# Effects of Radial and Circumferential Flows on Power Density Improvements of Tubular Solid Oxide Fuel Cells

^{*}

## Abstract

**:**

## 1. Introduction

## 2. Model Description

#### 2.1. Geometry Description

#### 2.2. Numerical Model

#### 2.2.1. Electrochemical Reactions

_{OCV}is the equilibrium potential at a current density equal to zero. The subscript a is for the anode electrode and c is for the cathode electrode. η

_{act}represents the activation overpotential, and the η

_{ohmic}represents the ohmic overpotential. The open-circuit voltage potential is expressed as

_{2}, and CO

_{2}denote the specific gas components. P

_{atm}is the ambient pressure [41]. The Butler–Volmer equations are used to calculate the activation overpotential:

_{a}and i

_{c}are the anodic and cathodic current density at the reference temperature T

_{ref}, respectively. n is the number of electrons transferred per υ molar reactant in the electrochemical reaction. α

_{1}and α

_{2}are the cathodic direction transfer coefficient defined in an early study [42], which are 1 and 2, respectively. The Faraday constant is F. i

_{0}is the reference exchange current density (A/m

^{2}). This can be stated in the form of Equations (8) and (9) using the Arrhenius expression [43]:

_{eff}is the effective conductivity, and ϕ is the potential. The subscript l is for the electrolyte, and s is for the electrodes. The effective conductivities of the porous electrodes and the structural factors are defined as follows:

#### 2.2.2. Momentum Transport

**u**denote the density, dynamic viscosity, and gas mixture velocity. P represents pressure. We can calculate the gas mixture density from the ideal gas law:

_{ij}is defined as

_{m}that appears in the mass and momentum equations is a result of the electrochemical reactions, as shown in Equation (20).

#### 2.2.3. Mass Transport

_{i}is the source term as a result of the generated species due to electrochemical reactions. The molecular diffusion coefficients can be calculated as

^{−1}) and v is the diffusion volume (cm

^{3}).

_{p}is the effective pore radius of the electrodes.

#### 2.3. Numerical Model

_{2}concentration to the outer surface of the porous cathode, a uniform CO mass flow rate at the fuel inlet, and the ambient pressure at the fuel outlet. The CO mole fraction was fixed as 0.95 at the fuel inlet for all cases investigated. The operating potential was applied to the cathode surface, while the zero potential is applied to the anode surface. At the insert walls, we used a no-slip boundary condition. The temperature is assumed to be 800 ℃ everywhere in the SOFC. The carbon deposition inside the porous electrode is not considered in this simulation. Table 2 summarizes the parameters used in this study.

#### 2.4. Model Validation

**u**are the fluid density and velocity.

## 3. Results and Discussion

#### 3.1. Effect of Radial Flows

_{gap}, is defined as the clearance between the cone-insert and the anode-fuel channel interface, while the width, C

_{width}, is the thickness of the cone base and the pitch, C

_{pitch}, is the distance between two successive cones.

_{gap}” on flow and mass transfer characteristics in a micro-tubular SOFC. Figure 5 plots the predicted streamlines and CO mole fractions in the mid-plane throughout the central axis of the SOFC. The streamlines in the flow channel were parallel to the anode-fuel channel interface (z-direction) for the case without the insert. The streamlines within the anode were oriented radially, showing the direction of fuel diffusion. The streamlines exhibited tiny fluctuations at the corners of the cone-inserts for the case with a 1 mm gap, as illustrated in Figure 5. The influence of these fluctuations on fuel diffusion is negligible. As the gap was further decreased from 1 mm to 0.1 mm or 0.01 mm, the fuel was forced into the porous anodes, which significantly affects the fuel diffusion, as observed from the streamlines in the anodes.

_{r}is the characteristic velocity in the r-direction, r

_{p}is the effective pore radius, and D

_{coco2,eff}is the effective diffusion coefficient. Figure 6a compares the local Péclet numbers as a function of axial locations for the cases with gaps of 1 mm, 0.1 mm, and 0.01 mm. The local Péclet number for the no-insert case is at the order of magnitude of 10

^{−5}, indicating a small convective mass transfer at the anode-fuel channel interface. The case with a 1 mm gap is comparable to the no-insert case, showing insignificant peaks at the order of 10

^{−5}, which indicates a low r-direction velocity at the anode-fuel channel interface. This is in part because most of the fuel can flow through the 1 mm gap between the cone-insert and the anode-fuel channel interface instead of entering the anode. As the gap is decreased from 1 mm to 0.1 mm, the local Péclet number is increased, and it reaches the maximum value at the base of the cones, indicating that the fuel enters the anode pores at a relatively large r-direction velocity. The case with a 0.1 mm gap shows a maximum local Péclet number of 0.005. As the gap is further decreased to 0.01 mm, these peaks become significantly higher and wider, indicating that more fuel is delivered into the porous anode. The positive and negative peaks highlight that the fuel enters or leaves the anode, respectively. The local Péclet number peaks with a higher order of magnitude, indicating that convective and diffusive mass transfer of fuel in the radial direction are comparable at the anode-fuel channel interface. The convective mass transfer along the radial direction enables more fuel to reach the triple-phase boundaries and therefore has the potential to improve the SOFC power density.

_{n}, which is the ratio of the power density of the micro-tubular SOFC with inserts to that with no insert, expressed as

_{in}represents the power density from the micro-tubular SOFC with an insert, while P

_{0}represents the power density from the conventional micro-tubular SOFC without an insert. The gas-phase pressure drop is calculated from the difference between the fuel inlet and outlet divided by the total length of the SOFC. A smaller gap between inserts and the anode-channel interface drives more fuel to enter the porous anode, leading to a higher CO concentration and a large concentration gradient inside the porous anode. This facilitates the fuel diffusion to the triple-phase boundaries and improves the output power density. A 0.01 mm gap could improve the output power density of SOFC by 24%. However, the increase in the SOFC power density coexists with the increase in pressure drop. As more CO fuel is forced to travel through the porous anode, a higher pump power is required to overcome the friction between the gas and porous electrode. A cone-insert could lead to a pressure drop of around 300 Pa/mm for the case with a 0.01 mm gap.

_{width}” on the mass transfer of fuel in the micro-tubular SOFC. Figure 7 shows the effect of width variations of the cone-insert base on the streamlines and CO mole fractions distributions in micro-tubular SOFCs. In these cases, the gap between cone-inserts and the anode-fuel channel interface was fixed at 0.01 mm, and the widths of the base of cone-inserts were 0.1 mm, 1 mm, and 5 mm, respectively. It can be seen from the streamline plots that the wavy patterns become wider as the width of the cone base increases. This is due in part to the fact that the fuel is forced to travel a longer distance for the case with a larger width. It also affects the radial variation of the CO mole fraction inside the porous anode. Compared to the case with 1 mm and 0.1 mm width, the cone-insert case of a 5 mm width shows a lower average molar concentration at the SOFC outlet, meaning more CO is consumed. This leads to the conclusion that forcing the fuel to travel inside the anode at a longer distance could result in a higher output power density.

_{pitch}” on flow and mass transfer characteristics in a micro-tubular SOFC. Figure 9 compares the streamlines and the CO mole fractions at the plane across the center axis for no insert and three cone-insert cases with a fixed gap of 0.01 mm and different pitches of 12 mm, 18 mm, and 24 mm. Figure 9 demonstrates that the wave interactions caused by the presence of the cone-insert become weaker as the pitch increases, so increasing the pitch causes a decrease in the radial concentration. The number of wavy patterns is inversely proportional to the insert pitch. A smaller insert pitch could enable fuel to enter the anode repeatedly.

_{ave}, is expressed as

_{r}is the local fuel velocity perpendicular to the anode surface at the anode-fuel channel interface, and A is the surface area of the interface between the fuel channel and porous anode. Figure 11a shows that, in general, an increase in the Péclet number leads to an increase in the SOFC power density. This is associated with the fact that radial flow facilitates fuel diffusion inside the porous anode, which then improves the electrochemical reactions and increases the power density. It should be noted from data group 2 that the normalized power densities are different for a similar average Péclet number. This proves that the distance fuel traveling inside the porous anode is another important factor for power density improvement, since a larger width of the cone base forces fuel to travel a longer distance in the anode. This is consistent with the observations in Figure 11b, which compares the current density distributions for the micro-tubular SOFCs with different cone-inserts. We demonstrate that the local current densities of the SOFC with inserts are systematically higher than that of the conventional micro-tubular SOFC, indicating that the electrochemical reactions are enhanced by the high local fuel concentration caused by radial flows. A smaller gap causes a higher current density peak, while a larger width also leads to a wider current density peak due to the long traveling distance of fuel in the anode. A decrease in the pitch leads to an increase in the number of peaks, which could effectively improve the output power density. However, it should be noted that increasing the local current density could also result in higher heat generations from electrochemical reactions. This may lead to local hotspots or high-temperature gradients, which might accelerate the degradation of the micro-tubular SOFC.

#### 3.2. Effect of Circumferential Flow

_{gap}, is the distance between the helical screw insert and the anode-fuel channel interface. The pitch, H

_{pitch}, is the distance measured between two points on the same plane of the helix tape that is one turn apart and parallel to the axis. The width, H

_{width}, is the thickness of the helix tape, and it was fixed as 0.2 mm in our study.

_{gap}on the flow and mass transfer characteristics of micro-tubular SOFC in anode and fuel channels. Figure 13c illustrates the simulated streamlines and CO mole fractions in the mid-plane along the central axis of the micro-tubular SOFC. As can be observed, for all the examined cases, the streamlines in the fuel channel are almost parallel to the anode-fuel channel interface (z-direction). For the cases with 1 mm, 0.1 mm, and 0.01 mm gaps, the streamlines are nearly identical. The CO mole fraction in the radial direction inside the anode seems uniform in all cases, indicating that the helical screw insert cannot force the flow to penetrate into the porous anode. This confirms that the circumferential flow field generated by the helical screw insert, regardless of gap or pitch variation, cannot increase the amount of fuel diffused into anode pores and has a negligible impact on the power density.

^{−5}, indicating a small radial velocity induced by the helical screw inserts. The non-zero velocity at the anode-channel interface is due in part to the flow rotation caused by helical screw inserts in the fuel channel. Figure 14a shows the Péclet numbers for the cases with three different gaps. A 1 mm gap has a negligible impact on the Péclet number distribution, while the gaps of 0.1 and 0.01 mm slightly increase the local Péclet number, which is mainly due to the increase in radial velocities with the decreasing gap. However, the maximum radial velocity induced by helical screw inserts is still three orders of magnitude smaller than that induced by cone-inserts. This is because the helical screw inserts generate and enhance the flow velocity in the circumferential direction, while the cone-inserts generate and enhance the flow velocity in the radial direction. Figure 14b shows that the oscillation amplitude of the Péclet number increases with the decreasing pitch of the helical screw insert. This is in part because the smaller pitch leads to a stronger flow interaction between two subsequent helical ridges. It should be noted that the location of the Péclet number peaks depends entirely on the helical ridge locations.

## 4. Conclusions

## Author Contributions

## Funding

## Institutional Review Board Statement

## Informed Consent Statement

## Conflicts of Interest

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**Figure 1.**Schematic of micro-tubular solid oxide fuel cells with (

**a**) cone-insert and (

**b**) helical screw insert.

**Figure 3.**The comparison between (

**a**) the applied voltage and current density of a micro-tubular SOFC, (

**b**) friction factor as a function of Reynolds numbers from our simulation and the results from previous study.

**Figure 4.**Schematic of the porous anode and cone-insert inside the fuel channel of micro-tubular SOFC.

**Figure 5.**The streamlines (

**left**) and CO mole fractions (

**right**) of the micro-tubular SOFCs, without an insert and with inserts of three different gaps.

**Figure 6.**The influence of cone-insert of different gaps on (

**a**) local Péclet number variation in the anode-fuel channel interface along the axial direction, (

**b**) the output power density and the gas-phase pressure drop of the micro-tubular SOFC.

**Figure 7.**The streamlines and CO mole fractions of the micro-tubular SOFCs, without an insert and with cone-inserts of different widths (the thickness of the cone base).

**Figure 8.**The influence of cone-insert of different widths on (

**a**) local Péclet number variation in the anode-fuel channel interface along the axial direction, (

**b**) the output power density and gas-phase pressure drop of micro-tubular SOFC.

**Figure 9.**The streamlines and CO mole fractions of the micro-tubular SOFCs, without an insert and with cone-inserts of different pitches.

**Figure 10.**The influence of cone-insert of different pitches on (

**a**) local Péclet number variation in the anode-fuel channel interface along the axial direction, (

**b**) the output power density and gas-phase pressure drop of micro-tubular SOFC.

**Figure 11.**Comparison of (

**a**) the power density increasement as a function of the average Péclet numbers for all the cases examined in Section 3.1. Data group 1, 2, and 3 respectively denotes the cases with gap, width, and pitch variations. (

**b**) axial current density distributions for the micro-tubular SOFCs with and without cone-inserts.

**Figure 12.**(

**a**) A schematic of the porous anode and helical screw insert inside the fuel channel and (

**b**) the predicted streamlines in the fuel channel of a micro-tubular SOFC with a helical screw insert.

**Figure 13.**The streamlines (

**left**) and CO mole fractions (

**right**) of the micro-tubular SOFCs (

**a**) without an insert; and with helical screw inserts of different (

**b**) pitches (

**c**) gaps.

**Figure 14.**Local Péclet number variation in the anode-fuel channel interface along the axial direction with helical screw insert for three different (

**a**) gap cases and (

**b**) pitch cases.

**Figure 15.**The influence of helical screw insert of different (

**a**) pitches and (

**b**) gaps on normalized power density and pressure drop of micro-tubular SOFC.

**Table 1.**The geometry details of the micro-tubular solid oxide fuel cells investigated in this study.

Parameters | Symbol | Value | Unit |
---|---|---|---|

Fuel channel diameter | D_{0} | 7 | mm |

Internal diameter | D_{i} | 1 | mm |

Total length | L_{t} | 50 | mm |

Anode thickness | t_{a} | 2 | mm |

Electrolyte thickness | t_{e} | 50 | µm |

Cathode thickness | t_{c} | 250 | µm |

Parameters | Symbols | Values | Units | Ref. |
---|---|---|---|---|

Activation energy for the anode reaction | E_{a} | 120 | kJ mol^{−1} | [47] |

Activation energy for the cathode reaction | E_{c} | 130 | kJ mol^{−1} | [47] |

Electrode porosity | ε | 0.35 | - | [43] |

Specific surface area of the anode | A_{υ,}_{a} | 2.33 × 10^{5} | m^{−1} | [48] |

Specific surface area of the cathode | A_{υ,}_{c} | 2.46 × 10^{5} | m^{−1} | [48] |

Permeability | k | 1 × 10^{−11} | m^{2} | [49] |

Electrode tortuosity | τ | 4 | - | [50] |

Viscosity, CO | μ_{CO} | 4.1877 × 10^{−5} | Pa·s | [44] |

Viscosity, O_{2} | μ_{O2} | 5.1343 × 10^{−5} | Pa·s | [44] |

Viscosity, CO_{2} | μ_{CO2} | 4.1904 × 10^{−5} | Pa·s | [44] |

Viscosity, N_{2} | μ_{N2} | 4.3529 × 10^{−5} | Pa·s | [44] |

Electrical conductivity, anode | σ_{Ni} | 30,316 | S·m^{−1} | [51] |

Electrical conductivity, cathode | σ_{LSM} | 12,793 | S·m^{−1} | [51] |

Ionic conductivity, electrolyte | σ_{YSZ} | 2.2669 | S·m^{−1} | [51] |

Diffusion volume, CO | υ_{CO} | 18.0 | cm^{3} | [52] |

Diffusion volume, O_{2} | υ_{O2} | 16.3 | cm^{3} | [52] |

Diffusion volume, CO_{2} | υ_{CO2} | 26.7 | cm^{3} | [52] |

Diffusion volume, N_{2} | υ_{N2} | 18.5 | cm^{3} | [52] |

Effective radius of the pores | r_{p} | 0.5 | µm | [53] |

Ambient pressure | P_{atm} | 101.325 | kPa | [41] |

Inlet mass flow rate at the anode | Q_{fuel} | 6.27 × 10^{−6} | kg s^{−1} | - |

Inlet mole fraction, CO | x_{CO} | 0.95 | - | - |

Inlet mole fraction, O_{2} | x_{O2} | 0.21 | - | - |

Reference temperature | T_{ref} | 800 | °C | - |

Operating potential | E | 0.6 | V | - |

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**MDPI and ACS Style**

Essaghouri, A.; Zeng, Z.; Zhao, B.; Hao, C.; Qian, Y.; Zhuge, W.; Zhang, Y. Effects of Radial and Circumferential Flows on Power Density Improvements of Tubular Solid Oxide Fuel Cells. *Energies* **2022**, *15*, 7048.
https://doi.org/10.3390/en15197048

**AMA Style**

Essaghouri A, Zeng Z, Zhao B, Hao C, Qian Y, Zhuge W, Zhang Y. Effects of Radial and Circumferential Flows on Power Density Improvements of Tubular Solid Oxide Fuel Cells. *Energies*. 2022; 15(19):7048.
https://doi.org/10.3390/en15197048

**Chicago/Turabian Style**

Essaghouri, Abdellah, Zezhi Zeng, Bingguo Zhao, Changkun Hao, Yuping Qian, Weilin Zhuge, and Yangjun Zhang. 2022. "Effects of Radial and Circumferential Flows on Power Density Improvements of Tubular Solid Oxide Fuel Cells" *Energies* 15, no. 19: 7048.
https://doi.org/10.3390/en15197048