Page 71 - Membranes for Industrial Wastewater Recovery and Re-Use

Membrane technology 5 1

0 viscosity of the fluid,
0 diffusivity of the solute, and
0 permeability of the cake.

Some of the analytical expressions developed for the equilibrium length-
averaged flux are included in Table 2.12. The solution varies according to the
approach taken and the assumptions made, but the general trend is for transport
of solute away from the membrane to be much higher than that predicted by
Brownian diffusion. This changes the degree of dependency of flux on both cross-
flow velocity (and so shear rate, according to Equations (2.18) and (2.19)) and
particle size changes significantly from the Brownian diffusion LevEque model if
either shear-induced diffusion or inertial lift are included. For example, flux
dependency on cross-flow changes from U0.33 proportionality to direct
proportionality for shear-induced diffusion (Zydney and Colton, 1986) or U2 for
inertial lift (Drew et al., 1991). Kim and Park (1999) based their prediction of
critical flux conditions on shear-induced diffusion.
A comprehensive, and complex, solution for equilibrium flux in CFMF has
been presented by Romero and Davis (1988). This model accounts the effects of
shear-induced diffusion on a non-uniform filter cake whose thickness increases
with axial membrane (channel) distance. This is considerably more complex
than the model for thin cake deposits (Zydney and Colton, 1986), where a
uniform cake deposit over the whole membrane area is envisaged, and requires
that the solute concentration dependency of viscosity and diffusivity be
predetermined. More recently, the Romero and Davis model has been slightly
simplified by basing the cake layer resistance on the Kozeny-Carman equation
(Ould-Dris et al., 2000).

Practml verification of modified models
Experimental studies on model colloidal or particulate systems have shown close
agreement between experimentally measured steady-state flux data and those
predicted from shear-induced diffusive mass transport theory for ideal systems,
such as latex, blood, bacterial and fractionated clay suspensions (Zydney
and Colton, 1986). The model of Kim and Park (1999) for predicting critical
flux, again based on shear-induced diffusion, appears to corroborate well
with experimental data from calcium carbonate filtration, the critical flux
increasing linearly with particle size. Inertial lift, on the other hand, would
appear to be restricted in importance to high shear rates and/or large particles
(Davis, 1992).
The shear-induced diffusion model for thick cake layers (Romero and Davis,
1988, 1990) was found to give a reasonable representation of both dynamic and
steady-state behaviour for both rectangular channels and ceramic tubular
membranes challenged with homodispcrsed spheres of 0.45-1.3 7 pm (Romero
and Davis, 1991). Ould-Dris et al. (2000), using a slightly simplified adaptation
of the Romero and Davis model, also found reasonable agreement between
theoretical and experimental steady-state flux for their trials on the less idealised
system of granular calcium carbonate. Agreement was only obtained, however,