Page 366 - Environmental Nanotechnology Applications and Impacts of Nanomaterials

Membrane Processes 351

membranes used in electrodialysis have a polymeric support structure
with fixed charged sites and water-filled passages. Charged functional
groups on these membranes attract ions of opposite charge (counterions).
This is accompanied by a deficit of like-charged ions (co-ions) in the
membrane and results in a so-called Donnan potential and the exclu-
sion of ions from ion exchange membranes with like-charged functional
groups. When an electrical potential is applied across these membranes,
ions migrate to the electrode of opposite charge. However, the ion
exchange membrane rejects co-ions, resulting in boundary layers on
either side of the membrane (referred to as concentration polarization
layers) that are either enriched in co-ions (the feed side) of the membrane
or have a deficit of these ions (the permeate side or “dialysate”). Because
there are fewer ions on the dialysate side, there is an increase in elec-
trical resistance that leads to an increase in power consumption to
achieve separation.
A similar phenomenon occurs in RO where salts are rejected by the
membrane, leading to a concentration polarization layer near the
membrane. The concentration polarization layer increases the local
osmotic pressure, resulting in the need for a higher pressure to over-
come this osmotic pressure, as well as a lower rejection of salt by the
membrane. The concentration profile of rejected species can be cal-
culated from a mass balance on solute in a differential volume in the
concentration polarization layer. For a simplified mass balance around
the concentration polarization layer, the advective flux of solutes
toward the membrane is balanced by diffusive back transport of
solute:

'c
v c 52D B 'y (31)
w

where v w is the fluid velocity in the y direction (perpendicular to the
membrane), and D B is the Brownian diffusion coefficient of the solute.
This expression can be integrated to yield an expression for the limit-
ing permeate flux as a function of the bulk concentration c bulk , the
limiting wall concentration, c wall , the diffusion coefficient for the solute,
and the concentration-polarization layer thickness, d cp ,

D c wall,lim
J lim 5 v w,lim 5 ln a b (32)
d cp c bulk

Eq. 32 predicts that the limiting permeate flux should decrease with
decreasing D. Because the diffusion coefficient increases as particle size
decreases, we can expect that when membranes are used to separate
nanoparticles, the limiting permeate flux for membrane operation