552
M. Spinola et al.
degrees of freedom significantly such that gradient-free optimization methods lead
to large computation times.
Thus, here, a numerical tool relying on the method of characteristics which provides a numerical approximation of the development of the synthesis process with
high accuracy was combined with a sensitivity-based optimization ansatz in which
a fast computation of gradient information is provided. By this method, optimal
process conditions within interconnected process chains were determined. It further reduced the mentioned computational times, also in the case of time-dependent
process parameters.
In the following, the proposed optimization method, which can be applied for a
variety of optimization problems within chemical engineering, will be pointed out
in detail. At first, the underlying mathematical model for processes in which seeded
growth occurs, will be presented and the numerical scheme will be outlined. This
is based on the method of characteristics which depicts the evolution of the PSD
in a natural way due to the incorporated conservation property and the hyperbolic
behavior of the corresponding PBE.
Relying on this scheme for the simulation of the state equation, i.e., the considered
PBE, a so-called adjoint equation was provided to require gradient information for
the here used optimization method instead of using the gradient by commonly-used
finite differences methods. Compared to finite differences methods and to gradientfree approaches, applying this method to reaction plants, this enables the calculation
of optimal time-varying process conditions by far faster since only two equations
have to solved, the state equation and the adjoint equation. [42–44] also provided an
optimization method relying on adjoint equations. However, no explicit applications
are outlined in [42, 43]. In opposite to this contribution, no optimization of complex
process chains was performed in [44].
To show the broad applicability of this approach, a range of synthesis processes
with different growth dynamics, optimization goals and also containing process
chains with recycling streams were considered. These examples justify the usage
of the here presented model-based optimization technique in the context of particle
synthesis, especially when only time-varying process conditions yield optimal PSDs.
2 Mathematical Model and Gradient-Based Optimization
Framework
This work aims to determine several time-varying process conditions simultaneously
which will yield optimal results for the corresponding synthesis processes. Before
proposing a mathematical formulation of the optimization approach, first, the mathematical model for the development of PSDs will be stated, which is done in the
following primal equation (1):
M. Spinola et al.
degrees of freedom significantly such that gradient-free optimization methods lead
to large computation times.
Thus, here, a numerical tool relying on the method of characteristics which provides a numerical approximation of the development of the synthesis process with
high accuracy was combined with a sensitivity-based optimization ansatz in which
a fast computation of gradient information is provided. By this method, optimal
process conditions within interconnected process chains were determined. It further reduced the mentioned computational times, also in the case of time-dependent
process parameters.
In the following, the proposed optimization method, which can be applied for a
variety of optimization problems within chemical engineering, will be pointed out
in detail. At first, the underlying mathematical model for processes in which seeded
growth occurs, will be presented and the numerical scheme will be outlined. This
is based on the method of characteristics which depicts the evolution of the PSD
in a natural way due to the incorporated conservation property and the hyperbolic
behavior of the corresponding PBE.
Relying on this scheme for the simulation of the state equation, i.e., the considered
PBE, a so-called adjoint equation was provided to require gradient information for
the here used optimization method instead of using the gradient by commonly-used
finite differences methods. Compared to finite differences methods and to gradientfree approaches, applying this method to reaction plants, this enables the calculation
of optimal time-varying process conditions by far faster since only two equations
have to solved, the state equation and the adjoint equation. [42–44] also provided an
optimization method relying on adjoint equations. However, no explicit applications
are outlined in [42, 43]. In opposite to this contribution, no optimization of complex
process chains was performed in [44].
To show the broad applicability of this approach, a range of synthesis processes
with different growth dynamics, optimization goals and also containing process
chains with recycling streams were considered. These examples justify the usage
of the here presented model-based optimization technique in the context of particle
synthesis, especially when only time-varying process conditions yield optimal PSDs.
2 Mathematical Model and Gradient-Based Optimization
Framework
This work aims to determine several time-varying process conditions simultaneously
which will yield optimal results for the corresponding synthesis processes. Before
proposing a mathematical formulation of the optimization approach, first, the mathematical model for the development of PSDs will be stated, which is done in the
following primal equation (1):