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A Short Method to Calculate Residue Curve Maps in Multireactive And ARTICLE pubs.acs.org/IECR A Short Method To Calculate Residue Curve Maps in Multireactive and Multicomponent Systems Marcelino Carrera-Rodríguez,† Juan Gabriel Segovia-Hernandez,*,† and Adrian Bonilla-Petriciolet‡ †Departamento de Ingeniería Química, Division de Ciencias Naturales y Exactas, Universidad de Guanajuato, Campus Guanajuato, Noria Alta S/N, Guanajuato, Guanajuato, 36050, Mexico ‡Departamento de Ingeniería Química, Instituto Tecnologico de Aguascalientes, Avenida Lopez Mateos 1801, Aguascalientes, Aguascalientes, 20256, Mexico ABSTRACT: Reactive residue curve maps (RRCMs) are useful for the design of reactive distillation columns as a tool to establish feasible zones of reaction-separation. However, the calculation of an RRCM usually involves great computational effort due to the nonlinearity of the model equations and its iterative nature for the determination of reactive phase equilibrium. In this study, a simplified method for the generation of RRCMs is presented. This method is based on the application of reaction-invariant composition variables and assumes that the phase equilibrium constants (i.e., the relative volatilities) are independent of the temperature. Specifically, the phase equilibrium constants are calculated using a suitable estimation of the bubble temperature obtained from pure-component boiling temperatures and the reaction-invariant composition of liquid phase. These assumptions avoid iterative phase equilibrium calculations for obtaining a good approximation of RRCMs. Several reactive systems are used to identify the capabilities and limitations of the proposed method. 1. INTRODUCTION numerical difficulties are involved in the modeling and design of ffi Within the broad range of topics that are related to reactive RD systems. These di culties have their origin mainly in the distillation (RD), the design of these processes has received multicomponent nature of the reactive systems, the nonlinearity significant attention from many researchers due to the techno- of the thermodynamic models caused by the presence of simultaneous chemical and physical equilibria, and also by the logical and economical advantages obtained from the simulta- fi neous occurrence of distillation and reaction. Specifically, this type of variables involved in de ning the problem, which are ff fi fi generally composition variables in molar units and extents of technology o ers signi cant bene ts over conventional pro- 5 cesses such as the elimination of a reaction vessel, fewer separa- reaction. In particular, the use of composition variables in molar tion units, high conversion of reactants, the improvement of units is not suitable for modeling reactive systems because these product selectivity, the effective separation of complex mixtures variables do not have the same dimensionality as the number of degrees of freedom given by the Gibbs phase rule for reactive (e.g., azeotropic mixtures), and reduced reboiler duty in the case 6 of exothermic reactions, among others.1 systems. Based on this fact, some approaches for the transfor- mation of composition variables have been introduced in the The RD process has been long known in the chemical 5,6 industry. However, it is only during the past decade that there literature and their aim is to provide a simpler thermodynamic has been a significant interest and an increase in the number of framework for treating systems subject to chemical reactions. 1 These approaches are generally based on transformation of the publications on this subject. Recently, this process has become fi very important in the production of fuel additives such as methyl physical compositions, and the principal bene t is that the tert-butyl ether (MTBE), ethyl tert-butyl ether (ETBE), and tert- chemical and physical equilibrium model in the reactive mixture amyl methyl ether (TAME), and also in the production of many is very similar to a strictly physical equilibrium model. In this other chemicals such as esters and alcohols.2 context, the premise of using the concept of transformed Because most of these reactive systems may contain azeo- composition variables for obtaining RRCMs is that the equations tropes, the reactive residue curve maps (RRCMs) are an that characterize a RD system are expressed mathematically in important tool in the initial stage of the process design for the same form as those reported for nonreactive distillation sys- tems.3,5 In particular, the reaction-invariant composition vari- identifying, in a fast form, the infeasible sequences. A residual 6 curve represents the change of the liquid composition with ables proposed by Ung and Doherty are attractive for the respect to time during a simple distillation.3 Thus, RRCMs simulation of separation process and favor the study of complex provide the possibility of determining the existence of distillation multireactive and multicomponent systems. Using this approach, boundaries and, as a consequence, determining different poten- tial zones of operation.4 Once the zone of feasible operation is Received: June 22, 2010 established, depending on the feed composition, it is possible to Accepted: December 20, 2010 predict the different components obtained as distillate and Revised: November 12, 2010 bottom products. However, it is important to note that several Published: January 11, 2011 r 2011 American Chemical Society 2157 dx.doi.org/10.1021/ie101335z | Ind. Eng. Chem. Res. 2011, 50, 2157–2166 Industrial & Engineering Chemistry Research ARTICLE P C-R the reaction-invariant composition variables depend only on the mole fractions must equal unity: i=1 Xi = 1. For the initial composition of each independent chemical species (i.e., transformation procedure X f x, which is necessary to evaluate they are independent of the amount of reaction), the solution thermodynamic properties, the reference mole fractions are space is restricted to compositions that are already at chemical calculated using eq 1 and the equilibrium constants for each 3,6 equilibrium, and the problem dimension is also reduced. These reaction Keq,r by solving a system of R nonlinear equations advantages allow study of a variety of real and complex reactive given by YC systems, because there are several combinations between the r ¼ vi ¼ ::: ð Þ number of reactions (R) and the number of components (C) that Keq, r ai r 1, , R 2 can be analyzed in ternary diagrams (i.e., C - R = 3). Therefore, i ¼ 1 the analysis of RRCMs can be performed in the same form as in r simple distillation without chemical reactions. Until now, only a where ai is the activity of component i and vi is the stoichiometric few methods have used transformed composition variables for number of component i in reaction r, respectively. When we the calculation of RRCMs. However, these methods may show know the reference mole fractions, the remaining mole fractions significant computer time for the calculation of RRCMs due to are calculated using eq 1. the iterative calculations involved in the modeling of reactive On the other hand, in a simple distillation process with or phase equilibrium.3,5,6 without chemical reactions, a liquid is vaporized and the vapor is removed from contact with the liquid as it is formed (see In this study, we take advantage of the characteristics of ff reaction-invariant composition variables to develop a short Figure 1). Each di erential mass of vapor is in equilibrium with method for calculating RRCMs. Specifically, we propose a the remaining liquid. The composition of the liquid will change simplified approach for determining RRCMs by discarding the with time, since in general the vapor formed is richer in the more effect of temperature on the phase equilibrium constants, but volatile components. Therefore, the locus of the liquid composi- ff tions remaining from a simple distillation process defines a preserving the composition e ect. Although this assumption has 8 been used for nonreactive mixtures,7 it has not been applied to residue curve. For homogeneous mixtures with multiple chem- ical reactions, the calculation of a reactive residual curve (RRC) the study of reactive mixtures. Our results indicate that the use of fi 3,5,8 reaction-invariant composition variables and the application of is based on a modi cation of the Rayleigh expression. For a simplified phase equilibrium constants (i.e., not dependent on system of C components subject to R independent chemical temperature) avoid the iterative calculation of the reactive bubble reactions and using reaction-invariant composition variables, the ff following set of ordinary differential equations is used to describe temperature, resulting in an e ective and faster strategy for 3,5,8,9 estimating RRCMs. Finally, the performance of our short the dynamics of simple distillation processes method is compared with those reported using a rigorous dXi ¼ ¼ ::: ð Þ method in several reactive systems. Xi - Yi i 1, , C - R - 1 3 dτ 2. DESCRIPTION OF REACTIVE RESIDUE CURVE CALCU- where Xi represents the transformed composition in the liquid LATION USING REACTION-INVARIANT COMPOSITION phase of component i, Yi is the transformed composition in the VARIABLES vapor phase of component i, and τ is the dimensionless time, respectively. Equation 3 is obtained from a mass balance applied In a reactive system, the thermodynamic property functions to a distillation unit and by introducing a dimensionless time behave as in a nonreactive system if reaction-invariant composi- variable.3,5,8,9 To obtain physical consistency using these trans- tion variables are used instead of
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