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The dynamics of a fluid flowing across a thin permeable interface (i.e. a membrane) is an intrinsically multiscale phenomenon, owing to very different scales at play, rendering its description complex from a physical and computational point of view. A clear explanation of the mechanisms at the basis of membrane processes is then needed. Thanks to multiscale homogenization, we develop a reduced-order, intuitive, robust, and computationally cheap model to simulate the interactions between a rigid membrane and the surrounding flow of a dilute solution. The model describes the flow through the membrane via a jump in the solvent velocity and stresses and the solute concentration and fluxes where the membrane acts as a macroscopic discontinuity surface. We compare the macroscopic solution with the fully-resolved Navier-Stokes and advection-diffusion equations in diverse flow configurations and the case of different chemical and geometrical membrane properties. The macroscopic model well reproduces the flow structures observed in the full-scale case. Finally, we introduce some applications of the model in membrane characterization and design.