A Robust Numerical Framework for Hollow-Fiber Membrane Module Simulation and Solver Performance Analysis

Robust numerical frameworks are essential for the simulation, design, monitoring, and control of membrane-based separation units, particularly under highly nonlinear and industrially relevant operating conditions. In this context, a comprehensive phenomenological and numerical framework is proposed for the simulation of hollow-fiber membrane modules, incorporating coupled mass, momentum (through pressure drop), and energy transport equations. The governing equations are discretized using a rigorous orthogonal collocation formulation, and the performances of two numerical solution strategies are systematically investigated for the first time to allow the in-line and real-time implementation of the model: a steady-state approach based on the Newton-Raphson method with careful treatment of initial estimates, and a pseudotransient formulation. Particularly, an original and consistent numerical treatment is introduced for the energy balance at boundaries where the permeate flow vanishes, enabling the stable incorporation of thermal effects and Joule-Thomson phenomena. The results clearly show that the steady-state Newton-Raphson approach provides the best overall performance in terms of computational efficiency, numerical robustness, and accuracy when physically consistent initial profiles are employed. In particular, the combination of a linear initial guess and a numerical mesh constituted of four collocation points yielded the most favorable balance between convergence speed, numerical robustness, and accuracy for the base-case sensitivity analysis. For monitoring-oriented applications, the numerical choice should be weighted primarily toward computational performance once physical consistency and convergence criteria are satisfied, rather than toward maximum mesh-refinement accuracy. In this context, small differences in internal-fiber profiles can be compensated through real-time permeance estimation and are negligible when compared with measurement uncertainty in real industrial processes. Under extreme operating conditions involving low concentrations, low flow rates, and highly permeable species, the pseudotransient formulation proved to be a reliable auxiliary strategy, enabling robust convergence when suitable initial guesses were not readily available. The proposed framework is validated against experimental data from the literature and subjected to extensive convergence and sensitivity analyses, providing a reliable basis for simulation and for assessing computational feasibility in in-line and real-time monitoring-oriented applications. A full demonstration of digital-twin integration, online parameter updating, reduced-order coupling, and closed-loop control is beyond the scope of the present study and will be addressed in future work.