Characterization of time-varying macroscopic electro-chemo-mechanical behavior of SOFC subjected to Ni-sintering in cermet microstructures

Abstract

In order to perform stress analyses of a solid oxide fuel cell (SOFC) under operation, we propose a characterization method of its time-varying macroscopic electro-chemo-mechanical behavior of electrodes by considering the time-varying geometries of anode microstructures due to Ni-sintering. The phase-field method is employed to simulate the micro-scale morphology change with time, from which the time-variation of the amount of triple-phase boundaries is directly predicted. Then, to evaluate the time-variation of the macroscopic oxygen ionic and electronic conductivities and the inelastic properties of the anode electrode, numerical material tests based on the homogenization method are conducted for each state of sintered microstructures. In these homogenization analyses, we also have to consider the dependencies of the properties of constituent materials on the temperature and/or the oxygen potential that is supposed to change within an operation period. To predict the oxygen potential distribution in an overall SOFC structure under long-period operation, which determines reduction-induced expansive/contractive deformation of oxide materials, an unsteady problem of macroscopic oxygen ionic and electronic conductions is solved. Using the calculated stress-free strains and the homogenized mechanical properties, both of which depend on the operational environment, we carry out the macroscopic stress analysis of the SOFC.

Appendices

Appendix 1

Usual Ginzburg–Landau type energy can be calculated as follows:

$$\begin{aligned} F = \int _V \left[ g(\rho )+\frac{\alpha }{2} ||\nabla \rho ||^2 \right] \mathrm{d}V \end{aligned}$$

(71)

The variation of Eq. (71) is calculated as the following [53].

(72)

(73)

(74)

(75)

(76)

Substituting

$$\begin{aligned} g= & {} \displaystyle {\sum _i} A_i\phi _i^2(1-\phi _i)^2 + \displaystyle {\sum _i} B_i\phi _i^2 \nonumber \\&+ \displaystyle {\sum _i}\left( \displaystyle {\sum _i}\frac{\beta _{ij}}{4}\phi _i^2\phi _j^2 \right) + \frac{\gamma }{2}\phi _1^2\phi _2^2\phi _3^2 \end{aligned}$$

(77)

into Eq. (76), the following equation is derived.

$$\begin{aligned} \frac{\delta F}{\delta \phi _i}= & {} A_i\left( 2\phi _i - 6\phi _i^2 + 44\phi _i^3 \right) \nonumber \\&+\, 2B_i\phi _i + \beta _{ij}\phi _j^2\phi _i + \gamma \phi _j^2\phi _k^2\phi _i - \alpha _i\nabla ^2\phi _i \end{aligned}$$

(78)

Appendix 2

Phase-field model simulation is carried out based on finite differential method. The microstructures of anode is made by voxel mesh, and the grids correspond to the voxels. Discretizing

$$\begin{aligned}&\left( \frac{\delta F}{\delta \phi _i} \right) ^{l,m,n} = A_i \left( 2\phi _i^{l,m,n}-6\left( \phi _i^{l,m,n} \right) ^2 + 4\left( \phi _i^{l,m,n} \right) ^3 \right) \nonumber \\&\quad + 2B_i\phi _j^{l,m,n} + \beta _{ij}\left( \phi _j^{l,m,n} \right) ^2\left( \phi _k^{l,m,n} \right) ^2\phi _i -\alpha _i\nabla ^2\phi _i^{l,m,n} \nonumber \\&\quad -\alpha _i \left( \frac{\phi _i^{l+1,m,n} \,{+}\, \phi _i^{l-1,m,n} \,{+}\, \phi _i^{l,m+1,n}\,{+}\,\phi _i^{l,m-1,n} \,{+}\, \phi _i^{l,m,n+1}\,{+}\,\phi _i^{l,m,n-1}}{\left( \Delta x \right) ^2 } \right) \nonumber \\ \end{aligned}$$

(79)

$$\begin{aligned}&\left( \frac{\partial \phi _i}{\partial t} \right) ^{l,m,n} \,{=}\, M_i^{l,m,n} \left[ \frac{D^{l,m,n}}{\left( \nabla x \right) ^2} \left\{ \left( \frac{\delta F}{\delta \phi _i} \right) ^{l-1,m,n} {+}\left( \frac{\delta F}{\delta \phi _i} \right) ^{l+1,m,n} {+}\left( \frac{\delta F}{\delta \phi _i} \right) ^{l,m-1,n} \right. \right. \nonumber \\&\quad \left. +\,\left( \frac{\delta F}{\delta \phi _i} \right) ^{l,m+1,n} +\left( \frac{\delta F}{\delta \phi _i} \right) ^{l,m,n-1} +\left( \frac{\delta F}{\delta \phi _i} \right) ^{l,m,n+1} -6\left( \frac{\delta F}{\delta \phi _i} \right) ^{l,m,n} \right\} \nonumber \\&\quad +\,\frac{1}{4\Delta x} \left\{ \left( \left( \frac{\delta F}{\delta \phi _i} \right) ^{l+1,m,n} -\left( \frac{\delta F}{\delta \phi _i} \right) ^{l-1,m,n} \right) \left( D^{l+1,m,n} - D^{l-1,m,n} \right) \right. \nonumber \\&\quad + \left( \left( \frac{\delta F}{\delta \phi _i} \right) ^{l,m+1,n} - \left( \frac{\delta F}{\delta \phi _i} \right) ^{l,m-1,n} \right) \left( D^{l,m+1,n} - D^{l,m-1,n} \right) \nonumber \\&\quad \left. \left. +\, \left( \left( \frac{\delta F}{\delta \phi _i} \right) ^{l,m,n+1} + \left( \frac{\delta F}{\delta \phi _i} \right) ^{l,m,n-1} \right) \left( D^{l,m,n+1} - D^{l,m,n-1} \right) \right\} \right] \nonumber \\ \end{aligned}$$

(80)

Discritizing the time with forward differential method, the phase-field simulation is conducted with (79).