[Show abstract][Hide abstract] ABSTRACT: First principle modeling of chemical processes very often leads to a mixed system of partial differential equations (PDEs) and differential algebraic equations (DAEs) which must be preprocessed for use in standard DAE numerical simulation or optimization tools. This contribution presents the symbolic preprocessing tool SyPProT developed for the simulation environment Diva in order to apply DAE numerics also to PDEs. The method-of-lines (MOL) approach for the required PDE discretization is implemented in SyPProT by configurable finite-difference and finite-volume schemes. The model as well as the MOL parameters are represented in a tailor-made Mathematica data structure (MDS). The preprocessing of a PDE model is illustrated by the example of a circulation-loop-reactor (CLR).
Mathematics and Computers in Simulation 05/2001; 56(2):157–170. DOI:10.1016/S0378-4754(01)00287-7 · 0.95 Impact Factor
[Show abstract][Hide abstract] ABSTRACT: This chapter discusses the symbolic discretization of population models for process simulation. In this connection, it describes the architecture of the symbolic preprocessing for the simulation environment. In addition, population balance equations are also discussed in detail in the chapter. The population balance approach characterizes particles of the dispersed phase by internal coordinates such as the particle length in crystallization processes. This approach allows integration of submodels for the microscopic phenomena on a macroscopic scale into a model for the overall process unit. Because of the contained integrals and the hyperbolic nature of the population balance equations, standardized discretization schemes can only be applied on fine grids for satisfactory simulation results. Therefore, advanced discretization methods have been investigated with the objective to reduce the computational effort—essentially nonoscillatory schemes and the robust upwind scheme. The main advantage of both methods is the maintenance of sharp profiles during dynamic simulations.
Computer Aided Chemical Engineering 12/2000; 8:547-552. DOI:10.1016/S1570-7946(00)80093-0
[Show abstract][Hide abstract] ABSTRACT: In chemical engineering, detailed process modeling, simulation, nonlinear analysis, and optimization of single process units as well as integrated production plants are issues of growing importance. As a software tool addressing these issues, the simulation environment DIVA is introduced. DIVA comprises tools for process modeling, preprocessing and code generation of simulation models. Furthermore, its simulation kernel contains several advanced methods for simulation, parameter continuation, and dynamic optimization which can be applied to the same process model. These numerical methods require a model description in form of differential algebraic equations (DAE). However, many models of chemical processes derived from first principles lead to partial differential equations (PDE) for distributed parametersystems, to integro partial differential equations (IPDE) for population balances of dispersed phases, and to DAE for lumped parameter systems. In order to transform the PDE and IPDE models into the required DAE model formulation, DIVA employs the "Method-of-Lines" (MOL) approach for one space coordinate. The wide variety of distributed parameter models of chemical processes requires on one hand conventional discretization methods like, e.g.,finite-difference schemes, and on the other hand more sophisticated methods to obtain reliable results in an acceptable computation time. Another approach of so called high-resolution methods uses adaptive approximation polynomials. High-resolution methods are developed forhyperbolic conservation laws with steep moving fronts. Examples areessentially-non-oscillatory (ENO) schemes or the robust upwindkappa-interpolation scheme. In order to support the user of the simulation environment diva, the symbolic preprocessing tool SYPPROT for MOL discretization is developed by means of the computer-algebra-system MATHEMATICA. This tool provides different discretization schemes to transform PDE, IPDE and the related boundary conditions on 1–dimensional spatialdomains into DAE. Discretization options concern fixed spatial grids and moving gridsbased on an equidistribution principle. The toolbox architecture of SYPPROT allows fast testing of various discretization schemes without redefinition of the model equations. This is regarded as an im portant feature to minimize the overall effort for modeling and simulation. A further characteristic that is key to theeasy exchange of discretization schemes is the separation between mode lequations and MOL parameter definitions by means of the mathematica datastructure (MDS). The resulting overall DAE are written in symbolic form as an inputfile for the DIVA code generator, which automatically generates the DIVA simulation files representing a process unit model in the modellibrary. In the following section 13.2 the architecture of the simulation environment DIVA is presented. This architecture consists of four layers, i.e.the DIVA simulation kernel, the code generator, the symbolic preprocessing tool SYPPROT, and the process modeling tool PROMOT. The MOL discretization of PDE and IPDE as the main preprocessing feature is the focus of Section 13.3. Standard discretizationschemes like finite-difference and finite volume schemes as well as adaptive approaches like high-resolution methods and an equidistribution principle based moving grid method are explained for distributed parameter models with one space coordinate. The application of these discretization methods within the symbolic preprocessing tool SYPPROT follows in Section 13.4, where the PDE and IPDE model representation as well as the implemented MOL discretization capabilities are described.