Effect of branching bonds

The yield over temperature at isothermal operation is plotted in Figure 5.5 for the linear polymer as well as the branched polymer. It is very surprising that the yield is higher for the branched polymer.

The branching bonds should reduce the activity of the endo-active enzyme. And limit dextrines, which cannot be attacked and therefore should decrease the yield, for the attack on end-chains by the exo-active enzyme are possible if branching bonds exist. The contribution of the enzymes to the rate of yield increase over time is shown in Figure 5.7. One can see that the ECS enzyme increases the yield much more than the RCS enzymes. This is due to ECS producing directly monomer and dimer that is counted as product whereas RCS enzyme produces smaller polymer that are still larger than the product. Initially, the rates are higher for the branched polymer because each polymer has more than one end-chain. The enzymes are therefore more likely to create small polymers by each attack. How-ever, if one compares the contribution of the ECS enzyme for the linear and branched polymer, one can see that the rate increases for the branched polymer whereas it decreases for the linear polymer. The rate decreases for the linear polymer because the effect due to denaturation of enzyme and increased product inhibition is larger than the effect from having more attackable polymers due to the action of the RCS enzyme. This should also occur for the branched polymer. However, instead the rate increases up to80 hrs. This is likely due to each attack of the RCS enzyme on an inner chain creating one end-chains with a reducing end and a non-reducing end. In the model it was assumed that the chain carrying the

0 1 2 3 4

Rate of change of temperature in mK/min

Yield Φ in %

Base case Sweet potato starch

Figure 5.4: The yield over the temperature change rate with fixed average temperature.

30 40 50 60 70 80

Figure 5.5: The yield of temperature at isother-mal operation for the simplified product distribu-tion funcdistribu-tion, the product distribudistribu-tion funcdistribu-tion pre-dicted by the Subsite Theory, and the branched polymer.

Figure 5.6: The fully refined temperature profile optimal for operating the reactor for 7 d for the simplified product distribution function, the prod-uct distribution function predicted by the Subsite Theory, and the branched polymers.

non-reducing end could be handled like a chain carrying a reducing end because it was only one chain out of many chains. However, due to this assumption every attack of RCS increases the amount of ECS attackable chain by 2 instead of by 1. This is a shortcoming of the model which should be redeemed.

The validation performed in the preceding chapter did not include an enzyme that performs mainly ECS and therefore did not catch this. However, the increased production of maltotriose, which is another short oligomer, might also be explained by having too many chains with reducing ends.

The temperature profile maximizing conversion computed for the branched polymer is shown in Fig-ure 5.6. This shows that the temperatFig-ure profile depends on the model used and also that the algo-rithm is able to deal with branched polymer. No further investigation into the effect of branching on the predicted optimal control profiles was undertaken because the underlying model is not correct.

5.5 Conclusion and outlook

A model-based optimization study of depolymerization described by a population balance approach was performed. The enzyme mixture amount and composition were predicted to have the greatest influence on economic feasibility. But, temperature has also a major influence on process performance.

Therefore, those parameters should be optimized. Furthermore, non-isothermal and non-linear temper-ature profiles have the potential to increase profitability. However, using the optimal linear tempertemper-ature already realizes much of the saving potential. Even though the optimization problem is non-linear, multi-dimensional, and dependent on the starting material, it could be solved in a simple sequential manner in this study with only a small loss of saving potential.

Model-based optimization of enzyme mixture composition and amount and temperature should be ap-plied to real processes and other biopolymers. This requires including e.g, the dissolution of polymer particles using models such as proposed by Griggs et al. [42], Lebaz et al. [46] or deactivation by ir-reversible unproductive binding [30]. Furthermore, experimentally obtained parameters are needed.

Then there is only a small extra effort for additionally exploiting temperature trajectory optimization.

It should be studied whether an (easy-to-implement) linear temperature profile would also be a good approximation for these more realistic cases.

Additionally, the objective functional should be extended to include more cost factors such as the energy consumption due to heating. If required, it can also be extended to include boundary conditions such as a given starting or end temperature. Furthermore, a more sophisticated refinement algorithm, e.g.

[240], may be used to speed up computation. Optimal control can also be applied to optimal enzyme dosage [160] and substrate feeding [23].

0 20 40 60 80 100 120 140 160 0

0.05 0.1 0.15 0.2 0.25 0.3 0.35 0.4

Time in hrs

Yield increase rate in %/hrs

Branched RCS enzyme Linear RCS enzyme Branched ECS enzyme Linear ECS enzyme

Figure 5.7: Contribution of the enzymes to the rate of yield increase for the branched and linear polymer at the optimal isothermal temperature for conversion of the branched polymer.

Chapter 6

Conclusion and outlook

In this dissertation a Population Balance Model (PBM) to describe the enzymatically catalyzed depoly-merization of branched starch was developed. Three techniques (Direct Quadrature Method of Mo-ments (DQMOM), Cell Average Technique (CAT), and Fixed Pivot technique (FP)) to solve such PBMs were adapted to the problem and a convergence study was performed. The adaption for DQMOM made the application of a domain decomposition technique necessary which was the first time such a decom-position was used for non-monovariate problems and with DQMOM. As FP was the best technique an optimal control algorithm was developed using this technique to find the optimal temperature profile.

6.1 Population Balance Model

For several mechanisms (chain scission (RCS), End-chain scission (ECS), and Random-debranching scission (RDS)) the Product Distribution Function (PDF) was derived for the first time for a branched non-crosslinking polymer. This PDFs can be used whenever usage of these mechanisms can be justified. The PBM developed in Chapter 2 is the first model that accounts for the distributed nature of the polymer as well as the temperature dependence of the enzyme. Both effects are described us-ing established and mechanistically sound models (the Subsite Theory [209, 216] and the Equilibrium Model [205]). For the first time a Population Balance Equation (PBE) that extends the Subsite Theory to a branched polymer was developed. Previously, only kinetic Monte Carlo (kMC) solvers could be used [32, 33].

The developed model was tested in Chapter 4 and found to describe experimental data as good as the only other currently available mechanistic model [32]. However, in Chapter 5 the model was found to predict likely false reaction rates for enzymes that perform mainly exo-active attacks. Accordingly, the assumption causing this effect should not be used and the model should be changed. This is not held to be a difficult task. Afterwards, the model should be validated more rigorously by comparison with re-sults obtained using the kMC algorithm proposed by Marchal et al. [33] at precisely defined conditions.

Furthermore, one can extract more information from the kMC simulations than just the evolution of the amount of polymers and some species. Rather, one can directly obtain the reaction rate and PDF as a

function of branching density and the amount of monomer units. This information can then be used to validate the PBM. Additionally, one could investigate the PBM separately from a solution technique for the PBE.

The model developed in this work neglects several effects which are relevant to obtaining a truly uni-versal model. Pressure [243], pH-value [243, 244], and ion concentration [243, 245] are known to have an influence on the activity and stability of starch degrading enzymes. If those values vary between or during production runs, then they need to be included in the model. This is possible for a PBM and has been done by Obersteiner [246] for a version of the DQMOM code used in this work. If inhibitors, e.g.

cellulose [247], are present, they should also be modeled using an appropriate model (see e.g. [248]).