Efficiency of ideal fuel cell and Carnot cycle from a fundamental perspective

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1 245 Efficiency of ideal fuel cell and Carnot cycle from a fundamental perspective H Hassanzadeh and S H Mansouri Shahid Bahonar University of Kerman, Kerman, Iran The manuscript was received on 8 June 2004 and was accepted after revision for publication on 13 January DOI: / X28571 Abstract: In this paper, we accept the fact that fuel cell and heat engine efficiencies are both constrained by the second law of thermodynamics and neither one is able to break this law. However, we have shown that this statement does not mean the two systems should have the same maximum thermal efficiency when being fed by the same amounts of chemical reactants. The intrinsic difference between fuel cells (electrochemical systems) and heat engines (combustion engines) efficiencies is a fundamental one with regard to the conversion of chemical energy of reactions into electrical work. The sole reason has been shown to be due to the combustion irreversibility of the latter. This has led to the statement that fuel cell efficiency is not limited by the Carnot cycle. Clarity is achieved by theoretical derivations and several numerical examples. Keywords: fuel cells, thermodynamics, Carnot cycle, efficiency 1 INTRODUCTION As environmental concerns receive increasing attention, the need for developing new technologies that address the conflicting issues of energy production and protection of the environment becomes evident. The extraordinary environmental quality and high efficiency of fuel cells make them a potential alternative energy source for both stationary and transportation applications. A fuel cell is an electrochemical device that converts the chemical energy of a reaction directly into electrical energy through a controlled chemical reaction. In a recent paper that appeared in the International Journal of Hydrogen Energy, Lutz et al. [1] have made the following statement: A commonly stated advantage of fuel cells is that they are more efficient than an equivalent process of combusting the fuel and converting the heat into electricity. Often, this advantage is explained by saying fuel cells are not limited by the Carnot efficiency, sometimes with the added comment, because a fuel cell is not a heat engine. While the latter is true, the Corresponding author: Shahid Bahonar University of Kerman, PO Box , Kerman, Iran. mansouri@alum. mit.edu Current affiliation: University of Birjand, Birjand, Iran. former is a myth. Fuel cells and heat engines are both constrained by the same maximum efficiency. The limit is established by the second law of thermodynamics, and neither process is able to break this law. In this paper, we present detailed thermodynamics derivations of the thermal efficiency of a fuel cell and a Carnot heat engine. We accept the fact that the fuel cell and the Carnot heat engine efficiencies are both constrained by the second law of thermodynamics, and neither one is able to break this law. However, this statement does not mean the two systems should have the same maximum thermal efficiency when being fed the same amounts of chemical reactants. Hence, the above statement is not a myth. The intrinsic difference between the fuel cells (electrochemical systems) and heat engines (combustion engines) efficiencies is a fundamental one in regard to the conversion of chemical energy of reactions into electrical work. The sole reason for this is due to the combustion irreversibility of the latter [2]. A century of delays in utilizing the electrochemical energy conversion were due to technological difficulties. Not recognizing its potential, this gave rise to a consumption of more than twice the fuel that would have been used electrochemically to produce the same electrical energy; many of the negative consequences of our present technology such as A07404 # IMechE 2005 Proc. IMechE Vol. 219 Part A: J. Power and Energy

2 246 H Hassanzadeh and S H Mansouri smog, smoke, vibration and noise could have been avoided. We review and comment on the analysis presented by Lutz et al. [1]. Their derivation is based on an ideal combustion process; they have also failed to take into account combustion irreversibility and the effect of chemical equilibrium criteria, which propelled them to reach a wrong conclusion. They have ignored the fact that a combustion process, whether complete or in equilibrium condition, could not be made reversible by replacing a high-temperature reservoir with a combustion reactor as depicted by them, as compared with the original heat engine, because the combustion process is inherently irreversible. A first-law computation based on the ideal combustion equation is often quite misleading, since the actual products frequently do not match those predicted by the theoretical reaction. A correct theoretical prediction of possible behaviour can be achieved through the use of the second law of thermodynamics; chemical equilibrium criteria are of major importance in the study of processes that involved chemical reactions. 2 FUEL CELL EFFICIENCY The fuel cell efficiency derivation is the same as that presented by Wark [3]. As shown in Fig. 1, the steadyflow device produces electric power. Oxidation of the fuel occurs at the anode, with release of electrons to the external circuit. At the cathode the oxidizer is reduced, with the consumption of electrons. The circuit is completed by the movement of ions in the electrolyte. In order to derive an expression for the work done by an ideal fuel cell, we consider an open system (control volume) as denoted in Fig. 1. Assuming an isothermal system, an energy balance for the fuel cell system can be written as Q W elec ¼ DH R ¼ X hi (T, P) (1) where, DH R is the difference between molar enthalpies of products and reactants (fuel and oxidizer). To keep the fuel cell temperature constant during an isothermal process, it may require that heat be added or removed. In this case, Q ¼ TDS R ; that is, the heat transfer to or from the system is assumed to be reversible. Therefore, the reversible work is done by the system when all internal and external interactions with the system are reversible. W elec,rev ¼ DH R þ Q ¼ DH R þ TDS R ¼ DG R (T, P) (2) DG R ¼ X g i (T, P) ¼ X n i g i (T, P) (3) where g i are the specific Gibbs functions and n i are the stoichiometric reaction coefficients. Hence, the maximum work is the difference between the entering and leaving values of the Gibbs function for the process. The total reversible work of the process can be achieved by the system when the final state of the system is in equilibrium with the environment, Fig. 1 Schematic of a fuel cell Proc. IMechE Vol. 219 Part A: J. Power and Energy A07404 # IMechE 2005

3 Efficiency of ideal fuel cell and Carnot cycle 247 that is, at the standard reference temperature and pressure (, P 0 ). W max ¼ W elec,rev (, P 0 ) (4) W max ¼ X (h 0 S 0 ) i ¼ X g i (, P 0 ) W max ¼ X n i Dg 0 f,i,298 (5) ¼ G P (, P 0 ) þ G R (, P 0 ) (6) W max ¼ X n i g 0 i,298 ¼ X n i m 0 i,298 ¼ DG R (, P 0 ) (7) where g i (, P 0 ) is the specific Gibbs function of formation, Dgf 0 ;298, at standard reference condition and g i ¼ m i, where m i is the chemical potential of the ith species. The thermal efficiency of an ideal fuel cell at reference temperature and pressure is the ratio of the maximum reversible work done by the system to the heating value of the fuel. provide other formulations that avoid reference to specific applications. One of these is due to Hatsopoulos and Keenan [4], who define a stable equilibrium state of a system such that a finite change of state cannot occur without a corresponding finite change occurring in the state of the environment. Making use of this definition, they proposed the following second-law statement: A system having specified allowed states and an upper bound in volume can reach from any given state a stable state and leave no net effect on the environment. This statement, which is not deducible from the definition of a stable state, simply predicts the existence in nature of stable equilibrium states. Based on the above statement, the thermal efficiency of unity for an ideal fuel cell is in complete compliance with the second law [3]. Thus the basic advantages of the fuel cell are twofold: the possibility of an approach to 100 per cent conversion of the free energy, and direct conversion of chemical energy into electricity. h FC ¼ W max Q In ¼ DG R(, P 0 ) DH R (, P 0 ) (8) 3 CARNOT EFFICIENCY where Q In ¼ DH R (, P 0 ). By definition, thermal efficiency should not exceed unity. However, based on the second-law analysis where heat addition is required by the process, the fuel cell efficiency might be greater than unity provided that the heat addition by the surroundings was excluded from Q In. When heat from the surroundings is included in Q In, the maximum thermal efficiency of an ideal fuel cell becomes unity as follows. h FC ¼ W max Q In ¼ DG R(, P 0 ) DH R þ DS R ¼ DG R(, P 0 ) DG R (, P 0 ) ¼ 1 (9) The unity of thermal efficiency for an ideal fuel cell does not violate the Kelvin Planck statement of the second law, which may be paraphrased as: It is impossible to construct a heat engine which produces no other effects than the extraction of heat from a single source and the production of an equivalent amount of work. This recognizes the fact that the fuel cell is not a heat engine. The above statement of the second law relies heavily on observations made with respect to cyclic devices such as refrigerators and heat engines. Since the second law has general applicability to a wide range of phenomena, it should be possible to At the present, the common method of obtaining electrical energy is by first converting the heat released during a combustion process into mechanical energy. The efficiency of this conversion is less than unity, even assuming a frictionless system. The ideal efficiency for the entire process was derived first by Carnot and expressed by h Carnot ¼ 1 T L T H (10) where, T H is the temperature of hot gases entering the engine and T L is the temperature of the cold gases leaving the system and entering the environment. This expression also shows that the thermal efficiency of a Carnot heat engine will become unity as T H! 1 or as T L! 0. 4 PHYSICAL INTERPRETATION OF THE INTRINSIC DIFFERENCES BETWEEN THE FUEL CELL AND CARNOT CYCLE EFFICIENCIES In the chemical reaction carried out in a combustion engine, there is heat transfer, where in a fuel cell, there is charge transfer. In charge transfer, the available energy of the electrons can be used up; that is, the electrons fall through the total potential difference, as in the case of the potential energy of a falling object provided that the internal resistance of the A07404 # IMechE 2005 Proc. IMechE Vol. 219 Part A: J. Power and Energy

4 248 H Hassanzadeh and S H Mansouri fuel cell and friction on the falling object are negligible. As a result of the potential difference between the two electrodes in the fuel cell, the extra energy of the electrons is completely used up by the time they get around the work circuit. On the contrary, since the temperature of the environment is fixed and is not zero, the efficiency of a heat engine is always less then unity. 5 WORK FROM THE ADIABATIC COMBUSTION PROCESS In a simplified model, we consider the overall process within the system to occur in two steps. First, as shown in Fig. 2, combustion occurs with the product gases reaching the temperature T ad.in the next step, heat removed from the hot-product gases is supplied to a heat engine, which produces work W and rejects heat Q 0 to the environment at. In the first step both W and Q are zero, and the energy balance leads to the adiabatic flame temperature T ad. The available energy of the combustion products at flame temperature is W rev ¼ H P (T ad, P) S P (T ad, P) H P (, P 0 ) þ S P (, P 0 ) (11) Therefore, the irreversibility of the combustion process is the difference between the available energy of the fuel and the combustion products I ¼ DS R ¼ W max,isotherm W rev (12) Of more significance are the numerical values of I obtained for typical hydrocarbon fuels when compared to the W rev values for the isothermal reaction. The available energy loss during adiabatic combustion could be per cent of the reversible work capability during an isothermal reaction [3]. Hence, a large loss in work capability occurs in the combustion process even before the hot gases reach the energy conversion portion of the device due to its inherent irreversibility. However, the second step, unlike the first step, can theoretically be made reversible by a series of Carnot-type heat engines (in theory, the number of heat engines required would be infinitely large). Each engine in the series operates at a slightly lower temperature on the high temperature side of engine, and each rejects heat at. The sum of the work outputs from the series of engines is equal to W rev for the product gases as they cool from T ad to T P. Hence, the reversible work produced is W rev,carnot ¼ H P (T ad, P 0 ) H P (T P, P 0 ) ½S P (T P, P 0 ) S P (T ad, P 0 )Š (13) The second law efficiency h SL is defined as h SL ¼ W rev,carnot W max,isotherm ¼ h Carnot h FC (14) Where T P ¼, the maximum reversible work occurs. The ideal fuel cell efficiency is only a function of fuel and oxidant type, and the temperature of the environment. However, the second-law efficiency of work transfer from the adiabatic combustion process is governed by a Carnot cycle efficiency working between the hot combustion products and the environment, and therefore is less than unity. One may wonder whether the Carnot efficiency can be increased by rising to the highest temperature possible in the combustion process. This question has been answered erroneously by Lutz et al. [1] in the following way: Fig. 2 Schematic of a heat engine The maximum temperature for which the reaction will proceed is where the change in Gibbs free energy is zero. As has been stated correctly by Appleby and Foulkes [5]: if it were possible in the limiting case to carry out the combustion process in a controlled, reversible manner (which is impossible), then the change in Gibbs free energy is zero Proc. IMechE Vol. 219 Part A: J. Power and Energy A07404 # IMechE 2005

5 Efficiency of ideal fuel cell and Carnot cycle 249 and the combustion temperature, T C, would be defined as DG R ¼ DH R TDS R ¼ 0 (15) T C ; DH R DS (16) R TC This definition is not correct because the combustion process is irreversible (for both complete and equilibrium assumption). If we assume the amount of entropy generation in an irreversible combustion process to be DS gen, then temperature can be defined as in Appendix 2 DH R T C ; DS R þ DS T C (17) gen DS gen can be calculated from the definition of irreversibility of the process, that is, the difference between the amount of available work of the fuel and oxidant at the environment temperature and the combustion products at the adiabatic flame temperature. That is, I ¼ W rev,fuel oxidants W rev,combustion products ¼ DS gen (18) This definition of T C is very meaningful, especially for the methane reaction. Lutz et al. [1] have found T C to be around K, which would be an impossible temperature for any possible combustion process. Based on the corrected definition given by equation (17), T C is found to be around 2014 K for complete combustion of methane. Therefore, the reason that Lutz et al. [1] have obtained extra large numbers for T C was not due to the fact that the entropy change for methane oxidation was negligible compared to the enthalpy change, but it was due to their incorrect definition of T C that propelled them to reach the wrong conclusion so as to show that the maximum efficiency approached unity for either a fuel cell or a Carnot heat engine. Using the corrected definition of T C as the hightemperature reservoir for the Carnot cycle will give us the same efficiency as the second-law efficiency h SL defined by equation (14). This conclusion indicates that T C is equivalent to the log mean enthalpy temperature difference of the combustion products as they cool down from T ad to (see Appendix 3) in the series of Carnot-type heat engines of Fig. 3, where each engine in the series operates at a slightly lower temperature on the high-temperature side of the engine, and rejects heat at. In calculating the thermo-chemical properties of the combustion products, Lutz et al. [1] have Fig. 3 assumed complete combustion; and by doing so they have implicitly ignored the effect of dissociation on the combustion products. They failed to recognize the very fact that the combustion process by no means could be made complete and reversible, especially at high temperatures, and the chemical equilibrium criteria were of major importance in the study of processes that involved chemical reaction and the second-law analysis. The following section will demonstrate this fact. 6 NUMERICAL EXAMPLES In this section, we review the examples presented by Lutz et al. [1]. We take into account the effect of combustion irreversibility and the effect of chemical equilibrium criteria, because their derivation was based on an ideal (complete) and reversible combustion process, which propelled Luiz et al. to reach the wrong conclusion. 6.1 Hydrogen oxygen Schematic of a reversible heat engine Consider the reaction of pure hydrogen and oxygen. The overall reaction is: H 2 þ 1 2 O 2! H 2 O (19) Using the equilibrium thermo-chemical properties for this reaction at 298 K from references [6] A07404 # IMechE 2005 Proc. IMechE Vol. 219 Part A: J. Power and Energy

6 250 H Hassanzadeh and S H Mansouri Fig. 4 Enthalpy of H 2 and O 2 mixture (2:1) and their combustion products Fig. 6 Gibbs free energy of H 2 and O 2 mixture (2:1) and their combustion products and [7], the maximum fuel cell efficiency is h FC ¼ DG R DH R ¼ 94:5 per cent (20) which agrees well with the value of 94 per cent given by Appleby and Foulkes [5] (and 94.5 per cent given by Wark [3]) for this reaction. To evaluate the efficiency of the Carnot heat engine operating on this same reaction the adiabatic flame temperature is calculated as 3077 K and the thermal efficiency is calculated as 82.9 per cent and T C ¼ 1743 K. The reason for loss in thermal efficiency is due to irreversibility of the combustion process. Lutz et al. [1] have argued that the variations in DH R and DS R with temperature were 7 and 28 per cent. While these might seem significant, they did not prevent the change in Gibbs function from being linear. The enthalpy and entropy of reaction were generally not strong functions of temperature, that is DH(T L ) DH(T C ) and DS(T L ) DS(T C ) (21) assuming the product water in the gaseous phase is the same as basing the efficiency on the lower heating Fig. 5 Entropy of H 2 and O 2 mixture (2:1) and their combustion products Fig. 7 Available energy of H 2 and O 2 mixture (2:1) and their combustion products Proc. IMechE Vol. 219 Part A: J. Power and Energy A07404 # IMechE 2005

7 Efficiency of ideal fuel cell and Carnot cycle 251 value of the fuel. Figures 4 7 show the enthalpy, entropy, Gibbs energy, and available energy for the reactants and products (gaseous water) of the reaction, and their differences. The equilibrium thermochemical properties are obtained from references [6] and [7]. In Fig. 4, we have compared the results of enthalpy (reactants, products and DH R ) calculations based on complete combustion (Lutz et al. [1]) and the chemical equilibrium assumption (present study). This figure shows that as temperature nears the adiabatic flame temperature the behaviour of the above properties deviate from the Lutz et al. [1] results, and the effect of temperature becomes very significant as a result of dissociation due to chemical equilibrium. It also shows that as the reactants preheat temperature increases, the enthalpy of reaction is almost constant up to the corrected T C, and for temperatures greater than T C, DH R decreases sharply. This behaviour cannot be seen by using the complete combustion assumption. In Fig. 5, we have compared the results of the entropy (reactants, products and DS R ) calculation based on complete combustion (Lutz et al. [1]) and the chemical equilibrium assumption (present study). The changes in entropy are almost constant up to the corrected T C, and, for temperatures greater than T C, DS R increases; again, this behaviour cannot be seen by using the complete combustion assumption. In Fig. 6, we have compared the results of the Gibbs function (reactants, products and DG R ) calculation based on complete combustion (Lutz et al. [1]) and the chemical equilibrium assumption (present study). The changes in the Gibbs function are always negative and nonlinear, and never reach zero, therefore the definition of T C base on zero DG R at high temperature is completely wrong. In Fig. 7, we have compared the results of available energy calculation (reactants, products and W rev ) based on the complete combustion (Lutz et al. [1]) and chemical equilibrium assumption (present study). This figure shows that preheating the reactants has a small effect on the available energy up to the corrected T C, and beyond T C the temperature effect will become more pronounced by lowering the available energy due to dissociation of the combustion products and available energy vanishes for temperature around 4000 K. Therefore, these figures clearly show that the Lutz et al. [1] argument would have been very much different if they had considered equilibrium data. If we take the corrected definition of T C, which is very much lower than the adiabatic flame temperature, then equations (21) hold. Therefore, based on the corrected definition of T C, the efficiency of the Carnot cycle will be lower than the ideal fuel cell efficiency. 6.2 Other examples In this section, we have considered several reactions, and the results of those calculations are shown in Tables 1 and 2. The tables compare the adiabatic flame temperature, T ad, the corrected combustion temperature, T C, the efficiency of ideal fuel cell, h FC, the efficiency of the Carnot heat engine, h Carnot, the ratio of Carnot to ideal fuel cell efficiency, h SL, and the ratio of the lost work to available work, I=W rev, as the result of the irreversible combustion process for both complete and equilibrium Table 1 Calculation of T ad, T C, h FC, h Carnot, h SL, and I=W rev based on the complete combustion process Reaction T ad (K) T C (K) h FC (%) h Carnot (%) h SL (%) I=W rev (%) H 2 þð1=2þo 2! H 2 O (gas) H 2 þð1=2þo 2 þ 2N 2! H 2 O þ 2N 2 (gas) CH 4 þ 2O 2! CO 2 þ 2H 2 O (gas) CH 3 OH þð3=2þo 2! CO 2 þ 2H 2 O (gas) C 2 H 4 þ 3O 2 þ 11.28N 2! CO 2 þ 2H 2 O þ 11.28N 2 (gas) Table 2 Calculation of T ad, T C, h FC, h Carnot, h SL and I/W rev based on equilibrium composition Reaction T ad (K) T C (K) h FC (%) h Carnot (%) h SL (%) I=W rev (%) H 2 þð1=2þo 2! Equilibrium composition H 2 þð1=2þo 2 þ 2N 2! Equilibrium composition CH 4 þ 2O 2! Equilibrium composition CH 3 OH þð3=2þo 2! Equilibrium composition C 2 H 4 þ 3O 2 þ 11:28N 2! Equilibrium composition A07404 # IMechE 2005 Proc. IMechE Vol. 219 Part A: J. Power and Energy

8 252 H Hassanzadeh and S H Mansouri combustion process assumptions. These tables show that ideal fuel cell efficiency is always greater than the Carnot cycle efficiency. 7 CONCLUSIONS As optimal design and analysis of fuel cells require thorough understanding of their second-law limitations, in this paper, we have criticized the argument promoted by Lutz et al. [1] that the maximum thermal efficiency of an ideal fuel cell is the same as a Carnot heat engine. Their derivation is based on the ideal combustion process. They have also failed to take into account combustion irreversibility and the effect of chemical equilibrium criteria, which propelled them to reach a wrong conclusion. They have ignored the fact that the combustion process, whether complete or in an equilibrium condition, could not be made reversible by replacing a high-temperature reservoir with a combustion reactor as depicted by them, as compared with the original heat engine, because the combustion process is inherently irreversible. We accept the fact that the fuel cell and the Carnot heat engine efficiencies are both constrained by the second law of thermodynamics, and neither one is able to break this law. However, we have shown that this statement does not mean the two systems should have the same maximum thermal efficiency when being fed the same amounts of chemical reactants. The intrinsic difference between the fuel cells (electrochemical systems) and heat engines (combustion engines) efficiencies is a fundamental one with regard to the conversion of chemical energy of reactions into electrical work. The sole reason has been shown to be due to the combustion irreversibility of the latter. This has led to the statement that fuel cell efficiency is not limited by the Carnot cycle. The unity of thermal efficiency for an ideal fuel cell does not violate the Kelvin Planck statement of the second law, recognizing the fact that the fuel cell is not a heat engine. Based on the statement of the second law proposed by Hatsopoulos and Keenan [4], the thermal efficiency of unity for an ideal fuel cell is in complete compliance with it. Thus the basic advantages of the fuel cell are twofold: the possibility of an approach to 100 per cent conversion of the free energy; and direct conversion of chemical energy into electricity. Clarity was achieved by theoretical derivations and several numerical examples. A first-law computation based on the complete combustion equation is often quite misleading, since the actual products frequently do not match those predicted by the theoretical reaction. A correct theoretical prediction of possible behaviour should be achieved through the use of the second law of thermodynamics and chemical equilibrium criteria. Hence, in our analysis and numerical calculations, we have taken into account the effect of combustion irreversibility and chemical equilibrium criteria. It is not possible to make the combustion process reversible by replacing a high-temperature reservoir with a combustion reactor as depicted by Lutz et al. [1], as compared with the original heat engine, because the combustion process is inherently irreversible. REFERENCES 1 Lutz, A. E., Larson, R. S., and Keller, J. O. Thermodynamics comparison of fuel cells to Carnot cycle. Int. J. Hydrogen Energy, 2002, 27, Haynes, C. Clarifying reversible efficiency misconceptions of high temperature fuel cells in relation to reversible heat engines. J. Power Sources, 2001, 92, Wark, K., Jr. Advanced Thermodynamics for Engineers, 1995 (McGraw-Hill, New York). 4 Hatsopoulos, G. N. and Keenan J. H. Principles of General Thermodynamics, 1965 (John Wiley & Sons, Inc., New York). 5 Appleby, A. J. and Foulkes, F. R. Fuel Cell Handbook, 1989 (Van Nostrand Reinhold, New York). 6 Sanford, G. and McBride, B. J. Computer program for calculation of complex chemical equilibrium compositions and applications, I. Analysis, NASA Reference Publication 1311, October McBride, B. J. and Sanford, G. Computer program for calculation of complex chemical equilibrium compositions and applications, II. Users manual and Program Description, NASA Reference Publication 1311, June APPENDIX 1 Notation C P G g H h I N P Q R S T W D m n specific heat at constant pressure Gibbs free energy specific Gibbs free energy enthalpy specific enthalpy irreversibility number of moles pressure heat transfer reaction entropy temperature work a finite difference of a property chemical potential stoichiometric reaction coefficient Proc. IMechE Vol. 219 Part A: J. Power and Energy A07404 # IMechE 2005

9 Efficiency of ideal fuel cell and Carnot cycle 253 Subscripts ad adiabatic flame temperature C combustion temperature Carnot Carnot cycle CV control volume elec electrical work FC fuel cell f, 298 final state value at 298 K f, i, 298 final state value of ith species at 298 K gen entropy generation H high temperature In heat input i ith chemical species i, 298 ith chemical species at 298 K L low temperature max maximum max, isothermal maximum at isothermal condition P products of combustion P R difference between products and reactants SL second-law 0 standard temperature and pressure condition APPENDIX 2 In this appendix, we extend the derivation of T C given by Lutz et al. [1], and present its corrected definition. In their analysis, a modified heat engine has been considered with a high-temperature reservoir replaced by a combustion reactor. Fuel and oxidizer enter the reactor, and the products of combustion leave it. This operation must be in the steady state in order to supply heat to the working fluid at a constant temperature consistent with the Carnot cycle. In order to obtain the maximum Carnot efficiency, the reactor temperature should be the highest temperature possible. This can be done by preheating of the fuel and oxidizer to the temperature of combustion products for which the reaction will proceed where the work transfer by the reactor is zero. As the combustion process is irreversible, zero work transfer does not mean the change in Gibbs free energy is also zero. Therefore by writing the energy and entropy balances for the above reactor (Fig. 8), we can derive an expression for T C as follows. By writing the first law analysis, we have: Fig. 8 and by writing the second law analysis, we obtain DS gen X s i (T C, P) Q H ¼ 0 T C (A2) Combining the two equations we obtain the following relation for the reactor work transfer. X W CV ¼ DH R þ T C s i (T C, P) T C DS gen ¼ DH R þ T C DS R T C DS gen (A3) As the work transfer for the reactor is zero, the value of the Gibbs free energy is shown to be nonzero. or W CV ¼ DG R T C DS gen ¼ 0 DG R ¼ T C DS gen (A4) (A5) By rewriting the above equation we can obtain the correct definition of T C as T C ¼ Schematic of a heat engine and its combustion reactor DH R j T C DS R þ DS gen (A6) APPENDIX 3 Q H W CV ¼ DH R ¼ X hi (T C, P) (A1) Referring to Fig. 3, each engine in the series operates at a slightly lower temperature on the A07404 # IMechE 2005 Proc. IMechE Vol. 219 Part A: J. Power and Energy

10 254 H Hassanzadeh and S H Mansouri high-temperature side of engine, and rejects heat at. The work done by each engine is and the overall thermal efficiency of the series of engines is dw ¼ dq H dq L ¼ X P C pi dt X P C pi dt T (A7) h Carnot ¼ 1 Ð Tad PP N ic pi (dt/t) Ð Tad PP N ¼ 1 ic pi dt T C (A11) where dq H ¼ X p C pi dt (A8) By rearranging the above equation, we have the log mean enthalpy temperature difference of the combustion products as they cool down from T ad to. and dq L /dq H ¼ /T (A9) The sum of the work outputs from the series of engines is equal to W rev for the product gases as they cool from T ad to. Hence, the reversible work produced as the product gases are cooled is W rev ¼ ð Tad dw ¼ ð Tad X C pi dt P ð Tad X P C pi dt T (A10) T C ¼ Ð Tad PP C pi dt Ð Tad PP C pi (dt/t) (A12) T C can be written again as equation (A6) by rearranging the above equation using the equations of Appendix 2. For the special case of complete combustion (constant mole fraction of species) and constant specific heat assumptions for the combustion products, we can show T C is the log mean temperature difference of the product gases as they cool down from T ad to, we have T C ¼ (T ad )/ ln (T ad / ) (A13) Proc. IMechE Vol. 219 Part A: J. Power and Energy A07404 # IMechE 2005