This work investigates the spatiotemporal dynamics and performance of reverse osmosis (RO) desalination under periodic oscillatory conditions using a partial differential equation (PDE) model. Assuming constant membrane transport properties, the model shows that sinusoidal variations in flow and pressure yield cycle-averaged performance metrics that closely align with those of steady-state operation under equivalent mean conditions. Increasing the oscillation frequency attenuates temporal fluctuations in permeate concentration. Notably, rectified sinusoidal variations increase the permeate production rate, a result attributed to the significantly higher energy input. While these findings are consistent with experimental trends, the analysis uncovers a key divergence from prior literature: rectified sinusoidal variations are the least energy-efficient strategy due to their disproportionate energy demands, whereas steady-state operation remains the most energy-efficient. Additionally, the potential correlation between rapid frequency switching between sub-osmotic and super-osmotic conditions under rectified sinusoidal conditions and accelerated membrane degradation warrants further exploration in future studies.
