Page 365 - Environmental Nanotechnology Applications and Impacts of Nanomaterials

350 Environmental Applications of Nanomaterials

per unit area of membrane per unit time) is proportional to the differ-
ence in pressure across the membrane. This is typically expressed as:

p
J f,vol 5 (29)
mR m

where p is the pressure drop across the membrane (the TMP), is the
absolute viscosity of the fluid, and R is the hydraulic resistance of the
m
membrane, with dimensions of reciprocal length. Membrane perform-
ance is often expressed as the ratio of permeate flux, J, to the pressure
drop across the membrane, p. This quantity is called the specific per-
meate flux, with an initial value equal to 1 .
mR m
If each pore is modeled as a capillary, permeate flux can be repre-
sented as Poiseuille flow through a large number of these capillaries
in parallel. In each pore, the velocity of the fluid is assumed to be zero
at the wall of the pore (termed the “no-slip” condition), and at a max-
imum value in the center of the pore. The no-slip condition at the pore
wall is ultimately a consequence of an affinity between fluid mole-
cules and with those of the membrane pore and leads to a parabolic
velocity profile.
Using the Hagen-Poiseuille equation to describe flow through cylin-
drical membrane pores (idealized as such, or perhaps truly cylindrical)
the following expression is obtained for the permeate flux through a
membrane characterized by an effective pore radius of r pore :

2
A pore pore P P
r
J 5 5 (30)
A membrane 8mud m mR m
A 8ud
where R 5 membrane 2 m , A pore /A is the ratio of the open pore area (A pore )
m
m
A pore r pore
to the entire area of the membrane surface (A ), is the pore tortu-
m
osity factor, and is the effective thickness of the membrane. Note that
m
Eq. 30 predicts that flux should decrease with the square of decreasing
pore size. If the assumptions of the Hagen-Poiseuille model hold, very
high pressures would be required to induce flow through membranes
with nanometer-sized pores.

Polarization phenomena and
membrane fouling
The rejection of materials by a membrane leads to the accumulation of
these materials near, on, or sometimes within the membrane. This can
lead to a decrease in membrane performance. For example, ion exchange