When liquid permeates through membrane, particles (or solutes) are rejected by membrane and form a layer with high particle/solute concentration near the membrane surface. Due to the high particle concentration in the layer, particles tend to diffuse back to the bulk as long as they are not fixed in gel/cake layer. The concentration profile settles at the equilibrium between convective particle transport to membrane surface and diffusive particle back transport to the bulk solution as illustrated in Fig. 1. This is so called concentration polarization phenomenon that fundamentally limits the membrane performanceÂ (Porter, 1972).

The flux, *J*, can be calculated by balancing the convective particle transport toward membrane and the diffusive particle back transport as shown in equation (1).

Â Â Â Â Â Â Â Â Â —————————————– (1)

where

*J**Â *Â Â Â = Â water flux at steady state (m/s)

*C**Â *Â Â = Â particle concentration (mg/L)

*x**Â *Â Â Â = Â distance from membrane surface (m)

*D _{eff}*Â = Â effective diffusion coefficient (m

^{2}/s)

The minus sign in above equation is used to reflect the negative concentration gradient along the x-axis. The effective diffusion coefficient, *D _{eff}*, conceptually includes the effects from thermodynamic diffusion, shear induced diffusion, and all other hydrodynamic forces that moves particles away from membrane surface. The particle back transport velocity is discussed in detail here.

The equation (1) can be integrated using the boundary conditions at steady state, *i.e*. (x=0, C=C_{G}) and (x=d, C=C_{B}), where dis a boundary layer thickness (m),Â *C _{G}*Â and

*C*Â are particle concentration in gel layer and in bulk, respectively.

_{B}

Â Â Â Â Â Â Â Â —————————————– (2)

According to this equation, steady state flux, *J _{SS}*, increases when boundary layer thickness, d, decreases and the effective diffusion coefficient,

*D*, increases. The thinner boundary layer and higher

_{eff}*D*can be achieved in a given condition by increasing cross-flow velocity on membrane surface in general (Bian, 2000).

_{eff }

Â© Seong Hoon Yoon