(403g) Modeling of Solute-Coupled Transport in Osmotically-Assisted Membrane Separations with Multicomponent Solution-Diffusion Theory

Osmotically assisted membrane processes (OAMP), such as forward and counter-flow reverse osmosis, are a class of separation technology that leverage osmotic pressure gradients for water purification. Accurate modeling of the solute-coupling effects for transmembrane transport is integral to the development, and subsequent optimization, of OAMP unit operations. Theoretically, in binary mixtures, species separation is achieved due to the solution-diffusion mechanism, which results from the combination of selective species partitioning and the relative diffusive rates within the membrane matrix.

To model membrane separations involving multicomponent mixtures, the transport is commonly linearized with the binary solution-diffusion. Referred to as the method of superposition, the transport between multicomponent mixtures is decoupled into a series of binary transport phenomena, where the resultant species fluxes are computed as the linear combinations of the fluxes from individual binary solution-diffusion models. This method benefits from the usage of binary membrane parameters, such as the support structural parameter and the water and solute permeability coefficients, which are easily characterized with established experimental protocols.

However, recent publications highlighted that large errors of up to 66% were obtained for water and solute fluxes when multicomponent transport was modeled with superposition. The deviations were attributed to solute-solute and solute-solvent coupling effects, which are significant in multicomponent mixtures at moderate to high concentrations, or in mixtures with large excess Gibbs energy of mixing.

In this study, we develop a new multicomponent solution-diffusion model by combining the frameworks of multicomponent Fickian diffusion and solution-diffusion theory. The derived model introduces multicomponent membrane parameters, which are coined as the diagonal and cross solute permeabilities, to incorporate the impact of species interactions on the chemical potentials. These multicomponent membrane parameters are highly concentration dependent, but recover their binary limits as the concentrations of the counter solutes tend towards infinite dilution. For multi-electrolyte mixtures, the diagonal and cross solute permeabilities can be computed from the multicomponent diffusion coefficients and the membrane’s binary solute permeabilities.

By incorporating solute coupling effects with multicomponent solution-diffusion, our study demonstrates significant improvements in model-to-experiment agreement for species fluxes. The model was evaluated relative to experimental data for 7 unique combinations of forward osmosis processes involving ternary electrolyte mixtures. The average absolute deviation (AAD) of the solution-diffusion model decreased from 21.0 % to 3.0% when solute coupling was incorporated.

In the absence of multicomponent diffusion coefficients, we explore an alternative method to regress coupling phenomena. We propose the modeling of the effective membrane permeabilities as a linear combination of the binary permeability with an excess permeability. Analogous to solution thermodynamics, the latter parameter quantifies the extent of departure from ideality arising from solute-solute interactions within the membrane matrix. The proposed method introduces one additional regression parameter for each solute present in the separation. Using H₂O-NaCl-EtOH forward osmosis processes as a case study, we demonstrate the model’s robustness in regressing water and solute fluxes across a wide range of concentrations. The AAD of the solution-diffusion model reduced from 66.1 % to 7.2% over a range of NaCl concentrations from 0 to 1.5 M and EtOH mass fractions of 0 to 0.5.

In essence, our study accentuates that significant improvements in multicomponent species fluxes can be obtained when solute interactions are incorporated. With the improved model, we envision that more accurate energetic and techno-economic performance of desalination systems can be predicted, leading to better representation of the viabilities of the technologies.