4.2 Development of a Mathematical Model
75
sustain a steady biofilm growth). The second flowchart (Fig. 4.5) utilized the value
of S w as obtained from the first flowchart to calculate L e following the concept of
Runge–Kutta method of analysis. The convergence of iteration is attributed at the
liquid–biofilm interface when computed substrate concentration (S w ) becomes equal
to the assumed one.
4.2.4 Modality of Application of the Developed Model
The solution method derived from the proposed model can be applied to determine
the exiting substrate concentration, i.e., bulk liquid substrate concentration (S w ) and
the average substrate flux (J avg ) by running the FORTRAN program (Program 1)
based on the flowchart as shown in Fig. 4.4. After the determination of the values of
S w and J avg , the FORTRAN program (Program 2) based on the flowchart as shown
in Fig. 4.5 can be run for evaluating both the total and effective biofilm thicknesses
L f and L e . The sequential steps of algorithm for Program 1 are stated in the Excel
Worksheet as shown in Fig. 4.6.
Similarly, the sequential steps of algorithm for Program 2 are also stated in the
Excel Worksheet as shown in Fig. 4.7.
4.2.5 Novelty of the Developed Model
In this mathematical model, simultaneous utilization of the substrate by suspended
and attached biomass in a competitive manner is considered, which is very much
essential for the process design of a fixed-bed hybrid bioreactor. A simplified model
(with computer programming and Excel Worksheet) for hybrid bioreactor finds its
importance for predicting the reliable outputs for the sake of process design. There
is a flexibility of the present model making it a versatile one to find out the exiting
substrate concentration along with substrate flux both in hybrid bioreactor and in a
completely mixed biofilm reactor. No approximation is considered in this mathematical model depicting its uniqueness compared with the existing methods of numerical
analysis. To maintain the steady-state biofilm model, it is necessary to check whether
the value of effluent substrate concentration is more than minimum substrate concentration, i.e., S min or not. This has been taken into consideration while developing the
mathematical modeling of the hybrid bioreactor.
Apart from that, the developed model determines the substrate flux at any layer
of the biofilm matrix along with the effective and total biofilm thickness. The model
can determine the effective biofilm thickness which contains the effective biomass
actively metabolizing the substrate beyond which the substrate flux ceases to get
utilized further. In fact, by the effective thickness of biofilm, it can be ascertained
whether the biofilm is deep or shallow. The application of Runge–Kutta method in
this mathematical model is found, the only numerical analysis method by which the
75
sustain a steady biofilm growth). The second flowchart (Fig. 4.5) utilized the value
of S w as obtained from the first flowchart to calculate L e following the concept of
Runge–Kutta method of analysis. The convergence of iteration is attributed at the
liquid–biofilm interface when computed substrate concentration (S w ) becomes equal
to the assumed one.
4.2.4 Modality of Application of the Developed Model
The solution method derived from the proposed model can be applied to determine
the exiting substrate concentration, i.e., bulk liquid substrate concentration (S w ) and
the average substrate flux (J avg ) by running the FORTRAN program (Program 1)
based on the flowchart as shown in Fig. 4.4. After the determination of the values of
S w and J avg , the FORTRAN program (Program 2) based on the flowchart as shown
in Fig. 4.5 can be run for evaluating both the total and effective biofilm thicknesses
L f and L e . The sequential steps of algorithm for Program 1 are stated in the Excel
Worksheet as shown in Fig. 4.6.
Similarly, the sequential steps of algorithm for Program 2 are also stated in the
Excel Worksheet as shown in Fig. 4.7.
4.2.5 Novelty of the Developed Model
In this mathematical model, simultaneous utilization of the substrate by suspended
and attached biomass in a competitive manner is considered, which is very much
essential for the process design of a fixed-bed hybrid bioreactor. A simplified model
(with computer programming and Excel Worksheet) for hybrid bioreactor finds its
importance for predicting the reliable outputs for the sake of process design. There
is a flexibility of the present model making it a versatile one to find out the exiting
substrate concentration along with substrate flux both in hybrid bioreactor and in a
completely mixed biofilm reactor. No approximation is considered in this mathematical model depicting its uniqueness compared with the existing methods of numerical
analysis. To maintain the steady-state biofilm model, it is necessary to check whether
the value of effluent substrate concentration is more than minimum substrate concentration, i.e., S min or not. This has been taken into consideration while developing the
mathematical modeling of the hybrid bioreactor.
Apart from that, the developed model determines the substrate flux at any layer
of the biofilm matrix along with the effective and total biofilm thickness. The model
can determine the effective biofilm thickness which contains the effective biomass
actively metabolizing the substrate beyond which the substrate flux ceases to get
utilized further. In fact, by the effective thickness of biofilm, it can be ascertained
whether the biofilm is deep or shallow. The application of Runge–Kutta method in
this mathematical model is found, the only numerical analysis method by which the