Robust Optimization in Nanoparticle Technology Exemplified by Means of a Residence Time Reactor

Knowledge-based determination of the best-possible experimental setups for nanoparticle design is highly challenging and currently often out-of-reach. Although production processes do not follow idealized lab conditions, it is necessary that quality requirements are met in order to receive high-quality end products. However, in particular large-scale processes in chemical engineering are accompanied by noticeable uncertainty, for instance in physico-chemical material properties, but also temperature profiles and residence time distributions. Therefore, production needs to be protected against uncertainties that are inherent in the process. Robust mathematical optimization can help determining such best possible processes that are hedged against uncertainties. The latter guarantees that quality requirements are met regardless of how the uncertainties manifest themselves within predefined ranges. As an example, in this work we model a particle synthesis process with seeded growth by population balance equations and study different growth kinetics. The optimization task consists of determining the mean residence time that maximizes the product mass subject to a guaranteed yield. The resulting model is a nonlinear optimization problem (NLP). Protecting against uncertainties is a crucial task in this context, since the total mass necessarily has to be disposed if the particles grow too large for further use. We hedge against uncertain growth rates and derive an equivalent and algorithmically tractable reformulation for the robust counterpart of the NLP. In particular the latter amounts to solving a convex optimization problem that can efficiently be solved to global optimality. We evaluate our optimization approach for the seeded growth synthesis of zinc oxide quantum dots. We demonstrate computationally that a guaranteed yield is indeed met for all growth rates that manifest themselves within previously defined regions. The protection against uncertainties only reduces the maximum amount of product that can be obtained by a negligible margin, i.e. the ”price of robustness” is affordable.