Theory of concentration polarization in cross-flow ultrafiltration: gel-layer model and osmotic-pressure model

Concentration polarisation in tubular membranes —a numerical approach

“…Most classical models used in the description of concentration polarisation phenomena during membrane crossflow filtration require the knowledge of a mass transfer coefficient, such as the boundary layer resistance model [2,15,16], the osmotic pressure model [13,14] and the gel layer model [13,14]. Such an expression for the mass transfer coefficient should be able to represent the effect of changing conditions in the different systems that are used for membrane filtration.…”

“…j.,=kc(Cw-Co) (14) where k c is the mass transfer coefficient, C~ is the concentration at the membrane wall and Co is the concentration of the feed stream. Sherwood [23] analysed the mass transfer in order to find a similarity condition between different flow cases.…”

“…Different theoretical models such as the gel polarisation model [13,14], the osmotic pressure model [13,14] and the resistance series model [2,15,16] have been widely used to describe the limiting phenomena and to predict permeate flux. Nevertheless, filtrations permeate flux as well as the limitations encountered such as fouling and concentration polarisation are still difficult to predict.…”

“…The local Sherwood number can be obtained from the precedent local mass flux equations,(14) and(15): Sh_Dac ac_*l ar ~ Or* c -Co c -i…”

Permeate Flux in Crossflow Ultrafiltration under Intermediate Pressures

“…Equation [14] is a linear equation and the equation does not change when c is replaced by (c Ϫ c 0 ). Therefore the boundary conditions can be rewritten as…”