Maximum Thermodynamic Electrical Efficiency of Fuel Cell System and Results for Hydrogen,Methane,and Propane Fuels

Rui-chao Mao,Xiao Ru,Zi-jing Lin

Hefei National Laboratory for Physical Sciences at the Microscale and CAS Key Laboratory of Strongly-Coupled Quantum Matter Physics,Department of Physics,University of Science and Technology of China,Hefei 230026,China

I.INTRODUCTION

The electrical power generation in a heat engine involves the processes of converting fuel chemical energy to heat,heat to mechanical energy,and mechanical energy to electricity.In comparison,the chemical energy of fuel is converted directly into electricity through electrochemical reactions in a fuel cell.Naturally,the electrical efficiency of a fuel cell is expected to be higher than that of a heat engine.The promising high electrical energy conversion efficiency is the major reason for the broad interest of the fuel cell technology.However,while the maximum efficiency of a heat engine has been clearly stated by the Carnot theorem,the maximum electrical efficiency of a fuel cell is only vaguely understood.A clear understanding of the maximum electrical efficiency of a fuel cell is of high scientific importance and technology implications.

Numerous efforts have been made towards the understanding of the fuel cell electrical efficiency[1–14].Lutz et al.performed a thermodynamic analysis and concluded that the maximum fuel cell efficiency is equivalent to the Carnot cycle with the high temperature reservoir set at the fuel combustion temperature[3].The work of Lutz et al.is enlightening,but involves the unrealizable condition that the fuel cell operates with no load while the fuel is utilized 100%.Sidwell and Coors developed a numerical model to study the electrical efficiency of a solid oxide fuel cell(SOFC)fueled by hydrocarbon[6].By setting the cell voltage at the Nernst potential of the consumed fuel and oxidant compositions,the dependence of the electrical efficiency on the cell voltage is determined and the maximum efficiency is then computed.Zhu and Kee carried out a general thermodynamic analysis that recovered the well known expression for the fuel cell electrical efficiency[7].That is,the fuel cell electrical efficiency(ηFC)is the product of a fuel reversible efficiency(ηRev),a cell voltage efficiency(ηV),and a fuel utilization ratio(ηfuel),i.e.,ηFC=ηRev·ηV·ηfuel[1,6,7].Both the work of Sidwell and Coors and the work of Zhu and Kee did not consider the energy required by heating the fuel and air streams to the working temperature and by reforming the hydrocarbon fuel.Besides,the oxygen utilization ratio was assumed to be negligibly small.As is well known,ignoring the heat of reformation may result in the use of some unphysical value of ηRev>1 for fuels such as methane and methanol[1,3].The cell efficiency may thus be overestimated.In addition,the maximum cell efficiency cannot be reliably determined without considering the heating requirement of the fuel and air streams or with the assumption of a negligible oxygen utilization[8,11].The fuel and air streams heating requirement has been considered in a recent study[14],but the study is limited to hydrogen fuel and does not consider the waste heat reuse that is bound to underestimate the fuel cell efficiency.

This work aims at establishing a general analytical theory for the maximum thermodynamic electrical efficiency of a fuel cell system using a general fuel and considering the waste heat reuse.Based on the operations of a fuel cell system,a definition of the electrical efficiency of the fuel cell system is introduced.A general theoretical model for the maximum electrical efficiency of the fuel cell system is derived through the applications of thermodynamic principle.The general theory is further developed to yield analytical expressions specific to alkane and hydrogen fuels.Numerical examples about the maximum electrical efficiency are shown for methane,propane,and hydrogen fuels with representative steam contents and operating temperatures.

II.THEORY

A.Problem setting for the electrical efficiency of a fuel cell system

The electrical efficiency of a fuel cell system,ηe,is defined as:

where I is the current output at the fuel cell operating voltage of V,Hfuelis the heating value of the fuel input to the fuel cell system during the operating time period of t.A fuel cell system here means a complete power plant,including the fuel cell stack and all balance-ofplant components.For the study of the maximal electrical efficiency,the parasitic power losses in the balanceof-plant components are kept at the minimum.Nevertheless,the energy requirement of heating the fuel and air streams from T0(the room temperature)to T(the fuel cell working temperature)is considered as it is absolutely necessary in principle for the operation of a fuel cell system.Besides,the fuel cell stack is assumed to operate under the isobaric and isothermal conditions.The isobaric and isothermal conditions can be realized if the fuel and air streams flow very slowly.Moreover,the isobaric condition also eliminates the power requirement of the gas blower,consistent with the need of achieving the maximal electrical efficiency.

FIG.1 Schematic of the thermodynamic processes of a fuel cell system:(a)the operating processes,(b)the energy balance diagram of the operation processes.

The overall thermodynamic processes of such a fuel cell system are schematically illustrated in FIG.1.As shown in FIG.1(a),the fuel and air streams are heated from T0to T.The amounts of heat required are?Hfueland?Hair,respectively.Chemical reactions such as the steam reforming and water gas shift reactions may be involved to convert the fuel into chemical species suitable for the fuel cell electrochemical reactions.The amount of heat required by the overall chemical reactions is?Hcr.The fuel cell outputs an electrical energy of W=IV t and releases an amount of heat,Hcell.The unused fuel exiting from the fuel cell stack can be burned to produce an amount of heat,Hef,at the temperature of T.Cooling the exhausted fuel and air streams from T to T0yields an amount of residual heat,?Hex.Through heat exchanger or other means,Hcell,Hefand?Hexare utilized to provide the required?Hfuel,?Hairand?Hcr.To be general,the amount of heat utilized is expressed as Hcell+Hef+α?Hex.Here α is an effective parameter denoting the capability of the fuel cell system of utilizing the waste heat.For an ideal fuel cell system,α=1.To be practical,α<1 is generally expected.It is noted that,as an effective parameter,α<0 is allowed in principle.For example,α<0 can be used to account for the effect of various kinds of heat losses,e.g.,thermal radiation of the fuel cell stack and/or incomplete combustion of the exiting fuel,on the system efficiency.A properly chosen value of α<0 may also account for the effect of the energy consumed by the balance-of-plant components on the system efficiency.

Denoting the heat released by burning fuel when both reactants and products are at the temperature of T as Qfuel(T).When Qfuel(T)is at the standard state of T=T0,it is equivalent to the conventional definition of the fuel heating value,i.e.,Hfuelin Eq.(1).To be specific,Hfuelis denoted below as Hfuel(T0)that corresponds to Qfuel(T0).Based on the nomenclature,the overall energy changes of the fuel cell operating processes described above can be summarized into FIG.1(b).

B.Maximum thermodynamic electrical efficiency

To be thermally sustainable,the amount of usable heat should exceed the amount of heat necessary for heating fuel and air streams as well as maintaining the required chemical reactions,i.e.,

According to the energy balances shown in FIG.1(b),one has:

The current production can be expressed in term of ηfueland the oxygen utilization ratio,ηO2,as:

where m is the number of electrons released by fully oxidizing a fuel molecule,e.g.,m=2 for H2and m=8 for CH4.For a supplied fuel consisting of H2and CH4as well as other fuel species,m can be calculated by considering the molar fractions of the fuel species.Nfuel(NO2)is the moles of the fuel(O2)input to the fuel cell during the time period of t,and F is the Faraday constant.Notice that no assumption is made here about the relationship between Nfueland NO2,i.e.,Nfueland NO2,or the corresponding ηfueland ηO2,are independent variables in the mathematical derivations.

With Eq.(6),only molar fuel and oxygen equivalent quantities are required for evaluating V of Eq.(5):

where?hfuel(T,T0)and?hair(T,T0)are the molar quantities of?Hfueland?Hair,respectively.Notice that the content of non-fuel species should be considered in evaluating?hfuel(T,T0).For example,for wet fuel with steam fuel ratio of y=NH2O/Nfuel,one has:

Similarly,

Using Eq.(6)and Eq.(7),the electrical efficiency Eq.(1)is found to be:

Apparently,the maximum electrical efficiency is found when“≤”in Eq.(10)is changed to“=”.However,the maximum V,Vmax,for current production is limited by the reversible potential of the fuel and air streams exiting from the fuel cell[6,7].The reversible potential,ENernst,is calculated with the equilibrium contents of the species in the exiting fuel stream.Notice that,according to the thermodynamics principle,ENernstis the same for all oxidation species at the equilibrium.For example,the reversible potentials for both H2and CO oxidations are the same.When H2or CO oxidation and other possible oxidation processes are involved,it suffices to compute either ENernst(H2+0.5O2→H2O)or ENernst(CO+0.5O2→CO2),with the contents of fuel species determined by the equilibrium condition.To be general,the reversible potential of the exiting gas streams is denoted as ENernst(T,ηfuel,ηO2).Using the ideal gas assumption,the maximum electrical efficiency is found for:

To summarize,there are only two steps in the computation of the maximum electrical efficiency of a fuel cell system.First,for a given ηfuel,the value of ηO2is obtained by solving the maximum operating voltage equation:

The rightmost hand side of Eq.(13)is useful and informative by showing explicitly the connection betweenand the thermal characters of the fuel cell system and the fuel,α,hfuel(T0),?hfuel(T,T0),etc.

Notice that,according to Eq.(12),ηO2should approach zero when α approaches 1,with(1-α)/ηO2approaching a finite value:

Consequently,the determination offor α≈1 is simpler than the two-step approach described above.for an ideal fuel cell system involves only one variable,ηfuel,and can be determined directly by:

The maximum electrical efficiency for an ideal fuel cell system fueled by H2is particularly simple and given by:

Eq.(16)is the closest analogue of the Carnot theorem for the fuel cell system.The analogue of the Carnot theorem for an ideal fuel cell system using a general fuel is given by Eq.(15).Unlike the Carnot efficiency that depends on T only(or T/T0,to be precise),ηmaxegiven by Eq.(15)depends also on the fuel type and the H2O content in the fuel supplied to the fuel cells.

Except for H2fuel,the determination of ηmaxefor any α requires the evaluation of ENernst(T,ηfuel,ηO2)based on the fuel equilibrium compositions.Generally,the equilibrium compositions of a fuel mixture can be computed numerically by using some free or commercial codes.

C.Maximum electrical efficiency for hydrogen or alkane fuel

Alkane,CnH2n+2,is a class of widely used fuel.Explicit result on the maximum electrical efficiency of alkane fuel is of practical significance.Attempt is made here to derive an analytical expression for ENernst(T,ηfuel,ηO2).For simplicity,the fuel stream is assumed to consist of H2,H2O,CO,CO2,and CnH2n+2only.The approximation should be accurate in practice due to:(i)coke formation is detrimental to the fuel cell operation and should be minimized by a choice of steam fuel ratio[15];(ii)the equilibrium contents of other fuel species encountered when n>1 are usually quite low and have a limited effect on the contents of H2and H2O.Nevertheless,it is cautioned that the equilibrium theory below is built upon the assumption that can be problematic under undesirable conditions.The results are rigorous for H2fuel and CH4fuel when the coke formation is prevented thermodynamically.

TABLE I Fuel compositions of the initial and final states and the composition changes due to the SR and WGS reactions and the overall electrochemical reaction.

The equilibrium contents of fuel species can be determined by the equilibrium conditions of two chemical reactions:(i)the steam reforming(SR)reaction,CnH2n+2+nH2O=nCO+(2n+1)H2;(ii)the water gas shift(WGS)reaction,CO+H2O=CO2+H2.

Assuming the species mole ratio of an input fuel is CnH2n+2:CO:CO2:H2:H2O=1:c1:c2:c3:c4.For 1 mol CnH2n+2,x1mol CnH2n+2is reformed by the SR reaction,x2mol CO is shifted by the WGS reaction.In addition,a fuel utilization of ηfuelcorresponds to a consumption ofηfuelmol H2for the electricity production.The fuel compositions for the initial and final states and the composition changes due to the SR and WGS reactions and the current production can be summarized into Table I.

The equilibrium SR and WGS reactions yield respectively:

where?GSR(T)and?GWGS(T)are the Gibbs free energy changes of the SR and WGS reactions,respectively.is the partial pressure of reacting species i(i=H2,H2O,CO,CO2or CnH2n+2)and P is the total pressure of the fuel stream.Eq.(17)and Eq.(18)are used to determine the results of the two variables,x1and x2.

With x1and x2obtained by solving Eq.(17)and Eq.(18),ENernst(T,ηfuel,ηO2)can be calculated as:

Inserting Eq.(19)into Eq.(12)and Eq.(13),the maximum electrical efficiency for alkane fuel can be determined.Eq.(19)is also valid for H2fuel that corresponds to n=0,x1=1 and c1=c2=c3=x2=0.

For the special case of c1=c2=c3=0(and c4=y and m=2(3n+1)),i.e.,the input fuel consists of CnH2n+2and H2O only,Eq.(17)?Eq.(19)are reduced to:

FIG.2 and the corresponding ηfueland Vmaxas functions of S/C of methane fuel at different temperature T.(a)for α=1,(b)ηfuelfor α=1,(c)Vmaxfor α=1,(d)for α=0,(e)ηfuelfor α=0,(f)Vmaxfor α=0.

III.NUMERICAL RESULTS

The maximum electrical efficiency theory developed above is applied to fuel cell systems fueled by methane,propane,and hydrogen,assuming both the total pressures of fuel and air are both 1 atm.In most literatures the low heating value(LHV)of a fuel for the fuel cell efficiency was used[3,6,7,13],the numerical results presented here are also based on hfuel(T0)=LHV.Conversion to high heating value(HHV)efficiency only requires a scaling factor of LHV/HHV.(LHV,HHV)in kJ/mol are(802,890),(2043,2219),and(241,285)for CH4,C3H8,and H2,respectively[16].

A.Maximum electrical efficiency of methane fuel

FIG.2 shows the variations of the maximum electrical efficiency,the corresponding optimal fuel utilization and maximal cell operating voltage with the steam carbon ratio of S/C ranging from 0 to 3,of methane fuel for fuel cell systems of α=1 and α=0.The results are obtained based on Eqs.(12),(13),(20),(21),(22)using the thermodynamic property data in Ref.[16].Low S/C methane is prone to coke formation,but the results are shown to indicate the potential benefit of reducing S/C to the fuel cell efficiency.Meanwhile,the results may be meaningful in practice as there are reports that the coke formation may be prevented when S/C is as low as 3%[17–20].Similarly,results for low temperature are presented to illustrate the effect of operating temperature on the fuel cell efficiency.

FIG.2(a)shows thatdecreases slowly and approximately linearly with the increase of S/C.The weak dependence ofon S/C for α=1 is due to that the residual heat of the exiting fuel stream is utilized for heating the input fuel.That is,an increase of S/C does not necessarily mean an increase of the heating requirement.Correspondingly,ηfuelis nearly independent of S/C,as shown in FIG.2(b).Due to the constant ηfueland the adverse effect of the steam content on the Nernst potential,Vmaxdecreases with the increase of S/C(FIG.2(c)).

For the temperatures examined,decreases by about 1%when S/C is increased by 1.The benefit of increasingby decreasing S/C is rather limited for an ideal fuel cell system,which suffers the drawback of high propensity of coke formation.Therefore,a relatively high S/C methane that safely inhibits the coke formation is a natural choice of fuel for fuel systems with α≈1.

Unlike the weak effect of S/C,the influences of T on,ηfuel,and Vmaxare much more substantial,as seen in FIG.2(a)?(c).Vmaxdecreases with the

higher T(Eq.(21)),ηfuelthat maximizes ηfuelENernst(T,ηfuel,ηO2=0)is lower for higher T.The combined effect of the reduced ENernst(T,ηfuel,ηO2=0)and ηfuelresults in a sharp decline ofwith the increase of T.ForS/C=2,is 91.5%,85.8%,77.1%,and 68.2%for T=300,500,700,and 900?C,respectively.

TABLE II ηmaxeof methane and propane fuels for representative operating parameters of the fuel cell system.

Contrary to the common perception concerning methane fuel,reducing the fuel cell operating temperature is highly beneficial for improving the electrical efficiency of an ideal fuel cell system.For a realistic working temperature of 700?900?C,is in the range of 68%?77%.

For α=0,i.e.,the residual heat of the exhausted gases at the working temperature is not utilized,exhibits a strong dependence on S/C,as seen in FIG.2(d).On average,decreases by about 6%with an increase of S/C by 1.Novel anodes designed to prevent the coke formation so that a low S/C methane fuel may be used[16,17]are highly desirable for obtaining high.The decrease ofwith the increase of S/C is primarily associated with the decrease of ηfuel(FIG.2(e))so that the increased need of heating the fuel stream can be met by Hef.Due to the sharp decline of ηfuel,Vmaxshows a slight increase with the increase of S/C(FIG.2(f)).

Expectably,is strongly dependent on T for α=0.Interestingly,for a given S/C,the temperature dependence offor α=0 is only slightly stronger than that for α=1.For example,for S/C=0.03,for α=0 decreases from 71.3%to 62.1%when T is increased from 700?C to 900?C,i.e.,a decrease of 9.2%.In comparison,the corresponding change for α=1 is 8.8%.

The combined effect of S/C and Tonof α=0 is very significant.For a practical operating condition of S/C=2 and T=700?900?C,(α=0)is in the range of 50%?59%.The result of(α=1)?(α=0)≈18%for the operating condition is startling,indicating the extreme importance of the thermal management for the electrical efficiency of the fuel cell system.

FIG.3 shows the effect of α onfor S/C=2 and T=500,700,and 900?C.increases rapidly with α for α≤0.5,but the rate of increase slows down substantially thereafter.This is attributed to the fact that once more than half of?Hexis utilized,the heating requirement can be met without a much loss of ηfuel.For S/C=2 and T=700,900?C,(α=1)?(α=0.5)≈3%is observed.That is,over 80%of the loss inwith α=0 is recovered with α=0.5.Compared to the impossible system design of α=1,α=0.5 appears to be a worthwhile design goal.

FIG.3 for methane of S/C=2 as a function of α for T=773,973,and 1173 K.

While α=1 is too idealistic,α≥0 might be achievable in practice.For S/C=2,T=700?C,and α=0,=59%,ηfuel=72%,and Vmax=0.85 V are found.ηfuel=72%is quite practical.Together with a realistic operating voltage of Vop=0.75 V,ηeis found to be 52%.As Hcellis higher for Vop=0.75 V than that for Vmax=0.85 V,the heating requirement met with Vmaxis automatically met with Vop.It is therefore concluded that an overall electrical efficiency of more than 50%is realistic for a fuel cell system.

B.Maximum electrical efficiency of propane fuel

FIG.4 shows the variations of,ηfuel,and Vmaxwith S/C of propane fuel at different temperature T for α=1 and α=0.Comparison of FIG.2 and FIG.4 shows that the variation patterns for methane and propane fuels are basically the same.The discussion made above for methane applies equally well to propane.To see their subtle differences more clearly,representative data offor methane and propane fuels are displayed in Table II.

As can be seen in Table II,of methane is larger than that of propane in most cases.For S/C=2 and 700?C≤T≤900?C,of CH4is higher than that of C3H8by 0.9%?1.6%.However,of CH4with low S/C shows a stronger α dependence than that of C3H8counterpart.For S/C=0.03,α=0 and 700?C≤T≤900?C,of CH4is lower than that of C3H8by 1.2%?1.5%.As S/C≈2 is more practical,however,it is roughly correct to say thatof CH4is about 1%higher than that of C3H8.

FIG.4 and the corresponding ηfueland Vmaxas functions of S/C of propane fuel at different temperature T.(a)for α=1,(b)ηfuelfor α=1,(c)Vmaxfor α=1,(d)for α=0,(e)ηfuelfor α=0,(f)Vmaxfor α=0.

FIG.5 for propane fuel of S/C=2 as a function of α for T=773,973,and 1173 K.

FIG.5 shows the effect of α onof propane with S/C=2 and T=500,700 and 900?C.As can be seen by comparing FIG.5 and FIG.3,the α dependences ofof propane and methane are basically the same and over 80%of the loss inwith α=0 is recovered with α=0.5.Similarly,it is found that an overall electrical efficiency of over 50%is expectable for a propane fueled fuel cell system.

C.Maximum electrical efficiency of hydrogen fuel

Hydrogen is the simplest fuel suitable for all fuel cell operating temperatures.FIG.6 shows the variations ofand the corresponding ηfuelas functions of H2molar fraction β,β=NH2/(NH2+NH2O)for α=1 and α=0 and 100,300,500,700,and 900?C.

Considering T=100?C,FIG.6(a)and(c)show that the effect of the steam content onis very limited.For α=1,for β=97%is only 0.6%higher than that for β=60%.For α=0,the corresponding difference inincreases to 1.4%,still a relatively small amount.The difference between(α=1)and(α=0)is a negligible amount of 0.1%for β=97%.For T<100?C,as encountered in PEM fuel cell operations,neither the water content nor the thermal management is important factor affecting.

Expectably,the effect of β onincreases with the temperature T.For α=0 and T=900?C,(β=97%)?(β=60%)=8%is observed.However,the result is likely purely theoretical as only high β(～97%)H2fuel may be used when T?100?C[1,21].For β=97%,the difference between(α=1)and(α=0)increases with the temperature T,but remains relatively small.For example,(α=1)?(α=0)is 1.6%and 2.3%for T=700 and T=900?C,respectively.Overall,the difference between(α=1)and(α=0)is limited for H2fuel operating at all temperatures.

Although the effect of α onis moderate for all temperatures,the effect of T onis much morelinearly with the increase of T.For an increase of T of 100 K,decreases by 3.5%for α=1 and 3.7%for α=0.

FIG.6 and the corresponding ηfuelas functions of the hydrogen molar fraction β in hydrogen fuel at different temperature T.(a)for α=1,(b)ηfuelfor α=1,(c)for α=0,(d)ηfuelfor α=0.

Considering 700?C≤T≤900?C,for H2fuel with β=97%and α=1 is in the range of 58%?64%.In comparison,for CH4fuel with S/C=2 and α=1 is in the range of 68%?77%.That is,(α=1,H2,β=97%)is less than(α=1,CH4,S/C=2).However,(α=0,H2,β=97%)is in the range of 55%?63%and is more than(α=1,CH4,S/C=2)that is in the range of 50%?59%.

Notice that α=0 means that the heat value of Hcell+Hefhas been utilized,though?Hexis wasted.That is,α=0 corresponds to a reasonable level of thermal management in a fuel cell system.Therefore,except for a fuel cell system with a high quality thermal design,H2fuel is expected to have a higherthan that of CH4fuel for a system operated at 700?C≤T≤900?C.

IV.CONCLUSION

Explicit expressions for the maximum electrical efficiency of a fuel cell system are derived by applying the fundamental thermodynamics principle.The result for an ideal fuel cell system is analogous to the Carnot theorem for an ideal heat engine.Unlike the Carnot efficiency,is fuel specific.

For an ideal fuel cell system fueled by hydrogen,is given by a single equation.For a realistic fuel cell system characterized by an effective heat exchange parameter,for hydrogen fuel is given by solving a set of two equations.Chemical equilibrium calculations are required to determinefor a general fuel.Other than the equilibrium calculations,for a general fuel is similar to the case of hydrogen fuel and given by one or two equations for an ideal or non-ideal fuel cell system,respectively.Chemical equilibrium is considered for a class of widely available fuel sources,alkane.Analytical solutions of the chemical equilibrium are obtained by assuming the fuel stream only consisting of CnH2n+2,H2,H2O,CO,and CO2

The theoretical models can be used to conveniently analyze the impacts of various factors on the maximum electrical efficiency.Numerical results are presented for methane,propane,and hydrogen fuels.decreases with the increase of T for all fuels examined and for both α=1 and α=0.For an ideal system,the steam content does not affectappreciably.For α=0,however,decreases substantially with the increase of S/C of the alkane fuel.The thermal management is critically important for the obtainable electrical efficiency of a fuel cell system fueled by methane or propane.Nevertheless,most of the efficiency loss with α=0 can be recovered with α=0.5 and a system design with α≈0.5 is therefore recommended.For H2fuel,however,the thermal management is less critical and α=0 is quite acceptable.

For 700?C≤T≤900?C and a representative steam content,S/C=2 for CH4and C3H8and β=97%for H2,for CH4,C3H8,and H2with α=1(α=0)are in the range of 68%?77%(50%?59%),66%?76%(49%?57%)and 58%?64%(55%?63%),respectively.That is,for CH4and C3H8are higher than that of H2for an ideal system.For a realistic system of α=0,however,hydrogen fuel can have a higher efficiency than that of methane and propane for the operating temperature of SOFCs.

V.ACKNOWLEDGMENTS

This work was supported by the National Natural Science Foundation of China(No.11574284 and No.11774324),the National Basic Research Program of China(No.2012CB215405)and Collaborative Innovation Center of Suzhou Nano Science and Technology.And Prof.Qing-quan Lei is thanked for his stimulation of this work.

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