2000 Distillation Columns. I. the Ponchon-Savarit Method

published 8 August 2000,, doi: 10.1098/rspa.2000.0596456 2000 Proc. R. Soc. Lond. A

J. W. Lee, S. Hauan, K. M. Lien and A. W. Westerberg

methoddistillation columns. I. The Ponchon-Savarit A graphical method for designing reactive

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10.1098/rspa.2000.0596

A graphical method for designingreactive distillation columns

I. The Ponchon{Savarit method

By J. W. Lee1, S. Hauan1, K. M. Lien2 and A. W. Westerberg1

1Department of Chemical Engineering, Carnegie Mellon University,5000 Forbes Avenue, Pittsburgh, PA 15213-3890, USA

2Department of Chemical Engineering, Norwegian University ofScience and Technology, N-7491 Trondheim, Norway

Received 25 February 1999; revised 19 October 1999; accepted 20 January 2000

We show how to construct a Ponchon{Savarit diagram for a binary reactive distilla-tion column, illustrating it speci cally for isomerization and decomposition reactions.We rst show the properties needed for points to lie on a straight line in composi-tion/enthalpy space. Then, for the isomerization reaction, we show how to step othe stages using a reactive cascade dierence point. In the Ponchon{Savarit diagram,the reactive cascade dierence point has two elements. One is the composition coordi-nate formed as a linear combination of stoichiometric coe cient vectors and the topproduct composition. The other is the enthalpy coordinate formed by combining thetop product molar enthalpy, the condenser molar duty and the molar heat of reac-tion. Finally, and in a similar manner, we construct the Ponchon{Savarit diagramfor a decomposition reaction.

Keywords: Ponchon{Savarit method; reactive distillation;reactive cascade di erence point

1. Introduction

Reactive distillation is a promising technology that integrates the functions of reac-tion and separation. In special cases (DeGarmo et al . 1992; Siirola 1995), this inte-grated unit can reduce investment and operating costs dramatically. As yet, there arefew design methods for reactive distillation. Barbosa & Doherty (1988a,b) introducedsingle- and double-feed distillation columns under the assumptions of constant molaloverow (CMO) and reaction equilibrium in a column. Okasinski & Doherty (1998)proposed a design method for a reactive distillation column considering the heateect from reaction and the reaction residence time in terms of the vapour-to-feedratio and the Damkohler number. In terms of graphical methods, Espinosa et al .(1993) developed Ponchon{Savarit diagrams in a transformed enthalpy-compositionspace, and Perez-Cisneros (1997) proposed McCabe{Thiele diagrams for designingreactive distillation columns based on the element balance approach. However, untilnow, there have been no easy graphical methods for designing reactive distillationsuch as Ponchon{Savarit and McCabe{Thiele diagrams in terms of untransformedmole fraction coordinates.

Proc. R. Soc. Lond. A (2000) 456, 1953{1964

1953

c 2000 The Royal Society


1954 J. W. Lee, S. Hauan, K. M. Lien and A. W. Westerberg

The Ponchon{Savarit diagram (Ponchon 1921; Savarit 1922) for binary distillationallows one to account for enthalpy eects that cause a varying molar overow whenstepping o stages graphically. It is a diagram that plots vapour and liquid enthalpycurves versus composition. Through the use of tie-lines to express vapour/liquidequilibrium, one can step o the stages required in a binary column as a function ofproduct purities, feed quality and condenser duty. It, like the McCabe{Thiele dia-gram (McCabe & Thiele 1925), provides very powerful visualization of the behaviourof binary distillation columns. Our goal is to extend these visualizations to reactivedistillation. In this paper we show how to formulate and then construct a reactivecascade dierence point, which is a useful concept to extend into higher dimensionssuch as ternary and quaternary mixtures (Lee et al . 2000; Hauan et al . 1999, 2000),to reect the reaction eect in terms of energy and material balances on a Ponchon{Savarit diagram. We employ two types of reaction|when there is no net change inthe number of moles caused by the reaction, as for an isomerization reaction, andwhen there is a net change, as for a decomposition reaction.

2. Theory and implications

The lever rule is ubiquitous in chemical engineering. It states that, if we have twomulti-component mixtures M1 and M2, then the composition resulting from mixingM1 with M2 to form M3 must lie on the straight line connecting the compositions ofM1 and M2. Furthermore, it says that the lengths of the lines connecting the com-position of M1 to M3 and connecting the composition of M2 to M3 are proportionalto the amounts of M2 and M1, respectively. Hauan et al . (2000) present a very con-cise proof of this rule. We show here that this rule is readily extended and that thePonchon{Savarit diagram is a special case of this extension.Suppose we can relate three vectors, v1, v2, v3, with the following linear relation-

ship, where ai (i = 1; 2; 3) are scalars:

a3v3 = a1v1 + a2v2: (2.1)

If the components of each vj add to a constant b, then, when we plot the pointscorresponding to v1, v2 and v3 in the space spanned by them, these points will lieon a straight line. The proof, based on that in Hauan et al . (2000), is as follows. Addall relationships with each element in each vj on the right- and left-hand sides of theabove equation to get the following:

a3b = a1b + a2b or a3 = a1 + a2: (2.2)

Substituting for a3 in (2.1) and rearranging, we get

v3 = v1 + (1 )v2; (2.3)where = a1=(a1 + a2). This last equation says that v3 is the stated linear combi-nation of v1 and v2 with the properties claimed.We can generalize this result as follows. Suppose we can write another equation

with the same constants a1, a2, a3,

a3z3 = a1z1 + a2z2: (2.4)

Proc. R. Soc. Lond. A (2000)


A graphical method for designing reactive distillation columns. I 1955

Then we can extend the space spanned by v1, v2, v3 by adding to each vector thecomponents z1, z2, z3, respectively, and, in this augmented space, the points will stilllie on a straight line. In other words, we form the vectors

v0Tj = [vTj ; zj ] (2.5)

for each v0Tj , where each has one additional element zj in it. The proof is simple:equation (2.3) still holds for this augmented vector.What we see is that, as long as any subset of the components of a set of vectors

satisfying (2.1) sums to a constant, then the points represented by these vectors willlie on a straight line in the space spanned by those vectors. For the Ponchon{Saravitdiagram, as we show momentarily, we write material balances around the top of anon-reactive distillation column,

Vn + 1yn+ 1 = Lnxn +DxD : (2.6)

Here, the composition vectors have terms that always add to unity. Thus, thesecomposition vectors are on a straight line in composition space. We can also writean energy balance around a non-reactive rectifying section,

Vn+ 1Hn+ 1 = Lnhn + D(h D + QC=D): (2.7)

Thus we can augment the composition vector with these three enthalpies, Hn + 1, hnand h D +QC=D, where Hn+ 1, hn, h D and QC are the vapour enthalpy at stage (n+1), the liquid enthalpy at stage n, the distillate enthalpy and the condenser duty,respectively, and nd that these points also lie on a straight line, which is the basisfor construction of the Ponchon{Savarit diagram.To construct Ponchon{Savarit diagrams for reactive distillation in composition/en-

thalpy space, we will consider reactive distillation columns in which isomerizationand decomposition reactions take place within a reaction zone above the feed stagein gure 1.

(a) Isomerization reaction

Here, we consider the isomerization reaction regardless of the reaction phasein (2.8) between R1 and P1 components, where R1 is assumed to be less volatile thanP1. The reaction takes place in the rectifying section of a staged distillation columnwith a partial reboiler and a total condenser. The unconverted R1 exits at the bottomof the column and the product P1 exits at the top through a total condenser. If wetake the total and component-wise material balance equations around the rectifyingsection, including the reaction zone in gure 1, we have the following equations (2.9)and (2.10). An energy balance around the rectifying section yields (2.11), as follows:

R1, P1; (2.8)Vn+ 1 = Ln +D T n; (2.9)

Vn + 1yn+ 1 = Lnxn + Dx D n; (2.10)Vn+ 1Hn+ 1 = Lnhn + Dh D + QC + hR n: (2.11)

In (2.9), the total sum of stoichiometric coe cients, T, is zero for the isomerizationreaction (2.8). Because we can think of n as a owrate in units of, for example,




D

B

F

L

QC = qC D

n - 1x

nx

D hR

VS+1

V

LS

LS+1VB

QB = qBB

n+1 n

Figure 1. A schematic of a reactive distillation column.

mol s1, we shall call n a molar turnover owrate of reaction. It is the sum ofthe reaction turnovers occurring on stages n and above. ( n is often termed anextent of reaction.) If the reaction always proceeds in the forward direction from areactant to a product on each reacting stage, then n is a sum of positive numbersor zero and will thus be monotone increasing as one moves down the column. is the stoichiometric coe cient vector [ 1; 1]T and can be written as the dierencebetween the product coe cient vector cP = [0; 1]

T and the reactant coe cient vectorcR = [1; 0]

T in (2.10). In (2.11), QC is the condenser duty and hR is the heat ofreaction, which is negative for an exothermic reaction and positive for an endothermicreaction. We will illustrate for an exothermic reaction here. We can rearrange (2.9){(2.11) as follows:

Vn + 1 Ln = D; (2.12)Vn + 1yn+ 1 Lnxn = Dx D cP n + cR n = D rR;n; (2.13)Vn+ 1Hn+ 1 Lnhn = DhrR;n; (2.14)

where

rR;n = (Dx D cP n + cR n)=D; (2.15)hrR;n = (h D + qC + hR n=D); (2.16)

qC = QC=D: (2.17)

Equations (2.12){(2.14) have the same forms as (2.1), (2.2) and (2.4). Hence thesewill produce straight lines for material and energy balances in composition/enthalpy




space. We call rR;n the composition coordinate and hrR;n the enthalpy coordinate

for the reactive cascade dierence point ( rR;n; hrR;n) in the rectifying section. We

can step o stages on a Ponchon{Savarit diagram using this dierence point for areactive cascade. Note that the Ponchon{Savarit diagram plots enthalpy versus thecomposition of only one of the two species, generally the more volatile one. Thus,while (2.13) and (2.15) are actually two equations each, we use only the rst equationfor each in constructing this plot.We assume reaction only proceeds in the forward direction, so the reaction molar

turnover owrate n increases as we go down the column. Thus the compositioncoordinate ( rR;n) in (2.15) will move from the distillate composition toward thereactant coe cient vector (cR). For an exothermic reaction, the enthalpy coordinate(hrR;n) in (2.16) will decrease. Thus the reactive cascade dierence point (

rR;n; h

rR;n)

will move toward the reactant and downward as we go down the column.With no reaction in the stripping section, the material and energy balances are as

follows:

Vs + 1 Ls = B; (2.18)Vs + 1ys+ 1 Lsxs = BxB; (2.19)Vs+ 1Hs + 1 Lshs = B(hB qB); (2.20)

qB = QB=B: (2.21)

Here, QB is reboiler duty. Thus the construction of straight lines in the strippingsection is the same as for an ordinary distillation column (reproduced in King 1980;Henley & Seader 1981).Figure 2 illustrates how to step o stages in both the reactive rectifying and the

non-reactive stripping sections. First, we x the top dierence point (h D + qC; x D )and bottom dierence point (hB qB; xB). Arbitrarily here, we shall assume thatstages 1 and 2 from the top are non-reactive (no catalyst present). Thus we step othe stages until stage 3 by alternating between using the operating lines emanatingfrom the top dierence point and equilibrium tie lines. This stepping o of stages isexactly as we would do for a non-reactive column. We assume an arbitrary but realis-tic extent of reaction on stage 3, which will move the reactive cascade dierence point( rR;3; h

rR;3) toward the left (i.e. toward the reactant) and downward (for an exother-

mic reaction). The vapour composition at stage 4 (y4) lies at the intersection of thevapour enthalpy line and the straight line connecting the reactive cascade dierencepoint ( rR;3; h

rR;3) to the liquid composition at stage 3 (x3). We follow the phase equi-

librium tie line from y4 to nd the liquid composition (x4) of stage 4. Assuming morereaction occurs on stage 4, we obtain the vapour composition at stage 5 from theoperating line connecting the reactive cascade dierence point ( rR;4; h

rR;4) and the

liquid composition at stage 4. The liquid composition at stage 5 (x5) is also availableon the other end of the equilibrium tie line.Assume the reaction turns o at stage 5. Thus the nal reactive cascade dierence

point of the rectifying section is ( rR;4; hrR;4). We would continue to use this dierence

point for all subsequent stages above the feed stage when stepping down the column.Using the bottom dierence point, we step up the stages from the bottom in the sameway as in non-reactive distillation, since there is no reaction in the stripping section.The feed stage can be stage 4 through stage 6, as we cannot move below stage 6from the nal reactive dierence point of the rectifying section ( rR;4; h

rR;4) and cannot




3 2 1*

5 6 7

4

yP1

cRd rR, 4 d

r xDcP

hD+qC

y1 = xD

y4, H4

L3

(x3, h3)

xP1, yP1

hB - qB

(zF , hF)xB

hr

hr

Hv, sat

H, h

hL, sat

yB

R,3

R,4

R,3

D

Figure 2. A Ponchon{Savarit diagram for reactive distillation when an isomerization reactionoccurs. Dot-dash line, equilibrium tie line; solid line, balance line; arrow, construction line forthe V=L equilibrium relationship; asterisk, stage number.

move up from stage 4 using the bottom dierence point. We nd in each of these casesthat the operating lines coincide with the equilibrium tie lines, creating pinch points.We determine the optimal feed stage in the same way as in non-reactive distilla-

tion (Henley & Seader 1981). Connecting the bottom dierence point and the naldierence point of the rectifying section by a straight line to satisfy total materialand energy balances, we obtain the intersection point of the total balance line and anequilibrium tie line, as seen in gure 2. Below this line, we use the bottom dierencepoint and above the top dierence points, which makes stage 5 the optimal feedstage. The intersection point is the feed point with its composition and the enthalpy.We can construct equilibrium tie lines by using a y{x equilibrium diagram placed

above the H; x diagram, as shown in gure 2 (Wankat 1988). As shown, the ow of liq-uid leaving the third stage L3 relates to the distance from (

rR;3; h

rR;3) to (y4;H4) and

the distillate ow D to the distance from (y4; H4) to (x3; h3) in terms of a lever rule.




We note, in this example, that reaction changes the slopes of the balance lines inreactive stages 3 and 4. With the amount of reaction we show occurring, the slopeshave even become negative. In this way, reactive distillation may allow us to stepacross an azeotropic composition or a pinch point. Also, the reaction turnover weused here has signi cantly reduced the number of stages in the column.

(b) Decomposition reaction

Here, R1 decomposes to form P1. We assume R1 and P1 form an ideal mixture, andR1 is less volatile than P1. In a reactive distillation column, unconverted reactant R1exits at the bottom, and we recover product P1 at the top. If we take the follow-ing decomposition reaction, the material and energy balance equations around therectifying section are the same as (2.9){(2.11), except here the sum of the stoichio-metric coe cients, T, is 1 rather than zero for reaction (2.22). Thus we introducethe reaction dierence point, R equal to = T, [ 1; 2]T for R1 and P1 (instead ofdecomposing the stoichiometric coe cient vector as in the previous section to avoidthe in nite reaction dierence point (Lee et al . 2000)). While the composition ele-ments of the reaction dierence point add to one, this point lies on the right extensionof the pure P1 and outside a conventional composition space, as shown in gure 3.We rearrange (2.9){(2.11) to give the following equations (the material and energybalance equations for the stripping section are the same as (2.18){(2.21)):

R1, 2P1; (2.22)Vn + 1 Ln = D T n = rR;n; (2.23)

Vn+ 1yn + 1 Lnxn = Dx D T nR = rR;n rR;n; (2.24)Vn+ 1Hn+ 1 Lnhn = D(h D + qC) +hR n = rR;nhrR;n; (2.25)

rR;n = D T n; (2.26) rR;n = (Dx D T nR)= rR;n; (2.27)hrR;n = [D(h D + qC) +hR n]=

rR;n: (2.28)

Here, rR;n is the dierence in the vapour and liquid owrates between stages (n+1)and n in the rectifying section. rR;n is the composition coordinate and h

rR;n the

enthalpy coordinate for the reactive cascade dierence point in the rectifying section. T n is a owrate of new moles produced by reaction. Here, it has a maximum valuein the system equal to the owrate of species R1 in the feed, since each mole of R1converted produces one additional mole of material handled by the column. If werecover in the distillate more than half of the product P1 produced and if the reactiononly proceeds in the forward direction anywhere in the column, then the distillateowrate D must be larger than T n. We shall arbitrarily assume reaction occurs onlyon stage 2. As shown in gure 3, the composition coordinate of the reactive cascadedierence point starts at xD .

rR;2 moves to the left of the distillate composition away

from the reaction dierence point, since 2 is positive. If our decomposition reactionis exothermic, the enthalpy dierence point also goes down. We draw a balance linefor reactive stage 2 in gure 3 using the reactive dierence point ( rR;2; h

rR;2). The

length of the line between ( rR;2; hrR;2) and (y3;H3) is denoted by L2 and the length

from (y3; H3) to (x2; h2) is represented by rR;2, in the sense of a lever rule. We can

easily complete tray-by-tray calculations in the same way as in the previous case of




yP1

hr

Hv, sat

H, h

hL, sat

hB - qB (zF , hF)

xP1, yP1

xB

d r

xP1

8

2 1

y1 = xD

hD + qC

xD d R

L2

y3

x3

D rR,2

R,2

R,2

Figure 3. A Ponchon{Savarit diagram for reactive distillation whenthe decomposition reaction takes place.

gure 2. The feed stage can be any of stages 3 through 5, but the optimal location isstage 4, as the total balance line connecting ( rR;2; h

rR;2) and (xB; hB qB) intersects

with the equilibrium tie line of stage 4. We require eight stages, including a partialreboiler, to obtain the desired top and bottom products with only one reactive stage,as shown in gure 3.

(c) Alternative way to construct a Ponchon{Savarit diagram

If we want to relate D and Ln geometrically, and to obviate a reactive cascadedierence rR;n with a zero value (in which case the composition coordinates for thecorresponding reactive cascade dierence point is at in nity due to the low recoveryof the product at the top), we can reformulate (2.24) and (2.25) using the followingequations:

Vn + 1 (Ln + D) = T n; (2.29)Vn + 1yn + 1 (Lnxn + Dx D ) = T nR; (2.30)

Vn + 1Hn + 1 [Lnhn + D(h D + qC)] = T n( hR= T): (2.31)




D

1 2 3

+

yP1

Hv, sat

H, h

hL, sat

xP1, yP1

xP1

xD = y1

L2hD+qC

d R

V3

- D hR / vT

vT 2x

Figure 4. A Ponchon{Savarit diagram for reactive distillation when we take a di erent linearcombination in the decomposition reaction. +, Linear combination of (x2 ; h2 ) and (xD ; hD +qC ).

Here, we take stage 2 as a reactive tray for the decomposition reaction. For decom-position in a reactive distillation column, we rst draw a straight line between(xD ; h D + qC) and (x2; h2), as we illustrate in gure 4. Since we know (x2; h2) fromthe equilibrium relationship with (y2;H2) (as indicated in the construction of the linefor equilibrium by the arrow lines in gure 4), we can determine the combined point(+). The lengths of (x D ; h D + qC) and (x2; h2) from the combined point (+) reectthe relative amounts of L2 and D, respectively. At the second step, we construct astraight line connecting (y3; H3), the combined point (+) and (R; hR= T). Thedistance between (y3;H3) and the combined point (+) denotes the relative amountof T 2, and the length from the combined point (+) to (R; hR= T) that of V3.

3. Conclusions

In this work, we illustrate how to construct a Ponchon{Savarit diagram for binaryreactive distillation. One important point is that the number of stages and the feedstage location strongly depend on the extent of reaction in a reactive distillationcolumn. The Ponchon{Savarit method will provide an accurate description of a reac-tive distillation column if enthalpy data and the heat of reaction are correct. Wecan also obtain many design implications, such as the proper reaction distributionwithin a column and the determination of optimal feed stage location, using theproposed diagram. Historically, the Ponchon{Savarit method (Ponchon 1921; Savarit




1922) preceded the McCabe{Thiele method (McCabe & Thiele 1925). We also devel-oped the Ponchon{Savarit diagram for a reactive distillation column earlier than theMcCabe{Thiele method. Thus subsequent papers will deal with McCabe{Thiele dia-grams for reactive distillation columns and apply these two graphical methods toactual processes.

The authors thank the NSF (grant CTS9710303), the Eastman Chemical Company and theNorwegian Institute of Science and Technology for their support of this work.

Nomenclature

a1, a2, a3, b vector coe cient or constantB bottom product molar owrate (mol s1)cP product coe cient vectorcR reactant coe cient vectorD top product (distillate) molar owrate (mol s1)Hn + 1 saturated vapour molar enthalpy at stage n + 1 of the rectifying

section (J mol1)Hs + 1 saturated vapour molar enthalpy at stage s + 1 of the stripping

section (J mol1)hB bottom product molar enthalpy (J mol

1)h D top product molar enthalpy (J mol

1)h F feed molar enthalpy (J mol

1)hn saturated liquid molar enthalpy at stage n of the rectifying

section (J mol1)hs saturated liquid molar enthalpy at stage s of the stripping

section (J mol1)hrR;n enthalpy coordinate at stage n of the rectifying section in

equations (2.16) and (2.28) (J mol1)Ln liquid molar owrate at stage n of rectifying section (mol s

1)Ls liquid molar owrate at stage s of stripping section (mol s

1)R1, P1 a reactant and a productQB reboiler heat duty (J s

1)qB reboiler molar heat duty (J mol

1)QC condenser heat duty (J s

1)qC condenser molar heat duty (J mol

1)Vn+ 1 vapour molar owrate at stage n + 1 of the rectifying

section (mol s1)Vs+ 1 vapour molar owrate at stage s + 1 of the stripping

section (mol s1)v1, v2, v3 vectorsv0j augmented vectorxB bottom product molar composition vectorx D top product molar composition vectorxn liquid molar composition vector at stage n of the

rectifying sectionxs liquid molar composition vector at stage s of the

stripping section




yn+ 1 vapour molar composition vector at stage n + 1 of therectifying section

ys+ 1 vapour molar composition vector at stage s + 1 of thestripping section

z1, z2, z3 elements of a vectorz F feed molar composition vector

Greek letters

parameter used in equation (2.3) rR;n dierence ow in vapour and liquid owrates in

equation (2.23) (mol s1)R reaction dierence point rR;n composition coordinate for a reactive cascade dierence point in

equations (2.15) and (2.27)hR molar heat of reaction, negative for exothermic and positive for

endothermic (J mol1) stoichiometric coe cient vector T sum of stoichiometric coe cients n sum of reaction molar turnover owrate or extent of

reaction (mol s1) from the top to stage n

Denition

( rR;n; hrR;n) reactive cascade dierence point at stage n in Ponchon{Savarit

diagram (subscript `R denotes `reaction and superscript `rrepresents rectifying section)

References

Barbosa, D. & Doherty, M. F. 1988a Design and minimum-re ux calculations for single-feedmulticomponent reactive distillation columns. Chem. Engng Sci. 43, 1523{1537.

Barbosa, D. & Doherty, M. F. 1988b Design and minimum-re ux calculations for double-feedmulticomponent reactive distillation columns. Chem. Engng Sci. 43, 2377{2389.

DeGarmo, J. L., Parulekar, V. N. & Pinjala, V. 1992 Consider reactive distillation. Chem. EngngProg. 88, 43{50.

Espinosa, J., Scenna, N. & Perez, G. 1993 Graphical procedure for reactive distillation systems.Chem. Engng Commun. 119, 109{124.

Hauan, S., Westerberg, A. W. & Lien, K. M. 1999 Phenomena based analysis of xed points inreactive separation systems. Chem. Engng Sci. 55, 1053{1075.

Hauan, S., Ciric, A. R., Westerberg, A. W. & Lien, K. M. 2000 Di erence points in extractiveand reactive cascades. I. Basic properties and analysis. Chem. Engng Sci. 55, 3145{3159.

Henley, E. J. & Seader, J. D. 1981 Equilibrium-stage separation operations in chemical engineer-ing, pp. 372{409. Wiley.

King, C. J. 1980 Separation processes, 2nd edn, pp. 273{283. McGraw-Hill.

Lee, J. W., Hauan, S., Lien, K. M. & Westerberg, A. W. 2000 Di erence points in extractiveand reactive cascade. II. Generating design alternatives by the lever rule for reactive systems.Chem. Engng Sci. 55, 3161{3174.




McCabe, W. L. & Thiele, E. W. 1925 Graphical design of fractionating columns. Ind. EngngChem. 17, 605{611.

Okasinski, M. J. & Doherty, M. F. 1998 Design method for kinetically controlled, staged reactivedistillation columns. Ind. Engng Chem. Res. 37, 2821{2834.

Perez-Cisneros, E. S. 1997 Modelling, designing and analysis of reactive separation processes.PhD thesis, Technical University of Denmark.

Ponchon, M. 1921 Tech. Moderne 13, 20{55.

Savarit, R. 1922 Arts et Metiers, pp. 65, 142, 178, 241, 266, 307.

Siirola, J. J. 1995 An industrial perspective on process synthesis. AIChE Symp. Ser. 304, 222{233.

Wankat, P. C. 1988 Separations in chemical engineering; equilibrium staged separations, pp. 9{34. Elsevier.



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2000 Distillation Columns. I. the Ponchon-Savarit Method

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