Journal of Power Sources 166 (2007) 376–385 Adjoint method for solid-oxide fuel cell simulations S. Kapadia ∗ , W.K. Anderson, L. Elliott, C. Burdyshaw University of Tennessee SimCenter at Chattanooga, 701 East M.L. King Boulevard, Chattanooga, TN 37403, USA Received 14 December 2006; received in revised form 25 January 2007; accepted 26 January 2007 Available online 6 February 2007 Abstract Adjoint methods suitable for obtaining sensitivity derivatives for numerical simulations of solid-oxide fuel cells are presented. The adjoint method is derived, and the implementation is discussed, including a methodology for accurately obtaining all the linearizations necessary for correct implementation. Results are included for a one-dimensional anode model that includes diffusion, permeation, and relevant chemical reactions. Using this model, the accuracy of the sensitivity derivatives is demonstrated for design variables describing geometric and material properties of the anode. Finally, the adjoint method is demonstrated for a three-dimensional fuel cell geometry where sensitivity derivatives are obtained for approximately 180,000 design variables. The results are used to modify the upper and lower walls of the plenum to obtain significantly improved distribution of fluid amongst the channels. © 2007 Elsevier B.V. All rights reserved. Keywords: SOFC; Fuel cell; Design; Adjoint; Sensitivity analysis 1. Introduction Numerical simulations of solid-oxide fuel cells (SOFC) have been used to gain understanding of important physical phenom- ena and to supply guidance in the continuing development of improved fuel cells [1–8]. To date, the simulations have been primarily focused on analysis of fuel cells or fuel cell compo- nents, without strong emphasis on utilizing the simulations in a design optimization environment. Because of the emphasis on analysis instead of design, sensitivity information to determine the effects of variations in design parameters on performance has been primarily implemented by simply changing the param- eter of interest, re-running the simulation, and comparing the results with those from the original simulation [1,5,6,8]. While this approach can be used to determine the effects of parameter variations on fuel cell performance, a more rigorous approach toward optimization would likely lead to better designs, and can also provide improved insight into the parameters affecting the performance of the fuel cell. For SOFC problems, exam- ∗ Corresponding author. Tel.: +1 4234255513; fax: +1 4234255517. E-mail addresses: Sagar-Kapadia@utc.edu (S. Kapadia), Kyle-Anderson@utc.edu (W.K. Anderson), Louie-Elliott@utc.edu (L. Elliott), Chad-Burdyshaw@utc.edu (C. Burdyshaw). ple cost functions that can be used for improving performance include minimizing temperature variations, obtaining equal dis- tribution of fuel in each of the channels, or maximizing power. Although not included in this study, time-dependent formula- tions are also possible that may be useful for minimizing start-up and short-term transient times. Design variables may be related to the shape/size of the fuel channels, electrodes, electrolyte, and interconnect, but may also be coupled to the stoichiometric composition of fuel or material properties such as the porosity or tortuosity of the electrodes. In refs. [9,10], optimization algorithms have been used to improve the performance of a polymer-electrolyte-membrane fuel cell (PEM) using four design variables, where the sen- sitivity derivatives used for the optimization algorithm have been obtained using a finite-difference approach. While finite differences are often a viable means for computing sensitiv- ity derivatives, this method can be computationally restrictive when a sufficiently large number of design variables are present. In addition, accurate derivatives can sometimes be difficult to obtain using finite differences because of subtractive cancella- tion errors [11,12], which occur when the function evaluations in the numerator become computationally indistinguishable when very small perturbations are used. To date, there has not been extensive research targeted at providing accurate sensitivity derivatives that may be used in conjunction with optimization 0378-7753/$ – see front matter © 2007 Elsevier B.V. All rights reserved. doi:10.1016/j.jpowsour.2007.01.085