- Email: [email protected]
An available method exploiting the waste heat in a proton exchange membrane fuel cell system
i n t e r n a t i o n a l j o u r n a l o f h y d r o g e n e n e r g y 3 6 ( 2 0 1 1 ) 6 0 9 9 e6 1 0 4
Available at www.sciencedirect.com
journal homepage: www.elsevier.com/locate/he
An available method exploiting the waste heat in a proton exchange membrane fuel cell system Xiaohang Chen, Liwei Chen, Juncheng Guo, Jincan Chen* Department of Physics, Xiamen University, Xiamen 361005, People’s Republic of China
article info
abstract
Article history:
Based on the models of a proton exchange membrane (PEM) fuel cell working at steady
Received 22 December 2010
state and a semiconductor thermoelectric generator, a hybrid system consisting of a PEM
Received in revised form
fuel cell, a semiconductor thermoelectric generator, and a regenerator is originally put
31 January 2011
forward. Expressions for the efficiencies and power outputs of the fuel cell, thermoelectric
Accepted 3 February 2011
generator, and hybrid system are derived. The relation between the operating electric
Available online 25 March 2011
currents in the fuel cell and thermoelectric generator is obtained. The maximum power output of the hybrid system is numerically given. The optimally operating electric currents
Keywords:
in the fuel cell and thermoelectric generator are calculated, and consequently, the optimal
Hybrid system
region of the hybrid system is determined. The results obtained here will provide some
Proton exchange membrane fuel cell
guidance for further understanding the performance and operation of practical PEM fuel
Thermoelectric generator
cellethermoelectric generator hybrid systems.
Performance characteristic
Copyright ª 2011, Hydrogen Energy Publications, LLC. Published by Elsevier Ltd. All rights
Optimum analysis
1.
Introduction
Proton exchange membrane (PEM) fuel cell has many advantages such as the low cost of construction materials, relative simplicity of design and operation, environmental clean, and so on [1,2]. It has been one of the most promising fuel cells. On the other hand, PEM fuel cell may become a strong alternative as a portable power source in widespread applications including automotives, laptop computers, and other electronic devices because of its low operating temperature, quick start, light weight, and high power density [3e9]. Although the performance of a PEM fuel cell has been investigated widely and deeply [10e18], one never consider and research the hybrid system composed of a PEM fuel cell and a heat engine because the working temperature of a PEM
reserved.
fuel cell is much lower than that of a solid oxide fuel cell [19e22] and the waste heat produced in the PEM fuel cell cannot be effectively used to drive a heat engine. Thus, it is a new investigative task how to availably utilize the waste heat produced in a PEM fuel cell, which will be very significant for the further performance improvement of the PEM fuel cell. In the present paper, we will construct a new hybrid system composed of a PEM fuel cell and a semiconductor thermoelectric generator which can conveniently utilize the waste heat produced in a PEM fuel cell. Based on the current models of the PEM fuel cell and thermoelectric generator, expressions for some key parameters in the hybrid system are derived. The performance characteristics of the hybrid system are revealed and the optimum criteria of some main performance parameters are determined. Some significant results are obtained.
* Corresponding author. Tel.: þ86 592 2180922; fax: þ86 592 2189426. E-mail address: [email protected] (J. Chen). 0360-3199/$ e see front matter Copyright ª 2011, Hydrogen Energy Publications, LLC. Published by Elsevier Ltd. All rights reserved. doi:10.1016/j.ijhydene.2011.02.018
6100
i n t e r n a t i o n a l j o u r n a l o f h y d r o g e n e n e r g y 3 6 ( 2 0 1 1 ) 6 0 9 9 e6 1 0 4
2. A proton exchange membrane fuel cellethermoelectric generator hybrid system The hybrid system considered here is composed of a PEM fuel cell, a semiconductor thermoelectric generator, and a regenerator, as shown in Fig. 1. In the hybrid system, the PEM fuel cell acts as the high-temperature heat reservoir of the thermoelectric generator for a further production of power and the role of the regenerator in the hybrid system is to preheat the incoming fuel and air by means of the relative high-temperature exhaust water produced in the PEM fuel cell. By using such a hybrid system, the waste heat produced in the fuel cell can be availably utilized, and consequently, the performance of the system can be improved. Below, we will analyze the performance of three composing parts, the PEM fuel cell, thermoelectric generator, and regenerator, respectively, and then synthetically investigate the performance characteristics of the hybrid system.
2.1. The efficiency and power output of a proton exchange membrane fuel cell PEM fuel cell mainly consists of an anode and a cathode electrode with a proton-conducting membrane as the electrolyte sandwiched in between the electrodes [1,3,8,23], as shown in Fig. 2. 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 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 [12,24] 2Hþ þ (1/2)O2 þ 2e / H2O þ heat. The overall electrochemical reaction is H2(gas) þ (1/2)O2(gas) / H2O (liquid) þ heat þ electricity. When the fuel cell is operated at a constant temperature T and 1 atm, the basic thermodynamic relationship is
DH ¼ DG TDS,
(1)
where DG is the Gibbs free energy change of the reaction, DS is the entropy production of the reaction,
Fig. 1 e The schematic diagram of a PEM fuel cellethermoelectric generator hybrid system.
Fig. 2 e The schematic diagram of a PEM fuel cell.
DH_ ¼ IDh=ðne FÞ;
(2)
I is the operating electric current of the fuel cell, Dh is the molar enthalpy change, ne is the number of electrons, and F is Faraday’s constant. Eq. (1) shows that even in a reversible electrochemical reaction, one part energy TDS cannot be converted to electric energy and is released as heat. When no current is required by the external load, the PEM fuel cell achieves its theoretical maximum potential, which can be expressed as [3,25,26] pffiffiffiffiffiffiffi V0 ¼ 1:2298:5104 ðTT0 Þþ4:3085105 Tln PH2 PO2 ;
(3)
where T0 is the environment temperature, and PH2 and PO2 are the partial pressures of hydrogen and oxygen, respectively. Obviously, the logarithmic function in Eq. (3) has been simplified and the partial pressures in the logarithmic function only indicate some numerical values with “atm” as the unit [24,25,27]. Similar treatment also appears in the following exponential function. The actual output voltage of a PEM fuel cell is always lower than its reversible potential and may be expressed as V ¼ V0 Vactivation Vconcentration Vohmic ¼ 1:2298:5 pffiffiffiffiffiffiffi 104 ðTT0 Þþ4:3085105 T ln PH2 PO2 b lA þlC RT i i 2 i b1 ln itmem lA lC ne F i0 im 1 1 1 ð0:005139lmem 0:003260Þexp 1268 303 T
ð4Þ
where Vactivation ¼ ((lA þ lC)/lAlC)(RT/neF )ln(i/i0) is the activation overpotential [28], i0 ¼ 1.08 1021exp(0.086 T ), lA and lC represent, respectively, the anode and cathode charge transfer coefficients of the electrodes, i ¼ I/A is the current density, and A is the surface area of the polar plate in the fuel cell; Vconcentration ¼ iðb1 ði=im ÞÞb2 is the concentration overpotential [29], im is the limiting current density, b1 is a parameter depending on PO2 and T, and b2 is a constant; Vohmic ¼ i(tmem/ smem) is the ohmic overpotential [30,31], tmem is the membrane thickness, smem ¼ (0.005139lmem 0.003260)exp[1268((1/303) (1/T ))] is the membrane conductivity, and lmem is the membrane humidity. By the way, some specified units of the parameters in the above equations are often adopted [24e31] and listed in Tables 1 and 2.
i n t e r n a t i o n a l j o u r n a l o f h y d r o g e n e n e r g y 3 6 ( 2 0 1 1 ) 6 0 9 9 e6 1 0 4
6101
Table 1 e Parameters used in the model of a PEM fuel cell. Parameter
Value
Number of electrons, ne Faraday constant, F(C mol1) Operating temperature, T (K) Environment temperature, T0 (K) 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 overpotential constant, b2
2 96,485 373 298 6.013 0.018 8.314 1 1 2.2 2.0
From Eqs. (2) and (4), one can derive the efficiency and power output of a PEM fuel cell as hf ¼
Pf
DH_
¼
ne FV Dh
(5)
and (6)
2.2. The efficiency and power output of a thermoelectric generator The semiconductor thermoelectric generator in the hybrid system is composed of many n- and p-type semiconductor legs that are connected electrically in series by metal strips and thermally in parallel, as shown in Fig. 3, which can be designed more simply and operated more conveniently than the traditional heat engines. The thermoelectric generating elements in Fig. 3 are assumed to be insulated, both electrically and thermally, from their surroundings, except at the junction-reservoir contacts. The current is assumed to flow in one dimension, as along the arm of the generation device. When the thermoelectric generator operates stably, the boundary conditions are determined by T1(0) ¼ T2(0) ¼ / ¼ Ti(0) ¼ / Tn(0) ¼ T and T1(L) ¼ T2(L) ¼ / ¼ Ti(L) ¼ / Tn(L) ¼ T0, where L is the arm length of the generation device. According to Fig. 3, we can derive equations of heat conduction of a multi-couple thermoelectric generator as follows [32e34] 1 q1 ¼ aIg T I2g R þ KðT T0 Þ 2
(7)
and (8)
where a, R, K, and Ig are the Seebeck coefficient, electrical resistance, thermal conductance, and operating electrical current of a multi-couple thermoelectric generator, respectively.
Table 2 e The values of some parameters under different temperatures. Dh (kJ mol1)
PH2 ðatmÞ
PO2 ðatmÞ
b1
284.3 284.0 283.7
2.530 2.305 1.995
0.4464 0.4068 0.3525
0.3236 0.3298 0.3374
Pg ZðT T0 Þj j2 ¼ q1 ZðT T0 Þ þ ZTj j2 =2
(9)
and Pg ¼ q1 q2 ¼ KðT T0 Þj Kj2 =Z;
(10)
where j ¼ aIg/K and Z ¼ a2/(KR) is the figure of merit of the semiconductor thermoelectric device. It is clearly seen from Eqs. (9) and (10) that when 0 < j < Z(T T0), hg > 0 and Pg > 0.
2.3.
The function of a regenerator
As shown in Fig. 1, the regenerator in the hybrid system works as a heat exchanger, heating the inlet reactants from the ambient temperature to the cell temperature by using the hightemperature outlet waster of the fuel cell. With the help of the expressions of the molar heat capacities Cp,i(i ¼ H2, O2 or H2O) of species i at 1 atm pressure [35e37]
Cp;i
8 < 27:28 þ 0:00326T þ 50; 000=T2 ; H2 ðgasÞ ¼ 29:96 þ 0:00418T 167; 000=T2 ; O2 ðgasÞ ; : 75:44; H2 OðliquidÞ
(11)
one can prove ZT
ZT Cp;H2 O dT > T0
1 q2 ¼ aIg T0 þ I2g R þ KðT T0 Þ; 2
353 363 373
It can be seen from Eqs. (7) and (8) that the internal irreversible losses in the thermoelectric generator come from Joule’s heat I2g R due to the electrical current and the heat leak K (T T0) due to the temperature difference between the hot and the cold junctions of the device. According to the model mentioned above, the efficiency hg and power output Pg of the thermoelectric generator can be, respectively, expressed as hg ¼
Pf ¼ VI.
T (K)
Fig. 3 e The schematic diagram of a multi-couple thermoelectric generator.
Cp;H2 þ
Cp;O2 dT: 2
(12)
T0
Eq. (12) means that the waste heat included in H2O is enough to heat the inlet reactants in the regenerator so that the export temperature of the inlet reactants is ensured to attain the working temperature T of the fuel cell. In this case, the regenerative process may be taken as to be ideal.
2.4. The efficiency and power output of the hybrid system As shown in Fig. 1, one part of the waste heat produced in the fuel cell is directly released to the environment, which is called the heat leak qL, and the other part is transferred to the thermoelectric generator in the hybrid system. The heat leak qL may be expressed as
6102
i n t e r n a t i o n a l j o u r n a l o f h y d r o g e n e n e r g y 3 6 ( 2 0 1 1 ) 6 0 9 9 e6 1 0 4
qL ¼ aL AL ðT T0 Þ;
(13)
where aL is the convective and/or conductive heat-leak coefficient and AL is the effective heat-transfer area. According to the first law of thermodynamics, one can derive the input rate of heat from the PEM fuel cell to the thermoelectric generator as q1 ¼ DH_ Pf qL :
(14)
It is seen from Eqs. (7) and (14) that only when DH_ Pf > qL þ KðT T0 Þ, can the thermoelectric device in the hybrid system begin to work as a generator with the power output Pg > 0. In such a case, the dimensionless current j is determined by j2 2ZTj 2ZðTT0 Þþð2Z=KÞA
a A Dh L L i 1 hf ðTT0 Þ ¼ 0: A ne F (15)
Using Eqs. (5), (6), (9), and (10), we can derive the efficiency and power output of the hybrid system as i Pg þ Pf ne F KT0 h ne F h¼ ¼ ðT=T0 1Þj j2 =ðZT0 Þ þ _ A Dhi Dh D H 8 > > < pffiffiffiffiffiffiffi 1:229 8:5 104 ðT T0 Þ þ 4:3085 105 T ln PH2 PO2 > > :
b lA þ lC RT i i 2 i b1 ln lA lC ne F i0 im
9 > > =
itmem 1 1 > > ; ð0:005139lmem 0:003260Þexp 1268 303 T
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. (5), (9), (15), and (16) and the data in Tables 1 and 2 [25,28,29,38], one can generate the curves of the efficiencies of the PEM fuel cell, thermoelectric generator, and hybrid system varying with the current density i of the fuel cell, as shown in Fig. 4, where ic is the critical current density that the thermoelectric generator begins to work. Similarly, using Eqs. (6), (10), (15), and (17) and the data in Tables 1 and 2, one can generate the curves of the power outputs of the PEM fuel cell, thermoelectric generator, and hybrid system varying with the current density i of the fuel cell, as shown in Fig. 5, where P* ¼ P/A is the power output density, Pmax is the maximum power output density of the hybrid system, and iP is the current density at the maximum power output. It is seen from Fig. 5 that the state of the hybrid system at the maximum power output is different from that of the PEM fuel cell or thermoelectric generator at the maximum power output. It is seen from Figs. 4 and 5 that when i > iP, the efficiency and power output of the hybrid system will decrease as the current density i in the fuel cell is increased. Using the data in Figs. 4 and 5, one can directly generate the curve in Fig. 6, where hP is the efficiency of the hybrid system at the maximum power output. Figs. 4e6 indicate clearly that the region of i > iP is not the optimally operating region of the hybrid system. Obviously, the optimal current density in the fuel cell is not arbitrary and should be located in the region of i iP :
ð16Þ and h i P ¼Pg þ Pf ¼ KT0 ðT=T0 1Þj j2 =ðZT0 Þ ( þ iA 1:229 8:5 104 ðT T0 Þ þ 4:3085 b pffiffiffiffiffiffiffi lA þ lC RT i i 2 i b1 105 T ln PH2 PO2 ln lA lC ne F i0 im ) itmem : ð17Þ 1 1 ð0:005139lmem 0:003260Þexp 1268 303 T
(18)
It shows that iP is an important parameter of the hybrid system because it determines the upper bound of the optimized current density of the fuel cell. Figs. 4e6 indicate that when the current density is operated in the optimal region determined by Eq. (18), the power output of the hybrid system will increase as the efficiency is decreased, and vice versa. According to Fig. 6, the optimal efficiency of the hybrid system should be located in the region of h hP ;
(19)
It should be pointed out that i and j in Eqs. (16) and (17) are not independent of each other and the relation between j and i can be determined through Eq. (15).
3. Maximum power output and parametric optimum criteria Eqs. (16) and (17) show clearly that the performance of the hybrid system depends on a series of thermodynamic and electrochemical parameters such as the working temperature T, current density i, surface area A of the polar plate, and heatleak coefficient aL of the fuel cell, the figure of merit Z, thermal conductance K, and dimensionless current j of the thermoelectric generator, and so on. Below, numerical calculations are carried out, based on the parameters summarized in Table 1 [1,24,25,27e29]. The values of these parameters are kept
Fig. 4 e The curves of the efficiencies of the PEM fuel cell (dash line), thermoelectric generator (dash-dot line), and hybrid system (solid line) varying with the current density i of the fuel cell, where ZT0 [ 1, aLALT0/A [ 0.01 and A/(KT0) [ 0.25.
i n t e r n a t i o n a l j o u r n a l o f h y d r o g e n e n e r g y 3 6 ( 2 0 1 1 ) 6 0 9 9 e6 1 0 4
Fig. 5 e The curves of the power output densities of the PEM fuel cell (dash line), thermoelectric generator (dashdot line), and hybrid system (solid line) varying with the current density i of the fuel cell. The values of the relevant parameters are the same as those used in Fig. 4.
where hP determines the lower bound of the optimized efficiency. Using Eq. (15), we can plot the curve of the dimensionless current j of the thermoelectric generator varying with the current density i of the PEM fuel cell, as shown in Fig. 7. It is seen from Fig. 7 and Eq. (15) once again that only when i > ic ¼
n F a A K e L Lþ ðT T0 Þ; A A Dh 1 hf
(20)
j > 0 and the thermoelectric generator in the hybrid system begins to exert its function. According to Eq. (18) and Fig. 7, it is easily found that the optimal region of the dimensionless current of the thermoelectric generator should be j jP :
(21)
It shows that jP is also an important parameter of the hybrid system because it determines the upper bound of the optimized dimensionless current of the thermoelectric generator.
6103
Fig. 7 e The curve of the dimensionless current in the thermoelectric generator varying with the current density in the PEM fuel cell. The values of the relevant parameters are the same as those used in Fig. 4.
It should be pointed out that these important parameters iP, jP, Pmax, and hP are dependent on the thermodynamic and electrochemical parameters of the hybrid system. It is significant to note that the results presented in Figs. 4e7 are directly dependent on two synthetic parameters aLALT0/A andKT0/A and the merit of figure Z of the thermoelectric device. It is well known that the larger the merit of figure Z is, the better the performance of the thermoelectric generator [32e34]. The further analysis shows that the smaller the values of the parameters aLALT0/A and KT0/A are, the larger the optimal region of the dimensionless current j of the thermoelectric generator, the larger the efficiency and power output of the thermoelectric generator, and the more the performance improvement of the hybrid system. Thus, one should make every effort to reduce the values of the two synthetic parameters in the design of the hybrid system so that the waste heat produced in the PEM fuel cell can be availably utilized.
4.
Conclusions
We have established a novel model of the hybrid system consisting of a PEM fuel cell, a thermoelectric generator, and a regenerator. The main parameters in the hybrid system are analyzed. The relationship between the current density of the PEM fuel cell and the dimensionless current of the thermoelectric generator is obtained. The optimally operating regions of the hybrid system, PEM fuel cell, and thermoelectric generator are determined. The results obtained here show that such a hybrid system can availably increase the maximum power output of the system and will be conducive to developing real PEM fuel cellethermoelectric generator hybrid systems.
Acknowledgments Fig. 6 e The power output density versus efficiency curve of the hybrid system. The values of the relevant parameters are the same as those used in Fig. 4.
This work has been supported by the National Natural Science Foundation (No. 51076134), People’s Republic of China.
6104
i n t e r n a t i o n a l j o u r n a l o f h y d r o g e n e n e r g y 3 6 ( 2 0 1 1 ) 6 0 9 9 e6 1 0 4
references [19] [1] Yilanci A, Dincer I, Ozturk HK. Performance analysis of a PEM fuel cell unit in a solarehydrogen system. Int J Hydrogen Energy 2008;33:7538e52. [2] Hasani-Sadrabadi MM, Emami SH, Ghaffarian R, Moaddel H. Nanocomposite membranes made from sulfonated poly (ether ether ketone) and montmorillonite clay for fuel cell applications. Energy Fuels 2008;22:2539e42. [3] Mert SO, Dincer I, Ozcelik Z. Exergoeconomic analysis of a vehicular PEM fuel cell system. J Power Sources 2007;165: 244e52. [4] Zhang ZH, Huang XH, Jiang J, Wu B. An improved dynamic model considering effects of temperature and equivalent internal resistance for PEM fuel cell power modules. J Power Sources 2006;161:1062e8. [5] El-Sharkh MY, Rahman A, Alam MS. Neural networks-based control of active and reactive power of a stand-alone PEM fuel cell power plant. J Power Sources 2004;135:88e94. [6] El-Sharkh MY, Rahman A, Alam MS, Byrne PC, Sakla AA, Thomas T. A dynamic model for a stand-alone PEM fuel cell power plant for residential applications. J Power Sources 2004;138:199e204. [7] Sammes NM, Boersma R. Small-scale fuel cells for residential applications. J Power Sources 2000;86:98e110. [8] Gencoglu MT, Ural Z. Design of a PEM fuel cell system for residential application. Int J Hydrogen Energy 2009;34:5242e8. [9] Yalcinoz T, Alam MS. Improved dynamic performance of hybrid PEM fuel cells and ultracapacitors for portable applications. Int J Hydrogen Energy 2008;33:1932e40. [10] Ramousse JL, Lottin O, Didierjean S, Maillet D. Heat sources in proton exchange membrane (PEM) fuel cells. J Power Sources 2009;192:435e41. [11] Zhang Y, Mawardi A, Pitchumani R. Numerical studies on an air-breathing proton exchange membrane (PEM) fuel cell stack. J Power Sources 2007;173:264e76. [12] Cheng CH, Lin HH, Lai GJ. Design for geometric parameters of PEM fuel cell by integrating computational fluid dynamics code with optimization method. J Power Sources 2007;165: 803e13. [13] Wishart J, Dong Z, Secanell M. Optimization of a PEM fuel cell system based on empirical data and a generalized electrochemical semi-empirical model. J Power Sources 2006; 161:1041e55. [14] Zong Y, Zhou B, Sobiesiak A. Water and thermal management in a single PEM fuel cell with non-uniform stack temperature. J Power Sources 2006;161:143e59. [15] Midilli A, Dincer I. Development of some exergetic parameters for PEM fuel cells for measuring environmental impact and sustainability. Int J Hydrogen Energy 2009;34: 3858e72. [16] Hatti M, Tioursi M. Dynamic neural network controller model of PEM fuel cell system. Int J Hydrogen Energy 2009;34: 5015e21. [17] Obara S, Tanno I. Exergy analysis of a regional-distributed PEM fuel cell system. Int J Hydrogen Energy 2008;33:2300e10. [18] Wu JF, Yuan XZ, Wang HJ, Blanco M, Martin JJ, Zhang JJ. Diagnostic tools in PEM fuel cell research: part 1
[20]
[21]
[22]
[23]
[24] [25] [26] [27]
[28]
[29]
[30] [31]
[32]
[33]
[34] [35]
[36] [37] [38]
electrochemical techniques. Int J Hydrogen Energy 2008;33: 1735e46. Zhao Y, Ou C, Chen J. A new analytical approach to model and evaluate the performance of a class of irreversible fuel cells. Int J Hydrogen Energy 2008;33:4161e70. Zhao Y, Chen J. Modeling and optimization of a typical fuel celleheat engine hybrid system and its parametric design criteria. J Power Sources 2009;186:96e103. Granovskii M, Dincer I, Rosen MA. Performance comparison of two combined SOFC e gas turbine systems. J Power Sources 2007;165:307e14. Yang WJ, Park SK, Kim TS, Kim JH, Sohn JL, Ro ST. Design performance analysis of pressurized solid oxide fuel cell/gas turbine hybrid systems considering temperature constraints. J Power Sources 2006;160:462e73. Maher AR, Sadiq A, Haroun AK, Shahad A. Influence of the design parameters in a proton exchange membrane (PEM) fuel cell on the mechanical behavior of the polymer membrane. Energy Fuels 2007;21:2258e67. Rowe A, Li XG. Mathematical modeling of proton exchange membrane fuel cells. J Power Sources 2001;102:82e96. Ay M, Midilli A, Dincer I. Exergetic performance analysis of a PEM fuel cell. Int J Energy Res 2006;30:307e21. Baschuk JJ, Li X. A general formulation for a mathematical PEM fuel cell model. J Power Sources 2004;142:134e53. Springer TE, Zawodzinski TA, Gottesfeld S. Polymer electrolyte fuel cell model. J Electrochem Soc 1991;138: 2334e42. Berning T. Three-dimensional computational analysis of transport phenomena in a PEM fuel cell. Victoria: University of Victoria; 2002. Pukrushpan JT, Stefanopoulou AG, Peng H. Modeling and control for PEM fuel cell stack system. In: Proceedings of the American control conference, Anchorage, Alaska, U.S.A; 2002. p. 3117e22. Ge SH, Yi BL. A mathematical model for PEMFC in different flow modes. J Power Sources 2003;124:1e11. Zawodzinski TA, Derouin C, Radzinski S, Sherman RJ, Smith VT, Springer TA, et al. Water uptake by and transport through Nafion 117 membranes. J Electrochem Soc 1992;140: 1041e7. Chen X, Lin B, Chen J. In: Sachs KG, editor. Semiconductor research trends. New York: Nova Science Publishers Inc.; 2007 [chapter 11]. Chen J, Lin B, Wang H, Lin G. Optimal design of a multicouple thermoelectric generator. Semicond Sci Technol 2000; 15:184e8. Holman JP. Thermodynamics. 3rd ed. New York: McGrawHill; 1980. Shin Y, Park W, Chang J, Park J. Evaluation of the high temperature electrolysis of steam to produce hydrogen. Int J Hydrogen Energy 2007;32:1486e91. Dean JA. Lange’s handbook of chemistry. 13th ed. New York: McGraw Hill Book Company; 1985. O’Hayre R, Cha SW, Colella W, Prinz FB. Fuel cell fundamentals. 1st ed. New York: John Wiley & Sons; 2006. Zhang X, Guo J, Chen J. The parametric optimum analysis of a proton exchange membrane (PEM) fuel cell and its load matching. Energy 2010;35:5294e9.
Available at www.sciencedirect.com
journal homepage: www.elsevier.com/locate/he
An available method exploiting the waste heat in a proton exchange membrane fuel cell system Xiaohang Chen, Liwei Chen, Juncheng Guo, Jincan Chen* Department of Physics, Xiamen University, Xiamen 361005, People’s Republic of China
article info
abstract
Article history:
Based on the models of a proton exchange membrane (PEM) fuel cell working at steady
Received 22 December 2010
state and a semiconductor thermoelectric generator, a hybrid system consisting of a PEM
Received in revised form
fuel cell, a semiconductor thermoelectric generator, and a regenerator is originally put
31 January 2011
forward. Expressions for the efficiencies and power outputs of the fuel cell, thermoelectric
Accepted 3 February 2011
generator, and hybrid system are derived. The relation between the operating electric
Available online 25 March 2011
currents in the fuel cell and thermoelectric generator is obtained. The maximum power output of the hybrid system is numerically given. The optimally operating electric currents
Keywords:
in the fuel cell and thermoelectric generator are calculated, and consequently, the optimal
Hybrid system
region of the hybrid system is determined. The results obtained here will provide some
Proton exchange membrane fuel cell
guidance for further understanding the performance and operation of practical PEM fuel
Thermoelectric generator
cellethermoelectric generator hybrid systems.
Performance characteristic
Copyright ª 2011, Hydrogen Energy Publications, LLC. Published by Elsevier Ltd. All rights
Optimum analysis
1.
Introduction
Proton exchange membrane (PEM) fuel cell has many advantages such as the low cost of construction materials, relative simplicity of design and operation, environmental clean, and so on [1,2]. It has been one of the most promising fuel cells. On the other hand, PEM fuel cell may become a strong alternative as a portable power source in widespread applications including automotives, laptop computers, and other electronic devices because of its low operating temperature, quick start, light weight, and high power density [3e9]. Although the performance of a PEM fuel cell has been investigated widely and deeply [10e18], one never consider and research the hybrid system composed of a PEM fuel cell and a heat engine because the working temperature of a PEM
reserved.
fuel cell is much lower than that of a solid oxide fuel cell [19e22] and the waste heat produced in the PEM fuel cell cannot be effectively used to drive a heat engine. Thus, it is a new investigative task how to availably utilize the waste heat produced in a PEM fuel cell, which will be very significant for the further performance improvement of the PEM fuel cell. In the present paper, we will construct a new hybrid system composed of a PEM fuel cell and a semiconductor thermoelectric generator which can conveniently utilize the waste heat produced in a PEM fuel cell. Based on the current models of the PEM fuel cell and thermoelectric generator, expressions for some key parameters in the hybrid system are derived. The performance characteristics of the hybrid system are revealed and the optimum criteria of some main performance parameters are determined. Some significant results are obtained.
* Corresponding author. Tel.: þ86 592 2180922; fax: þ86 592 2189426. E-mail address: [email protected] (J. Chen). 0360-3199/$ e see front matter Copyright ª 2011, Hydrogen Energy Publications, LLC. Published by Elsevier Ltd. All rights reserved. doi:10.1016/j.ijhydene.2011.02.018
6100
i n t e r n a t i o n a l j o u r n a l o f h y d r o g e n e n e r g y 3 6 ( 2 0 1 1 ) 6 0 9 9 e6 1 0 4
2. A proton exchange membrane fuel cellethermoelectric generator hybrid system The hybrid system considered here is composed of a PEM fuel cell, a semiconductor thermoelectric generator, and a regenerator, as shown in Fig. 1. In the hybrid system, the PEM fuel cell acts as the high-temperature heat reservoir of the thermoelectric generator for a further production of power and the role of the regenerator in the hybrid system is to preheat the incoming fuel and air by means of the relative high-temperature exhaust water produced in the PEM fuel cell. By using such a hybrid system, the waste heat produced in the fuel cell can be availably utilized, and consequently, the performance of the system can be improved. Below, we will analyze the performance of three composing parts, the PEM fuel cell, thermoelectric generator, and regenerator, respectively, and then synthetically investigate the performance characteristics of the hybrid system.
2.1. The efficiency and power output of a proton exchange membrane fuel cell PEM fuel cell mainly consists of an anode and a cathode electrode with a proton-conducting membrane as the electrolyte sandwiched in between the electrodes [1,3,8,23], as shown in Fig. 2. 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 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 [12,24] 2Hþ þ (1/2)O2 þ 2e / H2O þ heat. The overall electrochemical reaction is H2(gas) þ (1/2)O2(gas) / H2O (liquid) þ heat þ electricity. When the fuel cell is operated at a constant temperature T and 1 atm, the basic thermodynamic relationship is
DH ¼ DG TDS,
(1)
where DG is the Gibbs free energy change of the reaction, DS is the entropy production of the reaction,
Fig. 1 e The schematic diagram of a PEM fuel cellethermoelectric generator hybrid system.
Fig. 2 e The schematic diagram of a PEM fuel cell.
DH_ ¼ IDh=ðne FÞ;
(2)
I is the operating electric current of the fuel cell, Dh is the molar enthalpy change, ne is the number of electrons, and F is Faraday’s constant. Eq. (1) shows that even in a reversible electrochemical reaction, one part energy TDS cannot be converted to electric energy and is released as heat. When no current is required by the external load, the PEM fuel cell achieves its theoretical maximum potential, which can be expressed as [3,25,26] pffiffiffiffiffiffiffi V0 ¼ 1:2298:5104 ðTT0 Þþ4:3085105 Tln PH2 PO2 ;
(3)
where T0 is the environment temperature, and PH2 and PO2 are the partial pressures of hydrogen and oxygen, respectively. Obviously, the logarithmic function in Eq. (3) has been simplified and the partial pressures in the logarithmic function only indicate some numerical values with “atm” as the unit [24,25,27]. Similar treatment also appears in the following exponential function. The actual output voltage of a PEM fuel cell is always lower than its reversible potential and may be expressed as V ¼ V0 Vactivation Vconcentration Vohmic ¼ 1:2298:5 pffiffiffiffiffiffiffi 104 ðTT0 Þþ4:3085105 T ln PH2 PO2 b lA þlC RT i i 2 i b1 ln itmem lA lC ne F i0 im 1 1 1 ð0:005139lmem 0:003260Þexp 1268 303 T
ð4Þ
where Vactivation ¼ ((lA þ lC)/lAlC)(RT/neF )ln(i/i0) is the activation overpotential [28], i0 ¼ 1.08 1021exp(0.086 T ), lA and lC represent, respectively, the anode and cathode charge transfer coefficients of the electrodes, i ¼ I/A is the current density, and A is the surface area of the polar plate in the fuel cell; Vconcentration ¼ iðb1 ði=im ÞÞb2 is the concentration overpotential [29], im is the limiting current density, b1 is a parameter depending on PO2 and T, and b2 is a constant; Vohmic ¼ i(tmem/ smem) is the ohmic overpotential [30,31], tmem is the membrane thickness, smem ¼ (0.005139lmem 0.003260)exp[1268((1/303) (1/T ))] is the membrane conductivity, and lmem is the membrane humidity. By the way, some specified units of the parameters in the above equations are often adopted [24e31] and listed in Tables 1 and 2.
i n t e r n a t i o n a l j o u r n a l o f h y d r o g e n e n e r g y 3 6 ( 2 0 1 1 ) 6 0 9 9 e6 1 0 4
6101
Table 1 e Parameters used in the model of a PEM fuel cell. Parameter
Value
Number of electrons, ne Faraday constant, F(C mol1) Operating temperature, T (K) Environment temperature, T0 (K) 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 overpotential constant, b2
2 96,485 373 298 6.013 0.018 8.314 1 1 2.2 2.0
From Eqs. (2) and (4), one can derive the efficiency and power output of a PEM fuel cell as hf ¼
Pf
DH_
¼
ne FV Dh
(5)
and (6)
2.2. The efficiency and power output of a thermoelectric generator The semiconductor thermoelectric generator in the hybrid system is composed of many n- and p-type semiconductor legs that are connected electrically in series by metal strips and thermally in parallel, as shown in Fig. 3, which can be designed more simply and operated more conveniently than the traditional heat engines. The thermoelectric generating elements in Fig. 3 are assumed to be insulated, both electrically and thermally, from their surroundings, except at the junction-reservoir contacts. The current is assumed to flow in one dimension, as along the arm of the generation device. When the thermoelectric generator operates stably, the boundary conditions are determined by T1(0) ¼ T2(0) ¼ / ¼ Ti(0) ¼ / Tn(0) ¼ T and T1(L) ¼ T2(L) ¼ / ¼ Ti(L) ¼ / Tn(L) ¼ T0, where L is the arm length of the generation device. According to Fig. 3, we can derive equations of heat conduction of a multi-couple thermoelectric generator as follows [32e34] 1 q1 ¼ aIg T I2g R þ KðT T0 Þ 2
(7)
and (8)
where a, R, K, and Ig are the Seebeck coefficient, electrical resistance, thermal conductance, and operating electrical current of a multi-couple thermoelectric generator, respectively.
Table 2 e The values of some parameters under different temperatures. Dh (kJ mol1)
PH2 ðatmÞ
PO2 ðatmÞ
b1
284.3 284.0 283.7
2.530 2.305 1.995
0.4464 0.4068 0.3525
0.3236 0.3298 0.3374
Pg ZðT T0 Þj j2 ¼ q1 ZðT T0 Þ þ ZTj j2 =2
(9)
and Pg ¼ q1 q2 ¼ KðT T0 Þj Kj2 =Z;
(10)
where j ¼ aIg/K and Z ¼ a2/(KR) is the figure of merit of the semiconductor thermoelectric device. It is clearly seen from Eqs. (9) and (10) that when 0 < j < Z(T T0), hg > 0 and Pg > 0.
2.3.
The function of a regenerator
As shown in Fig. 1, the regenerator in the hybrid system works as a heat exchanger, heating the inlet reactants from the ambient temperature to the cell temperature by using the hightemperature outlet waster of the fuel cell. With the help of the expressions of the molar heat capacities Cp,i(i ¼ H2, O2 or H2O) of species i at 1 atm pressure [35e37]
Cp;i
8 < 27:28 þ 0:00326T þ 50; 000=T2 ; H2 ðgasÞ ¼ 29:96 þ 0:00418T 167; 000=T2 ; O2 ðgasÞ ; : 75:44; H2 OðliquidÞ
(11)
one can prove ZT
ZT Cp;H2 O dT > T0
1 q2 ¼ aIg T0 þ I2g R þ KðT T0 Þ; 2
353 363 373
It can be seen from Eqs. (7) and (8) that the internal irreversible losses in the thermoelectric generator come from Joule’s heat I2g R due to the electrical current and the heat leak K (T T0) due to the temperature difference between the hot and the cold junctions of the device. According to the model mentioned above, the efficiency hg and power output Pg of the thermoelectric generator can be, respectively, expressed as hg ¼
Pf ¼ VI.
T (K)
Fig. 3 e The schematic diagram of a multi-couple thermoelectric generator.
Cp;H2 þ
Cp;O2 dT: 2
(12)
T0
Eq. (12) means that the waste heat included in H2O is enough to heat the inlet reactants in the regenerator so that the export temperature of the inlet reactants is ensured to attain the working temperature T of the fuel cell. In this case, the regenerative process may be taken as to be ideal.
2.4. The efficiency and power output of the hybrid system As shown in Fig. 1, one part of the waste heat produced in the fuel cell is directly released to the environment, which is called the heat leak qL, and the other part is transferred to the thermoelectric generator in the hybrid system. The heat leak qL may be expressed as
6102
i n t e r n a t i o n a l j o u r n a l o f h y d r o g e n e n e r g y 3 6 ( 2 0 1 1 ) 6 0 9 9 e6 1 0 4
qL ¼ aL AL ðT T0 Þ;
(13)
where aL is the convective and/or conductive heat-leak coefficient and AL is the effective heat-transfer area. According to the first law of thermodynamics, one can derive the input rate of heat from the PEM fuel cell to the thermoelectric generator as q1 ¼ DH_ Pf qL :
(14)
It is seen from Eqs. (7) and (14) that only when DH_ Pf > qL þ KðT T0 Þ, can the thermoelectric device in the hybrid system begin to work as a generator with the power output Pg > 0. In such a case, the dimensionless current j is determined by j2 2ZTj 2ZðTT0 Þþð2Z=KÞA
a A Dh L L i 1 hf ðTT0 Þ ¼ 0: A ne F (15)
Using Eqs. (5), (6), (9), and (10), we can derive the efficiency and power output of the hybrid system as i Pg þ Pf ne F KT0 h ne F h¼ ¼ ðT=T0 1Þj j2 =ðZT0 Þ þ _ A Dhi Dh D H 8 > > < pffiffiffiffiffiffiffi 1:229 8:5 104 ðT T0 Þ þ 4:3085 105 T ln PH2 PO2 > > :
b lA þ lC RT i i 2 i b1 ln lA lC ne F i0 im
9 > > =
itmem 1 1 > > ; ð0:005139lmem 0:003260Þexp 1268 303 T
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. (5), (9), (15), and (16) and the data in Tables 1 and 2 [25,28,29,38], one can generate the curves of the efficiencies of the PEM fuel cell, thermoelectric generator, and hybrid system varying with the current density i of the fuel cell, as shown in Fig. 4, where ic is the critical current density that the thermoelectric generator begins to work. Similarly, using Eqs. (6), (10), (15), and (17) and the data in Tables 1 and 2, one can generate the curves of the power outputs of the PEM fuel cell, thermoelectric generator, and hybrid system varying with the current density i of the fuel cell, as shown in Fig. 5, where P* ¼ P/A is the power output density, Pmax is the maximum power output density of the hybrid system, and iP is the current density at the maximum power output. It is seen from Fig. 5 that the state of the hybrid system at the maximum power output is different from that of the PEM fuel cell or thermoelectric generator at the maximum power output. It is seen from Figs. 4 and 5 that when i > iP, the efficiency and power output of the hybrid system will decrease as the current density i in the fuel cell is increased. Using the data in Figs. 4 and 5, one can directly generate the curve in Fig. 6, where hP is the efficiency of the hybrid system at the maximum power output. Figs. 4e6 indicate clearly that the region of i > iP is not the optimally operating region of the hybrid system. Obviously, the optimal current density in the fuel cell is not arbitrary and should be located in the region of i iP :
ð16Þ and h i P ¼Pg þ Pf ¼ KT0 ðT=T0 1Þj j2 =ðZT0 Þ ( þ iA 1:229 8:5 104 ðT T0 Þ þ 4:3085 b pffiffiffiffiffiffiffi lA þ lC RT i i 2 i b1 105 T ln PH2 PO2 ln lA lC ne F i0 im ) itmem : ð17Þ 1 1 ð0:005139lmem 0:003260Þexp 1268 303 T
(18)
It shows that iP is an important parameter of the hybrid system because it determines the upper bound of the optimized current density of the fuel cell. Figs. 4e6 indicate that when the current density is operated in the optimal region determined by Eq. (18), the power output of the hybrid system will increase as the efficiency is decreased, and vice versa. According to Fig. 6, the optimal efficiency of the hybrid system should be located in the region of h hP ;
(19)
It should be pointed out that i and j in Eqs. (16) and (17) are not independent of each other and the relation between j and i can be determined through Eq. (15).
3. Maximum power output and parametric optimum criteria Eqs. (16) and (17) show clearly that the performance of the hybrid system depends on a series of thermodynamic and electrochemical parameters such as the working temperature T, current density i, surface area A of the polar plate, and heatleak coefficient aL of the fuel cell, the figure of merit Z, thermal conductance K, and dimensionless current j of the thermoelectric generator, and so on. Below, numerical calculations are carried out, based on the parameters summarized in Table 1 [1,24,25,27e29]. The values of these parameters are kept
Fig. 4 e The curves of the efficiencies of the PEM fuel cell (dash line), thermoelectric generator (dash-dot line), and hybrid system (solid line) varying with the current density i of the fuel cell, where ZT0 [ 1, aLALT0/A [ 0.01 and A/(KT0) [ 0.25.
i n t e r n a t i o n a l j o u r n a l o f h y d r o g e n e n e r g y 3 6 ( 2 0 1 1 ) 6 0 9 9 e6 1 0 4
Fig. 5 e The curves of the power output densities of the PEM fuel cell (dash line), thermoelectric generator (dashdot line), and hybrid system (solid line) varying with the current density i of the fuel cell. The values of the relevant parameters are the same as those used in Fig. 4.
where hP determines the lower bound of the optimized efficiency. Using Eq. (15), we can plot the curve of the dimensionless current j of the thermoelectric generator varying with the current density i of the PEM fuel cell, as shown in Fig. 7. It is seen from Fig. 7 and Eq. (15) once again that only when i > ic ¼
n F a A K e L Lþ ðT T0 Þ; A A Dh 1 hf
(20)
j > 0 and the thermoelectric generator in the hybrid system begins to exert its function. According to Eq. (18) and Fig. 7, it is easily found that the optimal region of the dimensionless current of the thermoelectric generator should be j jP :
(21)
It shows that jP is also an important parameter of the hybrid system because it determines the upper bound of the optimized dimensionless current of the thermoelectric generator.
6103
Fig. 7 e The curve of the dimensionless current in the thermoelectric generator varying with the current density in the PEM fuel cell. The values of the relevant parameters are the same as those used in Fig. 4.
It should be pointed out that these important parameters iP, jP, Pmax, and hP are dependent on the thermodynamic and electrochemical parameters of the hybrid system. It is significant to note that the results presented in Figs. 4e7 are directly dependent on two synthetic parameters aLALT0/A andKT0/A and the merit of figure Z of the thermoelectric device. It is well known that the larger the merit of figure Z is, the better the performance of the thermoelectric generator [32e34]. The further analysis shows that the smaller the values of the parameters aLALT0/A and KT0/A are, the larger the optimal region of the dimensionless current j of the thermoelectric generator, the larger the efficiency and power output of the thermoelectric generator, and the more the performance improvement of the hybrid system. Thus, one should make every effort to reduce the values of the two synthetic parameters in the design of the hybrid system so that the waste heat produced in the PEM fuel cell can be availably utilized.
4.
Conclusions
We have established a novel model of the hybrid system consisting of a PEM fuel cell, a thermoelectric generator, and a regenerator. The main parameters in the hybrid system are analyzed. The relationship between the current density of the PEM fuel cell and the dimensionless current of the thermoelectric generator is obtained. The optimally operating regions of the hybrid system, PEM fuel cell, and thermoelectric generator are determined. The results obtained here show that such a hybrid system can availably increase the maximum power output of the system and will be conducive to developing real PEM fuel cellethermoelectric generator hybrid systems.
Acknowledgments Fig. 6 e The power output density versus efficiency curve of the hybrid system. The values of the relevant parameters are the same as those used in Fig. 4.
This work has been supported by the National Natural Science Foundation (No. 51076134), People’s Republic of China.
6104
i n t e r n a t i o n a l j o u r n a l o f h y d r o g e n e n e r g y 3 6 ( 2 0 1 1 ) 6 0 9 9 e6 1 0 4
references [19] [1] Yilanci A, Dincer I, Ozturk HK. Performance analysis of a PEM fuel cell unit in a solarehydrogen system. Int J Hydrogen Energy 2008;33:7538e52. [2] Hasani-Sadrabadi MM, Emami SH, Ghaffarian R, Moaddel H. Nanocomposite membranes made from sulfonated poly (ether ether ketone) and montmorillonite clay for fuel cell applications. Energy Fuels 2008;22:2539e42. [3] Mert SO, Dincer I, Ozcelik Z. Exergoeconomic analysis of a vehicular PEM fuel cell system. J Power Sources 2007;165: 244e52. [4] Zhang ZH, Huang XH, Jiang J, Wu B. An improved dynamic model considering effects of temperature and equivalent internal resistance for PEM fuel cell power modules. J Power Sources 2006;161:1062e8. [5] El-Sharkh MY, Rahman A, Alam MS. Neural networks-based control of active and reactive power of a stand-alone PEM fuel cell power plant. J Power Sources 2004;135:88e94. [6] El-Sharkh MY, Rahman A, Alam MS, Byrne PC, Sakla AA, Thomas T. A dynamic model for a stand-alone PEM fuel cell power plant for residential applications. J Power Sources 2004;138:199e204. [7] Sammes NM, Boersma R. Small-scale fuel cells for residential applications. J Power Sources 2000;86:98e110. [8] Gencoglu MT, Ural Z. Design of a PEM fuel cell system for residential application. Int J Hydrogen Energy 2009;34:5242e8. [9] Yalcinoz T, Alam MS. Improved dynamic performance of hybrid PEM fuel cells and ultracapacitors for portable applications. Int J Hydrogen Energy 2008;33:1932e40. [10] Ramousse JL, Lottin O, Didierjean S, Maillet D. Heat sources in proton exchange membrane (PEM) fuel cells. J Power Sources 2009;192:435e41. [11] Zhang Y, Mawardi A, Pitchumani R. Numerical studies on an air-breathing proton exchange membrane (PEM) fuel cell stack. J Power Sources 2007;173:264e76. [12] Cheng CH, Lin HH, Lai GJ. Design for geometric parameters of PEM fuel cell by integrating computational fluid dynamics code with optimization method. J Power Sources 2007;165: 803e13. [13] Wishart J, Dong Z, Secanell M. Optimization of a PEM fuel cell system based on empirical data and a generalized electrochemical semi-empirical model. J Power Sources 2006; 161:1041e55. [14] Zong Y, Zhou B, Sobiesiak A. Water and thermal management in a single PEM fuel cell with non-uniform stack temperature. J Power Sources 2006;161:143e59. [15] Midilli A, Dincer I. Development of some exergetic parameters for PEM fuel cells for measuring environmental impact and sustainability. Int J Hydrogen Energy 2009;34: 3858e72. [16] Hatti M, Tioursi M. Dynamic neural network controller model of PEM fuel cell system. Int J Hydrogen Energy 2009;34: 5015e21. [17] Obara S, Tanno I. Exergy analysis of a regional-distributed PEM fuel cell system. Int J Hydrogen Energy 2008;33:2300e10. [18] Wu JF, Yuan XZ, Wang HJ, Blanco M, Martin JJ, Zhang JJ. Diagnostic tools in PEM fuel cell research: part 1
[20]
[21]
[22]
[23]
[24] [25] [26] [27]
[28]
[29]
[30] [31]
[32]
[33]
[34] [35]
[36] [37] [38]
electrochemical techniques. Int J Hydrogen Energy 2008;33: 1735e46. Zhao Y, Ou C, Chen J. A new analytical approach to model and evaluate the performance of a class of irreversible fuel cells. Int J Hydrogen Energy 2008;33:4161e70. Zhao Y, Chen J. Modeling and optimization of a typical fuel celleheat engine hybrid system and its parametric design criteria. J Power Sources 2009;186:96e103. Granovskii M, Dincer I, Rosen MA. Performance comparison of two combined SOFC e gas turbine systems. J Power Sources 2007;165:307e14. Yang WJ, Park SK, Kim TS, Kim JH, Sohn JL, Ro ST. Design performance analysis of pressurized solid oxide fuel cell/gas turbine hybrid systems considering temperature constraints. J Power Sources 2006;160:462e73. Maher AR, Sadiq A, Haroun AK, Shahad A. Influence of the design parameters in a proton exchange membrane (PEM) fuel cell on the mechanical behavior of the polymer membrane. Energy Fuels 2007;21:2258e67. Rowe A, Li XG. Mathematical modeling of proton exchange membrane fuel cells. J Power Sources 2001;102:82e96. Ay M, Midilli A, Dincer I. Exergetic performance analysis of a PEM fuel cell. Int J Energy Res 2006;30:307e21. Baschuk JJ, Li X. A general formulation for a mathematical PEM fuel cell model. J Power Sources 2004;142:134e53. Springer TE, Zawodzinski TA, Gottesfeld S. Polymer electrolyte fuel cell model. J Electrochem Soc 1991;138: 2334e42. Berning T. Three-dimensional computational analysis of transport phenomena in a PEM fuel cell. Victoria: University of Victoria; 2002. Pukrushpan JT, Stefanopoulou AG, Peng H. Modeling and control for PEM fuel cell stack system. In: Proceedings of the American control conference, Anchorage, Alaska, U.S.A; 2002. p. 3117e22. Ge SH, Yi BL. A mathematical model for PEMFC in different flow modes. J Power Sources 2003;124:1e11. Zawodzinski TA, Derouin C, Radzinski S, Sherman RJ, Smith VT, Springer TA, et al. Water uptake by and transport through Nafion 117 membranes. J Electrochem Soc 1992;140: 1041e7. Chen X, Lin B, Chen J. In: Sachs KG, editor. Semiconductor research trends. New York: Nova Science Publishers Inc.; 2007 [chapter 11]. Chen J, Lin B, Wang H, Lin G. Optimal design of a multicouple thermoelectric generator. Semicond Sci Technol 2000; 15:184e8. Holman JP. Thermodynamics. 3rd ed. New York: McGrawHill; 1980. Shin Y, Park W, Chang J, Park J. Evaluation of the high temperature electrolysis of steam to produce hydrogen. Int J Hydrogen Energy 2007;32:1486e91. Dean JA. Lange’s handbook of chemistry. 13th ed. New York: McGraw Hill Book Company; 1985. O’Hayre R, Cha SW, Colella W, Prinz FB. Fuel cell fundamentals. 1st ed. New York: John Wiley & Sons; 2006. Zhang X, Guo J, Chen J. The parametric optimum analysis of a proton exchange membrane (PEM) fuel cell and its load matching. Energy 2010;35:5294e9.