# Coupled Transport Effects in Solid Oxide Fuel Cell Modeling

^{*}

## Abstract

**:**

^{2}, 64.44% of the total losses occur in the electrolyte. The exergetic efficiency for this operating point is 0.42. Since lower entropy production rates can be assumed in the GDL, the primary need is to investigate alternative electrolyte materials.

## 1. Introduction

_{2}emissions, making it environmentally friendly, also with regard to the Paris Climate Agreement.

## 2. Materials and Methods

#### 2.1. The Thermodynamic System—Solid Oxide Fuel Cell (SOFC)

#### 2.2. Theory

#### 2.2.1. Mass and Energy Balance Equations for the Gas Diffusion Layers (GDL)

#### 2.2.2. Transport Equations for the Homogeneous Phase

#### 2.2.3. Modeling of the Gas Diffusion Layers (GDL)

#### 2.2.4. Mass Transport by Stefan–Maxwell Diffusion (1)

#### 2.2.5. Extension by Knudsen Diffusion (2)

#### 2.2.6. Extension by Convection (3)

#### 2.2.7. Extension through Thermal Diffusion (4)

#### 2.2.8. Mass Transfer Approach According to NET

#### 2.2.9. Modeling of the Reaction Layers

#### 2.2.10. Modeling of the Electrolyte

#### 2.2.11. Exergy Analysis

#### 2.3. Simulation

^{®}version R2020a software is used to perform these calculations. The operating conditions of the cell are fixed, which include the operating temperature T, the operating pressure p, the composition of the inlet gases, and the electric current density j. These operating conditions also determine the boundary conditions of the cell. The cell temperature agrees with the operating temperature, both when $y=0$ and when $y=\Delta {y}^{\mathrm{a}}+\Delta {y}^{\mathrm{e}}+\Delta {y}^{\mathrm{c}}$. At the anode side of the cell, the electric potential $\varphi \phantom{\rule{0.166667em}{0ex}}(y=0)$ is set to zero. The other boundary conditions are determined by the partial pressures of the inlet gases. As start values for the iteration, the heat flux density ${J}_{\mathrm{q}}\phantom{\rule{0.166667em}{0ex}}(y=0)$ and the partial pressures ${p}_{{\mathrm{O}}_{2}}^{\mathrm{R}}$ and ${p}_{{\mathrm{N}}_{2}}^{\mathrm{R}}$ of the cathode gases in the reaction zone are estimated first. The differential equation systems for the anode GDL, the electrolyte, and the cathode GDL, which result from the transport and balance equations, are solved step by step using the Runge–Kutta method. The profiles at the reaction layers are calculated using the Equations from Section 2.2.9. If the boundary conditions are not met after the first calculation round, the starting values are varied, and the cell is calculated again. The starting values are then adjusted using Newton’s method to start the next iteration step.

#### 2.3.1. General Parameters

#### 2.3.2. Parameters for Mass Transport

## 3. Results and Discussion

#### 3.1. Validation

#### 3.2. Partial Pressures

#### 3.3. Heat Transport, Temperature Gradient, and Potential Field

#### 3.4. Entropy Production Rate

#### 3.5. Exergy Analysis

## 4. Conclusions

## Author Contributions

## Funding

## Institutional Review Board Statement

## Informed Consent Statement

## Acknowledgments

## Conflicts of Interest

## Abbreviations

DGM | Dusty-Gas model |

GDL | Gas diffusion layer |

HOR | Hydrogen oxidation reaction |

NET | Non-equilibrium thermodynamics |

ORR | Oxygen reduction reaction |

PEMFC | Proton exchange membrane fuel cell |

SOFC | Solid oxide fuel cell |

SOEC | Solid oxide electrolyzer cell |

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**Figure 1.**Model of the solid oxide fuel cell (SOFC), including the electric current density j, all heat fluxes ${J}_{\mathrm{q}}$, and molar fluxes ${J}_{\mathrm{i}}$.

**Figure 2.**$U,j$-characteristics of an SOFC of the type KeraCell III with an electrolyte of 8YSZ at $T=1123.15$ K.

**Figure 3.**Partial pressure of (

**a**) hydrogen and (

**b**) water in the anode reaction layer, (

**c**) oxygen and (

**d**) nitrogen in the cathode reaction layer, as a function of the electric current density j.

**Figure 4.**Partial pressure of (

**a**) hydrogen and (

**b**) water in the anode reaction layer, (

**c**) oxygen and (

**d**) nitrogen in the cathode reaction layer, as a function of the electric current density j for different imposed temperature gradients $\Delta T$ between cathode and anode.

**Figure 5.**Simulated profile of (

**a**) heat flux density and (

**b**) temperature along spatial coordinate y.

**Figure 6.**Simulated profile of (

**a**) heat flux density and (

**b**) temperature along spatial coordinate y for a imposed temperature gradient $\Delta T=5$ K between cathode and anode for $j=8000$ A/${\mathrm{m}}^{2}$.

**Figure 8.**Simulated curves of the local entropy production rate in the anode GDL, the cathode GDL, and the electrolyte along the spatial coordinate y.

**Figure 9.**Simulated curve of the local entropy production rate in the electrolyte due to (

**a**) heat flux density and (

**b**) electrical potential along the spatial coordinate y for different current densities j and temperatures T.

**Figure 10.**Simulated exergetic efficiency as a function of temperature for $j=2000$ A/${\mathrm{m}}^{2}$ and $j=8000$ A/${\mathrm{m}}^{2}$.

Parameter | Value | Reference |
---|---|---|

Dimensions | $\Delta {y}^{\mathrm{a}}=40\phantom{\rule{0.166667em}{0ex}}\mathsf{\mu}$m | [44] |

$\Delta {y}^{\mathrm{c}}=50\phantom{\rule{0.166667em}{0ex}}\mathsf{\mu}$m | [44] | |

$\Delta {y}^{\mathrm{e}}=150\phantom{\rule{0.166667em}{0ex}}\mathsf{\mu}$m | [44] | |

Thermal conductivity | ${\lambda}^{\mathrm{a}}=2\phantom{\rule{0.166667em}{0ex}}$ W/m·K | [17] |

${\lambda}^{\mathrm{c}}=2\phantom{\rule{0.166667em}{0ex}}$ W/m·K | [17] | |

${\lambda}^{\mathrm{e}}=1.98\phantom{\rule{0.166667em}{0ex}}$ W/m·K | [45] | |

Entropy of the electrons | ${S}_{\mathrm{m},{\mathrm{e}}^{-},\mathrm{Pt}}=-1.81$ J/K·mol | [24] |

${S}_{\mathrm{m},{\mathrm{e}}^{-},\mathrm{Ni}}=-3.24$ J/K·mol | [25] | |

Entropy of the ions | ${S}_{{\mathrm{O}}^{2-}}$ calculated | [14] |

Peltier coefficient | ${\pi}^{\mathrm{a}}$ calculated with Equation (23) | |

${\pi}^{\mathrm{c}}$ calculated with Equation (24) | ||

${\pi}^{\mathrm{e}}$ calculated with Equation (66) | ||

Ionic conductivity | ${\sigma}^{\mathrm{e}}=2\phantom{\rule{0.166667em}{0ex}}$ Wm·K | [17] |

Pre-exponential factors | ${r}^{\mathrm{a},0}=\frac{1}{95\xb7{10}^{6}}$ $\mathsf{\Omega}\xb7$mK | [22] |

for electrical resistances | ${r}^{\mathrm{c},0}=\frac{1}{42\xb7{10}^{6}}$ $\mathsf{\Omega}\xb7$mK | [22] |

Activation energy | ${E}_{\mathrm{A},\mathrm{r}}^{\mathrm{a}}=1150$· R${}_{\mathrm{m}}\xb7$ K | [22] |

${E}_{\mathrm{A},\mathrm{r}}^{\mathrm{c}}=1200$· R${}_{\mathrm{m}}\xb7$ K | [22] | |

${E}_{\mathrm{A},{\mathrm{j}}_{0}}^{\mathrm{a}}=\mathrm{72,364}$ J/mol | [12] | |

${E}_{\mathrm{A},{\mathrm{j}}_{0}}^{\mathrm{c}}=\mathrm{156,306}$ J/mol | [12] | |

Pre-exponential factors | ${\gamma}^{\mathrm{a}}=2,4\xb7{10}^{7}$ A/m${}^{2}$ | [12] |

for exchange current densities | ${\gamma}^{\mathrm{c}}=1.82\xb7{10}^{11}$ A/m${}^{2}$ | [12] |

Penetration coefficients | ${\alpha}^{\mathrm{a}}=0.5$ | [15] |

${\alpha}^{\mathrm{c}}=0.3$ | [15] |

Parameter | Value | Reference |
---|---|---|

Diffusion volume | ${v}_{{\mathrm{H}}_{2}}=6.12$ | [29] |

${v}_{{\mathrm{H}}_{2}\mathrm{O}}=13.1$ | ||

${v}_{{\mathrm{O}}_{2}}=16.3$ | ||

${v}_{{\mathrm{N}}_{2}}=18.5$ | ||

Lennard-Jones | ${\sigma}_{{\mathrm{H}}_{2}}=2.827$ Å, ${\u03f5}_{{\mathrm{H}}_{2}}/{k}_{\mathrm{B}}=59.7\phantom{\rule{0.166667em}{0ex}}\mathrm{K}$ | [36] |

parameter | ${\sigma}_{{\mathrm{H}}_{2}\mathrm{O}}=2.641$ Å, ${\u03f5}_{{\mathrm{H}}_{2}}/{k}_{\mathrm{B}}=809.1\phantom{\rule{0.166667em}{0ex}}\mathrm{K}$ | |

${\sigma}_{{\mathrm{O}}_{2}}=3.467$ Å, ${\u03f5}_{{\mathrm{H}}_{2}}/{k}_{\mathrm{B}}=106.7\phantom{\rule{0.166667em}{0ex}}\mathrm{K}$ | ||

${\sigma}_{{\mathrm{N}}_{2}}=3.798$ Å, ${\u03f5}_{{\mathrm{H}}_{2}}/{k}_{\mathrm{B}}=71.4\phantom{\rule{0.166667em}{0ex}}\mathrm{K}$ | ||

Pore diameter for GDL | ${d}_{\mathrm{p}}^{\mathrm{i}}=5\phantom{\rule{0.166667em}{0ex}}\mathsf{\mu}$m | [1] |

Porosity for GDL | ${\u03f5}^{\mathrm{i}}=0.3$ | [1] |

${\tau}^{\mathrm{a}}=3.15$ | [51] | |

Tortuosity for GDL | ${\tau}^{\mathrm{c}}=2.9$ | [52] |

**Table 3.**Binary diffusion coefficients ${\u0110}_{\mathrm{kj}}^{\mathrm{eff}}$, effective Knudsen diffusion coefficients ${\mathit{D}}_{\mathrm{k}}^{\mathrm{Kn},\mathrm{eff}}$, Fick’s diffusion coefficients ${\mathit{D}}_{\mathrm{k}}^{\mathrm{eff}}$, and thermal diffusion coefficients ${\mathit{D}}_{\mathrm{k}}^{\mathrm{T}}$ at $T=1123.15$ K.

Diffusion coefficients | Value |
---|---|

${\u0110}_{{\mathrm{H}}_{2}-{\mathrm{H}}_{2}\mathrm{O}}^{\mathrm{eff}}$ | $0.004$ m${}^{2}$/s |

${\u0110}_{{\mathrm{O}}_{2}-{\mathrm{N}}_{2}}^{\mathrm{eff}}$ | $9.78\xb7{10}^{-4}$ m${}^{2}$/s |

${\mathit{D}}_{{\mathrm{H}}_{2}}^{\mathrm{Kn},\mathrm{eff}}$ | $5.45\xb7{10}^{-4}$ m${}^{2}$/s |

${\mathit{D}}_{{\mathrm{H}}_{2}\mathrm{O}}^{\mathrm{Kn},\mathrm{eff}}$ | $1.82\xb7{10}^{-4}$ m${}^{2}$/s |

${\mathit{D}}_{{\mathrm{O}}_{2}}^{\mathrm{Kn},\mathrm{eff}}$ | $1.49\xb7{10}^{-4}$ m${}^{2}$/s |

${\mathit{D}}_{{\mathrm{N}}_{2}}^{\mathrm{Kn},\mathrm{eff}}$ | $1.59\xb7{10}^{-4}$ m${}^{2}$/s |

${\mathit{D}}_{{\mathrm{H}}_{2}}^{\mathrm{eff}}$ | $4.79\xb7{10}^{-4}$ m${}^{2}$/s |

${\mathit{D}}_{{\mathrm{H}}_{2}\mathrm{O}}^{\mathrm{eff}}$ | $1.74\xb7{10}^{-4}$ m${}^{2}$/s |

${\mathit{D}}_{{\mathrm{O}}_{2}}^{\mathrm{eff}}$ | $1.29\xb7{10}^{-4}$ m${}^{2}$/s |

${\mathit{D}}_{{\mathrm{N}}_{2}}^{\mathrm{eff}}$ | $1.36\xb7{10}^{-4}$ m${}^{2}$/s |

${\mathit{D}}_{{\mathrm{H}}_{2}}^{\mathrm{T}}$ | $-2.74\xb7{10}^{-7}$ m${}^{2}$/Ks |

${\mathit{D}}_{{\mathrm{H}}_{2}\mathrm{O}}^{\mathrm{T}}$ | $2.74\xb7{10}^{-7}$ m${}^{2}$/Ks |

${\mathit{D}}_{{\mathrm{O}}_{2}}^{\mathrm{T}}$ | $6.48\xb7{10}^{-9}$ m${}^{2}$/Ks |

${\mathit{D}}_{{\mathrm{N}}_{2}}^{\mathrm{T}}$ | $-6.48\xb7{10}^{-9}$ m${}^{2}$/Ks |

**Table 4.**Reference values of the local entropy production rate at $y=0$ (anode GDL), $y=0.4\xb7{10}^{-4}\phantom{\rule{0.222222em}{0ex}}\mathrm{m}$ (electrolyte), and $y=1.9\xb7{10}^{-4}\phantom{\rule{0.222222em}{0ex}}\mathrm{m}$ (cathode GDL).

Anode GDL | Electrolyte | Cathode GDL | |
---|---|---|---|

${\dot{\sigma}}_{\mathrm{q},0}^{\mathrm{a}}\phantom{\rule{0.222222em}{0ex}}\left|\phantom{\rule{0.222222em}{0ex}}{\dot{\sigma}}_{\mathsf{\varphi},0}^{\mathrm{a}}\phantom{\rule{0.222222em}{0ex}}\right|\phantom{\rule{0.222222em}{0ex}}{\dot{\sigma}}_{\mathrm{k},0}^{\mathrm{a}}$ | ${\dot{\sigma}}_{\mathrm{q},0}^{\mathrm{e}}\phantom{\rule{0.222222em}{0ex}}|\phantom{\rule{0.222222em}{0ex}}{\dot{\sigma}}_{\mathsf{\varphi},0}^{\mathrm{e}}$ | ${\dot{\sigma}}_{\mathrm{q},0}^{\mathrm{c}}\phantom{\rule{0.222222em}{0ex}}\left|\phantom{\rule{0.222222em}{0ex}}{\dot{\sigma}}_{\mathsf{\varphi},0}^{\mathrm{c}}\phantom{\rule{0.222222em}{0ex}}\right|\phantom{\rule{0.222222em}{0ex}}{\dot{\sigma}}_{\mathrm{k},0}^{\mathrm{c}}$ | |

j in $\mathrm{A}/{\mathrm{m}}^{2}$ | in $\mathrm{W}/{\mathrm{m}}^{3}\mathrm{K}$ | in $\mathrm{W}/{\mathrm{m}}^{3}\mathrm{K}$ | in $\mathrm{W}/{\mathrm{m}}^{3}\mathrm{K}$ |

2000 | $-0.04\phantom{\rule{0.222222em}{0ex}}\left|\phantom{\rule{0.222222em}{0ex}}0.17\phantom{\rule{0.222222em}{0ex}}\right|\phantom{\rule{0.222222em}{0ex}}4.84$ | $0.18\phantom{\rule{0.222222em}{0ex}}|\phantom{\rule{0.222222em}{0ex}}613.95$ | $-0.15\phantom{\rule{0.222222em}{0ex}}\left|\phantom{\rule{0.222222em}{0ex}}0.95\phantom{\rule{0.222222em}{0ex}}\right|\phantom{\rule{0.222222em}{0ex}}4.18$ |

8000 | $0.64\phantom{\rule{0.222222em}{0ex}}\left|\phantom{\rule{0.222222em}{0ex}}1.31\phantom{\rule{0.222222em}{0ex}}\right|\phantom{\rule{0.222222em}{0ex}}77.40$ | $4.40\phantom{\rule{0.222222em}{0ex}}|\phantom{\rule{0.222222em}{0ex}}9822.7$ | $-0.98\phantom{\rule{0.222222em}{0ex}}\left|\phantom{\rule{0.222222em}{0ex}}17.33\phantom{\rule{0.222222em}{0ex}}\right|\phantom{\rule{0.222222em}{0ex}}68.15$ |

**Table 5.**Reference values of the local entropy production rate at $y=0.4\xb7{10}^{-4}\phantom{\rule{0.222222em}{0ex}}\mathrm{m}$ (electrolyte) for different operating temperatures at low and high current density.

j in $\mathbf{A}/{\mathbf{m}}^{2}$ | T in $\mathbf{K}$ | ${\dot{\mathit{\sigma}}}_{\mathbf{q},0}^{\mathbf{e}}$ in $\mathbf{W}/{\mathbf{m}}^{3}\mathbf{K}$ | ${\dot{\mathit{\sigma}}}_{\mathsf{\varphi},0}^{\mathbf{e}}$ in $\mathbf{W}/{\mathbf{m}}^{3}\mathbf{K}$ |
---|---|---|---|

2000 | 1073.15 | 0.19 | 933.10 |

2000 | 1273.15 | 0.16 | 233.78 |

8000 | 1073.15 | 5.70 | 14,928.61 |

8000 | 1273.15 | 3.06 | 3740.38 |

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Gedik, A.; Lubos, N.; Kabelac, S.
Coupled Transport Effects in Solid Oxide Fuel Cell Modeling. *Entropy* **2022**, *24*, 224.
https://doi.org/10.3390/e24020224

**AMA Style**

Gedik A, Lubos N, Kabelac S.
Coupled Transport Effects in Solid Oxide Fuel Cell Modeling. *Entropy*. 2022; 24(2):224.
https://doi.org/10.3390/e24020224

**Chicago/Turabian Style**

Gedik, Aydan, Nico Lubos, and Stephan Kabelac.
2022. "Coupled Transport Effects in Solid Oxide Fuel Cell Modeling" *Entropy* 24, no. 2: 224.
https://doi.org/10.3390/e24020224