On dimensionless modeling of two-stage reverse osmosis: Thermodynamic and economic insights

This study develops and applies a comprehensive dimensionless framework for the design and optimization of two-stage reverse osmosis (RO) desalination systems. Building upon earlier single-stage formulations, the framework reformulates the governing transport equations into scale-independent form and introduces new performance indices, including the maximum pressure index (MPI), to capture hydraulic safety alongside energy and water-quality constraints. The dimensional model was validated against established literature, while the dimensionless formulation was verified against its dimensional counterpart. System-level analyses highlight the influence of temperature, staging ratio, and dimensionless active membrane area on recovery, energy consumption, and permeate quality. Among seven tested allocations, a 5:3 distribution of eight membranes across two stages consistently outperformed the single-stage alternative, especially at second-stage pressure ratios near two, where flux variance and energy use are minimized. When applied across brackish and seawater salinities, the framework delineated feasible operating envelopes bounded by energy, cost, and quality constraints. Finally, differential evolution optimization established the combinations of dimensionless membrane area and staging ratio that maximize recovery while keeping permeate salinity and hydraulic pressures within limits. The resulting framework provides a scalable, physically consistent tool for guiding the design of efficient, cost-effective, and practical multi-stage RO systems.