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Concentration polarisation factor in a membrane channel

Salt rejected by the membrane piles up in a thin boundary layer, so the wall sees a higher concentration than the bulk. That raises osmotic pressure, lowers rejection and brings scaling closer. Film theory quantifies it.

Superficial velocity in the feed channelm/sTypical 0.02 to 0.2 m/s in spiral-wound elements.

Feed channel (spacer) heightmm0.4 to 1.2 mm.

Permeate fluxL/m²·h

Solute diffusivity10⁻⁹ m²/sSodium chloride about 1.35; natural organic matter about 0.1.

Temperature°C

Rejection%

Mass transfer correlation

Polarisation factor β

1.138

Manufacturers usually limit β to about 1.2 at the element outlet.

Mass transfer coefficient

5.3610⁻⁵ m/s

Reynolds number

257

Schmidt number

744

Hydraulic diameter

1.72mm

How it works

Film theory relates the wall concentration to the bulk through the flux and the boundary-layer mass transfer coefficient:

β = \frac{C_{M}}{C_{F C}} = (1 - R e j) + R e j \exp (J_{W} / k_{C P})

(1)

The mass transfer coefficient from channel velocity, height and solute diffusivity, with the hydraulic diameter of a thin slit:

k_{C P} = λ \frac{D}{d_{H}} R e^{0.5} S c^{1 / 3} d_{H} = 2 h R e = \frac{ρ v d_{H}}{μ} S c = \frac{μ}{ρ D}

(2)

with λ between 0.40 and 0.54, or alternatively the spacer-corrected form

k_{C P} = 0.023 \frac{D}{d_{H}} R e^{0.875} S c^{0.25}

(3)

where

$β$: concentration polarisation factor
$C_{M}, C_{F C}$: concentration at the wall and in the bulk channel, mg/L
$R e j$: rejection, fraction
$J_{W}$: permeate flux, m/s
$k_{C P}$: boundary layer mass transfer coefficient, m/s
$λ$: correlation coefficient, 0.40 to 0.54
$D$: solute diffusivity, m²/s
$d_{H}$: hydraulic diameter, m
$h$: channel height, m
$R e$: Reynolds number
$S c$: Schmidt number
$ρ$: water density, kg/m³
$v$: channel velocity, m/s
$μ$: water viscosity, kg/m·s

β rises with flux and falls with crossflow velocity, and it varies along the element because both change. The same coefficient feeds the limiting-salt recovery calculation, where β multiplies the concentrate-end concentration. Viscosity is taken at the entered temperature.

Concentration polarisation factor in a membrane channel

How it works

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