Staged Diafiltration Cascades Provide Opportunities to Execute Highly Selective Separations

Abstract

Improved solute selectivity, an issue commonly approached through the development of advanced membrane materials, can be achieved through staged diafiltration processes. A mathematical model that describes the solute concentration profile within a single diafiltration module is developed and experimentally validated. For this module, where the diafiltrate is introduced uniformly over its length, a critical diafiltrate flux that results in the retentate concentration remaining constant during operations is identified. The model is then extended to examine two configurations of multistage diafiltration cascades: stripping sections and rectifying sections. The two configurations differ based on the connectivity between stages. Namely, stripping sections connect multiple modules by utilizing the retentate from one stage as the feed to the subsequent stage while the permeate is repurposed as the diafiltrate. In contrast, rectifying sections utilize the permeate from one stage as the feed to the subsequent stage and the retentate becomes the diafiltrate. For cascades that operate with a constant diafiltrate to feed flow ratio for all stages, the analysis demonstrates that stripping sections can reduce the diafiltrate consumed when separating low molar mass impurities from larger, impermeable molecules. On the other hand, cascades in a rectifying section configuration can improve the separation of two solutes with finite sieving coefficients between 0 and 1. Finally, asymmetric cascades, i.e., systems in which each stage has a unique diafiltrate to feed flow ratio, are shown to be capable of improving the recovery and purity of solutes in effluent streams relative to systems that operate at constant a diafiltrate to feed ratio. As a whole, the study highlights that the continued advancement of membrane separations will rely equally on thoughtful module and process design as well as the development of new materials.

Optimal Diafiltration Membrane Cascades Enable Green Recycling of Spent Lithium-Ion Batteries

“…These relationships, along with empirical correlations (e.g., those encoded within the permeability-selectivity trade-off) that constrain the values of the permeability coefficients or capture their variations in complex multicomponent solutions, can be incorporated in future iterations of the superstructure optimization framework. Similarly, concentration polarization is not considered explicitly because the system is assumed to operate at constant J w . − This phenomenon would enter into the model by simply changing the assumed value of the sieving coefficients and would yield results similar to the sieving coefficient sensitivity analysis presented in this study. , Lastly, this work focuses on the simplest case of binary fractionation of Li + and Co 2+ in an idealized feed. Extending the optimization framework to consider many components requires specifying additional sieving coefficients and possibly augmenting the superstructure to consider removal of other products, purge streams, or additional recycle streams.…”

“…As demonstrated by prior experiments, a key advantage of staged designs is that they facilitate the continuous operation of diafiltration processes. ,, Therefore, within the superstructure (Figure ), we assume the system operates continuously at steady state. Additionally, we assume the membrane performance is adequately characterized by a constant water flux J w , which is supported by prior experimental validation. , Likewise, we assume the sieving coefficients and are constant, which holds for some nanofiltration membranes but breaks down for others. ,, It is easy mathematically to accommodate concentration-dependent sieving coefficients in the optimization framework, although experimentally developing and validating such correlations is left for future work. The goal of this study is to understand module and cascade design trade-offs as a function of these membrane performance metrics.…”

“…Additionally, we assume the membrane performance is adequately characterized by a constant water flux J w , which is supported by prior experimental validation. 33,34 Likewise, we assume the sieving coefficients S Li + and S Co 2+ are constant, which holds for some nanofiltration membranes 33 but breaks down for others. 19,69,70 It is easy mathematically to accommodate concentration-dependent sieving coefficients in the optimization framework, although experimentally developing and validating such correlations is left for future work.…”

“…71−73 This phenomenon would enter into the model by simply changing the assumed value of the sieving coefficients and would yield results similar to the sieving coefficient sensitivity analysis presented in this study. 34,73 Lastly, this work focuses on the simplest case of binary fractionation of Li + and Co 2+ in an idealized feed. Extending the optimization framework to consider many components requires specifying additional sieving coefficients and possibly augmenting the superstructure to consider removal of other products, purge streams, or additional recycle streams.…”

“…This management of the concentration profiles allows for the staging of membrane modules to form diafiltration cascades that enable the recovery of retentate and permeate streams enriched in the less permeable and more permeable solute, respectively. Yet, since development starting in the 1960s, − diafiltration has only expanded to niche applications that require high-purity, high-value products such as protein purification and in the food and beverage industry . In this study, we propose a novel diafiltration membrane process to recover lithium and cobalt in high-purity from a leach liquor without extreme operating conditions or harsh solvents.…”