Analysis of thermal and water management with temperature-dependent diffusion effects in membrane of proton exchange membrane fuel cells

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Analysis of thermal and water management with temperature-dependent

diffusion effects in membrane of proton exchange membrane fuel cells

Wei-Mon Yan

a,∗

, Falin Chen

b

, Hung-Yi Wu

b

, Chyi-Yeou Soong

c

, Hsin-Shen Chu

d a_{Department of Mechatronic Engineering, Huafan University, Shih Ting, Taipei 223, Taiwan, ROC}

b_{Institute of Applied Mechanics, National Taiwan University, Taipei 106, Taiwan, ROC}

c_{Department of Aerospace and System Engineering, Feng Chia University, Seatwen, Taichung 407, Taiwan, ROC} d_{Department of Mechanical Engineering, Chiao Tung University, Hsin-Chu 300, Taiwan, ROC}

Received 12 August 2003; accepted 21 November 2003

Abstract

In the present work, the detailed thermal and water management in the membrane of proton exchange membrane fuel cells (PEMFC) is investigated numerically. The coupling effects of mass diffusion and temperature gradient on the water distribution in the membrane are taken into account with consideration of the temperature-dependent diffusivity. Thermal and water transport equations with various boundary conditions are solved by the control volume finite difference method. Predictions show that under the conditions of fixed water concentration at the cathode side, the effect of cathode temperature, Tc, on the water concentration is significant. Increases in Tcmay lead to an increase in membrane dehydration. At the water-flux condition on the cathode side, the influence of the operating temperature on the water distribution in the membrane shows a similar trend. The effects of the anode temperature, Ta, on the water management in the membrane are also examined. It is found that Tahas considerable impact on the water content in the membrane. In addition, high current density may cause non-uniformity of the temperature distribution in the membrane.

Keywords: Diffusion effects; Proton exchange membrane fuel cells; Thermal and water management

1. Introduction

Recent interests in proton exchange membrane fuel cell (PEMFC) systems have caused extensive studies on thermal and water management. During (PEMFC) operation, water molecules can be carried from the anode side to the cathode side of the membrane by electro-osmosis, and if the trans-port rate of water is higher than the back-diffusion rate from the anode to the cathode, the membrane will become dehy-drated and too resistive to conduct high current. At the cath-ode side of the membrane, where water molecules are not only transported from anode side but also generated by the cathodic reaction, electrode flooding occurs when the water removal rate fails to keep up with its transport rate out of the electrode. On the other hand, the temperature gradient in the membrane may influence the fuel-cell performance by affecting the transport of water and gaseous species as well as the electrochemical reactions in the electrode. Therefore,

∗_{Corresponding author. Tel.:}_{+886-2-2663-3847;}

fax:+886-2-2663-3847.

E-mail address: [email protected] (W.-M. Yan).

it is appealing to have a theoretical model which can pro-vide detailed understanding of the governing phenomena in-side the membrane. This motivates the present study, which examines the water concentration and temperature within membrane of PEMFCs.

In past decades, there have been numerous studies of transport phenomena in PEMFCs. Bernardi[1]proposed a one-dimensional model of water management with consid-eration of the membrane thickness. By using this model, it was found that the diffusion in the water production and evaporation rate in the PEMFC can result in the flooding of the electrode or the membrane dehydration, and there-fore affect the performance of the fuel cells. In addition, the effects of the humidification on the current–voltage curves of the fuel cells under various operating conditions were presented. Springer et al.[2]developed an isothermal, one-dimensional, steady-state model for the PEMFC with Nafion®117[2]. Diffusion, electro-osmotic drag and mem-brane conduction were all taken into account. The results showed that the net water-flux ratio under a typical operat-ing condition is much less than that within a fully-hydrated membrane. It was also found that the membrane

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Nomenclature

Ca water concentration per unit volume at

the anode side (mol cm−3)

Cc water concentration per unit volume at

the cathode side (mol cm−3)

CH2O water concentration in the membrane per

unit volume (mol cm−3)

C_p,l specific heat of liquid water (J kg−1K−1)

d density of the membrane (g cm−3)

D diffusion coefficient of water in the membrane (cm2s−1)

Da diffusion coefficient of water at the

anode side (cm2_s−1₎

Dc diffusion coefficient of water at the

cathode side (cm2s−1)

F Faraday’s constant 96487 (C mol−1)

i operating current density (A cm−2)

K thermal conductivity (W cm−1K−1)

˙mH2O molecular flux of water (mol cm−1s−1)

M molecular weight of water (kg mol−1)

R ohmic resistance per unit volume ( cm−1)

T temperature (◦C)

V volume of the membrane (cm3)

w water transfer coefficient

Greek letters

κ flux of water into membrane by concentration gradient (m s−1)

λ membrane hydration or water content (moles water/moles charge sites)

ν rate of water entry the membrane proportional to the current density

Subscripts

a anode side of the membrane c cathode side of the membrane m membrane

tance is significantly enhanced as the current density is in-creased. By comparison, the resistance is reduced for a thin membrane.

Fuller and Newman[3]examined experimentally the wa-ter transport number in Nafion® 117. The relationship be-tween transport number and electro-osmotic coefficient was presented. It was demonstrated that the transport number decreases slowly as the membrane is dehydrated, but falls quickly to zero when the water concentration approaches to zero. Nguyen and White[4]performed modelling of the wa-ter and heat management in PEMFC. The model included the effect of electro-osmosis, diffusion of water; heat trans-fer from solid phase to gas phase and latent heat as water evaporation and condensation. It was found that the ohmic loss is noticeable at high current density. The voltage loss is

twice amount of that at the cathode electrode. The reactant gas at the anode needs to be humidified since the membrane is dehydrated at high current densities. Fuller and Newman

[5]proposed a two-dimensional mathematical model for the water and thermal management and the utilization of the fuel of a PEMFC. Due to the water sorption depending strongly on the temperature, the waste heat is a critical parameter in the design of the proton exchange membrane fuel cells.

In the numerical analysis of Mosdale and Srinivasan[6], it was clearly seen that the large current density limit of fuel cell is more for pure oxygen than for air used at the cathode side. Voss et al.[7]proposed a new technique for water man-agement, by which it was found that if the back-diffusion rate and the water concentration are increased, the water at the cathode could be removed via the anode stream. Xie and Okada [8] showed that the water transfer coefficient of Nafion® 117 membrane in the H+ form was 2.6. The Nafion®117 membrane has good performance for HCl so-lutions with a concentration that ranges from 0.003 to 1 N. Additionally, it was also shown that the water transport be-haviour is related to the surface-change density, the hydra-tion enthalpy and the water content in the membrane.

By using a linear transport equation for water in the PEMFC, detailed transport phenomena of the PEMFC, in-cluding diffusion and electro-osmotic drag effects, were an-alytically solved by Okada et al. [9,10]. In these studies, both semi-finite and finite boundaries were considered. The predicted results showed that the current density, the wa-ter penetration paramewa-ters, the membrane thickness and the diffusion coefficient of water are the key factors in determi-nation of the water content in the membrane. Foreign impu-rities such as NaCl will cause a serious impact on the water depletion at the anode side. Water supplied from the anode side of the membrane is needed. Okada extended the mod-elling to account of the effect of impurity ions at both the anode and the cathode side of the membrane [11,12]. The results indicated that both the current density and the mem-brane thickness are important parameters in the water man-agement of the membrane, especially when the membrane surface has impurity ions. The distribution of contaminant ions degrades the membrane and the performance of the fuel cell. Deterioration of cell performance in the presence of non-uniform impurities in the membrane is more serious than in the case of non-uniform impurities distribution.

Thermal management in the direct methanol fuel cell (DMFC) was investigated by Argyropoulos et al.[13,14]. A model was developed to investigate the effects of various op-erating parameters (feed and oxidant temperatures, flow rate and pressure, operating current density) and system design (active area, material properties and geometry) on the per-formance of the DMFC. The mathematical model includes the gas-diffusion layer, the catalyst layer and the membrane. It can also be used to predict the steady-state performance of the DMFC stacks. The diffusion flux across a Nafion® mem-brane can be accurately predicted by using Fick’s diffusion coefficient. Motupally et al.[15]showed that increasing the

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cell pressure will decrease the water activity and reduce the diffusion coefficient.

Baschuk and Li [16] developed a mathematical model with variable degrees of water flooding in the PEMFC. Phys-ical and electrochemPhys-ical processes occurring in the mem-brane electrolyte, the cathode catalyst layer, the electrode backing layer and the flow channel were considered. Com-pared with experimental results, it was found that when air is used as the cathode fuel, the flooding phenomena are similar for different operating conditions of the pressures and tem-peratures. When the cell pressure is increased significantly, the water flooding in the electrode becomes serious. This will significantly reduce the power output. Recently, Rowe and Li[17]carried out a two-dimensional simulation of wa-ter transport in the PEMFC without exwa-ternal humidification. This model calculated the fraction of product water leaving the anode side of the fuel cell. The results indicated that the amount of water leaving the anode depends on the hydrogen stoichiometry, oxygen stoichiometry, current density, and cell temperature. One of the most recent PEMFC models was proposed by Djilali and Lu[18]for analysis of fuel-cell performance and water transport. The thermodynamic equa-tion was determined by the Nernst equaequa-tion and the reac-tion kinetics were calculated by the Butler–Volmer equareac-tion. Analysis showed that the water requirement to prevent the membrane from dehydrating or flooding is important.

From the literature reviews presented above, it is con-cluded that the effects of the temperature gradient on water management in the PEMFCs are not well defined. In fact, the water content in the membrane can be influenced by the local temperature distribution since the diffusivity in water transport is temperature-dependent. On the other hand, the energy balance is also closely related to the water content or local water concentration in the membrane. The objective of the present study is to explore the coupling mechanisms of thermal–mass-transport phenomena in the membrane of PEMFC systems.

2. Analysis

Consideration is given to a PEMFC in which the polymer electrolyte membrane made from Nafion®. Its thickness is smaller than its length and width, as shown schematically in Fig. 1. Therefore, it can treat it as a one-dimensional problem. To simplify the analysis, the following assumptions are made.

(i) The transports are steady-state and one-dimensional. (ii) The pressure is constant.

(iii) An ideal gas mixture is assumed.

(iv) Liquid water flux is only determined in the membrane. (v) The volume of the membrane is constant.

(vi) The convective effects are negligible for a small Reynolds number.

x

y z x=o x=d Anode catalyst layer Cathode catalyst layer Membrane H+ H2O(drag) H2O(diffusion)

Fig. 1. Schematic diagram of physical system.

(vii) Heat loss to the surrounding environment is small and can be neglected.

(viii) Joule-heating is considered to be to the membrane ohmic resistance.

With the above assumptions, the governing equations for the water balance can then be formulated as follows.

2.1. Water transfer equation

In the membrane of a PEMFC, the water flux is com-posed of two components, namely, a diffusion flux and an electro-osmosis flux[4,5]. The latter is proportional to the current density, i. The total water flux can then be described by: ˙mH2O= −DH2O dCH2O dx + i FwH2O , (1)

where: ˙mH2Ois the molar flux of the water;DH2Ois the

dif-fusion coefficient of water in the membrane;CH2Ois the

wa-ter concentration in the membrane; i is the current density;

F is the Faraday constant;wH2Ois the water transfer

coeffi-cient. Therefore, the rate of water concentration is given by:

∂CH2O ∂t = − ∂ ˙mH2O ∂x = ∂ ∂x DH2O ∂CH2O ∂x − i FwH2O (2)

For steady-state conditions, the above equation becomes: d dx DH2O dCH2O dx − i FwH2O = 0 (3) dDH2O dx dCH2O dx + DH2O d2CH2O dx2 − i FwH2O= 0 (4)

Generally, the water transfer coefficient is a function of water concentration, for example:

wH2O = w(0)1 + w(1)1 CH2O+ w21C 2

H2O+ · · · (5)

To simplify the analysis, only the first two terms,w(0)₁ and

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coefficients with respect toCH2O. The water transfer

coeffi-cient can then be expressed as:

wH2O= w(0)₁ + w(1)₁ CH2O (6)

The water transfer coefficient for Nafion®membrane is cal-culated by the following equation[3,9]:

wH2O=

1100wmVwet

22dVdry

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where: the volume ratio for dry to wet, Vwet/Vdry, is 16.2, and

the density of the membrane, d, is 2.02 g cm−3. In addition, the water transfer coefficientwm is 3.2 at 80◦C.

The diffusion coefficient for liquid water in the membrane is determined as a function of temperature (in K) and mem-brane hydration[2], i.e.,

DH2O= exp 2416 1 303 − 1 T (2.563 − 0.33λ + 0.0264λ2_{− 0.000671λ}3_{) × 10}−10 _(8a)

If the membrane hydration parameterλ is taken to be 14, as given in[17], then the above equation reduces to:

DH2O= G exp −ξ T (8b) Here: ξ = 2416 (8c) G = 2.903 × 10−7f(λ) (8d) f(λ) = 2.563 − 0.33λ + 0.0264λ2_{− 0.000671λ}3 _(8e) 2.2. Energy equation

The energy equation is based on Fourier’s law of heat conduction; i.e., Km d2T dx2 + d dx( ˙mH2OCp,lT) + i 2_{R = 0} ₍₉₎

where: Kmis the membrane thermal conductivity, Cp,lis the

specific heat of liquid water, and R is the ohmic resistance per unit volume. The first term represents the diffusion term of the heat, the second term expresses the energy flux due to the convection, and the third term stands for the joule-heating owing to the membrane ohmic resistance.

2.3. Combination of water transport and energy equations

At first, the molar flux of water is changed into the mass flux of water. ThenEq. (1)becomes:

˙mH2O= −DH2O dCH2O dx + i FwH2O M (10)

where M is the molecular weight of water. Substituting the above equation intoEq. (9)gives:

Km d2T dx2 + −2DH2O dCH2O dx MCp,l+ 2i Fw(1)1 CH2OMCp,l ×dT dx + i 2_{R +} −dDH2O dx dCH2O dx MCp,l − DH2O d2CH2O dx2 MCp,l+ i Fw(1)1 dCH2O dx MCp,l T = 0 (11) By combiningEqs. (8) and (11), the above equation can be simply expressed as:

d2T dx2 + A exp _−ξ T dCH2O dx + 2BCH2O dT dx = H (12a)

Here the constant, A, B, and H are:

A = −2GMC_K p,l m (12b) B = iw (1) 1 MCp,l FKm (12c) H = −i2R Km (12d)

Similarly, the water transfer equation,Eq. (4), can be sim-plified as: d2CH2O dx + ξ T2 dT dx − N exp ξ T dCH2O dx = 0 (13a) where: N = iw(1)1 FG (13b) 0.7 0.75 0.8 0.85 0.9 0.95 1 0 0.2 0.4 0.6 0.8 1

x/d

C/Co

i=0.1A/cm2 0.5 Present Predictions Okada et al. [9] 0.3

Fig. 2. Comparison of present predictions with those of Okada et al.[15] under conditions ofi = 0.1 A m−2,Ta= 60◦C,Tc= 60◦C and constant cathode concentrationCc= 1.59 × 10−4mol m−3.

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Table 1

Physical parameters and corresponding values used in this work

Parameter Symbol Value

Constant term of water transference coefficient at anode side of membrane as expressed by a series expansion ofCH2O w(0)a 0 Constant term of water transference coefficient at cathode side of membrane as expressed by a series expansion ofCH2O w

(0)

c 0

First order term of water transfer coefficient at anode side of membrane as expressed by a series expansionCH2O w (1)

a 1.28 × 10−4

First order term of water transfer coefficient at cathode side of membrane as expressed by a series expansionCH2O w (1)

c 1.28 × 10−4

Current density (A cm−2) i 0–3.1

Coefficient characterizing water flux into anode side of membrane νa 0–1.0 Coefficient characterizing water flux into cathode side of membrane νc 0–1.0

Specific conductivity at anode side of membrane (cm s−1) κa 1× 10−3to 1

Specific conductivity at cathode side of membrane (cm s−1) κc 1× 10−3to 1

Thickness of membrane (cm) d l× 10−2

Thermal conductivity of membrane (W cm−1K−1) Km 0.0014

Specific heat of liquid water (J kg−1K−1) C_p,l 4180

Faraday constant (A s mol−1) F 96487

Molecular weight (kg mol−1) M 0.018

Ohmic resistance per unit length ( cm−1) R 0.000945

0.94 0.95 0.96 0.97 0.98 0.99 1 0 0.2 0.4 0.6 0.8 1

C/Co

x/d

60 70 80 90 Ta=100oC 0.94 0.95 0.96 0.97 0.98 0.99 1 0 0.2 0.4 0.6 0.8 1

C/Co

x/d

Ta=100oC 60 (a) (b)

Fig. 3. Water concentration distribution in membrane withi = 0.1 A cm−2,Tc= 100◦C, and constant cathode concentrationCc= 1.59 × 10−4mol cm−3: (a) constant diffusion coefficient; (b) variable diffusion coefficient.

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2.4. Boundary conditions

To solve the governing equations formulated in the last section, the following boundary conditions are specified.

2.4.1. Concentration conditions at anode-membrane interface

At the anode-side membrane interface, the condition of water-flux balance[9–12]is imposed, namely:

νai F + κa[Ca− CH2O(0)] = −D(0)a ∂CH2O(0) ∂x + i F[w(0)a + w(1)a CH2O(0)] (14) 0.94 0.95 0.96 0.97 0.98 0.99 1 0 0.2 0.4 0.6 0.8 1 C/Co x/d Tc=60oC 100 70 80 90 0.93 0.94 0.95 0.96 0.97 0.98 0.99 1 0 0.2 0.4 0.6 0.8 1 C/Co x/d Tc=60oC 100 70 80 90 0.93 0.94 0.95 0.96 0.97 0.98 0.99 1 0 0.2 0.4 0.6 0.8 1 C/Co x/d Tc=60oC 100 70 80 90 (a) (b) (c)

Fig. 4. Effect of Tc on water concentration distribution with

i = 0.1 A cm−2 _{and constant cathode concentration} _C

c = 1.59 × 104 mol cm−3: (a)Ta= 60◦C; (b)Ta= 80◦C; (c)Ta= 100◦C.

where:νais a factor expressing the rate of water entry at the

anode side of the membrane and is proportional to the cur-rent density;κais a factor characterizing the

concentration-gradient-driven water flux into or out of the membrane; Ca

is the concentration of water at the anode-membrane inter-face;CH2O(0) is the water concentration in the membrane

atx = 0.

2.4.2. Concentration conditions at membrane-cathode interface

Two types of boundary condition for the water concen-tration at the membrane–cathode interface are studied. One is the constant water concentration:

CH2O(d) = C0 (15) 0.92 0.93 0.94 0.95 0.96 0.97 0.98 0.99 1 0 0.2 0.4 0.6 0.8 1 x/d C/C(d) Tc=60oC 100 70 80 90 0.92 0.93 0.94 0.95 0.96 0.97 0.98 0.99 1 0 0.2 0.4 0.6 0.8 1 x/d C/C(d) Tc=60oC 100 70 80 90 0.92 0.93 0.94 0.95 0.96 0.97 0.98 0.99 1 0 0.2 0.4 0.6 0.8 1 x/d C/C(d) Tc=60oC 100 70 80 90 (a) (b) (c)

Fig. 5. Effect of Tc on water concentration distribution with

i = 0.1 A cm−2 _{and water-flux condition at cathode side: (a)}_T

a= 60◦C; (b)Ta= 80◦C; (c)Ta= 100◦C.

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the other is a water-flux condition: νci F + κc[Cc− CH2O(d)] = D(0)c ∂CH2O(d) ∂x − i F[w(0)c + w(1)c CH2O(d)] (16)

where νc is a factor expressing the rate of water entry at

cathode side of the membrane proportional to the current density; CH2O(d) is the water concentration at x = d in

Eq. (16);D(0)c is the diffusion coefficient of water at cathode

side of the membrane.

2.4.3. Thermal conditions at anode and cathode sides

In this study, the thermal conditions at the anode and cathode sides of the membrane are constant temperatures,

0.93 0.94 0.95 0.96 0.97 0.98 0.99 1 0 0.2 0.4 0.6 0.8 1 C/Co x/d Ta=60oC 100 70 80 90 0.94 0.95 0.96 0.97 0.98 0.99 1 0 0.2 0.4 0.6 0.8 1 C/Co x/d Ta=60oC 100 70 80 90 0.94 0.95 0.96 0.97 0.98 0.99 1 0 0.2 0.4 0.6 0.8 1 C/Co x/d Ta=60oC 70 80 90 100 (a) (b) (c)

Fig. 6. Effect of Ta on water concentration distributions with

i = 0.1 A cm−2 _{and constant cathode concentration} _C

c = 1.59 × 10−4 mol cm−3: (a)Tc= 60◦C; (b)Tc= 80◦C; (c)Tc= 100◦C.

Ta and Tc, respectively, i.e.,

T(0) = Ta (17)

T(d) = Tc (18)

3. Numerical method

The system of the governing equations mentioned above is non-linear and is difficult to obtain an analytical solution. In this work, the control volume finite difference method is adopted to solve the non-linear, coupled ordinary differential equations. The detailed solution scheme has been published elsewhere[19]. To check the grid independence, solutions on various grid systems are examined. In the separate numerical runs, it is found that there are no differences among the

0.94 0.95 0.96 0.97 0.98 0.99 1 0 0.2 0.4 0.6 0.8 1 C/Co x/d Tc=60oC 100 80 70 90 0.65 0.7 0.75 0.8 0.85 0.9 0.95 1 0 0.2 0.4 0.6 0.8 1 C/Co x/d Tc=60oC 100 70 80 90 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1 0 0.2 0.4 0.6 0.8 1 C/Co x/d Tc=60oC 100 70 80 90 (a) (b) (c)

Fig. 7. Effect of Tcon water concentration distribution withTa= 60◦C and constant cathode concentration Cc = 1.59 × 10−4mol cm−3: (a)

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solutions with three grid arrangements of 1000, 2000 and 3000 points. In order minimize the calculating time, 1000 grids are adopted for the present problem. Additionally, it is important to compare the predicted results with existing numerical or experimental data. In the comparison shown in

Fig. 2, it is apparent that the present predictions agree well with those of Okada et al.[9]. Through these preliminary tests, it is found that the numerical method is suitable for the present study.

4. Results and discussion

In Section 2, several parameters appear in the formula-tion. The physical parameters and their corresponding val-ues are presented in Table 1. To disclose the effects of

0.93 0.94 0.95 0.96 0.97 0.98 0.99 1 0 0.2 0.4 0.6 0.8 1 C/Co x/d Ta=60oC 100 0.65 0.7 0.75 0.8 0.85 0.9 0.95 1 0 0.2 0.4 0.6 0.8 1 C/Co x/d Ta=60oC 100 0.4 0.5 0.6 0.7 0.8 0.9 1 0 0.2 0.4 0.6 0.8 1 C/Co x/d Ta=60oC 100 (a) (b) (c)

Fig. 8. Effect of Ta on water concentration distribution withTc= 60◦C and constant cathode concentration Cc = 1.59 × 104mol cm−3: (a)

i = 0.1 A cm−2_{; (b)}_{i = 0.5 A cm}−2_{; (c)}_{i = 1.1 A cm}−2_.

the temperature-dependent diffusion coefficient on the wa-ter concentration distribution, Fig. 3(a) and (b)shows, re-spectively, the distribution of water concentration with or without consideration of a variable diffusion coefficient. It is seen that the water concentration increases with x/d. In addition, a large water concentration is noted for a system with a lower anode temperature Ta. It is also found that

these are noticeable differences between the results with or without consideration of variable diffusion coefficient. This implies that the effects of a variable diffusion coefficient on the water content in the membrane are of importance.

For thermal and water management in PEMFCs, the ther-mal effects of the anode and cathode temperatures (Ta and

Tc) on the water concentration in the membrane may be

im-portant. The effects of Taand Tcon the water concentration

at a current density i = 0.1 A cm−2 and a water

concen-0.3 0.4 0.5 0.6 0.7 0.8 0.9 1 0 0.2 0.4 0.6 0.8 1 x/d C/Co 1.1 0.3 0.5 0.7 0.9 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1 0 0.2 0.4 0.6 0.8 1 C/Co x/d 1.1 0.3 0.5 0.7 0.9 0.4 0.5 0.6 0.7 0.8 0.9 1 0 0.2 0.4 0.6 0.8 1 x/d C/Co 1.1 0.3 0.5 0.7 0.9 (a) (b) (c) i=0.1A/cm2 i=0.1A/cm2 i=0.1A/cm2

Fig. 9. Effect of current density i on water concentration distribution with

Ta= 60◦C and constant cathode concentrationCc= 1.59×104mol cm−3: (a)Tc= 60◦C; (b)Tc= 80◦C; (c)Tc= 100◦C.

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tration on cathode side of Cc= 1.59 × 10−4mol cm−3are

shown inFig. 4. That data show that the water concentra-tion at the anode side of the membrane decreases with in-crease in Ta. This can be explained by the fact that, as Ta

is increased, the diffusion coefficient becomes larger (see

Eq. (8a)). Therefore, water diffusion from the anode side of the membrane is enhanced. This means that an increase in

Tacauses dehydration of the anode. At a fixed Ta, a higher

water concentration within the membrane can be found in a system with a higher cathode temperature Tc due to strong

back-diffusion from the cathode to the anode.

The effects of cathode temperature on the water concen-tration distribution with water-flux conditions are shown in

Fig. 5. As inFig. 4, three sub-plots with different anode tem-peratures Ta are presented. It is noteworthy that the

dimen-sionless water concentration, C/C(d), is presented, where

60 60.2 60.4 60.6 60.8 61 0 0.2 0.4 0.6 0.8 1 x/d T( o C) 0.1 1.1 60 60.5 61 61.5 62 0 0.2 0.4 0.6 0.8 1 x/d T( o C) 0.1 1.1 60 60.5 61 61.5 62 62.5 63 0 0.2 0.4 0.6 0.8 1 x/d T( o C) (a) (b) (c) i=2.1A/cm2 i=2.1A/cm2 i=2.1A/cm2 1.1 0.1

Fig. 10. Effect of current density i on temperature distribution with

Ta= 60◦C, constant cathode concentrationCc= 1.59 × 104mol cm−3: (a)Tc= 61◦C; (b)Tc= 62◦C; (c)Tc= 63◦C.

C(d) is the water concentration at the cathode side of the

membrane. An overall inspection inFig. 5indicates that, for water-flux conditions at the cathode side, the water concen-tration increases with the normalized depth from the anode side. In the region near the anode side (i.e., at small values of x/d), a larger normalized water concentration, C/C(d), is noted for a system with a lower Tc. By contrast, in the region

away from the anode side (i.e., at large values of x/d), C/C(d) increases with an increase in Tc. In fact, the local water

con-centration, C(x), is a function of the operating temperatures,

Tcand Ta. As Tcis raised, membrane dehydration occurs at

the anode side, but hydration occurs at the cathode side. The dependence of the water concentration profiles on the temperature at cathode side of the membrane (Tc= 60

to 100◦C) is shown inFig. 6. Here the water concentration at the cathode side of the membrane is kept constant. The results show that at fixed Taa higher water concentration at

the anode side of the membrane is found in a system with a higher Tc. This is due to the fact that increasing Tc will

markedly enhance the membrane hydration. That is, the back diffusion of water to the anode side is significant at a high Tc.

In order to realize how the current density affects the wa-ter content in the membrane,Fig. 7 presents the effects of

1.5 1.52 1.54 1.56 1.58 1.6 10-3 0.01 0.1 1 κa (cm/s) C(mol/cm 3 ) Ta=60oC 70 80 90 1.61 1.62 1.63 1.64 1.65 1.66 1.67 1.68 1.69 10-3 0.01 0.1 1 κa (cm/s) C(mol/cm 3 ) Ta=60oC 70 80 90 (a) (b) ( 104 ) ( 104 ) × ×

Fig. 11. Effect of humidification factor κa on water concentration dis-tribution withTa= 60◦C,i = 0.1 A cm−2 and water-flux condition at cathode side under Tc: (a) concentration at anode side; (b) concentration at cathode side.

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the current density i on the water concentration distribution withTa = 60◦C and a constant cathode concentration of

Cc= 1.59×104mol cm−3. The influence of i on water

con-centration at the anode side is similar for different Tc.

Care-ful inspection of the data shows that there is a smaller water concentration at the anode side at a large current density. This can be explained by noting that an increase in current density causes the membrane to be seriously dehydrated due to water drag by electro-osmosis. As for the results men-tioned above, at a fixed x/d and i, the water concentration increases with an increase in Tc.

The effect of Ta on the water concentration distribution

is shown in Fig. 8 with Ta = 60◦C and Cc = 1.59 ×

104mol cm−3under different i. The water concentration pro-file has a parabolic form. As the current density is increased, however, the deviation in the water concentration distribu-tion at different Ta becomes small. Therefore, the

temper-ature at the anode side, Ta has only a small impact on the

water concentration in the membrane at high current density. The influence of current density i on the water concentra-tion distribuconcentra-tions at different anode operating temperatures are presented inFig. 9. By comparing the results inFig. 9(a), it is found that the anode side of the membrane tends to come dehydrated as the current density is raised. This is be-cause that the electro-osmotic drag effect becomes stronger as the current density is higher. It is also found in the sepa-rate numerical runs that the membrane is much wetter for the system with a higher Tcthan that with a lower Tc. This is due

to the temperature-dependence of the diffusion coefficient. The relationship between the current density and the tem-perature distribution is shown inFig. 10. It is clearly shown inFig. 10(a)that when the current density is raised, the tem-perature changes sharply at the anode side of membrane. For example, when it is necessary to speed up a car, the current density must go up. This will cause dehydration of the mem-brane, which, in turns, causes the temperature to increase and become more non-uniform. Thermal expansion of the membrane may become serious and lead to the breakdown of the membrane. Therefore, the strength of the membrane is a key factor for fuel cells operating under high current density conditions.

The effect of the humidification parameterκa on the

wa-ter concentration at the anode and cathode sides withTa =

60◦C and i = 0.1 A cm−2 are shown in Fig. 11. When

κa is increased, water vapour enters the membrane more

freely from the anode gas-diffusion electrode through the anode-membrane interface which, in turn, results in an in-crease in the water content. A careful inspection ofFig. 11

indicates that the water concentration changes sharply when

κa is increased from 10−3 to 10−1cm s−1. But, forκa >

10−1, the effect ofκaon the water content in the membrane

is insignificant.

The influence of the parameters of the electro-osmotic drag at anode side (νa) on the water concentration at the

an-ode and cathan-ode sides is presented inFig. 12. It is observed that the water concentration increases linearly with increase

1.5 1.52 1.54 1.56 1.58 1.6 1.62 1.64 0 0.2 0.4 _ν 0.6 0.8 1 C(mol/cm 3 ) Tc=60oC 100 70 80 90 a 1.62 1.64 1.66 1.68 1.7 1.72 1.74 1.76 0 0.2 0.4 0.6 0.8 1 ν Tc=60oC 100 C(mol/cm 3 ) a 70 80 90 ( 104 ) ( 104 ) × × (a) (b)

Fig. 12. Effect of humidification factorνaon water concentration distri-bution withTa= 60◦C,i = 0.1 A cm−2 and water flux at cathode side under different Tc: (a) concentration at anode side; (b) concentration at cathode side.

inνa. Whenνais increased, the water enters easily the

mem-brane from the anode gas-diffusion electrode through the anode–membrane interface and thus results in an increase in the water content within the membrane.

5. Conclusions

A detailed analysis of the thermal and water manage-ment in the PEMFC membrane with coupling effects of mass diffusion and temperature gradient have been per-formed by using a one-dimensional mathematical model. The thermal–mass diffusion coupling effects are taken into account with consideration of the temperature-dependent diffusivity. The model can predict the water distribution in the membrane under different operating conditions. This is useful for selecting the optimal membrane material and es-timating the gas-inlet temperature or working density in de-signing a PEMFC. The major findings in this study are sum-marized as follows.

(i) Increasing the temperature at the anode side of the membrane can cause dehydration of the membrane.

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(ii) Increasing the current density will increase dehydration of the anode side of the membrane. This is attributed to the strong electro-osmotic drag effect under the op-erating conditions of high current density.

(iii) At high current density, the temperature effect on the water concentration becomes smaller. The current den-sity effect dominates the water concentration distribu-tion.

(iv) Temperature distribution changes sharply in the mem-brane at high current densities. This can damage the membrane.

(v) Increasing the humidification factor κa augments the

water concentration at both the anode and the cathode sides of the membrane. Never the less, increase inκa

above 10−1cm s−1has little influence on the water con-centration.

(vi) At fixed current density, the effects of the parameters of electro-osmotic drag,ν, on the water concentration is considerable. The dependence of the water content onν is almost linear.

Acknowledgements

The authors are grateful for financial support from the National Science Council of Taiwan, NSC 92-2212-E-21 1-001 and NSC 92-2623-7-002-006-ET.

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