164
Model equations in the case of dead-end mode of filtration through a flat sheet
membrane with a constant volume of feed side (diafiltration mode) can be developed by the differential mass balance of the contaminating species as follows:
j
j
j
j
¶
¶
+
¶
¶
=
¶
¶
- -
( )
¶
¶
c
t
v
c
x
D
c
x
c
t
w
s
2
2
1
(6.14)
where c s , kmol/m
3
is the concentration of solute adsorbed on the solid phase. Equation
6.14 is the expression of Eq. 6.10 in rectangular coordinate system. The first term on
the left side is the time derivation of the solute concentration in bulk phase, and the
second one defines the axial convective flux. The solute diffusive component and the
solute adsorption/desorption onto porous surface of the solid phase with respect to
time are corresponded to the first and the second terms of the right-hand side of the
expression. The initial and boundary conditions can be expressed as
at
and
t
x
c
=
³
=
0
0
0
,
at
and
t
x
c s
=
³
=
0
0
0
,
at
and
t
x
K v c
K D
c
x
k A c c
i C w
i D
x
f
f i x
>
=
-
¶
¶
æ
è
ç
ö
ø
÷
=
-
(
)
=
=
0
0
0
0
,
,
,
,
j
at
and
t
x L K v c
K D
c
x
k A c
c
i C w
i D
xL
p
p i
x L
>
=
-
¶
¶
æ
è
ç
ö
ø
÷
=
-
(
)
=
=
0
,
,
,
,
j
where K i,c , k f , and k p are solute convective hindrance factor, feed side, and permeate
side mass transfer coefficients, respectively, and they can be estimated by the following equations:
K i C
i
i
i
i
,
.
.
.
= -
(
) +
-
+
(
)
2
1 0 054
0 988
0 441
2
3
Ø
l
l
l
k d
D
Re Sc
f
p
i
×
×
=
¥
2
0 567
0 33
,
.
.
a
k
D
p
i
p
=
¥
,
d
where Reynolds number Re
D N
p
w
w
=
2
r
m
, Scmidt number Sc
D
w
w
i
=
- ¥
m
r
,
, and water
2.6 is appropriate for the value of α. In those expressions, D p , N, and δ p reflect the
impeller diameter, mixing rate as rpm, and boundary layer thickness of the permeate
side, respectively (Yurekli and Altinkaya 2011). The model equations include equilibrium and dynamic parameters that can be estimated from experimental run. With
Y. Yurekli
Model equations in the case of dead-end mode of filtration through a flat sheet
membrane with a constant volume of feed side (diafiltration mode) can be developed by the differential mass balance of the contaminating species as follows:
j
j
j
j
¶
¶
+
¶
¶
=
¶
¶
- -
( )
¶
¶
c
t
v
c
x
D
c
x
c
t
w
s
2
2
1
(6.14)
where c s , kmol/m
3
is the concentration of solute adsorbed on the solid phase. Equation
6.14 is the expression of Eq. 6.10 in rectangular coordinate system. The first term on
the left side is the time derivation of the solute concentration in bulk phase, and the
second one defines the axial convective flux. The solute diffusive component and the
solute adsorption/desorption onto porous surface of the solid phase with respect to
time are corresponded to the first and the second terms of the right-hand side of the
expression. The initial and boundary conditions can be expressed as
at
and
t
x
c
=
³
=
0
0
0
,
at
and
t
x
c s
=
³
=
0
0
0
,
at
and
t
x
K v c
K D
c
x
k A c c
i C w
i D
x
f
f i x
>
=
-
¶
¶
æ
è
ç
ö
ø
÷
=
-
(
)
=
=
0
0
0
0
,
,
,
,
j
at
and
t
x L K v c
K D
c
x
k A c
c
i C w
i D
xL
p
p i
x L
>
=
-
¶
¶
æ
è
ç
ö
ø
÷
=
-
(
)
=
=
0
,
,
,
,
j
where K i,c , k f , and k p are solute convective hindrance factor, feed side, and permeate
side mass transfer coefficients, respectively, and they can be estimated by the following equations:
K i C
i
i
i
i
,
.
.
.
= -
(
) +
-
+
(
)
2
1 0 054
0 988
0 441
2
3
Ø
l
l
l
k d
D
Re Sc
f
p
i
×
×
=
¥
2
0 567
0 33
,
.
.
a
k
D
p
i
p
=
¥
,
d
where Reynolds number Re
D N
p
w
w
=
2
r
m
, Scmidt number Sc
D
w
w
i
=
- ¥
m
r
,
, and water
2.6 is appropriate for the value of α. In those expressions, D p , N, and δ p reflect the
impeller diameter, mixing rate as rpm, and boundary layer thickness of the permeate
side, respectively (Yurekli and Altinkaya 2011). The model equations include equilibrium and dynamic parameters that can be estimated from experimental run. With
Y. Yurekli