Page 363 - Environmental Nanotechnology Applications and Impacts of Nanomaterials

348 Environmental Applications of Nanomaterials

Eqs. 19 and 21 indicate the amount of available energy that is released
(dissipated) per mole of water and solute, respectively, that cross from
the feed to the permeate side of the membrane. If J is the molar flux of
i
indicates the rate of energy dissipation
species i, the product J i E i
associated with the transport of species i per unit area of membrane.
Therefore, the overall rate of energy dissipation per unit area of mem-
brane, , considering the flux of both water and solute is:

5 J E 1 J E s (22)
w
s
w
lnc s
V s P 2 pd 1 J aV P 1 pb
5 J w w s s
c s
Grouping terms in and P, we obtain:
J lnc s
s
5 sJ V 1 J V d P 1 a 2 J V b p
w
w
s
w
w
s
c s
5 J P 1 J p (23)
D
V
Eq. 23 describes the rate of energy dissipation in our simple system as
the sum of two terms. The first term on the right-hand side of Eq. 23 is
, of the components (water and
the product of the total volume flux, J V
solute) and a driving force, the difference in mechanical pressure across
the membrane ( P). The second term is the product of the diffusive flux
of solute relative to the flux of water, J , and a second driving force for
D
transport, the osmotic pressure. Thus, in this two-component system, the
rate of energy dissipation is described by two fluxes and their two
corresponding conjugate driving forces. This is an important result since
it reveals the appropriate fluxes and conjugate driving forces to be sub-
stituted into Eq. 13 yielding the result for reverse osmosis performance [3]:
J 5 L p 1 L VD p (24a)
V
V
J 5 L DV p 1 L p (24b)
D
D
where L V corresponds to L 11 , etc. Since we have two components, we end
up with two equations and four phenomenological coefficients, however,
by the Onsager reciprocal relationship (Eq. 25), L VD L DV . In a three-
component system, applying the Onsager reciprocal relationship we
would have three equations and six coefficients, and so on. Numerous
variations, expansions, and simplifications on Eqs. 24a and 24b have
been developed in the literature. However, they virtually all share the
feature that volume flux is proportional to a net pressure drop ( P )
while solute flux is proportional to the concentration difference across
the membrane, which in turn is proportional to p.