Reverse osmosis transport model
The solution-friction model (SF model) is a mechanistic transport model developed to describe the transport processes across porous membranes, such as reverse osmosis (RO) and nanofiltration (NF). Unlike traditional models, such as those based on Darcy’s law, which primarily describes pressure-driven solvent (water) transport in homogeneous porous mediums, the SF model also accounts for the coupled transport of both solvent (water) and solutes (salts).
Overview.
The solution-friction model is derived on a pore-flow or viscous flow mechanism, but extends its applicability by incorporating the force balances on the species transporting through the membrane. This inclusion allows for a detailed understanding of the interdependent fluxes of water and salt, influenced by interactions between salt ions and water molecules. The SF model has been able to successfully describe the transport of water and salt in RO membranes, showing good agreement with experiments. The development of the SF model also corrects the misconception that RO water transport is a diffusion-based process.
Ion transport.
Ion transport through the RO membrane is driven by the gradient of chemical potential within the membrane. The solution-friction model describes this transport by considering the frictions between ions, ions and water, and ions and membrane. The force balance for an ion is given by the equation:
formula_1
Note that the membrane is stationary and its velocity formula_10 is therefore set to zero. By considering only the coordinate perpendicular to the membrane surface, the ion flux (formula_6) governed by diffusion, electromigration, and advection can be expressed as:
formula_12
Water transport.
Water transport is governed by the gradient of total pressure, counterbalanced by water-membrane and ion-water frictions. The balance is expressed as:
formula_20
Substituting the expression of ion velocity into water velocity, we arrive at the following expression for the force balance on water:
formula_30
When ion-membrane friction is negligible (i.e.,formula_31), this equation can be written as
formula_32
The equation indicates that the water permeance is influenced by the electrical potential gradient inside the membrane, which has been verified by salt permeation through highly charged Nafion membranes. Due to the interactions between ions and water, increasing salt concentration decreases the water permeance. Nevertheless, a simplification can be made when a membrane has a low volumetric charge density (i.e., within the membrane), like in typical RO membranes. Therefore, the electrical potential gradient can be neglected as it is relatively small compared to the concentration gradient. The equation for water flux can be eventually simplified as:
formula_33
Defining formula_36 and formula_37, the water permeability velocity is obtained as:
formula_38
This equation is identical in form to the Spiegler-Kedem-Katchalsky equation, a classic model in irreversible thermodynamics for water transport through semipermeable membranes. This ensures that the SF model aligns with basic thermodynamic principles.
References.
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