Trajectory optimization for temporally multi-staged batch counterflow reverse osmosis

Temporal multi-staging greatly simplifies the equipment needs for osmotically assisted multi-stage reverse osmosis (RO) processes, by using the same apparatus and multiple tanks to cycle concentrate and diluate. Temporally multi-staged batch counterflow reverse osmosis (BCFRO) has been claimed to be among the most energy efficient and economical high-salinity RO processes, but its unsteady nature makes its design and optimization difficult. Past work on BCFRO used high-rejection membranes to simplify mass balance equations for time-varying stream salinities and tank volumes, which led to a surplus or deficit of salt and water mass between stages. However, more sophisticated modeling was needed to enable temporal multi-staging in RO. In this work, we use trajectory optimization to obtain optimal flux profiles and feasible trajectories over time for stream salinity and tank volume to minimize energy consumption. We use a reduced-order model for BCFRO and trapezoidal collocation to discretize dynamics along the optimal trajectory, with pressure constrained by membrane burst pressure. For a membrane with perfect rejection concentrating seawater to 84.39 % recovery, the optimization method achieves a specific energy consumption of 4.39 kWh / m 3 and a counterflow RO flux of 2.44 LMH, an improvement of 4.1 % and 42.7 % respectively compared to operation with constant water flux . • Treats batch counterflow reverse osmosis as an open loop optimal control problem. • Uses minimal hardware to reach extreme salinities via ambient pressure tanks. • Models unsteady salt concentration boundary layers, and non-ideal salt rejection. • Optimizes flux profile to reduce energy consumption with pressure constraints. • Achieves perfect matching of diluate salinity and volume between temporal stages.