Ind. Eng. Chem. Res. 1993,32,243&2445
2438
Minimum Reflux and Minimum Reboil in Ternary Distillation J o h a n n Stichlmair,’*+Hugo Offersf and Richard
W.Potthoffr
Lehrstuhl A fiir Verfahrenstechnik, Technical University of Munich, Arcisstrasse 21, 80290 Munich, Germany, Thermische Verfahrenstechnik, University of Essen, Uniuersitaetsstrasse 15,45141 Essen. Germany, BOC Group, Technical Center, 1W Mountain Avenue, Murray Hills. New Jersey 07076 Minimum reflux und minimum reboil ratio are key parameters for any distillation process because they determine the minimum energy requirement of the column. This paper presents a novel method for minimum reflux and reboil determination that can be applied to nonazeotropic as well as t o azeotropic ternary mixtures. The physical foundation for the limitation of reflux and reboil ratio lies in the intersection of the operating line and the equilibrium line. This intersection is called the pinch point. Here, vapor and liquid reach their equilibrium state. Three different cases have t o be considered in ternary distillation. In the simplest case the pinch has the concentration of the feed, as for the wellknown case of binary distillation. The energy demand has its lowest value in this case, but no pure products can be regained from the ternary mixture. In the two other cases, the separation of a pure low or a pure high boiler, the pinch concentration differs from feed concentration. The key contribution of the paper lies in locating the pinch concentration for these two cases. The novel method is much faster than any other method published before now. The method should be implemented in a rigorous column simulation program to generate starting values for minimum reflux and minimum reboil ratio. Introduction Distillation has been, until now, the standard method for separating homogeneous liquid mixtures in the process industry, and the consensus is that it will remain so for years to come. The only disadvantage of distillation is its high energy requirement. In modem distillation processes columns are operated using just a little more than the minimum energy necessary. Hence, easy and accurate determination of the minimum energy requirement is of great importancefor process design and column operation. The minimum energy demand of a distillation column depends primarilyonminimumrefluxandminimumreboil ratio. Operation of a distillation column with minimum reflux or reboil is characterized by the existence of a pinch withinthecolumn. Atthe pinchthevaporisinequilibrium with the liquid and, consequently, mass transfer between the two phases comes to an end. Approaching the pinch requires an infinite number of equilibrium stages. In binary distillation there exists a double pinch immediately above and below the feed point (except in the rare cases where a tangential pinch occurs) (King, 1980). As will be shown below, on the situation is more complex in multicomponent mixtures. Two restrictions are made in the following analysis: the liquid to gas ratio L/G (slope of operating line) is assumed to he constant in each column section and tangential pinches are not taken into account. In this paper ternary systems are primarily considered because they represent the simplest general case of multicomponentdistillation. All effects of multicomponent distillation are encountered in ternary distillation but not in binary distillation. Pinches in Ternary Distillation
Threeclaeaesofseparationscanbeestablishdintemmy and multicomponent distillation (see Figure 1). In the first class of separations, here called preferred separations, the low boiler and high boiler are separated primarily. The *To whom correspondence should be addressed. t Technical University of Munich. t University of Essen. 8 BOC Group. 08885885/93/26322438$04.00/0
121
T
BJ,,
Bx.T
Bx.T
lowboilerrich highboilerrich separation distillate bottoms Figure 1. Classes of multicomponent separations. Shaded column sections are operated with either minimum reflux or reboil. preferred
intermediate boiler occurs in the overhead fraction as well as in the bottom fraction. In this class of separations a double pinch exists immediatelyabove and below the feed point. The pinch concentration x p i is equal to the feed concentration XH. The upper and lower column sections are operatedwithminimum reflux and reboil, respectively. The situation is nearly identical to the well understood binary distillation. In the second class of separations an overhead fraction enriched inlow boileraisrecovered. Therectifyingsection is operated with increased reflux to provide the purer product. The stripping section is operated with minimum reboil. There exists a single pinch in the stripping section immediately below the feed point. Here the pinch concentration is different from the feed concentration. In the third class of separations a bottom fraction is recovered that is richer in high boiler than in the preferred separation. In order to get the purer bottom product, the stripping section is operated with increased reboil, but the rectifying section is operated with minimum reflux. In this case there exists a single pinch in the rectifying section immediately above the feed point. The pinch concentration here is also different from the feed concentration.
0 1993 American Chemical Society
Ind. Eng. Chem. Res., Vol. 32,No. 10, 1993 2439
lowboiler
The minimum reflux ratio RL can be calculated from knowledge of the pinch concentration x p by the equations derived for binary distillation. According to Mass (1979),
.
with x p = x F and YP* = YF* (1)
The minimum reboil ratio follows from with xp = xF and
yp* = y p *
(2)
The interrelation between reflux and reboil ratio is
(D/F)(RL+ 1) + (P  1) (3) 1 D / F Here, q describes the caloric state of the feed (for boiling liquid feed q = 1, and for saturated vapor feed q = 0). Boiling liquid feed is considered in the following, but all relations can be applied with small modifications to other caloric states of the feed. R, =
\
(4 highboiler
Box&
(b)
mediumboiler
Figure 2. Preferredseparation.AUproduetshave tolieouastraight line through the feed zrj and its equilibriumvapor m*.Shaded area is the separation region at total (or very high) reflux.
lowboiler
Preferred Separations
(0)
A
Equation 1 for the determination of minimum reflux ratio, RL, of a binary mixture can be directly applied to ternary mixtures.
Equation 4 is valid for each constituent i of the mixture. In ternary distillation it represents a set of three linear equations. By setting the distillate concentration x ~ ofi one component, e.g., the high boiler c (in sharp separation x& = O), the minimum reflux ratio is given by
(4 highboiler The distillate concentrations of components a and b follow from eq 4 after rearrangement: X D= ~ RL.WF.*
 XFJ + YF.*
 R Lmm. Wpb*  xpb) + ypb*
XDb
(6)
These linear equations are represented in Figure 2 by a straight line through xn and ypi* on which the feed F and the distillate D have to lie. The material balance around the column requires that the bottom fraction B lies on the same straight line through D and F. Under minimum reflux operation with the pinch at the feed concentration, the separation region of a ternary system shrinks to a straight line that lies within the separation region a t total (or very high) reflux (Stichlmair and Herguijuela, 1992; Wahnschafftetal., 1992). Thisspecialclassofseparations is called “preferred separation” in this paper. Preferred separationsrepresent certain natural separations between high and low boiler with the intermediate boiler distributed in both fractions. Here, the reflux and reboil ratio have their lowest value of all separations possible in ternary systems. Figure 3displaysthe internal liquid concentration proffie for a sharp separation (i.e., no high boiler in the distillate
B.&
XT*
(b)
mediumboiler
Figure% Internal liquidconcentrationprofiefor asharppreferred separation computed hy a rigorous column simulation program.
and no low boiler in bottoms) of an ideal mixture with temperature independent relativevolatilities a. From the feed point upward the high boiler cis primarily removed from the vapor. The righthand side of the triangular diagram is reached at the transition point x n where the ternary mixture transfers into a binary mixture. From this point upward the componentb is removed, now being the highest boiling constituent. From the feed point downward the low boiler a is separated primarily (but not exclusively) from the liquid. The base of the triangular diagram is reached at the transition point XT* From there on the substance b is separated from the liquid until the bottom fraction is established. The internal liquid concentrationproffie is characterized by the double pinch point located at the feed and, in case of sharp separations, the transition points. The concentrations of the transition points are easily determined in the McCabeThiele diagrams for the low boiler a and the high boiler c. They represent the intersection of the operating line of preferred separation with the equilibrium line of the binary mixtures ab (see Figure 4) and bc, respectively.
2440 Ind. Eng. Cham. Res., Vol. 32, No. 10,1993 lowboiler
xo

(4
Figure 4. Determination of transition point x n in the rectifying
section.
Xll (b) highboiler mediumboiler Fiure 6. Internal liquid concentration protide for a high boiler rich separation. There exists only a single pinch locatsd above the f e d point in the rectifying section. T h e transiton point x n is the same
as in preferred separation.
4. The stripping section of the column is operated with minimum reboil whereas therectifyingsection isoperated above theminimumrefluxratio. The feedinto thissection is a mixture of liquid from the rectifying section and the column feed. The concentration of this mixture is the pinch concentration.
Separation of a High Boiler Rich Bottom Fraction
(4
highboiler
B.X,i
(b) mediumboiler
Figure 5. Internal liquid concentration profile for a low boiler rich separation. There exists only a single pinch located below the feed point in the stripping section. The tramition point x n is the eame as in preferred Separation.
Separation of a Low Boiler Rich Distillate
The internal liquid concentration profile calculated by rigorous column simulation is plotted in Figure 5. The concentration of the low boiler in the distillate is higher than in preferred separation. Again, a sharp separation with no low boiler in the bottom fraction and no high boiler in the overhead fraction is considered. The rectifying section is operated with a reflux higher than minimum reflux in order to establish the purer overhead product. There are several effects here not encountered in binary and preferred separations: 1. The internal liquid concentration profile never reaches the concentration of the feed. Hence, a concentration jump is caused by admixingthe boiling liquid feed. 2. Theconcentrationprofiie immediatelyabovethe feed point is identical with the profile at preferred separation. The location of the transition point XT. remains unchanged. 3. There exista a single pinch immediately below the feed point. The concentration of the pinch is different from feed concentration. The pinch lies on a line between transition point XT, and feed point XF~.
It is well known that a high boiler rich bottom fraction (or even pure high boiler) can he recovered from a ternary mixture. In order to perform such a separation the reboil ratio has to be increased. The internal profile of liquid concentrations is plotted in the triangular diagram of Figure 6. There exist some noteworthy effects: 1. The internal concentration profile isalwaysdifferent from the feed concentration. Here, as above, a concentration jump is caused by admixing the feed. 2. Theconcentrationprofileimmediately belowthe feed isidentical with theconcentration profileof the preferred ~ the same location separation. The transition point X T has for both separations. 3. There exists a single pinch immediately above the feed point. The concentrationof the pinch,xpi, is different from the feed concentration. The pinch lies on the line between transition point xn and feed *pi. 4. The rectifying section is operated with minimum reflux wheras the stripping section is operated above the minimumreboilratio. Thefeedinto that sectionisavapor which is in equilibrium with the pinch concentration. Locating t h e Pinch Point Concentration In preferred separations the concentration of the pinch isequal to that of the feed as in binary distillation. In the general case of multicomponent distillation, however, the pinch concentration is different from the feed concentration. Determinationofthe pinch concentration makes use of the fact that the pinch lies on a line between transition point XT and feed XF (Levy et al., 1985). This line is determined by theconcentration profdeimmediately above or below the feed point. As can be seen in Figures 3,5,and6 thisconcentration profileis thesameinpreferred separation and in low boiler and high boiler separation, respectively. This is rather surprising since reflux and reboil are different.
Ind. Eng. Chem. Res., Vol. 32, No. 10,1993 2441 lowboiler
lowboiler (0)
A
(4
x0i
(b)
mediumboiler F i r s 7. Geometrical representation of the lever arm rule, eq I, here s h o w lor the rectifying section. hiqhboiler
The internal concentration profiles show, for sharp separations, three noteworthy points; the pinch point x p , , the transition point in the rectifying section x n and the transition point in the stripping section x n . At the pinch point all three components in liquid and vapor are in the equilibrium state. At the transition point in the rectifying section, x n , the vapor and liquid concentrations of substances aand b are in the equilibrium state (but not the high boilerc, which is present insmallconcentrations). Byanalogy,at thetransition point in thestrippingsection, X T ~ ,the vapor and liquid concentration of the substances b and c are in equilibrium (but not the low boiler a). The lever arm rule gives for the transition point (see Figure 7)
(4
%i
highboiler
xTr
(b)
mediumboiler
Loeiofpm~pom~foridsalsyaemswithconst volatilities. The concentration profde between feed and tranaition point is a straight line on which the pinch has to lie.
F8.
After rearrangement, For the pinch point Solution of the quadratic equation gives The liquid to vapor ratio L/G(=slope of operating line) is the same for both points; hence, For ideal systems the parameters C1, C2, Cg, and C4 are: C, = or,(l+ (a,
The ratios of the distances ddd2 have to be the same at the transition point and the pinch point. This condition is met in ideal systems with temperatureindependent relative volatilities when the line of equilibrium concentrationsyi* is parallel to a line of liquid concentrations xi. In ideal systems the equilibrium concentrations yi* for any straight line of liquid concentrations xi also form a straight line. These lines are not, in the general case, parallel to eachother. Thereexist, however, twoduections for each point where the vapor equilibrium concentrations are parallel to liquid concentrations (Figure 8). The concentration profiles starting from the feed have to follow this special direction. The condition is
The total derivatives of vapor equilibria yi* are
c,
 l)xpd/Na
= (1  a&xFaaJ2
C, = (1  am)xma&P c4 = a,(l
+ (sac  l)Xp.)/@
(15)
with
N = 1 + (aac l)x,
+ (a,  l)x,
The concentration profie between transition point xm and feed xpi is straight and is the same for all low boiler rich products in ideal systems (Figure 8). In nonideal systemsvaporliquid equilibrium is a nonlinear function, and thus no direction exists where liquid concentrations and equilibrium vapor concentrations are parallel straight lines. In such systemsthe curved concentration profile is linearized near the feed point (Figure 9). Since the pinch always lies close to the feed, this is quite a good approximation for the pinch point curve. For nonideal systems
2442 Ind. Eng. Chem. Res., Vol. 32, No. 10, 1993
lowboiler (a)
acetone (a) 56.2 'C
78.3 'C
xBi
xTs
ethanol (c)
64.6 'C methanol (b)
Figure 9. Loci of pinch points for nonideal systems. The concentration profiie between pinch and transition point is curved, but vapor and liquid concentrationsare equidietant. The line of liquid concentration is linearized near the feed by tangents.
(4 highboiler
(b) mediumboiler
Figure 11. Determination of pinch point for low boiler separations.
lowboiler (a)
lowboiler (0)
A
(4
highboiler
(4 highboiler
xTs
(b) mediumboiler
Figure 10. Loci of pinch points.
the partial derivatives have to be found numerically.
(b) mediumboiler
Figure 12. Determination of pinch point for high boiler separations.
section has to lie on a straight line through the pinch x p i and ita vapor equilibrium y p i * (Figures 11 and 12). It is important that the low boiler rich distillate be binary; otherwiseno transition point occurs. The bottom fraction, however, might be ternary. Analogously, the high boiler rich bottom fraction must be binary whereas the overhead fraction might be ternary (Figure 13). Determination of Minimum Reboil and Reflux Ratio
This requires the determination of vaporliquid equilibrium at the feed XFi and two additional concentrations. Recommended are AXa = o . l ( X ~ ~Xp,*) 
and &b = O.l(X,yfi*) (17) As a consequence, the pinch point lies either at the feed concentration or at one of the straight lines that meet the condition of parallelism near the feed; see Figure 10. The exact location on the pinch point lines is determined using the fact that one column section is operated with minimum reflux (or reboil). The product compositionof this column
There exist two strategies to determine the minimum reboil and minimum reflux ratio. The fiist strategy is based on the pinch point; the second one is based on the transition points (Stichlmair,1988). Here, the pinch point is used because it is physically more sound. From knowledge of the pinch point the minimum reflux ratio follows:
The minimum reboil ratio is
Ind. Eng. Chem. Res.,Vol. 32, No. 10,1993 2443
lowboiler
lowboiler (a)
rnediumboiie; highboiler
Figure 14. Triangular diagram of an ideal system.
rnediurnboiler
Figure 13. Range of validity of the novel method. Top and bottom fractions have to lie on a straight line through the feed.
RG+
XP~ XBi  ypi* 
for any component i
(19)
The first equation is used for preferred separations and for low boiler separations. The second equation is applied to preferred separations and to high boiler separations. Both equations can be applied to ideal and nonideal (and even azeotropic) mixtures. However, distillation borders have to he kept in mind in setting the product concentrations x~ and X I of azeotropic systems (Stichlmair and Herguijuela, 1992). For ideal mixtures with temperatureindependent relative volatilities and sharp low boiler separations ( x a = O), eq 18 becomes
For sharp high boiler separation
Tahle I. Minimum Reflux and Reboil for Several Distillations of an Ideal System with TemmentureIndependent Relative Volatilities Separation of High Boiler Rich Bottom
( 2 ~= . O),
R w
no.
(exact)
la
2.066
2a
3.79
ARdRw*% Underwood 4.0 0.0
novel method 0.7 0.8
pinch at feed 39.4 41.2
Senaratiou of Low Boiler Rich Distillate
no.
lb 2h
Rcphl (exact) 1.593 1.054
Underwood 4.0
4.0
MolRod, % novel method