New tools for the design of kinetically controlled reactive distillation columns for ternary mixtures

New tools for the design of kinetically controlled reactive distillation columns for ternary mixtures

~ Computers chem. Engng Vol. lY. No.4. pp. W5-4I1i'1. lY95 Pergamon 0098-1354(94)00068-9 Copyright © 1995 Elsevier Science Ltd Printed in Great Bri...

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Computers chem. Engng Vol. lY. No.4. pp. W5-4I1i'1. lY95

Pergamon 0098-1354(94)00068-9

Copyright © 1995 Elsevier Science Ltd Printed in Great Britain. All rights reserved 0098-1354/95 $9.50 ...Of X)

NEW TOOLS FOR THE DESIGN OF KINETICALLY CONTROLLED REACTIVE DISTILLATION COLUMNS FOR TERNARY MIXTURES G. BUZAD and M. F. DOHERTY Department of Chemical Engineering, University of Massachusetts, Amherst, MA 01003. U.S.A. (Received 28 October 1993; final revision received 16 May 1994; received for publication 21 June 1994)

Abstract-Reactive distillation is an emerging technology that has great potential as a process alternative for carrying out liquid phase chemical reactions. Systematic design methods for reactive distillation systems have only recently begun to appear, and so far these have been restricted to systems at chemical equlibrium. It is not always economical, however, to operate reactive distillation columns close to reaction equilibrium conditions. In this paper we present a design procedure for kinetically controlled reactive distillation columns for ternary mixtures, and demonstrate the concepts for reactions of the type 2C¢:}A + B. We show that the amount of liquid holdup per stage has a significant effect on the design, and that there is a sharp minimum in the total column holdup. at fixed product compositions. as the liquid holdup on each stage is varied.

INTRODUCTION

Distillation with chemical reaction has received increasing attention in recent years as an alternative to conventional processes (Agreda and Partin, 1984; Agreda et al., 1990; Agreda and Lilly, 1990; Amiet, 1988; Belson, 1990; Clearly and Doherty, 1985; DeGarmo et al., 1992; Grosser et al., 1987; Smith, 1981, 1990). A more extensive survey of the literature is given by Doherty and Buzad (1992). Reactive distillation systems offer certain advantages that cannot be matched by conventional processing, e.g. the suppression of unwanted byproducts, and freedom from some of the restrictions imposed by equilibrium. The increasing interest in reactive distillation processes has been accompanied by numerous algorithms to model these systems. Chemical process modeling is normally carried out in one of two different approaches: simulation, and design. In simulation the input and operating variables of a process are specified and the task is to solve for the resulting outputs. In design the input and (selected) output variables are specified and the task is to determine the optimal process configuration and the optimal design parameters that achieve the given product specifications. Several good simulation algorithms are currently available for modeling reactive distillation processes (e.g. Bogacki et al., 1989; Bondy, 1991; Chang and Seader, 1988; Venkataraman et al., 1990). Chang

and Seader (1988) provide a summary of additional procedures for simulating reactive distillation systems. Typically, simulation of reactive distillation columns involves specifying the feed composition and quality, the column pressure. reflux and reboil ratios, total number of stages, feed plate location and liquid-phase volumes on each stage. The resulting composition profiles are then determined. However. there is no good way to estimate what value(s) of the specified parameters will give rise to a reactive column with desirable product stream compositions. In recent years the design approach has been successfully applied to nonreactive distillation. For example, Levy et al. (1985) developed a boundaryvalue procedure for the design of nonideal ternary distillation columns with no reaction. Julka and Doherty (1990) extended the method to nomdeal multicomponent distillation. and Barbosa and Doherty (1988) extended the concepts to the design of reactive distillation columns for which chemical equilibrium prevails (i.e. large liquid holdup per stage). It is not always economical to operate reactive distillation columns close to reaction equilibrium conditions. In this paper we present a procedure for the design of kinetically controlled reactive distillation columns for ternary mixtures. Although this is a restricted class of systems, it does include commercially interesting chemistries such as the synthesis of MTBE from isobutene and methanol. the synthesis 395

G.

396

BUZAD

and M. F.

of tert butyl alcohol from isobutene and water, and the formation of cumene from benzene and propylene etc. Conceptual design methods for the general case of multiple chemical reactions involving multicomponent (~4) mixtures have only recently been developed for the special case of equilibrium chemical reactions (Ung, 1994). It is beyond the capability of current conceptual design procedures to treat such a general case with finite reactions rates. DERIVATION OF THE EQUATIONS

Consider a liquid-phase reaction of the type: 2C{:::}A+B,

(1)

which is carried out in an ideal solution and for which the rate of reaction per mole of mixture is given by: (2) Components 1, 2 and 3 correspond to A, Band C, respectively, K is the reaction equilibrium constant and k, is the forward rate constant. We use mole fractions in the rate expression, rather than the conventional approach of using concentrations, because they conform more naturally to the usual method of writing material balances and vaporliquid equilibrium equations for distillation problems. This approach also has the advantage of giving the rate constant units of reciprocal time regardless of the order of reaction. This in turn allows for a universal definition of the Damkohler number for all reaction orders. For simplicity, we assume a single feed column, a saturated liquid feed, constant molar overflow (implies negligible heat of reaction), a total nonreactive condenser, and that the molar liquid holdup is the same on each stage. It is possible to relax all these assumptions but only at the expense of obscuring the overall design strategy. The main purp?se of this paper is to introduce a design framework for kinetically controlled reactive distillation columns, and to explore the trade-offs in the context of simple examples. The example used throughout the paper is motivated by the toluene disproportionation (or transalkylation) reaction: 2 toluene {:::}benzene + xylene, (C)

(A)

DOHERTY

volatilities vary with temperature, the effect is not strong and constant values for the volatilities reproduce the experimental y-x data very well. The volatilities of benzene, and toluene relative to xylene are 4.8 and 2.3, respectively. We estimate that the reaction equilibrium constant does not vary very much over the temperature range 298-850 K, and a value of K ~ 0.1 seems reasonable. This reaction is normally carried out in the gas phase at high temperature and pressure (T-785 K, P-20 bar) above the conditions where distillation can be performed. Although there may be an opportunity to perform this chemistry at milder conditions in a reactive distillation column we do not explore that possibility here. There are other interesting equilibrium reactions that also follow the stoichiometry of equation (1), such as the formation of dimethyl ether and water from two molecules of methanol. More realistic design case studies will be published separately. Overall material balance

The overall and component material balances for the distillation column give:

O=F-D-B

(3)

and

0= FXF.i- DXO.i- BXB.i NT (

+HkfL j~l

2 }.1 t.: Xj.3---

x x ) ,

i= 1,2,

(4)

K

where, for simplicity, we assume that the molar holdup is the same for each stage. For realistic design calculations it is necessary and straightforward to remove this restriction. Removing it here, however, would make the design procedure and the results less transparent. Dividing equations (4) by F gives:

D O=XF.i

B

D+Bxo.i- D+BxB.i

i= 1,2,

(5)

or

DlB 1 O=XF.i DIB+ 1 XD.i- DIB+ 1 XB.i

(B)

where component A is the lightest, B is the heaviest and C is intermediate boiling. At low pressure the liquid mixture is ideal and the relative volatilities may be treated as constants, i.e. even though the

(6) where Da=HkrlF. The Damkohler number, Da, is dimensionless and is the ratio of a characteristic

397

New tools for design

liquid residence time (HI F) to a characteristic reaction time (11 kr). We can solve for DI B in terms of the feed and product compositions by eliminating the common reaction term in equations (6) to give:

D

(XB. I - XF. I) - (X B.2 -XF. 2)

B

(XF. I- XO.I ) - (XF. 2-X O.2)

(7a)

-=

and

respectively. Dividing equations (14) by L B gives: V

DIB

+ L LJ

Hk r", .;'-,

( X2p _Xr.K,Xi. 2)

B )= I

'

1

O=XF '

" DIB + l

XD . .s

DIB+l

B

oxn + l., - -V +B - y n ,',- -V+B - x H. i -

In what follows it will be convenient to select equation (7a) together with one of equations (6) as the pair of overall component ba lances, thu s:

'

._ 1-

1,2,

(15)

XB

NT (X32 - -Xi. l}.2 X) +Da L t. K'

.i

or

i= 1, or 2. (7b)

i~ 1

s I Ox - .-1' +-1 y - s+1 x H.I n+1.1 n./ + !.. Da

Rectifying section

L

B

The overall and component material balance s for the rectifying section are: (8) and

~ (X2J..', _XJ.'Xr.~) K '

LJ

i~ I

I -I')

-

,- .

( 16)

wher e .I' = VI B , and

F

1 + DI B

1+ BID

LB

1+ .1'

l +r +BID

(17)

Overall energy balance

~ +HkrLJ

( X!3-T Xi 'Xi 2) '

i= 1,2,

(9)

i~ 1

respectively. Dividin g equations (9) by V gives;

LT

D

D s+ l-q

Oy m + I .1 -L-+ -Dxm. i - L - - - XD · + D .1 T

r

r~ ( 2 _ + Hk V LJ xj.3

Xi .

f.2)'

,X

K

Th e reflux rat io , reboil ratio and D I B are not independ ent variahles . An overall energy balan ce together with the constant molar overflow assumption results in the following relationship betw een them:

B '=

I

1,2,

(10)

i-I

or

r 1 Oym +I . 1·- -r+ -1 xm . 1·- -r+ -1 x 0 .1. i= 1,2, (11) where r = L TID , and

r+q

(18)

wher e q represents the feed quality . This equation is not unique to reac tive distillation, and applies to nonreactive mixtur es also . The column mod el the refore consists of the ove rall balances. equations (7) and (18), together with the rectifying and stripping equations (11) and (16). In Appendix A we study the asympto tic limits for equations (11) and (16) when rand s approach infinity. DEGREE S OF FREEDOM AND DESIGN PROCED URE FOR

F

1 + BID

1+ DI B

V

1 +r

s

(12)

Stripping section The overall and component material balances for the stripping section are : (13)

TERNARY MIXTURES

For design problems we expect the column pressure , the feed composition and the feed quality to be specified . There are eight remaining variables in the preceding equ at ions. These are: the distillat e and bottoms comp osition s (two independent compositions in eac h strea m) . the reflux and reboil rat ios. the Damkohler number , and the ratio of distillat e to

398

G.

BUZAD

and M. F.

bottoms flowrate, DIB. There are two independent overall material balances, equation (7) , and one equation relating r , sand DIB , equation (18). This results in five degrees of freedom. The boundary-value design procedure involves specifying the composition of the product streams and solving the finite difference model equations inwards towards the middle of the column. A necessary (but not sufficient) condition for a feasible reactive distillation column is the intersection of the two composition profiles. Mathematically this criterion can be written as; (19) where N, and N, are the number of stages in the rectifying and stripping sections, respectively. With this criterion, there are two additional constraints and two additional variables , N, and N, so the degrees of freedom remain a total of five. Compared to nonreactive distillation there is one extra degree of freedom when reaction takes place. This is due to the fact that the magnitude of the liquid holdup on each stage influences the compositions when reaction occurs, but do es not when reaction is absent. The nonreactive problem can be viewed as a special case of the reactive problem where Da=O. Our design procedure for reactive columns consists of specifying des ired values for the distillate and bottoms compositions as well as the Damkohler number , and solving the model equations for the remaining variables . An algorithm for solving this reactive distillation problem is: (1) Specify the feed composition and quality, and the column pressure. (2) Specify the desired distillate and bottoms compositions. (3) Specify a value for the Damkohler number (or reflux ratio) . (4) Solve for DIB [equation (7a)]. (5) Guess a value for the reflux ratio (or Damkohler number). (6) Solve for the summation term in equation (7b) . (7) Solve for the reboil ratio [equation (18)]. (8) Solve for FIV [equation (12)1 and FI L B [equation (17)]. (9) Compute the composition profile for the recti fying section [equations (11)]. Step down the column until one of the following conditions occurs: (a) the summation term in equations (11) gets larger than the summation term in equation (7b) [from step (6)]; or (b) the

DOHERTY

composition profile has gone outside the composition space (allow at most one step outside the composition triangle). (10) Repeat step (9) for the stripping section composition profile. (11) Repeat steps (5)-(10) until the composition profiles intersect. (12) Count the number of stages in each section of the column at each intersection point of the profiles. (13) Intersection of the profiles is a neces sary but not sufficient condition for a solution to exist. In order for a solution to occur the summation term in the rectifying section [equations (11)] plus the summation term in the stripping section [equations (16)] must equal the summation term in the overall material balance [from step (6)], i.e. we must ensure that just the right amount of reaction has taken place. Note that not all values of Xo and XB lead to a feasible column . Values for Xo and XB must lie within the feasible product composition regions otherwise the column composition profiles will not intersect. This behavior is similar to that for nonreactive mixtures, as described by Stichlmair and Herguijuela (1992), Wahnschafft et al. (1992) , and Fidkowski et al. (1993) . DESIGN CALCULATIONS

We demonstrate the design technique by considering ternary mixtures that undergo a reaction of

B (Heavy) 1.0

Component

Feed

A B

0.0 0.0 1.0

C

Distillate 0.9900 0.0050 0.0050

Bottoms 0.0050 0.9900 0.0050

r = 0.7667 Da = 3.0

0.8

0.6 x2

0.4

0.2

O.O + - -0.0 C

(Intermediate)

---.-- -n , - - - - . - - - - - . - -0.2 0.8 ~ 06 XI

--"l 1.0 A

(Light)

Fig. I. Design calculations for the irreversible reaction 2C~A+B at Da=3 .0.

New tools for design B (Heavy)

Component

Feed 0.0 0.0 1.0

A B

1.0

C

Distillate 0.98981 0.00515 000504

B (Heavy)

Bolloms 0.00518 0.98985 000497

1.0

Feed

Distillate

Buttoms

A B

0.0 0.0 1.0

0.9900 0.0050 0.0050

0.0050 0.9900 00050

C

r= 0.7667 Da = 3.0

0.8

Component

r = 0,9148 Da = 12

0.8

0.6

0.6

X2

(a)

x2

0.4 0.4

0.2 0.2 O.O+-------,----,___~·--·-T---,-----u,

QO

02

O~

Q4

1.0

0.8

x,

C (Intermediate)

A

0.0

(Light)

I

I

0.2

0.4

' - " ' - " 1'--'-'0.6 (J.8

C

(Intermediate)

Fig. 2. Simulation of column designed in Fig. 1.

the type given by equation (1) and for which the volatility of the reactant is between the volatilities of the products. The objective is to obtain components A and B from the reactive distillation column in reasonably high purity from a feed containing pure C. In other words, we would like to obtain near complete conversion of the reactant to products, and separate these two products using only one reactive distillation column to carry out this combined reaction-separation process. In this section we discuss three examples. In all cases the volatilities of components A and C relative to component Bare 5.0 and 3.0, respectively, i.e. A is the lightest component and B is the heaviest. The

B (Heavy)

1.0

10 A H-ight)

Component

Feed

Distillate

Buttums

A B

0.0 0.0 1.0

0.98749 0.00719 0.00532

0.00682 0.98712 0.00606

C

t = 09148

0.8

Da= 1 :

0.6

0.4

0.2

O.O--!----r------.---,---'-1'-----I I 08 0.0 0.2 0.4 0.6 C (Intermediate)

x,

1.0 A

(Light)

Fig. 4. (a) Design calculations for the irreversible reaction 2C-->A + B at Da = 1.2. (b) Simulation

feed is taken to be a saturated liquid of pure component C. In the first example the reaction is irreversible; in the other two cases the reaction is reversible. The second and third examples differ only III the target product purities-the desired purity of the distillate and bottoms streams is higher in the latter.

Example I

O+----~---y---_.·

o

10

15

20

25

Da

Fig. 3. Reflux ratio vs Da at fixed product compositions for Example 1.

Consider a column with pure reactant C as feed. The reactant undergoes an irreversible reaction: 2C~A+B,

(20)

G . BUZAD and M. F.

400 B (Heny)

Component

Feed

DbtUlate

Bottoms

A

0.0 0.0 1.0

0.9900 0.0050 0.0050

0.0050

B

1.0

C

B (Heavy)

0.9900

1.0

= 0.7765 = 5.0

CompoDent

Feed

Distillate

Bottoms

A B

0.0 0.0 1.0

0.9900 0.0050 0.0050

0.0050 0.9900 0.0050

C

0.0050

r Da

0.8

DOHERTY

r .. 0.8954 Da= 7.0

0.8

0.6

0.6

(a)

(a)

Xz 0.4

0.4

0.2

0.2

O.04----,--+--,---,----.,---:::n,

0.0+---....,---....,---....,---....,---"'1 0.0

0.2

0.4

C (Intermediate)

0.6

0.8

1.0 A (LIght)

XI

B (Heavy)

CompoDeDt

Feed

Dbturate

Bottoms

A

0.0 0.0 1.0

0.98906 0.00558 0.00536

0.00561 0.98908 0.00531

B

1.0

C

0.0

0.4

0.6

0.8

(IDtermedlate)

B (Heavy)

1.0

=

r 0.7765 Da = 5.0

0.8

0.2

C

0.8

1.0 A (Light)

Component

Feed

Distlll.te

Bottoms

A

0.0 0.0 1.0

0.98882 0.00579 0.00539

0.00565 0.98870 0.00565

B C

r = 0.8954 Da = 7.0

0.6

0.6

(b)

Xz

(b)

Xz

0.4

0.4

0.2

0.2

O.O+---.,.....---r----.-----r--~

0.0

C (I_terlDedlate)

0.2

0.4

0.6

0.8

1.0 A (Light)

Fig. 5. (a) Design calculations for the irreversible reaction 2C-A+B at Da=5.0. (b) Simulation.

(i.e. K is set equal to a large number in equations (7), (11) and (16)]. For design purposes we specify a target distillate composition of XD.l =0.99 , xo.2=0.OO5, and a bottoms composition of XB. I= 0.005, XB.Z = 0.99. The remaining variable to be specified is Da . What value(s) of Da, if any, might make the desired separation/conversion feasible? At large Da the summation term in equation (7b) is small and so in computing the rectifying and stripping section profiles the stopping criterion [step (9a) of the algorithm] occurs a few stages away from each end of the

O.O+----,-----,---,----.,---:::n, 0.0 0.2 0.4 0.6 0.8 1.0 C (Intermediate)

A

(Light)

Fig. 6. (a) Design calculations for the irreversible reaction 2C-A+B at Da=7.0. (b) Simulation.

column-in other words, the composition profiles do not span much of the composition triangle and thus are unlikely to intersect. Therefore, we expect the desired separation/conversion to be infeasible at large values of Da. As Da approaches zero the desired conversion/separation is not feasible either because the reaction is almost extinguished and the points Xp, Xo and XB are not collinear . This leads us to try values of Da away from the extremes of its range . To begin, we consider a column with Da=

3.0.

401

New tools for design B (Heavy)

Component

Feed

Distillate

Bottoms

A

0.0 0.0 1.0

0.9900 0.0050 0.0050

0.0050 0.9900 0.0050

B

1.0

C

r= Da=

0.8

------"---_.-

100

• Stripping o Rectifying • Total

80 60 '" .... " 40

1.1508 10.0

E,

:l

'S

0.6

~

i

E

X2

z

I I

0.4

0.2

I

·- · 1

15

0.0 0.0

0

0.4

C (Intermediate)

A (Light)

XI

B (Heayy)

Component

Feed

Distillate

Bottoms

A

0.0 0.0 1.0

0.98793 0.00656 0.00551

0.00597 0.98733 0.00670

B C

1.0

Da

1.0

0.8

0.6

r= Da=

0.8

sections, respectively. The liquid phase composition profiles for the distillation column are shown on the triangular diagram in Fig. 1. Notice that the profiles intersect twice. The first intersection is not a solution since not enough conversion of reactant to products (relative to the total amount of reaction required by overall material balance) takes place by the time the profiles first cross. The existence of multiple intersections of the composition profiles implies that the function in step (13) of the algorithm will have multiple (connected) branches with singularities, such as turning points. Therefore. the initialization of the root-finding problem [in step (5)

1.1508 10.0

0.6

(b)

X2

Fig. H. Number of stages (NT. N, and N, vs Da for Example 1.

0.4

0.2

200j·------··--T--l.

O.O+-------r---.----,----r---~

0.0

0.2

0.4

C

0.6

0.8

1.0 A (Light)

xi

(Intermediate)

Fig. 7. (a) Design calculations for the irreversible reaction 2C...... A + B at Da = 10.0. (b) Simulation.

~ 180

Table 1. Design variables for the separations in Figs 4(b). 2. 5(b). 6(b) and 7(b) Figure

Da

4(b)

1.2 3.0 5.0 7.0 10.0

2 5(b) 6(b) 7(b)

0.9148 0.7667 0.7765 0.8954 1.1508

1.9148 1.7667 1.7705 1.8954 2.1508

NT

f

III 28 16

16 9 7

13

()

11

5

I

I

I

~

..

~

~

Implementing the design algorithm described in the last section we find the reflux ratio is equal to 0.77, the reboil ratio is equal to 1.77, and there are 19.0 and 8.8 stages in the rectifying and the stripping

J!

i

IW

I I

J

I

I,

"

~

o

:

I I

I

1~

"

f7

::l

'3

120

i

100

~

~

8JI----'---,---.--,---.-.~-.-,-----J o ..

10

IS

20

2S

Da

Fig. 9. Dimensionless total liquid holdup in the column (Oa x N I ) vs Da for Example I.

402

G. BUZAD and M. F. DOIIERTY

20 , - - - - - --

-

-

-

- · --

- - - ---,

B (Heavy)

Component

Feed

Distillate

Bottoms

A

0.0 0.0 1.0

0.8000 0.0025 0.t975

0.0100

B

1.0

C

0.8000 0.1900

16

0.8

I

r = 2.2327 Da = 0.0838

12 \

0.6

\ \

(a)

\

\

Xz

8

\ \

0.4

\ \ \ \

4

,, "

0.2 O+----~----,___---____,---__I

0.0

0.2

0.4

0.6

---

0.8

Da

Fig. 10. Reflux ratio vs Da at fixed product compositions for Example 2.

of the algorithm) must be done with care . In Appendix B we pro vide more details about selecting initial guesses for r in step (5) of the design algorithm. In order to check the results obtained using the boundary-valve design procedure (Fig . 1) we simulated the column using a steady-state simulator. As with the design approach , the feed composition, feed quality and column pre ssure are given, leaving five degrees of freedom to be specified . The se are taken to be the reflux and reb oil rat ios, the total number of stages in the column NT' the feed stage location f and Da. The task then is to solve the steady-state material balances for the resulting composition profiles, including the distillate and bottoms compositions. This was done by integrating a dynamic model out to steady-state using the ordinary differ ential equation solver LSODES. Figure 2 shows the simulated column profiles at r = 0.77, s= 1.77, NT = 28,/=9 and Da = 3.0. As can be seen , this figure agrees well with the result s obtained via the boundary-value design procedure . The target product compositions selected for this exampl e are also fea sible at oth er values of Da . We can use the informati on obtained at Da = 3.0 to search for solutions at oth er values of Da . For instance , we know that the reflux ratio equals 0.77 at Da = 3.0 , so we can use this value of r as the initial guess for calculating the profiles at say Da = 3.1. In other word s we can use a paramet er continuation method to move from one value of Da to another. Figure 3 shows a plot of r vs Da at fixed product compositions. As mentioned previously, there are no solutions in the vicinity of Da equal to zero, nor

0.0 0.2 0.0 C (Intermediate)

B (HeaYy)

0.4

0.6

Component A B

1.0

0.8

XI

C

Feed 0.0 0.0 1.0

Distillate 0.8010 0.0024 0.1966

1.0 A (Light)

Bottoms 0.0099 0.8009 0.1892

r = 2.2327 Da = 0.0838

0.8 \ \

0.6

\

\ \

xz

(b)

\

\

0.4

\ \ \ \

\

0.2

,,

-,

O.O+--.....-----.----,:::::::::~~=-=~~ 0.0 0.2 0.4 0.6 1.0 0.8 C A Xl (Intermediate) (Light)

Fig. I I. (a) Design calculations for the reaction 2C¢;}A + B at Da=0.0838. (b) Simulation. Dashed line represents the reaction equilibrium curve.

at large values of Da (above about 20.0 for this particular example). The liquid phase composition profiles for the distillation column at various values of Da are shown in Figs 4(a) , 5(a ), 6(a) and 7(a) for value s of Da equ al to 1.2,5 .0,7.0 and 10.0, resp ectively . The column designs are given in Table 1 and the corresponding simulations are shown in Figs 4(b ), 5(b) , 6(b ) and 7(b) . In general there is good agreement between the solutions obtained by the design procedure and the simulations. The agreement becomes

403

New tool s for design B (Hea vy)

Component

Feed

D1stlllate

Bottoms

A B

0.0 0.0 1.0

0.8000 0.0025 0.1975

0.0100 0.8000 0.1900

1.0

C

0.8

\

B (Heavy)

r = 2.5142 Da= 0.0950

r = 3.3954 Da = 01 305

\ \

(a)

\ \ \

\ \

0.4

\

0.4

\

\ \

\ \

\ \

,

\

"-

0 .2

-,

-,

-,

0,6

0.4

,0.2

C (Inte rmediate)

0.8

1.0

A (Light)

Xl

B (Hea vy)

C (Intermediate)

D1stlll ate

Bottoms

0.0 0,0 1.0

0.7982 0.0025 0. 1993

0.0102 0 .7984 0,1914

1.0

r= 2.5142 Da= 0.0950

0.8

\ \

O.R

1.0 A

Xl

(Light)

B (Heavy)

Feed

A B

C

0.0

0.0

Component

1.0

Component

Feed

Distill ate

Botl oms

A B

0.0 0 .0 1.0

0.7997 0.0025 0.1978

0 .0099 0.7997 0 .1904

C I

r = 33954 Da = 0 1305

0.6

\

(b)

\

\

(b)

X2

\

0.4

.~

0.0 0.0

X2

,,

0.2

0.0

0.6

0.0100 0.8000 01900

0.8 \

\

X2

Bottoms

0.8000 0.0025 0 .1975

C

(a)

\

Distillate

0.0 0.0 1.0

B

0 .6

\

Feed

A

1.0

\ \

0.6

Co mponen t

\

0.4

\ \ \

,

-,

-,

"

0 ,2

-,

-,

-,

0.0 0.2

-,

-,

0,0 0 .0 C (Intermediate)

-,

0.2

0.4

0.6 Xl

0.8

A

(Ligh t)

Fig , 12. (a ) Design ca lculatio ns for the reaction 2C<:;> A+ B at Da=0.0950 . (b) Simulatio n . Dashed line repres ents the reaction equilibrium curve .

bett er as the number of stages increases (i.e. as Da deereases ). Figur e 8 shows a plot of th e num ber of sta ges (NT' N, and N,) vs Da at fixed produ ct compositions. At sma ll va lues of D a man y stages (eac h with low liquid holdu p) are required to achieve the de sired co nve rsion / sep arat ion becau se the reactio n is almos t ext inguished . Th e curve flattens out at larger valu es of Da. Figure 9 is a plot of the dimen sionle ss tot al liquid holdup requirem ents in the column [(Da) x (NT)] vs the dim en sionl ess liquid holdup per stage , Da . As can be seen th er e is a sha rp minimum

I

0,0

1.0

C (Intermediat e)

0 .2

0.4

0.0 XI

O.R

1.0 A

(Light)

Fig. 13. (a) Design ca lculatio ns for the re actio n 2C{::}i\ + B at Da =O .1305. (h) Simulatio n . Da shed line represe nts the reactio n eq uilibr ium curve .

th is curve. indica ting tha t the re is an opti mum value of Da for th e specified target co mpos itio ns . Fro m Fig. 9 we see th at ( Da) x (Nd re aches a minimum at values of Da bet ween :2 and h . To design the co lumn at values of Da less tha n 4 is disadvantageou s beca use fro m Fig. X we see that in this regime man y stages are req uired to achieve the de sired conversion /sep aration . T his leaves values of Da betw een 4 and 6 as the region of opti ma l Da . An exa mple of a composition pr ofile th at lies 111 this region is sho wn in Fig. 5 wher e Da = S.U

In

404

G. B

Component A B

(HeBVy)

1.0

C

Feed 0.0 0.0 1.0

BUZAD

Distillate 0.8000 0.0025 0.1975

and M. F.

55

Bottoms 0.0100 0.8000 0.1900

45

fa

35

rJl

30

S

\

. 1 'lS

\

0.6

\

\

(a)

\

Xl

• Stripping o Rectifying • Total

50

40

r = 7.8322 Da= 0.3

0.8 1

DOHERTY

z"

\

25 20

\

0.4

15

\ \ \

10

\ \ \

0.2

-,

-,

-,

,,

0 0.0

0.2

0.4

C

0.6

0.8

1.0 A (Light)

XI

(Intermediate) B

Component

Feed

Distillate

Bottoms

(Heavy)

A B C

0.0 0.0 1.0

0.8006 0.0025 0.1969

0.0091 0.7997 0.1912

1.0

\

\

(b)

\ \ \

0.4

\ \ \ \ \

\

0.2

0.8

(21)

with an equilibrium constant K = 0.25, which is assumed to be independent of temperature. Pure reactant C is fed to a reactive distillation column, and the relative volatilities of components A, Band C remain the same as above. We begin with a low purity separation in which the target product compositions are specified to be XD,1 =0.8 and XD,2= 0.0025 for the distillate, and XB, I = 0.01 and XB,2 = 0.8 for the bottoms. Design calculations were carried out for a range of values for Da, and a plot of the reflux ratio vs Da at constant product compositions is shown in Fig. 10. Composition profiles at Da values of 0.084, 0.095,

\ \

0.6

0.6

Fig. 15. Number of stages (NT. N, and N s ) vs Da for Example 2. 2C~A+B

r = 7.8322 Da = 0.3

0.8 1

0.4 Da

o.o-!---.-----.,-------==;::::::::IIE=~=---~

0.0

0.2

, -, '"

6.0,-------------------,

O.O-!----,-----,-------.:::;::::::::::iI==........--:::., 0.0

0.2

0.4

0.6

0.8

1.0 A (Light)

C

(Intermediate)

]

0=

Fig. 14. (a) Design calculations for the reaction 2C~A + Bat Da=0.3. (b) Simulation. Dashed line represents the reaction equilibrium curve.

5.0

.~

~

... 4.0

.g

~

Example 2

1

As a second example consider a mixture undergoing a reversible reaction:

!

~

~

.

z Table 2. Design variables for the separations in Figs ll(b). 12(b). 13(b) and 14(b) Figure

Da

ll(b) 12(b) 13(b) 14(b)

0.0838 0.0950 0.1305 0.3000

2.2327 2.5142 3.3954 7.8322

s

NT

f

3.2023 3.4811 4.3541 8.7491

49 23 15 9

36 13 8 3

3.0

2.0

Q

1.0+-------,----,------,---------"1 0.0 0.2 0.4 0.6 0.8 Da

Fig. 16. Dimensionless total liquid holdup in the column (Da x NT) vs Da for Example 2.

405

New too ls for design B (Heny)

Component

Feed

Distillate

A

0.0

0.9900 2.0 x 1lt·6 0.009996

1.0

B

o.o

C

1.0

Bottoms 2.499 x 10-5 0.9900 0.009950

r = 4.4

0.8 \

Da = 0.0346

\ \

\ \

0.6

\

\

Xz

(a)

\

\ \

0.4

\ \

-,

"" -, 0.2

" -,

Example 3 <,

" 0.0 0.4

0 .0

C (I nter mediate)

0.6

0.8

1.0 A (Lig ht)

XI

Component

B (Heavy) 1.0

Feed

Distillate

0.0 0.0 1.0

0.9890 2.4 x 10. 6 0.0110

Bottom s 2.9 x 10-5 0.9891 0.0109

r = 4.4

0.8 \

Da = 0.0346

I \ \

\

0.6

in the pre vious example where the reaction was irrev e rsible . Figure 15 shows a plot of the number of stages vs Da at fixed product composition s. As in the previous example the number of stages gets large as Da get s small. Figure 16 is a plot of th e dimension less total liquid holdup in the column vs Da at fixed product compositions. Once again an optimum value of Da exists where (Da) x (NT) is a minimum . For this example . optimum values of Da occur between 0.12 and 0.25. An example of a co mpo sition profile that lies in this region is shown in Fig. 13 where Da =0 . 13

We now repeat Exa mple 2 for higher product puri ties than those specified in the last case . We specify a distillate composition of XIl .1 = 0.99, x" : = 2.0 x 10 6, and a bo tt oms composition of Xli . I = 2.499 x 10' \ Xll.2=O.99 . Consider a column with a reflux rati o equal to 4.4 (in this example we choose a value for r and calcula te the corresponding value of Da from the algorithm) . Figure 17(a} shows the solutio n to the de sign equations. Th e calcul ated value of Da is equal to 0.0346, the reboil ra tio is equ al to 5.4. and there are 24.4 and 51.X stages in the rectifying and the stripping sections. respecti vely. Figure l7 (b } shows the simulated column pro files. which agree well with th e results shown in Fig. 17(a ) .

\

(b)

\ \ \

\

0.4

\

\

"" "-,

0.2

-,

-,

-,

"' " o.o+--~--~:::!:::::!!!il::fil==<>;'~""""-e-e-i~ 0.0

0.2

0.4

0 .6

0.8

1.0

C

A

(Intermediate)

(Light)

Fig. 17. (a) Design calculations for the reaction 2C <=> A + B at Da = 0.0346. (b) Simulation . Dashed line represents the reaction equilibrium curve.

0.13 a nd 0.30 are shown in Figs 11-14, respectivel y. Figures 11-14(a) correspond to solutions computed using the boundary-val ue design procedure, and Figs 11-14(b) are the correspo nding simu lation resu lts. Design vari able s for these separations, as dete rmined by th e design procedure. a re give n in Table 2 (in this tabl e we report the Damkohler number to mo re significant figures so that our results may be reproduced more ea sily) . There is generally quite reasonable agreement between the design pro cedure calcu lations and the simulatio ns. The com positio n profiles show similar behavior to the profiles CACE 19-4-C

The conv ersion of component C achie ved in this reacti ve distillation column is equal to 99% . In a conv entional single-phase reactor with pur e C as feed the highest conversion possible , fo r a reaction with K=0 .2S. is 50'Yo. This occurs at reaction equ ilibrium and requires a large hold up. The reactor effluent contains 25%, A. 25% Band SO'X, C. In order to obtain high-purity products from this exit stream two distillation co lumns are required downstream. The desired products are withdr awn and any unreacted C is recycled to the reactor. Thu s a convention al plant requires a rea ctor. two distillation columns a nd a recycle stream to acco mplish what can be don e in a single combin ed reactor se pa ra to r unit. CONCLUS ION

A proc edure for the design of reactive distillation column s has been de veloped . For a reactive distillation column which has the same molar liquid holdu p on ea ch stage and for which the feed composition , the feed quality and the co lumn pre ssure are given there are five degrees of freedom . Compared to nonreactive dist illation there is one extra deg ree of fre ed om when reaction takes place. Thi s is due to th e influence of the liqui d holdup, as represented by

G. BUZAD and M. F. DOHERTY

406

the Damkohler number, Da , which is the ratio of the liquid residence time to the reaction time . We demonstrate the technique for the distillation of a ternary mixture that undergoes a reaction of the form 2C~A + B, and for which the volatility of the reactant is between the volatilities of the products. Our results show that high-purity A as distillate and high purity B as bottoms is achievable from a column with pure C feed. We tested our design results against column simulations and found them to be in good agreement. At fixed product compositions there is a sharp minimum in the total liquid holdup in the column as the liquid holdup on each stage is varied . The techniques proposed in this paper provide a framework for creating a more general design procedure that is capable of handling other types of reaction stoichiometry, as well as more sophisticated holdup distribution policies. The main features of the model that need modification before it is useful for systems of commercial interest are its ability to handle more components, nonideal mixtures and reactions with significant heat effects. In principle, there is nothing to prevent these improvements and we expect to report on them in the near future. Acknowledgements-We are grateful to Z . T. Fidkowski for permission to use and adapt his simulation program to reactive distillation columns . We are also grateful for financial support from the Petroleum Research Fund and the National Science Foundation (Grant No. crS-91l3717) .

NOMENCLATURE

A,B,C = Generic chemi cal species B = Bottoms product flowrate (mol/time) D = Distillate flowrate (mol/time) Da = Damkohler number [Hkj/F, dimensionless) f= Feed stage location, counting from the bottom up F= Feed flowrate (mol/time) H = Liquid holdup per stage (mol) K = Reaction equilibrium constant kf=Forward reaction rate constant (l/time) La = Liquid flow rate in the stripping section (mol/time) L r = Liquid flow rate in the rectifying section (mol/time) N, = To tal number of reacti ve stages in the rectifying section N, = Total number of reactive stages in the stripping section NT = Total number of reactive VLE stages in the column q = Feed quality r= Reflux ratio s = Reboil ratio V= Vapor flowrate (mol/time) X; = Mole fraction of component i in the liquid phase y, = Mole fraction of component i in the vapor phase

Greek letters b,um = Defined in eqution (B I) of Appendix B, it has a value of zero when the desired conversion/separation is feasible Sub scripts A ,B ,C=Components A, Band C B= Bottoms D;; Distillate F;;Feed i;; Component i rn ;; Stage index for the rectifying section, increasing down the column n;; Stage index for the stripping section, increasing up the column 1,2,3 = Components A , Band C Superscripts r ;; Rectifying section s ;; Stripping section REFERENCES

Agreda V. H. and L. R. Partin, Reactive distillation process for the production of methyl acetate . U.S. Patent No . 4,435,595 (1984) . Agreda V. H ., L. R . Partin and W. H . Heise, High-purity methyl acetate via reactive distillation. Chern. Engng Prog. 86, (2), 40-46 (1990) . Agreda V. H . and R. D . Lilly, Preparation of ultra high purity methyl acetate. U.S . Patent No . 4,939,294 (1990) . Amiet L., Process for the preparation of methyltrifluoroacetate. U.S. Patent No. 4,730,082 (1988) . Barbosa D. and M. F. Doherty , Design and minimum reflux calculations for single-feed multicomponent reactive distillation columns . Chern. Engng Sci. 43, 15231537 (1988) . Belson D . 1., A distillation method of aromatic nitration using azeotropic nitric acid. Ind. Engng Chern. Res. 29, 1562-1565 (1990). Bogacki M. B., K. Alejski and J. Szymanowski , The fast method of the solution of a reacting distillation problem. Computers chem. Engng 13, 1081-1085 (1989) . Bondy R. W ., Physical continuation approaches to solving reactive distillation problems. Paper presented at AIChE Meeting; Los Angeles (1991) . Chang Y . A . and J. D. Seader, Simulation of continuous reactive distillation by a homotopy-continuation method. Computers chem. Engng 12,1243-1255 (1988). Cleary W. and M. F. Doherty, The separation of closely boiling mixtures by reactive distillation 11. Experiments Ind. Engng Chern. Process Des. Deo, 24, 1071-1073 (1985) . DeGarmo J. L., V. N. Parulekar and V. Pinjala , Consider reactive distillation . Chern. Engng Prog. 88, (3) ,43-50 (1992) . Doherty M. F . and G. Buzad, Re active distillation by de sign . Trans IChernE 70, 448-458 (1992) . Fidkow ski Z. T . , M. F. Doherty and M. F . Malone, Fea sibility of separations for distillation of nonideal ternary mixtures. AIChE JI 39, 1303-1321 (1993) . Grosser J. H ., M. F. Doherty and M. F . Malon e , Modeling of reactive distillation systems . Ind . Engng Chern. Res. 26,983-989 (1987) . Julka V. and M. F. Doherty, Geometric behavior and minimum flows for nonideal multicomponent distillation . Chern. Engng Sci. 45, 1801-11\22 (1990) . Levy S. G. , D . B. Van Dongen and M. F. Doherty, Design and synthesis of homogeneous azeotropic distillations. 2. Minimum reflux calculations for non ideal and azeotropic columns. Ind . Engng Chern. Fundam . 24, 463-474 (1985) .

407

New tool s for design Smith L. A. , Method for th e pr ep ar at ion of meth yl tertiary but yl e th er. U .S. Pat ent No . 4 ,978,807 ( 1990). Smith L. A . , Ca talytic dist illat ion pr ocess. U.S. Patent No . 4,307,254 ( 1981). Stiehlm air J. G. and J. R . Herguijuela, Separation region s and processes o f zeotropi c and azeo tro pic te rna ry distillation . A ICh E 1/38,1 523- 1535 ( 1992). Ung S. . Doctoral Dissert ati on . U niversity o f Massachuse tts, A mherst ( 1994) . Ve nka ta ra ma n S., W . K. C ha n and J. F . Bosto n, Re acti ve d ist illati on using ASPEN PL US . Chem . Engng Progr. 86 , (8), 45-54 ( 1990). Wahn scha fft O. M. . J . W . Koehl e r. E . Blass and A . W. Westerber g , Th e product co mposition region s o f singlefeed azeotropic dist illati on co lumns . Ind . Engng Chem . Res. 31 , 2345-2362 ( 1992).

cu rve (i.e . co mpo nen ts A a nd B but no t co mpone nt C o r reacti ve azeotropes (see Barbosa and Doh e rty. 1988).

(b) Finite external flows and large in/ernul flow! T he operati ng line for th e rectifying section is given by equa tions (J 1). T he reflux ra tio appea rs in three te rms : r/(r + I), I/(r + I ) a nd PlV [as defined by equatio ns (12)]. At lar ge reflux ratios r/(r+ J) is equa l to uni ty. l/(r + I) is e qua l to ze ro . and F/V is equal to zero ; th er efor e , equation s (1 1) become :

1=

As we will show next , th e ter m Da 2:;'~ I (x f. 1 .•. X, boun ded and so equatio n (A 5) becom es:

1,2. ( A5)

,X, :I K ) is

APPENDIX A

(A6 )

Total Reflux and Reboil It is helpful to study the asy mpto tic limit s of the design equations derived in th e tex t when r an d s approach infinit y. The reflux ratio approa ches infinity either as L T approaches infinit y or as D goes to zero . To tal reb oil occurs e ithe r as V goes to infinity o r as B goes to ze ro . In this A ppe ndix we st udy th e following two case s: (a) ra nd s approach infinity via zero externa l flows a nd finite internal flows; and (b) r and s approach infinity via finite ex te rna l flows and large inte rna l flows.

Thi s is the classical condition for infinit e re flux in nonreac tive co lumns which also applies to reactive colu mns when th e infinite reflux limit is obtained by letting L , ge t large while D re mains finite . We will no w sho w that the ter m Da 2:i': \ (xi , ... x,. I X, ~ / K) is bou nded . Th e ove ra ll materi al balan ces , equatio n (6). ca n be wr itten as:

- ( x~ . ) .

(aJ Zero external flows and finite internal flows

In th is case th e distill at e ftowrate D and the bottom s flowrat e B ar e equal to zero. Since no thing le aves thc column , it follows tha t F must be eq ual to zero under steady state ope ratio n. Th e compo ne nt mater ial bal an ces over the e ntire co lumn. equatio ns (4), becom e : N, (

X

X

D

)

O= Hk f "' . ) , _ ...L:..!....l2 L.J x,.. K '

i '~' l,2

IA7)

Th e maximum value th is qu ant ity ca n att ain is whe n the first term in bracke ts on the right -han d side of eq uat io ns (A 7) is a maximu m and the second term is a minimu m. Th e latt er occurs at Xf = (OJl,1) . T he forme r occu rs at Xo . ) =, X O..' = 0, i.e. X H . ~= I - X B. I and X J) . ~ = I - X i> \ . For this specia l case , eq uat ion (7a ) becom es: I + 2tB.i

Ii= 1+ 2xlJ

(A I)


I

,- I

i.e . the re is no over all ge nera tion of co mpo ne nts A and B thr ou ghout the system . This occurs when chemical equilibrium is achieved everywhere . i.e. th e reaction rate on each sta ge is eq ual to zero. For the D = B = 0 column configuration there is no distinction betw een rectifying and stripping sections. Therefore, we choose to mod el the column with th e stripping section equ ati on s ( 13)-( 17). Since B = 0, equation (13) gives L u = V , a nd e qua tio ns ( 14) become:

H/LB ~ ( , Xi. IX;. ))· L.J Xi J- --;;:- .

O=Xn + l.i-Y" , ,+~

. 1=

1,2.

(A2)

i= I

[ . ~ ( ~ - ~)] Da L..

(A 3) Thi s e qua tio n along with the reaction eq uilibrium co nd ition : (A 4)

represent s the stea dy sta te co nditions in a co lumn whe re D = B = O. Thi s mod el is iden tical to the che mical equ ilibrium model of Barbosa a nd Doh e rty ( 1988). For the specia l case where the num ber of stages approaches infinity th e co mpositio ns at the ends of th e column approach either pure components that lie on the rea ction equilibrium

t"

I

2

K

( A9 )

n1. H

Similar ana lysis sho ws that Da tim es the summa tio n te rm. in equa tions (A5) , a lso has a lower bou nd , and th at thi s lower bo und is eq ual to - ( 1/ 2 ). Since m is less tha n o r eq ual 10 N I , it follows that : _ _I <

2"

Since we kn ow , from above. tha t every stage is at chem ical equ ilibrium we mu st put thi s co nstrai nt into these eq uation s. Th is is done by subtrac ting the two equations (A2) to elim inate the common reaction te rm , giving:

0 - ) _ X,. IX,. ) - x,..' K '

Replacing D I B in the ov era ll mater ial ba lance bv this equatio n lead s 10:

r

1

I Xi. ~ ~~ (X ~I ; _ x, . K ) ",-~ ~2'

. '\'

D.l

t A W)

and eq uatio ns (All ) a rc the va lid limits for the recti fying sect io n equa tions at large reflu x ratios. Similar ana lysis for the str ipp ing sectio n shows th at at lar ge reboil rat ios the result is the sa me as tha t for the rect ifying section. nam ely tha t passing strea ms have equa l co mpos ition. T his result is valid when r and .I' ap proach infinit y via small exte rna l flows and lar ge inte rna l flows, For the case whe n r and .I' app roac h infinit y via zero exte rna l flows equa tio ns ( A3) and (A4) a rc the co nditions in the co lumn at stcadv-sta te . For reac tive columns we sec that differe nt res ults arc obtai ned de pending o n the way in which r and .I' actually beco me infinite . T his is not the case with nonreact ive co lumns where th e sa me result is obta ined (nam el y. passing strea ms have eq ua l composition) regardl ess of the wav . in which r and .I' app roach infinit y

408

G.

B UZAD

and M. F. DOH ERTY

APPENDIX 8 Selecting an Initial Guess for the Reflux Ratio For design purposes we specify values for the distillate and bottoms compo sitions as well as for Da. T he task then is to find a value of r, if any, that will make the desired conversion/separation feasible. In step (5) of the design algor ithm we must make an initial guess for the value of r . Not every initial guess for r gives a solution. In this App endix we show how the initial guess of r is selected for Example 1. The first criterion for feasibility is the intersection of the rectifying and the st ripping section profiles. For the exa mple under consideration (Fig. I) Da is set equal to 3. At this value of Da we solve the rectifying and the stripping section mate rial balan ce equations at several values of r. The composit ion profiles intersect at values of r between 0.65 and 1.9. For r between 0.65 and 1.1 the profiles inte rsect more than once . Intersection of the profiles is a necessary but not sufficient condition for a solution to exist. In order for a solution to occur the summation term in the rectifying section plus the summation term in the stripping section must equal the summation term in the overall material balance-i.e . we must ensure that just the right amount of reaction has taken place. One way of achieving this is to make . osum in the following equa tion eq ual to zero:

[ ,LJ~~ I (2

Xi. 1Xi . 2) ] -

Xi , J- -

6...m

K

-

[ LJ ~ (XJ2 ,3,=1

", .'''r ~

lIP) .~ ( 2

IX 2 i. - )]

K

K

-Xi. -



~

;=1

.1:;. 3- "- J.

( 2 x,. IX;.2) X} ' } - -

K-

}. \

(B t)

The summation term over NT stages is solved for in step (6) of the algorithm. It is only a function of the target compositions and Da. In ste ps (9) and (to) of the algorithm we solve for the composition pro files of the rectifying and the stripping sections. If the profiles intersec t we can solve for

0.0

JE -1.0

-2.0

. 3.0 - t - -- - . ------r0.0

Fig. 8 I.

0.5

o sum

1.0

--.--I.S

-J

.- -.. 2.0

2.5

vs reflux ratio at Da = 3.0 for Example 1.

the numb er of stages , N, and N" where the inte rsectio n occurs. Knowing N, and N, o ne can eva luate the summation ter m in each column section. Figure 81 shows a plot of o sum vs r for Da= 3.0. We can see from this figure that a solution exists at a reflux ratio in the vicinity of 0.75. Notice that for r in the vicinity of 0.75 there are two values of o sum, indicating tha t the profiles intersect twice. We use 0.75 as the initial guess of r in a roo t-finding program that uses IMSL subroutine DN EQ NF to searc h for a zero. We find that the value o f the reflux ratio at the solution is equal to 0.77. The corr espond ing composition profiles arc shown in Fig. 1.