17 A Framework for Dynamic Simulation of Interconnected …
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To enable the calculation of complex process structures with recycle streams, partitioning and tearing algorithms are introduced at the stage of topological analysis
(Sect. 3.2). The main goal of this step is to prepare the process structure for the
sequential calculation of units. To do this, all present recycles must be initialized with
some values, and units within the closed circuits must be associated with partitions.
Partitions are such groups of units for which there are no recycle streams between
them. Depending on the type of partition, one of the branches of the algorithm is executed (Fig. 5). In the simplest case, when all units within the partition are connected
sequentially, the left part is chosen: units are calculated one by one on the whole
simulation time interval [t START ; t END ]. For partitions with recycles, a more complex
algorithm is applied: all units are calculated iteratively until the convergence in tear
streams is reached. To improve the performance of the simulation system in this
case, dynamic calculations are based on the waveform relaxation method (Sect. 3.3),
supplemented by convergence (Sect. 3.4) and extrapolation (Sect. 3.5) methods.
3.1 Sequential-Modular Approach
There exist two commonly used calculation approaches applied to flowsheet simulation [38, 51, 14]: simultaneous (equation-oriented) [24] and sequential-modular [16].
The first one assumes combining all mathematical models of units, as well as the
topological structure of the flowsheet into a single system of equations and its further simultaneous solution using a suitable integration method (Fig. 6, bottom). The
homogenization of all equations and considering the information about the topology
result in a good convergence rate, especially for flowsheets with complex structures
and, therefore, in relatively fast computations [51]. This approach is usually applied
in flowsheeting software developed for fluid processes.
However, the processes occurring in devices in the area of solid-phase process
technology are usually described by a variety of heterogeneous mathematical models. Their numerical solution may require equation solvers of various types [38].
Therefore, it may be difficult or even impossible to formulate a single generalized
system of equations for the entire flowsheet and there may be no suitable solver for it.
To overcome this problem, the sequential-modular approach (SMA) can be applied.
Here, each unit is treated as an individual entity, and can utilize its own equation
solver, calculation algorithm and time step [16, 22, 24]. In this case, the main task
of the modelling system is transferring data between units through material streams
according to the flowsheet structure and ensuring the convergence of the entire solution [53]. This method leads to relative independence between units and provides
high flexibility of the simulation system itself. The main advantages of this approach
are conceptual simplicity, correspondence to the physical structure of processes,
possibility to use different numerical methods to calculate models, greater flexibility
during the development of units and simple initialization procedure. Moreover, this
approach greatly simplifies the development of new models, which is one of the key
591
To enable the calculation of complex process structures with recycle streams, partitioning and tearing algorithms are introduced at the stage of topological analysis
(Sect. 3.2). The main goal of this step is to prepare the process structure for the
sequential calculation of units. To do this, all present recycles must be initialized with
some values, and units within the closed circuits must be associated with partitions.
Partitions are such groups of units for which there are no recycle streams between
them. Depending on the type of partition, one of the branches of the algorithm is executed (Fig. 5). In the simplest case, when all units within the partition are connected
sequentially, the left part is chosen: units are calculated one by one on the whole
simulation time interval [t START ; t END ]. For partitions with recycles, a more complex
algorithm is applied: all units are calculated iteratively until the convergence in tear
streams is reached. To improve the performance of the simulation system in this
case, dynamic calculations are based on the waveform relaxation method (Sect. 3.3),
supplemented by convergence (Sect. 3.4) and extrapolation (Sect. 3.5) methods.
3.1 Sequential-Modular Approach
There exist two commonly used calculation approaches applied to flowsheet simulation [38, 51, 14]: simultaneous (equation-oriented) [24] and sequential-modular [16].
The first one assumes combining all mathematical models of units, as well as the
topological structure of the flowsheet into a single system of equations and its further simultaneous solution using a suitable integration method (Fig. 6, bottom). The
homogenization of all equations and considering the information about the topology
result in a good convergence rate, especially for flowsheets with complex structures
and, therefore, in relatively fast computations [51]. This approach is usually applied
in flowsheeting software developed for fluid processes.
However, the processes occurring in devices in the area of solid-phase process
technology are usually described by a variety of heterogeneous mathematical models. Their numerical solution may require equation solvers of various types [38].
Therefore, it may be difficult or even impossible to formulate a single generalized
system of equations for the entire flowsheet and there may be no suitable solver for it.
To overcome this problem, the sequential-modular approach (SMA) can be applied.
Here, each unit is treated as an individual entity, and can utilize its own equation
solver, calculation algorithm and time step [16, 22, 24]. In this case, the main task
of the modelling system is transferring data between units through material streams
according to the flowsheet structure and ensuring the convergence of the entire solution [53]. This method leads to relative independence between units and provides
high flexibility of the simulation system itself. The main advantages of this approach
are conceptual simplicity, correspondence to the physical structure of processes,
possibility to use different numerical methods to calculate models, greater flexibility
during the development of units and simple initialization procedure. Moreover, this
approach greatly simplifies the development of new models, which is one of the key