#
Impact of Multi-Causal Transport Mechanisms in an Electrolyte Supported Planar SOFC with (ZrO_{2})_{x−1}(Y_{2}O_{3})_{x} Electrolyte

^{*}

## Abstract

**:**

## 1. Introduction

_{2}needed, in comparison to PEMFCs [4,5]. SOFC systems with pre- or internal reforming can work with common fuels in the actual infrastructure and also work in a future hydrogen network [6,7,8]. Depending on the electrolyte material and thickness, SOFCs are operated at a temperature between $700\text{}\xb0\mathrm{C}$ and $1100\text{}\xb0\mathrm{C}$ [9,10].

- We expand the gas diffusion layers’ (GDLs’) NET models of S. Kjelstrup and D. Bedeaux [20] accounting for diffusion.
- We give a detailed description of the transport mechanisms in YSZ electrolytes based on NET using the phenomenological coefficients calculated with MD.
- Our NET model is validated with our own experimental data.
- We discuss in detail the influence of the coupled mechanisms and calculate each contribution to the entropy production rate in each layer.

## 2. Materials and Methods

#### 2.1. Materials and Experimental Methods

#### 2.2. Theoretical Methods

#### 2.2.1. SOFC-System

_{2}O

_{3}in ZrO

_{2}to form YSZ results in vacancies [46]. This defect formation can be described by the following reaction in Kröger–Vink notation:

_{2},H

_{2}O-ideal gas mixture at the anode side and an O

_{2},N

_{2}-ideal gas mixture at the cathode side. The thermodynamic state point is given by $T=1300\text{}\mathrm{K}$ and $p={p}^{o}=1\text{}\mathrm{bar}$, and we only consider steady state conditions.

#### 2.2.2. Mathematical Description of the GDLs

#### 2.2.3. Mathematical Description of the CLs

_{2}and H

_{2}O and at ${y}^{R}=\Delta {y}_{a}+\Delta {y}_{c}$ for O

_{2}, and using the transformation ${x}_{k}^{R}={c}_{k}\left({y}^{R}\right)RT/p$. We define following Butler–Volmer equations for the ORR (4) and the HOR (6):

#### 2.2.4. Mathematical Description of the Solid Oxide Electrolyte

#### 2.2.5. Computational Details

^{®}Version R2017a. We solve the equations consecutively from the anode to the cathode using the Runge–Kutta (4,5) method [53]. We assume as initial conditions $T\left(0\right)=1300\text{}\mathrm{K}$ and $\varphi \left(0\right)=0\text{}\mathrm{V}$. To reproduce the isothermal operational modus, we iterate numerically using the Newton–Raphson method. Thereby, the heat flux ${J}_{q}$ is a variable with a start value of $1000\text{}\mathrm{W}/{\mathrm{m}}^{2}$ until the temperature at $y=\Delta {y}_{a}+\Delta {y}_{e}+\Delta {y}_{c}$ reaches $T\left(0\right)$ with an accuracy of ${10}^{-34}\text{}\mathrm{K}$.

## 3. Results and Discussion

#### 3.1. EIS Measurements

#### 3.2. Model Validation

#### 3.3. Simulation Results

_{2}O

_{3}concentration result in higher absolute values for the temperature gradient in all of the bulk phases. Furthermore, a minimum at the anode and a maximum at the cathode can be observed. To explain these effects, we plot in Figure 5a the results of a simulation for YSZ08 and YSZ20 at $j=8000\text{}\mathrm{A}/{\mathrm{m}}^{2}$, where we neglect all Peltier (coupling) mechanisms. Here, all absolute values of the temperature gradients are lower than for the original conditions. In the YSZ20 electrolyte case, the one in the anode is positive and the one in the electrolyte does not progress linearly, experiencing a maximum at $y=1.8\times {10}^{-4}\text{}\mathrm{m}$. From a comparison with the previous results, the high absolute values for the temperature gradients may result from a Seebeck mechanism. While ${J}_{{O}^{-2}}^{e}$, ${L}_{Oq}$ and ${z}^{*}$ are negative in Equation (41), ${L}_{OO}$ is positive. A high potential loss in the electrolyte results in a high negative value of the potential gradient and the negative Ohmic term decreases. To counter this and to assure a constant anion flux, the temperature gradient has to have a high positive value. This may explain not only the pronounced temperature progression, but also its dependency on the current density, since a higher anion flux gives a higher counter effect. A calculation without these coupled mechanisms results in a significantly different temperature progression.

_{2}O

_{3}concentration in the electrolyte. Since the potential losses in the anode GDL and in the cathode GDL are too small, the progression of the curve is mostly defined by the losses in the electrolyte. To give a more detailed discussion, Figure 5b also includes results for simulations neglecting all Peltier mechanisms. The progression of the potential in the electrolyte is not substantially different from the original results and the losses are mainly Ohmic. However, the potential losses in the GDLs are mainly given by the temperature gradient rather than by the electric resistance. To justify this statement, we can argue analogous to the discussion of the pronounced temperature gradient by using Equation (12). We can calculate the effective specific resistance ${r}^{e}\left(1300\text{}\mathrm{K}\right)\approx 0.05621\text{}\mathsf{\Omega}\mathrm{m}$ using the potential gradient across the electrolyte and the current density. This value deviates only $~0.2\%$ from the one calculated with the conductivity ${L}_{OO}$. Therefore, the potential losses due to the coupled effects do not significantly affect the progression of the $E$-$j$ charactersitic and the direct comparison between ${L}_{OO}$ and ${r}^{e}$ in Section 3.1 is appropriate.

_{2}O

_{3}concentration, contradicting Fourier conduction behavior. In Figure 6, the heat flux resulting from a simulation without any coupled effects is given for the cases of an YSZ08 and an YSZ20 electrolyte at $8000\text{}\mathrm{A}/{\mathrm{m}}^{2}$. Due to energy conservation, the progression of these curves is almost equal to the progression of the original curves. However, the amount and sign of the heat flux is different. If the coupled mechanisms are considered, the heat transported is higher in all layers. On one hand, the higher heat flux at the GDLs results from the higher temperature gradient, since the Peltier-effect may reduce it (see Equation (11)). On the other hand, the higher heat flux in the electrolyte results from the potential gradient, since the positive temperature gradient may diminish it (see Equation (40)). This effect is most pronounced for a higher Y

_{2}O

_{3}concentration in the electrolyte and a higher current density due to the higher potential losses. As it can be seen from the YSZ20 case, if the coupled mechanisms are neglected, not only the amount of heat transported may be underestimated, but also the direction of the heat flux. This may lead to the wrong decisions regarding the material design or heating/cooling strategies of SOFCs, such as which gas inlet flow may be used for cooling the cell to minimize thermal stress.

_{2}O

_{3}content. Due to the Peltier effect, most of the values are negative across the electrolyte. These negative contributions are smaller than the potential contribution and a positive local entropy production rate is always ensured. The heat flux contributions are only positive for YSZ08 and, clearly, for the cases assuming no Peltier effect. For these latter cases, the absolute values for ${\dot{\sigma}}_{Q,0}$ are smaller than for the original cases. The values for ${\dot{\sigma}}_{Q,0}$ are proportional to the temperature gradient (see Equations (20) and (42)) and, thus, all these dependencies can be explained following the above discussion. The absolute values for ${\dot{\sigma}}_{Q,0}$ are higher across the GDLs than across the electrolyte in each case, resembling the results for the heat flux in Figure 6. Looking at Figure 7, the value of the heat flux contribution ${\dot{\sigma}}_{Q}$ in the GDLs increases slightly in the $y$-direction. If we assume a constant temperature gradient, the temperature and the heat flux progression may define the progression of ${\dot{\sigma}}_{Q}$. Here, the dependency on the Y

_{2}O

_{3}concentration is not well defined and, thus, we cannot find a direct correlation to the curves for the temperature or the heat flux. Therefore, the resulting behavior may be given from a combination of both effects. This may also explain the low changes in the progression of the curves for lower current densities and for the cases neglecting all coupled mechanisms. However, these changes are smaller than $0.2\%$ of the initial value. As already discussed, the negative potential gradient in the electrolyte and a high current density result into an increasing heat flux due to energy conservation. Together with the increasing temperature appearing in the denominator of the term ${\dot{\sigma}}_{Q}$ in Equation (43), the negative progression of ${J}_{q}^{e}$ results in a decreasing progression of ${\dot{\sigma}}_{Q}$ for YSZ08. This trend changes for higher Y

_{2}O

_{3}concentrations because the heat flux becomes increasing positive due to the Peltier effect. This behavior is observed for both current densities studied here. For an YSZ08 electrolyte, the value can decrease by $30\%$ across the electrolyte and, for higher compositions, the value can increase by $150\%$ depending on the current density. Additionally, the progression for ${\dot{\sigma}}_{Q}$ is not well predicted if the coupled effects are neglected, resulting in a strong decaying tendency independent of the electrolyte.

_{2}O

_{3}composition in all bulk phases, but its dependency on the electrolyte composition is not given in the GDLs if the Peltier effect is neglected. The dependency on the current density and on the Y

_{2}O

_{3}composition is given by Equation (20), where ${\dot{\sigma}}_{\varphi ,0}$ is negatively proportional to the product between $j$ and the potential gradient divided by the temperature. The low values for ${\dot{\sigma}}_{\varphi ,0}$ in the GDLs for the cases assuming no Peltier effect result from the low potential gradients. As expected from the similar progression of the potential with and without any Peltier mechanisms, all ${\dot{\sigma}}_{\varphi ,0}$ values in the electrolyte are similar for both cases. Furthermore, this contribution is the highest across the cell. For all curves in Figure 7, the slightly increasing or decreasing progression of ${\dot{\sigma}}_{\varphi}$ is given by the temperature. This statement is justified, since the dependency on the spatial dimension, on the current density, on the Y

_{2}O

_{3}composition and on whether the Peltier mechanisms are taken into account, is in agreement with the results depicted in Figure 5.

_{2}O

_{3}concentration of the electrolyte. To make a comparison between the entropy production at the CLs and at the bulk phases, non-specific values are needed. If we consider an YSZ12 cell with a $4\text{}\mathrm{cm}$ $\times $ $4\text{}\mathrm{cm}$ active area working at $8000\text{}\mathrm{A}/{\mathrm{m}}^{2}$ and use the values given in Table 3 as an estimate for the entropy production rate contributions, it results in an entropy production rate of $4.60\times {10}^{-5}\text{}\mathrm{W}/\mathrm{K}$ in the GDLs, of $1.08\times {10}^{-3}\text{}\mathrm{W}/\mathrm{K}$ in the electrolyte, and of $3.44\times {10}^{-4}\text{}\mathrm{W}/\mathrm{K}$ at the CLs. This means that $3\%$ of the losses would originate in the GDLs, $73\%$ in the electrolyte, and $24\%$ at the CLs. These values are expected for an ESC like the one studied here.

## 4. Summary and Conclusions

_{2}O

_{3}content. We describe the CLs using the Butler–Volmer ansatz. We conduct EIS measurements in a YSZ08 SOFC to derive the parameters to describe the exchange current densities ${j}_{0}^{i,0}$. Moreover, we measure the $E$-$j$-characteristic and use this data to validate our model. Here, we show that the Butler–Volmer kinetics is well parametrized and implemented. Furthermore, the relative deviation of $~6.4\%$ between the ASR of both curves is mainly due to the deviation of the electrolyte resistance calculated with ${L}_{OO}$ to the experimental value of $~10\%$. For a current density of $8000\text{}\mathrm{A}/{\mathrm{m}}^{2}$ the relative deviation between both $E$-$j$-characteristics is only $~1\%$, so that our approach effectively describes the SOFC. The temperature gradient across each layer is higher for a higher Y

_{2}O

_{3}electrolyte concentration and a higher current density. This effect results from the coupled mechanisms and, thus, if the Peltier effect is neglected, the temperature profile across the cell is not properly described. From the potential profile, we conclude that the highest contribution to the electric potential losses are in the electrolyte. These losses are higher for a higher Y

_{2}O

_{3}concentration and no substantial differences arise, if the simulation does not consider any coupled effects. Using the entropy balance for the calculation of the heat flux at the CLs, it results in an endothermic half reaction at the anode and an exothermic half reaction at the cathode. Moreover, the description of the transport mechanisms across a SOFC using NET cannot be avoided, since the heat flux in the electrolyte is mainly given by the Peltier effect. If this effect is neglected, the heat flux will not be properly described. Finally, we calculate the heat flux contribution, the potential contribution, and the diffusion contribution to the local entropy production rate. The entropy production rate across the electrolyte is mostly given by the ionic conduction, since the Peltier heat transfer from lower to higher temperature would reduce the entropy production. While the temperature and the heat flux define the progression of ${\dot{\sigma}}_{Q}$ across the electrolyte, the potential contribution profile is mainly given by the temperature. Despite the appreciable increasing or decreasing tendency of the contributions to the entropy production rate, a substantial change along the $y$-direction can be observed only for the heat flux contribution in the electrolyte. If we neglect the Peltier effect, the values for the heat flux and potential contribution are underestimated in the GDLs and a different profile of the heat flux contribution to the local entropy production rate is predicted across the electrolyte. Finally, the entropy production rate of a cell is the highest in the electrolyte ($73\%$) followed by the contribution in both GDLs with ($24\%$).

## Author Contributions

## Acknowledgments

## Conflicts of Interest

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**Figure 2.**Solid oxide fuel cell (SOFC) including the electric current density $j$, all heat fluxes ${J}_{q}^{i}$ and all molar fluxes ${J}_{k}^{i}$. The spatial coordinate $y$ is defined perpendicular to the cell area.

**Figure 3.**(

**a**) Impedance spectrum at a cell temperature of 1117 K and (

**b**) Arrhenius plot for the specific resistance $r$ for the electrolyte and the exchange current density ${j}_{0}$ for the anode and the cathode.

**Figure 4.**Experimental (Exp) and simulated voltage ($E$)-current($j$)-characteristic (Sim1) at ${T}_{cell}=1287\text{}\mathrm{K}$ (upper diagram), and simulated $E$-$j$-characteristics (Sim2 and Sim3) at ${T}_{cell}=1300\text{}\mathrm{K}$ (lower diagram). We use the value ${r}^{e}\left(1287\text{}\mathrm{K}\right)=0.0667\left(44\right)\text{}\mathsf{\Omega}\mathrm{m}$ in Sim1 and the value ${r}^{e}\left(1300\text{}\mathrm{K}\right)=0.0622\left(40\right)\text{}\mathsf{\Omega}\mathrm{m}$ in Sim2. Both values are calculated from the experimental Arrhenius curve. In Sim3, we use the phenomenological coefficients for YSZ08 from Valadez Huerta et al. [40].

**Figure 5.**(

**a**) Simulated temperature and (

**b**) electric potential profile across the SOFC at $1300\text{}\mathrm{K}$ for various YSZ-electrolytes and the current densities $j=1000\text{}\mathrm{A}/{\mathrm{m}}^{2}$ and $j=8000\text{}\mathrm{A}/{\mathrm{m}}^{2}$. For the simulated curves marked with *, we neglect all Peltier type coupled mechanisms.

**Figure 6.**Simulated heat flux ${J}_{q}\left(y\right)$ across the SOFC at 1300 K for various YSZ-electrolytes and the current densities $j=1000\text{}\mathrm{A}/{\mathrm{m}}^{2}$ and $j=8000\text{}\mathrm{A}/{\mathrm{m}}^{2}$. For the simulated curves marked with *, we neglect all Peltier type coupled mechanisms.

**Figure 7.**Simulated local entropy production rate contributions ${\dot{\sigma}}_{Q}$ (heat flux contribution), ${\dot{\sigma}}_{\varphi}$ (potential contribution) and ${\dot{\sigma}}_{c}$ (diffusion contribution) in the anode GDL, in the cathode GDL and in the electrolyte. We normalized the values with ${\dot{\sigma}}_{i,0}={\dot{\sigma}}_{i}\left(y=0\right)$ for the anode, ${\dot{\sigma}}_{i,0}={\dot{\sigma}}_{i}\left(y=2\times {10}^{-4}\text{}\mathrm{m}\right)$ for the cathode and (c) ${\dot{\sigma}}_{i,0}={\dot{\sigma}}_{i}\left(y=0.4\times {10}^{-4}\mathrm{m}\right)$ for the electrolyte (see Table 3). The curves are given for a SOFC at $1300\text{}\mathrm{K}$, for various YSZ-electrolytes and two current densities, $j=1000\text{}\mathrm{A}/{\mathrm{m}}^{2}$ and $j=8000\text{}\mathrm{A}/{\mathrm{m}}^{2}$. For the simulated curves marked with *, we neglect all Peltier type coupled mechanisms.

Parameter | Anode GDL | Anode CL | Cathode CL | Cathode GDL | Reference |
---|---|---|---|---|---|

$\Delta {y}_{i}$ in ${10}^{-6}\text{}\mathrm{m}$ | $40$ | - | - | $40$ | - |

${r}_{0}^{i}$ in ${10}^{-6}\text{}\mathsf{\Omega}\mathrm{m}/\mathrm{K}$ | $1/95$ | - | - | $1/42$ | [25] |

${E}_{A,r}^{i}/\mathrm{R}$ | $1150$ | - | - | $1200$ | [25] |

${\lambda}^{i}$ in $\mathrm{W}/\mathrm{mK}$ | $2$ | - | - | $2$ | [20] |

${\pi}^{i}/{T}_{cell}$ in $\mathrm{J}/\left(\mathrm{CK}\right)$ | $-5.4\times {10}^{-4}$ | - | - | $-6.2\times {10}^{-4}$ | [20] |

${D}_{{H}_{2}}={D}_{{H}_{2}O}$ in ${\mathrm{m}}^{2}/\mathrm{s}$ | $2.1\times {10}^{-5}$ | - | - | [24] | |

${D}_{{O}_{2}}$ in ${\mathrm{m}}^{2}/\mathrm{s}$ | - | - | $5.4\times {10}^{-6}$ | [24] | |

${\alpha}^{i}$ | - | $0.5$ | $0.3$ | - | [24] |

**Table 2.**Parameters used in the phenomenological equations to describe the electrolyte. The values are taken from the MD study of Valadez Huerta et al. [40].

YSZ | ${\mathit{L}}_{\mathit{O}\mathit{O}}$$\text{}\mathbf{in}\text{}{10}^{-7}\mathbf{m}\mathbf{o}{\mathbf{l}}^{2}\mathbf{K}{\left(\mathbf{Jms}\right)}^{-1}$ | ${\mathit{L}}_{\mathit{q}\mathit{q}}$$\text{}\mathbf{in}\text{}{10}^{6}\text{}\mathbf{W}/\mathbf{m}$ | $\mathit{R}{\mathit{L}}_{\mathit{o}\mathit{q}}$$\text{}\mathbf{in}\text{}\mathbf{W}/\mathbf{m}$ |
---|---|---|---|

YSZ08 | $6.22\left(79\right)$ | $6.21\left(25\right)$ | $-0.81\left(69\right)$ |

YSZ12 | $4.10\left(57\right)$ | $5.37\left(27\right)$ | $-0.77\left(75\right)$ |

YSZ16 | $2.85\left(36\right)$ | $4.85\left(34\right)$ | $-0.55\left(56\right)$ |

YSZ20 | $1.95\left(55\right)$ | $4.54\left(29\right)$ | $-0.48\left(36\right)$ |

**Table 3.**Values of the entropy production rate used for the normalization in Figure 7. For the values marked with *, we neglect all Peltier type coupled mechanisms.

YSZ|$\mathit{j}$ | Anode GDL ${\dot{\mathit{\sigma}}}_{\mathit{i}}\left(\mathit{y}=0\text{}\mathbf{m}\right)$ $\text{}\mathbf{W}/\left({\mathbf{m}}^{2}\mathbf{K}\right)$ $\mathit{i}=\mathit{Q}$|$\mathit{\varphi}$|$\mathit{C}$ | Electrolyte ${\dot{\mathit{\sigma}}}_{\mathit{i}}\left(\mathbf{y}=0.4\times {10}^{-4}\mathbf{m}\right)$ $\mathbf{W}/\left({\mathbf{m}}^{2}\mathbf{K}\right)$ $\mathit{i}=\mathit{Q}$|$\mathit{\varphi}$ | Cathode GDL ${\dot{\mathit{\sigma}}}_{\mathit{i}}\left(\mathit{y}=2\times {10}^{-4}\mathbf{m}\right)$ $\mathbf{W}/\left({\mathbf{m}}^{2}\mathbf{K}\right)$ $\mathit{i}=\mathit{Q}$|$\mathit{\varphi}$|$\mathit{C}$ |
---|---|---|---|

YSZ08|$1000\text{}\mathrm{A}/{\mathrm{m}}^{2}$ | $0.03$|$0.17$|$4.60$ | $0.007$|$43.24$ | $0.074$|$0.25$|$5.37$ |

YSZ20|$1000\text{}\mathrm{A}/{\mathrm{m}}^{2}$ | $0.28$|$0.29$|$4.60$ | $-0.12$|$138.6$ | $0.40$|$0.39$|$5.37$ |

YSZ08|$8000\text{}\mathrm{A}/{\mathrm{m}}^{2}$ | $1.02$|$10.26$|$294.20$ | $0.70$|$2767$ | $5.77$|$16.58$|$369.73$ |

YSZ12|$8000\text{}\mathrm{A}/{\mathrm{m}}^{2}$ | $6.11$|$13.54$|$294.20$ | $-2.04$|$4204$ | $14.65$|$20.91$|$369.73$ |

YSZ16|$8000\text{}\mathrm{A}/{\mathrm{m}}^{2}$ | $7.33$|$14.16$|$294.20$ | $-2.82$|$6033$ | $18.27$|$22.35$|$369.73$ |

YSZ20|$8000\text{}\mathrm{A}/{\mathrm{m}}^{2}$ | $13.55$|$16.84$|$294.20$ | $-6.64$|$8871$ | $30.77$|$26.55$|$369.73$ |

YSZ08*|$8000\text{}\mathrm{A}/{\mathrm{m}}^{2}$ | $0.001$|$0.001$|$294.20$ | $0.20$|$2760$ | $0.90$|$0.003$|$369.73$ |

YSZ20*|$8000\text{}\mathrm{A}/{\mathrm{m}}^{2}$ | $0.06$|$0.001$|$294.20$ | $0.57$|$8846$ | $1.86$|$0.003$|$369.73$ |

© 2018 by the authors. Licensee MDPI, Basel, Switzerland. This article is an open access article distributed under the terms and conditions of the Creative Commons Attribution (CC BY) license (http://creativecommons.org/licenses/by/4.0/).

## Share and Cite

**MDPI and ACS Style**

Valadez Huerta, G.; Flasbart, V.; Marquardt, T.; Radici, P.; Kabelac, S.
Impact of Multi-Causal Transport Mechanisms in an Electrolyte Supported Planar SOFC with (ZrO_{2})_{x−1}(Y_{2}O_{3})_{x} Electrolyte. *Entropy* **2018**, *20*, 469.
https://doi.org/10.3390/e20060469

**AMA Style**

Valadez Huerta G, Flasbart V, Marquardt T, Radici P, Kabelac S.
Impact of Multi-Causal Transport Mechanisms in an Electrolyte Supported Planar SOFC with (ZrO_{2})_{x−1}(Y_{2}O_{3})_{x} Electrolyte. *Entropy*. 2018; 20(6):469.
https://doi.org/10.3390/e20060469

**Chicago/Turabian Style**

Valadez Huerta, Gerardo, Vincent Flasbart, Tobias Marquardt, Pablo Radici, and Stephan Kabelac.
2018. "Impact of Multi-Causal Transport Mechanisms in an Electrolyte Supported Planar SOFC with (ZrO_{2})_{x−1}(Y_{2}O_{3})_{x} Electrolyte" *Entropy* 20, no. 6: 469.
https://doi.org/10.3390/e20060469