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4 Development of a Mathematical Model for an Aerobic Fixed-Bed …
• The model essentially considered the substrate mass transport into the biofilm as
per Fick’s law.
• Average substrate flux (J avg ) has been determined considering a variation in
substrate flux from biofilm layer to layer on account of substrate gradient.
• 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 a rational approach of calculation of the overall substrate flux, an area-weighted
average value of individual fluxes in various layers has been taken.
• J 0 , the substrate flux corresponding to S min at attachment surface, is considered
zero.
4.1.2 Fundamental Concept
The mathematical model for fixed-bed hybrid bioreactor is based on steady-state
mass balance of both carbonaceous substrate and biomass under suspended and
attached growth simultaneously along with substrate mass transport into the biofilm.
Monod kinetics is followed for the utilization of carbonaceous substrate assuming
no inhibition. Average substrate flux on account of varying substrate concentration
profile inside the biofilm is considered for determining the value of exiting substrate
concentration (S w ), which is a key output variable for evaluating the performance
of the reactor. Substrate concentration S 0 becomes exiting substrate concentration
S w in the bulk liquid due to complete mixing in the reactor. This concentration S w
is the homogeneous bulk liquid concentration up to the liquid–biofilm interface,
which further decreases to S min , i.e., the minimum substrate concentration at biofilm
attachment surface. Firstly, the combination of mass balance equation of substrate
under both the suspended and the attached growth is derived. In this combined mass
balance equation of substrate, the expression of suspended biomass concentration is
substituted by the term derived from the biomass balance equation of the suspended
growth accounting for biomass shear loss from biofilm. Average substrate flux is
computed from the solution of the equation of substrate mass balance into biofilm.
Finally, exiting substrate concentration (S w ) is obtained substituting the value of
average substrate flux in the combined mass balance equation of substrate, which is
initially derived. Applying the Runge–Kutta method of analysis of the substrate mass
balance equation of biofilm and applying the boundary conditions, effective biofilm
thickness (L e ) is obtained. L e is determined by iteration process, where convergence
of iteration is attributed at the liquid–biofilm interface when computed substrate
concentration (S w ) becomes equal to the assumed one. The effective biofilm thickness
contains those biomasses actively metabolizing the substrate. Beyond this effective
biofilm thickness, the substrate flux ceases to get utilized further. In fact, by the
effective biofilm thickness of biofilm, it can be ascertained whether the biofilm is
deep or shallow [1].
4 Development of a Mathematical Model for an Aerobic Fixed-Bed …
• The model essentially considered the substrate mass transport into the biofilm as
per Fick’s law.
• Average substrate flux (J avg ) has been determined considering a variation in
substrate flux from biofilm layer to layer on account of substrate gradient.
• 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 a rational approach of calculation of the overall substrate flux, an area-weighted
average value of individual fluxes in various layers has been taken.
• J 0 , the substrate flux corresponding to S min at attachment surface, is considered
zero.
4.1.2 Fundamental Concept
The mathematical model for fixed-bed hybrid bioreactor is based on steady-state
mass balance of both carbonaceous substrate and biomass under suspended and
attached growth simultaneously along with substrate mass transport into the biofilm.
Monod kinetics is followed for the utilization of carbonaceous substrate assuming
no inhibition. Average substrate flux on account of varying substrate concentration
profile inside the biofilm is considered for determining the value of exiting substrate
concentration (S w ), which is a key output variable for evaluating the performance
of the reactor. Substrate concentration S 0 becomes exiting substrate concentration
S w in the bulk liquid due to complete mixing in the reactor. This concentration S w
is the homogeneous bulk liquid concentration up to the liquid–biofilm interface,
which further decreases to S min , i.e., the minimum substrate concentration at biofilm
attachment surface. Firstly, the combination of mass balance equation of substrate
under both the suspended and the attached growth is derived. In this combined mass
balance equation of substrate, the expression of suspended biomass concentration is
substituted by the term derived from the biomass balance equation of the suspended
growth accounting for biomass shear loss from biofilm. Average substrate flux is
computed from the solution of the equation of substrate mass balance into biofilm.
Finally, exiting substrate concentration (S w ) is obtained substituting the value of
average substrate flux in the combined mass balance equation of substrate, which is
initially derived. Applying the Runge–Kutta method of analysis of the substrate mass
balance equation of biofilm and applying the boundary conditions, effective biofilm
thickness (L e ) is obtained. L e is determined by iteration process, where convergence
of iteration is attributed at the liquid–biofilm interface when computed substrate
concentration (S w ) becomes equal to the assumed one. The effective biofilm thickness
contains those biomasses actively metabolizing the substrate. Beyond this effective
biofilm thickness, the substrate flux ceases to get utilized further. In fact, by the
effective biofilm thickness of biofilm, it can be ascertained whether the biofilm is
deep or shallow [1].