Page 64 - Membranes for Industrial Wastewater Recovery and Re-Use

44 Membranes for Industrial Wastewater Recovery and Re-use

(2.10)

where, in the case of dead-end operation, R, is time dependent. However, this
simple equation is only useful if there is prior knowledge concerning the
hydraulics of the cake or fouling layer. In the model case of the development of an
incompressible cake of homodispersed granular material the cake resistance
follows the same form as Equation (2.6):
K’( 1 - E,)~S;~~
R, = (2.11)
E:
where the symbols refer to the same parameters as before with reference to the
filter cake. In this case, however, K’ takes a value of 5 for spherical (or neo-
spherical) geometry (Grace, 1956), and the equation - or more usually the
pressure gradient form derived from it - represents an expression of the well-
known Kozeny-Carman equation.
Whilst the above equations may be used to calculate the total resistance to
filtration, their use is constrained by a number of simplifying assumptions:

0 flow is considered only in the direction orthogonal to the membrane:
tangential movement is ignored,
0 suspended particles are assumed homodispersed (i.e. all the same size,
shape) and neutrally buoyant (i.e. non-sedimenting),
0 particles are considered incompressible,
0 the filter cake is also considered incompressible,
0 cake porosity, hence permeability, is assumed to be independent of time,
and
0 migration of particles through the cake with time is ignored.

In fact, filtration can be characterised on the basis of more than just
cake filtration behaviour. Four filtration models, originally developed for dead-
end filtration (Grace, 1956), have been proposed to describe the initial flux
decline. All models imply a dependence of flux decline on the ratio of the particle
size to the pore diameter (Table 2.8). The standard blocking and cake filtration
models appear most suited to predicting initial flux decline during colloid
filtration (Visvanathan and Ben Aim, 1989) or protein filtration (Bowen et al.,
1995). According to Bowen and co-workers, four consecutive steps had been
defined: (1) blockage of the smallest pores, (2) coverage of the larger pores inner
surface, (3) superimposition of particles and direct blockage of larger pore and
(4) creation of the cake layer. All of the models contain empirically derived
parameters (A and B in Table 2.8), although some have been refined to
incorporate other key determinants (Table 2.9). On the other hand, a number of
empirical and largely heuristic expressions have been proposed for particular
matrices and/or applications, for example the filtration of activated sludge by
membrane bioreactors (Table 2.10). Classical dead-end filtration models can be