# Fractal-Like Flow-Fields with Minimum Entropy Production for Polymer Electrolyte Membrane Fuel Cells

^{1}

^{2}

^{3}

^{4}

^{*}

## Abstract

**:**

_{h}), cross-sectional area (D

_{Σ}) and hydraulic resistance (D

_{Z}). The analytical expressions allowed us to obtain exact solutions to the optimization problem (TEP→min, TV=const). It was shown that the optimal design corresponds to a non-uniform width and length scaling of consecutive channels that classifies the flow field as a quasi-fractal. The depths of the channels were set equal for manufacturing reasons. Recursive formulae for optimal non-uniform width scaling were obtained for 1D circular D

_{h}-, D

_{Σ}-, and D

_{Z}-based tubes (Cases 1-3). Appropriate scaling of the fractal system providing uniform entropy production along all the channels have also been computed for D

_{h}-, D

_{Σ}-, and D

_{Z}-based 1D models (Cases 4-6). As a reference case, Murray’s law was used for circular (Case 7) and rectangular (Case 8) channels. It was shown, that Dh-based models always resulted in smaller cross-sectional areas and, thus, overestimated the hydraulic resistance and TEP. The D

_{Σ}-based models gave smaller resistances compared to the original rectangular channels and, therefore, underestimated the TEP. The D

_{Z}-based models fitted best to the 3D CFD data. All optimal geometries exhibited larger TEP, but smaller TV than those from Murray’s scaling (reference Cases 7,8). Higher TV with Murray’s scaling leads to lower contact area between the flow-field plate with other FC layers and, therefore, to larger electric resistivity or ohmic losses. We conclude that the most appropriate design can be found from multi-criteria optimization, resulting in a Pareto-frontier on the dependencies of TEP vs TV computed for all studied geometries. The proposed approach helps us to determine a restricted number of geometries for more detailed 3D computations and further experimental validations on prototypes.

## 1. Introduction

_{2}pollution levels in the atmosphere, as replacements technologies like solar, thermal, geothermal, tidal, and photovoltaics are not sufficient. Methanol, hydrogen, and other types of fuel cells (FCs) are direct convertors of chemical to electric energy, and play a central role in the vision of the hydrogen society [2]. The polymer electrolyte membrane fuel cells (PEMFC) are also attractive because of their portability and scalability, low operating temperature and high current density (i.e., high performance). A major aim for PEMFC development is thus to increase the efficiency and reduce the production costs. The latest report of the US Department of Energy (25 April 2018), target a price of $40/kW at 500,000 systems per year, including 80 kW automobiles and 160 kW trucks [3].

_{2}and O

_{2}) to the cell and remove excess water from the catalytic layer (Figure 1). Quite a big number of different designs of FFPs have been proposed and tested, using computational fluid dynamics (CFD) and experimental methods (see review [4]). The FFP comprises >60% of the weight and 30% of the total cost of the FC stack [5,6] and its optimization could improve the FC performance significantly [7].

## 2. Tree-Like Branching Networks for Distribution of Fluids to the Catalytic

#### 2.1. Properties of Natural and Man-Made Distribution Systems

- 1)
- Open-side channels which are in direct contact with a gas distribution layer (GDL);
- 2)
- Closed channels which are in direct contact with GDL via the branches of the last generation only.

#### 2.2. Flow in Fractal-Like Fluid Delivery Systems

## 3. System and Case Studies

- (1)
- Flow in rectangular channels computed from (14)–(16);
- (2)
- Flow in cylindrical tubes with equivalent hydraulic diameters ${D}_{hj}$ determined by (8);
- (3)
- Flow in cylindrical tubes with the same cross-sectional areas ${\Sigma}_{j}=h{w}_{j}$ as of the rectangular channels, i.e., with diameters$${D}_{\Sigma j}=\sqrt{\frac{4h{w}_{j}}{\pi}},$$
- (4)
- Flow in cylindrical tubes with the same hydraulic resistivity ${Z}_{j}$ as the rectangular channels, i.e., with diameters$${D}_{Zj}=\sqrt[4]{\frac{8\mu {L}_{j}}{\pi {Z}_{j}}},$$

## 4. The Optimization Problem

#### 4.1. The State of Minimum Entropy Production

- Case 1): The equivalent hydraulic diameters (8)—scaling by (22);
- Case 2): The equivalent cross-sectional area diameters (17)—scaling by (25);
- Case 3): The equivalent hydraulic resistivity diameters (18)—scaling by (26).

#### 4.2. An Approximation to the State of Minimum Entropy Production: Constant Pressure Gradient

- Case 4): the equivalent hydraulic diameters (8)—scaling by (29);
- Case 5): the equivalent cross-sectional area diameters (17)—scaling by (30);
- Case 6): the equivalent hydraulic resistivity diameters (18)—scaling by (31).

## 5. Method of Calculation

#### 5.1. Transport Properties

^{3}and $\mu =2.1\times {10}^{-5}$ Pa∙s [36,79]. The flow rates Q $=5.71\times {10}^{-7}$ m

^{3}/s and Q = $5.71\times {10}^{-6}$ m

^{3}/s correspond to current densities j = 1000 A/m

^{2}and j = 10,000 A/m

^{2}. These values are typical in FC use. The temperature, often used in FC experiments, was set to T = 353 K. The width values for the 0-generation inlet channel have been taken as ${w}_{0}=1.5;2;3;4;5$, that were used in 3D CDF computations of the same fractal flow field [36] in order to validate the analytical 1D solution.

#### 5.2. State of Minimum Entropy Production

## 6. Results and Discussion

^{2}. The variation from the inlet (j = 0) to the open end of the last generation (j = 6) is shown. The smallest pressure-decrease in the channels was obtained for Murray law’s scaling, and the steepest decrease was when the equivalent area was used to constrain the entropy production.

## 7. Conclusions

## Author Contributions

## Funding

## Acknowledgments

## Conflicts of Interest

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**Figure 1.**Components of PEM FC. A schematic representation showing different layers. The channels for gas transport are engraved in the flow field, see white cross-sectional area of the channels.

**Figure 2.**A symmetrical fractal-like flow field for uniform delivery of reactants to a catalytic layer. The numbering of generations of branches is shown.

**Figure 3.**Pressure variation P(x) along the fractal-like flow field for five flows with ${w}_{0}$ = 1.5 mm (

**a**), ${w}_{0}$ = 2 mm (

**b**), ${w}_{0}$ = 3 mm (

**c**), ${w}_{0}$ = 4 mm (

**d**), ${w}_{0}$ = 5 mm (

**e**), respectively. The lines correspond to Case (1)–(8) (see explanations in Section 4.1 and Section 4.2).

**Figure 4.**Pressure profiles $P(x)$ along the fractal flow field for five channel widths, ${w}_{0}$ = 1.5 mm (

**a**), ${w}_{0}$ = 2 mm (

**b**), ${w}_{0}$ = 3 mm (

**c**), ${w}_{0}$ = 4 mm (

**d**), and ${w}_{0}$ = 5 mm (

**e**); for the cases 1–8.

**Figure 5.**Dependencies TEP(${w}_{0}$) for the cases 1–5,7,8 computed on the rectangular ducts and their equivalent circular models.

**Figure 6.**Total entropy production (TEP) as function of total volume (TV) for 1D and 2D systems; the symbols correspond to the cases 1–8 computed on (21), (24), (25), (28)–(30) for the same w

_{0}.

N | Fuel Cell Pane | FFP Design | Compared to | Verification Method | Heat Effects | Pressure Drop and Pumping Power | Maximum Power Density/Efficiency | Fuel Conversion Rate | FoM* | Wall T, Heat Resistivity | T Uniformity | Reference |
---|---|---|---|---|---|---|---|---|---|---|---|---|

1 | Circular | Fractal (n=4, $\alpha <\pi /2$), | 1D | + | 60% ↓ | - | - | - | 30 °C ↓ | - | [39] | |

2 | Fractal (n=4, $\alpha =\pi $) | 1D+ experiment | + | 54–64% ↓ | - | - | - | - | 2.17–2.78 ↑ | [40,41] | ||

3 | Rectangle | Fractal (n=5, $\alpha <\pi $, smooth) rectangle cross-sections, combined with parallel | Serpentine, parallel | Experiment PEMFC, DMFC | - | similar performance to parallel designs | - | - | - | - | [42] | |

4 | Circular | Fractal (n=4, $\alpha <\pi /2$), | 3D CFD | + | 10% ↓ | - | - | - | - | 75 ↑ | [43] | |

5 | Rectangle | Fractal, rectangle cross-sections | 1D | + | 8.6–15% ↓ | 1.7–26% ↑ | - | - | - | - | [44] | |

6 | Rectangle | Fractal, rectangle cross-sections | 2D | + | ↓ | ↑ | - | - | - | - | [45] | |

7 | Rectangle | Fractal (n=5, $\alpha =\pi $) | 1D | + | ↓ for turbulent ↑ for laminar | - | - | - | - | - | [46] | |

8 | Rectangle, Square, Circular | Fractal $\alpha =\pi $ $\alpha =\pi /2$ $\alpha <\pi $ + parallel | ↓ | - | - | - | - | - | [47] | |||

9 | Rectangle | Fractal (n=2,3, $\alpha =\pi $) | 3D CFD | + | ↓ | - | - | - | ↓ | ↑ | [37,48,49,50,51,52] | |

10 | Rectangle | Fractal (n=4, $\alpha \le \pi $) | 1D | + | ↑ or ↓ | - | - | - | ↑ or ↓ | - | [53] | |

11 | Square | Fractal (n=6, $\alpha =\pi $) | 3D CFD | + | ↓ | - | - | - | ↓ | ↑ | [54] | |

12 | Rectangle | Fractal (n=5, $\alpha =\pi $) | 1D | + | ↑ in 5 times | - | - | - | - | ↑ | [55] | |

13 | Rectangle | Fractal (n=6, $\alpha =\pi $) | 3D CFD | + | ↓ | - | - | - | ↓ | ↑ | [56,57] | |

14 | Rectangle | Fractal (n=6, $\alpha =\pi $) | 1D | + | ↓ | ↑ | - | - | - | ↑ | [58] | |

15 | Rectangle | Fractal (n=6, $\alpha =\pi $), h=const | 3D CFD for DMFC | - | - | 10% ↑ | - | - | - | [59] | ||

16 | Rectangle | Fractal (n=1,2, $\alpha =\pi $) | 3D CFD | + | ↑ | - | - | - | ↓ | ↑ | [60] | |

17 | Rectangle | Fractal (n=1, $\alpha =\pi /3\xf7\pi $) | 3D CFD | + | ↑ | 30–60% ↑ | - | - | ↓ | ↑ | [61,62] | |

18 | Rectangle | Fractal (n=4, $\alpha =\pi /2$), h=const | 3D CFD for DMFC | - | - | ↑ | ↑ | - | - | [63] | ||

19 | Rectangle | Fractal (n=2, $\alpha ={37}^{\circ},{74}^{\circ}$) | 3D CFD | - | ↓ | ↓ | - | - | - | - | [64] | |

20 | Square | 3D 5-layered lung-inspired fractal (n=4, $\alpha =\pi /2$ + $\alpha =\pi $) | Serpentine | 10 cm^{2} FEMFC | ↓↓ | ↑, max in fractal with n=4, better than in serpentine (at 50% and 75% RH**) | [33] | |||||

21 | Square | 3D 5-layered lung-inspired fractal (n=4, $\alpha =\pi /2$ + $\alpha =\pi $) | Serpentine | Experiment, PEM FC | + | + | ↓, high water accumulation; less stable performance than the serpentine | - | - | - | + | [34] |

22 | Rectangle | Fractal (n=6, $\alpha =\pi $) | - | 1D, 2D, 3D CFD | - | ↓ and close to linear | - | - | - | - | - | [36] |

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## Share and Cite

**MDPI and ACS Style**

Kizilova, N.; Sauermoser, M.; Kjelstrup, S.; Pollet, B.G.
Fractal-Like Flow-Fields with Minimum Entropy Production for Polymer Electrolyte Membrane Fuel Cells. *Entropy* **2020**, *22*, 176.
https://doi.org/10.3390/e22020176

**AMA Style**

Kizilova N, Sauermoser M, Kjelstrup S, Pollet BG.
Fractal-Like Flow-Fields with Minimum Entropy Production for Polymer Electrolyte Membrane Fuel Cells. *Entropy*. 2020; 22(2):176.
https://doi.org/10.3390/e22020176

**Chicago/Turabian Style**

Kizilova, Natalya, Marco Sauermoser, Signe Kjelstrup, and Bruno G. Pollet.
2020. "Fractal-Like Flow-Fields with Minimum Entropy Production for Polymer Electrolyte Membrane Fuel Cells" *Entropy* 22, no. 2: 176.
https://doi.org/10.3390/e22020176