ฮฒ g g-1 h-1 0 0.002
๐๐๐, ๐๐๐,๐๐2 g g-1 h-1 0 1
A, B, C (-) 0 1
4.4 Results and discussion
4.4.1 Model calibration and validation 4.4.1.1 Parameter sensitivity analysis
Sensitivities of different parameters with respect to all the state variables are tabulated in Table 4.4. The sensitivity analysis was performed to understand the effect of different parameters on the state variables and estimate the range of the parameters for model tuning, that is, lower bound and upper bound (Shiue et al., 1995; Zhong and Tang, 2004).
The result of positive perturbation in all the parameters concluded that the oxygen yield coefficient in the batch phase was the most sensitive parameter for biomass concentration compared to all other parameters. Similarly, the negative perturbation study indicated that the maintenance coefficient of oxygen is found to be the most sensitive parameter for biomass concentration. This observation, in turn, implies that the oxygen-related parameters are sensitive towards biomass concentration. In the case of substrate concentration, biomass yield coefficients are found to be more sensitive. A similar analysis for other state variables can be established, as shown in Table 4.4.
Table 4.4. Sensitivity ranking of the parameters for the different state variables of the developed model
State variable
Rank of parameters in the order of sensitivity
+10% perturbation in parameters -10% perturbation in parameters X ๐ฆ๐/๐2,๐ > ๐ฆ๐/๐2,๐ > ยต๐๐ > ยต๐๐>
๐ฆ๐/๐๐> ๐ฆ๐/๐๐> ๐๐๐>ฮฒ > ฮฑ > ๐๐๐
> ๐๐๐> ๐๐๐ > ๐๐2 > (A, B, C) >
๐๐2
๐๐2> ๐๐2> (A, B, C) > ๐๐๐> ๐๐๐>
๐๐๐ > ๐๐๐> ฮฒ > ฮฑ > ๐ฆ๐/๐๐ > ยต๐๐>
๐ฆ๐/๐๐> ยต๐๐ > ๐ฆ๐/๐2,๐ > ๐ฆ๐/๐2,๐
Sb ๐ฆ๐/๐๐> (A, B, C) > ๐๐2> ๐๐2> ๐๐๐
> ๐๐๐ > ฮฒ > ฮฑ > ๐ฆ๐/๐2,๐> ๐ฆ๐/๐๐>
๐๐๐> ยต๐๐ > ยต๐๐> ๐๐๐ > ๐ฆ๐/๐2,๐
๐ฆ๐/๐2,๐> ยต๐๐> ๐๐๐> ๐๐๐>ฮฒ > ฮฑ >
๐ฆ๐/๐2,๐ > ๐ฆ๐/๐๐ > ๐๐๐ > ยต๐๐ >๐๐๐
> ๐๐2> ๐๐2> (A, B, C) > ๐ฆ๐/๐๐
CHAPTER 4 Sf ๐ฆ๐/๐๐> ๐๐2 > (A, B, C) > ๐๐2 >
๐ฆ๐/๐๐> ๐๐๐>๐๐๐> ฮฒ > ฮฑ > ๐๐๐>
ยต๐๐ > ยต๐๐ > ๐ฆ๐/๐2,๐> ๐๐๐ >
๐ฆ๐/๐2,๐
๐ฆ๐/๐2,๐> ๐๐๐> ๐ฆ๐/๐2,๐> ยต๐๐> ยต๐๐
> ๐๐๐> ๐๐๐> ฮฒ > ฮฑ > ๐๐๐> ๐ฆ๐/๐๐>
(A, B, C) > ๐๐2> ๐๐2> ๐ฆ๐/๐๐
P ฮฒ > ฮฑ > ๐ฆ๐/๐๐ > ยต๐๐ > ยต๐๐ >
๐ฆ๐/๐2,๐> ๐ฆ๐/๐2,๐> ๐ฆ๐/๐๐> ๐๐๐> ๐๐๐
> ๐๐๐> ๐๐2> ๐๐2 > (A, B, C) >
๐๐๐
๐๐๐> ๐๐2> ๐๐2> ๐๐๐> ๐๐๐> ๐๐๐>
๐ฆ๐/๐๐ > ยต๐๐> ยต๐๐> ๐ฆ๐/๐2,๐> ฮฑ >
(A, B, C) > ๐ฆ๐/๐๐ > ฮฒ > ๐ฆ๐/๐2,๐
DO ๐ฆ๐/๐2,๐> ๐๐๐ > ๐ฆ๐/๐2,๐> (A, B, C)
> ยต๐๐ > ยต๐๐ > ๐๐๐ > ๐ฆ๐/๐๐>๐๐๐
> ๐๐๐> ฮฒ > ฮฑ > ๐๐2> ๐ฆ๐/๐๐> ๐๐2
๐ฆ๐/๐๐> ๐๐2> (A, B, C) > ๐๐2 >
๐ฆ๐/๐๐> ๐๐๐ > ๐๐๐> ฮฒ > ฮฑ > ๐๐๐>
ยต๐๐> ยต๐๐> ๐๐๐> ๐ฆ๐/๐2,๐ > ๐ฆ๐/๐2,๐ O2 ๐ฆ๐/๐2,๐> ๐ฆ๐/๐2,๐> ๐๐๐ > (A, B, C)
> ยต๐๐> ๐๐2> ๐๐๐> ๐ฆ๐/๐๐> ๐๐๐ >
ฮฒ > ฮฑ > ๐๐๐> ยต๐๐> ๐ฆ๐/๐๐> ๐๐2
๐ฆ๐/๐๐> ๐๐2 > ยต๐๐ > (A, B, C) >
๐ฆ๐/๐๐> ๐๐๐> ๐๐๐> ๐๐2> ๐๐๐> ฮฒ >
ฮฑ > ยต๐๐> ๐๐๐> ๐ฆ๐/๐2,๐> ๐ฆ๐/๐2,๐
4.4.1.2 Parameter estimation and model validation
Model calibration was performed by tuning the parameters within the defined bounds, as mentioned in section 4.3.3.2. The comparison of experimental and model-predicted values is presented in Figure 4.3. It is observed that the model is able to predict the respective experimental values with an average error of 12.64%. Figure 4.3 (A) shows the variation in the input variables with respect to time, and the same was used as an input for the model. Also, it could be noticed that a constant substrate flow rate was adapted for this experimental study. The residual substrate concentration (Sf) of the process was maintained near zero, as observed in Figure 4.3 (D). The offline analysis of biomass, substrate, and product concentration is represented as dotted points, as shown in Figure 4.3 (C) and (D). These data were used for the model calibration and validation studies.
The values were plotted as an average of two reactor runs.
The comparison of experimental and model-predicted values for DO and O2 is shown in Figure 4.3 B. The model predicted the experimental DO and O2 within an error value of 28.21% and 3.41%, respectively. The comparison of experimental and model-predicted values for biomass, substrate, and product concentration is shown in Figure 4.3 (C) and (D). The model predicted the experimental results within error values of 11.97%, 7.23%, 15.81%, and 9.20% for X, Sb, Sf and P, respectively. It can be concluded that the parameter
4.4 Results and discussion
estimation procedure has to ensure that all the mass balances are satisfied and the parameters are within their bounds in order to obtain a reliable calibrated model that could be applied further for optimization strategy implementation.
The estimated values for the parameters are summarized in Table 4.5. The parameters were constrained to be positive and also fall within the defined bounds. Following the model calibration, validation was carried out, and the results are plotted in Figure 4.4.
The model predicted the experimental values of the validation dataset with an average error value of 14.97%.
The reactor studies carried out for model development and validation were open-loop experiments with constant feed rates throughout the fed-batch phase. The input signals can be subjected to disturbance/excitation (5 or 10%) to verify the model predictions. The initial parameter values for the search were chosen randomly to fall within the bounds presented in Table 4.3. Start values for some parameters like yield coefficients were chosen to be roughly near the values calculated from offline analysis.
Compared to the model predictions using initial parameter values, the calibrated parameters resulted in improvements greater than 50%. The error value for validation set was ~14%, which is close enough to the error value of model calibration (~12%). This shows that the model is reliable and valid within the given range of input parameters.
Further, this error could be reduced by one of the following means: a. Choosing different kinetic models for biomass estimation; b. Smoothening/ filtering of experimental DO data, since the error value of DO was highest among the different state variables, owing to high signal noise; c. Acquiring more experimental data points (especially offline analysis) and implementing online estimation methods.
CHAPTER 4
Figure 4.3. Model calibration and parameter estimation: (A). Input process variables; (B).
Experimental (continuous lines) and model-predicted (dashed lines) comparison for dissolved oxygen and gas phase oxygen; (C). Experimental and model-predicted (dashed lines) comparison for biomass concentration (online (continuous line) & offline (black filled circle)) and product concentration (blue filled circle); (D). Experimental and model- predicted (dashed lines) comparison for substrate concentration in batch (black filled circle) and fed-batch phase (blue filled circle).
4.4 Results and discussion
Table 4.5. Estimated values for the parameters after model calibration Parameter
symbol Parameter name Value
๐๐2 Monod saturation constant for oxygen (g L-1) 0.002 ยต๐๐ Maximum specific growth rate of batch phase (h-1) 0.750 ยต๐๐ Maximum specific growth rate of fed-batch phase (h-1) 0.555 ๐๐๐ Monod saturation constant for substrate in batch phase (g L-1) 0.075 ๐๐๐ Monod saturation constant for substrate in fed-batch phase (g
L-1) 0.008
๐ฆ๐/๐๐
Biomass yield coefficient from substrate in batch phase (g
biomass g-1 substrate Sb) 0.666
๐ฆ๐/๐๐ Biomass yield coefficient from substrate in fed-batch phase (g
biomass g-1 substrate Sf) 0.699
๐ฆ๐/๐2,๐
Biomass yield coefficient from oxygen utilized in batch phase
(g oxygen g-1 biomass) 0.931
๐ฆ๐/๐2,๐ Biomass yield coefficient from oxygen utilized in fed-batch
phase (g oxygen g-1 biomass) 0.976
ฮฑ
Growth associated product yield coefficient (g product g-1
biomass) 0.010
ฮฒ Non-growth associated product yield coefficient (g product g-1
biomass h-1) 0.0002
๐๐๐
Maintenance coefficient of substrate in batch phase (g
substrate g-1 biomass h-1) 0.235
๐๐๐ Maintenance coefficient of substrate in fed-batch phase (g
substrate g-1 biomass h-1) 0.227
๐๐2 Maintenance coefficient of oxygen (g oxygen g-1 biomass h-1) 0.069 A
B C
Correlation constants for volumetric mass transfer coefficient (-)
0.451 0.013 0.195
CHAPTER 4
Figure 4.4. Model validation: (A). Input process variables; (B). Experimental (continuous lines) and model-predicted (dashed lines) comparison for dissolved oxygen and gas phase oxygen; (C). Experimental and model-predicted (dashed lines) comparison for biomass concentration (online (continuous line) & offline (black filled circle)) and product concentration (blue filled circle); (D). Experimental and model-predicted (dashed lines) comparison for substrate concentration in batch (black filled circle) and fed-batch phase (blue filled circle).