Real option theory from finance to batch distillation

Abstract

Batch distillation processes have gained renewed interest because of the recent development in small-scale industries producing high-value-added, low-volume specialty chemicals. The flexibility and unsteady state nature of batch distillation constitute a challenge for the designer. A particularly difficult problem is the optimal control problem involving open loop solution for the reflux ratio profile. This is because of the complexity of the formulation and the large computational effort associated to its solution. The mathematical and numerical complexities of the optimal control problem get worse when uncertainty is present in the formulation. In this work, by applying the optimality conditions from the real option theory based on the Ito's Lemma [Investment under uncertainity (1994); Memoirs Am. Math. Soc. 4 (1951) 1; Appl. Math. Opt. 4 (1974) 374], the mathematical tools needed to solve optimal control problems in batch distillation columns when uncertainties in the state variables are present have been developed. Furthermore, the coupled maximum principle and NLP approach developed by Diwekar [Am. Inst. Chem. Eng. J. 38 (1992) 1551] has been extended for solving the optimal control problem in the uncertain case. This new algorithm has been implemented in the MultiBatchDS batch distillation process simulator. Finally, a numerical case-study is presented to show the scope and application of the proposed approach.