## Reduced order modeling of distillation systems

Master Thesis

2007

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University of Cape Town

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The concept of distillation separation feasibility is investigated using reduced-order models. Three different models of nonequilibrium rate-based packed distillation columns are developed, each with progressive levels of complexity. The final model is the most complex, and is based on the Maxwell-Stefan theory of mass transfer. The first and second models are used as building blocks in the approach to the final model, as various simplifying assumptions are systematically relaxed. The models are all developed using orthogonal collocation. The order reduction properties of collocation are well documented. A low order model is desirable as the subsequent generation of data required for assessing the separation feasibility is fast. The first model is the simplest as constant molar overflow is assumed. This assumption is relaxed in the subsequent models. The second and third models differ in their respective mass and energy transfer. The second model uses a constant bulk phase approximation for an overall gas phase transfer coefficient. The third model uses rigorous Maxwell-Stefan mass transfer coefficients, which vary throughout the column. In all models, the bootstrap equation for the energy balance across the two-phase film is used after the appropriate modifications are made based on the system assumptions. Starting point solutions and minimum height and flows analysis are presented for all models. The first model is used to develop an azeotropic methodology for identifying and characterizing pinches. Different numerical techniques are also compared, and the accuracy of orthogonal collocation is verified. Ternary and pseudo McCabe-Thiele diagrams are used to represent the result$ for the multicomponent models 2 and 3. The results for models 2 and 3 are similar. This is expected as they differ only in the mass and heat transfer definitions. An argument is made for a specific definition of an objective function for models 2 and 3, which is subsequently used to generate separation surfaces. This function is defined such that there will always be a solution and for this reason is deemed superior to any alternatives. Feasible regions are identified using a grid projection of the relevant sections of the separation surfaces. The data set contained within the feasible region will be used in an optimizer in future work. In general, this work involves an understanding and application of the collocation mathematics to distillation systems. A further understanding of distillation systems, the associated mathematics and degrees of freedom is essential. A large section of this work is devoted to explaining and manipulating the available degrees of freedom, such that the desired end result of a feasible region for a specific separation can be obtained. Other complicating factors include the use of the collocation boundary conditions, and the relationship between these and the overall degrees of freedom for the system. In the literature, collocation is largely applied to staged columns. The resulting feed stage discontinuities are smoothed out using various interpolation routines. Both of these approaches are incorrect. It is shown that the use of collocation in staged columns is fundamentally flawed due to the underlying theory of staged distillation and the implications of collocation assumptions. Further, the feed discontinuities present in all the results are intrinsic features of the system and should be preserved. It is further concluded that Models 2 and 3 were correct in comparison with each other. Finally it was shown that the separation feasibility was successfully determined using the optimal objective function. This success was based on the accuracy and order reduction achieved through the use of collocation. Further work will involve optimizing the data found in the feasible region using Non-Linear Programming.

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Welz, M. 2007. Reduced order modeling of distillation systems. University of Cape Town.