Thermal Stress Intensity Factors of Crack in Solid Oxide Fuel Cells

Khairul Anam1, 2, a, Chih-Kuang Lin2, b 1Departement of Mechanical Engineering, Brawijaya University, Malang 65145, Indonesia

2Department of Mechanical Engineering, National Central University, Jhong-Li 320, Taiwan

akhairul.anam27@ub.ac.id, bt330014@cc.ncu.edu.tw

Keywords: PEN, Thermal Stress Intensity Factor, Principal Direction.

Abstract. Structural durability is the main focus of solid oxide fuel cells (SOFCs) development

which is affected by the thermal stress caused by considerable CTE mismatch between components

and thermal gradient. In this paper we investigate the thermal stress intensity factor for mode I, mode

II and mode III of positive electrode-electrolyte-negative electrode (PEN) at room temperature and

steady stage for an initial crack size of 10 m. A commercial finite element analysis (FEA) was used

to find the highly stressed regions in PENs and calculate the thermal stress intensity factors. The

stress distributions are calculated at uniform room temperature and at steady stage with a

non-uniform temperature profile. The thermal stress intensity factors are calculated for various

principal directions at the location having the greatest maximum principal stress at room temperature

and steady stage. The critical stress regions are identified based on the maximum principal stress at

room temperature and steady stage. The maximum principal stress is of 53.45 MPa and 45.12 MPa in

principal direction of -43.97 and -42.37 at room temperature and steady stage, respectively. The

mixed-mode stress intensity factor including mode I, mode II, and mode III is calculated due to

multi-axial thermal stresses. However, the stress intensity factor for mode I have a highest value

compared to those for modes II and III. The principal direction has an effect on the thermal stress

intensity factor for the critical region with the greatest maximum principal stress. All the calculated

stress intensity factors in the present study are less than the corresponding fracture toughness given in

the literature, ensuring the structural integrity for the given planar SOFC stack.

Introduction

One of the most efficient of fuel cells is solid oxide fuel cells (SOFCs), which converts chemical

energy directly into electrical energy without an intermediate step of conversion to heat because

SOFCs operate at high temperatures. The most popular SOFC is planar SOFC (pSOFC) due to a

lower fabricating cost, greater robustness, and higher current density. Durability and reliability of

pSOFC depend on electrochemical performance and thermal stresses of their components that arise

during operation. Thermal stresses in pSOFC are mainly caused by coefficient of thermal expansion

(CTE) mismatch between cell components and thermal gradient [1,2]. Thermal stress distribution in

pSOFC components are subjected to their distribution of strengths. When the thermal stresses in a

pSOFC stack are sufficiently high, large deformation and cracking might occur in the components

resulting in gas leakage or failure. On the other hand, the SOFC efficiency will decrease due to this

phenomenon. For a unit cell of pSOFC, it consists of a positive electrode-electrolyte-negative

electrode (PEN) assembly, interconnects, sealing materials, and porous nickel. The ceramic PEN

assembly is a critical electromechanical component of the pSOFC. The compromise between high

mechanical strength and low electrical resistance results in a rather fragile PEN having the lowest

thickness consistent with acceptable mechanical strength. When tensile stresses develop in PEN,

cracking of a ceramic PEN assembly is possible.

The main objective of this present is to investigate the cracking tendency of PENs in a multi-cell

pSOFC stack by using a commercial finite element analysis (FEA). Firstly, FEA modeling is applied

to obtain the stress distribution in a prototypical pSOFC stack at room temperature and operating

temperatures. Secondly, the stress in each component of the pSOFC stack, particularly PEN, is

evaluated according to proper failure criteria to identify the highly stressed regions. Thirdly, principal

direction is calculated to define the crack direction at room temperature and steady stage. Finally,

stress intensity factors including mode I, mode II, and mode III are calculated at these regions in

PENs for various crack direction. The mixed-mode stress intensity factor including mode I, mode II,

and mode III is calculated due to multi-axial thermal stresses.

Numerical Procedure

A commercial FEA was used to analyze the thermal stress distributions and stress intensity factors

in a 3-cell pSOFC stack at room temperature and steady stage. The FEA model was constructed based

on a prototypical stack design which was developed at the Institute of Nuclear Energy Research

(INER), Taiwan. Fig. 1 shows the exploded view of a half of the 3-cell pSOFC stack model. Because

the structural geometry is symmetric, only one half of the stack configuration is needed for building

up this 3-D FEA model. In Fig. 1, each unit cell of the 3-cell stack is composed of a PEN plate with a

supporting window frame, two interconnects with gas channels, and a nickel mesh. As described

above, a PEN assembly consists of anode, electrolyte, and cathode. For bonding the periphery of a

PEN assembly with a supporting window frame, a glass-ceramic sealant is applied. Between each

supporting window frame and the adjacent interconnect of the 3-cell pSOFC stack, a mica gasket is

used to function as a compressive seal.

Fig 1. Exploded view of a half of a 3-cell planar SOFC stack.

The overall dimensions of the FEA model have a length of 170 mm, a width of 150 mm, and a

height of 27.2 mm. The nominal thickness of the frame and interconnect is 2.4 mm with gas channels

in a depth of 0.8 mm. The size of the PEN is 99 x 99 x 0.6 mm3. The thickness of the mica layer is 0.2

mm in order to make the PEN uniformly contact with the gas channels. Note that in Fig. 1, the fuel

inlet for this counter-flow pSOFC is located at the left side of the stack while the air inlet is located at

the right side of the stack.

The materials used in the current pSOFC stack design include anode-supported PEN, Crofer 22

APU (PEN-supporting window frames and interconnect), G-9 glass-ceramic (rigid sealant), mica

(compressive sealant), and nickel mesh. The Poissons ratio of each material is assumed constant at all temperatures. The CTEs as a function of temperature for these five materials are given in Refs.

[1,2]. The anode-supported PEN is assumed to deform elastically at all given temperatures. As the

material properties of an anode-supported PEN are dominated by the anode, the PEN is treated as a

single material, i.e. Ni/YSZ. The elastic properties of Ni/YSZ listed in Table 1 [3].

Table 1. Elastic properties of PEN. [3]

Temperature [C] Youngs modulus [GPa]

25 120

100 116

200 110

300 105

400 99

500 94

600 89

700 83

800 78

850 75

Plastic deformation of the Crofer 22 APU was allowed at both room temperature and steady stage

following the tensile properties of Crofer 22 APU is given in Ref. [1]. The G-9 glass is assumed to

behave in a linear elastic manner at room temperature but inelastic behavior was allowed at 800C.

The Youngs moduli of the G-9 glass obtained by four-point bending tests are given in Ref. [4]. Mica sealant and nickel mesh were assumed to deform elastically. The elastic properties of the mica sealant

are orthotropic and assumed to be equal to those of single-crystal phlogopite mica. The elastic

properties of the phlogopite mica and nickel mesh are given in Refs. [5,6], respectively.

(a) (b)

Fig 2. Non-uniform temperature profile at the steady stage in (a) a 3-cell planar SOFC stack model

and (b) middle PEN

Temperature profiles are needed to calculate the thermal stress distribution at steady stage. The

temperature profiles at steady stage generated by a thermo-electrochemical computational fluid

dynamics (CFD) code, STAR-CD, at INER were imported into the FEA model in this study. Fig. 2

shows the temperature profiles calculated for (a) all components in the 3-cell pSOFC stack with a

range of 700C-758C and (b) middle PEN with a range of 741C-758C at steady stage. Thermal

stresses are calculated using a uniform temperature distribution at room temperature and this

non-uniform temperature profile at steady state.

Cell components are usually assembled together to form a multiple-cell pSOFC stack at a high

temperature above 800C by using glass-ceramic sealants. At such a high temperature in assembling

the pSOFC stack, the glass-ceramic becomes viscous flow and stress can be quickly relaxed, so that

800C, which is much higher than the glass transition temperature of glass-ceramic, is defined as the

initial stress-free condition. After the assembling process, the multiple-cell stack is then slowly

cooled down to room temperature for future operation. In the following thermal stress analyses, the

residual stresses in the pSOFC stack at room temperature before operation will be first calculated by

considering the temperature drop from 800C of the stress-free condition to room temperature. Then,

the thermal stresses after start-up of the cell operation are evaluated with consideration of such

existing residual stresses. Therefore, the sequence of temperature profiles imported into the FEA

model takes the following order: (i) uniform distribution at 800C, (ii) uniform distribution at room

temperature, and (iii) non-uniform temperature profile (Fig. 2a) at steady stage. Appropriate material

properties, boundary conditions, and temperature profiles are thus applied to calculate the thermal

stress distributions at room temperature and operation stage.

The stress intensity factor (K) including KI, KII, and KIII are parameter that is proportional to the

stress occurring near the crack tip due to multi-axial thermal stresses. It is a theoretical parameter

usually applied to a linear elastic homogeneous material and is useful for providing a failure criterion

for brittle materials such as PEN which operates with thermal gradient. PEN is made of ceramics with

many inherent pores, defects, and surface flaws. In this study, the stress intensity factors will be

calculated at the highly stressed regions in the middle PEN. The middle PEN is chosen as an example

to calculate the K values as the differences in the temperature profiles at steady stage among the three

PENs are not significant.

(a) (b)

Fig 3. (a) The initial crack size and (b) positions selected for calculation of stress intensity factor

along the crack front for c = a = 10 m.

The various crack direction at room temperature and steady stage are considered in the present

work for the fixed value of initial crack size of c = a = 10 m, as shown in Fig. 3a. The crack direction

is defined as a principal direction of maximum principal direction at room temperature and steady

stage. The principal direction was calculated using the following equation,

tan 2 = 2xy/(xx-yy) (1)

where xx is normal stress in x direction, yy is normal stress in y direction, and xy is shear stress. Fig. 3b shows the positions of which stress intensity factor is calculated along the crack front for surface

cracks of initial crack size of c = a = 10 m. These positions are selected in calculation of K at various

crack direction.

Results and Discussion

Fig. 4(a) shows the distribution of maximum principal stress in PEN at room temperature after

cooling down from the assembling temperature 800oC. The critical maximum principal stress in the

middle PEN plate appeared at the coordinate of (43.5, -43.5, 15). This point, designated as Point A, is

close to the bonding area between PEN and glass-ceramic sealant. It indicates that CTE mismatch

between PEN and glass-ceramic sealant has an effect on the maximum principal stress. The highest

maximum principal stress of Point A in the middle PEN at room temperature is 53.45 MPa and the

principal direction is equal to -43.97 from Eq. (1). Fig. 4(b) shows the distribution of maximum principal stress in PEN at steady stage after heating up from room temperature. The highest

maximum principal stress in PEN at steady stage takes place at the area around Point A, which

has been identified in the room-temperature distribution. The magnitude of maximum principal

stress of Point A at steady stage is 45.12 MPa and the principal direction is equal to -42.37. Point A is the position of initial crack which applied in this study, as shown in Fig. 3(b).

Point A

(a) (b)

Fig. 4. Distribution of maximum principal stress in the middle PEN at (a) room temperature and (b)

steady stage.

(a) (b)

Fig. 5. Stress intensity factors along the crack front at Point A for principal direction of -43.97 at (a)

room temperature and (b) steady stage.

Fig. 5 shows the stress intensity factor calculated along the crack front at Point A for for principal

direction of -43.97 at room temperature and steady stage. As shown in Fig. 5, the mixed-mode K

including KI, KII, and KIII is occured in pSOFC due to multi-axial thermal stresses. However, the KI of

0.2 MPam1/2

have a highest value compared to those for KII and KIII. At room temperature, Fig. 5(a)

the value of KI and KIII was almost similar along the crack front due to small crack size. Fig. 5(a) also

shows that KII values at position 1-31 and 31-61 are positive and negative, respectively. Its indicates

that CTE mismatch between PEN plate and G-9 glass and thermal gradient took a place in this

phenomena. During cooled down from stress free condition to room temperature, the area of crack

which is close to G-9 glass was shrinkage and the other area was expansion. Shear stress is occured at

the area which is expansion. At steady stage, Fig. 5(b), the value of KI and KIII have a linier trend to

value of KI and KIII at room temperature. However, the value of KI and KIII at steady stage is smaller

than at room temperature. This is evidenced by a decrease in the stress intensity factor with increasing

temperature.

Fig. 6 shows the stress intensity factor calculated along the crack front at Point A for for principal

direction of -42.37 at room temperature and steady stage. By changing the principal direction, the

value of KI, KII, and KIII was also changed. At room temperature, Fig. 6(a) shows that the value of KI

is smaller that KIII and the negative area of KII values was moved from position 31-61 to 18-61. This is

evidenced with a change in the principal direction, the cracking is not in purely opening mode. At

steady stage, Fig. 6(b) shows that the value of KII and KIII is very small (almost zero) compare to value

of KI which has a highest value. Its indicates that the cracking is in purely opening mode at steady

stage for principal direction of -42.37.

(a) (b)

Fig. 6. Stress intensity factors along the crack front at Point A for principal direction of -42.37 at (a)

room temperature and (b) steady stage.

For a porous material such as PEN, typical defects/cracks might be produced during fabrication.

From these results, it could be concluded that defect/crack type would affect the thermal stress

intensity factor. The stress fields and principal direction also play an important role in determination

of stress intensity factor. All the KI, KII, and KIII values calculated at Point A described above for the

given pSOFC stack are significantly less than the fracture toughness values given in the literature.

The reported fracture toughness for PEN is 1.65 MPam1/2

at room temperature and 1.51 MPam1/2

at

800C [7]. It indicates that the PEN in the current study has a good structural reliability under the

given SOFC operating conditions.

Summary

The critical stress regions are identified based on the maximum principal stress in middle PEN.

The maximum principal stress is of 53.45 Mpa and 45.12 MPa and of principal direction of -43.97

and -42.37 at room temperature and steady stage, respectively. The mixed-mode stress intensity

factor including mode I, mode II, and mode III for various principal direction is occured due to

multi-axial thermal stresses. However, the stress intensity factor for mode I have a highest value

compared to those for modes II and III. The principal direction has an effect on the thermal stress

intensity factor for the critical region with the greatest maximum principal stress. All the calculated

thermal stress intensity factors in the present study are less than the corresponding fracture toughness

given in the literature, ensuring the structural integrity for the given planar SOFC stack.

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