Solving nonlinear equation systems as they occur in process simulation simultaneously, often fails due to ill-conditioned models or bad initialization. To counteract these issues two methods have been implemented in the web-based platform MOSAICmodeling. The Dulmage-Mendelsohn algorithm forms a block diagonal Jacobian matrix. In addition, the bordered block transformation identifies variables that need to be carefully initialized due to their great influence on the system. Both methods are applied on two process models and their convergence and sensitivity towards initialization is analyzed. Motivation and IntroductionIn process simulation tools for chemical engineering applications, such as Aspen Plus, Aspen Hysis, ChemCAD, gPROMS, and Pro/II, the initialization and solution procedure is still mostly governed by the so-called sequential modular approach. This implies the selection of so-called tear streams, which act as entry points to flowsheets and the reconciliation of recycle streams. Starting with these tear streams, unit operations are solved sequentially and the whole flowsheet is solved repeatedly until the properties of the tear stream remain constant between iterations. Within each unit operation, similar approaches are combined with heuristic rules and algorithms on how to solve each independently, e.g., a sequential tray by tray calculation of a distillation column. These methods typically feature an internal modularization regarding property calculations, thermodynamics, etc., all of which are yet again solved with dedicated algorithms. Some of these process simulation tools have been under development for more than 35 years and major advancements have been achieved regarding the size and complexity of flow sheets, the incorporation of thermodynamic models, numerical simulation, and graphical user interfaces [1].Despite this progress, process simulation of flow sheets with many recycle streams still requires some form of manual intervention, implying a certain level of user interaction to steadily improve starting values for tear streams [2]. Such interaction typically requires detailed engineering knowledge. Mere guessing of starting values of temperatures, pressures, compositions, or flows is of course highly undesirable and should be minimized as far as possible. This becomes especially relevant in view of the increasing size of flowsheets with up to several millions of variables.If one chooses an equation-oriented approach instead of the sequential modular approach, this issue becomes even more imminent. The equation-oriented mode attempts a simultaneous solution of the whole process system, usually with some quasi-Newton-type solver. To ensure convergence, a good initial point close to the desired solution is required. In practice, equation-oriented tools, e.g., gPROMS, use sequential-modular techniques as initialization methods before switching over to the equation-oriented mode, Pantelides et al. [3].In general, the nonlinear models underlying any process simulation in chemical engineering ...

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Modelling and simulating intensified processes can be a rather demanding task, since convergence problems occur frequently and force the numerical algorithm to abort before a solution for the related equation system is found. These systems are mostly large-scaled, they contain a high number of nonlinear terms and their equations are strongly coupled through certain variables. A promising way to eliminate the convergence problems is presented here in the form of two decomposition methods alias Dulmage-Mendelsohn decomposition and bordered block transformation that arise from linear algebra. They divide the overall equation system into lower dimensional subsystems which can be solved separately in sequence. This is said to reduce the numerical costs and improve the condition in most cases. To check these assertions the decomposition methods were applied on the equation system of a reactive distillation column. Subsequently the original and the decomposed versions were iterated by a simultaneous solver. Their performance during iteration concerning condition, numerical costs and convergence were compared and validated by the condition number, the function counts and the number of successful iterations starting from systematically chosen initial values out of the solution environment. Based on the results we discuss use cases for the decomposed and the original versions and close this work by presenting some ideas about how the efficiency of the decomposition methods could be further improved.

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