Flexible modeling and control of capacitive-deionization processes through a linear-state-space dynamic Langmuir model

Abstract

While black-box models such as neural networks have been powerful in many applications, direct physical modeling (white box) remains crucial in many fields where experimental data are difficult or time-consuming to obtain. Here, we demonstrate with an example from desalination by capacitive deionization (CDI), how an existing physical model could be strengthened by combining a general modeling framework with physical insights (gray box). Thus, a dynamic Langmuir (DL) model is extended to a linear-state-space DL model (LDL). Results obtained show the new LDL model could incorporate general structural and operational modes, including membrane CDI and constant-current operation. The formulation removes the need for direct measurements of detailed device properties without adding model complexity, and MATLAB code for automatically implementing the model is provided in the Supplementary Information. We conclude the new LDL model is widely applicable, offering great flexibility in calibration data, and enabling prediction over general operating modes.

Sodium to cesium ions: a general ladder mechanism of ion diffusion in prussian blue analogs

“…36 That study also included simulations of analogous cations, such as Li + . That work also shows that such a model can be combined with macroscopic, 37–39 finite-element (FEM) 16,40–42 simulations to successfully predict device-level performance. Still, previous studies have argued that larger ionic size changes the diffusion pathways, and therefore it is important to also investigate how well the proposed mechanism translates to the diffusion of larger ions.…”

Langmuir-Based Modeling Produces Steady Two-Dimensional Simulations of Capacitive Deionization via Relaxed Adsorption-Flow Coupling

“… 24 The work presented the surprising results that Langmuir-based modeling could generate both accurate and physically meaningful results, at equilibrium 24 , 30 , 33 and in the 0D setting. 23 , 70 …”

“…Investigating this, we recently introduced a dynamic Langmuir (DL) model . The work presented the surprising results that Langmuir-based modeling could generate both accurate and physically meaningful results, at equilibrium ,, and in the 0D setting. , …”

“…Thus, we derive and implement a DL model in 2D. While earlier authors on 2D modeling have noted that such models can be “unsteady”, the early DL approach had a decoupled framework that enhanced the stability in lower dimensions. ,, Thus, we develop two decoupled approaches that could enhance the stability of spatial CDI modeling to demonstrate the underlying stabilizing principles. The first is based on an older circuit-modeling framework, while the other uses the new DL model.…”

A new automated model brings stability to finite‐element simulations of capacitive deionization

“…Different modeling approaches has been built to describe the CDI processes including molecular dynamics, [53] transmission line, [29,54,55] dynamic Langmuir, [40,41,[56][57][58][59][60] and modified Donnan (mD) [2,11,23,61,62] models. While some macroscopic studies simplify treating the entire cell as a single 0D entity, [63] others consider spatiotemporal information through spatially resolved finite-element modeling (FEM) using software packages like COMSOL.…”