質子交換膜燃料電池中陰極層汽液界面位置對性能之影響

國立交通大學

機械工程學系

博士論文

質子交換膜燃料電池中陰極層汽液界面位置對

性能之影響

Effects of the Gas-Liquid Interface Location in the Cathode

Layer on the Performance of a PEM Fuel Cell

研 究 生 : 李純怡

指導教授 : 陳俊勳 教授

曲新生 教授

質子交換膜燃料電池中陰極層汽液界面位置對

性能之影響

Effects of the Gas-Liquid Interface Location in the Cathode

Layer on the Performance of a PEM Fuel Cell

研 究 生 : 李純怡 Student : Chun-I Lee

指導教授 : 陳俊勳、曲新生 Advisor: Chiun-Hsun Chen

Hsin-Sen Chu

國 立 交 通 大 學

機 械 工 程 學 系

博 士 論 文

A Thesis

Submitted to Department of Mechanical Engineering National Chiao Tung University

in partial Fulfillment of the Requirements for the Degree of

Doctor of Philosophy in

Mechanical Engineering

June 2007

Hsinchu, Taiwan, Republic of China

質子交換膜燃料電池中陰極層汽液界面位置對性能之影響

研究生 : 李純怡 指導教授 : 陳俊勳、曲新生

摘 要

本論文主要是探討三維、多物種、二相混合模式下之質子交換膜燃料電池 中，其陰極增濕、陰極氣體擴散層的孔隙度、電池溫度以及增濕溫度的變化對燃 料電池的傳輸現象及電池性能之影響。本研究中主要是利用沿著流道方向之汽液 界面的位置來說明在這些操作條件下，液態水出現的位置對電池性能的影響。在 研究過程中，首先先建立描述質子交換膜燃料電池內部各種傳輸及電化學現象之 數學模式，其中以質量、動量、物種、能量及電流守恆方程式做為模式的主要方 程式。在電化學反應中，主要的驅動力是利用活化過電位來作為陰極觸媒層中電 子相電位與質子相電位的聯繫。而且當電池發生電化學反應時會產生熱能，其熱 能的來源包括因電化學反應所造成之不可逆的熱及熵、因質子和電子傳輸所造成 的焦耳熱以及因水的蒸發及凝結所產生的潛熱。因為本論文中有考慮二相流動的 傳輸，因此在此部分是利用M2模式來說明其各物種之間的相互關係。 本研究探討的議題分為二個部份：第一部份是以等溫系統為主，即是不考慮 能量方程式。主要是探討在陰極的流道及擴散層中，陰極的增濕情況以及陰極氣 體擴散層的孔隙度改變時對汽液界面位置的影響。數值模擬的結果顯示當陰極的 增濕程度逐漸增加以及電池的操作電壓逐漸降低時，汽液界面的位置會逐漸往流

道的入口方向移動。這是因為當陰極增濕度增加時，會有較多的液態水產生，而 液態水會造成多孔隙材質中的孔穴被液態水佔據，造成燃料氣體傳輸的阻礙，因 此電池性能降低。相同的，當電池電壓降低時，即是電流密度較大時，電化學反 應較為快速，因此汽液界面的位置也會逐漸往流道的入口方向移動因而造成相同 的情況。再者，當陰極的氣體擴散層孔隙度逐漸增加時，無論是水或是燃料氣體， 皆較容易通過多孔隙材質，因此，當陰極氣體擴散層的孔隙度增加時，電池性能 較佳。另外，沿著流道方向之氧氣分率、水分率以及液態水在陰極流道及擴散層 中的分佈及變化也一併提出及探討。 第二部份主要是以非等溫系統為主，因為液態水的蒸發與凝結皆與操作溫度 有著密切的關係，因此本部分考慮在不同的操作溫度時其對汽液界面位置的影 響。操作溫度主要是分成電池溫度以及增濕溫度來探討。數值模擬的結果顯示當 增濕溫度等於或大於電池溫度時，其汽液界面的位置會隨著電池溫度的降低而逐 漸往流道的入口處移動，進而造成電池性能的降低。另外也分別探討在薄膜內的 溫度分佈、在流道入口處陰極氣體擴散層內溫度以及液態水的分佈。並且清楚的 指出燃料氣體在氣體擴散層中是利用擴散方式由流道往勒條的方向移動，反之， 液態水在氣體擴散層中則是利用毛細作用力由肋條往流道出口處移動。

Effects of the Gas-Liquid Interface Location in the Cathode Layer on

the Performance of a PEM Fuel Cell

Student: Chun-I Lee Advisor: Chiun-Hsun Chen

Hsin-Sen Chu

ABSTRACT

This dissertation presents a three-dimensional, multi-component, two-phase

model to investigate the transport phenomena and performance of proton exchange

membrane fuel cell as the liquid water forms under various cathode humidification

conditions, gas diffusion layer porosities, cell temperatures, and humidification

temperatures. In this study, the location of the gas-liquid interface along the channel

direction is extracted to explain the effects of liquid water appearance that cause the

cell performance change. A mathematical model, coupled with the electrochemical

process, two-phase flows, species transfer, and heat transfer is developed at first. In

the electrochemical reaction of the cathode catalyst layer, the solid phase potential and

the electrolyte phase potential is connected by the activation overpotential.

Furthermore, thermal energy release and transport is accompanied with the

electrochemical reaction and is considered in modeling. The sources of thermal

energy accounts for irreversible heat and entropic heat generated due to

latent heat of water condensation and/or evaporation. The multiphase mixture

formulation (M2 model) is adopted as it is particularly suitable for two-phase flow

modeling in PEM fuel cells.

Our quest to the effects of the liquid-water interface location involves the

following two parts in the dissertation. First, we considered the isothermal system in

the PEM fuel cell, namely the energy conservation equation is not included in it. The

objective of this part is to investigate the effects of the location of the gas-liquid

interface along the channel direction under various cathode humidification conditions

and gas diffusion layer porosities in the conventional flow field. Numerical simulation

results indicate that the gas-liquid interface location approaches to the gas flow

channel inlet region and cell performance declines gradually as the relative humidity

of the cathode is increased. This is because of liquid water may occupy the pores in

the porous media, reducing the amount of fuel gas that can reach the cathode catalyst

layer to cause the cell performance diminished. Meanwhile, when the cell operating

voltage decreases and the current density gets larger, the electrochemical reaction

becomes more quickly. Hence, the gas-liquid interface location also moves to the

channel inlet region to cause the same consequence. Furthermore, as the gas diffusion

layer porosity is increased, the transport of liquid water and fuel gas becomes easier

fraction, the water fraction and the liquid water saturation field along the flow channel

direction in the flow channel and the gas diffusion layer are presented.

The investigation of the location of the gas-liquid interface along the channel

direction at various operating temperatures is conducted in the second part of the

dissertation, owing to the consideration that the condensation and/or evaporation of

the water is related with operating temperature. The effects of two model parameters,

namely cell temperature (Tcell) and humidification temperature (Th), on the

gas-liquid interface location and cell performance are presented. Simulation results

indicate that when the anode and cathode humidification temperatures are equal to or

higher than the cell temperature, the gas-liquid interface location moves toward the

flow channel inlet region and the cell performance decreases as the temperature is

decreased. Additionally, the membrane temperature distribution and the distributions

of the liquid water and temperature in the cross-section of the cathode gas diffusion

layer in the inlet region are presented. Simulation results indicate that gas-phase fluid

diffuses from the channel to the land and that the capillary-driven liquid water is

誌 謝

在交大的求學過程中，首先先感謝恩師 曲新生及陳俊勳博士，

恩師除了在學術上悉心的指導我之外，在待人處世方面更是值得我學

習的榜樣。其次，感謝顏維謀、陳發林、宋齊有及鄭金祥博士在台大

讀書會時給予我莫大的幫助，使我的博士論文能進行的更加順利。也

很感謝口試委員陳朝光、洪哲文、楊文美及顏維謀諸位教授對於論文

的建議及指導，使得本論文更加的嚴謹及完整。

此外，特別感謝木勝、輝忠及弘祥學長不厭其煩的幫助我修正論

文及解決疑惑。也感謝研究室的學長及學弟們為我的生活帶來了許多

歡樂的回憶。

最後，感謝我的家人－老爸、老媽、大哥、大嫂、二哥以及我的

丈夫 Francis，在我求學的過程中，陪伴我經歷了許多挫折及挑戰，

也給予我許多的幫助及鼓勵，使我能更堅定要完成學業的決心，僅以

此論文獻給所有關心、愛護及幫助我的人，非常非常的感謝!

TABLE OF CONTENTS

ABSTRACT (in Chinese) ………... i

ABSTRACT (in English)……… iii

LIST OF TABLES……….. ix

LIST OF FIGURES……… x

NOMENCLATURE……… xiv

CHAPTER 1 INTRODUCTION………... 1

1.1 Background……….. 1

1.1.1 Transport Phenomena in a PEMFCs……….. 2

1.2 Literature Survey………. 4

1.3 Objectives………... 14

1.4 Outlines………. 15

CHAPTER 2 MATHEMATICAL MODELING……….. 20

2.1 Model Description……… 21 2.2 Basic Assumptions……… 22 2.3 Governing Equations……… 23 2.4 Boundary Conditions………..…. 30 2.5 Numerical Procedures………..… 32 2.5.1 Numerical Method….………... 32 2.5.2 Calculation Procedure………. 33 2.5.3 Model Validation………... 33

CHAPTER 3 EFFECTS OF CATHODE HUMIDIFICATION AND POROSITY OF THE GDL ON THE GAS–LIQUID INTERFACE LOCATION IN A PEM FUEL CELL…….… 40

3.1 Introduction………..…… 40

3.2 Effects of cathode humidification scheme ……….. 41

3.3 Effects of cell operating voltage………..……. 43

3.4 Three-dimensional species field……….….. 44

3.5 Two-phase mixture velocity field………..….….. 47

3.6 Effects of the cathode gas diffusion layer porosity………..… 47

3.7 Summary………... 51

CHAPTER 4 EFFECTS OF TEMPERATURE ON THE LOCATION OF THE GAS–LIQUID INTERFACE IN A PEM FUEL CELL………... 67

4.1 Introduction……….. 67

4.2 Effects of temperatures scheme………..………. 68

4.3 Effects of cell temperatures scheme………..……….. 69

4.4 Three-dimensional species field………..………… 71

4.5 Temperature field in the membrane………... 73

4.6 Summary……….. 74

CHAPTER 5 CONCLUSIONS AND FUTURE PERSPECTIVES ... 84

5.1 Concluding Remarks………... 84

5.2 Future Perspectives..……… 87

LIST OF TABLES

LIST OF FIGURES

Fig. 1.1. Basic components of a single PEMFC……….. 17 Fig. 1.2. Operating principle of a single PEMFC……… 18 Fig. 1.3. The dominant mechanisms of cell performance……… 19 Fig. 2.1. Physical and computational domains considered in this study……. 36 Fig. 2.2. Numerical flow diagram of the solution procedure.………….…… 37 Fig. 2.3. Comparison of predictions on the three different grid systems..…... 38 Fig. 2.4. Comparison of the predicted I–V curve and the experimental data

of Squadrito et al. [44]………... 39 Fig. 3.1. Effects of relative humidity of cathode on the location of the

interface where liquid water begins to condense along the conventional flow channel at a cell operating voltage of 0.7 V. (a) RHca = 20~100 % and (b) RHca = 20~60 %... 53

Fig. 3.2. Effect of relative humidity of the cathode on cell performance with conventional flow fields. (a)I-V curves and (b) I-P curves... 54 Fig. 3.3. Effect of cell operating voltage on the location of the interface

where liquid water begins to condense along the conventional flow channel at a relative humidity of the cathode of 80 %. (a) V= 0.5~0.9 (b) V= 0.8~0.9……….………. 55 Fig. 3.4. Oxygen fraction in the cathode gas channel and gas diffusion layer

along the conventional flow channel at a cell voltage of 0.7 V and cathode relative humidity of 80%.………..…... 56 Fig. 3.5. Water fraction in the cathode gas channel and the gas diffusion

and cathode relative humidity of 80%... 57 Fig. 3.6. Liquid water saturation field in the cathode gas channel and gas

diffusion layer along the conventional flow channel at a cell voltage of 0.7 V and cathode relative humidity of 80 %... 58 Fig. 3.7. Oxygen mass fraction contours at the gas diffusion layer at the

cathode humidification of (a) 20% (b) 60% (c) 100% for cell voltage of 0.4 V... 59 Fig. 3.8. Water mass fraction contours at the gas diffusion layer at the

cathode humidification of (a) 20% (b) 60% (c) 100% for cell voltage of 0.4 V... 60 Fig. 3.9. Liquid saturation contours at the gas diffusion layer at the cathode

humidification of (a) 20% (b) 60% (c) 100% for cell voltage of 0.4 V... 61 Fig. 3.10. Two-phase mixture velocity field in the gas diffusion layer and

flow channel of the cathode at the cell voltage of 0.7 V and cathode relative humidity of 80%... 62 Fig. 3.11. Effect of porosity of the gas diffusion layer on the location of the

interface where liquid water begins to condense along the conventional flow channel at a cell voltage of 0.7 V and cathode relative humidity of 80%... 63 Fig. 3.12. Effect of porosity of the gas diffusion layer on cell performance

with conventional flow fields at cathode relative humidity of 80%. (a) I-V curves and (b) I-P curves... 64 Fig. 3.13. Oxygen fraction at various porosities of the gas diffusion layer at

%... 65 Fig. 3.14. Water fraction at various porosities of the gas diffusion layer at an

operating voltage of 0.7 V and cathode relative humidity of 80 %... 66 Fig. 4.1. Polarization curves at various cell temperatures with equal

humidification temperature……… 76 Fig. 4.2. Effect of cell temperature on the location of the interface where

liquid water begins to condense along the flow channel at a cell operating voltage of 0.7 V……….……..…….. 77 Fig. 4.3. Polarization curves for various cell temperatures at (a) Th = 323 K

(b) Th = 343 K.……….…… 78 Fig. 4.4. Effects of cell temperature on the location of the interface where

liquid water begins to condense along the flow channel at a cell operating voltage of 0.7 V and a humidification temperature of 343 K. (a) Z = 0 to 7.112 cm, (b) Z = 0 to 0.08 cm…………...…… 79 Fig. 4.5. Liquid water saturation field in the cathode gas channel and

diffusion layer along the flow channel at a cell voltage of 0.7 V and a humidification temperature of 343 K. (a) Tcell = 323 K, (b) Tcell = 333 K, (c) Tcell = 343 K………..…. 80 Fig. 4.6. Liquid water saturation distributions in a cross-section of the

cathode gas diffusion layer in the inlet region at a cell voltage of 0.7 V and a humidification temperature of 343 K. (a) Tcell = 323 K, (b) Tcell = 333 K, (c) Tcell = 343 K.……….… 81 Fig. 4.7. Temperature distributions in a cross-section of the cathode gas

humidification temperature of 343 K. (a) Tcell = 323 K, (b) Tcell = 333 K, (c) Tcell = 343 K.……….…………. 82 Fig. 4.8. Temperature contours in the membrane at a cell voltage of 0.7 V

and a humidification temperature of 343 K. (a) Tcell = 323 K, (b) Tcell = 333 K, (c) Tcell = 343 K.……….…….. 83

NOMENCLATURE

A specific electrochemically active area

a water activity

C molar concentration (mol m−3)

CF quadratic drag factor

CP constant-pressure heat capacity (J kg−1 K−1)

Dc capillary diffusion coefficient

Dij binary diffusion coefficient (m2 s−1)

F Faraday constant (C mol−1)

g gravitational acceleration (m s−2)

h enthalpy (J kg−1)

i current density (A m−2)

j transfer current density (A m−3)

K permeability (m2)

k thermal conductivity (W m−1 K−1)

M molecular weight (Kg mol−1)

Pc capillary pressure

R universal gas constant (J mol−1 K−1)

r rate constant

S source term

s liquid water saturation

T temperature (K)

Tcell cell temperature (K)

Th humidification temperature (K) uK fluid velocity (m s−1) w mass fraction x molar fraction Greek α transfer coefficient ε porosity φ phase potential (V) η overpotential (V) κ proton conductivity (S m−1) λk mobility of phase k μ dynamic viscosity (N s m−2)

ν kinematic viscosity (m2 s−1)

θc contact angle

ρ density (kg cm−3)

ρκ kinetic density (kg m−3)

σ electrical conductivity (S m−1)

τ tortuosity of the porous medium

ξ stoichiometric flow ratio

ψ membrane water content

ζ surface tension (N m−1) Subscripts a anode c cathode con condensation e electron eva evaporation

eff effective value

g gas phase

p proton

sat saturation value

w water

Superscripts

CHAPTER 1

INTRODUCTION

1.1 Background

The depletion of fossil fuels and the degradation of the environment are key

issues in modern society. There is an urgent need to substitute fossil fuels by

renewable fuels and at the same time find new energy conversion methods that

preserve the environment. Because of these problems, much effort has been devoted

in recent years to the development of different types of fuel cells [1]. Among them,

the polymer electrolyte membrane fuel cells (PEMFCs) have attracted much attention.

Proton exchange membrane fuel cell is a type of lower-temperature fuel cells

and has been regarded as promising candidate of future power sources for both

stationary and automotive applications. Especially for automotive applications, it is

one of the most probable types of fuel cells which are able to replace conventional

power conversion devices such as internal combustion engines because of its many

distinctive features such as high energy efficiency, low noise, low emission, and

especially low operating temperature (i.e., quick startup under environmental

this system becomes commercially available.

1.1.1 Transport Phenomena in a PEMFC

Figure 1.1 shows the geometry of a single fuel cell, which consists of a

membrane, two catalyst layers, two gas diffusion layers, two gas channels, and two

collector plates. The major functions and characteristics of these components are:

collector plates with flow channels are used for reactants and products transport,

electron conduction and heat removal; gas diffusion layers are for reactant

distributions, electron conduction, and liquid water removal; catalyst layers are used

to promote electrochemical reactions where reactants are consumed, and products and

heat are generated; and the membrane is used to conduct protons from the anode

catalysts layer to the cathode catalyst layer. The electrochemical reaction occurring in

the PEMFC is illustrated in Fig. 1.2. At the anode catalyst layer, hydrogen is

consumed to produce protons and electrons, i.e.,

2H2 → 4H+ + 4e− (1.1)

Electrons pass through an external circuit to the cathode thus providing electrical

power, while the protons transport through the membrane to the cathode. At the

cathode catalyst layer, oxygen combines with the protons and electrons to produce

water, i.e.,

Although chemical reaction principles are rather simple, the transport processes inside

a fuel cell are quite complex. First, the reactant species must be able to effectively

transfer to different parts of the catalyst layers. Second, the electrical resistance to

both protons and electrons must be low. And third, the product water and heat must be

removed out of the fuel cell from the catalyst layers through the gas diffusion layers

and other cell components. Further complications come from the duel effects of

temperature and water.

A PEM fuel cell simultaneously involves electrochemical reactions, current

distribution, hydrodynamics, multi-component transport, and heat transfer. Hence, a

comprehensive mathematical model is needed to gain a fundamental understanding of

the interacting electrochemical and transport phenomena. Nevertheless, polarization

occurs during operation of PEM fuel cells and this influences dramatically both the

performance and the commercialization of the technology. Accordingly, the

development of a theoretical model of the PEM fuel cell, as well as corresponding

analyses, are crucial to gain a good understanding of the effect of the operating

conditions on the cell potential so that the polarization level can be well controlled.

Performance of a fuel cell is determined by several factors, including the

geometry and morphology of transport components, and the operating conditions such

by its current-voltage relation (i.e., the polarization curve), as shown in Fig. 1.3. At a

particular current, the voltage drop is mainly caused by (i) overpotentials of

electrochemical reactions (mainly on the cathode), (ii) the ohmic drop across the

ionomeric membrane, and (iii) the mass transport limitations of reactants and products.

At high current densities of special interest to vehicular applications, excessive water

is produced within the cathode in the form of liquid, thus leading to a gas-liquid

two-phase flow in the porous electrode. The ensuing two-phase transport of gaseous

reactants to the reaction surface becomes a limiting mechanism for cell performance,

particularly at high current densities.

1.2 Literature Survey

Water and thermal management is essential for proper operation of PEM fuel

cells. The polymer membrane in the PEM fuel cell must be in a highly hydrated state

to facilitate proton transport. If there is not enough water, the membrane becomes

dehydrated and its resistance to proton conduction increases sharply. On the other

hand, if too much water is present, flooding may occur resulting in the pores of the

gas diffuser filled by liquid water, which will block the transport of reactants to the

reaction sites. In practice, humidification of anode fuels and/or cathode oxidants is

cathode catalyst layer and water also tends to migrate from the anode side to the

cathode side under the electro-osmotic drag, it becomes a key issue to avoid flooding

in the cathode in the design and operation of PEM fuel cells.

The influence of temperature on fuel cell operation has two important

consequences. On one hand, higher temperature is favorable for electrochemical

reactions. On the other hand, higher temperature also leads to a reduction in reactant

partial pressure, cell open circuit voltage, and water activity, which results in an

increase of membrane ionic resistance. In addition, non-uniform temperature

distributions cause transient effects and create hot spots that may lead to cell failures.

If the temperature or temperature gradient is too high, the fragile membrane could be

damaged.

Modeling and computer simulation of PEM fuel cell began with the pioneering

work of Bernardi and Verbrugge [2, 3] and Springer et al. [4, 5], whose models are

essentially one-dimensional that provided good preliminary foundations for PEM fuel

cell modeling. Bernardi and Verbrugge [2, 3] presented a one-dimensional model that

included water transport. However, their model did not account for surface tension

forces on the transport of the liquid phase. Also, the model was isothermal, and did

not take into account energy interactions due to phase change. Springer et al. [4, 5]

flux ratio across the membrane in PEM fuel cells. However, the one-dimensional

model cannot simulate the decrease of reactants and the accumulation of products in

the flow direction. In the pseudo two-dimensional model proposed by Fuller and

Newman [6] and Nguyen and White [7], the effects of reactant consumption along the

flow channels were considered. Nguyen and White [7] developed a two-dimensional

PEM fuel cell model with one-dimensional heat transport in the flow direction. The

model considered phase change of water in the flow channel as the only heat source.

Both of these models assumed a well-mixed concentration in the flow channel,

diffusion was the only mechanism for oxygen transport, the interaction of the flow

with the species field in the channel and gas diffusion layer was not included.

Subsequently, Yi and Nguyen [8] modified the previous models to include both the

liquid and gas phase along the flow channel of both anode and cathode sides of PEM

fuel cell. The results indicated that anode humidification improved the conductivity of

the membrane, and that the cell performance is enhanced by the injection of liquid

water and the increase in humidification temperature. Gurau and Liu [9] presented a

two-dimensional model by coupling the flow and transport governing equations in the

flow channel and the gas diffusion layer, but the density change due to species

consumption appeared to be neglected. Yi and Nguyen [10] developed a

cathode. However, this model is valid only in the absence of liquid water, and it does

not account for water condensation and evaporation phase change within porous

electrodes.

Several models have been developed to demonstrate the influence of liquid

water on water management. Wang et al. [11] first studied two-phase flow and liquid

water transport on the cathode side of a PEM fuel cell based on the multiphase

mixture model (M2 model) originally developed by Wang and Cheng [12].The model

encompassed both single- and two-phase regimes corresponding to low and high

current densities and was capable of predicting the transition between the two regimes.

The half cell domain including the gas channel, gas diffusion, and catalyst layer on

the cathode side of a PEM fuel cell was considered in the article. Later, this model

was further developed to include a complete cell by You and Liu [13]. They presented

a two-dimensional two-phase mixture model to analyze flow and transport in the

cathode of the PEM fuel cell, and to predict phase change and water/oxygen

distribution in both the flow channels and the porous diffusion layer. Also, such a

two-phase model provides actual characteristics of the flow and the transport of

reaction gas for thermal/water management design in the future. In Refs. [14, 15],

Mazumder and cole developed a three-dimensional model to predict the effect of the

polarization behavior was overestimated, based on the assumption that no liquid water

was formed. In contrast, when liquid water was present, the condensation and/or

evaporation of water via equilibrium phase transformation proceeded as described in

Ref. [15]. They presented a numerical study, also based on the M2 model of Wang and

Cheng. Since water condensation can obstruct pores and reduce cell performance, the

polarization behaviors of PEM fuel cell must be accurately predicted to investigate

such a situation. Pasaogullari and Wang [16] developed a model to explore the

two-phase flow physics in the cathode gas diffusion layer. The simulations revealed

that flooding of the porous cathode reduced the rate of oxygen transport to the cathode

catalyst layer. Furthermore, they indicated that the humidification level and the flow

rate of reactant streams are key parameters controlling PEM fuel cell performance and

two-phase flow and transport characteristics. He et al. [17] proposed a

two-dimensional, two-phase, multi-component transport model for PEM fuel cell with

interdigitated flow field, in which they included capillary transport of liquid water in a

completely wetted gas diffusion layer. They studied the effects of various electrode

and flow field design parameters on the performance of the cathode of a PEM fuel cell.

Natarajan and Nguyen [18] published a two-dimensional model that considered the

dimension normal to the reactive surface and the electrode width. In this work, gas

while Darcy’s law was adapted to model liquid water transport in unsaturated porous

gas diffusion layers. Here the effect of liquid water accumulation in the diffusion

layer was accounted for in terms of gas transport restrictions and reactive surface

coverage. Meng and Wang [19] improved their three-dimensional computational fluid

dynamics model based on recent visualization experiments, more accurately

investigated the two-phase behavior under different gas utilizations. Their model was

applied to successfully predict liquid water flooding dynamics although the energy

transport was ignored. Zhang et al. [20] experimentally and theoretically studied the

liquid water transport and removal from the gas diffusion layer and gas channel of the

PEM fuel cells. In situ observations of the distribution of liquid water on the gas

diffusion layer surface and inside the gas channel were made. They characterized the

formation and emergence of liquid water from the gas diffusion layer surface and

identified two modes of liquid water removal from the gas diffusion layer surface.

Thermal management in a PEM fuel cell is inherently coupled with water

management, and the two factors combine to ensure high performance and durability

of a PEM fuel cell. It is thus of paramount importance to understand the thermal

behavior under various design and operating conditions and furthermore develop a

capability to accurately predict the temperature distribution in PEM fuel cells. Rowe

Including entropic, irreversible, and phase change heats, they further took account of

Joule heating in the membrane and anode/cathode catalyst layers. This work predicted

the temperature variation in the through-membrane direction under the various current

densities and electrode thermal conductivities. A major drawback of this model is the

fact that the cathode catalyst layer is assumed to be fully hydrated and the membrane

water content is linearly interpolated, indicating inapplicability of the model in low

humidity operation. Baschuk and Li [22] developed a mathematical model with

variable degrees of water flooding in the PEMFC. Physical and electrochemical

processes occurring in the membrane electrolyte, the cathode catalyst layer, the

electrode backing layer and the flow channel were considered. Compared with

experimental results, they found that when air was used as the cathode fuel, the

flooding phenomena are similar at different operating conditions of the pressures and

temperatures. When cell pressure is increased significantly, the water flooding in the

electrode becomes serious and leads to a noticeable reduction in the power output. To

examine the effects of temperature and gas pressure gradients on fuel cell

performance and water management, a one-dimensional, non-isothermal model was

proposed by Djilali and Lu [23]. It was found that the temperature distribution in the

PEM fuel cell is affected by the water phase change in the electrode. Simpalee and

temperature distribution inside a straight channel PEMFC and to study the effect of

heat produced by the electrochemical reactions on fuel cell performance. Source terms

for transport equations, heat generation, and a phase change model were presented to

facilitate their modeling work. They treat the liquid water as a component of the gas

mixture, transported by convection by gas pressure and density gradients. The results

showed that the temperature profiles depend on heat generation produced by

electrochemical reactions and phase change of water inside the fuel cell. Further, the

cell performance does not merely influenced by the inlet humidity condition, cell

voltage, and membrane thickness but also by the temperature rise inside fuel cells.

Natarajan and Nguyen [25] developed a pseudo three-dimensional model by

extending a two-dimensional isothermal model for conventional flow fields. Their

results indicated that water removal by evaporation increases with higher temperature

and higher stoichiometric flow rates, which results in better cell performance.

However, if the water produced by the electrochemical reaction could not effectively

removed from the cathode side, then flooding of the electrode will occur, reducing

access of the reactants to the catalyst layer. Yang et al. [26] reported the mechanics of

the liquid water transport, starting from emergence on the gas diffusion layer surface,

droplets grow and move toward the main stream flow in gas channels. Furthermore,

channel walls and the subsequent channel clogging affects cell performance.Liu et al.

[27] experimentally studied the two-phase flow of reactants and products as well as

water flooding situation in the gas flow channel of a cathode in a PEM fuel cell with

three flow fields. Their study addressed the effects of flow field, cell temperature,

cathode gas flow rate, operation time on water build-up, and cell performance. The

results indicated that liquid water forms in columns that accumulate in the cathode

flow channels and clogs the porous media, liming mass transfer, resulting in degraded

cell performance. Further, the amount of water in flow channels at high temperature is

much less than that at low temperature. Ju et al. [28] presented the first work that

systematically derived mathematical model of the energy equation from

thermodynamics. The thermal model is further coupled with the electrochemical and

mass transport to deal with the heat and water management in PEM fuel cell. Later,

Wang and Wang [29] expanded the M2 model to investigate the interaction between

the two-phase flow and thermal transport by non-isothermal effect. The results reveal

that the vapor-phase diffusion enhances water removal and provides a new

mechanism for heat removal through a phase change process. This new heat removal

mechanism is similar to the heat pipe effect.

Much effort has been expended in recent years upon development of

complexities. Current development in PEM fuel cell modeling is in the direction of

applying computational fluid dynamics (CFD) to solve the complete set of transport

equations governing mass, momentum, species, energy, and charge conservation.

Dutta et al. [30, 31] developed a complete three-dimensional simulation of PEM fuel

cell using Fluent, a commercial CFD package. Their modeled domain covered the

entire fuel cell sandwich employing conventional gas distributors on both the anode

and the cathode sides. This model treats the MEA as an interface without thickness

and ignores the membrane water storage capability; thus this model cannot be used to

simulate transient phenomena. However, they fail to account for liquid water in the

gas diffusion layers. Yan et al. [32] conducted a numerical study of the effects of fuel

channel width and GDL porosity on the cross-cell transport of reactant gas and the

performance of a PEM fuel cell. They [33, 34] subsequently presented a

two-dimensional numerical model to carry out the reactant gas transport phenomena

and cell performance with baffle effects in the flow channel of bipolar plate. Effects of

the baffle width are explored. Additionally, the effects of liquid water formation on

the reactant gas transport are taken into account in the modeling and examined in the

analysis. A comprehensive non-isothermal, three-dimensional computational model of

a PEM fuel cell has been developed by Berning et al. [35, 36]. The model

channels. With the exception of phase change, the model accounts for all major

transport phenomena. The model is implemented in a CFD code, and simulations are

presented with an emphasis on the physical insight and fundamental understanding

afforded by the detailed three-dimensional distributions of reactant concentrations,

current densities, temperatures and water fluxes. Um and Wang [37] presented a

computational fuel cell dynamics model to elucidate three-dimensional interactions

between mass transport and electrochemical kinetics in polymer electrolyte fuel cells

with straight and interdigitated flow fields, respectively. The model results indicate

that forced convection induced by the interdigitated flow field substantially improves

mass transport of oxygen to, and water removal from, the catalyst layer, thus leading

to a higher mass-transport-limiting current density as compared to that of the straight

flow field. Chiang and Chu [38, 39] presented a three-dimensional, multi-species and

two-phase model by CFD code to investigate the effects of temperatures,

humidification levels, channel aspect ratio and gas diffusion layer thickness on the

cell performance.

1.3 Objectives

The above literature review clearly indicates that water and thermal

formation of liquid water depends on the saturation vapor pressure, which is a strong

function of temperature. Therefore, the temperature factor is inevitable in water

management investigation because the phase change of water such as condensation

and/or evaporation closely relates to the corresponding saturation pressure. However,

the discussions of the effect of gas-liquid interface location on the cell performance

and transport process are rarely seemed. Therefore, this study investigates the

formation and influence of the gas-liquid interface location along the flow channel

direction at various operation conditions with a three-dimensional, multi-component,

and two-phase model of a PEM fuel cell.

1.4 Outlines

The scope of this dissertation is mainly focused on four parts. Chapter 2

introduces the homogeneous model of the multiphase mixture formulation (M2 model)

development as it is particularly suitable for two-phase flow modeling and applying

computational fluid dynamics (CFD) to solve the complete set of transport equations

governing mass, momentum, species, energy, and charge conservation in a PEM fuel

cell. Furthermore, the proposed model incorporates the basic assumptions is also

presented. Chapter 3 investigates the effects of cathode humidification and gas

condense along the flow channel of a PEM fuel cell; its contribution to the cell

performance is also discussed. Additionally, the resulting oxygen and water fraction

distributions and liquid water saturation fields at fixed cathode humidity are obtained

to validate the simulation results. Chapter 4 reports the formation and influence of the

gas-liquid interface location along the flow channel direction at various cell

temperatures and humidification temperatures with a non-isothermal,

three-dimensional, multi-component, and two-phase model of a PEM fuel cell. The

temperature distribution in the cell domain membrane and the distributions of

temperature and liquid water saturation in the cross-section of the cathode gas

diffusion layer in the inlet region are investigated. In Chapter 5, the conclusions of

Figure 1.1 Basic components of a single PEMFC End plate Current collector Bipolar plate Diffusion media Membrane Current collector End plate Current collector Bipolar plate Diffusion layer Catalyst layer Membrane Diffusion layer Bipolar plate Current collector End plate End plate

Figure 1.2 Operating principle of a single PEMFC

Load

e

-

H

+

H

2

e

-e

-Anode

_Cathode

O

2

H

2

Electron

Proton

O

2

H

2

O

Figure 1.3 The dominant mechanisms of cell performance Open-circuit potential

Electrode Kinetic Dominate

Ohmic Polarization Dominate

Mass Transport Dominate

Cell V

oltage

CHAPTER 2

MATHEMATICAL MODELING

Traditionally, macroscopic problems of two-phase flow and transport in

porous media have been modeled using a two-fluid approach. However, this approach

results in a large number of primary variables for each phase, and highly nonlinear

equations. Therefore, exact solutions of two-phase problems with two-fluid models

are limited to a very limited number of problems with many simplifying assumptions.

Furthermore, the two-fluid models require explicitly tracking the irregular and moving

interface between two phases, increasing the numerical complexity of the problem.

Particularly in PEM fuel cells, the gas-liquid interfaces, i.e., the condensation and

evaporation fronts, are expected as well as the coexistence of single- and two-phase

regions. Therefore, a convenient model capable of describing both single- and

two-phase regions without a need to track the irregular, a priori unknown interface is

required. For these reasons, the multiphase mixture (M2) formulation of Wang and

Cheng [12] is particularly suitable for two-phase PEM fuel cell modeling. The

multiphase mixture model is a mathematical reformulation of the classical two-phase

the multiphase flow is then described in terms of a mass-averaged mixture velocity

and diffusive flux, representing the difference between the mixture velocity and

individual phase velocity. One major advantage of the M2 model over the classical

two-fluid models is that it eliminates the need for tracking phase interfaces, thus

simplifying the numerical complexity of two-phase flow and transport modeling.

Another salient feature of the M2 model for PEM fuel cell is that all model equations

are valid in all three types of regions possible in a PEM fuel cell: single-phase (gas),

liquid-gas (two-phase), and single-phase (liquid). Finally, the M2 model is

mathematically equivalent to two-fluid models without invoking any additional

approximations. These aforementioned advantages render the M2 model to be a

suitable and widely adopted two-phase flow and transport modeling framework for

PEM fuel cells [11, 13, 15].

In this chapter, a three-dimensional, two-phase, multi-component model of a

PEM fuel cell is presented and the present analytical study focus on two-phase

transport taking place in the cathode due to the production of water in the cathode

catalyst layer.

2.1 Model Description

bipolar plates, gas flow channels, gas diffusion layers, and catalyst layers on both the

anode and cathode sides, and a polymer membrane that is sandwiched between them,

as illustrated in Fig. 2.1. Symmetry is assumed, and a single straight flow channel is

therefore considered herein.

2.2 Basic Assumption

Utilizing the homogeneous model of M2 formulation for two-phase transport,

the proposed model incorporates the following assumptions. These basic assumptions

are made to simplify actual cell conditions in the theoretical model and thus facilitate

the modeling approach of transport component influence on transport phenomena and

cell performance.

z The gaseous phase of the working fluid behaves as an ideal gas and the liquid

water is incompressible;

z The Reynolds number of the fluid is below 100 because the velocity of the

mixture is low, and the flow is considered to be laminar;

z The properties of the porous medium are isotropic and homogeneous.

z The system operates in a steady state;

2.3 Governing Equations

The three-dimensional, multi-component, and two-phase model of a PEM fuel

cell includes five nonlinear coupled conservation equations of mass, momentum,

energy, species, and charge, which are described as follows.

Mass conservation equation

Since the gaseous and liquid water are present simultaneously in the control

volume, the conservation equation of mass for a multiphase mixture is:

(

)

=0 ⋅

∇

ε

_eff

ρ

uK (2.1)

where ρ represents the density of the mixture and is defined as the volume-weighted

average of the phase mass concentration in two-phase flow [11]. When the liquid

water is present, the effective porosity is given by

ε

_eff = 1

ε

(

−s

)

.

Momentum conservation equation

The general form of the Navier–Stokes equation is used with source terms that

describe the drag forces in the porous medium. The equation is:

(

effuu

)

= − eff∇P+∇⋅

(

eff eff∇u

)

+ eff kg+Sm

⋅

∇

ρε

KK

ε

ε

μ

K

ε

ρ

(2.2)

where μ_eff is the effective viscosity of the mixture, ρ_k is the kinetic density, and Sm is the sum of the Darcy and the Forchheimer drag forces,

)

( _{Darcy Forch}

m F F

S =− G + G (2.3)

Darcy drag force =

K u

F_Darcy eff eff

G

G

ε

2

μ

= (2.4)

Forchheimer drag force = uu

K C

FG_Forch

ε

eff F

ρ

G G

3

= (2.5)

The parameters CF andK represent the quadratic drag factor and the permeability.

Species conservation equation

The species conservation equation for the gas mixture is

(

effuCk

)

=∇⋅

(

Dkeff∇Ck

)

+Sc

⋅

∇

ε

G _, (2.6)

where k represents the chemical species, including hydrogen, oxygen, nitrogen, and

water. Dk,eff =

D

k

ε

τrepresents the effective diffusion coefficient of the k-th component

of the fuel reactant [17]. The exponent τon the porosity ε is the tortuosity of the

porous medium. The source term Sc defines the production or consumption of the k-th

species in the gas phase and is given by:

a a j FC H 2 1 : 2 − (2.7) c c j FC O 4 1 : 2 − (2.8) c c j FC O H 2 1 : 2 (2.9)

The terms j_aand j_care defined as,

( ) ( ) ⎥⎦⎤ ⎢⎣ ⎡ ₋ ⎟ ⎟ ⎠ ⎞ ⎜ ⎜ ⎝ ⎛ = α F/RT η _{α F/RT η} ref H H ref a a _c e e C C Aj j 1 2 2 0

(2.10)

( ) ( ) ⎥⎦⎤ ⎢⎣ ⎡ ₋ ⎟ ⎟ ⎠ ⎞ ⎜ ⎜ ⎝ ⎛ = α F/RT η _{αF/RT η} ref O O ref c a _c e e C C Aj j 1 2 2 0

(2.11)

The above conservation equations of mass, momentum, and species are derived on

the basis of the M2 model. The constitutive relationships of mixture parameter and

variables are all dependent on the liquid saturation, defined as the ratio of the liquid

volume to the pore volume:

pore l V V s= (2.12) Density:

ρ

=

ρ

_ls+

ρ

_g

(

1−s

)

(2.13) Concentration:

ρ

C =

ρ

_lC_ls+

ρ

_gC_g

(

1− s

)

(2.14)

Effective porosity:

ε

_eff = 1

ε

(

−s

)

(2.15)

Kinetic density:

ρ

k =

ρ

l

λ

l

( )

s +

ρ

g

λ

g

( )

s (2.16) Effective viscosity:

₍

₎

₍

(

)

₎

g rg l rl g l eff k k s s

ν

ν

ρ

ρ

μ

+ − + = 1 (2.17) Relative permeability:

₍

₎

phase phase gas liquid s k s k rg rl ⎪⎩ ⎪ ⎨ ⎧ − = = 3 3 1 (2.18) Relative mobility:

( )

( )

( )

phase phase gas liquid s s k k k s l g g rg l rl l rl l ⎪ ⎩ ⎪ ⎨ ⎧ − = + =

λ

λ

ν

ν

ν

λ

1 (2.19)

Liquid water transport

water vapor exceeds the saturation vapor pressure. The liquid water thus formed may

occupy the pores and thereby prevent the diffusion of fuel, causing mass transport

overpotential in the porous medium. Hence, the effect of liquid water is taken into

account. Additionally, capillary forces dominate the transport of liquid water on the

hydrophilic surfaces because the pores in the porous medium are extremely small.

Therefore, the generalized Richards equation, developed by Wang and Beckermann

[12, 40] to elucidate the two-phase flow transport in capillary porous media, is

applied:

( )

(

)

(

)

l l rg g rl g l rg rl c eff l eff S k k g k Kk s D s u _⎟⎟+ ⎠ ⎞ ⎜ ⎜ ⎝ ⎛ + − − ∇ ⋅ ∇ = ⋅ ∇

ν

ν

ρ

ρ

ε

λ

ρ

ε

K (2.20)

whereDcandPcrepresent the capillary diffusion coefficient and the capillary pressure,

respectively:

(

)

l rg g rl c rg rl c k k ds dP k Kk D

ν

ν

+ − = (2.21)

( )

s J K Pc _c eff 2 1 cos _⎟⎟ ⎠ ⎞ ⎜⎜ ⎝ ⎛ =

ς

θ

ε

(2.22)

and where J(s) is the Leverett function, which takes the following form [41, 42]:

( )

(

)

(

)

(

)

⎭ ⎬ ⎫ ⎩ ⎨ ⎧ > < + − − + − − − = _{2 3 0}0 3 2 90 90 263 . 1 12 . 2 417 . 1 1 263 . 1 1 12 . 2 1 417 . 1 c c s s s s s s s J

θ

θ

(2.23)

In Eq. (2.22), the contact angle, θc, of the gas diffusion layer is dependent upon the

hydrophilic (0° < θc < 90°) or hydrophobic (90° < θc < 180°) nature of this layer, and

varies with the Teflon content. We assume here that the gas diffusion layer is a

hydrophilic medium. Further, the surface tension, ζ, for the liquid water-air system is

taken as 0.0625 N/m [41].

The source term S_l is a simplified switch function between condensation

and/or evaporation of liquid water under these non-equilibrium conditions [15]. When

the partial pressure of water vapor exceeds the saturation pressure of water, liquid

water may form and occupy the pores in the porous medium. Conversely, the liquid

water will evaporate if the partial pressure of water vapor is less than the saturation

pressure of water: ⎪ ⎩ ⎪ ⎨ ⎧ < − > − = sat w O H w l eff eva sat w sat w w eff con l l P P x if P P x s r P P x if P P x RT x r M S ), ( , ) ( 2

ρ

ε

ε

(2.24)

where rcon and reva are the condensation and evaporation rate constants, respectively;

w

x is the molar fraction of water vapor, and P_satis the saturation pressure of water,

which varies with the temperature [4].

3 7 2 5 10 2.1794 0.02953 9.1837 10 1.4454 10 log P_sat =− + T − × − T + × − T (2.25)

The heat generation sources in a PEM fuel cell account for the irreversible

heat and entropic heat that is generated by electrochemical reactions, Joule heating

that arises from proton/electronic resistance, and the latent heat of water condensation

and/or evaporation. A generalized energy conservation equation is:

(

)

T n i n j i j ij eff p eff S i i S h w D T k T u C _⎟⎟− + ⋅ + ⎠ ⎞ ⎜⎜ ⎝ ⎛ ∇ + ∇ ⋅ ∇ = ⋅ ∇

∑∑

= − =

σ

ρ

ρ

ε

η 1 1 1 G (2.26)

where k represents the effective thermal conductivity. On the right-hand side of _eff

the equation, the first two terms represent the conduction energy and the reactant

enthalpy flux; the third and fourth terms represent electrical-related thermal effects,

and the last term is a source term, associated with the phase change.

Charge conservation equation

In a fuel cell, the potential gradient effect causes electrons and protons to

move along individual paths. Solid phase potential controls the movement of electrons.

Electron transport generally occurs only in the bipolar plates, the diffusion layers, and

the catalyst layers. However, ionomer phase potential controls the motion of protons,

which occurs in the catalyst layer and the membrane. Potential fields in these two

media are described as follows.

(

∇

)

+ =0 ⋅ ∇

σ

_eff

φ

_e S_φe (2.27) c a j j S_φe =− (2.28) For protons:

(

∇

)

+ =0 ⋅ ∇

κ

_eff

φ

_p S_φp (2.29) c a j j S _p − = φ (2.30)

where σ_eff , κ_eff , φ_e , φ_p,

e

S_φ and

p

S_φ denote electron conductivity, proton

conductivity, electronic phase potential, electrolyte phase potential, and consumption

rates of charge and product in the electrochemical reaction in the catalyst layer,

respectively.

The membrane conductivity is strongly related to the temperature and the

water content λ . It is defined as the ratio of the number of water molecules to the

number of charge sites [4]:

(

)

_⎥ ⎦ ⎤ ⎢ ⎣ ⎡ ⎟ ⎠ ⎞ ⎜ ⎝ ⎛ ₋ − = T T eff 1 303 1 1268 exp 00326 . 0 05139 ) (

ψ

κ

(2.31)

The water content of the membrane surface depends on the activity of the water vapor,

which also depends on the partial pressure of the water. Therefore, the empirical

relationship between them can be applied: At anode At cathode

At anode At cathode

⎩ ⎨ ⎧ ≤ < − + ≤ ≤ + − + = 3 1 ) 1 ( 4 . 1 14 1 0 0 . 36 85 . 39 81 . 17 043 . 0 2 3 a a a a a a

ψ

(2.32) sat w P P x a= (2.33) 2.4 Boundary Conditions

Boundary conditions are necessary and crucial for solving the above equations.

They describe the operating conditions as well as the model geometry characteristic of

the PEM fuel cell. Due to the single-domain formulation, boundary conditions are

required only at the external surfaces of the computational domain. The most

important ones are as follows.

Inlet boundaries: The fuel and oxidant flow rates along the flow channel can be

described by a stoichiometric flow ratio,

ξ

, which is defined as the ratio of the

amount of reactant supplied to the amount of reaction to generate the specified

reference current density Iref. [37]. At the gas channel, the temperature and gas species

concentrations are assumed to be uniform. The inlet velocities are specified by

ch O in a in a m ref a in a A x P RT A F I u 1 1 2 2 , , , =

ξ

(2.34) ch H in c in c m ref c in c A x P RT A F I u 1 1 4 , ₂ , , =

ξ

(2.35)

respectively. Am is the geometrical area of the membrane and Ach is the cross-sectional

area of the gas channel.

Outlet boundaries: Fully developed flow is applied. At the outlets, both anode and

cathode channels assumed sufficiently long so that velocity and species concentration

fields are fully developed.

Walls:Neumann conditions and no-slip conditions are applied.

Symmetric boundaries:Mass flux or momentum flux have zero gradients.

Electronic phase potential boundaries: Fixed total cell overpotential at the outer

boundary of the cathode is specified.

plate plate bipolar bipolar otherwise cathode anode at at y e cell e e ⎪ ⎪ ⎩ ⎪ ⎪ ⎨ ⎧ = ∂ ∂ = = 0 0

φ

η

φ

φ

(2.36)

The cell potential can be obtained from the following expression

tot oc l ce V

V = +η (2-37)

where ηtot andVocare total cell overpotential and pen circuit voltage, respectively

[17]. T 0025 . 0 2329 . 0 V_oc = + (2-38)

2.5 Numerical Procedures

2.5.1 Numerical Method

The solution to the governing equations is performed by employing a finite

volume scheme with the model domain divided into a number of cells as control

volumes. The governing equations are numerically integrated over each of these

computational cells or control volumes. The method exploits a collocated

cell-centered variable arrangement with the local or cell-averaged values of the

physical quantities evaluated and stored at each cell center.

The governing equations can be expressed in the form of a generalized

convection-diffusion type of transport equation:

(

ρ

u

ϕ

−

Γ

ϕ∇

ϕ

)

=Sϕ

⋅

∇ (2-39) where ϕ denotes the general dependent variable, Γ the exchange coefficient, ϕ S ϕ

the source term, uG velocity vector, and ρ the density. With the discretization of the governing equations, the coupled finite-difference equations can be expressed in

the form of φ ϕ ϕ ϕ ϕ ϕ a a a a S a_{P P} = _{E E} + _{W W} + _{N N} + _{S S} + (2-40)

where ϕ is the value of ϕ at the current point P, _P ϕ …_E ϕ stand for the values of _S

the grid points adjacent to the point P, and a_P…a_S are known as the link

2.5.2 Calculation Procedure

The governing equations with their related boundary conditions are solved

using a commercial code based on the SIMPLE algorithm for convection-diffusion

problems. The numerical flow diagram of present investigation is shown in Fig. 2.2.

As a convergence criterion it is imposed that the normalized residual for each model

variable is smaller than 10−4 [43].

2.5.3 Model Validation

In the simulation, a uniform grids distribution is used to calculate the complex

electrochemical reaction and physical phenomena in the fuel cell. Three mesh

systems- 41 x 13 x 47, 51 x 16 x 58, and 61 x 21 x 67 are constructed to explore

numerical result dependence on computational cell numbers. Table 2.1 presents

geometrical and operating parameters of the base model in the PEM fuel cell. The

results of the polarization curve by the base model under different grid systems are

shown in Fig. 2.3. Considering both accuracy and economics, the grid system of 51 in

the z-direction, 16 in the x-direction, and 58 in the y-direction was selected for present

research.

To further check the adequacy of the numerical scheme, it is clearly seen from

Fig. 2.4 that the present predictions agree reasonably with the experimental data of

the numerical method used are generally appropriate in analysis of the present

3 [14] Tortuosity of the membrane

0.0625 [41] Surface tension, ζ

100 [17] Evaporation rate constant

1.5 [14] Tortuosity of the diffusion and catalyst layers

0.5 / 1.5 [14] Transfer coefficient at anode/cathode

100 [17] Condensation rate constant

1 / 1 atm Anode/cathode pressure

1.534 / 71.12 mm Fuel cell height/length

0.03 mm Membrane thickness

0.01 mm Catalyst layer thickness

0.3 mm GDL thickness

0.381 mm Shoulder width

0.762 / 0.762 mm Gas channel depth/width

100 / 100 % Relative humidity of anode/cathode inlet

1.5 / 3 [44] Stoichiometry, at / at 1.0

0.28 [45] Porosity of membrane

0.4 / 0.4 [45] Porosity of diffusion and catalyst layers

[14] Permeability of membrane

/ [14] Permeability of diffusion and catalyst layers

3 [14] Tortuosity of the membrane

0.0625 [41] Surface tension, ζ

100 [17] Evaporation rate constant

1.5 [14] Tortuosity of the diffusion and catalyst layers

0.5 / 1.5 [14] Transfer coefficient at anode/cathode

100 [17] Condensation rate constant

1 / 1 atm Anode/cathode pressure

1.534 / 71.12 mm Fuel cell height/length

0.03 mm Membrane thickness

0.01 mm Catalyst layer thickness

0.3 mm GDL thickness

0.381 mm Shoulder width

0.762 / 0.762 mm Gas channel depth/width

100 / 100 % Relative humidity of anode/cathode inlet

1.5 / 3 [44] Stoichiometry, at / at 1.0

0.28 [45] Porosity of membrane

0.4 / 0.4 [45] Porosity of diffusion and catalyst layers

[14] Permeability of membrane

/ [14] Permeability of diffusion and catalyst layers

Table 2.1. Geometrical and operating parameters

Quantity Value 11 10 76 . 1 _× − _1.76×10−11 18 10 8 . 1 _× − 2 / cm A a ξ ξc 2 m 2 m 1 − s 1 1 − −_s atm Sources 1 − Nm

Cathode BP Anode BP Cathode Channel Anode Channel Cathode GDL Cathode CL Membrane Anode CL Anode GDL X Y Z Cathode BP Anode BP Cathode Channel Anode Channel Cathode GDL Cathode CL Membrane Anode CL Anode GDL X Y Z

Begin

Solve momentum eq. obtain u*, v*, w*

Obtain pressure and velocity correction Guess P*

Set P*=P Stop

Solve for scalar equations of species, enthalpy, potential field Obtain mass imbalances

Correct pressure and velocity fields

Converged ?

Yes No

Set up grid system

Set up BCs

Post processing

Begin

Solve momentum eq. obtain u*, v*, w*

Obtain pressure and velocity correction Guess P*

Set P*=P Stop

Stop

Solve for scalar equations of species, enthalpy, potential field Obtain mass imbalances

Correct pressure and velocity fields

Converged ? Converged ?

Yes No

Set up grid system

Set up BCs

Post processing

0 0.2 0.4 0.6 0.8 1 1.2 0 0.2 0.4 0.6 0.8 1 1.2 41 x 13 x 47 51 x 16 x 58 61 x 21 x 67

Vo

lt

a

g

e (

V

)

Current Density ( A/cm

2

)

0.3 0.4 0.5 0.6 0.7 0.8 0.9 1 0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 Squadrito et al. [44] Present result

Vo

lt

a

ge (

V )

Current Density ( A/cm

2

)

Figure 2.4 Comparison of the predicted I–V curve and the experimental data of Squadrito et al. [44].

CHAPTER 3

EFFECTS OF CATHODE HUMIDIFICATION AND POROSITY OF THE GDL ON THE GAS–LIQUID INTERFACE LOCATION IN A PEM FUEL CELL

3.1 Introduction

A PEM fuel cell is prone to gas-liquid two-phase formation due to its low

operating temperature, particularly under highly humidified or high current density

conditions. When the gas diffusion layer and the catalyst layer become saturated with

water vapor, the product water starts to condense and block open pores, reducing the

available paths for oxygen transport. This phenomenon is termed “flooding” and

becomes a major limiting factor of PEM fuel cell performance. Hence, it is critical to

understand the two-phase flow and transport in a PEM fuel cell, and a mathematical

model is useful to improve this understanding. In practice, humidification of anode

fuels and/or cathode oxidants is often used to provide sufficient membrane hydration.

Since water is generated in the cathode catalyst layer from electrochemical reaction

and it also tends to migrate from the anode side to the cathode side by the

electro-osmotic drag, it becomes a key issue in the design and operation of PEM fuel