70
4 Development of a Mathematical Model for an Aerobic Fixed-Bed …
h is the distance between z = 0 (at the attachment surface) and z = L e (at the
biofilm–water interface)
L2 = h ∗ f
z 0 +
h
2
,
dS n
dz
= h ∗
d
2 S fl
dz 2 =
k X f (S f0 + 0.5K 1 ∗ h)
D f (K + S f0 + 0.5K 1 ∗ h)
dS f2
dz
=
dS f0
dz
+ 0.5 ∗ L2 ∗ h = K 3
L3 = h ∗ f
z 0 +
h
2
,
dS f2
dz
= h ∗
d
2 S f2
dz 2 =
k X f (S f0 + 0.5K 2 ∗ h)
D f (K + S f0 + 0.5K 2 ∗ h)
dS B
dz
=
dS f0
dz
+ L3 ∗ h = K 4
L4 = h ∗ f
Z 0 + h,
d S f 3
dz
= h
∗ d
2 S f 3
dz 2 =
k X f
S f 0 + K 3
∗ h
D f
K + S f 0 + K 3 ∗ h
Y (1) =
h
6
∗ (K 1 + 2 ∗ K 2 + 2 ∗ K 3 + K 4)
(4.18)
Y (2) =
h
6
∗
L1 + 2 ∗ L2 + 2 ∗ L
3
+ L4
(4.19)
Here, Y (1) stands for increment of substrate concentration, and Y (2) stands
for
dS f
dz
.
Therefore,
S w = S min + Y (1)
(4.20)
J 5 = D f ∗ Y (2)
(4.21)
L f =
J ∗ Y
X f ∗ b t
, where L f = total biofilm thickness(cm)
(4.22)
In order to calculate S w , J and L e using Eqs. (4.5)–(4.21), i.e., to solve the hybrid
bioreactor model, two flowcharts were constructed as shown in Fig. 4.4 and Fig. 4.5.
Consequently, two detailed FORTRAN programs as shown in Program 1 and
Program 2 below have been developed on the basis of those flowcharts.