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M. Spinola et al.
1. In Sects. 4.1 and 4.2, the resulting PSD in the considered FBSG module was
exactly driven to desired mono- and bimodal PSDs, respectively.
2. In Sect. 4.3 the standard deviation for a PSD with a desired mean diameter and
obtained by Ostwald ripening was minimized.
3. In Sect. 4.4, within a cycled process chain with a FBSG module, two screens and
one crusher simultaneously, the yield of the total product was maximized and the
energy consumption due to milling was minimized.
In every example, it was elaborated that time-varying process conditions led to particulate products with a quality superior to the outcome generated by constant process
conditions or, more general, with a low number of discretization points such that
they justify the usage of the here used adjoint equations. Summarized, comparing to
optimization schemes applied in the literature, gradient-based optimization methods
relying on adjoint equations are suitable approaches to compute sensitivities fast also
for a wide range of optimization tasks for particle synthesis processes.
To conclude this contribution, some possible topics for future research associated
to the optimization approach described in this work are mentioned. Since the focus
lies on the applicability to real-world processes and optimization within chemical
engineering, it is required to validate numerical results obtained by the here used
scheme to experimental data such that in case of deviations refinements of the models can be performed. Especially then when not single reaction modules but rather
process chains occur, a high degree of interconnectivity of different reactors could
lead to the case that the ansatz within this work is not fully applicable and thus it has
to be adapted properly. Moreover, due to wall effects, different types of flow velocity
profiles emerge and thus residence time distributions occur. Residence time reactors
play a significant role within chemical engineering such that the mathematical model
should be extended to include these. There, one can also include the residence time
distribution itself as a process condition one aims to adjust to enhance the quality
of the final particulate products. Finally, due to the fact that the shape of reactors
influences their residence time distribution, this shape can thus be subject for optimization. By shape optimization techniques, desired residence time distributions can
be achieved and thus also the field of reactor optimization (cf. [58]) can be connected
to this work.
Acknowledgements The authors gratefully acknowledge financial support from the German
Research Foundation within the priority program SPP 1679 “DYNSIM-FP” LE 595/30-2 and the
travel funding provided by the “Bavaria California Technology Center”.
M. Spinola et al.
1. In Sects. 4.1 and 4.2, the resulting PSD in the considered FBSG module was
exactly driven to desired mono- and bimodal PSDs, respectively.
2. In Sect. 4.3 the standard deviation for a PSD with a desired mean diameter and
obtained by Ostwald ripening was minimized.
3. In Sect. 4.4, within a cycled process chain with a FBSG module, two screens and
one crusher simultaneously, the yield of the total product was maximized and the
energy consumption due to milling was minimized.
In every example, it was elaborated that time-varying process conditions led to particulate products with a quality superior to the outcome generated by constant process
conditions or, more general, with a low number of discretization points such that
they justify the usage of the here used adjoint equations. Summarized, comparing to
optimization schemes applied in the literature, gradient-based optimization methods
relying on adjoint equations are suitable approaches to compute sensitivities fast also
for a wide range of optimization tasks for particle synthesis processes.
To conclude this contribution, some possible topics for future research associated
to the optimization approach described in this work are mentioned. Since the focus
lies on the applicability to real-world processes and optimization within chemical
engineering, it is required to validate numerical results obtained by the here used
scheme to experimental data such that in case of deviations refinements of the models can be performed. Especially then when not single reaction modules but rather
process chains occur, a high degree of interconnectivity of different reactors could
lead to the case that the ansatz within this work is not fully applicable and thus it has
to be adapted properly. Moreover, due to wall effects, different types of flow velocity
profiles emerge and thus residence time distributions occur. Residence time reactors
play a significant role within chemical engineering such that the mathematical model
should be extended to include these. There, one can also include the residence time
distribution itself as a process condition one aims to adjust to enhance the quality
of the final particulate products. Finally, due to the fact that the shape of reactors
influences their residence time distribution, this shape can thus be subject for optimization. By shape optimization techniques, desired residence time distributions can
be achieved and thus also the field of reactor optimization (cf. [58]) can be connected
to this work.
Acknowledgements The authors gratefully acknowledge financial support from the German
Research Foundation within the priority program SPP 1679 “DYNSIM-FP” LE 595/30-2 and the
travel funding provided by the “Bavaria California Technology Center”.