16 Modeling, Simulation and Optimization of Process Chains
573
It has to be mentioned that the choice of the weights in the cost functional determine the preferences between different optimization goals. For instance, a by far
higher weight of the costs of P can lead to an (almost) vanishing milling power
whereas the maximization of the total mass will be taken less into account. Thus, in
actual applications these weights have to be selected problem-specifically. Finally,
these results, especially Fig. 14, show that a larger number of discretization points of
the process conditions are required to obtain significantly better outcomes compared
to constant optimization which again leads to the advantage of adjoint equations
when computing gradients.
5 Conclusions
The mathematical framework illustrated in this publication consists out of two major
aspects. On the one hand, an on the method of characteristics relying numerical
discretization scheme was used to obtain a sufficiently precise approximation of
PSDs even for a rather coarse discretization grid. On the other hand, based on these
simulations, an optimization tool was presented which can calculate optimal process
conditions for a variety of applications within mechanical process engineering and
powder technology. The numerical simulation consists of applying the method of
characteristics on the here considered nonlocal PBEs and due to the conservation
of number of particles incorporated in the scheme, the evolution of the PSD can be
tracked with high accuracy.
To demonstrate its broad usability, it was shown that it is capable to handle PSDs
with different numbers of modes, growth rates and even process chains with recycling
streams. Due to the fact that process conditions can be adjusted in a time-varying
manner, a sensitivity-based optimization scheme was chosen. This ensures that also
these complex, interconnected reaction modules can be improved in small time in
contrast to gradient-free approaches due to their high computational costs. Here, an
adjoint equation was derived and with its solution a representation of the gradient of
the cost functional with respect to the process conditions was obtained. By this adjoint
method, only two partial differential equations, namely the state and the adjoint
equation, have to be solved. This decreases the number of function evaluations and
thus the computational effort by far compared to commonly used finite differences
methods.
In this contribution four cases demonstrating the applicability of the presented
methodology were considered. For the following real-world examples within chemical engineering, numerical optimization results were derived and discussed also
concerning computational times:
573
It has to be mentioned that the choice of the weights in the cost functional determine the preferences between different optimization goals. For instance, a by far
higher weight of the costs of P can lead to an (almost) vanishing milling power
whereas the maximization of the total mass will be taken less into account. Thus, in
actual applications these weights have to be selected problem-specifically. Finally,
these results, especially Fig. 14, show that a larger number of discretization points of
the process conditions are required to obtain significantly better outcomes compared
to constant optimization which again leads to the advantage of adjoint equations
when computing gradients.
5 Conclusions
The mathematical framework illustrated in this publication consists out of two major
aspects. On the one hand, an on the method of characteristics relying numerical
discretization scheme was used to obtain a sufficiently precise approximation of
PSDs even for a rather coarse discretization grid. On the other hand, based on these
simulations, an optimization tool was presented which can calculate optimal process
conditions for a variety of applications within mechanical process engineering and
powder technology. The numerical simulation consists of applying the method of
characteristics on the here considered nonlocal PBEs and due to the conservation
of number of particles incorporated in the scheme, the evolution of the PSD can be
tracked with high accuracy.
To demonstrate its broad usability, it was shown that it is capable to handle PSDs
with different numbers of modes, growth rates and even process chains with recycling
streams. Due to the fact that process conditions can be adjusted in a time-varying
manner, a sensitivity-based optimization scheme was chosen. This ensures that also
these complex, interconnected reaction modules can be improved in small time in
contrast to gradient-free approaches due to their high computational costs. Here, an
adjoint equation was derived and with its solution a representation of the gradient of
the cost functional with respect to the process conditions was obtained. By this adjoint
method, only two partial differential equations, namely the state and the adjoint
equation, have to be solved. This decreases the number of function evaluations and
thus the computational effort by far compared to commonly used finite differences
methods.
In this contribution four cases demonstrating the applicability of the presented
methodology were considered. For the following real-world examples within chemical engineering, numerical optimization results were derived and discussed also
concerning computational times: