On the dynamics of distillation processes—VI. batch distillation

ChemkdEnaineering Science. Vol. 40, No. Printed in Chat Britain.

ON

THE

I I. pp. 2087-2093.

1985.

ooos-2509/85 53.00 + 0.00 Perspmon Press Ltd.

DYNAMICS OF DISTILLATION BATCH DISTILLATION

PROCESSES-VI.

DAVID B. VAN DONGEN Union Carbide Corporation, P.O. Box 670, Bound Brook, NJ 08805. U.SA.

and Department

of Chemical

MICHAEL F. DOHERTY Engineering, Goessmann Laboratory, University of Massachusetts, MA 01003, U.S.A.

Amherst,

(Received 27 December 1983) Abstract-We have developeda simple model whichexplainsthe behavior of azeotropie batch distillations. The model is in agreement with known experimental results and explains featuresof the distillationprocess which have previously been considered anomalous. The model can be used to design hatch distillation processes operating at either infinite or finite reflux ratios.

INTRODUCTION

Batch distillation remains an important separation technique in the manufacture of small-volume speciality chemicals. It is also used widely in the chemical industry for recovering valuable fractions from mixed waste-solvent storage tanks. Typically, the components involved form multicomponent azeotropic mixtures_ These processes are considered to be quite complex and to date there is no simple theory which explains their behavior. In this paper we develop a simple model for multicomponent azeotropic batch distillations which operate under the industrially realistic conditions of high reflux ratio and a large number of plates. The model is successfully applied to a ternary azeotropic mixture which has been studied experimentally by Ewe11 and Welch Cl]. One of the most important conclusions from our analysis is that it is possible to draw the exact trajectories followed by the liquid composition in a batch still and to predict the exact sequence of constant boiling vapor distillates which appear overhead without solving a single equation, provided the simpledistillation residue curve map is known. DESIGN The

simple-distillation

EQUATIONS

equation

dx = x-y(x) dT can be applied to a batch distillation process in order to examine how the composition of the liquid in the still changes with time. In a batch distillation, a column with many trays is placed on top of a pot containing the mixture to be boiled, and this column usually operates at a high reflux ratio (see Fig. 1). By mass balance, we can write a differential equation to describe the change in the composition of the liquid in the still as 2087

Fig. 1. Schematic diagram of a batch distillation process.

and by a suitable nonlinear change in the time scale (see [2]), we find dx dr=x-yo where yD is the composition of the vapor leaving the condenser. This equation is different from the simpledistillation equation in that yn is not in equilibrium with x. Rather, we must compute ya using the design equations for the column. The vapor distillate will be in equilibrium with the liquid in the condenser with composition xr,. Following the arguments given in [3] we know that the liquid composition profile in a column approaches the shape of a simple-distillation residue curve as r increases. When r = co, the column

2088

DAVID

B. VAN

DONGEN

profile is exactly the same as a simple-distillation residue curve. Therefore, in batch distillations operating at high reflux ratios (say, r > 7) we can assume, with very little error, the xD and xSri,,lie on the same simpledistillation residue curve. Thus, xD is related to the instantaneous still composition, x(e), by the approximate equation dx’ = y(x’)-xx’ dh x’(h = 0) = x(c)

WI

O
=

x’(h

W = N)

WV

where x’ denotes the liquid composition at level h in the column and N is the number of theoretical stages. Equation (4) is approximate in two ways: it assumes that the reflux ratio in the column is high and it represents the algebraic, tray-by-tray material balances in the column by a differential approximation_ These assumptions can easily be relaxed and equation (4) replaced by more exact equations. However, our results are insensitive to these approximations provided the reflux ratio is greater than about 7. It is convenient to write eq. (4) in integral form

and

MICEMEL

F.

DOHERTY

posite from the position of the low-boiling azeotrope on either the methanol/chloroform face or the methanol/acetone face, depending on the location of the initial point. The change in the liquid-phase composition is almost linear because the 20 trays in the column cause the composition of the vapor distillate to be approximately constant at a value near the lowboiling azeotrope. It is precisely this constant-valued composition oft& vapor distillate which makes batch distillation with large r and N industrially important. SPECIAL

CASE:

LARGE

r AND

N

batch-distillation residue curves shown in Fig. 5 are characteristic of those displayed by stills with large (but not necessarily infinite) r and N. In this section we will study the characteristic features of the distillate The

Acetone 1.0

N=l I

Armotropa

+

Initial

compmilian

0.e

&-----+

\

N

x’(h

= 0)+

= N) = x’(h *MO

s

(I

- x’)dh,

(5)

X* 0.4

-

x(S)

XD

i.e.

XD

= I x(C) +

0

N

0.2 -

(Y w - x’)dh_

(6)

Equations (3) and (6) can be combined to form a design equation for a batch distillation process

N X

D

=

x(t)+

0.2

0.9

0.6

X, Chloroform

0.6

I.0

Methanol

Fig. 2. Batch-distillation residue curves with a one-tray column.

dx = x - y,(x,) d< xf< = 0) = x0 (given)

o0

(7b)

(y(x’)-x’)dh

where N appears as a parameter. Equation (7), an implicitly nonlinear integro-differential equation, has been solved for various values of N and for several initial compositions, and the results are shown in Figs 2-5 for increasing values of N. The mixture used was the methanol/acetone/chloroform system. Note that for N = 0 we will recover the simple-distillation equations. The residue-curve map for this mixture is shown in Fig. 6 together with the boiling temperatures of the pure components and azeotropes. When the number of trays in the column in small, the batch residue curves look very similar to the simple-distillation residue curves (compare Figs 2, 3,4 and 6). The most interesting and industrially significant case is when N is large, i.e. N = 20 as in Fig. 5. The batch-distillation residue curves appear to move directly away from the initial point in a direction op-

X.

Chloroform

X*

Methar~ol

Fig. 3. Batch-distillation residue curves with a three-tray

column.

Dynamics

2089

of distillation processes--VI (56.lYl ACetOn

x3

Chkoroform

Fig. 4. Batch-distillation

residue curves with an eight-tray column.

Acetone 1.0

0.2

0.4

(53.491 0.6

0.8

x,

Chloroform (61.2W

1.0

Methonol (B4.VC)

Fig. 6. Simple-distillation residuwurve map for a mixture of methanol, acetone and chloroform. Ternary szeotropic tem-

perature is 57.5”C.

N=20 I

Areotrcw

+ lnitiil

wmprmitim

0.8

Chloroform

Fig.

0

Method

5. Batch-distillation

**

residue

Methonol

curves

with

a

20-tray

column.

composition, yD, and the liquid composition in the batch still, x, as they change with time for batch stills operating under conditions of large r and N. To begin, let us summarize the main features of the simple-distillation residue curve map shown in Fig. 6 since we will inevitably compare our batch still to this case. The mixture, methanol (1) + acetone (2) +chloroform (3), has two low-boiling binary azeotropes, one high-boiling binary azeotrope and a ternary saddle azeotrope. The saddle azeotrope has four separatrices, entering and leaving it, which divide the map into two distinct simple-distillation regions. In the left-hand region, all residue curves are attracted to the high-boiling acetone/chloroform azeotrope.. In the right-hand region, they are attracted to the pure methanol vertex. These two distillation regions each contain one stable separatrix (i.e. residue curves con-

verge towards it as they flow forward in time, en route to the stable singular points). The stable separatries play no special role in simple distillation but they do have signiiicance in batch distillations, as we shall see. The locus of vapor compositions in equilibrium with each instantaneous liquid composition along a simpledistillation residue curve also forms a curve, called a vapor boil-off curve. Each residue curve, then, has its own unique vapor boil-off curve. The special relationship between a residue curve and its vapor boil-off curve is that the tie lines between them are tangent to the residue curve. This can be demonstrated quite easily from eq. (1). Equation (1) can be rearranged to show that the slope of a residue curve, at any point x = (xi, x2), is dxz x2 -Y2 -=-* (8) dxl x1 -_yl Referring to Fig. 7, we note that the slope of the tie-line connecting point x on a residue curve to the corresponding point y(x) on its vapor boil-off curve is x2 -Y2 slope of tie-line = Xl -Y1

k,. $1

.

(9)

\ \ \ ’

(v,(a). Y,(A)) \ \

\

Reside

t3rva

\

Vapcar boil-oft

curve

Fig. 7. Residue curve with its corresponding vapor boil-off curve.

2090

DAVID B. VAN DONGEN and MICML

Hence, the tie-lines between a residue curve and its vapor boil-off curve are always tangent to the residue curve. Finally, we will note that separatrices in a residue-curve map have their own vapor boil-off curves, just as any other residue curve does. We will now consider an initial liquid-phase composition in the lower left-hand region of the composition triangle (see Fig. 9) and examine how the stillpot composition, x, and the distillate composition, yo, move with time during a batch distillation. For each

Batch-distillation residue curve

Fig. 8. Batch-distillation residue curve and its associated distillate curve. Acetone

se.*

Detail

Chloroform

Methanol

Fig. 9. Batch-distillation residue curve for the special case of large r and N.

F. DOHERN

batch-distillation residue curve there will be a corresponding distillate curve which denotes the locus of distillate compositions, yn. as they change with time during the course of the distillation. The relationship between these two curves is precisely the same as the relationship between a simple-distillation residue curve and its vapor boil-off curve. This is easily seen by comparing eqs (1) and (3). Therefore, the vapor distillate, y,, corresponding to any particular instantaneous still-pot liquid composition, x, in a batch distillation will lie on the tangent line to the batchdistillation residue curve at x (see Fig. 8). In the early stages of the batch distillation from our chosen initial condition, the effect of the column (at high r and N) is to maintain ye constant at the methanol/chloroform azeotrope. This occurs because under conditions of high r and N, the composition profile in the column follows the reverse residue curve through the point x on the hatch residue curve. All simple-distillation residue curves in the lower left-hand region come from the methanol/chloroform azeotrope. Since yr, is constant, the batch-distillation residue curve must move in a straight line, away from the methanol/chloroform azeotrope, in the direction of the stable separatrix (see Figs 5 and 9). Just before the liquid in the batch still, x, hits the stable separatrix (at point A, Fig. 9) the points x, ye and the column profile connecting them are as shown in Fig. 10(a). The still composition, x, is driven further towards the stable separatrix. If the still composition were to move across the stable separatrix it would find itself in the upper left-hand simple-distillation region. The column profile would precipitously switch to follow the residue curves in this new region and y,, would switch to the methanol/acetone azeotrope, as shown in Fig. 10(b). However, the still composition would again find itself being driven back towards the stable separatrix. Consequently, when x hits point A in Fig. 9, the batch residue curve must follow the stable separatrix. During this part of its travels, the batch residue curve will be curved, hence y,, will not be constant since it lies on a tangent to the batch residue curve. The relationship between x, ye and the column profile connecting them is shown in Fig. 10(c). Notice that the column profile will follow a simple-distillation residue curve passing infinitesimally close to the ternary saddle azeotrope. As N + co, a pinch zone in the column occurs in the vicinity of the ternary azeotrope. In Fig. 10(a) and (b) the pinch zone occurs in the vicinity of the binary azeotropes, as noted in the figure legends. The batch distillation follows a path identical to the simple-distillation residue curve along the stable separatrix until it reaches point B (Fig. 9) where the stable separatrix is tangent to a line from the methanol/acetone azeotrope. At this point, the residue curve followed by the column composition profile leads to the acetone/methanol azeotrope. Therefore, as the batch residue curve goes from A to B, the distillate composition moves continuously from some interior composition in the general vicinity of the ternary

Dynamics of distillation processesV1

2091

Composition profile in OoIUrm

i

.‘

--\ ‘

I I

I

\

-Stable ‘

reparotrix

I Unstable separatrix

Fig. 10. Batch-distillation residue curves along a stable separatrix.

axeotrope to the methanol/acetone binary azeotrope. At point B, the batch residue curve leaves the stable separatrix. This occurs because y. remains constant at the methanol/acetone azeotrope. It does so because the resulting straight-line batch residue curve from B to C (Fig. 9) lies entirely within the upper left simpledistillation region. Therefore, all simple-distillation residue curves (i.e. column profiles) emanating from the straight line BC ultimately lead to the methanol/acetone axeotrope, which is where yD will remain. It should be clear that the sudden straightening out of the batch residue curve beyond point B is due entirely to the sharp curvature of the stable separatrix in that area. If the curvature was less pronounced so that the straight-line batch residue curve through B attempted to cross the stable separatrix into the lower simple-

distillation region, then the batch residue curve should again be forced to follow the stable separatrix just like it was from A to B. The straight-line batch residue curve finally intersects the acetone/chloroform axis at C. Here, the distillate composition swings abruptly to the pure acetone vertex. All the methanol has been exhausted from the system and the column profile follows the binary residue curve (i.e. the side of the triangle) which leads to a distillate composition of pure acetone. The batch residue curve now follows the acetone/ chloroform axis until it reaches the high-boiling acetone/chloroform azeotrope. We can summa rize the course of the batchdistillation residue curve in Table 1. This example demonstrates how it is possible to separate a ternary

Table 1. Section

Characteristic of x

Characteristic of yD

1

Straight line

Constant at methanol/chloroform (acetone-free distillate)

2

Curved along the stable separatrix

Ternary mixture of variable composition, moving toward the methanol/acetone azeotrope

3

Straight line

Constant at methanol/acetone (chloroform-free distillate)

4

Straight line along acetone/chloroform (methanol-free residue)

5

Constant at high-boiling azeotroOe

edge

acetone/chloroform

azeotrope

azeotrope

Constant at pure acetone Constant at acetone/chloroform

azeotrope

DAVID B. VAN

2092

DONGEN

mixture of acetone, chloroform and methanol into several fractions. Only one of the fractions is a pure component (i.e. pure acetone in fraction 4). The other fractions would need further processing in order to recover the remaining components in pure form. Figure 11 shows how the distillate temperature varies with time. As the distillate composition shifts from the methanol/chloroform azeotrope to the vicinity of the ternary azeotrope and then on toward the methanol/acetone azeotrope, the distillate temperature rises and then falls. This behavior has been called anomalous by Ewe11 and Welch [l], Lang [4] and Wilson et al. [S]. However, according to our analysis, there is nothing anomalous about this behavior and it can be expected to occur quite naturally in batch distillations operating at high r and N. Finally, let us state that the foregoing analysis can be applied to any batch residue curve for this or any other mixture. We also observe that, provided the position of the simple-distillation separatrices are known, it is easy to draw the exact trajectories followed by the liquid composition in a batch still operating at high r and N without solving a single equation. This is because batch trajectories are composed of straight lines and known curves (i.e. the simple-distillation stable separatrices). In addition, it is also possible to predict the exact sequence of constant-boiling vapor distillates which appear overhead. These results are in agreement with the experimental work of Ewe11 and Welch [l], Lang [4] and Wilson et al. [S].

LOW

REFLUX

RATIOS

In columns operating at low reflux ratios, the problem becomes much more difficult to analyze, regardless of whether N is large or small. The column composition profile no longer follows a residue curve and eq. (4) must be replaced by dx’ dh-

rtl --~(x’)-x’+;Y,(x,) r x’(h

(I&)

= 0) = x(e)

DOHERN

Equation (7) now becomes dx dS=X

-Y&n)

x(< = 0) = x0

(given)

r+l 7

xD = x(c) +

+ ; Ydx,)

1

y(x’)

D=x’(h=N)

The difficulty here is that eq. (1 lc) is implicit in xD. When r -P co, the term yn/r drops out, eq. (1 lc) is explicit in xD and hence can be integrated directly. Equation (1 lc) must be solved iteratively by guessing a value for xe, calculating yn from the equilibrium relation and then performing the integration. The next estimate of xn is adjusted according to the size of the residual. The process is repeated until a convergent value for xD is obtained. We have not attempted to solve eq_ (11) and to our knowledge there has been no systematic study of azeotropic batch distillation with finite reflux reported in the literature. CONCLUSIONS

We have developed a simple model which explains the behavior of axeotropic batch distillations operating at high r and N. The model is in agreement with known experimental results and explains features of the distillation process which have previously been considered anomalous. One of the most important conclusions from our analysis is that it is possible to draw the exact trajectories followed by the liquid composition in the batch still and to predict the exact sequence of constant-boiling vapor distillates which appear overhead without solving a single equation, provided the simple distillation residuecurve map is known. The model extends to batch distillations operating at low reflux ratios but no attempt has been made to study this case.

C

D H h N ; t X

(OR

-x’

dh.

NOTATION

TIME

(1 lb)

Acknowldgement-The authorswish to thank the Petroleum Research Fund administered by the ACS for providing financialsupport.

O
and MKXUEX.F.

VOLUME

DISTILLED)

Fig. 11. History of distillate temperature.

X0 xD

Y

number of components distillate flow rate liquid hold-up in still tray number total number of trays reflux ratio temperature (boiling point) time liquid-phase composition vector x = (X1.X2,. . . )X,_1)= initial composition vector condensate composition vector vapor phase composition vector

Dynamics YD

e

of distillation processes-VI

distillate composition vector nonlinear transformation of t REFERENCES

[l]

Ewell R. H. and Welch L. M., hd. Engng Chem. 1945 37 1224. \

2093

[2] Doherty M. F. and Perkins J. D., Chem. Engng Sci. 1978 33 281. [3] VanDongen D. B. and Doherty M. F., Ind. Engng Chem. Fundam. 1985 in press.

[4] Lang H., 2. Phys. Chem. 1950 1% 278. [S]

Wilson R. Q.. Mink W. H., Munger H. P. and Clegg J. W., A.I.Ch.E. J. 1955 2 ?20.