The parametric optimum analysis of a proton exchange membrane (PEM) fuel cell and its load matching

Energy 35 (2010) 5294e5299

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The parametric optimum analysis of a proton exchange membrane (PEM) fuel cell and its load matching Xiuqin Zhang, Juncheng Guo, Jincan Chen* Department of Physics, Xiamen University, Xiamen 361005, People’s Republic of China

a r t i c l e i n f o

a b s t r a c t

Article history: Received 30 March 2010 Received in revised form 19 July 2010 Accepted 22 July 2010 Available online 15 September 2010

Based on the irreversible model of a PEM fuel cell working at steady state, expressions for the power output, efficiency and entropy production rate of the PEM fuel cell are analytically derived by using the theory of electrochemistry and non-equilibrium thermodynamics. The effects of multi-irreversibilities resulting from electrochemical reaction, heat transfer and electrical resistance on the key parameters of the PEM fuel cell are analyzed. The curves of the power output, efficiency and entropy production rate of the PEM fuel cell varying with the electric current density are represented through numerical calculation. The general performance characteristics of the PEM fuel cell are revealed and the optimum criteria of the main performance parameters are determined. Moreover, the optimal matching condition of the load resistance is obtained from the relations between the load resistance and the power output and efficiency. The effects of the leakage resistance on the performance of the PEM fuel cell are expounded and the optimally operating states of the PEM fuel cell are further discussed. Ó 2010 Elsevier Ltd. All rights reserved.

Keywords: PEM fuel cell Irreversible loss Performance characteristic Parametric optimum Load matching

1. Introduction Proton exchange membrane (PEM) fuel cells are one of the most promising fuel cells because of their advantages such as low cost construction materials, its relative simplicity of design and operation, being environmentally clean, highly efficient, and so on [1e3]. It is more attractive that they are a strong alternative as a portable power source in widespread applications including automotive, laptop computers, cellular phones, and other electronic devices owing to their low operating temperature, quick start, light weight and high power density [4e12]. In order to improve the performance of PEM fuel cells and reduce various irreversible losses that occur during the fuel cell operation, many significant researches have been devoted to PEM fuel cells. These researches include mathematical modeling, performance and exergy analyses in power generation and auxiliary systems [12e19], entropy production with Ohmic heating and concentration polarization [20], water and thermal management [21e24], and a lot of experiment investigations [25e32], on the basis of fluid dynamics, electrochemical reaction, and heat transfer theory [5,33e39]. However, it is still necessary to further consider some additional important issues related to the PEM fuel cell performance such as the entropy production rate of the PEM fuel

* Corresponding author. Fax: þ86 592 2189426. E-mail address: [email protected] (J. Chen). 0360-5442/$ e see front matter Ó 2010 Elsevier Ltd. All rights reserved. doi:10.1016/j.energy.2010.07.034

cell caused by irreversible losses, the power output versus efficiency characteristics, the optimally operating region of the system, and the optimal matching of the load, so that one can deeply understand the characteristics of PEM fuel cells and effectively control them to be operated in the optimal states. In the present paper, the concrete contents are arranged as follows. In Section 2, a typical irreversible model of a PEM fuel cell is established and electrochemical and thermodynamic description on the PEM fuel cell is given. In Section 3, the expressions of several key parameters of the PEM fuel cell are derived. In Section 4, the general performance characteristics of the PEM fuel cell are revealed and the optimum criteria of some main performance parameters are determined. In Section 5, the load matching problem of the PEM fuel cell system is expounded. In Section 6, the influence of the leakage resistance on the performance of the PEM fuel cell is further discussed. Some significant results are obtained. 2. General model of a PEM fuel cell PEM fuel cell is an electrochemical system that converts the chemical energy of a reaction such as hydrogen and oxygen directly into the electrical and thermal energy. It mainly consists of an anode and a cathode electrodes with a proton-conducting membrane as the electrolyte sandwiched in between the electrodes [1,6,8,40], as shown in Fig.1. At the anode, hydrogen is oxidized into electrons and protons and the reaction is H2 / 2Hþ þ 2e. At the cathode, oxygen is reduced to oxide species. Depending on the electrolyte, only

X. Zhang et al. / Energy 35 (2010) 5294e5299

2e Fuel Gas Channel

-

Air Channel

Membrane +

H H2O

H2O Ileak

Anode Electrode

H2O

H2

5295

Cathode Electrode

+

O2 N2

H

Fig. 3. The output voltage versus current density curves of the PEM fuel cell for RI/ RL ¼ 0.05.

Catalyst Layers Fig. 1. A schematic diagram of a PEM fuel cell.

protons are transported through the proton-conducting membrane but electronically insulating electrolyte to combine with oxide to generate water and electric power, and the reaction at the cathode is [35,41] 2Hþ þ 1=2O2 þ 2e /H2 O þ heat. For the PEM fuel cell mentioned above, the overall electrochemical reaction is [42]

1 H2 ðgasÞ þ O2 ðgasÞ/H2 OðliquidÞ þ heat þ electricity 2

(1)

Fuel input is a product of hydrogen consumption rate and its energy content is usually given as enthalpy DH. Hydrogen consumption rate in the electrochemical reaction is determined by Faraday’s Law as

q_ H2 ¼

I ne F

(2)

where I is the operating electric current of the fuel cell, ne is the number of electrons, and F is Faraday’s constant. Therefore, the maximum possible energy (both electrical and thermal) is

Dh DH_ ¼ q_ H2 Dh ¼  I ne F

(3)

where Dh is the molar enthalpy change. In practice, the maximum net work is never completely utilized owing to the existence of thermodynamic and electrochemical irreversibilities caused by various overpotentials and finite-rate heat transfer. It is often assumed that the fuel cell is operated at a constant temperature T and 1 atm. The basic thermodynamic relationship is

DH ¼ DG  T DS

Fig. 2. The equivalent circuit of a PEM fuel cell.

(4)

where DG is the Gibbs free energy change of the reaction and DS is the entropy production of the reaction. Eq. (4) shows that even in a reversible electrochemical reaction, one part energy TDS can’t be converted to electric energy and is released as heat. When no current is required by the external load and the leak electric current is negligible, the PEM fuel cell achieves its theoretical maximum potential, which can be expressed as

V0 ¼ 

DgðT; pÞ

(5)

ne F

where DgðT; pÞ ¼ ne F=I DG_ ¼ Dgf0  ðT  298:15ÞDs þ RT ln pffiffiffiffiffiffiffiffi ðpH2 pO2 =pH2 O Þ, Dgf0 is the standard molar Gibbs free energy change at T ¼ 298 K and p ¼ 1 atm, Ds is the standard molar entropy change, R is universal gas constant, and pH2 , pO2 , and pH2 O are partial pressures of reactants H2, O2 and H2O, respectively [41e44]. It is assumed that water is produced in liquid phase [33], so pH2 O ¼ 1. The actual output voltage is always lower than the reversible potential of a PEM fuel cell owing to the existence of irreversibilities resulting mainly from the following three overpotentials: (a) Activation overpotential losses are caused by low reaction rates in both anode and cathode by losing some of the energy while driving the reactions for transferring electrons and given by [8,45,46]

 Vactivation ¼



 

lA þ lC RT i ; ln lA lC ne F i0

(6)

where lA and lC represent anode and cathode charge transfer coefficients of the electrodes, i is the current density, i0 is exchange current density in the electrodes of the PEM fuel cell. (b) Ohmic overpotential losses are caused by membrane resistance and contact resistance. In the most of model, the contact electrical losses at the interfaces between different fuel cell elements are neglected [47]. Thus, Ohmic overpotential losses [47,48]

Fig. 4. The power output versus current density curves of the PEM fuel cell, where P* ¼ P/A and iP is the current density at the maximum power output Pmax.

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X. Zhang et al. / Energy 35 (2010) 5294e5299

Fig. 5. The efficiency versus current density curves of the PEM fuel cell, where ih is the current density at the maximum efficiency hmax.

Fig. 7. The power output versus efficiency curves of the PEM fuel cell, where hP and Ph* are the efficiency at the maximum power output Pmax and the equivalent power output at the maximum efficiency hmax , respectively.

Vactivation þ Vohmic þ Vconcentration I     lA þ lC RT i ¼ ln lA lC Aine F i0    1 i b2 b1 þ þtmem Að0:005139lmem  0:003260Þexp A iL  1 1 1 ½1268ð (10)  Þ 303 T

RI ¼ Vohmic ¼ IRohmic ¼ i

tmem

smem

:

(7)

Where tmem is membrane thickness, smem ¼ ð0:005139 lmem  0:003260Þexp½1268ðð1=303Þ  ð1=TÞÞ is the membrane conductivity, and lmem is membrane humidity. (c) Concentration overpotential losses [49]

  i b2 Vconcentration ¼ i b1 im

(8)

where im is the limiting current density, b1 is a parameter depending on pO2 and T, and b2is a constant. Using the above equations, one can calculate the output voltage of a PEM fuel cell as

V ¼ V0  Vactivation  Vohmic  Vconcentration  pffiffiffiffiffiffiffiffi Dgf0 pH2 pO2 Ds RT ¼   ðT  298:15Þ þ ln pH2 O ne F ne F ne F   b2     l þ lC RT i i  i b1 itmem ln  A lA lC ne F i0 im 1 1  ð0:005139lmem  0:003260Þexp½1268ð  Þ 303 T

is the equivalent internal resistance resulting from three overpotentials, where A is the surface area of electrodes of the fuel cell. 3. Expressions of several key parameters Except for the output voltage, the power output, efficiency and entropy production rate are also some important parameters of a PEM fuel cell. From Fig. 2, one can easily derive the power output of a PEM fuel cell, which is dependent on the electric current, output voltage and leakage resistance and can be expressed as

P ¼ VI 

V2 RL

(11)

From Eqs. (3) and (11), one can derive the efficiency and total entropy production rate of a PEM fuel cell, which are, respectively, expressed as

1 (9)

For a practical fuel cell, the measured open-circuit potential is usually lower than V0. One way to handle this behavior is to assume some electronic current leakage through the electrolyte [50]. In this case, a leakage resistance RL can be introduced which is in parallel with the external load [51], as shown in Fig. 2, where IL is the electric current through the equivalent leakage resistance RL, RLoad is the load resistance, and

Fig. 6. The entropy production rate versus current density curves of the PEM fuel cell, where s* ¼ s=A.

h¼

P DH_

¼

  ne FV V 1 Dh RL I

(12)

and

  nDehF  V 1  IRVL DH_  P s¼ ¼ I T0 T0

(13)

where T0 is the environmental temperature. It may be found by the further analysis that the total entropy production rate comes from

Fig. 8. The power output versus load resistance curves of the PEM fuel cell, where R* ¼ ARload and R*P is the value of R* at the maximum power output Pmax .

X. Zhang et al. / Energy 35 (2010) 5294e5299

5297

Table 2 The values of some parameters under different temperatures [42,45,49].

Fig. 9. The efficiency versus load resistance curves of the PEM fuel cell, where R* ¼ ARload and R*h is the value of R* at the maximum efficiency hmax .

the following three parts. The first part is DS_ produced in the electrochemical reaction. The second part is DQ_ =T produced by the Joule heat in the internal and leakage resistances, where DQ_ ¼ I2 RI þ IL2 RL . The third part is ðT DS_ þ DQ_ Þðð1=T0 Þ  ð1=TÞÞ produced by the irreversible heat transfer of the finite temperature difference between T0 and T. Thus, the total entropy production rate is also expressed as

s ¼ DS_ þ ¼

 

1 1  þ  T DS_ þ DQ_ T0 T T

DQ_

_ DQ_  T DSþ T0



T(K)

Dh (kJ mol1)

pH2 (atm)

pO2 (atm)

b1

i0 (A cm2)

353 363 373

284.3 284.0 283.7

2.530 2.305 1.995

0.4464 0.4068 0.3525

0.3236 0.3298 0.3374

1.651  108 3.902  108 9.220  108

It is seen from Figs. 4 and 5 that there exist a maximum power output Pmax and a maximum efficiency hmax for the PEM fuel cell and the corresponding current densities are iP and ih, respectively. It shows that iP and ih are two important parameters of the PEM fuel cell because they determine the upper and lower bounds of the optimized current density, respectively. Thus, the optimized current density is determined by

ih  i  iP

When the current density is operated in the optimal region, the power output of the fuel cell will increase as the efficiency is decreased, and vice versa, as shown in Fig. 7. One can find from Fig. 7 that the optimally operating regions of the power output and efficiency of the PEM fuel cell are, respectively, determined by

Ph  P  Pmax

T DS_ þ I 2 RI þ IL2 RL ¼ T0

(14)

Using Eqs. (3), (4) and (11), one can further prove that Eqs. (13) and (14) are identical to each other. 4. General performance characteristics and optimum criteria Eqs. (9)e(13) show clearly that the performance of a PEM fuel cell depends on a set of thermodynamic and electrochemical parameters such as operating temperature, current density, partial pressures of reactants, anode and cathode charge transfer coefficients, leakage resistance, membrane thickness, and so on. Below, numerical calculations are carried out, based on the parameters summarized in Tables 1 and 2, which are derived from the data available in literature [1,41,42,44,45,49,52]. These parameters are kept constant unless mentioned specifically. The fuel composition is taken as 97% H2 þ 3% H2O, and the typical oxygen composition in the ambient air, i.e., 21% O2 þ 79% N2, is used as oxidant. Using Eqs. (9)e(13), one can generate the curves of the output voltage V, power output P, efficiency h, and total entropy production rate s varying with the current density i, as shown in Figs. 3e6, respectively, where RI/RL ¼ 0.05 is chosen. Fig. 3 shows that the larger the current density is, the larger the effect of temperature on the output voltage. The output voltage will increase as temperature rises.

(16)

and

hmax  h  hP

(17)

It shows that hmax, Pmax, hP and Ph are also four important parameters of the PEM fuel cell, where hmax and Pmax give the upper bounds of the efficiency and power output, while hP and Ph determine the lower bounds of the optimized values of the efficiency and the power output. It is seen from Fig. 6 that in the optimally operating regions of the PEM fuel cell, the entropy production rate will increase as the current density increases or temperature rises. 5. Optimal matching of the load resistance In order to make the PEM fuel cell operate in the optimal region, the load resistance Rload shown in Fig. 2 may not be chosen arbitrarily. From Fig. 2, one can easily find that the load resistance RLoad is determined by

Rload ¼

VRL ðIRL  VÞ

(18)

and that the relations between the load resistance and the power output and efficiency are, respectively, determined by

P ¼ Table 1 Parameters used in the model of a PEM fuel cell [1,41,42,44,45,49,52].

(15)

Rload R2L V02 V2 ¼ Rload ½ðRload þ RL ÞRI þ Rload RL 2

(19)

Rload R2L V02 ne F ¼  Dh ðRL þ Rload Þ½ðRload þ RL ÞRI þ Rload RL  DH_

(20)

and

Parameter

Value

Number of electrons, ne Faraday constant, F (C mol1) Standard molar entropy change, Ds (J mol1) Standard molar Gibbs free energy change Dgf (kJ mol1) membrane humidity lmem membrane thickness tmem (cm) Universal gas constant, R (J mol1 K1) Transfer coefficient of the anode, lA Transfer coefficient of the cathode, lC Limiting current density, im (A cm2) Concentration overvoltage constant, b2

2 96485 163.4 237.3 6.013 0.018 8.314 0.5 1 2.2 2

h¼

P

Combining Eqs. (5), (9), (10), (18)e(20), one can plot the curves of the output power and efficiency of the PEM fuel cell varying with the load resistance, as shown in Figs. 8 and 9. It is seen from Figs. 8 and 9 that there are a maximum power output Pmax and a maximum efficiency hmax for the PEM fuel cell and the corresponding load resistances are RP and Rh, respectively. Using Eqs. (19) and (20), one can further prove

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X. Zhang et al. / Energy 35 (2010) 5294e5299

  20 RP ¼ ½RI RL =ðRI þ RL ÞP ¼ RI 21 P " #      20 lA þ lC RT iP 1 iP b2 tmem b þ ¼ þ ln lA lC AiP ne F i0 Asmem 21 A 1 iL

(21)

from the view point of thermodynamics, one can further choose the rational values of the parameters according to the practical requirements. In the most cases, the optimized current density should be chosen to be ih << i  iP so that the power output of the PEM fuel cell is equal to or close to Pmax.

and

  pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffii 20 Rh ¼ RL RI =ðRI þ RL Þ ¼ pffiffiffiffiffiffiRI h 21 h # "      lA þ lC ih ih b2 tmem 20 RT 1 þ b1 ¼ pffiffiffiffiffiffi þ ln lA lC Aih ne F A i0 iL Asmem 21

7. Conclusions

h

(22)

where RI/RL ¼ 0.05. It is also seen from Figs. 8 and 9 that in the region of RLoad Rh, the power output and efficiency of the PEM fuel cell will decrease as the load resistance Rload is increased. Thus, the regions of Rload < RP and Rload > Rh are not the optimally operating regions of the PEM fuel cell. Obviously, the optimal region of the load resistance should be situated in between RP and Rh, i.e.,

RP  Rload  Rh

(23)

We have investigated the effects of multi-irreversibilities on the performance of a PEM fuel cell. Expressions of the output potential, output power, efficiency, entropy production rate and load resistance are derived and used to discuss the optimal performance of the PEM fuel cell. The optimally operating regions of some important parameters including the current density, power output, efficiency, and load resistance are determined from the view point of thermodynamics. The influence of the leak electric current on the performance of PEM fuel cell is analyzed. The optimally operating states are further discussed from the view point of applications. It is pointed out that the optimized current density should be chosen to be much larger than ih and to be equal to or small than iP. The results obtained here may provide a theoretical basis for both the optimal design and operation of real PEM fuel cells.

It shows that RP and Rh are two important parameters of the PEM fuel cell system because they determine the upper and lower bounds of the optimized load resistance, respectively. When the load resistance is operated in the optimal region determined by Eq. (23), the power output of the PEM fuel cell will increase as the efficiency is decreased, and vice versa.

Acknowledgments

6. Discussion

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When the leakage resistance RL /N, the leak current between two electrodes of the PEM fuel cell tends to zero. In such a case, Eqs. (11), (12), (21), and (22) can be, respectively, simplified into

P ¼ VI

h¼

(24)

ne F V Dh

(25) 

RP ¼ ðRI ÞP ¼



 



lA þ lC RT i 1 i b P ln P þ lA lC AiP ne F i0 A 1 iL

b2

 þtmem Að0:005139lmem  0:003260Þexp 2

!3 1 1 1 41268 5  303 T

(26)

and

Rh /N

(27)

It can be proved from Eqs. (9) and (25) that the efficiency of the PEM fuel cell is a monotonically deceasing function of the current density i. When i ¼ 0, the efficiency attains its maximum and the power output is equal to zero. For a practical PEM fuel cell, the leakage resistance compared with the internal resistance is very large although it is not infinite, and consequently, both the current density ih and the power output Ph at the maximum efficiency hmax are very small. In the practical application of PEM fuel cells, one always hopes to obtain a large power output. Thus, with the help of the optimal relations obtained

This work was supported by the National Natural Science Foundation, People’s Republic of China. References

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