Page 70 - Membranes for Industrial Wastewater Recovery and Re-Use

50 Membranes for Industrial Wastewater Recoverg and Re-use

Table 2.12 Governing equations: steady-state expressions for the length-averaged
permeate flux
Model Equationa Reference

Leveque solution: Porter (1972). after
laminar Levique (1928)
flow, Brownian diffusive
transport, J i U
Similarity solution for Davis and Sherwood (1 990)
laminar flow, Brownian Romero and Davis (1 988)
diffusive transport, J + U

Fully developed turbulent Porter(1972)
flow
LBveque solution for ZydneyandColton (1986).
shear-induced diffusion after Ecstein et al. (19 77)
(based on TIs = 0.03r2y0)
Similarity solution for Davis and Sherwood (1990)
shear-induced diffusion
(basedon C* - 0.6 by
volume and C < 0.1 by
volume)
Integral model for Romero andDavis (1988)
shear-induced diffusion
from thick layers
(based on Ds(C))
0.036r3yZ
Inertial lift velocity J=- Drew et al. ( 199 1)
(basedon thin layers, 16~0
such that J = vL
Surface transport (I) = 2.4ryo(rZRL)215 cot0 Song and Elimelech (1 99 5)
a DB, Ds, diffusion coefficient for Brownian and shear-induced diffusion respectively: yo. maximum shear
rate; C*, C. concentration at the membrane-solution interface and the bulk retentate solution
respectively: c. dimensionless concentration: L. membrane elemental length U, cross-flow velocity: u,
kinematic viscosity (q/p); r. particle radius; q(C), concentration-dependent dimensionless viscosity:
Q,,(C), concentration-dependent excess particle-flux: R’c, specific cake resistance (RJcake thickness): vL,
inertial lift velocity: cot 0. surface morphology parameter.

Modified concentration polarisation model
Complications arise in the concentration polarisation model when it is applied to
systems in which colloidal and/or suspended material is present and
accumulates in the hydrodynamic boundary layer. In such cases Newtonian
behaviour cannot be assumed within or at the outer boundary of the stagnant
film, and the deviation from classic film theory increases with increasing solute
concentration in the boundary layer. This implies that a correction for non-
Newtonian behaviour is needed to account for the local solute concentration-
dependent changes in: