4.1 Basic Consideration
63
4.1.3 Justification of Current Methodology
The present mathematical model employed the kinetics of suspended and attachedgrowth biomass in integrated manner considering their simultaneous growth. Monod
kinetics is followed for the utilization of carbonaceous substrate considering no
inhibition in the application of this mathematical model for the treatment of municipal
wastewater.
J avg (average substrate flux) is considered in mathematical model, because there is
always a variation in substrate flux from biofilm layer to layer on account of substrate
gradient. The substrate gradient is exerted within individual biofilm layer, when the
substrate incoming due to diffusion gets utilized by the respective biofilm layer.
In fact, the substrate gradient becomes a driving force to allow for the movement
of the substrate through individual biofilm layer. Thus, the substrate concentration
gradually decreases from S w at the biofilm–liquid interface to S min at the attachment
surface.
From the definition of substrate flux (J) which essentially equals
l f
0 (−rut)dz
(Rittmann and Mccarty 2001), the overall utilization of substrate throughout the
biofilm needs to be taken into account. Therefore, as a rational approach of calculation
of the overall substrate flux, an area-weighted average value of individual fluxes in
various layers has been taken.
The accuracy of result for calculating the average substrate flux would be better
if the biofilm thickness is divided into several layers (since it is not a linear curve).
Therefore, calculations were made for evaluating the J avg , with 3, 4, 5, 6, 7 and
8 layers in the biofilm. Interestingly, the values of effluent substrate concentration
(S w ) as well as substrate flux (J) were converging for five (5) layers, and thereafter,
there was hardly any deviation of results when the number of layers increased to
five (5) and onward. In view of that, five (5) layers have been considered in the
mathematical model. In this mathematical model, efforts have been put for removing
of carbonaceous organic matter in the treatment of municipal wastewater. Thus, the
output variable S w in the model actually indicates the effluent substrate concentration
in the treatment of wastewater.
The output variable, effective biofilm thickness (L e ), is also considered in the
model as it contains those biomasses actively metabolizing the substrate beyond
which the substrate flux ceases to get utilized further. Runge–Kutta numerical analysis method is found most appropriate for determining the effective biofilm thickness
and is used for the solution of substrate mass balance equation in the biofilm.
4.2 Development of a Mathematical Model
A mathematical model for hybrid bioreactor is developed with a computer programming in FORTRAN and excel worksheet to easily calculate the output parameters
63
4.1.3 Justification of Current Methodology
The present mathematical model employed the kinetics of suspended and attachedgrowth biomass in integrated manner considering their simultaneous growth. Monod
kinetics is followed for the utilization of carbonaceous substrate considering no
inhibition in the application of this mathematical model for the treatment of municipal
wastewater.
J avg (average substrate flux) is considered in mathematical model, because there is
always a variation in substrate flux from biofilm layer to layer on account of substrate
gradient. The substrate gradient is exerted within individual biofilm layer, when the
substrate incoming due to diffusion gets utilized by the respective biofilm layer.
In fact, the substrate gradient becomes a driving force to allow for the movement
of the substrate through individual biofilm layer. Thus, the substrate concentration
gradually decreases from S w at the biofilm–liquid interface to S min at the attachment
surface.
From the definition of substrate flux (J) which essentially equals
l f
0 (−rut)dz
(Rittmann and Mccarty 2001), the overall utilization of substrate throughout the
biofilm needs to be taken into account. Therefore, as a rational approach of calculation
of the overall substrate flux, an area-weighted average value of individual fluxes in
various layers has been taken.
The accuracy of result for calculating the average substrate flux would be better
if the biofilm thickness is divided into several layers (since it is not a linear curve).
Therefore, calculations were made for evaluating the J avg , with 3, 4, 5, 6, 7 and
8 layers in the biofilm. Interestingly, the values of effluent substrate concentration
(S w ) as well as substrate flux (J) were converging for five (5) layers, and thereafter,
there was hardly any deviation of results when the number of layers increased to
five (5) and onward. In view of that, five (5) layers have been considered in the
mathematical model. In this mathematical model, efforts have been put for removing
of carbonaceous organic matter in the treatment of municipal wastewater. Thus, the
output variable S w in the model actually indicates the effluent substrate concentration
in the treatment of wastewater.
The output variable, effective biofilm thickness (L e ), is also considered in the
model as it contains those biomasses actively metabolizing the substrate beyond
which the substrate flux ceases to get utilized further. Runge–Kutta numerical analysis method is found most appropriate for determining the effective biofilm thickness
and is used for the solution of substrate mass balance equation in the biofilm.
4.2 Development of a Mathematical Model
A mathematical model for hybrid bioreactor is developed with a computer programming in FORTRAN and excel worksheet to easily calculate the output parameters