(474b) From Graphical to Optimization-Based Distillation Column Design: A Mccabe-Thiele-Inspired Math Program

The design and optimization of distillation columns have drawn considerable attention in the chemical engineering literature for more than hundred years. In general, there are three types of design methods: (1) graphical methods such as the McCabe-Thiele diagram [1]; (2) short-cut methods such as the Underwood equation [2]; and (3) rigorous tray-by-tray methods [3, 4]. While the graphical methods are straightforward and intuitive, they are not suitable for math-programming-based models. On the other hand, equation-based shortcut methods can be easily implemented in an optimization model, but they are based on various assumptions and simplifications, the most important being the approximation of the vapor-liquid equilibrium (VLE) behavior. Finally, the rigorous methods rely on minimum assumptions and can be formulated as mathematic programming models, but the energy balances rely on detailed calculations of thermodynamic properties (e.g. enthalpy), which can be computationally demanding. Not surprisingly, model accuracy comes at a steep price: complexity and computational intractability.

While graphical methods have some important limitations, such as limited degrees of freedom, they also have two important advantages: (1) they are intuitive and offer significant insights, and (2) “complex thermodynamics” can be readily represented graphically. As a result, they are still used nowadays for preliminary design. The natural question then becomes: can we develop flexible math-programming-based approaches than maintain the advantages of graphical methods? To the best of our knowledge this seemingly straightforward question has not been addressed. Accordingly, the goal of this work is to address it.

First, we propose a mixed integer nonlinear programming (MINLP) model for distillation column design based on the concepts and equations underpinning the McCabe-Thiele method as well as ideas used in rigorous methods. Specifically, the model involves tray-by-tray concentration calculations (as in rigorous models), which are based however on material balances written around different sections of the distillation column (as in the McCabe-Thiee method). Furthermore, similar to the McCabe-Thiele method, the concentration relationship between vapor and liquid is described by a vapor-liquid equilibrium (VLE) line, which can be approximated using piecewise linear functions or polynomial functions. Going beyond the graphical methods, the model employs binary variables to determine the optimal number of trays and optimal feed locations.

The proposed model is extended to account for multicomponent mixtures and complex columns (e.g., multiple feeds and multiple side streams). In that respect, it is a generalization of the McCabe-Thiele method and its representation as a math program. We also show that, in some cases, major assumptions (e.g., equimolar overflow) can be relaxed.

The decision variables of the model include not only column design specifications (e.g. number of trays and diameter) and operating conditions (e.g. reflux ratio and heat duty), but also flowrate and composition of streams entering (e.g., feed) and leaving (e.g., top and bottom) the column. Therefore, the proposed model can be readily used as a sub-model for optimization-based process synthesis. Finally, the applicability of our model is demonstrated through a cryogenic air separation example.

Reference

[1] W. L. McCabe and E. Thiele, "Graphical design of fractionating columns," Industrial & Engineering Chemistry, vol. 17, pp. 605-611, 1925.

[2] A. Underwood, "Fractional distillation of multicomponent mixtures," Industrial & Engineering Chemistry, vol. 41, pp. 2844-2847, 1949.

[3] J. Viswanathan and I. E. Grossmann, "Optimal feed locations and number of trays for distillation columns with multiple feeds," Industrial & engineering chemistry research, vol. 32, pp. 2942-2949, 1993.

[4] H. Yeomans and I. E. Grossmann, "Optimal design of complex distillation columns using rigorous tray-by-tray disjunctive programming models," Industrial & engineering chemistry research, vol. 39, pp. 4326-4335, 2000.