A generalized multi-dimensional mathematical model for charging and discharging processes in a supercapacitor S. Allu a, * , B. Velamur Asokan b , W.A. Shelton c, d , B. Philip a , S. Pannala a, * a Oak Ridge National Laboratory, Computer Science and Mathematics Division,1 Bethel Valley Road, Oak Ridge, TN 37831, USA b Exxonmobil Upstream Research Company, Houston, TX 77098, USA c Cain Department of Chemical Engineering, Louisiana State University, Baton Rouge, LA 70803, USA d Center for Computation and Technology, Louisiana State University, Baton Rouge, LA 70803, USA highlights  Generalized 3D computational model of an electric double layer supercapacitor.  3D microstructural aspects do not have a significant impact on the performance.  Specific capacitance, ionic conductivity, and tortuosity are critical.  State-of-the-art numerical methods provide accurate and robust solutions. article info Article history: Received 18 November 2013 Received in revised form 11 January 2014 Accepted 13 January 2014 Available online 23 January 2014 Keywords: Supercapacitors Computer modeling Electrochemical modeling Multidimensional simulations Energy storage abstract A generalized three dimensional computational model based on unified formulation of electrode eelectrolyte system of an electric double layer supercapacitor has been developed. This model accounts for charge transport across the electrode-electrolyte system. It is based on volume averaging, a widely used technique in multiphase flow modeling ([1,2]) and is analogous to porous media theory employed for electrochemical systems [3e5]. A single-domain approach is considered in the formulation where there is no need to model the interfacial boundary conditions explicitly as done in prior literature ([6]). Spatio-temporal variations, anisotropic physical properties, and upscaled parameters from lower length- scale simulations and experiments can be easily introduced in the formulation. Model complexities like irregular geometric configuration, porous electrodes, charge transport and related performance char- acteristics of the supercapacitor can be effectively captured in higher dimensions. This generalized model also provides insight into the applicability of 1D models ([6]) and where multidimensional effects need to be considered. A sensitivity analysis is presented to ascertain the dependence of the charge and discharge processes on key model parameters. Finally, application of the formulation to non-planar supercapacitors is presented. Ó 2014 Elsevier B.V. All rights reserved. 1. Introduction Ultracapacitors or supercapacitors are charge storage devices that operate on the principle of electrochemical double layer (ECDL) capacitance wherein, electrical energy can be stored and released by nanoscale charge separation at the interface between the electrode and the electrolyte. In paper [7] advances in electrode materials for the supercapacitor are summarized. There has been considerable activity in the recent years to exploit high surface area offered by Nanotubes [8e14] and more recently Graphene [15]. Most of the projected gains in energy densities (getting closer to that of current Li-ion batteries) are due to increase in capacitance through increase in surface area and exploiting the nano-pore and ion interactions. However, considerable development needs to be done at the system level to ascertain the true performance in a practical supercapacitor [16]. In this work, we developed a macroscopic model that can rapidly use the microscale properties to assess the overall performance of the supercapacitor device and possibly aid in transition of this rapid progress in supercapacitor electrode materials into high performance supercapacitors without lot of iterations at the device level where one needs to balance the cathode, anode, electrolyte, and current collectors to maximize energy and power density. * Corresponding authors. E-mail addresses: allus@ornl.gov (S. Allu), badri.velamur.asokan@exxonmobil. com (B. Velamur Asokan), shelton@lsu.edu (W.A. Shelton), philipb@ornl.gov (B. Philip), pannalas@ornl.gov, pannalas@gmail.com (S. Pannala). Contents lists available at ScienceDirect Journal of Power Sources journal homepage: www.elsevier.com/locate/jpowsour 0378-7753/$ e see front matter Ó 2014 Elsevier B.V. All rights reserved. http://dx.doi.org/10.1016/j.jpowsour.2014.01.054 Journal of Power Sources 256 (2014) 369e382