product yield (ϕXV) could be calculated from the optimal Pareto obtained in this study. A sample representation of optimal point where profit function ( fprofit) is calculated using assumed cost values with respect to broth volume is outlined in Figure 5.8. The assumed cost values were as follows: Cproduct = $25000/g of product, Cbiomass = $0.1/g of biomass,
sin Downstreamproces g
C = $23/ L of broth volume. It could be observed from Figure 5.8 that for the given cost values, the optimum point with the highest profit function value was observed at a broth volume of 2.9 L.
Figure 5.8. Optimal profit function for given cost factors and respective broth volume 5.4.2 Case study (2): Predicting optimal fed-batch harvest time
5.4.2.1 Multiobjective optimization with different fed-batch harvest time
As discussed in section 5.3.4, the case study (2) was formulated to explore the influence of different values of fed-batch harvest time upon solving the previously defined multiobjective optimization. The comparison of the objective function values obtained at λ values of 0.995 and 0.999 respectively, from different fed-batch harvest time (tend) values are presented in Figure 5.9. It could be observed from Figure 5.9 that beyond a tend
5.4 Results and discussions
value of 16 h, there was no significant reduction in the value of objective function f for a λ value of 0.995, and a similar observation could be observed after tend value of 20 h in the case of 0.999. The corresponding values of the objective functions f(1) and f(2) at the two λ values of 0.995 and 0.999 is presented in Figure 5.10 (A) and (B), respectively. It could be observed that at higher λ values, more significance is provided for the total biomass maximization, and therefore, the broth volume reaches the value closer to 5 L.
The maximum total biomass, XV value, was observed at a tend value of 20 h and 24 h respectively for the λ values 0.995 and 0.999, as presented in Figure 5.10. Beyond the value of 24 h, increasing the harvest time further up to 30 h did not improve the objective function f, and additionally, it could be observed that the total biomass value declined for a comparable value of broth volume at 24 and 30 h, as visible from Figure 5.10 (B). This emphasizes the significance of implementing optimization studies that include fed-batch harvest time as one of the objective functions.
Figure 5.9. Comparison of objective function f for different values of fed-batch harvest time (tend) at λ values of 0.995 and 0.999.
CHAPTER 5
Figure 5.10. Comparison of objective function f(1) and f(2) for different values of fed- batch harvest time (tend). (A). Objective functions at λ value of 0.995 and (B). Objective functions at λ value of 0.999.
The Pareto points at different tend values are presented in Figure 5.11. It could be observed from Figure 5.11 that for the fed-batch harvest time up to 20 h, the asymptotic value for the Pareto was observed after 0.995, and for higher tend values, asymptote was observed after 0.999. This observation could be interpreted as follows: for higher tend values, the algorithm results in a feeding profile that could increase the total biomass until the λ value gets closer to 1. The same could be corroborated since the broth volume increases closer to the bounds at higher tend values due to the availability of more feeding time with increasing harvest time. However, it was observed that at a tend value of 30 h, the Pareto points were lesser, signifying that the objective function might not significantly improve beyond this harvest time. Further investigation into the corresponding feeding profile and the respective state variables will provide the reason behind this decrease.
Accordingly, two significant regions were highlighted from this Pareto plot as represented by two quadrants in Figure 5.11. The two regions correspond to the asymptotic value of the Pareto functions observed at 0.995 and 0.999 for different tend values. It could be concluded that the reactor can be operated at a suitable harvest time from the two shaded regions according to the desired volume of operation and corresponding downstream processing cost, which could be decided by the operator. For instance, if the reactor can be operated at a higher volume close to 5 L, the feeding profiles from the gold-shaded quadrilateral and the corresponding harvest time can be chosen, resulting in enhanced total biomass of up to 250 g. On the other hand, if the major focus is on reducing the
5.4 Results and discussions
downstream processing cost, it signifies that operating the reactor at minimum broth volume is beneficial, and therefore, the operator can choose a point from the grey-shaded quadrilateral.
Another significant observation from Figure 5.11 is that increasing fed-batch harvest time might not necessarily enhance productivity. It was observed that the Pareto at a tend value of 30 h was below that of 24 h, and no significant improvement in the value of objective function f was observed either. Therefore, it could be concluded that for the case study (2) under assumed constraints, a fed-batch harvest time of 24 h might significantly be beneficial and can yield up to 250 g of total biomass within a total broth volume of 4.5 L as observed from the Pareto points.
Figure 5.11. Pareto points at different values of fed-batch harvest time (tend) obtained from changing the λ values. The two highlighted regions (grey and gold quadrilateral) represent two significant regions for operating the reactor at the desired volume.
5.4.2.2 Pareto Front for two objectives f(1) and f(2)
The two objectives f(1) and f(2), which are of conflicting nature as deciphered from the simulation studies presented in section 5.3.3.1, can also be solved using a Pareto search
CHAPTER 5