514
R. Ahrens et al.
Fig. 17 and indicates an under-prediction over the entire time period, similar to A
(3,3)
in the bottom right. The error does not exceed 0.6% and 2.3%, respectively. Even
though there is an error in the prediction of higher order moments, they are predicted
with a very good accuracy given the grid with N = 512
2 degrees of freedom.
4 Summary and Outlook
The model experiment in the HCT crystallizer led to interesting and new results
concerning the particle fluid interaction. Crystal size distributions change from the
inlet to the outlet, already without growth, as crystals of varying size have a different
residence time in the HCT. This newly found effect can be used to change the width
of the CSD; or can be used under growth conditions to keep the typically observed
broadening of the CSD very low. Some first insights on the shape dependence of this
effect are encouraging to investigate this topic in more detail in our future work.
For the numerical simulation of PBSs, a deterministic-stochastic algorithm was
developed that enables the simulation of PBSs in rather complex spatial domains.
Here, it was utilized for the simulation of a crystallization process in a fluidized bed
crystallizer. To this end, two codes, one with deterministic finite element methods for
the CFD equations and one with stochastic methods for the crystals, were coupled.
Furthermore, the stochastic code was extended by all features that are due to the
transport of the crystals in the three-dimensional domain.
Future work concerning the numerical simulations comprises algorithmic and
modeling issues. From the algorithmic point of view, the code with the SPS methods should be MPI parallelized (the finite element CFD code is already) such that
simulations of PBSs can be performed on clusters of computers to enhance the efficiency. From the point of view of the model, different agglomeration kernels and
sedimentation models should be implemented and the breakage of crystals should be
included. In addition, the SPS method should be extended to multi-variate crystals.
This method provides a natural framework for the simulation of crystals that are
modeled in a more complex way than with one internal coordinate.
The computational bottleneck of the expensive computation of agglomeration
terms has been overcome by the exploitation of fast Fourier transformation (FFT) for
the evaluation of convolution sums and tensor trains (TT) for the representation and
arithmetic of multi-variate density distributions. In a sense, numerical simulations
have become feasible for a much higher resolution than experimental results are
available. Future work might include the identification of models that include multivariate agglomeration, e.g., the determination of physically relevant agglomeration
kernels. Another point of interest would be the extension of tensor trains to the entire
population balance model, for example their adaptation to the breakage term. Future
work could also focus on a parallelization of the introduced algorithms.
The agglomeration phenomenon in the fluidized bed crystallizer clearly shows
the potential of such a device for a selective product removal, which is an important
aspect for the post-processing of solids under industrial settings. As agglomeration
R. Ahrens et al.
Fig. 17 and indicates an under-prediction over the entire time period, similar to A
(3,3)
in the bottom right. The error does not exceed 0.6% and 2.3%, respectively. Even
though there is an error in the prediction of higher order moments, they are predicted
with a very good accuracy given the grid with N = 512
2 degrees of freedom.
4 Summary and Outlook
The model experiment in the HCT crystallizer led to interesting and new results
concerning the particle fluid interaction. Crystal size distributions change from the
inlet to the outlet, already without growth, as crystals of varying size have a different
residence time in the HCT. This newly found effect can be used to change the width
of the CSD; or can be used under growth conditions to keep the typically observed
broadening of the CSD very low. Some first insights on the shape dependence of this
effect are encouraging to investigate this topic in more detail in our future work.
For the numerical simulation of PBSs, a deterministic-stochastic algorithm was
developed that enables the simulation of PBSs in rather complex spatial domains.
Here, it was utilized for the simulation of a crystallization process in a fluidized bed
crystallizer. To this end, two codes, one with deterministic finite element methods for
the CFD equations and one with stochastic methods for the crystals, were coupled.
Furthermore, the stochastic code was extended by all features that are due to the
transport of the crystals in the three-dimensional domain.
Future work concerning the numerical simulations comprises algorithmic and
modeling issues. From the algorithmic point of view, the code with the SPS methods should be MPI parallelized (the finite element CFD code is already) such that
simulations of PBSs can be performed on clusters of computers to enhance the efficiency. From the point of view of the model, different agglomeration kernels and
sedimentation models should be implemented and the breakage of crystals should be
included. In addition, the SPS method should be extended to multi-variate crystals.
This method provides a natural framework for the simulation of crystals that are
modeled in a more complex way than with one internal coordinate.
The computational bottleneck of the expensive computation of agglomeration
terms has been overcome by the exploitation of fast Fourier transformation (FFT) for
the evaluation of convolution sums and tensor trains (TT) for the representation and
arithmetic of multi-variate density distributions. In a sense, numerical simulations
have become feasible for a much higher resolution than experimental results are
available. Future work might include the identification of models that include multivariate agglomeration, e.g., the determination of physically relevant agglomeration
kernels. Another point of interest would be the extension of tensor trains to the entire
population balance model, for example their adaptation to the breakage term. Future
work could also focus on a parallelization of the introduced algorithms.
The agglomeration phenomenon in the fluidized bed crystallizer clearly shows
the potential of such a device for a selective product removal, which is an important
aspect for the post-processing of solids under industrial settings. As agglomeration