68
4 Development of a Mathematical Model for an Aerobic Fixed-Bed …
Equation (4.7) can be solved to find out ‘J’ at different points of substrate
concentrations as follows:
J 1 =
2k X f D f [(S 1 − S min ) + K ln[(K + S min )/(K + S 1 )]
(4.8)
J 2 =
2k X f D f [(S 2 − S 1 ) + K ln[(K + S 1 )/(K + S 2 )] + (J 1 ) 2
(4.9)
J 3 =
2k X f D f [(S 3 − S 2 ) + K ln[(K + S 2 )/(K + S 3 )] + (J 2 ) 2
(4.10)
J 4 =
2k X f D f [(S 4 − S 3 ) + K ln[(K + S 3 )/(K + S 4 )] + (J 3 ) 2
(4.11)
J 5 =
2k X f D f [(S w − S 4 ) + K ln[(K + S 4 /(K + S w )] + (J 4 ) 2
(4.12)
where J 1 , J 2 , J 3 , J 4 and J 5 are substrate fluxes corresponding to substrate concentrations S 1 , S 2 , S 3 , S 4 and S w , respectively. It is obvious that the substrate flux (say J 0 )
corresponding to S min is zero. Therefore, the arithmetic average of J can be calculated
as
J avg = (J 0 + J 1 + J 2 + J 3 + J 4 + J 5 )/6.
(4.13)
Now, from the steady-state mass balance of active microorganisms in a biofilm, as
well as under suspended-growth state (Referring to the conceptual diagram of hybrid
bioreactor with recirculation as shown in Fig. 5d, Annexure) II-b, we get,
Y a J avg V
b s
b t
− Q w X − (Q − Q w )X e + pV X
Y K S w
K + S w
− b d
= 0
(4.14)
In Eq. (4.8), shearing losses from the biofilm, i.e., Y a J avg V
b s
b t
, also enter the
suspended-growth regime.
Now, dividing Eq. (4.8) by pVX,
Y a J avg
pX
b s
b t
−
1
θ c
+
Y kS w
K + S w
− b d
= 0,
where, θ c =
pV X
Q w X +(Q−Q w )X e
i.e., X =
Y a J avg
b s
pb t
1
θ c
+b d −
Y kS w
K +S w
,
(4.15)
where,
4 Development of a Mathematical Model for an Aerobic Fixed-Bed …
Equation (4.7) can be solved to find out ‘J’ at different points of substrate
concentrations as follows:
J 1 =
2k X f D f [(S 1 − S min ) + K ln[(K + S min )/(K + S 1 )]
(4.8)
J 2 =
2k X f D f [(S 2 − S 1 ) + K ln[(K + S 1 )/(K + S 2 )] + (J 1 ) 2
(4.9)
J 3 =
2k X f D f [(S 3 − S 2 ) + K ln[(K + S 2 )/(K + S 3 )] + (J 2 ) 2
(4.10)
J 4 =
2k X f D f [(S 4 − S 3 ) + K ln[(K + S 3 )/(K + S 4 )] + (J 3 ) 2
(4.11)
J 5 =
2k X f D f [(S w − S 4 ) + K ln[(K + S 4 /(K + S w )] + (J 4 ) 2
(4.12)
where J 1 , J 2 , J 3 , J 4 and J 5 are substrate fluxes corresponding to substrate concentrations S 1 , S 2 , S 3 , S 4 and S w , respectively. It is obvious that the substrate flux (say J 0 )
corresponding to S min is zero. Therefore, the arithmetic average of J can be calculated
as
J avg = (J 0 + J 1 + J 2 + J 3 + J 4 + J 5 )/6.
(4.13)
Now, from the steady-state mass balance of active microorganisms in a biofilm, as
well as under suspended-growth state (Referring to the conceptual diagram of hybrid
bioreactor with recirculation as shown in Fig. 5d, Annexure) II-b, we get,
Y a J avg V
b s
b t
− Q w X − (Q − Q w )X e + pV X
Y K S w
K + S w
− b d
= 0
(4.14)
In Eq. (4.8), shearing losses from the biofilm, i.e., Y a J avg V
b s
b t
, also enter the
suspended-growth regime.
Now, dividing Eq. (4.8) by pVX,
Y a J avg
pX
b s
b t
−
1
θ c
+
Y kS w
K + S w
− b d
= 0,
where, θ c =
pV X
Q w X +(Q−Q w )X e
i.e., X =
Y a J avg
b s
pb t
1
θ c
+b d −
Y kS w
K +S w
,
(4.15)
where,