# Modified Ponchon-Savarit and McCabe-Thiele methods for distillation of two-phase feeds

**ABSTRACT** Modified versions of the Ponchon-Savarit and McCabe-Thiele methods are presented which incorporate a correct analysis of the feed region in the case of partly vaporized feeds. The methods proposed are fully graphical, require no additional information or calculations, and do not complicate significantly the traditional procedures. It is shown that for optimal effect, the liquid and vapor parts are introduced to separate but contiguous stages. Numerical examples confirm that a small improvement in column performance is achieved, but the influence on the number of stages or reflux ratio required for a given separation is negligible, except in very special cases.

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Page 1

Ind. Eng. Chem. Process Des. Dev. 1904, 23, 1-6

1

Modified Ponchon-Savarit and McCabe-Thiele Methods for

Distillation of Two-Phase Feeds

Jean Marle Ledanols and Claudlo Ollvera-Fuentes

Departamento de Termodinsmica y Fen6menos de Transferencia, Unlversidad Sim6n Bohar, Apartado 80659,

Caracas 1080, Venezuela

Modified versions of the Ponchon-Savarit and McCabe-Thiele methods are presented which incorporate a correct

analysis of the feed region in the case of partly vaporized feeds. The methods proposed are fully graphical, require

no additional information or calculations, and do not complicate significantly the traditional procedures. It is shown

that for optimal effect, the liquid and vapor parts are introduced to separate but contiguous stages. Numerical

examples confirm that a small improvement in column performance is achieved, but the influence on the number

of stages or reflux ratio required for a given separation is negligible, except in very special cases.

Introduction

Despite the widespread availability of computer facilities

and software for the design and calculation of distillation

processes, traditional graphical methods such as the

McCabe-Thiele and Ponchon-Savarit diagrams continue

to be used, either as means to obtain quick preliminary

estimates or as valuable teaching tools which allow the

interrelationship of the several process variables to be

easily grasped and understood.

Most, if not all, presentations of the subject, however,

are based on the concept of a "feed stage", in effect as-

suming that a feed stream, whatever ita physical condition,

is dways introduced in bulk to a single tray, where it mixes

with liquid from the tray above and vapor from the tray

below, before any phase separation can occur. This model

obviously cannot be correct in the practically important

case of a partially vaporized feed which, upon entering the

column at some point between two trays, splits sponta-

neously into a vapor that flows to the tray immediately

above, and a liquid that falls to the tray immediately be-

low. In such cases, there are actually two feed stages, and

the only way for the traditional model to be realized would

be for the feed stream to be flash-separated, at the feed

tray pressure outside the column, and for the resultant

phases to be introduced separately, the liquid portion

above and the vapor portion below the intended feed stage

(Bennet and Myers, 1962).

Cavers (1965) identified this problem, and accurately

pointed out that a correct treatment of partly-vapor feeds

implies a discontinuity in the McCabe-Thiele method of

stepping off stages. He failed, however, to propose a

modified, fully graphical technique, suggesting only that

the discontinuity be resolved by material balances around

the feed section. It is the purpose of the present paper

to show that all balance equations can be incorporated into

both the McCabe-Thiele and the Ponchon-Savarit con-

structions to yield more general diagrams which, at the

expense of negligible extra effort, will deal with two-phase

feeds correctly.

Modified Ponchon-Savarit Method

Consider the distillation of a binary mixture in a con-

ventional column with a total condenser and an internal

reboiler. The feed stream, introduced between stages "a"

and "b", separates into mutually saturated liquid and va-

por, as shown schematically in Figure 1. Trays are num-

bered starting at the column top, and are assumed to be

ideal, including the reboiler. Additional assumptions in-

0196-4305/84/ 1123-0001$01.50/0

clude steady-state operation and negligible heat losses.

The following convention is adopted, for economy and

compactness of notation. Any process stream, e.g., of mass

flow rate P, composition of the more volatile component

xp, and enthalpy h, is represented by a vector

(PI (P, PX,, Ph,)

so that the set of total mass, component mass, and energy

balance equations for any control volume can be summa-

rized in a single vector relation. The process stream is also

represented by a point of the enthalpy-composition dia-

gram

P (xp, hp)

Thus, the balance equation for a column section ex-

tending from any tray of the rectifying section up to the

condenser is simply

{V)j+l- V 4 j = AD)^' = 1, -., a)

(1)

(AD) E (D7 DXD, D ~ D

+ Q D ~

where (AD) is the fictitious net upward flow

(2)

which defines the rectifying-section difference point AD

Of coordinates (XD, h D + &/D).

Similarly, the balance equation

(Qk-1 - (Qh

(A,)

(k = b, ..., n)

(3)

is valid for any tray of the stripping section and defines

the fictitious net downward flow

(Awl {W, Wxw, Whw - Qwl

to which corresponds the stripping-section difference point

A, of coordinates (xw, hw - Qw/W). From the overall

balance for the entire column

(4 = (Awl + (AD)

(4)

(5)

the feed point F is found to lie on the straight line which

joins the two difference points, AD and Aw.

The streams immediately above and below the feed are

related by the equations

(4 = (LFl + (vF)

{ v a + l = ( q 3 b + (VF)

(L.)b+l = (L)a + &F)

(6)

(7)

(8)

only two of which are linearly independent, in view of eq

0 1983 American Chemical Society

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Ind. Eng. Chem. Process Des. Dev., Vol. 23, No. 1, 1984

F, zF, 1-0

Plate ~OVP

Feel?

Figure 1. Two-phase feed introduction to a distillation column.

1, 3, and 5. The simplest way to connect the upper and

lower column sections is to define a “changeover” net flow

stream

E f q b - &la

(94

Then from eq 1 and 7 there follows

lac1 = {Va+1- {VF) 4Lla = {AD) - {VF)

whereas from eq 5 and 6 and the above

(9b)

IACI = IF) - - {VFI = (LFI -

(gc)

Hence, the changeover difference point is the intersec-

tion point of two straight lines, one which passes through

points AD and VF, the other one passing through points

LF and Aw.

Equations 2,5,6, and 9 determine the basic appearance

of the modified Ponchon-Savarit diagram, as illustrated

in Figure 2. With the main points in place, stages may

be counted in the usual manner, starting at the top: (i)

A point Lj-, is joined with one of the A points by a straight

line; this and the saturated vapor curve intersect at point

Vi. (ii) The equilibrium tieline is drawn through V,; this

defines Lj on the saturated liquid curve.

This sequence is repeated for j = 1,2, ..., n. The starting

point is Lo at (xD,IzD), and the procedure is terminated

when x, 5 xw for the first time. Upon reaching this con-

dition, the total number of ideal stages is given by m - 1

C n I m .

Unless the stage numbers a and b are assigned a priori,

in which case the introduction of the feed stream in

probably nonoptimal, the difference point to use at each

step (i) of the above sequence should be that one among

AD, A,, and Ac which yields the lowest composition y j

(equivalently, the leftmost point V,, or the largest slope

of the line L,-,Vj) for the same L;-,. Initially, Lj-, lies to

the right of P, the point at which the line ADVFAc inter-

sects the saturated liquid curve. Difference point AD

should be used (Figure 3a), giving Vj also to the right of

VF.

Eventually, a tieline VjL, will cross the line ADVFAc,

giving Lj to the left of P, but necessarily to the right of LF.

%

( X r Y )

XD

Figure 2. Basic Ponchon-Savarit diagram for two-phase feed.

Difference point

from eq 9b means that the vapor stream, VF, has been

introduced at its optimal position, a = j. Hence Vj+l is

i ; 6 and falls to the left of VF. The use of A,, not Aw, at

this point clearly indicates that the liquid portion of the

feed should not be fed to the same tray that receives the

vapor portion. Otherwise, the optimal changeover would

be from AD to Aw directly, skipping A,.

The equilibrium tieline through i;6 next fixes & on the

saturated liquid curve, necessarily to the left of LF. Dif-

ference point Aw should now be used (Figure 3c). This

means that the liquid stream, LF, has been introduced at

its optimal position, b = a + 1; i.e., the liquid portion of

the feed is best input to the tray immediately below that

which receives the vapor portion.

As shown in Figure 4, the straight line which joins La

and A, directly gives v b at its intersection with the vapor

locus. Points V,+, and Lb-1 are not essential for continu-

ation of the method. If wanted, V,,, can be found at the

intersection of the two lines, VFvb and LaAD, cf. eq 1 and

7, and Zb-1 can similarly be found at the intersection of

LFLa and i;6Aw, cf. eq 3 and 8. It follows that V,,,

Lbl will not be saturated phases unless the dew-point and

bubble-point curves are perfectly straight lines. The rest

of the procedure reverts to the traditional method, A,

being used until the bottom composition xw is attained.

should now be used (Figure 3b), which

and

Modified McCabe-Thiele Method

Although the McCabe-Thiele diagram can always be

generated trivially once the Ponchon-Savarit diagram is

available, we specifically consider here the special case

where the “usual simplifying assumptions” (Treybal, 1980)

leading to constant molar overflow and vaporization are

satisfied exactly. In such instances, material balance re-

lations corresponding to the first two components of the

vector eq 1 through 8 and the y-x diagram are sufficient

for a description of the distillation process, with no need

to resort to enthalpy-related constructions.

In the rectifying section, the constant liquid and vapor

flow rates are denoted simply by L and V. Equations 1

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Ind. Eng. Chem. Process Des. Dev., Vol. 23, No. 1, 1984 3

( X . Y )

Figure 3. Optimal use of difference points in the Ponchon-Savarit diagram.

( to hC 1

Figure 4. Detail of the feed region in the Ponchon-Savarit diagram.

and 2 then provide an equation for the operating line, or

locus of counterflowing liquid and vapor compositions.

On introducing the external reflux ratio, R = LID, eq

10a becomes

XD

(j = 1, ..., a)

(lob)

R

Yi+l = R + 1 X ; + R+1

i.e., the rectifying line is a straight line of slope R/(R +

1) that passes through point (xD, xD).

Similarly, in the stripping section where flow rates are

denoted by

and v, eq 3 and 4 yield the relation

which, when combined with eq 5, 6, and 8 becomes

b, ..., n) (llb)

Thus, the stripping line is a straight line that passes

through point (xw, xw), and whose slope can be calculated

from known problem data. More usually, though, the

liquid fraction of the feed, q = LF/F, is defined as a

measure of its thermal condition, and the feed q line is

introduced from eq 6

This is the equation of a straight line which passes

through point (zF, zF), has a slope -q/(l - q), and intersects

the equilibrium curve at (xF, YF). Since it may be shown

that the three lines, eq 10, 11, and 12 meet in a single

intersection point, the rectifying line and feed q line are

traditionally drawn first, and their intersection point is

located. The stripping line is then the straight line joining

this point and (xw, xw).

It should be clear, however, that the joint solution of eq

10 and 11 can never represent actual conjugate vapor and

liquid compositions at any point along the column, since

by definition both equations apply to separate column

sections. The feed region between plates a and b must be

represented by a third, “changeover” operating line,

equivalent to the intermediate difference point Ac of the

revised Ponchon-Savarit method.

From eq 1, 2, and 7 total and light-component mass

balances may be written for a column section extending

from the condenser down to the vapor feed point, but

excluding the liquid portion LF. An equation for the

changeover line is thus derived.

While this straight line can be constructed readily from

problem data, it is easier and more instructive to consider

Page 4

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Ind. Eng. Chem. Process Des. Dev., Vol. 23, No. 1, 1984

I

I

I

I

I

I

I

_ _ -

2

--I

2

( x

X F

Figure 5. Basic McCabe-Thiele diagram for two-phase feed.

its intersections with the two traditional operating lines.

Solving eq 10b and 13 together for the x coordinate gives

the implicit result

\r F

i.e., the rectifying and changeover lines intersect at a point

which also lies on the horizontal line y = yF. By analogy

to eq 12, the latter line is simply the q line for the saturated

vapor feed VF (q = 0, z = YF).

Next, solving eq llb and 13 together gives the simple

result

= XF

(15)

i.e., the stripping and changeover lines intersect at a point

which also lies on the vertical line x = xF. By eq 12 this

line can be interpreted as the q line of the saturated liquid

feed LF (q = 1, z = XF).

The basic appearance of the modified McCabe-Thiele

diagram is illustrated in Figure 5. The rectifying and

stripping lines are drawn as previously described. The

vapor feed q line is drawn horizontally at y = yF, crossing

the rectifying line at point T. The liquid feed q line is

drawn vertically at x = xF, crossing the stripping line at

point U. Line TU is then the changeover operating line.

With the main lines thus in place, stages may be counted

by the usual staircase construction, starting at the top from

an initial point (xD, XD): (i) Through a point (x,-~, y,) on

the current operating line, a horizontal line is drawn to

obtain (x,, y,) at the intersection with the equilibrium

curve. (ii) Through point (x,, y,) a vertical line is drawn

to obtain (x,, Y / + ~ )

at the intersection with one of the op-

erating lines.

At each step (ii) of this sequence, optimal advantage is

taken of the operating lines only by selecting the line that

yields the lowest composition Y,,~ for the same x,. Initially,

this corresponds to the rectifying line, which should con-

tinue to be used for as long as (x,, y/) falls to the right of

T in Figure 5, i.e., while xI 1 TF.

Eventually, a point (x,, y,) will be generated to the left

of T, but necessarily to the right of U (or at most, on it)

since by construction T and U span exactly one ideal stage.

Thus yF > xJ 2 xF, and the changeover line should be used

_ ~ _ __ I

._._

I - - - - /

I

5

.n

'a-1

xS-l

Figure 6. Detail of the feed region in the McCabe-Thiele diagram.

x ( x )

to obtain (x,, Y,+~). This implies that the vapor stream VF

has been introduced at its optimal location, a = j. The

graphical situation is depicted in more detail in Figure 6.

It follows from the above that the next equilibrium

point, (E,+~, Y,+~), necessarily falls to the left of U,

xF Here, the stripping line should be used, and this means

that the liquid stream LF is optimally located at b = a +

1. In this way, we confirm the results obtained from the

preceding Ponchon-Savarit analysis regarding introduction

of the feed streams.

It should be noted that here, too, the intermediate points

(xa,

and (&, gb) are not essential for the construction.

If wanted, (xa, ya+l) can be located at the point where a

vertical line x = x, intersects the rectifying line. Similarly,

(Xbl, Yb) is the intersection of a horizontal line y = Y b with

the stripping line. As shown in Figure 6, these two points

do not coincide, a fact that was first noted by Cavers

(1965).

The remainder of the construction follows the traditional

method. The stripping line is repeatedly employed until

x , 2 x, for the first time.

Comments and Discussion

The methods described in the present work are easily

shown to reduce to the traditional constructions when

either the vapor or the liquid portion of the feed is neg-

ligibly small. Thus, for instance, if the feed is a saturated

liquid at its bubble point (VF = 0, F = LF, q = l), points

F and LF in the Ponchon-Savarit diagram of Figure 2 will

merge, and AD, A, become a single difference point, as

expected from eq 9b. Transition from AD to A, is irrele-

vant, and change from A,, (A,) to Aw is effected when a

tieline fist crosses the AD F(LF) A, line. Similarly, in the

McCabe-Thiele diagram of Figure 5, points S and U be-

come a single point, and the changeover line is identical

with the rectifying line, as expected from eq 19 and 13.

The stripping line is brought into operation when x, I

and the feed stage straddles point S.

The modified methods cease to be applicable when the

feed is a subcooled liquid or a superheated vapor, which

flows entirely onto a single stage. The traditional proce-

dures are correct here, even if the McCabe-Thiele con-

struction must, for consistency, still be based on the con-

cept of a liquid-in-feed fraction q that takes physically

impossible values, either q > 1 or q < 0.

<

xF,

Page 5

Ind. Eng. Chem. Process Des. Dev., Vol. 23, No. 1, 1984 5

Table I. Comparison of Traditional and Modified

McCabe-Thiele Calculations, Based on ZF = 0.50

(XF =0.4142,y~=O.5858),q=0.50,R=4.0,~=2.0,

XD = 0.95, xw = 0.05.

traditional method

stage

Y

X

1

2

3

4

5

6

7

8

9

10

11

12

13'

0.0691

0.0358

' Internal reboiler.

Whenever the feed stream is an actual two-phase mix-

ture at column conditions, however, the modified methods

proposed here constitute the correct, fully graphical

treatment of binary distillation. Inasmuch as they are

based on a model of two feeds, VF and LF to separate

stages, these methods are special cases of available pro-

cedures for multiple-feed columns. The distinguishing

feature of the present analysis is that, being mutually

saturated phases, VF and LF are shown to require intro-

duction to adjacent stages for optimal effect.

Cavers (1965) stated that the separate introduction of

the liquid and vapor portions should have but a small

influence on the computed number of trays, and this is

probably the main reason why the extremely simple

modifications described have not been attempted before.

That the difference is indeed minimal can be verified from

a typical design calculation, e.g., the separation of an

equimolar mixture into a distillate with xD = 0.95 and a

bottom product with xw = 0.05. The number of ideal

stages was computed using the McCabe-Thiele method

with q = 0.50, R = 4.0, and an assumed constant relative

volatility, a = 2.0. For greater precision, analytical versions

of both the traditional and the revised constructions were

programmed into an HP-41C portable calculator. The

computed compositions at each tray are reported in Table

I, and the combined diagrams of the two methods are

shown in Figure 7. The number of stages is the same in

both cases, n = 13. Composition differences do not exceed

0.007 mole fraction, being understandably greater in the

feed region and decreasing toward the reboiler as the

driving forces become smaller.

The example given is a somewhat extreme one, in that

Figure 7 shows point &,, x,) to lie roughly in the middle

of the changeover line, where the distance to the rectifying

and stripping lines is appreciable. Thus, in most cases one

should expect even smaller differences in composition and

number of stages between the textbook and the revised

methods. In the above example, for instance, changing the

reflux to R = 3.6838 results in a vapor composition from

stage 7 which exactly matches the vapor feed composition,

thereby leading to complete coincidence between the two

approaches, because point &, x,) becomes identical with

point T of Figure 5.

It should be noted that the modified procedures do not

alter the definition of the minimum reflux ratio. Referring

to the McCabe-Thiele diagram of Figure 5, as the reflux

ratio is decreased, and a pinch is approached in the feed

region, points S, T, and U slide respectively along the

combined feed, vapor feed, and liquid feed q lines until

this work

Y

X

0.9048

0.8413

0.7591

0.6629

0.56 29

0.4709

0.3908

0.3123

0.2330

0.1619

0.1049

0.0631

0.0343

0.9500

0.9138

0.8630

0.7973

0.7203

0.6403

0.5667

0.4818

0.3841

0.2847

0.1949

0.1225

0.9048

0.8413

0.7591

0.6629

0.5629

0.4709

0.3954

0.3173

0.2377

0.1659

0.1080

0.0652

0.9500

0.9138

0.8630

0.7973

0.7203

0.6403

0.5620

0.4760

0.3779

0.2787

0.1899

0.1186

0.0663

Figure 7. Comparison of traditional and modified McCabe-Thiele

constructions, (design problem): (-) this work; (- - -) traditional

method.

Table 11. Comparison of Traditional and Modified

McCabe-Thiele Calculations, Based on ZF = 0.50

(XF = 0.3090, YF = 0.6910), q = 0.50, fl= 4,a = 4.0,

XD = 0.95, xw = 0.05

traditional method,

R = 6.1763

this work,

R = 4.9560

stage

1

2

3

4a

' Internal reboiler.

they meet at the single point (xF, yF), in agreement with

the traditional version. At reflux ratios near the minimum,

the composition difference between the changeover line

and either operating line will be negligible, even if their

slopes differ sharply, and both procedures will tend to give

the same results.

For a noticeable contrast between the traditional and

the proposed methods, one must turn to systems of high

relative volatility, where each equilibrium stage (hence also

the feed region) involves substantial changes in compo-

sition. This is most clearly seen from a typical rating

calculation, in which the reflux ratio is computed for

specified product compositions in a prescribed number of

stages. The separation of an equimolar mixture into a

distillate with xD = 0.95 and a bottom product with xw =

0.05 was again considered. The reflux ratio was computed

by the McCabe-Thiele method with q = 0.50, n = 4, and

an assumed constant relative volatility, a = 4.0. The

previously mentioned HP-41C programs were used in an

iterative process assuming a reflux ratio, counting 4 stages

down from the known distillate composition and checking

whether the resultant bottom product matched the spec-

ified purity. The calculated reflux ratios and tray com-

positions are reported in Table 11, and the combined di-

agrams of the two methods are shown in Figure 8. The

reflux ratio is nearly 20% smaller by the modified proce-

dure, R = 4.9506 as against 6.1763 in the usual construc-

tion, even if composition differences again do not exceed

0.008 mole fraction.

Y

X

Y

X

0.9500

0.8137

0.5337

0.2083

0.7917

0.4663

0.1863

0.0500

0.9500

0.8183

0.5262

0.2083

0.7917

0.4738

0.1817

0.0500

Page 6

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Ind. Eng. Chem. Process Des. Dev., Vol. 23, No. 1, 1984

LI

' 3

' A

I F

Figure 8. Comparison of traditional and modified McCabe-Thiele

onndrtiot;ena ( v o t i n n nmhlnml.

,,"llu"rulrr".lu,

\'U""'6

Y'""'U&.*,.

method.

thin runrlr. (-l

, ,

trnrlitinnsl

Y L . . U .. "...( , , " . . . u . " - " . . ~ .

The above illustration also constitutes an extreme case.

Differences in reflux ratio will decrease rapidly as the

relative volatility decreases and the number of stages in-

creases. The same separation, for example, but with n =

10 and C Y = 2.0 requires a reflux ratio only 2% smaller in

the modified method, R = 3.0376 compared with 3.1068

for the single-phase feed. This discrepancy is undoubtedly

closer to what may be expected in most practical cases.

From the viewpoint of the present work, the traditional

diagrams represent nonoptimal locations of the feed

streams, with either early introduction of the liquid phase

or delayed introduction of the vapor phase. The small

differences encountered between the two methods are

therefore probably typical of any analysis of the influence

of feed position. Considering that the proposed methods

correspond to a conceptually correct, and physically more

realistic model, that they require no extra information and

a veritable minimum of additional graphical work, and that

they actually help to clarify the importance of feed loca-

tion, it seems beneficial that they be included in treatments

and applications of the subject.

Acknowledgment

The authors wish to thank their colleague, Mr. Julio

Bastida, for helpful discussions during the preparation of

this work.

Nomenclature

Most symbols are also used a s subscripts, to denote properties

calculated for the appropriate stream or stage.

a = stage immediately above feed point

b = stage immediately below feed point

D = distillate

F = feed

h = specific enthalpy of liquid

H = specific enthalpy of vapor

j = general stage, rectifying section

k = general stage, stripping section

4 = liquid flow, rectifying section

L = liquid flow, stripping section

LF = liquid portion of feed

m = final stage of graphical construction

n = number of ideal stages in column

P = general process stream

q = fraction of liquid in feed

QD = condenser heat load

Qw = reboiler heat input

R = external reflux ratio

S = intersection of stripping and rectifying lines

T = intersection of changeover and rectifying lines

U = intersection of changeover and stripping lines

V = vapor flow, rectifying section

V = vapor flow, stripping section

VF = vapor portion of feed

W = residue

x: = liquid composition, more volatile component

y = vapor composition, more volatile component

z = overall feed composition, more volatile component

Ac = net upward flow from stripping to rectifying section

AD = net upward flow in rectifying section

Aw = net downward flow in stripping section

Literature Cited

Bennet, C. 0.; Myers, J. E. "Momentum, Heat and Mass Transfer": McGraw-

Hill: New York, 1962; p 627.

Cavers, S. D. Ind. Eng. Chem. Fondam. 1965, 4 , 229.

Treybal. R. E. "Mass Transfer Operations", 3rd 4.; McGraw-Hill: New York.

1980; p 403.

Received for review April 12, 1982

Revised manuscript received December 28, 1982

Accepted January 29, 1983