batch reactive art 1

8/8/2019 Batch Reactive Art 1

1/12

Simpl i fi ed De s ign o f Ba tch Reac t ive Dis t i ll a t ion Co lumns

Maria E. Huer ta-Garrido and Vicente Rico-Ramirez*

Departamento de Ingenieria Quimica, Instituto Tecnologico de Celaya, Av. Tecnologico y Garcia Cubas S/ N,Celaya, Gto., CP 38010, Mexico

Salvador Hernandez-Cast ro

Facultad de Ciencias Quim icas, Universidad de Guanajuato, Col. Noria A lta S / N, Guanajuato, Gto.,CP 36050, Mexico

This work proposes a simplified methodology for the analysis and design of reactive batchdistillation column s based on th e McCabe - Thiele method for reactive continuous colum ns an don th e concept of a r eactive differen ce point. To exten d t he application of the concept of a r eactivedifference point for reactive bat ch distillation column s, expressions for t he McCabe - Thieleoperat ing line and for th e dynam ics of the reboiler in th e rea ctive case h ave been der ived; twocases ar e considered d epending up on the pha se in which reactions occur (i.e., liquid or va por).We also provide an appr oach t o the derivat ion of an expression to calculate th e molar tu rn overflow rate by considering each reactive distillation plate as a combined system of a conventionalequilibrium stage linked to a chemical reactor (phenomena decomposition). Furthermore, wepresent our derivation of th e Un derwood equa tions (minimu m r eflux) for continuous a nd ba tch

reactive distillation columns. Finally, five illustrative examples allow us to show that our resultspresent r easonable agreement with t hose obtained th rough BatchFrac of AspenPlus.

1. In t roduc t ion

Reactive distillation has recently emerged as a prom-ising technology because of its integrated functionalityof separation and reaction. The general advantages of reactive disti llation are 1 (1) i t can achieve higherconversion rat es for an equilibrium-limited r eactionbecause of the continuous remotion of products beingformed and (2) the heat of reaction can reduce the h eatload of condensers or reboilers. Its most serious limita-tion, h owever, is th at the temperature of the reactionmust coincide with the distillation ran ge. 2 Despite t hisweakness, potential savings in term s of investment andoperation costs of this synergistic unit operation havemotivated considerable resear ch effort a imed to gener -ate efficient design strategies.

Barbosa and Doherty 3,4 developed reactive residuecurves in transformed coordinates and then proposedan a lgor i thm to ca lcula te the minimum reflux onreactive disti llation columns by using tran sformedvariables. Espinosa et al. 5 developed Ponchon - Savaritdiagrams in a transformed enthalpy - composition space,and Perez-Cisneros 6 proposed McCabe - Thiele diagra msfor rea ctive distillation column s ba sed on the elementbalance approach. Following these init ial attempts,Westerbergs group at Carnegie Mellon University

developed the concept of the reactive difference point7

and used this concept to provide simplified designstrategies and visual and graphical insights for reactivedistillation columns. 8- 12 This reactive difference pointis similar to the difference point for extractive distilla-tion columns an d can be u sed as a basis for performingnum erical calculations and geometric visualizations of reactive columns.

Despite the extensive body of literature reported onthe topic of reactive distillation, m uch work rema ins t obe done. As yet, there are few works concerning thedesign of reactive batch distillation columns. 13 Given th ecurrent applications of batch distillation in small-scaleindust ries producing high-value-added specialty chemi-cals (pharmaceutical industry, fertilizers, etc.), attentionshould also be focused on tha t d irection. Fu rth ermore,very few reports provide results with respect to th etheoretical limiting conditions of reactive distillationcolumns (such as minimum reflux, minimum numberof plates, etc.). This paper intends to contribute to bothof the topics mentioned above.

First, we pr ovide a simplified met hodology for theanalysis and design of reactive batch distillation col-umns based on the McCabe - Thiele method and on theconcept of the reactive difference point. To extend theapp licat ions of the rea ctive difference point for r eactivebatch disti llation columns, we propose a simplifiedapproach t o the calculation of the molar tur nover flowrate. By using the idea of phenomena decomposition,we consider each reactive distillation stage as a com-bined system of a conventional distillation plate linkedto a chemical reactor. Also, expressions for the dynamicsof the reboiler and the McCabe - Thiele operat ing line(quasi-steady-state assumed) have been rederived for

reactive batch distillation columns.With respect to the limiting operation conditions of

reactive disti llation columns, we have derived theUnder wood equations (minimum reflux) for the reactivecase (for both continuous and batch distillation).

The paper has been divided into six sections. Section2 presents some previous work re levant to our ap-proach: (1) it briefly explains th e grap hical design of acontinuous reactive distillation column using the Mc-Cabe - Thiele method 9 an d (2) it describes the McCabe -Thiele m ethod for batch distillation columns for th enonreactive case. 13 The contributions of the paper are

* To whom correspondence should be addressed. Tel.: + 52-(461)-6117575 ext. 156. F ax: + 52-(461)-6117744. E-ma il:vicent e@iqcelaya .itc.mx.

4000 Ind. Eng. Chem. Res. 2004, 43 , 4000 - 4011

10.1021/ie030658w CCC: $27.50 2004 American Chem ical SocietyPublished on Web 06/08/2004


2/12


3/12

3. S impl i fied Method for Analyz ing a ndDes igning Reac t ive Batch Dis t i l l a t ion Columns

This section provides the derivation of the expressions

needed to simulate the performan ce and/or to designreactive batch distillation columns using a simplifiedmethod. The solution stra tegy can be directly used forbinary systems, but an approximation is also proposedfor the case of multicomponent systems (see section 5.3).Our approach is based on the McCabe - Thiele m ethodfor r eactive column s an d on th e concepts of the rea ctivedifference point and molar turnover flow rate. Weconsider separ ately th e cases in which reaction occurse ither in the liqu id phase or in the vapor phase ,although t he a nalysis could be integrated. As a simpli-fied approach, the following are the assum ptions madethrough our derivations for kinetically controlled reac-tive batch columns:

1. Heat of mixing, sta ge heat losses, and sen sible hea tchanges of both liquid a nd vapor ar e n egligible.

2 . The pressure on the column remains constantthrough the operation.

3. Sta rtu p conditions will not be considered.4. The stages of the column will be considered in quasi

steady state.5. Reaction takes place on all of the stages but the

reboiler and condenser.6. Reaction is irreversible because of the continuous

removal of the pr oduct.If assumption 1 holds, then the molar flow rate of the

phase in which reaction does not occur can be consideredas a constant . Seeking simplicity, in the followingderivations we will obviate most of the algebraic stepsand provide only the main results. The analysis consid-ers two cases depending upon the phase in which thereactions occur. The derivations for the liquid-phasereactions are presented first.

3.1 . Reac t ive Batch Dis t i l l a t ion Column: Reac-t ion in the Liquid Phase . Consider the batch reactivedistillation column represented in Figure 3. k , j repre-sents the sum of the molar turnover flow ra te forcomponent k (reactant defined as the base component)in the stage j, the disti llate product is represented as D , and the reflux ratio as R ) L 0 / D . The subtractionsindicated in Figure 3 a re used t o emphasize the contr i-butions of reaction on each stage because k , j i s anaccumu lated su m. For a binary system involving com-ponents k (heavy reactant) and i (light product), the

overall a nd component ( i) balance equations aroundsection 1 of Figure 3 a re

where represents the stoichiometric coefficients of reaction (negat ive for reacta nt k and positive for producti), T is th e tota l sum of stoichiometr ic coefficient s, an d( i / - k )k , j represents the production of i due to reactionof k on stage j. The last of the terms in eq 5 representsthe overa ll change in the number of moles due to

reaction. Notice t hat, al though we are using a binarysystem as the basis for our derivations here, eqs 4 and5 can actually be generally applied, and that is the basisfor the extension of th e approach to m ulticomponentsystems.

Because for this case the reaction is taking place inthe l iquid phase, the vapor flow rate can be assumedas constant , but the l iquid flow ra te might changedepending on the stoichiometric coefficients. Changesin t he liquid flow rate could be calculated as

Observe tha t if i ) - k , then T ) 0 and L j ) L j- 1, sothat the assumption of constant molar flow will also hold

for the l iquid phase. Equations 4 - 6 can be used toobtain the McCabe - Thiele operating line for reactivebatch distillation:

an d for th e case in which t here is no change in t he liquidflow rate:

Also note that, if there is no change in the liquid flow

Figure 2 . McCabe - Thiele diagram for batch distillation underconsta nt reflux policy.

Figure 3 . Reactive batch distillation column.

yi , j+ 1 ) L jV

xi , j + DV

xi,D - i

- k k , jV

(4)

V ) L j + D - Tk , j (5)

L j ) L j- 1 + T(k , j - k , j- 1) (6)

yi , j+ 1 ) L jV

xi, j +1

R + 1 xi ,D - i

- k k , jV

(7)

yi, j+ 1 )R

R + 1 xi, j +1

R + 1 xi,D - i

- k k , jV

(8)

4002 Ind. Eng. Chem. Res., Vol. 43, No. 14, 2004


4/12

rate, i ) - k , eq 8 is basically the sa me as t he operat ingline obta ined for continuous distillat ion (eq 1). Furt her-more , by compar ing eqs 3 and 8 (nonreact ive andreactive batch distillation), we see that the contributionof reaction is given through the term - ( i / - k )(k , j / V ),or - (k , j / V ) if there is no chan ge in th e nu mber of moles.

3 .1 .1 . D y n a m i c s o f t h e R e b o i le r. To derive thedynamics of t he reboiler, consider the overall andcomponent balance for section 2 of Figure 4:

where k ,T is the t otal molar tur nover flow rat e ( k beingthe base component). The overall ma ss balan ce for thecondenser is

By combining eqs 9 - 11, one can show tha t t hedifferential equations representing the behavior in thereboiler and condenser are

Observe again tha t, when compar ed to the nonrea ctivecase for batch distillation, eqs 12 and 14 include an extraterm to account for changes due to reaction. F urther,notice th at in eq 14 th e separa te effects of reaction an dsepara tion are well defined.

3.1 .2 . Exten t o f the Reac t ion : Molar TurnoverFlow Rate . Hauan e t a l .7,11 provided a generalizationof the concept of a difference point for reactive cascades.The authors assume that a reactive separation cascade

can be considered as a separation cascade linked to achemical reactor having an equivalent conversion; suchan approach is known as phenomena decomposition andhas been explained in detail. 7,11 Basically, Hauan et al. 7show that the phenomena occurring in some combinedsystems of separation and reaction (mixing, separation,and reaction) can be separated a nd analyzed indepen-dently from each other. Moreover, they show tha t, for

the case of reactive cascades, given the extent of th ereaction, the decomposition does not affect the massbalance equations. 11

In a similar way, to derive an expression for thecalculation of the molar tur nover flow ra te k , j, in thispape r we a s sume tha t each sepa ra t ion s t age of areactive batch distillation column can be considered asa nonreactive equilibrium separation stage linked to acontinuous stirred tank reactor (CSTR; see Figure 5).If we were a ble to know a pr iori th e exact value of themolar turnover flow rate for each stage, then the massbalance equat ions obtained th rough the phenomenadecomposition assum ption would hold. H owever, th esequent ial approach (reaction - separation) applied hereis, in fact, used for the calculation of such a value of

the extent of the reaction. Hence, it is evident that inthis case a sequential approach will not exactly repre-sent a s imul taneous phenomenon in natu re an d thatthe assumptions m ade, such as r epresenting the reac-tion volume as a CSTR, are approximations that maylead to err or. A CSTR is used becau se we a re consider-ing perfect mixing in the liquid and vapor phases. Also,the CSTR is placed above the separation stage justbecau se of numerical implicat ions in t he solution scheme.

The component mass balance equation for the reactorof Figure 5 is given by

where is the reaction volume (liquid holdup), r k , j isthe rat e of reaction of component k in stage j, L is theliquid flow rate, and x is the liquid composition. Inprinciple, the calculation of would require the use of column hydraulic analysis, which could provide a rela-tionship between L j- 1 and .

The reactor out let l iquid s t ream, L j, ca n be e x-pressed in terms of the molar conversion fraction of thereactant, C j:

so that eq 15 becomes

Figure 4 . Mass balance for the dynamics of the reboiler.

Tk ,T - D ) d B /dt (9)

i- k

k ,T - Dx i,D )dd t

( B xi ,B) (10)

V - L0 - D (11)

d Bd t

) Tk ,T -V

1 + R (12)

D ) V 1 + R

(13)

d xi,Bd t

) 1 B{ V 1 + R( xi,B - xi ,D) + [ i- k - T xi,B]k ,T}(14)

Figure 5 . Stage on a rea ctive bat ch distillation column: reactionin th e l iquid phase.

(r k , j) ) L j- 1 xk , j- 1 - L j xk , j (15)

L j xk , j ) L j- 1 xk , j- 1(1 - C j) (16)

(r k , j) ) L j- 1 xk , j- 1C j (17)

Ind. Eng. Chem. Res., Vol. 43, No. 14, 2004 4003


5/12

Assumin g an element al irreversible reaction, rR f pP,the rate of reaction can be represented by an expressionsuch as

where the term inside the parentheses represents themolar concentration of the reactant in the reactor andk j is the kinetics parameter given by the Arrheniusequation. 0 is the volumetric flow rate entering thestage. 0 can be calculated from

where F is the density of the liquid strea m. Knowing 0would a lso allow t he evaluation of the residence t ime, j ) / 0; hence, if the liquid holdup is known or can beevaluated, the use of 0 or j is equivalent.

When eqs 16 - 18 are combined, an implicit equationfor the molar conversion of the rea ctant can be obtained:

Once eq 20 is solved to obta in C j (analytically ornum erically for k greater than 2), one can finally

calcula te the molar turnover flow ra te through theexpression

3.2. Reactive Batch Dist i l lat ion Columns: Reac-t i o n i n t h e Va p o r P h a s e . If the reaction takes placein the vapor phase, der ivat ions s imi lar to the onespresented in the previous subsection can be obtained.Hence, t he component and overall mass balances forsection 1 of the column presented in Figure 6 result inthe operating line for r eactive batch distillation columns(quasi-steady-state approximation):

where changes in the vapor flow ra te due to reactionare given by

If ther e are n o chan ges in th e vapor flow rat es, then eq22 reduces to eq 8.3.2 .1 . Dynamics of the Reboi le r. The overall and

component mass balances for section 2 of Figure 6allow the derivation of the differential equations thatrepresent the dynamics of the reboiler as describedearlier:

3.2 .2 . Exten t o f the Reac t ion : Molar TurnoverF lo w R a te . Our approach to ca lcula te the molarturnover flow rate is based again on phenomena de-composition as represented in Figure 7. Calculations inthe reactor are performed before the calculations on theequilibrium st age. The consideration of having a sepa-ra tion sta ge connected to a CSTR in t he case of rea ctionin th e vapor pha se could seem ina ppropriate; however,more than the need of a CSTR, our approach works byassuming that the compositions of reactive and nonre-active streams are homogeneous as if the mixing onth em wa s perfect. A derivation similar t o tha t of section3.1.2 results in the following expressions:

where the conversion fraction of the reactant can beobtained from t he implicit eq 25:

3.3 . Solu t ion Approach . The resulting model is awell-posed differential - algebraic system of equations(DAE). Table 1 presents a summary of th e modelspecifications and the equations used for the calcula-t ions of the var iables involved in each s tage . Thesolut ion p rocedur e for ea ch time st ep of a r eactive batchdistillation column is similar to the procedure for solvingthe rectifying section of a continuous reactive column.The equations involved in the calculation could be solvedeither simultaneously with an appropriate numericalmethod or sequentially by using a method such as thedirect substi tution method. We describe next the se-quen tial procedure for th e case in which reaction occursin the liquid phase (constant molar flow in the vaporphase) in all of the st ages but the reboiler; an iterat iveprocedure for the calculation of the distillate composi-tions has to be completed.

Considering a constant reflux policy, we assume thatthe number of stages ( N ), the reflux ratio ( R ) , and theinitial conditions (feed load and still compositions) are

F i g u r e 6 . Derivation of expressions for the case of reaction inthe vapor phase.

r k , j ) k j( L j xk , j 0 )k

(18)

0 ) L j- 1 / F (19)

L j- 1 xk , j- 1C j ) k j[ L j- 1 xk , j- 1(1 - C j) 0 ]k

(20)

k , j ) j) 1

n

( L j- 1 xk , j- 1C j) (21)

yi, j+ 1 )V 1

V j+ 1

R R + 1 xi , j +

V 1V j+ 1

1 R + 1 xi,D -

i- k

k , jV j+ 1

(22)

V j+ 1 ) V j + T(k , j - k , j- 1) (23)

d Bd t

) Tk ,T -V 1

1 + R

d xi,Bd t

) 1 B{V 11 + R ( xi ,B - xi,D) + [ i- k - T i ,B]k ,T}

k , j ) j) 1

n

(V j yk , jC j

1 - C j) (24)

V j yk , jC j ) k j[V j yk , j 0 ]k

(1 - C j) (25)



6/12

given. Also, a typical specification in this case is thevapor boilup flow rate ( V ), which will remain consta ntthrough the column. As one of the main limitations of our approach, assumed values of the liquid holdup, ,and entering volumetric flow rate, 0 (or r esidence time, ), are used in this work (see section 5 for a n explana -tion).

Hence, we start by assuming the values of the dis-tillate compositions. Then, by using a bubble-point cal-culation, the temperature of the distillate can be com-puted. When the temperatures of the reboiler (whichcan also be computed through a bubble-point calculationbecau se th e compositions in th e reboiler ar e kn own) andthe condenser are known, a linear temperature profileis assumed. The temperature of each stage is requiredbecause the computations of the kinetics parameter andthe relative volatility of the mixture depend on it. Then,a m ass ba lance in th e condenser allows the calculationof L 0; for the condenser k ,0 ) 0 (no reaction in thecondenser) and eq 5 reduces to V ) L 0 + D , which isequivalent to L 0 ) V [ R /( R + 1)]. Also notice that thedistillate compositions are equal to the compositions of the vapor flow entering the condenser ( yk ,1).With that,we proceed to perform the ca lcula t ion on s tage 1 .Equations 20 and 21 are used for the calculation of C 1and the m olar tur nover flow rate, k ,1 , respectively. Theliquid molar flow rate on stage 1, L 1, is obtained fromeq 6. Equation 16 and the summation equation for thecompositions are used to find the values of the liquidflow ( L 1) and the compositions ( xk ,l) entering the equi-l ibrium separation stage of Figure 5. Then, the equi-librium curve (equilibrium in terms of relative volatility)and the operating line (eq 8) allow the calculation of the rest of the compositions for stage 1 ( xk ,1 and yk ,2).The procedure cont inues for the res t of the s tagesmoving down th e column un til we evaluate the composi-tions in t he reboiler. Those values provide the conver-gence criteria: If the comput ed values of the reboiler

compositions are equal to t he known values, then weproceed to the n ext tim e step; other wise, we modify ourguess for the distillate compositions and repeat the cal-culations in the column. Notice that, with respect to thenum ber of degrees of freedom, the model is squa re be-cause t he d istillat e compositions can be indirectly foundfrom th e independent equations used t o calculate t hestill compositions (sta ge-by-stage p rocedure ). Before onemoves on to th e next time step, integra tion of eqs 12 -14 is needed to find the values of the amount of mixturein the still, the distillate collected, and the bottom com-positions for t he following t ime st ep. N , R , and V remainas constants so that the simulation continues for thenext t ime steps and ends when a specified stoppingcriterion is met .

4. Limi t ing Condi t ions for Reac t ive Dis t i l la t ionColumns

In this section we derive the expressions to calculatethe minimum reflux (Underwood equa tions) for continu -ous and batch reactive disti l lation columns; further-more, a discussion regarding th e Fensk e equat ion (mini-

mum number of stages) for continuous reactive distil-lation columns is also provided.

4.1 . Underwood Equat ions . To derive the expres-sion for the minimum reflux in the case of continuousreactive distillation columns, a system such a s th e oneshown in Figure 8 is assumed. Our approach is basedon the derivation for t he n onrea ctive case pr esented byKing. 14 The section with an infinite number of stagesis placed in the rectifying section above the feed plate.A further assumption in our derivation is that there isno reaction in the section with an infinite number of plates (to avoid depletion of the reactan t).

The mass balance equation for component i aroundsection 1 of Figure 8 is

Figure 7 . Extent of the reaction: reaction in the vapor phase.

Ta bl e 1 . D e g r e e s o f F r e e d o m A n a l y s is : S p e c i fi c a t i o n sa n d C o m p u t a t io n s o n S t a g e j

va ria ble s pecified/ca lcu la ted fr om

N , R , V , B 0, xB specified 0, specified L0 eq 5 L j eq 6C j eq 20k , j eq 21 xk , j equilibrium (relative volat ility) yk , j+ 1 eq 8 xk , j L j eq 16 (plus sum mat ion

of compositions) xD itera tively from independent

equations used to calculate xBFigure 8 . Reactive distillation column with an infinite numberof stages.



7/12

where the index has been ass igned to the s t reamscoming from the section with an infinite number of plates. The overall mass balance for the condenser is

It is a lso considered th at the liquid - vapor equilibriumon th e last stage of section 1 is given by

When eq 28 is subst i tu ted in to eq 26 and wi th rear-rangement:

When the definition of relative volatility (with respectto the key component r ) is incorporated:

Here a new var iable, ) (1/ K r ,)( L / V ), called theabsorption factor, is introduced. Also, because thesummation of the compositions in the vapor streamgoing into the section with an infinite number of stages

has to be 1, eq 30 reduces to eq 31. where NC is the

number of components. Finally, the substitution of eq27 an d t he definition of the reflux r atio in eq 31 r esultin

which is the fi rst of the Underwood equat ions forcalculating the minimum reflux. Following a similarprocedure for section 2 (nonreactive stripping section)of Figure 8, one can obtain

Combining eqs 31 and 33 and t aking into account th at ) and Ri , ) R i , result in

Finally, by intr oducing t he definition of q

and the component mass balance equation for the wholecolumn

one can get

which is the second of the Underwood equations. Equa-tion 35 can be u sed for calculating and eq 32 for

calculating R min for continu ous reactive distillationcolumns.4.1 .1 . Extens ion of the Un derwoo d Equat ions to

Reac t ive Batch Dis t i l l a t ion Columns . A direct ex-tension to th e Un derwood equations derived above canbe done for the case of react ive batch dis t il la t ioncolumns. An approach similar to the one proposed byDiwekar 13 serves that purpose (Figure 9).

In Figure 9, a reactive batch distillation column isrepresented as a continuous column where the feedenters the column a t t he bottom sta ge with a temper-atu re equa l to its bubble point a nd with a compositionequal to xi ,B . Such a representa tion a llows rewriting of eqs 32 and 35 as

which correspond to the Underwood equations forreactive bat ch disti llation columns. Observe th at theminimum reflux computation not only depends on thedisti llate and bottom compositions, as i t does for anonreactive column, but also depends on the extent of the reaction. Therefore, a value of the extent of thereaction has to be given in order to make eq 37 usefulin practical terms. It might be difficult to provide anapproximation to the extent of the reaction a priori, anda parametr ic search might be necessary to obta in afeasible value. For a specific reaction - separation sys-tem, a r easonable value of th e extent of the rea ction canbe estimated, for instance, by assuming that the molarturnover flow rate is equally distributed among thereact ive s tages of th e column and by defin ing theamount of product obtained through the reaction (usinga value of the molar conversion fraction based on thekinetics of the reaction involved and the amounts of thereactants initially fed to the column).

Vyi, ) L xi , + Dx i,D - i

- k k ,T (26)

V ) L0 + D (27)

xi, ) yi, / K i, (28)

yi,V D

) xi ,D -

i- k

k ,T D

1 - 1K i,

LV

(29)

yi,V D

)Ri,( xi ,D - i- k

k ,T D )

Ri , -1

K r ,

LV

(30)

V

D)

i) 1

NC[Ri,( xi ,D - i

- k

k ,T

D )Ri, - ] (31)

Rmi n + 1 i) 1

NC

[Ri,( xi ,D - i- k

k ,T

D )Ri, - ]

(32)

-V h B

) i) 1

NC

[Ri, xi,BRi, - ] (33)

V - V h ) i) 1

NC[Ri ,( Dx i,D - i

- k k ,T + Bxi ,B)

Ri , - ](34)V - V h ) F (1 - q )

Fz i,F ) Dx i ,D - i

- k k ,T + Bxi,B

1 - q ) i) 1

NC

[Ri , zi,FRi , - ] (35)

i) 1

NC

[Ri, xi,BRi, - ]) 0 (36)

R mi n + 1 ) i) 1

NC[Ri ,( xi ,D - i

- k

k ,T

D )Ri, - ] (37)



8/12

4 .2 . C o mm e n ts o n t h e F e n sk e E qu a ti on fo r

Reac t ive Dis t i l l a t ion . The derivation of the Fenskeequation for nonreactive continuous columns assum esoperation at total reflux. Interesting conclusions can bedrawn with respect to the corresponding derivation inthe reactive case. Let us assume that an irreversibleisomerization reaction (binary mixture) is t aking placein the column and that the product of the reaction isthe most volatile component.

The case of total reflux is then the first one that weana lyzed for the r eactive case, as shown in F igure 10a.A problem arises, however, in such a scheme. Becausethe r eaction t akes place regardless of whether productsare being obta ined or not , as t ime and the react ionprogress, the r eactant will get depleted and the column

behaves just like a chemical reactor. Hence, the mini-mum number of stages in a reactive distillation columndoes not necessarily correspond to total reflux operation.An outlet product stream , such a s the distillate streamin Figure 10b, makes the problem worse because thelight product will be continuously removed an d thecolumn will become empty if no inlet stream is incor-porated; a feed to the column is therefore required.Figure 10c shows a column s imi lar to th e schemeconsidered for r eactive batch d istillat ion colum ns in th ederivation of th e Under wood equa tions becau se th e feedenters the column just above the reboiler. We believethat such a configuration could be used for obtainingthe Fenske equation in reactive distillation. However,

our experience of u sing the methodologies for thederivation of the Fenske equation reported by King, 14Henley a nd Seader, 15 and Hohmann 16 is t h at t h eappearance of the molar turnover f low ra te and thereflux ratio makes the equation more complicated as wemove down the column. Also, most of the reasonableassumptions used to simplify it introduce significanterrors in the numerical calculation. Moreover, thesummations involved do not seem to correspond to aconverging series of any kind. This topic should be thesubject of further investigation. Sundmacher and Kien-le 17 provide a simple expression for calculating theminimum number of stages for the case of a reactiontaking place in the reboiler of the column.

5. Numer ica l Examples and Analys i s o f theResul t s

This section describes the numerical examples per-formed to i llustrate the application of t he proposedmeth odology for reactive ba tch distillation columns.Because of the dynamic natur e of batch columns, thecompositions of the disti llate product change withrespect to time for a constant reflux operation policy.Hence, the results of the simulations are reported in

terms of the average distillate composition profiles of the key component versus time. The average distillatecomposition xj i ,D is calculated through integration asfollows:

When possible, the results were compared to t hoseobtained through the use of BatchFrac from AspenPlus.The Antoine parameters (for the vapor - liquid equilib-rium) were taken from Reid et al. 18 To keep th e designmethod simple enough, we did n ot include columnhydraulic equations in our approach; that means t hatvalues for the l iquid holdup have to be assumed.Fur ther more, the kinetic para meters (frequency factorand activation energy) were provided and adjusted toachieve an accumulated value of the extent of thereaction equal to the one obtained through AspenPlussimulations. Note th at our goal in th is work was not toperform realist ic calculations but to show that oursimulation results present reasonable agreement withthose of the state of the art process simulators; as amatter of fact, the assumed values for the liquid holdupand t he kinetic parameters were the sam e as th e onesused in Ba tchFra c. Finally, we also assumed a consta ntvalue of the volumetric flow rate 0; however, such a

simplification can easily be removed by evaluating theliquid density on each sta ge as a function of temperatu reand composition and then using eq 19.

5.1 . Reac t ion Causes No Chang e in the Nu mberof Moles . For a scenario in which the assumption of constant molar flow holds for both the liquid and vaporphases, two cases were solved. In the first one, thereaction takes place in the liquid phase; in the secondexample, the rea ction is assum ed to ta ke place only inthe vapor phase.

5.1 .1 . Reac t ion in the Liquid Phase . The examplecorresponds to an isomerization reaction in the liquidphase: o-xylene (C 8H 10 ) f ethylbenzene (C 8H 10 ). Thelightest component is ethylbenzene. The dat a used forthis example are shown in Table 2.

The McCabe - Thiele diagram for the initial time stepof this example is shown in Figure 11. As expected, thecontribution of the reaction is greater for stages closerto th e reboiler (larger r eactant concentra tions). Figure11 does not show the last of the operating lines becausethe difference point is negative. Figure 12 presents thr eeinstances of the calculation of the average disti l latecomposition by changing the value of the reflux ratio.Because the reaction shows high conversion values, theeffect of th e reflux ra tio is not significan t. Observe a lsotha t, because of reaction, th e a verage distillat e composi-tion increases with time; this result is contrary to thebehavior observed in nonreactive batch disti llationcolumns. This effect can be explained by looking at the

Figure 9 . Reactive batch column as a continuous column.

xj i,D(t ) ) 0

t Dx i ,D d t

0t D d t

(38)



9/12

right-hand side of eq 14. Because xB < xD, the first t ermis always n egative; the st ill composition (and, t her efore,the disti llate composition) will only increase if th esecond term (reaction) of eq 14 is positive and largertha n t he first one (separ at ion). Finally, Figure 13 sh owsa comparison between the results of our approach andthe results obtained through BatchFrac of AspenPlusfor a reflux ratio equal to 2 and a production of 73.83mol of ethylbenzene. An excellent agreement is ob-served.

5.1 .2 . Reac t ion in the Vapor Phase . The isomer-ization reaction n -butane (C 4H 10 ) f isobuta ne (C 4H 10 )tak es place in th e vapor phase; isobuta ne is the lightestcomponent. Table 3 shows the parameters used in thesimulation for this example. The McCabe - Thiele dia-gram for a simu lation t ime of 0.6 h is shown in F igure14. Once again, the last of th e difference points isnegative, and th e last of the operat ing lines ha s not beendrawn. The diagram corresponds to a r eflux rat io equalto 2 . Figure 15 shows the behavior of the averagedistillate composition of isobutane; the trend is similar

Figure 10 . Fenske equation for reactive distillation.

F i g u r e 11 . McCabe - Thie le d iagram for the in it ia l s tep inexample 1.

Table 2 . Da ta fo r the Reac t ion - Separa t ion Sys tem o-Xylene - Ethylbenzene

parameter va lue

no. of st ages 4 + reboilerfeed 100 kmolfeed composit ion of e thylbenzene 0 .1vapor boilup flow ra t e 140 kmol h - 1column pressure 1 a tmba tch t ime 1.4 horder of react ion 1.0

act iva t ion energy 30 000 kJ kmol- 1

frequency factor 1.6 10 9 h - 1volumet r ic flow ra t e 2 m 3 h - 1liquid holdup 4 10 - 4 m 3

Figure 12. Average distillate composition profiles of eth ylbenzenein example 1.

Figure 13 . Our results a s compared to t hose of BatchFrac forethylbenzene.



10/12

to the one on th e previous example because t he a veragedistillate composition increases with time because of reaction. Also observe in Figure 15 th at the reflux ra tiohas very little effect on the average product composition.Compar ison of this case to BatchFr ac was not possiblebecause BatchFrac does not support the case wherereaction occurs in t he vapor phase in a kineticallycontrolled column.

5 . 2 . R e a c t i o n C a u s e s a C h a n g e i n t h e N u m b e rof Moles. This situation is illustrated with the systeminvolving the reaction isobutyl - isobutyra te (C 8H 16 O2)f 2-isobutyraldehyde (C 4H 8O) in t he liquid phase. Thelightest component is isobutyraldehyde. Kinetics andequilibrium data are shown in Table 4. Figure 16 showsthe McCabe - Thiele diagram for the operation time of 0.6 h. The disti llate composition is very close to 1.Changes in the number of moles present an impact onthe difference points as well as on the slope of the

operating lines. Observe that the movement from onedifference point to an other is very notorious a s we moveto the left from sta ge 2 to stage 3. Figure 17 sh ows th eaverage distillate composition profiles for three valuesof the reflux ratio; because the l iquid flow rates arechan ging from one sta ge to another, the reflux rat io hasa significant effect in the composition. Also, in thisexample separation predominates over reaction and theaverage distillate composition of isobutyraldehyde de-creases with t ime. Finally, Figure 18 compares ourresults to those obtained through BatchFrac for a refluxrat io of 2.

5.3. McCabe - T h ie l e Me t h o d f o r M u l t ic o m p o -n e n t M i x t u r e s . The McCabe - Thiele method as de-scribed in section 3 of th is pap er can also be app lied formulticomponent mixtur es (NC componen ts) becau se th eequat ions developed for t he r eaction eith er on t he liquidphase or the vapor phase remain valid. To solve theproblem for multicomponent mixtures, a procedure

Figure 14 . McCabe - Thiele diagram for example 2.

Figure 15 . Average distillate composition profiles for isobutanein example 2.

Table 3 . Da ta fo r the Reac t ion - Separa t ion Sys tem n -Butane - I sobutane

pa rameter va lue

no. of st ages 4 + reboilerfeed 100 kmolfeed composition of n -but ane 0.1va por boilup flow ra t e 140 km ol h - 1column pressure 1 a tmba t ch t ime 1.4 horder of react ion 1.0act iva t ion energy 40 000 kJ kmol - 1frequency factor 2 10 3 h - 1volumet r ic flow ra t e 2 m 3 h - 1react ion volume 4 10 - 4 m 3

Figure 16 . McCabe - Thiele diagram for example 3.

Figure 17 . Average distillate composition for isobutyra ldehydein example 3.

Table 4 . Da ta fo r the Reac t ion - Separa t ion Sys temIsobuty l I sobutyra te - I sobutyra ldehyde

pa rameter va lue

no. of stages 3 + reboilerfeed 100 kmolfeed composition of isobutyraldehyde 0.1vapor boilup flow ra te 140 kmol h - 1column pressure 1 a tmbat ch t ime 1.4 horder of r eact ion 1.0act iva t ion energy 30 000 kJ kmol - 1fr equency factor 1 10 3 h - 1volumet r ic flow ra te 2 m 3 h - 1liquid holdup 4 10 - 4 m 3



11/12

consisting of the simultaneous calculation of NC - 1components is needed. A furt her a ssumption is tha t t hera te of react ion can be expressed in terms of theconcentra tion of just one of the r eactant s (if more th anone). Hence, the molar turnover flow rate for the restof the components in terms of the base component isgiven by a stoichiometric relationship

Also, equilibrium can be calculated in terms of theequilibrium constant instead of relative volatilities asfollows.

An example for t he r eaction - separ at ion of mu lticom-ponent mixtu res is illustra ted with th e system involvingthe reaction (no change in the number of moles) in thel iquid phase: 2 -toluene (C 7H 8) f benzene (C 6H 6) +o-xylene (C

8H

10), where the order in relative volatility

is benzene > toluene > o-xylene. The dat a u sed for t hiscase are shown in Table 5.

The McCabe - Thiele solution approach was used forbenzene and to luene. Figure 19 shows the averagedistillat e composition of the th ree component s when th ereflux ratio is equal to 2. This is an interesting casebecause reaction presents a higher contribution thanseparat ion for benzene and o-xylene, so that theirdistillat e compositions increase with time. H owever, forthe case of to luene, the t rend is d i fferent and thecomposition decreases with time.

5.4. Minimu m Reflux Calculat ion . The calculat ionof the minimum reflux rat io by using the Underwoodequations derived in this work (eqs 32 and 35) for

continuous reactive distillation columns was tested forthe case of th e isomerization reaction isobutane fn -buta ne. The system considered is a column with ninestages where the feed enters the column as a saturatedliquid in stage number 8. The column pressure is 1 atm.The feed is 1000 kmol h - 1 with a composition of 0.04 of n -buta ne; th e distillate composition of n -butane is 0.98.

I t is a s su m ed t h a t 3 80 k m ol h - 1 of n -butane areproduced in the column a nd th at su ch a molar t urn overflow rate is equally distributed in the stages of thecolumn. Solving eqs 32 a nd 35 results in R mi n equal to0.518. After th e solution approach using t he McCabe -Thiele meth od is used for a ra nge of values of the refluxrat io, a value of 0.51 was found for t he m inimum valueof the reflux ratio that allows the reaction and separa-tion specified. Such a result indicates that the Under-wood equations derived in this work present a reason-able agreement with the simulations.

6. Concluding Remarks

This paper provides the mathematical derivationsneeded for the simplified an alysis and design of reactivebatch disti l lation columns. In our opinion, the maincontr ibut ions of the paper are as fol lows: (1) Thedevelopment of the t heoretical a nd mat hema tical t oolsthat a l low the use of a McCabe - Thiele strategy forreactive batch distillation columns. This includes thederivation for the dynamics of the reboiler and theoperating l ine under the assumption of quasi steadysta te in the p la tes . Fur thermore , an approach to thecomputation of the molar turnover flow rate of reactivedistillation stages has been proposed. Such an approachis based on phenomena decomposition and is needed inorder t o apply th e concept of the r eactive differen ce pointto the case of batch distillation. (2) The derivation of the Underwood equations for continuous and batchreactive distillation columns.

The ma ny assum ptions considered in th is work h avebeen extensively explained, an d we expect that ournumerical examples involving various reactive systemsshow the scope and confi rm the usefulness of ourapproach. Through our analys is and numerical ex-amples we have (1) shown that our results present areasonable agreement with th ose of the st ate of the artprocess simulators for reactive bat ch distillation col-umns, (2) emphasized the advantages of the combina-tion of separation and reaction for batch distillation,where th e product purity can even be improved as timeprogresses , and finally (3) proved that the resultsobtained for R min by the derived Underwood equa tions

F i g u r e 1 8 . Comparison in example 3 for the composition of isobutyraldehyde.

Table 5 . Da ta fo r the Reac t ion - Separa t ion of theMul t icomponent Mix ture o f Example 4

pa rameter va lue

no. of st ages 2 + reboilerfeed 100 kmolfe ed com pos it ion of b en ze ne 0 .0 06feed composition of o-xylene 0.485va por boilup flow r at e 140 km ol h - 1

column pressure 1 a tmba t ch t ime 1 horder of react ion 1.0act iva t ion energy 30 000 kJ kmol - 1frequency factor 8 10 2 h - 1volumet r ic flow r a t e 2 m 3 h - 1liquid holdup 4 10 - 4 m 3

i, j ) ik

k , j (39)

xi , j ) K i, j yi, j (40)

Figure 19 . Average distillate composition for a reflux ratio equalto 2 in example 4.



12/12

for reactive columns are approximated enough to theresults obtained t hrough par ametric search.

A c k n o w l e d g m e n t

M.A.H.-G. and V.R.-R. are thankful for the financialsupport provided by th e Consejo del Sistema Naciona lde Educacion Tecnologica (CoSNET) and by the ConsejoNacional de Ciencia y Tecnologia (CONACYT). V.R.-R.and S.H.-C. also thank the Consejo de Ciencia y Tec-nologia del Est ado de Guan ajuat o (CONCyTEG).

Li te ra ture Ci ted

(1) Lee, J. W.; Westerberg, A W. Visua lization of Sta ge Calcula-tions in Ternary Reacting Mixtures. Comput. Chem. Eng. 2000 ,24 , 639.

(2) Lee, J. W.; Hauan, S.; Westerberg A. W. Graphical Methodsfor Rea ction Distr ibution in a Reactive Distillation Column . AIChE J . 2000 , 46 , 1218.

(3) Barbosa, D.; Doherty, M. F . A New Set of CompositionVariables for the Representation of Reactive Phase Diagrams.Proc. R. Soc. London 1987 , 413 , 459.

(4) Barbosa, D.; Doherty, M. F. Design and Minimum-RefluxCalculat ions for Single-Feed Mult icomponent Reactive Distillat ionColumns. Chem. Eng. S ci. 1988 , 43 , 1523.

(5) Espinosa, J .; Scenna , N.; Perez, G. Graphical Pr ocedure forReactive Distillation Systems. Chem. Eng. Commun. 1993 , 119 ,109.

(6) Perez-Cisneros, E. S. Modelling, Designing and Analysis of Reactive Separation Processes. Ph.D. Thesis, Technical Universityof Denmark, Lyngby, Denmark, 1997.

(7) Hauan, S.; Westerberg, A. W.; Lien, K. M. PhenomenaBased Analysis of Fixed Points in Reactive Separation Systems.Chem. Eng. S ci. 1999 , 55 , 1053.

(8) Lee, J . W.; Hau an, S.; Lien, K. M.; Westerberg, A. W. AGraphical Method for Reaction Design Reactive Disti llationColumn s. 1. The Ponchon - Savarit Method. Proc. R. Soc. London2000 , 456 , 1953.

(9) Lee, J . W.; Hau an, S.; Lien, K. M.; Westerberg, A. W. AGraphical Method for Reaction Design Reactive Disti llationColumns. 2. The McCabe - Thiele Method. Proc. R. Soc. London2000 , 456 , 1965.

(10) Lee, J. W.; Hauan, S.; Westerberg, A. W. Circumventingan Azeotrope in Reactive Distillation. Ind. Eng. Chem. Res . 2000 ,39 , 1061 - 1063.

(11) Hauan, S.; Ciric, A. R.; Westerberg, A. W.; Lien, K. M.Difference Points in Extractive and Reactive Cascades. I. BasicProperties and Analysis. Chem. Eng. S ci. 2000 , 55 , 3145.

(12) Lee, J . W.; Hauan, S.; Lien, K. M.; Westerberg, A. W.Difference Points in Extractive and Reactive Cascades. II. Gen-erating Designing Alternatives by the Lever Rule for ReactiveSystems. Chem. Eng. S ci. 2000 , 55 , 3161.

(13) Diwekar, U. M. Batch Distillation ; Taylor and Francis:Washington, DC, 1995.

(14) King, C. J. S eparati on Processes ; McGraw-Hill: New York,1980.

(15) Henley, E. J.; Seader, J. D. Equilibrium Stage SeparationOperations in Chemical En gineering ; Wiley: New York, 1981.

(16) Hohman n, E. C. Analytical McCabe - Thiele Method (Smok-ers Equation). AIChE J . 1980 , B1.8 , 42.

(17) Sundmacher, K.; Kienle, A. Reactive Distillation ; Wiley-VCH: Weinheim, Germany, 2003.

(18) Reid, R. C.; Prau snitz, J . M.; Poling, B. E. The Propertiesof Gases and Liquids ; McGraw-Hill: New York, 1987.

Received for review August 11, 2003 Revised m anu script received May 5, 2004

Accepted May 5, 2004

IE030658W


batch reactive art 1

Documents