Page 62 - Membranes for Industrial Wastewater Recovery and Re-Use

42 Membranes for Industrial Wastewater Recovery and Re-use

Table 2.7 Dense membrane rejection expressions based on different models (based on
Bhattacharyya and Williams, 1992)
Approach Rejectiond Reference
1 -I
Solution-diffusion Lonsdale et al. (1 965)
(1 + 5 (-)
Sherwoodetd.
Solution-diffusion K, 1 - 2 (A)) (1967)
imperfection AP- AR
Surface force pore Matsuura and Sourirajan
flow (1981): Sourirajan and
Matsuura (1985)
1 - ( yY (;;) - (
Donnanequilibrium ;)itiZ) lir Bhattacharyya and
Chen (1986)
a b, dimensionless frictional force (frictional force acting on solute in pore vs. that acting on solute in
ACs,
bulk solution): Cw concentration of water in the membrane: Cs, exp(y,)/[l + (expv, - l)(b&xp&)]: ~
solute concentration difference across the membrane = C,1 - Cs2: Csl.sl, solute concentration on feed,
permeate side of membranc: Cz, concentration of co-ion in bulk solution; &, (Brownian) diflusion
coefficient of water in the membrane; Ds. (Brownian) diffusion coefficient of solute in the membrane: Kw,
water permeability in the membrane, &C,VdRT; Ks, solute permeability in the membrane, DS+; K3,
empirical coefficient: Qm, charge capacity of membrane: R, gas constant: ~p dimensionless radial
distance (r/rp): T, absolute temperaturc: yr, dimensionless radial velocity (v&v(~=,,& V,, partial molar
volume of water: y. 2, charge on species y (the co-ion) and z (counter-ion) respectively:Y,Z, ( Yz+ + Zy-: y,
ym, activity coefficients in solution and membrane respectively: membrane thickness: K~, solute
distribution coefficient: p, osmotic pressure: &, dimensionless electrostatic or van der Waals forces
parameter.

0 extended Nernst-Planck equation, and
0 irreversible thermodynamics.

Pure solution-diffusion is based upon the assumption that both the solute and
solvent (i.e. water) dissolve in a homogenous non-porous membrane and then
diffuse across it as discrete (uncoupled) materials. The overall rate of transfer (i.e.
flux) of each, and thus the purity of the permeate product, depends upon their
relative solubilities and diffusivities (Lonsdale et a]., 196 5):
Kw
Solvent (water) flux: - (AP - An) (2.7)
A

KS
Solute (salt) flux: - (AC,)
A

where AP and Ax, respectively, represent the applied transmembrane and
osmotic pressures, ACs is the solute concentration difference across the
membrane, Kw and Ks represent the water and solute permeabilities in
the membrane and A is the membrane thickness. The ratios Kw/A and Ks/L are