- Email: [email protected]
heat transfer and structural stress in a planar solid oxide fuel cell
i n t e r n a t i o n a l j o u r n a l o f h y d r o g e n e n e r g y x x x ( 2 0 1 7 ) 1 e1 0
Available online at www.sciencedirect.com
ScienceDirect journal homepage: www.elsevier.com/locate/he
Numerical investigation of flow/heat transfer and structural stress in a planar solid oxide fuel cell Young Jin Kim a,1, Min Chul Lee b,* a
Fluid System Engineering Department, KEPCO Engineering and Construction Company, 111, Daedeok-daero 989 Beon-Gil, Yusung-Gu, Taejeon, 34057, Republic of Korea b Department of Safety Engineering, Incheon National University, 119 Academy-ro, Yeonsu-gu, Incheon, 22012, Republic of Korea
article info
abstract
Article history:
The present work investigates the effects of the temperature and thermal stress distri-
Received 18 January 2017
butions in a planar solid oxide fuel cell (SOFC) unit cell. A computational fluid dynamic
Received in revised form
(CFD) analysis of a planar anode-supported SOFC that considers electrochemical reactions
16 March 2017
is performed, and the thermal stresses are calculated. The static friction coefficients are
Accepted 18 April 2017
assumed to range from 0.05 to 0.3, and conservatively, a perfectly bonded condition is
Available online xxx
assumed. The results show that the electrolyte is the weakest component and has the maximum stress because the electrolyte is the thinnest and the Young modulus is the
Keywords:
highest. Thus, the contact between the anode electrode and the electrolyte, and between
Solid oxide fuel cell
the cathode electrode and the electrolyte, would be the perfectly bonded condition. As a
Temperature
result, this research showed that the stresses induced by constraint forces with various
Stress
contact conditions were dominant for the structural stability in a SOFC. Therefore, static
Friction coefficient
friction coefficients on operative high temperature conditions are important to predict the structural integrity in a SOFC, and they will be investigated in future works in order to improve the structural stability in a stack design as well as in a SOFC. © 2017 Hydrogen Energy Publications LLC. Published by Elsevier Ltd. All rights reserved.
Introduction Fuel cells are an electricity generation system that converts chemical energy from oxidation of a fuel directly into electric energy. Solid oxide fuel cells (SOFCs) are more efficient compared with other fuel cells (such as molten carbonate fuel cells and phosphoric acid fuel cells), have little pollution, and do not require a fuel reformer. In addition, the fuel cell can be combined with a gas turbine or a steam turbine. SOFCs can be classified as planar, cylindrical, or integral according to the geometric shape.
Briefly, an SOFC, in which the hydrogen supplied to the anode electrode, is separated into hydrogen ions and electrons. The electrons move to the cathode electrode through an external circuit, and the oxygen supplied to the cathode electrode becomes oxygen ions by receiving electrons through an external circuit. After the oxygen ions migrate to the anode electrode through the electrolyte, the oxygen ions bond with the hydrogen ions and generate water. Thus, the final reaction produces the electricity and water by which the hydrogen and the oxygen bond. The power generation efficiency of the SOFC is 50%e60%, and the efficiency increases to more than 70% when the SOFC
* Corresponding author. Fax: þ82 32 835 0779. E-mail address: [email protected] (M.C. Lee). 1 Fax: þ82 42 861 4859. http://dx.doi.org/10.1016/j.ijhydene.2017.04.140 0360-3199/© 2017 Hydrogen Energy Publications LLC. Published by Elsevier Ltd. All rights reserved. Please cite this article in press as: Kim YJ, Lee MC, Numerical investigation of flow/heat transfer and structural stress in a planar solid oxide fuel cell, International Journal of Hydrogen Energy (2017), http://dx.doi.org/10.1016/j.ijhydene.2017.04.140
2
i n t e r n a t i o n a l j o u r n a l o f h y d r o g e n e n e r g y x x x ( 2 0 1 7 ) 1 e1 0
Nomenclature SOFC CFD E E0 Erev F FEM . g hi DH i Icell Ji k keff kf ks mi p R Ri Rirr T ! u Vcell Wi Yi ha hc hohm r
Solid oxide fuel cell Computational fluid dynamics Total energy [J] Standard state potential [volts] Nernst potential [volts] Faraday constant [96,485.3 C mol1] Finite element method Gravitational force [m/s2] Enthalpy of species i [J/kg] Enthalpy change of reaction or adsorption [kJ/ mol] Electrical current density [amps/m2] Total cell electrical current [amps] Diffusion flux of species i [kg/m2$s] Thermal conductivity [W/m$K] Effective thermal conductivity [W/m$K] Thermal conductivity of fluid [W/m$K] Thermal conductivity of solid [w/m$K] Mass fraction of species i Pressure [N/m2] Universal gas constant [8.314 J/mol$K] Gas reaction rate of species i [mol/m3$s] Total cell irreversible resistance [Um2] Temperature [K] Velocity [m/s] Cell Voltage [volts] Molecular weight of species i [kg/mol] Mole fraction of species i Anode activation impedance [Um2] Cathode activation impedance [Um2] Cell ohmic resistance [Um2] Density [kg/m3]
temperature distribution can aggravate the destruction and leakage. Typical numerical studies for SOFCs were carried out by Wanga et al. [1], Thinh el al. [2], Hussaina [3], Achenbach [4], and Ferguson et al. [5]. A useful formula for the performance in the molten carbonate fuel cell has been derived from an electric circuit model [6,7]. The present work was carried out to investigate the effects of the temperature and thermal stress distributions in a planar SOFC. A computational fluid dynamics (CFD) analysis of a planar anode-supported SOFC considering electrochemical reactions was performed under operating conditions in which the average current density varied from 0 to 2000 A/m2. Based on the temperature distributions obtained from the CFD analysis, structural stress analysis using a finite element method (FEM) was also performed using one-way FluideStructure Interaction (FSI) analysis. An uncertain factor during the structural stress analysis is determining the bonding conditions between the anode or cathode electrode and the electrolyte. To consider these bonding conditions, this work used specific static friction coefficients and perfect bonding.
Numerical models To facilitate the present numerical simulation of the planar SOFC, the simulation work is divided into three parts. The first part corresponds to the gas flow in a current collector and a gas channel. The second part is the electrochemical reaction in the area that contains the electrodes and electrolyte. The third part is the thermal stress problem within the SOFC unit cell induced by the non-uniform temperature distribution. A separate numerical model is formulated for each part.
Governing momentum, heat and mass transfer equations is combined with a gas turbine or a steam turbine. In addition, natural gas, coal gas, and waste gas can be used for various fuels. NOx or SOx as the environmental pollution factor of exhaust gas is very small, there is little dust, and the CO2 emissions generated are 60% less than that generated by pulverized coal-fired power. Because an SOFC can be installed in the vicinity of the city center due to the cell's quiet operation, the transmission and distribution power loss are less. In addition, an SOFC can be used for various applications depending on the power supply system, such as the local mounting, decentralized, or centralized. However, because the SOFC production cost is high, combining SOFCs with peripheral devices is difficult due to the high temperature (~1000 C), and durability is an issue during long operations, much research is in progress. Especially, high-temperature operation can result in nonuniform temperature distribution in a solid oxide fuel cell. The metallic separator and the ceramic components, such as the anode electrodes, cathode electrodes, and electrolytes, can be destroyed and leak due to the different thermal expansion coefficients induced by the high temperature. Therefore, it is important to calculate the temperature distribution and the associated thermal stress distribution induced by the thermal expansion because the critical
The conservation equations for fluid mass, momentum, energy, and species are as follows. Continuity equation : Momentum equation :
V$ðr! uÞ ¼ 0
(1) . .
. V$ðr! u! u Þ ¼ V$p þ V t þ r g
Energy equation : V$ð! u ðrE þ pÞÞ X X . . þ hi Ri ¼ V$ keff VT þ hi J i þ u $teff
(2)
(3)
where; keff ¼ fkf þ ð1 fÞks Species equation :
. V$ðr! u mi Þ ¼ V$ J i þ Ri
(4)
where the state equation is given by, State equation :
p ¼ rRT
X Yi Wi
(5)
The separator, designed as a current collector and a gas channel, is equipped with periodic step-shaped supports on the inner surface as illustrated in Fig. 1 that depicts the channel configuration.
Please cite this article in press as: Kim YJ, Lee MC, Numerical investigation of flow/heat transfer and structural stress in a planar solid oxide fuel cell, International Journal of Hydrogen Energy (2017), http://dx.doi.org/10.1016/j.ijhydene.2017.04.140
3
i n t e r n a t i o n a l j o u r n a l o f h y d r o g e n e n e r g y x x x ( 2 0 1 7 ) 1 e1 0
Electrochemical reactions
Cell Voltage :
The following electrochemical reactions occur at the fuel cell electrodes: Anode :
H2 þ O2 /H2 O þ 2e
Vcell ¼ Erev iðha þ hc þ hohm Þ ¼ Erev iRirr
Here, Erev is the Nernst potential that accounts for the various species concentrations. It is evaluated by
(6) Nernst potential :
Cathode :
Overall :
1 O2 þ 2e /O2 2 1 H2 þ O2 /H2 O 2
(7)
(8)
The half-cell reactions take place in the anode and the cathode of the SOFC. Since the electrolyte ions are consumed in the anode and are generated in the cathode at equal rates, they are conserved within the cell. The real potential of the fuel cell depends on the thermodynamic reversible potential, the reactant species concentrations at the electrodes, and the irreversible losses of the cell due to the electrical current and kinetic rate limits. An equation for calculating the real cell potential is given by
(9)
Erev
0 qffiffiffiffiffiffiffi 1 PaH2 PcO2 RT 0 @ A þ RT ln P0 ¼E þ 2F 4F PaH2 O
(10)
Activation polarizations, ha and hc , in Eq. (9) represent a decrease from the ideal voltage due to the limit on the rate of chemical kinetics at the electrode surface. The ohmic loss,hohm , in Eq. (9) represents a decrease from the ideal voltage due to the electrical resistance caused by the current flow. Correlation equations available for SOFC [4,5] are. Ohmic losses: hohm ¼
dc da de dsep þ þ þ sc sa se ssep
where;
4:2 107 1200 exp T T 9:5 107 1150 exp sa ¼ T T 10300 se ¼ 3:34 104 exp T sc ¼
ssep ¼
(11)
1 1:07 106
Activation losses in the anode reactions me 1 2F PH2 Ea ¼ ka exp 0 ha RT P RT
(12)
where; Ea ¼ 110 kJ=mol Activation losses in the cathode reactions: me 1 4F PO2 Ec kc ¼ exp hc RT P0 RT
(13)
where; Ea ¼ 160 kJ=mol Charge conservation must be satisfied with the current generated by electrochemical reactions, which is expressed by the electric circuit model given in Ref. [6]. In Fig. 3, Erev ðx; yÞ and Rirr ðx; yÞ are the electromotive force and the irreversible losses, respectively. Each grid element through the electrodeseelectrolyte is connected to a parallel circuit, where the current passes along the vertical direction [4]. Current profiles are calculated by applying Kirchhoff's law that, neglecting the transverse current flow, considers the parallel connections within the fuel cell plates. The cell voltage must be the same in each grid element, which is derived from Eq. (10) as follows. Voltage Current Relation :
Vcell ¼
ðx;yÞ ∬ ERrevðx;yÞ dxdy Icell irr
1 ∬ Rirr ðx;yÞ dxdy
(14)
where; Icell ¼ ∬ iðx; yÞdxdy
Structural analysis Fig. 1 e SOFC unit cell frame. (a) Schematic diagram (b) Control volume for CFD analysis.
The relationship between the stress tensor considering the thermal deformation and the stain tensor is as follows:
Please cite this article in press as: Kim YJ, Lee MC, Numerical investigation of flow/heat transfer and structural stress in a planar solid oxide fuel cell, International Journal of Hydrogen Energy (2017), http://dx.doi.org/10.1016/j.ijhydene.2017.04.140
4
i n t e r n a t i o n a l j o u r n a l o f h y d r o g e n e n e r g y x x x ( 2 0 1 7 ) 1 e1 0
Fig. 2 e Constraints for FEM modeling.
2
Table 1 e Parameters for the thermal stress analysis of an SOFC [8]. Component
Thermal expansion coefficients ðaÞ [1/ C] [dimension of k]
Anode Electrolyte Cathode Separator (current collector)
3g
1.26 1.09 1.23 1.70
105 105 105 105
¼ f3 th þ ½D1 fsg
s : stress
½D1
nxy Ex 1 E y nzy Ez 0 0 0
nxz =Ex nyz Ey 1=Ez 0 0 0
0 0 0 1 Gxy 0 0
0 0 0 0 1 Gyz 0
3 0 0 7 7 0 7 7 0 7 7 0 5 1=Gxz
The temperature distribution obtained with the CFD analysis was used to calculate the thermal stress distribution in the SOFC unit cell using the finite element method (FEM) [9,10]. The temperature distribution was applied to the FEM grid data package to assign the thermal loads. The reference temperature was assumed to be 22 C. The SOFC materials in this research were assumed to be representative components as follows: - Cathode material: Lanthanum Strontium Manganite (LSM) - Electrolyte material: Yttria-Stabilized Zirconia (YSZ) - Anode material: Ni/YSZ cermet
: strain
o f3 th ¼ DT ax
87.5 145 50 193
(15)
Where,
3
Young's modulus ðEÞ [GPa]
1=E x 6 nyx Ey 6 6 nzx =Ez ¼6 6 0 6 4 0 0
ay
az
0 0 0
T
a : thermal expansion coefficient ½D1 : flexibility or compliance matrix
The properties of the SOFC components in the structural analysis are shown in Table 1 [8]. The weight of the upper cathode separator and uniform pressure (0.2 MPa) in the top of the cathode separator were applied. The bottom of the lower anode separator and each
Fig. 3 e Electric circuit model of a solid oxide fuel cell. Please cite this article in press as: Kim YJ, Lee MC, Numerical investigation of flow/heat transfer and structural stress in a planar solid oxide fuel cell, International Journal of Hydrogen Energy (2017), http://dx.doi.org/10.1016/j.ijhydene.2017.04.140
5
i n t e r n a t i o n a l j o u r n a l o f h y d r o g e n e n e r g y x x x ( 2 0 1 7 ) 1 e1 0
Fig. 4 e Calculation flow chart (one way FSI). contact between the separators and the electrodes were assigned for frictionless contacts. All side surfaces were set up to weak spring constraints. The applied constraints for FEM modeling were depicted in Fig. 2. However, one uncertain parameter during the thermalstructural analysis was to determine the type of contact between the components (anodeeelectrolyte and cathodeeelectrolyte). These types of contact in terms of the Coulomb friction can be analyzed as the following equation: F ¼ mN
Geometry - Anode channel thickness - Cathode channel thickness - Anode electrode thickness - Electrolyte thickness - Cathode electrode thickness Operating - Active area - Normal operating current density - Fuel utilization - Oxidant utilization Input for heat/mass analysis - Anode inlet mass flow rate - Cathode inlet mass flow rate - Anode inlet temperature - Cathode inlet temperature - Anode inlet component H2 H 2O - Cathode inlet component O2 N2 - Wall temperature
(17)
Where, F: friction force N: normal force ms: static friction coefficient md: dynamic friction coefficient b: exponential decay coefficient V: The relative velocity at the contact
(16)
Table 2 e Parameters for the heat and mass transfer analysis of an SOFC. Parameter
m ¼ md þ ðms md ÞebV
Unit
Value
mm mm mm mm mm
2 2 1.4 0.02 0.03
m2 A/m2
0.0081 (0.09 0.09) 2000
% %
46.7 20
kg/cm2 kg/cm2 C C
4.636 106 2.883 105 800 800
% %
97 3
% % C
21 79 800
Chiang et al. assumed the static friction coefficient of frictional contact is 0.16 [9]. This research assumed that the static friction coefficients varied from 0.0 to 0.3, and the perfect bond was the ideal case.
Computational code The commercial code Ansys including FLUENT (Ansys, Inc.) [10] was used to calculate the steady-state heat and mass transfer and the thermal stress in the SOFC unit cell. Continuity, momentum, energy, and species balance equations
Fig. 5 e Performance curve for a solid oxide fuel cell unit cell.
Please cite this article in press as: Kim YJ, Lee MC, Numerical investigation of flow/heat transfer and structural stress in a planar solid oxide fuel cell, International Journal of Hydrogen Energy (2017), http://dx.doi.org/10.1016/j.ijhydene.2017.04.140
6
i n t e r n a t i o n a l j o u r n a l o f h y d r o g e n e n e r g y x x x ( 2 0 1 7 ) 1 e1 0
Fig. 6 e Current density distribution for the solid oxide fuel cell unit cell. Average current density ¼ 2000 A/m2, fuel utilization ¼ 46.7%, oxidant utilization ¼ 20%.
were solved in three dimensions with user-defined functions describing the electrochemical reactions. The temperature distribution obtained with the CFD results was used to calculate the thermal stress distribution in
the SOFC using the FEM. A flow chart for the sequential calculation method is shown in Fig. 4. The calculation domain was based on a cross-flow 9 9 cm2 unit cell. A schematic diagram of the SOFC unit
Fig. 7 e H2 mole fraction distribution for the solid oxide fuel cell unit cell. Average current density ¼ 2000 A/m2, fuel utilization ¼ 46.7%, oxidant utilization ¼ 20%. Please cite this article in press as: Kim YJ, Lee MC, Numerical investigation of flow/heat transfer and structural stress in a planar solid oxide fuel cell, International Journal of Hydrogen Energy (2017), http://dx.doi.org/10.1016/j.ijhydene.2017.04.140
i n t e r n a t i o n a l j o u r n a l o f h y d r o g e n e n e r g y x x x ( 2 0 1 7 ) 1 e1 0
7
Fig. 8 e O2 mole fraction distribution for the solid oxide fuel cell unit cell. Average current density ¼ 2000 A/m2, fuel utilization ¼ 46.7%, oxidant utilization ¼ 20%.
cell for the present calculation is depicted in Figs. 1 and 2. In this numerical analysis, the calculating scheme of the pressure and velocity coupling was the SIMPLE method, and the second order up-wind difference approximation is applied for
the momentum, energy and the species conservation equation. For the boundary conditions, pressure outlet conditions were assumed and Neumann conditions were given for temperature and specie concentrations at the outlet. The used
Fig. 9 e Temperature distribution for the solid oxide fuel cell unit cell. Average current density ¼ 2000 A/m2, fuel utilization ¼ 46.7%, oxidant utilization ¼ 20%. Please cite this article in press as: Kim YJ, Lee MC, Numerical investigation of flow/heat transfer and structural stress in a planar solid oxide fuel cell, International Journal of Hydrogen Energy (2017), http://dx.doi.org/10.1016/j.ijhydene.2017.04.140
8
i n t e r n a t i o n a l j o u r n a l o f h y d r o g e n e n e r g y x x x ( 2 0 1 7 ) 1 e1 0
total grid elements were 60,554. The fuel cell is assumed to operate at a steady state, and the surroundings are assumed to be the fixed temperature at 800 C.
Results and discussion The input parameters for the present simulation of a SOFC unit cell are summarized in Table 2. First, the performance curve is calculated from 0 to 2000 A/m2 as shown in Fig. 5. When the average current density is 2000 A/m2, fuel utilization is 46.7%, and oxidant utilization is 20%. It was predicted that at an average current density of 2000 A/m2 the cell voltage calculated from Eq. (14) is 0.642 V, and the power density is 1284 W/m2. The current density distributions within the electrolyte obtained by using the electric circuit model are shown in Fig. 6. The peak current density is located near the fuel inlet because the supplied oxidant is enough to cover the active area except for the rib regions and the fuel component is dominant for the local current density. However, if the oxidant utilization is higher, then the oxidant component will affect the current density distribution. The hydrogen concentration within the anode electrode is shown in Fig. 7. The hydrogen concentration decreases rapidly from the inlet of the anode flow toward its outlet side. The decreasing rate is highest in the region where the current density is the highest (Fig. 6). The pattern of the change in the hydrogen concentration is well correlated with the current density distribution in Fig. 6. The oxygen concentration in the cathode electrode is displayed in Fig. 8. The change in the oxygen concentration is small compared to the change in the hydrogen concentration due to oxidant utilization. The oxygen concentration slowly decreases toward the outlet of the cathode flow. An important objective of this research was to predict the temperature distribution and to determine stack design that can investigate thermal stresses in the SOFC stack. The computed temperature distribution in the middle of the SOFC unit cell is shown in Fig. 9. The peak cell temperature
Fig. 11 e Maximum Von Mises stress in the perfectly bonded condition: (a) anode; (b) cathode; (c) electrolyte.
Fig. 10 e Maximum Von Mises stress corresponding to various static friction coefficients.
calculated as 810 C appears near the fuel and the oxidant outlet face. The maximum temperature difference is 10 C, and the temperature deviation in the SOFC unit cell is nearly negligible.
Please cite this article in press as: Kim YJ, Lee MC, Numerical investigation of flow/heat transfer and structural stress in a planar solid oxide fuel cell, International Journal of Hydrogen Energy (2017), http://dx.doi.org/10.1016/j.ijhydene.2017.04.140
i n t e r n a t i o n a l j o u r n a l o f h y d r o g e n e n e r g y x x x ( 2 0 1 7 ) 1 e1 0
Using the calculated temperature distribution, the thermal stress was calculated with the various static friction coefficients and the perfect bonded condition. The maximum Von Mises stresses for static friction coefficients that ranged from 0.05 to 0.3 are presented in Fig. 10 for the cathode electrode, the anode electrode, and the electrolyte. The results showed that the difference in the temperature distribution (△T~10 C) does not affect the thermal stress. Instead of the temperature distribution, the results showed that the static friction coefficients have a dominant effect on the structural stability. The results showed that the maximum Von Mises stresses increased as the static friction coefficients increased. Especially, the electrolyte has the maximum stress because
9
the electrolyte is the thinnest and its Young's modulus is the highest. For the 0.05e0.3 static friction coefficients, the maximum Von Mises stresses of the electrolyte ranged from 25.87 MPa to 110.07 MPa. Thermal stresses occur when there are constraints on the thermal deformations. If there are no constraints, frictionless contact, between the anode electrode and the electrolyte, and between the cathode electrode and the electrolyte, thermal stresses would not exist although the temperature is high. For the most fatal case, when the perfect bonded condition is applied, the Von Mises stress distribution is presented in Fig. 11. The maximum Von Mises stresses of the cathode
Fig. 12 e Simulated residual stress distribution in the electrolyte at room temperature: (a) for the single stack; (b) for the single cell [11]. Please cite this article in press as: Kim YJ, Lee MC, Numerical investigation of flow/heat transfer and structural stress in a planar solid oxide fuel cell, International Journal of Hydrogen Energy (2017), http://dx.doi.org/10.1016/j.ijhydene.2017.04.140
10
i n t e r n a t i o n a l j o u r n a l o f h y d r o g e n e n e r g y x x x ( 2 0 1 7 ) 1 e1 0
Table 3 e Yakabe et al.'s [11] measured and simulated data for SOFCs. Data
Unit
Value
Simulated residual stress at room temperature Measured residual stress at room temperature Modified residual stress at room temperature
MPa MPa MPa
670 471 612
electrode, electrolyte, and anode electrode are 28.6 MPa, 262.6 MPa, and 15.1 MPa, respectively. To validate these results, Yakabe et al.'s [11] measured and simulated stress data are presented in Fig. 12 and Table 3. The estimated result in Fig. 11 (c) in this study is very similar to the tested and analyzed data shown in Fig. 12. In detail, Yakabe et al.'s [11] measured and analyzed the residual stresses at room temperature and Yakabe et al.'s [11] maximum stress was measured at 471 MPa and simulated at 696 MPa that act as a compressive stress. Compared with 262.6 MPa in this research, the deviation is negligible considering the estimation at hot conditions (~800 C) and the measurement at room temperature (~20 C). However, the static friction coefficients are the dominant factors in the estimate of thermal stresses in a SOFC. When the static friction coefficient was 0.05, the maximum stress was calculated at about 25 MPa. In addition, when the perfectly bonded condition was applied, then the maximum stress was approximately 260 MPa. Thus, compared with the measured/simulated stress data at room temperature by Yakabe et al. [11], in the present research, the type of contact between the anode electrode and the electrolyte, and between the cathode electrode and the electrolyte, is the perfectly bonded condition according to the calculation results.
Conclusion To estimate the internal operating conditions and the overall thermal performance for a planar SOFC unit cell, threedimensional mass, momentum, energy and species equations were solved by using the commercial software Ansys including FLUENT (Ansys, Inc.). Electrochemical reactions in the flow field were modeled, and their model equations were added to the code through several user-defined functions. From this flow field simulation, the distributions of the current density, heat flux, temperature, hydrogen, oxygen, and water concentrations in the electrolyte, anode, and cathode electrodes were obtained. Based on the CFD calculation results, the temperature distribution was very uniform within approximately 10 C. Using the computed temperature distribution, the thermal stresses in the electrodes and the electrolyte were calculated by employing the structural analysis technique. Since the static friction coefficients are unknown, they were assumed to range from 0.05 to 0.3, and a perfectly bonded condition was assumed. The stress results showed that the electrolyte is the weakest component and has the maximum stress because the thickness of the electrolyte is the thinnest and Young's modulus is the highest. When the static friction coefficient was 0.05, the maximum stress in the
electrolyte was calculated at about 25 MPa. In addition, when the perfectly bonded condition was applied, then the maximum stress was calculated at approximately 260 MPa. Thus, similar to Yakabe et al.'s [11] measured/simulated stress data at room temperature, the results in the present research showed that the type of contact between the anode electrode and the electrolyte, and between the cathode electrode and the electrolyte, is a perfectly bonded condition. As a result, this research showed that the stresses induced by constraint forces with various contact conditions were dominant for the structural stability in a SOFC. Therefore, static friction coefficients on operative high temperature conditions are important to predict the structural integrity in a SOFC, and they will be investigated in future works in order to improve the structural stability in a stack design as well as in a SOFC.
Acknowledgements This work was supported by Incheon National University Research Grant in 2016. This research was supported by Korea Electric Power Corporation through Korea Electrical Engineering & Science Research Institute (grant number: R15XA0313).
references
raa MC, Steinerb N, Marrac D, [1] Wanga K, Hissela D, Pe Sorrentinoc M, et al. A review on solid oxide fuel cell models. Int J Hydrogen Energy 2011;36:7212e28. [2] Ho TX, Kosinski P, Hoffmann AC, Vik A. Numerical analysis of a planar anode-supported SOFC with composite electrodes. Int J Hydrogen Energy 2009;34:3488e99. [3] Hussaina MM, Lia X, Dincerb I. A numerical investigation of modeling an SOFC electrode as two finite layers. Int J Hydrogen Energy 2009;34:3134e44. [4] Achenbach E. Three-dimensional and time-dependent simulation of a planar solid oxide fuel cell stack. J Electrochem Soc 1994;142:3792e800. [5] Ferguson JR, Fiard JM, Herbin R. Three-dimensional numerical simulation for various geometries of solid oxide fuel cells. J Power Sources 1996;58:109e22. [6] Yoshiba F, Abe T, Watanabe T. Numerical analysis of molten carbonate fuel cell stack performance: diagnosis of internal conditions using cell voltage profiles. J Power Sources 2000;87:21e7. [7] Kim YJ, Chang IG, Lee TW, Chung MK. Effects of relative gas flow direction in the anode and cathode on the performance characteristics of a molten carbonate fuel cell. Fuel 2010;89:1019e28. [8] Giraud S, Canel J. Young's modulus of some SOFC materials as a function of temperature. J Eur Ceram Soc 2008;28:77e83. [9] Chiang L-K, Liu H-C, Shiu Y-H, Lee C-H, Lee R-Y. Thermoelectrochemical and thermal stress analysis for an anodesupported SOFC cell. Renew Energy 2008;33:2580e8. [10] Ansys WorkBench V12. Theory reference for the mechanical APDL and mechanical applications. 2010. [11] Yakabe H, Baba Y, Sakurai T, Satoh M, Hirosawa I, Yoda Y. Evaluation of residual stresses in a SOFC stack. J Power Sources 2004;131:278e84.
Please cite this article in press as: Kim YJ, Lee MC, Numerical investigation of flow/heat transfer and structural stress in a planar solid oxide fuel cell, International Journal of Hydrogen Energy (2017), http://dx.doi.org/10.1016/j.ijhydene.2017.04.140
Available online at www.sciencedirect.com
ScienceDirect journal homepage: www.elsevier.com/locate/he
Numerical investigation of flow/heat transfer and structural stress in a planar solid oxide fuel cell Young Jin Kim a,1, Min Chul Lee b,* a
Fluid System Engineering Department, KEPCO Engineering and Construction Company, 111, Daedeok-daero 989 Beon-Gil, Yusung-Gu, Taejeon, 34057, Republic of Korea b Department of Safety Engineering, Incheon National University, 119 Academy-ro, Yeonsu-gu, Incheon, 22012, Republic of Korea
article info
abstract
Article history:
The present work investigates the effects of the temperature and thermal stress distri-
Received 18 January 2017
butions in a planar solid oxide fuel cell (SOFC) unit cell. A computational fluid dynamic
Received in revised form
(CFD) analysis of a planar anode-supported SOFC that considers electrochemical reactions
16 March 2017
is performed, and the thermal stresses are calculated. The static friction coefficients are
Accepted 18 April 2017
assumed to range from 0.05 to 0.3, and conservatively, a perfectly bonded condition is
Available online xxx
assumed. The results show that the electrolyte is the weakest component and has the maximum stress because the electrolyte is the thinnest and the Young modulus is the
Keywords:
highest. Thus, the contact between the anode electrode and the electrolyte, and between
Solid oxide fuel cell
the cathode electrode and the electrolyte, would be the perfectly bonded condition. As a
Temperature
result, this research showed that the stresses induced by constraint forces with various
Stress
contact conditions were dominant for the structural stability in a SOFC. Therefore, static
Friction coefficient
friction coefficients on operative high temperature conditions are important to predict the structural integrity in a SOFC, and they will be investigated in future works in order to improve the structural stability in a stack design as well as in a SOFC. © 2017 Hydrogen Energy Publications LLC. Published by Elsevier Ltd. All rights reserved.
Introduction Fuel cells are an electricity generation system that converts chemical energy from oxidation of a fuel directly into electric energy. Solid oxide fuel cells (SOFCs) are more efficient compared with other fuel cells (such as molten carbonate fuel cells and phosphoric acid fuel cells), have little pollution, and do not require a fuel reformer. In addition, the fuel cell can be combined with a gas turbine or a steam turbine. SOFCs can be classified as planar, cylindrical, or integral according to the geometric shape.
Briefly, an SOFC, in which the hydrogen supplied to the anode electrode, is separated into hydrogen ions and electrons. The electrons move to the cathode electrode through an external circuit, and the oxygen supplied to the cathode electrode becomes oxygen ions by receiving electrons through an external circuit. After the oxygen ions migrate to the anode electrode through the electrolyte, the oxygen ions bond with the hydrogen ions and generate water. Thus, the final reaction produces the electricity and water by which the hydrogen and the oxygen bond. The power generation efficiency of the SOFC is 50%e60%, and the efficiency increases to more than 70% when the SOFC
* Corresponding author. Fax: þ82 32 835 0779. E-mail address: [email protected] (M.C. Lee). 1 Fax: þ82 42 861 4859. http://dx.doi.org/10.1016/j.ijhydene.2017.04.140 0360-3199/© 2017 Hydrogen Energy Publications LLC. Published by Elsevier Ltd. All rights reserved. Please cite this article in press as: Kim YJ, Lee MC, Numerical investigation of flow/heat transfer and structural stress in a planar solid oxide fuel cell, International Journal of Hydrogen Energy (2017), http://dx.doi.org/10.1016/j.ijhydene.2017.04.140
2
i n t e r n a t i o n a l j o u r n a l o f h y d r o g e n e n e r g y x x x ( 2 0 1 7 ) 1 e1 0
Nomenclature SOFC CFD E E0 Erev F FEM . g hi DH i Icell Ji k keff kf ks mi p R Ri Rirr T ! u Vcell Wi Yi ha hc hohm r
Solid oxide fuel cell Computational fluid dynamics Total energy [J] Standard state potential [volts] Nernst potential [volts] Faraday constant [96,485.3 C mol1] Finite element method Gravitational force [m/s2] Enthalpy of species i [J/kg] Enthalpy change of reaction or adsorption [kJ/ mol] Electrical current density [amps/m2] Total cell electrical current [amps] Diffusion flux of species i [kg/m2$s] Thermal conductivity [W/m$K] Effective thermal conductivity [W/m$K] Thermal conductivity of fluid [W/m$K] Thermal conductivity of solid [w/m$K] Mass fraction of species i Pressure [N/m2] Universal gas constant [8.314 J/mol$K] Gas reaction rate of species i [mol/m3$s] Total cell irreversible resistance [Um2] Temperature [K] Velocity [m/s] Cell Voltage [volts] Molecular weight of species i [kg/mol] Mole fraction of species i Anode activation impedance [Um2] Cathode activation impedance [Um2] Cell ohmic resistance [Um2] Density [kg/m3]
temperature distribution can aggravate the destruction and leakage. Typical numerical studies for SOFCs were carried out by Wanga et al. [1], Thinh el al. [2], Hussaina [3], Achenbach [4], and Ferguson et al. [5]. A useful formula for the performance in the molten carbonate fuel cell has been derived from an electric circuit model [6,7]. The present work was carried out to investigate the effects of the temperature and thermal stress distributions in a planar SOFC. A computational fluid dynamics (CFD) analysis of a planar anode-supported SOFC considering electrochemical reactions was performed under operating conditions in which the average current density varied from 0 to 2000 A/m2. Based on the temperature distributions obtained from the CFD analysis, structural stress analysis using a finite element method (FEM) was also performed using one-way FluideStructure Interaction (FSI) analysis. An uncertain factor during the structural stress analysis is determining the bonding conditions between the anode or cathode electrode and the electrolyte. To consider these bonding conditions, this work used specific static friction coefficients and perfect bonding.
Numerical models To facilitate the present numerical simulation of the planar SOFC, the simulation work is divided into three parts. The first part corresponds to the gas flow in a current collector and a gas channel. The second part is the electrochemical reaction in the area that contains the electrodes and electrolyte. The third part is the thermal stress problem within the SOFC unit cell induced by the non-uniform temperature distribution. A separate numerical model is formulated for each part.
Governing momentum, heat and mass transfer equations is combined with a gas turbine or a steam turbine. In addition, natural gas, coal gas, and waste gas can be used for various fuels. NOx or SOx as the environmental pollution factor of exhaust gas is very small, there is little dust, and the CO2 emissions generated are 60% less than that generated by pulverized coal-fired power. Because an SOFC can be installed in the vicinity of the city center due to the cell's quiet operation, the transmission and distribution power loss are less. In addition, an SOFC can be used for various applications depending on the power supply system, such as the local mounting, decentralized, or centralized. However, because the SOFC production cost is high, combining SOFCs with peripheral devices is difficult due to the high temperature (~1000 C), and durability is an issue during long operations, much research is in progress. Especially, high-temperature operation can result in nonuniform temperature distribution in a solid oxide fuel cell. The metallic separator and the ceramic components, such as the anode electrodes, cathode electrodes, and electrolytes, can be destroyed and leak due to the different thermal expansion coefficients induced by the high temperature. Therefore, it is important to calculate the temperature distribution and the associated thermal stress distribution induced by the thermal expansion because the critical
The conservation equations for fluid mass, momentum, energy, and species are as follows. Continuity equation : Momentum equation :
V$ðr! uÞ ¼ 0
(1) . .
. V$ðr! u! u Þ ¼ V$p þ V t þ r g
Energy equation : V$ð! u ðrE þ pÞÞ X X . . þ hi Ri ¼ V$ keff VT þ hi J i þ u $teff
(2)
(3)
where; keff ¼ fkf þ ð1 fÞks Species equation :
. V$ðr! u mi Þ ¼ V$ J i þ Ri
(4)
where the state equation is given by, State equation :
p ¼ rRT
X Yi Wi
(5)
The separator, designed as a current collector and a gas channel, is equipped with periodic step-shaped supports on the inner surface as illustrated in Fig. 1 that depicts the channel configuration.
Please cite this article in press as: Kim YJ, Lee MC, Numerical investigation of flow/heat transfer and structural stress in a planar solid oxide fuel cell, International Journal of Hydrogen Energy (2017), http://dx.doi.org/10.1016/j.ijhydene.2017.04.140
3
i n t e r n a t i o n a l j o u r n a l o f h y d r o g e n e n e r g y x x x ( 2 0 1 7 ) 1 e1 0
Electrochemical reactions
Cell Voltage :
The following electrochemical reactions occur at the fuel cell electrodes: Anode :
H2 þ O2 /H2 O þ 2e
Vcell ¼ Erev iðha þ hc þ hohm Þ ¼ Erev iRirr
Here, Erev is the Nernst potential that accounts for the various species concentrations. It is evaluated by
(6) Nernst potential :
Cathode :
Overall :
1 O2 þ 2e /O2 2 1 H2 þ O2 /H2 O 2
(7)
(8)
The half-cell reactions take place in the anode and the cathode of the SOFC. Since the electrolyte ions are consumed in the anode and are generated in the cathode at equal rates, they are conserved within the cell. The real potential of the fuel cell depends on the thermodynamic reversible potential, the reactant species concentrations at the electrodes, and the irreversible losses of the cell due to the electrical current and kinetic rate limits. An equation for calculating the real cell potential is given by
(9)
Erev
0 qffiffiffiffiffiffiffi 1 PaH2 PcO2 RT 0 @ A þ RT ln P0 ¼E þ 2F 4F PaH2 O
(10)
Activation polarizations, ha and hc , in Eq. (9) represent a decrease from the ideal voltage due to the limit on the rate of chemical kinetics at the electrode surface. The ohmic loss,hohm , in Eq. (9) represents a decrease from the ideal voltage due to the electrical resistance caused by the current flow. Correlation equations available for SOFC [4,5] are. Ohmic losses: hohm ¼
dc da de dsep þ þ þ sc sa se ssep
where;
4:2 107 1200 exp T T 9:5 107 1150 exp sa ¼ T T 10300 se ¼ 3:34 104 exp T sc ¼
ssep ¼
(11)
1 1:07 106
Activation losses in the anode reactions me 1 2F PH2 Ea ¼ ka exp 0 ha RT P RT
(12)
where; Ea ¼ 110 kJ=mol Activation losses in the cathode reactions: me 1 4F PO2 Ec kc ¼ exp hc RT P0 RT
(13)
where; Ea ¼ 160 kJ=mol Charge conservation must be satisfied with the current generated by electrochemical reactions, which is expressed by the electric circuit model given in Ref. [6]. In Fig. 3, Erev ðx; yÞ and Rirr ðx; yÞ are the electromotive force and the irreversible losses, respectively. Each grid element through the electrodeseelectrolyte is connected to a parallel circuit, where the current passes along the vertical direction [4]. Current profiles are calculated by applying Kirchhoff's law that, neglecting the transverse current flow, considers the parallel connections within the fuel cell plates. The cell voltage must be the same in each grid element, which is derived from Eq. (10) as follows. Voltage Current Relation :
Vcell ¼
ðx;yÞ ∬ ERrevðx;yÞ dxdy Icell irr
1 ∬ Rirr ðx;yÞ dxdy
(14)
where; Icell ¼ ∬ iðx; yÞdxdy
Structural analysis Fig. 1 e SOFC unit cell frame. (a) Schematic diagram (b) Control volume for CFD analysis.
The relationship between the stress tensor considering the thermal deformation and the stain tensor is as follows:
Please cite this article in press as: Kim YJ, Lee MC, Numerical investigation of flow/heat transfer and structural stress in a planar solid oxide fuel cell, International Journal of Hydrogen Energy (2017), http://dx.doi.org/10.1016/j.ijhydene.2017.04.140
4
i n t e r n a t i o n a l j o u r n a l o f h y d r o g e n e n e r g y x x x ( 2 0 1 7 ) 1 e1 0
Fig. 2 e Constraints for FEM modeling.
2
Table 1 e Parameters for the thermal stress analysis of an SOFC [8]. Component
Thermal expansion coefficients ðaÞ [1/ C] [dimension of k]
Anode Electrolyte Cathode Separator (current collector)
3g
1.26 1.09 1.23 1.70
105 105 105 105
¼ f3 th þ ½D1 fsg
s : stress
½D1
nxy Ex 1 E y nzy Ez 0 0 0
nxz =Ex nyz Ey 1=Ez 0 0 0
0 0 0 1 Gxy 0 0
0 0 0 0 1 Gyz 0
3 0 0 7 7 0 7 7 0 7 7 0 5 1=Gxz
The temperature distribution obtained with the CFD analysis was used to calculate the thermal stress distribution in the SOFC unit cell using the finite element method (FEM) [9,10]. The temperature distribution was applied to the FEM grid data package to assign the thermal loads. The reference temperature was assumed to be 22 C. The SOFC materials in this research were assumed to be representative components as follows: - Cathode material: Lanthanum Strontium Manganite (LSM) - Electrolyte material: Yttria-Stabilized Zirconia (YSZ) - Anode material: Ni/YSZ cermet
: strain
o f3 th ¼ DT ax
87.5 145 50 193
(15)
Where,
3
Young's modulus ðEÞ [GPa]
1=E x 6 nyx Ey 6 6 nzx =Ez ¼6 6 0 6 4 0 0
ay
az
0 0 0
T
a : thermal expansion coefficient ½D1 : flexibility or compliance matrix
The properties of the SOFC components in the structural analysis are shown in Table 1 [8]. The weight of the upper cathode separator and uniform pressure (0.2 MPa) in the top of the cathode separator were applied. The bottom of the lower anode separator and each
Fig. 3 e Electric circuit model of a solid oxide fuel cell. Please cite this article in press as: Kim YJ, Lee MC, Numerical investigation of flow/heat transfer and structural stress in a planar solid oxide fuel cell, International Journal of Hydrogen Energy (2017), http://dx.doi.org/10.1016/j.ijhydene.2017.04.140
5
i n t e r n a t i o n a l j o u r n a l o f h y d r o g e n e n e r g y x x x ( 2 0 1 7 ) 1 e1 0
Fig. 4 e Calculation flow chart (one way FSI). contact between the separators and the electrodes were assigned for frictionless contacts. All side surfaces were set up to weak spring constraints. The applied constraints for FEM modeling were depicted in Fig. 2. However, one uncertain parameter during the thermalstructural analysis was to determine the type of contact between the components (anodeeelectrolyte and cathodeeelectrolyte). These types of contact in terms of the Coulomb friction can be analyzed as the following equation: F ¼ mN
Geometry - Anode channel thickness - Cathode channel thickness - Anode electrode thickness - Electrolyte thickness - Cathode electrode thickness Operating - Active area - Normal operating current density - Fuel utilization - Oxidant utilization Input for heat/mass analysis - Anode inlet mass flow rate - Cathode inlet mass flow rate - Anode inlet temperature - Cathode inlet temperature - Anode inlet component H2 H 2O - Cathode inlet component O2 N2 - Wall temperature
(17)
Where, F: friction force N: normal force ms: static friction coefficient md: dynamic friction coefficient b: exponential decay coefficient V: The relative velocity at the contact
(16)
Table 2 e Parameters for the heat and mass transfer analysis of an SOFC. Parameter
m ¼ md þ ðms md ÞebV
Unit
Value
mm mm mm mm mm
2 2 1.4 0.02 0.03
m2 A/m2
0.0081 (0.09 0.09) 2000
% %
46.7 20
kg/cm2 kg/cm2 C C
4.636 106 2.883 105 800 800
% %
97 3
% % C
21 79 800
Chiang et al. assumed the static friction coefficient of frictional contact is 0.16 [9]. This research assumed that the static friction coefficients varied from 0.0 to 0.3, and the perfect bond was the ideal case.
Computational code The commercial code Ansys including FLUENT (Ansys, Inc.) [10] was used to calculate the steady-state heat and mass transfer and the thermal stress in the SOFC unit cell. Continuity, momentum, energy, and species balance equations
Fig. 5 e Performance curve for a solid oxide fuel cell unit cell.
Please cite this article in press as: Kim YJ, Lee MC, Numerical investigation of flow/heat transfer and structural stress in a planar solid oxide fuel cell, International Journal of Hydrogen Energy (2017), http://dx.doi.org/10.1016/j.ijhydene.2017.04.140
6
i n t e r n a t i o n a l j o u r n a l o f h y d r o g e n e n e r g y x x x ( 2 0 1 7 ) 1 e1 0
Fig. 6 e Current density distribution for the solid oxide fuel cell unit cell. Average current density ¼ 2000 A/m2, fuel utilization ¼ 46.7%, oxidant utilization ¼ 20%.
were solved in three dimensions with user-defined functions describing the electrochemical reactions. The temperature distribution obtained with the CFD results was used to calculate the thermal stress distribution in
the SOFC using the FEM. A flow chart for the sequential calculation method is shown in Fig. 4. The calculation domain was based on a cross-flow 9 9 cm2 unit cell. A schematic diagram of the SOFC unit
Fig. 7 e H2 mole fraction distribution for the solid oxide fuel cell unit cell. Average current density ¼ 2000 A/m2, fuel utilization ¼ 46.7%, oxidant utilization ¼ 20%. Please cite this article in press as: Kim YJ, Lee MC, Numerical investigation of flow/heat transfer and structural stress in a planar solid oxide fuel cell, International Journal of Hydrogen Energy (2017), http://dx.doi.org/10.1016/j.ijhydene.2017.04.140
i n t e r n a t i o n a l j o u r n a l o f h y d r o g e n e n e r g y x x x ( 2 0 1 7 ) 1 e1 0
7
Fig. 8 e O2 mole fraction distribution for the solid oxide fuel cell unit cell. Average current density ¼ 2000 A/m2, fuel utilization ¼ 46.7%, oxidant utilization ¼ 20%.
cell for the present calculation is depicted in Figs. 1 and 2. In this numerical analysis, the calculating scheme of the pressure and velocity coupling was the SIMPLE method, and the second order up-wind difference approximation is applied for
the momentum, energy and the species conservation equation. For the boundary conditions, pressure outlet conditions were assumed and Neumann conditions were given for temperature and specie concentrations at the outlet. The used
Fig. 9 e Temperature distribution for the solid oxide fuel cell unit cell. Average current density ¼ 2000 A/m2, fuel utilization ¼ 46.7%, oxidant utilization ¼ 20%. Please cite this article in press as: Kim YJ, Lee MC, Numerical investigation of flow/heat transfer and structural stress in a planar solid oxide fuel cell, International Journal of Hydrogen Energy (2017), http://dx.doi.org/10.1016/j.ijhydene.2017.04.140
8
i n t e r n a t i o n a l j o u r n a l o f h y d r o g e n e n e r g y x x x ( 2 0 1 7 ) 1 e1 0
total grid elements were 60,554. The fuel cell is assumed to operate at a steady state, and the surroundings are assumed to be the fixed temperature at 800 C.
Results and discussion The input parameters for the present simulation of a SOFC unit cell are summarized in Table 2. First, the performance curve is calculated from 0 to 2000 A/m2 as shown in Fig. 5. When the average current density is 2000 A/m2, fuel utilization is 46.7%, and oxidant utilization is 20%. It was predicted that at an average current density of 2000 A/m2 the cell voltage calculated from Eq. (14) is 0.642 V, and the power density is 1284 W/m2. The current density distributions within the electrolyte obtained by using the electric circuit model are shown in Fig. 6. The peak current density is located near the fuel inlet because the supplied oxidant is enough to cover the active area except for the rib regions and the fuel component is dominant for the local current density. However, if the oxidant utilization is higher, then the oxidant component will affect the current density distribution. The hydrogen concentration within the anode electrode is shown in Fig. 7. The hydrogen concentration decreases rapidly from the inlet of the anode flow toward its outlet side. The decreasing rate is highest in the region where the current density is the highest (Fig. 6). The pattern of the change in the hydrogen concentration is well correlated with the current density distribution in Fig. 6. The oxygen concentration in the cathode electrode is displayed in Fig. 8. The change in the oxygen concentration is small compared to the change in the hydrogen concentration due to oxidant utilization. The oxygen concentration slowly decreases toward the outlet of the cathode flow. An important objective of this research was to predict the temperature distribution and to determine stack design that can investigate thermal stresses in the SOFC stack. The computed temperature distribution in the middle of the SOFC unit cell is shown in Fig. 9. The peak cell temperature
Fig. 11 e Maximum Von Mises stress in the perfectly bonded condition: (a) anode; (b) cathode; (c) electrolyte.
Fig. 10 e Maximum Von Mises stress corresponding to various static friction coefficients.
calculated as 810 C appears near the fuel and the oxidant outlet face. The maximum temperature difference is 10 C, and the temperature deviation in the SOFC unit cell is nearly negligible.
Please cite this article in press as: Kim YJ, Lee MC, Numerical investigation of flow/heat transfer and structural stress in a planar solid oxide fuel cell, International Journal of Hydrogen Energy (2017), http://dx.doi.org/10.1016/j.ijhydene.2017.04.140
i n t e r n a t i o n a l j o u r n a l o f h y d r o g e n e n e r g y x x x ( 2 0 1 7 ) 1 e1 0
Using the calculated temperature distribution, the thermal stress was calculated with the various static friction coefficients and the perfect bonded condition. The maximum Von Mises stresses for static friction coefficients that ranged from 0.05 to 0.3 are presented in Fig. 10 for the cathode electrode, the anode electrode, and the electrolyte. The results showed that the difference in the temperature distribution (△T~10 C) does not affect the thermal stress. Instead of the temperature distribution, the results showed that the static friction coefficients have a dominant effect on the structural stability. The results showed that the maximum Von Mises stresses increased as the static friction coefficients increased. Especially, the electrolyte has the maximum stress because
9
the electrolyte is the thinnest and its Young's modulus is the highest. For the 0.05e0.3 static friction coefficients, the maximum Von Mises stresses of the electrolyte ranged from 25.87 MPa to 110.07 MPa. Thermal stresses occur when there are constraints on the thermal deformations. If there are no constraints, frictionless contact, between the anode electrode and the electrolyte, and between the cathode electrode and the electrolyte, thermal stresses would not exist although the temperature is high. For the most fatal case, when the perfect bonded condition is applied, the Von Mises stress distribution is presented in Fig. 11. The maximum Von Mises stresses of the cathode
Fig. 12 e Simulated residual stress distribution in the electrolyte at room temperature: (a) for the single stack; (b) for the single cell [11]. Please cite this article in press as: Kim YJ, Lee MC, Numerical investigation of flow/heat transfer and structural stress in a planar solid oxide fuel cell, International Journal of Hydrogen Energy (2017), http://dx.doi.org/10.1016/j.ijhydene.2017.04.140
10
i n t e r n a t i o n a l j o u r n a l o f h y d r o g e n e n e r g y x x x ( 2 0 1 7 ) 1 e1 0
Table 3 e Yakabe et al.'s [11] measured and simulated data for SOFCs. Data
Unit
Value
Simulated residual stress at room temperature Measured residual stress at room temperature Modified residual stress at room temperature
MPa MPa MPa
670 471 612
electrode, electrolyte, and anode electrode are 28.6 MPa, 262.6 MPa, and 15.1 MPa, respectively. To validate these results, Yakabe et al.'s [11] measured and simulated stress data are presented in Fig. 12 and Table 3. The estimated result in Fig. 11 (c) in this study is very similar to the tested and analyzed data shown in Fig. 12. In detail, Yakabe et al.'s [11] measured and analyzed the residual stresses at room temperature and Yakabe et al.'s [11] maximum stress was measured at 471 MPa and simulated at 696 MPa that act as a compressive stress. Compared with 262.6 MPa in this research, the deviation is negligible considering the estimation at hot conditions (~800 C) and the measurement at room temperature (~20 C). However, the static friction coefficients are the dominant factors in the estimate of thermal stresses in a SOFC. When the static friction coefficient was 0.05, the maximum stress was calculated at about 25 MPa. In addition, when the perfectly bonded condition was applied, then the maximum stress was approximately 260 MPa. Thus, compared with the measured/simulated stress data at room temperature by Yakabe et al. [11], in the present research, the type of contact between the anode electrode and the electrolyte, and between the cathode electrode and the electrolyte, is the perfectly bonded condition according to the calculation results.
Conclusion To estimate the internal operating conditions and the overall thermal performance for a planar SOFC unit cell, threedimensional mass, momentum, energy and species equations were solved by using the commercial software Ansys including FLUENT (Ansys, Inc.). Electrochemical reactions in the flow field were modeled, and their model equations were added to the code through several user-defined functions. From this flow field simulation, the distributions of the current density, heat flux, temperature, hydrogen, oxygen, and water concentrations in the electrolyte, anode, and cathode electrodes were obtained. Based on the CFD calculation results, the temperature distribution was very uniform within approximately 10 C. Using the computed temperature distribution, the thermal stresses in the electrodes and the electrolyte were calculated by employing the structural analysis technique. Since the static friction coefficients are unknown, they were assumed to range from 0.05 to 0.3, and a perfectly bonded condition was assumed. The stress results showed that the electrolyte is the weakest component and has the maximum stress because the thickness of the electrolyte is the thinnest and Young's modulus is the highest. When the static friction coefficient was 0.05, the maximum stress in the
electrolyte was calculated at about 25 MPa. In addition, when the perfectly bonded condition was applied, then the maximum stress was calculated at approximately 260 MPa. Thus, similar to Yakabe et al.'s [11] measured/simulated stress data at room temperature, the results in the present research showed that the type of contact between the anode electrode and the electrolyte, and between the cathode electrode and the electrolyte, is a perfectly bonded condition. As a result, this research showed that the stresses induced by constraint forces with various contact conditions were dominant for the structural stability in a SOFC. Therefore, static friction coefficients on operative high temperature conditions are important to predict the structural integrity in a SOFC, and they will be investigated in future works in order to improve the structural stability in a stack design as well as in a SOFC.
Acknowledgements This work was supported by Incheon National University Research Grant in 2016. This research was supported by Korea Electric Power Corporation through Korea Electrical Engineering & Science Research Institute (grant number: R15XA0313).
references
raa MC, Steinerb N, Marrac D, [1] Wanga K, Hissela D, Pe Sorrentinoc M, et al. A review on solid oxide fuel cell models. Int J Hydrogen Energy 2011;36:7212e28. [2] Ho TX, Kosinski P, Hoffmann AC, Vik A. Numerical analysis of a planar anode-supported SOFC with composite electrodes. Int J Hydrogen Energy 2009;34:3488e99. [3] Hussaina MM, Lia X, Dincerb I. A numerical investigation of modeling an SOFC electrode as two finite layers. Int J Hydrogen Energy 2009;34:3134e44. [4] Achenbach E. Three-dimensional and time-dependent simulation of a planar solid oxide fuel cell stack. J Electrochem Soc 1994;142:3792e800. [5] Ferguson JR, Fiard JM, Herbin R. Three-dimensional numerical simulation for various geometries of solid oxide fuel cells. J Power Sources 1996;58:109e22. [6] Yoshiba F, Abe T, Watanabe T. Numerical analysis of molten carbonate fuel cell stack performance: diagnosis of internal conditions using cell voltage profiles. J Power Sources 2000;87:21e7. [7] Kim YJ, Chang IG, Lee TW, Chung MK. Effects of relative gas flow direction in the anode and cathode on the performance characteristics of a molten carbonate fuel cell. Fuel 2010;89:1019e28. [8] Giraud S, Canel J. Young's modulus of some SOFC materials as a function of temperature. J Eur Ceram Soc 2008;28:77e83. [9] Chiang L-K, Liu H-C, Shiu Y-H, Lee C-H, Lee R-Y. Thermoelectrochemical and thermal stress analysis for an anodesupported SOFC cell. Renew Energy 2008;33:2580e8. [10] Ansys WorkBench V12. Theory reference for the mechanical APDL and mechanical applications. 2010. [11] Yakabe H, Baba Y, Sakurai T, Satoh M, Hirosawa I, Yoda Y. Evaluation of residual stresses in a SOFC stack. J Power Sources 2004;131:278e84.
Please cite this article in press as: Kim YJ, Lee MC, Numerical investigation of flow/heat transfer and structural stress in a planar solid oxide fuel cell, International Journal of Hydrogen Energy (2017), http://dx.doi.org/10.1016/j.ijhydene.2017.04.140