Transient behavior of water transport in the membrane of a PEM fuel cell

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Transient behavior of water transport in the membrane

of a PEM fuel cell

Falin Chen

a,*

, Yu-Guang Su

a

, Chyi-Yeou Soong

b

, Wei-Mon Yan

c

, Hsin-Sen Chu

d

a_{Institute of Applied Mechanics, National Taiwan University, Taipei 106, Taiwan, ROC} b_{Department of Aeronautical Engineering, Feng Chia University, Seatwen, Taichung 40745, Taiwan, ROC}

c

Department of Mechatronic Engineering, Huafan University, Shih-Ting, Taipei 22305, Taiwan, ROC d

Department of Mechanical Engineering, National Chiao Tung University, Hsin-Chu 300, Taiwan, ROC Received 10 August 2003; received in revised form 14 October 2003; accepted 7 November 2003

Abstract

The dynamic performance of a PEM fuel cell is one of the most important criteria in the design of fuel cells, especially when the application of the fuel cell in mobile systems is concerned. To attack this issue, we extend the theoretical model developed by Okada (J. Electroanal. Chem. 465 (1999) 1,18) to an unsteady state model and investigate the transient behavior of water transport across the membrane as well as the influences of several physical parameters on the characteristic time to reach the steady state. We also consider the influence due to the presence of foreign impurity ions, which turn out to be a crucial factor affecting the unsteady state features of water transport across the membrane. The results suggest that a higher initial current density to start the operation, a higher water flux from the anode side, a smaller operational current density, and a lower level of contamination in the membrane (especially at the cathode side) can all result in a shorter time for the water transport to reach the steady state, and thus a better dynamic performance of the fuel cell can be obtained.

Keywords: PEM fuel cell; Membrane; Water management; Dynamic performance; Unsteady state

1. Introduction

In applications of PEM fuel cells in mobile systems, various characteristic times inﬂuence the dynamic per-formance of the power system. Of these, the time in-terval from the start of operation to the achievement of the steady state of water transport across the membrane is one of the important characteristic times. To seek this time interval and other potential time constants, the unsteady state water management of the fuel cell will be considered. This, in fact, in the last two decades has become one of the crucial issues in PEM fuel cell re-search, mainly because the electric conductivity depends signiﬁcantly on the water content in the fuel cell, espe-cially in the membrane [1–6].

Many investigations have developed theoretical models governing water transportation in the compo-nents of the fuel cell, such as the fuel gas channel, the gas diffusion layers, the catalyst layers, and the membrane. Together with the boundary conditions on both boundaries and interfaces, nevertheless, the whole mathematical formulation becomes too complicated and tedious to be solved. To avoid the complication and heavy computational loading, a one-dimensional model simulating water transport in the direction across the membrane was developed, see for example [7–11]. The authors of these papers concluded that, under a proper water-balance operation, the important operation pa-rameters could be reasonably estimated. They indicated further that the ohmic loss becomes significant in the high current density regime, while in a moderate current density regime, humidification at the cathode may not be necessary because the water generated at the cathode may be sufficient to counteract the water depletion in the membrane. By considering the non-isothermal effect,

Journal of Electroanalytical Chemistry 566 (2004) 85–93

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Electroanalytical Chemistry

*

Corresponding author. Tel.: 5692; fax: +886-2-3366-5691.

E-mail address:falin@spring.iam.ntu.edu.tw(F. Chen).

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they pointed out that a high operational temperature might lead to a more serious dehydration of the mem-brane, increasing its internal resistance.

Later, many other investigations [12–16] extended the model to two dimensions. It was found that the two-dimensional simulation results in a lower output voltage than the one-dimensional model. It was indicated fur-ther that for a high current density, the ohmic loss in the membrane accounts for a majority part of the overall voltage loss of the cell, and the water diffusing from cathode to anode is not sufficient to supply the need for water at the anode. It was also found that the pressure-gradient driven water transportation across the mem-brane has a significant influence on the water content of the cell and there is an obvious thermal gradient across the cell, which has a significant effect on the cell performance.

Instead of considering all the physical variations in the fuel cell, many studies focused on water management in the whole physical domain including the gas diffusion layer, catalyst layer, and the membrane [17–23], while some work was restricted to the membrane [24,25]. Recently, Okada [26,27] proposed a one-dimensional water transport model in the membrane, in which spe-cial attention was paid to the influences imposed by the contamination at both the anode and cathode. Okada pointed out that the water transportation is influenced significantly by the presence of foreign impurity ions, and that, at the cathode, these influences have a greater effect than at the anode. The contaminant may either exist in the membrane before use or enter into the membrane with the fuel due to the corrosion of piping or stack materials. In their series of studies, Okada and coworkers [28–35] found that for a NafionR_membrane,

the attraction of some impurity ions is larger than that of the Hþ cation, and the sequence is Fe3þ>Ca2þ>Ni2þ>Cu2þ>Naþ, which implies that an ion of higher valence has a greater attraction to the membrane. Since the conductivity of the membrane is linearly proportional to the number of Hþ _{ions in the}

membrane, the conductivity of the membrane decreases as the number of impurity ions increases. Okada and coworkers indicated further that, since the water con-tent, the number of Hþ _{cations and the water}

perme-ability all decrease with increasing number of impurity ions, the contamination could actually influence the fuel cell performance seriously. The electro-osmotic effect of the membrane becomes more significant when foreign impurity ions are present and are increasing. Accord-ingly water transport from the anode to cathode in-creases. However, on the other hand, the water diffusion decreases due to the presence of impurity ions and water transport from the cathode to anode decreases. Conse-quently, when the membrane is contaminated, water depletion at the anode will become more serious, leading to a decrease of the cell performance.

In their series of studies [24–35], Okada and co-workers investigated only the steady state water distri-butions in the membrane. In the application of a fuel cell for a dynamic system, such as a vehicular power system, the transient behavior of the fuel cell operation may be crucial, and this behavior has a close bearing on some of the characteristic times of the system, such as the time needed for water transport in the membrane to reach a steady state. Accordingly, in the present paper, we ex-tend the theoretical model developed by Okada [26,27] and investigate the unsteady state features of the water transportation in the membrane. Special attention is paid to the inﬂuence of the relevant physical and oper-ational parameters on the characteristic time for the water transport across the membrane to reach the steady state.

The paper is organized as follows. In Section 2 the mathematical model and relevant boundary conditions regarding the unsteady state water transport across the membrane are presented. In Section 3 the values of the relevant physical parameters are given. In Section 4 the results from analyses regarding the transient be-havior of water transport across the membrane and the time needed for the water distribution to reach the steady state are discussed. Finally, in Section 5 the conclusions are presented.

2. Theoretical model of unsteady water transport in membrane

Consider a membrane of PEM fuel cell of thickness d (Fig. 1). We investigate the unsteady state water trans-port across the membrane. Although the theoretical model is essentially an extension from the model of Okada [26,27], we still present the details of the math-ematical formulation and relevant physical parameters in this section for the convenience of the following dis-cussion. Firstly, we assume that the interfaces between the membrane and the two electrodes are ﬂat planes and all the physical properties are uniform on the plane. The volume of the membrane is assumed to be unchanging

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with the variation of hydration, so that the thickness d remains constant. The reactions in electrodes are as-sumed to be much faster than the water transport in the membrane, so that the current density across the elec-trodes and the membrane is a step function. Since the combination of the diffusion and the electro-osmosis effects accounts for the flux of water [24], namely, J¼ Doc

oxþ j

Fw; ð1Þ

so that the water transport equation in the membrane can be written as oc ot¼ oJ ox¼ o ox D oc ox j Fw ; ð2Þ

in which J is the flux of water across the membrane, D is the diffusion coefficient of water in the membrane, c is the water concentration in the membrane, i is the cur-rent density across the membrane, w is the water transference coefficient, and F is the Faraday constant. To account for the fact that D and w change with the water concentration, these two coefficients are expressed by the following functional relations [26,27]

D¼ Dð0Þ_{þ D}ð1Þ_c_{þ D}ð2Þ_c2_{þ ;} _ð3Þ

w¼ wð0Þ_{þ w}ð1Þ_c_{þ w}ð2Þ_c2_{þ :} _ð4Þ

After substituting (3) and (4) into (2), we obtain oc ot¼ D ð0Þo 2_c ox2 j Fw ð1Þoc ox þ Dð1Þ_co2c ox2 " þ Dð1Þ oc ox 2 2j F w ð2Þ_coc ox # þ : ð5Þ We neglect the higher order terms on the right hand side of (5) and obtain the ﬁnal water transport equation as follows oc ot¼ D ð0Þo2c ox2 j Fw ð1Þoc ox: ð6Þ

As shown in Fig. 1, the membrane is divided into three regions. The left region extends from x¼ 0 to d1,

in which the contaminant ion designated as Qnþ comes from the anode, and is called the left infected zone (or the Q-type membrane). The middle region extends from x¼ d1to d2, in which no contaminant ion appears, and

is called the non-infected zone (or the H-type mem-brane). The right region extends from x¼ d2 to d, in

which the contaminant ion comes from the cathode, and is called the right infected zone. In the present study, the distributions of Qnþ in the two infected zones are as-sumed to be uniform, so that the ionic fractions of Qnþ and Hþ, denoted by X_Qnþand X

Hþ, respectively, of these

three zones can be expressed by

X_Qnþ ¼ q_a; X_Hþ ¼ 1 q_a ð0 < x < d₁Þ; ð7aÞ

XQnþ¼ 0; X_Hþ¼ 1 ðd₁< x <d₂Þ; ð7bÞ

X_Qnþ¼ q_c; X_Hþ¼ 1 q_c ðd₂< x < dÞ; ð7cÞ

in which qa is the ionic fraction of Qnþ at the anode

(0 < qa<1) and qc is the ionic fraction of Qnþ at the

cathode (0 < qc<1), and XQnþ X_Hþ are expressed by

[26,27,35] XQnþ¼ nc_Qnþ nc_Qnþþ c_Hþ ; XHþ¼ cHþ nc_Qnþþ c_Hþ ; ð8Þ

in which cQnþ and c_Hþ are respectively the concentration

of Qnþ and Hþ _{and n is the valence of the contaminant}

ion.

In the two infected zones, the diffusion coefficient of water D is modified by the presence of the contaminants as follows [26,27,31,32]

D¼ XHþDHMþ XQnþDQM; ð9Þ

in which DHMis the diﬀusion coeﬃcient of water of the

H-type membrane and DQM is the water diﬀusion

coef-ficient of the Q-type membrane. Similarly, with con-tamination, the water transference coefficient is modified into [26,27,31–35]

w¼ NHþwHMþ NQnþwQM; ð10Þ

where NHþ and N_Qnþ are respectively the ionic

transfer-ence numbers of Hþ _{and Q}nþ

ions in the membrane, deﬁned as NHþ ¼ u_HþX_Hþ uHþX_Hþþ u_QnþX_Qnþ ¼ X_Hþ XHþþ pX_Qnþ ; NQnþ¼ 1 N_Hþ ¼ pX_Qnþ XHþþ pX_Qnþ : ð11Þ

In the above equations, wHM and wQM are the

water transference coeﬃcients of the H-type and Q-type membrane, respectively, u_Hþ and u

Qnþ are the

mobility of Hþ and Qnþ, respectively, and p¼ u_Qnþ=

u_Hþ <1.

Due to the presence of contaminants, the water transport equations of the three regions are also chan-ged into the following

oc ot ¼ D ð0Þ qa o2_c ox2 j Fw ð1Þ qa oc ox ð0 < x < d1Þ; ð12aÞ oc ot ¼ D ð0Þ HM o2_c ox2 j Fw ð1Þ HM oc ox ðd1< x <d2Þ; ð12bÞ oc ot ¼ D ð0Þ qc o2_c ox2 j Fw ð1Þ qc oc ox ðd2< x < dÞ; ð12cÞ in which Dð0Þ qa, D ð0Þ

HM and Dð0Þqc are respectively zero-order

coefficients with respect to c of the water diffusion co-efficients (Eq. (3)) of the left, the middle, and the right zones (see Fig. 1), and wð1Þ

qa, w

ð1Þ

HM and wð1Þqc are the

first-order coefficients with respect to c of the water transference coefficients (Eq. (4)) of these three zones,

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respectively. According to Eqs. (3), (7a), (7c) and (9), Dð0Þ qa and D ð0Þ qc can be expressed by Dð0Þ_q a ¼ XHþD ð0Þ HMþ XQnþDð0Þ_QM ¼ ð1 qaÞDð0ÞHMþ qaDð0ÞQM ð0 < x < d1Þ; ð13aÞ Dð0Þ_q_c ¼ XHþDð0Þ_HMþ X_QnþDð0Þ_QM ¼ ð1 qcÞDð0ÞHMþ qcDð0ÞQM ðd2< x < dÞ; ð13bÞ

in which Dð0Þ_QMis the zeroth-order coefficient of the water diffusion coefficient of the Q-type membrane. From Eqs. (4), (7a), (7c), (10) and (11), wð1Þ qa and w ð1Þ qc can be re-written as wð1Þ_q a ¼ 1 1 qaþ paqa ð1 n qaÞwð1ÞHMþ paqawð1ÞQM o ð0 < x < d1Þ; ð14aÞ wð1Þ_q c ¼ 1 1 qcþ pcqc ð1 n qcÞw ð1Þ HMþ pcqcw ð1Þ QM o ðd2< x < dÞ; ð14bÞ

in which wð1ÞQM is the ﬁrst-order coeﬃcient of the water

transference coeﬃcient of the Q-type membrane and pa

is the ratio between the mobilities of Qnþ _{and H}þ _{in the}

membrane close to the anode and pcis another mobility

ratio in the membrane close to the cathode.

To examine the unsteady state features of the water transport in the membrane, we assume initially that the water concentration is

cðx; 0Þ ¼ c0ðxÞ ð0 < x < d; t ¼ 0Þ; ð15Þ

in which c0ðxÞ is the water concentration corresponding

to the initial current density j0. In addition, there are

four boundaries in the physical domain (Fig. 1), and the relevant boundary conditions are illustrated in the following:

(1) The interface between the anode and the left infected zone (x¼ 0). At this interface, the continuity of water ﬂux is satisﬁed so that

k c½ a cð0; tÞ ¼ Dð0Þqa ocð0; tÞ ox þ j F w ð0Þ qa h þ wð1Þ qacð0; tÞ i ; ð16Þ in which k is the rate constant of HenryÕs law ac-counting for the water ﬂux entering into or emerging from the membrane driven by the water concentra-tion gradient in the membrane, called the humidiﬁ-cation parameter, ca is the water concentration in

the membrane being in equilibrium with the water vapor in the anode gas.

(2) The interface between the left infected zone and the non-infected zone (x¼ d1). At this interface the

con-tinuities of both water ﬂux and water concentration are satisﬁed, namely,

Dð0Þ qa ocðd1; tÞ ox þ j F w ð0Þ qa h þ wð1Þ qacðd1; tÞ i ¼ Dð0ÞHM ocðd1þ; tÞ ox þ j F w ð0Þ HM h þ wð1ÞHMcðd1þ; tÞ i ; ð17Þ cðd1; tÞ ¼ cðd1þ; tÞ; ð18Þ

where d1þ and d1 account for the positions in the

right- and left-hand vicinities of the interface, re-spectively.

(3) The interface between the non-infected zone and the right infected zone (x¼ d2). At this interface, again,

the continuities of both water ﬂux and water concen-tration are satisﬁed. They are

Dð0ÞHM ocðd2; tÞ ox þ j F w ð0Þ HM h þ wð1ÞHMcðd2; tÞ i ¼ Dð0Þ qc ocH2Oðd2þ; tÞ ox þ j F w ð0Þ qc h þ wð1Þ qccH2Oðd2þ; tÞ i ; ð19Þ cðd2; tÞ ¼ cðd2þ; tÞ; ð20Þ

where d2þ and d2 account for the positions in the

right- and left-hand vicinities of the interface, re-spectively.

(4) The interface between the right infected zone and the cathode (x¼ d). At this interface, the water con-centration is constant, namely,

cH2Oðd; tÞ ¼ c0; ð21Þ

where c0 is the initial water concentration of the

membrane at the interface x¼ d, which is in equilib-rium with the saturated vapor. This condition is jus-tiﬁed because during the operation of a fuel cell the water is constantly generated from the reaction at the cathode so that the water concentration at the cathode can be maintained constant.

3. The parameters used in the present analyses

We consider the membrane made of NaﬁonR_{117 and}

assume the contaminant ion to be Naþ _{[26,27]. The}

water diﬀusion coeﬃcients of the H-type membrane are DHM¼ Dð0Þ_HM¼ 1:25 105cm2/s (at k¼ 14, T ¼ 80 °C)

and those of the Na-type membrane are DNaM ¼

Dð0Þ_NaM¼ 5:0 106 _cm2_{/s (at k}_{¼ 14, T ¼ 60 °C) [25,36–}

39]. The water transference coeﬃcient of the H-type membrane is wHM¼ 3:2 (at k ¼ 22, T ¼ 80 °C), of the

Na-type is wNaM¼ 10:2 (at k ¼ 17, T ¼ 50 °C) [25,29].

Since wH2O¼ w

ð0Þ_{þ w}ð1Þ_{c, and w}ð0Þ_{¼ 0 and w}ð1Þ _¼

MVexwM=22ddry, so that for a H-type membrane one has

wM ¼ wHM and for a Na-type membrane one has

wM ¼ wNaM, where M is the molar mass of the

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rate of the membrane from dry to wet and is taken as 1.62, and ddryis the density of the dry membrane and

is taken to be 2.02 g/cm3 _{[24,30,36–39]. The water}

concentration in the membrane can be expressed by c¼ kddry=MVex. By assuming that the initial water

concen-tration c0 is in equilibrium with the water vapor in the

membrane, or the vapor entering into the cathode is saturated, one has k¼ 14 [24,25]. The mobility of Hþ_in

the H-type membrane is u0_Hþ¼ 1:49 107 m2/V s and

that of Naþ _{in the Na-type membrane is u}0

Naþ ¼ 2:7

108 _m2_{/V s. The presence of other ions will reduce the}

mobility of the ion itself, so that the above two mobil-ities can be modiﬁed into u_Hþ ¼ u0

Hþð1 0:2X_NaþÞ and

u_Naþ ¼ u0

Naþð1 0:2X_HþÞ [34].

Accordingly, the ratio between the mobilities of Naþ

and Hþ can be expressed by pa¼ u_Naþ uHþ ¼ u0 Naþð1 0:2XHþÞ u0 Hþð1 0:2XNaþÞ ¼u 0 Naþ½1 0:2ð1 qaÞ u0 Hþð1 0:2qaÞ ð0 < x < d1Þ; ð22aÞ pc¼ u_Naþ uHþ ¼ u0 Naþð1 0:2XHþÞ u0 Hþð1 0:2XNaþÞ ¼u 0 Naþ½1 0:2ð1 qcÞ u0 Hþð1 0:2qcÞ ðd2< x < dÞ; ð22bÞ

in which XNaþ is the ionic fraction of Naþ in the

mem-brane [34]. There are some other routine parameters used in the present analyses but these are not shown here. For the details of their values the reader is referred to Table 1 of Ref. [26].

4. Results and discussion

Based on the theoretical model and boundary and initial conditions shown in Section 2, we implement a systematic parametric study to examine the transient behavior of the water content across the membrane under the inﬂuence of various physical parameters. Special attention will be paid to the time needed for the water distribution across the membrane to reach the steady state, accounted for by tss , and the inﬂuence on

tss of diﬀerent physical parameters is discussed. In the

present paper tss is deﬁned as the time at which the

following relation is satisﬁed

absfMax½ynþ1ði; jÞ ynði; jÞ=ynþ1ði; jÞg < 104; ð23Þ

in which ynði; jÞ is any physical variable at the nth time

step on nodeði; jÞ. These control parameters, include the current density j, the initial current density j0, the ionic

fraction of the contaminant at cathode (qc) and anode

(qa), and the humidiﬁcation parameter k. The values of

these parameters employed in the theoretical model corresponding to the base case are T ¼ 80 °C, p ¼ 1

atm, d ¼ 100 lm, and d1=d¼ ðd d2Þ=d ¼ 0:25, which

are j0¼ 0 A/cm2 j¼ 1 A/cm2, k¼ 0:001 cm/s, and

qa¼ qc¼ 0. As the physical eﬀect of one parameter is

considered, the value of this parameter is varied while the others are ﬁxed at those shown above.

4.1. The eﬀect of operational current density j

Fig. 2 illustrates the change of tsswith varying current

density (j) for two diﬀerent initial current densities j0¼ 0 and 0.5 A/cm2. The results of both cases show

that tss increases exponentially with j and reaches a

maximum (tss 9:8s for j0¼ 0 A/cm2, tss 8:2 s for

j₀¼ 0:5 A/cm2_{) after the current density is larger than 1}

A/cm2_{. The increase of t}

sswith j arises because the water

depletion in the membrane becomes more serious when the current density is larger, leading to the water content across the membrane changing more dramatically and thus a larger tss is needed. When the current density is

suﬃciently high, say j > 1 A/cm2_{, the chemical reaction}

is so strong that the water generated from the cathode is suﬃcient to equilibrate the water depletion in the membrane, implying that the water content in the membrane has reached a steady state so that tssdoes not

change signiﬁcantly with j. In the same ﬁgure, one can also see that tssdecreases as j0increases. This is because

for a larger j0 the initial water content of the membrane

is smaller, so that the time needed to reach the steady state is shorter.

Fig. 3 shows the distribution of water concentration across the membrane under three diﬀerent operational current densities, j¼ 0:1, 0.5 and 1.0 A/cm2_{. It is seen}

that the membrane becomes much drier at the anode when the current density is larger. This is because, for a larger j, the electro-osmotic effect, dragging water mol-ecules from anode to cathode is more significant than the water diffusion effect pumping water molecules from

0 2 4 6 8 10 12 0 0.2 0.4 0.6 0.8 1 0.5 0 j=0.0 A/cm2 j/Acm-2 tss /s

Fig. 2. The inﬂuence of operational current density j on the time to reach the steady state tss. The two curves correspond to diﬀerent initial

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cathode to anode. This also explains why tssis larger for

a higher j, as is shown in Fig. 2. 4.2. The eﬀect of contaminants

Fig. 4 illustrates the variations of tss with j for the

case with contamination (curve B) and without con-tamination (curve A). The contaminants present at the anode and cathode are of the same concentration. It is seen that, for all the operational current densities con-sidered, tssfor the case with contamination, is invariably

larger than that without contamination, indicating that the contaminant enhances water depletion in the mem-brane so that the time needed to reach the steady state is longer. That the water depletion in the membrane is enhanced by the presence of contaminant is due to the fact that the electro-osmotic coeﬃcient of the contami-nant is larger than that of Hþ. As a result, the amount of water dragged from anode to cathode increases, while the water diﬀusing from cathode to anode decreases,

leading to water depletion in the membrane. Fig. 5 il-lustrates the distribution of the water concentration across the membrane at different times. Without con-taminant, as shown in Fig. 5(a), the water concentration decreases with time in the whole membrane, the water depletion is more serious at the anode than at the cathode and the time to reach the steady state is about 9.78 s. With contaminant, as shown in Fig. 5(b), the water concentration also decreases with time in virtually the whole membrane, and the time to reach the steady state also increases to 11.7 s. Note please that the water concentration gradient in the infected zone is greater than that in the non-infected zone, which is due to the fact that the presence of the contaminant enhances the electro-osmotic effect while it inhibits the diffusion of water. As a result, the water concentration gradient at the cathode is larger than that at the anode, implying that the impurity effect on the electro-osmosis is greater on the cathode than on the anode. We have also inves-tigated the cases of different levels of contamination, qa¼ qc¼ 0:3 and 0.8, and obtained the same conclusion

that a higher contaminant concentration leads to a

0.5 0.6 0.7 0.8 0.9 1 1.1 0 0.2 0.4 0.6 0.8 1 0.5 1.0 x/d c/c 0 j =0.1 A/cm2

Fig. 3. Three water distributions across the membrane corresponding to three diﬀerent operational current densities.

0 2 4 6 8 10 12 14 0 0.2 0.4 0.6 0.8 1 A B j/A cm-2 tss /s

Fig. 4. The inﬂuence of the operational current density j on the time to reach the steady state tss. Curve A: without contamination qa¼ qc¼ 0;

Curve B: with contamination qa¼ qc¼ 0:5.

0.5 0.6 0.7 0.8 0.9 1 1.1 0 0.2 0.4 0.6 0.8 1 1.0 3.0 9.78 (a) (b) 0.4 0.5 0.6 0.7 0.8 0.9 1 1.1 0 0.2 0.4 0.6 0.8 1 1.0 3.0 11.7 t =0.101 s x /d c/c 0 x /d c/c 0 t =0.101 s qa=qc=0.0 qa=qc=0.5

Fig. 5. The water distribution across the membrane at diﬀerent times. (a) Without contamination qa¼ qc¼ 0, corresponding to Curve A of

Fig. 4. (b) With contamination qa¼ qc¼ 0:5, corresponding to Curve

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lower water content across the membrane, so that the time needed to reach the steady state is longer.

4.3. The eﬀect of initial current density j0

The initial current density corresponds to the water content of the membrane at the beginning of operation. As the fuel cell becomes operational, the water content in the membrane should reach the steady state in a shorter time when the initial current density is closer to the current density at the steady state. This scenario is conﬁrmed by the results of Fig. 6, in which two curves regarding the relation between tss and j0 under two

diﬀerent operational current densities j¼ 0:5 and 1 A/ cm2_{are shown. It is seen that, as implied previously, a j}

0

closer to j leads to a smaller tss. When the membrane is

contaminated, as shown in Fig. 7, the presence of the

contaminant enhances the electro-osmotic eﬀect, leading to a more serious water depletion at the anode. As a result, it needs a longer time for the water distribution in the membrane to reach the steady state. For both cases shown in Fig. 7, the operational current density j at the steady state is approximately 1 A/cm2_.

4.4. The eﬀect of humidiﬁcation parameter k

The humidiﬁcation parameter k accounts for the amount of water entering the anode; namely, a larger k implies a larger amount of water entering the anode, increases the water content of membrane, and anneals the water depletion in the membrane. This scenario is conﬁrmed by the results of Fig. 8 that the distribution of water concentration increases for a larger k, so that the time (tss) needed to reach the steady state of the water

0 2 4 6 8 10 12 0 0.2 0.4 0.6 0.8 1 0.5 j=1.0 A/cm2 j/A cm-2 tss /s

Fig. 6. The inﬂuence of initial current density j0on the time to reach

the steady state tss. The two curves correspond to diﬀerent operational

current densities. 0 2 4 6 8 10 12 14 0 0.2 0.4 0.6 0.8 1 A B j/A cm-2 tss /s

Fig. 7. The inﬂuence of the initial current density i0 on the time to

reach the steady state tss. Curve A: without contamination qa¼ qc¼ 0;

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 0.0001 0.0005 0.001 0.005 k=0.01 cm/s x /d c /c 0

Fig. 8. The water distributions across the membrane for various hu-midiﬁcation parameters k. 0 2 4 6 8 10 12 0 0.2 0.4 0.6 0.8 1 0.005 0.01 k=0.001 cm/s j/A cm-2 tss /s

Fig. 9. The inﬂuence of the operational current density j on the time to reach the steady state tss. The three curves correspond to three diﬀerent

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distribution in the membrane is shortened, as shown in Fig. 9. When the membrane is contaminated, as shown in Fig. 10, tss becomes larger because the contaminant

enhances the water depletion by way of a stronger electro-osmotic eﬀect.

4.5. The effect of contaminants at different locations As we mentioned previously, the presence of con-taminants will not only influence the electro-osmotic effect so that the drag of water from anode to cathode is affected, but will also inhibit the diffusion of water from cathode to anode. Since it influences the transportation of water in both directions, from anode to cathode or from cathode to anode, it is expected that the presence of contaminant in either the cathode or anode would have different levels of influence. We examine this sce-nario by considering four cases of different contamina-tion situacontamina-tions: (a) without contaminacontamina-tion qa¼ qc¼ 0;

(b) with contamination at the anode qa¼ 0:5, qc¼ 0; (c)

with contamination at the cathode qa¼ 0, qc¼ 0:5; (d)

with contamination at both the anode and cathode qa ¼ qc¼ 0:5. The results show that the presence of the

contaminant at the anode (case B, tss¼ 9:86 s) increases

tss only slightly above tss for the case without

contami-nation (case A, tss¼ 9:78 s), implying that the inﬂuence

of contaminant on the electro-osmotic eﬀect (the mechanism driving the water from anode to cathode) is insigniﬁcant. On the other hand, the presence of con-taminant at the cathode (case C, tss¼ 11:5 s; case D,

tss¼ 11:7 s) increases tssto a greater extent from the case

with contamination at the anode, suggesting that the contaminant reduces the diﬀusion of water (the mecha-nism driving the water from cathode to anode) more signiﬁcantly. The above conclusions can be reasoned clearly from the distributions of the water concentration of these four cases as shown in Fig. 11.

5. Concluding remarks

The transient behavior of water transport across the membrane has been investigated on the basis of the theoretical model developed by Okada [26,27] and the inﬂuence of the relevant physical parameters on the time (tss) needed for the water distribution to reach the

steady state is investigated systematically. Results from the parametric study give the following conclusions: (1) During the fuel cell operation, water depletion at the

anode is more marked than that at the cathode be-cause the electro-osmotic effect, dragging water from anode to cathode, is usually higher than the diffu-sion effect pumping the water from cathode to an-ode. This phenomenon is more obvious when the current density is higher. As a result, tss increases

with the current density, while the increase is an-nealed when the current density is suﬃciently high so that the water generated from the cathode is suf-ﬁciently large that the water depletion at the anode can be balanced by the water supply from the cathode.

(2) When foreign impurity ions are present in the mem-brane, the contaminants enhance the electro-os-motic effect but inhibit the diffusion effect. Thus the drag of water from anode to cathode becomes larger while the diffusion of water from cathode to anode becomes smaller, both lead to a more marked depletion of water at the anode and therefore the va-lue of tss increases.

(3) A higher initial current density j0 means the water

content in the membrane is smaller at the beginning of the fuel cell operation, so that the water distribu-tion is closer to that of the steady state. Accordingly, a higher j0 leads to a smaller tss.

2 4 6 8 10 12 14 16 18 0 2 4 6 8 10 A B 103k/cm s−1 tss /s

Fig. 10. The inﬂuence of humidiﬁcation parameter k on the time to reach the steady state tss. Curve A: without contamination qa¼ qc¼ 0;

0.4 0.5 0.6 0.7 0.8 0.9 1 0 0.2 0.4 0.6 0.8 1 A B C D x /d c /c 0

Fig. 11. The distributions of the water concentration across the membrane of the four cases (A) Without contamination qa¼ qc¼ 0.

(B) With contamination at the anode qa¼ 0:5, qc¼ 0. (C) With

con-tamination at the cathode qa¼ 0, qc¼ 0:5. (D) With contamination at

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(4) To anneal the water depletion at the anode, one can increase the humidiﬁcation parameter k at the anode so that the supply of water from the anode is in-creased. This scheme is particularly useful when the membrane is contaminated, and can be easily implemented by either increasing the humidity of the fuel at the anode side or by choosing a mem-brane of higher k.

(5) The presence of contaminant at the cathode has a greater effect on the water transport across the mem-brane than that at the anode, suggesting that the in-fluence of contamination on the electro-osmotic effect is less significant than that on the diffusion ef-fect, leading to a more marked water depletion in the membrane.

From the above concluding remarks, one can clearly see that, to have a fuel cell with an eﬃcient dynamic response, which can be accounted for by a shorter tssfor

the water transport to reach a steady state in the membrane, one should either increase the humidity at the anode side, or choose a membrane of larger k at the anode, or start the operation with a larger j0, or prevent

to a large extent the contamination in the membrane, especially at the cathode side.

Acknowledgements

The authors would like to express their grateful ap-preciation for ﬁnancial support from the National Science Council of Taiwan through the research grants: NSC 92-2623-7-002-006-ET and NSC 91-2218-E-211-001.

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