4.2 Development of a Mathematical Model
65
r [Q(S 0 − S w )]
pV k X S w
K + S w
= 0
(4.1)
where
S 0 = entering substrate concentration (mg/cm
3 ).
S w = exiting substrate concentration (mg/cm
3 ) in the bulk liquid. Q = inflow rate
(m
3 /s), r = fraction of substrate used by suspended growth, p = porosity of hybrid
reactor, V = empty-bed volume of hybrid reactor (m
3 ), k = maximum specific rate
of substrate utilization (per day), X = concentration of suspended biomass in hybrid
reactor (mg/cm
3 ), K = half-velocity coefficient (mg/cm
3 ).
Dividing Eq. (4.1) by Q, the following equation is obtained:
r [(S 0 − S w )]
pθ k X S w
K + S w
= 0
( 4 . 2 )
where θ = empty-bed hydraulic detention time (h) =
V
Q
.
Now, the steady-state substrate balance for the attached growth is,
(1 − r )[Q(S 0 − S w )] − a J avg V = 0
(4.3)
where a = specific surface area of supporting media (cm
−1 ), J avg = average substrate
flux into the biofilm (mg/cm
2 /day). Again, dividing Eq. (4.3) by Q, the following
equation is obtained:
(1 − r )(S 0 − S w ) − a J avg θ = 0
( 4 . 4 )
Now, combining the two substrate balance Eqs. (4.2) and (4.4) for both the
suspended growth and attached growth the following equation is obtained:
S 0 − S w −
pk X S w θ
K + S w
− a J avg θ = 0.
(4.5)
The substrate profile within the biofilm accounting for its utilization by the
attached biomass can be represented in Fig. 4.2.
To calculate the average value of “substrate flux” (J avg ) into the biofilm, 5 (five)
divisions (based on equal interval of substrate concentration) inside the biofilm have
been considered as shown in Fig. 4.3.
Such an arrangement of divisions is conceptualized within the biofilm for determining the accurate value of individual substrate flux at respective division. It has
also been observed that there is hardly any deviation of results in case number of
divisions more than five (Sarkar and Mazumder 2016).
Now, J avg = (J 0 + J 1 + J 2 + J 3 + J 4 + J 5 )/6 [2], 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 ,
65
r [Q(S 0 − S w )]
pV k X S w
K + S w
= 0
(4.1)
where
S 0 = entering substrate concentration (mg/cm
3 ).
S w = exiting substrate concentration (mg/cm
3 ) in the bulk liquid. Q = inflow rate
(m
3 /s), r = fraction of substrate used by suspended growth, p = porosity of hybrid
reactor, V = empty-bed volume of hybrid reactor (m
3 ), k = maximum specific rate
of substrate utilization (per day), X = concentration of suspended biomass in hybrid
reactor (mg/cm
3 ), K = half-velocity coefficient (mg/cm
3 ).
Dividing Eq. (4.1) by Q, the following equation is obtained:
r [(S 0 − S w )]
pθ k X S w
K + S w
= 0
( 4 . 2 )
where θ = empty-bed hydraulic detention time (h) =
V
Q
.
Now, the steady-state substrate balance for the attached growth is,
(1 − r )[Q(S 0 − S w )] − a J avg V = 0
(4.3)
where a = specific surface area of supporting media (cm
−1 ), J avg = average substrate
flux into the biofilm (mg/cm
2 /day). Again, dividing Eq. (4.3) by Q, the following
equation is obtained:
(1 − r )(S 0 − S w ) − a J avg θ = 0
( 4 . 4 )
Now, combining the two substrate balance Eqs. (4.2) and (4.4) for both the
suspended growth and attached growth the following equation is obtained:
S 0 − S w −
pk X S w θ
K + S w
− a J avg θ = 0.
(4.5)
The substrate profile within the biofilm accounting for its utilization by the
attached biomass can be represented in Fig. 4.2.
To calculate the average value of “substrate flux” (J avg ) into the biofilm, 5 (five)
divisions (based on equal interval of substrate concentration) inside the biofilm have
been considered as shown in Fig. 4.3.
Such an arrangement of divisions is conceptualized within the biofilm for determining the accurate value of individual substrate flux at respective division. It has
also been observed that there is hardly any deviation of results in case number of
divisions more than five (Sarkar and Mazumder 2016).
Now, J avg = (J 0 + J 1 + J 2 + J 3 + J 4 + J 5 )/6 [2], 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 ,