Optimal control for mixing enhancement in boundary layers at membrane walls

This paper proposes a scheme for mixing enhancement in the boundary layers of pressure-driven membrane systems. This scheme uses an external electric field to activate the ions in the area adjacent to the membrane surface and generate an electro-osmotic flow. This scheme should reduce fouling and concentration polarization close to the membrane surface and may increase productivity of membrane systems. The objective of the feedback control design for this system needs to determine the voltage (and waveform) applied to the electrodes so that the electric field can effectively increase the mixing in the vicinity of membrane surface, while saving control power. This paper uses a mixing index in terms of the spatial gradients of the perturbation velocity field, which describes the mixing caused by both length stretching and folding. An optimal control problem is defined to maximize mixing in the area adjacent to the membrane and achieve control energy efficiency. In addition, the efficacy of the feedback scheme is validated by Computation Fluid Dynamics (CFD) simulation. The given control law not only solves the optimal problem but also provides the desired waveform for such applications.