A Robust Method To Solve Mass Balances in Reversible Column Sections

AGUIRRE, PÍO; ESPINOSA, JOSÉ

INDUSTRIAL & ENGINEERING CHEMICAL RESEARCH

AMER CHEMICAL SOC

Año: 1996 vol. 35 p. 559 - 572

0888-5885

Reversible composition profiles in azeotropic and reactive distillation were used as pieces of new methods for the prediction of product composition regions and minimum reflux in singleand multiple-feed columns. A simple and robust algorithm to solve the nonlinear equation system corresponding to mass and energy balances in a reversible distillation column section for highly nonideal mixtures is proposed. If a distillate (bottom) is specified, one additional variable has to be set to calculate a cross section in the rectifying (stripping) part. We initially use the concentration of the component to be eliminated in the section as the variable to be fixed. A former direct successive substitution (DSS) procedure used in a previous paper is here analyzed, and the conditions for convergence are discussed. A damped Newton-Raphson method (DNR) is an alternative also presented. The new algorithm involves an improved direct successive substitution (IDSS) outer loop and a bubble point temperature (BUTE) loop included. By assuming the liquid phase composition at the reversible section, solving BUTE, and using geometric considerations, a related “hypothetical” product composition can be calculated. The difference between the real and the “hypothetical” product composition is used to recompute the liquid phase. This method can be considered robust in view that it does not uses derivative calculations that cause numerical drawbacks in cases that maxima, minima, or turning points are reached. The initialization variables are the liquid mole fractions instead of temperatures and component flow rates, resulting in a more general case-independent method. The geometrical characteristics allow that the algorithm can be initialized in any arbitrary region of the concentration simplex. For constant relative volatility (CRV) mixtures, a small number of iterations are needed despite the start point selected. The convergence was found always satisfactory independently of the start point, even when inversions in volatility order of some components take place. Finally, a variant (secant) of imbedding homotopy continuation is proposed to efficiently trace the complete path of the reversible profile, using the IDSS algorithm as the corrector. This combination becomes necessary when multiple disjoint branches of a product pinch point curve must be computed in highly nonideal mixtures. Other approaches to the problem are mentioned and their results compared.-Raphson method (DNR) is an alternative also presented. The new algorithm involves an improved direct successive substitution (IDSS) outer loop and a bubble point temperature (BUTE) loop included. By assuming the liquid phase composition at the reversible section, solving BUTE, and using geometric considerations, a related “hypothetical” product composition can be calculated. The difference between the real and the “hypothetical” product composition is used to recompute the liquid phase. This method can be considered robust in view that it does not uses derivative calculations that cause numerical drawbacks in cases that maxima, minima, or turning points are reached. The initialization variables are the liquid mole fractions instead of temperatures and component flow rates, resulting in a more general case-independent method. The geometrical characteristics allow that the algorithm can be initialized in any arbitrary region of the concentration simplex. For constant relative volatility (CRV) mixtures, a small number of iterations are needed despite the start point selected. The convergence was found always satisfactory independently of the start point, even when inversions in volatility order of some components take place. Finally, a variant (secant) of imbedding homotopy continuation is proposed to efficiently trace the complete path of the reversible profile, using the IDSS algorithm as the corrector. This combination becomes necessary when multiple disjoint branches of a product pinch point curve must be computed in highly nonideal mixtures. Other approaches to the problem are mentioned and their results compared.