550
M. Spinola et al.
of a process chain. It is described by population balance equations relying on the
method of characteristics for the numerical simulation and on the usage of gradient
information to enhance the performance of the optimization. The presented scheme to
optimize time-dependent process conditions in a time efficient manner is applicable
for a wide range of particle syntheses.
Keywords Ripening · Method of characteristics · Population balance equation ·
Gradient-based optimization · Sensitivity · Adjoint
1 Introduction
Process conditions of most nanoparticle syntheses can be adjusted over time. In
order to achieve high quality particulate products, the question arises how to optimize
synthesis processes by their process conditions. Model-based optimization of process
conditions and an efficient, gradient based concept for this will be focused in this
work.
For example, in the last decades, the class of quantum dots emerged. Due to their
semiconductor properties, they play an important role in the field of nanoparticle technology. There, many issues regarding process design and the production on higher
scales arised which are still not fully covered [1–3]. Several contributions such as
[4–10] showed that the development of disperse properties during a chemical process
can mathematically be described by population balance equations (PBEs). Due to
their general formulation, PBEs are suitable to appropriately describe the evolution
of disperse properties of the mentioned sub-10 nm particles due to ripening [11–16].
Originally, PBEs were used to model growth mechanisms operating outside of the
nanoregime. For instance, fluidized bed spray granulation (FBSG), crystallization
and aggregation [17–19], a.o., can be depicted by this modeling. PBEs also allow to
track the dispersity in all five dimensions, i.e., size, shape, surface, composition and
structure, during the process (cf. [20]). Even though growth dynamics of the particles
can be described appropriately by means of PBEs, in general the modeling of process
chains with interconnected reaction modules is a complex issue to solve (cf. [21]).
This becomes especially challenging when process conditions are considered which
are varied in course of time, such as feeding rates of new nuclei into reaction systems or process-temperature profiles (see Fig. 1). The corresponding mathematical
model consists of systems of nonlinear, fully coupled differential equations. Thus,
on the one hand, numerical methods are strongly required not only to simulate the
development of the synthesis process and to predict its further development but also
to enable the path towards a model-based optimization (cf. [22]).
In literature, essentially three categories of methods for the numerical approximation of balance laws are discussed [23]. The first and most widespread are finite
volume type methods or in more generality, discontinuous Galerkin type approaches
on a—with respect to process time—fixed discretization of the spatial as well as
disperse coordinates [24–28]. The second category is a class of methods which rely
M. Spinola et al.
of a process chain. It is described by population balance equations relying on the
method of characteristics for the numerical simulation and on the usage of gradient
information to enhance the performance of the optimization. The presented scheme to
optimize time-dependent process conditions in a time efficient manner is applicable
for a wide range of particle syntheses.
Keywords Ripening · Method of characteristics · Population balance equation ·
Gradient-based optimization · Sensitivity · Adjoint
1 Introduction
Process conditions of most nanoparticle syntheses can be adjusted over time. In
order to achieve high quality particulate products, the question arises how to optimize
synthesis processes by their process conditions. Model-based optimization of process
conditions and an efficient, gradient based concept for this will be focused in this
work.
For example, in the last decades, the class of quantum dots emerged. Due to their
semiconductor properties, they play an important role in the field of nanoparticle technology. There, many issues regarding process design and the production on higher
scales arised which are still not fully covered [1–3]. Several contributions such as
[4–10] showed that the development of disperse properties during a chemical process
can mathematically be described by population balance equations (PBEs). Due to
their general formulation, PBEs are suitable to appropriately describe the evolution
of disperse properties of the mentioned sub-10 nm particles due to ripening [11–16].
Originally, PBEs were used to model growth mechanisms operating outside of the
nanoregime. For instance, fluidized bed spray granulation (FBSG), crystallization
and aggregation [17–19], a.o., can be depicted by this modeling. PBEs also allow to
track the dispersity in all five dimensions, i.e., size, shape, surface, composition and
structure, during the process (cf. [20]). Even though growth dynamics of the particles
can be described appropriately by means of PBEs, in general the modeling of process
chains with interconnected reaction modules is a complex issue to solve (cf. [21]).
This becomes especially challenging when process conditions are considered which
are varied in course of time, such as feeding rates of new nuclei into reaction systems or process-temperature profiles (see Fig. 1). The corresponding mathematical
model consists of systems of nonlinear, fully coupled differential equations. Thus,
on the one hand, numerical methods are strongly required not only to simulate the
development of the synthesis process and to predict its further development but also
to enable the path towards a model-based optimization (cf. [22]).
In literature, essentially three categories of methods for the numerical approximation of balance laws are discussed [23]. The first and most widespread are finite
volume type methods or in more generality, discontinuous Galerkin type approaches
on a—with respect to process time—fixed discretization of the spatial as well as
disperse coordinates [24–28]. The second category is a class of methods which rely