598
V. Skorych et al.
by the one-dimensional population balance equation given in [47]. Verification of
the simulation results from Dyssol showed very good agreement (Fig. 8, right) with
the results obtained in SolidSim-Dynamics [16], which was used as the reference
system.
Various test studies were performed to analyse the influence of the waveform
relaxation method as applied to dynamic flowsheet simulation, as well as to estimate
the contribution of different convergence and extrapolation methods. In one of them,
the granulation process, whose flowsheet is shown in Fig. 9, was used as a reference.
Its transient behaviour in an unstable operation mode, caused by fine milling [47],
can be seen by the changes in the mass flow of the material leaving the granulator
and by the changes in Sauter diameter of bed material inside the granulator as shown
in Fig. 10a.
Figure 10b represents the sizes of the time windows, as well as the number of
iterations required to achieve convergence at each moment of the process time. It can
be seen that with monotonic changes in the process parameters the window size of the
waveform relaxation method dynamically increases, while the number of iterations
remains low. When significant changes in the process parameters occur at a certain
time interval, the number of iterations increases, and the size of the time window
decreases accordingly.
Convergence methods are involved in each iteration of the waveform relaxation
method, so they significantly affect the performance of the calculations. For the
granulation process, this effect can be seen in Fig. 10c. It shows the relationship
between the model time and the cumulative number of iterations required to achieve
convergence using different convergence methods and a fixed time window size
(10 s). The methods of Wegstein and Steffensen require significantly less iterations
to achieve convergence for the process under consideration compared to the basic
direct substitution approach. Thus, it can be concluded that both advanced methods
can provide higher convergence rates and potentially significantly shorter simulation
times, despite their greater computational complexity.
To analyse the effectiveness of extrapolation methods that are used to initialize the
first iteration at each time window, WRM-frames were also fixed at 10 s. Figure 10d
shows the dependence of the cumulative number of necessary iterations at each
moment of simulation time for three applied extrapolation algorithms. The most
basic extrapolation using the nearest-neighbour method leads to a greater number of
iterations compared to linear and cubic spline algorithms, which provide almost the
same convergence rate in this case.
It should be noted, however, that the final choice of convergence and extrapolation
methods in the general case should depend on the dynamics of the respective process.
4 Built-in Equation Solvers
In accordance with the applied sequential-modular approach, each model in the flowsheet can use its own equation solver. Therefore, models may have their own built-in
mechanisms for calculation, depending on the type of algorithms used. However,
V. Skorych et al.
by the one-dimensional population balance equation given in [47]. Verification of
the simulation results from Dyssol showed very good agreement (Fig. 8, right) with
the results obtained in SolidSim-Dynamics [16], which was used as the reference
system.
Various test studies were performed to analyse the influence of the waveform
relaxation method as applied to dynamic flowsheet simulation, as well as to estimate
the contribution of different convergence and extrapolation methods. In one of them,
the granulation process, whose flowsheet is shown in Fig. 9, was used as a reference.
Its transient behaviour in an unstable operation mode, caused by fine milling [47],
can be seen by the changes in the mass flow of the material leaving the granulator
and by the changes in Sauter diameter of bed material inside the granulator as shown
in Fig. 10a.
Figure 10b represents the sizes of the time windows, as well as the number of
iterations required to achieve convergence at each moment of the process time. It can
be seen that with monotonic changes in the process parameters the window size of the
waveform relaxation method dynamically increases, while the number of iterations
remains low. When significant changes in the process parameters occur at a certain
time interval, the number of iterations increases, and the size of the time window
decreases accordingly.
Convergence methods are involved in each iteration of the waveform relaxation
method, so they significantly affect the performance of the calculations. For the
granulation process, this effect can be seen in Fig. 10c. It shows the relationship
between the model time and the cumulative number of iterations required to achieve
convergence using different convergence methods and a fixed time window size
(10 s). The methods of Wegstein and Steffensen require significantly less iterations
to achieve convergence for the process under consideration compared to the basic
direct substitution approach. Thus, it can be concluded that both advanced methods
can provide higher convergence rates and potentially significantly shorter simulation
times, despite their greater computational complexity.
To analyse the effectiveness of extrapolation methods that are used to initialize the
first iteration at each time window, WRM-frames were also fixed at 10 s. Figure 10d
shows the dependence of the cumulative number of necessary iterations at each
moment of simulation time for three applied extrapolation algorithms. The most
basic extrapolation using the nearest-neighbour method leads to a greater number of
iterations compared to linear and cubic spline algorithms, which provide almost the
same convergence rate in this case.
It should be noted, however, that the final choice of convergence and extrapolation
methods in the general case should depend on the dynamics of the respective process.
4 Built-in Equation Solvers
In accordance with the applied sequential-modular approach, each model in the flowsheet can use its own equation solver. Therefore, models may have their own built-in
mechanisms for calculation, depending on the type of algorithms used. However,