# A Microscale Modeling Tool for the Design and Optimization of Solid Oxide Fuel Cells

^{1}

^{2}

^{3}

^{4}

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## Abstract

**:**

^{2}, a typical SOFC stack cell should consist of a rib width of 0.9 mm, a cathode current collection layer thickness of 200–300 μm, a cathode functional layer thickness of 20–40 μm, and an anode functional layer thickness of 10–20 μm in order to achieve optimal performance.

## 1. Introduction

## 2. Method

_{channel}and d

_{rib}are one half of the channel width and one half of the interconnect rib width, respectively. The typical geometric, material and operational parameters are shown in Table 1. The air and the fuel are supplied to the reaction sites in the functional layers (CFL and AFL) from the gas channels through the outside porous layers (CCCL and ASL). Two important electrochemical reactions occur in the functional layers when cell operates with hydrogen fuel and air oxidant as illustrated in Figure 2. At the CFL reaction site, each half of an oxygen molecule gets two electrons conducted from the nearest cathodic interconnect rib and is reduced to an oxygen ion. The oxygen ion is then transported from the CFL reaction site to the AFL reaction site through the electrolyte (YSZ). Finally at the AFL reaction site, the oxygen ion reacts with a hydrogen molecule and produces a water molecule and two electrons and the electrons are conducted to the nearest anodic interconnect rib.

_{cell}) can be formally expressed as [8]:

_{c}

_{onc}

_{;a}and η

_{c}

_{onc}

_{;c}are the concentration overpotentials in the anode and cathode, respectively, η

_{act}

_{;a}and η

_{act}

_{;c}the anode and cathode activation overpotentials, η

_{ohm;e;a}, η

_{ohm;e;c}the electronic ohmic overpotentials in the anode and cathode, η

_{ohm;i;a}, η

_{ohm;el}and η

_{ohm;i;c}the ionic ohmic overpotentials in the anode, electrolyte and cathode, η

_{ASR;a}and η

_{ASR;c}the anode-rib and cathode-rib interface overpotentials due to the contact resistance at the material boundaries.

_{2}, H

_{2}O and O

_{2}are all at 1 atm (${E}_{0}^{0}$ = 1.01 V at 973 K or 700 °C), R the universal gas constant, T the temperature at Kelvin, F the Faraday constant. ${p}_{{\text{H}}_{2}\text{,AFL}}$, ${p}_{{\text{H}}_{2}\text{O,AFL}}$ and ${p}_{{\text{O}}_{2}\text{,CFL}}$ are the partial pressure of H

_{2}and H

_{2}O at the electrochemically active site in the AFL and the partial pressure of O

_{2}at the active site in the CFL, respectively. ${\eta}_{a}^{0}$ and ${\eta}_{c}^{0}$ are respectively the anode and cathode concentration balance potentials and are calculated as:

_{pitch}is the pitch width and d

_{pitch}= d

_{channel}+ d

_{rib}. The x and y axis are along the horizontal and vertical directions in the 2D model, respectively. j

_{y}is the y component of the current density flux vector.

Parameters | Value |
---|---|

Cell temperature, T (°C) | 700 |

Inlet fuel/air pressure, P_{atm} (Pa) | 1.013 × 10^{5} |

Cell output voltage, V_{op} (V) | 0.7 |

ASL thickness, l_{ASL} (μm) | 1,000 |

AFL thickness, l_{AFL} (μm) | 20 |

Electrolyte thickness, l_{YSZ} (μm) | 8 |

CFL thickness, l_{CSL} (μm) | 20 |

CCCL thickness, l_{CCCL} (μm) | 50 |

Porosity, $\epsilon $ | 0.48 (ASL), 0.23 (AFL), 0.26 (CFL), 0.45 (CCCL) |

Ni volume fraction, ${\phi}_{\text{Ni}}$ | 0.55 (ASL, AFL) |

LSM volume fraction, ${\phi}_{\text{LSM}}$ | 0.475 (CFL), 1 (CCCL) |

Tortuosity, $\tau $ | 3 |

Angle of particle contact, $\theta $ (^{o}) | 30 |

Bruggeman factor, m | 1.5 |

Mean particle diameter, d_{p} (μm) | 1 (ASL, CCCL), 0.5 (AFL, CFL) |

Electrical conductivity of Ni, ${\sigma}_{\text{Ni}}$ (s m^{-1}) | $3.27\times {10}^{6}-1065.3T$ |

Electrical conductivity of LSM, ${\sigma}_{\text{LSM}}$ (s m^{-1}) | $8.855\times {10}^{7}/T*\mathrm{exp}(-1082.5/T)$ |

Ionic conductivity of YSZ, ${\sigma}_{\text{YSZ}}$ (s m^{-1}) | $3.34\times {10}^{4}\mathrm{exp}(-10300/T)$ |

Diffusion volume of H_{2}, ${v}_{{\text{H}}_{2}}$ (m^{3} mol^{-1}) | 7.07 × 10^{-6} |

Diffusion volume of H_{2}O, ${v}_{{\text{H}}_{2}\text{O}}$ (m^{3} mol^{-1}) | 12.7 × 10^{-6} |

Diffusion volume of O_{2}, ${v}_{{\text{O}}_{2}}$ (m^{3} mol^{-1}) | 16.6 × 10^{-6} |

Diffusion volume of N_{2}, ${v}_{{\text{N}}_{2}}$ (m^{3} mol^{-1}) | 17.9 × 10^{-6} |

Molar mass of H_{2}, ${M}_{{\text{H}}_{2}}$ (kg mol^{-1}) | 2 × 10^{-3} |

Molar mass of H_{2}O, ${M}_{{\text{H}}_{2}\text{O}}$ (kg mol^{-1}) | 18 × 10^{-3} |

Molar mass of O_{2} , ${M}_{{\text{O}}_{2}}$ (kg mol^{-1}) | 32 × 10^{-3} |

Molar mass of N_{2} , ${M}_{{\text{N}}_{2}}$ (kg mol^{-1}) | 28 × 10^{-3} |

Permeability of anode (m^{2}) | 7.93 × 10^{-16} |

Permeability of cathode (m^{2}) | 3.06 × 10^{-16} |

Viscosity of fuel (Pa s) | 2.8 × 10^{-5} |

Viscosity of air (Pa s) | 4 × 10^{-5} |

Knudsen diffusion coefficient of H_{2} (m^{2} s^{-1}) | 4.37 × 10^{-4} |

Knudsen diffusion coefficient of H_{2}O (m^{2} s^{-1}) | 1.46 × 10^{-4} |

Knudsen diffusion coefficient of O_{2} (m^{2} s^{-1}) | 7.64 × 10^{-5} |

Knudsen diffusion coefficient of N_{2} (m^{2} s^{-1}) | 8.17 × 10^{-5} |

#### 2.1. Electrochemical Reactions in AFL and CFL

_{i}the partial pressure of species i.

^{3}) can be calculated as ${i}_{\text{TPB}}{\lambda}_{\text{TPB}}^{\text{V}}$. Here ${\lambda}_{\text{TPB}}^{\text{V}}$ is the volume specific TPB length (mm

^{-3}) and can be expressed [17,21]

_{p}is the particle diameter, n

_{t}($=6(1-\epsilon )/\pi {d}_{p}^{3}$) the number density of all particles, ${\mathrm{\Phi}}_{j}$ the volume fraction of particle type j (el: electron conducting particle, io: ionic conducting particle or YSZ), ${P}_{j}$ (= ${\left[(1-{((4.236-6{\mathrm{\Phi}}_{j})/2.472)}^{2.5}\right]}^{0.4}$) the probability of particle type j to form percolated or globally continuous clusters, $\theta $ the angle of particle contact.

^{2}) can be calculated as ${i}_{\text{TPB}}{\lambda}_{\text{TPB}}^{\text{A}}$. Here ${\lambda}_{\text{TPB}}^{\text{A}}$ is the area specific TPB length (mm

^{-2}) and may be expressed [17]

_{act,a}and η

_{act,c}, are related to the electric and balance potentials by [8]

_{e,AFL}and V

_{i,AFL}are respectively the electronic and ionic potentials at the electrochemically active site in the AFL, V

_{i,CFL}and V

_{e,CFL}the ionic and electronic potentials at the electrochemically active site in the CFL.

#### 2.2. Gas Transport in Porous Electrode

#### 2.2.1. Dusty gas model

_{i}is the total molar flux of species i, x

_{i}($={c}_{i}/{\displaystyle \sum _{j}{c}_{j}}$) the molar fraction, k the permeability, µ the viscosity, p the total gas pressure, ${D}_{iK}^{eff}$ (= $\epsilon {D}_{iK}/\tau $) the effective Knudsen diffusion coefficients and ${D}_{ij}^{eff}$ (= $\epsilon {D}_{ij}/\tau $) the effective binary diffusion coefficients, D

_{ij}(= $3.198\times {10}^{-8}\times \frac{{T}^{1.75}}{p{({v}_{i}^{1/3}+{v}_{j}^{1/3})}^{2}}{[\frac{1}{{M}_{i}}+\frac{1}{{M}_{j}}]}^{1/2}$) the binary diffusion coefficient, ε and τ the porosity and tortuosity, c

_{i}, ν

_{i}and M

_{i}the molar concentration, diffusion volume and molar mass of species i, respectively [24,25]. The required parameters are shown in Table 1.

_{i}, are given by

#### 2.2.2. Governing equations

_{i}is the reaction rate of species i (mol m

^{-3}s

^{-1}) in the functional layers and ${R}_{i}=0$ in ASL and CCCL because there is no reaction, ${R}_{{\text{H}}_{2}}=-{i}_{\text{TPB,a}}{\lambda}_{\text{TPB}}^{\text{V}}/2F$ for the hydrogen consumption and ${R}_{{\text{H}}_{2}\text{O}}={i}_{\text{TPB,a}}{\lambda}_{\text{TPB}}^{\text{V}}/2F$ for the water steam production in AFL and ${R}_{{\text{O}}_{2}}=-{i}_{\text{TPB,c}}{\lambda}_{\text{TPB}}^{\text{V}}/4F$ for the oxygen consumption in CFL.

#### 2.3. Electrical Conduction

_{e}the electric potential in the electrode. $-{\sigma}_{\text{e}}^{eff}\nabla {V}_{\text{e}}$ is the flux vector of the electronic current density. ${Q}_{\text{e}}$ is the electronic current source (A/m

^{3}) and ${Q}_{\text{e}}=0$ in ASL and CCCL because there is no electrochemical reaction, ${Q}_{\text{e}}=-{i}_{\mathrm{TPB}\text{,a}}{\lambda}_{\text{TPB,a}}^{\text{V}}$ in AFL and ${Q}_{\text{e}}={i}_{\mathrm{TPB}\text{,c}}{\lambda}_{\text{TPB,c}}^{\text{V}}$ in CFL. The electronic potential differences along the electrical current flux paths yield the ohmic overpotentials, η

_{ohm;e;a}and η

_{ohm;e;c.}

_{i}the electric potential. $-{\sigma}_{\text{i}}^{eff}\nabla {V}_{\text{i}}$ is the flux vector of the ionic current density. ${Q}_{\text{i}}$ is the oxygen ionic current source (A/m

^{3}) and ${Q}_{\text{i}}=0$ in ASL and CCCL as there is no electrochemical reaction, ${Q}_{\text{i}}={i}_{\mathrm{TPB}\text{,a}}{\lambda}_{\text{TPB,a}}^{\text{V}}$ in AFL and ${Q}_{\text{i}}=-{i}_{\mathrm{TPB}\text{,c}}{\lambda}_{\text{TPB,c}}^{\text{V}}$ in CFL. The ionic potential differences along the ionic current flux paths yield the ohmic overpotentials, η

_{ohm;i;a}, η

_{ohm;i;c}and η

_{ohm;el}.

_{j}is the conductivity of species j, m Bruggeman factor considering the effects of tortuous conduction paths and constricted contact areas between particles.

_{e,I/a}and V

_{e,a/I}are respectively the interconnect and ASL electric potentials at the anode-interconnect boundary, V

_{e,c/I}and V

_{e,I/c}the CCCL and interconnect electric potentials at the cathode-interconnect boundary, ASR

_{contact}the area specific contact resistance.

#### 2.4. Boundary Conditions (BCs)

Equations | Boundary | ASL/channel interface | AFL/Electrolyte interface | All others | ||||

Fuel transfer | BC Type | H_{2} molar concentration | H_{2}O molar concentration | H_{2} inward molar flux | H_{2}O inward molar flux | Insulation/Symmetry | ||

BC | ${c}_{{\text{H}}_{2}}^{0}$ | ${c}_{{\text{H}}_{2}\text{O}}^{0}$ | $-{i}_{\text{TPB,a}}{\lambda}_{\text{TPB}}^{\text{A}}/2F$ | ${i}_{\text{TPB,a}}{\lambda}_{\text{TPB}}^{\text{A}}/2F$ | ||||

Boundary | CCCL/channel interface | CFL/Electrolyte interface | All others | |||||

Air transfer | BC Type | O_{2} molar concentration | N_{2} molar concentration | O_{2} inward molar flux | N_{2} inward molar flux | Insulation/Symmetry | ||

BC | ${c}_{{\text{O}}_{2}}^{0}$ | ${c}_{{\text{N}}_{2}}^{0}$ | $-{i}_{\text{TPB,c}}{\lambda}_{\text{TPB}}^{\text{A}}/4F$ | 0 |

Equations | Boundary | Rib/CCCL interface | Rib/ASL interface | CFL/Electrolyte interface | AFL/Electrolyte interface | All others |

Electronic transfer | BC Type | Reference potential | Reference potential | Inward current flow | Inward current flow | Electric insulation |

BC | V_{cell}-${E}_{0}^{0}$ | 0 | ${i}_{\text{TPB,c}}{\lambda}_{\text{TPB}}^{\text{A}}$ | $-{i}_{\text{TPB,a}}{\lambda}_{\text{TPB}}^{\text{A}}$ | ||

Ionic transfer | BC Type | Interior current source | Interior current source | Electric insulation | ||

BC | $-{i}_{\text{TPB,c}}{\lambda}_{\text{TPB}}^{\text{A}}$ | ${i}_{\text{TPB,a}}{\lambda}_{\text{TPB}}^{\text{A}}$ |

#### 2.5. Numerical Method

^{®}Version 3.4 [26] was used in the present study to solve the required partial differential equations (PDEs) such as the mass diffusion and convective equations, the electronic and ionic charge transfer equations. The Butler–Volmer equations are integrated into the PDEs as sources or boundary settings. The COMSOL stationary nonlinear solver uses an affine invariant form of the damped Newton method [26] to solve the discretized PDEs with a relative tolerance of 1 × 10

^{-6}. Free triangle meshes with a maximum element size of 5 × 10

^{-6}m for electrolyte and AFL areas were used.

## 3. Results and Discussion

#### 3.1. Model Fitting of the Experimental I-V Curve

^{2}and the boundary hydrogen/oxygen molar fraction is 0.97/0.21, the theoretical curve shown in Figure 3 agrees well with the experimental data. It should be noted here that when the pitch width is small, the current through the electrode/rib interface ${j}_{\text{I}\to \text{a}}$ and ${j}_{\text{c}\to \text{I}}$ are uniform [8], then the effective contact resistance is approximately $AS{R}_{\text{contact}}^{\text{eff}}\approx 2AS{R}_{\text{contact}}{d}_{\text{pitch}}/{d}_{\text{rib}}$. The effective contact resistance is an effective value assuming that the contact resistance was distributed over the entire pitch width instead of the rib width. Accordingly, $AS{R}_{\text{contact}}^{\text{eff}}$ is found here to be about 0.096 Ωcm

^{2}, in good agreement with the experimental estimation that the total ohmic loss is 0.19 Ωcm

^{2}at 700 °C and half the total ohmic loss is due to the contact resistance [3]. The agreement between the theoretical and experimental results validates the numerical model described here. Moreover, it is striking to see that the interconnect-electrode contact resistance is magnified by a factor of 2d

_{pitch}/d

_{rib}and even a very small contact resistance may substantially affect the fuel cell testing result.

^{2}. The current densities generated by the functional layer body reaction and the functional layer/electrolyte boundary reaction are respectively 0.591 A/cm

^{2}and 0.049 A/cm

^{2}in the cathode side, and 0.439 A/cm

^{2}and 0.201 A/cm

^{2}in the anode side. This means that the body reaction is the dominant contributor for the fuel cell current generation and the model picture with only boundary reactions may be too simplified.

#### 3.2. Distributions of Physical Quantities in Stack Cell

_{CCCL}, 50 μm and 200 μm, are used to show the physical quantities in stack cell. The pitch width is 2 mm and the rib width is 0.8 mm. The boundary hydrogen and oxygen molar fraction are 0.7 and 0.21 respectively. The area specific contact resistance on the anode/interconnect and the cathode/interconnect boundaries are uniform (0.025 Ωcm

^{2}). Figure 5 shows the physical quantity distributions for l

_{CCCL}of 50 μm. The average output current density is 0.374 A/cm

^{2}. Figure 6 shows the distributions for l

_{CCCL}of 200 μm and the average output current density is 0.409 A/cm

^{2}.

**Figure 5.**The distributions of physical quantities when the CCCL layer thickness is 50 μm: (a) hydrogen and oxygen molar fractions; (b) total electronic current density and current flux vector; (c) the y component of the current flux vector.

**Figure 6.**The distributions of physical quantities when the CCCL layer thickness is 200 μm: (a) hydrogen and oxygen molar fractions; (b) total current density and current flux vector; (c) the y component of the current flux vector.

^{4}s m

^{-1}) is much largely than that of CFL (5307 s m

^{-1}), and the current flux in CCCL has a much larger x-component than that in CFL to carry the current to the interconnect rib. Understandably, the current density distribution is more uniform and with a less ohmic polarization for a l

_{CCCL}of 200 μm than that for a l

_{CCCL}of 50 μm.

_{CCCL}= 50 μm (Figure 5c) while it is quite uniform for l

_{CCCL}= 200 μm (Figure 6c), also indicating that the latter is more favorable.

#### 3.3. Optimization of Geometry Parameters

#### 3.3.1. Optimization of the CCCL thickness

_{CCCL}. The current density increases by 9.4% when l

_{CCCL}increases from 50 to 200 μm, but the difference is less than 0.4% when the l

_{CCCL}changes between 200 and 400 μm. Considering the performance benefit and material cost, the optimized CCCL layer thickness should be 200–300 μm. Unless stated otherwise, the layer thickness of 200 μm is used for optimizing other parameters described below.

#### 3.3.2. Optimization of the rib width

_{ASR;a}and η

_{ASR;c}. The optimal current density decreases rapidly with the increase of the contact resistance for two main reasons: a larger contact resistance leads to a higher ohmic loss and the increased optimal rib width leads to a smaller active area. For the reference contact resistance of 0.025 Ωcm

^{2}, the optimized rib width is 0.9 mm.

**Figure 8.**The optimized rib width and the optimal output current density vs the area specific contact resistance.

#### 3.3.3. Optimization of the CFL thickness

_{CFL}). Significant improvement is observed when l

_{CFL}is increased from 5 μm to 20 μm and may be attributed to the increase of CFL active area as indicated in Figure 4. The output current density is quite similar when l

_{CFL}is increased from 20 μm to 40 μm as neither the active CFL thickness is extended nor the gas transport is affected appreciably. As shown in Figure 9, further increase of l

_{CFL}may cause reduced output current density due to the increased concentration polarization. Therefore, the optimal l

_{CFL}is between 20 μm and 40 μm.

#### 3.3.4. Optimization of the AFL thickness

_{AFL}.

_{AFL}varies from 5 μm to 20 μm as the concentration polarization is not significant for such thin AFL layer. An AFL thickness of less than 20 μm may be considered to be optimal.

## 4. Conclusions

## Acknowledgements

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**MDPI and ACS Style**

Liu, S.; Kong, W.; Lin, Z.
A Microscale Modeling Tool for the Design and Optimization of Solid Oxide Fuel Cells. *Energies* **2009**, *2*, 427-444.
https://doi.org/10.3390/en20200427

**AMA Style**

Liu S, Kong W, Lin Z.
A Microscale Modeling Tool for the Design and Optimization of Solid Oxide Fuel Cells. *Energies*. 2009; 2(2):427-444.
https://doi.org/10.3390/en20200427

**Chicago/Turabian Style**

Liu, Shixue, Wei Kong, and Zijing Lin.
2009. "A Microscale Modeling Tool for the Design and Optimization of Solid Oxide Fuel Cells" *Energies* 2, no. 2: 427-444.
https://doi.org/10.3390/en20200427