Collocation methods for distillation design.

In this second paper on collocation methods for distillation design, we address the minimum reflux problem. Existing minimum reflux techniques do not work for nonsharp splits, as they use approximate geometric criteria to simplify modeling of minimum reflux conditions to a set of pinch point calculations for sharp splits. Collocation allows exact calculation of nonsharp minimum reflux and provides the missing link that geometric criteria replace. We examine the behavior of distillation near minimum reflux and get results consistent with prior work. We also show that sensitivity of trace species drops as less sharp splits for the key components are sought. We have found that collocation can be applied to sharp split minimum reflux problems, but we cannot exactly reproduce Underwood's values for constant relative volatility systems due to breakdown of the model near saddle pinches. We discuss the modeling of minimum reflux conditions with collocation and discuss the difficulties we have encountered. We can accurately calculate minimum reflux for nonsharp splits for nonideal systems, even when approaching a sharp split. We overestimate the minimum reflux for the sharp split case, using nonsharp calculations at very low impurities. We encourage discussion of the problems we have encountered. This work has been partially supported by the Engineering Design Research Center, a NSF Engineering Research Center. Introduction Several researchers have explored and developed collocation for distillation column modeling. In the first paper in this series, we presented a collocation model which expands on prior models, addressing the problems specific to steady-state, continuous columns. In this paper we discuss use of the collocation method for minimum reflux determinations. Minimum reflux determination is useful during the early stages of distillation design and for approximate design methods. Several methods of varying complexity and accuracy exist for determining minimum reflux. Underwood's [1945,1946,1948] method is very quick but is accurate only for ideal systems and only for sharp splits. As described in the motivation section of paper I in this series, Doherty and coworkers,[1990,1991] have done a great deal of work on distillation design and minimum reflux. Their methods rely on approximate geometrical conditions to remove the need for infinite column section calculations. The development of their techniques demonstrates how nonsharp splits should be handled, but their minimum reflux algorithms were developed only for sharp splits. For sharp splits, Koehler et al. [1991] developed a minimum reflux determination method that uses a reversible distillation model. They have shown that for nonideal and azeotropic systems their method provides accurate results, based on comparisons to rigorous column models with a large number of trays. However, their method requires the nonkey components to be nondistributing and is based on the same geometric considerations in Doherty and coworkers. We present an approach for minimum reflux calculations that does not require approximate geometric constraints. We discuss some interesting insights as one moves from nonsharp to sharp splits. We also discuss computational difficulties we found in realizing the theory with collocation models. By presenting both the strong and weak points of collocation for distillation modeling we hope to initiate discussions leading to better methods. Pinch Points When a distillation section has a very large number of trays in it, a region of constant composition occurs. The liquid and vapor passing each other approach being in equilibrium with each other, i.e., they approach being pinched. Minimum reflux occurs when at least one column section is pinched. When modeling minimum reflux conditions, "pinch point" calculations can sometimes remove the need for tray-by-tray models of column sections. A pinch point is calculated by requiring that the liquid and vapor passing each other are in equilibrium and in mass and heat balance with the other end of the distillation section. Therefore, if one half of a column has a large number of trays and is pinched adjacent to the feed tray, the whole section can be modeled with only a pinch point calculation. This simplification can be exploited for calculation of minimum reflux. Nonsharp Splits A separation where all the components are distributed in some amount to both products is nonsharp. In the case of a nonsharp split, there is a feasible intersection region where the tray by tray trajectories from the top and bottom product can intersect [Wahnschafft 1992]. Figure 1 shows a ternary diagram for a nonsharp minimum reflux problem. The regions between the infinite reflux (distillation) and minimum reflux (pinch point) curves for both the top and bottom products are the only feasible regions where the trajectories can go. The darkened region that shows the intersection of the feasible space for the top and bottom trajectories shows the only feasible region for the feed tray. At minimum reflux, the bottom section will pinch in this region. Figure 2 shows a blowup of this region. Any trajectory from the bottom of the column will terminate on the pinch point curve as the number of trays increases. As the trajectory from the bottom of the column moves to the right, the reboil ratio (R) increases. However,