Page 61 - Membranes for Industrial Wastewater Recovery and Re-Use

Membrane technology 41

forces become important. Instead, simple experimental measurements are used
to assess the fouling propensity of feedwaters for specific membrane processes,
and dense membrane processes in particular (Section 2.4.3).
Notwithstanding the difficulties involved, a number of models have been
presented to define the operational determinants of various membrane processes.
To describe the derivation of each of these in detail would be beyond the scope of
this review since the outcomes, i.e. the ultimate analytical expressions
generated, are often specific to the process under consideration and the
assumptions made. The starting point for the theoretical development of water
and solute flows through a dense membrane varies according to the relative
importance placed of solution-diffusion, sorption, pore flow and electrical charge.
Most of the mathematical derivations of system hydrodynamics for cross-flow
operation are based on fiZm theory (which incorporates the concentration
polarisation model). Film theory assumes the interfacial region mass transport to
be determined by the degree of concentration polarisation, which can then be
calculated from the fluid cross-flow, the permeate flux and the solute diffusivity.

2.3.1 Membrane mass transfer control
Under the simplest operational conditions, the resistance to flow is offered
entirely by the membrane. For porous membrane systems, the flux can be
expressed as:

where J is the flux in m s-l, Ap is the transmembrane pressure, p is the fluid
viscosity and R, is the resistance of the membrane in m kg-’. For microporous
membranes, specifically those used for microfiltration, the Hagen-Poiseuillc
equation may be considered applicable for a permeate undergoing laminar flow
through cylindrical pores. The resistance R, then equates to:

(2.6)

where E is the porosity (or voidage), S, the pore surface area to volume ratio and
I, the membrane thickness. K is a constant equal to 2 for perfectly cylindrical
pores but changes for other geometries. It is apparent from Equation (2.5) that
temperature has a profound impact upon the flux through the viscosity, which
increases by around 3% for each degree drop in temperature below 2 5°C.
For dense membranes a number of expressions have been developed (Table
2.7) which are derived from a variety of approaches:

0 solution-diffusion,
0 sorption-capillary flow,
0 Donnan equilibrium,