Membranes - Principles of Chemical Separations with Environmental Applications - page 266

Flux

Time
Figure 9.15 Flux behavior as a function of time.

Flux

Concentration
polarization

Fouling

Time
Figure 9.16 Flux as a function of time. Both concentration polarization and
fouling are shown [2]. Reproduced with kind permission of Kluwer Academic
Publishers.

Membrane

Boundary
layer

Bulk feed

C m

J . C p

J . C

C b

dC
dx

C p

D .

x 0

d

Figure 9.17 Concentration polarization; concentration profile under steady-state
conditions [2]. Reproduced with kind permission of Kluwer Academic Publishers.

Using Figure 9.17: steady-state differential mass balance on the retained solute in the
boundary layer is:

D d 2 C

− v p d C

d x = d x 2 ,

(9.25)

where v p = permeate velocity.