Simple Theoretical Results on Reversible Fouling in Cross-Flow Membrane Filtration

. 2019 Apr 3;9(4):48. doi: 10.3390/membranes9040048

$C_{i n}$	inlet feed concentration (mol × m $^{- 3}$ )
$C_{d}$	critical concentration for deposition (mol × m $^{- 3}$ )
$N_{d}$	deposit number ( $N_{d} \equiv C_{d} C_{i n}^{- 1}$ )
C	dimensionless concentration (in units of $C_{i n}$ )
$P_{i n}$	transmembrane pressure at inlet (TMP) (Pa)
$I_{0}$	membrane resistance to pure solvent filtration (kg × m $^{- 2} \times$ s $^{- 1}$ )
$U_{i n}$	filtration velocity for pure solvent feeding ( $U_{i n} \equiv P_{i n} I_{0}^{- 1}$ ) (m × s $^{- 1}$ )
$ρ_{0}$	fluid density (uniform) (kg × m $^{- 3}$ )
$μ_{0}$	dynamical viscosity of the fluid (uniform) (kg × m $^{- 1} \times$ s $^{- 1}$ )
d	channel half-spacing (m)
$R_{i n}$	transverse Reynolds number of pure solvent filtration [ $R_{i n} \equiv (P_{i n} ρ_{0} d) {(μ_{0} I_{0})}^{- 1}$ ]
$P e_{i n}$	tranverse Péclet number of pure solvent filtration [ $P e_{i n} \equiv (P_{i n} d) {(D_{0} I_{0})}^{- 1}$ ]
$S c$	Schmidt number of the feed [ $S c \equiv μ_{0} {(D_{0} ρ_{0})}^{- 1}$ ]
$D_{0}$	solute diffusivity in the feed (uniform) (m $^{2} \times$ s $^{- 1}$ )
$W_{i n}$	axial mean velocity at inlet (m × s $^{- 1}$ )
$L_{d e}$	“clear water” exhaustion length ( $L_{d e} \equiv I_{0} P_{i n}^{- 1} W_{i n} d$ ) (m)
L	channel length (m)
$λ$	dimensionless channel length (in units of $L_{d e}$ )
$Δ P_{H P}$	pressure drop along the channel (Pa)
$α$	Regirer number, $α \equiv W_{i n} P_{i n}^{- 1} \sqrt{μ_{0} I_{0} d^{- 1}}$
$P_{i n}^{o s m}$	osmotic pressure in the absence of polarization ( $P_{i n}^{o s m} \equiv i R T C_{i n}$ ) (Pa)
i	number of dissociated entities per molecule of solute
T	temperature of the solution (K)
R	perfect gas constant (J × K $^{- 1} \times$ mol $^{- 1}$ )
$N^{o s m}$	osmotic number, defined as $N^{o s m} \equiv P_{i n}^{o s m} P_{i n}^{- 1} = i R T C_{i n} P_{i n}^{- 1}$
$F_{l}$	fouling number, defined as $F_{l} \equiv P e_{i n} N_{d}^{- 1} = (P_{i n} C_{i n} d) {(I_{0} D_{0} C_{d})}^{- 1}$
$I_{f} (z)$	local resistance to permeation due to fouling (kg × m $^{- 2} \times$ s $^{- 1}$ )
$R_{e c}$	recovery factor
$R_{f}$	fouling rate