118
6 Modelling of Packed Bed Column for the Adsorption …
where R s is bead radius (m), D eff is effective diffusivity of metal within the bead,
(m
2 /s) is the bead density (dry mass/wet volume) (kg/m
3 ), C C is the metal concentration in the core (mol/m
3 ) and R c is the core radius (m). Equating 6.1 and 6.2, we
obtain the net rate of adsorption; hence,
r D = r C =
dq
dt
=
3η C K C S
R
3
S
R
3
S
+
η C Ø
2
S (RS −R C )
R C
(6.3)
But diffusion in a single particle spherical pore, following heterogeneous
catalyzed first-order kinetics has effectiveness factor (η o ) as given in 6.4. The
effectiveness factor for the unloaded core, c, is equally defined by 6.5.
η o =
1
Ø S
1
tan h3Ø S
−
1
3Ø S
(6.4)
η C =
1
Ø C
1
tan h3Ø C
−
1
3Ø C
(6.5)
where Ø is a dimensionless group called Thiele modulus which can also be defined
based on the adsorbent particle spherical radius (6.6) and on the active core radius
as given in 6.7
Ø S =
R S
3
K p
D eff
(6.6)
Ø C =
R C
3
K p
D eff
(6.7)
θ =
q
q max
(6.8)
If adsorption loading fraction (θ ) is defined by the expression (1 – θ =
R
3
c
R 3
s
), which
on substituting into 6.3, we obtain 6.9.
dq
dt
=
η c K C s
1
1−θ
+
3η c ∅ 2
s
1−(1−θ )
1
3
(1−θ )
1/3
(6.9)
dq
dt
= η o F K C s
(6.10)
F =
η c
η o
1
1−θ
+
3η c Ø 2
s (1−(1−θ )
1/3
(1−θ )
1/3
(6.11)
6 Modelling of Packed Bed Column for the Adsorption …
where R s is bead radius (m), D eff is effective diffusivity of metal within the bead,
(m
2 /s) is the bead density (dry mass/wet volume) (kg/m
3 ), C C is the metal concentration in the core (mol/m
3 ) and R c is the core radius (m). Equating 6.1 and 6.2, we
obtain the net rate of adsorption; hence,
r D = r C =
dq
dt
=
3η C K C S
R
3
S
R
3
S
+
η C Ø
2
S (RS −R C )
R C
(6.3)
But diffusion in a single particle spherical pore, following heterogeneous
catalyzed first-order kinetics has effectiveness factor (η o ) as given in 6.4. The
effectiveness factor for the unloaded core, c, is equally defined by 6.5.
η o =
1
Ø S
1
tan h3Ø S
−
1
3Ø S
(6.4)
η C =
1
Ø C
1
tan h3Ø C
−
1
3Ø C
(6.5)
where Ø is a dimensionless group called Thiele modulus which can also be defined
based on the adsorbent particle spherical radius (6.6) and on the active core radius
as given in 6.7
Ø S =
R S
3
K p
D eff
(6.6)
Ø C =
R C
3
K p
D eff
(6.7)
θ =
q
q max
(6.8)
If adsorption loading fraction (θ ) is defined by the expression (1 – θ =
R
3
c
R 3
s
), which
on substituting into 6.3, we obtain 6.9.
dq
dt
=
η c K C s
1
1−θ
+
3η c ∅ 2
s
1−(1−θ )
1
3
(1−θ )
1/3
(6.9)
dq
dt
= η o F K C s
(6.10)
F =
η c
η o
1
1−θ
+
3η c Ø 2
s (1−(1−θ )
1/3
(1−θ )
1/3
(6.11)