Journal of Chemistry

Volume 2015, Article ID 876821, 17 pages

http://dx.doi.org/10.1155/2015/876821

## A Comprehensive Review on Measurement and Correlation Development of Capillary Pressure for Two-Phase Modeling of Proton Exchange Membrane Fuel Cells

^{1}State Key Laboratory of Alternate Electrical Power System with Renewable Energy Sources, North China Electric Power University, Beijing 102206, China^{2}Beijing Key Laboratory of Multiphase Flow and Heat Transfer for Low Grade Energy, North China Electric Power University, Beijing 102206, China^{3}Department of Energy and Refrigerating Air-Conditioning Engineering, National Taipei University of Technology, Taipei 10608, Taiwan^{4}Mathematics and Physics Department, North China Electric Power University, Beijing 102206, China

Received 21 October 2014; Revised 23 March 2015; Accepted 24 March 2015

Academic Editor: Tingyue Gu

Copyright © 2015 Chao Si et al. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.

#### Abstract

Water transport and the corresponding water management strategy in proton exchange membrane (PEM) fuel cells are quite critical for the improvement of the cell performance. Accuracy modeling of water transport in porous electrodes strongly depends on the appropriate constitutive relationship for capillary pressure which is referred to as - correlation, where is the capillary pressure and is the fraction of saturation in the pores. In the present PEM fuel cell two-phase models, the Leverett-Udell - correlation is widely utilized which is proposed based on fitting the experimental data for packed sands. However, the size and structure of pores for the commercial porous electrodes used in PEM fuel cells differ from those for the packed sands significantly. As a result, the Leverett-Udell correlation should be improper to characterize the two-phase transport in the porous electrodes. In the recent decade, many efforts were devoted to measuring the capillary pressure data and developing new - correlations. The objective of this review is to review the most significant developments in recent years concerning the capillary pressure measurements and the developed - correlations. It is expected that this review will be beneficial to develop the improved PEM fuel cell two-phase model.

#### 1. Introduction

Proton exchange membrane (PEM) fuel cells are one kind of the most promising power generators which can offer clean power source for mobile and stationary applications due to their high efficiency, low start-up temperature, portability, and near-zero emissions [1–4]. The performance of PEM fuel cells is strongly related to the material properties, structure design, and operating conditions. By optimizing the structure design and operating conditions, the reactant transport to the porous electrodes can be significantly enhanced due to the improved water distribution in the cell.

Water management is essential to improve the cell performance [5–11]. The polymer exchange membranes currently used in PEM fuel cells require well hydration to maintain large proton conductivity. Lower membrane water content reduces the cell performance due to the increase of ohmic resistance in the membrane. To avoid membrane dehydration, the reactants usually need to be humidified. On the other hand, the cathode catalyst layer produces water vapor due to electrochemical reactions. The vapor will condense into liquid water when the local partial pressure is higher than the saturation pressure, so that the produced liquid water may accumulate in the pores of the porous electrodes. Moreover, the electroosmosis effect also leads to water transport from the anode to the cathode. Thus, if liquid water cannot be removed effectively from the cathode porous electrode, its pores will be blocked, which will significantly increase the mass transfer resistance of the reactants and lead to serious concentration polarization.

In the recent two decades, numerical modeling and simulations have already become powerful tools to predict the cell performance and optimize the cell structure and operation conditions [1, 5–7]. Numerical simulations can provide the local transport characteristics and distributions of reactants and water anywhere in the fuel cell; however, this information is very difficult to be observed and measured by experiments. Accurate description of liquid water transport and contribution in the porous electrodes is especially important to improve the accuracy of PEM fuel cell models. Two kinds of PEM fuel cell two-phase models have been developed. One is referred to as the multiphase mixture model and the other is the two-fluid model [1]. The two kinds of models both adopt the so-called volume-average method to treat the porous electrode, so that the real pore structure is not considered. Due to the limitation of the volume-average approach, how to describe the complex two-phase interactions in the pores of the porous electrodes is greatly challenging. Since the capillary force and the viscous drag are the main forces governing the liquid water transport in the porous electrodes, Wang et al. [12, 13] for the first time introduced a concept of capillary pressure to connect the pressures of reactants and liquid water in the porous electrodes and treated the capillary pressure as a function of the liquid water saturation. The introduction of the correlation of capillary pressure versus liquid water saturation ( correlation) greatly simplifies the complexity of two-phase modeling in the porous electrodes and this concept has been extensively adopted by other research groups [6, 7, 14–30]. Up to now, the two kinds of PEM fuel cell two-phase models all use the correlation and Wang’s idea is considered to be the most feasible approach to model the full-scale PEM fuel cell.

Unfortunately, since accurate experimentally determined correlation for the gas diffusion layer (GDL) was lacking, Wang et al. [12, 13] used a Leverett-Udell correlation, proposed by Leverett [31] and Udell [32] based on the experimental data of packed sands, in their model. The pore size, structure, and wettability of packed sands are greatly different from those of real carbon cloth or carbon paper GDLs; thus, this correlation may be improper to describe the liquid water transport in GDLs of PEM fuel cells [33–35].

In the last decade, many researchers measured the data for various commercially used GDLs [36–60]. Some studies are also devoted to modifying the Leverett-Udell correlation [33, 37–39] or developing a new correlation [34, 35]. Up to now, a large number of papers on the data have been published in open literature and the development is reaching a plateau. A comprehensive review on it is urgently necessary to generate knowledge in this field and make further breakthrough for the fuel cell two-phase modeling.

The aim of this work is to summarize current status and recent advance of capillary pressure measurement and modeling for PEM fuel cells. The review is organized as follows. Section 2 briefly describes the definition of the capillary pressure and the reasons for introduction of correlation in the PEM fuel cell two-phase modeling. The Leverett-Udell correlation has been extensively used in the PEM fuel two-phase models, and hence its disadvantages are also presented in Section 2. Section 3 reviews the measurement methods of the capillary pressure. Section 4 reviews the experimental and simulated data of the capillary pressure for a variety of commercial GDLs. Section 4 also reviews the new correlations based on fitting the measured experimental data. Section 5 reviews the comparative studies of Leverett-Udell correlation and new correlation for predicting water transport and distribution in the fuel cell. Finally, the further development directions for the correlation and PEM fuel cell two-phase modeling are presented in Section 6.

#### 2. Leverett-Udell Correlation for Capillary Pressure

For a real porous medium, it is very difficult to model the flow and heat transfer in an individual pore since the pore scale is far less than the scale of the porous medium. Moreover, the pore structure is generally irregular. Therefore, a volume-average method is adopted, which is based on the assumption that there coexist solid matrix and pores for each space point in the porous medium, so that the volume-averaged parameters such as porosity and permeability can be introduced to characterize the pore structure and porous flow feature. In the current PEM fuel cell two-phase modeling, the volume-average method is also widely adopted to model the porous electrodes [1]. In fact, however, the transport of gaseous reactants and liquid water occurs inside the pores in the real porous electrodes and there exist many interfaces between the two phases. Thus, the mass, momentum, and energy may exchange through the interface, and hence these exchanges in the porous electrodes should be taken into account to construct a reasonable two-phase model.

Unfortunately, in the volume-average method, the concepts, such as porosity, , and permeability, , are introduced to characterize the porous structure. The solid matrix and pores coexist everywhere in the porous media, and the volume fractions for them are and , respectively. Besides, in the modeling based on the volume-average method, the saturation, , is also introduced to characterize the volume fractions of gaseous phase and liquid phase for each space point, with the fraction of for the liquid phase and of for the gaseous phase. From this point of view, the interfaces between the two phases are neglected.

The above neglect leads to an issue that the exchanges for mass, momentum, and energy through the two-phase interfaces cannot be described. One alternative way to address the issue is to add source terms into the liquid and gas governing equations. For the two-phase modeling in PEM fuel cells, the generation mechanisms for mass source terms in the continuity equation and heat source terms in the energy equation are clear. However, momentum source terms in the momentum equation are very complex; they are not only determined by the phase fractions of the two phases but also influenced by the phase distributions of the two phases. For example, when the liquid phase exists in the form of liquid films or discrete drops, the interactions between the two phases are entirely different. It is difficult to accurately measure the phase distribution of the liquid phase in real porous electrodes. Consequently, an accurate description of force source terms in the momentum equation is almost impossible.

Due to the difficulty for accurately modeling the force source terms, Wang et al. [12, 13] introduced a concept of capillary pressure which correlated the local pressure difference between the liquid and gaseous phases in the porous electrodes to the local liquid water saturation. The capillary pressure is defined as [12, 13]where is the capillary pressure, is the pressure of the gaseous phase, and is the pressure of the liquid phase. Because there were not any experimental data of the capillary pressure versus saturation for porous GDLs, Wang et al. [12, 13] introduced a correlation for packed sands into the modeling of the fuel cell. This correlation can be expressed as [31]where is the water surface tension and is the contact angle of water on the pore walls. Equation (2) is also referred to as the Leverett correlation, and is the Leverett -function. It is noted that, by introducing a scaling factor into the Leverett correlation, the experimental data of measured in different porous media and with different fluid pairs gather together into a single curve [54]. Based on Leverett’s experimental data, Udell adopted a polynomial fitting to obtain the Leverett -function [32]: For case with , Pasaogullari and Wang [61] proposed a modified function expressed as follows:Equation (2) with functions expressed by (3a) or (3b) is called the Leverett-Udell correlation in the PEM fuel cell modeling.

Until now, almost all PEM fuel cell two-phase models adopted the Leverett-Udell correlation to characterize the liquid water transport in the porous electrodes. However, this correlation has some obvious drawbacks. The contact angle, , in (2) should exactly reflect the local wettability of water on the pore walls in the GDL material. However, the contact angle in the pores is very difficult to measure, so that some early studies specified [8, 9, 13–19] in the Leverett-Udell correlation. To improve the removal of liquid water, commercial carbon paper or carbon cloth GDL is generally loaded with hydrophobic materials such as PTFE (polytetrafluoroethylene) and FEP (fluorinated ethylene propylene) [62]. Based on the hydrophobic characteristics of GDL, subsequent studies adopted the contact angles greater than 90°, such as [22], 110° [20], 120° [24, 63], 91–120° [23], or 91–130° [25].

It is noted that the contact angles adopted in above studies are all hypothetical due to lack of experimental data. Therefore, many efforts were carried out to determine the contact angle of water on GDL surface by sessile drop or capillary rise methods [64, 65]. For example, Mathias et al. [65] measured the contact angles of Toray-TGP-060 carbon paper by the sessile drop method. They found that the values were 135°, 156°, and 164° for carbon papers with 0, 9, and 23 wt% PTFE, respectively. Unfortunately, the contact angles measured by these methods are the external contact angles, which are not the contact angles of water on the pore walls in the porous material. Gurau et al. [26] for the first time evaluated the internal contact angles by the Washburn technique and the Owens-Wendt two-parameter theory. They presented that the internal contact angle ranged from 88 to 101° for water in various GDLs, which was slightly lower than that on a smooth surface of pure PTFE (108°) and much lower than the external contact angle measured on the GDL surface by the sessile drop method.

It is worth noting that hydrophobic pores and hydrophilic pores coexist in the GDLs due to the heterogeneous distribution of the loaded hydrophobic materials, and hence the pore walls of GDLs have a mixed wettability. Consequently, using a single contact angle in the Leverett-Udell correlation may be improper. In addition, when liquid plugs and/or liquid films transport in the GDL pores, a dynamic contact angle should be adopted to characterize the wettability of the pore walls [66–68]. This phenomenon adds further complexity for wettability of GDL pores. More importantly, the pore size, pore structure, and wettability of packed sands are found to be greatly different from those of real carbon paper or carbon cloth GDLs, as shown in Figure 1 [37, 69]. The Leverett-Udell correlation may be intrinsically inappropriate for the GDLs even though the internal contact angle could be determined accurately. Several measurements of the liquid water distributions in GDLs using neutron radiography techniques [70, 71] and pressure drop methods [72] have shown that the PEM fuel cell two-phase model incorporated with the Leverett-Udell correlation underestimates the liquid water saturation.