On the Formation of Unstirred Layer in Osmotically Driven Flow

TL;DR

This paper analyzes the formation and growth of the unstirred layer in osmotically driven flow across membranes, providing approximate solutions and discussing thermodynamic implications of the process.

Contribution

It introduces mathematical techniques for solving nonlinear boundary problems in osmosis and characterizes the rapid formation and unbounded growth of the unstirred layer over time.

Findings

Layer forms rapidly in time scale ~ O(√t)

Layer thickness grows without upper limit over infinite time

Discusses thermodynamic work in osmotic engine

Abstract

Osmotically driven flow across a semi-permeable membrane under a constant static pressure difference is revisited with referring to the previous reports for reverse osmosis. A few mathematical techniques for obtaining the approximate solution, such as that for inverse problems used in the field of heat transfer, are presented with an emphasis on the nonlinear boundary condition and the time-dependent solvent flow-rate. It is concluded that the layer is spontaneously formed by osmosis rapidly in the time scaled by $\sim {\rm O}\bigl(\sqrt{t}\bigr)$, and that the layer thickness grows with no upper limit in an infinite time interval. Based on the obtained solution, we will also discuss the thermodynamical output work in an irreversible process which is extracted from the system as an osmotic engine.