Filtration Membranes Containing Nanoparticles of Hydrated …
765
Table 3 Filtration of liquids
Membrane
Solution containing
Ca 2+ , Mg 2+
OVA
BSA
J × 10 7 m 3
m −2 s −1
ϕ (%) J × 10 7 m 3
m −2 s −1
ϕ (%) J × 10 7 m 3
m −2 s −1
ϕ (%)
PAN-2
12.9
–
10.9
92
6.8
97
PAN-2–HZD–GO 3.3
14.28 4.1
98
3.5
98
PAN-3
35.0
–
14.4
–
15.1
–
PAN-3–HZD–GO 9.6
9.74
7.3
95
4.9
98
which improves diffusion of alkaline ions. This indicates the effect of surface charge
of the composite on the rejection of divalent ions. Besides dissociated and protonated
–OH groups of HZD, functional groups of GO make sufficient contribution to the
modifier surface charge.
The thickness of the secondary active layer caused by inorganic particles within
the pores (l) can be calculated from the Kozeny–Carman equation [67]:
P
1
=
180μ(1 − ε
2
)v
Φ 2 d 2 ε 2
,
(5)
where ε is the porosity (0.33 for compact packing of globules), is the sphericity
of the particles (we can assume = 1), d is the diameter of the particles, μ the
dynamic viscosity (9 × 10
−3 Pa at 298 K), and v is the liquid velocity. Since the
effective membrane area was 21 cm
2 , v = 1.6 × 10
−4 (PAN-2–HZD–GO) and 4.8 ×
10
−4 (PAN-3–HZD–GO). Equation (5) can be rewritten as follows:
l
d 2 =
Φ
2
ε
2
P
180μ(1 − ε 2 )v
,
(6)
Here, l/d
2 is the parameter, which characterizes the retarding ability of ion exchanger.
This value is 1.66 × 10
7 (PAN-2–HZD–GO) and 5.2 × 10
8 m
−1 (PAN-3–HZD–GO),
or 166 and 52 nm
−1 , respectively. Nanoparticles of insoluble zirconium hydroxocomplexes, a size of which is 6 nm, dominate in sol [34]. It is possible to assume no change
of the size of primary nanoparticles after deposition in polymers. Thus, the thickness
of the ion exchanger layer, which determines rejection, is 12 (PAN-2–HZD–GO) and
3% (PAN-3–HZD–GO) of the total membrane thickness. It means the formation of
thicker ion exchanger layer in the membrane containing smaller pores.
The permeate flux is related to the hydrodynamic resistance of the membrane
according to the Darcy equation [7, 67]:
J =
P
η R m
,
(7)
765
Table 3 Filtration of liquids
Membrane
Solution containing
Ca 2+ , Mg 2+
OVA
BSA
J × 10 7 m 3
m −2 s −1
ϕ (%) J × 10 7 m 3
m −2 s −1
ϕ (%) J × 10 7 m 3
m −2 s −1
ϕ (%)
PAN-2
12.9
–
10.9
92
6.8
97
PAN-2–HZD–GO 3.3
14.28 4.1
98
3.5
98
PAN-3
35.0
–
14.4
–
15.1
–
PAN-3–HZD–GO 9.6
9.74
7.3
95
4.9
98
which improves diffusion of alkaline ions. This indicates the effect of surface charge
of the composite on the rejection of divalent ions. Besides dissociated and protonated
–OH groups of HZD, functional groups of GO make sufficient contribution to the
modifier surface charge.
The thickness of the secondary active layer caused by inorganic particles within
the pores (l) can be calculated from the Kozeny–Carman equation [67]:
P
1
=
180μ(1 − ε
2
)v
Φ 2 d 2 ε 2
,
(5)
where ε is the porosity (0.33 for compact packing of globules), is the sphericity
of the particles (we can assume = 1), d is the diameter of the particles, μ the
dynamic viscosity (9 × 10
−3 Pa at 298 K), and v is the liquid velocity. Since the
effective membrane area was 21 cm
2 , v = 1.6 × 10
−4 (PAN-2–HZD–GO) and 4.8 ×
10
−4 (PAN-3–HZD–GO). Equation (5) can be rewritten as follows:
l
d 2 =
Φ
2
ε
2
P
180μ(1 − ε 2 )v
,
(6)
Here, l/d
2 is the parameter, which characterizes the retarding ability of ion exchanger.
This value is 1.66 × 10
7 (PAN-2–HZD–GO) and 5.2 × 10
8 m
−1 (PAN-3–HZD–GO),
or 166 and 52 nm
−1 , respectively. Nanoparticles of insoluble zirconium hydroxocomplexes, a size of which is 6 nm, dominate in sol [34]. It is possible to assume no change
of the size of primary nanoparticles after deposition in polymers. Thus, the thickness
of the ion exchanger layer, which determines rejection, is 12 (PAN-2–HZD–GO) and
3% (PAN-3–HZD–GO) of the total membrane thickness. It means the formation of
thicker ion exchanger layer in the membrane containing smaller pores.
The permeate flux is related to the hydrodynamic resistance of the membrane
according to the Darcy equation [7, 67]:
J =
P
η R m
,
(7)