**Overview**

When particles are moving near membrane surface, various forces act on them. Drag forces associated with the axial (tangential) and lateral (permeation) components of fluid flow tend to carry particulate species along streamlines. At the same time, particles are subject to a number of forces that cause them to cross streamlines. These include thermodynamic (or Brownian) diffusion, inertial lift, van der Waals attractions, charge repulsion, shear induced diffusion, etc. as shown in Fig. 1. Since sedimentation velocity is negligible comparing to convective flows, it can be neglected. The effective particle deposition velocity can be estimated from the difference between the permeation velocity (or flux) and overall back transport velocity.

The back transport phenomena in open channel system with a two-phase flow such as submerged membranes are not well theorized, but they are extensively studied for closed channel systems with a single-phase flow such as tubular and plate and frame membranes. Although the magnitude of each component of hydrodynamic forces are different in open and closed channel systems, the basic principles must remain same. It is worth to take a look at the back transport theories developed for closed channel system to understand the basic principles of submerged membrane filtration processes.

Fig. 1. Forces and torques acting on a charged, spherical particle suspended in a viscous fluid undergoing laminar flow in the proximity of a flat porous surface (reproduced from Yoon, 1999).

*Total back transport velocity and critical flux*

It is very hard to know actual back transport velocity in submerged membrane system due to the difficulties of defining hydrodynamics of two-phase flow moving over membrane surface. However, it is worthwhile to take a look at how particle back transport phenomenon appears in the more defined system with closed slit channels. The underlying principle in particle back transport should be identical in both systems except the irregular shear stress imposed by air bubbles.

The back transport velocity induced by Brownian diffusion decreases with particle size. On the contrary, shear induced diffusion and inertial migration increase with particle size as can be seen in Fig. 1. Overall, it is apparent that particles around 0.5 have the lowest back transport velocity under the given condition. If liquid velocity increases, all the lines move up based on the equations in the following sub-sections on Brownian diffusion, shear induced diffusion, and lateral migration. Since the lines for shear induced diffusion and lateral migration move up more than that for diffusion, the lowest back transport occurs in the particle size less than 0.5.

By definition, the total back transport velocity exactly matches with the critical flux. For instance, if mono-dispersed particles with 1 diameter are filtered, no particle will deposit under the flux at or below 15 LMH because back transport velocity is higher than permeation velocity (Fig. 1). If no particles deposit, the TMP required to obtain the flux equates to the TMP required in clean water to get an identical flux.

Fig. 1. Back transport velocity of iron oxide particles as a function of particle size in a slit channel. =0.24m/s, T=298K,=0.001kg/m/s, channel height = 3×10^{-3} m, =5,745 kg/m^{3}, and =1.38×10^{-23} J/K. (reproduced from Yoon, 1999)