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Research Papers

# An Analytical Study of Diffusion, Chemical Reaction and Voltage Loss in High-Temperature Solid Oxide Fuel Cells

[+] Author and Article Information
John B. Young

Hopkinson Laboratory, Cambridge University Engineering Department,  Cambridge University, Cambridge, CB2 1PZ, UKjby@eng.cam.ac.uk

J. Fuel Cell Sci. Technol 9(2), 021002 (Mar 19, 2012) (11 pages) doi:10.1115/1.4005413 History: Received June 13, 2011; Revised September 30, 2011; Published March 07, 2012; Online March 19, 2012

## Abstract

The paper describes a mathematical analysis of multi-component diffusion with chemical reaction in the porous materials of high-temperature solid oxide fuel cells. The objectives are to clarify the underlying physics, to investigate different modeling approaches and to establish expressions for the cell voltage loss. The description proceeds from the simplest non-reactive binary diffusion process, through a multi-component analysis with non-reactive diluent gases present, to diffusion in the presence of the water-gas shift chemical reaction. Using a single average diffusion coefficient, an analytical solution can be found, not only for the limiting cases of frozen and equilibrium water-gas shift chemistry but also for the general non-equilibrium situation. A Damköhler number is identified and it is shown that shift equilibrium is not necessarily preserved in the anode flow. The non-equilibrium analysis also reveals unusual behavior whereby the molar fluxes become discontinuous in the equilibrium limit while the mole fractions and cell voltage loss approach the limit continuously. A physically more realistic model based on two diffusion coefficients provides a more detailed description for frozen and equilibrium chemistry but does not yield an explicit non-equilibrium solution. In all, the analysis provides fundamental insight and quantitative predictions for many of the flow phenomena occurring in the porous materials of SOFCs.

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## Figures

Figure 1

Boundary conditions and fluxes for diffusion of an O2 -N2 mixture in the cathode

Figure 2

Cathode voltage loss due to diffusion of a binary O2 -N2 mixture with T = 900 °C, i = 3000 Am−2 , and XO2,0=0.21

Figure 3

Boundary conditions and fluxes for non-reactive diffusion in the anode

Figure 4

Anode diffusion of a non-reactive H2 -H2 O-N2 mixture with T = 900 °C, i = 3000 Am−2 , β = 0.10, XH2,0=0.25, XH2O,0=0.25, XN2,0=0.50; (a) mole fraction variations, (b) voltage loss

Figure 5

Temperature variation of the shift reaction equilibrium constant Kps

Figure 6

Boundary conditions for diffusion of a reacting mixture of H2 , H2 O, CO and CO2 in the anode or support material

Figure 7

Variation of the Damkhöler number Φ with material thickness L for various pressures. Calculations based on T = 900 °C, β = 0.1, D = 5.0 × 10−4 m2 s−1 (at 1 bar pressure) and kfs  = 2.5 × 10−7 mol.m−3 Pa−2 s−1 .

Figure 8

Model 1. Frozen and equilibrium solutions for an H2 -H2 O-CO-CO2 mixture with Kps  = 1 (825 °C), λ = 0.2 and XH2,0e=XH2O,0e=0.25. Variation with z¯ of (a) H2 and H2 O mole fractions, (b) CO and CO2 mole fractions.

Figure 9

Model 1. Variation of voltage loss with material thickness. Frozen and equilibrium solutions for an H2 -H2 O-CO-CO2 mixture with Kps  = 1 (825 °C), i = 3000 Am−2 , n = 10.0 mol.m−3 , D = 3.9 × 10−4 m2 s−1 , β = 0.1 and XH2,0e=XH2O,0e=0.25.

Figure 10

Model 1. Non-equilibrium solution for an H2 -H2 O-CO-CO2 mixture with Kps  = 1 (825 °C), λ = 0.2, XH2,0e=XH2O,0e=0.25 and various Φ from 0 (frozen) to ∞ (equilibrium). Variations with z¯ of (a) dimensionless molar fluxes Ji/J0, (b) H2 and H2 O mole fractions, and (c) CO and CO2 mole fractions.

Figure 11

Model 1. Variation of voltage loss with material thickness. Non-equilibrium solutions for an H2 -H2 O-CO-CO2 mixture with Kps  = 1 (825 °C), i = 3000 Am−2 , n = 10.0 mol.m−3 , D = 3.9 × 10−4 m2 s−1 , β = 0.1 and XH2,0e=XH2O,0e=0.25.

Figure 12

Model 2. Frozen and equilibrium solutions for an H2 -H2 O-CO-CO2 mixture with Kps  = 1 (825 °C), λ1  = 0.127, λ2  = 0.471 and XH2,0e=XH2O,0e=0.25. Variation with z⇀ of (a) H2 and H2 O mole fractions, (b) CO and CO2 mole fractions.

Figure 13

Model 2. Variation of voltage loss with material thickness. Frozen and equilibrium solutions for an H2 -H2 O-CO-CO2 mixture with Kps  = 1 (825 °C), i = 3000 Am−2 , n = 10.0 mol.m−3 , D1  = 6.14 × 10−4 m2 s−1 , D2  = 1.66 × 10−4 m2 s−1 , β = 0.1 and XH2,0e=XH2O,0e=0.25.

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