## 1 Introduction
Finally, energy stored in an all-iron flow battery using MWCNT slurry (both positive and negative) electrodes exhibited an improved state-of-charge at current densities higher than 200 mA cm\({}^{-2}\)[32]. Mathematical modeling and numerical simulations of electrochemical energy systems such as redox flow batteries and supercapacitors play vital roles in design improvements and performance estimation. Newman's porous electrode theory [33; 34] predicts the electrodes' adsorption capacity connected with EDL charging and the interfacial area of electrode per unit volume. The active mass transport of electrodes and electrolytes in the FEs corresponds to the diffusion, migration, and convection processes without affecting electroneutrality [35]. Continuum theory is mainly used to investigate the electrode-electrolyte interface, porous material theories, and geometrical effects on the device's performance. Figure 1: Schematic architecture of an EFC for charging slurry by EDL formation. However, in such an approach, the interface is considered a continuous phase, having numerous challenges, including false numerical diffusion [36] and accurate consideration of the particle shape [37; 38; 39], chemical reactions at the surface [40], and interactive forces such as the Brownian and drag forces [41; 42]. The standard Poisson-Nernst-Planck model simulates the effect of spherical particle size on the interactive forces at a water-oil interface [43]. Particles at close proximity and their geometries determine the localized interaction forces since the particle can distort the electric field around it [43]. The Stokesian simulation model predicts the effect of particle concentration and the ratio of the particle charging time to interaction time for cluster formation and improvement in electric conductivity [44]. The particle-particle interaction driven by the electric field contributes to the dielectrophoretic interaction near the particle surface and forms connection chains depending upon the concentration of the particles [45]. The role of computational fluid dynamics (CFD) to predict carbon-loaded FEs hydrodynamics has increased considerably. The calculation of charge redistribution and interactive forces between particles is a time-consuming CFD simulation, considering the large number of particles in the system. Particle-in-cell is a common approach to studying the high-particle-count behavior by considering a large computational superparticle (SP) representing thousands of real particles [46; 47; 48]. The present work investigates the charging of slurry electrodes in an EFC device using a novel stochastic particle model. The model uses the computational superparticle as a basic unit, predicting the dynamic behavior of real carbon particles in the background fluid (water). Furthermore, the mathematical model is optimized based on critical parameters of the model, and their sensitivity is validated against experimental measurements of the EFC device. The model also allows observing the dynamics of SPs charging in both flow channels side by side. The aim is to improve the working understanding and estimate the efficiency of EFCs, capacitive deionization, and similar systems.