Microsoft Word - 476hernandez.docx CHEMICAL ENGINEERING TRANSACTIONS VOL. 43, 2015 A publication of The Italian Association of Chemical Engineering Online at www.aidic.it/cet Chief Editors: Sauro Pierucci, Jiří J. Klemeš Copyright © 2015, AIDIC Servizi S.r.l., ISBN 978-88-95608-34-1; ISSN 2283-9216 An Alternate Formulation for Calculating Reactive Residue Curve Maps Karim Alloula Université de Toulouse, Institut National Polytechnique de Toulouse (INPT) – Ecole Nationale Supérieure des Ingénieurs en Arts Chimiques et Technologiques (ENSIACET) 4, allée Emile Monso, BP 44362, 31030 Toulouse Cedex 4, France Karim.Alloula@ensiacet.fr Until now, reactive residue curve maps (RRCM) are computed for a liquid phase where either only kinetically controlled reactions, or only chemical equilibrium reactions, take place. We try to model a liquid phase where both kinds of reactions may occur. Using two new transformed composition vectors and the temperature as unknown variables, we provide an alternate formulation for calculating RRCM. The main benefit is a unified and extended framework where one can easily add or remove one or more reactions, and obtain the dynamic model to be integrated. The computational cost of the suggested approach may be less than the previous ones. 1. Introduction During the last decades, professor Michael F. Doherty and his colleagues provide the main advances, both in the formulation of the differential algebraic equations (DAE) modelling the (reactive) Rayleigh distillation, and in the algorithms for solving them. First, (Doherty and Perkins, 1978) modify the original equations of the simple distillation by replacing the time independent variable, , by a dimensionless variable, , when integrating the partial mass balances under the constraints of thermodynamic equilibria. The resulting dynamic model for calculating a RCM no longer includes the energy balance: in the non-reactive case, it is proved that the residue curve map does not depend on the heating policy. A dynamic model for RRCM calculation appears in (Barbosa and Doherty, 1988). The DAE, and the graphical representation obtained from its numerical integration, prove to be efficient for providing a “deeper understanding of distillation phenomena” and “the tools for the design and synthesis of reactive distillation columns”. This dynamic model is restricted to one chemical equilibrium reaction. (Venimadhavan et al., 1994) build a DAE modelling a diphasic system with one kinetically controlled chemical reaction in the liquid phase. The Damköhler number, “a dimensionless ratio of a characteristic liquid residence time to the characteristic reaction time” is introduced and the influence of this parameter on the RRCM profiles is discussed. More general frameworks for simulating several chemical reactions are published later. (Ung and Doherty, 1995a, 1995b) introduce a “new set of composition variables” reducing the dimensionality of the RRCM when a vapor-liquid equilibrium with multiple chemical equilibrium reactions is considered. For such a case, they define a new independent variable , and write a DAE using the new set of composition variables. Compared with the non-reactive case, the significance of the independent variable changes, and the new composition variables play the role of the molar fractions, but the differential equations look the same. (Venimadhavan et al., 1999) generalize to multiple reactions the model for a single kinetically controlled reaction. Once again, a Damköhler number is defined and allows parametric studies of the RRCM. Apart from the Doherty papers, some extra publications are worth analyzing. (Almeida-Rivera and Grievink, 2004) summarize the dynamic model for multiple chemical equilibrium reactions as it is commonly adopted since (Ung and Doherty, 1995a). Under the chemical equilibrium assumption, (Sanchez-Daza et al., 2006) introduce an alternate formulation based on the element concept, an element being an atom, a molecule or a fragment of molecule. This element approach leads to RRCM where the vertices are associated to elements. It allows the representation on ternary diagrams of reactive systems with more than three species. (Alloula et DOI: 10.3303/CET1543258 Please cite this article as: Alloula K., 2015, An alternate formulation for calculating reactive residual curve maps, Chemical Engineering Transactions, 43, 1543-1548 DOI: 10.3303/CET1543258 1543 al., 2013) apply an index reduction technique to an initial DAE in order to provide a DAE more suitable for numerical integration. From the previous papers, it appears that adequate formulations for modelling either multiple kinetically controlled reactions, or multiple chemical equilibrium reactions exist. However, a formulation for handling both kinds of reactions is missing. In order to build such a framework, part 2 collects the three main DAE models (non-reactive case and two reactive cases). Even if those models can be retrieved in the previously cited papers, a common notation is established for further comparison with the suggested approach. In part 3, this paper introduces an alternate formulation which can be substituted to any of the three usual formulations and, furthermore, can model reactive systems with both kinds of reactions. Part 4 lists the expected benefits of such a new formulation from the points of view of modelling and numerical integration. 2. Usual formulations for calculating residue curve maps 2.1 Non-reactive residue curve maps Throughout this paper, we consider that two phases exist, there is no vapour retention and no chemical reactions in the vapour phase. The vapour phase is considered to be ideal. Heat and pressure policy remain free. Table 1 gives a list of all the symbols appearing in the forthcoming model. According to the previous assumptions, and following the given notations, a mathematical model of the non-reactive Rayleigh distillation may be written as: ∀ ∈ 1,…, ; ∙∙ = − ∙ℎ ( , , ) ∙∙ = − ∙ ( , , )∀ ∈ 1,…, ; = ( , , ) ∙∑ = 1∑ = 1 (1) This model consists of partial mass balances, 1 energy balance, thermodynamic equilibria, and 2 definitions of molar fractions. Derivatives are with respect to the independent variable . The unknown vector can be defined as ( , , , , ) . and remain user-defined variables. The overall mass balance = − can be deduced from ∑ = 1 by adding all the partial mass balances. Then, combining the overall mass balance with each partial mass balance, one obtains the following differential equations: ∀ ∈ 1,…, ; ∙ ∙ = − (2) These equations suggest the introduction of a dimensionless variable , satisfying: = (3) From this variable change one obtains the differential algebraic system usually integrated for producing a RCM: ∀ ∈ 1,…, − 1 ; = −∀ ∈ 1,…, ; = ( , , ) ∙∑ = 1∑ = 1 (4) In (4) the unknown vector can be defined as ( , , ) . From the initial equation system, and have been eliminated, and two equations –one partial mass balance and the energy balance- have been dropped. 2.2 Kinetically controlled reactions only For kinetically controlled reactions only, a reactive Rayleigh distillation model may be written as: ∀ ∈ 1,…, ; ∙∙ = ∑ , Δ ( , , ) − ∙ℎ ( , , ) ∙∙ = − ∙ ( , , ) − ∑ ( , ) ∙ Δ ( , , )∀ ∈ 1,…, ; = ( , , ) ∙∑ = 1∑ = 1 (5) Applying the same variable change and elimination process as in the non-reactive case, one obtains a reduced differential algebraic system modelling the reactive Rayleigh distillation: 1544 ∀ ∈ 1,…, − 1 ; = − + ∙ ∑ , − ∑ , ∙ ∙ ( , , )∀ ∈ 1,…, ; = ( , , ) ∙∑ = 1∑ = 1 (6) Assuming that some heating policy can maintain the ratio to its initial value ( )( ) , and evaluating one of the forward reaction rate constants ( for example) at a reference temperature (Venimadhavan et al., 1999), the kinetically controlled residue curve map is computed from: ∀ ∈ 1,…, − 1 ; = − + ∙ ∑ , − ∑ , ∙ ∙ ( , , )( )∀ ∈ 1,…, ; = ( , , ) ∙∑ = 1∑ = 1 (7) In the multiple kinetically controlled reaction case, the Damköhler number can be defined as =( )( ) ( ) , ratio of a characteristic residence time ( )( ) to a characteristic reaction time ( ). The RRCM depends on , that is to say on the conditions offered to the chemical reactions. When goes to 0, the dynamic model (7) goes to the non-reactive model (4). When goes to +∞, all the reactions can be viewed as a chemical equilibrium reaction. 2.3 Chemical equilibrium reactions only For chemical equilibrium reactions only, a reactive Rayleigh distillation model may be written as: ∀ ∈ 1,…, ; ∙∙ = ∑ , ( , , ) − ∙ℎ ( , , ) ∙∙ = − ∙ ( , , ) − ∑ ( , ) ∙ ( , , )∀ ∈ 1,…, ; = ( , , ) ∙∀ ∈ 1,…, ; ( , ) = ∏ ( , , ) ,∑ = 1∑ = 1 (8) , ,…, no longer have an explicit expression. They become additional variables. (8) is saturated thanks to chemical equilibrium equations. (Ung and Doherty, 1995a) details how , ,…, are eliminated, and how new variables , ,…, are defined in terms of ( , ) and , ,…, . New variables , ,…, are also defined in terms of ( , ) and , ,…, . These “transformed composition variables” satisfy ∑ = 1, ∑ = 1, and − − 1 partial mass balances: ∀ ∈ 1,…, − − 1 ;f(ν) ∙ ∙ = − (9) where f(ν) depends only on stoichiometric coefficients. (9) suggests the introduction of the dimensionless variable , satisfying: = ∙ ( ) (10) As mentionned in (Ung and Doherty, 1995b), the differential equation “has exactly the same functional form as the simple distillation equation for non-reacting mixtures in terms of mole fractions”: ∀ ∈ 1,…, − − 1 ; = −∀ ∈ 1,…, ; = ( , , ) ∙∀ ∈ 1,…, ; ( , ) = ∏ ( , , ) ,∀ ∈ 1,…, − ; = ( , )∀ ∈ 1,…, − ; = ( , )∑ = 1 ∑ = 1∑ = 1 ∑ = 1 (11) 1545 3. Alternate formulation for calculating residue curve maps Assuming that a mixture contains components, a DAE modelling the non-reactive Rayleigh distillation may consist of partial mass balances, 1 energy balance, thermodynamic equilibria, and 2 molar fraction definitions. In the reactive case, , ,…, modify the partial mass balances, and additional equations, either algebraic (chemical equilibrium reaction), or differential (kinetically controlled reaction) are added. (12) is a generic DAE modelling a system with both kinetically controlled and chemical equilibrium reactions: ∀ ∈ 1,…, ; ∙∙ = ∑ , ( , , ) − ∙ℎ ( , , ) ∙∙ = − ∙ ( , , ) − ∑ ( , ) ∙ ( , , )∀ ∈ 1,…, ; = ( , , ) ∙∀ ∈ 1,…, ; ( , ) = ∏ ( , , ) , ( , , ) = Δ ( , , )∑ = 1∑ = 1 (12) The alternate formulation for calculating a RRCM from (12) is based on the elimination of , ,…, . The elimination process is very similar to the one detailed in (Ung and Doherty, 1995a). However, here we write the set of partial mass balances using , the numbers of moles of each species in the liquid, instead of using the liquid molar fractions . Adopting a vectorial notation, the partial mass balances may be written: = . − . (13) We start by isolating partial mass balances, for example the last ones. ∙ = . − . (14) In (14), stands for the last lines of the vector, stands for the last lines of the vector, and is the submatrix built from the last lines of the stoichiometric matrix . Provided that is not singular: = . ∙ + . (15) Using a notation analoguous to the previous one, the first − partial mass balances may be written: ∙ = . − . (16) or, after replacing by its value from (15): ∙ = . . ∙ + . − . (17) In (17) variables can be grouped in the following way: − . .∙ = − .( − . . ) (18) These partial mass balances may be viewed as the generalization to the reactive case of the partial mass balances in the non-reactive case. With the following definition: = − . .= − . . (19) the first − partial mass balances take the form: ∙ = − . (20) If we generalize the definition (19) of and with: == (21) we obtain two new transformed composition vectors, and , which are in bijective correspondence with and . There is not such a bijective correspondence for the usual “transformed composition variables” and from (Ung and Doherty, 1995b). Using and , and taking into account the fact that + 1 variables , ,…, , are replaced by variables ,…, , (12) becomes: ∀ ∈ 1,…, − ; ∙ = − .ℎ ( , , ) ∙∙ = − ∙ ( , , ) − ∑ ( , ) ∙ . ∙ + .∀ ∈ 1,…, ; = ( , , ) ∙∀ ∈ 1,…, ; ( , ) = ∏ ( , , ) , . ∙ + . = Δ ( , , )∑ = 1 (22) 1546 Finally, defining from = . , and assuming that some heat policy can make the vapour flowrate vary in the same rate as the liquid retention = ∑ , lead to the DAE to be integrated for generating the RRCM: ∀ ∈ 1,…, − ; = −∀ ∈ 1,…, ; = ( , , ) ∙ ∀ ∈ 1,…, ; ( , ) = ∏ ( , , ) ,. + = ∙ ( ) ∙ ∏ , − ∏ ,∑ = 1 (23) The heat policy assumption removes one degree of freedom in the physical system and makes it possible to drop one of the equation -the energy balance- from (22). In (23), there are only 2 + 1 unknown variables. and appear in several equations, but (19) and (21) give the bijective correspondence between and , and between and . Consequently, (23) may be solved with respect to , and . 4. Expected benefits 4.1 Single modelling framework This alternate formulation unifies and extends the three usual ones. A generic DAE (23) consists of − partial mass balances written in terms of new transformed composition variables, thermodynamic equilibria, and 1 definition for the vapour molar fractions. Each chemical equilibrium reaction adds 1 algebraic constraint. Each kinetically controlled reaction adds 1 differential equation. Instead of providing a DAE whose size may be 2 + 1 or 4 − 2 + 1, the suggested approach always leads to fixed size (2 + 1) system. Instead of defining either the molar fractions or the “transformed composition variables” as dependent variables, the suggested approach always integrates the new transformed composition variables and . Instead of using different definitions for the independent variable , the suggested approach always uses the same definition: = . . The alternate formulation is more versatile than the usual ones. According to (23) an arbitrary number of kinetically controlled chemical reactions and/or chemical equilibrium reactions can be modelled within a single framework. 4.2 Reduced computational effort Finally, the numerical integration based on this alternate formulation may be less costly. With the usual formulation, in the chemical equilibrium case, 2( − ) definitions of the “transformed composition variables” are integrated by a DAE solver as algebraic equations, or may be part of a non linear equation system defining in terms of . Whatever integration method is selected, the computation time in the usual approach should be greater than the one obtained with the new formulation. Here , and are the only integrated variables; the reactive liquid composition is only used after numerical integration, for producing graphical representations. 5. Conclusions After reviewing the different DAE models used for generating RCM and RRCM, we introduce an alternate formulation. Within this new modelling framework, the number of equations (2 + 1) depends only on the number of components, and the unknown variables are always the temperature and a set of new transformed composition variables and . From a generic DAE, one can easily build a specific DAE in order to model a liquid phase with an arbitrary number of chemical equilibrium reactions and/or kinetically controlled reactions. When simulating this resulting DAE, we take the view that the new approach will prove to be not only versatile, but also cost-effective due to its reduced computational complexity. 1547 Nomenclature Symbol name Meaning Unit Number of components Number of chemical reactions Number of reactants Time, independent integration variable Dimensionless independent integration variable Liquid retention Vapour flow rate . Liquid molar fraction of component Vapour molar fraction of component Transformed composition variable in the liquid phase (Ung and Doherty, 1995b) Transformed composition variable in the vapor phase (Ung and Doherty, 1995b) New transformed composition variable in the liquid phase New transformed composition variable in the vapor phase Temperature Pressure ℎ Molar liquid enthalpy . Molar vapour enthalpy . Heating power . Thermodynamic equilibrium constant for component , Stoichiometric coefficient for component in reaction Extent of reaction Heat of reaction . Activity of component Chemical equilibrium constant of reaction Reaction rate constant of reaction Damköhler number , − 1 , Δ ∙ . Acknowledgements This work is part of the LEDA -Logistique des Equations Différentielles Algébriques- project, supported by the French National Research Agency (ANR-10-BLAN 0109). References Alloula K., Monfreda F., Thery Hétreux R., Belaud J.-P., 2013, Converting DAE models to ODE models: application to reactive Rayleigh distillation, Chemical Engineering Transactions, 32, 1315-1320. Almeida-Rivera C., Grievink J., 2004, Feasibility of equilibrium-controlled reactive distillation process: application of residue curve mapping, Computers and Chemical Engineering, 28, 17-25 Barbosa D., Doherty M. F., 1988, The simple distillation of homogeneous reactive mixtures, Chemical Engineering Science, 43, 3, 541-550. Doherty M. F. and Perkins J. D., 1978, On the dynamics of distillation processes – I, The simple distillation of multicomponent non-reacting, homogeneous liquid mixtures, Chemical Engineering Science, 33, 281-301. Sanchez-Daza O., Vidriales Escobar G., Morales Zarate E., Ortiz Munoz E., 2006, Reactive residue curve maps A new study case, Chemical Engineering Journal, 117, 123-129. Venimadhavan G., Buzad G., Doherty M. F., Malone M. F., 1994, Effects of kinetics on residue curve maps for reactive distillation, AiChE J., 40, 11, 1814-1824. Venimadhavan G., Malone M. F., Doherty M. F., 1999, Bifurcation study of kinetic effects in reactive distillation, AiChE J., 45, 3, 546-556. Ung S. and Doherty M. F., 1995a, Vapor-liquid phase equilibrium in systems with multiple chemical reactions, Chem. Ing. Sci., 50, 23-48. Ung S. and Doherty M. F., 1995b, Calculation of residue curve maps for mixtures with multiple equilibrium reactions, Ind. Eng. Chem. Res., 34, 3195-3202. 1548