Page 68 - Membranes for Industrial Wastewater Recovery and Re-Use

48 Membranes for Industrial Wastewater Recovery and Re-use

J =%In(:)
D
(2.12)

where D is the diffusion coefficient in m2 s-l and C* and C are the respective
concentrations at the membrane surface and in the bulk solution. The ratio C*/C
is a critical term, sometimes denoted (p or b, and referred to as the concentration
polarisation parameter or index. Note that Equation (2.12) includes no pressure
term, although the transmembrane pressure is inferred by the flux value. In the
case of gel layer formation, the term C* equates to the concentration of solute in
the gel layer, for which some experimentally determined values have been
reported for specific components (Cheryan, 1998).
Determination of the flux from Equation (2.12) relies on knowledge of the
solute diffusivity, the boundary layer thickness and the solute concentration at
the membrane surface. If the solute comprises dissolved ions or small molecules,
as would be the case for pressure-driven dense membrane processes, then D is
simply given by the Stokes-Einstein equation:

(2.13)

where kB is the Boltzmann constant, Tis absolute temperature and rp is the solute
radius.
Conventionally, the ratio D/6 is assigned the mass transfer coefficient k and the
various operational and performance determinants expressed in terms of
the non-dimensionalised groups of Sh (Sherwood number), Re (Reynolds
number) and Sc (Schmidt number):
J = k ln(C*/C) (2.14)

k = ShD/d (2.15)
Sh = aRebSc(d/L)" (2.16)
Re = pUd/p (2.17)

SC = p/pD (2.18)

where p is the fluid density, U its velocity, d and L the hydraulic dimension and
length of the membrane and a, b, c and n are constants. The hydraulic dimension
d is equal to four times the ratio of the channel cross-section to its perimeter, and
is thus equal to the diameter of a tube or twice the height of a wide parallel flow
channel.
Appropriate values for a, b, c and n, based on Newtonian behaviour, are given
in Table 2.1 1. The solution for laminar flow and Brownian diffusive transport is
attributed to LevEque (Levtque, 1928; Porter, 1972). The LCvCque solution
assumes channel flow with completely impermeable boundaries, and is thus
strictly only applicable to membrane permeation systems with a permeate flux