European Symposium on Computer Aided Process Engineering- 13 A. Kraslawski and I. Turunen (Editors) © 2003 Elsevier Science B.V. All rights reserved.
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Generalized Modular Framework for the Representation of Petlyuk Distillation Columns p. Proios and E.N. Pistikopoulos* Centre for Process Systems Engineering, Department of Chemical Engineering, Imperial College, London SW7 2BY, U.K.
Abstract In this paper the Generalized Modular Framework (Papalexandri and Pistikopoulos, 1996) is used for the representation of the Petlyuk (Fully Thermally Coupled) column. The GMF Petlyuk representation, which avoids the use of common simplifying assumptions while keeping the problem size small, is validated for a ternary separation, by a direct comparison of its results to those obtained by a rigorous distillation model.
1. Introduction The Petlyuk column (Petlyuk et al, 1965) is an energy efficient distillation system which, along with its thermodynamically equivalent Dividing Wall Column (Wright, 1949), has been reported of being able to lead to energy savings of even up to 40% when compared to conventional simple column arrangements (Glinos and Malone, 1988 and Schultz et a/., 2002). The importance of this complex distillation column has compelled the development of numerous methods for its design and analysis. These methods can be classified into two main categories, namely, those using simplified (shortcut) models and those using rigorous (detailed) models. Petlyuk et a/. (1965) used shortcut calculations for the minimum reflux based on constant relative volatilities and internal flowrates. Cerda and Westerberg (1981) developed a shortcut model for the minimum reflux assuming sharp separations for the Petlyuk column. In Fidkowski and Krolikowski (1986) the Petlyuk column was studied for ternary mixtures and sharp calculations through a shortcut model for the minimum vapour flowrate based on the Underwood method. Glinos and Malone (1988) and Nikolaides and Malone (1988) designed the Petlyuk column using shortcut calculations under constant relative volatilities and equimolar flowrates. Carlberg and Westerberg (1989) and Triantafyllou and Smith (1992) used a three-simple-column approximation of the Petlyuk column. The former proposed a shortcut model for minimum vapour flowrate for nonsharp separations whilst the latter based their design on the Fenske-Underwood-Gilliland shortcut techniques. Halvorsen and Skogestad (1997) used a dynamic shortcut model based on assumptions of equimalr flowrates and constant relative volatilities for their Petlyuk/Dividing Wall Column model. Agrawal and Fidkowski (1998) used Underwood's method for their Petlyuk design and Fidkowski and Agrawal (2001) proposed a shortcut method for the separation of quarternary and higher mixtures in To whom correspondence should be addressed. Tel.: (44) (0) 20 7594 6620, Fax: (44) (0) 20 7594 6606, E-mail:
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264 Petlyuk arrangements extending the Fidkowski and Krolikowski (1986) method. Shah and Kokossis (2001) designed the Petlyuk columns in their framework based on the Triantafyllou and Smith (1992) shortcut procedure. Finally, Amminudin et ah, (2001) proposed a shortcut method for the design of Petlyuk columns based on the equilibrium stage composition concept. It must be noted that the above methods provide fast and simple ways of designsing and analysing the performance of the Petlyuk column. However, the fact that they are based on simplifying assumptions can place a limitation on their accuracy and applicability, notably for the cases where these assumptions do not hold. This limitation can be overcome through the use of rigorous methods, not relying on simplifying assumptions. Chavez et al. (1986) examined the multiple steady states of the Petlyuk column through a detailed tray-by-tray model under fixed design, which was solved with a differential arc-length homotopy continuation method. Dtinnebier and Pantelides (1999) designed Petlyuk columns using a detailed tray-bytray distillation model based primarily on the rigorous MINLP distillation model of Viswanathan and Grossmann (1990). Also based on the latter, Yeomans and Grossmann (2000) proposed a disjunctive programming model for the design of distillation columns including Petlyuk arrangements. These methods are based on detailed and accurate models with general applicability. However, they do generate considerably larger nonlinear programming problems which lead to an increase of the computational effort. The scope of the presented work is twofold: a) to provide a valid method for representing and analyzing the performance of the Petlyuk column with respect to its energy efficiency potential at a conceptual level and b) based on this, to put the foundations for the extension of the method to the synthesis level, that is, for the generation and evaluation of all column arrangements for this separation problem, involving simple and also (partially) thermally coupled columns. These will be realized in an integrated way, from a process synthesis point of view, and without generating a large optimization problem (as the rigorous methods), while avoiding the common limiting assumptions, characteristic of the shortcut methods.
2. The Generalized Modular Framework In this work the Petlyuk column is represented through the Generalized Modular Framework (GMF) (Papalexandri and Pistikopoulos, 1996), which is an aggregation framework for process synthesis/representation. The GMF is based on the fact that since a large number of process operations are characterized by mass and heat transfer phenomena (for instance the mass and heat exchange between liquid and vapour streams in distillation), using a generalized method for capturing those, the process operations in question can be systematically represented in a compact and unified way. The GMF through its generalized mass and heat exchange modelling, aims towards that direction. In brief, it can be stated that the GMF is a superstructure optimization method and, alike most of the methods belonging to this class, consists of a Structural Model, responsible for the generation of the (structural) process alternatives and a Physical Model, responsible for the evaluation of the latters' performance/optimality.
265 Cooler
Heater
6
-D
a—o Pure Heat Module
Pure Heat Module
Figure 1: GMF Building Blocks (Ismail et al 2001)' The Structural model consists of: (i) the GMF building blocks and (ii) their interconnection principles. The GMF building blocks (Figure 1) are representations of higher levels of abstraction and lower dimensionality where mass/heat or pure heat exchange take place. The existence of the building blocks is denoted mathematically through the use of binary (0-1) variables. The Interconnection Principles define the way the various building blocks should be connected to each other for the generation of physically meaningful alternative units and their resulting flowsheets. The mathematical translation of these principles is realised through a set of mixed and pure integer constraints, which define the backbone of the GMF structural model. The GMF Physical Model is employed for the representation of the underlying physical phenomena of the generated structures. Each building block is accompanied by its physical model, which is based on fundamental (and thus general) mass and heat exchange principles at the blocks' boundaries, consisting of mass and energy balances, molar fraction summation corrections and appropriate Phase Defining and Driving Force Constraints arranging the mass and heat transfer. The complete GMF mathematical model, as a combination of the structural and physical models is a Mixed Integer Nonlinear Programming problem (MINLP), and can be found in detail in Papalexandri and Pistikopoulos (1996) and Ismail et al. (2001). For the GMF representation of the Petlyuk column a minimum number of 6 mass/heat and 2 pure heat modules are employed (Figure 2). The connectivities of the building blocks are appropriately arranged so that the complex structure of the Petlyuk column is obtained. This is done by fixing the corresponding binary variables to 0 or to 7 for the respective nonexistence or existence of building blocks and their interconnections. For the Petlyuk column representation, each mass/heat module represents a column section (aggregation of trays), where a separation task takes place. The pure heat exchange modules represent the condenser (cooler) and the reboiler (heater) of the Petlyuk column. It must be noted that since a tray-by-tray model is not employed, the equilibrium constraints are being replaced by Driving Force Constraints at the two ends of each one of the six mass/heat modules, according to the type of contact (countercurrent for distillation). These constraints along with the Phase Defining Constraints and the conservation law constraints ensure mass and heat transfer feasibility and define the distribution of the components in the existing building blocks. However, the main motivation for representing the Petlyuk column through the GMF lies on the latter's main representational advantages, which for the examined case are summarised below: (i) the GMF physical model captures efficiently the underlying
266
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Figure 2: Petlyuk Column (Conventional and GMF representation). mass/heat transfer phenomena, since it is not based on simplifying and limiting assumptions such as sharp splits, equimolar flowrates, constant volatilities and it does not involve any shortcut calculations, (ii) the GMF physical model can accomodate any thermodynamic model, (iii) the GMF structural and physical models allow the represention of the Petlyuk column in an aggregated way, leading to a smaller and easier to solve optimization problem (iv) the framework can be potentially extended to the evaluation of other (distillation) systems through a superstructure based on the existing six mass/heat modules and by allowing more interconnections. In the following section the above advantages and the framework's validity and representational merit will be demonstrated through a GMF/Petlyuk column case study.
3. Numerical Results - Validation The GMF representation of the Petlyuk column is employed for the separation of the ternary mixture of Benzene, Toluene and 0-xylene. The problem data was taken from Chavez et al (1986) and it involves the separation of a saturated liquid feed of 211.11 mol/s, with molar fractions of Benzene, Toluene and 0-xylene, 0.2, 0.4 and 0.4, respectively, into three product streams with molar fractions of 0.95, 0.9 and 0.95, in the above components. The objective is the minimization of the utility cost. For a fixed (Petlyuk) structure, the corresponding GMF mathematical problem is a nonlinear programming problem (NLP) which was solved in GAMS (Brooke at al, 1992) using the solver C0N0PT2. Due to the inherent stream mixing and splitting terms the problem is nonconvex which is solved only to local optimality. However a systematic procedure has been employed with appropriate initial guesses and bounds for the stream flowrates, temperatures and molar fractions in order to find a local optimal point which represents the potential (energy consumption levels) of the examined Petlyuk column. From the optimization runs for the mixture and feed composition examined, the GMF provided the energy consumption levels (heater duty of 9,026.3 kW) and the operating conditions of the Petlyuk column, using the mass/heat exchange principles of the GMF physical model.
267 However, since the GMF physical model is an aggregated (and, thus, nonconventional) model, the validity of the GMF results for the Petlyuk representation was evaluated by comparing these results quantitatively and qualitatively, to those derived from a conventional tray-by-tray model. For these purposes, the rigorous model of Viswanathan and Grossmann (1990) was used for the minimization of the operating cost, with the problem definition and the column design taken also from Chavez et al. (1986). From the results of the optimization, the two models are found to be in quantitative agreement, since the reboiler heat duty in the rigorous model was 10530 kW, which is very close to that of the GMF heater, indicating that the GMF model predicted correctly the energy consumption of the Petlyuk column. The small divergence between the two is possibly due to the fact that in the GMF the bottoms product stream is removed before the heater (with less liquid entering it, Figure 2). However, such a quantitative agreement needs to be the product of a qualitative agreement (that is, in the components' distribution over the various column sections). Since the GMF does not provide information at the tray level, in order to enable a comparison of the composition and temperature profiles of the two models, the points of the feeds, interconnections and side streams of the GMF representation were placed on the corresponding points (tray locations) of the tray-by-tray model, in a common x-axis. In Figure 3 are shown the profiles of the Toluene composition and of the temperature in the main column of the Petlyuk arrangement. From these it is apparent that the two models are also in qualitative agreement in the main column (similar results were derived for the prefractionation column, as well). This qualitative agreement shows that the GMF provided insights on the performance of the Petlyuk column, with respect to its energy consumption, based on a sound physical model which is capable of capturing efficiently the mass and heat transfer phenomena of the examined system. Another point of importance is related to the size of the generated optimization problem. Due to the aggregated nature of the GMF representation (where variables and equations are accounted for only at the building blocks' boundaries and not at the tray level, as in the rigorous models), a size reduction of 75% in the number of variables and constraints, respectively, has been noted when using the GMF instead of the trayed model (depicted
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Figure 3: Qualitative Comparison of GMF and Rigorous Models (Petlyuk Column).
268 in Figure 3 with the fewer GMF points), with direct effects in the computational effort. Of course, as it can be observed in Figure 3, the GMF does not provide detailed results and profiles, as the rigorous model does, but this is beyond the aim of the framework, which is not a simulation but a synthesis/representation tool, at a conceptual level.
4. Conclusions As shown, the GMF provides a sound and useful tool for the representation and evaluation of the Petlyuk column and its underlying physical phenomena providing valid information about the energy consumption levels of the fully thermally coupled column. Moreover, the GMF results, which were derived using an aggregated physical model and thus generating a significantly reduced optimization problem, were evaluated for their consistency and validity through their comparison with a well-established rigorous distillation model. Finally, having validated the GMF physical model for the examined system, the complete GMF model, with its physical model, as it was used in the presented work, and with its full structural model (without incorporating a fixed structure but with an adequate number of building blocks and their interconnections to be determined by the optimizer) can now be used for the synthesis problem, i.e. the generation and evaluation of all the alternatives of interest (simple and complex) for the examined separation problem, which is the scope of our current research.
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