1. Date of download: 5/31/2016 Copyright © ASME. All rights reserved. From: Mechanistic Three-Dimensional Analytical Solutions for a Direct Liquid Fuel Cell Stack J. Fuel Cell Sci. Technol. 2015;12(6):061003-061003-11. doi:10.1115/1.4031958 Schematic of procedure used to obtain closed-form analytical solutions for a DMFC stack. (a) The stack comprising of n cells is first decoupled; (b) each cell, cell j, is considered individually. (c) Symmetry is used to reduce the computational domain to a representative volume cell; (d) the anode of the representative volume cell is decoupled from the membrane and cathode based on high stoichiometry of the latter. (e) Spatial smoothing is used to reduce the remaining computational domain to two dimensions while retaining salient features from the third dimension; (f) leading-order asymptotics is employed to simplify the set of PDEs, and the approximate analytical solutions for cell j is found. The solutions for each individual cell can then be recoupled to yield the closed-form analytical solution for a stack. Figure Legend:

2. Date of download: 5/31/2016 Copyright © ASME. All rights reserved. From: Mechanistic Three-Dimensional Analytical Solutions for a Direct Liquid Fuel Cell Stack J. Fuel Cell Sci. Technol. 2015;12(6):061003-061003-11. doi:10.1115/1.4031958 Schematic of a single cell depicting the various functional layers and their dimensions for the (a) 3D model and (b) spatially smoothed model Figure Legend:

3. Date of download: 5/31/2016 Copyright © ASME. All rights reserved. From: Mechanistic Three-Dimensional Analytical Solutions for a Direct Liquid Fuel Cell Stack J. Fuel Cell Sci. Technol. 2015;12(6):061003-061003-11. doi:10.1115/1.4031958 Polarization curves for decoupled cells with anode inlet velocity 2.19 × 10 −3 (cell 1, ), 4.38 × 10 −3 (cell 2, ), and 7.3 × 10 −3 (cells 3–10, ) m s −1 as well as overall ten cell stack. Individual cell potentials along a prespecified stack current can be summed to obtain the stack potential ; this process can be repeated multiple times using different currents to yield the stack polarization curve. Figure Legend:

4. Date of download: 5/31/2016 Copyright © ASME. All rights reserved. From: Mechanistic Three-Dimensional Analytical Solutions for a Direct Liquid Fuel Cell Stack J. Fuel Cell Sci. Technol. 2015;12(6):061003-061003-11. doi:10.1115/1.4031958 Polarization curves for the best performing single cell, worst performing single cell, ten cell stack, and limiting stack current Figure Legend:

5. Date of download: 5/31/2016 Copyright © ASME. All rights reserved. From: Mechanistic Three-Dimensional Analytical Solutions for a Direct Liquid Fuel Cell Stack J. Fuel Cell Sci. Technol. 2015;12(6):061003-061003-11. doi:10.1115/1.4031958 Individual cell potentials for a perturbed ten cell stack operating at V stack = 3.5 V. Due to the difference in inlet velocities, each cell is operating at a different potential in order to sustain an equal current density within each cell. Cells 1, 2, 3–10 and the normalized stack potential are 0.320, 0.347, 0.354, and 0.350 V, respectively. Figure Legend:

6. Date of download: 5/31/2016 Copyright © ASME. All rights reserved. From: Mechanistic Three-Dimensional Analytical Solutions for a Direct Liquid Fuel Cell Stack J. Fuel Cell Sci. Technol. 2015;12(6):061003-061003-11. doi:10.1115/1.4031958 Local current densities of a ten cell stack operating at V stack = 3.5 V for cell 1, cell 2, and cells 3–10. Cells 3–10 are operated with the highest anode inlet velocity; hence, they have the most uniform streamwise local current density profile. Figure Legend:

7. Date of download: 5/31/2016 Copyright © ASME. All rights reserved. From: Mechanistic Three-Dimensional Analytical Solutions for a Direct Liquid Fuel Cell Stack J. Fuel Cell Sci. Technol. 2015;12(6):061003-061003-11. doi:10.1115/1.4031958 Parasitic current densities of a ten cell stack operating at V stack = 3.5 V for cell 1, cell 2, and cells 3–10. Cells 3–10 are operated with the highest anode inlet velocity; therefore, they experience the highest parasitic current due to high methanol crossover via diffusion. Figure Legend:

8. Date of download: 5/31/2016 Copyright © ASME. All rights reserved. From: Mechanistic Three-Dimensional Analytical Solutions for a Direct Liquid Fuel Cell Stack J. Fuel Cell Sci. Technol. 2015;12(6):061003-061003-11. doi:10.1115/1.4031958 Methanol mass fraction within the anode flow channel and diffusion layer of cells 1, 2, and 3–10 of the perturbed stack. The analytical stack solution is able to resolve the boundary layer formed near the catalyst surface in each cell. Figure Legend: