A bottom-up, multi-scale theory for transient mass transport of redox-active species through porous electrodes beyond the pseudo-steady limit

New theory is presented for the dynamic response of redox-active electrolyte flowing through porous electrodes under time-dependent applied current. This is done by introducing certain frequency-dependent transfer functions (TFs) that incorporate pore-scale transport physics. One TF – dubbed the spectral Sherwood number – extends the film law of mass transfer (FLoMT) to transient conditions. Another TF captures the acceleration/suppression of solute advection that results from pore-scale velocity/concentration gradients. Numerical results are shown for the frequency-dependent TFs of solid cylinders in crossflow to represent porous electrodes commonly used in flow batteries (FBs). Spectral regions are observed where a transition from lagless response to semi-infinite Warburg response occurs with increasing frequency. The embedding of these TFs into an up-scaled model is also formulated to obtain the time-domain response of FBs. Without adjustable parameters this model predicts polarization in agreement with transient FB experiments, despite systematic overprediction by the conventional FLoMT model. Analysis of concentration polarization and reactant concentration is also used to construct non-dimensional maps of operational space. These predictions show that fast current fluctuations are sustained even when current exceeds the limiting current expected from the time-invariant FLoMT without solute advection suppression/acceleration, suggesting implications for electrochemical conversion and separations devices in addition.