Geometric visualisation of reactive fixed points

Pergamon

Computerschem.EngngVol.20, Suppl.,pp. SI33-SI38, 1996 Copyright© 1996 ElsevierScienceLtd S0098.1354(96)00033-6 Printed in GreatBritain.All rightsre.fred 0098-1354/96 $15.00+0.00

G E O M E T R I C VISUALISATION OF REACTIVE FIXED POINTS Steinar Hauan and Kristian M. Lien* Laboratory of Chemical Engineering, The Norwegian Institute of Technology The University of Trondheim, N-7034 Trondheim, Norway.

A b s t r a c t Fixed points in reactive distillation are due to a balance between the elementary phenomena reaction, separation and mixing. By use of a new vector representation, this paper develops the mathematical and graphical framework for easy understanding of such points.

KEYWORDS Design, reactive distillation, reactive fixed points, azeotropic behaviour

INTRODUCTION The presence of azeotropes in a mixture is known to have a major impact on the design of separation systems based on distillation. The concept of distillation boundaries and how to deal with them in an efficient manner has been subject to intensive research throughout the century. In order to avoid difficult and expensive separations, introducing chemical reaction in a distillation column may in some applications, as put by Doherty [Doherty and Buzad, 1992], "react away some of the aseotropes ... and greatly simplify the phase behaviour". Reactive distillation combines several process tasks into a single unit, but introduces new phenomena compared to conventional structures. From simulation and experiments, Bravo [Bravo et al., 1993] concludes: "There appears to be some interaction between the reaction and the VLE that produces fiat concentration profiles for large portions of the column under certain conditions." Rev [R~v, 1994] defined the term reactive azeotropy to describe simultaneous reaction and separation where no change in composition occurs. In the limit of chemical equilibrium, Doherty [Venimadhavan et al., 1994] used the concept of reactive azeotropes. The scope of this paper is based on the view that the fixed points occurring in reactive distillation are composite phenomena that may be better understood if decomposed. Thus, the present paper will first decompose the composite phenomena into individual elements, then define terminology and present the requirements for reactive fixed points and finally derive some general design implications. The examples are selected as simple as possible in order to illustrate the central issues of the proposed approach. *Author to whom correspondence should be addressed, email lienOkjemi.unit.no

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European Symposiumon Computer Aided Process Engineering---6. Part A

THEORY In a reactive distillation column, there are three independent phenomena occurring simultaneously: reaction, s e p a r a t i o n and mixing. In the composition space, these elementary phenomena are represented by v e c t o r s with given directions and lengths. The overall change of composition in the liquid phase of a given point is:

= ~

+ ~

+ mi~

(1)

The direction of the mixing vector is trivially defined as a straight line towards the local feed composition. The length is determined by the relative amount of material in the two streams being mixed: m=x

-=

:

HO + H F

(2)

X N C , F -- XNC,O

where NC is the total number of components. The definition of feed in this context is illustrated in Figure 1. In a distillation column the local feed consists of entering liquid and vapour streams in addition to a possible external feed.

Li. 1 Vi

I

IV, Y I

Li

Vi, l

Fig. 1: Local feed stream

Fig. 2: Simple distillation setup

For a single reversible reaction: C,-

Cp

~-"~lv,,,IA,,,

~

m----1

~-~lv,,IA,,

(3)

rl--1

where vi is the stoichiometric coefficient, Ai is the reacting species, C~ is the number of reactants and Cp is the number of products, the corresponding reaction vector is derived in Appendix A: b'l -- Xl,O " ~-~i----1 //i =

y

(4)

:

UNC

NC

- -

ZNC,O " ~"~-i=l Ui

In the mole fraction space, the reaction vector .direction is a function of local composition due to volume change by chemical reaction. The scalar .T only affects the vector length and is calculated from catalyst activity, hold-up, temperature and pressure.

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The extension to systems with multiple chemical reactions is trivial: the overall reaction vector is the sum of all individual reaction vectors. A mass balance on the simple distillation process in Figure 2 gives: dxi -~

or =

v_ (1 - K i ) . xi H

(5)

at p h a s e equilibrium

where the second form incorporates the phase equilibrium equation y~ = Ki • zi. The separation vector is thus defined as:

:

in g e n e r a l

XNC -- YNC

=

(6)

o~

[(1 - ICl)...

(1 - K~c)l

r

:

equilibrium

XNC where ~ is a scalar reflecting average mass transfer rates and the second form again assumes phase equilibrium.

FIXED

POINTS

From equation 1 it is clear that the necessary condition for a reactive fixed point in the composition space is given by: +

~

+

mzx -~"

=

~

(7)

There are however several s t r u c t u r a l l y different types of fixed points: • No non-zero vectors: - Equation 7 is trivially satisfied if all three vectors are zero. This corresponds to a mixer where all feeds have the same composition, a batch reactor where chemical equilibrium is reached or an ordinary, non-reactive azeotrope in a system without reaction and mixing. • Two non-zero vectors: - In a CSTR reactor, there is no separation caused by mass transfer from one phase to another. At steady state, the liquid composition is still kept constant because the net result of mixing with feed and reaction is zero. -

A simple distillation configuration with chemical reaction has no feed stream and thus by nature a mixing vector equal to zero. In order to satisfy equation 7, the composition change resulting from reaction and separation must exactly cancel each other.

- The "system" in Figure 3 consists of a non-reactive reboiler and a total condenser. The liquid composition in unit # 1 remains constant because the net effect of separation and mixing is zero. In practise, this configuration is of little interest and is only indicated for completeness. • T h r e e non-zero vectors:

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European Symposium on Computer Aided Process Engineering---6. Part A - The most general type of fixed point occurs when all vectors in equation 7 are non-zero with no change in composition. This may for instance be possible in a reactive distillation column with simultaneous separation, mixing and separation and is illustrated in Figure 4.

UNIT #2

.,,~"-"''.

°'x 1 .I.

U N I T #1

rx

V

Fig. 3: Separation and mixing

Fig. 4: A general reactive fixed point

E~amples For simple, reactive distillation, Rev [R~v, 1994] constructed the extended line of kinetic azeotropy given by the solution to a system of differential equations. From a phenomena-oriented point of view, this line corresponds to the solution of equation 7 with no mixing vector. The physical interpretation is simple: along this line, the separation and reaction vectors are parallel and opposite, thus a fixed point is possible. The actual location of this fixed point is determined by the relative rate of reaction and separation 1, in other words the vector lengths. Example 1 Assume a reactive flash-tank (Figure 2) with a single reversible chemical reaction

2A~B+C

(8)

The VLE is described by constant relatives volatilities, a = [ 1.0 3.0 5.0 IT, and the local feed is zero. As ~-':~vi = 0, all reaction vectors are parallel to [ - 2 1 1 ]T, and a fixed point may occur if the separation vector points in the exactly opposite direction. For instance, such a point is found at X = [ 0.60 0.30 0.10 ]7",y = [ 0.30 0.45 0.25 IT which may trivially be confirmed by hand calculations. Example 2 Introducing a feed of pure A, the general type of fixed point with three non-zero composition change vectors may be found. For a liquid composition X = [ 0.68 0.20 0.12 ]T the corresponding vapour phase is Y = [ 0.36 0.32 0.32 ]T. The separation vector becomes ~ = Wsep "[ 0.32 - 0 . 1 2 - 0 . 2 0 ]T and the mixing vector ~ = Wm~' [ 0.32 -0.20 - 0 . 1 2 ]T where W, ep and Wmi~ are the weighting factors for separation and mixing respectively. In order to keep a constant holdup, the amount of leaving vapour must in this particular case equal the amount of fresh feed. To satisfy the material balance, Wmi~ and W, ep must thus be equal. As the sum (se-~ + m--~x) = W.[ 0.64 -0.32 -0.32 ]T is parallel and opposite to the reaction vector, the reactive fixed point shown in Figure 4 is possible. i by Doherty [Venimadhavanet al., 1994] incorporated through a Damkohler number

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IMPLICATIONS

The basic idea behind multi functional units is to combine two or more process tasks inside a single physical shell. A reactive distillation column with a flat column profile may work reasonably well as a reactor, but is obviously not ideal from a separation point of view. When a combined unit exhibits locally inefficient behaviour with respect to one or more process tasks, it would b y i t s e l f suggest the need for structural changes. There are however several ways to move away from a fixed point in the composition space. The local mixing vector is a function of reflux and reboil policies, external feed and the holdups; all process variables which may easily be altered in the design phase. Reaction vector length may be manipulated through process variables like catalyst density and holdup or structurally by introducing non-reactive stages in the reactive zone as indicated in Figure 5. Another option would be to use side streams to conventional reactors as illustrated in Figure 6 to avoid a counterproductive separation vector.

,'

',

i

FEED

>,

1:

- -

i

.......

_ _ , , L _ _ _-~ I

FEED non-reactive

V

~

reacti

.

l

/

Fig. 5: Middle, non-reactive zone

.

.

.

~"

.

, - - ->

>,

I '

. . . .

r - - ->

~

,

l

.

,,-)/--I"

f

"~

i '

• _,

i '

CSTR

,. . . . . . . . . . . . . . . .

t. _ _ -~

Fig. 6: Side stream to conventional reactor(s)

CONCLUSION A constant column profile in a reactive distillation column does not represent a separation boundary like in ordinary, azeotropic distillation, but may indicate a weakness in design or operation. A reactive fixed point occurs if and only if the individual phenomena reaction, separation and mixing balance each other exactly. The proposed vector representation may be used as a tool to extract information from parts of the composition space where operation is desirable. Future work will be directed toward development of practical design methods and insights based on the composite vector representation introduced in this paper.

~

ZO:13(k)-V

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European Symposium on ComputerAided Process Engineering--6. Part A

NOTATION

Yi

r e a c t i n g species, c o m p o n e n t i n u m b e r of r e a c t a n t s n u m b e r of p r o d u c t s t o t a l feed to s t a g e i, F, = V,+I + L,-1 + EF, /moles/time/ external feed to stage i /moles/time/ liquid holdup [moles] Vapour-liquid equilibrium ratio, c o m p o n e n t i liquid flow from s t a g e i /moles/time/ n u m b e r of c o m p o n e n t s n u m b e r of independent chemical reactions v a p o u r flow from stage i /moles/time/ liquid mole fraction, c o m p o n e n t i time derivative of liquid composition [time -1 ] v a p o u r mole fraction, c o m p o n e n t i

e vi

extent of reaction stoichiometric coefficient, c o m p o n e n t i

A i

Cr Cp F,

EFi H Ki Li NC NR

V, x,

scalar reaction p a r a m e t e r scalar s e p a r a t i o n parameter subscript F subscript i subscript 0

feed composition (Figure 3) component number initial s t a t e

REFERENCES [Bravo et al., 1993] Bravo, J., Pyhalathi, A., and J£rvelin, H. (1993). Investigations in a catalytic distillation pilot plant: Vapor/liquid equilibrium, kinetics and mass-transfer issues. Ind. Eng. Chem. Res., 32(10):2220-2225. [Doherty and Buzad, 1992] Doherty, M. and Buza~l, G. (1992). Reactive distillation by design. Trans IChemE, 70:448-458. [R@v, 1994] Rdv, E. (1994). Reactive distillation and kinetic azeotropy. Ind. Eng. Chem. Res., 33:2174-2179. [Venimadhavan et al., 1994] Venimadhavan, G., Buzad, G., Doherty, M., and Malone, M. (1994). Effect of kinetics on residue curve maps for reactive distillation. A IChE Journal, 40(11): 18141824.

APPENDIX

A: Reaction vector n,

=

n =

ni,o +,.u~ ~n, i ni

z,

-

dx d-7

df(¢)

d,

(9)

= ~ n , . o +,~--~v, i ni,o + , . v i

-

d

(10)

i =

Of(,)

d,

0,

' d-t

=

d-t [ f ( e ) ] =

=

v , . ~~, n,,o - ~"], . , . n,,o

f(,)

(II)

(12) (13)

(E,-,0 +,B,

Locally, ~ reflects the scalar reaction rate, thus: dx de 1 l i m -d'[ = dt )-'~i - - ni,o " (ui - xi,o ~ d,-*O

v , ) = .~" • (ui - x,,o B i

vi) i

(14)