This study investigates the non-equilibrium transport dynamics and macroscopic thermodynamic efficiency of a binary gas mixture traversing a tapering, selectively permeable cascade operating strictly within the Knudsen regime. By employing a Lagrangian test particle Monte Carlo framework alongside a Fokker-Planck formalism, discrete stochastic trajectories driven by particle-boundary interactions are coupled with ensemble macroscopic concentration profiles. The separation process is thermodynamically evaluated by balancing the separation work gain against two primary costs: the microscopic entropic penalty of momentum erasure via diffuse wall collisions, and the macroscopic transport penalty induced by geometric backscattering. To formalize this, the specific separation thermodynamic efficiency is introduced, a metric that normalizes overall performance against the intrinsic material transmission probability. The results reveal a critical morphological transition in optimal cascade architecture. It is demonstrated that low-affinity membranes fundamentally require moderate geometric constriction to mechanically force boundary collisions and maximize the integrated probability of permeation, optimally balancing permeation against induced backscattering. On the other hand, in high-affinity systems, the active species is rapidly extracted near the inlet, localizing maximum thermodynamic dissipation and rendering severe tapering physically detrimental. Consequently, highly selective membranes strictly favor uniform channel geometries to mitigate irreversible transport losses. Finally, this framework establishes that optimal geometric design is not static but must be dynamically tailored to the intrinsic surface affinity to maximize macroscopic thermodynamic efficiency.
Antonio F. Miguel (Mon,) studied this question.