Results

Besselink et al. [32] used the Dextrose Equivalent (DE) to compare simulation and experiment. It is computed here as

DE=100· (MH₂O+MGlu)·P

iw˘i

P

iw˘i·(MH₂O+k˘i·MGlu). (4.40) Furthermore, the mass fraction was used. The mass fractionx_j of the j-th pivot is

xj=100· (MH₂O+MGlu·k˘j)·w˘j

P

iw˘i·(MH₂O+k˘i·M_Glu). (4.41)

1Here, a chain is defined as the amount of linearly connected monomer units starting from a branching bond [10].

Table 4.5: Parameters used to compare the developed Population Balance Model to data from Besselink et al. [32].

Parameter Symbol Value

Subsites below the catalytic center^a z˘− 6 Subsites above the catalytic center^a z˘₊ 4

Acceleration factor^d ∆G_a 1.55 kJ mol⁻¹

Molar mass of water^a MH₂0 18.02 g mol⁻¹ Molar mass of anhydro-glucose^a M_Glu 162.14 g mol⁻¹ Inhibition by chain starting branching bond^b b_I 0.1 Inhibition by branching bond towards the reducing end^b bII 0.2 Inhibition by branching bond towards the non-reducing end^b bIII 0.4

Temperature^c T 50^◦C

Initial concentration of starch^c 100 g l⁻¹

Mass percentage of starch as amylopectin^c 75 %

Mean molar mass by weight of amylopectin^e 2.27×10⁸g mol⁻¹

Polydispersity of amylopectin^e 1.13

Mean degree of branching of amylopectin^c 5 %

Average polymerization degree of amylose^f 1220

90 cutoff of amylose^f 3130

10 cutoff of amylose^f 190

Mesh growth factor 1.1

Amount of exactly represented monomer units k˘_c 14

Maximal amount of monomer units 4.20×10⁶

Maximal amount of branching bonds 2.12×10⁵

aRef: Besselink et al. [32] originally from Allen and Thoma [217]

bRef: Besselink et al. [32] originally from Marchal et al. [33]

cRef: Besselink et al. [32]

dRef: Allen and Thoma [217]

eRef: Rolland-Sabate et al. [184]

f Ref: Hanashiro and Takeda [183]

The Dextrose Equivalent is plotted over time in Figure 4.12. There is good qualitative agreement between kMC results and the PBE results. The difference between the two methods can be either attributed to the assumptions made in deriving the PBE model or the to different initial conditions. How-ever, the agreement between the experimental and the PBE results is slightly better than for kMC re-sults. The mass fraction of the monomer and the linear trimer is plotted in Figure 4.13. One can see that the overprediction of Maltotriose is greater for the PBE model than for the kMC results. Furthermore, the underprediction of the glucose is also sightly worse. The total mass fraction of the heptamers and hexamers (both branched and linear) is shown in Figure 4.14. The agreement between experimental data and PBE model is better than for the kMC results.

4.6 Conclusion and outlook

In this chapter the Fixed Pivot technique (FP) and Cell Average Technique (CAT) were adapted to a discrete Population Balance Equation (PBE). For several test cases the results of FP and CAT were compared with kinetic Monte Carlo (kMC) simulations based on the PBE. Both methods were found to have comparable accuracy but FP was found to be significantly faster. Therefore, FP should be used to describe depolymerization processes of the kind studied in this work.

It was found that End-chain scission (ECS) is solved with a significant error, unless very fine meshes were used. This has already been reported by Ho et al. [48]. For Random-chain scission (RCS) and Random-debranching scission (RDS) alone the solution of the products and the amount of polymers were found to be in excellent agreement for quite coarse meshes. However, the combination of those two mechanisms resulted in a deviation which did only slowly disappear with increasing mesh fineness.

For linear polymer the dependence of the reaction rate on the chain length and the complicated Product Distribution Function (PDF) predicted by the subsite theory did not pose any issues. However, no rigor-ous convergence study was performed. Also for the branched polymer using the reaction rate and the PDF of the subsite model developed in Chapter 2 did not pose any problems, but rather the agreement between kMC and FP was excellent.

For one case the results of the model developed in Chapter 2 were compared to literature data. The agreement between FP data and experimental data was slightly better than between the kMC results and the experimental data. There was a significant deviation between the FP results and the kMC results. This does not allow a definite statement on the validity of the developed PBE. But the re-sults raise confidence that a Population Balance Model (PBM) approach can be used to describe this process. One could validate the PBM by comparison with either experiments or kMC simulations. If experimental validation is chosen, the initial state and the parameters of the enzymes need to be deter-mined. The effort for experimental validation is high but would result in a model able to describe reality.

If validation by kMC simulation is chosen, further simulations need to be performed, because not all input data for the PBE model was provided by Besselink et al. [32]. Because the initial conditions do not necessarily agree, one cannot state whether the deviations between the kMC and the PBE results

0 50 100 150

Figure 4.12: Dextrose equivalent over time.

0 50 100 150

Figure 4.13: Mass fraction of the monomer (glu-cose) and linear trimer (maltotriose) over time.

0 50 100 150

Figure 4.14: Mass fraction of all heptamers and all hexamers over time.

are due to invalid assumptions or different starting conditions. One should therefore implement the model used by Besselink et al. [32] and for example extract the branching distribution directely from the structures generated by the kMC algorithm.

It was found that increasing the exactly represented region increased the computation time by a signif-icant amount but only had a minor benefit for the accuracy. The simulations times and the accuracy of FP are acceptable for usage in optimal control which is therefore performed in the following chapter.

Chapter 5

Optimal control

Here, first the necessary conditions for optimality are derived for one specific case. Then possible numerical techniques to solve optimal control problems are reviewed and the implemented algorithm is described. The algorithm is applied to a case study and the effect of branching is investigated. Last, a short conclusion is drawn.

5.1 Necessary conditions for optimality

In this work an optimal profile in time is desired and, therefore, the necessary conditions are derived for such a profile. The necessary conditions are required input for an indirect solver.