD and office-lattice pseudo particle simulations are needed. 9.2. Molecular dynamics and office-lattice pseudo particle simulations Fig. 45. Liquid water distribution in GDL obtained from PN model when a steady state is reached [279]. The micro-structure of CL depends on particle size of catalyst, catalyst loading, ionomer fraction, addition of other materials (such as PTFE to increase hydrophobicity), temperature, and so on. Due to the fact that the electrochemical reactions only occur at the threephase contact interface, the CL micro-structure becomes critically important in optimizing cell performance. Investigation of reaction mechanisms and morphology in CL is therefore demanded, which 270 K. Jiao, X. Li / Progress in Energy and Combustion Science 37 (2011) 221e291 Fig. 46. Schematics of agglomerate arrangements and corresponding computational domains of cathode CL [271]. can be achieved by MD and office-lattice pseudo particle simulations. Quantum mechanical MD and MC simulations have been performed to study the elementary reaction processes on catalyst surfaces [293e298] by focusing on the understanding of carbon monoxide (CO) adsorption processes [293,294] and the reaction kinetics of on different metal surfaces [295e297]. CGMD simulations have also been carried out to evaluate the key factors for CL fabrication [299]. The simulations were performed with the presence of carbon/platinum particle, water, ionomer and solvent (used for CL fabrication) [299]. Fig. 48 shows the equilibrium structure of a catalyst blend composed of carbon/platinum, ionomer, water and implicit solvent [299,346]. It was found that the hydrophobic backbones are attached to the catalyst surfaces, and the polymer side chains tend to leave from the catalyst surfaces. Water and ionomer are clustered together. Generally, the CGMD simulation was demonstrated as a powerful tool to provide valuable microstructural information of CL to help design optimization. 9.3. Summary The different models for water transport in CL are reviewed in Section 9. Water transports in both the ionomer and pores in CL, and the models for water transport in ionomer and pores are introduced in Sections 7 (in ionomer) and 8 (in pores). However, the top-down VOF K. Jiao, X. Li / Progress in Energy and Combustion Science 37 (2011) 221e291 271 model cannot be applied with CL micro-structures due to the small length scale, and the rule-based models (the FM and PN models) based on top-down formulation may not be able to be directly applied to CL micro-structures as well. Knudsen diffusion also needs to be considered in the small pores of CL. Top-down formulation can still be used in agglomerate models with simplified CL structures that neglect the nanometre holes in CL. The MD and off-lattice pseudo particle simulations have been conducted for investigating the reaction mechanisms and material micro-structures. 10. Modeling water transport in flow channel The multi-component gas transport in flow channel can be modeled by using the top-down approach described in Section 6 by fully accounting for the convective and diffusive mass transfer. The multi-component two-phase transport involving gas and liquid water in flow channel can be simulated by using the two-fluid and mixture models presented in Section 6, or by using the VOF and LB models described in Section 8. The two-fluid and mixture models are able to predict liquid water fraction, however, the detailed liquid water transport behaviours cannot be simulated because the interface between gas and liquid cannot be tracked. Better correlations between gas and liquid velocities are also needed to predict liquid water transport more accurately. Due to the numerical instability, the LB model often assumed unit density ratio between gas and liquid. This assumption may be reasonable for flow in GDL and CL when the capillary effect dominates. However, for the convection dominated flow in flow channel, the inertia force becomes significant, and the unit (or low) density ratio assumption fails. The most suitable model for simulating liquid water transport so far is the VOF model, which can account for all the major flow effects such as inertia force, viscous force, surface tension and gravity and track the interface between gas and liquid. By using the VOF model, liquid water transport behaviour in flow channel has been extensively investigated in single serpentine [262,265e267], serpentine-parallel [11], and straight-parallel [12] flow channels. The effect of surface wettability of flow channel on liquid water transport was also studied by using VOF model [263,264]. 10.1. Volume-of-fluid model Corresponding to the liquid water injection condition for membrane hydration, Figs. 49 and 50 show the liquid water transport in a three-cell stack with serpentine flow channel design 600 Uni-directional staggered agglomerate arrangement Reaction Rate, A cm -3 500 400 300 Bi-directional staggered agglomerate arrangement 200 In-line agglomerate arrangement 100 0 0 2 4 6 8 10 Distance in CL from Membrane to GDL, μm Fig. 47. Reaction rates at a current density of 0.6 A cm to the agglomerate arrangements in Fig. 46 [271]. 2 in cathode CL corresponding Fig. 48. Equilibrium structure of a catalyst blend composed of carbon/platinum (black), ionomer (red), water (green) and implicit solvent [299,346]. (For interpretation of the references to colour in this figure legend, the reader is referred to the web version of this article.) [11] obtained from VOF model. When small liquid water droplets are supplied (Fig. 49), all the droplets hit the end wall of the inlet manifold driven by gas flow, and then they all flow into the last cell (cell 3). When a large amount of liquid water is supplied (Fig. 50), the supplied liquid water moves slower than the small droplets, and it is able to be distributed into the three cells, however, most of the supplied liquid water flows into cell 3. Largely uneven distributions of supplied liquid water among the different cells are observed for the two liquid water injection modes, suggesting that liquid water injection may cause water flooding and membrane dehydration in different single cells of PEMFC stack. Liquid water sticking at the dead ends of the inlet and outlet manifolds can also be observed at the end of the water transport processes in Figs. 49 and 50. Fig. 51 presents the liquid water transport in a cross section of the inlet manifold corresponding to the water transport process in Fig. 50 [11]. It can be observed that the liquid water is first pushed to the end wall of the inlet manifold, with a vortex formed there, the liquid water is pushed back towards to the inlet again, and then it is moved to the three cells by the inlet gas flow. Such unordered liquid water movement may affect reactant transport. Figs. 52 and 53 shows the pressure drops for each single cell and the whole stack corresponding to the water transport processes in Figs. 49 and 50, respectively. In Figs. 49 and 52, when the liquid water droplets enter cell 3 (in the first 3 ms), the pressure drop in this cell increases significantly, and then decreases to its normal state. The pressure drop variations shown in Fig. 53 with a large amount of liquid water supply is more significant than in Fig. 52, it can be noticed from Fig. 53 that the pressure drops in all the three cells all change dramatically until the liquid water is removed out of the stack. The results shown in Figs. 52 and 53 indicate that liquid water movement leads to unordered reactant transport, which may cause unstable cell performance. Recall the experimental observations shown in Figs. 23 and 24, different cell voltages were reported at the same current density for the same cell due to the presence of liquid water. In fact, the variations of pressure drop caused by the presence of water can also be used as useful indicators of the water content within the cell, as the experimental 272 K. Jiao, X. Li / Progress in Energy and Combustion Science 37 (2011) 221e291 Fig. 49. Liquid water transport in a three-cell stack with serpentine flow channel design when liquid water droplets are injected [11]. study in [382] showed that maintaining back pressure at the cathode exit is an indication that the product water is removed properly. 10.2. Other aspects Even though the various numerical simulations have been carried out to investigate liquid water transport in flow channel, it is worthwhile to be noticed that most of the studies neglected the effects of GDL. In fact, the flow characteristics in flow channel with and without GDL can be significantly different. Fig. 54 shows the pressure drops from inlet to outlet of a serpentine flow channel with a GDL compressed to different thicknesses and without GDL [189]. The pressure drops with GDL were experimentally measured by inserting various metal shims parallel to the GDL to control the thickness, and the pressure drop without GDL was calculated based on known flow resistances. For the most uncompressed condition, the porosity and permeability of the GDL are all the highest, allowing more gas flow through it. By further compressing the GDL, it is harder for gas flowing through it, and gas can only flow in flow channel if without GDL. It can be observed in Fig. 54 that the pressure increases with the decrement of GDL thickness for the same inlet Reynolds number, and it is the highest without GDL. The differences among the different compressions and without GDL are Fig. 51. Liquid water transport in a cross section of inlet manifold corresponding to the water transport process in Fig. 50 [11]. very significant, especially at high inlet Reynolds number. The observation in Fig. 54 indicates that GDL has to be considered for studying the flow characteristics in flow channel, especially when strong flow through GDL is possible. However, as mentioned in Section 8, the small pores in GDL need very fine grids, and therefore usually only small samples of GDL were simulated to investigated liquid water behaviours. For the computational domain of real GDL micro-structure with a dimension corroding to the length of flow channel (e.g. on the level of 10 or 102 mm), extremely large number of grid is needed, which is not affordable with most of the current computational power. However, with the porous formulation by assuming homogenous GDL (as described in Section 6), it is still possible to simulate flow channel and GDL simultaneously, which may result in unrealistic liquid water behaviour in GDL. It is also worthwhile to mention that a new concept of “porous media flow field” has been proposed for fuel cell water management, and promising potential in performance improvement has also been demonstrated through numerical modeling [383,384]. This concept is to fill porous media in the flow channel region, allowing a simultaneous transport of gas, liquid, heat and electron through the porous media flow channel. With such porous flow channel, the transport in flow channel can be modeled in a similar way as in GDL [383,384]. 10.3. Summary Fig. 50. Liquid water transport in a three-cell stack with serpentine flow channel design when a large amount of liquid water is supplied [11]. The different models for water transport in flow channel are reviewed in Section 10. The previously described full cell models 273 K. Jiao, X. Li / Progress in Energy and Combustion Science 37 (2011) 221e291 1.4 1.2 Pres s ure D rop, kPa a Cell 1 Cell 2 Cell 3 Overall 30 Cell 1 Cell 2 Cell 3 Overall 25 Pres s ure D rop, kPa a 1 0.8 0.6 20 15 10 5 0.4 0 0.2 0 0.5 1 1.5 2 2.5 0 3 0.5 1 0.6 Pressure D rop, kPa 0.5 Cell 1 Cell 2 Cell 3 Overall 0.4 b 1 Pres s ure D rop, kPa b 1.5 2 2.5 3 Time, ms Time, ms 0.8 0.3 Cell 1 Cell 2 Cell 3 Overall 0.6 0.4 0.2 0.2 0 10 20 30 40 50 0 60 10 20 11. Cold start In winter conditions, it is unavoidable for vehicles driving below the freezing point of water (0  C), therefore, for successful commercialization of PEMFC in automotive applications, rapid startup from subzero temperatures must be achieved, which is referred to as “cold start” of PEMFC. The major problem of PEMFC cold start is that the product water freezes when the temperature inside PEMFC is lower than the freezing point of water. If the CL is fully covered by ice before cell temperature rises above freezing point, the electrochemical reaction may be stopped due to the blockage of the reaction site (as shown in Fig. 3a). In addition, ice formation may also result in serious damage to the structure of the MEA. Therefore, for PEMFC in automotive applications, successful cold start is of paramount importance. PEMFC cold start capability 50 60 70 Fig. 53. Pressure drops for each single cell and the whole stack corresponding to the water transport process in Fig. 50 (a: for the first 3 ms; b: for the whole process) [11]. still needs significant improvement, especially for unassisted cold start, this is because assisted cold start might increase the volume and weight of the system, as well as the operation complexity and installation costs. 160 140 Pressure D rop, kPa (Section 6), and the VOF and LB models (Section 8) can all be used for modeling water transport in flow channel. Due to the numerical instability, the LB model often assumes unit density ratio between gas and liquid. This assumption may be reasonable for flow in GDL and CL when the capillary effect dominates. However, for the convection dominated flow in flow channel, the inertia force becomes significant, and the unit (or low) density ratio assumption fails. The most suitable model for simulating liquid water transport in flow channel so far is therefore the VOF model. 40 Time, ms Time, ms Fig. 52. Pressure drops for each single cell and the whole stack corresponding to the water transport process in Fig. 49 (a: for the first 3 ms; b: for the whole process) [11]. 30 120 100 80 60 8 μm 10 μm 12 μm No GDL 40 20 0 0 1000 2000 3000 4000 5000 6000 Inlet Reynolds Number Fig. 54. Pressure drops from inlet to outlet of a serpentine flow channel with a GDL compressed to different thicknesses and without GDL [189]. 274 K. Jiao, X. Li / Progress in Energy and Combustion Science 37 (2011) 221e291 11.1. Experimental work Experimental studies for PEMFC cold start have been carried out by focusing on the effects of cell design and operating conditions on cold start performance, material degradations, property measurement at subzero temperatures, and so on. Tajiri et al. [385,386] designed experimental procedures to mainly investigate the effects of MEA characteristics on PEMFC cold start, and the initial water content in Nafion membrane was controlled by different purging methods. They found that the best cold start capability is achieved with the lowest initial water content, because more product water can be taken by the membrane, resulting in less water freezing in the CL. Hou et al. [387,388] investigated the performance degradations after freezeethaw cycles, and they indicated that membrane degradation is not a major issue for PEMFC cold start. Cho et al. [389,390] studied the freezeethaw cycles with a subfreezing temperature of 10  C, they reported that the unpurged initial condition resulted in severe performance degradation, and the solution purge (purged with anti-freezes such as methanol) might be an effective way to achieve fast startup. Similarly, other experimental studies that focused on the effects of purging methods [391], startup temperatures [392], and purging temperatures [393] on cold start performance were also carried out. Thompson et al. [394] experimentally determined the ORR kinetics for PEMFC operating at subzero temperatures, and no significant change in the ORR mechanism was found. Ge and Wang [395,396] conducted the first two in situ visualizations of ice formation on the cathode CL surface by cooling the transparent cells with coolant recirculation, and they have reported that the freezing point of water in the CL is around 1  C. Ishikawa et al. [397] also reported that the freezing temperature of liquid water in the CL changes between 0.1 and 2.2  C. All these studies [395e397] showed that liquid water freezes at the temperatures lower than 0  C so-called the freezing point depression, which is mainly due to the wettability and extremely small pore size in CL, known as the GibbseThomson undercooling [398]. Sun et al. [399] described a new method called catalytic hydrogen/oxygen reaction assisted cold start by supplying mixed hydrogen and air (or oxygen) into the fuel cell, however, further investigations are needed for this method. McDonald et al. [400] conducted an ex situ freezeethaw study on MEA, and they reported that no degradation was observed for dry MEA. Cappadonia et al. [65,66] performed experimental measurements of Nafion membrane conductance at various subzero temperatures, and Thompson et al. [64] reported the conductivities of Nafion membrane at different subzero temperatures based on the measured membrane conductance and sample size. Chacko et al. [401] measured the HFR of a PEMFC before, during and after cold start. The measurements in [64e66,401] all observed decrements of membrane conductance/conductivity at subzero temperatures. In [64], it was also found that such decrements occurred when the water phase changes were observed, and the amount of non-frozen membrane water content at different subzero temperatures were also estimated. 11.2. Numerical model Numerical models for PEMFC cold start have also been developed. Sundaresan and Moore [402] conducted an analytical model for cold start of PEMFC stacks. This one-dimensional model could predict the temperature for each of the single cells by performing energy balance and heat transfer analysis, and this model could also reveal the effects of the endplate thermal mass and the internal heating method for PEMFC cold start. Khandelwal et al. [403] also conducted a one-dimensional thermal model, and similar to Sundaresan and Moore [402], the cold start capability of PEMFC stacks can be evaluated. They reported that adjusting the startup current density, coolant heating, isolation of stack endplates are all effective ways to optimize the cold start performance. Mao and Wang [404] developed an analytical model, not only for temperature, this onedimensional model can also predict the amount of ice formation in CL, water transport, changes of cell voltage and current density etc. Wang [405,406] also conducted analytical studies and defined some important parameters affecting PEMFC cold start performance, and a three-step electrode process was also defined. The analytical models in [402e406] can only roughly predict the PEMFC cold start performance, and in order to investigate the fundamental physics of PEMFC cold start, multi-dimensional and multiphase models are needed. However, not many literatures are related to this field [258e260,407e409]. Ahluwalia and Wang [407] conducted a simple two-dimensional cold start model for single PEMFCs, and they reported that high startup current density is favourable for rapid cold start, and they also investigated the effects of feed gas temperatures, operating pressure, and electrical heating on cold start performance. Mao et al. [258] developed a threedimensional, multiphase model, Meng [408] conducted a twodimensional multiphase model for PEMFC cold start simulations, and based on the work in Mao et al. [258,404], Jiang et al. [259,260,409] further conducted non-isothermal cold start simulations for PEMFC, and evaluated the effects different design and operating parameters on cold start performance. However, all the numerical models in [258e260,407e409] assumed instantaneous desublimation of water vapour to ice, and the reaction product is water vapour. Since no liquid water is considered, these models can only provide reasonable results when the temperature is lower than the freezing point of water. It is worthwhile to mention that several correlations for proton conductivity of Nafion membrane at subzero temperatures have been developed from available experimental data. These correlations can be found in [62,259,406]. Development of multi-dimensional, multiphase PEMFC cold start model with the capability to consider the detailed phase changes of water in both the membrane electrolyte and the pore volumes of the GDL and CL needs to be carried out for better predictions of cold start performance. Recently, a three-dimensional multiphase cold start model has been developed with the capability to account for all kinds of water (vapour, liquid water, ice, non-frozen water in ionomer and frozen water in ionomer) [62]. This model was then used to evaluate different design and operating parameters on cold start performance [63,261]. This model can be briefly represented by the following conservation equations. Mass of gas mixture (solved in flow channel, GDL and CL): v

3 1 vt slq  
 ! sice rg þ V$ rg u g ¼ Sm (104) Momentum of gas mixture (solved in flow channel, GDL and CL): ! ! ! rg ! rg ! ug ug ug v þ V$ vt 3 1 slq sice 32 1 slq sice 2 ! ! ug ¼ Vpg þ mg V$ V þV 3 1 slq sice 3 1 !! ! ug 2 m V V$ þ Su 3 g 3 1 slq sice !T ug slq sice !! ð105Þ Gas species (solved in flow channel, GDL and CL, i represents hydrogen, oxygen or vapour): v

3 1 vt slq  

 ! sice rg Yi þ V$ rg u g Yi ¼ V$ rg Deff i VYi þ Si (106) 275 K. Jiao, X. Li / Progress in Energy and Combustion Science 37 (2011) 221e291 Liquid water (solved in GDL and CL): can be represented by Equation (64) Ice (solved in GDL and CL): vð3sice rice Þ ¼ Sice vt rmem EW vt In Membrane Non-frozen Membrane Water In CL Non-frozen Membrane Water (107) Non-frozen membrane water (solved in membrane and CL): can be represented by Equation (65) Frozen membrane water (solved in membrane): 
v ulf Normal Operating Condition: In GDL Vapour Vapour Liquid Liquid Cold Start: ¼ Sfmw In Membrane Non-frozen Membrane Water Frozen Membrane Water In CL Non-frozen Membrane Water Vapour (108) Energy (solved in whole computational domain): can be represented by Equation (66) Electronic potential (solved in CL, GDL, BP): can be represented by Equation (67) Ionic potential (solved in CL, membrane): can be represented by Equation (68) Apparently, this cold start model is essentially based on the framework of the top-down two-fluid model described in Section 6, by adding two more conservation equations for ice in pores of CL and GDL and frozen water in membrane (Equations (107) and (108)). The phase changes between all different water are accounted for in the source terms, in a similar approach to the two-fluid model. It is also worthwhile to be mentioned that another three-dimensional multiphase cold start model was developed based on the mixture model framework described in Section 6 by replacing liquid water by ice [258e260] (liquid water and frozen water in ionomer were neglected in this model). So far the models based on the two-fluid model framework [62,63,261] and based on the mixture model framework [258e260] are the only two three-dimensional multiphase cold start models, and the cold start model based on the two-fluid model framework [62,63,261] is the only model that fully accounts for all kinds of water (vapour, liquid water, ice, non-frozen water in ionomer and frozen water in ionomer) with non-equilibrium phase change processes. Since the two-fluid model is described in Section 6, therefore the details of the cold start model based on the two-fluid model framework are not presented here. As shown in Fig. 4, the phase change of water is very complicated and therefore hard to be implemented. Due to the fact that non-humidified inlet gases are often supplied during PEMFC cold start to avoid ice formation. Therefore the non-equilibrium water transfer into and out of ionomer can be assumed only between non-frozen water in ionomer and vapour in pores. In CL, for simplicity, it was also assumed that the non-frozen water in ionomer freezes to ice in pores [62,63,261]. Based on these simplifications, the water phase change processes in different components of PEMFC for both the normal operating condition and cold start with partially or non-humidified inlet gases are shown in Fig. 55 [62], which was implemented in the cold start model in [62,63,261]. Similar to the two-fluid model described in Section 6, this cold start model also assumes that the non-frozen water in ionomer is equivalent to liquid water [137], and therefore the latent heats of the non-frozen water in ionomer and liquid water were assumed to be the same, as it was shown that the difference is very small [410]. For modeling water freezing and melting in ionomer, Equation (9) can be used to calculate the maximum allowed non-frozen water content in ionomer, and the phase change function can be implemented in the source terms of the water conservation equations as described in Section 6. The experiments in [395e397] all observed that the fusion and desublimation of water in CL take place at the temperatures below the In GDL Vapour Liquid Liquid Ice Ice Fig. 55. Schematics of water phase change in different components of PEMFC for both normal operating condition and cold start (with partially or non-humidified gases supplied) [62]. normal freezing point of water (273.15 K), which is mainly due to the wettability and extremely small pore size in CL, known as the GibbseThomson undercooling [398]. The difference between the freezing point in small pores and normal freezing point of water (TN ¼ 273.15 K) is defined as the freezing point depression, TFPD (K), and can be calculated as TFPD ¼ TN sj273:15K cosq rice hfusn rCL;GDL (109) where hfusn (J kg 1) is the latent heat of fusion for water, rCL,GDL (m) the pore radius of CL or GDL, s (N m 1) the surface tension coefficient between liquid water and gas, and rice (kg m3) the density of ice. By using typical values of these parameters, it was calculated that TFPD is about 1 K in CL and about 0 K in GDL [62]. Also note that the ice formation in flow channel was neglected in all the models to the best of the authors’ knowledge, because ice is formed in CL and GDL first, and sticks on the solid materials so that hard to move. Details of the cold start model can be found in [62,63]. 11.3. Cold start characteristics Based on the cold start model developed in [62], extensive analyses of different colds start processes were carried out [62,63,261]. Fig. 56 shows the evolutions of current densities, ice volume fraction in cathode CL and cell temperature for both the 3  C) cold start failed (from 10  C) and successful (from processes [62]. For the failed cold start process shown in Fig. 56a, the current density increases quickly at the beginning due to the fast electrochemical double layer charging and discharging process. Then the variation of the current density becomes the minimum for a period of time, owing to the combined effects of the membrane hydration, temperature increment and ice blockage. Finally the current density drops fast indicating that the cold start process is failed. The fast drop of current density is caused by the ice blockage on the active catalyst surface, and it occurs when the ice volume fraction in the cathode CL reaches unity (higher than 0.9). The ice volume fraction in cathode CL increases almost linearly in Fig. 56a. During a cold start process, the product water is absorbed by ionomer, taken by gas streams and freezes to ice. Before ice formation can take place in CL pores, the water content in the ionomer of CL must reach the saturation value to freeze (Equation (9)). Therefore the ice volume fraction in cathode CL in Fig. 6 remains unchanged in the first second of the cold start process because the product 276 Current D ens ity, A cm -2 a 0.2 1 0.18 0.8 0.16 0.6 0.14 0.4 0.12 0.2 Ice V olume Fraction in Catho de CL K. Jiao, X. Li / Progress in Energy and Combustion Science 37 (2011) 221e291 0.1 0 2 4 6 8 10 12 14 10 9 8 7 6 5 4 3 2 1 0 -1 -2 -3 0.25 0.2 0.15 0.1 0.05 0 0 5 10 15 20 25 30 o Current D ens ity, A cm -2 b Cell Temperature, C Time, s 35 Time, s Fig. 56. Evolutions of current densities, ice volume fraction in cathode CL and cell temperature for a failed (a: from 10  C) and a successful (b: from 3  C) cold start processes [62]. water is absorbed by ionomer. This indicates that purging the cell to ensure dry ionomer (more capacity to store product water) is critical to reduce ice formation. For the successful cold start process shown in Fig. 56b, the current density increases with the increment of the cell temperature, and the increment of the cell temperature becomes the lowest at around 0  C, because heat is needed for ice melting. Corresponding to the failed cold start process shown in Fig. 56a, the transient ice volume fraction in a cross section of cathode CL, non-frozen water content in a cross section of membrane and CLs, and temperature in a cross section of all cell components are presented in Figs. 57e59, respectively [62]. In Fig. 57, the ice first appears under the land because the temperature is low in those areas (close to the surrounding walls), which result in low saturation water content. In the areas close to the membrane, more water in the ionomer of CL could diffuse into the membrane. Therefore, the ice is first generated at the locations away from the membrane interface. As time passes, more ice is formed and the cathode CL is fully blocked (12 s), indicating that no further electrochemical reaction can take place. In Fig. 58, it can be observed that the nonfrozen water content in the cathode CL increases fast and the changes of the water content in the membrane and anode CL are relatively slow. The reason is that the water production rate is higher than the water diffusion rate in the ionomer (low diffusivity at subzero temperatures). Fig. 58 even shows that the water content difference across the membrane is still very significant even at 35 s (23 s after the electrochemical reaction stops), suggesting that increasing the ionomer fraction in the cathode CL may have more significant effects than increasing the thickness of the membrane layer in reducing the amount of ice formation. In Fig. 59, the highest temperature is very close to the membrane due to the ohmic heating, a slight shift to the cathode CL due to the other heating sources (activational heat, reversible heat and latent heat). The temperature is also the highest under the flow channel rather than under the land because the heat is lost at the surrounding walls. It is also observed that the temperature in the anode is slightly higher than in the cathode, and the reasons are: 1) the ohmic heat is the highest in the anode CL (a low water content results in a low membrane conductivity); 2) the heat transfer rate is higher in the anode, due to the higher thermal conductivity of hydrogen by comparing with oxygen and the blockage of ice in the cathode CL. Corresponding to the successful cold start process shown in Fig. 56b, the transient ice and liquid water volume fractions in a cross section of CLs and GDLs are shown in Figs. 60 and 61, respectively [62]. In Fig. 60, the ice first melts in the CLs under the land (t ¼ 1 s), then the whole CL and the GDL under the land, and finally the whole area. It is also observed that the ice melting in the anode is slightly faster than in the cathode, due to the faster heat transfer in the anode, as mentioned earlier. In Fig. 61, the locations of the liquid water formation matches the locations of the ice melting in Fig. 60. It should be noticed that the liquid water in the cathode CL becomes the maximum at 3 s, and at 30 s, the liquid water in the anode is still decreasing and it remains almost unchanged in the cathode, because water is produced in cathode. The transport phenomena during PEMFC cold start are investigated in Figs. 56e61. The key parameters affecting PEMFC cold start performance also need to be identified. The effects of membrane thickness on cold start performance for both the potentiostatic and galvanostatic cold start processes from 20  C are shown in Fig. 62 [261]. During a potentiostatic cold start process, the cell voltage is controlled while the current density varies and mainly depends on the cell voltage, temperature increment/decrement, ionomer hydration/dehydration and water freezing/melting. On the other hand, the current density is controlled during a galvanostatic cold start process and therefore the cell voltage changes. For the potentiostatic cold start processes in Fig. 62a, it can be noticed that the highest current densities achieved during the cold start processes are about 0.19, 0.25 and 0.49 A cm 2 with Nafion 117, 115 and 112, respectively. Such significant difference among the three membranes is mainly attributed to the different ohmic losses, which are mainly caused by the difference in the membrane thickness. Since the membrane conductivities at subzero temperatures are much lower than at normal operating temperatures (e.g. the membrane conductivity with a water content of 15 at 20  C is only about 10% of at 80  C), the effects of membrane thickness on the cell performance are therefore more significant at subzero temperatures. Due to the fact that the water production rate is proportional to the current density, the ice formation process is the slowest with Nafion 117 and the quickest with Nafion 112, and the failure times of the cold start processes are about 8, 7, and 4 s with Nafion 117, 115 and 112, respectively. For the galvanostatic cold start processes in Fig. 62b (the water production rates are same), different from the potentiostatic condition, the thinnest membrane results in the slowest ice formation process for the galvanostatic condition, the reason is that the Nafion 112 membrane can absorb water more quickly than the other two membranes (to be explained with Fig. 64). The evolutions of various heat generation rates, heat loss rates through BPs, and cell temperatures when Nafion 112 is used for both the potentiostatic and galvanostatic cold start processes from K. Jiao, X. Li / Progress in Energy and Combustion Science 37 (2011) 221e291 277 Fig. 57. Transient ice volume fraction in a cross section of cathode CL corresponding to the failed cold start process in Fig. 56a [62]. 20  C are presented in Fig. 63 [261]. During a cold start process, heat is generated from the electrochemical reactions (activational heat and reversible heat), electron and ion transport (ohmic heat), and water phase change (latent heat). For the potentiostatic cold start process in Fig. 63a, it can be observed that the largest heating source is the ohmic heat, mainly due to the low membrane conductivity at subzero temperatures. The activational heat is the second largest heating source, followed by the reversible heat and latent heat. The heat loss is caused by the heat transfer at the outer surfaces of the BPs and by the outflow of the gas streams, and it has been found that the heat loss due to the outflow of the gas streams can be neglected by comparing with the heat loss from the BPs [62]. For the galvanostatic cold start process in Fig. 63b, the activational heat generation rates increase when the ice volume fractions are high, due to the fact that higher activational energy is needed to maintain the current density when the reaction area becomes smaller due to the ice blockage. The largest heating source is the activational heat, since the current density is lower than the potentiostatic condition in Fig. 63a. The latent heat is the lowest. The heat losses to surroundings all increase with the increment of cell temperature for both the potentiostatic and galvanostatic cold start processes in Fig. 63, and the cell temperature increment becomes slower during the cold start process. To further examine the effects of the membranes on the ice formation processes, the evolutions of amounts of ice formation, amounts of water absorbed by ionomer, and amounts of water taken by the gas streams for the potentiostatic (cell voltage is 0.3 V) and galvanostatic (current densities are 0.15 and 0.05 A cm 2) cold start processes from 20  C are shown in Fig. 64 [261]. All the water amounts shown in this figure are normalized by the amounts of the water production. It should be noticed that the summation of the amount of water absorbed by the ionomer, amount of ice formation and amount of water taken by the gas streams is equal to the amount of the water production. Therefore, the normalized amounts of water in Fig. 64 represent the percentages of the product water in different forms. In Fig. 64a for the potentiostatic cold start process, it can be observed that the amount of water taken by the gas streams can be neglected, due to the fact that the saturation pressures of water vapour are very low at subzero temperatures. Initially most of the product water is absorbed by the ionomer, and most of the absorbed water is in the cathode CLs (shown in Fig. 58). With the increments of the nonfrozen water content in the ionomer of the cathode CLs, the saturation levels are reached, and then most of the product water becomes ice. Fig. 64a shows that the Nafion 117 membrane is able to absorb the largest portions of the product water among the 278 K. Jiao, X. Li / Progress in Energy and Combustion Science 37 (2011) 221e291 Fig. 58. Transient non-frozen water content in a cross section of membrane and CLs corresponding to the failed cold start process in Fig. 56a [62]. three membranes before the CLs are severely blocked by ice. However, for the galvanostatic (0.15 A cm 2) cold start process in Fig. 64b, the water production rates are the same for the different membranes (same current density), therefore it is more meaningful to compare the effects of membrane thickness on ice formation. It can be noticed that by using Nafion 112 the ionomer can absorb about 10% more of the product water than by using Nafion 115 and 117, and the reason is that a thinner membrane results in a larger water gradient across the membrane, hence faster water absorption. Advantage of Nafion 112 in absorbing product water is more apparent in Fig. 64c when the current density is lower (0.05 A cm 2). In this case, the water production is also lower, allowing more water being absorbed by ionomer before ice fully plugging CL. Even the thinnest membrane (Nafion 112) shows the greatest effect on reducing ice formation, it should be mentioned that the thickness of membrane still needs to be sufficiently high to maintain enough water capacity to store product water. With the same membrane (Nafion 117), the effects of ionomer volume fraction on ice formation are investigated in Fig. 65 [261]. It can be observed that increasing the ionomer volume fraction in the CLs from 0.2 to 0.4 can absorb 20% more of the product water, which is a significant improvement in reducing the ice formation. One of the reasons is that the higher ionomer volume fraction increases the water capacity of the ionomer in the CLs, and another is that it also provides wider paths for the water transport in the CLs, therefore the membrane can absorb the product water in the cathode CLs more effectively. Fig. 66 presents the evolutions of current densities and cell temperatures for the self and assisted potentiostatic (cell voltage is 0.3 V) cold start processes from 20  C when Nafion 117 is used [261]. It can be noticed that cell insulation, heating the outer surface and inlet air can all accelerate the temperature increment, however, since all the cold start processes failed at about the same time (about 9 s), the ice formation rate is not effectively reduced. It was reported that a heating power of 0.04 W cm 2 is needed to increase the cell temperature for 1  C in 9 s when heating the outer surface, however, a heating power of only 0.01 W cm 2 is needed for the same effect when heating the inlet air [261]. This indicates that heating the inlet air is more efficient than heating the outer surface, because heating the inlet air results in higher temperature increment in CL, which increases the reaction rate and therefore the heat generation. However, heating on the outer surface may be easier than heating the inlet air, because the inlet air must be heated to very high temperatures to carry enough amount of heat, or the inlet air flow rate needs to be very high. Another effective method to accelerate the heating-up process is reducing the total thermal mass of the cell [261]. Generally, based on the present understanding of PEMFC cold start, a thinner membrane is more favourable in reducing the ice formation since it can result in a larger water gradient across the membrane thus accelerating the water absorption from the cathode CL into the membrane. However, the membrane thickness still needs to be kept sufficiently high to ensure a sufficient amount of water capacity to store the product water. The potentiostatic condition is K. Jiao, X. Li / Progress in Energy and Combustion Science 37 (2011) 221e291 279 Fig. 60. Transient ice volume fraction in a cross section of CLs and GDLs corresponding to the successful cold start process in Fig. 56b [62]. Fig. 59. Transient temperature in a cross section of all cell components corresponding to the failed cold start process in Fig. 56a [62]. more favourable than the galvanostatic condition for PEMFC cold start. Even though larger portions of the product water can be absorbed by the ionomer at lower current densities to reduce the ice formation rate, operating at high current densities is still favourable due to the faster heating-up. The gas streams can only take negligible portion of the product water due to the low saturation pressures of water at subzero temperatures, and therefore has negligible improvement in reducing the ice formation. Optimizing the ionomer volume fraction in the CLs is an effective way to decelerate the ice formation by increasing the water capacity of the ionomer in the CLs and accelerating the water diffusion into the membrane. The external heating on the outer surfaces of PEMFCs results in direct improvements in raising the cell temperature, however, with negligible improvement in reducing the ice formation. Heating up the inlet air can increase the cell temperature more effectively than applying heat on the outer surfaces of PEMFCs. However, heating on the surfaces of PEMFCs may be easier to implement than heating the inlet air, because the inlet air must be heated to very high temperatures to carry enough amount of heat, otherwise the inlet air flow rate must be very high. Heating up the inlet air also has negligible improvement in reducing the amount of ice formation. Reducing the total thermal mass of the cell has significant improvements in accelerating the heating up of the cell. 11.4. Summary The previous studies on PEMFC cold start are reviewed in Section 11. It has been identified that ice formation that hinders Fig. 61. Transient liquid water volume fraction in a cross section of CLs and GDLs corresponding to the successful cold start process in Fig. 56b [62]. 280 K. Jiao, X. Li / Progress in Energy and Combustion Science 37 (2011) 221e291 0.4 0.7 0.35 0.6 0.5 Nafion 117 Nafion 115 Nafion 112 0.25 0.4 0.3 0.2 0.2 0.15 0.1 Current 1 2 3 4 5 6 7 8 -15 Activational heat 0.6 0.3 Reversible heat Cell V oltage, V -19 Heat loss Latent heat 0.1 -20 1 2 3 4 Time, s b 0.9 0.8 0.7 0.65 0.6 0.6 0.5 0.55 0.4 0.5 0.3 0.45 0.2 0.4 Voltage 0.3 0.1 0 0 1 2 3 4 5 6 7 8 9 10 11 12 13 -15 -16 0.2 o 1 Ice 0.7 0.35 -18 0.2 0 Heat Generation and Los s Rates , W 0.8 0.75 Nafion 117 Nafion 115 Nafion 112 -17 Cell temperature 0.4 9 Ice V olume Fractio n in Catho de CL 0.85 -16 0.5 Time, s b -14 0.7 0 0 0 Ohmic heat 0.8 Activational heat Cell temperature Fig. 62. Effects of membrane thickness on cold start performance for potentiostatic (a: cell voltage is 0.3 V) and galvanostatic (b: current density is 0.15 A cm 2) cold start processes from 20  C [261]. reactant delivery and damages cell materials is the major issue for PEMFC cold start. Enhancing water absorption by membrane electrolyte is the most effective way to reduce the ice formation for self cold start. Cell purging, increasing ionomer fraction in CL, proper control of membrane thickness and startup current (or voltage), and differential pressurization are all proper ways to enhance membrane electrolyte water absorption. Reducing cell thermal mass is useful for accelerating temperature increment. Assisted cold start methods mainly involving external heating have also been approved for fast heating-up. 12. High temperature polymer electrolyte membrane fuel cell HT-PEMFCs with operating temperatures higher than 100  C have attracted growing interests in the past decade. By comparing with conventional PEMFCs operating at around 80  C, HT-PEMFCs with elevated operating temperatures feature faster electrochemical kinetics, simpler water management (presence of liquid water can be neglected, and membrane hydration may not be needed), higher CO tolerance (e.g. >1% CO at 150  C [411]), and easier cell cooling and waster heat recovery. Although HT-PEMFCs have many attractive features, technical challenges still remain and are mostly related to the membrane. PFSA polymer membranes (e.g. Nation membranes) widely used in -17 Ohmic heat 0.1 -18 Reversible heat Heat loss -19 Latent heat 0 -20 0 1 2 3 4 5 6 7 8 9 10 11 12 13 Time, s Time, s b Cell Temperature, C 0.3 0.9 o -2 0.8 -13 1 Cell Temperature, C Ice Heat Generation and Los s Rates , W 0.9 0.45 Current D ensity , A cm a 1 0.5 Ice V olume Fractio n in Catho de CL a Fig. 63. Evolutions of various heat generation rates, heat loss rates through BPs, and cell temperatures when Nafion 112 is used for potentiostatic (a: cell voltage is 0.3 V) and galvanostatic (b: current density is 0.15 A cm 2) cold start processes from 20  C [261]. conventional PEMFCs suffer significant decrement in mechanical strength at the high operating temperature of HT-PEMFCs, and the much lower relative humidity in HT-PEMFCs than in conventional PEMFCs due to the significantly increased vapour saturation pressure with temperature also results in severe reduction of proton conductivity of the PFSA polymer membranes. Therefore, developing membranes with high mechanical strength at the temperatures higher than 100  C and with high proton conductivity in anhydrous environments becomes the major challenge, and most of the previous HT-PEMFC related researches focused on this important issue [412]. PBI membranes first proposed by Aharoni and Litt [413] have been investigated in the previous studies and recognized as a promising membrane when doped with a strong oxo-acid (e.g. phosphoric acid or sulphuric acid) for HT-PEMFCs [414e416]. Moreover, phosphoric acid doped PBI membrane first suggested for fuel cell applications by Wainright et al. [417] has attracted most of the attentions due to its relatively higher proton conductivity and mechanical strength by comparing with the other types of acid doped PBI membranes. Therefore, in this section, only the HT-PEMFCs with PBI membranes are focused on, and excellent discussions on the development of different high temperature membranes can be found in [412]. 12.1. Experiential work The proton conductivity measurements of phosphoric acid doped PBI membranes have been carried out [411,417e420] and it 281 K. Jiao, X. Li / Progress in Energy and Combustion Science 37 (2011) 221e291 a 100 110 90 100 90 A mount of Water, % 80 A mount of Water, % Ionomer volume fraction in CL = 0.2 Ionomer volume fraction in CL = 0.4 70 Ice formation 60 Nafion 117 Nafion 115 Nafion 112 50 40 Ionomer absorption 30 20 10 Ice formation 80 70 60 50 40 30 20 Ionomer absorption 10 0 0 Taken by gas streams 0 1 2 3 4 5 6 7 8 Taken by gas streams 0 9 1 2 3 4 b 6 7 8 9 Fig. 65. Evolutions of amounts of ice formation, amounts of water absorbed by ionomer, and amounts of water taken by the gas streams when Nafion 117 is used for potentiostatic (cell voltage is 0.3 V) cold start processes from 20  C with different ionomer volume fractions in CLs (water amounts are all normalized by amounts of the water production) [261]. 100 90 80 A mount of Water, % 5 Time, s Time, s 70 Ice formation 60 Nafion 117 Nafion 115 Nafion 112 50 40 Ionomer absorption 30 20 10 0 Taken by gas streams 0 1 2 3 4 5 6 7 8 9 10 11 12 13 conductivity. The experimental study in [421] reported that the thermal stability of a PBI membrane with a doping level of 4.8 (4.8 phosphoric acid molecules per PBI repeat unit) is more than enough for use as a membrane in HT-PEMFCs. Weng [422] et al. concluded that the EOD effect is negligible in PBI membranes, which could further simplify the water management of HT-PEMFCs. In situ tests of HT-PEMFCs were also conducted and promising cell performances were obtained under various operating conditions with good CO tolerance and acceptable performance degradation [419,423e425]. Time, s 12.2. Numerical model c 100 Nafion 117 Nafion 115 Nafion 112 Ionomer absorption 90 70 60 50 40 0.2 30 0.19 Taken by gas streams 0 10 20 30 40 50 60 70 Time, s Fig. 64. Evolutions of amounts of ice formation, amounts of water absorbed by ionomer, and amounts of water taken by the gas streams for potentiostatic (a: cell voltage is 0.3 V) and galvanostatic (b: current density is 0.15 A cm 2; c: current density is 0.05 A cm 2) cold start processes from 20  C (water amounts are all normalized by amounts of the water production) [261]. -8 0.18 Current 0.17 0.16 -12 0.15 -14 0.14 0.13 a b c d e 0.12 0.11 has been found that the temperature, phosphoric acid doping level and surrounding relative humidity all have significant effects on the proton conductivity. It was shown that the proton conductivity of phosphoric acid doped PBI membranes increases with temperature by following the Arrhenius Law [417], and the experimental measurements in [411,418e420] also observed that increasing both the phosphoric acid doping level and surrounding relative humidity all have significant improvements on the proton -10 Temperature 0.1 0 1 2 3 4 5 6 7 -16 o Ice formation 0 Current D ensity , A cm 10 -6 Cell Temperature, C 20 -2 A mount of Water, % 80 Numerical models for HT-PEMFCs with phosphoric acid doped PBI membranes have also been developed in the previous studies [425e430]. Cheddie and Munroe developed a one-dimensional model [426] and then further extended their model to threedimensional [427]. A three-dimensional model similar to [427] was introduced in [425] as well. Both the steady and unsteady three- -18 -20 8 9 Time, s Fig. 66. Evolutions of current densities and cell temperatures for self and assisted potentiostatic (cell voltage is 0.3 V) cold start processes from 20  C when Nafion 117 is used (a: self cold start without cell insulation; b: cold start with cell insulation; c: cold start with 0.045 W cm 2 heating on outer surface and with cell insulation; d: cold start with 0.18 W cm 2 heating on outer surface and with cell insulation; e: cold start with inlet air heated to 80  C without cell insulation) [261]. 282 K. Jiao, X. Li / Progress in Energy and Combustion Science 37 (2011) 221e291 f1 f2 exp T  Eact RT  (110) where f1 and f2 are the two different pre-exponential factors, T (K) the temperature, R (8314 J kmol 1 K 1) the universal gas constant, and Eact the activation energy (J mol 1) can be calculated as [430] Eact ¼ 619:6DL þ 21; 750 (111) where DL is the phosphoric acid doping level of PBI membranes, which is defined as the number of phosphoric acid molecules per PBI repeat unit. The pre-exponential factors f1 and f2 in Equation (110) further accounts for the effect of phosphoric acid doping level and humidification on the membrane proton conductivity, and can be calculated as [430] f1 ¼ 168DL3 6324DL2 þ 65; 760DL þ 8460 (112) 8 < 1 þ ð0:01704T 4:767Þa; if 373:15 K  T  413:15 K f2 ¼ 1 þ ð0:1432T 56:89Þa; if 413:15 K < T  453:15 K : 1 þ ð0:7T 309:2Þa; if 453:15 K < T  473:15 K 1.2 1 (113) psat ¼ 0:68737T 3 732:39T 2 þ 263; 390T 31; 919; 000 (114) 0.3 Cell V oltage, V where the unit of T is K, a is the corresponding surrounding water activity. In addition, for temperatures higher than 100  C, Equation (11) cannot accurately predict the water saturation pressure, and the following correlation was developed to calculate the water saturation pressure (psat, Pa) in the temperature range of 100 to 200  C [430]. 0.4 0.8 0.6 0.2 0.4 where the unit of T is K. 0 12.3. Performance The effects of operating temperature, membrane doping level, inlet relative humidity and feed gas characteristics on HT-PEMFC 0.1 Cell temperature = 190 oC Cell temperature = 150 oC Cell temperature = 110 oC 0.2 0 0.2 0.4 0.6 0.8 1 -2 kion ¼ performance with PBI membrane are shown in Figs. 67e70, respectively [430]. In Fig. 67, the cell operates with hydrogen and air without humidification at atmospheric pressure, the stoichiometry ratio is 2 for both the anode and cathode for a reference current density of 1.5 A cm 2, and the phosphoric acid doping level for the PBI membrane is 6. No apparent concentration loss is observed for all the operating temperatures due to the high stoichiometry ratios and the avoided liquid water formation. The peak power densities are obtained at a cell voltage of 0.4 V for all the operating temperatures. An increment of the peak power density of 0.065 W cm 2 (from 0.213 to 0.278 W cm 2) is obtained by increasing the operating temperature from 110 to 150  C, and the increment is 0.062 W cm 2 (from 0.278 to 0.34 W cm 2) from 150 to 190  C. The almost linear and significant increment of the peak power density with temperature indicates that operating the cell at high temperatures is favourable, and the main reasons are the enhanced electrochemical kinetics and the membrane proton conductivity at high operating temperatures. In Fig. 68, the operating condition is similar to in Fig. 67 and the operating temperature is fixed at 190  C. The results in this figure indicate that increasing the phosphoric acid doping level of PBI membrane have significant improvement on the cell performance. However, it should be noticed that the phosphoric acid doping level of 9 is still not feasible with an operating temperature of 190  C [430], and therefore further development to increase the thermal stability while keeping the phosphoric acid doping level for PBI membranes is needed. In Fig. 69, the operating condition is similar to in Fig. 67 and the operating temperature is fixed at 190  C. It should be noticed that the relative humidities of 0.25% and 3.8% at 190  C are equivalent to 100% relative humidities at 25  C and 80  C, respectively, meaning that the feed gases are fully humidified at room temperature and at 80  C. It can be noticed that humidifying the feed gases at room temperature has almost negligible improvement on the peak power density (2%, from 0.34 to 0.346 W cm 2), and the peak power density is increased by about 14% (from 0.34 to 0.386 W cm 2) by increasing the relative humidity from 0 to 100% at 80  C. However, obtaining a relative humidity of 100% at room temperature is much easier than achieving a 100% relative humidity at 80  C. To obtain a 100% relative humidity at 80  C, liquid water injection is needed if humidified at room temperature, otherwise the temperature of the humidifier needs to be increased to at least 80  C. Both of the humidification methods require more complex system design as well as extra power consumption. Therefore, humidifying the feeds gases may not be a very maneuverable way to improve the cell performance. Despite that, Pow er D ens ity, W cm dimensional models were presented by Peng et al. [428,429]. However, the numerical models in [425e427] assumed constant proton conductivities of the membranes, and only the temperature dependence of the membrane proton conductivity was considered in [428,429]. As mentioned earlier, temperature, phosphoric acid doping level and surrounding relative humidity all have significant effects on the membrane proton conductivity, therefore, these effects need to be fully accounted for in numerical models. A three-dimensional non-isothermal model of HT-PEMFCs with phosphoric acid doped PBI membranes has been developed with a semi-empirical correlation based on the Arrhenius Law and previously reported experimental data to fully account for the effects of temperature, phosphoric acid doping level and surrounding relative humidity on the membrane proton conductivity, and therefore these effects on the cell performance are all considered in this model [430]. For modeling HT-PEMFC, since the operating temperature is higher than conventional PEMFCs (negligible liquid water formation), and the relatively humidities of inlet gases are low (negligible membrane hydration), the liquid water transport and water transport in membrane can be safely neglected. Therefore, HT-PEMFC models are much simpler than conventional PEMFC models. Since the full cell models for conventional PEMFC are described in Section 6, therefore the simpler HT-PEMFC model by neglecting the transport of liquid water and water in membrane is not presented here. It is worthwhile to be mentioned that a semi-empirical correlation was first formulated based on the Arrhenius Law and previously reported experimental data to fully account for the effects of temperature, phosphoric acid doping level and surrounding relative humidity on the membrane proton conductivity (kion, S m 1) in [430]: 0 1.2 Current Density, A cm-2 Fig. 67. Effects of operating temperature on HT-PEMFC performance with PBI membrane [430]. 283 K. Jiao, X. Li / Progress in Energy and Combustion Science 37 (2011) 221e291 0.3 0.6 0.2 0.4 Doping level = 9 Doping level = 6 Doping level = 3 0.2 0 0 0.2 0.4 0.6 0.8 1 1.2 0.1 0.3 0.8 0.6 0.2 0.4 0.1 Air, St = 2, 1 atm Air, St = 4, 1 atm Air, St = 2, 2 atm Oxygen, St = 2, 1 atm 0.2 0 0 1.4 0 -2 0.2 0.4 0.6 0.8 1 1.2 -2 1 Cell V oltag e, V 0.8 0.4 -2 0.4 Pow er D ens ity, W cm 1 Cell V oltag e, V 1.2 0.5 Pow er D ens ity, W cm 1.2 0 1.4 -2 Current Density, A cm Current Density, A cm Fig. 68. Effects of PBI membrane doping level on HT-PEMFC performance [430]. Fig. 70. Effects of stoichiometry ratios of feed gases, operating pressure, and air/ oxygen on HT-PEMFC performance with PBI membrane [430]. promising cell performance can be obtained without humidification, as shown in this Figure. Fig. 70 compares the effects of stoichiometry ratios of the feed gases, operating pressure, and air/ oxygen on the cell performance. In this figure, the operating condition is similar to in Fig. 67 and the operating temperature is fixed at 190  C. It can be noticed that increasing the stoichiometry ratio from 2 to 4 for both the anode and cathode has almost negligible improvement on the cell performance, and pressurizing the cell from 1 to 2 atm results in an increment in the peak power density of 11%, and the increment is 15% by replacing the supplied air with oxygen at 1 atm. Since the other design and operating parameters are kept the same, the changes of the cell performance are only attributed to the changes of the concentrations of the reactants. Generally, the cell performances by running with hydrogen and air shown in Figs. 67e70 are very promising even without any external humidification and pressurization. The negligible liquid water formation and water transport in membrane greatly simplify the water management. The easy thermal management and high CO tolerance are also very attractive features of HT-PEMFCs. Nevertheless, the high operating temperature results in more difficulty for startup from both the normal and subzero temperatures. Therefore, so far HT-PEMFCs are mostly likely considered to be suitable for stationary applications. Further developments of high temperature 1.2 0.4 0.2 0.6 0.4 0.1 Inlet RH = 0% Inlet RH = 0.25% Inlet RH = 3.8% 0.2 0 0 0.2 0.4 0.6 0.8 1 1.2 Pow er D ens ity, W cm Cell V oltag e, V 0.3 0.8 -2 1 0 1.4 -2 Current Density, A cm Fig. 69. Effects of inlet relative humidity on HT-PEMFC performance with PBI membrane [430]. membranes as well as the materials for CL and GDL are critically important for future practical applications of HT-PEMFC. 12.4. Summary The previous studies on HT-PEMFC operating at between 100  C and 200  C are reviewed in Section 12. With simplified water and thermal management, HT-PEMFCs with acid doped PBI membranes have attracted many attentions in the past decade. Promising performances of HT-PEMFCs have already been demonstrated. The main drawback of HT-PEMFC by comparing with conventional PEMFC is the more difficult startup from both the normal and subzero temperatures. Development of membranes those are able to offer excellent proton conductivity and stability in dry and hot environment is the key for the succession of HT-PEMFC. Measurements of water and proton transport properties in the different high temperature membranes are necessary, and the measured properties are pivotal for HT-PEMFC modeling. 13. Summary and outlook Water management to ensure both effective membrane hydration and fast reactant delivery has become one of the most important issues for polymer electrolyte membrane fuel cell (PEMFC). Even though many water management strategies have been developed in the past twenty years, understanding water transport in PEMFC is of paramount importance to implement these strategies properly according to design and operating conditions. In this article, the previous studies related to water transport in PEMFC have been comprehensively reviewed. In PEMFC, water exists in forms of vapour, liquid and ice (during cold start) in pores of catalyst layer (CL) and gas diffusion layer (GDL) and in flow channel, and water in ionomer of membrane and CL can be categorized into non-freezable, freezable and free types based on how tight water molecules are bound to proton exchange sites. Understanding water transport in PEMFC is therefore an intricate study, and it requires multi-discipline knowledge that involves fluid mechanics, thermodynamics, materials science, and so on. Presently available experimental techniques are excellent tools that can predict water, various gas species, temperature and current distributions in different layers of PEMFC. Difficulties for simultaneous measurements of the various parameters, modifications of cell and system design required by measurements, and cost for materials and building testing apparatus are the disadvantages of experimental observations when compared with numerical 284 K. Jiao, X. Li / Progress in Energy and Combustion Science 37 (2011) 221e291 models. Simultaneously measuring more parameters with minimum modification of cell and system design is the primary target of future experimental observations. Modeling water transport in PEMFC involves developing rulebased and first-principle-based models. Rule-based models relying on physical rules for certain flow types have been mainly used for investigating liquid water transport in porous media and membrane electrolyte. Development of first-principle-based models relying on solving a set of governing equations has become a multi-scale work. The top-down models with homogeneous material assumption have been extensively developed for full cell modeling, which are excellent tools for investigating the various transport phenomena in different PEMFC components simultaneously, and therefore are useful for design optimization. Other top-down models have also been developed to investigate the transport phenomena in pores of GDL and CL (CL structure has to be simplified) and in flow channel and membrane electrolyte with fewer assumptions and more comprehensive treatments than the full cell models. The bottom-up models from atomistic to nanoscales involving molecular dynamics (MD) and off-lattice pseudo particle methods have been developed to study the mechanisms of water/proton transport in membrane electrolyte, self-organization of membrane electrolyte at different hydration levels, mechanisms of elementary electrochemical reaction processes in CL, material morphology in CL, and so on. The bottom-up models based on lattice pseudo particle methods, often referred to as the lattice Boltzmann (LB) model, have also been adopted to simulate gas and liquid water transport in GDL and CL as well as in simplified structures of membrane electrolyte. The different models have been adopted for different purposes to provide a comprehensive view of water transport in PEMFC. With the improvement of computational power, development of full cell models involving the volume-of-fluid (VOF) method for simulating multiphase dynamics and the chemical potential method for water/proton transport in membrane electrolyte and with GDL and CL (simplified to avoid nanometre pores) micro-structures is the ultimate goal for top-down modeling of PEMFC in the future. Increasing both the size of computational system and time scale to account for more complex transport phenomena is the trend of developing bottomup models in the future for PEMFC. An important water management related issue, cold start, has attracted many attentions in recent years. It has been identified that ice formation that hinders reactant delivery and damages cell materials is the major issue for PEMFC cold start, and enhancing the water absorption by membrane electrolyte has been identified to be the most effective to reduce the ice formation for self cold start. Cell purging, increasing ionomer fraction in CL, proper control of membrane thickness and startup current (or voltage), and differential pressurization have all been identified as effective ways to enhance membrane electrolyte water absorption. Reducing cell thermal mass has been confirmed to be useful for accelerating temperature increment. Assisted cold start methods mainly involving external heating have also been approved for fast heating-up, other methods such as supplying anti-freezes have also been investigated. Measurements of material and transport properties at subzero temperatures, and experimental and bottom-up modeling studies for understanding the water phase change processes all need to be carried out in the future researches. The information obtained from the experimental and bottom-up modeling work is critically important for developing more accurate and comprehensive top-down models for PEMFC cold start. With simplified water and thermal management, high temperature PEMFCs (HT-PEMFCs) operating at temperatures higher than 100  C have attracted many attentions in the past decade. Promising performances of HT-PEMFCs with polybenzimidazole (PBI) membranes have already been demonstrated. The main drawback of HT-PEMFC by comparing with conventional PEMFC is the more difficult startup from both the normal and subzero temperatures. Development of membranes those are able to offer excellent proton conductivity and stability in dry and hot environment is the key for the succession of HT-PEMFC. Measurements of water and proton transport properties in the different high temperature membranes are necessary, and the measured properties are pivotal for HTPEMFC modeling. Acknowledgments The financial support by the Natural Sciences and Engineering Research Council of Canada (NSERC) via a strategic Project Grant (Grant No. 350662-07) and by Auto21 is greatly appreciated. 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