Revista Mexicana de Ingeniería Química · two ideal reactive systems. The results of the reactive...

Vol. 15, No. 1 (2016) 193-207Revista Mexicana de Ingeniería Química

CONTENIDO

Volumen 8, número 3, 2009 / Volume 8, number 3, 2009

213 Derivation and application of the Stefan-Maxwell equations

(Desarrollo y aplicación de las ecuaciones de Stefan-Maxwell)

Stephen Whitaker

Biotecnología / Biotechnology

245 Modelado de la biodegradación en biorreactores de lodos de hidrocarburos totales del petróleo

intemperizados en suelos y sedimentos

(Biodegradation modeling of sludge bioreactors of total petroleum hydrocarbons weathering in soil

and sediments)

S.A. Medina-Moreno, S. Huerta-Ochoa, C.A. Lucho-Constantino, L. Aguilera-Vázquez, A. Jiménez-

González y M. Gutiérrez-Rojas

259 Crecimiento, sobrevivencia y adaptación de Bifidobacterium infantis a condiciones ácidas

(Growth, survival and adaptation of Bifidobacterium infantis to acidic conditions)

L. Mayorga-Reyes, P. Bustamante-Camilo, A. Gutiérrez-Nava, E. Barranco-Florido y A. Azaola-

Espinosa

265 Statistical approach to optimization of ethanol fermentation by Saccharomyces cerevisiae in the

presence of Valfor® zeolite NaA

(Optimización estadística de la fermentación etanólica de Saccharomyces cerevisiae en presencia de

zeolita Valfor® zeolite NaA)

G. Inei-Shizukawa, H. A. Velasco-Bedrán, G. F. Gutiérrez-López and H. Hernández-Sánchez

Ingeniería de procesos / Process engineering

271 Localización de una planta industrial: Revisión crítica y adecuación de los criterios empleados en

esta decisión

(Plant site selection: Critical review and adequation criteria used in this decision)

J.R. Medina, R.L. Romero y G.A. Pérez

REACTIVE McCABE-THIELE: SHORT CUT METHOD INCLUDING REACTIVEVAPOR - LIQUID EFFICIENCY

McCABE-THIELE REACTIVO: METODO CORTO INCLUYENDO EFICIENCIAVAPOR LIQUIDO REACTIVO

R.A. Flores-Estrella1*, G.A. Iglesias-Silva21IPICyT, Division de Matematicas Aplicadas, Camino a la Presa de San Jose Apdo. Postal 2055, Col. Lomas 4a Seccion, San

Luis Potosı, S.L.P., Mexico.2Departamento de Ingenierıa Quımica, Instituto Tecnologico de Celaya, Celaya, Gto. CP 38010, Mexico.

Received March 19, 2015; Accepted February 8, 2016

AbstractA graphic McCabe-Thiele method is developed for application to fully reactive column configuration. The extended methodtakes in to account several aspects as classic McCabe-Thiele method under transformed reaction-invariant compositionvariables as: minimum number of reactive theoretical stages, number of reactive theoretical stages, minimum reflux ratio,feed tray location, top and / or bottom flow. Reaction equilibrium is assumed on each stage, vapor-liquid equilibrium isassumed on all stages and is also considered a reactive Murphree vapor tray efficiency. The methodology is illustrated intwo ideal reactive systems. The results of the reactive McCabe-Thiele method agree with those obtained using a rigorousmethod in commercial simulator AspenONE Aspen Plus (Radfrac module).Keywords: reactive McCabe-Thiele, reactive distillation, transformed composition, reactive Murphree efficiency, reactivesystem.

ResumenUn metodo grafico basado en el metodo McCabe-Thiele es desarrollado para la aplicacion del diseno de columnasreactivas. El metodo toma en cuenta varios aspectos del metodo clasico McCabe-Thiele bajo variables reactivastransformadas: numero mınimo de etapas reactivas teoricas, numero de etapas reactivas, reflujo mınimo, ubicacion delplato de alimentacion, flujos de destilado y fondos. Se asume: equilibrio reactivo para cada etapa, equilibrio vapor lıquidoy es considerado una eficiencia reactiva Murphee para la fase de vapor. La metodologıa es ilustrada en dos sistemas reativosideales. Los resultados del metodo McCabe-Thiele reactivo concuerdan con los obtenidos usando un metodo riguroso enun simulador comercial AspenONE Aspen Plus (Radfrac module).Palabras clave: McCabe-Thiele reactivo, destilacion reactiva, composiciones transformadas, eficiencia reactiva Murphee,sistema reactivo.

1 Introduction

Reactive distillation process integrates functionalityof reaction and distillation in combined unit instead ofconventional unit process that has single functionality.Combining chemical reaction with vapor-liquidequilibrium provides advantage in the sense ofchemical reactions modify the phase behavior.Because of these interactions the modeling and designof reactive distillation (RD) is complex (Segovia-Hernandez et al. 2015).Roughly speaking, the design methods forRD can be classified in three main areas:

graphical/topological, optimization-based andheuristics-based. Graphical/topological methods aresuitable for feasibility studies but in some casesinefficient for a detailed column design. Heuristics-based methods are useful for post-design analysis.Optimization-based techniques are mathematical-based, involve both physical constraints and appear tobe the most flexible and rigorous approach to identifythe optimal RD column configurations (Domingues etal. 2014). The solution of these complex modelsis computationally demanding. Despite of todayrobust simulation computer equipment and robustoptimization-based techniques are available, several

*Autor para la correspondencia. E-mail: [email protected]

Publicado por la Academia Mexicana de Investigacion y Docencia en Ingenierıa Quımica A.C. 193

Flores-Estrella and Iglesias-Silva/ Revista Mexicana de Ingenierıa Quımica Vol. 15, No. 1 (2016) 193-207

optimal design methods can not guarantee the globaloptimality (Segovia-Hernandez et al. 2015). Often,convergence to local optima can only be achievedif good initial guess is available (Avami et al.2012). Then, as an alternative graphical/geometricmethods are used in the initial stages of solution(Carrera-Rodrıguez et al. 2014). Indeed, despiteof conventional distillation process is considereda mature technology, conceptual approach are stilldeveloped (Mendoza and Riascos 2015).Models and design tools for RD were developedduring the last decade (Barbosa and Doherty 1987,Barbosa and Doherty 1988, Espinosa et al. 1995,Bessling et al. 1997, Okasinski and Doherty 1998).Indeed, several short cut graphical methods has beenproposed for reactive distillation processes towardspreliminary design and visualization of mass balancein reactive column (Carrera et al. 2011). Neverthelessthe implementation still remain in solving a set ofdifferential equations or in-out tray by tray calculus.Graphical McCabe-Thiele method was an importanttool in the preliminary design of column distillationof nonreactive systems. Several contributions ofreactive McCabe-Thiele method have been reportedfor reactive systems. Similar a binary distillationcolumn, in reactive distillation column three importantkey variables must be calculate: operation minimumreflux ratio, the number of theoretical reactivestages and feed stage. Espinosa et al. (1995)reported a Ponchon-Savarit using a set of transformedvariables proposed by Barbosa and Doherty (1988)including limiting behavior as minimum reflux ratioand minimum reactive stages. Reactive McCabe-Thiele was reported by Lee et al. (2000a, 2000b)for binary mixture assuming reaction equilibrium(isomerization and decomposition) and limitingbehavior analysis as total reflux. Conceptual designwas developed by Sundmacher and Qi (2003) in aMcCabe-Thiele method for limiting behavior, simplereversible reaction in ideal binary mixture and severalconfiguration (i.e. external recycle loop, pre-reactorwith nonreactive column on top of a reactive reboiler).A reactive McCabe-Thiele method was developed bySanchez-Daza et al. (2003) based on the elementmass-balance concept assuming reaction equilibriumon every stage. Graphical design methodology wasdeveloped for nonhybrid (Groemping et al. 2004) andextended for hybrid reactive columns for systems withtwo degrees of freedom (according to Gibbs phaserule) under the concept of reactive stage compositionlines (Dragomir and Jobson 2005a, 2005b). Dragomirand Jobson generated multiple designs avoiding in-

out tray by tray calculations assuming constant molaroverflow.Based on Dragomir and Jobson’s method(2005a), another graphical reactive McCabe-Thieledesign methodology has been proposed includingthermodynamical rigorous reactive vapor liquidequilibrium calculation (Flores-Estrella 2007).Although graphical reactive McCabe-Thiele designare available, alternative reliable graphical designmethod including deviations of equilibrium vapor-liquid in tray calculations are still needed. Indeed,it has not been applied to the study of reactivemixtures. The present work introduces a designmethod for RD column, this approach is similar inconcept to nonreactive distillation design method.The present contribution derive some analytical shortcut graphical reactive McCabe-Thiele method usingtransformed composition (Ung and Doherty 1995a)and compare these implications with the analogousbinary nonreactive McCabe-Thiele method. A fullextension of reactive McCabe-Thiele is restricted tosystems with two degrees of freedom (in the sense ofGibbs phase rule systems) and the reaction equilibriumon every stage is assumed. The implication ofdeviation of equilibrium vapor-liquid is included,we propose a new concept of reactive Murphreeefficiency for vapor phase in RD column. To the bestof our knowledge, reliable short cut methods for RDare not available yet for deviations of equilibriumvapor-liquid in reactive tray calculations. Finally, theperformance of our graphical method is comparedwith those obtained with the commercial simulatorAspenONE Aspen Plus using a rigorous method(Radfrac module) for several reactive systems. Resultsshow the capabilities of this method for the design ofRD columns.

2 Description of a reactiveMcCabe-Thiele method

2.1 Transformed compositions

Ung and Doherty introduced a set of compositionvariables for treating phase equilibria inmulticomponent, multi-reaction systems with orwithout inert components (Ung and Doherty 1995a).For a system of R independent reactions among creacting components with I inert components for atotal of C = c+I species andΠ phases; the transformed

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composition variables are denote by:

Xi =xi − vT

i N−1xre f

1− vTTOT N−1xre f

i = 1, . . . ,C −R (1)

where xre f is the column vector of r referencecomponent mole fractions, vT

i is the row vector ofstoichiometric coefficients of component i for allreactions, vT

TOT is a row vector of the sum of thestoichiometric coefficients for each reaction, N isa square matrix of the stoichiometric coefficientsof the reference components in the R reactions.The transformed compositions satisfy the followingproperties:

Xi(0) = Xi(ξ), ∀ξ, i = 1, . . . ,C −RC−R∑i=1

Xi = 1(2)

The first condition is related with the conservationof mass at each value of the extent of reaction,indeed the transformed variables can be thought ofas reaction-invariant compositions. For a Π = 2(vapor-liquid phases), only C − R− 1 are independenttransformed composition variables. This matchesthe degrees of freedom (C − R − 1) given by Gibbsphase rule for isobaric reactive mixtures (Ung andDoherty 1995b). Therefore, for a isobaric systemwith C − R = 2 only have one degree of freedom.Then, for a temperature range (similar to nonreactivesystems for two components) where reactive vapor-liquid equilibrium exists, it is possible to calculatea two dimensional equilibrium curve in terms oftransformed composition variables. Defining thetransformed composition in the liquid phase by (1) andvapor phase by:

Yi =yi − vT

i N−1yre f

1− vTTOT N−1yre f

i = 1, . . . ,C −R (3)

it is possible obtain (analogous to system withoutreaction) a continuous isobaric vapor-liquidequilibrium curve in a range of temperature wherereactive equilibrium exist. It is important to mentionthat the range of transformed composition variablesvalue it is not necessary the unity.

2.2 Design equations

According to Barbosa and Doherty’s method (1987,1988) it is possible describe the steady-state operationof reactive distillation columns in terms of transformedcompositions. For these equations the followingassumption has to be account:

• The heat losses from the column walls arenegligible.

• The molar molar heat of vaporization for themixture is constant.

• The increase in sensible heat with increase intemperature through the column is negligible.

• The heat of reaction is negligible compared tothe enthalpy of the vapor phase.

• The feed is saturated liquid.

• Vapor-liquid equilibrium is achieved on eachplate.

• The column operates with a total condenser.

Under such assumptions, systems without reactionsguarantee: (i) a constant molar overflow and (ii)vapor-liquid flow rates in each section of a distillationcolumn are constant. Nevertheless, due to changes inthe total number of moles by reaction, only the vaporflow rate is constant in each section of the reactivedistillation column and the liquid flow rate in eachsection of the column only change due to changes inthe total number of moles by reaction.Then, as in the transformed composition, flow rates aredefined as (Ung and Doherty 1995c):

F = F(1− vTTOT N−1xre f ,F)

D = D(1− vTTOT N−1xre f ,D)

B = B(1− vTTOT N−1xre f ,B)

(4)

According to (1) and (4) the general and componentmass balances are defined (Ung and Doherty 1995c):

F = D + BFXi,F = DXi,D + BXi,B

(5)

Assuming m plates for a reactive stripping section anda single feed, the operating line is defining by:

Xi,m+1 =S m

S m + 1Yi,m +

1S m + 1

Xi,B i = 1, . . . ,c + 1

S m =Vm

B=

V0

B=

V0(1− vTTOT N−1yre f ,0)

B(1− vTTOT N−1xre f ,B)

= S ext(1−vT

TOT N−1yre f ,0)(1−vT

TOT N−1 xre f ,B)

S ext =V0

B(6)

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Similarly, assuming n plates in the reactive rectifyingsection and a single feed, the operating line for therectifying section is:

Yi,n+1 =Rn

Rn + 1Yi,n +

1Rn + 1

Yi,D i = 1, . . . ,c + 1

Rn =Ln

D=

L0

D=

L0(1− vTTOT N−1xre f ,0)

D(1− vTTOT N−1xre f ,D)

= Rext(1−vT

TOT N−1 xre f ,0)(1−vT

TOT N−1 xre f ,D)

Rext =L0

D(7)

Note 1. Under operating condition of total condenser(xre f ,0 = xre f ,D) Rn = Rext.It is important to mention that Rn and S m are functionof the overall reaction stoichiometry and dependsupon the change in the internal flows due to reaction(Dragomir and Jobson 2005a):

Rn = Rext1− vT

TOT N−1xre f ,0

1− vTTOT N−1xre f ,n

S m = S ext1− vT

TOT N−1yre f ,0

1− vTTOT N−1yre f ,m

(8)

Considering isobaric full reactive column for systemwith C − R = 2 and defining operation conditionssimilar to a nonreactive distillation column; for a trayby tray calculations the Eq. (6) and (7) are not a uniqueline because are function of xre f ,n and yre f ,n as Eq. (8)depicts (even if we consider a sutured liquid feed anda total condenser). In order to extend a full McCabe-Thiele in to reactive system we consider vTOT = 0(only for operating lines), then R = Rext and S =

S ext for all reactive stripping and rectifying sectionrespectively. Moreover, under such consideration itis possible to ensure a unique operation line for allreactive stripping and rectifying section even if weconsidered a saturate vapor feed or a q term (feedvapor fraction). As a calculation of equilibrium doesnot depend of tray by tray calculus, the previousconsideration allows us extend a graphic McCabe-Thiele for isobaric full reactive column with C−R = 2.Then under vTOT = 0 assumption the Eq (4, 7, 6, 8)are:

F = F, D = D, B = B,FXi,F = DXi,D + BXi,BRn = Rext, S m = S ext

Xi,m+1 =S ext

S ext + 1Yi,m +

1S ext + 1

Xi,B i = 1, . . . ,c + 1

Yi,n+1 =Rext

Rext + 1Yi,n +

1Rext + 1

Yi,D i = 1, . . . ,c + 1

(9)

Note 2. Similar to the classic McCabe-Thiele method;for nonreactive system to ensure a internal constantLn, Vm, Ln, and Vm the vTOT = 0 criterion arisein order to avoid change in moles due to reactionin operating lines for each stage of the column in areactive McCabe-Thiele method.Under the previous assumptions it is possible comparethe classic McCabe-Thiele method for nonreactivesystem (Seader 1998) and a proposed reactiveMcCabe-Thiele method, even considering limitingbehavior (Minimum number of equilibrium stagesand Minimum reflux ratio). The Table 1 showed thecomparative between these two methods.

As extension of Murphee plat efficiency (Seader1998) for reactive McCabe-Thiele a vapor efficiencyis proposed to describe individual tray performanceunder transformed composition. Applied to the vaporphase, it can be defined by:

EMV =Yn −Yn+1

Y∗n −Yn+1(10)

where EMV is the reactive Murphree vapor efficiencyfor stage n, Y∗n is the transformed composition in thehypothetical vapor phase in equilibrium with the liquidtransformed composition leaving stage n, Yn+1 is thetransformed vapor phase in equilibrium with the liquidtransformed composition leaving below stage n.The general procedure for design is as follows:

1. Specify compositions for the feed, top andbottoms products, set ratio of reflux to minimumreflux, feed flow rate and phase condition. Forthe overall mass balance (to satisfy the numberof degrees of freedom) specify D/F or B/F.

2. Calculate the vapor-liquid equilibrium curve intransformed composition at column operatingpressure.

3. Calculate the minimum number of equilibriumstages and minimum reflux ratio.

4. Calculate the reflux ratio and the set of slopesfor rectifying and stripping section operatinglines taking in to account the q-line.

5. Set reactive stages for rectifying section until thecriterion of optimum feed be accomplished.

6. Set reactive stages for stripping section untilthe criterion of composition of bottoms beaccomplished.

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Table 1. Specifications for and results from the McCabe-Thiele and Reactive McCabe-Thiele method.Specifications Classic McCabe Reactive McCabeTotal feed rate F FFraction composition of the feed zF ZFFraction composition of the distillate xD XDFraction composition of the bottoms xB XBRatio of reflux to minimum reflux R/Rmin R/RminColumn operating pressure P PPhase condition of the feed at column pressure q qVapor-liquid equilibrium curve xi vs yi Xi vs YiResultsDistillate flow rate D DBottoms flow rate B BMinimum number of equilibrium stages Nmin NminMinimum reflux ratio Lmin/D Lmin/DReflux ratio L/D L/DBoilup ratio V/B ˆV/BNumber of equilibrium stages N stages N reactive stagesOptimal feed-stage locationStage vapor and liquid compositionsStage temperature profile

7. For a reactive Murphree efficiency, set reactivestages for rectifying and stripping section fromreboiler to total condenser until the criterion ofcomposition of distillate be accomplished.

Despite of our method does not depend of thethermodynamical model, it is important to mentionthat high nonlinear behavior (i.e. near of pinchpoint, reactive azeotropes) are not predicted by linearequations (9).

The detailed design procedure is shown in Figure1 and Figure 2. It is important to point out that stagesextends from the top section of the RD column to thebottom. Then, the partial reboiler is equilibrium stageN. Indeed, the number of reactive stages obtainedinvolves a partial reboiler and considers the use ofa total condenser. As reactive Murphree efficiencyis defined only for tray vapor-liquid efficiency, it isnot applied to stage N. It is convenient to remarkthat the operating minimum reflux is determineduntil a single calculation related to intersection ofoperating rectifying line with q-line is performed.Notice that the calculations for RD design doesnot involve an iterative method. In this sense, theconstruction of operating lines are done in a similarform of nonreactive McCabe-Thiele method. Thus,this approach ensure the intersection of compositionprofiles without iterations. It is important to mention

that the design approach involves a set of linearequations, such as the method can be performancewithout the necessity of robust simulation computerequipment.

3 Results and discussionTwo systems are considered for the proposed graphicaldesign method. The first example is a designof reactive distillation column for the metathesisof 2−pentene (MP) (Okasinski and Doherty 1998).The reversible reaction of 2−pentene (x1) from2−butene (x2) and 3−hexene (x3) has ideal vapor-liquid equilibrium at atmospheric pressure over thetemperature range of [4,65.5]◦C. The reactionequilibrium constant is considered equal Keq,re f =

0.25 [@T = 25◦C] with temperature dependenceunder the Van’t Hoff expression. The transformedcomposition with 2−pentene as reference componentare:

X1 = x1 + 2x3X2 = x2 − x3 = 1− X1

(11)

In the second case, we consider the synthesisof Methyl tert-butyl ether (MTBE). The reversiblereaction of MTBE (x3) from isobutene (x1) andmethanol (x2) without inert was reported by Barbosaand Doherty (1988).

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Fig. 1. Detailed design procedure for Reactive McCabe-Thiele method.

Fig. 2. Detailed design procedure for Reactive McCabe-Thiele method including reactive Murphree vapor efficiency.

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Table 2. Reactive mixtures.System Model Thermodynamical data

MP

Ideal gas - Ideal liquid

Kre f = 0.252C5H10↔C4H8 + C6H12 ∆H◦ = −2410[J/mol] and T[K]

2(2- Pentene)↔ Keq = Kre f exp(−∆H◦R )( 1

T −1

Tre f)

2-Butene + 3-HexeneMTBE

Ideal gas - Wilson model

∆G/R = −4254.05+

C4H8↔CH4O + C5H12O 10.0982T + 0.2667Tln(T )isobutene + methanol↔ T[K]methyl-tert-butyl-ether

Table 3. Specifications and results for limiting behavior for Reactive McCabe-Thiele method.Specifications Results

Case System Feed n R/Rmin Xi XF XD XB Nmin Rmin R1 MP Liq. Sat. 100% 1.70 X2 0.0 0.98 −0.98

80.90 1.532 MP Liq. Sat. 80% 1.70 X2 0.0 0.98 −0.98

3 MP Vap. Sat. 100% 1.70 X2 0.0 0.98 −0.98 2.07 3.524 MP Vap. Sat. 80% 1.70 X2 0.0 0.98 −0.985 MTBE Liq. Sat. 100% 1.60 X1 0.85 0.99 0.55

30.07 0.116 MTBE Liq. Sat. 80% 1.60 X1 0.85 0.99 0.55

7 MTBE Vap. Sat. 100% 1.50 X1 0.90 0.99 0.55 0.33 0.508 MTBE Vap. Sat. 80% 1.50 X1 0.90 0.99 0.55

The transformed composition with MTBE asreferent component are defined by:

X1 =x1 + x3

1 + x3

X2 =x2 + x3

1 + x3= 1− X1

(12)

It is important to notice that the method proposeddoes not depend of the thermodynamical model usedfor the calculus of reactive vapor-liquid equilibrium.The thermodynamic data for both systems are givenin Table 2. An algebraic method that includestransformed compositions and transformed Gibbsenergy minimization was used for determine phasecompositions in reactive systems (Iglesias et al. 2006).The numerical procedure used equal area principle(Eubank and Hall 1995, Ung and Doherty 1995b) todetermine the composition in reactive systems.Although the proposal method does not include thedesign of hybrid column (i.e. nonreactive stages),as alternative, a hybrid method can be used. In thissense, some stages can be calculated with the rigorousmethod and the rest of the reactive stages can becalculated with the short-cut method.

Eighth cases were explored including vaporsaturated feed, liquid saturated feed and Murphreeefficiency for vapor phase. The graphical designmethod for binary transformed composition systems

have been developed and compared with resultsobtained by a rigorous method in a commercialstandard simulator (Radfrac module of AspenONEAspen Plus simulator). In order to ensure similarcondition calculus in commercial simulator, samethermodynamical model and Gibbs minimization forreaction equilibrium calculus were used. In order tocompare the results obtained from Murphree vaporefficiency in reactive McCabe-Thiele, a modificationof: N reactive stages, feed-stage location andMurphree efficiency for vapor phase was updated tothe RadFrac module.

Table 3 shows the cases for MP and MTBE systemin terms of transformed composition space. For allcases in the reactive McCabe-Thiele is considered atotal condenser and partial reboiler; these conditions isalso amused in rigorous simulation (RadFrac module).Fig. 3 shows the reactive equilibrium curve of MP andMTBE systems and the graphical solution for limitingbehavior. Then, under specification in Table 3: (i) theminimum equilibrium stages at infinity reflux ratio isobtained and, (ii) the minimum reflux ratio (Lmin/D) iscalculated through the scope of straight line containingpinch point and XD in 45◦ line. After the operatingminimum reflux ratio has been determined underlimiting behavior, an operating reflux ratio can befixed and operating lines are obtained.

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Fig. 3. Limiting behavior for MP and MTBE system.

Four cases are explored for MP system and thespecification design are depicted in Table 3. Fig. 3shows the reactive equilibrium curve of MP system intransformed composition. Distillate specification ofXD,1 = 0.98 indicates rich composition of 2−butene(x1 = 0.02, x2 = 0.98) and bottom specification ofXB,1 = −0.98 connotes rich composition of 3−hexene(x1 = 0.02, x3 = 0.98). The Feed is fixed to XB,1 = 0.0which corresponds x1 = 0.5, x2 = x3 = 0.25 inmolar composition. Results in the sense of limitingbehavior: (i) the minimum reactive stage Nmin = 8,(ii) calculus of Rmin,sat−liq = 0.90 (case 1-2), and(iii) Rmin,sat−vap = 2.07 (case 3-4) are shown in Fig.3. Then, using the specification of R/Rmin (Table3) the operating reflux ratio is calculated. With thespecification of XD, XF , XB and R calculated, theoperating lines are obtained. Finally, the stages aredrawn for rectifying and striping reactive section.The change between reactive rectifying and stripingoperating line is done according to the optimum feedstage criteria as the classical McCabe-Thiele method.

If tray efficiency n , 100% is considered, stagesare drawn from the bottom (last stage obtained inthe first design of n = 100% tray efficiency) to thedome with the reactive equilibrium curve modified byconsidering the equation 10. Then, stages are drawnfor striping and rectifying reactive section. The case 1and 2 illustrates the procedure for full reactive columndesign with liquid saturated feed. The case 2 takesin to account the same specifications given in case 1with the difference of the reactive vapor tray efficiencyof n = 80%. The case 3 and 4 takes the specificationof vapor saturated feed and the reactive vapor trayefficiency of n = 100% (case 3) and n = 80% (case 4).The results for MP system are shown in Fig. 4.

Fig. 5 shows the reactive McCabe-Thieleresults for or MTBE system. The highest obtainedconcentration of MTBE is near of X1 = 0.5 inthe binary reactive equilibrium curve when theconcentration objective in bottoms column is fixed atX1 = 0.55 which corresponds xMT BE = 0.75.

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Fig. 4. Reactive McCabe-Thiele for MP system.

Fig. 5. Reactive McCabe-Thiele for MTBE system.

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The distillate concentration is rich in isobutene and itis fixed at xiso = 0.99 which corresponds X1 = 0.99.Case 5 and 6 studies the vapor and liquid saturatedfeed under 100% tray vapor efficiency, the case 7and 8 takes the specification of previous cases (5and 7 respectively) and assume reactive Murphee platefficiency of 80%. The results for MTBE system areshown in Fig. 5.

With the results obtained in reactive McCabe-Thiele method, the rigorous method was simulatedin RadFrac module. In order to homogenize results,the plates are counted from top to bottom and for alldesigns the total condenser is not considered a reactivestage. Eight cases were compared under specificationsdepicted in Table 4. Fully feed specification,number of stages, feed stage, reflux ratio, distillationratio and vapor plat efficiency were specified inRadFrac module. The bottom and distillate productcomposition for MP system (case 1 − 4) obtainedfrom proposed method are in good agreement with

those obtained from commercial simulator using arigorous method. The approach proposed predictsimilar results for product composition in distillatefor MTBE system (case 5 − 8); nevertheless theapproach might be less accurate in predict resultsfor molar fraction composition in bottom productbecause rigorous simulation (RadFrac module) usesmass and energy balances on tray by tray calculations.Our results show that, it is observed a satisfactoryagreement between the short cut method and rigoroussimulation, nevertheless the molar profile of MTBEsystem is in less agreement than MP system. Thisbehavior is attributed to equimolar assumption.Because MTBE reaction present change in numberof molecules, the proposal method might be presentmayor discrepancies with results obtained by rigorousmass and energy balances. Then, we expect that ourmethodology are not accurate in high non-equimolarsystems.

Table 4. Specifications and results for limiting behavior for Reactive McCabe-Thiele Method.Specifications Results

Case xi Feed Distillate BottomsMcTM Aspen McTM Aspen

1NTOT = 15 1. Cis-2-Pentene 0.50 0.02 0.0096 0.03 0.0096NFeed = 6 2. Cis-2-Butene 0.25 0.98 0.9903 0.00 0.0000D = 50.00 3. Cis-3-Hexene 0.25 0.00 0.0001 0.97 0.9903




5NTOT = 5 1. Isobutileno 0.82 0.99 0.9949 0.20 0.4546NFeed = 3 2. Methanol 0.00 0.00 0.0000 0.04 0.0001D = 60.18 3. MTBE 0.18 0.01 0.0051 0.76 0.5453




Note: the R is specified in Table 3

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Temperature profiles results are presented in Fig.6. Despite of the fact that energy balances are nottake into account in reactive McCabe-Thiele method,temperature profiles obtained from equilibrium stagesare in good agreement with those obtained fromcommercial simulator using a rigorous method. Thebest agreement between the proposed method andAspenONE Aspen Plus is obtained for temperatureprofiles of MP systems (case 1-4); this agreementmight be less for bottoms temperature stage of MTBEsystems (case 5-8) because our method assumesconstant molar flows, uses only total mass balancesand heat effects along the column are not considered.On the other hand, the rigorous method considerschange in numbers of moles, mass and energybalances for modeling each stage. Despite of ourmethod is not recommended for rigorous optimizationdesign including heat requirements, it provides a pre-desing for a rigorous design method, such as operatingparameters, concentration and temperature profile asinitial iteration.

Many designs are possible under the proposedmethodology; to illustrate the implementation, resultsof vapor saturate for MP system and liquid suturedfor MTBE are shown in Fig. 7 and Fig. 8respectively. Comparative between compositionsprofiles (component in the vapor and liquid phases)of MP and MTBE system obtained from commercialsimulator and that obtained from the proposeddesign methodology present satisfactory agreement(Fig. 7). Fig. 7 show less agreement betweencomposition bottoms stage of MTBE system (similarto temperature profile of MTBE system in Fig. 6),obtained from commercial simulator and that obtainedfrom the proposed method.

4 Concluding remarksA new graphical design methodology has beendeveloped for full reactive columns for proposedreactive distillation processes.

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The methodology is restricted to systems withtwo degrees of freedom according to the Gibbsphase rule and single-feed columns. A directcorrespondence is made between binary reactivetransformed composition and conventional binarynonreactive distillation. Based upon the theoryof transformed coordinates, a necessary conditionfor reactive McCabe-Thiele emerge as similarcondition in nonreactive McCabe-Thiele method. Themethodology could be extended to more complexconfigurations, such as double-feed and columns withside-draws.Application of these methodology is illustrated byconsidering ideal and nonideal reactive mixtures,became the approach does not depend of thethermodynamic model.

It is assumed that reactive equilibrium vapor-liquid areachieved at each tray and reactive vapor tray efficiencyallows us to take into account nonideal behavior invapor-liquid tray equilibrium. The minimum refluxratio Rmin and minimum number of reactive stagesof this methodology can be easily estimated withoutiteration. In addition, our method optimizes the feedlocation by simple inspection similar of nonreactivedistillation method.The present work proposes a reactive Murphree vaporefficiency in order to include deviations in vapor-liquidequilibrium during the calculation of reactive stages.It should be noted that such procedure is performedunder transformed compositions variables.

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Our extension of the McCabe-Theile method toreactive systems generates estimates of compositionand temperature profiles which are in good agreementwith those obtained from commercial simulator usinga rigorous method.The results obtained from the proposed methodprovides an excellent initialization for a rigoroussimulation. This graphical method allows us to obtaina pre-design of full reactive column with simplevisualization.

Nomenclature

c number of reacting componentsC total number of components◦C celciusH◦ standard enthalpy of formationI number of inert componentsN square matrix of the stoichiometric

coefficients of the referencecomponentes

R number of independent reactionsT temperature

vTi row vector of stoichiometric

coefficients of component ivT

TOT row vector of the sum of thestoichiometric coefficients for eachreaction

Xi transformed composition ofcomponent i

xi molar composition of component i

Greek letters∆G/R Gibbs energy / gas constantΠ number of phasesξ extend reaction

SuperscriptsT transpose vector−1 inverse of matrix◦ standard condition∗ reactive equilibrium

Subscriptsre f reference componentB bottomsD distillate

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eq equilibtiumext externalF feedm plate in the reactive stripping sectionmin minimumMV Murphee plat efficiencyn plate in the reactive rectifying section

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