4.2 Development of a Mathematical Model
67
⇒
dS f
dz
2
=
2k X f
D f
S f − K log(K + S f )
+ A
Now, when S f = S w ,
dS f
dz
= 0
⇒ 0 =
2k X f
D f
[S w − K log(K + S w )] + A
A = −
2k X f
D f
[S w − K log(K + S w )]
⇒
dS f
dz
2
=
2k X f
D f
[S f − K log(K + S f )]
−
2k X f
D f
[S W − K log(K + S W )]
⇒
dS f
dz
2
=
2k X f
D f
(S f − S W ) + K log
(K + S W )
(K + S f )
⇒
dS f
dz
=
2k X f
D f
(S f − S W ) + K log
(K + S W )
(K + S f )
Hence,
J =
dS f
dz
D f = D f
2k X f
D f
(S f − S W ) + K log
(K + S W )
(K + S f )
J =
2k X f D f
(S entry − S exit ) + K log
(K + S exit )
(K + S entry )
(4.7)
where S f = substrate concentration at any point in the biofilm (mg/cm
3 ),
X f = active biomass density within the biofilm(mg/cm
3 ),
D f = molecular diffusion coefficient of the substrate in the biofilm (cm
2 /day),
S entry = entering substrate concentration in a small segment in the biofilm (mg/cm
3 ),
S exit = exiting substrate concentration in a small segment in the biofilm (mg/cm
3 ),
S min = K
∗
b e
Y ∗ k − b t
,
where S min = minimum concentration of rate-limiting substrate at biofilm attachment
surface (mg/cm
3 ).