Computers and Chemical Engineering 28 (2004) 17�/25

www.elsevier.com/locate/compchemeng

Feasibility of equilibrium-controlled reactive distillation process:application of residue curve mapping

Cristhian Almeida-Rivera *, Johan Grievink

Process Systems Engineering Group, DelftChemTech, Delft University of Technology, Julianalaan 136, 2628 BL Delft, Netherlands

Abstract

The residue curve mapping technique (RCM) has been considered a powerful tool for the flow-sheet development and preliminary

design of conventional multi-component separation processes. It does not only represent a good approximation to actual

equilibrium behavior, but also it allows performing feasibility analysis of separation processes where non-ideal and azeotropic

mixtures are involved. Applications of RCM to reactive distillation processes have recently been reported, but a generalized and

systematic approach is still missing for the case of reactive feeds outside the conventional composition ranges and represents the aim

of this contribution. An RCM-based feasibility analysis has been applied to the homogeneous reactive distillation synthesis of

methyl tert -butyl ether (MTBE) at 11 atm from isobutene (IB) and methanol (MeOH) and in the presence of n -Butane (nC4). The

reaction space, defined in terms of transformed composition variables, has been divided into sub-regions characterized by separation

boundaries. A feasibility analysis of the reactive distillation process has been performed based upon the location of the reacting

mixture, defined by [xMeOHF , xIB

F , xn C4] � /[0, 1], and initial separation sequences have been generated according to the feed

transformed-composition. In all the cases, high purity MTBE has been obtained.

# 2003 Elsevier Ltd. All rights reserved.

Keywords: Reactive distillation; Residue curve mapping (RCM); Feasibility analysis; Graphical methods; Methyl tert -butyl ether; Non-conventional

feed compositions

1. Introduction

The art of process design involves finding equipment

sizes, configurations and operating conditions that will

allow an economical operation, only by specifying the

state of the feeds and the targets on the output streams

of a system. As reported by Fien and Liu (1994), residue

curve mapping (RCM) techniques may assist the design

task of*/particularly*/non-ideal systems.Traditionally, non-reactive residue curves have been

used to predict the liquid-composition trajectories in

continuous distillation units in the �/� case (i.e.

infinite number of stages and infinite reflux). Although

for finite columns those profiles differ slightly compared

with the residue curves under same isobaric conditions

(Fien & Liu, 1994), this difference might be considered

* Corresponding author. Tel.: �/31-15-278-4466; fax: �/31-15-278-

4370; http://www.dct.tudelft.nl/pse.

E-mail addresses: c.p.almeida@tnw.tudelft.nl (C. Almeida-Rivera),

j.grievink@tnw.tudelft.nl (J. Grievink).

0098-1354/03/$ - see front matter # 2003 Elsevier Ltd. All rights reserved.

doi:10.1016/S0098-1354(03)00168-6

negligible at the first stages of design. Analytically,

RCM’s are constructed based upon physical properties

of the system (i.e. VL equilibrium, LL equilibrium andsolubility data), wherein the composition of a non-

reacting liquid remaining in the system may be deter-

mined by performing an overall and component mate-

rial balances in the system (Westerberg, Lee & Hauan,

2000) (Fig. 1),

dL

dt��V ; (1)

d(xiL)

dt�L

xi

dt�xi

dL

dt��yiV i � [1; n]: (2)

where: L is the mass of liquid in the system, V is the

vapor flow, xi is the molar composition of component i

in the liquid phase, yi is the molar composition of

component i in the vapor phase and n is the number of

components present in the system.Replacing expression (1) in expression (2) and intro-

ducing a warped time (dt�/dt �/ V /L ) allow to rewrite

the RCM expression as,

Fig. 1. Simple batch still for the experimental determination of (non-)

reactive residue curves.

C. Almeida-Rivera, J. Grievink / Computers and Chemical Engineering 28 (2004) 17�/2518

dx

dt�x�y: (3)

where: x¯

is the composition vector in the liquid phase [x1

x2 x3 . . . xn ]T and y¯

is the composition vector in the

vapor phase [y1 y2 y3 . . . yn ]T .

Combining the previous [n�/1] independent expres-

sion (3) with and appropriate vapor�/liquid equilibrium

model enables to define the set of simultaneous equa-

tions that describe the residue curves,

dxi

dt�xi�yi i � [1; n�1];

yi� f (T ; P; xi) i � [1; n];

Xn

i�1xi�

Xn

i�1yi �1: (4)

When a mixture of n-components undergoes R simulta-

neous equilibrium chemical reactions, the RCM expres-

sion may be described in terms of transformed molarcompositions and a reaction-warped time,

dXi

dt�Xi�Yi i � [1; n�R�1]: (5)

with,

Xi�xi � vT

i (Vref )�1xref

1 � vTtotal(Vref )�1xref

Yi�yi � vT

i (Vref )�1yref

1 � vTtotal(Vref )�1yref

i � [1; n�R];

Vref �v(n�R�1);1 � � � v(n�R�1);R

n vi;r nvn;1 � � � vn;R

0@

1A;

dt�dt ��1 � vT

total(Vref )�1yref

1 � vTtotal(Vref )�1xref

�� V=L: (6)

where: Xi and Yi are the transformed composition

variables for the component i � /[1, n�/R ] in the liquid

and vapor phases respectively, yi/T is the row vector of

stoichiometric coefficients for component i � /[1, n�/R ]

in all the R reactions [yi ,1 yi ,2 . . . yi ,R ], vtotal/T is the row

vector of the total mole number change in each reactionan

i�1 vi;1 ani�1 vi;2 . . . an

i�1 vi;R

� ; ref denotes the re-

ference components for the R reactions, numbered from

n�/R�/1 to n and Vref is the square matrix ofstoichiometric coefficients for the R reference compo-

nents in the R reactions.

From the above-mentioned expressions it can be

inferred that:

. the new set of variables defines a sub-region of lower

dimension in the composition space*/reaction

space*/composed of the feasible product composi-tions for equilibrium controlled reactions,

. the transformed compositions keep the same value

for a liquid mixture before and after reaction

equilibrium is attained (i.e. Xi0�/Xi

e) and are normal-

ized (i.e. an�Ri�1 Xi �an�R

i�1 Yi �1);/. in the new composition space the reactive curves may

be described by a simplified expression, which has

exactly the same functional form as the distillationequation for non-reacting mixtures in terms of mole

fractions,

. ref can be a product or a reactant, depending upon

the value of ytotal B/0, and

. the reaction warped time increases monotonically

with the time (t jt0�/0; t jtL�0

�/�).

The reactive problem is completed by calculating the

chemical equilibrium constants for the multi-component

vapor�/liquid mixture,

keq;r�Yi�n

i�1

(gi � xi)vi �f (xi; gi(xi; T)): (7)

where: keq,r is the chemical equilibrium constant for

reaction r � /[1, R ].

The equations involved in the reactive problem

include [n�/R�/1] RCM expressions, [2n�/2R ] trans-

formed composition definitions, [n ] phase equilibrium

conditions, [R ] chemical equilibrium constant and [2]

normalization expressions, resulting in [4n�/2R�/1]

equations. On the other hand, the involved variablesare xi � [1,n ], yi � [1,n ], Xi � [1,n�R ], Yi � [1,n�R ], P and T ,

resulting in [4n�/2R�/2] variables. Since the problem

has only one degree of freedom, all the unknown

variables may be found by specifying a single variable

(e.g. P ) and solving simultaneously the following set of

differential and algebraic expressions,

Xi�

Xi�xi � vT

i (Vref )�1xref

1 � vTtotal(Vref )�1xref

i � [1; n�R];

C. Almeida-Rivera, J. Grievink / Computers and Chemical Engineering 28 (2004) 17�/25 19

Yi�yi � vT

i (Vref )�1yref

1 � vTtotal(Vref )�1yref

i � [1; n�R];

yi� f (T ; P; xi) i � [1; n];

keq;r�Yi�n

i�1(gi � xi)

vi r � [1; R]: (8)

2. Feasibility analysis: distillation-based approach

Residue curve maps provide a powerful tool to

represent relevant properties of the system, particularly

those aiming to predict feasible design sequences. In

addition, analytical material balances may be repre-

sented in a RCM, resulting in constraints to feasible

product compositions and convenient operating strate-

gies (e.g. direct or indirect distillation).

The presence of (reactive) singular points (i.e. purecomponents and azeotropes) in RCM’s allows to divide

the (transformed) composition diagram into separate

(reactive) distillation regions by introducing (reactive)

distillation separatrices, which connect two singular

points in the composition space. If a separatrice ends

at a saddle point, it is commonly referred to as boundary

curve. Crossing distillation boundaries has been exten-

sively reported for industrial cases and especially forthose with curved distillation boundaries. For those

systems, the boundaries can be crossed by simple recycle

methods provided an appropriate entrainer. However,

conventional design strategies require that the feed and

products remain in the same distillation region (Fien &

Liu, 1994).

Since sharp separations are not possible in the

industrial practice, short-cut methods have been devel-oped to assist in the design of distillation sequences. A

transparent example of those procedures is the so-called

Fig. 2. Bow-tie region construction. Feasible residues curves must go

through both shaded areas, so that top and bottom compositions line

on a material balance line passing through the feed.

bow-tie region method (Fig. 2) which intends to find a

set of feasible top and product compositions (in terms of

mass balance lines) for columns at finite reflux or when

distillation-boundaries are crossed. An extensive discus-sion on this topic can be found in Fien and Liu (1994).

For separations that involve entrainers, the selection

of them is without question the most determining task.

For reactive distillation, the entrainment agent should

react selectively with one of the azeotropic components

allowing to cross distillation boundaries.

3. Case study: synthesis of MTBE

3.1. Motivation

In the early 1990s methyl tert-butyl ether (MTBE)was considered one of the most promising clean burning

octane enhancers, being expected to be the second

largest organic chemical produced in United States by

2000, especially promoted by the increased demand for

premium-grade fuels and the elimination of lead gaso-

lines. Although its harmlessness has been lately ques-

tioned, its synthesis via RD still represents a major

development in the field and an open door to novel andinnovative processes.

3.2. Description of the system

MTBE is produced from methanol (MeOH) and

isobutene (IB) in an equilibrium-limited liquid-phasereaction (Jimenez, Wanhschafft & Julka, 2001) cata-

lyzed heterogeneously by a strong acid ion-exchange

resin (e.g. Amberlyst 15) or homogeneously by sulfuric

acid according to the following stoichiometry,

C4H8�CH3OHXC5H12O

isobutene�methanolXMTBE (9)

The IB stream is normally originated from FCC

crackers or steam cracking units, and therefore, contains

a complete range of butanes and butenes. For the sake

of simplifying the study and provided that the reaction is

highly selective for IB, all the organic components in the

IB stream are lumped together as nC4 (Guttinger, 1998).

3.3. Thermodynamics

The thermodynamic model used for this study corre-

sponds to the Gamma�/Phi formulation, in which the

fugacities of each component in the vapor phase equal

those in the liquid phase, according the following

expression,

yi � Fi � P�xi � gi � p0i i � [1; n]: (10)

where: Fi is the vapor fugacity coefficient of component

Table 1

Wilson interaction parameters for the system IB�/MeOH�/nC4 at 11

atm (Ung & Doherty, 1995; Guttinger, 1998

Component IB Methanol MTBE n C4

aij (�/)

IB 0.00000 �/0.74200 0.24130 0.00000

Methanol 0.74200 0.00000 0.98330 0.81492

MTBE �/0.2413 �/0.98330 0.0000 0.00000

n C4 0.00000 �/0.81492 0.00000 0.00000

bij (K )

IB 0.0000 �/85.5447 30.2477 0.0000

Methanol �/1296.719 0.0000 �/746.3971 �/1149.280

MTBE �/136.6574 204.5029 0.0000 0.0000

n C4 0.0000 �/192.4019 0.0000 0.0000

Table 2

Antoine’s coefficients for the pure components IB, methanol, MTBE

and n C4 (Guttinger, 1998)

Component Ai (�/) Bi (K) Ci (K)

IB 20.65556 �/2125.74886 �/33.16000

Methanol 23.49989 �/3643.31362 �/33.43400

MTBE 20.71616 �/2571.58460 �/48.40600

n C4 20.57070 �/2154.89730 �/34.42000

Table 3

Parameters for the chemical equilibrium constant for the MTBE

synthesis (Guttinger, 1998)

Parameter

a (�/) �/10.0982

b (K) 4254.0500

d (�/) 0.2667

C. Almeida-Rivera, J. Grievink / Computers and Chemical Engineering 28 (2004) 17�/2520

i , gi is the liquid activity coefficient of component i and

pi0 is the vapor pressure of component i .

In this study, the vapor phase has been assumed to

behave ideally (Fi �/1), whereas for the liquid phaseWilson activity coefficient equation has been used due to

its accuracy for modeling multicomponent mixtures that

do not form two liquids (Kooijman & Taylor, 2000).

The model adopts the following form,

ln(Lij)�aij�bij=T i; j � [1; n];

Si �Xj�n

j�1xj � Lij i; j � [1; n];

ln(gi)�1� ln(Si)�Xk�n

k�1

xk � Lki

Sk

i � [1; n]: (11)

where: Lij is the binary interaction parameter for the

pair i�/j and aij and bij are model parameters.

Pure component vapor pressures are expressed as a

function of absolute temperature by means of Antoine’s

equation,

ln(p0i )�Ai�Bi=(T�Ci) i � [1; n]: (12)

where: Ai , Bi and Ci are model parameters of compo-

nent i .

3.4. Reaction equilibrium

The reaction equilibrium and kinetics of MTBE

synthesis have been subject of intensive research during

the last decade (Colombo, Cori, Dalioro & Delogu,

1983; Rehfinger & Hoffmann, 1990; Zhang & Datta,1995; Izquierdo, Cunill, Vila, Tejero & Iborra, 1992).

Without loosing generality, the chemical equilibrium

constant may be expressed as an algebraic function of

temperature,

ln(keq)�a�b=T�d � ln(T)�o � T�h � T2: (13)

where: a , b , d , o and h are experimental constants for

the particular synthesis reaction.

3.5. Simulation data

The thermodynamic and kinetic parameters listed in

Ung and Doherty (1995) and Guttinger (1998) have

been used in this study, after being corrected to the

system pressure of 11 atm (Tables 1�/3).

3.6. Simulation problem

The simulation problem encompasses the solution of

the RCM expressions for two cases: non-reactive and

reactive mixture of IB�/MeOH�/MTBE�/nC4 system.According to the degree of freedom analysis of the

previous sections, only the total pressure has been

specified (P�/11 atm). The criterion for choosing this

value is determined by the solubility of IB in the liquid

phase and enables to carry out the reaction to a

desirable extent (Zhang & Datta, 1995).

In this study GPROMS programming software (Process

Systems Enterprise Ltd.) has been used to solve the set

of simultaneous differential and non-linear expressions.

The solving algorithm is composed of an DAE inte-

grator based on variable step-size/variable order back-

ward differentiation formulae. Initial molar

compositions in the non-reactive case and transformed

variables in the reactive case have been specified to

generate the residue lines, aiming*/by visual in-

spection*/to cover completely the composition space.

The simulations have been performed until singular

solutions have been found, which correspond to pure

component or (non-) reactive azeotropes (i.e. x¯�/y

¯and

X�/Y).

¯ ¯

C. Almeida-Rivera, J. Grievink / Computers and Chemical Engineering 28 (2004) 17�/25 21

3.7. Preliminary results

Three minimum boiling point azeotropes have been

reported for the non-reactive system at 11 atm: IB�/

MeOH at x1�/0.917, MeOH�/nC4 at x2�/0.8794 and

MeOH�/MTBE at x2�/0.577, which represent the

corners of a non-reactive distillation boundary plane

and divides the composition simplex into a MTBE-rich

and a MeOH-rich regions (Guttinger, 1998). As depicted

in Fig. 3 all the residue curves originate at the IB�/

MeOH azeotrope and end either at the MeOH or

MTBE pure component.

3.8. Feasibility analysis and sequence generation

When chemical reaction takes place all the reactive

curves start at either pure IB corner or at the MeOH�/

nC4 azeotrope (MNAz) and end predominantly at the

pure MeOH corner (Fig. 4). The remaining low-boiling

azeotropes did not survive the reaction, whereas an

additional saddle point (quaternary azeotrope QAz) has

appeared close to the pure nC4 corner. For highly inert-

containing mixtures, the residue curves have been found

to end at the nC4 pure corner.

A clearer representation of the reactive system is

possible when the conventional composition axis are

substituted by the Doherty’s composition transforma-

tions (Fig. 5). This composition space allows to easily

visualize the stable behavior of the quaternary azeotrope

(Fig. 6) and, furthermore, to identify a pseudo -reactive

azeotrope, PRAz, (Fig. 7) located in the hypotenuse of

the transformed composition simplex and correspond-

ing to an IB�/MeOH�/MTBE mixture at high conver-

sion.

Fig. 3. Residue curve map for the non-reactive sys

The PRAz is characterized by the fact that the residue

lines curve toward it but never converge to it (Ung &

Doherty, 1995) and for practical purposes it imposes the

limit beyond which distillation can not proceed (Ung &

Doherty, 1995). As depicted in Fig. 8, the singular

points of the reacting system define five reactive

distillation boundaries: (QAz:MNAz), (QAz:IB), (QAz:-

MeOH), (QAz:nC4) and (QAz:PRAz).

The composition space has been divided into five sub-

regions:

. region I: (QAz:MNAz:nC4),

. region II: (QAz:IB:nC4),

. region III: (QAz:MeOH:MNAz),

. region IV: (QAz:PRAz:IB),

. region V: (QAz:PRAz:MeOH).

For the MTBE synthesis the regions I (�/0.1%), II

(�/0.25%) and III (�/0.8%) might be neglected at the

first stage of design without incurring in measurable

errors. This assumption simplifies considerably the

feasibility of the process, since all but one of the

boundaries have been removed. The composition sim-

plex is then divided into two well-defined zones:

. zone a: residue curves that start at MNAz azeotrope

and end at pure MeOH corner with a saddle point at

the PRAz (i.e. region IV),

. zone b: residue curves that start at pure IB and

MNAz and end at the PRAz (i.e. region V).

Thus, the separation sequences may differ depending

upon the location of the feed in the composition

simplex. For zone (b) the sequence is composed of a

single column as depicted in Fig. 9. In this case the fresh

feed stream (IB, MeOH and nC4) is mixed with a recycle

tem: IB�/MeOH�/MTBE�/n C4 at P�/11 atm.

Fig. 4. Reactive curve map for the synthesis of MTBE at P�/11 atm. Reference component: MTBE.

C. Almeida-Rivera, J. Grievink / Computers and Chemical Engineering 28 (2004) 17�/2522

stream (enriched in IB after purging most of the existing

inert n-butane) before entering the column, while the

product stream (i.e. PRAz) is continuously removed at

the bottom in an indirect separation strategy.When the feed to the column is embedded within zone

(a) the sequence synthesis is not straightforward and

requires some additional insight originated from the

residue curve map (Fig. 10a). Thus, a distillation

sequence has been generated for the MTBE synthesis

using MeOH as an entrainer, according to the approach

presented by Fien and Liu (1994) for non-reactive

Fig. 5. Reactive residue curve for the MTBE synthesis a

mixtures. The sequence involves the following opera-

tions (Fig. 10b):

. the feed F (MeOH:IB:nC4) is mixed with the recycle

stream B2 to obtain the feed mixture M1,

. M1 is fed to column C1 and separated into distillate

D1 and bottoms B1 (pure MeOH),

. D1 enters column C2 and is divided into the stream

D2 and the recycle stream B2 (since the distillation

boundary is curved, the distillation balance line is

able to cross the boundary),

t 11 atm in terms of the transformed compositions.

Fig. 6. Quaternary azeotrope in the MTBE synthesis at 11 atm.

C. Almeida-Rivera, J. Grievink / Computers and Chemical Engineering 28 (2004) 17�/25 23

. the stream D2 is then separated following the

sequence designed for zone (b) and stream B1 may

be eventually recycled back to the first column.

It can be noticed that the PRAz is an intermediate

boiling reacting mixture and behaves like a tangent

pinch that prohibits designing the separation sequence

towards pure MTBE.

Fig. 7. Pseudo azeotrope in the M

4. Conclusions

Residue curve maps have shown to provide valuable

insights and design assistance for non-ideal systems,

particularly for reactive distillation.

Transforming the composition variables according to

Doherty’s approach allows to define a reaction space of

lower dimension, formed by attainable product compo-

TBE synthesis at 11 atm.

Fig. 8. Schematic representation of the distillation boundaries and

zones for the synthesis of MTBE in the ‘reaction space’.

Fig. 9. Separation sequence for zone (b), where high purity MTBE

(pseudo azeotrope) is recovered at the bottom of the column in an

indirect separation.

Fig. 10. Separation sequence for zone (a).

C. Almeida-Rivera, J. Grievink / Computers and Chemical Engineering 28 (2004) 17�/2524

sitions and where the conventional concepts for residue

curves can be applied.

For the MTBE synthesis two main subregions have

been defined in the modified composition simplex, each

of both characterized by singular points (nodes and

saddles) and delimited by separation boundaries.

Two different separation sequences have been gener-

ated, covering all the possible reacting mixture composi-

tions. In the generated sequences, high purity MTBE

has been obtained as a product, due to the appearance

of a pseudo reactive azeotrope, which imposes limitation

to the separation task.

As all geometrical methods, the RCM technique is

limited by its intrinsic graphical nature and requires

accurate thermodynamic data in order to correctly

describe reactive distillation processes.

Furthermore, RCM technique is suitable only for

equilibrium-based processes, which are relatively com-

mon in industrial practice (Fien & Liu, 1994).

The combinatorial number of possible separation

sequences for multicomponent mixtures is considerably

large and calls for the definition of heuristics andthermodynamic guidelines based upon RCM’s to find

convenient initial designs. Furthermore, operational

issues should be addressed in terms of, for instance,

predicting the controllability of the sequence and defin-

ing feasible reflux ratios. Operation pressure is also a

parameter that should be carefully analyzed since

bifurcation pressures have been reported (Fien & Liu,

1994; Okasinski & Doherty, 1997), at which thethermodynamic behavior of the reacting mixture

changes considerably.

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