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308 J. Paul Chen et al.
Feed Permeate (product)
Nw c 1
P 1
∏ 1
N s
where Nw is the water flux (kg water/s m 2 ), Ns is the salt flux (kg salt/s m 2 ), Aw is the water
permeability constant (kg water/s m 2 atm), As is the salt permeability constant (m/s), P1 is the
pressure applied on feed solution (atm), P2 is the pressure of permeate (product) solution
(atm), P1 is the osmotic pressure of feed solution (atm), P2 is the osmotic pressure of
permeate (product) solution (atm), c1 is the salt concentration in feed solution (kg salt/m 3 )
and c2 is the salt concentration in permeate (product) solution (kg salt/m 3 ).
Water permeability constant (Aw) is a function of the membrane material such as water
membrane permeability and membrane thickness. The presence of salt, however, is less
important for the constant. For CA membranes, Aw typically ranges from 1 10 4 to 5
10 4 kg water/s m 2 atm. Salt permeability constant depends on the types of salts in the
solution. For CA membranes, As values range from 1 10 7 to 6 10 7 m/s. Some of the
typical values are 4 10 7 m/s for NaCl, 2.2 10 7 m/s for MgCl2, 2.4 10 7 m/s for
CaCl2, 6 10 7 m/s for KCl and 4 10 7 m/s for Na2SO4, respectively.
The solution volumetric flow rate in the feed side is approximately equal to that in the
permeate side. Therefore, we have
Ns Nw
Am ¼ Am ;
c2 cw2
where cw2 is the concentration of water in permeate (product) (kg water/m 3 ).
Thus, we can have
Ns ¼ Nw
c2: (36)
cw2
If the salt concentration in permeate (product) (c2) is low, its osmotic pressure (P2) is
normally ignored and the concentration of water in permeate (product) (cw2) is approximately
the pure water density that is a function of temperature (Table 7.5).
Define the salt rejection (R) as
c 2
P 2
∏ 2
c w2
Fig. 7.21. Mass transfer in RO membrane.
R ¼ c1 c2
: (37)
c1