Page 364 - Environmental Nanotechnology Applications and Impacts of Nanomaterials

Membrane Processes 349

For example, assuming dilute solutions, Eqs. 24a and 24b, known as
the Kedem and Katchalsky model [3], can be expressed as:
J 5 L s p 1 s pd
V
V
(25)
J s 5 v p 1 s1 2 sdc lm J V
p L VD
where s 5 P 5 is the log mean
p J v 50 L is the reflection coefficient, c lm
V
2
average concentration across the membrane, and v 5 sL L 2 L VD d /L V .
V D
The reflection coefficient can be shown to be [4] a product of an equi-
librium term expressing relative affinity (or exclusion) of salt with
respect to the membrane, and a kinetic term that expresses the relative
mobilities of water and salt and their potentially coupled migration.
The parameter permeability, P, of a membrane to a given molecule also
reflects a multiplicative relationship between affinity and mobility. In the
solution-diffusion model [5], permeability is defined as the product of the
molecule’s solubility in the membrane, K , and its diffusivity, D:
s
P K D (26)
s
This leads to an equation for water flux that is again similar to that
obtained from irreversible thermodynamics:

J 5 A s p 2 pd (27)
w
w
w
D C m,1 V w w
w
where A is the hydraulic permeability, A 5 RTd , and D , c m,1 are
w
w
w
m
the diffusivity and concentration of water in the membrane, respectively,
and d m is the thickness of the membrane. For solutes, the solution-
diffusion model yields following expression for the flux of solute, J :
s
sC feed 2 C permeate d
J s 5 D s K s (28)
d m
is the solubility coefficient for the solute.
where K s
Application to porous membranes:
Fluid filtration
Materials with high diffusivities are not typically removed from fluids
by porous membranes such as MF and UF membranes operated under
a pressure differential. As a result, coupled transport of mass can
typically be ignored. (However, the development of a streaming cur-
rent when flow moves through a charged membrane may entail cou-
pling, particularly for smaller pore sizes.) As a result, we need only
consider the first term in Eq. 13, which we have shown reduces to a
form resembling D’Arcy’s law where fluid flux J f,vol (volume of fluid