Page 357 - Environmental Nanotechnology Applications and Impacts of Nanomaterials

342 Environmental Applications of Nanomaterials

as a force, F. The force is in the direction of decreasing energy so, for
example, for movement in the x direction:

dE
F 52 (2)
x
dx
Applying this idea to a membrane system, the decrease in total avail-
able energy of the system (concentrate, membrane, and permeate) as a
consequence of a substance moving across the membrane divided by the
distance moved can be interpreted as the driving force for that
movement.
In most membrane systems, transport is driven by an externally
imposed gradient in a single type of energy. Although each type of energy
is typically linked, in many systems, the imposed gradient in one type
of energy is the only one that need be considered and the linkages or cou-
pling with the other gradients can be ignored. For instance, pressure-
driven MF and UF membranes do not reject solutes to an appreciable
extent. In this case, the transmembrane gradient in pressure is the
only significant factor affecting the decline in available energy of water
and solutes as they cross from the feed to the permeate side of the
membrane.
If two or more types of energy affect transport of a single component
in the feed, then at any single moment in time the total force for trans-
port can be approximated by adding the corresponding energy gradients
for that component. The expressions for the available energy per mole
of a substance associated with pressure, solution composition, and elec-
trical energy are shown in Table 9.3.
In the table, V is the molar volume of i; G i and G i o are the molar Gibbs
i
free energy of i in the given system and at standard state, respectively; a i
is the chemical activity of i; R and T are the universal gas constant and
the absolute temperature; z is the charge on species i (including sign);
i
F is the Faraday constant; and is the electrical potential. G i
o 0
and G i are also commonly written as and , in which case they are
i
i
TABLE 9.3 Expressions of Available Energy per Mole of Chemical Species
Type of energy Expression for energy/mole of i
Mechanical (pressure-based) E p,i 5 V i P (9.3)
o
Chemical (concentration-based) E chem,i 5 G i 5 G i 1 RTln a i (9.4)
Electrical E elec,i 5 z i F (9.5)
Total available energy (a) E i 5 E p,i 1 E chem,i 1 E elec,i (9.6)
(a)
In a typical membrane system; in other systems, other types of energy would have to be
included