Energy Fuels 2009, 23, 6079–6084 Published on Web 09/04/2009
: DOI:10.1021/ef9006774
General Performance Characteristics and Parametric Optimum Criteria of a Braysson-Based Fuel Cell Hybrid System Xiaohang Chen,† Bihong Lin,‡ and Jincan Chen*,†,§ †
Department of Physics, Xiamen University, Xiamen 361005, People’s Republic of China, ‡College of Information Science and Engineering, Huaqiao University, Quanzhou 362021, People’s Republic of China, and §Institute of Theoretical Physics and Astrophysics, Xiamen University, Xiamen 361005, People’s Republic of China Received July 1, 2009. Revised Manuscript Received August 22, 2009
With the help of the current models of a solid oxide fuel cell and a heat engine with the Braysson cycle, a new theoretical model of a fuel cell/heat engine hybrid system is established, in which some main irreversible losses existing in real hybrid systems are taken into account. Expressions for the efficiency and power output of the hybrid system are analytically derived from the model. The effects of some irreversibilities on several important parameters of the hybrid system are discussed in detail. The general performance characteristics of the hybrid system are revealed. The optimally operating regions of not only the hybrid system but also the fuel cell and heat engine in the hybrid system are determined, and consequently, the optimum criteria of some important parameters are obtained.
there always exist the irreversibility of heat transfer and other irreversible effects. The temperature of the working substance in the heat engine is always lower than that of the fuel cell. The heat-exchange process between the heat engine and the fuel cell is not isothermal. On the other hand, there exist heat leak losses from the fuel cell to the surroundings. Thus, it is necessary to establish a new model of fuel cell/heat engine hybrid system, which is still simple for the theoretical analysis but more close to practice. In the present paper, the model of a hybrid system consisting of a SOFC and a Braysson heat engine13-15 will be used to investigate the influence of some main irreversible losses resulting from the fuel cell and heat engine on the performance of the hybrid system and to optimize the key parameters of the hybrid system.
1. Introduction Fuel cells convert a fuel and oxidizer into products through electrochemical reactions to produce electricity and are considered to be one of the best candidates1-5 for next-generation power sources because of their high efficiency and ultra-low emission of environmentally harmful gases. Among the different types of fuel cells, solid oxide fuel cells (SOFCs) are the most promising because of their less polluting, fuel flexibility, and high temperature of the exhaust heat, which can be used for the heat source of the heat engines for additional electricity generation. There are many research papers6-10 that are involved in the modeling of the fuel cell/heat engine hybrid systems as well as the practical development of such hybrid systems. It is wellknown that the efficiency of a fuel cell/heat engine hybrid system is always higher than that of a single fuel cell or a single heat engine.11,12 The simplest hybrid mode is composed of a fuel cell and a reversible Carnot heat engine. Although the reversible model of the Carnot cycle is very important in the theoretical analysis, the results obtained from this model are usually far away from the practical case. In practical systems,
2. Irreversible Braysson-Based Fuel Cell Hybrid System Figure 1 shows schematically a novel hybrid system composed of a SOFC and a heat engine with the Braysson cycle, where T0 is the ambient temperature, T and p are temperature and pressure in the fuel cell, Pc and Pe are the power outputs of the fuel cell and heat engine, respectively, q1 and q2 are the rates of heat transfer between the fuel cell and the heat engine and between the heat engine and the environment, respectively, and qL is the rate of the heat losses of the fuel cell. In Figure 1, the fuel cell acts as the high-temperature heat reservoir of the heat engine for a further production of power and the function of the regenerator in the hybrid system is to preheat the incoming fuel and air with the high-temperature exhaust gas of the fuel cell. Using such a hybrid system, the waste heat produced in the fuel cell can be availably used and, consequently, the efficiency of the system can be higher than that of either a single fuel call or a single heat engine.
*To whom correspondence should be addressed. Telephone: 0086592-2180922. Fax: 0086-592-2189426. E-mail:
[email protected]. (1) Singhal, S. C.; Kendal, K. High Temperature Solid Oxide Fuel Cell: Fundamentals, Design and Applications; Elsevier Ltd.: Oxford, U.K., 2003. (2) Yang, J. S.; Sohn, J. L.; Ro, S. T. J. Power Sources 2007, 166, 155– 164. (3) Shen, M.; Scott, K. J. Power Sources 2005, 148, 24–31. (4) Zhao, Y.; Ou, C.; Chen, J. Int. J. Hydrogen Energy 2008, 33, 4161– 4170. (5) Ni, M.; Leung, M. K. H.; Leung, D. Y. C. Energy Convers. Manage. 2007, 48, 1525–1535. (6) S anchez, D.; Chacartegui, R.; Torres, M.; Sanchez, T. J. Power Sources 2009, 192, 84–93. (7) Motahar, S.; Alemrajabi, A. A. Int. J. Hydrogen Energy 2009, 34, 2396–2407. (8) Zhao, Y.; Chen, J. J. Power Sources 2009, 186, 96–103. (9) Haynes, C. J. Power Sources 2001, 92, 199–203. (10) Suslu, O. S.; Becerik, I. Energy Fuels 2009, 23, 1858–1873. (11) Ro, S. T.; Sohn, J. L. J. Power Sources 2007, 167, 295–301. (12) Lutz, A. E.; Larson, R. S.; Keller, J. O. Int. J. Hydrogen Energy 2002, 27, 1103–1111. r 2009 American Chemical Society
(13) Frost, T. H.; Anderson, A.; Agnew, B. Proc. Inst. Mech. Eng., Part A 1997, 211, 121–131. (14) Zheng, J.; Sun, F.; Chen, L.; Wu, C. Int. J. Exergy 2001, 1, 41–45. (15) Zhou, Y.; Tyagi, S.; Chen, J. Int. J. Therm. Sci. 2004, 43, 1101– 1106.
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: DOI:10.1021/ef9006774
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Figure 1. Schematic diagram of a Braysson-based fuel cell hybrid system. Figure 2. Schematic diagram of a hydrogen-oxygen SOFC.
Below, we will first describe the performance of the fuel cell, regenerator, and heat engine, respectively, derive the efficiency and power output of the hybrid system, and then synthetically investigate the influence of some main irreversible losses existing in real systems on the performance of the hybrid system. 2.1. Efficiency and Power Output of a SOFC. The fuel cell model represented in ref 4 may be directly used in the hybrid systems mentioned above, as shown in Figure 2, which is based on a SOFC using hydrogen as the fuel and air as the oxidant. The overall electrochemical reaction in such a cell can be summarized as H2 þ
1 O2 f H2 O þ heat þ electricity 2
reaction at temperature T for the steady-state fuel cell, h is the molar enthalpy of the species at temperature T and pressure p0=1 atm, ne is the number of electrons transferred in the reaction, F=96 485 C mol-1 is Faraday’s constant, i is the current density, A is the surface area of the polar plate (supposing the bipolar plates have the same area), d1 = 2ne sinh-1(i/2i0,a) þ 2ne sinh-1(i/2i0,c) - ln(1 - (i/iL,a)) - ln(1-(i/iL,c)) þ (ineFLel/σ0R)exp(Eel/RT), R=8.314 J mol-1 K-1 is the universal gas constant, Lel is the thickness of the electrolyte, Eel is the activation energy for ion transport, σ0 is the reference ionic conductivity, iL,a and iL,c are the limiting current densities of the anode and cathode, respectively, i0,a and i0,c are the anode and cathode exchange current densities, respectively, m=-Δg(T) þ RT ln(pH2pO21/2/pH2O) - RTd1, k=Rint/Rleak is the ratio of the internal resistance Rint to the leakage resistance Rleak, Δg(T) is the molar Gibbs free-energy change at p0 = 1 atm and is, therefore, called the standard molar Gibbs free-energy change, which also depends upon the temperature,16,27-29 and pH2, pO2 and pH2O are the partial pressures of the reactants H2, O2, and H2O, respectively. 2.2. Function of a Regenerator. The regenerator in the hybrid system works as a heat exchanger, heating the inlet reactants from the ambient temperature T0 to the cell temperature T using the high-temperature outlet gas of the fuel cell. For the sake of simplicity, the regeneration process is assumed to be ideal. This assumption is reasonable, because the efficiency of regenerators with the values of 98-99% have already been reported.30-32 Because of perfect regeneration, it does not need the additional heat in the input process of the reactants and the export temperature of the inlet reactants can be ensured to attain the working temperature T of the fuel cell. 2.3. Efficiency and Power Output of an Irreversible Braysson Heat Engine. The Braysson heat engine working in the hybrid system is a new power cycle model based on a conventional Brayton cycle for the high-temperature heat addition and an Ericsson cycle for the low-temperature heat rejection, which
ð1Þ
As described in refs 16-22, the irreversible losses existing in the fuel cell originate primarily from activation overpotential, ohmic overpotential, and concentration overpotential. With the help of the results obtained in refs 22-26, the authors in ref 4 discussed in detail the influence of these irreversible losses on the performance of SOFCs and derived the efficiency and power output of SOFCs as Pc 1 k mð2Þ ¼ m2 ηc ¼ -Δh RTd1 -ΔH_ and iA k 2 Pc ¼ mm ne F RTd1
ð3Þ
where ΔH_ = (iA/neF)Δh is the rate of the enthalpy change between the products and reactants of the global electrochemical (16) Calise, F.; Palombo, A.; Vanoli, L. J. Power Sources 2006, 158, 225–244. (17) Noren, D. A.; Hoffman, M. A. J. Power Sources 2005, 152, 175–181. (18) Ji, Y.; Yuan, K.; Chung, J. N.; Chen, Y. C. J. Power Sources 2006, 161, 380–391. (19) Watowich, S. J.; Berry, R. S. J. Phys. Chem. 1986, 90, 4624–4631. (20) Wang, C.; Nehrir, M. H.; Shaw, S. R. IEEE Trans. Energy Convers. 2005, 20, 442–451. (21) Haddad, A.; Bouyekhf, R.; Moudni, A. E.; Wack, M. J. Power Sources 2006, 163, 420–432. (22) Zhu, H; Kee, R. J. J. Power Sources 2003, 117, 61–74. (23) Williams, M. V.; Kunz, H. R.; Fenton, J. M. J. Electrochem. Soc. 2005, 152, A635–644. (24) Dalslet, B.; Blennow, P.; Hendriksen, P. V.; Bonanos, N.; Lybye, D.; Mogensen, M. J. Solid State Electrochem. 2006, 10, 547–561. (25) Zhang, Z.; Huang, X.; Jiang, J.; Wu, B. J. Power Sources 2006, 161, 1062–1068. (26) Sorrentino, M.; Pianese, C.; Guezennec, Y. G. J. Power Sources 2008, 180, 380–392.
(27) Chan, S. H.; Khor, K. A.; Xia, Z. T. J. Power Sources 2001, 93, 130–140. (28) Campanari, S. J. Power Sources 2001, 92, 26–34. (29) Palsson, J.; Selimovic, A.; Sjunnesson, L. J. Power Sources 2000, 86, 442–448. (30) Blank, D. A.; Davis, G. W.; Wu, C. Energy 1994, 19, 125–133. (31) Bhattacharyya, S.; Blank, D. A. Int. J. Energy Res. 2000, 24, 539– 547. (32) Blanka, D. A. J. Appl. Phys. 1998, 84, 2385–2392.
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For a practical heat engine, Ri > 1. When the internal irreversibility of the heat engine is negligible, Ri=1 and the cycle is endoreversible. Using eqs 4, 5, and 8, the efficiency of the heat engine may be written as ηe ¼ 1 -
q2 Ri ln x T3 ¼ 1x -1 T1 q1
ð9Þ
Substituting eqs 5, 6, 8, and 9 into eq 4, the rate of the heat transfer q1 may be expressed as Figure 3. Schematic diagram of a Braysson cycle.
q1 ¼
13
was put forward in the 1990s and researched by several scholars.14,15,33,34 The cycle of the working substance in the heat engine consists of one isobaric branch, one isothermal branch, and two adiabatic branches, as shown in Figure 3, where Ti (i=1, 2, and 3) are the temperatures of the working fluid at three different states. When the working substance in the Braysson cycle is assumed to be an ideal gas and the heat transfer is assumed to obey Newton’s law,35-39 the rates of the heat transfer q1 and q2 absorbed from the heat reservoir at temperature T and released to the heat sink at temperature T0 are _ p T1 ðx -1Þ ¼ q1 ¼ mc
RA1 T1 ðx - 1Þ ln y
Using eq 10 and the external condition (∂ηe/∂T1)Ae,q1=0, we can prove that for given Ae and q1, the optimal efficiency of the heat engine is given by ηe ¼ 1 -
ð4Þ
ð5Þ
where d2=(((RRi/β)ln x)/(TT1(x - 1)/[(T - T1)(T - xT1)] ln y))1/2, (Ae/A1)=1 þ (RRi/βd2)(ln x/ln y), and T1 is determined by the following equation:
respectively, where R and β are the heat-transfer coefficients between the working substance and the heat reservoirs at temperatures T and T0, respectively, Ai (i = 1 and 2) are the corresponding heat-transfer areas, m_ and cp are the mass flow rate and specific heat of the working substance, respectively, y= (T-T1)/(T-xT1), and x = T2/T1 is the isobaric temperature ratio. The overall heat-transfer area of the heat engine is Ae ¼ A1 þA2
q1 ¼
qL ¼ RL AL ðT - T0 Þ
ð6Þ
ð13Þ
q1 ¼ -ΔH_ - Pc - qL ¼ -ΔH_ - Pc - RL AL ðT - T0 Þ
ð7Þ
ð14Þ
and consequently, eq 12 may be written as T1 ðx - 1Þ ¼ C1 ð1 - ηc Þi - C2 ðT=T0 - 1Þ i ln y þ RR βd2 ln x
βA2 ðT3 - T0 Þ=T3 _ p ln x mc
βA2 ½ðT3 -T0 Þ=T3 ln y ¼ RA1 ln x
ð12Þ
where RL is the convective and/or conductive heat-leak coefficient and AL denotes the effective heat-transfer area. According to Figure 1 and eq 13, one can obtain
From eqs 4 and 7, we may introduce an internal irreversible parameter Ri ¼
RAe T1 ðx - 1Þ i ln y þ RR βd2 ln x
As illustrated in Figure 1, a part of the waste heat produced in the fuel cell is directly released as heat leak (qL) to the environment,40-42 which may be expressed as43
In the investigation on the optimum performance of an irreversible Braysson heat engine, it is very significant to further consider the influence of irreversibility within the cycle. Using the second law of thermodynamics for this cycle model, we have _ p lnðxÞ -βA2 ðT3 -T0 Þ=T3 < 0 mc
Ri T0 ln x T1 ðx - 1Þð1 - d2 Þ T0 ð1 - d2 Þ½q1 =ðRRi Ae Þ½ln y=ln x þ ðRRi Þ=ðβd2 Þ ð11Þ
¼1 -
and q2 ¼ βA2 ðT3 - T0 Þ
RAe ð10Þ ln y ðR=βÞRi ln x þ T1 ðx - 1Þ T1 ðx - 1Þ - ½RT0 =ð1 - ηe Þln x
ð15Þ
where C1=-(AΔh/neFRAe) and C2=(RLALT0/RAe). Using eqs 11 and 14, the efficiency and power output of the heat engine may be, respectively, expressed as
ð8Þ
ηe ¼ 1 R i T0 ð1 - d2 Þ½C1 ð1 - ηc Þi - C2 ðT=T0 - 1Þ½ln y=ln x þ ðRRi Þ=ðβd2 Þ
(33) Zheng, J.; Chen, L.; Sun, F.; Wu, C. Int. J. Therm. Sci. 2002, 41, 201–205. (34) Zheng, S.; Chen, J.; Lin, G. Renewable Energy 2005, 30, 601–610. (35) Andresen, B. Finite-Time Thermodynamics; Phys. Lab. II, University of Copenhagen: Copenhagen, Denmark, 1983. (36) Bejan, A. Advanced Engineering Thermodynamics; Wiley: New York, 1988. (37) Chen, J.; Yan, Z. Phys. Rev. A: At., Mol., Opt. Phys. 1989, 39, 4140–4147. (38) De Vos, A. Endoreversible Thermodynamics of Solar Energy Conversion; Oxford University Press: Oxford, U.K., 1992. (39) Wu, C.; Chen, L.; Chen, J. Recent Advances in Finite-Time Thermodynamics; Nova Science Publishers, Inc.: New York, 1999.
ð16Þ (40) Chan, S. H.; Ding, O. L. Int. J. Hydrogen Energy 2005, 30, 167– 179. (41) Sanchez, D.; Chacartegui, R.; Mu~ noz, A.; Sanchez, T. J. Power Sources 2006, 160, 1074–1087. (42) Bavarsad, P. G. Int. J. Hydrogen Energy 2007, 32, 4591–4599. (43) Sanchez, D.; Mu~ noz, A.; Sanchez, T. J. Power Sources 2007, 169, 25–34.
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and
: DOI:10.1021/ef9006774
Chen et al. Table 1. Operating Conditions and Performance-Related Parameters18,44-47
( Pe ¼ q1 ηe ¼ RAe ½C1 ð1 - ηc Þi - C2 ðT=T0 - 1Þ 1 -
Ri T0 ð1 - d2 Þ½C1 ð1 - ηc Þi - C2 ðT=T0 - 1Þ½ln y=ln x þ ðRRi Þ=ðβd2 Þ
ð17Þ 2.4. Efficiency and Power Output of the Hybrid System. Combining eqs 1, 2, 14, 16, and 17 yields the following expressions of the efficiency and power output of the hybrid system as q1 Pe P η ¼ ¼ ηc þ -ΔH_ -ΔH_ q1 ( ¼ ηc þ ½1 - ηc - ðT=T0 - 1ÞC2 =ðC1 iÞ 1 -
Ri T0 ð1 - d2 Þ½C1 ð1 - ηc Þi - C2 ðT=T0 - 1Þ½ln y=ln x þ ðRRi Þ=ðβd2 Þ
ð18Þ and P ¼ Pc þ Pe iA Δh ηc þ ½1 ¼ ne F
parameter
value
operating temperature, T (K) ambient temperature, T0 (K) operating pressure, p0 (atm) fuel composition pressures, pH2;pH2O (atm) air composition pressures, pO2; pN2 (atm) number of electrons, ne anode-exchange current density, io,a (A m-2) cathode-exchange current density, io,c (A m-2) electrolyte thickness, Lel (μm) activation energy of O2-, Eel (J mol-1) prefactor of O2-, σ0 (S m-1) ratio of the internal resistance to the leakage resistance, k anode-limiting current density, iL,a (A m-2) cathode-limiting current density, iL,c (A m-2) Faraday constant, F (C mol-1) universal gas constant, R (J mol-1 K-1) standard molar enthalpy change at 1273 K, Δh (J mol-1) standard molar Gibbs free-energy change at 1273 K, Δg (J mol-1)
1273 298 1 0.97; 0.03 0.21; 0.79 2 1.3 103 5.6 103 20 8.0 104 3.6 107 1/100 2.99 104 2.16 104 96485 8.314 -249490 -177460
- ηc - ðT=T0 - 1ÞC2 =ðC1 iÞ 1 -
Ri T 0 ð1 - d2 Þ½C1 ið1 - ηc Þ - C2 ðT=T0 - 1Þ½ln y=ln x þ ðRRi Þ=ðβd2 Þ
ð19Þ From eqs 18 and 19, it is clearly seen that the efficiency and power output of the hybrid system are closely dependent upon the various losses resulting from the irreversibilities within the fuel cell and heat engine and originating from the heat transfer as a result of convection/conduction in the fuel cell/heat engine hybrid system. In the next section, numerical predictions will be studied to outline how the irreversible model based on the above analysis can provide a valuable tool for improving the system performance.
Figure 4. Curves of the efficiencies of three thermodynamic systems varying with the current density, where the parameters C1 = 0.01 m2/ A, C2 = 0.1 K, and R/β = 1 are chosen, iη is the current density at the maximum efficiency ηmax, ηc,η and ηe,η are the efficiencies of the fuel cell and heat engine in the hybrid system at the maximum ηmax, respectively, and curves 1, 2, and 3 correspond to the cases of the hybrid system, heat engine, and fuel cell in the hybrid system, respectively.
3. General Performance Characteristics and Optimum Criteria It can be seen from eqs 18 and 19 that the performance of the hybrid system depends upon a set of thermodynamic and electrochemical parameters, such as the operating temperature T and the current density i of the fuel cell, the parameters related to the heat transfer between the fuel cell and heat engine, and the heat leak from the fuel cell to the surroundings, i.e., C1 and C2. Numerical calculations are performed on the basis of the parameters summarized in Table 1, which are derived from data available in the literature.18,44-47 Their values 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 the oxidant. Using eqs 18 and 19 and Table 1, we can generate the curves of the efficiency and power density of the hybrid system
varying with the current density, as shown in Figures 4 and 5. It can be clearly seen from Figures 4 and 5 that for a hybrid system, there always exist a maximum efficiency ηmax and a maximum power output Pmax at which the corresponding current densities are iη and iP, respectively. In the region of i < iη, the efficiency and power output of the hybrid system will decrease as the current density i is decreased, while in the region of i > iP, the efficiency and power output of the hybrid system will also decrease as the current density i is increased. It is thus obvious that the regions of i < iη and i > iP are not optimal from the thermodynamic point of view, although the hybrid system may be operated in these regions. Therefore, the optimal region of the current density i for the fuel cell/heat engine hybrid system should be
(44) Costamagna, P.; Selimovic, A.; Del Borghi, M.; Agnew, G. Chem. Eng. J. 2004, 102, 61–69. (45) Cordiner, S.; Feola, M.; Mulone, V.; Romanelli, F. Appl. Therm. Eng. 2007, 27, 738–747. (46) Zhang, X.; Li, G.; Li, J.; Feng, Z. Energy Convers. Manage. 2007, 48, 977–989. (47) Yoon, K. J.; Zink, P.; Gopalan, S.; Pal, U. B. J. Power Sources 2007, 172, 39–49.
It shows that iη and iP are two important parameters of the hybrid system, which determine, respectively, the lower and upper bounds of the optimized current density. In the practical operation of the fuel cell/heat engine hybrid system, engineers should choose a reasonable current density according to eq 20 to ensure that the system is operated in the optimal region.
iη e i e iP
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ð20Þ
Energy Fuels 2009, 23, 6079–6084
: DOI:10.1021/ef9006774
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Figure 5. Curves of the power densities of three thermodynamic systems varying with the current density, where P* = P/A is the power density, the parameters C1 = 0.01 m2/A, C2 = 0.1 K, and R/β = 1 are chosen, iP is the current density at the maximum power output Pmax, Pc,P and Pe,P are the power outputs of the fuel cell and heat engine in the hybrid system at the maximum power output Pmax, respectively, and curves 1, 2, and 3 correspond to the cases of the hybrid system, heat engine, and fuel cell in the hybrid system, respectively.
Figure 6. Efficiency versus power density curves of three thermodynamic systems, where the values of the relevant parameters are the same as those used in Figures 4 and 5, Pm* and ηm are the power density at the maximum efficiency and the efficiency at the maximum power output, respectively, and curves 1, 2, and 3 correspond to the cases of the hybrid system, heat engine, and fuel cell in the hybrid system, respectively.
output Pmax are not, in general, equal to the maximum power outputs of the fuel cell and heat engine. It shows clearly that, in the investigation of fuel cell/heat engine hybrid systems, one must synthetically analyze the whole performance of these hybrid systems. To ensure that the hybrid system operates in the optimal region, the fuel cell and the Braysson cycle must work in their respective optimum regions. According to the optimum criterion of the current density and Figures 4-6, we can determine the optimal regions of some important parameters in the fuel cell and Braysson cycle as
To further understand the performance characteristics of the hybrid system, we give the efficiency versus power output curves of the hybrid system, as shown in Figure 6. According to the optimum criterion of the current density and Figure 6, one can further determine the optimum regions of the efficiency and power output as Pm e P e Pmax
ð21Þ
ηm e η e ηmax
ð22Þ
and
where Pm and ηm are the power output at the maximum efficiency ηmax and the efficiency at the maximum power output Pmax. When the hybrid system is operated in these optimum regions, the power output will increase as the efficiency is decreased and vice versa. The above results show that Pmax, ηmax, Pm, and ηm are also four important parameters of the hybrid system. Pmax and ηmax determine the upper bounds of the power output and efficiency of the system, while Pm and ηm determine the allowable values of the lower bounds of the optimized power output and efficiency. Obviously, the four important parameters closely depend upon some other parameters in the system and can be numerically calculated for the given values of these parameters. It is clearly seen from Figures 4-6 that, in the optimally working region, the efficiency and power output of the hybrid system are always larger than those of a fuel cell4,22-26 or a heat engine.13-15,48 It shows once again that the application of the hybrid system may availably improve the performance of the fuel cell system and enhance the used efficiency of energy sources. It is also seen from Figures 4-6 that the efficiencies ηc,η and ηe,η of the fuel cell and heat engine in the hybrid system at the maximum efficiency ηmax are not, in general, equal to the maximum efficiencies of the fuel cell and heat engine and the power outputs Pc,P and Pe,P of the fuel cell and heat engine in the hybrid system at the maximum power
minðPc, η , Pe, P Þ e Pc
ð23Þ
ηc, P e ηc e ηc, η
ð24Þ
Pe, η e Pe e Pe, P
ð25Þ
ηe, P e ηe e ηe, η
ð26Þ
and
respectively, where Pc,η and Pe,η are the power outputs of the fuel cell and heat engine in the hybrid system at the maximum efficiency ηmax and ηc,P and ηe,P are the efficiencies of the fuel cell and heat engine in the hybrid system at the maximum power output Pmax. The four parameters Pc,P, Pe,P, ηc,P, and ηe,P are dependent upon other parameters in the hybrid system and can be calculated from eqs 1, 2, 16, and 17. 4. Effects of Irreversible Losses It is seen from eqs 18 and 19 that the efficiency and power output of the hybrid system are monotonically decreasing functions of the parameters Ri, x, C1, and C2. The smaller the parameters Ri, x, C1, and C2, the better the performance of the hybrid system. The ideal hybrid system under the condition of R β corresponds to the case of Ri=1, x=1, C1=0, and C2= 0. It is worthwhile to point out that, when the different values of the parameters Ri, x, C1, and C2 are chosen, one can obtain different hybrid systems. For example, when Ri = 1, x = 1, C1 > 0, and C2 > 0, the hybrid system consists of a fuel cell and an endoreversible Canot cycle. Moreover, when C1 = 0
(48) Ust, Y.; Yilmaz, T.; Turkish, J. Energy Environ. Sci. 2005, 29, 271–278.
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Table 2. Several Different Hybrid Systems Included in the Present Model under the Condition of R β hybrid system
Ri
x
C1
C2
cycle mode of heat engine
model
I II III IV V VI VII
>1 1 1 >1 1 1 1
>1 >1 >1 1 1 1 1
>0 >0 >0 >0 >0 0 0
>0 >0 0 >0 >0 >0 0
irreversible Braysson cycle endoreversible Braysson cycle endoreversible Braysson cycle without heat leak losses irreversible Carnot cycle endoreversible Carnot cycle Carnot cycle including heat leak losses reversible Carnot cycle
general model simplified model 1 simplified model 2 simplified model 3 simplified model 4 simplified model 5 simplest model
under the condition of R β, it means R f ¥ and β f ¥ and, consequently, the endoreversible Canot cycle in the hybrid system becomes a Carnot cycle. Such a Carnot cycle may include the heat leak losses when C2 > 0 and be completely reversible when C2 = 0. Several representational hybrid systems are listed in Table 2. It is thus obvious that, as long as the values of the parameters Ri, x, C1, and C2 listed in Table 2 are adopted, the optimum performance characteristics of some simplied fuel cell/heat engine hybrid systems can be directly derived from the results obtained in the present paper.
finite-rate heat transfer, and internal irreversibility in the heat engine are synthetically taken into account. Expressions of the efficiency and power output are derived and used to discuss the optimal performance of fuel cell/heat engine hybrid systems. The general performance characteristics of the hybrid systems are revealed through numerical calculation. The maximum efficiency and power output are calculated. The optimally operating regions of some important parameters including the efficiency, power output, and current density are determined. The optimum performance of some special cases is directly deduced. The results obtained here are very general and useful. They may be used to analyze the optimal performance of a class of irreversible hybrid systems and to provide some theoretical basis for the performance improvement and optimal design of practical fuel cell/ heat engine hybrid systems.
5. Conclusions We have successfully established a general irreversible model of a Braysson-based fuel cell hybrid system, which may include some different simplified hybrid systems. Using the current models of SOFCs, regenerators, and Braysson heat engines, the irreversible losses resulting from the electrochemical reaction and electric resistances in the fuel cell, the heat leak losses from the fuel cell to the environment,
Acknowledgment. This work was supported by the National Natural Science Foundation, People’s Republic of China, and the Science Foundation, Xiamen, People’s Republic of China.
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