##
640
(Ion Power Poster Award Winner) Analytical Model for the Interdigitated Flow Field

^{1}. Of the four major flow field design categories (flow-through, parallel, serpentine, and interdigitated flow fields (IDFF)), the IDFF has demonstrated high current densities with only a moderate pressure drop. Although many studies have noted the performance enhancements offered by the IDFF, development has been limited to application-specific numerical or empirical studies

^{2–4}; no bottom-up design rules exist. This lack of design principles stems from an inability to analytically model species flux and flow profile through the complicated IDFF geometry. As such, this study offers the first analytical solution of the velocity and reactant distributions through a two-dimensional (2-D) IDFF.

The flux of a species through the IDFF channel can be described by the Nernst-Planck equation in porous media, which is a *conformally invariant* mathematical* *function^{5}. Thus, it is possible to achieve an analytical expression for reactant flux and / or streamlines through the IDFF via the method of *conformal mapping*^{5}. Through this strategy, the combined migration-diffusion-advection problem can first be solved in a straight channel, and then the straight-channel solution is mapped (Figure 1) to the more complicated IDFF geometry by exploiting the Schwarz-Christoffel transformation^{5}. The original numerical 2-D model by Kazim et al. is solved in this work, analytically^{2}; Figure 2 shows a representative velocity magnitude profile for an IDFF electrode cross section. Due to unwieldy equations which arise from the analytical construction, dimensionless numbers are developed to provide an intuitive understanding of transport relationships in the IDFF, and limiting cases of electrode thickness are considered. Ultimately a convenient framework for IDFF design rules is developed by analytically solving the transport equations, allowing for greater insight to system design.

**Acknowledgements**

We would like to acknowledge the financial support of the Joint Center for Energy Storage Research and the United States National Science Foundation Graduate Research Fellowship program.

**References**

1. X. Li and I. Sabir, *Int. J. Hydrog. Energy*, **30**, 359–371 (2005).

2. A. Kazim, H. T. Liu, and P. Forges, *J. Appl. Electrochem.*, **29**, 1409–1416 (1999).

3. Y. Wang, S. Basu, and C.-Y. Wang, *J. Power Sources*, **179**, 603–617 (2008).

4. R. M. Darling and M. L. Perry, *J. Electrochem. Soc.*, **161**, A1381–A1387 (2014).

5. M. Z. Bazant, *Proc. R. Soc. Math. Phys. Eng. Sci.*, **460**, 1433–1452 (2004).