A novel framework for forward osmosis in zero- and low-flow conditions: Applicability and fundamental differences...

Forward osmosis (FO) systems operating under low or zero cross-flow velocities present modeling challenges due to the limitations of conventional external concentration polarization (ECP) formulations, which often predict near-zero flux under stagnant conditions, contradicting experimental observations. To address this, we propose a revised ECP model incorporating an asymptotic Sherwood number that enables continuous mass transfer prediction as the Reynolds number approaches zero. The model accounts for both molecular diffusion and natural convection, allowing accurate flux prediction in spacer-free and low-flow environments. Model parameters were estimated from experimental data and validated through simulations of a hydration pack (zero flow) and a commercial FO module operating at 0-10 cm/s cross-flow velocity. Simulated results closely matched experimental trends and successfully reproduced water flux behavior across operating regimes. Sensitivity analysis revealed that baseline mass transfer parameters (Sh₀, a, b, c, d) had influence comparable to intrinsic membrane properties (A and S), particularly in no-spacer systems where diffusion and boundary layer resistance dominate. These findings confirm the critical role of mass transfer coefficients in FO performance. In the low cross-flow regime, analysis of the recovery-flux-velocity relationship demonstrated the feasibility of low-velocity operation and clarified key distinctions from RO. The model supports FO system design under minimal flow conditions, facilitating the development of compact modules suitable for portable and hybrid applications.