Energy Conversion and Management 86 (2014) 1102–1109

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Energy Conversion and Management

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Theoretical basis and performance optimization analysis of a solid oxidefuel cell–gas turbine hybrid system with fuel reforming

http://dx.doi.org/10.1016/j.enconman.2014.06.0680196-8904/� 2014 Elsevier Ltd. All rights reserved.

⇑ Corresponding author. Tel.: +86 592 2180922; fax: +86 592 2189426.E-mail address: jcchen@xmu.edu.cn (J. Chen).

Xiuqin Zhang a, Yuan Wang b, Tie Liu b, Jincan Chen b,⇑a Department of Physics, Jimei University, Xiamen 361021, People’s Republic of Chinab Department of Physics, Xiamen University, Xiamen 361005, People’s Republic of China

a r t i c l e i n f o a b s t r a c t

Article history:Received 16 March 2014Accepted 20 June 2014

Keywords:Hybrid systemSolid oxide fuel cellGas turbineFuel reformingOptimization criteria

A novel model of the solid oxide fuel cell–gas turbine hybrid system with fuel reforming is established,where the residual fuel from the fuel cell is further burned in a combustor and the solid oxide fuel cell(SOFC) and combustor act as the high-temperature reservoirs of the gas turbine (GT). The irreversibilitiesexisting in real systems including the overpotentials and heat leakage in the SOFC, the finite-rate heattransfer between the working substance of the gas turbine and the reservoirs, and the irreversible com-pression, expansion, and regeneration processes in the gas turbine are considered. By using the theoriesof electrochemistry and non-equilibrium thermodynamics, expressions for the power output andefficiency of the hybrid system are derived and the advantages of the hybrid system compared withthe pure SOFC are represented. The optimally operating regions of some of the important parametersincluding the power output and efficiency of the hybrid system and the rate of the fuel flowing intothe SOFC are determined. The rate of the air flowing into the cell at the optimal efficiency of the hybridsystem is also derived. The results obtained here may provide some theoretical bases and optimizationcriterion for the design and operation of practical syngas SOFC-based hybrid systems.

� 2014 Elsevier Ltd. All rights reserved.

1. Introduction

The solid oxide fuel cell (SOFC) can use hydrocarbons as fuelwith internal fuel reforming at the anode [1–3] due to its relativelyhigh operating temperature. Compared with other fuel cells [4],one of the advantages of the SOFC is the ability to handle varioushydrocarbons such as biogas, methane, methanol, ethanol, syngas,and propane [5–10]. The syngas as one of the fuels has attractedmuch attention because of its renewable status, widely availableresource, relatively low feedstock cost, reduced environmentalimpact, etc. [9,11–14]. The syngas is produced from the biomasswhich comes from the living biological organisms and it is consid-ered to be one of the renewable energy sources [12,15].

The effects of the materials of electrodes and electrolyte on thefuel reforming in the SOFC have been investigated [16–19], themathematical models to analyze the performance of the SOFC havebeen developed [20–22], and the thermodynamic analyses to eval-uate the key parameters have been carried out [23,24]. In order toefficiently utilize the residual fuel from the anode of the SOFC, acombustor may be introduced [25–28]. How to utilize the wasteheat from the SOFC and the heat from the combustor and improve

the energy conversion efficiency of the SOFC is a significant prob-lem. Here, a new model of indirect combined SOFC and gas turbine(GT) is established where not only the waste heat from the cell butalso the combustion heat of the residual fuel is absorbed by theworking substance in the GT. The optimal rates of fuel and air flow-ing into the SOFC are explored for the maximum efficiency of thehybrid system based on the conservation of energy. The generalperformance characteristics of the SOFC, and hybrid system aredescribed, respectively. The advantages of the hybrid system arerepresented by numerical results.

The concrete contents are organized as follows. In Section 2, amodel of the SOFC–GT hybrid system with fuel reforming is estab-lished, and the power output and efficiency of the hybrid systemare derived. In Section 3, some general performance characteristicsof the hybrid system are obtained and parametric optimal criteriaare given. Finally, some important conclusions are drawn.

2. A model of the SOFC–GT hybrid system with fuel reforming

The hybrid system based on hydrocarbon is primarily composedof a SOFC, a GT and a combustor, as shown in Fig. 1, where the un-reacted fuel from the SOFC is burnt in the combustor, the combus-tion products released from the combustor are used to preheat the

T0

TC

q2Air

6

HEX4

Air

HEX2

SOFCHEX1

C T

75

43

2

1

AB

HEX3

Fuel

q1

T

Fig. 1. A schematic diagram of the SOFC–GT hybrid system.

X. Zhang et al. / Energy Conversion and Management 86 (2014) 1102–1109 1103

fuel and the air in heat exchanger 1, the high-temperature heatsreleased from the fuel cell and combustor are, respectively, utilizedas the heat input of the GT through heat exchangers 2 and 3, q1 andq2 are the rates of the heat flow from the SOFC at temperature Tand from the combustor at temperature TC to the working substanceof the GT, respectively, and heat exchanger 4 is a regenerator in thegas turbine. The further analyses will be carried out on the basis ofthe following assumptions: (1) CH4 and CO in the fuel are reactedfully at the anode of the SOFC; (2) the fuel is burned completelyin the combustor; (3) the temperature and pressure are uniformand constant in the fuel cell; (4) the working substance of the GTis regard to be the ideal gas. In these assumptions, (1) and (2) canbe approximated very well by engineering efforts and (3) and (4)are some idealized assumptions often used in the theoreticalanalysis, which can availably capture the physical properties ofthe investigating problems. Below, the performance of the maincomponents will be, respectively, analyzed and then the efficiencyand power output of the hybrid system will be derived.

2.1. The SOFC with fuel reforming

In heat exchanger 1 in Fig. 1, the fuel and air are, respectively,preheated from the environment temperature T0 to the operatingtemperature T of the SOFC by using the high-temperature gasesfrom the combustor. According to the product of gasification ofbiomass [29–32], the fuel entering the anode of the SOFC mayconsist of H2, CH4, CO, CO2, H2O, and N2. The chemical reactionsin the cell can be summarized as

CH4 þH2O! COþ 3H2; ð1Þ

COþH2O! CO2 þH2; ð2Þ

and

H2 þ12

O2 ! H2O; ð3Þ

where Eqs. (1) and (2) are, respectively, the steam reforming andwater gas shift reactions which occur at the anode of the SOFC[33,34], and Eq. (3) is the overall electrochemical reaction in thecell.

2.1.1. The power output and efficiency of the SOFCAccording to Faraday’s law, the molar rate of hydrogen con-

sumed in Eq. (3) is iAc/(neF), where i is the current density of thecell, Ac is the surface area of mean active plates, ne is the numberof electrons transferred in reaction, and F is Faraday’s constant.

When studying the performance of a SOFC, one has to considerthe electrochemical irreversibilities which mainly result from theactivation overpotential (Vact), ohm overpotential (Vohm), andconcentration overpotential (Vcon). Considering the influence ofirreversible losses mentioned above on the performance of theSOFC, one can derive the power output and efficiency of the SOFCas [35–37].

Pe ¼ iAcðE� Vact � Vohm � VconÞ

¼ iAc

neF�DgðTÞ þ RT ln

pH2p1=2

O2

pH2O

!� 2RTsinh�1 i

2i0;a

� �"

� 2RTsinh�1 i2i0;c

� �� ineF

Xj

djaj expbj

T

� �þ RT ln 1� i

iL;H2

� �

þ RT2

ln 1� iiL;O2

� ��ð4Þ

and

ge ¼Pe

�iAcDhðTÞ=ðneFUaÞ

¼ UaRT�DhðTÞ

�DgðTÞRT

þ lnpH2

p1=2O2

pH2O

!� 2sinh�1 i

2i0;a

� �"

� 2sinh�1 i2i0;c

� �þ ln 1� i

iL;H2

� �þ 1

2ln 1� i

iL;O2

� �

� ineFRT

Xj

djaj expbj

T

� �#ð5Þ

respectively, where E is the open-circuit voltage of the SOFC[38–40], Dg(T) is the molar Gibbs free energy change of the electro-chemical reaction at temperature T and standard pressure, R is theuniversal gas constant, and pk(k = H2, H2O, O2) are the partialpressures of different components at electrodes, Dh(T) is the molarenthalpy change of the electrochemical reaction at temperature Tand standard pressure, i0,a and i0,c are, respectively, the exchangecurrent densities at the anode and cathode, iL;H2 and iL;O2 are, respec-tively, the limiting current densities at which the hydrogen and theoxygen are used up at a rate equal to their maximum supply speedsand assumed to be constant [41] because the expression of the con-centration overpotential adopted here is obtained by simplifyingthe Fick’s model, dj is the current flowing length in layer j (j = anode,cathode, electrolyte, and interconnectors), aj and bj are constants,and Ua is the hydrogen utilization factor of SOFC and defined asthe ratio of the rate of hydrogen consumed in the electrochemicalreaction to that supplied from the anode of the SOFC. Dh(T) andDg(T) can be expressed as [42,43]

DhðTÞ ¼ Dh0f ðH2OÞ þ Lm þ

Z T

T0

CH2O;mds

� Dh0f ðH2Þ þ

Z T

T0

CH2 ;mds� �

� 12

Dh0f ðO2Þ þ

Z T

T0

CO2 ;mds� �

ð6Þ

and

DgðTÞ¼DhðTÞ�T Ds0f ðH2OÞþ Lm

373þZ T

T0

CH2O;m

sds

�

� Ds0f ðH2Þþ

Z T

T0

CH2 ;m

sds

� ��1

2Ds0

f ðO2ÞþZ T

T0

CO2 ;m

sds

� ��; ð7Þ

respectively, where Dh0f ðkÞ and Ds0

f ðkÞ are, respectively, the molarenthalpy and entropy changes of component k (k = H2, H2O, O2) atstandard pressure and temperature T0 = 298 K, Ck,m is the corre-sponding molar heat capacity, and Lm is the molar latent heat of H2O.

2.1.2. The waste heat of the SOFCAccording to Eqs. (1)–(3), one can obtain

nf Ua xH2 þ xCO þ 4xCH4

� ¼ iAc

neF; ð8Þ

where nf is the molar flow rate of the fuel, xk (k = H2, CO, CH4, . . .) arethe molar fractions of different gases in the fuel stream. The fuel

1104 X. Zhang et al. / Energy Conversion and Management 86 (2014) 1102–1109

reforming reaction is endothermic [33,34], energy should be sup-plied for the steam reforming reaction to proceed, and the energycan be expressed as

D _Hr ¼ nf ðxCH4 þ xCOÞ Dh0f ðCO2Þ þ

Z T

T0

CCO2 ;mds� ��

þ 4xCH4 þ xCOÞ Dh0f ðH2Þ þ

Z T

T0

CH2 ;mds� ��

� xCH4 Dh0f ðCH4Þ þ

Z T

T0

CCH4 ;mds� �

� 2xCH4 þ xCOÞ Dh0f ðH2OÞ þ Lm þ

Z T

T0

CH2O;mds� ��

� xCO Dh0f ðCOÞ þ

Z T

T0

CCO;mds� ��

ð9Þ

by using Eqs. (1) and (2). For a SOFC, there inevitably exists heatleak, which is directly released to the environment. It is oftenassumed that the heat transfer between the fuel cell and the envi-ronment obeys the Newtonian law, and consequently, the heat leakmay be expressed as [44,45]

qL ¼ KlAlðT � T0Þ; ð10Þ

where Kl is the convective and/or conductive heat leak coefficient, Al

is the effective heat transfer area. It should be pointed out that thevalues of KlAl will affect the performance of the hybrid system. Forthe sake of convenience, the ratio of KlAl to other parameters, ratherthan the concrete value of KlAl, is adopted in the following calcula-tion. According to the first law of thermodynamics, one can obtainthe rate of the waste heat flowing from the SOFC to the GT as

q1 ¼ �iAcDhðTÞ

neF� Pe � D _Hr � qL: ð11Þ

2.2. The combustor

The residual hydrogen and air leaving the anode and cathode ofthe SOFC are mixed and burned in the combustor, where thehydrogen is further oxidized by oxygen. According to the energyconservation, one can obtain

_HinðTÞ ¼ _HoutðTCÞ þ q2; ð12Þ

where _HinðTÞ is the enthalpy of the gases at temperature T enteringthe combustor, _HoutðTCÞ is the enthalpy of the combustion productat temperature TC leaving the combustor, q2 ¼ kqLHV ðH2Þ½iAcð1� UaÞ=ðneFUaÞ� is the rate of the heat flow from the combustor[46] to the working substance of the GT, qLHV(H2) is the low heatingvalue of the hydrogen per molar, and k is a constant. According tomass conservation, _HinðTÞ and _HoutðTCÞ can be, respectively,expressed as

_HinðTÞ ¼ nf xCH4 þ xCO þ xCO2

� Dh0

f ðCO2Þ þZ T

T0

CCO2 ;mds� �

þ nf ðxH2O � 2xCH4 � xCOÞþ iAc=ðneFÞ� Dh0

f ðH2OÞ þ Lm þZ T

T0

CH2O;mds� �

þ iAcð1� UaÞ=ðneFUaÞ Dh0f ðH2Þ þ

Z T

T0

CH2 ;mds� �

þ ðnf xN2 þ nax0N2Þ Dh0

f ðN2Þ þZ T

T0

CN2 ;mds� �

þ iAcð1� UcÞ=ð2neFUcÞ Dh0f ðO2Þ þ

Z T

T0

CO2 ;mds� �

ð13Þ

and

_HoutðTCÞ ¼ nf ðxCH4 þ xCOþ xCO2 Þ Dh0f ðCO2Þþ

Z TC

T0

CCO2 ;mds� �

þ nf ðxH2O�2xCH4 � xCOÞþ iAc=ðneFUaÞ� Dh0

f ðH2OÞþ LmþZ TC

T0

CH2O;mds� �

þðnf xN2 þnax0N2Þ Dh0

f ðN2ÞþZ TC

T0

CN2 ;mds� �

þ iAcðUa�UcÞ=ð2neFUcUaÞ Dh0f ðO2Þþ

Z TC

T0

CO2 ;mds� �

; ð14Þ

where na ¼ iAc=ð2neFUcx0O2Þ is the rate of air flowing into the SOFC, Uc

is the oxygen utilization factor at the cathode of the SOFC, and x0O2and

x0N2are, respectively, the molar fractions of O2 and N2 in the air stream.

Eq. (14) clearly shows that the residual hydrogen can be completelyburned in the combustor under the condition of Ua P Uc . If not, theadditional air should be provided. Obviously, the combustiontemperature TC can be derived by using Eqs. (12)–(14).

2.3. The heat exchanger 1

To guarantee the SOFC working at steady, the temperature ofthe fuel and air entering the SOFC should reach the operating tem-perature of the SOFC. The heat absorbed by the fuel and air in heatexchanger 1 shown in Fig. 1 will be

qa ¼ nf xH2

Z T

T0

CH2 ;mdsþ xCH4

Z T

T0

CCH4 ;mdsþ xCO

Z T

T0

CCO;mds�

þ xCO2

Z T

T0

CCO2 ;mdsþ xH2O Lm þZ T

T0

CH2Ods� ��

þ ðnf xN2 þ nax0N2ÞZ T

T0

CN2 ;mdsþ nax0O2

Z T

T0

CO2 ;mds ð15Þ

and the heat provided by the combustion product may be expressedas

qp ¼ nf xCH4 þ xCO þ xCO2

� Z TC

T0

CCO2 ;mds

þ nf ðxH2O � 2xCH4 � xCOÞ þ iAc=ðneFUaÞ �

ðLm þZ TC

T0

CH2O;mdsÞ

þ ðnf xN2 þ nax0N2ÞZ TC

T0

CN2 ;mdsþ iAcðUa � UcÞ=ð2neFUcUaÞ

�Z TC

T0

CO2 ;mds: ð16Þ

It can be proved from Eqs. (15) and (16) that qp P qa if TC P T. Itwill ensure that the temperatures of the fuel and air entering theanode and cathode of the SOFC can attain the operating tempera-ture of the cell.

2.4. The gas turbine with two high-temperature reservoirs

The entropy-temperature diagram of the GT with two high-tem-perature reservoirs is shown in Fig. 2, where q1 and q2 are the ratesof the heat flow from the SOFC at temperature T and from the com-bustor at temperature TC to the working substance of the GT duringtwo isobaric processes 3-4 and 4-5, respectively, q0 is the rate of theheat flow from the substance of the GT to the environment duringprocess 7-1, 1-2S and 5-6S are two reversible adiabatic processes,1-2 and 5-6 are two irreversible adiabatic processes, qr is the rateof the heat flow from the isobaric regeneration process 6-7 to 2-3,Tj (j = 1, 2, 3, 4, 5, 6, 7) are the temperatures of the workingsubstance at state points j, and T1 = T0 .

TC

q2

7

T

6

5

6S2

Tem

pera

ture

Entropy

T01

2S

3

q0

q1 4

qR

Fig. 2. Temperature-entropy diagram of the GT model consisting of two isobaricand two adiabatic processes.

X. Zhang et al. / Energy Conversion and Management 86 (2014) 1102–1109 1105

Assuming that the heat transfers in the GT obey the Newtonianlaw [47–51], one can obtain the rates of the heat flow in the GT as[52,53].

q1 ¼K1A1ðT4 � T3Þ

ln½ðT � T3Þ=ðT � T4Þ�¼ _mcpðT4 � T3Þ; ð17Þ

q2 ¼K2A2ðT5 � T4Þ

ln½ðTC � T4Þ=ðTC � T5Þ�¼ _mcpðT5 � T4Þ; ð18Þ

q0 ¼ _mcpðT7 � T0Þ; ð19Þ

and

qr ¼ KrArðT6 � T3Þ ¼ KrArðT7 � T2Þ ¼ _mcpðT3 � T2Þ¼ _mcpðT6 � T7Þ; ð20Þ

respectively, where K1, K2, and Kr are, respectively, the heat transfercoefficients in processes 3-4, 4-5, and 6-7, A1, A2, and Ar are the cor-responding heat transfer areas, _m is the mass flowing rate, and cp isthe heat capacity of the working substance at constant pressure. Byusing Eqs. (17)–(20), the expressions of q1 and q2 can be, respec-tively, rewritten as

q1 ¼K1AðT4 � T3Þ

ln T�T3T�T4þ K1

Kr

T6�T7T6�T3

þ K1K2

ln TC�T4TC�T5

ð21Þ

and

q2 ¼K1AðT5 � T4Þ

ln T�T3T�T4þ K1

Kr

T6�T7T6�T3

þ K1K2

ln TC�T4TC�T5

; ð22Þ

where A = A1 + A2 + Ar is the total heat transfer area of the GT. Eqs.(21) and (22) show that in the following calculation, it is unneces-sary to choose the concrete values of the heat transfer areas A1, A2,and Ar, because the performance of the GT is directly dependent onA. From Eqs. (17)–(19), one can obtain the efficiency of the GT as

gg ¼ 1� q0

q1 þ q2¼ 1� T7 � T0

T5 � T3: ð23Þ

If further introducing the compression, expansion, and regenerationefficiencies [54–56] ec ¼ T2S�T0

T2�T0, ee ¼ T5�T6

T5�T6S, and er ¼ T3�T2

T6�T2of the GT

and using the relation T2ST0¼ T5

T6S¼ r1�1=c

p and Eq. (20), one can obtain

the efficiency of the GT

gg ¼ 1� T5Yð1� erÞ þ T0ðXer � 1ÞT5ð1� YerÞ þ T0Xðer � 1Þ ð24Þ

and the following constraint conditions

q1=ðK1AÞ ¼ T4 � T5Yer þ T0Xðer � 1Þln T�T5YerþT0Xðer�1Þ

T�T4þ K1

Kr

er1�erþ K1

K2ln TC�T4

TC�T5

; ð25Þ

and

q2=ðK1AÞ ¼ T5 � T4

ln T�T5YerþT0Xðer�1ÞT�T4

þ K1Kr

er1�erþ K1

K2ln TC�T4

TC�T5

; ð26Þ

where rp is the pressure ratio of two isobaric processes 2S-5 and6-1, c is the ratio of the specific heats of the working substance,X ¼ 1þ ðr1�1=c

p � 1Þ=ec , and Y ¼ 1þ ðr1=c�1p � 1Þee.

In order to derive the maximum efficiency of the GT under theconstraint conditions mentioned above, the Lagrangian functionmay be introduced as

L ¼ gg þ aq1=ðK1AÞ þ bq2=ðK1AÞ; ð27Þ

where a and b are the Lagrangian multipliers. Using the extremeconditions

@L@rp

� T4 ;T5

¼ 0

@L@T4

� rp ;T5

¼ 0

@L@T5

� rp ;T4

¼ 0

8>>>>><>>>>>:

ð28Þ

and Eqs. (25) and (26), one can obtain the optimum values rp,g, T4,g,and T5,g of rp, T4, and T5, respectively. Substituting rp,g, T4,g, and T5,ginto Eq. (24), one can obtain the maximum efficiency gg of the GTunder given values of q1/(K1A), q2/(K1A), c, ec, ee, er, K1/K2, andK1/Kr. By combining Eqs. (8)–(12), q1/(K1A) and q2/(K1A) can be,respectively, expressed as

q1=ðK1AÞ ¼ ia1 1� ge þD _Hr

nf

1UaðxH2 þ xCO þ 4xCH4 ÞDhðTÞ

" #

� a2ðT � T0Þ ð29Þ

and

q2=ðK1AÞ ¼ ia1ð1� UaÞkqLHV ðH2Þ�UaDhðTÞ ; ð30Þ

where a1 ¼ �AcDhðTÞneFK1A and a2 ¼ KlAl

K1A .

2.5. The efficiency and power output of the hybrid system

Using Eqs. (8)–(12), one can obtain the power output and effi-ciency of the SOFC–GT hybrid system as

P ¼ Pe þ ðq1 þ q2Þgg

¼ Pe þ �iAcDhðTÞ=ðneFÞ � D _Hr þ a2AcDhðTÞðT � T0Þ=ða1neFÞh

þ iAcð1� UaÞkqLHV ðH2Þ=ðneFUaÞ�gg ð31Þ

and

g¼ P�iAcDhðTÞ=ðneFUaÞ

¼geþgg UaþneFUaD _Hr

iAcDhðTÞ �a2ðT�T0ÞUa

ia1þð1�UaÞkqLHV ðH2Þ

�DhðTÞ

" #; ð32Þ

respectively. Eqs. (31) and (32) clearly show that the performanceof the hybrid system is directly affected by many parameters inthe SOFC and GT.

3. Performance optimization analysis and parametric design

It is considered here that some steam is mixed with the syngas[35] from the coal to avoid carbon deposition and the total amountof the steam in the syngas is expected twice that needed for thesteam reforming and water gas shift reactions. The expression ofactivation overpotential is derived from Butler–Volmer equation

Table 2Parameters used in the hybrid system.

Parameter Value

Mole fraction of H2 in the fuel stream, xH2 0.13Mole fraction of CH4 in the fuel stream, xCH4 0.01Mole fraction of CO in the fuel stream, xCO 0.16Mole fraction of CO2 in the fuel stream, xCO2 0.05Mole fraction of H2O in the fuel stream, xH2 O 0.36Mole fraction of N2 in the fuel stream, xN2 0.29Mole fraction of O2 in the air stream, x0O2

0.21

Mole fraction of N2 in the air stream, x0N20.79

Operating pressure, p0 (atm) 1Temperature of SOFC, T (K) 1173Temperature of environment, T0 (K) 298Number of electrons, ne 2Faraday constant, F (C mol�1) 96485Universal gas constant, R (J mol�1 K�1) 8.314Anode exchange current density, i0,a (A m�2) 6610Cathode exchange current density, i0,c (A m�2) 3550Anode: da (cm); aa (X cm); ba (K) 0.01; 0.00298; �1392Cathode: dc (cm); ac (X cm); bc (K) 0.22; 0.00811; 600Electrolyte: de (cm); ae (X cm); be (K) 0.004; 0.00294; 10350Interconnector: di (cm); ai (X cm); bi (K) 0.0085; 0.12; 4690Limiting current density of H2, iL;H2 (A m�2) 2.99 � 104

Limiting current density of O2, iL;O2 (A m�2) 2.16 � 104

Low heating value of H2, qLHV(H2) (kJ mol�1) 241.9Regeneration efficiency of the heat exchanger 4, er 0.85Compression efficiency of the gas turbine, ec 0.85Expansion efficiency of the gas turbine, ee 0.85Constant, K1/K2 1Constant, K1/Kr 1Specific heat ratio, c 1.4

0.2

0.3

0.4

0.5

0.6 λ=1.1λ=0.9λ=0.7

Ua=0.8

n* a /m

ols- 1

m-2

n*f /mols-1m-2

0.0 0.1 0.2 0.3 0.4

Fig. 3. The curves of n�a at the optimum efficiency of the hybrid system versus n�f forthe parameters a1 = 0.0025 and a2 = 0.001.

1106 X. Zhang et al. / Energy Conversion and Management 86 (2014) 1102–1109

with exchange current density [36] and the expression of ohmoverpotential of the tubular SOFC [37] is adopted. Thus, numericalcalculations can be carried out based on the parameters summa-rized in Tables 1 and 2 which can be derived from Refs.[35–37,57]. These parameters are kept constant unless otherwisementioned specifically.

On the basis of these given parameters mentioned above, byusing Eqs. (4)–(9), (12)–(14), and (24)–(32) and the constraintrelation, nf ðxH2 þ 4xCH4 þ xCOÞ 6 2nax0O2

, between nf and na obtainedfrom Eqs. (1)–(3), the curves of the rate of the air flowing into theSOFC at the optimum efficiency of the hybrid system varying withthe rate of the fuel flowing into the SOFC can be generated, asshown in Fig. 3, where n�f ¼ nf =Ac and n�a ¼ na=Ac . The curves inFig. 3 show that the rate of the air flow is not a monotonic functionof the rate of the fuel flow. When the rate of the fuel flow is aboutequal to 0.2 mol s�1 m�2, the rate of the air flow attains its mini-mum. When n�f < 0:2mol s�1m�2, the rate of the air flow is speededup to increase the enthalpy input and the combustion temperatureof the burner according to Eq. (12). The smaller the rate of the fuelflow is, the larger the rate of the air flow, so that the optimal effi-ciency of the hybrid system can be achieved. One can obtain thecurves of the optimum efficiency and power output at the optimumefficiency of the hybrid system varying with the rate of the fuelflowing into the SOFC, as shown in Figs. 4 and 5, respectively, whereP* = P/Ac and P�e ¼ Pe=Ac . Figs. 4 and 5 show that the performance ofthe SOFC–GT hybrid system is much better than that of the singleSOFC. The maximum efficiency of the hybrid system is about 1.5times of that of the single SOFC. The maximum power density ofthe hybrid system is about 3 times of that of the single SOFC andis just between the experiment result, 1.24, and the simulationresults, 4.64, obtained from Refs. [58–61]. It is seen from Fig. 4 thatthere is a maximum efficiency of the hybrid system with the corre-sponding n�fg and the corresponding n�a can be obtained from Fig. 3.In the region of n�f < n�fg, although the efficiency of the SOFCdecreases with the increase of the rate of the fuel flow, the effi-ciency of the GT driven by the waste heat from the fuel cellincreases quickly, and consequently, it results in the increase ofthe optimum efficiency of the hybrid system. Figs. 4 and 5 also showthat both the power output and the efficiency of the hybrid systemdecrease with the decrease of the rate of the fuel flow in the regionof n�f < n�fg. Thus, the optimal region of the rate of the fuel flowinginto the SOFC may not be arbitrarily chosen and should be situatedin the range of n�f P n�fg. However, in the region of n�f > n�fg, thepower output of the hybrid system increases but the efficiency ofthe hybrid system decreases as the rate of the fuel flowing intothe SOFC is increased, and consequently, how to further give con-sideration to both the power output and efficiency in the regionof n�f P n�fg becomes an important problem for the practical opti-mum design and operation of SOFC hybrid systems.

For this purpose, the product of the efficiency and power den-sity of the hybrid system, i.e., Z = gP*, may be introduced as anew objective function and generate the n�f � Z curves, as shownin Fig. 6, where n�fZ is the rate of the fuel flowing into SOFC at the

Table 1Thermodynamic parameters for the chemical components, where (g) and (l) refer to gas a

Component k Dh0f ðkÞ (J mol�1) Ds0

f ðkÞ (J mol�1 K�1)

N2(g) 0 191.598O2(g) 0 205.138CH4(g) �75,000 186.300CO2(g) �393,800 213.760CO(g) �1,105,00 198.016H2 (g) 0 130.695H2O (g) – 188.823H2O (l) �285,800 69.940

maximum value Zmax of Z and the corresponding n�a can beobtained from Fig. 3. In the region of n�f > n�fZ , both the efficiencyand Z will decrease with the increase of the rate of the fuel flowinginto SOFC. In the region of n�f < n�fg, both the efficiency and Z willdecrease with the decrease of the rate of the fuel flowing into SOFC.n�fg and n�fZ are the important parameters of the hybrid system.Substituting n�fg and n�fZ into Eq. (31), one can obtain the powerdensity P�g and P�Z corresponding to the maximum efficiency and

nd liquid phases, respectively.

Lv (J mol�1) Molar heat capacity(Ck,m) (J mol�1 K�1)

- 29.12- 25.8911 + 0.0129874T � 0.0000038644T2

– 14.1555 + 0.0755466T � 0.0000180032T2

– 26.0167 + 0.0435259T � 0.0000148422T2

– 26.8742 + 0.006971T � 0.0000008206T2

– 29.0856–0.0008373T + 0.0000020138T2

– 30 + 0.01071T + 33000/T2

40,700 75.44

0.5

0.6

0.7

0.8

0.9

1.0

λ=1.1λ=0.9λ=0.7

η max

Ua

0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9

Fig. 7. The maximum efficiency of the hybrid system versus hydrogen utilizationfactor curves for the parameters a1 = 0.003 and a2 = 0.001.

0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9

2000

3000

4000

5000

λ=1.1λ=0.9λ=0.7

Ua

Z max

/Wm

-2

Fig. 8. The maximum product of the power density and efficiency of the hybridsystem versus hydrogen utilization factor curves. The values of other parametersare the same as those used in Fig. 7.

0.0 0.1 0.2 0.3

0.10.20.30.40.50.60.70.80.9

ηmax

n*fη

λ=1.1λ=0.9λ=0.7

Ua=0.8η

n*f /mols-1m-2

ηe

Fig. 4. The efficiencies of the hybrid system and SOFC versus n�f curves. The valuesof other parameters are the same as those used in Fig. 3.

0.2

0.4

0.6

0.8

1.0

1.2

1.4 λ=1.1λ=0.9λ=0.7

Ua=0.8

n*f /mols-1m-2

P* /Wcm

-2

P*e

0.0 0.1 0.2 0.3

Fig. 5. The power densities of the hybrid system and SOFC versus n�f curves. Thevalues of other parameters are the same as those used in Fig. 3.

0.0 0.1 0.2 0.3

2000

3000

4000

5000

6000

7000

8000Zmax

λ=1.1λ=0.9λ=0.7

Ua=0.8

n*f /mols-1m-2

Z /W

m-2

n*fZ

Fig. 6. The product of the power density and efficiency of the hybrid system versusn�f curves. The values of other parameters are the same as those used in Fig. 3.

X. Zhang et al. / Energy Conversion and Management 86 (2014) 1102–1109 1107

Z of the hybrid system, respectively. It can be easily proved thatZmax ¼ gZP�Z > gmaxP�g. It shows that it is very significant to chooseZ as an objective function. In general, one always wants to obtaina large power output and efficiency for a practical SOFC–GT hybridsystem. To this end, the working states of the hybrid system maybe designed in the region between gmax and Zmax, and conse-quently, the optimal choice criteria of some key parameters willbe given as

n�fg 6 n�f 6 n�fZ ; ð33Þ

gmax P g P gZ ; ð34Þ

and

P�g 6 P� 6 P�Z : ð35Þ

Combining Eqs. (4)–(9), (12)–(14), and (24)–(32), one can obtain thecurves of the maximum efficiency and product of the efficiency andpower density of the hybrid system varying with the fuel utilization

factor, as shown in Figs. 7 and 8, respectively. It is clearly seen fromFigs. 7 and 8 that the maximum efficiency and product of the effi-ciency and power density of the hybrid system are both monoton-ically increasing functions of the hydrogen utilization factor in theSOFC. It shows that the larger the hydrogen utilization factor is,the more the fuel consumed by the SOFC, the less the fuel consumedby the GT, and the larger the efficiency and Z of the hybrid systembecause the energy conversion efficiency of the SOFC is larger thanthat of the GT. The maximum efficiency or product of the efficiencyand power density of the hybrid system may be the targets of thedesign and operation of the SOFC–GT hybrid system by controllingthe rates of fuel and air flowing into the SOFC. But the rates of fuelor air flowing into the SOFC at the maximum efficiency and productof the efficiency and power density of the hybrid system are differ-ent, and their values should be determined according to the resultsobtained above and the practical demands.

It is seen from the present model that a part of the waste heatfrom the SOFC is utilized by the GT and the residual hydrogencan be combusted in the burner. The burner can be availably usedas the second high-temperature reservoir of the GT because thecombustion temperature is higher than that of the SOFC. Theresults obtained here confirm the feasibility and excellence of themodel of the hybrid system.

It is worthwhile to point out that the hybrid system canenhance the efficiency and save the fuel, but it needs the additionalinvestment costs. Thus, one has to face a new problem: is it eco-nomically meaningful to structure a hybrid system? This problemwill be discussed below. When a hybrid system is used, one have toprovide the additional investment cost per unit time Fc = Ca/N, butcan save the fuel fee per unit time Ff = gfDnf, where Ca is the addi-tional investment cost of a hybrid system compared with a singlefuel cell, N is the life time of a hybrid system, gf is the price ofthe fuel per molar, and Dnf = nf(g/ge � 1) is the saved fuel per unit

1108 X. Zhang et al. / Energy Conversion and Management 86 (2014) 1102–1109

time. Only when Ff > Fc, the structure of a hybrid system can beeconomically meaningful. Thus, a criterion will be obtained as

g=ge � 1 > Fc=FT ; ð36Þ

where FT = gfnf is the total fuel fee of a hybrid system per unit time.Using such a criterion, one can evaluate the thermal-economic per-formance of a hybrid system.

4. Conclusions

A novel model of the SOFC–GT hybrid system with fuel reform-ing has been established, where both the waste heat from the SOFCand combustor is utilized by the GT. On the basis of the theories ofelectrochemistry and thermodynamics, the power output andefficiency of the hybrid system are derived. The advantages ofthe hybrid system compared with the SOFC are shown, and theoptimization criteria of the rate of the fuel flowing into the SOFC,the efficiency of the hybrid system and so on are determined.The conclusions obtained here may confirm the feasibility of thehydrocarbons for the SOFC and provide some optimization sugges-tions for the operation and design of SOFC–GT hybrid systems withfuel reforming.

Acknowledgements

This work has been supported by the Fujian Natural ScienceFoundation (No. 2012J05097), the National Natural Science Foun-dation (No. 51076134), People’s Republic of China.

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