# Transport Phenomena in a Banded Solid Oxide Fuel Cell Stack—Part 2: Numerical Analysis

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## Abstract

**:**

## 1. Introduction

## 2. Geometry Description

- (a)
- The electrolyte-supported design with 0.3 mm of electrolyte thickness. The stack is enclosed by two half-tubular covers, which make the fuel and air channels (initial prototype).
- (b)
- Electrolyte-supported stack is enclosed by rectangular covers. This solution reduces the stack volume and simultaneously increases the volumetric power density; this proposal has two subdesigns:
- (b1)
- supported on 0.3 mm electrolyte;
- (b2)
- supported on 0.1 mm electrolyte.

- (c)
- Extension of the system to multiple stacks. By combining two stacks, in which anodes are targeted face-to-face, a fuel channel is created, while cathodes facing the outer sides could be supplied with air without a separate channel, using, for example, a fan or placing the stack in a moving vehicle. This case consists of two stacks supported on a 0.1 mm electrolyte. This design allows reduction of the volume of the stack even more and, theoretically, double the power (because of the doubled number of cells).

**Figure 2.**Stack configuration options with schematic cross-sections. Stack supported on 0.3 mm of electrolyte thickness closed in half-tubular covers, marked as (

**a**). Stack with rectangular covers marked as (

**b**), prepared in two versions: 0.3 mm of electrolyte thickness, marked as (

**b1**), and 0.1 mm of electrolyte thickness, marked as (

**b2**). Multistack composition with 0.1 mm of electrolyte thickness, marked as (

**c**).

## 3. Mathematical Model

## 4. Numerical Analysis

#### 4.1. Gas Flow Rates Influence Study

#### 4.2. Geometric Improvements Study

## 5. Conclusions

## Author Contributions

## Funding

## Data Availability Statement

## Acknowledgments

## Conflicts of Interest

## Nomenclature

$\mathrm{F}$ | Faraday constant | (9.65 × 10^{4} $\mathrm{C}$ ${\mathrm{mol}}^{-1}$) |

h | specific enthalpy | ($\mathrm{J}\mathrm{k}\mathrm{g}{}^{-1}$) |

${h}_{\mathrm{react}}$ | enthalpy change | ($\mathrm{J}\mathrm{mol}{}^{-1}$) |

I | current | ($\mathrm{A}$) |

${i}_{\mathrm{an}}^{\mathrm{eq}}$ | anode equilibrium exchange current | ($\mathrm{A}\mathrm{m}{}^{-1}$) |

${i}_{\mathrm{cat}}^{\mathrm{eq}}$ | cathode equilibrium exchange current | ($\mathrm{A}{\mathrm{m}}^{{}^{-2}}$) |

i | current density flux | ($\mathrm{A}{\mathrm{m}}^{{}^{-2}}$) |

j | volumetric transfer current density | ($\mathrm{A}\mathrm{m}{}^{-3}$) |

${\overrightarrow{J}}_{i}$ | diffusion flux | ($\mathrm{k}\mathrm{g}\mathrm{m}{}^{-2}\mathrm{s}{}^{-1}$) |

k | thermal conductivity | ($\mathrm{W}\mathrm{m}{}^{-1}\mathrm{K}{}^{-1}$) |

M | molar mass | ($\mathrm{k}\mathrm{g}\mathrm{mol}{}^{-1}$) |

p | static pressure | ($\mathrm{Pa}$) |

P | power | ($\mathrm{W}$) |

${P}_{\mathrm{den}}\phantom{\rule{3.33333pt}{0ex}}$ | power density | ($\mathrm{W}{\mathrm{cm}}^{-2}$) |

$\dot{{q}_{\mathrm{V}}}$ | volumetric flux of species | (${\mathrm{m}}^{3}\mathrm{s}{}^{-1}$) |

$\mathrm{R}$ | universal gas constant | ($8.314$ $\mathrm{J}$ $\mathrm{K}{}^{-1}$ $\mathrm{mol}{}^{-1}$) |

$\overrightarrow{S}$ | source/sink term of momentum | ($\mathrm{k}\mathrm{g}\mathrm{m}{}^{-2}\mathrm{s}{}^{-2}$) |

${S}_{h}$ | source/sink term of heat | ($\mathrm{W}\mathrm{m}{}^{-3}$) |

${S}_{i}$ | source/sink term of species rates | ($\mathrm{k}\mathrm{g}\mathrm{m}{}^{-3}\mathrm{s}{}^{-1}$) |

${S}_{p}$ | source/sink term of mass | ($\mathrm{k}\mathrm{g}\mathrm{m}{}^{-3}\mathrm{s}{}^{-1}$) |

T | temperature | ($\mathrm{K}$) |

$\overrightarrow{v}$ | velocity vector | ($\mathrm{m}\mathrm{s}{}^{-1}$) |

${X}_{i}$ | local species concentration | (kmol $\mathrm{m}{}^{-3}$) |

${Y}_{i}$ | species mass fraction | (-) |

Greek letters | ||

$\alpha $ | transfer coefficient | (-) |

$\gamma $ | concentration dependence | (-) |

$\epsilon $ | porosity rate | (-) |

${\zeta}_{\mathrm{TPB}}$ | triple phase boundary length density | ($\mathrm{m}\mathrm{m}{}^{-3}$) |

${\zeta}_{\mathrm{DBP}}$ | double phase boundary length density | ($\mathrm{m}{}^{2}\mathrm{m}{}^{-3}$) |

$\eta $ | local surface overpotential | ($\mathrm{V}$) |

$\rho $ | density | ($\mathrm{k}\mathrm{g}\mathrm{m}{}^{-3}$) |

$\sigma $ | conductivity | ($\mathrm{S}\mathrm{m}{}^{-1}$) |

$\overline{\overline{\tau}}$ | stress tensor | ($\mathrm{Pa}$) |

$\varphi $ | electric potential | ($\mathrm{V}$) |

Sub- and superscripts | ||

$\mathrm{a}$ | anodic | |

$\mathrm{an}$ | anode | |

$\mathrm{c}$ | cathodic | |

$\mathrm{cat}$ | cathode | |

$\mathrm{den}$ | density | |

$\mathrm{DPB}$ | double phase boundary | |

$\mathrm{eff}$ | effective value | |

$\mathrm{el}$ | electronic | |

$\mathrm{eq}$ | equilibrium | |

i | reaction component | |

$\mathrm{ion}$ | ionic | |

$\mathrm{react}$ | reaction | |

$\mathrm{ref}$ | reference value | |

$\mathrm{s}$ | solid | |

$\mathrm{TPB}$ | triple phase boundary | |

Abbreviations | ||

3D | three-dimensional | |

CAD | computer-aided design | |

CFD | computational fluid dynamics | |

OCV | open circuit voltage | |

SOFC | solid oxide fuel cell |

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**Figure 4.**Top left: Characteristics of current–voltage (current–power) comparison of the model results and experimental study. Top right: Characteristics of current–voltage (current–power), obtained from the model. Two cases represent different gas flow rates at the operating temperature of $T=1173\mathrm{K}$. The circle marks indicate the maximum power points. Bottom: Potential (power) distribution of individual cells in the stack of two circles marked as (

**a**,

**b**). Cells are numbered according to the flow direction.

**Figure 5.**Hydrogen mass fraction distribution in the plane located 5 $\mathsf{\mu}$$\mathrm{m}$ above the anode surface. Local range of values. (

**a**) ${\dot{q}}_{\mathrm{V}}^{{\mathrm{H}}_{2}}=200$ mL/min; ${\dot{q}}_{\mathrm{V}}^{\mathrm{Air}}=200$ mL/min; $P=0.935\mathrm{W}$; $I=0.356\mathrm{A}$; $T=1173\mathrm{K}$, (

**b**) ${\dot{q}}_{\mathrm{V}}^{{\mathrm{H}}_{2}}=1000$ mL/min; ${\dot{q}}_{\mathrm{V}}^{\mathrm{Air}}=2000$ mL/min; $P=1.191\mathrm{W}$; $I=0.515\mathrm{A}$; $T=1173\mathrm{K}$.

**Figure 6.**Distribution of the oxygen mass fraction in the plane located 5 $\mathsf{\mu}$m above the cathode surface. Local range of values. (

**a**) ${\dot{q}}_{\mathrm{V}}^{{\mathrm{H}}_{2}}=200$ mL/min; ${\dot{q}}_{\mathrm{V}}^{\mathrm{Air}}=200$ mL/min; $P=0.935\mathrm{W}$; $I=0.356\mathrm{A}$; $T=1173\mathrm{K}$, (

**b**) ${\dot{q}}_{\mathrm{V}}^{{\mathrm{H}}_{2}}=1000$ mL/min; ${\dot{q}}_{\mathrm{V}}^{\mathrm{Air}}=2000$ mL/min; $P=1.191\mathrm{W}$; $I=0.515\mathrm{A}$; $T=1173\mathrm{K}$.

**Figure 7.**Distribution of the water vapor mass fraction in a plane located 5 $\mathsf{\mu}$$\mathrm{m}$ above the anode surface. Local range of values. (

**a**) ${\dot{q}}_{\mathrm{V}}^{{\mathrm{H}}_{2}}=200$ mL/min; ${\dot{q}}_{\mathrm{V}}^{\mathrm{Air}}=200$ mL/min; $P=0.935\mathrm{W}$; $I=0.356\mathrm{A}$; $T=1173\mathrm{K}$, (

**b**) ${\dot{q}}_{\mathrm{V}}^{{\mathrm{H}}_{2}}=1000$ mL/min; ${\dot{q}}_{\mathrm{V}}^{\mathrm{Air}}=2000$ mL/min; $P=1.191\mathrm{W}$; $I=0.515\mathrm{A}$; $T=1173\mathrm{K}$.

**Figure 8.**Velocity distribution located in the flow channel symmetry plane. Fixed values range. (

**a**) ${\dot{q}}_{\mathrm{V}}^{{\mathrm{H}}_{2}}=200$ mL/min; ${\dot{q}}_{\mathrm{V}}^{\mathrm{Air}}=200$ mL/min; $P=0.935\mathrm{W}$; $I=0.356\mathrm{A}$; $T=1173\mathrm{K}$, (

**b**) ${\dot{q}}_{\mathrm{V}}^{{\mathrm{H}}_{2}}=1000$ mL/min; ${\dot{q}}_{\mathrm{V}}^{\mathrm{Air}}=2000$ mL/min; $P=1.191\mathrm{W}$; $I=0.515\mathrm{A}$; $T=1173\mathrm{K}$.

**Figure 9.**Temperature distribution located in the flow channel symmetry plane. (

**a**) ${\dot{q}}_{\mathrm{V}}^{{\mathrm{H}}_{2}}=200$ mL/min; ${\dot{q}}_{\mathrm{V}}^{\mathrm{Air}}=200$ mL/min; $P=0.935\mathrm{W}$; $I=0.356\mathrm{A}$; $T=1173\mathrm{K}$, (

**b**) ${\dot{q}}_{\mathrm{V}}^{{\mathrm{H}}_{2}}=1000$ mL/min; ${\dot{q}}_{\mathrm{V}}^{\mathrm{Air}}=2000$ mL/min; $P=1.191\mathrm{W}$; $I=0.515\mathrm{A}$; $T=1173\mathrm{K}$. (

**c**) Difference between (

**a**,

**b**) distribution.

**Figure 10.**Comparison of current–voltage and current–power characteristics of four different designs, with a bar plot illustrating the power density peak of each. Design marks (a, b1, b2, c) are consistent with geometries shown in Figure 2.

**Figure 11.**The temperature distribution comparison of peak power point of each inspected geometry. Design marks (

**a**), (

**b1**), (

**b2**), and (

**c**) are consistent with geometries shown in Figure 2. The cases (

**a**), (

**b1**), and (

**b2**) are co-flow formed, and flow direction determines the x-axis. The (

**c**) case cross-flow is shown in Figure 3. Distribution is located along the channels’ symmetry plane.

**Figure 12.**The hydrogen mass fraction distribution comparison of peak power point of each inspected geometry. Design marks (

**a**,

**b1**,

**b2**,

**c**) are consistent with geometries shown in Figure 2. The cases (

**a**,

**b1**,

**b2**) are co-flow formed, and flow direction determines the x-axis. The (

**c**) case cross-flow is shown in Figure 3. Distribution is located along the channels’ symmetry plane.

**Figure 13.**The oxygen mass fraction distribution comparison of peak power point of each inspected geometry. Design marks (

**a**,

**b1**,

**b2**,

**c**) are consistent with geometries shown in Figure 2. The cases (

**a**,

**b1**,

**b2**) are co-flow formed, and flow direction determines the x-axis. The (

**c**) case cross-flow is shown in Figure 3. Distribution is located along the channels’ symmetry plane.

**Figure 14.**The water vapor mass fraction distribution comparison of peak power point of each inspected geometry. Design marks (

**a**,

**b1**,

**b2**,

**c**) are consistent with geometries shown in Figure 2. The cases (

**a**,

**b1**,

**b2**) are co-flow formed, and flow direction determines the x-axis. The (

**c**) case cross-flow is shown in Figure 3. Distribution is located along the channels’ symmetry plane.

**Table 1.**Boundary conditions applied for all cases studied in this section, with the distinction between cases studied in Section 4.1 and Section 4.2.

Parameter | Description | Value |
---|---|---|

Applied: Section 4.1 for case “(a)” | ||

Air inlet flow rate | constant value | 0.2 $\mathrm{L}{\mathrm{min}}^{-1}$ |

Fuel (hydrogen) inlet flow rate | constant value | 0.2 $\mathrm{L}{\mathrm{min}}^{-1}$ |

Applied: Section 4.1 for case “(b)” and all cases in Section 4.2 | ||

Air inlet flow rate | constant value | 2 $\mathrm{L}{\mathrm{min}}^{-1}$ |

Fuel (hydrogen) inlet flow rate | constant value | 1 $\mathrm{L}{\mathrm{min}}^{-1}$ |

Applied: all cases | ||

Inlets temperature | constant value | 1173 $\mathrm{K}$ |

Outer walls temperature | constant value | 1173 $\mathrm{K}$ |

Outer walls gas leakage | constant value | 0 $\mathrm{L}{\mathrm{min}}^{-1}$ |

Air outlet temperature | constant value | 1173 $\mathrm{K}$ |

Fuel outlet temperature | constant value | 1173 $\mathrm{K}$ |

Air outlet pressure | constant value | 1 atm |

Fuel outlet pressure | constant value | 1 atm |

Anode current collectors external contact | constant potential value | 0 $\mathrm{V}$ |

Cathode current collectors external contact | constant current value | range 0–0.63 $\mathrm{A}$ (case dependent) |

Solid bodies outer walls current leakage | constant value | 0 $\mathrm{A}$ |

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## Share and Cite

**MDPI and ACS Style**

Śreniawski, K.K.; Moździerz, M.; Brus, G.; Szmyd, J.S.
Transport Phenomena in a Banded Solid Oxide Fuel Cell Stack—Part 2: Numerical Analysis. *Energies* **2023**, *16*, 4512.
https://doi.org/10.3390/en16114512

**AMA Style**

Śreniawski KK, Moździerz M, Brus G, Szmyd JS.
Transport Phenomena in a Banded Solid Oxide Fuel Cell Stack—Part 2: Numerical Analysis. *Energies*. 2023; 16(11):4512.
https://doi.org/10.3390/en16114512

**Chicago/Turabian Style**

Śreniawski, Karol K., Marcin Moździerz, Grzegorz Brus, and Janusz S. Szmyd.
2023. "Transport Phenomena in a Banded Solid Oxide Fuel Cell Stack—Part 2: Numerical Analysis" *Energies* 16, no. 11: 4512.
https://doi.org/10.3390/en16114512