Membranes - Principles of Chemical Separations with Environmental Applications - page 267

Rearranging,

D d 2 C
d x 2

d C
d x =

+ v p 0 . (9.26)

Integrating,
D d C

d x + v p C

= constant = v p C p (9.27)

= solute flux across the membrane.

The boundary condition is:

x = 0; C = C m (9.28)

= membrane surface concentration.

Solution for concentration profile is:

C p )e − ( v p / D ) x C = ( C m − + C p . (9.29)

Rearranging,

C p
C m −

C −

e − ( v p / D ) x

C p =

. (9.30)

At x = δ (boundary-layer thickness), C = C b (9.31)

C b − C p

e − ( v p / D ) δ .

C p =

(9.32)

C m −

Note :

mass transfer coefficient
concentration profile is exponential
when

D /δ = k =

v p = 0 (process off ), effect disappears.

9.11 Effect of concentration polarization on membrane

performance

C p
C m

= = −

R rejection coefficient 1 (9.33)

C p
C b .

R obs = observed rejection coefficient = 1 − (9.34)

We want to obtain an expression to relate the solvent flux (typically water) to measurable
quantities. In general, we need to include the effect of osmotic pressure, especially for
reverse osmosis applications.

J = K [ P − ] (9.35)

= K [ P − ( m − p )] Note : m = b . (9.36)