Comparative Studies of Solutions of Homogeneous Electrochemical Capacitors Models

•The article presents homogeneous ECs models that incorporated self-discharges effects.•The homogeneous ECs models were solved both numerically and analytically.•Compares solutions of ECs models obtained using analytical and different numerical methods.•Insights into effects of self-discharge on ECs performance from modelling and simulation.•Electric double layers (EDLs) instability and side-reactions or redox species reactions were taken as sources of self-discharge The models of symmetric ECs with self-discharges term have been studied using a combination of various self-discharge mechanisms in devices’ conservation equations (mass transfer and charge) under charge and discharge processes. The models were solved both analytically and numerically for comparative study purpose, three numerical methods (Crank-Nicolson numerical method, fully implicit numerical method, and fully explicit numerical method) were employed via central finite differences for spatial derivatives. The profiles of capacitors with or without self-discharges using Crank-Nicolson numerical solution, fully implicit numerical and fully explicit numerical solution methods were 95%, 80% and 70% respectively in agreement with similar capacitors using analytical solution. Devices without self-discharges were charged from 0.00V to the expected voltage of 1.20V within expected charging time when analytical solution and Crank-Nicolson numerical method were employed, and from 0.00V to 1.15V and 0.00V to 1.00V (which are less than the target voltage) within target charging time by using fully implicit numerical and fully explicit numerical methods, respectively. The energy density of capacitors with electrode and electrolytes effective conductivities of 0.05S/cm without self-discharges using analytical solution, Crank-Nicolson numerical solution, fully implicit numerical and fully explicit numerical solution methods were 35.957Wh/kg, 35.757Wh/kg, 34.282Wh/kg and 24.953Wh/kg, respectively. The first cycle energy efficiency of devices using analytical solution, Crank-Nicolson numerical, fully implicit numerical and fully explicit numerical solution methods for capacitors with electrode and electrolytes effective conductivities of 0.05S/cm without self-discharges Vs were 84.24%, 84.04%, 72.33% and 38.13%, respectively. Simulation results obtained from Crank-Nicolson numerical solution method for certain device is more accurate than those from fully implicit numerical solution and fully explicit nume