161
In their model, transport equations in flow channel (Domain 1) and in porous
media (Domain 2) are related to each other at the membrane solution interface as
represented in Fig. 6.8. In Domain 1, velocity and concentration profiles with
respect to radial and length dimensions of the fiber are solved in unsteady-state
condition (0 ≤ r ≤ R; 0 ≤ x ≤ L), while only the convective+diffusive+adsorptive
species continuity equation is solved in Domain 2 (R ≤ r ≤ R + h) considering the
Darcy velocity which is a function of only z through the membrane. Symmetry axis
through cylindrical coordinate is selected. Mathematical model equations developed for Domain 1 are given below. The continuity equation is
1
0
r
rv
r
u
x
¶ ( )
¶
+
¶
¶
=
(6.5)
r-component of Navier–Stokes equation for an incompressible fluid ignoring the
¶
¶
2
2
v
z
term as the radial dimension is too small with respect to fiber length,
r
m
¶
¶
+
¶
¶
+
¶
¶
æ
è
ç
ö
ø
÷ = -
¶
¶
+
¶
¶
¶
¶
æ
è
ç
ö
ø
÷
é
ë
ê
ù
û
ú -
v
t
v
v
r
u
v
x
p
r
r r
r
v
r
v
r
1
2
(6.6)
Similarly, the z-component is
r
m
¶
¶
+
¶
¶
+
¶
¶
æ
è
ç
ö
ø
÷ = -
¶
¶
+
¶
¶
¶
¶
æ
è
ç
ö
ø
÷
é
ë
ê
ù
û
ú
u
t
v
u
r
u
u
x
p
x
r r
r
u
r
1
(6.7)
As shown in Eqs. 6.5–6.7, u, v, and p are the functions of r, z, and t. The appropriate boundary and initial conditions for those three equations are as follows:
(i) at r = 0, for all z: from symmetry,
¶
¶
=
¶
¶
=
¶
¶
=
u
r
v
r
p
r
0
r = 0
r = R
r = R + h
Filtrate
Filtrate
Retentate
Domain 1
Feed
X = 0
X = L
Line of symmetry
u
x
r
v
Fig. 6.8 A schematic illustration of the symmetric and cylindrical membrane filtration unit.
(Modified after Mukherjee et al. 2019)
6 Recovery of Heavy Metals by Membrane Adsorbers
In their model, transport equations in flow channel (Domain 1) and in porous
media (Domain 2) are related to each other at the membrane solution interface as
represented in Fig. 6.8. In Domain 1, velocity and concentration profiles with
respect to radial and length dimensions of the fiber are solved in unsteady-state
condition (0 ≤ r ≤ R; 0 ≤ x ≤ L), while only the convective+diffusive+adsorptive
species continuity equation is solved in Domain 2 (R ≤ r ≤ R + h) considering the
Darcy velocity which is a function of only z through the membrane. Symmetry axis
through cylindrical coordinate is selected. Mathematical model equations developed for Domain 1 are given below. The continuity equation is
1
0
r
rv
r
u
x
¶ ( )
¶
+
¶
¶
=
(6.5)
r-component of Navier–Stokes equation for an incompressible fluid ignoring the
¶
¶
2
2
v
z
term as the radial dimension is too small with respect to fiber length,
r
m
¶
¶
+
¶
¶
+
¶
¶
æ
è
ç
ö
ø
÷ = -
¶
¶
+
¶
¶
¶
¶
æ
è
ç
ö
ø
÷
é
ë
ê
ù
û
ú -
v
t
v
v
r
u
v
x
p
r
r r
r
v
r
v
r
1
2
(6.6)
Similarly, the z-component is
r
m
¶
¶
+
¶
¶
+
¶
¶
æ
è
ç
ö
ø
÷ = -
¶
¶
+
¶
¶
¶
¶
æ
è
ç
ö
ø
÷
é
ë
ê
ù
û
ú
u
t
v
u
r
u
u
x
p
x
r r
r
u
r
1
(6.7)
As shown in Eqs. 6.5–6.7, u, v, and p are the functions of r, z, and t. The appropriate boundary and initial conditions for those three equations are as follows:
(i) at r = 0, for all z: from symmetry,
¶
¶
=
¶
¶
=
¶
¶
=
u
r
v
r
p
r
0
r = 0
r = R
r = R + h
Filtrate
Filtrate
Retentate
Domain 1
Feed
X = 0
X = L
Line of symmetry
u
x
r
v
Fig. 6.8 A schematic illustration of the symmetric and cylindrical membrane filtration unit.
(Modified after Mukherjee et al. 2019)
6 Recovery of Heavy Metals by Membrane Adsorbers