Charging of flowable electrodes with bimodal distribution of carbon particles

Abstract

Carbon slurries as “flowable electrodes” have been used recently in a few electrochemical systems, e.g., electrochemical flow capacitors (EFCs) for energy storage and flow-electrode capacitive deionization (FCDI) for water treatment. These slurries typically have three parts: activated carbon particles, a conductive additive such as carbon black, and an aqueous electrolyte solution. Previously, a particle-based computational model that employs Stokesian dynamics was developed to describe the particle motion and interaction, while simultaneously solving for the charge transfer inside an electrical network of moving particles. In this work, we develop a unified expression of the dynamically varying electrical network. Furthermore, we incorporate a group of smaller particles as the conductive additive, whose effect on the charge transfer of the slurry is studied. The results suggest that at lower concentrations, the small particles may enhance charge transfer by filling interstitial spaces and bridging contacts of large particles; however, at higher concentrations, the benefits are not as clear since direct contacts of the large particles play the dominant role in charge transfer.

Appendix A

In addition to the verification in Karzar-Jeddi et al. [13], we have done further tests of the SD code against the BEM calculations using a code previously developed by Prof. Pozrikidis [26]. In this test, the particle ratio $a_2/a_1$ = 5 is used to match the bimodal particle interactions in the present SD simulation. A total of 1294 triangular elements with 7 Gaussian integration points (6 integration points for singular elements) were used in the BEM for each particle. Figure 6a displays the normalized wall-normal particle velocity versus the normalized gap distance between the two particles, while Fig. 6b displays the inline particle velocity. The calculations from the SD simulation we employ agree well with the BEM method, especially at a gap distance greater than $a_2$. At lower gaps, the inline velocity shows more pronounced differences for the smaller particle, but the error is still within 15%. Similar to Karzar-Jeddi et al. [13], further mesh refinement in the BEM shows that this discrepancy persists. Thus, the lubrication approximation in the Stokesian dynamics likely has caused the error.

Fig. 7

Time-averaged charge contour, $\langle Q(x, y)\rangle$, averaged from up to 8 simulations for each case. Here, the volume concentration is $\varphi$ = 0.10 (a–c), $\varphi$ = 0.15 (d–f), and $\varphi$ = 0.20 (g–i)

In order to verify the statistical results, a comparison of three, five, and eight simulations was performed on the $N_b$ = 25 case for the three volume concentrations. This comparison, as seen in Fig. 7, shows that the charge contours are consistent among these calculations and thus ensures that the number of simulations used to provide the data for Fig. 4 is sufficient.