Design of three-component kinetically controlled reactive distillation columns using fixed-points methods

Pergamon

Chemical Engheering

0009-2509(94)EOOOl-7

Science, Vol. 49, No. 12, pp. 1947-1963, 1994 Copyright Q 1994 Elsetier Scimcc Ltd Printed in Great Britain. All rights -cd ooos25G9/94 $7.00 + 0.00

DESIGN OF THREE-COMPONENT KINETICALLY CONTROLLED REACTIVE DISTILLATION COLUMNS USING FIXED-POINT METHODS GEORGE Department

of Chemical (First

BUZAD and MICHAEL

Engineering,

University

received 29 July 1993; accepted

F. DOHERTY+ of Massachusetts, Amherst, in revised

form 10

December

MA 01003, U.S.A. 1993)

Abstract-There is an increasing interest in reactive distillation, which has led to the development of several good simulation algorithms for modelling these systems. Tn this paper we present a procedure for the design of three-component kinetically controlled reactive distillation columns based on fixed-point methods. Using these methods we have developed an algorithm for calculating the minimum flows in a reactive distillation column. For the examples studied we show that the minimum reflux ratio in a kinetically controlled distillation is equal to that for an equilibrium reaction. We also show how to choose spccifications that will lead to feasible reactive distillation columns. We tested our design results against column simulations and found them to be in good agreement.

In recent years reactive distillation has received increasing attention as a process alternative for carrying out liquid phase chemical reactions (Smith, 1981; Grosser et al., 1987; Chang and Seader, 1988; Bogacki et al., 1989; Agreda et al., 1990, Smith, 1990, Venkataraman et al., 1990; Bondy, 1991; DeGarmo et al., 1992). A more extensive survey of the literature is given by Doherty and Buzad (1992). Although there is a significant number of publications, there is still no effective design technique for these systems. In the design of chemical processes, 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. This is in contrast to simulation where the input and operating variables of a process are specified and the task is to solve for the resulting outputs. The aim of this paper is to present a design procedure for kinetically controlled reactive distillation columns based on fixed-point methods. In the boundary value design approach for nonreactive columns, product stream compositions are specified and the finite difference material balance equations for the rectifying and the stripping sections are solved stage by stage away from the bottoms and distillate streams. As the number of stages gets large, a zone is reached where the composition on successive stages is essentially constant. This behavior indicates that a pinch (or fixed point) has been reached. In addition to the stable node fixed point located at the end of a profile, there are other fixed points which may influence the behavior of composition profiles. For a C-component mixture with no chemical reac-

‘Author to whom correspondence

should be addressed.

tions and no azeotropes, there are C isolated fixed points in both the rectifying map and the stripping map. The fixed points are either nodes (stable or unstable) or saddles (Julka and Doherty, 1990). At total reflux the fixed points of the distillation maps occur at the pure component compositions, as well as at the azeotropes present in the mixture. For this limiting case, the stage-to-stage composition profiles in the column closely follow the residue curves. As the reflux ratio is lowered, the fixed points move to new positions, some of them even go outside the composition triangle. The composition profiles, however, remain within the composition space. Using these concepts, it is possible to develop novel design procedures for nonideal and azeotropic mixtures (Fidkowski et al., 1991). In this paper we discuss the influence of kinetically controlled chemical reactions on the fixed points and on the composition profiles of a distillation column. We use these concepts to develop a new design procedure for kinetically controlled reactive distillation columns. For ease of presentation the method is illustrated in the context of a specific example but the ideas are not limited by the particular stoichiometry selected or the simplifying assumptions adopted. COLUMN MODEL Consider

a liquid-phase

reaction

2CoA+B

of the type (1)

which is carried out in an ideal solution where the rate of reaction per mole of mixture is given by

Components respectively, 1947

1,2 and 3 correspond to A, B and C, K is the reaction equilibrium constant

GEORGE BUZAD and MICHAEL F. DOHERTY

1948

and kf is the forward rate constant. For simplicity, we assume a single feed column, a saturated liquid feed, constant molar overflow (implies negligible heat of reaction), a total nonreactive condenser, a nonreactive reboiler, 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. Material balances around the entire column, the rectifying section, and the stripping section give (see Buzad and Doherty, 1994). Overall 0=

xf7.i

-

DIB

~ D,B + 1 xD,i

+ viDa E

x~+S

1 D/B+1

-

lxj.

xj.

--

I

pxB

’

2

i=

K

j=l

1,3

(3)

where Da = Hk,fF. The Damkohler number, Da, is dimensionless and is the ratio of a characteristic liquid residence time (H/F) to a characteristic reaction time

( 1IV. We can solve for D/B in terms of the feed and product compositions by eliminating the common reaction term in eq. (3) to give D -= B

(XF,I + :XF, 3) -

(xiv

(XD. 1 + :%I, 3) -

b,

+ :xs,3) 1 + :+.

3)

XF, i -

D/B •t 1

+v$az

xfj-j=l (

lXB,’

’ xi. I xj. K

2

i = 1 or 3. >

W

Rectifying section

i= where r = L,/D,

1,3

(5)

and

F 1 + B/D -z_=_, V 1+r

1 + DfB s

(6)

Stripping section

i=

1,3

(7)

where s = V/B, and F

-L

1 + D/B 1+s

l+ B/D 1 +r+ B/D‘

(9)

where q represents the feed quality. When Da = 0, the model reduces to the conventional model for nonreactive distillation, and when Da = co, it corresponds to reactive equilibrium distillation. Although the model agrees with the known limits of Da = 0 and Da = co, the algorithm described below may only be used for finite values of Da (i.e. for 0 < Da < co). This is because the degrees of freedom change when the limits are attained, and the structure of the algorithm is not appropriate for such cases. The column model therefore consists of the overall balances, eqs (4) and (9), together with the rectifying and stripping eqs (5) and (7). For given feed conditions and column pressure, the design model has five degrees of freedom, which we take to be the distillate and bottoms compositions, and the reflux ratio. The boundary value procedure involves specifying these quantities and solving the finite difference equations from the ends of the column towards the middle. If the target compositions are feasible, then for the given value of r there will be a value of Da that satisfies the design equations. FWED POINTS

1

D/B

-XDTDIB+

D s+l-q -Z.Z B r+9

there is the

(44

In what follows it will be convenient to select eq. (4a) together with one of eq. (3) as the pair of overall component balances, thus 0 =

In addition to the material balances following overall energy balance:

(8)

Before developing the design procedure, we begin by studying the behavior of the fixed points. We do this by specifying the five variables that were specified in the design problem formulation and examine the behavior of the column composition profiles as the Damkohler number is varied. Consider an ideal ternary mixture for which the reaction equilibrium constant is equal to one quarter, and for which the voIatilities of components A and C relative to component B are 5 and 3, respectively. Pure reactant C is fed to a reactive distillation column and we set the target distillate composition equal to x&1 = 0.803 and xD,s = 0.003 and the target bottoms composition equal to xs.i = 0.01 and xg.* = 0.81. Take the reflux ratio to be 2.1. From the overall material balance, eq. (4a), we solve for D/B in terms of G, xD, and xB, and obtain D/B = 1.0. From the energy balance, eq. (9), we calculate the reboil ratio to be 3.1. If we specify a value for Dn we are in a position to solve the stage-to-stage material balance equations by stepping up and down the column. Before looking at the entire column we will focus our attention on the stripping section. Figures l(a) and (b) show the stripping section composition profile at Da = 0.10348 - e and 0.10348 + E, respectively. Notice that the profiles have a sharp corner, and that at the point where they intersect the reaction equilibrium curve, there are a large number of stages with essentialIy the same composition. The point of intersection is a fixed point of the saddle type. We can solve for the composition of this fixed point in the following way.

Design of three-component Compoaerl

Feed

A

if:

:

io

Disllllak

0.803 0.194 0.003

Bottoms

0 = (x1 + $3, - &(Yl

0.01 0.81 0.18

0.0

C

(Inkrmed

0.2

0.4

0.6

2.1 0.10348-e

0.8

1.0 A (Light)

iale)

B (=-Y)

_._ , 0.0

I

0.2

014

0.6

0.8

1.0

+ iY3,

--&(XF,, +:xn,3).

rDa=

1949

kinetically controlled reactive distillation columns

(11)

Notice that in eqs (10) and (11) the stage index has been eliminated. We can solve these equations simultaneously using a root finder. For the parameter values defined above we take x1 = 0.3 and xz = 0.3 to be the initial guess for the composition of the fixed point and treat the vapor as in phase equilibrium with the liquid. We obtain a fixed point with the following composition (0.3207,O. 188 LO.49 12). This is indeed where the profiles intersect the reaction equilibrium curve in Fig. 1. Notice, from eqs (10) and (1 l), that the fixed points are not a function of Do. At Da = 00, the rate of reaction goes to zero and chemical equilibrium is achieved throughout the system. This is a singular case that requires special methods for distillation design [see Barbosa and Doherty (1998a)J Even though different models are required for designing kinetically controlled columns and equilibrium controlled columns it transpires that the fixed points are the same in both cases, and are found by solving eqs (10) and (11). Figure 2 shows the composition profiles for the mixture under consideration at several values of Dn. The fixed point, x = (0.3207,0.1881,0.4912), is the same for all values of Da, but the profile gets asymptotically close to it only at Da = Ci.10348. We call this the critical value of Da and denote it by DaC. At values of DA smaller than Da’ the profiles cross the reaction equilibrium curve, but do not for values of Da larger than DaC, as shown in Fig. 2. In order to solve for the critical value of DA we use the IMSL subroutine DZBREN. DZBREN finds

(Li;hl)

Fig. 1. Composition profiles for a mixture with X,., = 0.9, X a.1 =O.l, r=2.1, x&*= 0003 and xS.I = 0.01 at (a) Da = 0.10348 - E, (b) DU = 0.10348 + E. Dashed line represents the reaction equilibrium curve.

Fixed points in the stripping section correspond to solutions of eq. (7) as the index n approaches infinity. The only way for compositions on successive stages to become equal as the number of stages goes to infinity is for the summation term in eq. (7) to converge to a constant value, i.e. the reaction rate must become zero. In other words, all fixed points must lie on the reaction equilibrium curve. Therefore, the first criterion for calculating the composition of a fixed point is

0+-y). The second criterion is obtained by eliminating common reaction term from eqs (7) to get

0.0

1

,

I

I

0.2

0.4

0.6

0.8

C *lc~cdLld

the

-----____

I 0.0

XI

\

1.0 A &I*)

Fig. 2. Stripping section composition profiles for the mixture described in Fig. 1 at several values of Da. Dashed line represents the reaction equilibrium curve.

GEORGE

1950

and

BUZAD

a zero of a real function which changes sign in a given interval. We define our function to be a step function with a value of +1 if the profile crosses the reaction equilibrium curve, and -1 if it does not. The user must supply an upper and a lower guess for Da’. For the example under consideration we used 0.05 as the lower guess and 0.15 as the upper guess for Da’, and obtained Da’ equal to 0.10348. The fixed points are not a function of Da; they are, however, a function of (x~.~ + (1/2)x& which we denote by X,.,. Lines of constant X1 represent stoichiometric lines in the composition triangle [see Barbosa and Doherty (1988b), Fig. 2; Ung and Doherty (199411. So far we have studied a mixture with a bottoms composition equal to xg,, = 0.01 and ~a,~ = 0.8 1, which gives Xa, 1 = 0.1. The same value of X,,, can be obtained from mixtures with other bottoms compositions, for example XB., = 0.05 and 0.85. We can solve for the fixed points using = xB.2 eqs (10) and (11) and for both cases we obtain the fixed point composition shown in Fig. 2. Figure 3 shows the composition profiles for the latter case at several values of Da. Note that the profiles do not go near the fixed point, i.e. we are not able to find a-value for Da’. In order for a critical value of Da to exist, the bottoms composition must lie in the triangle such that at Da = 0 the composition profile crosses the reaction equilibrium curve. Figure 4 shows a plot of Da’ vs xBmlfor X,,, = 0.1. From this figure we see that DaC occurs only for values of xB.l smaller than 0.01389. Notice that Da’increases as x&i moves away from the B/C binary edge and approaches the reaction equilibrium curve. From eq. (11) we see that the fixed points are a function of the reboil ratio. At total reboil there are two fixed points: pure component A and pure com-

MICHAEL

F.

DOHERTY

ponent B. Figure 5 shows the fixed points at several other values of s.The fixed point originating at pure component B leaves the composition triangle as the reboil ratio is lowered. The fixed point originating at pure component A, on the other hand, moves along the reaction equilibrium curve in the direction of component B as s is lowered. It does not reach pure component B, rather at s = 0 the fixed point lies at the intersection of the straight line X, = X,,, and the equilibrium curve [see eq. (li)]. This is also what happens for a series of stirred tank reactors, which is functionally equivalent to a stripping section at zero reboil ratio. The reactive composition profiles shown in Figs 1-3 do not end inside the composition triangle. This is in contrast to the behavior of profiles for the two limiting cases of nonreactive distillation and for distillation of mixtures at chemical equilibrium. At Da = 0 and Da = co, a stable node (located inside the composition space) is reached as the number of stages gets large. With kinetically controlled reactive mixtures we have so far seen fixed points of the saddle type inside the triangle. In the Appendix we investigate the global behavior of fixed points both inside and outside the composition triangle. We find that in addition to saddles stable focii also exist. So far our study has been restricted to the stripping section. We will now briefly consider the rectifying section. In solving for the fixed points of the stripping section map we used eqs (10) and (11). For the rectifying map eq. (10) still holds but we need to replace eq. (11) by its counterpart, obtained by eliminating the common reaction term from eq. (5):

0 = (Yt + :y,, -

-&D,

Component

Feed

Dtstlllale

Bottoms

i?

0.0

O.CQ3 0.803

0.85 0.05

C

1.0

0.194

0.10

+x1 1 +

+:x31

:%J.3,.

(12)

0.4-

{

0.3,

0.2-

-.._._ ----_ 0.0 , 0.0 C <*atermrdt.te,

------_____ I

1

I

I

0.2

0.4

0.6

0.8

x1

O.l1.0 A www

0.0 0

Fig. 3. Stripping section composition profiles for a mixture with X,,, =0.9, Xa., =O.l, r = 2.1, XD,~ = 0.003 and x~,~ = 0.05 at several values of Da. Dashed line represents the reaction equilibrium curve.

I 0.004

I 0.008

I 0.012

I 0.016

0

20

w.1

Fig. 4. (Dd)~,.,,

for a system with = 0.1 and r = 2.1.

YS xs.I

X,,,

X,.,

= 0.9,

Design of three-component

Campcmnl

Feed

Dbtlllate

(He8vY)

A

0.0

0.803

0.01

‘.O7\

C B

0.0 1.0

0.194 0.003

0.18 0.81

B

Da = 0.11486.

\ \

0.6

hi

(IntermedIate)

A (LWW

DESIGN

PROCEDURE

FOR TERNARY

0;2

0.4

016

0.6

XI

(Inter:edhlc) B

Fig. 5. Stripping section fixed points for a mixture with XB,1 = 0.1 at several values of s. Dashed line represents the reaction equilibrium curve.

Consider the same mixture that was studied above with a distillate composition set equal to (0.803,0.003, 0.194) and a reflux ratio equal to 2.1. Solving equations (10) and (12), with x1 = 0.3 and x2 = 0.3 as the initial guess of the fixed point, gives a fixed point with the following composition (0.0948, 0.4789, 0.4263). The critical value of Da for this mixture is equal to 0.11486. Figures 6(a) and (b) show the composition profile at Da’ - E and Da’ + E, respectively.

L

\

‘.

0;o =I

Ekmttama

r= 2.1

0.8

C

1951

kinetically controlled reactive distillation columns

W=*r)

I= Da=

2.1 0.11486+~

(b)

MIXTURES

For a given column pressure and feed condition, the column model has five degrees of freedom. Our design procedure for reactive columns consists of specifying desired values for X,,,( = x~,~ + $x&. X R,, , I, x~,~ and xB, 1, and solving the model equations for the remaining variables. The reason we chose to specify these variables is because the fixed points are a function of Xo.l and I for the rectifying section map, and of X,, 1and s for the stripping section map. Before solving the difference equations we must guess a value for Da, In the previous section we saw that a critical value for Da exists and, as we will show below, Da’ is a good guess for Da. An algorithm for solving this reactive distillation problem is: (1) Specify the feed composition and quality, and the column pressure. (2) Specify desired values for (xgel + (1/2)x,.,) and b,l + (1/2)xB,d (3) Solve for D/B [eq. (4a)]. (4) Specify a value for the reflux ratio. (5) Solve for the reboil ratio [eq. (911. (6) Solve for the fixed points of the rectifying and the stripping section maps [eqs (10) and (ll), and eqs (IO) and (12)]. (7) Solve for F/V [eq. (6)] and F/Ls [eq. (8)].

010 :

0.2

OA

0.6

0.8

1.0

Fig. 6. Composition profiles for the mixture described in Fig. 1 at (a) Da = 0.11486 - e,(b) Da = 0.11486 + E. Dashed line represents the reaction equilibrium curve.

(8) Specify desired values for x~,~ and x~,~. (9) Solve for xD. 1and Q,,~ (from the values in steps (2) and (8)). (10) Solve for the critical value of the Damkohler number, Da’, in each column section. number (11) Guess a value for the Damkohler based on a knowledge of Da’. term in eq. (4b). (12) Solve for the summation profile for the rec(13) Compute the composition tifying section [eq. (511. Step down the column until one of the following conditions occurs: (a) the summation term in eq. (5) gets larger than the summation term in eq. (4b) (from step (12)), or(b) the composition profile has gone outside the composition space (allow at most one step outside the composition triangle). (14) Repeat step (13) for the stripping section composition profile.

GEORGE BUZAD and MICHAEL F. DOHERTY

19.52

(15) Repeat steps (ll)-(14) until the composition profiles intersect. (16) Count the number of stages in each section of the column for each time the profiles intersect. (17) 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 [eq. (5)] plus the summation term in the stripping section [eq. (7)] must equal the summation term in the overall material balance (from step (12)), i.e. we must ensure that just the right amount of reaction has taken place. The normalized difference of these terms is denoted by a,,,,, and defined in the Notation section. Some of the properties of this function are discussed by Buzad and Doherty (1994). Note that not all values of xD and 5s lead to a feasible column. Values for x_D and xs must lie within the feasible product distribution 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 will demonstrate the design technique by considering ternary mixtures that undergo a reaction of the form 2C-=A+B

(13)

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 reactionPseparation process. In this section we discuss two examples. In both cases the volatilities of components A and C relative to component B are 5.0 and 3.0, respectively, i.e. A is the lightest component and B is the heaviest. The feed is taken to be a saturated liquid of pure component C. In both examples the reaction is reversible with an equilibrium constant K = 0.25, which is assumed to be independent of temperature. The examples differ in the target product purities-the desired purity of the disMlate and bottoms streams is higher in the second example Example

1

There are three parts to this example. In all three cases X,., is taken to be 0.9, and X,,, is equal to 0.1. In parts (a) and (b) we illustrate the design procedure and explore the effect of varying the distribution of components in the product streams at a fixed reflux

ratio order value, target

of 2.1. In part (c) the reflux ratio is varied in to demonstrate the existence of a minimum below which it is not possible to achieve the compositions in the product streams.

Example l(a). As mentioned above X,,, = 0.9, -%I. 1 = 0.1, and the reflux ratio is equal to 2.1. We can use eqs (t 0) and (11) and eqs (10) and (12) to solve for the fixed points of the stripping and rectifying sections, respectively. We obtain fixed points with the following compositions (0.0948, 0.4789, 0.4263) and (0.3207,0.1881,0.4912) for the rectifying and the stripping sections, respectively. The remaining two variables to be specified are x~,~ and x~,~ which we take to be x,,~ = 0.003 and xtJ. 1 = 0.01. The task then is to find a value of Da, if any, that will make the desired conversion/separation feasible. Earlier in the paper we showed that a critical value of Da exists such that for values of Da smaller than Da’ the profiles cross the reaction equilibrium curve, but do not for vaIues of Da larger than Da’. We will use this information to help us design reactive distillation columns. Since the reflux ratio and the distillate and bottoms compositions are completely fixed we can solve for Da’ in each column section. We obtain (Dac)rec, = 0.11486 for the rectifying section, and (Dac),,,iP = 0.10348 for the stripping section. The composition profiles at (Dac)rsct - E and (Du’)~~~, + E are shown in Figs 6(a) and (b), respectively, while the composition profiles at (Dac)atrip - E and (Da’),,,i, + E are shown in Figs l(a) and (b). The composition profiles intersect in two of the four cases-at Da = (DGsct. + E [Fig. 6(b)] and at Da = (DaOstriP - E [Fig. l(a)]. In neither case, however, is the desired conversion/separation feasible since a,,,,, is not equal to zero at the intersection points. Let us consider the behavior of the composition profiles at other values of Da. For the mixture under (Dac)rec, ( = 0.11486) is larger than consideration (Dac)s,rir ( = 0.10348). The behavior of the profiles for values of Da greater than (Dac)rec, (Or z=-(Du’)~,~~~if it were larger) is distinct from their behavior at smaller values of Da. When Da > (Da’)..., [e.g. Fig. 6(b)] both composition profiles lie to the side of the reaction equilibrium curve where the rate of reaction for components A and B on each stage in the column is positive, i.e. components A and B are, as desired, being produced and not consumed. When Da < (Dae)lec, [e.g. Figs 6(a), and l(a) and (b)] one or both of the cqmposition profiles cross the reaction equilibrium curve, and so for certain stages in the column the rate of reaction for components A and B is negative. For the latter case the composition profiles may intersect [e.g. Fig. l(a)], but S,,, is not normally equal to zero. Our experience is that values of Da greater than (Dd),,,, (or (Dac),,riP if it is larger) is the regime where feasible column designs are to be found. The composition profiles for Da = (DcP)~~=, + E ( = 0.11486) are shown in Fig. 6(b). As Da is increased the profiles move as shown in Fig. 7. For values of Da greater than 0.11492 the profiles do not intersect. We

Design of three-component

kinetically

controlled

reactive distillation

0.11486

1953

columns

0.11488

0.11492

0.11490 DI

Fig. 8. a,.,

vs Da for the mixture described in Fig. 1.

Fig. 7. Composition profiles for the mixture described in Fig. 1 at several values of Da. The stripping profiles at Da = 0.11492 and 0.11486 + E lie on top of each other. Dashed line represents the reaction equilibrium curve.

Compoaeml Feed A

B C

0.0 0.0 1.0

Bottoms

Dbtillate

0.01 0.81 0.18

0.803 O.M3 0.194

I-

now check to see whether a solution to the design equations exists in the regime where the profiles intersect (step (17) of the design algorithm). Figure 8 shows a plot of IS,,, vs Da. As can be seen, a,,,.,, does go through zero at a calculated value of Da equal to 0.11487. The corresponding reboil ratio is equal to 3.1, and there are 14.01 and 11.52 stages in the rectifying and the stripping sections, respectively. The liquid phase composition profiles for the feasible column are shown in Fig. 9(a). Figure 9(b) shows the simulated column profiles at r = 2.1, s = 3.1, NT = 26, f= 12 and Da = 0.11487. As can be seen, this figure agrees well with the results obtained via the boundary-value design procedure. Notice that the solution to the design equations occurs at a value of Da very close to (DaOrce,. We will see the same behavior occuring in the examples that follow. Example l(b), In this example we keep X,., = 0.9, X,,, = 0.1 and r = 2.1 (as in the last example) and explore the designs at other values of x~,~ and x~,~. The desired conversion/separation is not feasible for all values of xDSz and x~,~. In Example l(a) XD,Zand XB,1 were carefully chosen so that the desired conversion/separation was feasible. Here, we will make use of our knowledge of fixed points to help us choose sensible values of x~,~ and x,,,~. For fixed values of Xn, 1 and X,, , the fixed points of the rectifying and stripping sections are independent of xD.* and x~.~. The critical values of Da ((Du~)~~~~ and (Dac)s,ri,,), however, are a function of the distillate and bottoms compositions. Earlier we solved for (Dcz’)~,~~,,as a function of xB, 1 and the results are shown in Fig. 4. We see that (Du~)~,~~~occurs only for values of xgVl smaller than 0.01389.

Repeating

the same

calculations

for the

m=

2.1 0.11487

(8)

0

r= 2.1 Da = 0.11487

0.0

Fig. 9. (a) Design

0.2

0.4

0.6

0.8

1.0

calculations for tbe mixture described in Fig. 1 (Da = 0.11487). (b) simulation. Dashed line represents .... . the reactlon equmbnum curve.

GEORGE

1954

BUZAD

and MICHAEL F. DOHERTY

rectifying section shows that (Du’),_~ occurs only for values of x~,~ smaller than 0.00604. Figure 10 shows a plot of (Dae)lco, versus x~,~ and (Dac)BIrlp vs xs, I. From Fig. 10 we see that the desired conversion/separation is notfeasible for values of x~,~ greater than 0.00604 and for values of x,,,~ greater than 0.01389. This is because a critical value for Da does not exist in these regimes. For other values of x~,~ and xs,l the desired conversion/separation may or may not be feasible. The key to making it feasible is to choose values for xg.2 and x~,~ such that (Du’),~~~ is nearly equal to (DU’)oaip. The feasible solution in example l(a) was constructed using this heuristic. For this example (Dd)..,, = 0.11486 and (Dac&, =

cmponcol

B

nl=d

; C

I.0

Fed

DistOl.tc

0.0 0.0 I.0

Bnttoml

D.8M5 D.Dm5 0.1950

0.003 Cl.803 0.194

0.10348.

We

now

demonstrate

typical

behavior when a mixture where xg.2 = 0.0025 and x~,~ = 0.003. Solving for the critical values of Da we obtain (Du’)~~~, = 0.0822166 (In the vicinity of Da’ the composition profiles are very sensitive to small changes in Da, which explains why so many digits are reported. It is not necessary to use this many significant figures in the simulation calculations) and (DCI’).lrip = 0.02071. The composi+ E are shown in Fig. 11. tion profiles at (Dd),.,, Notice that the profiles do not intersect. Increasing the value of Da will simply cause the profiles to move further away from the reaction equilibrium curve and so the profiles will not intersect for any value of Da larger than (DaC)r.Et. In other words there is no solution to the design equations. The desired conversion/separation was not feasible in the last example because (Du~),_~ was not close enough to (Dcz~)~,~~~.For fixed values of xD.2 and xg, 1 we can change the values of (Due),eE, and (Dac)nriP by introducing nonreactive stages at either end of the column. However, there is an upper bound on the number of nonreactive stages that can be used. Figure 12 shows the composition profiles at Da = 0. The (DcI’)~~~, is not close to (DUc)s‘rip. Consider

Fig. 1I. Composition profiles for a mixture with X,,, = 0.9, XB.1--01 .* r=Z.l, x~.~ = 0.0025 and xg., = 0.003 at Da = 0.0822166 + E. Dashed line represents the reaction equilibrium curve.

0;o

0.2

0.4

0.6

0.8

LO

Fig. 12. Composition profiles for the mixture described in Fig. 11 at Da = 0. Dashed line represents the reaction equilibrium curve.

0

0.004

O.W8

0.012

I

0.016

0.020

XB.l R “II. I

Fig. 10.

(Da’),,,, vs .x~.~ and (Dac)acrb vs xssI for a system with X,,, = 0.9, X,,, = 0.1 and r = 2.1.

maximum number of nonreactive stages possible for the stripping section is seven; this ensures that at least one of the reactive stages in the stripping section lies to the side of the reaction equilibrium curve where the rate of reaction for components A and B is positive (i.e. so that (Dcf&, exists). Solving for (Dd&, in the same manner as in the previous examples yields the results shown in Fig. 13. Observe that (Du~),,~,~ increases as the number of nonreactive stages increases. As we saw above (DcP)~~~, for the mixture under consideration is equal to 0.0822166. From Fig. 13 we

Design of three-component

kinetically

controlled

reactive distillalion

thmpmemt

W&, 1.0

1955

columns A a C

Feed Didlate

0.0 0.0 1.0

Bo~torm 0.8025 O.M3 0.0025 0.803 0.19SO 0.194

I" 2.1 Da- 0.0822172

0.8

I \ \ 0.6 '\ t\

0.4 0.24

0.2

0.0 0

Fig. 13. (Du~)~,,~~ vs number of non-reactive stages in the stripping section for the mixture described in Fig. 11.

Campenenl

A

Feed

0.0 0.0 1.0

Distillate Lttams O.SC017 0.0030 O.M)256 0.8007 0.19727 0.1963

r= 2.1

Da - 0.0822172 r= 2.1 Ds = 0.0822166

+E

0:o C Unwrmediate)

0:o

I

0:2

0:4

0.6

1:o

Fig. 14. Composition profiles for the mixture described in Fig. 11 at Do = 0.0822166 + E, for a column with five nonreactive stages in the stripping section. Dashed line represents the reaction equilibrium curve.

0:2

0:4

ok XI'

0:s

1:o (I&*)

Fig. 15. (a) Design calculations for the mixture described in Fig. 1 I (Da = 0.0822172), (b) simulation. Dashed line rcpresents the reaction equilibrium curve.

the event that (Dd),,,, is not close to (DLl’),l.ip we can try to make them equal, for fixed values of x,,~ and xg, 1, by introducing nonreactive stages at either end of the column thereby causing the value of Da' to

see that (Dac&, ( = 0.07291) is nearly equal to increase. For X,,, = 0.9, X,., = 0.1 and r = 2.1 we showed UWrect when there are five nonreactive stages in the stripping section. The composition profiles at solutions to the design equations at two different (Dd),.,, + E are shown in Fig. 14. As can be seen the combinations of x~.~,x~.~, and the number of nonprofiles now intersect. A solution to the design equareactive stages in each column section (Figs 9 and 15). tions occurs at Da = 0.0822172 and is shown in The desired conversion/separation is feasible at other Fig. 15(a). Figure 15(b) shows the simulated column combinations of these variables. The best set of variprofires at r = 2.1, s = 3.1, NT = 34, f= 15, and ables at which to operate the column is determined as Do = 0.0822172. Once again we see good agreement the solution of an optimization problem. One repeats between the solution obtained by the design procedthe calculations for various combinations of these ure and the simulation. variables in order to minimize an objective function For a column with fixed values for X,.,, X,., and such as: (1) the total annualized cost of the process r we showed that we ean make the desired conver[Douglas (1988)], or (2) the total number of stages sion/separation feasible by choosing x~,~ and required, or (3) the total liquid holdup requirements, .Y~,~such that (Dd)lccr is nearly equal to (Dcz~).,~~,,. In etc. In this paper, however, we focus our attention on

1956

GEORGE

how to choose

specifications distillation columns

reactive ization.

and

BUZAD

that will lead to feasible rather than on optim-

Example l(c). In parts (a) and (b) of this example X,,, and X,_, were taken to be 0.9 and 0.1, respectively, and the reflux ratio was equal to 2.1. In part (c), we will keep the same values for X& and X,,, and repeat the calculations for a different value of the

reflux ratio. Let us specify r to be equal to 1.0. Solving for the fixed points of the rectifying and the stripping

B

Weaw) L.0

component

Feed

Dlstilltte

Bottoms

A

0.0 0.0 1.0

0.803 0.003 0.194

O.cQ7 0.807 0.L8.5

MICHAEL

F. DOHERTY

sections gives the fixed point compositions shown in Fig. 16. The remaining two variables to be specified are xn,* and x~,~ which we take to be .x~,~ = 0.003 and x~.~ = 0.007. Solving for the critical values of Da gives UWrect = 0.05057 and (Dac)l,riP = 0.04459. The composition profiles at several values of Da are shown in Fig. 16. From this figure we see that at r = 1.0 the profiles never intersect to the side of the reaction equilibrium curve where the rate of reaction for components A and B is positive. They do intersect on the other side of the reaction equilibrium curve, but as mentioned previously feasible designs are rarely, if ever, to be found in this regime. In order for the profiles to intersect on the side of the reaction equilibrium curve where the rate of reacB

Component

Feed

Distillate

A

0.0 0.0 1.0

0.99 + 1.0 x 10’2 1.0 x IO= 0.01 2.0 x lct12

(HUT)

r= Da=

X2

Bottoms o.oooM 0.99002 O.OiJ9%

3.0 0.D

0

c Unlwmediate)

Fig. 16. Composition profiles for a mixture with XD., = 0.9, X,., = 0.1, r = 1.0, xnJ2 = 0.003 and xg,, = 0.007 at several values of Da. Dashed hne represents the reaction equilibrium curve.

Component

@I&) LO\

Mstlllate

BOaemIl

A

Fed

0.0

0.803

0.007

:

0.0 1.0

0.194 0.003

0.186 0.800

0.0 0.0 C (lntennedi.t*)

0.2

04

0.6 XI

0.8

1.0 (L&L,

Fig. 18. Composition profiles for Example 2 at Da = 0. Dashed line represents the reaction equilibrium curve.

4-

3B 2-

0.0 C (IntermedIate)

0.2

0.4

0.6 XI

0.8

1.0 (L&Q

Fig. 17. Fixed points for the mixture described in Fig. 16 at several values of P. The fixed points of the rectifying and stripping sections coalesce at r = rlnin ( = 1.4). Dashed line represents the reaction equilibrium curve.

Fig. 19. Da’ vs number of non-reactive stages in the rectifying and stripping sections for Example 2.

Design of three-component

kinetically

controlled

B

Component

(Heavy) 1.0

Example

columns

1957

be feasible depending on the values of the number of nonreactive stages in parts (a) and (b) of this example, r was which is 1.5 times the minimum value

2

We now repeat Example 1 for higher product purities than those specified in the last case. We specify X,,, = 0.995 and X,,, = 0.005. Consider a column having a reflux ratio equal to 3. The fixed points for this mixture are shown in Fig. 22(a). The two remaining variables to be specified are taken to be x,,~ = 1.0x lo-” and x~,~ =2.0x lo-‘.

Fred

A

reactive distillation

may or may not xD.1. .%I.1 9 and each section. In taken to be 2.1 of r.

for components A and B is positive the fixed points must cross each other, i.e. their order should be as in Fig. 15(a) and not as in Fig. 16. For fixed values of XD., and X,.,, the fixed points move along the equilibrium curve as a function of r. We can solve for a minimum value of r which occurs when the composition of the fixed point in the rectifying section is equal to the composition of the fixed point in the stripping section. For the mixture under consideration rmin is equal to 1.4. The fixed points coalesce at r = rrnin as shown in Fig. 17. For values of r smaller than rmin the desired conversion/separation is not feasible, while for values of r larger than rmlnthe desired conversion/separation tion

0.0 0.0 1 .o

DIstillate

Bottoms

0.99 + 1.0 x 10-12 1.0 x 10-12 0.01 - 2.0 x 10-12

0.8

o.ooca2 0.99002 0.00996

r= 3.0 Da = 4.10561

+ e

0.6

Component

Feed

Diattllatc

Bottoms

B

0.0 0.0 1.0

0.99 + 1.0 x 10-12 1.0 x lo-12 0.01 - 2.0 x 10-12

o.ooOo2 0.99002 0.00996

(HUT)

r=

Da=

3.0 0.14425

Component B” C

DiStlll~tc

0.0 0.0 1.0

0.99 + 1.0 x 10-12 1.0 x IO-12 0.01 - 2.0 x 10-12

I= +E

0.6

Bottoms

Feed

o.oooo2 0.99002 0.00996

3.0

Da = 0.03778 + E

0.6

0:2 (Intermedhte)

0:4

0:6 Xl (b)

0:8

1:o (L&O

C (Iatcrmcdlnte)

=I

(L&t)

k)

Fig. 20. Composition profiles for example 2 at Da = (Do’),,,, + e, for a column with (a) 19, (b) 18 and (c) 17 nonreactive stages in the rectifying section. Dashed line represents the reaction equilibrium curve.

1958

GEORGE

and MICHAEL F. DOHERTV

BUZAD

Since the product purities are very close to pure component A and pure component B we introduce some nonreactive stages at either end of the column so as to avoid the backward reaction from taking place. Let us first consider the rectifying section. Figure 18 shows the composition profiles at Da = 0. From this figure we see that the maximum number of nonreactive stages possible in the rectifying section is 19. Solving for (Dd),.,, as a function of the number of nonreactive stages in the rectifying section gives the results shown in Fig. 19. Figures 20(a)-(c) show the composition profiles at Da = (Da’),.,, + E for a column with 19, 18 and 17 nonreactive stages, respectively. For a column with 16 nonreactive stages, or fewer, the value of (Dac)res, is very close to zero, and we do not expect a feasible design since practically no reaction occurs. From Fig. 19 a sensible number of non-reactive stages in the rectifying section would be 18. We would now like to find the number of nonreactive stages in the stripping section, if any, that will give a value of (DcI~).,,~~ which is nearly equal to (Du’),~~, (which is equal to 0.14425 for this example). From Fig. 18 we see that the maximum number of nonreactive stages in the stripping section for this mixture is 10. Figure 19 shows a plot of(DaC)s,rip vs the number of nonreactive stages in the stripping section. We select eight nonreactive stages in the stripping section since it gives (Dac).,+ = 0.09481which is close to the value of (DcQce,_ The composition profiles at Da = V’c’),eot + E for a column with 18 and eight nonreactive stages in the rectifying and the stripping sections, respectively, are shown in Fig. 21. As can be seen the profiles intersect. The solution to the design

B VICWY)

Component

Peed 0.0 0.0

A

1.0

D,.UII~k

Bottoau

0.99+ 1.0x 10."

o.cKnm

equations occurs at a value of Da = 0.14426, which is again very close to (Du~),~~,. The composition profiles at this value of Da are shown in Fig. 22(a). Figure 22(b) shows the simulated column profile at r = 3, s = 4, Nr = 46, f = 12, and Da = 0,14426. As can be seen, the simulated result matches the solution obtained by the design procedure. The conversion of component C achieved in this reactive distillation column is equal to 99%. In a conventional single-phase reactor with pure C as feed the highest conversion possible, for a reaction with K = 0.25, is 50%. This occurs at reaction equilibrium and requires a large holdup. The reactor effluent contains 25% A, 25% B and 50% C. In order to obtain high purity products from this exit stream two distillation columns are required downstream. The desired

B (HemY)

Compoacat

1.0

ii C

Feed

DktUhk Battcmm 0.0 0.99+ 1.0x IO-12 0.oalo* 0.0 1.0x I*12 0.99002 1.0 0.01 -2.0x l&l2 0.00996

0.8n ' \

B Wc=ry) 1.0

0.99aJ2 1.0x 10’~ 0.01 -2.0 i IO-12 OX0996

r= 3.0 Da= 0.14426

Component Feed

DlStihk 0.0 0.985 0.0 1.7x IW'Z 1.0 0.0147

A B C

O.woO29 0.985291 0.014680

r= 3.0 Da= 014425+L 0.8h

r= 3.0 IJar 0.14426

I

1’1

\

(b)

012 C (I"knndl.tc)

014

016

0:s

I:0

0.0

0.2

0.4

0.6

0.8

1.0

II

2 at for l%Elmple Fig. 21. Composition profiles Da = 0.14425 + E, for a column with 18 nonreactive stages in the rectifying section and eight in the stripping section. Dashed line represents equihbnum curve.

Fig. 22. (a) Design calculations

for the mixture described in Fig. 21 (Da = 0.14426), (b) simulation. ._.. . Dashed line represents the reactlon equlllbrrum curve.

Design of three-component kinetically controlled reactive distillation columns products are withdrawn and any unreacted C is recycled to the reactor. Thus a conventional plant requires a reactor, two distillation columns and a recycle stream to accomplish what can be done in a single combined reactor-separator unit. GENERALIZATION TO OTHER SYSTEMS

Our method has been illustrated in the context of a specific example but we are confident that the main ideas will generalize to other ternary mixtures, and quite probably to mixtures with four or more components. Such generalizations have already been achieved in nonreactive distillation [see Julka and Doherty (1990)]. The algebraic eliminations needed to obtain the key eqs (4a) and (12) are guaranteed to extend to arbitrary stoichiometries by the generalized transforms introduced by Ung and Doherty (1994). One of the main features demanded of the rate expression is that it agrees with the reaction equilibrium limit. Most rate expressions in the literature for liquid phase reactions do not exhibit this feature and we are finding it necessary to develop new ones based on activities rather than concentrations in order to overcome this difficulty. Irreversible reactions are covered as a special case of this model in which the equilibrium constant, K, is set to a large value. So far we have not encountered any special difficulties with this approach. Nonideal liquid mixtures introduce the possibility of azeotropes and additional fixed points, but there is now a mature literature on how to handle these effects in nonreactive distillation and they are not expected to present any exceptional difficulties for reactive distillation. The main new feature of reactive distillation that may require special attention is the possibility of significant heat effects for exothermic reactions. We have explicitly ruled out such effects in the present work, but our ongoing research is focused in this direction and we expect to report results for realistic mixtures in the future.

making it feasible is to choose values for xD.2 and xg,, such that the critical values of Da in each column section are nearly equal. in the event that (&&=, is not nearly equal to (Du~)~,~~,,we can try to make them equal, for fixed values of xDv2and xg, 1, by introducing nonreactive stages at either end of the column thereby causing the value of Da’ to increase. Once the five degrees of freedom have been specified the task becomes one of finding a value of Da, if any, that will make the desired conversion/separation feasible. Before solving the material balance equations we must guess a value for Da. Da’ is a good guess for Da since. feasible designs occur at values of Da very close to Da’.

We demonstrate the technique for the distillation of a ternary mixture that undergoes a reaction of the form 2C e 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. are grateful to 2. T. Fidkowski for permission to use and adapt his simulation program to reactive distillation columns. Thanks also to Ms. Pam Stephan for preparing the figures. We are grateful for finanAcknowledgements-We

ciai support-from the Petroleum Research Fund and the National Science Foundation (Grant No. CTS-9113717).

NOTATION

A&C B D Da Da’ I F H b K

CONCLUSION

A procedure for the design of reactive distillation columns has been developed. For a reactive distillation column which has the same molar liquid holdup on each stage and for which the feed composition, the feed quality and the column pressure are given, there are five degrees of freedom. The fixed points of the rectifying section map are a function of X,., and r, and the fixed points of the stripping section map are a function of X,,, and s, so in our design procedure we choose to specify X,., , X,., , and r. For fixed values of X,,, and X,,, we can solve for a minimum value of T which occurs when the composition of the fixed point in the rectifying section is equal to the composition of the fixed point in the stripping section. The remaining two variables to be specified are taken to be xD,* and x~,~. The desired conversion/separation is not feasible for all values of xD.Z and xg.l The key to CES 49: 12-F

1959

r, LT

NS

Xl

Yi

generic chemical species bottoms product flow rate, mol/time distillate flow rate, mol/time Damkiihler number, dimensionless ( = H&,-/F) critical

value of Do

feed stage location, counting from the bottom up feed flow rate, mol/time liquid holdup per stage, moles forward reaction rate constant, l/time reaction equilibrium constant liquid flow rate in the stripping section, mol/time liquid flow rate in the rectifying section, mol/time total number of stages in the rectifying section total number of stages in the stripping section total number of VLE stages in the column feed quality reflux ratio reboil ratio vapor flow rate, mol/time mole fraction of component i in the liquid phase Xl + (l/2)x, mole fraction of component i in the vapor phase

GEORGE BUZAD and MICHAEL

1960 Greek

F. DOHERTY

letters.

s l”” =

it has a value of zero when the desired conversion/separation is feasible Subscripts

.4&C

components A, 8, and C

B

bottoms distillate

D F i m min n

rect strip 1,2,3

feed component i stage index for the rectifying section, increasing down the column minimum stage index for the stripping section, increasing up the column rectifying section stripping section components A, B, and C REFERENCES

Agreda, V. H., Partin, L. R. and Heise, W. H., High-purity methyl acetate via reactive distillation. Chem. Engng Progr. 84(2), 40-46 (1990). Barbosa, D. and Doherty, M. F., Design and minimum

reflux calculations for single-feed multicomponent reactive distillation columns. Chem. Engng Sci. 43, 1523-1.537 (1988a). Barbosa, D. and Doherty, M. F.. The simple distillation of homogeneous reactive mixtures. Chem. Engng Sci. 43, 541-550 (1988b). Bogacki, M. B., Alejski, K. and Szymanowski, J., The fast method of the solution of a reacting distillation problem. Comput. Chem. Engng 13, 1081-1085 (1989). Bondy, R. W., Physical continuation approaches to solving Paper presented at reactive distillation problems. A.1.Ch.E. Meeting, Los Angeles (1991). Buzad, G. and Doherty, M. F., New tools for the design of kinetically controlled reactive distillation columns (submitted) (1994). Chang, Y. A. and Seader, J. D., Simulation of continuous reactive distillation by a homotopy-continuation method. Compul. Chem. Engng 12, 1243-1255 (1988). DeGarmo, J. L., Parulekar, V. N. and Pinjala, V., Consider reactive distillation. Chem. Engng Progr. 88(3), 43-50 (1992). Doherty, M. F. and Buzad, G., Reactive distillation by design. Trans. Instn Chem. Engrs 70, 448-458 (1992). Douglas, J.M., Conceptual Design of Chemical Processes. McGraw-Hill, New York (1988). Fidkowski, 2. T., Malone,_M. F. and Doherty, M. F.,

C-

(Intermediate)

Fig. Al. (a).

Design

of three-component

kinetically

B

reactive

distillation

columns

1961

Smith, L.A., Catalytic distillation process. U.S. Patent NO. 4,307,254, December 22 (1981). Smith, L. A., Method for the preparation of methyl tertiary butyl ether, U.S. Patent No. 4.978.807, December 18 (1990). Stichlmair, J. G. and Herguijuela, J. R.. Separation regions and processes of motropic and azeotropic ternary distillation, A.1.Ch.E. J. 38, 1523-1535 (1992). Ung, S. and Doherty, M. F., Vapor-liquid phase equilibriuin in systems with multiple chemical reactions. Chem. Engng Sci., accepted (1994). Venkataraman, S., Chan, W. K. and Boston, J. F., Reactive distillation using ASPEN PLUS, Chem. Engng Progr. 86(S), 45-54 (1990). Wahnschalft, 0. M., Koehler, J. W., Blass, E. and Westerberg, A. W., The product composition regions of singlefeed azeotropic distillation columns, fnd. Engng Chem. Res. 31, 2345-2362 (1992).

cFrod=ct)

Fig. Al. (a) Reaction equilibrium curves, both inside and outside. the composition triangle+ at several values of K, (b) the curves inside the composition triangle at a larger scale.

distillation: use of bifurcation Nonideal multicomponent theory for design. A.I.Ch.E. J. 37, 1761-1779 (1991). Fidkowski, Z. T., Doherty, M. F. and Malone, M. F., Feasibility of separations for distillation of nonideal ternary mixtures. A.1.Ch.E. J. 39, 1303~1321 (1993). Grosser, J. H., Doherty, M. F. and Malone, M. F., Modeling of reactive distillation systems. Ind. Engng Chem. Rex 26, 983-989 (1987). Julka, V. and Doherty, M. F., Geometric behavior and minimum flows for nonideal multicomponent distillation, Chem. Engng Sci. OS, 1801-1822 (1990).

\

Fig. A2. Stripping

controlled

\

section

APPENDIX In this appendix we study the global behavior of composition profiles and fixed points both inside and outside the composition triangle. As shown above, all fixed points must lie on the reaction equilibrium curve. Figure 3 shows the reaction equilibrium curve for a mixture with an equilibrium constant equal to 0.25. The equilibrium curve extends outside the composition triangle and for K = 0.25 forms a hyperbola. This curve, together with curves at other values of K, is shown in Fig. Al (a). Figure Ai shows the curves inside the composition triangle at a larger scale. For values of K greater than 0.25 the equilibrium curves are closed (i.e. ellipses or for the unique case K = 0.5 it is a circle), while at values of K smaller than 0.25 the curves form hyperbolas. For K = 0, the hyperbola becomes the xL and x2-axes. At K = a~. the elliptical equilibrium curve collapses to the line x3 = 0 (i.e. x2 = 1 - x1). Consider a ternary mixture for which K = 0.5. The volatilities of components A and C relative to component B are 5 and 3, respectively. We will focus our attention on the stripping section. In order to solve the stripping section eq (7), the following variables must be known: X,, 1, s, xg,, , Da

0 saddk points OStabkfmii

fixed points for a mixture with X ll.l = 0. I, both inside and outside tion triangle, at several values of s. K = 0.5.

the composi-

1962

GEORGE BUZAD and MICHAEL F.

and D/E. Take Xa,, to be equal to 0.1. Figure A2 shows the fixed points for this mixture, both inside and outside the composition triangle. at several values of 8. At total reboil there are four fixed points, labelled a, b, c, d, that are located at the pure components A and B, and the two points where the line xr + xs = 2 (xs = -1) intersects the reaction equilibrium curve. As the reboil ratio is lowered, the fixed points all move; fixed points (1 and d move along the equilibrium curve in the direction of decreasing X1, until in the limit of s = 0 they lie at the intersection of the line X1 = Xs,, and the

.-.-.-.-.-.__ Xl (Wemtdtate)

DOHIX~

reaction equilibrium curve. The branches of fixed points beginning at b and.c are always inaccessible to the stripping section composition profiles. The shape of the stripping section profile, however, is governed by the two branches of fixed points, II and d. For the parameter values selected in the example below, the profile moves away from the bottoms composition, passes by a saddle on branch n and then moves in the direction of a stable focus on branch d. We now present some representative profiles. Consider a mixture with Xa,r = 0.1, s = 3.1, xs,# = 0.02279, and

(L1&

Fig. A3. (a)-(b).

Design of three-component

kinetically

controlled

reactive distillation

columns

1963

Fig. A3. Stripping section composition profiles for a mixture with XaI, = 0.1. s = 3.1, x,,, = 0.02279 and D/B = I al (a) Dn = 0.1, (b) Da = 0.5, (c) Do = 1.3, and (d) Da = 1.5.

D/B = 1. Figure A3 shows the stripping section composition profile for this mixture at several values of Da. The fined point located inside thetriangleis a saddle (similar to the one shown in Fig I). The fixed point locatedoutside the camposition triangle is a locally stable focus. At small values of Do the oscillation of the composition profile (at a large

number of stages) about the fixed point is damped. Increasing the value of Da increases the amplitude of the oscillations. At larger values of Da (e.g. Fii. A3(d)) the amplitude is large cnougb for the profile to divergtit heads in the direction of increasing x, (i.e. x1 and xz tend to negative infinity) as the number of stages gets large.