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Two-stage thermoelectric generators for waste heat recovery from solid oxide fuel cells
Accepted Manuscript Two-stage thermoelectric generators for waste heat recovery from solid oxide fuel cells Houcheng Zhang, Haoran Xu, Bin Chen, Feifei Dong, Meng Ni PII:
S0360-5442(17)30743-0
DOI:
10.1016/j.energy.2017.05.005
Reference:
EGY 10807
To appear in:
Energy
Received Date: 17 February 2017 Revised Date:
2 April 2017
Accepted Date: 1 May 2017
Please cite this article as: Zhang H, Xu H, Chen B, Dong F, Ni M, Two-stage thermoelectric generators for waste heat recovery from solid oxide fuel cells, Energy (2017), doi: 10.1016/j.energy.2017.05.005. This is a PDF file of an unedited manuscript that has been accepted for publication. As a service to our customers we are providing this early version of the manuscript. The manuscript will undergo copyediting, typesetting, and review of the resulting proof before it is published in its final form. Please note that during the production process errors may be discovered which could affect the content, and all legal disclaimers that apply to the journal pertain.
ACCEPTED MANUSCRIPT
Two-stage thermoelectric generators for waste heat recovery from solid oxide fuel cells Houcheng Zhang a, b, *, Haoran Xu a, Bin Chen a, Feifei Dong a, Meng Ni a, * Department of Building and Real Estate, The Hong Kong Polytechnic University, Hong Kong, China
b
Department of Microelectronic Science and Engineering, Ningbo University, Ningbo 315211, China
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a
Abstract: A novel hybrid system that couples a solid state two-stage thermoelectric generator (TTEG)
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to a solid oxide fuel cell (SOFC) is proposed to harvest the waste heat from SOFC for performance enhancement. The number of thermoelectric elements among the top and bottom stages is optimized
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to maximize the power output of the TTEG. A relationship between the operating current density of the SOFC and the dimensionless electric current of the optimized TTEG is derived. The operating current density interval that enables the operation of TTEG is determined. The analytical expressions for the power output and efficiency of the hybrid system are theoretically derived by considering the
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irreversible losses in SOFC, regenerative losses in regenerator, Joule heat and the heat-conduction losses in TTEG, and heat leakage from the SOFC to the environment. The proposed system is found to be more efficient than the stand-alone SOFC, SOFC-single stage TEG hybrid system and several
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other emerging SOFC-based hybrid systems. Comprehensive parametric studies are conducted to
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investigate the effects of operating and design parameters on the system performance. The results obtained may provide some theoretical bases for the design and operation of a real SOFC/TTEG hybrid system.
Key Words: Solid oxide fuel cell; two-stage thermoelectric generator; irreversible loss; waste heat recovery *
Corresponding authors.
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ACCEPTED MANUSCRIPT Email: [email protected] (H. Zhang).
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Email: [email protected] (M. Ni).
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ACCEPTED MANUSCRIPT Nomenclature Effective polar plate area of an SOFC (m2)
AL
Heat-transfer area between the SOFC and the low temperature reservoir (m2)
Are
Heat-transfer area of the regenerator (m2)
b1 ~ b7
Variables in Eq. (15)
c1 , c2
Composite parameters in Eq. (21 ) (W m-2 K-1)
Dp
Pore size (m)
Ds
Grain size (m)
Dyeff
Effective diffusion coefficient for specie y (m2 s-1)
d1 ~ d 4
Variables in Eq. (16)
E
Equilibrium potential of an SOFC (V)
Eact
Activation energy level (J mol-1)
F
Faraday’s constant (C mol-1)
f1 ~ f 7
Variables in Eq. (18)
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•
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A
Total energies supplied to the hybrid system per unit time (J s-1)
∆h
Molar enthalpy change of the electrochemical reactions (J mol-1)
I Ig
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(−∆ H )
Operating electric current through SOFC (A) Electrical current flowing through TTEG (A)
i
Dimensionless electric current flowing through TTEG
j
Operating current density of SOFC (A m-2)
jB
Lower bound operating current density (A m-2)
jM
Allowable maximum current density of the SOFC (A m-2)
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* Operating current density at Pmax (A m-2)
jS
Stagnation operating current density of the SOFC (A m-2)
j fc
-2 * Operating current density at PSOFC ,max (A m )
jlH2
Limiting current densities of H2 mass transfers (A m-2)
jlH2 O
Limiting current densities of H2O mass transfers (A m-2)
jtg
-2 * Operating current densities at PTTEG ,max (A m )
j0
Exchange current density (A m-2)
jη ,tg
Operating current densities at ηTTEG ,max (A m-2)
∆j
Effective current density interval (A m-2)
KL
Heat leak coefficient (J m-2 K-1 s-1)
Kg
Thermal conductance of a thermoelectric element (W K-1 m-1)
K re
Heat-transfer coefficient (J m-2 K-1 s-1)
L
Thickness for the components (i.e., anode, cathode or electrolyte) of SOFC (m)
m
Number of pairs thermoelectric elements in the top stage of TTEG
n
Number of pairs thermoelectric elements in the bottom stage of TTEG
P* * Pmax
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P
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jP
Power output of the hybrid system (W)
Power density (W m-2) Maximum power density ((W m-2))
PB*
Power density at jB (W m-2)
PM*
Power density at jM (W m-2)
PH2
Partial pressures H2 (atm)
PO2
Partial pressures O2 (atm) 4
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Partial pressures H2 O (atm)
p
Operating pressure (atm)
Q1
Heat-transfer rate from the SOFC to the TTEG (J s-1)
Q2
Heat-transfer rate from the TTEG to the low temperature reservoir (J s-1)
Qm
Heat-transfer rate between the top stage TEGs and the bottom stage TEGs (J s-1)
R
Universal gas constant (J mol-1 K-1)
S
Cross-sectional areas of semiconductor arms (m2)
T
Operating temperature of SOFC (K)
TL
Temperatures of the low temperature reservoir (K)
Tm
Temperature of the cold junction of the top stage TEGs or the hot junction of the
Output voltage of an SOFC (V)
V act
Activation overpotential (V)
V con
Concentration overpotential (V)
V ohm
Ohmic overpotential (V)
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Z
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V
x
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bottom stage TEGs (K)
X
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PH2O
Ratio of the length of the grain contact neck to the grain size
Ratio of thermoelectric element numbers between the top stage and the bottom stage
Figure of merit of a thermoelectric element (K-1)
Greek symbols
α
Seebeck coefficient
β
Effectiveness of the regenerator
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Exchange current density pre-exponential factor (A m-2)
ε
Electrode porosity
η
Efficiency
ηB
Efficiency at jB
ηM
Efficiency at jM
ηP
* Efficiency at Pmax
θ
Temperature ratio between the SOFC and the low temperature reservoir
κ
Thermally conductivity of semiconductor materials (W K-1 m-1)
ξ
Electrode tortuosity
ρ
Electrical resistivity of semiconductor materials (Ω m)
σ
Electrical conductivity of the components of SOFC (Ω-1 m-1)
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γ
Subscripts
Anode; Cathode; Electrolyte
max
Maximum
N
N-type semiconductor
opt
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ohm
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a; c; e
Ohmic
Optimum
P
P-type semiconductor
SOFC
Solid oxide fuel cell
TTEG
Two-stage thermoelectric generator
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1. Introduction Fuel cells are more efficient energy conversion devices than conventional heat engines as they can directly convert the chemical energy of a fuel into electricity through electrochemical reactions
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[1]. Among the various types of fuel cells, solid oxide fuel cells (SOFCs) are promising candidates for stationary power generation due to their high operating temperature (typically about 800oC), fuel flexibility, and relatively low cost [2].
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The high operating temperature of SOFCs (600 °C-1000 °C) not only offers high
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electrochemical reaction rates but also generates significant amount of high-quality waste heat which can be used in cogeneration or CCHP (combined cooling, heating, and power) systems for performance improvement [3-5]. In the previous studies, various hybrid systems have been proposed to recover the waste heat from SOFC by using various thermodynamic cycles, such as Stirling cycle
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[6, 7], gas turbine cycle [8], Kalina cycle [9], organic Rankine cycle [10] and etc [11]. As solid-state devices, thermoelectric generators (TEGs) are more suitable for recovering the waste heat from SOFC than other thermodynamic cycles. As TEGs can directly convert heat into usable electricity
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without moving parts via the Seebeck effect, they are stable in operation, quiet, compact, and
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flexibility to various heat sources [12, 13]. However, the poor energy conversion efficiency of TEG hinders its widespread applications [14]. One way to improve the efficiency of TEG is to increase the figure of merit ( ZT ) of thermoelectric materials [15, 16]. Alternatively, the TEG efficiency can be increased by good thermal design and management of the TEG system [17]. Two-stage thermoelectric generators (TTEGs) are effective in enhancing the TEG efficiency by adding a bottom-stage TEG to recover the waste heat from the top-stage TEG, which have attracted much attention in recovering waste heat from high-temperature heat sources with large temperature gaps
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ACCEPTED MANUSCRIPT [18-20]. Chen et al. [18] optimized the heat-transfer area at high-temperature side and the number of thermoelectric elements at the top stage to achieve maximum power output and thermal efficiency. Xiong et al. [19] built a physical and numerical model of a TTEG to harvest the waste heat contained
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in blast furnace slag water. It was found that the maximum power and maximum efficiency at 100 °C were 0.44 kW and 2.66%, respectively. Manikandan et al. [20] carried out an exergy analysis on an exoreversible and irreversible TTEG model in which the Thomson effect along with Peltier, Joule
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and Fourier heat conduction were considered. To now, although some pioneering study has been
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conducted to use TEG to harvest waste heat from high-temperature fuel cells such as SOFCs [21], phosphoric acid fuel cells [22] and molten carbonate fuel cells [23], all of them only considered single-stage TEGs. Tian et al. [24, 25] compared the two-stage and traditional single-stage TEGs in recovering the waste heat from high-temperature exhaust gas of internal combustion engines.
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Calculations showed that the maximum value of the output power of the serial TTEG was 10.9% larger than that of the single-stage TEG at 800 K, and the conversion efficiency of the serial TTEG was 12.4% larger than that of the single-stage TEG.
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The literature survey shows that the two-stage TEGs could be promising for recovering waste
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heat from SOFCs but no study has been done yet. As a result, it is still unknown how the number of TEGs in the top and bottom stage influence the system overall efficiency. To fill the research gap, this paper is purposely designed to explore a hybrid system that uses an output-oriented optimum TTEG to recover the waste heat from an SOFC for performance improvement. The performance of the hybrid system will be numerically evaluated by considering various irreversible losses such as electrochemical losses in SOFC, regenerative losses in regenerator, internal irreversible losses inside the TTEG, and heat losses from the SOFC to the surroundings. The feasibilities and advantages of
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ACCEPTED MANUSCRIPT the TTEG as an exhaust heat recovery approach for SOFC will be demonstrated by comparing with other SOFC-based hybrid systems. The optimum operating region for some important performance parameters will be determined. Finally, extensive parametric studies will be conducted to investigate
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the influences of some design parameters and operating conditions on the hybrid system performance.
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2. System description
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The proposed hybrid system consists of an SOFC, a TTEG and a regenerator, as shown in Fig. 1. The TTEG consists of two stages TEGs with an external resistance RL . The regenerator preheats the inlet reactants with the help of the heat contained in the outlet products. The SOFC electrochemically converts the fuel chemical energy into electrical power, PSOFC , and high-grade waste heat. The waste
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heat is collected and transferred for three uses: a part of the waste heat, QR (J s-1), is transferred to compensate the regenerative losses in the regenerator; another part, QL (J s-1), is directly leaked from the SOFC at temperature T into the low temperature reservoir at temperature TL ; the rest part,
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Q1 (J s-1), is transferred from the SOFC to the TTEG for additional electrical power production. In
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addition, Qm (J s-1) is the heat-transfer rate between the top stage TEGs and the bottom stage TEGs, Q2 (J s-1) is the heat-transfer rate from the TTEG to the low temperature reservoir.
For simplicity, the following assumptions are adopted [18, 21, 25,26]: •
Both the SOFC and the TTEG are under steady state conditions;
•
Uniform operating temperature and pressure are assumed in the SOFC;
•
All reactants are completely consumed in the SOFC;
•
Geometric configuration of the TTEG is in the optimum form;
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Newton’s law is used to describe the heat transfers within the system;
•
Thermoelectric elements are insulated both electrically and thermally from their surroundings; Electric current flows along the arm of the TTEG;
•
TTEG is closely attached to the SOFC, and the external irreversible losses between the TTEG and the heat reservoirs are neglected;
Temperature of the cold junction of the top stage TEGs is the same as that of the hot
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•
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junction of the bottom stage TEGs; •
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•
Physical properties of TTEG such as Seebeck coefficient, electrical resistance, thermal conductance and dimensional figure of merit are independent of temperature;
•
Thompson effect in a thermoelectric element is neglected.
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2.1. SOFC
The typical anode-supported SOFC shown in Fig. 1 is made of Ni | YSZ | LSM with hydrogen as fuel and air as oxidant. The electrochemical performance of SOFC is deteriorated by activation,
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concentration, and ohmic overpotentials, which can be respectively described by the Bulter-Volmer
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equation, dusty gas model (DGM), and Ohm’s law. Adopting the electrochemical model previously developed in Refs. [27-29], the power output PSOFC and efficiency η SOFC for an SOFC can be given by
PSOFC = VI = jA ( E − Vact ,a − Vact ,c − Vcon ,a − Vcon ,c − Vohm ) ,
(1)
and
η SOFC =
PSOFC •
,
(2)
−∆ H
where 10
ACCEPTED MANUSCRIPT RT j j ln + = 2j 2 j0,l F 0,l
2 + 1
(l = a, c) ,
Vcon , a =
RT 1 + j /jlH2 O ln , 2 F 1 − j /jlH 2
Vcon , c =
PO2 RT ln 4 F Pc / δ O − ( − Pc / δ O − PO ) exp RTLc jδ O / (4 FPc DOeff ) 2 2 2 2 2
(
)
PH 2O Eact ,a exp − , RT Pref
72 X [ D p − ( D p + Ds )ε ]ε PO2 = γc Ds2 D p2 (1 − 1 − X 2 ) Pref
and •
−∆ H = − I ∆h (2 F ) ,
0.25
−E exp act ,c RT
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j0,c
72 X [ D p − ( Dp + Ds )ε ]ε PH 2 Ds2 D p2 (1 − 1 − X 2 ) Pref
,
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L L L Vohm = j a + c + e , σa σc σe j0,a = γ a
(3)
(4)
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Vact , l
j RT = sinh -1 2j F 0,l
,
(5)
(6)
(7)
(8)
(9)
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V is the output voltage. I is the electric current flowing through the SOFC. j is the operating current density. A is the effective polar plate area of an SOFC. E is the equilibrium potential.
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Vact , Vcon and Vohm are, respectively, the activation, concentration and ohmic overpotentials, and the subscripts “a”, “c” and “e” stand for anode, cathode and electrolyte. R is the universal constant.
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T is the operating temperature. F is the Faraday’s constant. j0 is the exchange current density. jlH2O = 2FPH 2O DHeff2O /RTALa and jlH2 = 2FPH2 DHeff2 /RTALa are the limiting current densities of H2O and H2 mass transfers, respectively. PH2 O and PH2 are partial pressures of H2O and H2 at the electrode surfaces, respectively. DHeff2O , DHeff2 , and DOeff2 are, respectively, the effective diffusion coefficients of H2O, H2 and O2. δ O2 = DOeff2 , Kn
(D
eff O 2 , Kn
)
+ DOeff2 -N2 , and DOeff2 , Kn is the effective
Knudsen diffusion coefficient of O2. DOeff2 -N2 is the effective molecular diffusion coefficient for O2-N2 binary system. L and σ are, respectively, the thickness and electrical conductivity for the 11
ACCEPTED MANUSCRIPT components of the SOFC (i.e., anode, cathode or electrolyte). X is the ratio of the length of the grain contact neck to the grain size. ε is the porosity of the electrodes. D p and Ds are the pore size and grain size, respectively. γ is exchange current density pre-exponential factor. Eact is the
•
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activation energy level. Pc is the cathode operating pressure. Pref is the reference pressure. (−∆ H ) = − I ∆h (2 F ) is the overall released energy (both electrical and thermal) per unit time. ∆h is the molar enthalpy change for the hydrogen-oxygen reaction.
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2.2. TTEG
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As shown in Fig. 1, the TTEG is operated between the SOFC and the low temperature reservoir. The TTEG is composed of a top stage with m pairs thermoelectric elements and a bottom stage with n pairs thermoelectric elements, and the two stages are electrically connected in serials. Each thermoelectric element consists of a P-type and an N-type semiconductor legs which are joined by a
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thin copper. Neglecting the Thomson effect, the internal irreversible losses inside the TTEG are mainly from the Joule heat and the heat-conduction losses. The amount of Joule heat is I g2 Rg , where I g is the electrical current flowing through the TTEG, and Rg = ( ρ PlP S P + ρ N lN S N ) is the
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internal electrical resistance of a thermoelectric element, ρ is the electrical resistivity, l is the
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length of the semiconductor arms, S is the cross-sectional areas of semiconductor arms, subscripts “P” and “N” indicate P-type and N-type semiconductors, respectively. The heat-conduction losses for the top and bottom stages can be, respectively, expressed as mK g (T − Tm ) and nK g (Tm − TL ) , where K g = (κ P S P lP + κ N S N lN ) is the thermal conductance of a thermoelectric element, κ is the thermally conductivity of the semiconductor materials, Tm is the temperature of cold junction of the top stage or hot junction of the bottom stage. When heat flows from the SOFC to TTEG, the heat balance equations can be expressed as [18, 25, 30]
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(10)
Qm = α mI g Tm + 0.5mI g2 Rg + mK g ( T − Tm ) ,
(11)
Qm = α nI g Tm − 0.5nI g2 Rg + nK g (Tm − TL ) ,
(12)
Q2 = α nI g TL + 0.5nI g2 Rg + nK g (Tm − TL ) ,
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and (13)
where α P , α N and α = (α P − α N ) are the Seebeck coefficients for a P-type semiconductor leg, an
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N-type semiconductor leg and a thermoelectric element, respectively.
TTEG can be, respectively, expressed as
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Eliminating Tm in Eqs. (10) - (13), the power output PTTEG and efficiency ηTTEG of the
2 ( xθ − 1) i θ x + 1 ( x + 1) i ( 2ZTL ) + ( xθ + 1) i2 PTTEG = Q1 − Q2 = K g TL n ( x + 1) − + + , x + 1 2 ZT x + 1 x − 1 i − x + 1 ( ) ( ) L
and b + b i + b3i 2 + b4i 3 Q2 = 1− 1 2 , Q1 b1 + b5i + b6i 2 + b7 i 3
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ηTTEG = 1 −
(14)
(15)
where i = α I g / K g is the dimensionless electric current [23]. Z = α 2 / ( K g Rg ) is the figure of
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merit of a thermoelectric element [26]. θ = T / TL is the temperature ratio between the SOFC and
top
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the low temperature reservoir. x = m / n is the ratio of thermoelectric element numbers between the stage
and
the
bottom
stage,
b1 = (1 − θ ) x ,
b2 = −2 x ,
b3 = x − 1 − (1 + x ) ( ZTL ) ,
b4 = ( x − 1) ( 2ZTL ) , b5 = −2θ x , b6 = x ( x − 1) θ + x (1 + x ) ( ZTL ) , and b7 = − x ( x − 1) ( 2ZTL ) . For a given amount of heat from SOFC to TTEG Q1 and the total number of thermoelectric elements (m + n) , it is meaningful to determine the optimum allocation value for maximizing the power output of the TTEG. Substituting Eq. (14) into the extreme condition ∂PTTEG ∂x = 0 , one has
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i 3 − (2d1 − 1)i 2 − 2d1i + 2 i −i 4 + 2d1i 3 − d 2i 2 + d3i + 2d1d 4 = , i ( i − 1)( i − 2d1 )
(16)
where d1 = θ ZTL + 1 , d 2 = ZTL (θ + 1) , d 3 = ZTL ( 2θ 2 ZTL + 2θ ZTL + θ + 3) , d 4 = ZTL (θ − 1) . Replacing the symbol x in Eqs. (14) and (15) by xopt , the expressions for the maximum
PTTEG ,max = K g TL n ( xopt
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power output PTTEG ,max and the corresponding efficiency ηTTEG , m can be, respectively, given by 2 ( xoptθ − 1) i θ xopt + 1 ( xopt + 1) i ( 2 ZTL ) + ( xoptθ + 1) i2 + 1) − + + , 2ZTL xopt + 1 ( xopt − 1) i − ( xopt + 1) xopt + 1
where
f1 = (1 − θ ) xopt ,
f 2 = −2 xopt ,
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ηTTEG ,m
f1 + f 2i + f 3i 2 + f 4i 3 = 1− , f1 + f5i + f 6i 2 + f 7 i 3
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and
f 3 = xopt − 1 − (1 + xopt ) ( ZTL ) ,
f 5 = −2θ xopt , f 6 = xopt ( xopt − 1) θ + xopt (1 + xopt )
2.3. Regenerator
( ZTL ) , and
(17)
(18)
f 4 = ( xopt − 1) ( 2 ZTL ) ,
f 7 = − xopt ( xopt − 1) ( 2 ZTL ) .
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The regenerator in Fig. 1 works as a counter-flow heat exchanger that preheats the inlet reactants from temperature TL to the SOFC operating temperature T . As the regeneration process
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is not perfect, the rate of regenerative losses is often given by [23]
QR = K re Are (1 − ξ )(T − TL ) ,
(19)
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where K re , Are and ξ are the heat-transfer coefficient, heat-transfer area and effectiveness of the regenerator, respectively.
2.4. Performance of the hybrid system As the operating temperature of SOFC is higher than that of the low temperature reservoir, the rate of heat loss from the SOFC to the low temperature reservoir, QL , can be described by [21]
QL = K L AL (T − TL ) ,
(20)
where K L and AL are heat leak coefficient and the heat leak area, respectively. 14
ACCEPTED MANUSCRIPT Based on the energy conservation law, the heat-transfer rate Q1 from the SOFC to the TTEG can be calculated as •
Q1 = −∆ H − PSOFC − QR − QL = −
2 Fc1 (T − TL ) 2 Fc2 (T − TL ) A∆h − (1 − η SOFC ) j − , 2F −∆h −∆h
(21)
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where c1 = K re Are (1 − ξ ) A and c2 = K L AL A are temperature-independent composite constants which are associated with the regenerative losses and heat leak losses, respectively.
generate electricity only when the following formula is valid
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•
−∆ H − PSOFC > QR + QL + mK g (T − Tm ) .
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Considering the heat-conduction losses within the TTEG, the bottoming TTEG begins to
(22)
Combining Eq. (21), Eq. (22) could be further rewritten as
2F j > jB = ( c1 + c2 )(T − TL ) + nxopt K g (T − Tm ) / A , −∆ h 1 − η ( ) SOFC
(23)
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where jB is the lower bound operating current density from which the TTEG begins to work. Meanwhile, one may easily determine the maximum current density jM that allows the TTEG to work by combining Eqs. (14) - (21) and the condition of PTTEG > 0 . Thus, the TTEG in the hybrid
(24)
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j B < j < jM .
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system only works in the following region:
Therefore, the effective operating current density interval is given by ∆ j = jM − j B .
(25)
When the SOFC operates in the region of jB < j < jM , the numerical relationship between the operating current density of SOFC, j , and the dimensionless electric current of TEG, i , is determined by Eq. (26)
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xopt i −
=
xopt i
2
2ZT
+ xopt
xopt + 1 2 i + xopt + 1/ θ xopt 2ZT + ( xopt − 1) i − ( xopt + 1)
2 Fc1 (T − TL ) 2 Fc2 (T − TL ) − A∆h − (1 − η SOFC ) j − . 2 FnK g T −∆h −∆h
(26)
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Thus, the power output P and efficiency η for the hybrid system can be, respectively, expressed as
( jB < j <
(j≤
jM )
jB or j ≥ jM )
( jB < (j≤
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and PSOFC + PTTEG ,max = • η= −∆ H = η SOFC
(27)
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= PSOFC + PTTEG ,max P= = PSOFC
j < jM )
(28)
jB or j ≥ jM )
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3. Generic performance characteristics and comparisons The typical parameters for SOFC modeling are adopted from Refs. [27-29], the values for γ a and γ c are calculated based on the experimental data recommended by Chan et al. [27]. The other
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parameters are listed in Table 1 [23, 25, 28, 29].
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Figure 2 shows the generic performance characteristics and the performance comparisons for the SOFC, the TTEG and the proposed hybrid system. It is observed that the power densities of SOFC, TTEG and proposed hybrid system first increase and then decrease with increasing current density j , and the TTEG delivers electricity only in the region of jB < j < jM . As distinct from power densities, η SOFC decreases as j increases in the whole region of j , while ηTTEG first increases and then decreases in the region of jB < j < jM , η first rapidly decreases then somewhat * * is larger than PSOFC increases and finally decreases as j increases. It is also seen that Pmax ,max and
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ACCEPTED MANUSCRIPT * PTTEG ,max , and jP is always different from j fc as the SOFC and the TTEG achieve their maximum
power densities at different current densities. Furthermore, the hybrid system efficiency at jP is larger than the SOFC efficiency at j fc . Adopting the thermoeconomic optimization criterion [23],
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the optimum operating regions for current density, power density and efficiency are suggested to be * situated in jB < j ≤ jP , PB* < P* ≤ Pmax and η B > η ≥ η P , respectively.
The performance comparisons between the two-stage TEG and the single-stage TEG as
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bottoming thermal devices are shown in Fig. 3 (a), where both the numbers of thermoelectric
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elements in two-stage TEG and the single-stage TEG are n (1 + xopt ) . It is evidently observed that the maximum power density of the former is larger than that of the latter. For the parameters given in Table 1, the maximum power density of the TTEG integrated hybrid system is about 22.3% larger than that of the single-stage TEG integrated hybrid system. The values of jB and jM of the TTEG
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are larger than that of the single-stage TEG, and the value ∆j of the former is also larger than that of the latter. Moreover, the value of jP for the SOFC/TTEG hybrid system is larger than that for the SOFC/single-stage TEG hybrid system. Outside the region of jB < j < jM , the performance of the
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two hybrid systems is the same as that of the stand-alone SOFC.
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For further comparison purposes, the emerging solid state thermophotovoltaic cell (TPVC) [3] and thermionic generator (TIG) [31] are employed in waste heat recovery from the SOFC in Fig. 1. With reference to the stand-alone SOFC, the maximum power density improvements of SOFC-based * * * hybrid systems (i.e., [( Pmax − PSOFC ,max ) / PSOFC ,max ] × 100% ) are clearly compared in Fig. 3 (b). It is
observed that the improvement of the proposed hybrid system is smaller than that of the hybrid systems integrated with TPVC and TIG at high operating temperatures, but it is always greater than the improvement of the SOFC/single-stage TEG hybrid system. Since lowering the operating
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ACCEPTED MANUSCRIPT temperature of SOFCs will be an important R&D trend in the future [32-34], TTEG may be treated as a potential waste heat recovery candidate for SOFCs.
4. Parametric studies
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4.1. Effects of number of thermoelectric elements in bottom stage of TTEG When x is operated in the optimum form, a larger number of n means more thermoelectric elements are employed for additional electricity production, which improves the performance TTEG
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* and the overall performance of the hybrid system. As shown in Fig. 4, Pmax increases with the
increasing n , and simultaneously, the values of jB , jM , ∆j and jP shift to larger ones. The
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effect of n only occurs in the region of jB < j < jM and becomes more pronounced in the region of jP < j < jM . In practice, the thermoelectric elements should be as more as possible, and the thermoelectric elements are suggested to be cascaded in the multi-stage form.
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4.2. Effects of SOFC operating temperature
The operating temperature T not only affects the performance of the SOFC and TTEG but also affects the amount of heat transferred from the SOFC to the TTEG. It is seen from Fig. 5 that
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both P * and η increase as T increases, and this effects occur in the entire region of j and
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become more significant at an elevated T . The values of jP , jS , jB , jM and ∆ j increase with increasing T . A larger T not only improves the performance of the SOFC but also improves the performance of the TTEG. As described by Eqs. (19) and (20), a larger T also leads to larger thermodynamic losses. Because the performance enhancements in the SOFC and TTEG are more significant than the performance deterioration resulting from the thermodynamic losses, a larger T is always preferable. However, a high operating temperature of the SOFC may result in higher system costs as well as slow start-up and shutdown cycles.
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* * increase with p , and the values of Pmax , PSOFC jP , jS , jB , jM and ∆ j are increased ,max ,
with increasing p . The effects of p on the system performance become more significant at larger current densities. A larger operating pressure is more preferable for performance improvement, but it
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practice, as shown by the black solid lines in Fig. 6.
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also consumes some electricity to compress the inlet reactants, and thus 1.0 atm is usually chosen in
4.4. Effects of thermal conductance of a thermoelectric element
Different from T and p , the thermal conductance K g only affects the system performance in the region of jB < j < jM , as shown in Fig. 7. Outside this region, the curves of P* ~ j and
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* * η ~ j are overlapped with that of the PSOFC ~ j and η SOFC ~ j , respectively. The values of Pmax ,
jP , jB , jM and ∆ j increase as K g increases, the effects of K g on the system performance become more significantly at larger current densities. However, a larger K g needs higher
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requirement for materials fabrication, which often leads to higher manufacture cost.
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4.5. Effects of figure of merit of a thermoelectric element The performance of a TTEG depends on the figure of merit value of a thermoelectric element [35]. With the fast development of thermoelectric materials, the dimensionless figures of merit larger than 1.0 have been already reported [36]. It is necessary and meaningful to discuss the effects of the dimensionless figures of merit on the performance of the overall system. As shown in Fig. 8, both P* and η are increased with the increasing ZTL . The effect of ZTL only occurs in the region of * jB < j < jM , the values of Pmax , jB , and jP shift to larger ones as ZTL increases, while the value
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ACCEPTED MANUSCRIPT of jM almost keeps invariant and the value of ∆ j decreases as ZTL increases. In the literatures, numerous efforts have been made to achieve a greater Z [37].
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4. Conclusions A novel hybrid system consisting of a TTEG and an SOFC is proposed to recover the waste heat from the SOFC. A theoretical model is derived to evaluate the hybrid system performance by
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considering the various irreversible losses. An analytical expression is obtained to determine the
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optimal numbers of TEG in the top and bottom stage of the TTEG. A numerical relationship for the output electric currents of the SOFC and the TTEG is derived, and the current density interval of SOFC that enables the TTEG to work is determined. The performance parameters for the hybrid system are specified under different operating conditions, and the generic performance
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characteristics are demonstrated. From extensive comparisons with other SOFC-based hybrid systems, the proposed hybrid system using TTEG to recovery the waste heat from SOFC is found feasible and effective. Comprehensive parametric studies are conducted to discuss the effects of
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operating temperature and pressure of SOFC, number of the thermoelectric element, thermal
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conductance, and figure of merit of the TTEG on the hybrid system performance. The results obtained are useful for the design and optimization of SOFC-based cogeneration systems to achieve a better performance.
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Acknowledgments This work has been supported by the Natural Science Foundation of Zhejiang Province (Grant No. LQ14E060001), National Natural Science Foundation of China (Grant No. 51406091), and The
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Hong Kong Polytechnic University Research Project (Grant No. 1-YW1F).
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References [1] Sharaf OZ, Orhan MF. An overview of fuel cell technology: fundamentals and applications. Renew Sust Energ Rev 2014; 32: 810-53.
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[2] Mekhilef S, Saidur R, Safari A. Comparative study of different fuel cell technologies. Renew Sust Energ Rev 2012; 16:981-9.
[3] Liao T, Cai L, Zhao Y, Chen J. Efficient exploiting the waste heat in solid oxide fuel cell by
SC
means of thermophotovoltaic cell. J Power Sources 2016; 306: 666-73.
M AN U
[4] Saebea D, Magistri L, Massardo A, Arpornwichanop A. Cycle analysis of solid oxide fuel cell-gas turbine hybrid systems integrated ethanol steam reformer: Energy management. Energy (2017), doi:10.1016/j.energy.2017.03.105.
[5] AI-Sulaiman FA, Dincer I, Hamdullahpur F. Exergy analysis of an integrated solid oxide fuel cell
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and organic Rankine cycle for cooling, heating and power production. J Power Sources 2010; 195: 2346-54.
[6] Rokni M. Thermodynamic analyses of municipal solid waste gasification plant integrated with
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solid oxide fuel cell and Stirling hybrid system. Int J Hydrogen Energy 2015; 40:7855-69.
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[7] Chen L, Gao S, Zhang H. Performance analysis and multi-objective optimization of an irreversible solid oxide fuel cell-stirling heat engine hybrid system. Int J Electrochem Sci 2013; 8:10772-87.
[8] Buonomano A, Calise F, d’Accadia MD, Polombo A, Vicidomini M. Hybrid solid oxide fuel cells–gas turbine systems for combined heat and power: a review. Appl Energy 2015; 156:32-85. [9] Pierobon L, Rokni M. Thermodynamic Analysis of an Integrated Gasification Solid Oxide Fuel Cell Plant with a Kalina Cycle. Int J Green Energy 2015;12:610-9.
22
ACCEPTED MANUSCRIPT [10] Pierobon L, Rokni M, Larsen U, Haglind F. Thermodynamic analysis of an integrated gasification solid oxide fuel cell plant combined with an organic Rankine cycle. Renew Energy 2013;60:226-34.
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[11] Mehrpooya M, Dehghani H, Moosavian SMA. Optimal design of solid oxide fuel cell, ammonia-water single effect absorption cycle and Rankine steam cycle hybrid system. J Power Sources 2016; 306:107-23.
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[12] Date A, Date A, Dixon C, Akbarzadeh A. Progress of thermoelectric power generation systems:
M AN U
Prospect for small to medium scale power generation. Renew Sust Energ Rev 2014; 33: 371-81. [13] Song H, Cai K. Preparation and properties of PEDOT:PSS/Te nanorod composite films for flexible thermoelectric power generator. Energy 2017; 125:519-25.
[14] Gou X, Xiao H, Yang S. Modeling, experimental study and optimization on low-temperature
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waste heat thermoelectric generator system. Appl Energy 2010; 87: 3131-6. [15] Zhang Q, Sun Y, Xu W, Zhu D. Organic thermoelectric materials: Emerging green energy materials converting heat to electricity directly and efficiently. Adv Mat 2014; 26: 6829-51.
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[16] Zeier WG, Schmitt J, Hautier G, Aydemir U, Gibbs ZM, Felser C, Snyder J. Engineering
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half-Heusler thermoelectric materials using Zintl chemistry. Nat Rev Mat 2016; 1: 16032. [17] Xiao J, Yang T, Li P, Zhai P, Zhang Q. Thermal design and management for performance optimization of solar thermoelectric generator. Appl Energy 2012; 93: 33-8. [18] Chen L, Li J, Sun F, Wu C. Performance optimization of a two-stage semiconductor thermoelectric-generator. Appl Energy 2005; 82: 300-12. [19] Xiong B, Chen L, Meng F, Sun F. Modeling and performance analysis of a two-stage thermoelectric generator energy harvesting system from blast furnace slag water waste heat. Energy
23
ACCEPTED MANUSCRIPT 2014;77:562-9. [20] Manikandan S, Kaushik SC. The influence of Thomson effect in the performance optimization of a two stage thermoelectric generator. Energy 2016; 100: 227-37.
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[21] Chen X, Pan Y, Chen J. Performance and evaluation of a fuel cell-thermoelectric generator hybrid system. Fuel Cells 2010; 10: 1164-70.
[22] Chen X, Wang Y, Cai L, Zhou Y. Maximum power output and load matching of a phosphoric
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acid fuel cell-thermoelectric generator hybrid system. J Power Sources 2015;294:430-6.
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[23] Wu S, Zhang H, Ni M. Performance assessment of a hybrid system integrating a molten carbonate fuel cell and a thermoelectric generator. Energy 2016;112:520-7. [24] Sun X, Liang X, Shu G, Tian H, Wei H, Wang X. Comparison of the two-stage and traditional single-stage TEG in recovering the waste heat from high temperature exhaust gas of internal
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combustion engine. Energy 2014;77:489-98.
[25] Liang X, Sun X, Tian H, Shu G, Wang Y, Wang X. Comparison and parameter optimization of a two-stage thermoelectric generator using high temperature exhaust of internal combustion engine.
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Appl Energy 2014;130:190-9.
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[26] Zhao M, Zhang H, Hu Z, Zhang Z, Zhang J. Performance characteristics of a direct carbon fuel cell/thermoelectric generator hybrid system. Energy Convers Manag 2015; 89: 683-9. [27] Chan SH, Khor KA, Xia ZT. A complete polarization model of a solid oxide fuel cell and its sensitivity to the change of cell component thickness. J Power Sources 2001;93:130-40. [28] Ni M, Leung MKH, Leung DYC. Parametric study of solid oxide fuel cell performance. Energy Convers Manag 2007;48:1525-35. [29] Zhang H, Chen J, Zhang J. Performance analysis and parametric study of a solid oxide fuel cell
24
ACCEPTED MANUSCRIPT fueled by carbon monoxide. Int J Hydrogen Energy 2013;38:16354-64. [30] Temizer İ, İlkılıç C. The performance and analysis of the thermoelectric generator system used in diesel engines. Renew Sust Energ Rev 2016;63:141-51.
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[31] Wang Y, Cai L, Liu T, Wang J, Chen J. An efficient strategy exploiting the waste heat in solid oxide fuel cell system. Energy 2015;93:900-7.
[32] Wachsman ED, Lee KT. Lowering the temperature of solid oxide fuel cells. Science
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2011;334:935-9.
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[33] Dong F, Ni M, He W, Chen Y, Yang G, Chen D, Shao Z. An efficient electrocatalyst as cathode material for solid oxide fuel cells: BaFe0.95Sn0.05O3−δ. J Power Sources 2016; 326:459-65. [34] Lv X, Gu C, Liu X, Weng Y. Effect of gasified biomass fuel on load characteristics of an intermediate-temperature solid oxide fuel cell and gas turbine hybrid system. Int J Hydrogen Energy
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2016;41:9563-76.
[35] Fahrnbauer F, Souchay D, Wagner G, Oeckler O. High thermoelectric figure of merit values of
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germanium antimony tellurides with kinetically stable cobalt germanide precipitates. J Am Chem Soc 2015;137:12633-8.
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[36] Venkatasubramanian R, Siivola E, Colpitts T, O’Quinn B. Thin-film thermoelectric devices with high room temperature figures of merit. Nature 2001;413:597-602. [37] Ovik R, Long B D, Barma M C, Riaz M, Sabri MFM, Said SM, Saidur R. A review on nanostructures of high-temperature thermoelectric materials for waste heat recovery. Renew Sust Energ Rev 2016; 64:635-59.
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Table captions: Table 1. Parameters used in the modeling [23, 25, 28, 29].
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Figure captions:
Fig. 1. The schematic diagram of an SOFC/two-stage thermoelectric generator hybrid system. Fig. 2. Comparisons of (a) power densities and (b) efficiencies between the SOFC, TTEG and hybrid
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* * system, where PSOFC = PSOFC / A , PTTEG = PTTEG / A , P * = P / A are, power densities for
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* * * SOFC, TTEG and hybrid system, respectively; PSOFC ,max , PTTEG ,max and Pmax are the
corresponding maximum power densities; ηTTEG ,max is the maximum efficiency of the TTEG; * j P and η P are operating current density and efficiency at Pmax , respectively; j fc is the * operating current density at PSOFC jtg and jη ,tg are operating current densities at ,max ;
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* * PTTEG ,max and ηTTEG ,max , respectively; PB and η B are power density and efficiency of the
hybrid system at jB , respectively; jS is the stagnation current density from which the
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SOFC does not output electricity any more. Fig. 3. Performance comparisons between (a) two-stage and single-stage TEGs, and (b) some solid
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state thermal devices as bottoming thermal devices for SOFCs. Fig. 4. Effects of number of the thermoelectric element n on the performance of the hybrid system. Fig. 5. Effects of operating temperature T on the performance of the hybrid system. Fig. 6. Effects of operating pressure p on the performance of the hybrid system. Fig. 7. Effects of thermal conductance K on the performance of the hybrid system. Fig. 8. Effects of figure of merit of thermoelectric materials ZTL on the performance of the hybrid system. 26
ACCEPTED MANUSCRIPT Table 1 Value
Operating pressure, p (atm)
1.0
Operating temperature, T (K)
1073
Faraday’s constant, F (C mol-1)
96485
Anode interface gas compositions
95 % H2 +5 % H2O
Cathode interface gas compositions
79 % N2 +21% O2
Activation energy for anode, Eacta (J mol-1)
1.0×105 [28]
Activation energy for cathode, Eactc (J mol-1)
1.2×105 [28]
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Parameter
Electrode porosity, ε
0.48 [28]
Average pore diameter, Dp (m)
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5.4 [28]
Electrode tortuosity,ξ
3.0 ×10-6 [28] 1.5 ×10-6 [28]
Average grain size, Ds (m) Average length of grain contact, X
0.7 [28]
5.0×10-4
Anode thickness, La (m)
8.0×104 [29]
Cathode thickness, Lc (m)
5.0×10-5
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Anode electric conductivity, σa (Ω-1 m-1)
8.4×103 [29]
Electrolyte thickness, Le (m)
5.0×10-5
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Cathode electric conductivity, σc (Ω-1 m-1)
3.34×104 exp(-1.03×104/T) [29]
Effective surface area of the SOFC, A (m2)
4.0×10-2
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Electrolyte ionic conductivity, σe (Ω-1 m-1)
Heat conductivity of a thermoelectric element, K g (W -1
-1
K m )
0.09
Figure of merit of the thermoelectric materials, ZTL
1.0 [23]
Number of TEGs in the second stage, n
8
Sectional area of a thermoelectric element
5.0×10-3 [25]
Constants in Eq. (21), c1 ; c2 (W m-2 K-1)
0.1; 0.1
Temperature of low-temperature reservoir, TL (K)
298
27
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Fig. 1.
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Products Regenerator 28
ACCEPTED MANUSCRIPT Fig. 2.
(a)
P*max
SOFC TTEG Hybrid system
7500 6000 4500
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P*SOFC,max
P*B
P
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3000 * TTG,m ax
1500 jP 0
jtg
jB 0
5000
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P*SOFC , P*TTEG , P* (W m-2)
9000
jfc jM
jS
10000 15000 20000 25000 30000
j (A m-2)
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(b) 1.0
TTEG Hybrid system
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ηB 0.6
ηP
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ηSOFC , ηTTEG , η
0.8
SOFC
0.4
ηTTG,max 0.2 0.0
jB 0
5000
jη,tg jP
jM
jS
10000 15000 20000 25000 30000
j (A m-2) 29
ACCEPTED MANUSCRIPT Fig. 3.
(a)
1.0 Two-stage TEG Single-stage TEG
6000
0.6
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4000
0
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2000 jP
0
5000
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70
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50
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Improvement (%)
60
Single-stage TEG TTEG TPVC [3] TIG [31]
40 30 20 10 0
960
0.2
0.0 10000 15000 20000 25000 30000
j (A m-2)
(b)
0.4
1040
1120
T (K) 30
1200
1280
η
P* (W m-2)
0.8
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8000 P
* max
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10000
1.0 n=6 n=8 n = 10
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6000
0.6
2000
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4000
0
0.8
jP
0
5000
0.4 0.2
0.0 10000 15000 20000 25000 30000
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j (A m-2)
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η
P* (W m-2)
8000 P*max
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16000
1.0
12000
0.6
8000 P*max
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η
4000
0
jP
0
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P* (W m-2)
0.8
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T = 973 K T = 1073 K T = 1173 K
jS
0.4 0.2
0.0 10000 20000 30000 40000 50000 60000
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j (A m-2)
32
ACCEPTED MANUSCRIPT Fig. 6.
12500
1.0 p = 1.0 atm p = 2.0 atm p = 3.0 atm
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0.8
7500
0.6
η
2500 0
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5000
jP 0
10000
20000
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j (A m-2)
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P* (W m-2)
10000 P*max
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30000 jS
0.4 0.2
0.0 40000
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1.0
12000
Kg = 0.05 W K-1 m-1
10000
Kg = 0.13 W K-1 m-1 0.8
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Kg = 0.09 W K-1 m-1 Kg = 0.17 W K-1 m-1
P*max
8000
0.6
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6000
2000 0
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4000
jP
0
5000
0.4 0.2
0.0 10000 15000 20000 25000 30000
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j (A m-2)
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η
P* (W m-2)
Fig. 7.
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ZTL = 1.0
P*max 8000
ZTL = 1.2
0.8
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10000
ZTL = 1.4
0.6
6000
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η
4000 2000 0
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P* (W m-2)
1.0
jP
0
5000
0.4 0.2
0.0 10000 15000 20000 25000 30000
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j (A m-2)
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ACCEPTED MANUSCRIPT Two-stage thermoelectric generators are proposed to recover waste heat from SOFCs. Two-stage thermoelectric generators are optimized by power outputoriented criterion.
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Performance parameters for assessing the hybrid system are specified.
Multiple comparisons are conducted to verify the feasibility of the hybrid
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system.
Effects of some important parameters on the system performance are
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discussed.
S0360-5442(17)30743-0
DOI:
10.1016/j.energy.2017.05.005
Reference:
EGY 10807
To appear in:
Energy
Received Date: 17 February 2017 Revised Date:
2 April 2017
Accepted Date: 1 May 2017
Please cite this article as: Zhang H, Xu H, Chen B, Dong F, Ni M, Two-stage thermoelectric generators for waste heat recovery from solid oxide fuel cells, Energy (2017), doi: 10.1016/j.energy.2017.05.005. This is a PDF file of an unedited manuscript that has been accepted for publication. As a service to our customers we are providing this early version of the manuscript. The manuscript will undergo copyediting, typesetting, and review of the resulting proof before it is published in its final form. Please note that during the production process errors may be discovered which could affect the content, and all legal disclaimers that apply to the journal pertain.
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Two-stage thermoelectric generators for waste heat recovery from solid oxide fuel cells Houcheng Zhang a, b, *, Haoran Xu a, Bin Chen a, Feifei Dong a, Meng Ni a, * Department of Building and Real Estate, The Hong Kong Polytechnic University, Hong Kong, China
b
Department of Microelectronic Science and Engineering, Ningbo University, Ningbo 315211, China
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a
Abstract: A novel hybrid system that couples a solid state two-stage thermoelectric generator (TTEG)
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to a solid oxide fuel cell (SOFC) is proposed to harvest the waste heat from SOFC for performance enhancement. The number of thermoelectric elements among the top and bottom stages is optimized
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to maximize the power output of the TTEG. A relationship between the operating current density of the SOFC and the dimensionless electric current of the optimized TTEG is derived. The operating current density interval that enables the operation of TTEG is determined. The analytical expressions for the power output and efficiency of the hybrid system are theoretically derived by considering the
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irreversible losses in SOFC, regenerative losses in regenerator, Joule heat and the heat-conduction losses in TTEG, and heat leakage from the SOFC to the environment. The proposed system is found to be more efficient than the stand-alone SOFC, SOFC-single stage TEG hybrid system and several
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other emerging SOFC-based hybrid systems. Comprehensive parametric studies are conducted to
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investigate the effects of operating and design parameters on the system performance. The results obtained may provide some theoretical bases for the design and operation of a real SOFC/TTEG hybrid system.
Key Words: Solid oxide fuel cell; two-stage thermoelectric generator; irreversible loss; waste heat recovery *
Corresponding authors.
1
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Email: [email protected] (M. Ni).
2
ACCEPTED MANUSCRIPT Nomenclature Effective polar plate area of an SOFC (m2)
AL
Heat-transfer area between the SOFC and the low temperature reservoir (m2)
Are
Heat-transfer area of the regenerator (m2)
b1 ~ b7
Variables in Eq. (15)
c1 , c2
Composite parameters in Eq. (21 ) (W m-2 K-1)
Dp
Pore size (m)
Ds
Grain size (m)
Dyeff
Effective diffusion coefficient for specie y (m2 s-1)
d1 ~ d 4
Variables in Eq. (16)
E
Equilibrium potential of an SOFC (V)
Eact
Activation energy level (J mol-1)
F
Faraday’s constant (C mol-1)
f1 ~ f 7
Variables in Eq. (18)
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A
Total energies supplied to the hybrid system per unit time (J s-1)
∆h
Molar enthalpy change of the electrochemical reactions (J mol-1)
I Ig
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(−∆ H )
Operating electric current through SOFC (A) Electrical current flowing through TTEG (A)
i
Dimensionless electric current flowing through TTEG
j
Operating current density of SOFC (A m-2)
jB
Lower bound operating current density (A m-2)
jM
Allowable maximum current density of the SOFC (A m-2)
3
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* Operating current density at Pmax (A m-2)
jS
Stagnation operating current density of the SOFC (A m-2)
j fc
-2 * Operating current density at PSOFC ,max (A m )
jlH2
Limiting current densities of H2 mass transfers (A m-2)
jlH2 O
Limiting current densities of H2O mass transfers (A m-2)
jtg
-2 * Operating current densities at PTTEG ,max (A m )
j0
Exchange current density (A m-2)
jη ,tg
Operating current densities at ηTTEG ,max (A m-2)
∆j
Effective current density interval (A m-2)
KL
Heat leak coefficient (J m-2 K-1 s-1)
Kg
Thermal conductance of a thermoelectric element (W K-1 m-1)
K re
Heat-transfer coefficient (J m-2 K-1 s-1)
L
Thickness for the components (i.e., anode, cathode or electrolyte) of SOFC (m)
m
Number of pairs thermoelectric elements in the top stage of TTEG
n
Number of pairs thermoelectric elements in the bottom stage of TTEG
P* * Pmax
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P
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jP
Power output of the hybrid system (W)
Power density (W m-2) Maximum power density ((W m-2))
PB*
Power density at jB (W m-2)
PM*
Power density at jM (W m-2)
PH2
Partial pressures H2 (atm)
PO2
Partial pressures O2 (atm) 4
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Partial pressures H2 O (atm)
p
Operating pressure (atm)
Q1
Heat-transfer rate from the SOFC to the TTEG (J s-1)
Q2
Heat-transfer rate from the TTEG to the low temperature reservoir (J s-1)
Qm
Heat-transfer rate between the top stage TEGs and the bottom stage TEGs (J s-1)
R
Universal gas constant (J mol-1 K-1)
S
Cross-sectional areas of semiconductor arms (m2)
T
Operating temperature of SOFC (K)
TL
Temperatures of the low temperature reservoir (K)
Tm
Temperature of the cold junction of the top stage TEGs or the hot junction of the
Output voltage of an SOFC (V)
V act
Activation overpotential (V)
V con
Concentration overpotential (V)
V ohm
Ohmic overpotential (V)
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Z
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V
x
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bottom stage TEGs (K)
X
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PH2O
Ratio of the length of the grain contact neck to the grain size
Ratio of thermoelectric element numbers between the top stage and the bottom stage
Figure of merit of a thermoelectric element (K-1)
Greek symbols
α
Seebeck coefficient
β
Effectiveness of the regenerator
5
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Exchange current density pre-exponential factor (A m-2)
ε
Electrode porosity
η
Efficiency
ηB
Efficiency at jB
ηM
Efficiency at jM
ηP
* Efficiency at Pmax
θ
Temperature ratio between the SOFC and the low temperature reservoir
κ
Thermally conductivity of semiconductor materials (W K-1 m-1)
ξ
Electrode tortuosity
ρ
Electrical resistivity of semiconductor materials (Ω m)
σ
Electrical conductivity of the components of SOFC (Ω-1 m-1)
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γ
Subscripts
Anode; Cathode; Electrolyte
max
Maximum
N
N-type semiconductor
opt
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ohm
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a; c; e
Ohmic
Optimum
P
P-type semiconductor
SOFC
Solid oxide fuel cell
TTEG
Two-stage thermoelectric generator
6
ACCEPTED MANUSCRIPT
1. Introduction Fuel cells are more efficient energy conversion devices than conventional heat engines as they can directly convert the chemical energy of a fuel into electricity through electrochemical reactions
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[1]. Among the various types of fuel cells, solid oxide fuel cells (SOFCs) are promising candidates for stationary power generation due to their high operating temperature (typically about 800oC), fuel flexibility, and relatively low cost [2].
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The high operating temperature of SOFCs (600 °C-1000 °C) not only offers high
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electrochemical reaction rates but also generates significant amount of high-quality waste heat which can be used in cogeneration or CCHP (combined cooling, heating, and power) systems for performance improvement [3-5]. In the previous studies, various hybrid systems have been proposed to recover the waste heat from SOFC by using various thermodynamic cycles, such as Stirling cycle
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[6, 7], gas turbine cycle [8], Kalina cycle [9], organic Rankine cycle [10] and etc [11]. As solid-state devices, thermoelectric generators (TEGs) are more suitable for recovering the waste heat from SOFC than other thermodynamic cycles. As TEGs can directly convert heat into usable electricity
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without moving parts via the Seebeck effect, they are stable in operation, quiet, compact, and
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flexibility to various heat sources [12, 13]. However, the poor energy conversion efficiency of TEG hinders its widespread applications [14]. One way to improve the efficiency of TEG is to increase the figure of merit ( ZT ) of thermoelectric materials [15, 16]. Alternatively, the TEG efficiency can be increased by good thermal design and management of the TEG system [17]. Two-stage thermoelectric generators (TTEGs) are effective in enhancing the TEG efficiency by adding a bottom-stage TEG to recover the waste heat from the top-stage TEG, which have attracted much attention in recovering waste heat from high-temperature heat sources with large temperature gaps
7
ACCEPTED MANUSCRIPT [18-20]. Chen et al. [18] optimized the heat-transfer area at high-temperature side and the number of thermoelectric elements at the top stage to achieve maximum power output and thermal efficiency. Xiong et al. [19] built a physical and numerical model of a TTEG to harvest the waste heat contained
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in blast furnace slag water. It was found that the maximum power and maximum efficiency at 100 °C were 0.44 kW and 2.66%, respectively. Manikandan et al. [20] carried out an exergy analysis on an exoreversible and irreversible TTEG model in which the Thomson effect along with Peltier, Joule
SC
and Fourier heat conduction were considered. To now, although some pioneering study has been
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conducted to use TEG to harvest waste heat from high-temperature fuel cells such as SOFCs [21], phosphoric acid fuel cells [22] and molten carbonate fuel cells [23], all of them only considered single-stage TEGs. Tian et al. [24, 25] compared the two-stage and traditional single-stage TEGs in recovering the waste heat from high-temperature exhaust gas of internal combustion engines.
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Calculations showed that the maximum value of the output power of the serial TTEG was 10.9% larger than that of the single-stage TEG at 800 K, and the conversion efficiency of the serial TTEG was 12.4% larger than that of the single-stage TEG.
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The literature survey shows that the two-stage TEGs could be promising for recovering waste
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heat from SOFCs but no study has been done yet. As a result, it is still unknown how the number of TEGs in the top and bottom stage influence the system overall efficiency. To fill the research gap, this paper is purposely designed to explore a hybrid system that uses an output-oriented optimum TTEG to recover the waste heat from an SOFC for performance improvement. The performance of the hybrid system will be numerically evaluated by considering various irreversible losses such as electrochemical losses in SOFC, regenerative losses in regenerator, internal irreversible losses inside the TTEG, and heat losses from the SOFC to the surroundings. The feasibilities and advantages of
8
ACCEPTED MANUSCRIPT the TTEG as an exhaust heat recovery approach for SOFC will be demonstrated by comparing with other SOFC-based hybrid systems. The optimum operating region for some important performance parameters will be determined. Finally, extensive parametric studies will be conducted to investigate
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the influences of some design parameters and operating conditions on the hybrid system performance.
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2. System description
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The proposed hybrid system consists of an SOFC, a TTEG and a regenerator, as shown in Fig. 1. The TTEG consists of two stages TEGs with an external resistance RL . The regenerator preheats the inlet reactants with the help of the heat contained in the outlet products. The SOFC electrochemically converts the fuel chemical energy into electrical power, PSOFC , and high-grade waste heat. The waste
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heat is collected and transferred for three uses: a part of the waste heat, QR (J s-1), is transferred to compensate the regenerative losses in the regenerator; another part, QL (J s-1), is directly leaked from the SOFC at temperature T into the low temperature reservoir at temperature TL ; the rest part,
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Q1 (J s-1), is transferred from the SOFC to the TTEG for additional electrical power production. In
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addition, Qm (J s-1) is the heat-transfer rate between the top stage TEGs and the bottom stage TEGs, Q2 (J s-1) is the heat-transfer rate from the TTEG to the low temperature reservoir.
For simplicity, the following assumptions are adopted [18, 21, 25,26]: •
Both the SOFC and the TTEG are under steady state conditions;
•
Uniform operating temperature and pressure are assumed in the SOFC;
•
All reactants are completely consumed in the SOFC;
•
Geometric configuration of the TTEG is in the optimum form;
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ACCEPTED MANUSCRIPT •
Newton’s law is used to describe the heat transfers within the system;
•
Thermoelectric elements are insulated both electrically and thermally from their surroundings; Electric current flows along the arm of the TTEG;
•
TTEG is closely attached to the SOFC, and the external irreversible losses between the TTEG and the heat reservoirs are neglected;
Temperature of the cold junction of the top stage TEGs is the same as that of the hot
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•
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junction of the bottom stage TEGs; •
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•
Physical properties of TTEG such as Seebeck coefficient, electrical resistance, thermal conductance and dimensional figure of merit are independent of temperature;
•
Thompson effect in a thermoelectric element is neglected.
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2.1. SOFC
The typical anode-supported SOFC shown in Fig. 1 is made of Ni | YSZ | LSM with hydrogen as fuel and air as oxidant. The electrochemical performance of SOFC is deteriorated by activation,
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concentration, and ohmic overpotentials, which can be respectively described by the Bulter-Volmer
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equation, dusty gas model (DGM), and Ohm’s law. Adopting the electrochemical model previously developed in Refs. [27-29], the power output PSOFC and efficiency η SOFC for an SOFC can be given by
PSOFC = VI = jA ( E − Vact ,a − Vact ,c − Vcon ,a − Vcon ,c − Vohm ) ,
(1)
and
η SOFC =
PSOFC •
,
(2)
−∆ H
where 10
ACCEPTED MANUSCRIPT RT j j ln + = 2j 2 j0,l F 0,l
2 + 1
(l = a, c) ,
Vcon , a =
RT 1 + j /jlH2 O ln , 2 F 1 − j /jlH 2
Vcon , c =
PO2 RT ln 4 F Pc / δ O − ( − Pc / δ O − PO ) exp RTLc jδ O / (4 FPc DOeff ) 2 2 2 2 2
(
)
PH 2O Eact ,a exp − , RT Pref
72 X [ D p − ( D p + Ds )ε ]ε PO2 = γc Ds2 D p2 (1 − 1 − X 2 ) Pref
and •
−∆ H = − I ∆h (2 F ) ,
0.25
−E exp act ,c RT
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j0,c
72 X [ D p − ( Dp + Ds )ε ]ε PH 2 Ds2 D p2 (1 − 1 − X 2 ) Pref
,
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L L L Vohm = j a + c + e , σa σc σe j0,a = γ a
(3)
(4)
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Vact , l
j RT = sinh -1 2j F 0,l
,
(5)
(6)
(7)
(8)
(9)
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V is the output voltage. I is the electric current flowing through the SOFC. j is the operating current density. A is the effective polar plate area of an SOFC. E is the equilibrium potential.
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Vact , Vcon and Vohm are, respectively, the activation, concentration and ohmic overpotentials, and the subscripts “a”, “c” and “e” stand for anode, cathode and electrolyte. R is the universal constant.
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T is the operating temperature. F is the Faraday’s constant. j0 is the exchange current density. jlH2O = 2FPH 2O DHeff2O /RTALa and jlH2 = 2FPH2 DHeff2 /RTALa are the limiting current densities of H2O and H2 mass transfers, respectively. PH2 O and PH2 are partial pressures of H2O and H2 at the electrode surfaces, respectively. DHeff2O , DHeff2 , and DOeff2 are, respectively, the effective diffusion coefficients of H2O, H2 and O2. δ O2 = DOeff2 , Kn
(D
eff O 2 , Kn
)
+ DOeff2 -N2 , and DOeff2 , Kn is the effective
Knudsen diffusion coefficient of O2. DOeff2 -N2 is the effective molecular diffusion coefficient for O2-N2 binary system. L and σ are, respectively, the thickness and electrical conductivity for the 11
ACCEPTED MANUSCRIPT components of the SOFC (i.e., anode, cathode or electrolyte). X is the ratio of the length of the grain contact neck to the grain size. ε is the porosity of the electrodes. D p and Ds are the pore size and grain size, respectively. γ is exchange current density pre-exponential factor. Eact is the
•
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activation energy level. Pc is the cathode operating pressure. Pref is the reference pressure. (−∆ H ) = − I ∆h (2 F ) is the overall released energy (both electrical and thermal) per unit time. ∆h is the molar enthalpy change for the hydrogen-oxygen reaction.
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2.2. TTEG
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As shown in Fig. 1, the TTEG is operated between the SOFC and the low temperature reservoir. The TTEG is composed of a top stage with m pairs thermoelectric elements and a bottom stage with n pairs thermoelectric elements, and the two stages are electrically connected in serials. Each thermoelectric element consists of a P-type and an N-type semiconductor legs which are joined by a
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thin copper. Neglecting the Thomson effect, the internal irreversible losses inside the TTEG are mainly from the Joule heat and the heat-conduction losses. The amount of Joule heat is I g2 Rg , where I g is the electrical current flowing through the TTEG, and Rg = ( ρ PlP S P + ρ N lN S N ) is the
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internal electrical resistance of a thermoelectric element, ρ is the electrical resistivity, l is the
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length of the semiconductor arms, S is the cross-sectional areas of semiconductor arms, subscripts “P” and “N” indicate P-type and N-type semiconductors, respectively. The heat-conduction losses for the top and bottom stages can be, respectively, expressed as mK g (T − Tm ) and nK g (Tm − TL ) , where K g = (κ P S P lP + κ N S N lN ) is the thermal conductance of a thermoelectric element, κ is the thermally conductivity of the semiconductor materials, Tm is the temperature of cold junction of the top stage or hot junction of the bottom stage. When heat flows from the SOFC to TTEG, the heat balance equations can be expressed as [18, 25, 30]
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ACCEPTED MANUSCRIPT Q1 = α mI g T − 0.5mI g2 Rg + mK g ( T − Tm ) ,
(10)
Qm = α mI g Tm + 0.5mI g2 Rg + mK g ( T − Tm ) ,
(11)
Qm = α nI g Tm − 0.5nI g2 Rg + nK g (Tm − TL ) ,
(12)
Q2 = α nI g TL + 0.5nI g2 Rg + nK g (Tm − TL ) ,
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and (13)
where α P , α N and α = (α P − α N ) are the Seebeck coefficients for a P-type semiconductor leg, an
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N-type semiconductor leg and a thermoelectric element, respectively.
TTEG can be, respectively, expressed as
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Eliminating Tm in Eqs. (10) - (13), the power output PTTEG and efficiency ηTTEG of the
2 ( xθ − 1) i θ x + 1 ( x + 1) i ( 2ZTL ) + ( xθ + 1) i2 PTTEG = Q1 − Q2 = K g TL n ( x + 1) − + + , x + 1 2 ZT x + 1 x − 1 i − x + 1 ( ) ( ) L
and b + b i + b3i 2 + b4i 3 Q2 = 1− 1 2 , Q1 b1 + b5i + b6i 2 + b7 i 3
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ηTTEG = 1 −
(14)
(15)
where i = α I g / K g is the dimensionless electric current [23]. Z = α 2 / ( K g Rg ) is the figure of
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merit of a thermoelectric element [26]. θ = T / TL is the temperature ratio between the SOFC and
top
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the low temperature reservoir. x = m / n is the ratio of thermoelectric element numbers between the stage
and
the
bottom
stage,
b1 = (1 − θ ) x ,
b2 = −2 x ,
b3 = x − 1 − (1 + x ) ( ZTL ) ,
b4 = ( x − 1) ( 2ZTL ) , b5 = −2θ x , b6 = x ( x − 1) θ + x (1 + x ) ( ZTL ) , and b7 = − x ( x − 1) ( 2ZTL ) . For a given amount of heat from SOFC to TTEG Q1 and the total number of thermoelectric elements (m + n) , it is meaningful to determine the optimum allocation value for maximizing the power output of the TTEG. Substituting Eq. (14) into the extreme condition ∂PTTEG ∂x = 0 , one has
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ACCEPTED MANUSCRIPT xopt
i 3 − (2d1 − 1)i 2 − 2d1i + 2 i −i 4 + 2d1i 3 − d 2i 2 + d3i + 2d1d 4 = , i ( i − 1)( i − 2d1 )
(16)
where d1 = θ ZTL + 1 , d 2 = ZTL (θ + 1) , d 3 = ZTL ( 2θ 2 ZTL + 2θ ZTL + θ + 3) , d 4 = ZTL (θ − 1) . Replacing the symbol x in Eqs. (14) and (15) by xopt , the expressions for the maximum
PTTEG ,max = K g TL n ( xopt
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power output PTTEG ,max and the corresponding efficiency ηTTEG , m can be, respectively, given by 2 ( xoptθ − 1) i θ xopt + 1 ( xopt + 1) i ( 2 ZTL ) + ( xoptθ + 1) i2 + 1) − + + , 2ZTL xopt + 1 ( xopt − 1) i − ( xopt + 1) xopt + 1
where
f1 = (1 − θ ) xopt ,
f 2 = −2 xopt ,
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ηTTEG ,m
f1 + f 2i + f 3i 2 + f 4i 3 = 1− , f1 + f5i + f 6i 2 + f 7 i 3
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and
f 3 = xopt − 1 − (1 + xopt ) ( ZTL ) ,
f 5 = −2θ xopt , f 6 = xopt ( xopt − 1) θ + xopt (1 + xopt )
2.3. Regenerator
( ZTL ) , and
(17)
(18)
f 4 = ( xopt − 1) ( 2 ZTL ) ,
f 7 = − xopt ( xopt − 1) ( 2 ZTL ) .
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The regenerator in Fig. 1 works as a counter-flow heat exchanger that preheats the inlet reactants from temperature TL to the SOFC operating temperature T . As the regeneration process
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is not perfect, the rate of regenerative losses is often given by [23]
QR = K re Are (1 − ξ )(T − TL ) ,
(19)
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where K re , Are and ξ are the heat-transfer coefficient, heat-transfer area and effectiveness of the regenerator, respectively.
2.4. Performance of the hybrid system As the operating temperature of SOFC is higher than that of the low temperature reservoir, the rate of heat loss from the SOFC to the low temperature reservoir, QL , can be described by [21]
QL = K L AL (T − TL ) ,
(20)
where K L and AL are heat leak coefficient and the heat leak area, respectively. 14
ACCEPTED MANUSCRIPT Based on the energy conservation law, the heat-transfer rate Q1 from the SOFC to the TTEG can be calculated as •
Q1 = −∆ H − PSOFC − QR − QL = −
2 Fc1 (T − TL ) 2 Fc2 (T − TL ) A∆h − (1 − η SOFC ) j − , 2F −∆h −∆h
(21)
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where c1 = K re Are (1 − ξ ) A and c2 = K L AL A are temperature-independent composite constants which are associated with the regenerative losses and heat leak losses, respectively.
generate electricity only when the following formula is valid
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•
−∆ H − PSOFC > QR + QL + mK g (T − Tm ) .
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Considering the heat-conduction losses within the TTEG, the bottoming TTEG begins to
(22)
Combining Eq. (21), Eq. (22) could be further rewritten as
2F j > jB = ( c1 + c2 )(T − TL ) + nxopt K g (T − Tm ) / A , −∆ h 1 − η ( ) SOFC
(23)
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where jB is the lower bound operating current density from which the TTEG begins to work. Meanwhile, one may easily determine the maximum current density jM that allows the TTEG to work by combining Eqs. (14) - (21) and the condition of PTTEG > 0 . Thus, the TTEG in the hybrid
(24)
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j B < j < jM .
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system only works in the following region:
Therefore, the effective operating current density interval is given by ∆ j = jM − j B .
(25)
When the SOFC operates in the region of jB < j < jM , the numerical relationship between the operating current density of SOFC, j , and the dimensionless electric current of TEG, i , is determined by Eq. (26)
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ACCEPTED MANUSCRIPT
xopt i −
=
xopt i
2
2ZT
+ xopt
xopt + 1 2 i + xopt + 1/ θ xopt 2ZT + ( xopt − 1) i − ( xopt + 1)
2 Fc1 (T − TL ) 2 Fc2 (T − TL ) − A∆h − (1 − η SOFC ) j − . 2 FnK g T −∆h −∆h
(26)
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Thus, the power output P and efficiency η for the hybrid system can be, respectively, expressed as
( jB < j <
(j≤
jM )
jB or j ≥ jM )
( jB < (j≤
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and PSOFC + PTTEG ,max = • η= −∆ H = η SOFC
(27)
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= PSOFC + PTTEG ,max P= = PSOFC
j < jM )
(28)
jB or j ≥ jM )
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3. Generic performance characteristics and comparisons The typical parameters for SOFC modeling are adopted from Refs. [27-29], the values for γ a and γ c are calculated based on the experimental data recommended by Chan et al. [27]. The other
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parameters are listed in Table 1 [23, 25, 28, 29].
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Figure 2 shows the generic performance characteristics and the performance comparisons for the SOFC, the TTEG and the proposed hybrid system. It is observed that the power densities of SOFC, TTEG and proposed hybrid system first increase and then decrease with increasing current density j , and the TTEG delivers electricity only in the region of jB < j < jM . As distinct from power densities, η SOFC decreases as j increases in the whole region of j , while ηTTEG first increases and then decreases in the region of jB < j < jM , η first rapidly decreases then somewhat * * is larger than PSOFC increases and finally decreases as j increases. It is also seen that Pmax ,max and
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ACCEPTED MANUSCRIPT * PTTEG ,max , and jP is always different from j fc as the SOFC and the TTEG achieve their maximum
power densities at different current densities. Furthermore, the hybrid system efficiency at jP is larger than the SOFC efficiency at j fc . Adopting the thermoeconomic optimization criterion [23],
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the optimum operating regions for current density, power density and efficiency are suggested to be * situated in jB < j ≤ jP , PB* < P* ≤ Pmax and η B > η ≥ η P , respectively.
The performance comparisons between the two-stage TEG and the single-stage TEG as
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bottoming thermal devices are shown in Fig. 3 (a), where both the numbers of thermoelectric
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elements in two-stage TEG and the single-stage TEG are n (1 + xopt ) . It is evidently observed that the maximum power density of the former is larger than that of the latter. For the parameters given in Table 1, the maximum power density of the TTEG integrated hybrid system is about 22.3% larger than that of the single-stage TEG integrated hybrid system. The values of jB and jM of the TTEG
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are larger than that of the single-stage TEG, and the value ∆j of the former is also larger than that of the latter. Moreover, the value of jP for the SOFC/TTEG hybrid system is larger than that for the SOFC/single-stage TEG hybrid system. Outside the region of jB < j < jM , the performance of the
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two hybrid systems is the same as that of the stand-alone SOFC.
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For further comparison purposes, the emerging solid state thermophotovoltaic cell (TPVC) [3] and thermionic generator (TIG) [31] are employed in waste heat recovery from the SOFC in Fig. 1. With reference to the stand-alone SOFC, the maximum power density improvements of SOFC-based * * * hybrid systems (i.e., [( Pmax − PSOFC ,max ) / PSOFC ,max ] × 100% ) are clearly compared in Fig. 3 (b). It is
observed that the improvement of the proposed hybrid system is smaller than that of the hybrid systems integrated with TPVC and TIG at high operating temperatures, but it is always greater than the improvement of the SOFC/single-stage TEG hybrid system. Since lowering the operating
17
ACCEPTED MANUSCRIPT temperature of SOFCs will be an important R&D trend in the future [32-34], TTEG may be treated as a potential waste heat recovery candidate for SOFCs.
4. Parametric studies
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4.1. Effects of number of thermoelectric elements in bottom stage of TTEG When x is operated in the optimum form, a larger number of n means more thermoelectric elements are employed for additional electricity production, which improves the performance TTEG
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* and the overall performance of the hybrid system. As shown in Fig. 4, Pmax increases with the
increasing n , and simultaneously, the values of jB , jM , ∆j and jP shift to larger ones. The
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effect of n only occurs in the region of jB < j < jM and becomes more pronounced in the region of jP < j < jM . In practice, the thermoelectric elements should be as more as possible, and the thermoelectric elements are suggested to be cascaded in the multi-stage form.
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4.2. Effects of SOFC operating temperature
The operating temperature T not only affects the performance of the SOFC and TTEG but also affects the amount of heat transferred from the SOFC to the TTEG. It is seen from Fig. 5 that
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both P * and η increase as T increases, and this effects occur in the entire region of j and
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become more significant at an elevated T . The values of jP , jS , jB , jM and ∆ j increase with increasing T . A larger T not only improves the performance of the SOFC but also improves the performance of the TTEG. As described by Eqs. (19) and (20), a larger T also leads to larger thermodynamic losses. Because the performance enhancements in the SOFC and TTEG are more significant than the performance deterioration resulting from the thermodynamic losses, a larger T is always preferable. However, a high operating temperature of the SOFC may result in higher system costs as well as slow start-up and shutdown cycles.
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ACCEPTED MANUSCRIPT 4.3. Effects of SOFC operating pressure Similar to the effect of operating temperature T , the performance of the SOFC dramatically depends on the operating pressure p in the whole region of j . Figure 6 shows that P * and η
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* * increase with p , and the values of Pmax , PSOFC jP , jS , jB , jM and ∆ j are increased ,max ,
with increasing p . The effects of p on the system performance become more significant at larger current densities. A larger operating pressure is more preferable for performance improvement, but it
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practice, as shown by the black solid lines in Fig. 6.
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also consumes some electricity to compress the inlet reactants, and thus 1.0 atm is usually chosen in
4.4. Effects of thermal conductance of a thermoelectric element
Different from T and p , the thermal conductance K g only affects the system performance in the region of jB < j < jM , as shown in Fig. 7. Outside this region, the curves of P* ~ j and
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* * η ~ j are overlapped with that of the PSOFC ~ j and η SOFC ~ j , respectively. The values of Pmax ,
jP , jB , jM and ∆ j increase as K g increases, the effects of K g on the system performance become more significantly at larger current densities. However, a larger K g needs higher
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requirement for materials fabrication, which often leads to higher manufacture cost.
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4.5. Effects of figure of merit of a thermoelectric element The performance of a TTEG depends on the figure of merit value of a thermoelectric element [35]. With the fast development of thermoelectric materials, the dimensionless figures of merit larger than 1.0 have been already reported [36]. It is necessary and meaningful to discuss the effects of the dimensionless figures of merit on the performance of the overall system. As shown in Fig. 8, both P* and η are increased with the increasing ZTL . The effect of ZTL only occurs in the region of * jB < j < jM , the values of Pmax , jB , and jP shift to larger ones as ZTL increases, while the value
19
ACCEPTED MANUSCRIPT of jM almost keeps invariant and the value of ∆ j decreases as ZTL increases. In the literatures, numerous efforts have been made to achieve a greater Z [37].
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4. Conclusions A novel hybrid system consisting of a TTEG and an SOFC is proposed to recover the waste heat from the SOFC. A theoretical model is derived to evaluate the hybrid system performance by
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considering the various irreversible losses. An analytical expression is obtained to determine the
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optimal numbers of TEG in the top and bottom stage of the TTEG. A numerical relationship for the output electric currents of the SOFC and the TTEG is derived, and the current density interval of SOFC that enables the TTEG to work is determined. The performance parameters for the hybrid system are specified under different operating conditions, and the generic performance
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characteristics are demonstrated. From extensive comparisons with other SOFC-based hybrid systems, the proposed hybrid system using TTEG to recovery the waste heat from SOFC is found feasible and effective. Comprehensive parametric studies are conducted to discuss the effects of
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operating temperature and pressure of SOFC, number of the thermoelectric element, thermal
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conductance, and figure of merit of the TTEG on the hybrid system performance. The results obtained are useful for the design and optimization of SOFC-based cogeneration systems to achieve a better performance.
20
ACCEPTED MANUSCRIPT
Acknowledgments This work has been supported by the Natural Science Foundation of Zhejiang Province (Grant No. LQ14E060001), National Natural Science Foundation of China (Grant No. 51406091), and The
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Hong Kong Polytechnic University Research Project (Grant No. 1-YW1F).
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References [1] Sharaf OZ, Orhan MF. An overview of fuel cell technology: fundamentals and applications. Renew Sust Energ Rev 2014; 32: 810-53.
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[2] Mekhilef S, Saidur R, Safari A. Comparative study of different fuel cell technologies. Renew Sust Energ Rev 2012; 16:981-9.
[3] Liao T, Cai L, Zhao Y, Chen J. Efficient exploiting the waste heat in solid oxide fuel cell by
SC
means of thermophotovoltaic cell. J Power Sources 2016; 306: 666-73.
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[4] Saebea D, Magistri L, Massardo A, Arpornwichanop A. Cycle analysis of solid oxide fuel cell-gas turbine hybrid systems integrated ethanol steam reformer: Energy management. Energy (2017), doi:10.1016/j.energy.2017.03.105.
[5] AI-Sulaiman FA, Dincer I, Hamdullahpur F. Exergy analysis of an integrated solid oxide fuel cell
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and organic Rankine cycle for cooling, heating and power production. J Power Sources 2010; 195: 2346-54.
[6] Rokni M. Thermodynamic analyses of municipal solid waste gasification plant integrated with
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solid oxide fuel cell and Stirling hybrid system. Int J Hydrogen Energy 2015; 40:7855-69.
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[7] Chen L, Gao S, Zhang H. Performance analysis and multi-objective optimization of an irreversible solid oxide fuel cell-stirling heat engine hybrid system. Int J Electrochem Sci 2013; 8:10772-87.
[8] Buonomano A, Calise F, d’Accadia MD, Polombo A, Vicidomini M. Hybrid solid oxide fuel cells–gas turbine systems for combined heat and power: a review. Appl Energy 2015; 156:32-85. [9] Pierobon L, Rokni M. Thermodynamic Analysis of an Integrated Gasification Solid Oxide Fuel Cell Plant with a Kalina Cycle. Int J Green Energy 2015;12:610-9.
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ACCEPTED MANUSCRIPT [10] Pierobon L, Rokni M, Larsen U, Haglind F. Thermodynamic analysis of an integrated gasification solid oxide fuel cell plant combined with an organic Rankine cycle. Renew Energy 2013;60:226-34.
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[11] Mehrpooya M, Dehghani H, Moosavian SMA. Optimal design of solid oxide fuel cell, ammonia-water single effect absorption cycle and Rankine steam cycle hybrid system. J Power Sources 2016; 306:107-23.
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[12] Date A, Date A, Dixon C, Akbarzadeh A. Progress of thermoelectric power generation systems:
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Prospect for small to medium scale power generation. Renew Sust Energ Rev 2014; 33: 371-81. [13] Song H, Cai K. Preparation and properties of PEDOT:PSS/Te nanorod composite films for flexible thermoelectric power generator. Energy 2017; 125:519-25.
[14] Gou X, Xiao H, Yang S. Modeling, experimental study and optimization on low-temperature
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waste heat thermoelectric generator system. Appl Energy 2010; 87: 3131-6. [15] Zhang Q, Sun Y, Xu W, Zhu D. Organic thermoelectric materials: Emerging green energy materials converting heat to electricity directly and efficiently. Adv Mat 2014; 26: 6829-51.
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[16] Zeier WG, Schmitt J, Hautier G, Aydemir U, Gibbs ZM, Felser C, Snyder J. Engineering
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half-Heusler thermoelectric materials using Zintl chemistry. Nat Rev Mat 2016; 1: 16032. [17] Xiao J, Yang T, Li P, Zhai P, Zhang Q. Thermal design and management for performance optimization of solar thermoelectric generator. Appl Energy 2012; 93: 33-8. [18] Chen L, Li J, Sun F, Wu C. Performance optimization of a two-stage semiconductor thermoelectric-generator. Appl Energy 2005; 82: 300-12. [19] Xiong B, Chen L, Meng F, Sun F. Modeling and performance analysis of a two-stage thermoelectric generator energy harvesting system from blast furnace slag water waste heat. Energy
23
ACCEPTED MANUSCRIPT 2014;77:562-9. [20] Manikandan S, Kaushik SC. The influence of Thomson effect in the performance optimization of a two stage thermoelectric generator. Energy 2016; 100: 227-37.
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[21] Chen X, Pan Y, Chen J. Performance and evaluation of a fuel cell-thermoelectric generator hybrid system. Fuel Cells 2010; 10: 1164-70.
[22] Chen X, Wang Y, Cai L, Zhou Y. Maximum power output and load matching of a phosphoric
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acid fuel cell-thermoelectric generator hybrid system. J Power Sources 2015;294:430-6.
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[23] Wu S, Zhang H, Ni M. Performance assessment of a hybrid system integrating a molten carbonate fuel cell and a thermoelectric generator. Energy 2016;112:520-7. [24] Sun X, Liang X, Shu G, Tian H, Wei H, Wang X. Comparison of the two-stage and traditional single-stage TEG in recovering the waste heat from high temperature exhaust gas of internal
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combustion engine. Energy 2014;77:489-98.
[25] Liang X, Sun X, Tian H, Shu G, Wang Y, Wang X. Comparison and parameter optimization of a two-stage thermoelectric generator using high temperature exhaust of internal combustion engine.
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Appl Energy 2014;130:190-9.
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[26] Zhao M, Zhang H, Hu Z, Zhang Z, Zhang J. Performance characteristics of a direct carbon fuel cell/thermoelectric generator hybrid system. Energy Convers Manag 2015; 89: 683-9. [27] Chan SH, Khor KA, Xia ZT. A complete polarization model of a solid oxide fuel cell and its sensitivity to the change of cell component thickness. J Power Sources 2001;93:130-40. [28] Ni M, Leung MKH, Leung DYC. Parametric study of solid oxide fuel cell performance. Energy Convers Manag 2007;48:1525-35. [29] Zhang H, Chen J, Zhang J. Performance analysis and parametric study of a solid oxide fuel cell
24
ACCEPTED MANUSCRIPT fueled by carbon monoxide. Int J Hydrogen Energy 2013;38:16354-64. [30] Temizer İ, İlkılıç C. The performance and analysis of the thermoelectric generator system used in diesel engines. Renew Sust Energ Rev 2016;63:141-51.
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[31] Wang Y, Cai L, Liu T, Wang J, Chen J. An efficient strategy exploiting the waste heat in solid oxide fuel cell system. Energy 2015;93:900-7.
[32] Wachsman ED, Lee KT. Lowering the temperature of solid oxide fuel cells. Science
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2011;334:935-9.
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[33] Dong F, Ni M, He W, Chen Y, Yang G, Chen D, Shao Z. An efficient electrocatalyst as cathode material for solid oxide fuel cells: BaFe0.95Sn0.05O3−δ. J Power Sources 2016; 326:459-65. [34] Lv X, Gu C, Liu X, Weng Y. Effect of gasified biomass fuel on load characteristics of an intermediate-temperature solid oxide fuel cell and gas turbine hybrid system. Int J Hydrogen Energy
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2016;41:9563-76.
[35] Fahrnbauer F, Souchay D, Wagner G, Oeckler O. High thermoelectric figure of merit values of
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germanium antimony tellurides with kinetically stable cobalt germanide precipitates. J Am Chem Soc 2015;137:12633-8.
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[36] Venkatasubramanian R, Siivola E, Colpitts T, O’Quinn B. Thin-film thermoelectric devices with high room temperature figures of merit. Nature 2001;413:597-602. [37] Ovik R, Long B D, Barma M C, Riaz M, Sabri MFM, Said SM, Saidur R. A review on nanostructures of high-temperature thermoelectric materials for waste heat recovery. Renew Sust Energ Rev 2016; 64:635-59.
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Table captions: Table 1. Parameters used in the modeling [23, 25, 28, 29].
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Figure captions:
Fig. 1. The schematic diagram of an SOFC/two-stage thermoelectric generator hybrid system. Fig. 2. Comparisons of (a) power densities and (b) efficiencies between the SOFC, TTEG and hybrid
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* * system, where PSOFC = PSOFC / A , PTTEG = PTTEG / A , P * = P / A are, power densities for
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* * * SOFC, TTEG and hybrid system, respectively; PSOFC ,max , PTTEG ,max and Pmax are the
corresponding maximum power densities; ηTTEG ,max is the maximum efficiency of the TTEG; * j P and η P are operating current density and efficiency at Pmax , respectively; j fc is the * operating current density at PSOFC jtg and jη ,tg are operating current densities at ,max ;
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* * PTTEG ,max and ηTTEG ,max , respectively; PB and η B are power density and efficiency of the
hybrid system at jB , respectively; jS is the stagnation current density from which the
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SOFC does not output electricity any more. Fig. 3. Performance comparisons between (a) two-stage and single-stage TEGs, and (b) some solid
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state thermal devices as bottoming thermal devices for SOFCs. Fig. 4. Effects of number of the thermoelectric element n on the performance of the hybrid system. Fig. 5. Effects of operating temperature T on the performance of the hybrid system. Fig. 6. Effects of operating pressure p on the performance of the hybrid system. Fig. 7. Effects of thermal conductance K on the performance of the hybrid system. Fig. 8. Effects of figure of merit of thermoelectric materials ZTL on the performance of the hybrid system. 26
ACCEPTED MANUSCRIPT Table 1 Value
Operating pressure, p (atm)
1.0
Operating temperature, T (K)
1073
Faraday’s constant, F (C mol-1)
96485
Anode interface gas compositions
95 % H2 +5 % H2O
Cathode interface gas compositions
79 % N2 +21% O2
Activation energy for anode, Eacta (J mol-1)
1.0×105 [28]
Activation energy for cathode, Eactc (J mol-1)
1.2×105 [28]
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Parameter
Electrode porosity, ε
0.48 [28]
Average pore diameter, Dp (m)
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5.4 [28]
Electrode tortuosity,ξ
3.0 ×10-6 [28] 1.5 ×10-6 [28]
Average grain size, Ds (m) Average length of grain contact, X
0.7 [28]
5.0×10-4
Anode thickness, La (m)
8.0×104 [29]
Cathode thickness, Lc (m)
5.0×10-5
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Anode electric conductivity, σa (Ω-1 m-1)
8.4×103 [29]
Electrolyte thickness, Le (m)
5.0×10-5
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Cathode electric conductivity, σc (Ω-1 m-1)
3.34×104 exp(-1.03×104/T) [29]
Effective surface area of the SOFC, A (m2)
4.0×10-2
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Electrolyte ionic conductivity, σe (Ω-1 m-1)
Heat conductivity of a thermoelectric element, K g (W -1
-1
K m )
0.09
Figure of merit of the thermoelectric materials, ZTL
1.0 [23]
Number of TEGs in the second stage, n
8
Sectional area of a thermoelectric element
5.0×10-3 [25]
Constants in Eq. (21), c1 ; c2 (W m-2 K-1)
0.1; 0.1
Temperature of low-temperature reservoir, TL (K)
298
27
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Fig. 1.
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Products Regenerator 28
ACCEPTED MANUSCRIPT Fig. 2.
(a)
P*max
SOFC TTEG Hybrid system
7500 6000 4500
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P*SOFC,max
P*B
P
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3000 * TTG,m ax
1500 jP 0
jtg
jB 0
5000
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P*SOFC , P*TTEG , P* (W m-2)
9000
jfc jM
jS
10000 15000 20000 25000 30000
j (A m-2)
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(b) 1.0
TTEG Hybrid system
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ηB 0.6
ηP
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ηSOFC , ηTTEG , η
0.8
SOFC
0.4
ηTTG,max 0.2 0.0
jB 0
5000
jη,tg jP
jM
jS
10000 15000 20000 25000 30000
j (A m-2) 29
ACCEPTED MANUSCRIPT Fig. 3.
(a)
1.0 Two-stage TEG Single-stage TEG
6000
0.6
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4000
0
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2000 jP
0
5000
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70
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50
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Improvement (%)
60
Single-stage TEG TTEG TPVC [3] TIG [31]
40 30 20 10 0
960
0.2
0.0 10000 15000 20000 25000 30000
j (A m-2)
(b)
0.4
1040
1120
T (K) 30
1200
1280
η
P* (W m-2)
0.8
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8000 P
* max
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10000
1.0 n=6 n=8 n = 10
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6000
0.6
2000
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4000
0
0.8
jP
0
5000
0.4 0.2
0.0 10000 15000 20000 25000 30000
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j (A m-2)
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η
P* (W m-2)
8000 P*max
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16000
1.0
12000
0.6
8000 P*max
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η
4000
0
jP
0
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P* (W m-2)
0.8
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T = 973 K T = 1073 K T = 1173 K
jS
0.4 0.2
0.0 10000 20000 30000 40000 50000 60000
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j (A m-2)
32
ACCEPTED MANUSCRIPT Fig. 6.
12500
1.0 p = 1.0 atm p = 2.0 atm p = 3.0 atm
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0.8
7500
0.6
η
2500 0
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5000
jP 0
10000
20000
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j (A m-2)
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P* (W m-2)
10000 P*max
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30000 jS
0.4 0.2
0.0 40000
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1.0
12000
Kg = 0.05 W K-1 m-1
10000
Kg = 0.13 W K-1 m-1 0.8
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Kg = 0.09 W K-1 m-1 Kg = 0.17 W K-1 m-1
P*max
8000
0.6
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6000
2000 0
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4000
jP
0
5000
0.4 0.2
0.0 10000 15000 20000 25000 30000
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j (A m-2)
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η
P* (W m-2)
Fig. 7.
ACCEPTED MANUSCRIPT Fig. 8.
ZTL = 1.0
P*max 8000
ZTL = 1.2
0.8
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10000
ZTL = 1.4
0.6
6000
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η
4000 2000 0
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P* (W m-2)
1.0
jP
0
5000
0.4 0.2
0.0 10000 15000 20000 25000 30000
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j (A m-2)
35
ACCEPTED MANUSCRIPT Two-stage thermoelectric generators are proposed to recover waste heat from SOFCs. Two-stage thermoelectric generators are optimized by power outputoriented criterion.
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Performance parameters for assessing the hybrid system are specified.
Multiple comparisons are conducted to verify the feasibility of the hybrid
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system.
Effects of some important parameters on the system performance are
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discussed.