Complex domain distillation calculations

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The phrase 'complex column' is quite widely used in the literature on distillation and usually is intended to conjure up images of distillation columns with multiple feeds, sidedraws, pumparounds, and sidestrippers. We prefer to classify these operations as 'complicated columns.' In this paper, the phrase 'complex distillation' refers to distillation calculations carried out in the complex domain. Lucia and Xu ('Global Minima in Root Finding', In Recent Advances in Global Optimization, C.A. Floudas and P.M. Pardalos, Eds., Princeton Univ. Press, 543, 1992) studied the complex domain behavior of various fixed-point methods for solving the Soave-Redlich-Kwong equation of state and show that the bifurcation of a pair of roots into the complex plane coincide with a phase transition. They also show that the dogleg strategy of Powell (1970) can terminate at singular points and that these singular points are saddle points of the complex absolute value function. Based on this observation, they proposed an extension of the dogleg strategy for reliably finding real- or complex-valued roots. Lucia and Taylor ('Complex Iterative Solutions to Process Model Equations?', European Symp. Comp. Aided Process Eng., S387-S394, 1992) had shown that there are some advantages to carrying out equilibrium flash calculations in the complex domain. Although in that exercise, only real-valued solutions were obtained, Lucia et al. ('Process Simulation in the Complex Domain', A.I.Ch.E. J, 39 (3), 461-470, 1993) obtained complex solutions to dew point temperature calculations when activity coefficient models are part of the phase equilibrium equations. Complex solutions had been found by Lucia and Wang ('Complex Domain Process Dynamics', Ind. Eng. Chem Res., 202-208, 1995) for the TV flash of a system whose thermodynamics were modeled using a cubic equation of state. Sridhar and Lucia ('Process Analysis in the Complex Domain', A.I.Ch.E.J., 585-590, 1995) had shown that TP flashes involving cubic equations of state must have real-valued solutions and suggested an eigenvalue-eigenvalue decomposition for moving from a singular point to a solution. The objectives of this study were to see if there are any advantages to carrying out distillation calculations in the complex domain. We also wanted to know if it was possible to find complex solutions to the equations that model multicomponent separation process calculations. Such solutions, if they exist, may have a bearing on the feasibility of a desired separation. The starting point for our work was an existing computer program, written in Fortran 77, for performing multicomponent, multistage separation process calculations using Newton's method. The code was 'complexified', mainly by converting all double precision declarations and function calls to their complex equivalents. The code was used to solve a variety of problems that often are considered difficult with results summarized below. A complex code will converge to a real solution, even from a complex starting point if there is a real solution to be found. Complex distillation calculations take somewhat longer than corresponding calculations that are confined to the real domain when both codes solve the same problem in the same number of iterations. The complex code can solve certain problems far more easily than the real code. Specific examples we cited included a high-pressure reboiled absorber and part of an LNG plant with purity specifications on the bottoms products. For both of these examples the real-code converges only for certain values of the product specifications even though real solutions do exist. The complex-code converged without difficulty to real solutions over a much wider range of specification values. It is suggested that separations calculations at near critical conditions be carried out in the complex domain if calculations in the real domain show little progress after a modest number of iterations.

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Computers and Chemical Engineering