Efficiency of osmotic pipe flows

We present experiments and theory for flows of sugar or salt solutions in cylindrical tubes with semipermeable walls (hollow fiber membranes) immersed in water, quantifying the strength of the osmotic driving force in relation to the dimensionless parameters that specify the system. The pumping efficiency of these flows is limited by the presence of “unstirred” concentration boundary layers near the tube walls, and our primary aim is to understand and quantify these layers and their effect on the flow. We measure the outlet flow rate Q_out while varying the inlet flow rate Q*, concentration c*, and tube length L, and map out the dependence of the flow rate gain γ=Q_out/Q*-1 on these parameters. A theoretical analysis based on (1) the known velocity field for slow flow in cylindrical porous tubes and (2) a parabolic concentration profile allows us to compute analytically how the flow gain depends on the relative magnitude of radial diffusion and advection as well as the ratio of the osmotic velocity to pumping velocity, in very good agreement with experiments and with no adjustable parameters. Our analysis provides criteria that are useful for optimizing osmotic flow processes in, e.g., water purification devices.