Batch Distillation: The Forward and Inverse Problems O. Samimi Abianeh, C. P. Chen, and Ramon L. Cerro* Department of Chemical and Materials Engineering, University of Alabama in Huntsville, Huntsville, Alabama 35899, United States ABSTRACT: There are two basic theoretical and computational problems associated with batch distillation. The forward or direct problem consists on generating the distillation curve of a given mixture. This is an old problem, but new results are presented here to relate mathematical properties of the distillation curve with the physicochemical properties of the molecular species present in the mixture. The inverse problem consists of, given a distillation curve, finding a surrogate mixture that would accurately represent experimental data. There is more than one solution to the reverse problem because there are theoretically an infinite number of mixtures that will present very similar experimental distillation curves. The method developed in this paper requires the same number of molecular species in the surrogate mixture as the points of the distillation curve that will be matched precisely. The choice of exact points to match on the distillation curve allows to conform a square system of equations where the number of equations is equal to the number of unknowns. Other points of the distillation curve are satisfied within a prescribed small error tolerance. A surrogate mixture for gasoline was developed as an example. 1. INTRODUCTION Although distillation has been known for thousands of years, 1 the scientific principles of batch distillation can be traced back to the work of Rayleigh in the early 1900s. 2 Industry all but substituted the practice of using batch distillation by more energy and materially efficient continuous distillation systems attached to packing or tray counter-current separation columns. 3 There are two main aspects of batch distillation theory that are still subject of active current research: theoretical generation of accurate distillation curves for complex mixtures used to match existing experimental data 4,5 and the definition of surrogate mixtures to simulate the distillation curve of an unknown or very complex mixture. 4,5 Surrogate mixtures are necessary when complex reaction systems take place in the presence of a large number of molecular species, such as in combustion chambers, 14,15 or when a long and complex mass transfer problem must be simulated computationally with a large number of components as is the case of simulation of oil spills in the open sea. Experimental determination of distillation curves is closely related to petroleum production and refining where natural occurring mixtures can have tens of thousands of components. Theoretical prediction of distillation curves are used to predict the range of temperatures that will maximize the molar fraction of a particular component in a distillate cut. Theoretical and experimental characteristics of distillation curves are discussed in section 3. Surrogate mixtures are designed with the purpose of numerical simulation of complex mixtures using a small number of components. 6,7 Design of surrogate mixtures is usually focused on the ability of the mixture to reproduce a particular property of the original mixture. 4 In our research, surrogate mixtures are designed to reproduce distillation curves of unknown petroleum mixtures. Other thermodynamic properties like thermal conductivity, heat capacity, and latent heat of evaporation of surrogate are strong functions of the choice of surrogate components, especially if they are in same chemical family like linear or branched hydrocarbons and are out of the scope of this research. A novel mathematical methodology for finding the initial composition in the surrogate mixture is developed in section 4. A surrogate mixture for gasoline fuel was developed using our method. We designed the original mixture to include seven molecular species. Decreasing the points of the distillation curve that are identically satisfied, the number of molecular species in the mixture can be shortened while still showing a close correspondence with the original distillation curve. Finally, we developed a three-component mixture to satisfactorily model the original gasoline distillation curve. 2. MATERIAL BALANCES We analyze a simple batch distillation process without intermediate rectification. The basic distillation system consists of a heated vessel and a distillate recovery condenser. Assuming no chemical reactions, the basic equations for overall molar balance and balance of one component are ∫ ∮ ∫ ∮ + · = + · = = α α t cV c nA t c V c nA A N d d d v d 0 (a) d d d v d 0; 1, 2 , ..., (b) V A V A A A (1) where c is the total concentration in mol/volume, and c A is the concentration of individual components. The control volume as well as the exit surface are schematically shown in Figure 1. Figure 1 is similar to the figure shown in the textbook by Doherty and Malone, 3 and concurring with these authors, we assume that the vapor phase is in equilibrium with the well- mixed liquid remaining in the vessel. We will consistently define Received: March 16, 2012 Revised: August 20, 2012 Accepted: September 4, 2012 Published: September 4, 2012 Article pubs.acs.org/IECR © 2012 American Chemical Society 12435 dx.doi.org/10.1021/ie300710s | Ind. Eng. Chem. Res. 2012, 51, 12435-12448