Membrane and Desalination Technologies - TCE Moodle Website

292 J. Paul Chen et al.
Considering the membrane process as a binary system, the transport of solvent (e.g. water)
and solute is involved. Designating solute and solvent by subscripts A and B, Eq. (5) can be
written for solvent B as
rm B ¼ RTr ln aBm þ vBrp: (6)
Assuming vB is constant, integration over the membrane thickness yields
Dm B ¼ vB
RT
D ln aBm þ Dp : (7)
vB
(D is defined here as “permeate side–feed side”.)
Assuming further that thermodynamic equilibrium is established at the membrane–
solution boundaries, aBm = aB, where aB is the activity of solvent (mol/m 3 ) outside of the
membrane. This relationship should be valid on both sides of the membrane. Since the
osmotic pressure, P (Pa), is defined as
P ¼ RT
ln aB; (8)
Eq. (7) can be written as
vB
Dm B ¼ vBðDp DPÞ: (9)
For practical purposes, the osmotic pressure of a feed may be estimated by van’t Hoff’s
equation
or for NaCl,
P ¼ RðT þ 273Þ X Mi
(10a)
P ¼ 7;720ðT þ 273Þ X Mi; (10b)
where T is the temperature ( C) and P Mi is the sum of molalities or molarities of ions and
non-ionic compounds (mol/kg or mol/L).
As for solute A, we shall further assume that the activity coefficient remains constant, then
Eq. (5) becomes
rmA ¼ RTr ln aAm þ vArp
¼ RTr ln cAm þ vArp;
where aAm is the activity of solute in the membrane (mol/m 3 ), cAm is the concentration of
solute in the membrane (mol/m 3 ) and v A is the molar volume of solute (m 3 /mol).
Integration over the membrane thickness yields
(11)
Dm A ¼ RTD ln cAm þ vADp: (12)