Page 63 - Membranes for Industrial Wastewater Recovery and Re-Use

Membrane technology 43

normally denoted A and B, respectively. According to this approach, the
transport of solute is not directly related to transmembrane pressure.
The osmotic pressure l7 is given by the well-known van’t Hoff equation:

n = yRTCC,

where R is the gas constant, T the absolute temperature, CC, the total
concentration of individual solute ions and y the osmotic coefficient accounting
for non-ideal behaviour. The coefficient y normally takes a value between 0.7 in
concentrated solutions and unity at infinite dilution, and equates to the mean of
the activity coefficients of the dissolved ions and the degree of dissociation.
However, the relationship between l7 and C is rarely linear, as Equation (2.9)
suggests, due to departure from ideality at high concentrations.
The simple solution-diffusion model can be extended to account for pore flow
by adding an extra empirical coefficient to the water and solute flux expressions.
Use of the model is, however, constrained by the necessity of determining this
coefficient experimentally. The model can also be extended to allow coupling of
solute and solvent flows, although this also introduces empirical constants.
Pore flow models assume that the membrane is microporous with cylindrical
pores. The membrane material is assumed to preferentially adsorb water, as a
monolayer, both within the pores and at the surface. The water is then
transported across the membrane at a rate depending upon chemical
interaction, frictional and diffusive forces. The expression can be modified to
account for steric and electrostatic effects, and can take account of the pore size
distribution of the membrane. Charged pores tend to increase the effective
viscosity of the water in the pores due to electroviscous effects, but these effects
alone do not appear to account for reported charged membrane permeability
changes with ionic strength and pH.
The Donnan equilibrium model takes account of solute charge and membrane
charge which, through repulsion of co-ions and attraction of counter-ions,
creates a Donnan potential at the membrane surface. This has the effect of
repelling co-ions and, since electroneutrality must be preserved, counter-ions
are also rejected. This approach applies to both electrodialysis and nanofiltration
and can provide qualitative trends, but does not allow a quantitative
description of nanofiltration since it does not take account of diffusive and
convective transport. The extended Nernst-Planck equation has also been used
to describe transport in charged membranes, but with similar limitations.

2.3.2 Foulinglcake layer mass transfer control

Resistance model: cake filtration
The simplest way of accounting for the additional resistance offered by the
material accumulating in the interfacial region is to simply add its resistance R,
(the cake layer resistance) to that of the membrane, such that Equation (2.5)
becomes: