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Elements of
Fractional Distillation

CLARK SHOVE ROBINSON

AND

EDWIN RICHARD GLLLILAND

Revised and Rewritten

t>y

EDWIN RICHARD GILT.II.ANO

Professor of Chemical Engineering
Afassachusetts Institute of Technology

FOURTH EDITION
SKCOND

McGRAW-HILL BOOK COMPANY, INC.

NKW YORK TORONTO LONDON
1950

ELEMENTS OF FRACTIONAL DISTILLATION

Copyright, 1922, 1930, 1939, 1950, by the McGraw-Hill Book Company, Inc
Printed in the United States of America. All rights reserved. This book, or parti
thereof, may not be reproduced in any form without permission of the publishers

THE MAPLE PRESS COMPANY, YORK, PA.

PREFACE TO THE FOURTH EDITION

The firs^^ditipn.of-this book and the early revisions were the result
of the efforts of Professor Robinson, and he took an active part in
guiding the revision of the previous edition. His death t made it
necessary to prepare this edition without his helpful guidance and
counsel.

The present revision differs extensively from the previous edition.
The material has been modified to bring it more closely into line with
the graduate instruction in distillation at Massachusetts Institute of
Technology. Much greater emphasis has been placed on the measure-
ment, prediction, and use of vapor-liquid equilibria because it is
believed that this is one of the most serious limitations in design calcu-
lations. Greater emphasis has also been placed upon the use of
enthalpy balances, and the treatment of batch distillation has been
considerably expanded. Unfortunately, the design calculations for
this type of operation are still in an unsatisfactory status. Azeotropic
and extractive distillation are considered as an extension of conven-
tional multicomponent problems. The sections on column design
and column performance have been completely rewritten and increased
in scope. In all cases quantitative examples have been given because
it has been found that this greatly aids the student in understanding
descriptive material.

During the last 15 years a large number of design methods have been
proposed for multicomponent mixtures, some of which are reviewed in
Chapter 12. Most of these do not appear to offer any great advantage
over the conventional Sorel method, and it is believed that the law of
diminishing returns has been applying in this field for some time. It is
hoped that the present edition will stimulate some of these investi-
gators to transfer their efforts to more critical problems, such as
vapor-liquid equilibria, batch distillation, transient conditions within
the distillation system, and column performance.

EDWIN RICHARD GILLILAND
CAMBRIDGE, MASS.
July, 1960

PREFACE TO THE FIRST EDITION

The subject of fractional distillation has received but scant atten-
tion from, writers in the English language since Sidney Young published
his book " Fractional Distillation " in 1903 (London). French and
German authors have, on the other hand, produced a number of books
on the subject, among the more important of which are the following:

"La Rectification et les colonnes rectificatriccs en distillerie,"
E. Barbet, Paris, 1890; 2d ed., 1895.

"Der Wirkungsweise der Rcctificir — und Destillir— Apparate,"
E. Hausbrand, Berlin, 1893; 3d ed., 1910.

"Theorie der Verdampfimg und Verfliissung von gemischcn und der
fraktionierten Destination," J. P. Kuenen, Leipzig, 1906.

"Theorie der Gewinnung und Trennung der atherischen Olc durch
Destination," C. von Rechenberg, Leipzig, 1910.

"La Distillation fractione*e et la rectification," Charles Manlier,
Paris, 1917.

Young's "Fractional Distillation," although a model for its kind,
has to do almost entirely with the aspects of the subject as viewed
from the chemical laboratory, and there has been literally no work in
English available for the engineer and plant operator dealing with the
applications of the laboratory processes to the plant.

The use of the modern types of distilling equipment is growing at
a very rapid rate. Manufacturers of chemicals are learning that they
must refine their products in order to market them successfully, and
it is often true that fractional distillation offers the most available
if not the only way of accomplishing this. There has consequently
arisen a wide demand among engineers and operators for a book which
will explain the principles involved in such a way that these principles
can be applied to the particular problem at hand.

It has therefore been the purpose of the writer of this book to
attempt to explain simply yet accurately, according to the best ideas of
physical chemistry and chemical engineering, the principles of frac-
tional distillation, illustrating these principles with a few carefully
selected illustrations. This book is to be regarded neither as a com-
plete treatise nor as an encyclopedia on the subject but, as the title
indicates, as an introduction to its study.

Viii PREFACE TO THE FIRST EDITION

In general, it has been divided into five parts. The first part deals
with fractional distillation from the qualitative standpoint of the
phase rule. The second part discusses some of the quantitative
aspects from the standpoint of the chemical engineer. Part three
discusses the factors involved in the design of distilling equipment.
Part four gives a few examples of modern apparatus, while the last
portion includes a number of useful reference tables which have been
compiled from sources mostly out of print and unavailable except in
large libraries.

The writer has drawn at will on the several books mentioned above,
some of the tables being taken nearly bodily from them, and has also
derived much help from Findlay's " Phase Rule" (London, 1920) and
from "The General Principles of Chemistry" by Noyes and Sherrill
(Boston, 1917). He wishes especially to express his gratitude for
the inspiration and helpful suggestions from Dr. W. K. Lewis of the
Massachusetts Institute of Technology and from his other friends and
associates at the Institute and of the E. B. Badger & Sons Company.
Finally, he wishes to express his appreciation of the assistance of
Miss Mildred B. McDonald, without which this book would never
have been written.

CLARK SHOVE ROBINSON

CAMBRIDGE, MASS.
June, 1920,

CONTENTS

PREFACE TO THE FOURTH EDITION . v

PREFACE TO THE FIRST EDITION vii

INTRODUCTION 1

1. Determination of Vapor-Liquid Equilibria. . . . .3

2. Presentation of Vapor-Liquid Equilibrium Data 16

3. Calculation of Vapor-Liquid Equilibria . . .26

4. Calculation of Vapor-Liquid Equilibria (Continued) . . .79

5. General Methods of Fractionation . . . 101

6. Simple Distillation and Condensation. 107

7. Rectification of Binary Mixtures. . 118

8. Special Binary Mixtures . 192

/9/ Rectification of Multicomponent Mixtures . ... , 214

10. Extractive and Azeotropic Distillation ... . 285

i^l. Rectification of Complex Mixtures . .... 325

12. Alternate Design Methods for Multicomptonent Mixtures , 336

13. Simultaneous Rectification and Chemical Reaction. . . 361

14. Batch Distillation 370

15. Vacuum Distillation 393

16. Fractionating Column Design 403

17. Fractionating Column Performance 445

18. Fractionating Column Auxiliaries 471

APPENDIX 479

INDEX 481

INTRODUCTION

Definition of Fractional Distillation. By the expression fractional
distillation was originally meant the process of separating so far as it
may be feasible a mixture of two or more volatile substances into its
components, by causing the mixture to vaporize by suitable application
of heat, condensing the vapors in such a way that fractions of varying
boiling points are obtained, re vaporizing these fractions and separating
their vapors into similar fractions, combining fractions of similar boil-
ing points, and repeating until the desired degree of separation is finally
obtained.

Purpose of Book. Such a process is still occasionally met with in
the chemical laboratory, but it is a laborious and time-consuming
operation which has its chief value as a problem for the student, for the
purpose of familiarizing him with some of the characteristic properties
of volatile substances. It is possible to carry on a fractional distilla-
tion by means of certain mechanical devices which eliminate almost all
of this labor and time and which permit separations not only equal to
those obtained by this more tedious process but far surpassing it in
quality and purity of product. The purpose of this book is to indicate
how such devices may be profitably used in the solution of distillation
problems.

Origin of Fractional Distillation. Like all the older industries,
fractional distillation is an art that originated in past ages and that
developed, as did all the arts, by the gradual accumulation of empirical
knowledge. It is probable that its growth took place along with that
of the distilled alcoholic beverages, and to the average person today
the word "still" is synonymous with apparatus for making rum,
brandy, and other distilled liquors. To France, which has been the
great producer of brandy, belongs the credit for the initial development
of the modern fractionating still.

Physical Chemistry and Fractional Distillation. Fractional dis-
tillation has labored under the same sort of burden that the other
industrial arts have borne. Empirical knowledge will carry an
industry to a certain point, and then further advances are few and far
between. It has been the function of the sciences to come to the rescue

2 FRACTIONAL DISTILLATION

of the arts at such times and thus permit advancement to greater use-
fulness. The science that has raised fractional distillation from an
empirical to a theoretical basis is physical chemistry. By its aid the
study of fractionation problems becomes relatively simple, and it is on
this account that the subject matter in this book is based upon physical
chemistry.

CHAPTER 1
DETERMINATION OF VAPOR-LIQUID EQUILIBRIA

The separation of a mixture of volatile liquids by means of fractional
distillation is possible when the composition of the vapor coming from
the liquid mixture is different from that of the liquid. The separation
is the easier the greater the difference between the composition of the
vapor and that of the liquid, but separation may be practicable even
when the difference is small. The relation between the vapor and
liquid compositions must be known in order to compute fractional dis-
tillation relationships. Usually this is obtained from information
concerning the composition of the vapor which is in equilibrium with
the liquid. On this account a knowledge of vapor-liquid equilibrium
compositions is usually essential for the quantitative design of frac-
tional distillation apparatus. In most cases the study is made on the
basis of the composition of the vapor in equilibrium with the liquid.
However, this is not a fundamental requirement and any method that
would allow the production of a vapor of a different composition than
that of the condensed phase, whether equilibrium or not, could be used
for separation. However, most of the equipment employed depends
on the use of a vaporization type of operation, and the equilibrium
vapor is a good criterion of the possibilities of obtaining a separation.

The methods for obtaining vapor-liquid equilibrium compositions
can be considered in two main classifications: (1) the experimental
determination of equilibrium compositions and (2) the theoretical
relationships.

EXPERIMENTAL DETERMINATIONS OF VAPOR-LIQUID EQUILIBRIA

The measurement of vapor-liquid equilibrium compositions is not
simple. A highly developed laboratory technique is therefore needed
to obtain reliable data in any of the several methods described here.

Circulation Method. A common method for obtaining vapor-liquid
equilibrium (Refs. 11, 13, 16, 23, 27, 35) is by circulating the vapor
through a system and bringing it into repeated contact with the
liquid until no further change in the composition of either takes
place. A schematic diagram of such a system is shown in Fig. 1-1.
The vapor above the liquid in vessel A is removed, passed through

FRACTIONAL DISTILLATION

chamber 5, and recirculated by pump C through the liquid in A.
While the system appears simple, in actual practice it involves a num-
ber of complications:

1. The system must be completely tigkt; otherwise the total quan-
tity of material will continually vary and the equilibrium compositions
of the vapor and liquid will also change.

2. The quantities of liquid and of vapor when equilibrium is obtained
must remain constant and not vary during the recirculation. To keep

Heat-*
exchanger

Equilibrium I
cell

--Vapor sample

FIG. 1-1. Circulation apparatus for vapor-liquid equilibrium measurements.

them constant it is necessary for the system to remain isothermal and
for the total volume to remain constant. The chief variation in the
volume of the system is due to the fact that it is usually found expedient
to use a reciprocating pump. The error due to this variation is usually
minimized by making the displacement volume of the pump small.
The pumps are generally of a mercury-piston type; t.e., a mercury
column is forced up and down in a steel or glass cylinder serving
as the piston of the pump. This makes it possible to have an essen-
tially leakproof pump and allows the pumping operation to be carried
out with very little contamination of the circulating vapors.
3. This type of system has been used most successfully under con-

DETERMINATION OF VAPOR-LIQUID EQUILIBRIA 5

ditions where the vapor does not condense at room temperature. If
it were necessary to operate the pumping system at a high temperature
to avoid condensation of the vapor, difficulties might be encountered
due to the vapor pressure of the mercury, in which case other lower
vapor pressure metallic liquids should be suitable.

4. Another condition that could cause the relative volumes of vapor
and of liquid to vary is the rate of flow. The rate of recirculation
varies the pressure drop through the apparatus and thereby changes
the quantity, of vapor present. In most cases the rate of recirculation
is such that the pressure differential for recirculation is not great.
Both the volume variation due to pumping and the pressure changes
due to recirculation can be made less detrimental by making the vol-
ume of the liquid in vessel A large.

5. It is necessary to ensure that there is no entrainment of liquid
with the vapor leaving A. If liquid is carried over to vessel B, the
vapor sample will be contaminated. This entrainment is eliminated
by the use of low velocity and by efficient entrainment separators in
the upper part of A.

6. Another precaution is the necessity to prevent any condensation
of the vapor during recirculation. If any vapor condenses, the con-
densate will be of different composition and the results will be in error.

This type of apparatus has been used for a variety of systems.
It is particularly suitable for very low temperature studies such as
those involved in the equilibria associated with liquid air. In this
case vessel A is maintained in a low-temperature cryostat, and the
recycle vapor stream is heat-exchanged with the exit vapor; the rest
of the system is maintained at essentially room temperature. One of
the difficulties with the operation is the fact that the vapor sample is
obtained as a vapor and, unless the pressure is high, the quantity of
vapor obtained in vessel B may be so small as to offer difficulties in
analysis.

The system has the great 'ad vantage that a vapor can be repeatedly
bubbled through the liquid until equilibrium is obtained. Theoretically
exact equilibrium is not obtained because of the fact that there are
pressure differentials in the system. Thus the vapor entering at the
bottom of A must be under a pressure higher than the vapor leaving A,
at least by an amount equal to the hydrostatic head of the liquid in A.
Since the vapor-liquid equilibria depend on pressure, it is obvious that
there cannot be exact equilibrium. However, the change in the
composition of the equilibrium vapor due to this small change of pres-
sure is small in most cases. It could be serious in the critical region

FRACTIONAL DISTILLATION

where the vapor is very compressible. Basically this system is one of
the best for obtaining true equilibrium.

Bomb Method. In the bomb method (Refs. 3, 4, 12, 14, 36) the
liquid sample is placed in a closed evacuated vessel. It is then agitated
by rocking, or by other means, at constant temperature until equilib-
rium is obtained between the vapor and the liquid. Samples of the
vapor and the liquid are then withdrawn and analyzed.

The method appears simple, but it involves certain difficulties.
During sampling there are pressure changes due to the removal of
material, and these pressure changes can be large in magnitude. In
order to avoid them, it is customary to add some fluid, such as mercury,
to the system while the samples are being removed in order to prevent

Rocking
mechanism

- Sampling line

fjf - Constant temperature

FIG. 1-2. Bomb apparatus.

any vaporization or condensation. Another difficulty with the
method is the fact that in most cases it is necessary to use sampling
lines of small cross sections. These may fill up with liquid during the
initial part of the operation, and this liquid may never come to the true
equilibrium. It is necessary to purge the sampling lines to remove
such liquid. This liquid holdup is particularly serious in the case of
the vapor sample since in quantity it may be large in comparison to the
sample. A schematic diagram of the bomb-type apparatus is shown
in Fig. 1-2.

Dynamic Flow Method. Another method that has been widely
used (Refs. 10, 21, 25, 37) for the determination of vapor-liquid equi-
libria is one in which a vapor is passed through a series of vessels con-
taining liquids of a suitable composition. The vapor entering the
first vessel may be of a composition somewhat different from the
equilibrium vapor, but as it passes through the system it tends to
approach equilibrium, If all the vessels have approximately the same
liquid composition, the vapor will more nearly approach equilibrium
as it passes through the unit, The number of vessels employed should
be suoh that the vapor entering the last unit is of essentially equilib-
rium composition.

DETERMINATION OF VAPOR-LIQUID EQUILIBRIA 7

This system has the advantage that it is simple and, in certain cases,
it is possible to dispense with the analysis of the liquid sample, i.e., the
liquids can be made of a known composition, and since the change in
the last vessel is small, it is possible to assume that the composition of
the liquid in this case is equal to that originally charged. A schematic
diagram of such a system is shown in Fig. 1-3.

It is obvious that it cannot be an exact equilibrium system because
of the fact that a pressure drop is involved in passing the vapor through
the system; i.e., there are pressure variations which will affect equi-
librium. There is also the danger of entrainment, although this can
be minimized by low velocities.

In a great many cases, the gas introduced into the first vessel has
.been carrier gas of low solubility and not a component of the system.

FIG. 1-3. Dynamic flow method.

Thus, in the determination of the vapor-liquid equilibria for systems
such as ammonia and water, ammoniacal solutions are placed in the
vessels, and a gas such as nitrogen is bubbled into the first of these and
the resulting nitrogen-ammonia-water vapor mixture is passed through
the succeeding vessels obtaining a closer approach to equilibrium.
Equilibrium obtained in such a manner is not the true vapor-liquid
equilibria for the water vapor-ammonia system. It closely approaches
true equilibrium for the binary system under a total pressure equal to
the partial pressure of the ammonia and the water vapor in the gaseous
mixture. Even this is not exact. The carrier gas has some solubility
in the liquid phase, and the partial pressure of these added constituents
modifies the energy relations of the liquid and vapor phases. Usually
for low-pressure operation these errors are not large in magnitude, but
as the pressure becomes higher the errors are serious and the method
can give erroneous results if the true vapor-liquid equilibria for mix-
tures without the carrier gas are desired.

Dew and Boiling-point Method. In essence this technique consists
in introducing a mixture of known composition into an evacuated
equilibrium container of variable volume (Refs. 6, 7, 9, 15, 17, 18, 20,
28). The system is maintained at a constant temperature, and by
varying the volume the pressure is observed at which condensation

8 FRACTIONAL DISTILLATION

commences and is completed. The dew- and bubble-point curves of
pressure vs. temperature for a number of these prepared samples are
obtained and, by cross-plotting, conditions of phase equilibrium may
be found by locating points at which saturated liquid and saturated
vapor of different compositions exist at the same temperature and
pressure.

The pressures are determined in two ways. One involves the
measuring and plotting of the PV isotherm, the dew point and bubble
point being indicated by the discontinuities in the curve at the begin-
ning and the end of condensation. The other employs a glass or
quartz equilibrium cell, and the conditions are determined visually.

The advantages of this method are that it allows the critical condi-
tions to be determined, gives data on specific volumes of mixtures at
high pressures, and requires no analysis of the phases since the total
composition of the mixture is accurately determined gravimetrically
upon charging.

There are disadvantages, however. For certain conditions the dew
and bubble points are not sharply defined ; hence they require measure-
ments to be made with highly refined precision instruments. The
simpler units using mercury as a variable volume confining fluid cannot
be used below the freezing point of mercury. In addition, the mate-
rials used must be very pure and free in particular from traces of fixed
gases, for in the critical region the saturation pressure is quite sensitive
to small amounts of fixed gases. Further, a large amount of experi-
mental work must be done in order to define completely and accurately
the phase equilibria over all ranges of liquid and vapor composition.
The major limitation, however, is the fact that the method can be used
only on binary systems. As the phase rule dictates that more complex
systems are not a unique function of pressure and temperature, dew
and bubble points alone cannot define the composition of two phases in
equilibrium.

Dynamic Distillation Method. The four previous methods involved
repeated contact of the vapor with the liquid and thus afforded the
time necessary for the attainment of equilibrium. The dynamic dis-
tillation method (Refs. 2, 5, 11, 19, 24, 26, 34, 39) involves a different
procedure (see Fig. 1-4). In this system a distilling vessel is connected
to a condenser and a receiver.

In the simplest case, a small sample of distillate is taken, and the
compositions of this sample and the liquid in the still are determined.
During such a distillation the composition of the distillate and the
liquid in the still changes, and the samples represent average values.

DETERMINATION OF VAPOR-LIQUID EQUILIBRIA

To reduce this composition variation the quantity of liquid in the still
is made large in comparison to the quantity of distillate. Frequently
successive samples of condensate are obtained, and these are analyzed
and the composition plotted vs. the quantity of liquid that has been
distilled. An extrapolation of this curve back to zero quantity of
liquid removed is taken as the composition of the vapor in equilibrium
with the original liquid.

Top heater

Charge

FIG. 1-4. Dynamic distillation apparatus.

The method involves a new assumption, namely, that the vapor
obtained by boiling a liquid is in equilibrium with the liquid. There
has been no adequate proof of this assumption, and theoretical con-
siderations would tend to indicate that equilibrium should not be
obtained. The few experimental data that are available would indi-
cate that the difference in the composition between the vapor obtained
in this manner and the true equilibrium is not great in most cases, but
in a few systems significant differences have been found.

After the vapor leaves the liquid, any condensation in the upper part
of the distilling vessel will change the composition of the vapor and

10 FRACTIONAL DISTILLATION

therefore introduce errors. Such condensation is usually reduced or
eliminated by having the upper part of the system jacketed and at a
higher temperature than the condensation temperature of the vapor.
However, this higher temperature can introduce errors; for example,
in such a boiling system there is a certain amount of spray and splash-
ing. The spray that lands on the heated walls will tend to vaporize
totally and give a vapor of the composition of the liquid rather than of
the equilibrium composition.

The pressure involved in such a system is of course essentially that
prevailing in the receiver, and this method can be used either for nor-
mal pressures, high pressures, or vacuum. The exact temperature of
the operation is usually not known because the liquid is generally
superheated. The vapor and the liquid therefore are not in thermal
equilibrium, and it is doubtful whether they are in true composition
equilibrium. The apparatus has been extensively used because of its
simplicity, and the results are of sufficient accuracy to be of real value
in distillation calculations.

In order to obtain a closer approach to equilibrium, various com-
plicating arrangements have been used; for example, Rosanoff modified
the system to obtain a second contact of the vapor with the liquid.

Continuous Distillation Methods. Continuous distillation methods
involve distilling a liquid, condensing the vapor sample, and recycling
the condensate back to the still. A schematic drawing of such an
equilibrium still is given in Fig. 1-5. This system was developed by
Yamaguchi (Ref . 38) and Sameshima (Ref . 29) and has been modified
and improved by a number of other investigators (Refs. 1, 8, 22, 30, 31,
32, 33). This method has been widely used and has the great advan-
tage that it is simple, and the unit can be placed in operation and
allowed to come to a steady state without any great amount of atten-
tion. The same precautions relative to entrainment, condensation
and total vaporization of splashed liquid must be observed in the still
as was indicated for the dynamic distillation method. The condensate
collects until the level is high enough to flow over the trap and back to
the still. At the end of the distillation, this condensate is removed and
analyzed to determine the composition of the vapor, and a sample is
removed from the still to determine the still composition.

This method suffers from the same difficulties as the dynamic dis-
tillation method in that it is open to the question of whether the vapor
formed by boiling a liquid is in equilibrium with the liquid. It is also
difficult to obtain the true liquid temperature because of the super-
heating effects. The pressure is maintained by the pressure in the exit

DETERMINATION OF VAPOR-LIQUID EQUILIBRIA

tube, and in normal pressure determinations this is usually open to the
atmosphere. This theoretically offers the possibility of errors in that
it allows Oxygen and nitrogen to dissolve in the condensate sample,
which is then recycled back to the still. At low pressures the solu-
bility of such gases is usually small and the error is slight, but in high-
pressure operations the use of this gas system can lead to serious errors.

FIG. 1-5. Continuous distillation equilibrium still.

The gas pressuring system, however, is extremely desirable in that it
regulates the condenser cooling capacity so that it exactly balances
heat input to the still. At high pressures the errors become so serious
that this benefit must be foregone. Figure 1-6 indicates a type of
apparatus in which the heat input and removal are adjusted so that
the pressure remains constant without the necessity of a sealing gas.
Another source of error in the system is possible because the con-
densate returned to the still is of a different composition from the
liquid in the still and in general is of lower boiling point. If this
vaporizes before it is completely mixed with all of the liquid in the still,
this vapor composition will not be an equilibrium vapor.

FRACTIONAL DISTILLATION

Although the apparatus appears to be of the recirculation type and
it might be supposed that the successive contacts would tend to give
a closer approach to equilibrium, this is not the case. If the vapor
evolved from the liquid is not an equilibrium vapor, this type of recycle

Vent*.

Condenser,
Top heater

Relay controlling heat
supply to still

SHU
thermo •
couple

St/it
sample

Hot

Bottom heater
FIG. 1-6. Continuous distillation still for high-pressure operation.

system does not give a closer approach with repeated recycling since
new vapor is formed and the recycled material is not brought to equi-
librium by successive contacts. The recycling does give a steady-state
condition, but the approach to equilibrium is only that obtained by
boiling the liquid.

In order to eliminate some of the sources of error in continuous dis-
tillation systems, various modifications ^have been made. The most

DETERMINATION OF VAPOR-LIQUID EQUILIBRIA

important of these (Refs. 1, 8) would appear to be one in which the
condensate stream is re vaporized before it is returned to the still; i.e.,
the heat is added to revaporize the condensate stream instead of form-
ing a new vapor in the still. Such an apparatus is shown in Fig. 1-7,
In this case, the result is equivalent to recycling the vapor, and the
operation tends to be equivalent to the usual recycle system. It is
more difficult to operate than the conventional continuous distillation
system. The condensate must be completely vaporized. If any

Thermometer
well

FIG. 1-7. Continuous distillation still with re vaporized condensate.

liquid is allowed to return to the still, the purpose of the system is
defeated and the rate of distillation decreases; i.e., less vapor leaves the
still. If the vapor returned to the still is greatly superheated, it will
cause additional evaporation in the still and the operation will speed
up. By proper adjustment a satisfactory balance can be obtained.
It is believed that this apparatus is a definite improvement over the
regular continuous distillation system, and comparative data on the
same system taken with this and the usual continuous distillation sys-
tem show definite differences of the type that would be anticipated.

Both the continuous distillation system and the modifications of it
suffer from the difficulty that the vapor must be totally condensable
under the operating conditions^ This is usually not a serious difficulty,

14 FRACTIONAL DISTILLATION

It is also necessary that the condensate be a homogeneous mixture.
, Thus, if the condensate separates into two layers, the operation is not
satisfactory. The other vapor-liquid equilibrium methods are suitable
for multilayer systems either in the still or in the vapor sample.

ACCURACY OF VAPOR-LIQUID EQUILIBRIUM DATA

A satisfactory investigation of the accuracy of the various experi-
mental methods has not been made, and there is real question con-
cerning a large amount of the published experimental vapor-liquid equi-
librium data. The circulation and the bomb methods have the
potentiality of giving high accuracy, and the value of the results depends
on the care employed by the experimentalist in eliminating sources of
error. The dynamic distillation and the continuous distillation
methods involve the assumption that the vapor produced by boiling a
liquid is of a composition that is in equilibrium with the remaining
liquid. The adequacy of this assumption has not been proved experi-
mentally, but there are experimental results which cast doubt on its
validity for all cases. Analysis of the published data obtained by
employing the continuous distillation method would indicate that it
may give values for the differences of the vapor and liquid compositions
at a given liquid composition that are within 10 to 15 per cent of the
true values. When results for the same system are compared on this
basis, it is not uncommon to find deviations of ± 10 per cent between
different investigators employing essentially the same techniques. A
critical study of the methods of determining vapor-liquid equilibria is
needed.

References

1. ADEY, S. M. thesis M.I.T., 1941.

2. BALY, Phil. Mag., (V), 49, 517 (1900).

3. BEBGANTZ, Sc.D. thesis, M.I.T., 1941.

4. BOOMER and JOHNSON, Can. J. Research, 16B, 328 (1938).

5. BROWN, Trans. Chem. Soc., 35, 547 (1879).

6. CALINGAERT and HITCHCOCK, J. Am. Chem. Soc., 49, 750 (1927).

7. CAUDET, Compt. rend., 130, 828 (1900).

8. COLBURN, JONES, and SCHOENBORN, Ind. Eng. Chem., 35, 666 (1943).

9. CUMMINGS, Ind. Eng. Chem., 23, 900 (1931).

10. DOBSON, J. Chem. Soc., 127, 2866 (1925).

11. DODGE and DUNBAR, J. Am. Chem. Soc., 49, 591 (1927),

12. FEDORITBNKO and RUHEMANN, Tech. Phys., U.S.S.R., 4, 1 (1937).

13. FERGUSON and FUNNELL, /. Phys. Chem., 33, 1 (1929).

14. FREETH and VERSCHOYLB, Proc. Royal Soc. (London), A 130, 453 (1931).

15. HOLST and HAMBURG, Z. physik. Chem., 91, 513 (1919).

16. INGLIS, Phil. Mag. (VI), 11, 640 (1906).

DETERMINATION OF VAPOR-LIQUID EQUILIBRIA 15

17. KAY, Ind. Eng. Chem., 30, 459 (1938),

18. KTJBNEN, VERSCHOYLE, and VAN UBK, Communs. Phys. Lab., Univ. Leiden,
No. 161 (1922).

19. LEHFELDT, Phil. Mag. (V), 46, 42 (1898).

20. MEYEK, Z. physik. Chem., A 175, 275 (1936).

21. ORDOBFF and CARRELL, /. Phys. Chem., 1, 753 (1897).

22. OTHMER, Ind. Eng. Chem., 20, 743 (1928).

23. QUINN, Sc.D. thesis,- M.I.T., 1940.

24. RAYLEIGH, Phil. Mag. (VI), 4, 521 (1902).

25. RIGNAULT, Ann. chim. et phys., (3) 16, 129 (1845).

26. ROSANOFF, BACON, and WHITE, J. Am. Chem. Soc., 36, 1993 (1914).

27. ROSANOFF, LAKE, and BREITHUT, /. Am. Chem. Soc., 48, 2055 (1909).

28. SAGE and LACEY, Ind. Eng. Chem., 26, 103 (1934).

29. SAMESHIMA, J. Am. Chem. Soc., 40, 1482, 1503 (1918).

30. SCATCHARD, RAYMOND, and GILMANN, J. Am. Chem. Soc., 60, 1275 (1938).

31. SCHEELINE and GILLILAND, Ind. Eng. Chem., 31, 1050 (1939).

32. SIMS, Sc.D. thesis, M.I.T., 1933.

33. SMYTH and ENGEL, /. Am. Chem. Soc., 61, 2646 (1929).

34. TAYLOR, /. Phys. Chem., 4, 290 (1900).

35. TOROCHESNIKOV, Tech. Phys., U.S.S.R., 4, 337 (1937).

36. VERSCHOYLE, Trans. Roy. Soc. (London), A 230, 189 (1931).

37. WILL and BREDIG, Ber., 22, 1084 (1889).

38. YAMAGUCHI, /. Tokyo Chem. Soc., 34, 691 (1913).

39. ZAWIDSKI, Z. physik. Chem., 36, 129 (1900).

CHAPTER 2
PRESENTATION OF VAPOR-LIQUID EQUILIBRIUM DATA

It is usually desirable to present the experimental vapor-liquid
equilibrium data graphically. A number of methods of presentation
have been developed, but the most important are the temperature-
composition and the vapor-liquid composition diagrams.

Phase Rule. The method of presentation must be consistent with
the number of variables involved. For equilibrium conditions the
number of independent variables can be obtained from the phase rule
which states that the number of phases <j> ^lus the degrees of variance
F is equal to the number of components C plus 2.

0 + p = C + 2 (2-1)

In the usual vapor-liquid equilibria two phases are involved: liquid and
vapor. ?Iowever, in some systems more than one liquid phase may be
encountered. For the two-phase system the phase rule states that the
degrees of freedom or variance are equal to the number of components.
Thus a binary system has two degrees of freedom and can be repre-
sented by two variables * on rectangular coordinates. Three-com-
ponent systems involve three degrees of freedom and are usually
presented on triangular coordinates. Multicomponent systems with
more than three components are difficult to present, and special
methods are employed for such systems.

Temperature-Composition Diagrams. By fixing the total pressure
of a two-component system, a temperature-composition diagram can
be made. Figure 2-1 shows a temperature-composition curve for
carbon tetrachloride-carbon bisulfide at a total pressure of 760 mm.
Any point on the curve ABC gives the composition x, of a mixture of
CCU and 082, which boils at a pressure of 760 mm. at a temperature £,
where t is in degrees centigrade and x is the mol fraction of CS2. The
use of the "mol fraction " greatly facilitates calculations of vapor-
pressure phenomena. It is the ratio of the number of molecular
weights of one component in a mixture divided by the sum of the num-
ber of the molecular weights of all components. Mol per cent is equal
to 100 times the mol fraction. The line ADC represents the com-
position of the vapor that is in equilibrium with the liquid at any

PRESENTATION OF VAPOR-LIQUID EQUILIBRIUM DATA 17

given temperature. Thus a liquid with the composition xi will have
a vapor pressure of 760 mm. at the temperature t^ and the vapor in
equilibrium with it will have the composition yi = #2.

Starting with a mixture of the composition xi, at a constant total
pressure equal to 760 mm., and at a temperature below £2, there will be
but one phase present, the liquid mixture of CCU and CSa. As the
temperature is raised, only a liquid phase will be present until the
vertical line at Xi intersects the curve ABC, when a vapor phase of

80
A

: 70

^60

v
\X4

0 01 02 03 04 05 06 07 08 09 10
CC14 Mo! fraction C$2 CS2

FIG. 2-1. Boiling-point curve for CCh-CSa mixtures.

the composition x2 will appear. Since there are now two phases and the
pressure is fixed, there can be but one variable, temperature, and the
composition of the phases will depend upon it. Let the temperature
then be raised to some point tz and the liquid and vapor compositions,
being no longer independent variables, must change accordingly,
which they do along the curves ABC and ADC, respectively, the liquid
now having a composition x*, and the vapor in equilibrium with it a
composition y* = rc4. It should be remembered that the quantity of
CC14 and CS2 in the system has not changed during this process; there-
fore, the change in the compositions of the liquid and the vapor includes
such a corresponding change in the relative proportions of each phase
that the total composition of the system remains the same, x\. Fur-
thermore, the relative proportions of the liquid phase and the vapor
phase, at the temperature /3, are as the distances FG and EF. It will

FRACTIONAL DISTILLATION

be seen that, as the temperature is raised farther, the proportion of
liquid phase decreases until, when the temperature reaches a point
corresponding to the intersection of the vertical line x\ and the curve
ADC, which occurs at a temperature U, the vapor has the same com-
position as the original liquid, and the liquid phase disappears. At
higher temperatures, there is but one phase, and the system again
becomes trivariant, so that at constant pressure it is possible to vary

1.U

0.8
S.

5* °-6

t;
| 0.4
"o

OQ.2

(

/
'

l/_

) 0.2 0.4 0.6 0.8 1.

vug

Fia. 2-2. Equilibrium y,x data for CC14-CS2 mixtures.

both the temperature and the composition of the vapor. This is the
region of superheated vapor.

If the foregoing process is reversed, the steps can be followed in the
same way. Starting with superheated vapor of a composition 3/5 = xi
and at a temperature /6, condensation will first occur when the vertical
line 2/5 cuts the vapor line ADC, when liquid of a composition #5 will
separate out. Further cooling will change both the composition of the
liquid and the vapor along the lines ABC and ADC, respectively, until
the liquid has reached the composition x\ when all the vapor will have
disappeared.

Vapor-Liquid Equilibrium. It is possible to plot the same data as
were used in Fig. 2-1 as vapor composition vs. liquid composition at
either constant pressure or constant temperature. The data presented
in Fig. 2-1 were for constant pressure so they have been replotted in

PRESENTATION OF VAPOR-LIQUID EQUILIBRIUM DATA 19

Fig. 2-2 for the same conditions. In this curve the composition
yi = x2 is plotted as ordinate with composition x\ as abscissa, y* « x\
as ordinate with z3 as abscissa, and so forth. This particular relation
is very useful in distillation calculations. It does not give so much
information as Fig. 2-1, owing to the elimination of temperature.
However, in most distillation calculations it is desired to make a given
separation between the components and the temperatures are allowed

120

100

I 60

I - Benzene - toluene
TST- /sobufano/- wafer"

~ Acetone -chloroform
H -Acetone -carbonctisutfide

I I I I I I

3 02 0.4 06 0.8

x,Mol fraction in liquid

Fio. 2-3. Temperature-composition diagram.

1.0

to adjust themselves accordingly. On Pig. 2-2 the 45° line represents
a vapor of the same composition as the liquid. If the temperature is
important, this variable can be plotted vs. the liquid composition on
the same figure.

The curves given in Figs. 2-1 and 2-2 are termed the normal type.
However, there are several other common types of curves, In Fig.
2-3 temperature-composition diagrams for constant total pressure are
given for four different types of binary mixtures, and in Fig. 2-4 the
corresponding vapor-liquid diagrams are given for the four same

mixtures.

Type I is normal, i.e., the composition of the equilibrium vapor is
always richer in the same component than the composition of the
liquid, thus by repeated operations it is possible to obtain complete
separation.

FRACTIONAL DISTILLATION

In type II the temperature-composition diagram passes through a
minimum, and the vapor-liquid composition diagram crosses the
diagonal. Thus there are mixtures that have lower boiling points than
either of the pure components at the same pressure. In other words,
the mixture is the minimum boiling-point type. When such tempera-
ture-composition diagrams are encountered, the vapor-liquid com-
position curve will always cross the 45° line. In the region below this
intersection with the diagonal, the equilibrium vapor is richer in one

[ "Benzene-toluene
H - Acetone -carbondisutfide
HI - A cetone - ch/o ro fo rm
IF - Isobutanol- water

Q 0.2 * 0.4 0.6 0.8

x,Mo! fraction in liquid
FIG. 2-4. Vapor-liquid equilibrium curves.

1.0

component than the liquid; above this intersection, the vapor is poorer
in this component than the corresponding liquid from which it came.
Thus the volatilities have reversed. Where the vapor-liquid curve
crosses the 45° line, the vapor has the same composition as the liquid
and operations based on producing an equilibrium vapor from this
liquid would not be able to separate mixtures of this composition.
This particular composition is called a constant boiling mixture, or
azeotropic mixture, sincejt^will yaporize^without any change in Com-
position and, theipfore, without any change in temperature during the
evaporation. *~

TyplFHTTs the reverse of type II. In this case there are mixtures
that have boiling points higher than either of the pure components at
the same pressure. It will be noted in Fig. 2-4 that the curve of type

PRESENTATION OF VAPOR-LIQUID EQUILIBRIUM DATA 21

III also crosses the 45° line but curve II cuts the 45° line with the
slope less than 1 while curve III crosses the 45° line with the slope
greater than 1. Curve III is of the maximum boiling-point type, and
the particular composition at which the curve crosses the 45° line is
called a maximum constant boiling mixture or a maximum boiling
azeotrope.

The curve of type IV is similar to that of type II except that, for a
considerable range of composition, the temperature of the liquid phase
is constant. This curve type is characteristic of a partly miscible
liquid system. In the immiscible region, two liquid phases are present
and the phase rule indicates that the boiling temperature of the mix-
ture must be constant. In the diagram, the over-all composition of
the liquid is plotted as x regardless of whether one or two phases are
present. There is no single liquid phase that has a composition equal
to the value given in the two-phase region. The y,x data for this
system are given in Fig. 2-4. In this case the y,x curve crosses the
diagonal in the two-phase region; thus at this intersection the com-
position of the vapor is the same as that of the combined liquid phases.
Such a mixture can be evaporated to dryness at constant pressure
without change in composition or temperature. The mixture of this
particular composition is termed a pseudo-azeotrope. This terminol-
ogy is sometimes applied to any two-phase mixture, but the original
usage of the term azeotrope by Wade and Merriman (Ref . 50) implied
that the liquid could be evaporated to dryness without change in
composition. Only the two-phase mixture corresponding to the inter-
section of the y,x curve with the y = x line can be evaporated without
changing composition.

In other cases the y,x curve may not cross the diagonal in the two-
phase region, and such mixtures do not form pseudo-azeotropes, but
they may form, and usually do, true azeotropes in one of their single-
phase regions.

Literature Data. The vapor-liquid equilibria for a large number of
mixtures have been experimentally determined, and Table 2-1 lists
some of the more reliable and useful determinations.

The data given in the table represent a large amount of experi-
mental effort, and owing to the difficulties of making such determina-
tions a number of investigators have tried to develop theoretical and
empirical methods of predicting such vapor-liquid equilibria from the
physical properties of the pure components. While certain correla-
tions have been developed by this method, reliable experimental deter-
minations are still to be preferred to any such calculated values.

FRACTIONAL DISTILLATION
TABLE 3-1. VAPOB-LIQUID EQUILIBRIUM DATA

System

Constant
TorP

Tech-
nique *

Ref.

Acetaldehyde— water

760 mm.

Acetic acid-acetic anhydride
—acetone

750 mm.
760 mm.

C.D.
C.D.

27
55

-acetone— water .
—benzene

760 mm.
758 mm.

C.D,
C.D.

55
27

—ethyl acetate

760 mm.

C.D.

-methyl amyl ketone . .
-water

Acetone-carbon disulfide .

-chloroform
—ethyl ether ...

760 mm.
125, 250, 300, 760 mm.
760 mm.
35.2°C.
760 mm.
760 mm.
30°C.

C.D.
C.D.
C.D.
B.M.

D.D.
C.D.

29
14
55
17
19
38
39

— isopropanol

25°C.

D.F.

— methanol

755 mm.

C.D.

— w-butanol

760 mm.

C.D.

—nitrobenzene

20°C.

D.F.

— trichloroethylene . . .

755 mm.

C.D.

—water . ,

25°C.

B.M.

760 mm.
200, 350, 500, 760 mm.
50, 100, 200 p.sa.a.
150, 300, 760 mm.

C.D.
C.D.
C.D.
C.D.

5
29
31
30

1 to 20 atm.

Aniline— benzene

70°C.

C.D.

Benzene-carbon disulfide
~carbon tetrachloride

19.9°C.
40°C.

B.M.
C.D.

17
40

—chloroform

760 mm.
760 mm.

D.D.

— cyclohexane

740 mm.

C.D.

-ethyl bromide
-ethylene chlorohydrin .
-ethylene dichloride
—methanol

760 mm.
760 mm,
100, 200, 400, 760 mm.
40°C.

C.D.
C.D.
C.D.

48
44
4
22

— n-hexane

760 mm.
735 mm.

C.D.
C.D.

—nitrobenzene

20°C.

D.F.

—phenol , ,

70°C.

C.D.

— n-propanol

40°C.

C.D.

—2 2 3 trimethylbutane

740 mm.

C.D.

*B,M. «• bomb method
C.D. — continuous distillation
C.M. — circulation method

D.B. •• dew-point, bubble-point method
D.D. « dynamic distillation
D.F. « dynamic flow

PRESENTATION OF VAPOR-LIQUID EQUILIBRIUM DATA 23
TABLE 2-1. VAPOR-LIQUID EQUILIBRIUM DATA (Continued)

System

Constant
TorP

Tech-
nique*

Ref.

750 mm.
760 mm.
760 mm.
760 mm.
760 mm.
760 mm.
760 mm.
20°C.
19 8°C.
29.2°C.
20.0°C.
17 0°C.
20 0°C.
745 mm.
49.9°C.
745 mm.
760 mm.
760 mm.
762 mm.
100°C.
757 mm.
20°C.
760 mm.
740 mm.
25°C.
740 mm.
10, 50 mm.

0 1 mrn.
0 1 mm.
-25.5, -12, 0, 12,
23°C.
760 mm.
20°C.
760 mm.
30°C,
760 mm.
95, 190, 380 mm.
760 mm.
39.8°C.
20°C.
20°C.
120°C.

D.D.
C.D.
C.D.
D.D.
C.D.
D.D.
D.D.
B.M.
B.M.
B.M.
B.M.
B.M.
D.F.

D.D.

C.D.
C.D.
C.D.

D.F.
D.D.
C.D.
D.F.
C.D.
C.D.

C.D.
C.D.
C.M.

C.D.
D.F.
C.D.
B.M.
C.D.
C.D.
C.D.
D.D.
D.F.
D.F.
C.D.

35
7
44
45
44
45
38
17
17
17
17
17
52
48
56
48
48
34
8
20
48
52
37
16
51
16
4

33
33
26

13
10
15
12
23
2
21
5
52
52
20

Ti-Butanol— /t-butyl acetate

-ethylene chlorohydrin
—water

i-Butanol-ethylene chlorohydrin
—water . ....

Carbon disulfide-carbon tetrachloride
-chloroform
— cyclohexane
-ethyl ether. . ....
— isobutyl chloride

— isopentane

-nitrobenzene
Carbon tetrachloride-ethanol ... .
-ethyl acetate

—ethyl ether ....
-tetrachloroethylcne
-toluene . ...
Chlorobenzene-ethylenc bromide
Chloroform-methanol
—nitrobenzene

Cvclohexane—cyclohexane

-ethanol
223 trimethylbutanc

Diethvl benzene o-dichlorobenzene

Diethyl hexyl phthalate-diethyl hexyl seba-
cate

Dioctyl phthalate-diethyl hexyl sebacate
Ethane— ethylene ...

TfltKannl f»tlrv1 acetate

— ethyl ether .... . • • •

methanol— water

— 7^-heptane

FRACTIONAL DISTILLATION

TABLE 2-1. VAPOR-LIQUID EQUILIBRIUM DATA (Continued)

System

Constant
^orP

Tech-
nique *

Ref.

Ethylene chlorohydrin-toluene

760 mm.

C.D.

Formic acid— water

750 mm.

C.D.

Furfural— 2-methylfuran

738 mm.

C.D.

—water

760 mm.

w-Heptane-w-pentane . .

10.3, 20.6, 31.1atm.

D.B.

-toluene

760 mm.

C.D.

w-Hexane-methanol

45°C.

B.M.

Hydrochloric acid-water. . . .

751 mm.

C.D.

Isobutylene-propane ....

14.1, 21.1,28.2,

C.D.

35 2, 42.2atm.

Isopentane-propane .

0 to 180°C.

D.B.

Isopropanol— nitromethane-water

760 mm.

C.D.

—water

760 mm.

C.D.

760 mm.

C.D.

760 mm.

—

Methanol-methyl acetate .

39 8°C.

D.D.

-nitrobenzene. . . .

20°C.

D.F.

-water

59.4°C.

D.D.

60, 115, 165p.s.i.a.

C.D.

200, 350, 500, 760 mm.

C.D.

Methyl ethyl ketone-water

200, 350, 500, 760 mm.

C.D.

Nitric acid— water

760 mm.

Nitrogen-oxygen

0.4 to 45atm.

C.M.

Nitromethane-water

760 mm.

C.D.

n-Octane-toluene .

760 mm.

C.D.

Phenol— wi-cresol

760 mm.

D.D.

-o-cresol

760 mm.

D.D.

-water

40, 260, 760 mm.

D.D.

43.4, 58.4, 73.4,

C.D.

98.4°C.

n-Propanol-water

30.5°C.

D.D.

Pyridine-water . . . . ...

80.1°C.

C.D.

Sulfuric acid-water . .

760 mm.

-—

References

1. BAERE, Me VICAR, and FERGUSON, /. Phys. Chem., 34, 1310 (1930).

2. BEEBE, COULTER, LINDSAY, and BAKER, 2nd. Eng. Chem.t 34, 1501 (1942).

3. BOSNJAKOVIC, "Technische Termodynamik II," diagrams, T. Steinkopf,
Dresden, 1937.

4. BRAGG and RICHARDS, Ind. Eng. Chem., 34, 1088 (1942).

5. BREDIG and BAYER, Z. physik. Chem., 130, 15 (1927).

6. BROMILEY and QUIGGLE, Ind. Eng. Chem., 25, 1136 (1933).

7. BRUNJES and FURNAS, Ind. Eng. Chem., 27, 396 (1935).

8. CAREY, Sc.D. thesis in chemical engineering, M.I.T., 1929.

9. CXJMMINGS, STONE, and VOLANTE, Ind. Eng. Chem., 25, 728 (1933).

PRESENTATION OF VAPOR-LIQUID EQUILIBRIUM DATA 25

10. DESMAROUX, Mem. poudres, 23, 198 (1928).

11. DODGE and DUNBAR, Chem. Met. Eng., 36, 336 (1929).

12. FERGUSON, FREED, and MORRIS, Landolt-Bernstein Tabellen, eg IIIc, 2470,
and /. Phys. Chem., 7, 87 (1933).

13. FURNAS andLEiGHTON, Ind. Eng. Chem., 29, 709 (1937).

14. GILMONT and OTHMER, Ind. Eng. Chem., 36, 1061 (1944).

15. GRISWOLD and DINWIDDIE, Ind. Eng. Chem., 34, 1188 (1942).

16. HARRISON and BERG, Ind. Eng. Chem., 38, 117 (1946).

17. HIRSHBERG, Bull. soc. Mm. Beiges, 41, 163-195 (1932).

18. HOLDREN and HDCON, Ind. Eng. Chem., 38, 1061 (1946).

19. " International Critical Tables," McGraw-Hill Book Company, Inc., New
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20. LACKER and HUNT, J. Am. Chem. Soc., 63, 1752 (1941).

21. LANGDON and KEYES, Ind. Eng. Chem., 34, 938 (1942).

22. LEE, /. Phys. Chem., 36, 3569 (1931).

23. LEWIS and CAREY, Ind. Eng. Chem., 24, 882 (1932).

24. MARTIN and COLLE, J. Chem. Soc., 1932, 2658 (1932).

25. MARTIN and GEORGE, /. Chem. Soc., 1933.

26. MCDONALD and McMiLLAN, Ind. Eng. Chem., 36, 1175 (1944).

27. OTHMER, Ind. Eng. Chem., Anal. Ed., 4, 232 (1932).

28. OTHMER, Ind. Eng. Chem., 20, 743 (1928).

29. OTHMER and BENENATI, Ind. Eng. Chem., 37, 299 (1945).

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42. SCHUMACHER and HUNT, Ind. Eng. Chem., 34, 701 (1942).

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CHAPTER 3
CALCULATION OF VAPOR-LIQUID EQUILIBRIA

The calculation of vapor-liquid equilibria is important because of
the difficulty of obtaining experimental values and because it gives a
picture of the general behavior of liquid-vapor mixtures. The basic
thermodynamic relationships for such equilibria are complex, and in
most cases they involve unknown factors or quantities which limit
their usefulness until simplifying assumptions are made. It is the
limitations of the simplifying assumptions that restrict the applicabil-
ity of the thermodynamic relations. However, even for the cases in
which such approximations do not give satisfactory quantitative rela-
tionships, they do serve as valuable criteria for estimating the behavior
of a distillation process and they help to clarify and explain the
divergencies that are frequently noted.

VAPOR PRESSURES OF COMPLETELY MISCIBLE LIQUID MIXTURES
AT CONSTANT TEMPERATURE

Raoult's Law. Whe^<m& liquid is dissolved in another, the partial
pressure of each is decreased. Assume two liquids, the molecules of
which are the same size and which mix without the complicating effects
of molecular association, chemical combination, and the like. In an
equimolecular mixture of two such liquids, each unit of surface area of
the liquid mixture will have in its surface half as many molecules of
each component as exist in the liquid surface of that component in the
pure state. Hence the escaping tendency or partial pressure of each
component in the mixture will be half that of the same component in
the pure state. Similarly, in a mixture containing 25 mol per cent of
the first component and 75 mol per cent of the second, the first will
exert a partial pressure 25 per cent of that of this component in the
pure state. In more general terms, for any such mixture the partial
pressure of any component will equal the vapor pressure of that com-
ponent in the pure state times its mol fraction in the liquid mixture.
This generalization is known as Raoult's law (Ref. 19). It is expressed
in the relationship, pa = Paxa, where pa is the partial pressure of the
component A in the solution, xa is its mol fraction in the solution and

CALCULATION OF VAPOR-LIQUID EQUILIBRIA 27

Pa is the vapor pressure of the component A in its pure state, If P6 is
the vapor pressure of pure B, and pb the partial pressure in the mixture,
then pi = P&a;&. This relationship is shown graphically in Fig. 3-1,
where the abscissas are the mol per cent of the two components, A and
By in the liquid portion. The ordinates are pressures, C being the
vapor pressure of pure A, and D that of pure B. The lines AD and
BC represent the partial pressures of the components over any mixture,
while the line CD is the total pressure of the mixture.

Deviations from Raoult's Law. In view of the above assumptions
as to equal molecular size, absence of association, etc., it is not sur-
prising to find Raoult's law honored more in the breach than in the
observance. Nonetheless mixtures of some organic liquids, such as
benzene-toluene, deviate from it
but little. The deviations of mix-
tures of hydrocarbons of the same
series can usually be neglected for a
great deal of engineering work, and
even for mixtures of a number of
series this is often true. For mix- 0 Mol ent A

tures of aromatic and aliphatic com- FlG< 3_L gchematic diagram for
pounds, however, the deviations Raoult's law.
are often large, though never of the

order of magnitude of such mixtures as hydrochloric acid and water,
and the like. Organic stereoisomers obey it very closely as would be
expected from the considerations upon which it is based. However,
the data for the great majority of other liquids, when plotted as shown
in Fig. 3-1, deviate largely from the lines BC and AD. When very
near points C and D, the deviation for any component is slight if that
component is present in very large amount. This ordinarily is
expressed by saying that in dilute solution Raoult's law applies to the
solvent. Since the deviation from Raoult's law may be either positive
or negative, great or small, this graphical generalization serves as a
convenient standard of comparison.

Henry's Law. This relation is a modification of Raoult's law which
applies to the vapor pressure of the solute in dilute solutions, just as
Raoult's law applies to that of the solvent. Henry's law states that the
partial pressure of the solute is proportional to its concentration in the
solution. In analogy with Raoult's law it may be expressed by the
equation

Pa, = kxa (3-1)

FRACTIONAL DISTILLATION

where pa = partial pressure of the solute
xa — its mol fraction
k = an experimentally determined constant

Comparison with Raoult's law, pa = Paxa, shows that they differ only

in the constant that determines the slope of the line. This constant is

the vapor pressure of the pure com-
ponent in the one case, while it must
be experimentally determined in
the other. A typical partial pres-
sure curve for one component of a
liquid mixture is shown in Fig. 3-2
where BD is the range over which
Henry's law applies, while Raoult's
law holds over the section EC, where
C is the vapor pressure of pure A.

Dalton's Law. The most com-
monly used rule for relating the
composition of the vapor phase to
the pressure and temperature is
Dalton's law (Ref. 8). It states

that the total pressure is equal to the sum of the partial pressures of

the components present, i.e.,

Mol percent A.

Fia. 3-2. Schematic diagram
Henry's law.

100
for

Pi + P2 + Pa +

(3-2)

where pi, p2, pa = partial pressures of components 1, 2, and 3

IT = total pressure

For Dalton's law partial pressure is defined as the pressure that
would be exerted by a component alone at the same molal concentra-
tion that it has in the mixture. If the perfect-gas laws apply to each
of the components individually and to the mixture, it can be shown
that the partial pressure of any component is equal to the mol fraction
times the total pressure, i.e.,

pi =

(3-3)

where y\ is the mol fraction of component 1 in the vapor.

This is the most commonly used form of Dalton's law. For the pre-
diction of vapor-liquid equilibria, it is usually combined with Raoult's
law to give

2/iTT » xfi (3-4)

CALCULATION OF VAPOR-LIQUID EQUILIBRIA 29

This combination gives the composition of vapor as a function of the
composition in the liquid with the total pressure and vapor pressure as
proportionality constants. Thus, fixing the temperature and the
total pressure defines TT, PI, and the relationship between the vapor and
the liquid composition. It is to be noted that the assumption of
Raoult's and Dalton's laws leads to the conclusion that the relationship
between the vapor and liquid composition of a given component is a
function of the temperature and pressure only and is independent of
the other components present. The only influence of the other com-
ponents is the fact that they may be instrumental in determining the
relationship between the temperature and the total pressure.

Deviations from Dalton's Law. Both forms of Dalton's law are
satisfactory for engineering uses for most gas mixtures at pressures of
1 atm. or lower because deviations from the perfect-gas law are usually
small in this region. However, when higher pressures are encountered
and appreciable deviations from the perfect-gas law are found, Dal-
ton's law becomes unsatisfactory. A later section of this chapter will
consider methods of handling these deviations, but as a general rule
Dalton's law should not be employed for cases in which the deviations
from the perfect-gas law are large.

• Volatility. The term " volatility" is loosely used in the literature,
generally as equivalent to vapor pressure when applied to a pure sub-
stance; as applied to mixtures, its significance is very indefinite.
Because of the convenience of the term, the volatility of any substance
in a homogeneous liquid will be^defined as its partial pressure in the
vapor in^ equilibrium with tHat liquid, divided by its mol fraction in the
liquid nr the substance is in the pure state, its mol fraction is unity
and its volatility is identical with its vapor pressure. If the substance
exists in a liquid mixture that follows Raoult's law, its volatility as
thus defined is still obviously equal to its vapor pressure in the pure
state; i.e., its volatility is normal. If the partial pressure of the sub-
stance is lower than that corresponding to Raoult's law, e.g., that of
hydrochloric acid in dilute aqueous solutions, the volatility according
to this definition is less than that of the pure substance, i.e., is abnor-
mally low. Similarly, if the partial pressure is greater than that
indicated by Raoult's law, e.g., that of aniline dissolved in water,
the volatility is abnormally high. The volatility of a substance
in mixtures is therefore not necessarily constant even at con-
stant temperature but depends on the character and amount of the
components.

30 FRACTIONAL DISTILLATION

Relative volatility is the volatility of one component divided by that
of another. Since the volatility of the first component of a mixture,
0a, is its partial pressure, pa divided by its mol fraction xa, and that of
the second /?& = pb/x*, the volatility of the first relative to the second
is fta/ffb = paVb/pbXa. When Dalton's law applies, the relative amount
of any two components in the vapor (expressed in mols) is ya/yb =* Pa/p**

& = »-•£ = a (3-5)

& yb xa ^ J

Owing to the utility of the mol fraction ratio given in Eq. (3-5) for
distillation calculations, the group will be used as the definition of the
relative volatility for all cases whether or not Dalton's law applies.
Thus, in this text the relative volatilities of any two components in a
mixture will be defined as the ratio of y/x values for the two components.

In any constant-boiling homogeneous liquid mixture the composi-
tion of the liquid is identical with that of the vapor in equilibrium with
it, i.e., x = y] hence, the relative volatility a is unity.

Volatility, like vapor pressure, increases rapidly with rise in tem-
perature. The ratio of the pressures of pure substances does not
change rapidly with change in temperature, and the same is true of
relative volatilities; whereas vapor pressures always increase with
temperature, relative volatility may, in a given case, either rise or fall,
depending on the nature of the components. At constant temperature
the relative volatility is independent of the liquid composition for
systems that obey Raoult's law; however, for most systems a is a func-
tion of the liquid composition and frequently is greater than unity for
one range of concentrations and less than unity for another range.
Relative volatility is the most important factor in determining ease of
separation of components by distillation.

Raoidt-Dalton Laws; Example. The data of Rhodes, Wells, and Murray
(Ref . 20) indicate that the system phenol-o-cresol obeys Raoult's law. Using the
vapor pressure data tabulated at the top of page 31 together with Raoult's and
Dalton's laws, construct the following curves for 75 mm. Hg abs. pressure:

1. Temperature-mol fraction diagram giving both the vapor and the liquid
curves.

2. Mol fraction ,of phenol in vapor phase, y, vs. the mol fraction of phenol in
liquid, x.

3. Relative Volatility vs. the mol fraction of phenol in the liquid.

4. Mol fraction of phenol in the vapor vs. the mol fraction of phenol in the liquid,
using as an average value of the relative volatility, a, the arithmetic mean value
of a at x »» 0, and x « 1.0.

CALCULATION OF VAPOR-LIQUID EQUILIBRIA
DATA ON VAPOR PRESSTJRES

Tt °C.

Phenol, pres-
sure, mm.

o-Cresol, pres-
sure, mm.

113.7

75.0

57 8

114.6

78.0

59,6

115.4

81.0

61 6

116 3

84.0

63 8

117.0

87.0

65.7

117.8

90.0

68 0

118 6

93.0

70.5

1919.4

96.0

73.0

120 0

98 6

75.0

Solution

1. Let mol fraction of phenol in liquid •» xp.

2. Choose a temperature, T.

3. Read vapor pressures at T:PP, Poc.

4. Employ the combined form of Raoult's and Dalton's laws.

XpPp "f XocPoe « 75

Pp - Poc

TABLE 3-1

Tt °C.

Poo

yp (calc. on

a - 1.313)

113.7

57.8

1.0

1.295

1.0

114.6

59.6

0.837

0.87

1 31

0.87

115.4

61.6

0.691

0.746

1.315

0.748

116.3

63.8

0.555

0.622

1.315

0.622

117.0

65.7

0.437

0.507

1.32

0.506

117.8

0.318

0.383

1.323

0.382

118.6

70.5

0.2

0 248

1.32

0.247

119.4

73.0

96.7

0.084

0.109

1.325

0.108

120.0

99.8

1.33

ypxoc

• ___-_--~
' 1 4- (« - l)xp

FRACTIONAL DISTILLATION

These values are plotted in Fig. 3-3. An a of 1.313 was chosen to calculate
equilibrium curve. Within the accuracy of the calculations, this curve was coin-
cident with the curve obtained in Part 2.

0.2 0.4 06 08 1.0

Mol fraction phenol in liquid, %p

FIG. 3-3. Calculated vapor-liquid equilibria for system phenol — o-cresol.

Basic Thermodynamic Relations. Most mixtures are not ideal,
and the deviations from Raoult's law are large in all but a limited num-
ber of cases. For such mixtures, the basic thermodynamic relations
can be employed to formulate expressions for the equilibria involved.
It is not the purpose of this text to consider in detail the thermody-
namics of solutions, but an appreciation of this subject is necessary for
an understanding of the following sections on vapor-liquid equilibria.
For background in this field the reader is referred to any modern text
on chemical thermodynamics.

In the study of vapor-liquid equilibria, one of the most important
of the basic relations states that at equilibrium the fugacity of a given

CALCULATION OF VAPOR-LIQUID EQUILIBRIA 33

component is the same in all phases. Thus,

fy = /* (3-6)

where / is the f ugacity , F refers to the vapor phase, and L refers to the
liquid phase. This same equation applies to each component in the
mixture at equilibrium. To be useful for distillation calculations,
the fugacity terms in Eq. (3-6) must be related to the compositions of
the phases. For isothermal changes in conditions at constant composi-
tion, the change in fugacity is evaluated by the following equation :

RT d In / = v dp (3-7)

where R = gas law constant

T = absolute temperature
/ = fugacity

v = molal volume, i.e., volume of 1 mol of substance under
consideration = partial molal volume for a component in a
mixture
p = pressure

For mixtures that obey the perfect-gas law this relationship makes
the fugacity of a component in the vapor proportional to the partial
pressure of the component, i.e., the mol fraction of the component
times the total pressure. It is customary to define the standard state
for fugacity such that it is numerically equal to the pressure for a per-
fect gas; i.e., the proportionality factor is made equal to unity.

The fugacity of a component in an ideal liquid is defined as propor-
tional to the mol fraction of that component in the liquid times the
fugacity of the pure component under the same temperature and pres-
sure as the mixture. Thus for a perfect gas and an ideal solution,

fv = yi* = JL = foxi (3-8)

where y\ = mol fraction of component 1 in vapor
xi = mol fraction of component 1 in liquid
TT = total pressure of mixture
/£ = fugacity exerted by the pure liquid at temperature T under

total pressure, w

This equation is not identical with Raoult's law, and /£ is not in
general equal to PI, the vapor pressure of pure liquid 1 at temperature
r, because the liquid is under a pressure different from Pi. This
difference can be evaluated by Eq. (3-7).

(3'9)

34 FRACTIONAL DISTILLATION

where v = partial molal volume of liquid 1

PI « vapor pressure of pure component 1 at temperature T
R = gas-law constant

fpi = fugacity of component 1 at temperature T and pressure PI
Over the pressure range usually involved, v is essentially constant and

f*i = /pi«'(r-pl)/*r (3-10)

Equations (3-8) and (3-10) differ from Raoult's law by the expo-
nential term, and fpi instead of PI. These corrections are frequently
small and can be neglected. However, in high-pressure equilibria the
corrections become large, and even an ideal solution would not be
expected to obey Raoult's law because (1) the .components in the liquid
mixture at a given temperature are under a different total pressure
than they would be as pure components and (2) the fugacity of the pure
liquid is not equal to its vapor pressure. An equation similar to (3-8)
applies to each of the components in an ideal mixture.

Most mixtures do not obey Raoult's law or the corrected Raoult's
law given by Eqs. (3-8) and (3-10). The deviations from the ideal
solution laws can be due to the vapor phase, the liquid phase, or both.
These deviations are both chemical and physical in nature. The most
important factors involved in these deviations are believed to be (1)
the fact that the molecules have volume and (2) the fact that the mole-
cules exert forces on each other that may be attractions, or repulsions,
or actual chemical effects.

The problem for such mixtures becomes one of relating the fugacities
of the components in the vapor and the liquid to the composition and
the physical properties of the components, and it has been found desir-
able to consider separately the deviations in the liquid and the vapor
phase.

VAPOR PHASE

Pure gases or gaseous mixtures approach agreement with the perfect-
gas law if the pressures are low enough. l Under these conditions, the
volume of the gaseous mixture is so large that the volume of the mole-,
cules is a negligible percentage of the total, and the molecules on the

1 The reduction to zero pressure may cause a change in species. Thus, if the
pressure on pure CU gas is reduced, the gas may never obey the perfect-gas law
since, before exact agreement is reached, the CU will dissociate to atomic chlorine.
The atomic chlorine should agree with the perfect-gas law at zero pressure, and for
all practical engineering purposes diatomic chlorine gas agrees with the gas laws at a
pressure less than one-tenth of an atmosphere, and under these conditions, the
dissociation to atomic chlorine is negligible at moderate temperatures.

CALCULATION OF VAPOR-LIQUID EQVlLt&RtA 35

average are so far apart that the forces between them are small. As
the pressure increases, the effect of the volume of, and the attraction
between, the molecules become so great that deviations from the ideal
solution laws become large and it is necessary to employ the fugacity
instead of the pressure in vapor-liquid relationships. The isothermal
change in fugacity of a mixture of given composition can be calculated
by integrating Eq. (3-7). In this integration the absolute value of the
fugacity at any given state can be arbitrarily chosen. Defining the
fugacity equal to the pressure for a perfect gas, Eq. (3-7) can be
rearranged as follows:

RTdlnf=vdp (3-11)

RTdlnt^ (va-v^dp (3-12)

where va = actual molal volume

vt = perfect-gas law molal volume

Fugacity of Pure Gases. In order to utilize Eq. (3-12) it is necessary
to have information on the actual molal volume as a function of the
pressure at the temperature in question. The lack of these data limits
the utility of this equation. However, it has been found possible to
develop correction factors to the perfect-gas law that will apply to
almost all gases and gaseous mixtures. One method of representing
these deviation factors is to plot the compressibility factor ju, which is
equal to PV/RT, as a function of the reduced pressure at constant
reduced temperature, where reduced pressure PR is the pressure divided
by the critical pressure, and reduced temperature TR is the absolute
temperature divided by the absolute critical temperature. Such a
plot is given in Fig. 3-4; similar plots for higher temperatures and
pressures are available.

These plots give good agreement with the experimental data for
most vapors with the exception of hydrogen and helium, both of which
have low critical temperatures and pressures. It has been found possi-
ble to use the plot for these two gases by using modified critical con-
stants. However, neither gas is particularly important in vapor-
liquid equilibria.

Using the p, factor correction, the actual molal volume va becomes
When this is substituted in Eq, (3-12), it gives

(3-13)
or

rfln£-(f.-l)& (3-14)

P P

FRACTIONAL DISTILLATION

If this equation is integrated from zero pressure up to the pressure in
question, the limits on the left-hand side are from 1.0 to the ratio of
fugacity to pressure at the pressure in question. Thus In (f/p) equals
the integral from zero to p of (M — 1) dp /p.

Equation (3-14) can be modified and dPR/PR used instead of dp/p.
The limits of integration then become from zero to PR. At constant
temperature, it is therefore obvious that the integral is a function only
of PR, T/z, and the variables determining ju. If the M values are a func-

P-V-T RELATIONS FOR VAPORS
BELOW THE CRITICAL
PR= P/ Pc= Reduced Pressure -
TR =T/VReduced Temperature
R= 6ois Constant

065 TR=07TR=015TR*08TR-085 TR-0.9 TR

02 0.3 0.4 0.5 06 07

FIG. 3-4.

tion of PR and TR only, the ratio of fugacity to the pressure is also a
unique function of the reduced pressure and the reduced temperature.
Integration using the ju plots have been made, and one method of
presentation is given in Fig. (3-5).

With an accuracy suitable for engineering purposes these plots make
it possible to calculate the fugacity of a pure gas at any temperature
and pressure, assuming that the critical constants of the gas are known.
Even in cases where the critical constants are not known, highly satis-
factory methods have been developed for estimating these constants.

Fugacity of Mixtures. When applying the n plot and the fugacity
plot to mixtures, the question arises as to the proper values to be
employed for the critical temperature and the critical pressure. Mix-
tures have critical temperatures and critical pressures, but it has been
found that these values do not give satisfactory results when used for
calculating reduced temperatures and pressures to be used with the
charts, but pseudocritical constants can be calculated which give better
agreement. For these pseudo constants one of the best methods of

CALCULATION OF VAPOR-LIQUID EQUILIBRIA

calculation is the mol fraction average; i.e., the calculated pseudo-
critical temperature is equal to the sum of the products of the mol frac-
tion times the critical temperature for each of the pure components.
The pseudocritical pressure is calculated in an analogous manner.
When these values are employed for calculating the reduced tempera-

0.2 0.4 06 08 10

1.2 14 16 1.8 2.0
Pfl * Reduced pressure

FIG. 3-5 Fugacities of hydrocarbon vapors.

26 20 50

ture and pressure, satisfactory agreement is attained with /* charts.
It should be emphasized that these calculated values are not true
critical constants for the mixture.

Fugacity of a Component in a Mixture. In vapor-liquid equilibria,
vapor mixtures are generally involved, and Eq. (3-6) requires, not the
fugacity of the mixture, but the fugacity of the components in the
mixture. In order to estimate the fugacity of the components in the
mixture, Lewis and Randall (Ref. 16) suggested that this fugacity
was equal to the mol fraction of the component in the mixture times
the fugacity of the pure component at the same temperature and total
pressure as the mixture. Thus,

fvi « yi/n (3-15)

where fvi = fugacity of component in vapor

y = mol fraction in vapor

ffl = fugacity of pure component at same temperature and
pressure as mixture

38 FRACTIONAL DISTILLATION

The value of f* can be estimated from the fugacity plots, and this
fugacity rule has been widely used in vapor-liquid and other equilib-
rium calculations. It can be shown that this is true if the vapor mix-
ture obeys Amagat's law, which states that the volume of a mixture is
equal to the sum of the volumes of the pure components when these
are measured at the same temperature and total pressure as the mix-
ture. Experimental data have shown this rule to be a reasonable
approximation if the pressures are not too high; at high pressures, large
deviations are found to occur. However, in essentially all cases it is
better than Dalton's law.

A large number of additional methods of predicting the fugacity of a
component in a gaseous mixture have been proposed. In general these
have been based on different rules for the PVT relations of mixtures
used to evaluate the fugacity. None of them has been satisfactory in
all cases although some of them are better than the Lewis and Randall
rule, but they are more difficult to employ, and a number of them
require more experimental information than is usually available. This
complexity has greatly limited their use.

The Lewis and Randall fugacity rule can generally be used with
reasonable accuracy for vapor mixtures up to pressures approximately
one-half of the critical pressure; i.e., 300 to 500 p.s.i.g. in the case of
hydrocarbons. This is only an approximate limit, but large errors will
not generally be encountered at values of PR, based on the pseudo-
critical pressure of the mixture, less than one-half. If the components
in the mixture are of similar types, i.e., 02 and Ns, ethylene and ethane,
benzene and toluene, etc., satisfactory results can often be obtained for
values of PR as high as 0.9. The rule is not satisfactory at the critical
condition. Figure 3-5 gives values of the fugacity for conditions where
a pure component cannot exist as a vapor. These values were obtained
by using the Lewis and Randall rule with actual vapor-liquid equilibria
to calculate the fugacity values.

Instead of using the reduced correlations, it is thermodynamically
possible to calculate the r true fugacity of a component in a gaseous
mixture, if sufficient PVT data are available. The methods of cal-
culation are very laborious and require PVT data of high pre-
cision. This method has not had any significant engineering use
owing to the great difficulty in obtaining the extensive PVT data
necessary and the work involved in making the calculations from
such data. Its chief value is furnishing an exact basis to evaluate the
empirical rules.

CALCULATION OF VAPOR-LIQUID EQUILIBRIA 39

LIQUID PHASE

Deviations from ideal solution laws are more important for the
liquid phase than those for the vapor phase because they are encoun-
tered even at low pressures, and in general their magnitudes are
greater. The densities of the liquids are such that the volume of the
molecules and the forces between them are always significant. Devi-
ations for the liquid phase are of at least two main types: (1) Those due
to the fact that the vapor does not obey the perfect-gas law. Thus, if
one tries to define the fugacity of a component in the liquid phase as
mol fraction times the vapor pressure of the pure component, this can
be satisfactory only when the vapor at a pressure equal to the vapor
pressure is essentially a perfect gas. (2) The deviations that are due
to special phenomena associated with the liquid phase such as associa-
tions or chemical combinations.

Gas Law Deviation. In the ideal solution the partial pressure, or
activity, of a component was equal to the mol fraction times the vapor
pressure of the pure component at the temperature in question. If
the pressures are such that the vapor under these conditions does not
obey the perfect-gas law, then a fugacity correction should be applied.
For such a case, the Lewis and Randall rule would be

/LI = *i/*i (3-16)

where fLi = fugacity of component 1 in liquid phase

Xi = mol fraction of component 1 in liquid phase
/*! = fugacity of pure liquid component 1 at temperature and

pressure of mixture

In general the total pressure is different from the vapor pressure of
the pure components at the same temperature, and /*t is not equal to
the fugacity of the pure liquid under its own vapor pressure, fpi.
They are related:

or, assuming v = constant,

& - fPie't*-™'RT (3-17)

where ir = total pressure

v = partial molal volume of component
Pi = vapor pressure of component at temperature of mixture
T « absolute temperature
R = gas law constant

40 FRACTIONAL DISTILLATION

From equilibrium consideration, the fugacity of the pure liquid
under its own vapor pressure, fpi, is equal to the fugacity of the satu-
rated vapor at the same temperature and pressure. Thus, Fig. 3-5
can be used to evaluate fp, but it should be emphasized that the reduced
pressure is calculated at the vapor pressure of the pure component
instead of at the total pressure. This application of the fugacity to
the liquid phase corrects for the fact that the vapor is not a perfect gas,
but it does not correct for the special phenomena associated with the
liquid phase.

Combining Eqs. (3-15), (3-16), and (3-17) gives

yiAi = xi/pie**-™** (3-18)

In using Eq. (3-18) certain difficulties are encountered. Considering
the vapor phase, the temperature and pressure of the mixture usually
are intermediate between those of the pure components, and it is often
found that the temperature and pressure are such that one of the pure
components at these conditions would be superheated vapor and the
other a supersaturated vapor. The superheated vapor offers no diffi-
culty since it is easily possible to obtain PVT data for superheated
vapors, and this type of information was used to develop the /* plots.
However, it is essentially impossible to obtain PVT data on super-
saturated vapors since they tend to condense easily. In the liquid
phase such difficulties are not encountered at moderate temperatures.
If the temperature is not too high, the vapor pressure of the compo-
nents can be obtained and the fugacity can be determined. However,
at higher temperatures, it is possible for the temperature of the mix-
ture to be greater than the critical temperature of one of the com-
ponents and still have this component present in the liquid phase in
large amounts. The calculation of fp for such conditions is complicated
because data are not available on the vapor pressure of a liquid above
the critical temperature.

Empirical rules have been developed to handle these fugacity diffi-
culties for the vapor and liquid. In the case of the liquid phase, it has
been customary to plot the logarithm of the vapor pressure vs. the
reciprocal of the absolute temperature. It is well known that such
plots are remarkably straight, and for fugacity calculations the straight
line has been extrapolated past the critical to higher temperatures.
Such extrapolations have given satisfactory results. In the case of
the vapor, the problem is more complex, and the main solution has
been to calculate /* by Eq. (3-18), using the experimental data and the
estimated value of /p. This has been done for a number of mixtures,

CALCULATION OF VAPOR-LIQUID EQUILIBRIA
TABLE 3-2. K VALUES

Temp.,
°F.

Absolute pressures, atm.

CIU

-90

14'

10 5

8 5

7 1

5 4

4 3

2 3

1 7

1 3

1 2

-20

10 6

8 6

4 4

3 0

2 3

2 0

115

14 5

11 5

6 0

4 1

3 1

2 6

100

190

9 7

6 6

5 0

i.l

200

300

150

100

15 5

10 3

7.8

6.2

400

580

290

195

145

116

19.5

14.5

12.0

-60

7.7

3 9

2 6

2 0

1 6

1 4

1.1

0 9

0 51

0 42

0 38

0 3(

24 5

12 3

8 3

6 3

5 1

4 2

3 2

2 6

1 44

1 08

0 92

0 &

100

25 5

17 2

13 0

10 2

8 8

6 5

5 3

2 8

2.0

1 6

1 31

200

103

34 5

20 5

17 2

13 0

10 4

5 3

3 7

2 8

2 31

300

172

17 5

8 9

6 0

4 7

3 9

400

265

132

9 2

7 0

5 6

C2Hft

*-60

4 8

2 4

1.6

1 3

1 0

0 88

0 69

0 58

0 36

0 29

0 26

0.20

16 0

8 0

6 4

4 1

3 3

2 8

2 1

1 75

1 0

0 79

0 69

0 66

100

11 7

8 8

7 1

5 9

4 5

3 7

2 0

1 45

1 18

1.0

200

14 5

12 1

9 2

7 4

3 8

2 7

2.1

1 75

300

123

25 5

6 6

4 5

3 4

2 8

400

195

6 9

5 2

4 3

-60
20

'0 79
4 5

0.35
2 3

0 27
1 55

0 21
1 2

0 17
0.95

0 15
0 82

0 12
0 64

0 10
0 54

0 07
0 34

0 06
0 28

0 05
0 26

100

12 2

6 3

4 3

3 3

2 G

2 2

1 7

1 4

0 82

0 06

0 59

0.68

200

7 7

6 2

5 2

4 0

3 2

1 75

1 3

1 1

0 98

300

7 7

6 2

3 2

2 3

1 8

1 5

400

100

12 5

10 0

5 2

3.6

2 8

2.3

-60
20
100
200
300
400

0 67
3 7
11
25
51
85

0 34
1 9
5 5
13
25
43

0 23
1 3
3 7
8 6
17
29

0 18
0 98
2 8
6 5
13
22

0 15
0 8
2 3
5 2
10
17

0 13
0 69
1.9
4 4
8.7
14

0 10
0.54
1 5
3.4
6.5
11

0 09
0 45
1.2
2 8
5 3
8 8

0 06
0 29
0 75
1 5
2 8
4 5

0.05
0 24
0,6
1 1
2.0
3.1

0 05
0 23
0 54
0 95

1 6
2 4

0 06
0 24
0 54
0.85
1 3
2.0

t-CJIic

-60
20
100
200
400
600

0 18
1 2
4 5
13
51
125

0 09
0 63
2 3
6 7
25
63

0 07
0 43
1 6
4 5
17
42

0.05
0 34
1 2
3 5
13
32

0 04
0 28
1.0
2 8
10 5
25.5

0 04
0 24
0.86
2.4
8 8
21 5

0 03
0 19
0 67
1 8
6.6
16

0 03
0 16
0 57
1 5
5 3
13

0 02
0 11
0 36
0 91
2 9
6.6

0.02
0.10
0,31
0 74
2 0
4.5

0 02
0 09
0 30
0 68
1 6
3.4

0 02
0 11
0 32
0 67
1 4
2 9

FRACTIONAL DISTILLATION
TABLE 3-2. K VALUES (Continued)

Temp,,

Absolute pressures, atm.

-60

0 10

0 05

0 03

0 02

0 01

0 81

0 43

0 29

0 23

0 19

0 16

0 13

0 11

0 07

0 08

100

3 4

1 7

1 2

0 9

0 73

0.63

0 5

0 43

0 28

0 24

0 26

200

10 5

5.4

3 7

2 8

2 3

1.9

1.5

1 3

0 77

0 64

0 61

400

8 8

7.3

5 5

4.5

2 4

1 7

1 4

1 3

600

5.6

3 9

3 0

2 5

-20

0 09

0.05

0 03

0 02

0 01

0 63

0.32

0 22

0.17

0 14

0 13

0.10

0 08

0 06

0.06

0 07

100

1 3

0 67

0 47

0 36

0 30

0 26

0 21

0 18

0 12

0 11

0 12

0 14

200

5.2

2 7

1 8

1 4

1 2

1 0

0 77

0 65

0 43

0 37

0.37

0 40

400

25.5

8.8

6 7

5 4

4 5

3 4

2 8

1 6

1 2

1 0

0 94

600

8 7

7.0

3.7

2 6

2 1

1 72

-20

0 05

0 03

0 02

0.01

0 01

0.01

0 01

0.47

0 24

0 17

0 13

0,11

0 09

0 07

0 06

0 05

0 04

0 05

100

1.0

0.53

0 36

0 28

0 23

0 20

0 16

0 14

0 10

0 09

0 10

200

4.3

2.2

1 5

1 2

0 96

0 83

0 65

0 55

0 36

0 32

0 35

400

22 5

7.8

5 9

4 8

4 0

3 1

2 5

1 4

1.1

0 93

0 89

600

13 4

11 2

8 5

6.9

3 6

2 6

2 0

1 7

n-Cellu

0 13

0 07

0 05

0 04

0 03

0 02

0 01

0 02

100

0.35

0.18

0 13

0 10

0 08

0 07

0 06

0 05

0 04

0 05

200

1.9

0.97

0 67

0 51

0 43

0 37

0 30

0 26

0 18

0 17

0 18

0 22

400

13 6

7 0

4 7

3.6

2 9

2 5

1 9

1 6

0 98

0 80

0 77

600

8.6

7 2

5.5

4.4

2.4

1.8

1 5

1 3

100

0 11

0 05

0 04

0.03

0 03

0.02

0 02

0.02

0.01

0 01

0 02

0.02

200

0.83

0.43

0.29

0 23

0 20

0 17

•0.14

0.12

0 09

0,09

0.10

0 14

400

8.4

4 3

2 9

2 2

1 8

1.6

1 2

1 0

0 67

0.60

0 67

600

7.6

6.1

5.1

3.9

3.2

1.75

1.3

1.1

1.05

n-CsHis

100

0.04

0 02

0 01

200

0 39

0 20

0.14

0 11

0.09

0 08

0.07

0,06

0.04

0.05

0,06

0.09

400

5,3

2.6

1.8

1.4

1.15

1.0

0.78

0 67

0 47

0 45

0 49

0.57

600

19.5

6.7

5.1

4.2

3.5

2.7

2.25

1 3

1.1

0.95

0.91

CALCULATION OF VAPOR-LIQUID EQUILIBRIA 43

and these calculated values for the supersaturated region have been
plotted on the fugacity plot and used as a basis for extrapolating the
curves based on PVT relations. In most cases, the extrapolation is
not large, and the deviations should not be too great.

When equilibrium calculations are to be made a number of times for
the same components, it is convenient to simplify the equation and
express it the form of ^i = Kx\ where

K ~ fjp (3-19)

K values based on fugacity calculations have been prepared for the
common lower molecular weight petroleum hydrocarbons and are
presented in Table 3-2. On the whole, they are equivalent to results
obtained with Eq. (3-18) and the fugacity plots, but they are simpler
to use in vapor-liquid calculations. If such values have not been cal-
culated for the components in question, it is necessary to employ the
fugacity plot.

Equation (3-18) includes the exponential correction term, and this
term is appreciable at high pressure. However, it has been found that
in hydrocarbon systems up to about GOO p.s.i. the agreement is about
the same whether or not this factor is included. This is probably
because the values of ff in Fig. 3-5 for the supersaturated region were
evaluated from experimental data without the use of the exponential
term. If an exact method of computing the fugacity of the com-
ponents were available, the exponential correction should be applied.
However, in using values from this figure it is suggested that it be neg-
lected for distillation calculations.

Equation (3-18) has been found to give good agreement with the
experimental data on mixtures of similar compounds up to pressures
equivalent to a reduced pressure of about 0.5. It has been extremely
useful in high-pressure calculations associated with petroleum mix-
tures. The vapor-liquid equilibrium data for mixtures of oxygen and
nitrogen are well correlated by the equation. Basically, the equation
should be useful for any mixtures that agree reasonably well with
Raoult's law at low pressure. At pressures greater than PR « 0.5, the
agreement is not so good, but it is still a useful approximation. How-
ever, this method of calculation breaks down completely in the
neighborhood of the critical region of the mixtures. This condition
will be considered in a later section.

The fugacity equation can be applied to each component in the mix-

FRACTIONAL DISTILLATION

ture. It is suitable for multicomponent mixtures as well as for binary
mixtures.

Example Illustrating the Use of Fugacity Corrections. Scheeline (Ref. 23) has
studied the vapor-liquid equilibrium of propane-isobutylene mixtures at high
pressures. Using the data and notes given below, make a y,x plot for a total
pressure of 400 p.s.i.a. showing

1. Experimental data.

2. Curve calculated by Raoult's law.

3. Curve calculated by using fugacity corrections.

CRITICAL CONSTANTS

PC, p.s.i.a.

Tc, °F,

632

209

580

291

1. Experimental data:

Absolute
pressure,

Temp.,

xct

yc9

p.s.i.a.

400

242

0 086

0 140

400

222

0 286

0 413

400

206

0.438

0.572

400

190

0.648

0.754

400

184

0 734

0.827

400

175

0.847

0.895

Solution. Calculate the y,x values at 400 p.s.i.a. and 200°F.

2. Raoult's and Dalton's laws:

ZsPa 4- £4P4 = 400

400j-^P4

At 200°F. from Table 3-3 P3 - 569 p.s.i.a., P4 - 232 p.s.i.a., x* « 0.498, and
yz ** 0.498 (569/400) - 0.709.

3. Fugacity corrections :

2/s/7r3 •• Xzfpt x\ — \ — Xz

2/4/7T4 = ^4/p4 2/4 — 1—2/3

Combining and solving,

/7T4 — /P4

. 0.88;
« 0.986;

=» 0.403; Km ** 40%so «* 0.69
- 0.90; - *•*» - 40%32 - 0.633

CALCULATION OF VAPOR-LIQUID EQUILIBRIA
From Fig. 3-5,

(fs\ = 0.665 (&\ = 0.755

VP/3 V /S

(&} -0.78 (^ -0.6

VP/4 \7T /4

.64

/P4

0.665 X 569 - 379
0.78 X 232 - 181

181

- 0.755 X 400 - 302
= 0.64 X 400 - 256

0.53

2/3 = —

302
0.53(379)

302

- 181
- 0.665

In a similar manner, the data in Tables 3-3 and 3-4 were calculated.

TABLE 3-3. RAOULT'S LAW
(400 p.s.i.a.)

T, °F.

P4, p.s.i.a.

PS, p.s.i.a.

2/3

255

399

237

305

730

0.224

0.409

220

287

688

0.282

0.485

200

232

569

0.498

0 709

180

187.5

470

0.753

0.885

163

150

400

1.0

TABLE 3-4 FUGACITY CALCULATION

T, °R.

697

680

660

640

623

TRS

1.041

1.017

0.986

0.957

0.931

730

688

569

470

400

PR3

1.16

1.09

0.90

0.744

0.633

(A/p)3

0 64

0.65

0.665

0.68

0.71

466

447

379

319

284

(U M 3

0.8

0.78

0 755

0.74

0.71

/,3

320

312

302

296

284

TRI

0.929

0.906

0.88

0.853

0 830

305

287

232

187.5

150

P*4

0.526

0.495

0.40

0.324

0.258

(fp/P)<

0.76

0.77

0.78

0.79

0.82

/P4

232

221

181

148

123

(ArA)4

0.68

0.655

0.64

0.605

0.59

/r4

272

262

256

242

236

a;3

0.244

0.266

0 53

0.832

1.0

2/3

0.356

0.381

0.665

0.896

1.0

FRACTIONAL DISTILLATION

The experimental data and the calculated results are shown in Fig. 3-6. It will
be noted that the fugacity calculations for the ytx values are in excellent agreement
with the experimental results, but that Raoult's and Dalton's laws give values of
the vapor composition that are much too high. For average relative volatility
these laws give 2.45; fugacity, 1.77; and experimental, 1.71.

Solution Deviations. The corrections so far considered have been
limited to those associated with the fact that the vapor does not obey
the perfect-gas law. A large number of mixtures, in fact most of
them, do not obey the ideal solution laws even at very low pressure,
and the deviations cannot be predicted by the use of gas-phase fugacity
corrections. Thejleyj^c^jt^ejbhe result of the forces between the
molecules in the liquid phase, and these forces can Ee^veryTargely^ue

IJU

0.6
0.6
0.4
02

/ /

—— Experimental
Raoult's and
Dalton's law
X Fugacity

3 0.2 0.4

06 0.8 1.

Mol fraction propane in liquid,
Fia. 3-6. Vapor-liquid curves for system propane-isobutylene.

to the close packing in the dense phase. The theoretical method of
attack for the liquid phase is not so simple as for the vapor phase.
For the vapor-phase calculations a convenient basis was possible
because at low pressure all vapor mixtures obey the perfect-gas laws.
Thus the deviations could be calculated on the basis of the differences
between the mixture at low pressure and at high pressure. In the case
of the liquid phase no such convenient basis is possible. Thus a mix-
ture of ethyl alcohol and water does not agree with the ideal solution
rules under any practical conditions of temperature and pressure.

CALCULATION OF VAPOR-LIQUID EQUILIBRIA 47

Basic thermodynamic relations are available for the liquid phase,
but their practical application has not been so well developed as those
for the vapor phase. They are helpful in formulating general con-
cepts and are directly applicable in certain special cases. One of the
most useful relations follows:

+.-.=0(3-20)

For a binary mixture dxi — ~rf#2, Eq. (3-20) reduces to

(3-21)

If the pressure is such that the vapor satisfies the perfect-gas law, then
the equation can be modified as follows:

This equation is called the Duhem equation (Ref. 9). Equation
(3-20) is applicable to any system of any number of components, but
the Duhem equation is limited to a binary mixture under conditions
such that the perfect-gas law applies to the vapor.

Theoretically these equations apply only to a process carried out at
constant temperature and constant total pressure on the liquid phase. In
most mixtures encountered in distillation, if one varies the composition
at constant temperature, the total pressure also varies and Eqs. (3-20)
to (3-22) are not strictly applicable. The equations would apply for
this constant-temperature case if some method other than the vapor
pressure were employed to exert pressure on the liquid which was
adjusted to keep the total pressure constant; e.g., a gas insoluble in the
liquid could be added to the vapor to maintain constant total pressure.
Actually, these equations apply satisfactorily for most engineering
purposes if they are employed at constant temperature and a variable
total pressure equal to the vapor pressure. The error introduced is
that due to the change in the fugacity of the liquid with the total
pressure which can be calculated by Eq. (3-7). A more exact rela-
tionship for binary mixtures at constant temperature is

rainwpnl Faincp^g)]

L 0X1 Jr L ^2 Jr t

where d In pf is the fractional change in effective pressure of component

48 FRACTIONAL DISTILLATION

1 due to the change in total pressure on the liquid. It is calculated by
Eq. (3-7).

d ln Pi - JLL (3-24)

rfT " RT (6 M)

where TT = total pressure on liquid phase

v\ = partial molal volume of component 1

R = gas law constant

T = absolute temperature

The value of d In p$ is calculated in the same manner with v2 instead of
v\. In most cases these corrections at constant temperature are small
in comparison to the change in fugacity due to the change in composi-
tion; for this reason Eq. (3-22) is frequently accepted as applying to
mixtures under their own vapor pressure. It can be in serious error at
high pressure, and Eq. (3-23) would be more exact.

For constant pressure, variable temperature conditions, Eq. (3-22)
generally is unsatisfactory owing to the rapid change in fugacity or
vapor pressure of a liquid with changes in temperature. Equation
(3-22) can be modified to compensate for this effect as follows:

where d In P( is the fractional change in the partial pressure of com-
ponent 1 at a composition xi, with the change in temperature. It is
evaluated by

' ' (3.26)

^ }

dT RT*

where A£T' is the heat of vaporizing 1 mol of the component from the
solution into a vacuum.

Equations (3-23) and (3-25) can be combined and expressed in the
more exact fugacity form,

where d In /° is the fractional change of the fugacity of the component
in the liquid phase at the given composition and is calculated by Eq.
(3-26) for constant pressure changes and by Eq. (3-24) for constant
temperature changes.

The Duhem equation as such cannot be integrated, but if a relation-
ship between the pressure of one of the components and the mol frac-

CALCULATION OF VAPOR-LIQUID EQUILIBRIA 49

tion is available, it is possible to calculate the relationship between the
partial pressure of the other component and the mol fraction. This is
not of any real engineering utility for predicting vapor-liquid equilibria
since in general when the partial pressure of one component is known
that of the other component is also known. However, the Duhem
equation is useful in checking experimental data and also in guiding
the development of correlations. For example, Eqs. (3-22) to (3-27)
can be used to evaluate the accuracy of vapor-liquid equilibrium data.
Consider the case of a binary mixture at constant temperature for
which the liquid- vapor data are available. By Eq. (3-23),

j 1 Pi ^2 i * PZ

PI ~~ i - x2 n pf

p^ i M r ** ^ P«

^* ) ~ ln I -^ i = — I ^ a In —;:

J>*/6 VPl/a Ja l-a;2 p?

(3-28)

If one experimental value of Pi/p* is taken as a base, then the value
of this ratio can be determined at other compositions by the integration
of the right-hand side of the equation. This integration must usu-
ally be performed graphically, and a convenient method is plotting
#2/(l — #2) vs. In (pz/p*) and determining the area under the curve.
The experimental data for p2 as a function of x2 and one value of p\
allow the value of p\ to be predicted at other compositions. If correc-
tions for total pressure on p* and p* are to be included, they can be
evaluated from the experimental total pressure data or by summing
the calculated value of pi and p2; in the latter case, the integration
becomes trial and error.

Constant pressure conditions are more important in distillation cal-
culations than constant temperature, and for this condition Eq. (3-25)
can be modified to

The evaluation of the Pf terms involves the heat of vaporizing 1 mol
of the component from the mixture. At moderate pressure this is
equal to the latent heat of vaporization at the same temperature plus
the heat effects of mixing the liquids at this temperature and bringing
them to the total pressure. As an approximation the latent heat of
vaporization of the pure liquids can be employed in Eq. (3-26), but it
will not give satisfactory results if (1) the heat of mixing is large or (2)

50 FRACTIONAL DISTILLATION

the total pressure is so high that the pressure-enthalpy corrections for
the vapor are large.

One of the cases in which the Duhem equation is of real utility in
determining vapor-liquid equilibria is where the analysis of the com-
position of the two phases offers serious difficulties. In such a case, if
it is possible to prepare known mixtures of the two components and to
determine their equilibrium total pressure at a given temperature,
these data can be used with the Duhem equation to calculate the com-
position of the equilibrium vapor.

Activity Coefficient. Because there are no convenient conditions for
the liquid mixture upon which to base calculations, it is customary to
use the pure liquids before they are mixed as the basis, and then calcu-
late the deviations that result from the mixing operation. The devia-
tions in the liquid phase are summed up in what is termed an activity
coefficient. • Thus Raoult's law and the idealized fugacity law are
modified by the insertion of a factor on the right-hand side.

pi = yi* = yfixi (3-30)

/i = yi/n = yifp&i (3-31)

The value of the activity coefficient, 7, is the factor that will make
these equations correct for the case in question. All deviations other
than those associated with the gas law are lumped into the one value,
and the problem is thus made into one of predicting the activity coeffi-
cient. These factors will be different for each component, but they
are interrelated by Eq. (3-20) and, for a binary mixture, Eqs. (3-21),
(3-22), (3-30), and (3-31) become

(3-32)
V>T v

The methods that have been used for predicting the activity coeffi-
cient are either empirical or semitheoretical, the theoretical part being
the use of thermodynamic equations to direct the development of
empirical rules. A number of rules have been proposed (Refs. 3, 5, 13,
15, 28), but the two most commonly used methods of estimating
the activity coefficients for solutions of the type employed in distilla-
tion are the Margules and Van Laar equations.

Example Illustrating the Use of Duhem Equation. Data on the vapor-liquid
equilibria of benzene-rc-propanol and ethanol- water are given in the accompanying
tables. Using the Duhem equation, check the consistency of the data.

CALCULATION OF VAPOR-LIQUID EQUILIBRIA

EQUILIBRIUM DATA FOR THE SYSTEM BENZENE-n-PROPANOL
[From Lee (Ref. 14). Temp. - 40°C.]

Mol fraction in liquid

Partial pressure, mm. Hg

Benzene

n-Propanol

Benzene

w-Propanol

" 1.0

50.2

0.099

0.901

59.6

42.4

0.209

0.791

95.0

39.6

0.291

0 709

118.6

37.4

0.360

0.640

132.2

36,2

0.416

0.584

139.1

35.9

0.508

0.492

149.2

34.3

0.700

0.300

161.6

31.4

0.820

0.180

167 4

28.6

0 961

0.039

175 6

15.7

1 0

183 5

VAPOR PRESSURE DATA FOR SYSTEM KTHANOL- WATER
(Ref. 12)

Vapor pro HSU re, mm.

J. , 1^.

Ethanol

Water

693

301.4

750

327.3

812

355.1

877

384 9

950

416.8

1,026

450.9

1,102

487.1

1,187

525.76

1,280

566.99

1,373

610.9

1,473

657.6

1,581

707.3

100

1,693

760.0

FRACTIONAL DISTILLATION

ETHANOL-WATER EQUILIBRIUM DATA AT NORMAL BAROMETRIC PRESSURE

(Ref. 4)

T, °C.

Mol fraction of ethanol

Liquid x

Vapor y

95.7

0.0190

0.1700

90.0

0.0600

0.3560

86.4

0.1000

0 4400

84.3

0.1600

0.5040

83.3

0.2000

0 5285

82.3

0.2600

0.5570

81.8

0.3000

0 5725

81.2

0.3600

0.5965

80.7

0.4000

0 6125

80.2

0.4600

0.6365

79.8

0.5000

0.6520

79.4

0.5600

0.6775

79.13

0.6000

0.6965

—

0.6600

0.7290

78.6

0.7000

0.7525

—

0.7600

0.7905

78.3

0.8000

0 8175

—

0.8600

0.8640

78.17

0 8943

0.8943

Solution for Benzene-n-Propanol System at Constant Temperature. Using Eq.
(3-28) given on page 49,

The right-hand side of this equation requires graphical integration. At the low
pressures involved the variation in the values of p* and p* will be neglected. In
order to reduce the variation in the groups involved, the equation was rearranged to

Pi*

The values utilized for preparing the graphical integration plot are given in the
first five columns of Table 3-5. As a basis for calculations, propanol was taken as
component 1 and benzene as component 2. As the check point, the value for the
partial pressure of propanol of 28.6 mm. Hg at a mol fraction of propanol of 0.180
was chosen.

CALCULATION OF VAPOR-LIQUID EQUILIBRIA
TABLE 3-5.

1 -*,

io-V2

104s2

In Pl

Plcalo

plexp

2/2calo

2/2 oxp

100 Ay

(1 ~ xz)pl

28.6

y -x

1.0

50.2

0.099

0.901

59.6

0 356

0,309

0.554

49.8

42.4

0.545

0 585

9.2

0.209

0.791

95.0

0 903^

0 293

0.472

45.8

39.6

0.675

0.706

6.2

0.291

0.709

118.6

1.406

0.292

0.398

42.6

37.4

0.736

0.760

5.1

0.360

0.640

132.2

1.747

0.322

0.346

40.4

36.2

0.767

0.783

3.8

0.461

0 584

139.1

1.933

0.408

0.312

39.1

35.9

0.781

0.795

4 2

0.508

0.492

149.2

2.225

0 464

0.249

36.7

34.3

0.803

0.814

3.6

0 700

0.300

161.6

2.612

0.894

0.118

32.2

31.4

0.833

0.838

3.6

0 820

0.180

167 4

2 800

1.625

0.0

28 6

28.6

0.855

0 855

0.0

0 961

0.039

175.6

3 080

8.0

-0.673

14.6

15.7

0 923

0.917

11.4

1.0

183.5

3 367

—

1 0

—

The sixth column of the table gives the area on each side of the check point, and
these are of course equal to the left-hand side of the equation with the base value,
pib, equal to 28.6. The seventh column lists the calculated values for the partial
pressure of propanol, and the eighth column gives the experimental values. The
agreement is not so good as would be desired but is a fair correlation. When the
partial pressure values are converted to mol fractions in the vapor, the agreement
appears somewhat better. The ninth and tenth columns of the table give calcu-
lated and experimental values for mol fractions of benzene in the vapor phase.
The last column of the table gives the difference in these vapor compositions as a
percentage of the difference between the experimental vapor and liquid composi-
tions. Most of the calculated points are within 5 per cent of the experimental
points on this basis, and this is probably reasonably good accuracy for vapor-
liquid equilibrium measurement.

Solution for Ethanol-W ater at Constant Pressure. For this case, it is convenient
to use the mol fraction form of Eq. (3-29) instead of partial pressures.

1 - ;

"p;

/b
, r^r/In

For convenience in plotting, this equation was rearranged to

1/t

The integration of this equation requires a knowledge of the variations of P' with
temperature. Equation (3-26) indicates that the variation of the pressure with
the temperature could be calculated if the heat of vaporization of 1 mol of the
component from the solution into a vacuum were known. This heat of vaporiza-

FRACTIONAL DISTILLATION

tion differs from the true heat of vaporization of the liquid by two main effects :
(1) enthalpy effect due to mixing of the liquid phase and (2) enthalpy effects result-
ing from nonideal behavior of the vapor phase. In the present case the pressure is
low enough that the vapor-phase effects should be very small, and for illustration
purposes the enthalpy of the mixing in the liquid phase will be neglected. On the
basis of such an assumption d In P' becomes equal to d In P, where P is the vapor
pressure of the pure component at the temperature in question. With this modifi-
cation and the data given in the table, it is possible to carry out the integration.
The results of such calculations are given in Table 3-6. The first five columns of

TABLE 3-6. ETHANOL- WATER

X'l

2/2

«*(£)'

10-8^2

In Vl/Pl

103 Vi

2/2 oalc

100 Ay

[(1 ~ 32)(WP*)P

m 0.0003 18

10 Pi

V - x

0 019

0 170

1450

0 138

14 0

.428

1.325

647

0 856

0 144

17.2

0 060

0 356

1180

0 926

6 9

387

1 272

527

0 670

0 330

- 8.8

0 100

0 440

1035

1 808

6.1

.359

1 237

459

0 568

0 432

2 4

0 2

0.528

916

3 325

7 5

308

1.177

404

0 475

0 525

0 9

0.3

0.572

865

4 40

9.7

262

1 122

380

0.426

0.574

0.7

0.4

0.612

827

5 48

12.2

1 203

1 058

363

0.384

0 616

1.9

0.5

0 652

802

6 60

15 2

1.129

0.984

351

0 345

0 655

2.0

0.6

0 696

782

7.90

19 0

1 019

0.880

343

0.304

0 694

2 1

0.7

0 752

769

9 57

24 4

0 839

0 736

337

0.248

0 752

0 0

0.8

0.817

760

11.54

34.7

0.555

0 554

333

0 184

0 816

5 9

0.8943

0 8943

757

13.96

60 6

0.318

332

0.1057

0 8943

this table give the values of the groups used for preparing the graphical integration
plot. Ethanol was taken as component 2, and the check point for the system was
taken at the azeotrope. The sixth column gives the measured areas from the
azeotrope and is therefore equal to the left-hand side of the equation. The seventh
column gives calculated mol fraction in the vapor divided by vapor pressure for
the water component. The ninth column lists the calculated mol fractions of
water vapor. The tenth column gives the calculated values of alcohol in the
vapor and is obtained by subtracting the ninth column from 1. This column is
directly comparable to the second column and indicates that, except for the first
two points, the agreement is excellent. These first two points are at low composi-
tion and may be somewhat in error. The last column in the table gives the
percentage deviations.

Margules Equation. Margules (Ref. 17) developed an expression
for the activity coefficient of the components in a binary mixture by
taking empirical expressions for these coefficients as follows:

In 71 =
In 72 -

+ bx% + cx\
i + Vx\ + c'x\

When the Margules equations for the activity coefficients are sub-
stituted in Eq. (3-32), the following relations are obtained:

CALCULATION OF VAPOR-LIQUID EQUILIBRIA 55

Xl — ~ as — *Xl — -= = — (O^l + 2bXiXz 4

#2 — — — = ~#2 — - — * = — (a'#2 ~|- 2b'xiXz + Sc'xzxl)

By Eq. (3-32) the right-hand side of the two preceding equations
must be equal for all compositions, and this relation will be satis-
fied when the coefficients of corresponding terms are equal ; thus, using
xl = 1 — x<t and equating the components of the corresponding terms
gives the following solution :

a = 0
c' = -c
a' = 0

26 + 3c

0 - ~ 2
In 71
In 72

It will be noted that the equations have two independent constants.
However, only one experimental point is needed to evaluate both of
these, since one determination of the vapor-liquid equilibrium for a
binary mixture gives both 71 and 72, and these values can be used to
evaluate the constant. If more data are available, it is convenient to
plot the logarithm of the activity coefficient divided by the mol fraction
squared for the other component vs. the mol fraction of one of the
components. If the Margules equation agrees with the data, both of
the activity coefficients should give straight lines which are parallel
when plotted in this manner, and the slope should be equal to the value
of the constant c. The intercepts will be different but can be used for
evaluating the constant b. The two activity coefficient equations can
be arranged to plot as a single line.

=* U -p 1^2

(3-34)

^__Lf « 5 + C(Q.5 + x,

Thus if (In 71) /x\ is plotted vs. #2, and (In 72)/£? is plotted against
0.5 + xZj they will both give a line of slope c and intercept b. Such a
plot facilitates the evaluation of the constants.

The constants in the Margules equation are a function of tempera-
ture; thus, if one experimental point is used to evaluate them, then

56 FRACTIONAL DISTILLATION

assuming the equation to apply, they should be suitable for other com-
positions at the same temperature. A generalized relationship for the
effect of temperature on the activity coefficient is not available, but as
an approximation it is suggested that the constants be taken as
inversely proportional to the one-fourth power of the absolute tem-
perature, Thus,

T°-25 In

T2
#2

In 72

Van Laar Equation. Van Laar (Ref . 25) attempted to follow a more
theoretical approach than did Margules. He based his derivation on
the thermodynamic changes occurring on the mixing of pure liquids.
Two of the basic thermodynamic equations relating to isothermal

processes are

AF = A# - T AS (3-35)

d&F = RTdlnf (3-36)

where AF = partial molal change in free energy
&B = partial molal change in enthalpy

T = absolute temperature
A$ = partial molal change in entropy

/ = fugacity

For an ideal mixture, there is no change in volume when the mixing
is carried out at constant temperature and under a constant total
pressure, and there are no heat effects. Thus A# is equal to zero, and
the partial molal change in free energy is equal to — T A/S for each com-
ponent. For such an ideal solution, the partial molal entropy of mix-
ing is — R In x, and the fugacity is proportional to the mol fraction
(Lewis and Randall rule). However, in most actual solutions there
are changes of volume on mixing, heat effects, and the entropy of mix-
ing differs from that for an ideal solution.

The combination of Eqs. (3-30) and (3-31) with Eq. (3-36) gives

RT d In 7 = d AFa - d AF» = d AFe (3-37)

and

AFe = A#e - T A& (3-38)

where A^a = actual partial molal free energy

AFi = partial molal free energy of ideal solution of same com-
position = RT In x

Ape = excess partial molal free energy = AFa — APi
A5C = excess partial molal enthalpy = AJ?«
A& « excess partial molal entropy = AS« — R In x

CALCULATION OF VAPOR-LIQUID EQUILIBRIA 57

Thus in calculating the activity coefficient, 7, the value of the excess
partial molal free energy is desired. If methods were available for
calculating the partial molal entropy and enthalpy effects upon mixing,
the deviations from the ideal solution law could oe determined. As a
first approximation for systems containing molecules not too dissimi-
lar, several writers (Refs. 11, 22, 25) have recommended the assump-
tion of a solution in which only the internal energy change on mixing is
different from an ideal solution. In such solutions, which Hildebrand
terms "regular," the partial molal volume of each component remains
constant, and the change in entropy is equal to that of an ideal solu-
tion; i.e., complete " randomness" exists. Thus the only deviation of
a " regular " solution from an ideal solution is due to the fact that there
is an excess partial molal internal energy of mixing. In addition to
these assumptions, Van Laar attempted to calculate the internal
energy of mixing by the use of the van der Waals equation of state.
The assumptions used by Van Laar in deriving his equations are

1. A£ = Q',i.e., ASa = A£.

2. No volume change on mixing.

3. The van der Waals equation applies to each of the components
and to the mixture, both as liquids and as vapors.

4. The van der Waals constants of the mixture can be calculated
from the constants of the pure constituents.

For a van der Waals fluid, i.e., one that satisfies the relationship

„ ET a ., , , .,.
P = _ , — •=£, it can be shown that

(*l\ _±

\dV/T ~ V*

where E = internal energy

V = volume

a = van der Waals constant

T = absolute temperature

When this relationship is integrated from a vapor at zero pressure to the
liquid state,

(3"39)

Van Laar substituted the volume constant, 6, of the van der Waals
equation, for the molal volume of the liquid and used the values devel-
oped by other investigators (Refs. 2, 10) for evaluating the constants
of a mixture.

Otnix = (xi Voi + £2 Vosi + x* Vos + ' ' ' )2 1 (3-40)

femix = Xibi + Xj)2 + Xzbz + ' ' • J

58 FRACTIONAL DISTILLATION

where o»t*, &«** « van der Waals constants of mixture
ai,«2,a«, bifbzjb9 = van der Waals constants of pure components
For a binary mixture the internal energy of mixing the pure liquid
components at constant temperature per mol of mixture is

A.EL = ELM — X\E^ -- X%ELZ

where AEL = internal energy change in mixing per mol of mixture
ELM = molal internal energy of liquid mixture
ELI = molal internal energy of pure component 1 as a liquid

before mixing
EL» = molal internal energy of pure component 2 as a liquid

before mixing
By Eq. (3-39) and using 6 = F,

T H ( Emi* oo — X\Ei^ — #2/?2oe I

6mix 6l

The last term is the internal energy of mixing of the vapors at zero
pressure and is, therefore, equal to zero. The values of Eq. (3-40) give

From Eq. (3-41) the partial molal change in internal energy is

,» ^

(3-42)

_ . l6l +

and on the basis of Van Laar's assumptions :

lnTl « ^l'L "* ^ bl

combining constants,

where A * 61/62

CALCULATION OF VAPOR-LIQUID EQUILIBRIA 59

By combining Eq. (3-44) with Eq. (3-32),

' ln^=- AB/T

Scatchard and Hildebrand obtained similar expressions without the
use of a van der Waals fluid. They recommend evaluating the expres-
sion for (A£I)L as

where w = internal energy of vaporization

V = molal volume

This leads to equations of the same form as Eqs. (3-44) and (3-45),
except that

It has been pointed out by Cooper (Ref . 7) that the same relation-
ship can be obtained more simply than the method employed by Van
Laar. What is desired in calculating the activity coefficient is the
difference between the partial molal free energy of mixing of an actual
solution and that of an ideal solution. If the excess free energy of
mixing per unit volume of mixture is assumed proportional to the
product of the volume fractions of the two components, an expression
identical in form to the Van Laar equation is obtained. Thus,

(3.47)

where AFe = excess free-energy change of mixing = actual free-energy
change minus ideal free-energy change

m = mols of component 1

n2 = mols of component 2

Vi = molal volume of component 1

F2 = molal volume of component 2
But

(MB

60 FRACTIONAL DISTILLATION

and

In -n = S 7-^ x-2 • (3-49)

In 71 = y v2 (3-50)

where J5 = K'Vi/R
A « 7i/Fi

These equations are identical with those given by Van Laar, but the
assumptions are somewhat different. In the case of the Van Laar,
Scatchard, and Hildebrand derivations, both A and B should be posi-
tive, while in Cooper's equation B could be either positive or negative.

On the basis of the derivations, the two constants of the Van Laar
equations and the modifications of it are related to the physical proper-
ties of the pure components. When the best values of the constants
are chosen to fit the data, they usually do not agree with the predicted
values, although the trends are approximately the same. Generally
the constants are chosen to agree with the data, and the equations are
used empirically.

As was the case with the Margules equation, two constants are
involved. For a binary mixture one vapor-liquid equilibrium point
will give the activity coefficients of both components and thus define the
whole equation. The form of the equations are significantly different
from that of the Margules. It includes a temperature correction, and
the value of the constants should be independent of temperature.
Thus an experimental determination at one temperature should allow
equilibrium data to be calculated at other temperatures. There are
several ways in which Van Laar equations can be rearranged to plot
as a straight line in order that the data for more than one determination
can be easily correlated. One of the most convenient methods of
making such a plot is to use the reciprocal of the square root of the
temperature times the logarithm of the activity of component 1,
l/(r In 7i)H, vs. the ratio, x\/x*. A similar plot can be made for the
other component. The plots should both be straight lines; the slopes
and intercepts will be different but related because they are based on
the same constants.

Clark Equation. For the vapor-liquid equilibria of a binary mixture
at either constant pressure or constant temperature, Clark (Ref . 6) has
suggested that the ratio of the mol fractions in one phase is a linear

CALCULATION OF VAPOR-LIQUID EQUILIBRIA 61

function of the ratio of the mol fractions in the other phase, when the
ratios are utilized such that the component in largest amount appears
in the numerator.

Thus, when component 1 is present in largest amount,

8i-0£! + 6 (3-51)

2/2 o:2 v '

and, when component 2 is present in largest amount,

»!«a'£! + 6' (3-52)

y\ xi

Clark uses the value xi/xz = ^a'b/aV as the change-over point
between the two equations. The use of these equations requires three
experimental points. This greatly limits its utility.

Evaluation of Empirical Equations. Tucker (Ref. 24) and Mason
(Ref. 18) have studied the agreement of the various equations with
published experimental data. The data available were screened, and
only those that gave good agreement with the Duhem-type equations
were selected for the evaluation. The experimental data were plotted
to evaluate the equation constants, and the average constants so
obtained were used to recalculate the y,x curve. In distillation calcu-
lations the difference between the vapor and the liquid compositions
gives a better indication of the ease of separation than the absolute
value of the vapor composition. Mason therefore made the compari-
son on the basis of

Per cent deviation = \y^° ~ ^exp| 100
L (V - z)e*pj

As a qualitative standard, he classified average deviation of 0 to 5
per cent as good, 5 to 11 per cent as fair, and greater than 11 per cent
as poor. Some of the results are given in Table 3-7. The table gives
the average deviation and the maximum deviation. In certain cases,
the agreement between the experimental and the calculated values is
very good but very poor in other cases. It is difficult to determine any
definite types of mixtures that give good or poor agreement. However,
in all cases, mixtures approaching immiscibility, i.e., y>x curves that
are nearly horizontal over an appreciable concentration region, gave
poor agreement with the Margules and Van Laar equations. Good
agreement was obtained with all the maximum boiling mixtures
studied. These latter mixtures give negative values of the Van Laar

FRACTIONAL DISTILLATION
TABUS 3-7

Margulea

Scatchard

Van Laar

Clark

System

Max.

Av.

Corre-

Max.

Av.

Corre-

Max.

Av.

Corre-

Max.

Av.

Corre-

%
dev.

lation

%
dev.

lation

%
dev.

dev.

lation

dev.

%
dev.

lation

Ethanol-water

14 6

4.5

Good

5 8

3 1

Good

5 7

4 4

Good

10 6

2.4

Good

Methanol -water .

6 9

4 9

Good

6 2

5 1

Good

14 7

5 8

Good

2 3

1.0

Good

Kthylenebromide- 1-

23 8

6 8

Good

Vi ** V*,

Good

8 2

2 9

Good

nitropropano

see Mar-

gules

Carbon disulfide -ben-

zene.

13.9

5 3

Good

6 7

3 0

Good

11 1

4 3

Good

Benzene— aniline

1.5

0.5

Good

Vi -

V,.

Good

2.0

0.5

Good

see Mar-

gules

Carbondisulfido- nitro-

benzene*

2 5

1.8

Good

1 3

0 7

Good

7 5

2 7

Good

Chloroform— nitroben-

zene* ...

1 4

0 5

Good

0 8

0 2

Good

6 8

2 2

Good

Ethyl ether-nitroben-

0.7

0.3

Good

Vi ->i

Good

11 8

4 2

Good

zene*

see Mar-

gules

Methanol-nitroben-

fsene* . . .

4.1

1.7

Good

19 9

5 9

Good

16 2

5 5

Good

Carbon tetrachloride~

benzene

6.6

3.6

Good

6 6

3 6

Good

15 0

6 2

Fair

18 2

8 5

Fair

Acetone-chloroform . .

12 5

4 3

Good

17.6

4 8

Good

52 9

13 5

Fair

55.2

13 5

Fair

Ethyl ether-acetone. .

21.7

6.0

Good

12 1

4 7

Good

21 0

8.5

Fair

Benzene~phenol . . , , .

0.8

0 5

Good

Vi -

V»

Good

50 0

16 5

Poor

see Mar-

gules

Benzene-nitroben-

1.8

0.7

Good

1.8

0.8

Good

No corre-

Poor

zene*

lation

CS*— isobutylene- chlo-

ride

14 7

7 0

Fair

16 4

6 8

Fair

14 2

6 7

Fair

Isopentane~CS$

23.3

9 0

Fair

26 8

9.1

Fair

10 9

6 5

Fair

CCli-ethyl acetate. . .

40.0

11.8

Fair

40 0

11.1

Fair

54 5

32 8

Poor

40 0

10.1

Fair

n- Heptane-toluene . . .

53 5

9.6

Fair

53 4

9 8

Fair

15 9

8 9

Fair

8 7

4 2

Good

Benzene-n-propanol . .

53.3

8.6

Fair

68 4

11 4

Poor

18.8

17 8

Poor

16 3

4 7

Good

Methanol-benzene . . .

23 0

8 5

Fair

65 2

18 4

Poor

76 8

28 8

Poor

Water— pyridine . .

54.4

21 . 1

Poor

32 0

12 1

Poor

47 7

18.4

Poor

3 9

1.5

Good

GSj-chlorof orm . . . .

23 2

11.2

Poor

14 7

13.9

Poor

23 5

16 3

Poor

12 3

4.5

Good

Ethyl ether-CSa

65.0

32.1

Poor

50.0

21.3

Poor

40 0

15.5

Poor

CSt— acetone

29.1

16.0

Poor

33 7

18 8

Poor

94.4

59.2

Poor

CSr-cyolohexane. . . .

51 0

24.4

Poor

26.6

14.6

Poor

73 3

74.9

Poor

Cyoiohexane-ethanol .

48.2

11.8

Poor

50.0

12.8

Poor

29 1

11 0

Fair

18.8

3.8

Good

The per cent deviations are based on [(yealo — y«p)/(v —
These systems marked * are baaed on [(po*io — p«xp)/Pe*pl 100.

CALCULATION OF VAPOR-LIQUID EQUILIBRIA 63

constant, B, which is not consistent with the derivation, but the use of
the negative value gave satisfactory predictions. It was also found
that the use of one point for the estimation of the constants of the
Van Laar equation was not very satisfactory. In some cases the
single-point method worked well and in other cases very poorly, and
it depends upon the accuracy of the particular point in question.
However, if only one point is available, such as is frequently the case
when the azeotrope only is known, the Van Laar and Margules equa-
tions probably offer the best method of estimating the vapor-liquid
equilibria for other conditions. When data on the azeotropic condi-
tion are not available, they can be estimated from Figs. 3 to 5 of the
Appendix.

The Van Laar equation is also very useful for transferring data
obtained at constant temperature to constant pressure and the
reverse. It is also useful for transferring data from one temperature
to another. The unsatisfactory agreement in the cases of solutions
approaching immiscibility is not surprising. In these cases, the
entropy of mixing cannot be equal to that for an ideal solution, and
for complete immiscibility the entropy of mixing would be zero. It
is therefore not surprising that solutions approaching immiscibility
deviate from the Van Laar equation.

Other Applications of the Van Laar Equation. The Van Laar equa-
tion can be used to indicate qualitatively the type of phenomena
encountered in liquid mixtures. Thus, it can be arranged as follows:

This equation gives the logarithm of the ratio of the vapor pressures
divided by the relative volatility as a function of the Van Laar con-
stants and the concentrations. If the solution were ideal, the loga-
rithm term would be zero. Thus the real fact determining deviation
from ideal solution is the constant B. If B is equal to zero, the relative
volatility will be equal to the ratio of vapor pressures, and the y,x
values will be the same as those calculated by Raoult's and Dalton's
laws. If B is not equal to zero, the system is not ideal. The terms
involving A and the concentrations would have about the same varia-
tion independent of the value of B.

It is interesting to consider this equation for various limits. For
example, for Xi equal to zero, the logarithm term equals — B/T. For

64 FRACTIONAL DISTILLATION

most mixtures encountered, B is positive. (B is negative for solutions
having maximum boiling azeotropes and for certain other solutions.)
Thus the logarithm is negative, and the relative volatility is greater
than the ratio of the vapor pressures. In these cases, it is easier to
remove the component in low concentration than would be expected
from the ideal solution law. At the other extreme, i.e., #2 equals zero,
then Xi over x2 is infinity, and the logarithm becomes B/AT, and with
B positive (in all cases so far encountered A is positive) the relative
volatility is less than the ideal relative volatility. These conditions
are found in most common mixtures; i.e., the relative volatility at low
concentration is greater than that of an ideal solution and the relative
volatility at high concentration is lower. It is often expressed by
saying that the components in small amount are squeezed out. Thus,
at the low concentration of the volatile component, it is squeezed out
and the relative volatility is high. At high concentration of the
volatile component, the nonvolatile component is squeezed out, and
the relative volatility is low. For the mixture in which B is negative,
the reverse phenomena are true.

The Van Laar equation also would state that a mixture would agree
with Raoult's law, independent of the value of By when the ratio of the
mol fractions equals l/-\/~A. For most mixtures, the value of A is
somewhere between 0.5 and 2, which would indicate that the vapor-
liquid equilibria at some concentration in the middle range would
agree with Raoult's law. Thus, the Van Laar equation would imply
that the assumption of Raoult's law for the region around a mol
fraction of 0.5 would be more satisfactory than at the two ends of the
curve. It should be emphasized that this relation does not state that
Raoult's law is valid at this condition. It indicates that the relative
volatility is equal to the ideal relative volatility at this concentration,
but the temperature-total pressure relationships may be far from those
indicated by Raoult's law. Thus, in the system ethyl alcohol and
water, the total pressure is always higher than would be indicated by
Raoult's law, but the vapor-liquid equilibrium curve crosses the
Raoult's law curve. Below this intersection, ethyl alcohol is more
volatile than would be indicated by Raoult's law; above this concen-
tration, it is less volatile"than would be indicated by Raoult's law.

The Van Laar equation would indicate that the mixtures would
become more ideal as the temperature increased. In other words, the
ratio of B/T becomes smaller and nearer to the value for an ideal
solution.

The Van Laar equation gives interesting relationships for the condi-

CALCULATION OF VAPOR-LIQUID EQUILIBRIA 65

tion under which the relative volatility becomes unity, i.e., the forma-
tion of an azeotrope. For this condition, the equations can be
arranged as follows:

t = 5

and for azeotropic conditions:

A - !n <' (3-54)

In (T/P

COMBINED GAS LAW AND SOLUTION DEVIATION

In the preceding discussion the deviations of the vapor-liquid
equilibria due to gas law and solution abnormalities have been treated
separately; however, such deviations frequently occur simultaneously.
In such cases, the factors are combined into a single equation as
follows :

yf, = 7/pS (3-56)

In many cases, by combining the Van Laar and Margules equations
with the fugacity corrections, it is possible to correlate vapor-liquid
equilibrium data up to high pressure.

At very high pressures, it is probable that the density of the vapor
phase may be such that deviations of the type handled by the activity
coefficient equations also occur in this phase. Thus,

yvvf* = 7L/p3 (3-57)

The deviations encountered in yv would probably be of a lower degree
than those for the liquid phase, but equations of the type of the Van
Laar should be applicable. Two experimental points should be
sufficient to calculate the constants involved for the activity coeffi-
cients of the liquid and vapor phase.

Van Laar-Margules Equations; Example. The table of data below gives the
vapor-liquid equilibria for the system acetone-chloroform obtained by Rosanoff
and Easley (21), for a total pressure of 760 mm. Hg.

1. Using the Van Laar equation, calculate the y,x curve,

a. Employing the azeotrope data only to evaluate the constants.
6. Using all of Rosanoff' s points to obtain best average constants.

2. Repeat la and 16 using the Margules equation.

66 FRACTIONAL DISTILLATION

VAPOR-LIQUID EQUILIBRIUM VAPOR PRESSURE DATA

Temp., °C.

2/i

Pi, mm. Hg

P2, mm. Hg

57 45

0.9145

0.9522

780

675

58.34

0 8590

0.9165

805

695

59.44

0.7955

0.8688

835

745

60.42

0 7388

0.8235

865

725

61.60

0.6633

0.7505

900

775

62.84

0.5750

0.6480

935

805

63.91

0.4771

0.5170

970

835

64.6

0 3350

0.3350

995

855

64.36

0.2660

0.2370

990

850

63.84

0.2108

0.1760

965

830

63.08

0.1375

0.100

950

810

62 77

0 1108

0.0650

930

795

xi "• mol fraction of acetone in liquid

y\ «» mol fraction of acetone in vapor

Pi *** vapor pressure of acetone

P» » vapor pressure of chloroform

Solution of Part la. Employing azeotrope data only,
Constant boiling temperature « 64.6°C. or 337.6°K

At azeo tropic point, x

yir = yPx or
y, therefore, y ~ ir/P.

•

- 76%i>5 - 0.764; In Ti - -0.269

- 76%55 - 0.889; In ^2 - -0.118

, B/T

In yi -

In y* — y

xt - 0.335 #2 - 0.665
Solving the Van Laar equations, one obtains

A - 1.74 and B - -320

The following is the procedure used to calculate y at various values of x:
Let xi « 0.796 and xa — 0.204,

5i * 3.90

Assume T - 332°K.

CALCULATION OF VAPOR-LIQUID EQUILIBRIA

In 7i = M rn/o r^\ r~T75 ** —0.0159

In 71

[1.74(3.90) + lp
•• 0.998
1

In 72

In 72 - 1.74(3.90)2( -0.0159) - -0.421
72 - 0.685

2/2

+2/2

: 0-998(835) (0.796) _ Q g?1

0.685(725) (0.204)

, _ . 0.133

• 1.004

The fact that the sum of the mol fractions is greater than 1.0 indicates the assumed
temperature is too high, but the relative volatility is nearly constant with small
temperature changes, and

"•-S-"-""

0.133 _
2/2 " 1T004 -°-WJ

Other results are tabulated as follows:

2/calo

2/exp

100 A?/

(y - z)exp

0.9145

0.955

0 9522

7 9

0 7955

0 868

0.8688

1 1

0 6633

0.758

0.7505

8 6

0.4771

0.521

0.5170

10 0

0.3350

0 3350

0 0

0.2108

0 175

0.1760

2 9

0.1108

0 070

0.0650

10 9

Solution of Part Ib

In 71

B/T

(Tin-

Btt

Also,

i AB/T

h, 72 . -^^^^

(a)

(6)

FRACTIONAL DISTILLATION

By plotting
evaluated.

(T In

vs. ~i the constants A and B can be

TABLE 3-8

T, °K.

(-rin-yi)*

0.9145

0.0855

330

1.015

0.8590

0.1410

331

1 007

—

0,7955

0.2045

332

0 991

0.709

3.89

0.7388

0 2612

333

0.980

0.376

2.83

0.6633

0.3367

335

0.956

0.258

1 965

0.5750

0.4250

336

0.916

0.185

1 35

0.4771

0.5229

337

0.849

0.135

0.913

0 3350

0.6650

338

0.762

0.105

0 504

0.2660

0.7340

337

0 684

0.088

0 363

0.2108

0.7892

337

0.657

0.084

0 268

0.1375

0.8625

336

0.582

0.074

0.1596

0.1108

0.8892

336

0.479

0 063

0.1246

In this case the mixture is of the maximum boiling point type, and the activity
coefficients are less than unity, resulting in negative values for In 7. This negative
value is handled in the square roots by multiplying Eq. (a) by l/\/ — 1. The
intercept is then l/\/—jB, and the slope is A/^/ — B- A similar procedure is
employed for In 72. The first two values of 71, given in Table 3-8 are greater than

0.4
03
02
0.1
0

«^

0 ' v

. |i

^^x

f-^

" •TuT?, '

31234567
*i „ *2

FIG. 3-7. Van Laar plot for system acetone-chloroform.

CALCULATION OF VAPOR-LIQUID EQUILIBRIA

unity, while the values of 72 at the same point are less than unity, and it <;an be
shown that this is not consistent with the Duhem equation. These two values of
71 are so near to unity that a very small error would account for the discrepancy,
and the values were not used in the calculations. In all cases the values of the
activity coefficients near unity tend to be of little value for calculating the constants
of either the Van Laar or Margules equations because In 7 is small and subject to
large errors. Unfortunately, the method of plotting used in Fig. 3-7 emphasizes
these inaccurate points and gives less weight to the better values. The plotting
for the Margules equation is better in this respect and tends to weigh all the values
about equally.

Intercept of plot = 0.056
Slope of plot - 0.097 =

Thus, B = -319 and A - 1.73.
given in Table 3-9.

Similar values for the other component are

TABLE 3-9

£2

(-7Tln7*)H

0.9145

0 0478

0.610

0.078

0 093

0 8590

0 0835

0 643

0.083

0 164

0 7955

0 1312

0 675

0.087

0 294

0.7388

0 1765

0.690

0 090

0 354

0.6633

0 2495

0 724

0.096

0 410

0 5750

0 3520

0.782

0.098

0.740

0.4771

0.4830

0.841

0.131

0.095

0.3350

0.6650

0.884

0.155

1.987

0.2660

0.7630

0.925

0.195

2 76

0 2108

0.8240

0 957

0.257

3 62

0.1375

0 900

0 964

0.294

6 27

0 1108

0.9350

0.985

0 445

8 02

From Fig. 3-7,

Intercept is 0.074 = ( ~ g)
Slope is 0.0434 = _AB%

Thus B - -310 and A - 1.72. Average A = 1.725 and average B m -315.
The values of A and B obtained here are, within the accuracy of the method, the
same as those obtained from the azeotrope data alone, and the y,x calculations will
not be repeated.

70 FRACTIONAL DISTILLATION

Solution of Part 2a

NOTE. Some values used below are taken from Part 1.

In 71 « bx\ -f- cx\
In 72 - 6aJ + 3Acxl - cx\
Tl » 0.764, 72 = 0.889 (from Part 1)
Xi « 0.335, £2 « 0.665
-0.269 - 6(0.665)2 + c(0.665)3
-0.118 - 6(0.335)2 + ^c(0.335)2 - c(0.335)«

Thus b - -0.0106 and c - -0.894.

In Tl „ -(0.0106 + 0.894£s)g|
In 72 - -(1.351 - 0.894zi)z?

Let xi - 0.4771, xz - 0.5229,

In 71 = -(0.0106 + 0.894a?s)ajJ - -0.131

Tl - 0.878
In 72 - -(1.351 - Q.894*iX » -0.210

72 - 0.810

Assuming T - 63.91°C.,

yi . 0.878(970X0.4771) ^ Q 53g

0.810(835) (0.5229) _ n _-
yi » _ U.4b5

Sy - 1.000. If the sum of y is not equal to 1.000, then other values of P should
be used until y = 1.0.

Values of y corresponding to various other values of x may be similarly calcu-
lated. The results are given in Table 3-10.

TABLE 3-10

2/loalo

2/lexp

100 Ay

(y - X)exp

0.9145

0.951

0.9552

3.2

0.7955

0.869

0.8688

0.3

0.6633

0.753

0.7505

2.9

0.4771

0.519

0.5170

5.0

0 3350

0.3350

0.0

0.2108

0.173

0.1760

8.6

0.1108

0.072

0.0650

15.0

Solution of Part 26.

Since In 71 — bx%
In 71

CALCULATION OF VAPOR-LIQUID EQUILIBRIA
Also In 78 •• bx\ -H

or, substituting,

-cxl
In 72

In 72

b -f - c -

,6+ic

- cx*
•• b + c(0.5 -f X*]

(d)

(e)

By plotting Eqs. (c) and (e), a single line should be obtained with a slope — c,
and intercept at xz — 0 equal to 6. The values for these equations are given in
Table 3-11.

TABLE 3-11

In 71

In 72

111 71

In 72

0.9145

0.0855

-0.495

-0 591

0 8590

0.1410

—

-0.441

—

-0.599

0 7955

0.2045

-0.0091

-0.394

-0.217

-0 621

0.7388

0 2612

-0.0202

-0 371

-0.296

-0.680

0 6633

0.3367

-0 045

-0 324

-0.396

-0.737

0.5750

0.4250

-0.0878

-0.246

-0.486

-0.744

0.4771

0.5229

-0.164

-0.173

-0.600

-0.760

0 3350

0.6650

-0.271

-0.123

-0.613

-1.094

0.2660

0.7340

-0.380

-0.0780

-0.706

-1.101

0.2108

0 7892

-0 420

-0.0440

-0.675

-0.986

0.1375

0.8625

-0.540

-0.0366

-0.726

-1.937

0.1108

0 8892

-0.739

-0.0152

-0.936

-1.239

The values of Table 3-11 are plotted in Fig. 3-8.

Slope of plot = -0.74 = c
Intercept of plot « -0.11 « b

Thus, In 7j - -(0.11 + 0.74z2)z2 and In 72 - -(0.48 + 0.74x^x1
Values of y at various values of x are calculated as in Part 2a. The results are
given in Table 3-12.

TABLE 3-12

T/cfllc

•^exp

100 Ay

(y — #)«p

0.9145

0.959

0 9522

18 0

0.7955

0.871

0.8688

3.0

0.6633

0.753

0.7505

2.9

0.4771

0.520

0 5170

7.5

0 3350

0.336

0.3350

0.0

0.2108

0.175

0 . 1760

2.9

0.1108

0.076

0.0650

15.0

FRACTIONAL DISTILLATION

An appreciable portion of the variations in 100 &y/(y — x) is probably due to
the errors in the experimental calculation and the data. The agreement with both
the Van Laar and the Margules equations is fairly good for this system. The
azeotrope point for the Margules equation gave better agreement than the line

u
1.0
0.3
*06
0.4
0.2

)

Inxj
o Values of £|~~ vs *2

A Values of - •—£*• V3(x2 + 05)

V0 02 0.4 06 0.8 1.0 1.2 1,4 U
FIG. 3-8. Margules plot for system acetone-chloroform.

drawn through the points, but a line drawn on the basis of the azeotrope constants
would agree with the plotted points as well as the line employed. In drawing the
plot for the Margules evaluation no attempt was made to place the line to give the
minimum average deviation in 100 Ay/ (y — x). Mason studied the " optimum
line" and obtained constants that gave a maximum deviation of 12.5 per cent and
an average deviation of 4.3 for 100 A^/(y — x).

MULTICOMPONENT SYSTEMS

The ideal solution laws such as Raoult's law and Raoult's law cor-
rected for gas law deviation are applicable to binary or multicompo-
nent systems. They treat each component independently of any
other component present; i.e., the relationship between the mol frac-
tion in the vapor and in the liquid for a given component depends only
on the temperature and total pressure. In many cases, these simpli-
fied rules are not applicable, and there is interreaction between the
various components. It would be particularly desirable to have a
satisfactory theoretical approach to the problem of multicomponent
vapor-liquid equilibria since the experimental determination for this

CALCULATION OF VAPOR-LIQUID EQUILIBRIA 73

case is an order of magnitude more difficult than for binary mixtures.
The equations as originally given by Margules and Van Laar were
limited to binary mixtures, although Van Laar did indicate the method
for multicomponent problems. In recent years, there have been a
number of derivations of multicomponent, Van Laar-type equations
(Refs. 1, 7, 22, 26, 27). For example, Bonham (Ref. 1) obtained a
multicomponent Van Laar equation by using multicomponent mixture
constants for the van der Waals equation, amix and Zw, given by
Eq. (3-40).

A similar derivation to that employed for the binary Van Laar
equation leads to

(3-58)

T In 73 =

In the above equations, the values of A and B are similar to those
given for a binary, but in this case, considerable care must be exer-
cised with respect to the subscripts. Thus A 12 is equal to 61/62, A 32 is
equal to 63/62, and similarly for other subscripts. It should be pointed
out that for a ternary mixture, there are only two independent A
terms. Any other A terms can be calculated from these two by
multiplication or division. The three activity coefficient equations
contain only the values of A and B associated with the three binary
mixtures possible from the three components. If the Van Laar equa-
tion for multicomponent mixtures is applicable, the only information
needed is the vapor-liquid equilibrium data for the binary mixtures.

In the case of the value of 5, it should be noted that it occurs in the
multicomponent equation as a square root, and this immediately
raises the question of whether the value is positive or negative. This
question can be answered by considering the relationships given on
page 58. On the basis of the relations employed for amix and 6mi«,
the square root of Bn is equal to

and the value is negative or positive depending on whether

is greater or less than vW&2. There are at least two ways of deter-

74 FRACTIONAL DISTILLATION

mining the sign of the \/B. The first involves having sufficient data
on the multicomponent mixture in order that the three equations can
be forced to fit. This method is of little value since it requires a large
amount of data on the mixture, and the mathematical procedure is
complicated. It is useful, however, in cases where data are available,
and it is desirable to interpolate or extrapolate. The other method is
based on evaluating the sign of these terms independent of vapor-
liquid equilibria data and determining their magnitude from binary
data.

Applying the Van Laar to a binary mixture does not indicate the
sign of the -\/B. The basis for deciding the sign is the relative values
of the Va/6? which corresponds to the square root internal pressure of
the liquid. Thus polar compounds which have high internal pressures
would be expected to have high values of this group, while compounds
of low polarity would be expected to have low values. If the com-
pounds in the binary mixture are of widely different polarity, it is
fairly easy to determine whether the square root should be positive or
negative. For example, for a mixture of ethyl alcohol and water,
components 1 and 2, respectively, it is well established that water is
the more polar; therefore, if V^i is taken to be positive, \/#i2 would
be negative.

If the square roots of the £12/61, 523/&2, and Bsi/fes are added, it will
be found that, on the basis of the definition given on page 58, the
sum is zero. In fact, all that is necessary to obtain this conclusion is
to have each one of the subscripts appear first on one of the J5's and
last on another, and to have the subscript on the 6 correspond to the
first subscript. These relationships are extremely useful since any
two independent values of B together with the corresponding A's are
sufficient to evaluate the other. Thus, for a ternary mixture, it is
necessary to have data on only two of the binaries. This is useful in
several ways, e.g., predicting the three-component data from data on
two of the binaries or predicting the vapor-liquid equilibria for a
binary for which there are no data. In this latter case, it is necessary
to find data on two binaries which have a common component, the
other components being the ones for the desired mixture. For exam-
ple, if data are available on ethanol and water and methanol and
water, it is possible to calculate the vapor-liquid equilibria for methanol
and ethanol on the basis of these data, assuming that the systems
agree with the Van Laar-type equation.

The various relations of the A's and B's of the Van Laar equations
for a ternary mixture are summarized below:

CALCULATION OF VAPOR-LIQUID EQUILIBRIA

or, alternatively,

In the case of more than three components, the relationship is
obtained by the same procedure, but it is relatively easy to write the
activity coefficient equations simply by inspection. Returning to the
three component equations, Eq. (3-58), it is noted that the denominator
is always the same and simply involves the square of the sum of the
mol fraction times the A for the term in question relative to some given
base component. In the equations given, component 2 is used as the
base, and -4.22 was omitted since it is obviously 1. The numerator
involves terms for the components other than the one under considera-
tion. These terms are all of a general type and can best be explained
by considering the specific equations. Thus, for the T In 71, there
are terms involving #2 and #3, each term has the A value corresponding
to its component and the \/B for component 1 relative to it. The
same rules apply to T In 72 and Tin 73 with appropriate shifts of sub-
scripts. For n components, the equations become

-f

)2, CtC.

-f OV
\/Bn

(3-61)

Tlnyn

and the following relationships apply:

A large number of equations of this latter type can be written as
long as the number for each component appears first in the subscript
of one B and last in another subscript, and the A terms are made to
correspond to the first subscript of the associated B term. Depending

76 FRACTIONAL DISTILLATION

on what particular values of B are available, there may be a preference
for some particular series. The other relationships for the A's and
B's are the same'as listed for the ternary system.

To use these equations with the evaluation of the constants from
binary data, it is necessary to have information on n — 1 binary
mixtures.

As was shown in a previous section, the application of Van Laar's
equation to a binary mixture showing negative deviations from
Raoult's law, i.e., tendency to form a maximum boiling mixture, gives
negative values for the B term. If some of the binary mixtures give
negative values and some give positive values, they cannot be used in
Eq. (3-58). If all the B terms are either negative or positive, the
equation can be applied. If any of the binary systems involved does
not agree with Van Laar's equation, then the multicomponent relation-
ship should not be applied in the region near this binary. Multicom-
ponent equations of the Margules type have been presented, but they
involve so many independent constants that their engineering utility
is small.

Cooper's relationship (see page 59) can be applied to multicom-
ponent mixtures. Thus for a three-component mixture Eq. (3-47)
becomes

' f ,« QON

n\vi + UzVz + n&s (H^VI + n&z +

and, by partial differentiation and substituting for

T 1 = xigl2

71
T 1 - xlAltBti + xlAlzBw + xiXz(A*JBu +

+ x* +
, x\A\iBzi '

1 in *Y3 ==

: - ;

+ £2 +

(3-64)

In these equations the A values are related in the same manner
as for Eq. (3-58), but no assumption has been made relative to the
values of the B terms. To be consistent with the binary equations,
Bn = AwBzi, etc., and this leaves three independent B terms. If the
values of these terms are made to fit Eqs. (3-59) and (3-60), then Eq.
(3-64) reduces to Eq. (3-58). However, the values can be used inde-
pendently, and the equation then is more general than Eq. (3-58). If
binary data are used to evaluate the constants, information on all
three binaries must be available, and frequently this limits the useful-

CALCULATION OF VAPOR-LIQUID EQUILIBRIA 77

ness of Eq. (3-64) as compared to (3-58). Negative values of B may
lead to mathematical difficulties in the use of Eq. (3-58), but they can
be handled by Eq. (3-64). Because Eq. (3-58) requires less experi-
mental data to evaluate the constants and is easier to use, it has been
more widely applied than Eq. (3-64).

Nomenclature

A ,B « constants in Van Laar equation

E — internal energy

AF = partial molal change in free energy
AF0 = actual value of AF
AFt- = ideal value of AF

AFe =s excess partial molal free energy change » A/*0 — &Pi
AF€ = excess free-energy change on mixing

/ = fugacity

/L = fugacity of component in liquid phase
fp =s fugacity of pure liquid under its own vapor pressure
/* = fugacity of pure liquid under total pressure of mixture
fv = fugacity of component in vapor phase
fv as fugacity of pure vapor at pressure, TT
Aj? « partial molal change in enthalpy
AHe — excess partial molal change in enthalpy

A/T = enthalpy of vaporizating 1 mol of a component from a liquid into a vacuum
K as equilibrium constant = y/x
K' = proportionality constant
n =* mols

p = vapor pressure
p = partial pressure
R = gas law constant
AS = partial molal change in entropy
ASe — excess partial molal entropy change
T — absolute temperature
y = volume

t; = molal volume or partial molal volume
x =» mol fraction in liquid
y « mol fraction in vapor
a « relative volatility
0 =* volatility == p/x
y «• activity coefficient
TT = total pressure
fi = gas law correction factor » PV/RT

Subscripts:

a,6,l,2,3 refer to components
L refers to liquid phase
R refers to reduced conditions
V refers to vapor phase

78 FRACTIONAL DISTILLATION

References

1. BONHAM, M.S. thesis in chemical engineering, M.I.T., 1941.

2. BEBTHELOT, Compt. rend., 126, 1703, 1857 (1898).

3. BBOWN, J. Chem. Soc., 39, 304, 517 (1881).

4. CABBY, Sc.D. thesis in chemical engineering, M.I.T., 1929.

5. CAUBET, Compt. rend., 130, 828 (1900).

6. CLABK, Trans. Faraday Soc., 41, 718 (1945); 42, 742 (1946).

7. COOPEB, 10.90 Report, M.I.T., 1941

8. DALTON, Gilberts Ann. Physik, 12, 385 (1802).

9. DUHEM, Compt. rend., 102, 1449 (1886).

10. GALITZINE, Wied. Ann. Physik, 41, 770 (1890).

11. HILDEBBAND, "Solubility of Non-electrolytes," 2d ed., Reinhold Publishing
Corporation, New York, 1936.

12. "International Critical Tables," Vol. Ill, pp. 212-217, McGraw-Hill Book
Company, Inc., New York, 1928.

13. LECAT, "L'Azeotropisme," 1918.

14. LEE, /. Phys. Chem., 36, 3554 (1931).

15. LEHFELDT, London, Edinburgh Dublin Phil. Mag., 6, 246 (1895).

16. LEWIS and RANDALL, "Thermodynamics,0 p. 226, McGraw-Hill Book Com-
pany, Inc., New York, 1923.

17. MABQULES, " Sitzungsberichte der math-naturw," Classe der Kaiserlichen
Akademie der Wissenschaften (Vienna), 104, 1243 (1895).

18. MASON, M.S. thesis in chemical engineering, M.I.T., 1948.

19. RAOITLT, Compt. rend., 104, 1430 (1887).

20. RHODES, WELLS, and MUBBAY, Ind. Eng. Chem., 17, 1200 (1925).

21. ROSANOFP and EASLEY, /. Am. Chem. Soc., 31, 977 (1909).

22. SCATCHABD, Trans. Faraday Soc., 33, 160 (1937).

23. SCHEELINE, Sc.D. thesis in chemical engineering, M.I.T., 1938.

24. TUCKEB, M.S. thesis in chemical engineering, M.I.T., 1942.

25. VANLAAB, Z. physik. Chem., 72, 723 (1910); 83, 599 (1913).

26. WHITE, Trans. Am. Inst. Chem. Engrs., 41, 539 (1945).

27. WOHL, Trans. Am. Inst. Chem. Engrs., 42, 215 (1946).

28. ZAWIDSKJ, Z. physik Chem., 35, 129 (1900).

CHAPTER 4

CALCULATION OF VAPOR-LIQUID EQUILIBRIA

(Continued)

Critical Regions. At very high pressures special phenomena asso-
ciated with the critical region are encountered in vapor-liquid equi-
libria. If the vapor pressure of a pure component is plotted vs. the
temperature, a line concave upward is obtained. This line terminates
at the critical point. Conditions below the line in region A, Fig. 4-1,

I 25

120

220

245

145 170 195

Temperature, °£

FIG. 4-1. Typical pressure-temperature curves.

correspond to all vapor and no liquid. The line represents conditions
under which the vapor and liquid are in equilibrium. Conditions
above the line in region B represent all liquid. In region C the state
of the substance is in question since it is possible to obtain either vapor
or liquid without a change in phase. If a given binary mixture is
plotted in the same way, similar conditions are attained except that a
loop region is obtained for a mixture of given composition instead of a
single line, Fig. 4-1. The upper line of the loop represents the bubble-
point curve, i.e., the condition under which the mixture first forms
vapor. The lower side of the loop is the dew point curve, i.e., the
condition under which the mixture begins to condense. In the case

FRACTIONAL DISTILLATION

of the pure substance these two lines coincide, and the critical tem-
perature and pressure are the maximum for both variables that can
exist and have coexistence of liquid and vapor phase. In the case of
the mixture the maximum temperature does not coincide with the
maximum pressure.

For a pure component at the critical condition, the properties of the
vapor and the liquid phases become identical in all respects. In the
case of the loop curve for a binary mixture, the property of the vapor

40r

120

220

245

145 170 195

Temperature, °C
FIG. 4-2. Pressure-temperature curves for butane-hexane system.

and the liquid are in general different, both at the point of maximum
temperature and at the point of maximum pressure. There is a point
on the loop, usually between the maximum temperature and the maxi-
mum pressure points, at which the properties of the liquid and the
vapor are identical. This is taken as the critical of the mixture. This
critical point K is shown on Fig. 4-1. Figure 4-2 shows loop curves
for three different mixtures of butane and hexane, and a curve indi-
cating the loci of critical points is shown. If the conditions inside a
single loop curve are analyzed, it is found possible to plot lines of con-
stant fraction vaporized; such curves for 0, 20, 40, 60, 80, and 100 per
cent vapor have been drawn in Fig 4-3. All these curves converge to
a common point at the critical.

The curves of Fig. 4-3 indicate the phenomena of retrograde con-
densation, i.e., conditions under which an increase in pressure causes
vaporization instead of condensation, or in which a decrease in tem-
perature causes vaporization instead of condensation. Thus, start-

CALCULATION OF VAPOR-LIQUID EQUILIBRIA

ing at point A, Fig. 4-3, and increasing the temperature at constant
pressure, the bubble-point curve is first contacted and vapor begins
to form. An increase in temperature causes more vaporization up to
about 20 per cent. Higher temperatures then cause a decrease in
vapor until the mixture is again all liquid at point C. A similar type
of phenomenon is exhibited by curve EL. If at constant temperature
the pressure is lowered from point E, the conditions first reach the
dew-point curve; i.e., the mixture is all vapor. On lowering the pres-

Temperature
FIG. 4-3. Typical pressure-temperature loop curve.

sure, the mixture becomes more and more liquid until about 25 per
cent is condensed. In this region it has exhibited the retrograde
phenomena. At still lower pressures, the mixture behaves normally
and vaporizes with decrease in pressure.

In a binary mixture, if two of the loop curves intersect, i.e., if the
vapor curve of one crosses the liquid curve of the other, then the two
compositions determine a vapor-liquid equilibrium point. This is due
to the fact that, for a binary system of two phases, the phase rule
allows two degrees of freedom. However, the value may not be
unique, i.e., in the higher pressure region, particularly very near the
critical, it is possible for a given vapor to have two possible equilibrium
liquids of different compositions. These two conditions can be at the

FRACTIONAL DISTILLATION

0.2 0.4 06 08

Mol fraction C02 in liquid

(a)

0.2 0.4 06 OB 1.0

Mol fraction C02 in liquid

FIG, 4-4. a, Vapor-liquid equilibria for C02-S02 at constant pressure. 6. Vapor-liquid
equilibria for COa-SQj at constant temperature.

CALCULATION OF VAPOR-LIQUID EQUILIBRIA

same temperature, but at different total pressures. This is shown in
Fig. 4-4 which gives the vapor-liquid equilibrium data for the system
carbon dioxide-sulfur dioxide.

In general, it is found that the relative volatility decreases as the
total pressure increases, and there are several factors that combine to
give this effect. It is generally found that (1) the ratio of the vapor
pressures of two components becomes nearer to unity as the pressure
increases and (2) the deviations from the perfect-gas laws also tend to
reduce the relative volatility. Actually the decrease in relative

0.2 0.4 0.6 0.8 1.0

x= Mol Fraction €4 in Liquid
FIG. 4-5. Effect of pressure on vapor-liquid equilibria.

volatility is even larger than the result of these two factors. At the
critical pressure of a binary mixture, the vapor and liquid phases
become identical in all respects, and the relative volatility becomes
unity. No separation is possible, not only because of the relative
volatility effects but because differentiation between the vapor and
liquid is no longer possible. This decrease in relative volatility to the
value of 1 at the critical is a progressive effect, although a large part of
it occurs close to the critical condition. The Lewis and Randall
fugacity rule does not show the convergence of the relative volatility
to unity at the critical For example, in the case of a mixture like
ethane and propane, the rule would show a finite relative volatility not
only at the critical condition but at pressures much higher.

This effect of total pressure on the vapor-liquid equilibria of the

84 FRACTIONAL DISTILLATION

system rc-butane-ft-hexane (Ref. 1) is shown in Fig. 4-5. The decrease
in the relative volatility with increasing pressure is apparent, and the
y,x curves become discontinuous at pressures above 30 atm. Thus,
at 33.5 atm. the y,x curves exist only for liquids containing more than
24 mol per cent butane. Mixtures containing less than this amount of
butane are at pressures above the envelope curve of Fig. 4-2, and only
a single phase is present. At higher pressures the range of the dis-
continuity increases, and above 37.5 atm. the y,x curve becomes dis-
continuous at both ends because there are mixtures of butane and
hexane which have higher critical pressures than either of the pure com-
ponents. Above 38 atm. only one phase is present for all compositions.

From the viewpoint of the ease of separation, it is almost always
disadvantageous to operate at high pressure, but it is frequently
necessary to accept this more difficult separation in order to obtain
other desirable features of high pressure, such as higher condensation
temperatures and lower volume of apparatus.

The relative volatilities of all vapor-liquid systems do not decrease
with increasing pressure in all regions. Thus, it is possible for abnor-
mal mixtures to have an increase in relative volatility in some regions
for an increase in pressure. It is in general true that they will not
increase the relative volatility at all compositions. For example, in
the case of ethyl alcohol and water at atmospheric pressure, the con-
stant-boiling mixture is about 89 mol per cent alcohol. As the pressure
is increased, the composition of the constant-boiling mixture becomes
lower in alcohol, and the relative volatility of water to alcohol at a
composition of 89 mol per cent becomes greater than 1 as the pressure
is increased above 1 atm. and then decreases at still higher pressures.

The approach of the relative volatility to unity at the critical means
that the compositions of the vapor and liquid are identical. Thus the
K values for all components equal 1. The temperature and pressure
at which these values become unity are functions of the other com-
ponents present. Thus, a mixture of butane and ethane would have a
certain critical temperature and pressure, while a mixture of butane
with hexane would have different critical temperature and pressure,
but under both conditions the K value for butane would have to be
equal to unity. Thus, the values given in Table 3-2, which were taken
to be independent of the character of the other components and a
function of the temperature and pressure only, cannot apply in the
critical region. In most cases, these effects of the critical region are
not serious at total pressures less than 0.5 to 0.7 of the critical pressure.
Modifications of the method of utilizing the K values in the critical

CALCULATION OF VAPOR-LIQUID EQUILIBRIA 85

region have been suggested which allow for the effect of the other
components present (Ref. 2).

Immiscible Liquids. Immiscible liquids are not an important case
encountered in fractional distillation. It is much simpler to separate
two liquids which are insoluble in each other by simple decantation
than it is by fractionation. However, the physical-chemical laws
that apply to such cases are helpful in explaining certain of the phe-
nomena involved in the intermediate case of partially miscible liquids.

The various rules developed for the vapor phase for miscible liquids
apply equally well to this case. In the case of the liquid phase, if the
liquids are completely immiscible, at equilibrium each component
would exert its own vapor pressure independent of the others present.
For this case, the vapor-liquid equilibrium expression will be

pi **Vi* =Pi)

p* = wr - P2J (4'1}

at high pressures,

A = yiAi = /Pie"i<

where p = partial pressure

P = vapor pressure

TT = total pressure

y = mol fraction of vapor

/ = fugacity

/*. = fugacity of pure component at total pressure
fp = fugacity of pure liquid under its own vapor pressure

v = partial molal volume

R = gas-law constant

T = absolute temperature

These equations are similar to those for miscible liquids except that
the mol fraction in the liquid is omitted. It is to be expected that the
fugacity relationship would give satisfactory results up to a pressure
of approximately one-half of the critical pressure. At higher pres-
sures the Lewis and Randall fugacity rule for the vapor mixture would
tend to be less satisfactory. Actually it is doubtful whether absolute
immiscibility ever occurs. However, there are cases in which the
miscibility is so limited that each phase would act as essentially a
pure material, e.g., mercury and water.

Example for Immiscible Liquids. 1. A two-phase liquid mixture comprised of
30 mols of toluene and 70 mols of water existing as liquids at 1 atm. pressure and

FRACTIONAL DISTILLATION

60°C. is heated at constant pressure. Assuming that water and toluene arc
completely nonmiscible and that all the vapors formed were at all times in equilib
rium with the remaining liquids, and using the vapor pressure data given below
construct the following curves:

a. Mol per cent vaporized vs. temperature.

6. Mol fraction of toluene in vapor vs. temperature.

2. A two-phase liquid mixture consisting of 30 mols of toluene and 70 mols o
water is vaporized at a constant temperature of 85°C. by reducing the total pres

>
\ 1000

)

„

'
5

IOCJ

Watet
s.

\ \

Toluene "'

-•*\

.0 2.2 2.4 26 28 30 3.<
1000
"W

FIG. 4-6. Vapor pressures of toluene and water

sure. The initial pressure is 2 atm. absolute. Assuming that the vaporization
carried out such that equilibrium between vapor and liquid exists at all times, plol

a. Mol per cent vaporized vs. pressure.
6. Composition of vapor vs. pressure.

Solution of Part 1. System: 30 mols toluene, 70 mols H20. (Let subscripts
and W refer to toluene and water, respectively.)

Assuming complete immiscibility, from the phase rule it follows that for tl
three-phase-two-component system there exists only one degree of freedoi
Hence, vaporization will occur at constant temperature as long as two liquid phas
are present,

CALCULATION OF VAPOR-LIQUID EQUILIBRIA

Vaporization occurs when v « Pw + PT «• 760 mm. From Fig. 4-6 this
temperature » 84.4°C.

PT « 336 mm. yT - 3a^feo - 0.442
Pw «• 424 mm. TT « 760 mm.

Both water and toluene will vaporize, but the composition of the vapor will
remain constant at yr •» 0.442 as long as two phases are present. Since the ratio
of the vapor pressures of toluene to^water is greater than the ratio in the charge,
the toluene liquid phase will disappear first. When all of the toluene has just
vaporized, the water vaporized will be

30 X 424-is6 - 37.9 mois
Per cent vaporized » 37.9 4- 30 « 67.9

At 70 per cent vaporized, the vapor will contain 30 mols of toluene and 40 mols of
water.

Pw - 760(4?<f 0) - 434 mm. t - 85°C.
» 0.425

The following table was prepared in this manner:

r, °c.

Per cent
vaporized

—

84.4

—

84.4

67.9

0.442

0.429

87.4

0.375

0 333

90 3

100

0 300

These results are plotted in Fig. 4-7.

Solution of Part 2. Basis: 30 mols toluene, 70 mols HaO. Vaporization will not
occur until the total pressure is equal to the sum of the partial pressures. At 85°C.,

PT » 341 mm.

Pw - 434mm.

IT « 775 mm.

Reasoning as in Part 1, all of the toluene will be vaporized at this pressure
together with 43^4i X 30 = 38.2 mols of H20.

Per cent vaporized

30 + 38.2 - 68.2
30
68.2

0.440
In order to vaporize the remaining H20, the pressure must be lowered. Let

FRACTIONAL DISTILLATION

N «• per cent vaporized
- N ~ 3°. _ 30

yw ~~ • yT _ _

Using this relationship,

434JV

AT - 30 N - 30

AT (% vaporized)

TT, mm.

759

0 429

694

0.375

651

0.333

100

620

0.300

The results of these calculations are plotted in Fig. 4-8.

IUU

^ 60
c

'§_40
e

20
°6

/ % vapor/zed

0 70 80 90 10
Temperature, °C

Fio. 4-7.

Partially Miscible Liquids. There are a large number of systems in
which the components are miscible only over limited ranges of con-
centration. These mixtures form a very important group for frac-
tional distillation. The fact that the liquids are partially miscible
greatly alters the normal type of vapor-liquid relationships, and the
fact that two liquid phases are present requires that one less degree of
freedom be available than is normal for a system of a given number of
components. In general, the vapor-liquid equilibria of these systems

CALCULATION OF VAPOR-LIQUID EQUILIBRIA

are very abnormal and, by properly exploiting these abnormalities and
combining distillation and decanting operations, separations can easily
be made.

The types of systems in this category range from those which are
almost immiscible, such as water and benzene, to those that are miscible
except for very limited regions, such as phenol and water. Even
though benzene and water are essentially immiscible, distillation is
often employed for the purpose of drying benzene. Benzene saturated
with water contains a very small percentage of* the latter. By the

IUUS

»-
>%
o

2 60

O
T5

|40

0
62

^v,

*^n

% vaporized

^^,

•^-^

^^ — ^9

— "

^ — /

\< —

— -A—*

. — -A-

>0 640 660 680 700 720 740 760 780 800 82
Total pressure mmHg

FIG. 4-8.

proper distillation technique this water can be completely and easily
removed. This type of operation is employed chiefly for drying'mate-
rials that are partially miscible with water. If the solubility of water
in the material is low, distillation is an economical method of drying
the liquid and is effective down to the extremely low concentrations
of water. As an example of the method of estimating the vapor-liquid
equilibria for systems in which the mutual solubility is low, the mix-
ture ethyl-ether and water will be considered.

In Fig. 4-9 a constant-temperature diagram for this system has been
constructed in which the ordinate is the partial pressure of the com-
ponents, and the abcissa is the mol fraction of ether in the mixture.
This mol fraction is the mols of the ether in the combined liquid phases
divided by the total mols of ether and water in all of the liquid present.

FRACTIONAL DISTILLATION

It is a "pseudo" mol fraction. At the left-hand side of the diagram
the water partial-pressure curve starts at the vapor pressure of
water, and the ether curve starts at zero, since none is present. As
ether is added to the system, it first dissolves in the water, giving only
one liquid phase, until a concentration of 0.9 mol per cent ether is
reached, which is the solubility of ether in water at 60°C. In the

Water phase
See Fig. 4 -/O
for details

-Two phases

Ether phase *-*

2000
1600
1200
800
400

Tota)

pres.

$ure^

rtial

press

ure o

fetht

•rS

Raoult's law
for ether*. \ ,

67%

wate

r —

rrf/a/

pres*

wre

of wa

fer\

'enry

'$ la

y fort wah

v —

) 02 0.4 0.6 08 t.

XE t Mol fraction of ether in liquid *
* In the two-phase region mol fraction values are based on total of both phases.
FIG. 4-9. Estimation of vapor-liquid equilibria for system ethyl ether-water at 60°C.

region below 0.9 mol per cent ether, the water partial pressure will
agree closely with that predicted by Raoult's law since there is not
enough ether present in the liquid phase to alter significantly the water
properties. If one component of a binary mixture obeys Raoult's
law, the Duhem equation states that the other equation must obey
Henry's law, p = Hx. Thus in this region it is reasonable to assume
Henry's law for ether.

When the amount of ether present in the liquid exceeds 0.9 mol per
cent, two phases form. The water phase will consist of 99.1 mol per
cent water and 0.9 mol per cent ether. The ether phase will consist of
93.3 mol per cent ether and 6.7 mol per cent water. These are the
solubility relationship of these two components at a temperature of

CALCULATION OF VAPOR-LIQUID EQUILIBRIA

60°C. Thus, at a mol per cent of ether equal to 0.9, two liquid phases
will be formed. As more and more ether is added to the system, the
amount of the liquid ether phase increases, but the compositions of the
ether and the water phases remain constant. This is in agreement
with the phase rule which states that, for two-component systems
involving three phases, only one degree of freedom is available, and in

^er phase

< "- Wat

1600
1200

xt-

Total pressure

0.9%
ether -^^

-Par
of el

fial p
herfa

ressu
?nr/'s

400

*ial p
(Rat

ressu
wit's

<-e of
law)

wate

r 1

ran

) OOOZ 0004 0.006 0.008 0.(

xt , Mol fraction of ether in liquid *
* In the two-phase region mol fraction values are based on total of both phases.
FIG. 4-10. Estimation of vapor-liquid equilibria for system ethyl ether-water at 60°C,

this case the temperature has been fixed. Therefore, the composi-
tions of all phases are fixed, as well as the pressure, as long as the three
phases are present.

On adding more ether, a condition is finally reached at which suffi-
cient ether is present to dissolve all of the water, and the water phase
disappears. The concentration of the phases just at this condition is
the same as it has been throughout the two-phase region. Thus, at
mol per cents of ether from 0.9 to 93.3, two phases are present, and the
partial pressures of water and ether in the vapor are constant. Above
a mol percentage of ether of 93.3 only one liquid phase is present, and
the system then has two degrees of freedom; i.e., the vapor composition
then becomes a function of the liquid composition. For the ether
phase it is logical to assume Raoult's law for the ether and Henry's law

92 FRACTIONAL DISTILLATION

for the water, but the assumptions are probably not so good as those
made for the water phase because of the high solubility of water in
ether. With these assumptions, the pressure-composition diagram
can be completed.

At 60°C. the vapor pressures of ether and water are 1,730 and 149.4
mm. Hg, respectively. Thus for mol fractions of ether of 0.9 or less,
the partial pressure of water by Raoult's law is pw = 149.4#TF, and the
partial pressure of ether is PE = HE%E. For mol fractions of ether of
93.3 or greater, ps = l,730#j?, and pw = Hwxw, where HE and Hw are
the Henry's law constants for ether in water and water in ether,
respectively. The partial pressures of a component must be the same
in both liquid phases in the two-liquid-phase region and the above
equations can be equated.

For water,

149.4(0.991) - flV(0.067)

Hw = 2,210
For ether,

(0.009)#* = 1,730(0.933)
HE = 179,500

It is interesting to note that ether at a mol fraction of 0.009 in water
exerts a partial pressure equal to 93.3 per cent of that of pure ether.
Thus its volatility, p/x, is extremely high.

The results of such calculations are shown in Figs. 4-9 and 4-10.
The latter figure is an expansion of the left-hand side of Fig. 4-9.
Raoult's law for ether is shown as a straight line from the vapor pres-
sure of ether at the right-hand side of the diagram to zero at the left-
hand side. On the basis of the above assumptions, this line is used
only for values x* from 0.933 to 1.0. From xs = 0.009 to 0.933 the
partial pressure of ether is constant, and from XE = 0,009 the partial
pressure drops on a straight line to 0 at XE = 0. A similar construction
is used for water. The sum of the two partial-pressure curves is the
total pressure. This is also shown.

If these data are replotted as mol fraction of ether in the vapor vs.
mol fraction of ether in the liquid, assuming that the vapors obey the
perfect-gas law, one obtains the results given in Fig. 4-11. The value
of the mol fraction of ether in the vapor increases very rapidly with the
mol fraction in the liquid and becomes constant at 0.915 in the two-
phase-liquid region. The vapor-liquid curve crosses the 45° line at a
composition of 0.915 mol per cent ether. The mixture of this compo-
sition is a pseudo-azeotrope. For mol fractions of ether greater than

CALCULATION OF VAPOR-LIQUID EQUILIBRIA

0.915, water is more volatile than ether in spite of the fact that the
vapor pressure of ether is over eleven times that of water. If a liquid
corresponding to a composition in the ether phase region were distilled
at 60°C., water would tend to pass off in the vapor leaving ether in the
still. Thus, ether could be dried by fractionally distilling water
overhead.

This simplified analysis is probably of sufficient accuracy for most
distillation calculations, but it is not suitable for cases of partially

Twopha
'

phi

j ITTV fJIKJIVOO

\*f**?* (note break — >k Ether phase

I in scale)

001 092 094 096 098

x- Mol fraction ether in liquid
* In the two-phase region mol fraction values are based on total of both phases.

FIG. 4-11. Estimated vapor-liquid equilibria for ethyl ether-water at 60°C.

miscible liquids in which the mutual solubility is much greater. The
experimental data on such systems indicate that the partial pressure
vs. mol fraction curves pass from the one-phase region to the two-
phase region in a smooth type of curve; i.e., the corner at the end of
the horizontal line rounds into the Henry law region. The construc-
tion used for Fig. 4-9 indicated a sharp corner. This rounding effect
tends to make the Henry law constant larger. If one component
obeys Raoult's law, the Duhem equation indicates that the other com-
ponent must obey Henry's law, which in the special case may also be
Raoult's law. It follows that, in the region where the curvature

94 FRACTIONAL DISTILLATION

occurs and Henry's law does not apply, the other components cannot
agree with Raoult's law.

These effects are more clearly illustrated by mixtures in which the
immiscibility is limited to a narrow region. In such cases the straight-
line construction applied in Fig. 4-9 is entirely unsatisfactory. The
data of Sims (Ref. 3) for the system phenol-water for the constant-
temperature conditions of 43.4°C. are presented in Fig. 4-12. In this
ease the partial pressure of each component divided by the vapor pres-

|06
I

0.2
0

'Phenol

l/vf/,;_ ,

r_^_. nhr»*A "I

Raoulte law
for phenol - '

, -

U-

phases

Wat

/*-R,

loult
>f wa

s la*
ter

-/-

___

Phen

ol jo/

ase'

0 02 0.4 06 0.8 t.O

Mol fraction water in liquid*

* In the two-phase region mol fraction values are based on total of both phases.
FIG. 4-12. System, phenol-water at 43.4°C.

oire of the pure component at 43.4°C. is plotted vs. the mol fraction
n the liquid. This method of plotting is applied since the vapor
pressures of water and phenol are so greatly different that it is difficult
,o represent both of them on the same graph.

The limit of solubility of water in phenol corresponds to a mol frac-
ion of water of 0.74, while the solubility of phenol in water is 0.0225
nol fraction phenol. For mol fractions of water between 0.74 and
K9775, the components exist as two liquid layers. Throughout this
*ange in which two liquid phases are present and where the eomposi-
;ions of these phases are therefore constant, the partial pressures re-
nain constant and are represented by the horizontal line EF and CD.
The 45° lines corresponding to Raoult's law are drawn for both com-
)onents. The data for the water phase cover such a short region that

CALCULATION OF VAPOR-LIQUID EQUILIBRIA 95

deviations from Raoult's law for the water component are not obvious.
In the phenol phase a moderate deviation from Raoult's law is appar-
ent. It will be noted in the figure that the curves tend to approach
the immiscible region with rounded corners instead of with sharp
angles.

The two points of importance regarding the diagram are (1) con-
stancy of partial pressure so Iqng as two liquid phases are present and
(2) the character and extent of the deviation from Raoult's law. The
diagram shows that the partial pressure in phenol, when dissolved in
water is abnormally high; i.e., it is much greater than is called for by
Raoult's law, the line AB. Limited miscibility of two liquids implies
that the molecules of one find it difficult to force their way into the
other. Thus, it requires a relatively high pressure for phenol to force
a small amount of itself into water. This is equivalent to saying that,
when phenol has been dissolved in water, the volatility of phenol is
abnormally high. The less the mutual solubility, the more abnormal
the partial pressure; hence the greater the volatility of the dissolved
component. The practical results of these relationships are shown in
the following example.

Despite the fact that phenol boils almost 80° higher than water, in
certain regions, i.e., low concentrations of phenol, the volatility of
phenol is greater than that of water; i.e., the vapor given off by such a
solution is richer in phenol than the solution itself. If a solution in
this low concentration region is distilled, the water is discharged from
the bottom of the column essentially free of phenol, which is found in
the distillate.

These data for phenol and water are replotted in Fig. 4-13 as the
vapor-liquid equilibrium, i.e., the mol fraction of the phenol in the
vapor as a function of the mol fraction of phenol in the liquid. The
data curve is labeled ll experimental. " These data indicate that this
system forms a minimum boiling azeotrope at a concentration of
about 0.0073 mol fraction phenol. At concentrations lower than this,
phenol is more volatile than water. At all concentrations greater than
this, water is more volatile than phenol. This y,x curve indicates the
constancy of vapor composition in the two-phase region. The proce-
dure involving the use of Raoult's and Henry's laws employed for the
ethyl ether-water system is probably not suitable for the phenol-water
system because of the high mutual solubilities. However, such calcu-
lations were made for illustrative purposes and the results are shown
on Figs. 4-13 and 4-14. The latter figure gives the relative volatilities
corresponding to the vapor-liquid curve of Fig. 4-13. It is apparent

FRACTIONAL DISTILLATION

-if-

f
JC

t &

13*

\\\

"o 3

s 1

^. -^

S S

QSDLjd JodoA u; |ouei(d uoipcjj. |

CALCULATION OP VAPOR-LIQUID EQUILIBRIA

jfe

^
/

— "•"

/ f
/

*B *

y s

* r~

O Q P^

Vs.

'/ /

// S

-sj

/ /

•B t §

? S r^

S-5 °<

/ /

^-°

Y ,

S O +3

4 -a 2

c | g

.2 ° ^
•ft H <y

>
\

101
Xp,Mol fra<
s are based 01
4, Relativ

J 1

9 S rH

T "~

jf>

1 ^

•J-:

^ 6

r§

0 M

.2 &

•^

<*«

I-

o c

98 FRACTIONAL DISTILLATION

that this simplified method is not very satisfactory. In fact it would
indicate that water was more volatile than phenol at all compositions
and that no azeotrope was formed. The results obtained would not
be of much utility and would actually be very misleading. Thus, if
water containing a low concentration of phenol were to be distilled, the
calculations based on Raoult's and Henry's laws would indicate that
water was the more volatile component, while the actual data indicate
that phenol is the more volatile. This simplified procedure gives only
very approximate results if the mutual solubilities are over a few mol
per cent.

A better prediction of the actual vapor-liquid equilibria can be
obtained by the use of the Van Laar equation. It has been found that
in general this equation can be employed empirically to give satisfac-
tory correlation for each of the single-phase regions of a partially
miscible system. Because of the constancy of partial pressure over the
two-phase region, the Van Laar constants for two single phases should
be related to each other by the values of the mutual solubilities. How-
ever, in most cases it is found that the constants obtained for the two
single-phase regions do not correlate with each other and the solubility
limits in a manner theoretically required. This is undoubtedly
because the assumptions made by Van Laar are not satisfied in partially
miscible systems. For example, it is obvious that the entropy of mix-
ing for such a system would not be equal to that for the ideal case.

To apply the Van Laar equation independently to each of the two
single-phase regions requires experimental vapor-liquid data; where
such data are available, the equation is useful for interpolating, extra-
polating, and smoothing the results. However, if such experimental
data are not available, the Van Laar equation can be used with the
solubility limits to predict vapor-liquid equilibria. This method forces
the constants for the two single-phase regions to be identical and, as
has already been pointed out, the experimental data for a number of
systems give different constants for the two regions. However, in
general, this method is a better approximation of the vapor-liquid
equilibria than calculations based on Raoult's and Henry's laws. The
only information required is the solubility limits at the temperature
in question and the vapor pressure of the two pure components at this
temperature. Thus at constant temperature,

CALCULATION OF VAPOR-LIQUID EQUILIBRIA 99

where ww = water in water phase

wp = water in phenol phase

pw = phenol in water phase

pp = phenol in phenol phase
and by Van Laar relationships,

T In yww = B

T In ywp =
!T In ypu =

+ A *

R A (•*• /„

T In ypp =

BA(xwp/xpp)2

(

l+A

Using T = 316.4°K, a?wp = 0.74, and xww « 0.9775 gives six equa-
tions with six unknown quantities which can be solved to give A = 0.206
and B = 238. With these constants, the vapor-liquid equilibria were
calculated for the system phenol and water at a temperature of 43.4°,
and the calculated results are given in Figs. 4-13 and 4-14, labeled
"Van Laar." It will be noted in this case that the agreement with the
experimental results is satisfactory and would be of great utility for
actual distillation calculations. The water phase, i.e., the low phenol
concentration, shows a constant-boiling mixture very close to that
determined experimentally.

Where it is necessary to estimate vapor-liquid equilibria for partially
miscible systems for which such data are not available, it is believed
that the Van Laar equation combined with the solubility limits is of real
utility. This method of calculation becomes equivalent to the assump-
tion of Henry's and Raoult's laws if the mutual solubility of the two
components becomes extremely low.

In all cases, miscible, partially miscible, or immiscible, fugacity
should be used instead of partial pressure and vapor pressure if the
pressures are such that deviations from perfect-gas law are significant.

If more than two components are involved and are partially miscible,
the relationships become so complicated that the theoretical method of
attack in its present form is not particularly helpful. In such cases, it

100 FRACTIONAL DISTILLATION

is necessary to determine experimental values. However, even in these
cases the general principles that have been developed for the binary
mixture are useful in attaining a picture of phenomena to be expected.

Nomenclature

A,B •« constants of Van Laar equation
/ =* fugacity

fp « fugacity of pure liquid under its own vapor pressure
/*• a* fugacity of pure vapor under total pressure, IT
H ** Henry's law constant
P as vapor pressure
p «• partial pressure
T *» temperature *

x » mol fraction in liquid
y «* mol fraction in vapor
7 «s activity coefficient
TT •» total pressure

References

1. CUMMINGS, Sc.D. thesis in chemical engineering, M.I.T., 1933.

2. GILLILAND and SCHEELINE, Ind. Eng. Chem., 32, 48 (1940).

3. SIMS, Sc.D. thesis in chemical engineering, M.I.T., 1933.

CHAPTER 5
GENERAL METHODS OF FRACTIONATION

There are several methods by which fractionation can be obtained.
The more important among them are (1) successive distillation of con-
densed distillate, (2) fractional condensation, and (3) rectification.

Successive Distillation. The first method, successive distillation of
the condensed distillate, can be shown best by referring to Fig. 2-1.
Starting with a large amount of liquid of the composition #5 which boils
at 760 mm. pressure, temperature U, a small amount of vapor of the
composition x\ is removed from the apparatus and condensed, giving
a liquid of the composition x\. If this new liquid is again distilled,
the first portion of the distillate will have a composition #2. Continu-
ing this process, successive compositions of the distillates can be esti-
mated by following a series of steps which eventually approach the
point C, pure carbon disulfide, as a limit.

Removal of any vapor of the composition x\ from the liquid of the
composition rrs will change the composition of the liquid in the direction
of pure carbon tetrachloride. Therefore, if the distillation of the
liquid is continued, the composition will approach pure carbon tetra-
chloride as a limit, and the last of the liquid to be distilled would have
this composition.

It is therefore possible by a systematic series of distillations to
separate any mixture of carbon disulfide and carbon tetrachloride into
practically pure carbon disulfide and pure carbon tetrachloride. This
systematic fractionation may be shown diagramatically as in Fig. 5-1,
in which the original mixture (1) is divided into a distillate (3) and a
residue (2). (3) and (2) are then distilled separately and produce dis-
tillates and residue. The distillate from (2) and the residue (3) being
combined into a new liquid (5) which is again distilled with (4) and (6)
to continue the separation. This process is continued until practically
complete separation is obtained. Such a process is sometimes carried
out in the laboratory, but it is extremely tedious and the same results
usually can be obtained in other more convenient ways.

The procedure outlined in Fig. 5-1 would appear to result in a num-
ber of intermediate products and only a small amount of the desired

101

102

FRACTIONAL DISTILLATION

fractions. Actually by a sufficient number of distillations the original
mixture can be obtained essentially as the desired products. For
example, assume that fractions 11 and 15 represent the desired separa-
tion, then fractions 12, 13, and 14 can be redistilled as shown in Fig.
5-2 to give fractions 11' and 15' which can be obtained as the same

2 IRES i DUE | 3 PISTI

7\ /

RESIDUE | 5 |DIST.RE5| 6 [DISTILLA

8 IDIST RES 1 0 blSTREs! tO [DISTILLA

FIG. 5-1. Fractionation diagram.

Product

FIG. 5-2. Fractionation diagram.

composition as fractions 11 and 15. Samples 20, 21, and 22 can be
redistilled in the same manner, and by a repetition of the procedure
essentially all of the material will be given the desired separation.

The method outlined in the preceding paragraph can be made con-
tinuous. Thus by regulating the fractions vaporized in the various
distillations it is possible to have fractions 5 and 13 of the same com-
position as the original mixture, and fresh feed can be added to these
fractions before they are distilled.

Fractional Condensation. Instead of partially distilling a liquid
into a distillate and a residue, a vapor can be partly condensed into a
condensate and a residual vapor. The results obtained are exactly

GENERAL METHODS OF FRACTION ATION

103

analogous to those for the successive distillation and, by a similar series
of successive vaporization and partial condensation, similar separation
can be effected. In fact, successive distillation and successive frac-
tional condensation can be combined to increase the efficiency of the
operation.

Multiple Distillation. Suppose an apparatus as in Fig. 5-3 consist-
ing of a series of distilling kettles A, J3, C, etc., each kettle containing
a heating coil and necessary connection for vapors and liquid. Sup-
pose that kettle A contains a liquid mixture of carbon disulfide and
carbon tetrachloride of the composition x& as in Fig. 2-1; the kettle J5,

FIG. 5-3. Diagram of multiple distillation.

a liquid of composition x\; the kettle C, a composition of #2; and so on.
The liquid in A boils at t^ that in B at t^ and that in C at U. Since the
vapor leaving A is at a temperature U which is higher than the boiling
temperature t% in 5, then, if the vapor from A is led into the heating
coil of J5, it will give up its heat to the contents of JS, boiling the liquid
and itself being partly condensed. The vapor from B, if led into the
heating coil of C will in the same way boil the liquid in C, the vapor
being itself condensed as before. The condensed vapors in the coil
may be drawn off into receiver D, E, and F, etc. However, since the
composition of the liquid in B was selected to be the same as that of
the vapor coming from the kettle A, from Fig. 2-1, the condensed
v$,por in the coil can be allowed to mix with the contents of B instead
of being withdrawn into the receiver E. Now since the vapor from A
is being mixed with the liquid in B and since there is a heat inter-

104

FRACTIONAL DISTILLATION

change between the two, it is much simpler to blow the vapor directly
into the liquid thus dispensing with the coil.

The vapor leaving still B will be richer in carbon disulfide than the
vapor from still A, and the liquid in B will therefore tend to become
poor in carbon disulfide. The concentration of the liquid in B can
be maintained constant by adding liquid from C which is rich in carbon

-i-

FIG. 5-4. Schematic diagram of rectifying column.

disulfide, and removing liquid from B and adding it to A. By a similar
procedure the operation of the other stills can be maintained at a
steady state. A little consideration of Fig. 2-1 will show that to make
the system operate at the compositions indicated would require that
all the vapor from C be condensed and returned through line L to C and
that no liquid be withdrawn.

Rectification. An apparatus in which this direct interchange of
heat, condensation, and evaporation can take place is called a rec-
tifying tower, and the process carried on within it is called rectification.

Such a system is shown in Fig. 5-4, where 8 is the still body, or ket-
tle, Resting on the outlet of the still is a column divided into com-

GENERAL METHODS OF FRACTION ATION 105

partments by plates perforated with small holes. Each plate has an
overflow pipe discharging into a pool of liquid on the plate below.
The layer of liquid on each plate is prevented from passing down
through the holes by vapor which is rising up through these holes from
the compartment next below. Any excess liquid accumulated on the
plate flows down through the overflow pipe. The letters on the
apparatus correspond to those of Fig 5-3. The vapor from the still at
temperature i* and composition x\ passes up and exchanges heat and
molecules with the liquid in compartment B. A binary vapor of the
composition x% is produced which bubbles up through the liquid on the
next higher plate which is richer in carbon disulfide with the composi-
tion xz. Here again exchange between the vapor and liquid takes
place, and a vapor of composition x& even richer in the carbon disulfide
is produced. This can be repeated any number of times, and the vapor
finally issuing from the apparatus at the top and into the condenser is
practically pure carbon disulfide. As in the previous case, each one of
the compartments in the column may be considered a small still, in
which the source of heat is the hot vapor coming from below and the
cooling element is the cooler liquid from the plate above.

The quantitative relationships given are valid only in case the
molal ratio of liquid overflowing from plate to plate to the vapor flow-
ing through the plate is practically unity; i.e., the ratio of distillates to
liquid vaporized is exceedingly small. In practice, less overflow must
be employed to reduce the heat consumption, and the rate of enrich-
ment is less rapid than that indicated in the explanation.

The analogy between this fractionating column and the series of
kettles would be better if the vapor leaving the liquid on the plate
had the equilibrium composition assumed. But, unfortunately, no
design has been able to prevent some of the vapor from the plate
below from passing through the liquid on the plate without coming into
equilibrium with it. The vapor above any plate, therefore, will con-
tain less of the volatile component than would be the case if complete
equilibrium were reached. This ideal case is discussed here because it
brings out clearly the nature of the underlying phenomena, The more
practical cases will be considered in Chap. 7 on the Rectification of
Binary Mixtures.

The interchange between the vapor bubbles and the liquid o"n the
plate is a result of the fact that the two are not at equilibrium. Thus
in the ideal case considered a vapor of composition x\ was bubbling
through a liquid of the same composition. The vapor in equilibrium
with a liquid of composition x\ would have been #2. Thus, as the sys-

106 FRACTIONAL DISTILLATION

tern tends to approach equilibrium, carbon tetrachloride molecules
pass from the vapor to the liquid and carbon disulfide molecules will
pass from the liquid to the vapor. The number of molecules passing
in the two directions will be essentially equal since in most cases the
energy released when one molecule of carbon tetrachloride goes into
the liquid phase will be about equal to that required for vaporizing one
molecule of the carbon disulfide. Thus the total number of molecules
in the vapor tends to remain about constant. This interchange
between the vapor and the liquid is governed by the usual mass-trans-
fer mechanism, and the rate of exchange increases with the amount
of interfacial area and the turbulence involved. A close approach to
equilibrium is desired, and the equipment is designed to give intimate
contact between the two phases. Besides the bubbling action already
described, the process produces a considerable amount of spray, and
there is also an interchange between the vapor and the liquid droplets
above the main body of liquid that is helpful in obtaining a closer
approach to equilibrium.

This process of countercurrent contact of a vapor with a liquid
whicK has been produced bv partial condensation of the vapor is
termed rectification. Its result is equivalent to a series of redistilla-
tions with the consumption of no additional heat and is analogous in
this" respect to multieffect evaporation. However, it is only the result
that is similar and not the mechanism of obtaining it.

CHAPTER 6
-SIMPLE DISTILLATION AND CONDENSATION

Simple Distillation. Distillation without rectification can be
carried out by several methods. The two most generally considered
cases are (1) continuous simple distillation and (2) differential distilla-
tion. In continuous distillation, a portion of the liquid is vaporized
under conditions such that all the vapor produced is in equilibrium
with the unvaporized liquid. In differential vaporization, the liquid
is vaporized progressively, and each increment of vapor is removed
from contact with the liquid as it is formed and, although each incre-
ment of vapor can be in equilibrium with the liquid as it is formed, the
average composition of all of the vapor produced will not be in equi-
librium with the remaining liquid.

Continuous Simple Distillation. Distillations that approximate
this type are usually carried out on a continuous basis such that the
liquid feed is added continuously to a well-mixed still in which a definite
fraction is vaporized and removed and the excess unvaporized liquid is
withdrawn from the still. An alternate arrangement is to preheat the
feed and add it to a flash or disengaging section where vapor and liquid
are separated and removed without additional heat requirements. In
either case, assuming that the vapor and liquid leaving are in equi-
librium with each other, the two fractions are related to each other by
equilibrium constants and material balances. Thus for each com-
ponent in a mixture the following material balance can be written :

Vy + Lx = Fz (6-1)

where F, L, F = mols of vapor, unvaporized liquid, and feed, respec-

tively
y, x, z = mol fractions of components in corresponding

streams
By over-all material balance V + L = F and the fraction vaporized,

Thus if the fraction vaporized, the feed composition, and the relation

107

108 FRACTIONAL DISTILLATION

between y and x are known, both the vapor and the liquid compositions
can be calculated. In order to determine the equilibrium relation-
ships either the temperature or the total pressure must be known.
Given either of these, the other can be determined from the equilibrium
relationship for all the components involved.

Differential Distillation. This type of distillation is usually carried
out as a batch operation although continuous units may also- operate
in this manner. Considering first a batch distillation, if a mixture of
liquid is distilled, the distillate contains a greater portion of the more
volatile material than the residue, and as distillation proceeds both the
distillate and the residue become poorer in the more volatile compo-
nents. This change in composition may be estimated quantitatively if
the relation of the composition of vapor to that of the liquid is known.
Consider W parts of original mixture containing x0 fraction of com-
ponent A. Allow a differential amount — dW to be vaporized of a
composition, y, under such conditions that the vapor is continually
removed from the system.

By material balance,

-y dW = ~-d(Wx)

= -W dx - xdW
W dx

— ni .,_-. ff>

fwdW= /•'• dx

Jw. W Jx.y-;

In W I" dX to tt

ln W. = A, ?=-* (6"3)

This equation was developed by Rayleigh (Ref. 2) and is often
termed the Rayleigh equation. It can be used with W as weight and
x as weight fraction, or with W as mols and x as mol fraction. It is
usually applied on the basis that, at any given instant, y is in equi-
librium with x, but the derivation does not require this condition. A
similar equation applies to each component in a mixture.

The use of Eq. (6-3) requires the relationship between y and x and,
even if they are assumed to be in equilibrium with each other as the
vapor is formed, it is usually difficult to express the equilibria mathe-
matically for a general integration. The integration can be performed
graphically if the relationship between y and x is available.

Batch Distillation Example. As an illustration of the use of this equation, con-
sider the experiment performed by Rayleigh. 1,010 g. of a 7.57 mol per cent solu-
tion of acetic acid in water was distilled until the still contained 254 g. whose com-

SIMPLE DISTILLATION AND CONDENSATION

109

position was 11 mol per cent acetic acid. Assuming that the vapor leaves in
equilibrium with the liquid, calculate the final composition to be expected on the
basis of Eq. (6-3). Equilibrium data for the system acetic acid-water are given in
Table 6-1.

There are several approximate methods that may be followed in integrating Eq.
(6-3). First, for small temperature and composition ranges, the relation between
the vapor and liquid may be approximately represented by a straight line, or
y = cXj where c is a constant.

w_ _ f* dx _ /•*

JFo Jx0 co; - a? /*,

TABLE 6-1

Mol fraction
of acetic acid
in liquid, x

Mol fraction
of acetic acid
in vapor, y

c-«

Relative
volatility
of acetic acid
to water, a

0 0677

0 0510

0.75

0 74

0 1458

0.1136

0 2682

0.2035

0.76

0.70

0 3746

0.2810

0.4998

0.3849

0.77

0.625

0.6156

0.4907

0 7227

0 6045

0.84

0.59

0.8166

0.7306

0 9070

0.8622

0.95

0.64

Clearing of logarithms,

W (x V/'-i ^ /TFV"1

™r - (-) or :r*(ir)

rr o \^o/ Xo \ rr Q/

From the table a value of c •» 0.75 is appropriate and

254

(6-4)

)0 76-1

0.0757
x = 0.107 (10.7 mol per cent)

The calculated value is in good agreement with the experimental result and
indicates that the various assumptions made are reasonably satisfied.

Another method is by the use of the relative volatility which is denned by
Eq. (3-5).

VA/XA ^ a
VB/XB aAB

where <XAB is the relative volatility of component A to component B. For a large
number of mixtures, the variation of a. with composition is small, and an average
value may be employed. For a binary mixture, the expression can be rewritten

CfXA

1 + (a - l)xA

110 FRACTIONAL DISTILLATION

The use of this relation with Eq. (6-3) gives

. W f* da

In Tjnr =* I

W0 I ctx

JXo 1 + (« -

a - 1 x9(l — x) I — x

Using a » 0.74 (see Table 6-1) gives x - 0.106.

In other cases, it will be found that the variations in c and a are so great that
these methods are not satisfactory. In such cases they can be applied successively
over small concentration ranges but graphical integration is usually preferable.

There is an alternate form of the differential distillation equation
that is frequently more convenient to use than Eq. (6-3). Consider a
mixture containing A mols of one component and B mols of some other.
Let a differential quantity of vapor be produced containing — dA and
— dB mols of the two components, respectively, then assuming vapor-
liquid equilibria,

dA A ,c ,jv

-=dB = aB (6'6)

A similar equation can be written between any two components of the
mixture. If the relative volatility is constant, the equation can be
integrated directly,

M
) A.

where A0, B0 =* mols of the two components in still at some base time
A, J5 = mols in still at some later time

i A i B

In -- = a In --

A (0

If the vapor formed is not in equilibrium with the liquid, the value of
a in Eq. (6-6) will have to be modified to express the true relationship.

Batch Dehydration of Benzene. As another example of the use of Rayleigh's
equation, consider the dehydration of benzene. Benzene saturated with water at
20°C. contains 0.25 mol per cent water, and it is to be given a simple differential
distillation at a constant pressure of 1 atm. The operation is to proceed until the
mol per cent water in the liquid remaining in the still is 0.00025. The following
data and simplifying assumptions will be used in the calculations:

1. Over the temperature range involved it is assumed that the vapor pressure
of pure benzene is 2.1 times the vapor pressure of water. *

SIMPLE DISTILLATION AND CONDENSATION 111

2. At the distillation temperature, benzene saturated with water contains
1.5 mol per cent water, and water saturated with benzene contains 0.039 mol
per cent benzene.

3. It is assumed that the vapor leaves in equilibrium with the liquid.

4. No condensate will be returned to the still.

5. For the two single-phase regions of benzene containing water and water
containing benzene, it is assumed that Raoult's law applies to the component in
large amount, and Henry's law to the component in small amount.

Solution. By a procedure similar to that employed for ether and water in
Chap. 4, it is possible to calculate the relative volatility. The solubility of water
in benzene is higher at the distillation temperature than at 20°C., and only a
benzene phase will be present in the still. The partial pressure of benzene will
follow Raoult's law.

PB - XaPn

For water the partial pressure will follow Henry's law with a constant such that a
mol fraction of 0.015 will give a partial pressure equal to that over water saturated
with benzene. Thus,

«
Pw 0.015

and the relative volatility of water to benzene is

0.015P*
= 31.7

Even though the vapor pressure of benzene is over twice that of water at the
distillation temperature, the volatility of water in benzene is over 30 times that of
the benzene because of the abnormalities indicated by the low mutual solubilities.

The value of the relative volatility can be used with Eq. (6-5) and gives

W _ 1 0.0000025(0.9975) , 0.9975

Wo ~ 31.7 - 1 ln 0.0025(0.9999975) "*" 0.9999975

w. - °-796

Therefore, 20.4 per cent of the charge should be vaporized.

The example given assumed that no condensate would be returned to the still.
Actually the condensate will break in two layers, a water layer and a benzene
layer, and the benzene layer saturated with water could be returned continually
to the still for redistillation or it could be stored and added to a subsequent cycle.
The amount of heat required for the drying operation could be reduced by rectifica-
tion or partial condensation. These operations will be considered in a later
chapter.

Steam Distillation. A common example of simple distillation is the
so-called steam distillation. The term is generally applied to distilla-
tions carried out by the introduction of steam directly into the liquid

112 FRACTIONAL DISTILLATION

in the still and is usually limited to those cases in which the solubility
of the steam in the liquid is low at the temperature and pressure in
question. It is usually applied to relatively high-boiling organic
materials which would decompose if they were distilled directly at
atmospheric pressure or to liquids that have such poor heat-transfer
characteristics that excessive local superheating would result with
indirect heating. By steam distillation a volatile organic material
may be separated from nonvolatile impurities, or mixtures may be
separated with results about equivalent to those predicted by the Ray-
leigh equation.

The molal ratio of the organic material to the steam is the ratio of
the mol fractions in the vapor, and, assuming that the gas laws apply,

22 = Hi = P2 = _££_
nw yw Pw TT — po

where n = mols

y = mol fraction in vapor

p = partial pressure

TT = total pressure

Subscripts 0 and W refer to organic and water, respectively.
If the organic material is immiscible with water, and equilibrium is
attained, po would be the vapor pressure of the organic material, and
the partial pressure of water would be the total pressure minus p0. If a
liquid water phase were present, the partial pressure of water would
haVfc to be the vapor pressure of water, thus fixing the distillation
temperature at a given total pressure. For this case, the distillation
temperature will always be less than that corresponding to the boiling
point of water at the total pressure in question. This illustrates one
of the real advantages of steam distillation. Thus high-boiling
organic material can be steam-distilled in an atmospheric pressure
operation at a temperature below 100°C. If a liquid water phase
is not present, then both the total pressure and the temperature can be
arbitrarily chosen, and the partial pressure of water is the difference
between the total pressure and the vapor pressure of the organic
material. Of course, this partial pressure of water must not be greater
than the vapor pressure of pure water, or a liquid phase will form.

If water and the material being distilled are not immiscible, the

vapor-liquid equilibria for the system in question will have to be known

in order to determine the relation between the vapor and the liquid

composition.

Steam distillation can be carried out in several manners. It is possi-

SIMPLE DISTILLATION AND CONDENSATION 113

ble to pass steam directly through the liquid without any other source
of heat. Because heat must be supplied for the vaporization of the
organic material, stearn will condense and form a liquid phase unless it
is very highly superheated. If water does condense, the distillation
temperature will be less than the boiling point of pure water, and at
such a temperature the value of po will frequently be very small and
Eq. (6-8) would require a large number of mols of steam per mol of
organic material distilled. Actually the molal ratio given by this
equation is simply the overhead vapor ratio, and any condensation of
steam an the still would be in addition to the values so obtained.
Owing to the low molecular weight of steam relative to that of the
high-boiling organic material the consumption of steam may not be
excessive. However, in some cases it may be desirable to reduce the
steam consumption. This can be accomplished by indirectly heating
the still and maintaining the distillation temperature higher than that
obtained when a liquid water phase is present. For maximum steam
economy, the temperature should be as high as is possible without
undesirable thermal effects. This higher temperature increases the
value of po and reduces that of IT — p0 thus decreasing the ratio of the
steam required per unit of organic material distilled. Another
method of reducing the steam consumption is to reduce the total
pressure. Thus, if the total pressure was reduced to the value of p0,
no steam would be required and the organic material would boil
directly. Frequently this is not feasible because of the very high
vacuum required or the undesirable heat-transfer characteristics oHhe
liquid. However, by the use of reduced pressure the amount of
steam required can be made relatively small, and if the vacuum is
adjusted such that the vapor mixture will condense with the cooling
water available, the load on the vacuum pump will be low.

Steam Distillation Example. As an example of steam distillation, consider the
separation of a mixture of two high-boiling organic acids from a small amount of
nonvolatile carbonaceous material. The steam distillation is carried out at
100°C. under a total pressure of 150 mm. Hg. The organic acid mixture contains
70 and 30 mol per cent of the low- and high-boiling acids, respectively, and at
100°C. the vapor pressures of the two acids are 20 and 8 mm. Hg. It is assumed
that the mixture of the two acids obeys Raoult's law and that they are immiscible
with water. The nonvolatile carbonaceous material is assumed to have no effect
on the vapor-liquid equilibria. It will be assumed that the vapor leaves in equilib-
rium with the liquid in the still, and two cases will be considered. In the first, the
mixture of acids will be fed continuously to a still of small capacity, and it will be
assumed that steady-state conditions have been reached in which the composition
of the organic acids in the condensate is the same as in the feed. In the second

114

FRACTIONAL DISTILLATION

case, a batch distillation of the differential type will be carried out. It is assumed
that all the sensible and latent heat is supplied either externally or by superheat in
the steam. The calculations are to determine how many pounds of steam must
be used per mol of acid recovered in each case.

Solution of Part 1. Continuous Operation. In this case the ratio of the two
organic acids in the vapor is the same as in the feed and, since Raoult's law has been
assumed for the organic acids mixture, it is possible to calculate the ratio of the
two acids in the still. Thus,

Hi « P&\ » a 5!

2/2 PzXz Xz

where x « mol fraction based on the two acids only
At, Al — original mols of the more and less volatile acids, respectively
With xt « 1 - xij

Pi 4- P2 =» PlXi 4-

- l)a?i 4-

Pi(A? 4- AQ

where PI, p* =« partial pressures of acids

Pi, PS «• vapor pressures of acids
and

pHiO » IT - (pi 4- p8)

The pounds of water required per pound mol of acid is

18(ir - pi -
Pi *f Pa

18 ^ ^^- - 1

+r;)

- 178

Solution of Part 2. In this case, it is assumed that the still is charged with the
mixture of acids and that the distillation is continued until all the acids have been
vaporized.

Considering the acids only, the differential distillation Eq. (6-6) gives

where Ai «• mols of more volatile acid in still
At •» mols of less volatile acid in still

SIMPLE DISTILLATION AND CONDENSATION

and for constant a

4» dL4«

115

By steam distillation Eq. (6-8),
-dN -dN

dA\ -f- dAz / A i . .
where N = mols of steam

^ TT - Pl - P2 _
pi +p2

The pounds of steam per pound mol of acid is

1SN

- 1

This is identical to the relation found in Part 1, and the steam requirement is the
same. Less steam per mol of acids distilled would be required in the first part of
the batch distillation, but more would be required in the last portion. The above
analysis indicates that the two differences would just balance out.

Partial Condensation. It is frequently desirable to partially condense
a vapor. Such an operation can be used to produce a separation of
the components but, in general, it is employed for obtaining a portion
as condensate for some specific purpose.

The partial condensation of a vapor mixture can produce a wide
variation in degree of separation obtained. If the condensation is
carried out rapidly, the time for interchange between the condensate
and the vapor may be so short that essentially no selective mass trans-
fer of the components occurs. In this case, the composition of the
condensate will be the same as that of the vapor. If the condensation

116 FRACTIONAL DISTILLATION

is carried out at a slow rate, mass-transfer interchange will occur and
several different degrees of separation can be obtained. The operation
can be carried out such that the condensate is essentially in equilibrium
with the uncondensed vapor, and this type of operation will be termed
equilibrium partial condensation. Alternatively, the condensation can
be carried out such that the condensate as it is formed is in equilibrium
with the vapor, but the condensate is removed continuously and thus
the total condensate would not be in equilibrium with the uncon-
densed vapor. This type of operation will be termed 'differential par-
tial condensation. In a third distinct type of partial condensation, the
vapor passes through the condenser unit countercurrent to the con-
densate. Assuming that efficient countercurrent contact is obtained,
it should be possible to produce a higher degree of separation than is
possible with equilibrium partial condensation.

Equilibrium partial condensation is handled mathematically in a
manner completely analogous to that for equilibrium distillation. It
is a type of condition that is frequently encountered in the partial
condenser of a rectification unit in which the uncondensed vapor and
the condensate flow along together and reach a close approach to
equilibrium.

Differential partial condensation is not a common type of operation,
but the countercurrent version of partial condensation is probably
approached in condensers in which the vapor flows upward condensing
on the tube walls and the condensate flows down along the walls in
contact with the upward flowing vapor. Such a wetted-wall unit will
give mass transfer between the vapor and the liquid, but in general the
sizes of the tubes desirable for heat transfer and fluid flow are such that
the interchange between vapor and liquid is relatively poor, and the
separation obtained in practice is probably not much greater than that
equivalent to equilibrium partial condensation. While a condenser
could be designed such that it would give more efficient countercurrent
action, it has been found to be more economical to design a condenser
for the liquefaction function and to obtain the desired separation of the
components by more effective means.

Analysis of Partial Condenser Data. Gunness (Ref. 1) reports experimental
data obtained on the partial condenser of a rectifying column stabilizing absorption
naphtha. The vapor from the column was passed downward through the partial
condenser, and the uncondensed vapor and condensate from the bottom of the
condenser passed together to the reflux drum where they were separated. Owing
to the concurrent flow of the vapor and the liquid, it would be expected that this
system might approximate equilibrium partial condensation. The data for a test

SIMPLE DISTILLATION AND CONDENSATION

117

taken when the pressure was 254 p.s.i.a. and the temperature in the reflux drum
was 117°F. are given in the first three columns of Table 6-2. The fourth column
gives values of the equilibrium constant obtained from Table 3-2, page 41, at the
temperature and pressure corresponding to the reflux drum. If the uncondensed
vapor and the liquid were in equilibrium with each other, their composition should
be related by the equilibrium constants. As a method of making this comparison,
the last column in the table gives the values of the vapor composition divided by
the equilibrium constant for each of the components. If the vapor and liquid were
at equilibrium, the compositions so calculated should be the same as those given
for the liquid reflux. It will be noted that the experimental composition and the
calculated values are in good agreement. In fact it is probable that the agree-
ment i& within the accuracy of the experimental data and the equilibrium con-
stants. The close agreement indicates the reliability of the experimental data

and the applicability of the vapor-liquid equilibrium constant to this system.
v

TABLE 6-2

V V i t
Coirfp^nt

Residue gas, y

Liquid reflux, x

** - 1

CH4

0.053

0.007

0.0044

C2H4

0.011

0.002

3.6

0.003

C2H6

0.146

0.0618

2.55

0.057

C8H6

0.140

0.12

1.05

0.133

C8H8

0.537

0.580

0.94

0.572

t-C4

0.081

0.160

0.50

0.162

n-C4

0.032

0.069

0.40

0.080

Nomenclature

mols of A in still
mols of B in still
c = constant
F = mols of feed

« mols of unvaporized liquid
= mols of steam
•• mols

• partial pressure

• vapor pressure
« mols of vapor

W ~ mols of liquid in still
x = mol fraction in liquid
y » mol fraction in vapor
z « mol fraction in feed
a •» relative volatility
TT « total pressure

A
B

N .
n

P
P
V

References

1. GUNNESS, Sc.D. thesis in chemical engineering, M.I.T., 1936.

2. RAYLEIGH, Phil Mag., 6th series, 4, 521 (1902).

CHAPTER 7
RECTIFICATION OF BINARY MIXTURES

The separation of two liquids from each other by fractional distilla-
tion may be accomplished in two general ways: (1) the batch, or inter-
mittent, method and (2) the continuous method. In the former, the
composition and temperature at any point in the system are changing
continually; in the latter, conditions at any point are constant.

It will be recalled that a fractionating column consists of a system
up through which vapors are passing and down through which a liquid
is running, countercurrent to the vapor, the liquid and vapor being in
more or less intimate contact with each other. Furthermore, the
vapor and liquid tend to be in equilibrium with each other at any point
in the column, the liquid and vapor at the bottom of the column being
richer in the less volatile component than at the top. It is evident,
therefore, that the action of such a column is similar to that of a
scrubbing or washing column, where a vapor is removed from a gas
that is passing up through the column, by bringing into contact with it,
countercurrent, a liquid in which the vapor is soluble, and that will
remove it from the gas.

SorePs Method. Sorel (Ref . 17) developed and applied the mathe-
matical theory of the rectifying column for binary mixtures. He cal-
culated the enrichment, the ^change in composition .from plate to plate,
by making^energy and material balances around each plate and
assumecf that equilibrium was attained between the vapor and liquid
leaving the plate. He proceeded stepwise through the column T>y
applying this method successively from one plate to the next.

Owing to the steady-state condition involved in continuous dis-
tillation, its analysis is simpler than batch operation and so will be
considered first.

The equations for SorePs method will be derived for the case illus-
trated in Fig. 7-1. The column is assumed to be operating continu-
ously on a binary mixture with the feed entering on a plate between
the top and bottom. The column is provided with heat for reboiling
by conduction such as steam coils in the kettle; the case of the use of
live or open steam will be considered later. A simple total condenser

118

RECTIFICATION OF BINARY MIXTURES

119

is assumed where all of the overhead vapor is liquefied, this condensate
being divided into two portions, one of which is returned to the column
for refly ~c and the other withdrawn as overhead product. The bottoms
are cf atinuously withdrawn from the still or reboiler. (See end of
chapter for nomenclature.)

Consider the region bounded by
the dotted line in Fig. 7-1. , The
only material entering this section
is the vapor from the nth plate
Fn, while leaving the section is
the distillate D and the overflow
from the (n + l)th plate On+i.
By material balance,

Condenser

m+i

m-f

Vn - On+1 + D (7-1)

Considering only the more volatile
component, the mols entering this
section are the total mols of vapor
from the nth plate multiplied by
the mol fraction of the more vol-
atile component in this vapor
Vnyn. Likewise, the mols of the
more volatile component in the
distillate are DX&; and in the
overflow from the (n + l)th plate
are On+i£n-fi. A material balance
on the more volatile component for this section therefore gives

FIG. 7*1. . Diagram of continuous distills
tion column.

Vny« =
yn -

On+1

DxD

V n

(7-2)

(7-2a)

(7-26)

Thus, starting at the condenser, the composition of the reflux to the
tower, with the type of condenser employed, is the same as the com-
position of the distillate, which makes the composition of the vapor to
the condenser the same as that of the distillate. The mols of vapor
from the top plate are equal to OR + Z>, and the reflux to this plate is
OR, Sorel's assumption of the vapor and liquid leaving the plate being
in equilibrium, called a theoretical plate, makes it possible to calculate

120

FRACTIONAL DISTILLATION

the composition of the liquid leaving the top plate of the tower from
the composition of the overhead vapor and vapor-liquid equilibrium
data.

In the design of such a tower, it is generally customary to set or fix
certain operating variables such as the composition of the distillate
and of the bottoms, the reflux ratio OR/D, and the composition and
thermalljondition of the feed. With these values and a known quan-
tity of feed per unit time, by over-all material balances it is possible
to calculate D, 0Rl and W. To calculate the composition of the vapor

0.2 0.4 0.6 0.8 1.0

Mol Fraction Benzene in Liquid
Fia. 7-2. Equilibrium curve for benzene-toluene mixture.

from the plate below the top plate by Eq. (7-2a), it is necessary to
know the mols of overflow from the top plate and the mols of vapor
from the plate below as well as the known quantities D, XD, and the
mol fraction of the more volatile component in the liquid overflow from
the top plate. Sorel obtained the mols of overflow from, and the mols
of vapor to, the plate by heat (enthalpy) balances. Thus the heat
brought into the plate must equal that leaving.

0RhR + F<_i#f-i - VtHt + Otht + losses (7-3)

Equation (7-3) gives a relation between 7*_i and Ot\ if the enthalpies
are known, this equation can be solved simultaneously with Eq. (7-1)
to give Vt~i and Ot. This, in general, involves trial-and-error solu-
tions, since Ht-i is not known and must be assumed until the condi-
tions of the next plate are known. The values of F<-i and Ot obtained
in this manner are used in Eq. (7-2a) to give #t-~i. From this compo-

RECTIFICATION OF BINARY MIXTURES 121

sition the value of xt~\ is obtained from vapor-liquid equilibrium data
as well as the temperature on this plate. The value of Ht~i can then
be accurately checked, and the calculations corrected if necessary.
This operation is continued plate by plate down the tower to the feed
plate.

A similar derivation for the plates below the feed gives

Wxw

Wxw (7 . .

- (7'4a)

These equations are used in the same way as Eq. (7-2).

Because of the complexity of SqrePs method, it is usually modified
by certain simplifying assumptions. The heat supply to any section
of the column, above the feed is solely that of the vapor entering that
section. This supply of heat to the next plate goes to supply vapor
from this plate, to heat loss from the section of the tower that corre-
sponds to this plate, and to heating up the liquid overflow across this
plate. In a properly designed column, the heat loss from the column
should be reduced as far as is practicable and is generally small enough
to be a negligible quantity relative to the total quantity of heat flowing
up the column. Thus the enthalpy of the vapor per unit time tends
to be constant from plate to plate, and in order to simplify the calcu-
lations for such systems, Lewis (Ref . 9) assumed that the molal vapor
rate from plate to plate was constant except as changed by additions
or withdrawals of material from the column. This assumption also
leads to a constant overflow rate for such a section. In the case illus-
trated in Fig. 7-1, this simplifying assumption would give constant
vapor and overflow rates above and below the feed plate, but the rates
in the two sections of the tower would be different due to, the introduc-
tion of the feed. This assumption, together with the theoretical
plate concept, has been of great assistance in the analysis and design
of fractionating column. The validity of these two assumptions will
be considered in later sections.

On the basis of Lewis' assumption On+i and Vn are constant in the
section above the feed plate, and the relation between yn and x*+\
becomes a straight line with the slope equal to 0/7. Similarly, below
the feed, ym is linear in xm+i. On the basis of the operating variables
previously fixed, On+i, F», D, and XD are known, and the equation
between yn and #n+i is completely defined; likewise for ym and xm+i.

A plate on which SorePs conditions of equilibrium are attained is

122 FRACTIONAL DISTILLATION

defined as a "theoretical plate/' i.e.) a plate on which the contact
between vapor and liquid is sufficiently good so that ffie vapor leaving
the plate has the same composition as the vapor in equilibrium with
the overflow from the plate. For such~a pfate the vapor and liquid
leaving are related by the equilibrium y,x curve (see page 18). Rec-
tifying columns designed on this basis serve as a standard for compar-
ing actual columns. By such comparisons it is possible to determine
the number of actual plates equivalent to a theoretical plate and then
to reapply this factor when designing other columns for similar service.

Sorel-Lewis Method. As an illustration of the Sorel-Lewis method, consider the
rectification of a 50 mol per cent benzene and 50 mol per cent toluene mixture into a
product containing 5 mol per cent toluene and a bottoms containing 5 mol per cent
benzene. The feed will enter as a liquid sufficiently preheated so that its intro-
duction into the column does not affect the total mols of vapor passing the feed
plate; i.e., such that F» -» Vm. A reflux ratio On/A equal to 3, will be employed,
and the column wfll o'pef ate with a total condenser and indirect heat in the still.
The y,x equilibrium curve is given in Fig. 7-2.

Taking as a basis 100 Ib. mols of feed mixture, an over-all benzene material
balance on the column gives

0.5(100) - 0.95D -f 0.05TF

* 0.95D -f 0.05(100 - D)

gives

D » 50 Ib. mols

W « 50 Ib. mols
Since

On «

D 3

On - 150

Fft - On + D = 200

by Eq. (7-2a),

V* - (15%oo)*«+i + (5%oo)(0.95) - 0.75sn+i + 0.2375 (7-5)

Since a total condenser is used,

yt « XD = XR =» 0.95

from the equilibrium curve at"jT*= 0.95, x *» 0.88; i.e., xt in equilibrium with yt is
0.88,

Equation (7-5) then gives

yM - 0.76x« -f 0.2375 - 0.75(0.88) -f 0.2375 - 0.8975
by equilibrium curve, Xt-i at y^i ** 0.8975 is 0.77 and

yi_2 * 0.75(0.77) 4* 0.2375 - 0.8145

ak-i - 0.64

yt-3 - 0.75a;«-2 + 0.2375 - 0.75(0.64) -f 0.2375 - 0.7165

St., « 0.505

RECTIFICATION OF BINARY MIXTURES 123

Since thejraJue of Xt-s is close to the composition of the feed, this plate will be
taken asThe feed^E*!^ portion of

the tower must be used. Since the feed was preheated such that

Vn ** V

Y! - 200
W-SQ

.a.-*o

and

i - (5%oo)(0.05) - 1.25**+i - 0.0125

since xt-s « a?/ = 0.505

y/_i - 1.25(0.505) - 0.0125 - 0.615

aj/-1 » 0.392 from equilibrium curve

y/_2 - 1.25(0.392) - 0.0125 « 0.478

*/-« - 0.275

^/_8 - 1.25(0.275) - 0.0125 * 0.323

as/-, - 0.172

y/-4 - 1.25(0.172) - 0.0125 - 0.21

z/_4 - 0.100

y/_6 - 1.25(0.100) - 0.0125 - 0.122

z/_6 - 0.058

2//_8 - 1.25(0.058) - 0.0125 -

The desired strength of the bottoms was xw ~ 0.05; #/_6 is too high, and #/_e
is too low. Thus it is impossible to satisfy the conditions chosen and introduce
the feed on the fourth plate from the top with an even number of theoretical plates.
Hdwever, by slightly reducing the reflux ratio it would be possible to make s/-8
equal to xw , or by increasing the reflux ratio to make #/_6 equal to xw* In general,
such refinements are not necessary, and it is sufficient to say that between eight
and nine theoretical plates are required in addition to the still, three plates above
the feed plate, the feed plate, and four or five plates below the feed, and the still,
approximately 8M- The percentage difference between eight and nine is much
less than the accuracy with which the ratio of actual to theoretical plates is known;
whichever is used, a sufficient factor of safety must be utilized to cover the varia-
tion of this latter factor.

McCabe and Thiele Method (Ref . 11). By the Sorel-Lewis method,
the relation between yn and xn+i is a straight line, and the equation of
this line may be plotted on the y,x diagram. Thus, for the example
worked in the preceding section,

0.2375

This is a straight line of slope 0.75 *» On/Vn which crosses the y *» x
diagonal at yn » x«+i - 0.95 - XD. On the y,x diagram for benzene-
toluene, a line of slope 0.75 is drawn through y * x « XD (see line

124 FRACTIONAL DISTILLATION

AB, Fig. 7-3). Likewise, below the feed,
0

- 0-0125

This represents a straight line of slope Om/Vm = 1.25 and passes
through the y = x diagonal &t x — Xw ~ 0.05 (see line CD, Fig. 7-3)
These two lines are termed the operating lines, since they are deter-

0.2 0,4 0.6

x=Mol Fraction Benzene in Liquid

FIG. 7-3. McCabe and Thiele diagram.

0.8

1.0

mined by the tower operating conditions, AB being the operating line
for the enriching section and CD the operating line for the stripping,
or exhausting, section. To determine the number of theoretical
plates ty £ig. 7-3, start at XD] as before, yt = XD == 0.95, and the value
of Xt is determined by the intersection of a horizontal line through
yt ** 0.95 with the equilibrium curve at 1, giving xt = 0.88. Now,
irustead of using Eq. (7-2a) algebraically as in the Sorel-Lewis method,
it is used graphically as the line AB. A vertical line at xt = 0.88
intersects this operating line at 2, giving yt~i - 0.89. By proceeding

RECTIFICATION OF BINARY MIXTURES 125

horizontally from intersection 2, an intersection is obtained with the
equilibrium curve at 3. Since the ordinate of intersection 3 is yt-i, the
abscissa must be the composition of the liquid in equilibrium with this
vapor; i.e., Xt-\ = 0.77. As before, the intersection of the vertical
line through the point 3 with the operating line at 4 gives the y on the
plate below, or yt~2 = 0.815. This stepwise procedure is carried down
the tower. At intersection 8, xt-s is approximately equal to xy; and
at this plate, the feed will be introduced. The stepwise method is now
continued, using the equilibrium curve and the operating line CD.

Such a stepwise procedure must yield the same answer as the previ-
ous calculations, since it is the exact graphical solution of the algebraic
equation previously used. It has a number of advantages over the
latter method: (1) It allows the effect of changes in equilibrium and
operating conditions to be visualized. (2) Limiting operating coudi-
tion§~afe easily determined, and if a column contains more than two or
thTee~ptates, it is generally more rapid than the corresponding algebraic
procedure. Because of the importance of this diagram it will now be
considered in further detail.

Intersection of Operating Lines. In Fig. 7-3, the operating lines
intersected at x = XF- This intersection is not fortuitous, since the
positions of the two operating lines are not independent but are related
to each otherHby the composition and thermal condition of the feed.
This~relation is mtf^Teasity "showtf by : writing a Heat balance around
the feed plate. Let p be the difference between the mols of overflow
to and from the feed plate divided by the mols of feed.

P = °f+1~ °f (7-6)

A material balance gives

p + 1 = Vf ~FVf~l (7-6a)

Let Xi and yi be the coordinates of the intersection of the operating
lines. At this intersection, yn must equal ym, and xn must equal xm.
An over-all material balance on the more volatile component gives
DXD + Wxw = FzFj where ZF is the average mol fraction of this com-
ponent in the feed. Writmg Eqs. (7-2a) and (7-4) for the intersection
and using the values y% and x%J

- DxD (7-2a')

- Wxw (7-4')

126 FRACTIONAL DISTILLATION

and subtracting,

(7, - 7/-OW - (0/+i - 0,)a* + Dar^
« (0/+i - 0/)a* + FzF
(Vf - 7/-Jy, _ /Q/+! - O

- -

Substituting values of p and p + 1 gives the point on the diagram at
which the intersection must occur.

(p + l)yi - pxi + zf

Equation (7-7) together with Eq. (7-26) gives

— (7-8)

and

=
' (0/D) - p

This line of intersections crosses the y = x diagonal at

y. = Xi = 2j.

and has a slope of p/(p + 1). The effects of various values of p are
shown in Fig. 7-4 for a given slope of the operating line above the feed.
Thus, if p « 0, the mols of overflow above and below the feed are
equal, and the operating lines must intersect in a horizontal line
through the diagonal at Zr. A value of p = — 1, i.e., F/ = 7/-i,
would put the intersection on a vertical line at ZF.

The value of p is best obtained by an enthalpy balance around the
feed plate. However, when the molal enthalpy of the overflow from
the feed plate and the plate above is essentially the same and the
enthalpy of the vapor from the feed plate and the plate below is also
the same, then, by Eqs. (7-6) and (7-6a), — p becomes approximately
the heat necessary to vaporize 1 Ib. mol of the feed divided by the
latent heat of vaporization of the feed. Thus, an all-vapor feed at its
boiling point would have a value of p = 0, for an all-liquid feed at its
boiling point, p would equal — 1; p would be less than — 1 for a cold
feed, between — 1 and zero for a partially vaporized f eed7 and greater
than zero for a superheated vapor feed.

RECTIFICATION OF BINARY MIXTURES

127

A little study of Fig. 7-4 indicates that for a given 0/D fewer plates
are required for a given separation the colder the feed. This results
from the fact that the cold feed condenses vapor at the feed plate and
increases the reflux ratio in the lower portion of the column. This
higher reflux ratio is obtained at the expense of a higher heat con-
sumption in the still.

xw Mol Fraction in Liquid

FIG. 7-4. The effect of the thermal condition of the feed on the intersection of the
operating lines.

1, pis greater than 0 (superheated vapor feed)

2, p - 0 (O/+i - Of)

3, 0 > p > — 1 (partly vapor feed)
4fp - -KF/-1 - F/)

5, p < — 1 (cold feed)

Logarithmic Plotting. When the design involves low concentrations
at the terminals of the tower, it is necessary to expand this part of the
diagram in order to plot the steps satisfactorily. This may be done by
redrawing these regions of the y,x diagram to a larger scale. In some
cases, it may be necessary to make more than one expansion of suc-
cessive portions of the diagram. Alternately, the y,x diagram may be
plotted on logarithmic paper, and the steps constructed in the usual
manner. On this type of plot in the low-concentration region, the
equilibrium curve is generally a straight line, since, for small values of

becomes y — ax] however, the operating line

x:y

1 + (a - l)x

128

FRACTIONAL DISTILLATION

which is of the form ym = axm+i + 6 is a curved line unless 6 = 0.
The operating line is constructed from points calculated from the
operating-line equation.

Minimum Number of Plates. The slope of the operating line above
the feed is On/Vn, and as this slope approaches unity the number of
theoretical plates becomes smaller. When On/Vn is equal to 1, 0R/D
is equal to infinity, and only an infinitesimal amount of product can
be withdrawn from a finite column. Frequently it is assumed that
total reflux corresponds to the addition of no feed or to the removal of
no products. If such is the case, the tower is not meeting the design
conditions. It is better to visualize a tower with an infinite cross
section, which is separating the feed at a finite rate into the desired

products. Under such conditions

the column is said to operate at
total reflux or with an infinite re-
flux ratio, and both operating lines
have a slope of unity causing them
to coincide with the y = x diago-
nal. Since a higher reflux ratio
than this is not possible, the size of
the steps on the y,x diagram is a
maximum, and a minimum num-
ber of theoretical plates to give a
given separation is obtained.
This number is determined by
simply using the y = x diagonal
as the operating line and con-
A column with the minimum num-
ber of plates serves as a reference below which no column with fewer
plates can give the desired separation, but such a column would have
a zero capacity per unit volume and would require infinite heat con-
sumption per unit of product.

Minimum Reflux Ratio. In general, it is desired to keep the reflux
ratio small in order to conserve heat and cooling requirements. As
the reflux ratio 0R/D is reduced from infinity, the slope of the operating

x«Mol Fraction in Liquid
FIG. 7-5. Plot for minimum reflux ratio.

structing the steps from XD to xw .

linc 0* „ 0/D
Vn (0/D) + I

decreases from unity. Thus, in Fig. 7-5 a reflux

ratio of infinity would correspond to operating lines coinciding with
the diagonal as acb, and a lower reflux ratio would correspond to adb.
It is obvious that the average size of the steps between the equilibrium
curve and the line adb will be much smaller than the size of the steps

RECTIFICATION OF BINARY MIXTURES 129

between the equilibrium curve and the line acb. Thus, a reduction of
the reflux ratio requires an increase in the number ^TflEeoretical plates
to effect a given_sejgara|raa; S1the"re9ux ratio iifuriber decreased,
thcTsfze oTthe steps between the operating lines and the equilibrium
curve becomes still smaller, and still more theoretical plates are
required, until the conditions represented by aeb are encountered, when
the operating line just touches the equilibrium curve. In this final
case, the size of the step at the point of contact would be zero, and an
infinite number of plates would be required to travel a finite distance
down the operating line. The reflux ratio corresponding to this case
is called the minimum reflux ratio and represents the theoretical limit
below wMch^tMtf*^^ Be ireHuced and produce the desired

separation even if an infinite column is employed. This reflux ratio
is easily determined by laying out the operating line of the flattest
slope through XD that just touches but does not cut the equilibrium

curve at any point; the slope of this line ^ = (n/ ^ ._ . gives the
value of 0/D. Alternately, it may be calculated from the equation

(7-10)
'

D yc - xc

where xc and yc are the coordinates of the point of contact. For mix-
tures having normal-shaped equilibrium curves, such as benzene-
toluene, the point of contact of the operating line with the equilibrium
curve will occur at the intersection of the operating lines. For cases
that deviate widely from Ilaoult's law, the operating line may become
tangent to the equilibrium curve before the intersection of the operat-
ing lines touches the equilibrium curve, and in such cases it is usually
best to plot the diagram and determine the slope On/Vn>

Optimum Reflux Ratio. The choice of the proper reflux ratio is a
matter of economic balance. At the minimum reflux ratio, fixed
charges are infinite, because an infinite number of plates is required.
At total reflux, both the operating and the fixed charges are infinite.
This is due to the fact that an infinite amount of reflux and a column
of infinite cross section would be required for the production of a finite
amount of product. The tower cost therefore passes through a mini-
mum as the reflux ratio is decreased above the minimum. The costs
of the still and condenser both increase as the reflux ratio is increased.
The heat and cooling requirements constitute the main operating
costs, and the sum of these increases almost proportionally as the

130

FRACTIONAL DISTILLATION

reflux ratio is increased. The total cost, the sum of operating and
fixed costs, therefore passes through a minimum.

Optimum Reflux Ratio Example. The following estimates illustrate these
factors for the fractional distillation of a methanol-water mixture to produce 250
gal. of methanol per hour. In making the calculations, it was assumed that the
heat-transfer surface required was proportional to the vapor rate which is equal to

D ( £ + 1 ) and that the tower costs were proportional to the total square feet of

plate area. The charges on the equipment, including maintenance, repairs, depre-
ciation, interest, etc,, were taken at 25 per cent per year, and the heating costs were
based on the heat load. The costs per hour as a function of the reflux ratio are
summarized in Table 7-1 . Labor charges have been excluded since these should be

TABLE 7-1. ESTIMATED COST FOB THE FRACTIONATION OF A METHANOL- WATER

MIXTURE

Costs, dollars per hour

Charges
on tower

Charges on con-
denser and reboiler

Steam and
cooling water

Total

0 65

0.39

0.023

0.33

0.68

0.40

0.11

0.024

0.335

0.47

0 71

0.41

0 09

0.025

0.34

0 455

0.84

0 46

0.064

0.026

0.37

0.46

1.1

0 52

0 057

0.030

0.42

0 51

1.6

0.61

0.06

0.037

0.52

0.62

2.6

0.72

0.07

0.051

0.71

0.82

6.5

0.87

0.12

0.017

1.48

1.71

1.0

relatively independent of the reflux ratio. These results are plotted in Fig. 7-6.
It will be noted that the total of these costs passes through a minimum at a reflux
ratio, 0/V, equal to about 0.43, (0/D - 0.75). This is very close to the mini-
mum reflux ratio, 0.65 and is a result of the fact that the heating and cooling costs
are large and increase rapidly with the reflux ratio.

The calculated economic reflux ratio for most cases is so close to the minimum
reflux ratio that the accuracy of the latter becomes a critical matter. In the pres-
ent case, the most economical reflux ratio, 0/D, is only 15 per cent above the mini-
mum, and it is doubtful whether the equilibrium data available are sufficiently
accurate to make the calculation of the minimum reflux ratio better than ±10
per cent. For this reason, it is industrial practice to employ a reflux ratio some-
what higher than the most economical, and values of J.3 to 2 times the minimum
reflux ratio are common. For the case here considered, reflux ratios in this range
would give operating costs only slightly greater than the minimum. This small
increase in cost gives a design that will be less sensitive to slight inaccuracies in the
data employed.

RECTIFICATION OF BINARY MIXTURES

131

One of the important pieces of data needed for such an economic
study is the number of theoretical plates as a function of the reflux
ratio. Approximate methods for the rapid estimation of such data are
given in Chap. 12, page 348.

0.9
08
,07

•?0.6
S.

I05
•"0.4

02
O.I

°c

To o»

*/7

/ /i

f '

/ u —

Toe

a/V

r -O.v

***«..•»

-Cost
-Chat
-Chat
- To/a

sofs

*ges i

"ges c
r/of.

team
in fra
m sti
A*B*

Me/cooling watei
icfionafing co/urr

fl and conofenset
/%

C
n

'.§

^^j

\ — i

$ -0.39

— B~

.To oo at

to? •

7->

,3 0.4 0.5 06 07 0.8 09 1

0.
V
FIG. 7-6. Optimum reflux ratio.

Feed-plate Location. One step between the equilibrium curve and
the operating line for the enriching section corresponds to one the-
oretical plate in the enriching section above the feed, and one step
between the equilibrium curve and the other operating line corresponds
to one theoretical plate below the feed. Therefore the step that passes
from one operating line to the other corresponds to the feed plate.
Thus, in Fig.TjT^jr^^ the operating line

a6cj it is not possible toji£iLQp.-tihe- ^p^M^g^3^-^^ " ntil the yalue jpf
x IslesS tliiaKTHe^Sue correspondinjgj2,J^in^ e- However, as soon as

possible to shift to the other operating

line, but it is not necessary to do so at this value, since steps can be
continued down abc until they are pinched in at c, but a value less than
c cannot be obtained unless the shift is made. The step from one

132

FRACTIONAL DISTILLATION

operating line to the other must therefore occur at some value of x
between the values corresponding to c and e, and a change at any
value within this range will give an operable design. In general, for a
given reflux ratio, it is desired to carry out the rectification with as few
plates as possible in order to reduce the plant costs; i.e., the minimum
number of steps from a to d between the equilibrium curve and the
operating line is desired. This minimum number of steps for the
design conditions selected is obtained by taking the largest possible
steps at all points between a and d. It is obvious that for values
between e and b larger steps will be obtained between the equilibrium

curve and operating line abc than
would be obtained with operating
line dbe. Likewise, for values be-
tween c and b larger steps will be
obtained by using line dbe than by
using abc. Therefore it is desir-
able to use operating line abc for
values from a to 6, and line dbe for
values between d and 6; and by
making the feed plate, i.e.j the
shift from one line to the other,
straddle the value f>, the minimum
number of theoretical plates will
be obtained for the operating con-
ditions chosen. If a step happens
to fall directly on 6, then the feed may be introduced either at b or on
the plate below without changing conditions.

X Partial vs. Total Condenser, In the foregoing discussion the
column was assumed to be operating with a total condenser, i.e., a
condenser that completely liquefies the overhead vapor and returns a
portion of the condensate as reflux, removing the remainder as product.
However, partial condensers are quite frequently used in commercial
operations, especially where complete liquefaction of the overhead
would be difficult. In this case, only enough condensate for the reflux
to the column may be produced, and the product is withdrawn as a
vapor. In other cases, mixtures of vapor and liquid are withdrawn.
For example, in gasoline stabilizers employed by the petroleum indus-
try, where the overhead contains appreciable percentages of methane,
ethane, and ethylene, together with Cs and €4 hydrocarbons, in order
to condense the methane and C2 hydrocarbons, very low temperatures

FIG. 7-7. Diagram for limits of feed-plate
composition.

RECTIFICATION OF BINARY MIXTURES 133

would be required with resulting high refrigeration costs. However,
sufficient of the C8 and C4 hydrocarbons can be liquefied at moderate
temperatures and pressures to serve as reflux, and the remainder of the
overhead containing a large portion of the Ci and C% hydrocarbons can
be removed as vapor and sent to the gas lines.
A partial condenser may operate in any of several ways :

1. The cooling may be so rapid and the contact between condensate
and uncondensed vapor so poor that essentially no transfer of com-
ponents back and forth is obtained, with the result that the condensate
and uncondensed vapor are of the same composition. (This is possible
if part of the vapor condenses completely and the balance does not
condense at all.) In this case, the partial condenser is equivalent to
the total condenser with the exception that the product is removed as
vapor instead of as liquid.

2. The vapor product may be in sufficiently good contact with the
returning reflux for the two to be in equilibrium with each other, in
which case the partial condenser acts as a theoretical plate, and one
less theoretical plate may be used above the feed plate in the column
when this condition exists than when a total condenser is employed.
Such a condition can be approximated by requiring an overhead vapor
to bubble through a pool of reflux to the column.

3. The vapor is differentially condensed, and the equilibrium con-
densate continually removed, giving a differential partial condensation.
Alternately, the vapor may be condensed on vertical tubes such that
the condensate flows countercurrent to the rising vapor, and fractiona-
tion occurs between the vapor and condensate. Theoretically, such a
condenser can give a separation equal to a number of theoretical
plates; actually, such conditions are seldom employed, since to obtain
efficient transfer of components from vapor to liquid, low rates of con-
densation per unit area are required, thus necessitating large and costly
condensers, and, in general, it is found more satisfactory and cheaper
to obtain additional rectification by adding more plates to the column
and using a condenser to produce condensate rather than make it per-
form composite duties.

Actual partial condensers usually operate somewhere between Cases
1 and 2. For an absorption naphtha stabilizer, Gunness (Ref . 6) (see
page 116) found good agreement with Case 2. In actual design cal-
culation, the conservative assumption is to assume operation as in Case
1, and any fractionation that does occur will act as a factor of safety;
with ordinary condenser design, with the most optimistic assumption,

134

FRACTIONAL DISTILLATION

m-H

not more than one theoretical plate should be taken for the partial
condenser.

Open vs. Closed Steam. When rectifying
mixtures in which the residue is water and in
some cases where the mixture undergoing f rac-
tionation is immiscible with water, the steam
for heating may be introduced directly into the
still. Such a procedure may materially reduce
the temperature and pressure of the steam nec-
essary for the distillation by giving in effect a
steam distillation.

Considering the distillation of an ethyl alco-
hol-water mixture, the lower operating line when
a closed-steam heating is used was shown to
have a slope of (0/V)m and to pass through the

y = X line at X = Xw. In Fig. 7-8, a column Op-
erating ^^ g molg rf jjve gteam

Fia, 7-8. Diagram of
column using live steam.

material balance between the m and m + 1 plate gives

.M.I

3 - Vm + W

(7-11)

and with the usual simplifying assumptions, S would equal Fm, making
0 = W. An alcohol balance gives

y m

This is an operating line of slope Om/Vm; but at x = y, x is equal to

(w ciixw instead of xw] and at x equal to 3wr, y becomes zero
W - S/

corresponding to the composition of the vapor (steam) to the bottom
plate. For a given 0/D and feed condition, Om/Vm must be the same
whether closed or open steam is used, so that the lower operating line
must cross the y = x diagonal at the same x value in both cases, Xw for
the live steam being lower than xw for closed steam because of the
dilution effect of the steam. In stepping off theoretical plates, the
step must start at y = 0 and Xw] i.e., in Fig. 7-9, the bottom plate
corresponds to the step dbc. In such a case, the introduction of live
steam can eliminate the still, but it dilutes the bottoms and requires
more plates in the lower section of the column. Since the steps in the

RECTIFICATION OF BINARY MIXTURES

135

case of live steam start lower, it always requires at least a portion of a
step to come up to the intersection of the operating line and the y = #

100 mots Oil \ f

2,SSmols C3Hfl
4/no/s Steam

FIG, 7-9. y,x diagram for case of live steam.

diagonal, and more plates are required with live steam than with
closed steam. In Fig. 7-9, about 1J^ more plates would be necessary.
One plate is needed to replace the still,
and the additional "fraction of a
plate" is required to offset the dilu-
tion.

As an example of using open steam
to obtain a steam distillation, consider
the steam stripping of an oil containing
2.54 mol per cent propane at 20 p.s.i.
The temperature will be maintained
constant at 280°F. by internal heat-
ing. The molecular weight of oil may
be taken as 300, and 4 mols of steam
will be used per 100 mols of oil stripped.
It is desired to estimate the number of

4 mols Steam

FIG. 7-10.

fOOmo/s Oil
O.OSmols CjHg

Figure for illustration.

theoretical plates necessary to reduce the propane content of the oil to
0.05 mol per cent. The oil may be assumed nonvolatile, and the vapor-

136

FRACTIONAL DISTILLATION

liquid relation of the propane in the oil may be expressed as y = 33.4$.
It is obvious that the mols of vapor will increase up the tower, since
the steam does not condense under the conditions given, and the
propane vaporizes into it as it passes up the tower. This will cause
0/V to vary through the tower, and points on the operating line must

0.50

0.01
Mol Fraction

FIG. 7-11.

0.02
in Liquid

Steam stripping diagram.

be calculated, since it will not be a straight line on a y,x diagram.
This is easily done by taking a basis of 100 mols of entering oil, for
which the terminal conditions are given in Fig. 7-10. Now, assume
that the liquid flowing down the tower at some position contains 1.3
mols of CaHg. The vapor at this point must then contain 1.25 mols of
C3H8, giving xn+i = 1.3/101.3 = 0.0128 and yn - 1.25/5.25 « 0.238.

RECTIFICATION OF BINARY MIXTURES

137

tVn

fVs

In a similar manner, other values on the operating line are calculated
and plotted in Fig, 7-11 together with the equilibrium curve, and a
little more than six steps are required to give the desired stripping.

Side Streams. Side streams are removed from a column most
often in multicomponent mixtures;
however, they are occasionally used in
the distillation of binary mixtures.
Thus, a plant separating alcohol and
water might have uses for both 80 and
95 per cent alcohol mixtures, which of
course could be produced by making
only 95 per cent alcohol and diluting
with water to produce the required 80
per cent; or alternately liquid could be
tapped off of a plate in the column on
which the concentration was approxi-
mately 80 per cent. The proper plate
in the tower can be determined by con-
structing the usual operating lines.
Figure 7-12 illustrates the removal of
a liquid side stream L. Considering
this figure and making the usual sim-
plifying assumptions, the operating
line above the side stream is, as before,

FIG. 7-12. Diagram of continuous
column with side stream.

DxD

A material balance around the top of the column and some plate
between the feed plate and the side-stream plate gives

W. *•

Lxi

(7-12)

The operating line above the side stream passes through y = x = XD
and has a slope of (0/F)«, while the operating line below the side

stream passes through y = x = x* J*^ (i.e., the molal average

composition of the product and side stream) and has a slope (0/V)8.
Since XL is less than XD and 0, is less than On, (0« = On - L); this
latter operating line will cross the y = x diagonal at a lower value than

138

FRACTIONAL DISTILLATION

the upper operating line and will have a flatter slope. The two
operating lines will intersect at x = XL. Figure 7*13 illustrates these
lines.

Theoretical plates are stepped off in the usual manner, using the
operating line ac from a to c, the operating line bcf from c to
some value between e and /, and then the line edg from there to g.
Although the feed plate may have any composition between e and /
and the feed-plate step may be made at any value between these two,
the side-stream step must fall exactly on c. This is because the feed

FIG. 7-13. y,x diagram for column with side stream.

can be actually introduced into plates of different composition, but a
side stream has to be of the same composition as the plate from which
it was withdrawn unless a partial separation of the side stream is made
and a portion returned to the column. By altering the reflux ratio
slightly, the step can be made to fall very close to c.

Unequal Molal Overflow. 1. Above Feed Plate. The analysis
given in the preceding sections was based on constant molal overflow
rate, and it is necessary to consider the validity of this assumption.

Equations (7-1) and (7-2) can be combined with the following
enthalpy balance:

V«Hn

+ On+ihn+i + DhD + losses

RECTIFICATION OF BINARY MIXTURES 139

where Hn » molal enthalpy of vapor entering plate n + 1

Q0 = heat removed in condenser
hn+i = molal enthalpy of liquid leaving plate n + 1
hD « molal enthalpy of product

to give

Xp ~- Xn+l _ _Xj> "" J/n /7 jq%

~ MD - Hn ('~L6)

where MD equals -j? + ftx> + . °®fes or, in general, DMD equals the

total enthalpy removed from the section in question other than by
Vn and On+i. Equation (7-13) can be rearranged as

M D — H n i *fn kn+1 M /"7 i A\

n
/I ^n — hn+i\

-. i i _ - — i ^

\ MD "^ fan+l/

-- -_ -

MD ~

By comparison with Eq. (7-2) it is obvious that

fr~ — THF T

Kn MD — Aln-fl

and

0»-.l Af/) — ffn Hn —

Vn MD - An+i MD -

The condition for constant molal overflow rate is that the group,

tin — hn+i should be a constant. In general the losses are or should
M D — An+i

be small making MD a constant for a section having no additions or
withdrawals except at its ends, and the value of MD is usually large in
comparison to hn+i. Thus, a constant value of Hn — hn+i will lead to
essentially constant molal rates of vapor and overflow. The difference
Hn — An+i is not a conventional latent heat but is the difference in
molal enthalpy between the vapor entering and the liquid leaving a
plate. In order to analyze this difference, it is desirable to evaluate
Hn and hn+i*

In the case of .the liquid phase, the enthalpy will be calculated on
the basis of heating the pure components to the mixture temperature
and then mixing at this condition.

140 FRACTIONAL DISTILLATION

where (xi)n+i, (a^n+i = niol fractions of components 1 and 2 in liquid

leaving (n + l)th plate
hn+i = molal enthalpy of liquid
ci, c2 = molal specific heat

tn+i = temperature of liquid leaving (n + l)th plate

fa) fa = base temperatures for calculating enthalpy,

enthalpy of pure component 1 taken as zero

at fa, enthalpy of pure component 2 taken as

zero at £2&

Ahm = enthalpy change on mixing pure liquids to

give 1 mol of desired composition

In most cases it is desirable to choose the base temperatures reasona-
bly close to the distillation temperatures. For such cases Ci and c2
can be taken at constant average values and

fen+l = (3l)n+l(Cl)(*n+l ~ fa) + (Za)»+l(Cs)(*n+l — fa) + Akm (7-19)

In the case of the vapor enthalpy, several paths for calculating the
mixture value relative to the pure components are possible: (1) The
pure components can be heated as liquids from the base temperature
to their boiling point at the pressure under consideration, (2) the
liquids can be vaporized, and (3) the vapors can be mixed and then
heated or cooled to the desired mixing temperature. Alternately, the
pure components can be (1) heated from the base temperature to the
desired mixture temperature, (2) vaporized at this temperature, and
(3) mixed. Thus, for the two cases,

Hn = (yi)n ftlB Cl dt + (y2)n ('** C2 dt + foi)
Jtlb Jtti,

+ AHm + (yi)n f^ Ci dt + (y,)n C2 dt (7-20)
or

ci dt + (y2)n ^ c2 dt + (yi)n AH[

(7-21)
where Hn = molal enthalpy of vapor

2/i, ^2 ^ niol fractions
i, c2, fi6, fa = same as for Eq. (7-18)

tin, UB = boiling temperatures for pure components at pressure

in question

tn = temperature of vapor entering plate n *+ 1
i, AJEf2 = latent heats of vaporization of the pure components at

IB and t%B

i, &H'Z =« latent heats of vaporization of the pure components at tn
AHm, AH'm = enthalpy changes on mixing pure vapors
Ci, C2 « molal heat capacities of the pure vapors

RECTIFICATION OF BINARY MIXTURES 14

For most of the following discussion the second equation will b<
employed, and the heat capacities will be taken constant at averag<
values, giving

Hn = (yi)nCl(tn — tib) + (y^nC^(tn ~ *2&)

+ (Vl)n AH( + (y2)n AH't + AH'm (7-22

At low and moderate pressures, the enthalpy effects on mixini
vapors are small and AJET^ and AHm will be neglected, although at higi
pressures this procedure could lead to large errors.

Combining Eqs. (7-19) and (7-21),

Hn - hn+i = (yi)n &H{ + (y,)

(7-23

The values of tw and £2& can be arbitrarily chosen. For convenient
they will be taken as the boiling points of the pure components at th<
pressure in question. On this basis, (tn+i — tib) and (tn+i — £2&
seldom exceed 50 to 100°F. and (yn ~ xn+i) and (tn — tn+i) are small
As a result, the last two brackets of Eq. (7-23) are usually only a fev
per cent of the value of (Hn — hn+i) and are not of real significance ii
determining the constancy of this difference. The most importan
factors are (yi)n &H( and (y2)n AH'z, although Ahm may be large ii
some cases. As the calculations are made from plate to plate, (yi)
will vary, and (Hn — 7&n+i) will vary or remain constant depending 01
whether or not AH( equals A-H^. Thus for most cases, the criterion o
the constancy of (Hn — An+i) will be the difference in the latent hea
of vaporization of the pure components at the operating pressure.

Figure 7-14 gives the values of the molal latent heat divided by tin
absolute temperature plotted as a function of the ratio of the vapo
pressure divided by the absolute temperature. (See also page 479.
As a rule, the maximum variation in (yi AH( + y2 AH'2) is obtained ty
going from yi = 1.0 to yi = 0, although abnormal mixtures may shov
greater values at some intermediate point. As an example, con
SJder a mixture of ethanol and water at atmospheric pressure. Fo
ethanol P/T = 760/351.5 = 2.16, AH/T = 27, AH' = 9,470; fo
water P/T - 760/373 - 2.04, AH/T - 26.2,A H' = 9,750. Unles
this mixture has a very large heat of mixing for the liquid phase, i
would be expected that the value of (Hn — hn+i) would be essentiall;
constant and the assumption of constant rates of vapor and over
flow would be well justified. At higher pressures the variation i

142

FRACTIONAL DISTILLATION

of vaporization of ethanol. As another case, consider the distillation
of an ammonia-water mixture at 20 atm. abs. For ammonia

P
T

(20 X 760)

321.5
AH/T = 13.5, and AH' = 4,340; for water

P
T

47.3

(20 X 760) _
484.5 ""

AH/T = 17.0, and AJET = 8,230, and the variation in vapor and over-
flow rates from plate to plate would be large and calculations based on
the constancy of these rates would be appreciably in error.

44
40
36
32
28

20
16
12
8

0
01

III

IIMI 1 1

Nomenclature
1 -Molecular wejaht
- Latent heat,(5m, Cal/6m
" - Absolute temperatur*,°K.
"Vapor pressure Mm. Hg.

^*x

^s,

i!ll^>

•4

S|6

„ .

1 * • '^x

Sv. '

A ^5

»C

'^v

J '

sr*

*• lfev«-

vxS>

S J J

^ ^O

vs.

s|'v

"^N^

Leger
-Wbter
!-EmylA
i-EthylEI
-Ammo
i- Ethane
i-Benzer
7-n-Octa

cohol
tier
nia

le
nil

'v^s

' V

» ^L

S ^W

\ ^

i ^

1 >

\ V\

_7 1 ^

\ }

)l 0.1

1.0 to no ijooo

P/T
FIG. 7-14. Hildebrand chart for estimating latent heats of vaporization. (Ref. 19.)

It is possible to have mixtures of similar components that would
not satisfy the constant 0/V assumption even approximately and to
have mixtures of unlike components that give good agreement. Thus,
in a mixture of hydrocarbons the operating pressure could be such that
one of the components is near its critical pressure as a pure com-
ponent while the other could be at more normal conditions. Under
such conditions, the difference in latent heats of vaporization of the
two pure components could be very large. At high pressures, most
mixtures give large variation in (H n — /&w+i) because one of the com-
ponents will be nearer its critical conditions than the other.

In order to carry out more exact calculations, it is necessary to

RECTIFICATION OF BINARY MIXTURES 143

evaluate the other terms of Eq. (7-23). The specific heat" terms require
the heat capacity of the pure liquid components. Values for a number
of materials can be obtained from Fig. 2 in the Appendix. (Ref. 10.)
The value of Ahm is difficult to obtain, owing to the lack of published
data. Such data lire available in a few cases and can be calculated
if vapor-liquid equilibrium data are available at several temperatures
and pressures. Thus, by Eqs. (3-37) and (3-38),

RTd In 71 =
and

» RT In 7l
= ft?7 In T2

It can be shown that, at constant composition and total pressure,

d(*P./T) _ - _ _
d(l/T) ~~ ^e "
and

(7-24)

Thus if vapor-liquid equilibrium data are available to evaluate the
activity coefficients at constant composition as a function of the tem-
perature, the heat of mixing can be calculated. This heat of mixing,
so calculated, will be slightly in error due to neglecting the effect of
pressure on the enthalpy of liquids.

It is to be noted that Ahm is usually of such a sign as to make the
enthalpy of the liquid more nearly constant. In using Eq. (7-19)
with the base temperatures taken at the boiling point for each pure
component, the enthalpy of the liquid is equal to zero at both x\ = 1
and Xi = 0, but at intermediate values the enthalpy may be greater or
less than zero. For mixtures with positive deviations from Raoult's
law, i.e., activity coefficients greater than unity, the temperatures of
the mixtures at constant pressure are lower than for ideal solutions,
and the sum of the sensible heat terms of Eq. (7-23) is negative.
This is partly offset by the fact that (d In y/dT) is usually negative
for such mixtures, making Ahm positive. Mixtures with negative
deviations from Raoult's law give similar compensations. Thus, in
most cases the variation in the enthalpy of the liquid (with the base
temperature assumed) with composition is small.

If the Van Laar equation for the activity coefficient is used, Eq.

144 FRACTIONAL DISTILLATION

(7-24) becomes

Ahm » RT(xl In 71 + z2 In 72) (7-25)

and, using the Van Laar relations,

Ahm = **f*^ (7-26)

Comparison with experimental data for a number of mixtures of
organic liquids indicates that the Van Laar relations give high values
and that better results are obtained by

Ahm = Q.5RT(x! In 7! + x2 In 72)

The use of the modified Margules relations, Eq. (3-34a), would give a
similar relation with the constant equal to 0.25.
2. Below Feed Plate. A similar analysis using

Om+l - Vm + W

Om+1xm+1 = Vmym + Wxw
Om+ihm+i + Q* = VmHm + Whw

where Q8 is the heat added in the still, gives

ym - xw _ xm+1 - xw (?-27)

Hm - \hw - p~

9I - Mw — Hm Hm —

ym —

~ - - - m+i XT - z^ - •wr

Mw — /Im-fl MTT — km+1

- A - gm "~ ^+A r J_ -^m - Am+1

— II— J-f - jr - I Xm+l "f TTf - 1 - Xw

— —

where Mw * hw — TF?>

- hm+l

These equations are similar to Eqs. (7-16) and (7-17), and the same
considerations relative to the constancy of molal rates of vapor and
overflow apply in this case,

3. General Case. The derivations given in the two previous sections
were for the case with no side streams, but the general case is similar.

RECTIFICATION OF BINARY MIXTURES

145

In the system shown in Fig. 7-15, the over-all and component material
balances above plate a are

Fa = Oa+i + N (7-32)

F«2/a = Oa+iXa+i + NxN (7-33)

where Fa = vapor rate to plate

~D
~Li

Oa+i — overflow rate from

plate a + 1

N = net mols leaving sec-
tion = Fa — Oa+l

2/a> #a-f i — mol fractions

NXN = total mols of compo-
nent leaving section
other than with Fa
and 00+i

The enthalpy balances are

VaHa = Oa+iha+1 + NM (7-34)

where Ha = molal enthalpy of va-
por entering plate

ha+i = molal enthalpy of FIG. 7-15. Schematic diagram of frac-
liduid leaving! plate tionating system with side streams.

a + 1

NM = net enthalpy removed from the column above plate a + 1
For Fig. 7-15,

NhN = Li(hL)i + £2(^)2 + DhD — FhF + Qc + losses
These equations can be combined to give

If - Ha

M — Ha

M -

M -
= I 1 -

,
~\

where

M - h<

Hg — hg+1

M - ha+1

1 ~% r^ —

M - h

Xa-4-1 i

- ha.

'•XN

.+1

(7-35)

(7-36)

(7-37)
(7-38)
(7-39)

146

FRACTIONAL DISTILLATION

These equations reduce to Eqs. (7-14), (7-17), (7-27), and (7-31) for
the special cases considered.

Method of Ponchon and Savarit. Ponchon (Ref. 14) and Savarit
(Ref. 15) showed that Eqs. (7-13) and (7-27), or in general any equation

0 0.1 0.2 03 0.4 05 0.6 07 0.8 0.9 1.0

y or x , Mol fraction
FiQ, 7-16, Enthalpy-composition diagram.

of this type, could be easily solved by plotting the enthalpy (or other
property) of the saturated vapor and liquid vs. the mol fraction. For
example, in Fig. 7-16, if the value M D for the upper section is plotted
at XD, it is easily shown that any straight line drawn through the
point (M D, XD) will intersect the enthalpy lines to give values of x and

RECTIFICATION OF BINARY MIXTURES 147

y that will satisfy Eq. (7-13). Thus if yn is known, a line through the
vapor enthalpy curve at this composition and the point (M», XD) will
cut the liquid enthalpy at hn+i and xn+i. When xn+i is known, yn+i is
obtained from the equilibrium curve and xn+$ is determined by drawing
a new line through (yn+i, Jffn+i), (MD, xx>), etc.

Similarly if Mw for the lower section is plotted at Xw, then a straight
line through this point intersects the two enthalpy curves at values
that satisfy Eq. (7-27) and these straight lines give the relation
between ym and xm+i.

A heat and material balance around the whole column (no side
streams) gives

Fhr = Whw + DhD + QC-Q, + losses = WMW + DMD
and

Fzp = Wxw + DxD

By rearranging,

ZF — Xw

hF ~ Mw

(7-40)

Equation (7-40) is of the same type as Eqs. (7-13) and (7-27), and
similar reasoning leads to the conclusion that a straight line through
(ZP, hp) and (xw, Mw) will also pass through (XD, M&). In other
words, the point (ZF, hp) lies on a straight line between (XD, MD) and
(xw, Mw). This line will be termed the terminal tie line.

In general the same type of information given by the constant 0/V
method can be obtained by the use of the Ponchon and Savarit
method. For example, the cases of total reflux, minimum reflux
ratio, and optimum feed-plate location can be easily solved.

Total Reflux. In the case of total reflux the values of M D and M w
are infinite, and lines drawn through them and the enthalpy curves will
therefore be vertical. Thus, for this case the diagram shows that the
composition of the liquid leaving the plate is equal to the composition
of the vapor entering the plate, and the same number of theoretical
plates will be obtained by both the constant 0/V method and Ponchon-
Savarit methods, regardless of the value of the enthalpies.

Minimum Reflux Ratio. The case of the minimum reflux ratio
corresponds to conditions that require an infinite number of plates to
obtain the desired separation. As in the case of the y,x diagram, this
necessitates a region in which succeeding plates differ only differen-
tially in composition, i.e., a pinched-in region.

The step equivalent to a theoretical plate on the enthalpy diagram

148 FRACTIONAL DISTILLATION

involves going from the vapor below a plate to the liquid on a plate by
means of the enthalpy operating line through one of the terminal
enthalpy points and then proceeding from the composition of the
liquid on the plate to the vapor above the plate by the equilibrium
relationship, i.e., by an equilibrium tie line. If the composition of
this vapor above the plate is to be equal to the composition of the
vapor entering the plate, it is necessary for the enthalpy operating
line to coincide with the equilibrium tie line. In the general case it
is a trial-and-error procedure to determine the least value of MD that
can be employed. However, if the pinched-in region occurs at the
feed plate, the minimum reflux ratio can be easily determined by find-
ing the equilibrium tie line that passes through the point (hp, ZF) and
extrapolating this line until it intersects the vertical line at x&. The
enthalpy value at this intersection will correspond to the minimum
M &. In other cases, the equilibrium tie lines for a number of compo-
sitions above the feed plate can be extrapolated to the vertical line
through XD, and the maximum value of MD so obtained corresponds to
the minimum reflux ratio for this section. A similar procedure can be
used below the feed plate and the minimum value of Mw determined.
Since M /> and M w must fall on the line through the feed point, it can
be determined which of the two values is the limiting one, and thus
whether the pinched-in region is above or below the feed plate.

Optimum Feed-plate Location. The optimum feed-plate location
again corresponds to making the total number of theoretical plates
required for the operating conditions chosen a minimum, which is
equivalent to making the change in composition per plate a maximum
at all points. In general, it will be found that, when the enthalpy
operating line is on the XD side of the terminal tie line through (MD, XD)
and (Mw, xw), larger steps will be obtained by using enthalpy
operating lines drawn through (MD, XD) than those drawn through
(Mw, xw)9, on the xw side of the terminal tie line the reverse will be
true. The enthalpy operating lines should be drawn through Mw for
values of x less than the composition given by the intersection of the
terminal tie line with the liquid enthalpy curve,* and the (M&, XD)
point should be used for all operating lines corresponding to liquid
mol fractions greater than this value. It should be noted that the
value of the mol fraction at which the change-over is made is not equal
to the composition of the feed. It will be equal to the composition of
the feed when the feed enters with an enthalpy equal to that of a
saturated liquid. If the enthalpy of the feed is greater than that of a
saturated liquid, the change-over value will be at composition lower

RECTIFICATION OF BINARY MIXTURES 149

than that of the feed. If the enthalpy of the feed is less than that of
the saturated liquid, the reverse will be true. This is similar to the
conditions found for the y,x diagram.

General. It is frequently advantageous to use enthalpy diagrams
to determine a series of values of yn, xn+i, and ym, xm+i, which can be
plotted on the y,x diagram to give the actual operating lines which are
then employed in the usuaLstepwise manner. The y,x curves and the
enthalpy composition values can be plotted on the same diagram,
and the combined graphical procedure shown in Fig. 7-16 completes
both the y,x and the enthalpy diagrams. To illustrate the procedure,
(1) starting at (hw , xw), a vertical line is drawn to the equilibrium curve
giving yw, (2) this value is transposed to the H,y line by going horizon-
tally to y = x and then vertically to the // curve giving the point (Hw,
yw), (3) the line through this point and (Mw, xw) gives the value of xi on
the h curve, and (4) the process is repeated. The intersection of the
vertical lines with the horizontal lines on the y,x diagram corresponds
to points of the operating lines, and the triangles above the lines drawn
through these points are the usual steps on the y,x diagram.

Heat losses from the column can be taken into account by shifting
the value of MD from plate to plate by an amount equal to the heat
loss per plate divided by N. A side stream in the upper section of the
column is handled by locating a point (Ms, XN$) where

, , LhL + DhD + Qc , LxL + DxD

Ms - - —- - - and XM - ---

and drawing lines through this point.

In this case, (ZF, hr) lies on a straight line through (xw, Mw) and
(XNS, Ms), and (XNS, Ms) lies on a straight line through (XD, Mn) and
(XL, hL). In a similar way any number of side streams or feeds can be
handled.

In general the Ponchon-Savarit diagram is somewhat more difficult
to use than the constant 0/V diagram, but it is the exact solution for
theoretical plates assuming that the enthalpy data employed are
correct. This graphical procedure suffers because the absolute values
of M D and M w are frequently large, and to plot them on the diagram
.requires the use of an ordinate scale such that the enthalpy curves for
the liquid and vapor are crowded together, making it difficult to
obtain accurate results. Likewise when low concentrations are
encountered, it is impossible to obtain accurate results from the dia-
gram unless the plot is greatly expanded. For such regions Eqs.
(7-14) and (7-28) can be used algebraically, or in most cases the con-

150

FRACTIONAL DISTILLATION

t«150C,H«3°l30

.P-

<l4J5mols99.9%Oz,|
t'15°C H33IIO

{

"100 mols air

100 mols

t_ OAO/">

p*l

* i\J L

p=ZOdtmos

.—>

h=330

H*3I20

85.85 mols

t--!85°C.
94 3% Liquid

Refrigeration

o <

92% N2
p=1

5.7% Vapor

a
1

t=H94°C.
H = I630

Heat leakage in

<
4

100 mols

Air

+•- — IIA0^

14 15 mols 99 9% 02

h*330

p*l t=-!8295°C.

u «. mA

' /«

L£~t

!*SA

100 mols air
p=20atmos
H = I645
FIG. 7-17. Liquid-air distillation.

stant 0/V diagram is useful since in such regions the value of (Ha — ha+i)
is essentially constant.

EXAMPLES OF ENTHALPY-COMPOSITION METHOD

Liquid-Air Fractionation Example. The enthalpy-composition method will be
illustrated by several examples. Figure 7-17 gives a schematic diagram of a liquid-
air fractionating system. In this particular case the system employed involves

TABLE 7-2

Xn+i

2/ncalo

Constant 0/V

Heat balance

0.001

0.002

0.00218

0.004

0.0045

0.01

0.0116

0.02

0.0233

0 024

0.04

0.0468

0 048

0.1

0.1175

0.12

0.2

0.235

0.24

0.4

0.470

0.475

0.6

0.705

0.71

0.8

0.94

RECTIFICATION OF BINARY MIXTURES

151

2000

only a stripping column, and the enthalpy diagram (Fig, 7-18) is only for the lower

portion. The heat supply to the still

is from the ingoing air and, on the

basis given for Fig. 7-17, Q, is equal

to 100(1,645 - 330) - 131,500. The

molal enthalpy of the bottoms from

the still, Hw, is 1,730, and for Eq.

(7-27) the terminal enthalpy point

becomes

Bw-- -7,550

This construction is given in Fig. 7-18.

The enthalpy values employed are

those given by Keesom (Ref . 7) and,

since they were already available, they

were not recalculated to the basis sug-

gested in the foregoing section. The

diagram of Fig. 7-18 is not very suit-

able for a major portion of the con-

centration region under consideration.

This diagram and Eq. (7-28) were used

to calculate y and x values for the op-

erating line. The results of these cal-

culations are compared with those for

constant 0/V in Table 7-2. It will be

noted that within the accuracy of the

calculation the operating line coordi-

nates calculated by the two methods are in agreement, indicating that the constant

0/V method is satisfactory for this case.

The operating line values and the equilibrium data of Dodge (Ref. 4) are shown
in Fig. 7-19. Logarithmic plotting is employed so that steps can be satisfactorily
made in the low concentration region. Nine theoretical plates in addition to the
still give the desired separation.

TABLE 7-3. 02-NZ AT 10 ATM.

04 06
x ory

0.8

1.0

FIG. 7-18.
separation.

Heat diagram for liquid-air

2/noalo

X»+l

Heat balance,

Constant 0/V

data of Bosn-

jakovic (Ref. 2)

1.0

0.9

0.955

0.953

0.8

0.91

0.7

0.865

0.867

0.6

0.82

0.823

152

FRACTIONAL DISTILLATION

1.0
0.8

0.6

0.4

0.2

01
t 0.08

R 0.06

.£ 0.04

•j: 0.02
o

o 0.01
2 0.008

aooe

0002

•

^l"^H

'/*

^^x \/

x ^

.^y

.vVy^

\4^

f»

0,001

0.001 0.002 00040.006 001 0.02 0.04 0.06 01 02 04 0.6 1.0
Mol Fraction N2 in Liquid

FIG. 7-19. x,y diagram for liquid-air separation.

TABLE 7-4. NH3-H2O AT 10 ATM.

2/n oalc

Xn+l

Heat balance,

Constant 0/V

data of Bosn-

jakovic (Ref. 2)

0.1

0.701

0 705

0.09

0.624

0.625

0.08

0 547

0.547

0.07

0.47

0.46

0.06

0.393

0.385

0.05

0.316

0.305

RECTIFICATION OF BINARY MIXTURES

153

Oxygen-Nitrogen and Ammonia-Water Example. Tables 7-3 and 7-4 give similar
comparisons for other systems. In Table 7-3 the comparison is for the oxygen-
nitrogen system at 10 atm., and again the agreement between the two methods is
good. The other data are for ammonia-water at 10 atm., and even in this case the
two methods are in fair agreement.

Ammonia-Water Example. In order to illustrate the fact that the assumption of
constant molal overflow rate is not always justified, consider the following example
in which the molal latent heat of vaporization varies approximately twofold over
the tower. An aqueous solution containing 20 weight per cent ammonia is to be
separated into a distillate containing 98 mol per cent ammonia and a bottoms con-
taining 0.1 mol per cent ammonia. The tower and total condenser will operate
at an absolute pressure of 20 atm. The feed will enter the system at 20°C. and
be heated to 40°C in the condenser as shown in Fig. 7-20.
Using the data and notes given below, calculate:

1. The minimum reflux ratio 0/D, using the enthalpy-composition method.

2. The number of theoretical plates required at 0/D equal to 1.5 times the
minimum value found in Part 1, using the enthalpy-composition method.

3. The number of theoretical plates required at total reflux.

4. The minimum reflux ratio 0/D, assuming constant molal overflow rates.

5. The number of theoretical plates required for 0/D equal to 1.5 times the
value obtained in Part 4, using constant molal overflow rates.

Data and Notes. As a basis, the enthalpies of pure liquid water and pure liquid
ammonia are taken as zero at their boiling points. On this basis a 20 weight per
cent ammonia solution at 40°C. has an enthalpy equal to —3,180 cal per g. mol.
Other enthalpy and equilibrium data (Ref. 2) for 20 atm. follow.

Mol fraction NH3,
in liquid
or vapor

Enthalpy liquid
cal. per g. mol vs.
mol fraction
in liquid

Enthalpy vapor
cal. per g mol vs.
mol fraction
in vapor

T, °C. vs.
mol fraction
in liquid

0.0

8,430

211.5

0.1053

-500

7,955

182 5

0 2094

-970

7,660

165 5

0.312

-1,310

7,495

131 5

0 414

-1,540

7,210

113 0

0 514

-1,650

6,920

91 0

0.614

-1,580

6,620

76 0

0.712

-1,400

6,300

65 5

0.809

-960

5,980

58.0

0.905

-520

5,470

52.5

0.923

-430

5,350

51.6

0.942

-310

5,240

50.8

0.962

-210

5,050

50.0

0.981

-110

4,870

49.3

1.000

4,250

48.5

154

FRACTIONAL DISTILLATION

VAPOR-LIQUID EQUILIBBIA AT 20 ATM.

x Mol Fraction NH3 y Mol Fraction NH»

0.0529 0.262

0.1053 0.474

0.2094 0.742

0.312 0.891

0.414 0.943

0.514
0.614
0.712
0.809
1.00

Solution of Part 1. Basis; 100 mols of feed.
Feed composition

NHg balance,

0.977
0.987
0.990
0.995
1.000

0.209

0.209(100) - 0.98D -f 0.001(100 - D)
D - 21.25 and W - 78.75

An enthalpy balance on the exchanger between the bottoms and the feed is
WCP(tB - 45) - F(hr - h,0)
Feed

FIG. 7-20. Flowsheet for ammonia-water example.

RECTIFICATION OF BINARY MIXTURES

155

0.3 03 0.4 05 06 0.7 0.8 0.9 1.0
x or y , Mol fraction of ammonia
FIG. 7-20a. Enthalpy-composition diagram for ammonia-water example.

where IB ** temperature in reboiler «• 211.2°C.
hp — enthalpy of feed entering tower
CP « heat capacity of bottoms « 19.0 cal. per g. mol/°C, (used same as

Cp of H2O at 20 atm.)
hto - enthalpy of feed at 40°C. - -3,180
78.75(19.0) (211.2 - 45) - 100 (hr + 3,180)
hp a* —690 cal. per g. mol

This point is located on the enthalpy-composition diagram (Fig. 7-20a). Utiliz-
ing the y,x diagram it is found that the equilibrium tie line that passes through this
point corresponds to x » 0.193 and y - 0.70. The extrapolation of this tie line
to XD - 0.98 gives M D - 10,230 and at xw - 0.001, M w ~ -3,630.
Also

VTHT - 0RkR 4- Qc + DhD - Osh* +

156

FRACTIONAL DISTILLATION

Assuming that HR = ho ™ enthalpy of saturated liquid, and using VT «* OB -f D,
by a derivation similar to that for Eq. (7-16),

OR _ MD - HT
D ** H T - hn

0un 4,850 + 100
The above calculations assume that the minimum reflux ratio corresponded to a
pinched-in condition at the feed point. A check of the other tie line above and
below the feed plate indicated that this corresponded to the maximum value for
MD. It is apparent from Fig. 7-21 that the operating line on the ytx diagram would
have to be very curved in order to have the pinch occur at any place but the feed
plate.

0.1 08 0.3 04 0.5 06 07 0.8 09 10
Mol fraction NH3 in liquid

FIG. 7-21. y,x diagram for NHa-water example.,
Solution of Part 2

MD - 12,940 and Mw - -4,360

Using this value, the steps corresponding to theoretical plates are stepped off.
In this manner the lines on Fig. 7-20a were constructed for the top seven plates
giving the vapor entering the T — 6 plate equal to 0.073. Below this value the
diagram becomes difficult to use, and the calculations were completed by the use of
Eq. (7-31), using Hm - 8,400, and hm+l « -20. '

8,420

' ~ MW -
2.92xw+1 - 1.92(0.001)

-4,360 + 20

RECTIFICATION OF BINARY MIXTURES 157

Over the low concentration region involved, Fig. 7-21 indicates that the vapor-
liquid equilibrium curve can be expressed as

The liquid on plate T — 7 will be in equilibrium with a vapor composition of 0.073
and, from the equilibrium relationship just assumed, XT-I would equal 0.0145.
This lower section is to reduce this mol fraction to 0.001 and, by Eq. (7-77),

_5_ J A/00146 _ \
2.92 A 0.001 /

4/2.92

3.9 plates

In (5/2.92)

Thus a still and about eleven (7 + 3.9) theoretical plates are required.

Solution of Part 3. At total reflux the plates are stepped off on Fig. 7-21, and a
still plus four theoretical plates are required.

Solution of Part 4. The determination of the minimum reflux ratio on the basis
of the usual simplifying assumptions requires determining the difference Vm — Vn.
A given enthalpy for the feed does not determine this difference for the general
case because the feed-plate location is still a variable; however, in this case, a
definite value is obtained at the minimum reflux ratio, because the pinched-in
condition occurs at the feed plate.

By an enthalpy balance around the feed plate,

Fhp + F/_iF/-.i + Of+ihf+i = VfHf + Ofhf

where subscripts/, / -f 1, / — 1, refer to feed plate, plate above, and plate below,
respectively.

For the pinched-in condition,

H/-.I = Hf — H and hf = TI/+I « h
Also,

F + F/-I + Of+i **V/+Of

Solution of these equations gives

V - F -i - F (hp ~ h)

For a first approximation take the enthalpy values H and h, corresponding to a
mol fraction of ammonia in the liquid of 0.19; h = —900 and H = 6,300. From
Part 1, hp = -690.

F(-— 690 -f- 900)
Vf - F/-I = 6,300 + 900 " °'029F

The intersection of the operating lines for this condition falls on a line of slope
-(0.971/0.029) = -33.5, which passes through the y = x line at ZF = 0.209.
Such a line on Fig. 7-21 cuts the equilibrium curve at x — 0.193, y — 0.71 which is
close enough to the assumed value, and

0.98 - 0.71 _ 0_343

F/min 0.98 - 0.193

°'343 , 0.522

D /„,„ 0.657

158 FRACTIONAL DISTILLATION

It will be noted that this value is much lower than that obtained in Part 2, and
designing for an actual reflux ratio even twice this value would still give heat
requirements less than the true minimum.

Solution of Part 5. For this case, the actual value of (0/D) is to be 1 .5 times that
obtained in Part 4, and the plates are to be calculated with the usual simplifying
assumptions.

(<jL\ ** 1.5(0.522) - 0.783

On - 0.783(21.25) - 16.65
Vn - 37.9

In this case the value of Vn •— Vm will not be quite equal to that calculated in
Part 4 because H/ will not equal H/+I, and h/ will not equal h/+i, but the change
will not be large and the same difference will be used as an approximation.

Vm - Vn - -2.9
Vm - 35 Om - 113.75

(?).--*« -

These lines are drawn on Fig. 7-21, and the corresponding steps are shown. A still
plus approximately 12 theoretical plates are required. This is in close agreement
with the result of Part 2, and this is generally the case. Thus, if a tower is calcu-
lated on the basis of a certain factor times the minimum reflux ratio, the theoretical
plates required are usually approximately the same on either the (y,x) or enthalpy-
composition basis, provided the method chosen is used consistently throughout the
calculation. However, the heat and cooling requirements may be seriously in
error when calculated on the constant overflow basis.

Modified Latent Heat of Vaporization Method. The use of the
Ponchon-Savarit method has two major disadvantages: (1) The
enthalpy-composition relations for the vapor and liquid are required
over the whole operating range and (2) it is limited to binary mix-
tures. On page 141 it was shown that the variation in 0/V was due
mainly to a variation in the difference between the enthalpy of the
vapor entering and the liquid leaving a plate, and since the enthalpy
of either the vapor or liquid does not vary greatly over the concentra-
tion range corresponding to one plate, this difference in enthalpy is
essentially equal to the latent heat of vaporization for the concentra-
tion region involved. Neglecting heat of mixing effects, this heat of
vaporization can be approximated as

(H - h)n - [yi(H - k)i + y*(H - A),

where (H — A)i, (H — A) 2 are the latent heats of vaporization of the
pure components.

The values of this molal enthalpy difference vary chiefly because the
heats of vaporization of the individual components are different rather

RECTIFICATION OF BINARY MIXTURES 159

than because their values vary over the temperature range of the
column. Thus, if the latent heats of vaporization of all the com-
ponents were the same, the enthalpy difference would be approximately
constant, independent of the composition. It is pointed out on page
141 that this condition leads to essentially straight operating lines for
most cases.

If pseudo mol fractions, y', x', z', and flow quantities, F', 0', W, F',
could be defined such tha£ the following five relationships apply:

Vryf(H — K)B = Vy(H — h) for each component1

y\ + y'* + y* + • • • = 1
x( + xi + 4 + ' * • = 1

n = O'n+1 + D'

V'ny'n = #n+i<+i + D'XB for each component

then the design could be carried out with the pseudo quantities, and
the molal enthalpy difference (H — h) would be equal to (H — h)R for
all compositions and would make the usual simplifying assumptions a
good approximation. Such relations for the pseudo values in terms
of the normal variables are

where 2fty = ftiyi + pzy* + foys +

where %$x = 0iXi + j32x2 +

Dr == D20XD
Ff «

W -

In these equations the values of ft are equal to the (H — h) values for
the component in question divided by (H — h)R. If the (H — h)
values for individual components are constant over the operating
range involved, ft will be constant and a design problem in the new

1 Where (H — K)R « same constant for all components and (H - h) - enthalpy
difference value for the individual component and is assumed to be constant over
the temperature range involved,

160 FRACTIONAL DISTILLATION

coordinates will give straight operating lines. This method necessi-
tates calculating the equilibrium curves and all the flow quantities
over to the new coordinates. At the same temperature, the relative
volatility is the same for both coordinates. However, if the relative
volatility varies with the temperature the y',xf curve will be different
from the y,x curve because the temperature will be different for equal
values of x and xf. The design problems can then be carried out with
these new variables, making the usual simplifying assumptions. After
the number of theoretical plates and design conditions is obtained on
this pseudo basis, the results can be converted to the true flow quanti-
ties and mol fractions. The (H — h)R value can be arbitrarily chosen
although it is usually convenient to make it equal to the (// — h)
value for one of the components because this makes 0 for that com-
ponent equal to 1. The use of these pseudo variables is equivalent to
assigning a fictitious molecular weight to a component such that its
molal heat of vaporization on the new basis will be a specified value.
The use of this method will be illustrated by the ammonia-water
example already considered.

Ammonia -Water Example. From page 153 the latent heats of vaporization for
ammonia and water are 4,250 and 8,430 cal. per g. mol, respectively. Using
(H - h)R = 8,430 gives 0H2o = 1.0 and j3NH3 = 4,250/8,430 = 0.504. Recal-
culating the feed and terminal concentrations to the pseudo units,

" 0.504(0.98) + 0.02

0.504(0.209) _01175
* 0.504(0.209) + 0.791

Per 100 mols of original feed,

D' - 21.25(0.504(0.98) + 0.02] = 10.9
F' - 100(0.504(0.209) + 0.791] - 89.6
W = [0.504(0.001) + 0.999] - 78.7

The vapor-liquid equilibrium data were recalculated to the new basis and the
results are plotted in Fig. 7-2 la.

To determine the minimum reflux ratio, it is necessary to locate the intersection
of the operating line and the equilibrium curve. From page 155 the enthalpy of
the feed is —690 cal. per g. mol and the enthalpy of saturated liquid of the feed
composition is —970 cal. per g. mol (both values on original basis). Thus the
excess enthalpy of the feed above saturated liquid is 100(280) - 28,000 cal. The
latent heat on the new basis is 8,430 cal. per g. mol and the fraction vapor in the

RECTIFICATION OF BINARY MIXTURES

161

feed is

Fraction vapor -

0.037

The line of intersection of the operating line will cross the yf «• x* line at
= 0.1175 and will have a slope equal to -0.963/0.037 = —26. This line is

0 0.1 0.2 0.3 0.4 0.5 06 07 0.8 09 1.0

FIG. 7-21a.

shown in the figure. This line intersects the equilibrium curve at x' = 0.102.
y' ss 0.517 and the minimum reflux ratio corresponds to

QL\ = 0.961 - 0.517

F'/mln 0.961 - 0.104 U'&1/

0.517

(°*\ = °'517

VP'/nnx 0.483

1.07

Because the compositions of OR and D' are the same, the conversion factors to
OR and D will be the same and

(°*\ « 1.07

This value is close to one obtained on page 156, and the two values probably agree
within the accuracy of the calculation. For

(°*\ =1.5^) =1.605
\/>7.ct \D'Jma

1.605

= 0.616

^V'Jn 2.605
The corresponding operating lines are drawn on the" diagram, and the theoretical

162 FRACTIONAL DISTILLATION

plates stepped off in the usual manner. The diagram indicates that a still and
11 theoretical plates are required. This checks the result obtained by the
enthalpy-composition method.

This modified latent heat of vaporization method generally gives
good results and would be only appreciably in error if (1) the heats of
mixing were large or (2) the latent heats of vaporization of the com-
ponent varied appreciably over the temperature range involved. In
the latter case the results can be improved by using separate average
values for stripping and enriching section.

Theoretically the enthalpy-composition method is more exact than
the use of the modified latent heats of vaporization, but in most cases
the two agree within the accuracy of the calculation. For binary
mixtures, if the necessary enthalpy data are already available, the
enthalpy-composition method is the easier to apply; if the data are not
available and must be calculated, then the other method is the more
convenient. For multicomponent mixtures the modified latent heat
method is more convenient even if the complete enthalpy data are
available.

HEAT ECONOMY

In the section on the Optimum Reflux Ratio it was pointed out that
the operating costs, i.e., the heating and cooling charges, were fre-
quently the major cost of a distillation process and that fixed charges
on the equipment were small. In such cases, it would appear desirable
to decrease the heating and cooling requirement at the expense of
additional equipment, and this section will consider a number of
methods of accomplishing that result.

From a thermodynamic viewpoint the inefficiencies of a distillation
operation can be grouped into two main categories: (1) those that are
a function of the distillation process itself and (2) those that are related
to supplying the necessary energy to the materials being separated.

Separation Efficiency of a Distillation Column. The minimum
isothermal thermodynamic work required for separating 1 mol of a
liquid binary mixture into its pure liquid constituents is given by the
following equation:

Minimum work - AF = -RT (xi In |~ + x2 In ^ ) (7-41)

V ^i *V

(7-42)
-RT[xi In (y&i) + X* In (y&$] (7-43)

RECTIFICATION OF BINARY MIXTURES 163

where AF « minimum work of separation (free-energy change) per

mol of mixture
R = gas law constant
T = absolute temperature
Xi, £2 = mol fractions of components in mixture
71, 72 = activity coefficients of two components
IT *= total pressure -
p = partial pressure,

or fugacity of component in mixture
P = vapor pressure,

or fugacity of pure saturated liquid
2/i, 2/2 = mol fractions in equilibrium vapor

For an ideal solution, this expression has a maximum value at a mol
fraction of 0.5, and for this condition the minimum work required is
equal to 0.693flT'. Thus to separate 1 Ib. mol of an ideal mixture of
this composition at a temperature of 212°F. would require about
920 B.t.u.

In case the mixture is not completely separated, the minimum work
required per mol is obviously less and can be calculated from the
following equation:

AF = — RT[(xi In yiXi + x2 In 72z2) — D(XID In yiDXiD

+ XZD In 72i>z2z>) — W(xiw In 71^1^ + x^w In 72^2^)] (7-44)

where D refers to distillate and W refers to bottoms.

Mixtures with positive deviations from Raoult's law, i.e., solutions
with activity coefficients greater than 1, require lower minimum work
for separation than do ideal solutions; the reverse is true for those solu-
tions with negative deviations. For example, consider the minimum
work for separating a 3 mol per cent solution of ethanol and water at its
normal boiling point into pure water and 87 mol per cent ethanol. The
boiling point of such a solution is 173°F., and the activity coefficients
are 4.4 and 1.01 for the 3 mol per cent solution for ethanol and water,
respectively; the corresponding values for the 87 per cent solution are
1.01 and 2.2. On the basis of Eq. (7-44), the minimum work of sepa-
rating 1 mol of this mixture into the desired product wo.uld be about
106 B.t.u. Separating an ideal solution of the same composition into
the same products would require 153 B.t.u. The actual energy
requirement is lower than the theoretical because ethyl alcohol and
water have positive deviations from Raoult's law, indicating a tend-
ency to immiscibility. An immiscible system requires essentially no
work for separation. On the other hand, systems of negative devia-

164 FRACTIONAL DISTILLATION

tions from Raoult's law, i.e., those that tend to maximum boiling
azeotropes, would need more work for separation than an ideal
solution.

The energy required for separating a mixture is supplied by adding
heat to the fluid in the still and removing heat at a lower temperature
level in the condenser. The available work energy, based on an
isentropic process, in the heat supplied to the liquid in the still can be
calculated on the basis of the following equation:

Ws = available work = Q — ^-^ (7-45)

1 w

where Q = heat added

Tw — absolute temperature of liquid in still
To = absolute temperature at which heat can be discharged, i.e.,

temperature of cooling water

The available work equivalent to the heat removed from the con-
denser can be calculated by the following equation:

Wc - available work = Qc — ^-° (7-46)

JL c

where Qc = heat removed from condenser

Tc = temperature of condensation of distillate
In order to simplify the following discussion, it will be assumed that
the heat added to the still is equal to the heat removed in the con-
denser. This is essentially true when the feed enters as a liquid at its
normal boiling point and when the distillate and bottoms leave as
liquid at their boiling points. Other cases can be handled in a similar
manner but are more involved. For a system in which Q3 = Qc, the
net available work supplied in the heat to the distillation process itself
is equal to

The minimum heat that can be utilized in a distillation corresponds
to the minimum reflux condition and, for a binary mixture, can easily
be determined from the enthalpy-composition diagram.

It is interesting to compare the thermodynamic minimum work with
that required by the actual distillation process at the minimum reflux
condition. For ideal mixtures of close boiling compounds Eq. (7-47)
can be combined with Eqs. (7-44) and (7-56), and approximations for

RECTIFICATION OF BINARY MIXTURES

165

the relative volatility and latent heats as a function of temperature
can be made to give simplified expressions, but they are misleading for
most actual distillations. It is more instructive to study the ratio of
Eq. (7-44) to Eq. (7-47) for actual cases. Such calculations for the
systems benzene-toluene and ethanol-water at atmospheric pressure
are given in Tables 7-5 and 7-6. In the case of the first system, it was
assumed that complete separation was being obtained; in the second
case, that the distillate was 87 per cent mol alcohol and the bottoms
was pure water. In both cases, actual activity coefficients were
employed, minimum heat requirements were estimated from enthalpy-
composition diagrams, and pressure drop and heat losses were neglected.

TABLE 7-5. FRACTIONATION EFFICIENCY FOR ETHANOL-WATER MIXTURE
(Tw = 672°R., To = G33°H., T0 =» 546°R.)

Mol fraction
ethanol m
feed

Minimum heat
requirement,
B.t.u. per Ib.
mol of feed

Thermo-
dynamie work
equation (7-44),
B.t.u. per Ib.

Available work
equation (7-47),
B.t.u. per Ib.
mol of feed

Thermo-
dynamic
efficiency,
per cent

mol

0.03

2,620

106

J31

0.1

8,740

195

437

0.2

17,500

261

875

0.3

26,200

310

1,310

0.4

34,900

288

1,745

0.5

43,600

283

2,180

0.6

52,400

234

2,620

0.7

61,200

195

3,060

0.8

69,600

3,480

The benzene-toluene system shows a maximum thermodynamio
efficiency of about 80 per cent for a feed composition of 40 per cent
benzene. The efficiency decreases for feeds both weaker and stronger,
but in the range of feed compositions from 0.1 to 0.8 it is good. Equa-
tion (7-44) shows that the efficiency is zero both for XF approaching 0
and 1.0. In the case of the ethanol-water system, the heat require-
ment per mol of distillate is essentially independent of the feed compo-
sition over the range shown because the minimum reflux ratio condi-
tion corresponds to a tangency with the equilibrium curve at about
84 mol per cent alcohol Thus the reflux ratio is the same over the
whole concentration region considered. This reflux ratio corre-

166

FRACTIONAL DISTILLATION

spends closely to a pinched-in region for the 0.03 feed as well as to the
pinched-in condition at the mol fraction of 0.84. Thus the operating
line follows the equilibrium curve approximately, and the efficiency for
the 0.03 feed concentration is high. For the higher strength feeds,
the operating lines over most of the lower concentration regions are a
considerable distance from the equilibrium curve. This leads to low
efficiency.

TABLE 7-6. FRACTIONATION EFFICIENCY FOR BENZENE-TOLUENE MIXTURE
(TV - 692°R, Tc - 636°R., T0 - 546°R.)

Mol fraction
benzene in
feed

Minimum heat
requirement,
B.t.u. per Ib.
mol of feed

Eq. (7-43)
thermo dynamic
work, B.t.u.
per Ib. mol

Eq. (7-47)
net available
work, B.t.u. per
Ib. mol

Thermo-
dynamic
efficiency,
per cent

—

0.1

13,100

439

895

0.2

14,100

662

964

0.3

15,200

810

1,040

0.4

16,200

882

1,110

0.5

17,200

900

1,180

0.6

18,300

880

1,250

0.7

19,300

785

1,320

0.8

20,300

630

1,390

0.9

21,400

410

1,460

1.0

—

The values given in the tables were the maximum efficiencies and
correspond to the minimum reflux ratio. In an actual tower, the heat
supply would be greater and the efficiency values should be reduced.
The reduction factor is approximately

Efficiency of Supplying and Removing Energy from a Distillation
Tower, In a simple distillation tower, the greatest inefficiency
usually results from the methods employed in supplying energy to the
system rather than in the separation process. For example, consider
the benzene-toluene mixture of Table 7-6. Even allowing for the
actual reflux ratio being greater than the minimum reflux ratio, a
thermodynamic efficiency of 50 per cent or greater would be obtained
over a wide range of feed compositions. However, in such a case it

RECTIFICATION OF BINARY MIXTURES

167

would be common practice to use steam as a heating fluid with a con-
densation temperature of at least 250°F. This would give a tempera-
ture difference for heat transfer in the still of 18°F. There would then
be a temperature drop through the distillation system corresponding
to 56°F., and with cooling water available at 80°F. there would be tem-
perature loss equivalent to 96°F. in going from the overhead vapor to
the available heat sink. 'These temperatures and the corresponding
losses in availability are given in Table 7-7. It is evident that the

TABLE 7-7

Temp.,
°R.

Net available work, B.t.u. per
B.t.u. of heat

Per cent
of total

Steam ...

710

Steam to still, 0 02

Still

692

Still to distillate, 0 069

Distillate

636

Distillate to cooling water, 0.151

Cooling water

540

Total

0 24

100

major inefficiency is in adding and removing the heat. In this case
over 70 per cent of the total available work in the heat was dissipated
in these ways. In the case of a 3 per cent ethanol mixture distilled at
atmospheric pressure, the loss in available energy of the steam due to
heating and cooling could be over 80 per cent. These heat-transfer
processes are the source of the major thermal inefficiencies in most
industrial distillations. For these cases, the greatest improvement in
efficiency will be obtained by reducing these losses. The most obvious
way of accomplishing this result is by utilizing a higher fraction of the
temperature difference between the source and sink temperatures for
the actual distillation process and less for the heat transfer.

METHODS OF INCREASING EFFICIENCY

1. Separation Process. One method of reducing the irreversibility
in the fractionating tower is to make the vapor stream entering a plate
more nearly in equilibrium with the liquid on the plate. The extreme
in this case is for this vapor and the liquid to be in equilibrium, but this
obviously means no enrichment per plate. To accomplish this condi-
tion throughout the tower requires that the operating line coincide
with the equilibrium curve, and this in turn involves a different reflux
ratio for each plate, i.e., heat must be added to or removed from each
plate in the tower. It is obvious that such a system is impractical.

168 FRACTIONAL DISTILLATION

However, certain approaches have been made in this direction. For
example, in most systems the vapor load is limited by the region around
the feed plate. Actually a lower quantity of vapor would be suitable
for the regions near the top or bottom of the tower, and it would be
possible to operate a fractionating tower by supplying less heat at the
bottom than corresponds to the required reflux ratio and then to
supply additional heat at some point, or points, intermediate between
the still and the feed region. This would be advantageous where an
additional heat source was available which was not at a temperature
level sufficient to operate the still but was suitable for the intermediate
region. Likewise, above the feed plate, heat could be removed at
some intermediate position and used for other heating purposes.

For example, in the ethanol separation given in Table 7-5 high feed
concentrations gave low fractionating efficiency. This was because
lower quantities of heat could have been used for the concentration
region below a mol fraction of ethanol less than 0.8, and the high heat
requirement resulted from the pinched-in region at the top of the
tower. In this case, a high percentage of the heat could have been
added at a temperature level only a few degrees above that of the con-
denser rather than at the higher temperature of the still. This would
have given a much higher thermodynamic efficiency.

Such modifications result in a lower enrichment per plate and thus
require more plates, but they might be attractive in certain cases.
Actually, they are seldom advantageous.

2. Heat -transfer Process. In most cases, using a higher percentage
of the available work in the heat supplied to the still is more practical
than attempting to reduce the irreversibility in the distillation column.
The available work utilized is relatively small because the difference
in the absolute temperature in the still and the condenser is not large.
If the heat could be employed over a wider temperature differential,
the efficiency would be increased. The most common method of
accomplishing this result is by the multieffect principle or a modifica-
tion of it. For example, the mixture can be fractionated in two or
more towers arranged with operating pressures such that the distillate
condensing temperature of one tower is high enough to serve as the
heating medium for the succeeding towers. This is entirely analogous
to the use of multieffect evaporators. Figure 7-22 illustrates the
multieffect system, and Figs. 7-23 to 7-26 show modifications of it.

Multieffect. This system reduces the heat required roughly in pro-
portion to the number of columns employed. This is not quite true
because there will be changes in the relative volatility with pressure

RECTIFICATION OF BINARY MIXTURES

169

that may make some difference. It is possible to make further heat
savings by taking the bottoms from the higher pressure columns and
expanding them directly into the bottom of the succeeding columns,
thereby obtaining a certain amount of sensible heat. The sensible
heat of the condensate from a high-pressure stage can also be used in a
low-pressure stage.

FIG. 7-22. Multitower system.

If the relative volatility does not change with pressure, the total
volume of the distillation system will be essentially the same as for a
single column operating at about the average pressure of the multi-
effect system. Thus the column costs will be of the same order for the
single- and multieffect system. However, the heat-transfer surfaces
required will be larger for the latter due to the lower temperature
differences involved, and usually more cooling water will be required.
The control problem will be greater than for a single column, but it
would not be particularly difficult. Thus, the multieffect system
should be attractive where the heating and cooling costs are large
relative to the equipment costs.

The number of stages that can be employed is relatively limited
because of the high temperature drop per stage. For example, in
distilling the mixture ethanol and water, it would be difficult to
operate without having each stage cover a temperature differential of
the order of 60 to 80°F. Thus, a system having more than two stages
would require a high-temperature heat source and would involve rela-
tively high pressures.

In order to evaluate the heat requirement for the multitower system,

170

FRACTIONAL DISTILLATION

consider the case of the separation of a binary mixture in a single
tower, for which the relative volatility is independent of the pressure
and composition. Making the usual simplifying assumptions, the
minimum vapor for a system with Vn = Vm and xj> = 1, xw = 0 is

Kmln

a -I

(7-48)

where Fmi»
F

compressor

minimum vapor rate
feed rate

a = relative volatility
XF — feed composition
For a multieffect system based on the same assumptions

Vi _ 1 + (a .— l)xy

JJT "~" " / ^^ •* \

where Vi = minimum vapor generation required for first effect

n = number of effects

Vapor Recompression. One method that has been proposed and
utilized on a limited scale is vapor recompression, a schematic diagram

of which is shown in Fig. 7-23. In this
case, the overhead vapor is compressed to
a pressure such that its condensation
temperature will be suitable for heating
the still. Theoretically, such a method
appears attractive because the heat needs
to be pumped through a relatively small
temperature rise. However, in practice
the system has not been particularly at-
tractive due to the high cost of efficient
vapor compressors of the size required
for distillation units, and wheii. the ef-
ficiency of the compressor and the effi-
ciency for producing the power to drive
the compressor are included, a great deal of the theoretical advantage
is lost.

Vapor Rewe. A modification of the multieffect system, shown in
Fig. 7-24, has been termed the vapor reuse system (Ref. 12). In it,
the feed is introduced into a stripping tower, and the vapor from it is
used as the source of heat and the feed of the lower pressure column.
The two columns may not be in balance; i.e., the minimum heat
required for the stripping section may be more or less than that

w 0

7-23. Vapor recompres-

FIG.

aion. system.

RECTIFICATION OF BINARY MIXTURES

171

required for the fractionating column. This heat differential can
either be added or be removed from the still of the lower pressure
fractionating column. In case the
heat requirement of the low-pres-
sure tower is greater than' for the
stripper, it might be thought that
the additional heat could be employed
advantageously through the stripping
column as well as through the frac-
tionating column. However, this is
not the case because this additional
heat would require that more vapor
leave the top of the stripping column
and thus it must be a diluter vapor,
which makes the work of the frac-
tionating column more difficult. For FlG- 7"24' Vapor reuse sysfcem-
the same idealized case considered in the multieffect section, the
minimum vapor is

For the stripping tower,

(a - l)xr] (7-49)

Flmin = -

and, for the fractionating tower,

V 2 mm ==

a(a — - 1)

If XF < ox 1-\ ' Vl *"n > F2 mm

(7-60)

2(« - I)'
a - 2

'i... - F2min

If Xp > H7 :r~TY' ^l'»'n < ^2min

Split Tower. Figure 7-25 shows a split-tower system which partly
separates the feed in one fractionating column and completes the frac-
tionation in the second column. It is similar to the previous case
except that a fractionating column with reflux is employed instead of
the stripping column. This system can be adjusted so that the two
towers are in heat balance. The feed is shown entering the high-
pressure tower. This makes relatively pure bottoms but impure dis-
tillate, which is then fractionated in the second tower. This arrange-
ment is desirable for feeds with low concentrations of the more volatile
component. If the feed concentration is high, it is more desirable to

172

FRACTIONAL DISTILLATION

fractionate the feed first into pure distillate and impure bottoms and
refractionate the bottoms in the lower pressure column.

This system involves a lower over-all temperature drop than a two-
stage multieffect system and thereby makes the design of the heat-
transfer surfaces easier. However, the multieffect system has the

»l *t

FIG. 7-25. Split-tower system.

advantage that it uses all the heat over the whole concentration
region each time. This system utilizes the heat over a limited con-
centration region twice but over other regions only once. Thus, it
tends to give a higher reflux ratio in the central portion of the column
and lower reflux ratios at the ends. The minimum vapor requirement
for this system for the assumptions made in the other cases is given
byEq, (7-51):

?[1 + (a -

r mm —

(a - 1)[2 + (a -

(7-51)

This arrangement is very similar to that employed in the so-called
double towers for the separation of oxygen and nitrogen. In this case
the feed is relatively rich in the lower boiling component, nitrogen,
and the first tower is used to produce an overhead that contains a high
concentration of nitrogen and an impure bottoms containing 40 to 50
per cent oxygen. The liquid nitrogen overhead product from the high-
pressure tower is added to the top of the low-pressure tower and serves
as the only reflux for it. There is no other source of refrigeration to
produce additional reflux. The impure bottoms from the high-pres-
sure tower is introduced into the middle portion of the low-pressure
tower. Such a two-tower system will give high-purity nitrogen and
oxygen low in nitrogen although it will contain appreciable quantities

RECTIFICATION OF BINARY MIXTURES

173

of noble gases. It is common practice in this case to build the low-
pressure tower on top of the high-pressure tower, and the condenser-
reboiler is a common unit of both columns.

A further modification of the split-tower system is given in Fig. 7-26
in which the feed is rectified into impure bottoms and an impure dis-
tillate, and these are then retreated in the lower pressure tower. In
general, this system gives slightly lower heat requirements than the
previous system, and it utilizes the same principle, namely, reusing the
heat several times in the middle region concentrations where the heat
requirements are the highest. The systems of Figs. 7-22 to 7-26 can
be utilized with three or more towers to obtain still further heat
reductions.

FIG. 7-26. Modified split-tower system.

In order to show the comparison between the different systems, the
various equations have been plotted in Fig. 7-27 for the cases of feed
concentrations equal to 0 and 1.0. The equation for the multieffect
system has been plotted for n = 1, i.e., a single tower. If a two-stage
system is used, the values should be divided by 2, etc. For the vapor-
reuse system the maximum vapor requirement for the two columns is
given, and no credit has been applied for the low-temperature heat
that could be withdrawn for the cases with the relative volatility
greater than 2.0. It will be noted that none of the modifications is so
effective in reducing the heat requirements as the multistage system,
although some of them are equal to it for specialized conditions. Of
the specialized arrangements, vapor reuse would not appear to be so
attractive as the split-tower system shown although, in cases where the
waste heat was of real utility, it could be attractive. The relative

174

FRACTIONAL DISTILLATION

attraction of the systems will be shifted somewhat as they are com-
pared at reflux ratios lower than the minimum (or greater than total
reflux). It still appears that, for a given number of towers, the multi-
effect system is the most attractive from the heat viewpoint. Like-
wise, for a given total quantity of heat supplied to the system and for
the same number of towers, the total plate area required is less for the

V/F

1.0
0.6
06

0.4

Key

S - Single tower
VR. - Vapor reuse
JJ - Split tower
(For multiple towers ctmcte "
values for $ by number
of towers)

I 3 V

oc, Relative volatility
FIG. 7-27. Comparison of vapor requirements.

multieffect system than for the modifications. This greater efficiency
is due to the more efficient utilization of the heat over a wide tempera-
ture range.

ANALYTICAL EQUATIONS

For a few special cases of limited applicability, it has been possible
to develop mathematical solutions, some of the most useful of which are
considered in this section.

Total Reflux. Fenske (Ref . 5) developed an algebraic method of
calculating the minimum number of theoretical plates by utilizing the
relative volatility together with the fact that at total reflux the operat-
ing line becomes the y » x diagonal. Thus, considering the two

RECTIFICATION OF BINARY MIXTURES 175

components x' and x" and starting with the still,

and at total reflux the operating-line equation gives yw — x\, giving

Continuing, in the same manner,

(x'\ i v ^ (x'\

1-77 I = (an-i)(<Xn-2) ' ' ' aiawi—}

\X /n \X /w

In some cases the relative volatility does not vary widely, and an
average value for the entire column can be used, giving

where N is the number of plates in the column. This can be changed to

N = 10g (*/'")*('"/«% (7.52)

log aav v '

If a total condenser is employed, this becomes

N log (MWM*

log «av V '

or, if a partial condenser equivalent to one theoretical plate is employed,

N + 2 . ^ W/fWM* (7.54)

log aav

These equations offer a simple and rapid means of determining the
number of theoretical plates at total reflux and avoid the necessity
of constructing the y,x diagram. Principally an arithmetic average of
the relative volatility at the temperature of the still and the top of the
tower is employed. This is a satisfactory average only if the relative
volatility is reasonably constant over the concentration region involved.
For larger variations the geometric average would be more satisfactory,
i.e., aav = -VatOiW) or still better aav = 1 + V(«« — l)(«ir — 1), where
at is the relative volatility at the top of the column. In the cases of
abnormal volatility such as are exhibited ty ethyl alcohol and water,

176 FRACTIONAL DISTILLATION

the use of an average relative volatility is not satisfactory over an
appreciable concentration range; however, the equation may be
applied successively to small concentration ranges, but the operation
becomes more time-consuming than constructing the y}x diagram and
stepping off the plates. The use of the y,x diagram has the advantage
that it gives a picture of the concentration gradient, and after the
diagram has been constructed, the number of theoretical plates for
other reflux ratios can be easily determined. This method is not
applicable to conditions other than total reflux.

Minimum Reflux Ratio. The analytical equation for this case can
be easily derived in terms of the coordinates, yc and xe, of the point of
contact of the equilibrium curve and the operating line.

(7 w\

^ J

(a- l)(xc)(l -XC)(XD-XW)
Xp[l + (a — l)xe] - axc

_ F(xc - XW)(XD ~ ZF*)[l + (« - l)xc] /7 5?)

(a - l)(xc)(l - £)(XD - Xw) V "

when Vn == Vm and XD and xw are approximately 1.0 and 0.0,
respectively,

T7 £>[! + (a - l)xr] _ F(l + (a - l)xF]

Vmin = - (« --- "

and

These equations are exact for the case of constant molal flow rates
where the value of the relative volatility is taken at the composition
xc. Equations (7-55) to (7-57) involve so many factors that it is
usually easier to obtain the minimum reflux ratio values graphically
or by using yc and xc with Eq. (7-10).

Theoretical Plates Required for Finite Reflux Ratios. The equation
for the number of theoretical plates at total reflux is convenient for
mixtures in which a is relatively constant, and it would be useful to

RECTIFICATION OF BINARY MIXTURES

177

have a comparable method applicable at other reflux ratios. Two
basic equations were used in the total reflux derivation:

fa) . an fa) and fa) = fa)

\JlB/n \X*/n \3/B/n VWn+l

The first of these two equations is true at any reflux ratio, but the
latter is true only at total reflux. However, new variables may be
defined such that

and

-A

~ W

(7-60)

(7-61)

It is obvious that the equation developed from these will be identical
in form with Eq. (7-52). Thus, above the feed

\r _i_ i _ *°& (XA/XB)I>(XB/XA)P

log

(7-62)

It remains to relate the new variables to the actual compositions and
the usual operating variables. In addition it would be convenient,
but not necessary, to have the following relation,

xfA + x'B = 1.0

A number of solutions for the new variables have been obtained
(Refs. 16, 18), and one set is given here.
Above the feed,

8*0
aV

(*--7hnh) (7'63)

Below the feed,

^ / ( MW xw \
"__ 1 \XA» + 0 1$^ \)

(7-64)

Above the feed plate,

s - i -

= 1 — x'

i xAm

2(D/O)xp(cl - 1)

' (<*(V/0) - 1 - (D/Q)xD(a - I)]2 |
(7-65)

178 FRACTIONAL DISTILLATION

Below the feed the same equation is employed to calculate the value
of 8 except that — (Wxw/0m) is used instead of DxD/On. The value
of An is calculated as follows:

^« = ™ (7-66)

These equations are rather involved to use, and they become par-
ticularly difficult where the number of plates is large. The most effort
is involved in evaluating the term /S, and Fig. 7-28 is arranged to
facilitate this calculation.

The equation above the feed plate should be employed down to the
feed-plate conditions; i.e., for minimum plates at a given 0/D, it
should be applied down to the conditions given by the intersection of
the operating lines.

For special conditions found to apply in a great mg/ny design prob-
lems, it is possible to simplify the calculation of S. It is often found
that the value of S below the feed plate is equal to a(F/0), which
makes the calculation of S simple. It can be shown that this is the

W
exact value of S for the case of -^- xw(a — 1) equal to zero. For

W
values of -^ xw(a — 1) that are less than 1 per cent of a(V/O) — 1,

good results are obtained by using

(V\ W
~J+-~^(«- 1)

and

-» (7-68)

Above the feed plate similar approximations can be made giving,
S = «[l + ^ (1 - xAO)(a - 1)] (7-69)

« -1)]

= l+2(l -**„)(« -1) (7-70)

Lewis Method. Lewis (Ref. 8) expresses the rate of increase of
concentration of the liquid in the column from one plate to the next by
the differential dx/dn; therefore,

Xn + (7-71)

RECTIFICATION OF BINARY MIXTURES

179

FIG. 7-28. Plot for analytical equation.

180 FRACTIONAL DISTILLATION

The material-balance equation

yn = xn+i + ^ (XD - yn)

can be written, using V = 0 + D,

0
From Eq. (7-71),

dn 1

dx Xn+l - Xn _ _ D

(7-72)

n - Xn- (XD - yn)

The integration of this equation gives the number of theoretical plates
between the x limits chosen; thus, the number of theoretical plates
above the feed Nn is

CXD dx

Nn = / zn ~

/ yn — Xn — ~fi (XD "~ y*)

J xf V

Similarly, below the feed the theoretical plates Nm become

The equilibrium y}x curve gives the relation between yn and xn and
ym and xm- thus by assuming values for xn, values of yn can be obtained
from the equilibrium data, and

yn — Xn ~ 0 (XD — 2/n)

can be calculated and plotted vs. the assumed values of x. The area
under such a curve from x/ to XD is the number of theoretical plates by
the Lewis method.

Although this method is based on a continuous change of concentra-
tion instead of the actual step wise concentration increase, the error
involved is not serious when the change in concentration per plate is
small. This latter condition is generally true when the number of
plates involved is large.

When the relative volatility is reasonably constant over the range of
concentrations involved, the equilibrium curve may be approximated
by the relative-volatility relation (see page 30)

RECTIFICATION OF BINARY MIXTURES 181

and the foregoing equation may be integrated directly, avoiding the
necessity for the graphical integration. Thus, for total reflux at which
D/0 = 0 and (F - D)/(0 - pF) = 0, the foregoing equations give

Nn + Nm + 1 * (_!_) in *5 (1— *lY (7-75)

\Q! - I/ 0V \1 - #/>/ V J

where ATn + Nm + 1 is the total number of steps including the still
step and any condenser enrichment. Noting that In a for values of a
near to 1 is approximately equal to (a — 1) makes the foregoing equa-
tion approach the equation for total reflux. The values at total
reflux were calculated for an example with a equal to 1.07, XD = 0.96,
and xw = 0.033. Equation (7-75) gave 97 theoretical plates, while
Eq. (7-53) gave 94. Equation (7-53) is applicable only at total reflux,
while the Lewis equation can be integrated for the case of finite reflux
ratios and constant relative volatilities. Thus Eq. (7-73) can be
integrated to give

Nn . i+jvao. ln r A^JLA (*L±*\\

Vb* - 4ac l\xf + A) \x» + B/j

where a = 1 — a

b = (a - 1)[1 - (D/0)xD] + <xD/0
c = —(D/0)xD

A __ b - Vb* - 4ac 6 + V62 - 4ac

A _ ~—~2a and B -- 2^ -

A similar equation is obtained below the feed with — W/Om replacing
D/0, —(W/Om)xw replacing (D/0)xD, and using x/ instead of XD and
xw instead of x/.

The integrated equations are most useful in cases where the number
of plates is large and the graphical method would be long and tedious.

Example Illustrating the Use of Analytical Equations. As an example of the
use of the analytical equations, consider the benzene-toluene rectification problem
of page 122.
Total reflux [Eq. (7-53)]:

Using a relative volatility of 2.5,

loe 0.95

iog

N « 5.4 olates

182 FRACTIONAL DISTILLATION

Minimum reflux ratio [Eq. (7-56)):

(9\ 0.95[1 4 1.5(0.5)] - 2.5(0.5)

\DJmm " 1.5(6:5)(0.5) "

Theoretical plates at (0/D) - 3.0:
Above the feed plate, using Fig. 7-28,

(0.95) (1.5) -0.475

Q2 » 2.5 (%) - 3.333
4- 0.475 ~ 3.333

n _
°'65

0475

S - 3.333 - 1.65(0.475) - 2.55

1 - 0.9 - 0.1

3 f (1.5/2.5) (2.1

4 L 1.95 - 1

.55) 0.95

3(2.55 -
1 - 0.357 = 0.643

Below the feed plate, TF/0 - 0.2, 7/0 «• 0.8, and using Fig. 7-28 and the
equations gives S « 2.03, A = 2.06, x^F » 0.731, x'BF « 0.269, a?^ — 0.086,
and a;^ - 0.914.

o.269

Thus the total number of steps including the still is 8.9, corresponding to 7.9
theoretical plates plus the still. This compares with 8H plates estimated by the
stepwise method (page 123).

The values of S and A can be evaluated by the approximations, thus,
Above the feed plate,

S - 2.5(1 4- H(0.05) (1.5)1 * 2.55
A - 2.5(M)[1 4- ?i(0.05) (1.5)] - 1.95

Below the feed plate,

S - 2.5(«) 4 H (0.05) (1.5) - 2.02
A - 2.5(#) 4- % (0.05) (1.5) - 2.03

These values are so close to those obtained before that the result is essentially
the same.

RECTIFICATION OF BINARY MIXTURES

183

In certain cases the terminal compositions are small, and to obtain
any accuracy by the usual y,x method requires expanding the diagram.
However, in most of these cases, the number of steps between the
operating line and the equilibrium curve can be easily calculated,
utilizing the fact that the equilibrium curve and the operating line are
straight in this region. The equilibrium curve can be expressed as
y « Kx and, for values of a; -near zero, K is equal to the relative vola-
tility. The operating line in such a region will be straight for most
cases, even on an enthalpy basis, because the variation of composition
is so small that it makes essentially no difference in the enthalpy
values.

For the bottom of the tower when the two lines are straight, the
number of steps or theoretical plates required is given by

Nw =

(V/0)(K -

-'- + 1

(7-77)

In (VK/O)

where Nw = number of plates including plate m, but excluding the

still

x = mol fraction of most volatile component
This relationship is similar to Eq. (7-62) for small values of Xw
Above the feed plate, the corresponding relationship for a column and
total condenser is obtained by using a similar analysis for the least
volatile component,

Nn+ I =

0 -

1 - K

(7-78)

In (0/K'V)

where Nn = plates above plate n

y, x = mol fractions of least volatile component

Kr = equilibrium constant for least volatile component
Packed Towers. Packed towers can be used in fractional distilla-
tion as well as bubble-plate columns. Instead of bubbling through a
pool of liquid as in a bubble-plate tower, the interaction between
vapor and liquid can be obtained by causing the reflux to flow over the
surface of the packing material while the vapor flows up through the
voids. Thg iffie of packed towers isgenerally limited to towers of small
sizes or to special distillations Well an'MlB tiUUUtiZLU'UllOft olTaitrlFacid7
TiTsmall laboratory and pilot-plant size, the packed tower necessary

184

FRACTIONAL DISTILLATION

for a given separation is, in general, less expensive than a corresponding
bubble-plate tower; in large diameter, the reverse may be true. Aside
from the economic aspects, packed towers are ^asily constructed and
can be made of noncorrosive refractory earffienware. glass, and carBon

— ,_ _ , .uu, ~.— '~«—~>~~.| ,- -.-— V.-..- »^,ww j^^r*,^'*****™"- "*••"*«•»»•»-.,< ^^

as well as the usualmetals used in bubble-plate tower construction.
They have the disadvantage that it may be difficult to clean the tower
without completely dismantling the unit, and often they channel
badly; i.e., the liquid and vapor segregate from each other, and the
efficiency of contact between them is poor. The packed towers, in

general, have very low pressure drops
from top to bottom relative to an
equivalent bubble-plate tower.

For tower packing, a wide variety
of materials have been used, e.g.,
coke, stone, glass, earthenware, car-
bon and metal rings, wood grids, jack
chain, carborundum, and metal and
glass helices, as well as a large num-
ber of other packings including many
manufactured packings of special
shapes.

Design of Packed Tower. Ide-
ally, the interaction between va-
por and liquid in packed towers is
true counter current rather than the
stepwise-countercurrent propess of
a bubble-plate tower with theoreti-
cal plates. Instead of finite steps, the true countercurrent action
should be treated differentially. Consider the schematic drawing of
the packed tower in Fig. 7-29. Let 0 be the mols of overflow, V the
mols of vapor, x and y the average mol fraction in the liquid and vapor,
respectively, n distance above and m distance below the feed. Focus-
ing on the differential section dn, V(dy/dri) must equal 0(dx/dri) for
each component, and this transfer must be due to an exchange of
components back and forth between the liquid and vapor. This
transfer is due to the fact that the vapor and liquid at a given cross
section are not in equilibrium with each other, and the rate of transfer
will be a function of the distance from equilibrium; thus,

•w

FIG. 7-29. Schematic diagram for
packed tower.

• dy

= kA(y* - t/) - k'A(x -

(7-79)

RECTIFICATION OF BINARY MIXTURES 185

where &, kf = proportionality constants

A = area per unit height

y* = vapor in equilibrium with x

x* = liquid in equilibrium with y

Integration of Eq. (7-79) is difficult because very little information is
available on the values of and the factors involved in kA. Assuming
that this product is constant and making the usual simplifying assump-
tions, equation can be arranged as follows:

n _ Is [v=VT_dy__ = 0. I*'-" dx

Ic A I » ti ^ ~— ti 1f^ A / i i* -I.- '1**
A//I jy^yff y y K /i J x**Xf' ** •*"

Below the feed,

1^ A i tt^ — 11 Jf* A I / 'T — — o*

A/ /I Jy—ys y y ft/ /I J x=*XW •" *

where x/, t// are the liquid and vapor compositions in the tower at the
level at which the feed is introduced, and x'w is the liquid concentra-
tion at the bottom of the packed section.
A material balance above the feed gives

V = \?)* + \^)*» <7-82)

By assuming values of y, values of x can be calculated by Eq. (7-82),
and these used in Eq. (7-80) together with equilibrium data to evaluate
the integrals. A similar material balance below the feed can be used
with Eq. (7-81). The equilibrium curve and the material-balance
equation can be plotted on the y,x diagram, and x* — x or y — y*
read directly.

Consider the separation of an equimolal mixture of benzene and
toluene into a product containing 95 mol per cent benzene and a residue
containing 5 mol per cent benzene. An 0/D of 3 will be used, and the
feed will enter heated such that the mols of vapor above and below the
feed are the same. The usual simplifying assumptions will be made.
The equilibrium curve is given in Fig. 7-30.

The operating lines are identical with those for the stepwise dia-
gram. The vertical distance between the equilibrium line and the
operating line is y* — y, and the horizontal distance between the
operating line and the equilibrium line is x — x*. In general, the inte-
gration must be performed graphically, although in cases where the
equilibrium curve can be expressed as an algebraic relation between y

186

FRACTIONAL DISTILLATION

and x the integration can be carried out algebraically, but the resulting
equations are often complex and involved.

I.Or

0.2

0.4 0.6

FIG. 7-30.

0.8

1.0'

From Fig. 7-30, values of y* — y are read at various values of y.
Such values are tabulated in Table 7-8.

TABLE 7-8

y* -y

0.615

0.098

10 2

0.7

0.100

10.0

0.8

0.090

11.1

0.9

0.043

23.2

0.95

0.028

35.7

Values of l/(y* — y) are then calculated
and plotted vs. y. The area under this curve
from y = 0.615 to y = 0.95 is equal to
kAn/V] so that if kA is known, n can be cal-
culated. The values l/(y* — y) vs. y are
plotted in Fig. 7-31. The value of
dy/(y* — y) is the shaded area, which is
equal to 5.35. A similar procedure is used
below the feed, and alternately the x — x* values may be used.

The difficulty in using this procedure is in evaluating kA and k'A.

RECTIFICATION OF BINARY MIXTURES 187

There is a real question as to whether Eq. (7-79) is a proper formula-
tion for the rate of transfer. This doubt arises from several factors:

1. There is no adequate proof that the rate of transfer is directly
proportional to the difference in concentration, but there are theoretical
considerations that would indicate this is not the proper driving force.

2. The distribution and ratio of the liquid and gas are not the same
at all points of a given cross section resulting in differences of concen-
tration, and it is doubtful whether a rate equation of this type is satis-
factory with average values.

3. The wetted-area factor is a nebulous quantity because all the area
of the packing is not wetted and because the wet areas are not all
equivalent. It is a factor that varies widely with the system being
distilled and with the column construction.

As a result of these factors, no satisfactory correlations have been
presented for the kA and k'A. If data on a tower operating under
essentially identical conditions to the one being designed are available,
a reasonable extrapolation can usually be made. However, any major
extrapolation will make the results questionable.

A number of attempts have been made to relate the values of kA
obtained for one system with those for other systems. The coefficients
involve resistances to mass transfer for both the vapor and the liquid
phases, and it has been customary to apply relations based on the
Lewis and Whitman two-film theory. It is doubtful that such a
theory is applicable in this case since it is difficult to visualize the con-
ditions inherent in this theory for a liquid phase in a packed tower.
Further studies of the mechanism of mass transfer between the vapor
and the liquid for systems approximating the conditions in a packed
tower are needed to furnish a sound basis for correlating the over-all
mass-transfer coefficients.

Height Equivalent to a Theoretical Plate. In general, the graphical
calculation involved in the design of a packed tower is more tedious
and time consuming than the stepwise procedure used for plate towers.
Actually, the equilibrium curve and operating lines on the ytx diagram
are identical for the two cases, and one of the most common methods
of designing packed towers has been to determine the number of
theoretical plates required for the separation by the usual stepwise
method and then to convert to the height of the corresponding packed
tower, by multiplying the number of theoretical plates by the height of
packing equivalent to one theoretical plate. This is abbreviated to
H.E.T.P. (Ref. 13) and is a height of packing such that the vapor
leaving the top of the section will have the same composition as the

188 FRACTIONAL DISTILLATION

vapor in equilibrium with the liquid leaving the bottom of the section.
The use of the H.E.T.P. substitutes a stepwise countercurrent proce-
dure for the true countercurrent operation and is therefore theoretically
unsound; but when the concentration change between plates is small
and the number of plates is large, the error introduced by its use will
be small. Values of H.E.T.P. are determined experimentally by cal-
culating the number of theoretical plates necessary to be equivalent to
some actual packed tower; the height of the packed tower divided by
the number of theoretical plates is then the H.E.T.P. Values of
H.E.T.P. will be considered in Chap. 17 on Column Performance
(page 465).

Height of a Transfer Unit. Chilton and Colburn (Ref. 3) have
proposed that the integrals of Eqs. (7-80) and (7-81) be termed the
number of transfer units, N.T.U. The height of the packed section
divided by the number of transfer units is termed the height of a trans-
fer unit, H.T.U. This latter unit being defined above the feed plate as

TT «T> TT _ U

H.l.U. - j^rjr^

= JL = J5L

kA k'A

The transfer unit consists of a step on the operating line such that
the change in y or x is equal to the average difference between the
equilibrium curve and the operating line over the region of the step.
If the equilibrium and operating lines are parallel, the step will be
exactly equal to that for a theoretical plate, and the H.T.U. value will
equal the H.E.T.P. value. If the slope of the equilibrium curve is less
than that of the operating line, the two curves will tend to converge
with increasing concentration, and the initial difference, which corre-
sponds to the step taken for a theoretical plate, will be greater than the
average difference corresponding to the transfer unit. Thus, in this
case one equilibrium plate will give a greater concentration change
than one transfer unit. If the two curves diverge with increasing
concentration, the reverse is true.

Over the height of one transfer unit, the value of y* — y does not
ordinarily vary widely, and the arithmetic average may be used.
Baker (Ref. 1) has developed a relatively simple stepwise method for
estimating the number of transfer units under these conditions. In
the usual y,x diagram (Fig. 7-32), a line ab is drawn at the arithmetic
mean of the equilibrium curve and the operating line. The H.T.U.
corresponds to a step giving a change in y equal to the average value of

RECTIFICATION OF BINARY MIXTURES

189

* — y over the step. Starting at A, one proceeds not to B but to C,
uch that AD = DC, and then steps from C to the operating line,
f the curvature of the equilibrium curve is not too great, C(r, the
hange in y which is numerically equivalent to the EF, will be approxi-
nately equal to the average value of y* — y between A and (?; there-
ore, the steps correspond to one transfer unit. This stepwise proce-
lure is continued to the terminals of the tower, giving the number of
/ransfer units in the tower. Values of H.T.U. are multiplied by the
lumber of transfer units required to determine the height of packing
lesired.

FIG. 7-32. Diagram for H.T.U.

Because the transfer unit is defined on a differential countercurrent
basis, it is usually assumed to be more correct for the design of packed
towers than the stepwise countercurrent procedure of the theoretical
plate. This is probably true, but there is a serious question whether
the transfer unit is on a sound theoretical basis. Most distillation
operations are of a degree that requires a number of theoretical plates,
and for suph cases it is doubtful whether at the present stage of devel-
opment the transfer-unit concept has any advantage over the the-
oretical-plate basis for the design of packed towers. The latter is
easier to employ.

In practice the packed tower has been losing out relative to the
bubble tower. The development of corrosion-resistant alloys and of
bubble-plate columns made of ceramic, glass, and plastic has made it
possible to rectify corrosive mixtures in such units. The development
of efficient laboratory bubble-plate columns as small as 1 in. in diam-
eter has made it possible to carry out such distillation in the laboratory,
and experience has indicated that these columns give data that are

190 FRACTIONAL DISTILLATION

much more suitable for extrapolation in the design of large-scale bub-
ble-plate towers than those obtained in packed towers. These small
columns give plate efficiencies that are comparable to those of large
towers, and their effectiveness is not a major function of the wetting
characteristics of the liquid as it is in a packed tower. The one major
remaining advantage of the packed tower is low pressure drop; even
in this case, new types of columns are being developed which give
definite and controlled contact between the liquid and the vapor and
which have pressure drop as low as in packed towers. If large-sized
packed towers can be developed to give efficiencies commensurate with
those obtained in small laboratory packed towers and if their design
can be made reliable and reproducible such that their performance can
be predicted with reasonable accuracy, they could become a major
factor in vapor-liquid interchange processes.

Nomenclature

a •» relative volatility
C ** specific heat

D « mols of distillate withdrawn as overhead products per unit of time
F ** mols of mixture fed to column per unit of time
// « enthalpy, or heat content of vapor
h «• enthalpy, or heat content of liquid
H.K.T.P. » height equivalent to a theoretical plate
H.T.U. «« height of a transfer unit

m » number of plate under consideration, counting up from still
n = number of plate under consideration, counting up from feed plate
0 •» total mols of overflow from one plate to next, per unit of time
p as vapor pressure
p - (0/+1 - 0,)/P
Q w heat added or removed
T,t, » temperature
W •» mols of residue per unit of time
x «• rnol fraction of more volatile component in liquid
re7 «• pseudo mol fraction in liquid
y « mol fraction of more volatile component in vapor
yf ** pseudo mol fraction in vapor
z «• mol fraction in feed mixture
y a« activity coefficient

Subscripts

n refers to nth plate; *".«., 0« and Vn refer to mols of liquid and vapor leaving nth

plate, respectively
m refers to wth plate
/ refers to feed plate; i.e., Xf is mol fraction of more volatile component in overflow

from feed plate
F refers to feed; i.e., xp is mol fraction of more volatile component in feed mixture

RECTIFICATION OF BINARY MIXTURES 191

D refers to distillate
R refers to reflux
W refers to bottoms
L refers to liquid
F refers to vapor
t refers to top plate

References

1. BAKER, Ind. Eng. Chem., 27,~ 977 (1935).

2. BOSNJAKOVIC, "Technische Thermodynamik II," diagrams, T. Steinkopf,
Dresden, 1937.

3. CHILTON and COLBURN, Ind. Eng. Chem., 27, 255, 904 (1935).

4. DODGE, Chem. Met. Eng., 35, 622 (1928).

5. FENSKE, Ind. Eng. Chem., 24, 482 (1932).

6. GUNNESS, Sc.D. thesis in chemical engineering, M.T.T., 1936.

7. KEESOM, Bull. Intern. Inst. Refrig., 15 (1934).

8. LEWIS, Ind. Eng. Chem., 14, 492 (1922).

9. LEWIS, "Unit Operation Notes/' M.I.T., 1920.

10. McADAMS, "Heat Transmission/' 2d ed., McGraw-Hill Book Company, Inc.,
New York, 1942.

11. McCABE and THIELE, Ind. Eng. Chem., 17, 605 (1925).

12. OTHMER, Ind. Eng. Chem.t 28, 1435 (1936).

13. PETERS, Ind. Eng. Chem., 14, 476 (1922).

14. PONCHON, Tech. moderne, 13, 20 (1921).

15. SAVARIT, Arts et metiers, pp. 65, 142, 178, 241, 266, 307 (1922).

16. SMOKER, Trans. Am. Inst. Chem. Engrs., 34, 165 (1938).

17. SOREL, "La rectification de Falcool," Paris, 1893.

18. UNDERWOOD, /. Inst. Petroleum, 29, 147 (1943).

19. WALKER, LEWIS, MCADAMS, and GILLILAND, "Principles of Chemical Engi-
neering/' 3d ed., McGraw-Hill Book Company, Inc., New York, 1937.

CHAPTER 8
SPECIAL BINARY MIXTURES

This chapter covers special fractional distillation systems for binary
mixtures. The examples illustrate the flexibility of the fractional dis-
tillation process and the broad applicability of the design methods.
The first section will consider operating conditions that lead to unusual
operating lines, and the last section will consider the separation of
binary azeotropic mixtures.

Special Operating Lines. The operating lines so far considered
intersected the y ~ x line at values between x = 0 and x = 1.0 and,
with the exception of the bottom portion of the lower operating line
for the steam distillation case, they were between the equilibrium curve
and the y = x diagonal. Although most operating lines lie in this
region, it is not necessary that they do so. The operating line is a
combination of the over-all material balance and a component balance.
Consider the case of the section above the feed plate :

By over-all balance,

Vn = On+1 + R

By component balance,

Vnyn = On+iXn+i

where R is the net molal withdrawal from the section other than in Vn
and On+i, and RxR is the net molal withdrawal of the component from
the section other than in Fn and On+i>

In the cases considered in Chap. 7, it was assumed that -there was a
net withdrawal at the top of the column equal to Z), but it may be that
material is also added to the section such that R is positive, zero, or
negative. If R is positive, On+i/Vn will be less than 1.0; if R = 0,
On+i/Vn will equal 1.0; if R is negative, On+i/Vn will be greater than
1.0. In these cases XR may be positive or negative. The following
example will illustrate this case.

Leaky Condenser Example. An old ethyl alcohol distillation column has been
tested, and it is concluded that there is a leak into the condensate in the condenser
that amounts to 5 mols of water per 100 mols of feed. The tower is operating on a
feed containing 3.5 mol per cent ethanol and 96.5 mol per cent water and produces

192

SPECIAL BINARY MIXTURES

193

a distillate containing 70 mol per cent ethanol and a bottoms containing 0.001 mol
per cent ethanol. The feed enters preheated such that Vn ** Fw. Make the usual
simplifying assumptions.
Solution. As a basis take 100 mols of feed. Then, by over-all material balance,

By alcohol balance,

For the upper section,

D + W - 105

0.7D + 0.00001 W = 3.5

• D - 5.0
TF - 100

5 + Fn - On4.! -f D

# =o"

The alcohol-enriching line is

Vny« » On+lxn+l H-

» On+l*»+I + 3.5

Since Vn = On+i, this line intersects the y « x line at « •* eo and is parallel to the
diagonal. With Vn = Fm, this line will intersect the lower operating line at the
usual value, i.e., at x = 0.035.

0.1 0.2 03 0.4 05 06 0.7 0.8 0.9 1.0
Mol fraction ethanol in liquid

FIG. 8-1.

A possible position for the operating line is shown in Fig. 8-1. (The exact posi-
tion is not known since the reflux ratio was not specified.) While the line extends
outside of the diagram, it is utilized only in the region below the y,x equilibrium
curve. The composition of the top vapor and the liquid reflux corresponds to a
point on the operating line, and the step for the top plate should terminate at this
point.

194 FRACTIONAL DISTILLATION

For the case just considered, R *= D — leak. If the leak is less than Z>, R would
be positive, 0/V would be less than 1.0, and the intersection with y ** x would be
positive and greater than 0.7. As the size of the leak increased relative to £>,
0/V would become nearer to 1.0 and the intersection would increase and tend to
infinity. If the leak became greater than D, R would become negative, 0/V
would be greater than 1.0, and the intersection with the y *• x line would be
negative.

It appears that the leak makes the slope of the upper operating line more desir-
able (i.e., nearer to 1.0), but a little consideration will show that, when this line
keeps the same intersection with the lower operating line, the smaller the value of
(0/V)n, the fewer the plates required. Common sense also indicates that the leak
is undesirable.

In this example the operating lines were above the y = x diagonal
but had intersections with the diagonal outside of the usual diagram.
The following example illustrates a case for which the operating lines
are also below the diagonal.

Isopropyl Alcohol Stripping Example. In the manufacture of isopropyl alcohol,
propylene is dissolved in sulfuric acid to give an extract of mono-isopropyl sulfate.
To avoid excessive decomposition this extract must be diluted until the ratio

Lb.H2S04

Lb. H20 + lb. H2S04

before the alcohol can be distilled. In this ratio the H2S04 is the total acid,
whether free or combined with propylene, and H2O is the total water, free or com-
bined with propylene as isopropyl alcohol. A typical extract containing H2S04,
propylene, and H20 in equimolal proportions is first diluted with water, and is then
stripped of isopropyl alcohol. Usually live steam is used in the stripping, and this
results in dilute alcohol and very dilute bottom acid which must be reconcentrated
The stripped acid contains negligible alcohol.

It has been proposed to modify this operation by diluting the extract in the
stripping tower with the overflow and thereby produce more concentrated alcohol
and at the time obtain 45 per cent II 2804 as bottoms. To avoid polymerization
during dilution, it is estimated that the overflow with which the extract is mixed
must not contain over 10 mol per cent isopropyl alcohol. Isopropyl alcohol and
water form an azeotrope containing 68 mol per cent alcohol. The accompanying
diagram (Fig. 8-2) shows the flow sheet for the proposed modification to produce a
66 mol per cent alcohol product. The tower will operate with live steam and a
total condenser. It is assumed that the feed will enter such that Vn « Vm, and
the usual simplifying assumptions will be made. It is assumed that sulfuric acid is
nonvolatile, and the calculations will be made on a ' ' sulfuric acid-free " basis. It is
also assumed that all of the propylene is in the solution as isopropyl alcohol and
that the vapor-liquid equilibria for the isopropyl alcohol-water system will be used.

Solution. Basis : 1 mol of propylene in feed (also 1 mol each of HaO and

By a sulfuric acid balance the water in the bottom equals

98/0.55

SPECIAL BINARY MIXTURES

195

Less than

f r

fresh W
extract Mixer

66%
isopropanol

45%

FIG. 8-2. Isopropyl alcohol distillation unit.

Uncombined water in distillate = ^!~ = 0.515 mol

O.oo

Combined water as isopropyl alcohol in distillate — 1.0 mol
Water with feed » 1.0 mol

Mols of water added as steam S = 6.65 + 1.515 — 1.0 « 7.165 mols.
Operating lines:

By an alcohol balance,

Vnya « On+lXa

D l

On+iXa -f 1.0

1.515

' 0.66

Vn = Vm - S - 7.165
Om SB 6.65 (sulfuric acid-free basis)
On - 5.65

Operating line above feed plate,

5.65 , 1.0

" 7.165"

7.165

= 0.789z0 + 0.139

This line crosses the y » x line at 0.66 = XD.
Operating line below feed plate,

Wxw

The two operating lines intersect at y » 0.928, x « 1.0. These lines are shown
in Fig. 8-3. The equilibrium curve shown is for water-isopropanol and should be
suitable above the feed plate; below the feed plate, with the sulfuric acid present
the separation should be easier than shown. The steps on the diagram would start
at XD' "• 0.66 and continue down the upper operating line until the mol fraction of
alcohol in the liquid is less than 0.10; then the shift to the lower operating line would

196

FRACTIONAL DISTILLATION

be made and the step wise procedure carried on to the bottom. In the diagram as
drawn, the first plate with an overflow containing less than 10 mol per cent alcohol
has a concentration of 5 per cent alcohol. This would be mixed with the fresh
feed, and the mixture would be the liquid to the feed plate. As drawn in Fig. 8-3,
the liquid leaving the feed plate contains approximately 1 mol per cent alcohol,
and from this concentration on down, the steps would be made between the equilib-
rium curve and the lower operating line which is below the y = x line.

0 01 02 03 04 05 06 07 08 09 10
Mol fraction of isopropanol in liquid, sulfunc-acid -free basis

FIG. 8-3. y,x diagram for isopropyl alcohol example.

In this case the diagram is unusual in two respects: (1) The distillate has a lower
ratio of alcohol to water than the feed and (2), the feed-plate composition is
purposely fixed at a value different from that corresponding to the fewest plates.

Separation of Binary Azeotropic Mixtures. A large number of two-
component systems form azeotropic mixtures, and it is frequently
necessary to separate them into their components. Regular frac-
tional distillation will not separate such mixtures into the components
in high purity, but by suitable modifications it is frequently possible to
obtain the desired separation. At the azeotropic composition the
relative volatility is unity, and rectification is not possible. The
methods employed for separating such systems involve using either (1)
distillation plus other separation processes to get past the azeotropic
composition or (2) a modification of the relative volatility.

1. Distillation plus Other Separation Processes. The techniques
most commonly employed for moving by the azeotropic composition
are decantation, extraction, crystallization, or absorption. A compo-

SPECIAL BINARY MIXTURES

197

sition near the azeotrope can be produced by distillation and then
separated into two fractions having compositions on each side of the
azeotrope by one of these techniques. These two fractions can be
fractionally distilled separately to give the essentially pure components
and fractions of approximately azeotropic composition which can be
recycled through the process.

The method of utilizing Recantation to aid in the separation of the
azeotrope is illustrated by the system phenol-water which at atmos-
pheric pressure forms an azeotrope containing 1.92 mol per cent

Greater than 1.68%
phenol —

Less than 192%
phenol

JJ/% phenol
168% phenol

* Water (low in phenol)
FIG. 8-4. Fractionating system to concentrate dilute aqueous phenol.

phenol. If a mixture of phenol and water is subjected to a continuous
f ractionation, the overhead will tend to the azeotropic composition and
the bottoms will approach either phenol or water depending on the
composition of the feed. Thus, if a feed containing 1 mol per cent
phenol is given such a distillation, an overhead product approaching
the azeotropic composition can be made with water, low in phenol, as
bottoms. In this case, the azeotrope can be separated because phenol
and water are only partly miscible. For example, at 20°C. the two
saturated liquid phases contain 1.68 and 33.1 mol per cent phenol,
respectively. Thus if the fractionation gives an overhead vapor con-
taining more than 1.68 mol per cent phenol, it will break into two
liquid layers on cooling to 20°C., the water layer containing 1.68 mol
per cent phenol can be used as reflux, and the phenol layer can be
withdrawn as product. This system of fractionation, cooling, and
decantation is shown in Fig. 8-4. Such a system will separate dilute
solutions of phenol into water and 33 per cent phenol. The conden-
sate can be cooled to temperatures other than 20°C., but for the system
to be effective two liquid phases must be formed, one of which must

198

FRACTIONAL DISTILLATION

have a composition less than the azeotrope and the other a composition
greater. The system of Fig. 8-4 does not give complete separation of
the two components, but by a similar arrangement concentrated solu-
tions of phenol can be fractionated to give an overhead condensate that
will separate into two liquid phases. Figure 8-5 shows a schematic
flow sheet for a feed containing 50 mol per cent phenol. The column
operates to produce an overhead vapor containing between 1.92 and
33.1 per cent phenol and, on cooling to 20°C., the condensate would
give the same liquid compositions as obtained in Fig. 8-4. In this
case the water layer would be the product and the phenol layer would
be employed as reflux.

Less than 53 1 %
phenol — •

Greater than 132%
phenol

168% phenol
35.1% phenol

FIG. 8-5.

50%phenol
50% wafer

Phenol (low in water)
Fractionating system to dehydrate aqueous phenol.

The systems shown in Figs. 8-4 and 8-5 can be combined to give
complete separation. The arrangement of Fig. 8-6 shows a two-tower
system for a dilute phenol feed. Both towers give the same con-
densate compositions, and the overhead vapors are fed to the same
condenser and decanter. Tower 1 operates in the same manner as
the single tower of Fig. 8-4 producing an overhead product containing
33.1 per cent phenol and a bottoms low in phenol. Tower 2 is only
a stripping section because the reflux stream is its only feed. If the
fresh feed contains less than 1.68 per cent phenol, it should be intro-
duced into tower 1; from 1.68 to 33.1 per cent phenol, it should be
added to the decantation system; if the feed contains more than 33.1
per cent phenol, it should be introduced into tower 2. The arrange-
ment can give a high degree of separation between phenol and water.
The quantitative calculations for such a system are illustrated by the
following example.

SPECIAL BINARY MIXTURES

199

Phenol- Water Rectification Example. A plant using phenol as a solvent desires
to rectify a mixture containing 1.0 mol per cent phenol in water. As the bottoms
are to be discarded in a nearby river, they must not exceed 0.001 mol per cent
phenol. A system similar to that shown in Fig. 8-6 will be employed. The over-
head vapors from the bubble-plate columns to be used are condensed and cooled to
20°C. i Under these conditions the condensate separates into two saturated layers.
The water layer is reheated to its boiling point and refluxed to column 1, and the

Water Phenol

FIG. 8-6. Two-tower fractionating system to separate aqueous phenol.

phenol layer is reheated and sent to stripping tower 2, to recover a 99.99 per cent
phenol as bottoms. At 20°C. the saturated water layer contains 1.68 mol per cent
phenol, and the saturated phenol layer contains 33-ljnol per cent pEenol. Assum-
ing that Vn ~ Vm and making the usual simplifying assumptions, determine:

1. The minimum mols of vapor per 100 mols of feed for each tower.

2. The number of theoretical plates required, using a vapor rate % times the
minimum rate.

3. The minimum number of plates at total reflux. (Only the water layer is
refluxed to the water tower.)

Solution of Part 1. Basis : 100 mols of fresh feed. Referring to Fig. 8-6,

1 - 0.00001 Wi + 0.9999 W 2
W i + W* - 100; Wz - 1.0; Wl - 99.0

Operating lines for tower 1 :

Above the feed plate, a balance between plates n and n -f 1 and around tower 2,

Vn - ft,*, -f W*

+ 0.9999JT ,

This line intersects the y « x line at x « 0.9999.

200

FRACTIONAL DISTILLATION

Below the feed plate,

Om+l - Fm + Wi
Vmym - 0»+iafcH4 - 0.00001 Wi

which intersects the y « x diagonal at x «• 0.00001.

VAPOK-LIQUID EQUILIBRIUM DATA FOR PHENOL- WATER AT 1 ATM.

(Ref. 5)

0.10

0.029

0.001

0.002

0.20

0.032

0.002

0.004

0.30

0.038

0.004

0.0072

0.40

0.048

0.006

0.0098

0.50

0.065

0.008

0.012

0.60

0.090

0.010

0.0138

0.70

0.150

0.015

0.0172

0.80

0.270

0.017

0.0182

0.85

0.370

„ 0.018

0.0186

0.90

0.55

rO.019

0.0191

0.95

0.77

0.020

0.0195

1.00

Minimum vapor rate for tower 1 :

The minimum vapor corresponds to the minimum reflux ratio, and there are two
possibilities in this case: (1) the pinched-in region could occur at the feed plate,
or (2) it could occur at the top of the tower.

If the feed plate is the limiting condition, then

0.0138(Fn)mm
(F.U,

0.010n+i + 0.9999TF2
0.01 Vn +0.9899TT2

260TF, - 260

(0.0138 is the vapor in equilibrium with a liquid of 0.01).

If the top plate is the limiting condition, then the concentration on this plate
must be equal to the reflux concentration, and the top vapor will be in equilibrium
with this composition, thus,

0.0181 (Vn)mm

0.01680n+i + 0.9999TF2
• 0.0168 Vn + 0.9831 JF2
0.9831TF2

7K-
755

Since the top condition requires more vapor, it is the limiting condition and

(Fn)min - 755.

Minimum vapor rate for tower 2:

Tower 2 will pinch at top, the liquid on the top plate will have a composition of
0.331, and the equilibrium vapor composition will equal 0.0403.

SPECIAL BINARY MIXTURES 201

0.0403(F)min - 0.3310 - 0.9999JF2
- 0.331 F ~ 0.6689 JF2

Solution of Part 2

Tower 1:

(Fn)aet - ^(755) - 1,007; 0»+i - 1,006
Fw - 1,007; Om+i « 1,106

Above the feed plate, *

l,007?/n - 1,006*.+! + 0.9999
0.9999 - yn « 0.999(0.9999 - sn+i)

Below the feed plate,

l,007yw - l,10tew+i - 0.00001(99)
y« - 0.00001 = 1.098(3m+i - 0.00001)
Tower 2:

Fact - «(2.3) - 3.06

Operating line for phenol,

3.06?/m = 4.06zm+i - 0.9999
Operating line for water,

3.06yM » 4.06xw+i - 0.0001

These operating lines are plotted in Figs. 8-7 and 8-8. Logarithmic plotting is
used to facilitate the calculations at the low concentrations. In the case of tower
1, the operating lines for phenol are used; for tower 2, the operating line for water
is used.

According to the diagram, tower 1 requires a still and 15 theoretical plates.
Since the bottoms of this tower are essentially water, live steam could be used, but
if the same phenol recovery (99.99 per cent) were obtained, a total of 20 theoretical
plates would be required. The large increase in the number of plates is due to
dilution by the large amount of vapor used. If the same bottoms composition
had been maintained instead of the same recovery, 16 theoretical plates would be
required and the recovery would be 98.9 per cfent.

The steps for the region x = 0.00001 to 0.001 of this tower could be calculated
by Eq. (7-77) since the equilibrium curve is y » 2x.

In the case of tower 2 using mol fractions of water instead of phenol, the plates
are stepped up the operating line from xw - 0.0001 to XF ** 0.669. In Fig. 8-8
a still and six theoretical plates would give a reflux composition of 0.54, and seven
theoretical plates give 0.71. Thus a still and seven theoretical plates would give a
slightly better separation than desired.

Solution of Part 3. At total reflux, the plates for tower 1 correspond to the
steps between the equilibrium curve and the y — x line from xw » 0.00001 to the
reflux composition, XR « 0.0168. From the diagram it is found that a still and
between 11 and 12 theoretical plates are needed.

For tower 2, the steps at total reflux go from x « 0.0001 to 0.669. In this case,
a still and five theoretical plates are required.

202

FRACTIONAL DISTILLATION

It might be thought that the effectiveness of the fractionation would be improved
by refluxing to tower 1 not only the 1.68 per cent phenol layer but also a portion
of the 33.1 per cent layer. This would result in an increased overhead vapor com-
position but would require additional plates for the same vapor rate. Thus for
the same separation and vapor rate, more plates are required, indicating that only

O.I
0.08
0.06

0.04
0.02
001

^cj^r operating line ••

--^

irf

0.008

_^J_

^2|__^

0.006
|^ 0.004
§

£ 0002
i

"*• 0.00!

*jf

\\^

Z^W

?/•

operaf

//7^ ///7tf

Z__

./,'„. ..„

.1 0.0008
g 0.0006

A (\f\f\A

— j7

__ ^

/ ~~

TSr

|J/

ji/1

0 U.UUU4

0.0002
0.0001

•'7

£ _

0.00008

-7

t .

0.00006
0.00004

000002

0.00001
O.OC

•-•••-^

"™

00! 000004

0000! 0.0004

000! 0004

001 002

Mo! fraction phenol in liquid
FIG. 8-7. Diagram for tower 1.

the 1.68 per cent layer should be refluxed. For compositions between 1.68 and
33.1 per cent "phenol, decantation is a more efficient method of separation than
distillation.

This two-tower system can be used to separate a number of binary
systems, such as isobutanol-water, aniline-water, or benzene-water.
In the last system the solubility of water in benzene is so low that a
single-tower system is usually used for the dehydration of benzene,

SPECIAL BINARY MIXTURES

203

and the water layer is discarded without treating it in a stripping
tower.

Some partially miscible mixtures cannot be separated directly by
this method. For example, methyl ethyl ketone and water are par-
tially miscible and form an azeotrope at atmospheric pressure, but

1.0

-i4

"-X

^ ^

0.4

0.1

_ ?t

rrri

4r*

j- /\ t\i

,16

CL 0.04

k.
«

* 001

^ V

ra\

'/>3

^ ///5

.2
g

"""71

-r —

— -

*~t*

^ 0.004
"o

F-

0.001

0.0004

'v'

<^x

f\ ftOAl

0.0001
0.0

001

004

aooi

00!

Mol fraction water in liquid
FIG. 8-8. Diagram for tower 2.

the compositions of the two liquid phases do not bracket the constant-
boiling mixture composition. Cooling and decantation will not take
the composition across the azeotrope. In this case, if salt is added to
the decantation system, the solubility limits can be made to overlap
the azeotrope and the fractionating-decantation system will make the
separation.

2. Modification of Relative Volatility. The two most common meth-
ods of modifying the relative volatility bf azeotropic mixtures involve
(1) changing the total pressure and (2) adding other components to

204 FRACTIONAL DISTILLATION

the mixture. The effect of pressure on the azeotropic composition will
be considered in the following section and the second method will be
analyzed in Chap. 10.

The effect of pressure on the azeotropic composition is the result of
(1) the change in the ratio of the vapor pressures and (2) the change in
the activity coefficients. At a given composition the change in the
activity coefficients is usually small in comparison to the effect of the
Vapor-pressure ratio. Thus, the qualitative effect of pressure on the
azeotropic composition can be predicted from the vapor-pressure ratio.
In the case of ethanol and water mixtures at atmospheric pressure,
ethanol is the more volatile component for mixtures containing less
than 89.5 mol per cent alcohol, and the less volatile component for
more concentrated solutions. The ratio of the vapor pressure of
ethanol to water decreases with increasing temperature and, assuming
that the activity coefficients do not change, the azeotropic composition
should decrease in alcohol content as the total pressure increases.
This conclusion is in agreement with the experimental data.

A more quantitative prediction can be obtained by combining the
Margules equation for the activity coefficients with equations for the
vapor pressures.

At the azeotropic composition,

v =
7 Px

and Eq. (3-34a) becomes

T°-26(ln TT - In P«) = xftV + c'(xz + 0.5)] (8-2)

The variation of the azeotrope composition with temperature can
be obtained by subtracting Eq. (8-1) from Eq. (8-2).

yo.26 ln « v(2Xl - 1) - c'(l - 3aa + l.6xl) (8-3)

If cr and 6' have been determined for one temperature, the value of
#i can be calculated as a function of the ratio of the vapor pressures.
The total pressure can then be calculated by either Eq. (8-1) or (8-2).

Equations (8-1) and (8-2) can be combined with empirical equations
for the vapor pressure as a function of the temperature to give a rela-
tion between the total pressure and the azeotropic composition, but
the procedure outlined in the preceding paragraph will be found simpler
in general.

SPECIAL BINARY MIXTURES

205

Equation (8-3) was applied to the system ethanol-water, using T as
degree Kelvin, V = 0.605, and c' = 6.01. These values were obtained
by fitting Eq. (8-1) and (8-2) to the azeotrope data for atmospheric
pressure. The calculated results are compared with the experimental
data (Ref . 2) in Fig. 8-9. Some of the difference shown in this figure
is due to the fact that the constants used in the equation were based on

£WV

1000
a? 800

1 600
£ 500

1 40°

*• 300
</>

EOO

100

xf>
*C*

perit

1/CU/C

rjsnfi
v feet I

,/
*fi

8-3

)

/•

r o

) 1 2 3 4 5 6 7 8 9 10 11 12

Mol percent water in azeotrope
FIG. 8-9. Effect of pressure on azeotrope composition for system, ethanol-water.

an azeotropic composition at atmospheric pressure of 89.4 mol per cent
alcohol (Ref. 4), while the experimental data plotted give 90 per cent
alcohol.

Example of Fractionation at Two Pressures to Separate Azeotrope. Lewis
(Ref. 3) suggested rectification at two different pressures to produce absolute
alcohol from aqueous solutions. As an example of the use of this system, consider
the production of 99.9 mol per cent alcohol from an aqueous feed containing 30 mol
per cent alcohol. The azeotrope in this case increases in alcohol content as the
pressure is reduced. Thus, by operating at reduced pressure, an overhead product
can be produced which contains a higher percentage of alcohol than corresponds to
the azeotrope at some higher pressure. By redistilling this overhead product at a
higher pret <»re, it can be separated into a high-concentration alcohol as bottoms
and an overhead which can be recycled to the low-pressure tower. Figure 8-10
shows such an arrangement. The 30 mol per cent feed is introduced into the low-
pressure tower which operates at 95 mm. Hg abs. and produces an overhead con-
taining 95 mol per cent alcohol and a bottoms containing 0.0001 per cent alcohol.
The 95 per cent overhead product is pumped into the higher pressure tower which
will operate at atmospheric pressure and produce an overhead containing 92.5

206

FRACTIONAL DISTILLATION

per cent alcohol and a bottoms containing 99.9 per cent alcohol. The overhead in
this column will be returned to the low-pressure column for further rectification.
This stream may be returned either as vapor or as liquid; however, the condensing
temperature of the atmospheric column is sufficiently high that it will serve as the
heat supply for the low-pressure column, and in such a case heat economy will be
obtained by totally condensing the overhead and recycling the 92.5 per cent stream
as a liquid. In order to simplify the calculations, it is assumed that all three feed

30%
alcohol

95% alcohol

0.000/% alcohol

Fio. 8-10.

99.3 '% alcohol
Production of absolute alcohol.

streams enter such that there is no change in feed rate across the feed plates. The
95 per cent feed will be slightly below the temperature necessary for this condition,
but it could be preheated by countercurrent heat exchange with the steam con-
densate from the reboiler of the atmospheric tower. The 92.5 per cent stream
will be slightly superheated but will closely approximate the assumption made.
The usual simplifying assumptions are made, and it is assumed that the vapor rates
in the two towers are equal. The vapor rate will be calculated for each tower at a
reflux ratio, 0/D, equal to 1.5 times the minimum 0/D. The larger vapor rate
for the two towers will be employed.

The equilibrium data of Lewis and Carey (Ref. 4) will be used for the atmos-
pheric conditions and the data of Beebee et al (Ref. 1) for 95 mm. Hg. The frac-
tionation will be most difficult in the region containing more than 80 mol per cent

SPECIAL BINARY MIXTURES

207

alcohol, and the relative volatilities for this range are shown in Fig. 8-11. The
atmospheric pressure data were extrapolated past the azeotrope by the use of the
Margules equation. The data for 95 ram. Hg were drawn to agree with the
azeotrope composition reported in the literature (Ref. 2).

1.4

j Data
o Lewis and Carey
KBeebee etal. •

"0.7 0.8 09 1.0

x,Mol fraction alcohol in licjuid

FIG. 8-11. Equilibrium data for system, ethanol-water.

Solution. Basis: 100 mols of original feed. (See Fig. 8-10 for nomenclature.)
By over-all alcohol balances,

30 « 0.000001 Wi + 0.999 JF2

By total balance,

For tower 2:

By alcohol balance,

By total balance,

100 =
2 » 30,

0.95Z)i

i + Wz
ft7! - 70

0.925jD2

and

D2 * 60, Dl - 90
Minimum reflux ratios:

Tower 2: F« * Vm and, from a study of the equilibrium data, it is apparent that
the pinched-in condition will occur at the feed plate. At x ** 0.95, the equilibrium

208

FRACTIONAL DISTILLATION

vapor is 0.948 and by Eq. (7-10),

= 0.925 - 0.948
i 0.948 - 0.95
n _ 0 + D - 12.5D

- 11.5

750 mols

Tower 1 : The minimum reflux ratio in this case could be limited by either feed-
plate condition or by a tangent contact between the equilibrium curve and the
operating line. All three possibilities will be checked:

1. Pinch at 0.925 feed.

0.925, y in equilibrium
. 0.95 - 0.9277
" 0.9277 - 0.925

9.25(90) - 832 mols

0.9277

2. Pinch at 0.30 feed.

x = 0.30, y in equilibrium ~ 0.5875
Using the operating line below the feed plate,

0.5875 - 0.000001

0.3 - 0.000001
- 1.96

Om - Vm + 70
Vm - 72.9 mols

This is much less vapor than required for the feed at x = 0.925, and the minimum
reflux ratio will not be determined by a contact at # = 0.3.

3. Tangency. For this case, the 832 mols of vapor required for x = 0.925 will
be used to determine whether the operating line between x = 0.3 and 0.925 is
below the equilibrium curve. In this section the overflow will be

O - 832 - 90 + 60 - 802
V = 832

and the equation of the operating line will be

0 _ 802 _ 0.9277 - y
V ~ 832 ~ 0.925 - x

y values were calculated for a series of values of x and are given in Table 8-1.

TABLE 8-1

3/op line

^/equilibrium

0.925

0 9277

0.92

0.9229

0 923

0 91

0 9132

0.9139

0 90

0.9036

0 9048

0.88

0.8843

0 887

0.86

0.8651

0.87

0.84

0.8458

0.854

0.82

0 8265

0 838

SPECIAL BINARY MIXTURES

209

In all cases the equilibrium curve is above the operating line except at X' » 0.925.
Thus the minimum reflux ratio for tower 1 is determined by a pinched-in condition
at x = 0.925. The minimum vapor corresponding to this condition is 832 mols
which is greater than the 750 mols calculated for tower 2. This larger value will be
taken as the minimum vapor rate for the system. Thus,

Tower 1:

Operating lines i
Above x = 0.925.

(§)- = 8-25(1-5) = 12-38

OR - 1,115
VT - 1,205

1,115

- 0.9254zn+i + 0.071

•?J-

O.I

Hl^'TTyr
:>! 1^

O.Oi

itiil/b

rium cu

s
2

8 0.001
o

...J-

fflr i

*ope

rating

//>7

Mol fraction
<=>

.00001

_ .,_

... -,

Ssgs

==•

: %. ,, ..,,

.000001
QOO

> —

0001

0.00001

aoooi

0.001

0.01 at u

Mol fraction alcohol in liquid
Fia. 8-12.

210

FRACTIONAL DISTILLATION

SPECIAL BINARY MIXTURES

211

From x - 0.3 to x - 0.925,

' M7^'

90(0.95) - 60(0.925)

1,205 ~n+1 1,205

= 0.9751o£+1 + 0.0249

From x - 0.000001 to x - 0.3,

- ^275 - 70(0.000001)
^m ~" 1 205-a?m"1"1 1 205

- l'.058l£m+i - 5.81 'x 10-8

Tower 2:

Operating lines,
Above feed plate,

Below feed plate,

V - 1,205
0* - 1,145

1,145

60(0.925)

1^05 *w+1 ' 1,205
» 0.9502xn+i + 0.0461

iCn-fl

1,235 30(0.999)

l^OS3"^1 1,205

1.0249a;m+1 - 0.0249

0.91

088

0.88

0.89 OSO

Mol fraction alcohol in liquid
FIG. 8-14.

To Fig. 8-15

212

FRACTIONAL DISTILLATION

The various operating lines are plotted in Figs. 8-12 to 8-15. For the atmos-
pheric pressure tower the plates in the bottom of the tower up to 1 per cent water
were calculated by Eq. (7-77).

Equilibrium curve
atmospheric pressure

XF for tower No 2
xn for tower No /

XD for fewer No 2
xf for tower No I

0910 0915 0920 0925 0930 0935 0940 0945 0950 0955 0960 0965 0970 0975 0980 0985 0990 0995 10
Mol fraction alcohol in liquid
FIG. 8-15.

The low-pressure tower requires a still and 65 theoretical plates. The atmos-
pheric pressure tower needs a still plus 120 theoretical plates. In view of the high
heat consumption and the large number of plates required, other methods of
producing absolute alcohol are more economical. Pressures above atmospheric
would be advantageous for tower 2 and result in fewer theoretical plates.

Nomenclature

6', c' «* constants in Margules equation
D «• distillate rate, mols per unit time
Q as overflow rate, mols per unit time
P » vapor pressure

R « net molal withdrawal rate from section other than Vn and On+i
8 » steam rate, mols per unit time

SPECIAL BINARY MIXTURES 213

T «• temperature

V «• vapor rate, mols per unit time
W « bottoms rate, mols per unit time
x » mol fraction in liquid
# « mol fraction in vapor
7 « activity coefficient
IT «• total pressure

References

1. BEEBEE, COULTER, LINDSAY, and BAKER, Ind. Eng. Chem., 34, 1501 (1942).

2. "International Critical Tables," Vol. Ill, p. 322, McGraw-Hill Book Company,
Inc., New York, 1928.

3. LEWIS, U.S. Patent 1,676,700 (1928).

4. LEWIS and CAREY, Ind. Eng. Chem., 24, 882 (1932).

5 SIMS, Sc.D. thesis in chemical engineering, M.I.T., 1933.

CHAPTER 9
RECTIFICATION OF MULTICOMPONENT MIXTURES

Multicomponent mixtures are those containing more than two com-
ponents in significant amounts. In commercial operations, they are
encountered more generally than are binary mixtures, and as with
binary mixtures, they can be treated in batch or continuous opera-
tions, in bubble-plate or packed towers. Since the continuous opera-
tion is much more amenable to mathematical analysis, owing to the
steady conditions of concentration and operation, it will be considered
first.

Fundamentally, the estimation of the number of theoretical plates
involved for the continuous separation of a multicomponent mixture
involves exactly the same principles as those given for binary mixtures.
Thus, the operating-line equations for each component in a multicom-
ponent mixture are identical in form with those given for binary mix-
tures (see page 119). The procedure is exactly the same; i.e., starting
with the composition of the liquid at any position in the tower, the
vapor in equilibrium with this liquid is calculated; and then by apply-
ing the appropriate operating line for the section of the tower in ques-
tion to each component, the liquid composition on the plate above is
determined, and the operation repeated from plate to plate up the
column. However, actually the estimation of the number of theoreti-
cal plates required for the separation of a complex mixture is more
difficult than for a binary mixture. When considering binary mix-
tures, fixing the total pressure and one component in either the liquid
or vapor immediately fixes the temperature and composition of the
other phase; i.e., at a given total pressure, a unique or definite relation
between y and x allows the construction of the y,x curve. In the case
of a multicomponent mixture of n components, in addition to the
pressure, it is necessary to fix (n — 1) concentrations before the system
is completely defined. This means that for a given component in such
a mixture the y,x curve is a function not only of the physical character-
istics of the other components but also of their relative amounts.
Therefore, instead of a single y,x curve for a given component, there are
an infinite number of such curves depending on the relative amounts of

214

RECTIFICATION OF MULTICOMPONENT MIXTURES 215

the other components present. This necessitates a large amount of
equilibrium data for each component in the presence of varying pro-
'portions of the others, and, except in the special cases in which some
generalized rule (such as Raoult's law) applies, these are not usually
available, and it is very laborious to obtain them. One of the greatest
uses of multicomponent rectification has been in the petroleum indus-
try; for a large number of the hydrocarbon mixtures encountered in
these rectifications, generalized rules have been developed which give
multicomponent vapor-liquid equilibria with precision sufficient for
design calculations. Such data are usually presented in the form
y = Kxj where K is a function of the pressure, temperature, and com-
ponent. The use of equilibrium data in such a form requires a trial-
and-error calculation to estimate the vapor in equilibrium with a given
liquid at a known pressure. This results from the fact that the tem-
perature is not known, so a temperature is assumed, and the various
equilibrium constants at the known pressure jju4 .assumed temperature
are used to estimate the vapor composition. If the_sum of the mol
fractions of all the components in the vapor, so calculated, add up to 1».
the assumed temperature was correct. If the sum is not equal to 1, a
new temperature must be assumed, and the calculation repeated until
the sum is unity. Such a procedure is much more laborious than that
involved in a binary mixture where the composition of the liquid and
the pressure together with equilibrium data immediately gives the
vapor composition without trial and error.

In the foregoing discussion of multicomponent systems, it was
assumed that the complete composition of the liquid at some position
in the column was known as a starting point for the calculation. _The
determination of this complete composition as a starting point is often
t£e most difficult part of the whole multicomponent design. This
difficulty arises from the fact that there are a limited number of inde-
pendent variables which will completely define the distillation process;
therefore, it is not possible to select arbitrarily the complete composi-
tion of a liquid or vapor at some position in the distillation system.
The deggpgesjDO system can be

evaluated (Ref. 3) by applyinglTTthe law of conservation of matter,
(2) tEe law of conservation of energy, and (3) the second Jaw of thexmo-
. These laws together with the phase rule can be applied to
, the still, and the condenser in a distillation unit and the
over-M degrees of freedom for the system determined. For the case of
a reeling column consisting of a total condenser, a reboiler, a feed
plate,lf theoretical plates above the feed plate, and m theoretical

216 FRACTIONAL DISTILLATION

plates below the feed plate, the degrees of freedom for a system involv-
ing C components is

C + 2m + 2n + 10 (9-1)

The variables used for these degrees of freedom are usually chosen from
the ones summarized in Table 9-1. Theoretically, the choice of
variables is completely independent, but in practically all distillation
calculations certain of those given in the tables are ordinarily fixed.
For example, it is usual to define the composition and condition of the
feed, the operating pressure of each plate, and the heat gain or loss to
or from each plate and the condenser. Referring to the table, these
four items add up to C + 2m + 2n + 6, leaving four variables that
can still be assigned. In most cases, to facilitate the design calcula-
tions the reflux ratio is fixed, and in general it is desirable to carry out
the separation specified with the minimum number of theoretical
plates; i.e., the ratio of n/m is such that the total number of plates
shall be a minimum and this effectively fixes one additional variable.
There are thus only two remaining variables which can be fixed, and

TABLE 9-1. RECTIFYING COLUMN VARIABLES

No. of
Type of Variable Variables Fixed

Complete composition of feed (C — 1 )

Condition of feed . . 2

Operating pressure over each plate and in still and condenser . . w + n + 3

Operating temperature on each plate and in still and condenser . m + n -f 3

Heat gain or loss to or from each plate and condenser . . m + n -f 2

Heat supplied' to still . . 1

Composition of product streams . . . , 2(C — 1)

Relative quantity of two product streams . ... 1

No. plates above feed . . . .... 1

No. plates below feed ... 1

Relative quantity of liquid returned to top plate to overhead

product 1

the choice of these is dictated by the essential nature of the operation
to be performed in the column. In the case of a binary mixture, the
choice of these two independent terminal concentrations obviously
gives the complete compositions of the distillate and residue and makes
the design calculations easy and straightforward. However, in the
case of multicomponent mixtures, the problem is more complex and,
in general, the complete composition of neither the residue nor the dis-
tillate can be determined by using the two additional factors to fix two
terminal conditions. In this case, it is necessary to estimate the com-

RECTIFICATION OF MULTICOMPONENT MIXTURES 217

plete composition of either the product or the residue and then proceed
with the calculations as before until the desired degree of separation is
attained. If, then, the calculated product and residue compositions
satisfy a material balance for each component, the estimated composi-
tion was correct. However, if a material balance is not satisfied by
any one of the components, it is necessary to readjust the composition
and repeat the calculation until the material balances are all satisfied
simultaneously. This estimation is often simplified because the degree
of separation is so high that the heavier components will appear in the
product in quantities so small as to be negligible. The same will be
true for the lighter components in the residue.

In selecting the two terminal concentrations, it is desirable to choose
components that will give a significant control of the separation desired
and, at the same time, be components that appear in appreciable
amounts in both the bottoms and the distillate. Because these con-
trolling components are so important in determining the design calcu-
lations, they have been termed the "key components." In other
words, they are the key to the design problem.

In the development of design equations, it has been found convenient
to pick two key components: the light key component and the heavy
key component. The former is the more volatile component whose
concentration it is desired to control in the bottoms; the latter is the
less" volatile component whose concentration is specified in the distil-
late. Thus, in the stabilization of gasoline it is often desired to have
only a sniall concentration of propane in the bottoms in order that the
vapor pressure of the finished product will meet the desired specifica-
tions and also to limit the butane in the distillate so as to retain this
component in the gasoline. In such a case, propane would be the
light key component and butane the heavy key component.

TfheYerminal concentrations of the two key components are impor-
tant because most of the practical equations which have been developed
for the minimum number of theoretical plates at total reflux, the
optimum feed-plate location, and the minimum reflux ratio have
involved these concentrations. However, certain difficulties are
involved: (1) the design specifications may be such that the key com-
ponents are not^bvious and (2) these design equations often require
the concentrations of both key components in the distillate and bot-
toms as well as the concentration of some of the other components.
But as demonstrated in the foregoing analysis, only two of these
terminal concentrations are independent and can be arbitrarily fixed
as design conditions.

218 FRACTIONAL DISTILLATION

The difficulties of choosing the key components and estimating the
complete distillate and bottoms compositions are often the most diffi-
cult parts of a multicomponent design calculation. The problem can
generally be simplified if the design conditions are chosen with this
problem in mind. Thus, in cases where the separation between
adjacent components is essentially complete, the two independent
variables can be chosen as the concentration of the more volatile of
these two in the bottoms and as the concentration of the less volatile
component in the distillate. These adjacent components then become
the key components, and the composition of the distillate and bottoms
can be determined completely enough for design calculations by simple
material balances. Components more volatile than the light key com-
ponent will be almost negligible in the bottom, and components
heavier than the heavy key component will be negligible in the dis-
tillate. For rectifications in which there is an appreciable difference
in volatility between adjacent components and in which a fairly high
degree of separation is being carried out, the design condition can gen-
erally be specified in this manner and thereby simplify the problem.
If the degree of separation is low and/or there are several components
of nearly the same volatility in the range in which the separation is
being made, the selection of the two key terminal concentrations will
generally not give enough information to allow the complete terminal
compositions to be calculated by simple material balances. In such
a case, it is necessary to estimate the terminal concentration of the
other distributed components and then check this estimation by pro-
ceeding with the usual stepwise plate-to-plate calculations. If such
plate-to-plate calculations give a consistent over-all result, the esti-
mated values are satisfactory; if the results are inconsistent, new values
must be estimated and the calculation repeated. It should be empha-
sized that even in this latter case, although a large number of the
terminal concentrations may not be known, only two of these are
independent after the other variables selected have been fixed. , All
the other terminal concentrations are fixed when these two independent
ones are chosen and therefore cannot be given values arbitrarily. The
necessity of having the concentrations of these other components, that
are fixed but not known, offers the main difficulty in setting up a multi-
component distillation example.

The above procedure will give the optimum, design for the, operating
conditions chosen. However, occasionally it is desirable not only to
obtain the desired separation between the two components but at the
same time to control the amount of one of the other components in one
of the products. Frequently, it is possible to accomplish this result

RECTIFICATION OF MVLTICOMPONENT MIXTURES 219

by shifting the feed-plate location. Thus the tower is not designed for
the minimum number of plates to separate two key components but
will employ a larger number of plates to accomplish a desired result.

Lewis and Matheson Method. Several methods have been pro-
posed for the design of multicomponent mixtures, but fundamentally
they are based on Sorel's method. One of the best is that due to
Lewis and Matheson (Ref. 5). This is the application of Sorel's
method together with the usual simplifying assumptions to multicom-
ponent mixtures. The same operating lines as used on page 119 for
binary mixtures are employed to determine the relation between the
vapor composition and the composition of liquid on the plate above,
this calculation together with vapor-liquid equilibrium data being
sufficient for the determination of the number of theoretical plates for
given conditions. The use of this method will be illustrated by the
fractionation of a mixture of benzene, toluene, and xylene under condi-
tions where the separation will be sufficiently good so that the determi-
nation of the terminal conditions will not be difficult.

Benzene-Toluene-Xylene Example. Consider the rectification of a
mixture containing 60 mol per cent of benzene, 30 mol per cent of
toluene, and 10 mol per cent of xylene into a distillate or product con-
taining not over 0.5 mol per cent of toluen^ntfid a bottoms or residue
containing 0.5 mol per cent of benzene V^. reflux ratio 0/D equal to
2 will be used, and the feed will enter preheated so that the change in
mols of overflow across the feed plate will be equal to the mols of feed.
The usual simplifying assumptions will be made, and Raoult's law will
be used. The distillation will be carried out at 1 atm. abs. pressure.

Since the concentration of toluene in the distillate D is low, the
xylene will be practically zero and therefore will be essentially all in the
residue W. Taking as a basis 100 mols of feed, a benzene material
balance, input equals output, gives the following:

60 « DxDB + 0.005TF = (100 - W)xDB + 0.005TF

- WQxDB + (0.005 -
XDB = 0.995
60 « 99.5 - TF(0.99)

D - 60.1

where D = mols of product
W = mols of residue

mol fraction of benzene in liquid distillate

220

FRACTIONAL DISTILLATION

The terminal conditions are then

Distillate

Residue

Mols

Mol
per cent

Mols

Mol
per cent

Benzene

59 8
0 30
0

99 5
0 5

0 20
29 7
10.0

0 5

74 4
25 1

Toluene
Xylene
Total

60 1

100.0

39.9

100 0

Since 0/D = 2, in the top part of the tower
On « 2 X 60.1 - 120.2,

and Vn = On + D = 180.3, and Om = On + F = 220.2, giving Vm
= 180.3. ' :i' l ' * '

For the part of the column below the feed plate, the operating lines
are
For benzene:

0\ /W

ymB = ). X(m+ 1)B ~~ VT

= 1.221a?(m+i)B - 0.0011
For toluene :

/220.2\

= \i8o3y

180.3

(0.005)

For xylene :

ymx =

- 0.164

- 0.0555

Beginning with the composition of the liquid in the still, a tempera-
ture is assumed, and the partial pressure of each component is calcu-
lated using Raoult's law. If the sum of the partial pressure is 760
mm. Hg (the total pressure), the assumed temperature was correct.
The vapor pressure of these components is given in Fig. 9-1. Assume
T = 115°C.

C6
C7
C8

xwP

yw = xwP/ 2xwP

0 005
0.744
0.251

1990
850
390

10
632
98

0 0135
0 854
0 1325

740

1 0000

Since the total is 740 instead of 760, the assumed temperature was
too low, but a more nearly correct temperature is easily found by deter-
mining the temperature from Fig. 9-1 at which the vapor pressure of
toluene is (76!^4o)(850) * 873, giving T « 116.0°C. The calculation
is then repeated for 116°C.

RECTIFICATION OF MULTICOMPONENT MIXTURES 221

xwP

yw « xwPHW 4

0.005

2000

0 0131

0 744

873

650

0 855

0 261

400

100 4

0 132

760 4

1 0000

1800

1400
1300
1200
1100

E 900

fsoo

With this assumed temperature, the sum of xwP is seen to be very
close to 760, and the temperature is satisfactory. Actually, such a
recalculation is not necessary, since
the values desired are ywy and
these values may be obtained from
yw = XwP/^xwP where 2xwP is
the sum of the xwP values. Thus,
in the fourth column of the table
for the first assumed temperature
are given values of yw = x^P/740.
These are seen to correspond closely
to the values in the corrected table
and agree well within the accuracy
of such factors as the vapor pres-
sures, the applicability of Raoult's
law, etc. In general, such a simpli-
fied procedure is satisfactory when
the sum of XwP is within 10 per cent
of the desired value; however, at
times, such a simplification is not
justified, and a preliminary check
on the system in question should be
made to determine the satisfactory
limit of the sum of xwP.

The value of x: is obtained from yw by the use of the appropriate
operating-lme equation applied to each component, yi is then calcu-
lated using Raoult's and Dalton's laws at an assumed temperature of
115°C., and £2 is obtained from the values of y\ by the use of the
operating-line equation. The operation is repeated, making adjust-
ments of the assumed temperatures such that 2xP stays between 700
and 820. In making these adjustments of temperature, it is desirable
to continue using one temperature until the value o^SzP is about as
much greater than 760 as it was less than 760 on the first plate on which
the temperature was used. Thus, in the following table, the values of

90 100 no

Temperature, deg C.
FIG. 9-1.

Com-
ponent

T, °C.

as-
sumed

XWP

yw - XwP/2xwP

Ce
CT
C8

115

1990
850
390

0.005
0.744
0 251

10
632

0.0135
0.854
0.1325

ZxP - 740

2y * 1.0000

Ce
C7
C8

116

2000
873
400

0 005
0.744
0.251

10
650
100.4

0.0131
0.855
0 132

0 0116
0.835
0 153

VxP =760.4

2y - 1.0000

*lP

2/i

rc2

Ce
C7
Cs

115

1990
850
390

0.0116
0.835
0.153

23 1
709
59.7

0.0292
0.895
0.0755

0 0248
0 868
0 1065

2zP « 791 8

2y - 1

x*P

2/2

C7
C8

110

1740
740
330

0.0248
0.868
0.1065

43
642
35

0 0597
0 892
0.0486

0 0498
0 865
0.085

2zP - 720

x*P

2/3

C7
C8

110

1740
740
330

0 0498
0 865
0.085

86.7
640

0 115
0 848
0.037

0 095
0 830
0.075

S#P - 754 7

x*P

2/4

C«
CT
C8

110

1740
740
330

0 095
0 830
0.075

165
614
25

0.205
0 763
0.031

0.169
0.759
0.071

SarP - 804

3«

xj>

y« «

Ce
CT
Cs

105

1520
645
280

0.169
0.759
0.071

257
489
20

a. 336
0.638
0.026

0.276
0.657
0.067

SxP * 766

222

RECTIFICATION OF MULT I COMPONENT MIXTURES , 223

110°C. were continued from xf = 720 to xJP = 804, giving approxi-
mately an equal displacement on both sides of 760.

Com-
ponent

J. J O.

as-
sumed

K - P/760

y, - xK/ZxK

100

1 745

0 276

0.482

0.49

0.402

—

0 735

0 657

0 482

0 49

0 535

0 316

0.067

0.021

0 021

0.063

ZxK - 0.985

x,K

. 2/7

1 52

0.402

0.612

0.635

0.521

—

0 628

0 535

0 336

0 348

0 420

—

0.263

0.063

0 016

0.017

0.059

VxK ~ 0 964

If.

**,

1 52

0 521

0 793

0 738

0 605

—

0.628

0 420

0.264

0 246

0 336

—

0.263

0 059

0 016

0 015

0.058

VxK = 1 073

The table on page 222 carries these calculations up to the sixth plate,
As pointed out on page 215, the vapor-liquid data are more often
given as y = Kx rather than as Raoult's law. With equilibrium data
in such form, the method of calculation is similar. Given the values
of x in the liquid on any plate, the temperature is assumed, and a value
of K for each component is obtaijied from equilibrium data at the
assumed temperature and the operating pressure. The value of y in
equilibrium with this liquid is given by Kx. If the sum of the values
of y is equal to 1, the assumed temperature was correct, and the x
values on the plate above may be obtained from the y values just cal-
culated by using the operating-line equation. If the sum of the values
of Kx does not equal 1, the temperature should be readjusted; but as in
the previous case, this adjustment is usually unnecessary if the sum of
Kx is within 10 per cent of 1, in which case the values of y are calculated
by ya = KaXa/I,Kx.

The benzene-toluene-xylene calculations will be continued, using the
K method. For this particular mixture, where Raoult's and Dalton's
laws are assumed to apply, the equilibrium constant is equal to the
vapor pressure divided by the total pressure; e.g., yair » Xa

224

FRACTIONAL DISTILLATION

The ratio of xC6 to zC7 on plate 9 is approximately that in the feed,
so this plate will be used as the feed plate. The proper feed-plate loca-
tion for this column will be considered in a later section. Above the
feed plate, the procedure is the same, except that the equation for the
upper portion of .the tower is utilized.

The operating-line equations above the feed are
For Oe; " '""

= (^ )

\V /n

0.332

For C7,

For

ynT =

0.0017

Proceeding as before:

Com-
ponent

T, °C.

as-
sumed

y, = xKfZxK

-BIO

C6
C7
C8

1 33
0 533
0.221

0 605
0.336
0 058

0 805
0.179
0 013

0 807
0.180
0 013

0 712
0 267
0 020

ZxK = 0 997

2lO

XuK

2/io

C6
C7
Cs

1.15
0 452
0 184

0 712
0.'207
0.020
* *

0.819
0 121
0 004

0.867
0 128
0 004

0 802
0.189
0.006

VxK - 0 944

XnK

2/ii

#12

Ce
C7
C8

1 15
0 452
0.184

0 802
0 189
0.006

0 923
0 085
0 0012

0.914
0 084
0.0012

0 873
0 123
0.0018

2xK - 1.009

#12

XnK

2/12

#13

C7
C8

1.15
0 452
0.184

0.873
0.123
0.0018

1.005
0.055
0.0004

0.947
0 053
0.0004

0.922
0.0765
0.0006

2xK = 1.061

RECTIFICATION OF MULTICOMPONENT MIXTURES 225

Com-
ponent

as-
sumed

£-"•"

£10

xnK

2/13

c«

C7
C8

0.995
0.379
0.153

0 922
0 0765
0.0006

0 917
0 029
0 0001

0 968
0.032
0.0001

0.953
0.045
0.00015

SarJfC == 0 946

*14

XuK

014

#16

C8
C7
C8

0 995
0 379
0 153

0 953
0 045
0 00015

0 948
0 017
0.00002

0 982
0 018
0 00002

0 974
0.024
0 00003

#15

*i*

2/15

C6
C7

0 995
0 379
0 153

0 974
0 024
0 00003

0 969
0 0091
0 000005

0 99
0 0093
0 000005

0.988
0 0114
0.000007

*!,

*16*

2/16

C6
C7
C8

0 995
0.379
0.153

0 988
0.0114
7 X 10~6

0 983
0 0043

io-6

0 9956
0 0044
10-6

The vapor leaving the sixteenth plate, on being liquefied in the total
condenser, will give a product containing slightly more than 99.5 per
cent benzene. Thus, approximately 16 theoretical plates together
with a total condenser and still or reboiler are required to effect the
desired separation under the operating conditions chosen.

In general, it is instructive to plot the compositions vs. the plates.
This type of figure is shown in Figs. 9-2 and 9-3 for 'the example just
solved. The benzene is seen to rise on a smooth curve, and the con-
centration of toluene in the liquid passes through a maximum two
plates above the still and then falls off in a smooth curve with the
exception of a slight break at the feed plate; the xylene drops rapidly
above the still and then flattens out until the feed plate is reached and
then drops rapidly to a negligible value. The maximum in the toluene
curve is a result of the fact that, in the still, toluene and xylene are the
main components; and since toluene is the more volatile of the two, it

226

FRACTIONAL DISTILLATION

RECTIFICATION OF MULTICOMPONENT MIXTURES 227

228 FRACTIONAL DISTILLATION

tends to increase, and the xylene tends to decrease. This increase in
toluene concentration jefmtinues until the benzene concentration
becomes appreciable V^nd since this latter component is very volatile,
it increases rapidly and forces the toluene to decrease. This increase
of benzene relative to the toluene continues up to the condenser. The
xylene decreases up the column from the still because of its low vola-
tility but cannot decrease below a certain value, since the 10 mols of
xylene in the feed must flow down the column, and this sets a minimum
limit on the concentration of

10 10

OTm - 220 = 0'0455

actually the value will be slightly higher, since the small amount of
xylene that passes upward in the vapor must again pass down the
column. This is due to the fact that essentially no xylene leaves the
top of the column. Above the feed plate, the amount of xylene passing
with the vapor to a plate must be equal to the xylene in the overflow
from the plate, i.e., Vyn = Oxn+i, since DxD is essentially zero. How-
ever, yn = Kxn, giving xn+i = (VK/On)xn\ for a heavy component
such as xylene which does not leave the top of the column in appreciable
amount, the composition of the liquid on one plate is related to that on
the plate below VK/0. In general, K is very small for such compo-
nents and the concentration decreases rapidly as shown by the straight
line in Fig. 9-2.

These concentration-gradient curves are typical of those generally
obtained. The two main components between which the rectification
is taking place tend to increase and decrease up the column, much as in
a binary mixture. They are often called the key components. The
concentrations of the components heavier than the heavier key com-
ponent decrease rapidly as one proceeds from the still up the column,
but they tend to become constant because of the necessity of their
flowing down the overflows in order that they may be removed at the
still. These components then decrease rapidly above the feed plate,
usually dropping to negligible values a few plates above this plate.
The concentrations of components lighter than the light key com-
ponent give the same type of curves from the condenser down the
column as the heavier components do from the still upward. Thus, the
concentrations of these light components decrease rapidly for a few
plates down from the condenser but then flatten out, since by material
balance essentially all of the mols of these components that are in the
feed must flow up through the upper part of the column to be removed

RECTIFICATION OF MULTICOMPONENT MIXTURES

229

at the condenser, and this factor sets a lower limit on their concentra-
tion in this section. Below the feed plate, these light components
decrease rapidly and generally become negligible a few plates below the
feed plate.

Lewis and Cope Method. Lewis and Cope (Ref . 4) applied the same
method graphically, by constructing a separate y,x plot for each com-

0 I 0.02 0.04 0.06 0.08

' Mol Fraction in Liquid

FIG. 9-4.

0.10

ponent. On these plots, the y = x line and the operating lines are
drawn the same as for a binary mixture. The three plots for the
previous examples are given in Figs. 9-4 to 9-6. Only the lower por-
tions of the benzene and xylene curves are given in order to increase the
graphical accuracy. It is interesting to note that for xylene and
toluene the intersection of the operating line, which occurs at xj?, just
as in the case of a binary mixture, falls below the y = x diagonal.
This is due to the fact that xw is greater than XD, and the components
are both of lower volatility than the benzene. If unique equilibrium

230

FRACTIONAL DISTILLATION

0.2 0.4 0.6 0.8

Mol Fraction in Liquid

FIG. 9-5.

O.I 0.2 Q.3

Mol Fraction in Liquid
FIG. 9-6.

RECTIFICATION OF MULTICOMPONENT MIXTURES 231

curves could be drawn on the diagrams, the problem would become
similar to the stepwise procedure for a binary mixture. However, IB
general, such curves are not known. The Lewis and Cope method
was to draw a series of equilibrium curves of the type y ~ Kx which
at constant temperature are in general straight lines through the origin
of slope K. Thus, in the present example, K = P/760, where P of
any one component is a function of the temperature only. Such
equilibrium curves have been drawn in for the temperatures of 105,
110, and 115°C. Starting at xw on each plot, vertical lines are drawn
through this point cutting the equilibrium curves. By trial and error,
temperatures are tried until the sum of the y values at the intersection
of the vertical line through Xw and the equilibrium curve for the
assumed temperature adds up to unity. Thus, if 115°C. is tried, the
sum of the y values at the intersection of the 115°C. curve with the xw
lines is 0.013 + 0.837 + 0.13 = 0.98, indicating that 115°C. is too low.
By interpolation at 116°C., the sum becomes 0.013 + 0.855 4- 0.132
= 1.00, indicating that this is the correct temperature, and the y
values give the composition yw of the vapor in equilibrium with Xw.
Horizontal lines are then drawn through the yw values to the operating
line, the abscissa of the intersection with the operating line being x\.
Vertical lines are drawn through the #i's; and by using the same proce-
dure as for xw a temperature of 112.5°C. is found to give I>y equal to
unity, and the step is then completed to the operating line. In a like
manner, steps are taken up the column. The same operating line is
used until the feed plate is reached, and theh the change is made to the
operating line for the upper portion of the column simultaneously for
all three components.

A comparison of the values of these figures with those obtained in
the previous algebraic calculation shows the close agreement. Actu-
ally, they have to give the same result, since they both are solutions ol
the same set of equations, one being algebraic and the other graphical,
Both methodsjhiavejbheir advantages; in the algebraic method, as a
rule,TugEer accuracy can be obtained "than in the graphical method
this {^especially true in the low-concentration region where the graphi-
cal diagram must t>e greatly expanded or replotted on. logarithmic
paper, such as was utilized in the binary mixtures. The advantage o1
tte graphical method is that it gives a visual picture of the concentra-
tion gradients and operation of the tower. The amount of labor anc
time consumed i£ approximately the same for the two methods.

Numerous analytical methods based on the foregoing methods hav<
been proposed to simplify the trial and error required in the Lewis anc

232 FRACTIONAL DISTILLATION

Matheson method. Some of these methods will be considered in a
later section, but generally the stepwise method outlined above is more
satisfactory. By using y = Kx/^Kx instead of making the Kx's add
to exactly unity, the trial-and-error work of the Lewis and Matheson
method is practically eliminated. In the example just solved, when
became larger than 1, the temperature was dropped, making
less than 1; and this temperature was used until 2Kx again
became greater than unity, and then the temperature was again
dropped. Thus, no actual trial and error was required, but merely
successive drops of temperature of 5 to 10°. Such calculations require
only a few hours more than the simplest of the approximate methods
and only two tor three such stepwise calculations at different reflux
ratios together with the minimum number of plates at total reflux and
the minimum reflux ratio are required to allow the construction of a
curve of theoretical plates required vs. the reflux ratio. In general,
the added confidence that may be placed in the stepwise calculations
relative to the approximate methods more than justifies the extra work
involved.

In using the stepwise method with the simplification that
y = Kx/%Kx, the problem arises as to how much IZKx can differ from
unity and still not appreciably affect the values of y. The justification
of this simplification is that for moderate changes in temperature the
percentage change in the values of K for substances that do not differ
too widely is approximately the same. A little consideration will show
that if all the K values change the same percentage with temperature,
then the values of y calculated by such a method will be independent of
the temperature chosen. This relative variation in the K values is
best expressed in the relative volatility. Thus, if yA = KAxA and
yB = KBxB, then yt/yu = (KA/KB)(XA/XB), and (KA/KB) is the rela-
tive volatility of A to B, OMB (see page 30). If the percentage change
in both KA and KB is the same with temperature, aAB will be a constant
over this region, and a plot of &AB vs. temperature will give immediately
the region over which IZKx can vary without appreciably altering the
y value. Actually, the a's can be introduced into the equations, and
the K 'a eliminated. Thus,

yA + VB + yc + yo + • • • = l

using the relative volatility

<XAB%A , « , QtCBXc , XD

. -p 1 -f~ - ~ -f- Otj)B

RECTIFICATION OF MULTICOMPONENT MIXTURES 233
which can be rearranged to give

since

+ XB + otcsXc + otDBXp +
XB

Likewise,

A similar analysis starting with

XA + XB + Xc
leads to

90 100 110 120

Temperature, deg . C .
FIG. 9-7.

(9-2)

(9-3)

where all the relative volatilities are with respect to the B component;
and in the case of the y* equation, a relative volatility does not appear
with xB, since aBs is L Given the liquid composition on any plate, the

234

FRACTIONAL DISTILLATION

values of x are multiplied by the a corresponding to the component in
question, and the values of ax are totaled to give Sax, then the value of
y for any component is calculated by dividing ax for the component by
Sox. In general, it is desirable to take the volatilities relative to one
of the key components; this will cause a to be greater than 1 for the
components that are lighter and less than 1 for the heavier components.
This method will be most clearly brought out by its application to
actual problems. First it will be applied to the benzene, toluene, and
xylene problem previously solved. Figure 9-7 shows the volatilities
relative to toluene plotted as a function of temperature and also shows
the K for toluene as a function of the temperature. It will be noticed
that the variation in the relative volatilities with temperature is very
small and that for xylene in the lower part of the column a constant
value of a equal to 0.43 is well within the design accuracy. The ben-
zene volatility relative to toluene varies more, but even here the varia-
tion is small. In the previous example, starting at the still:

ano

aXw

yw - <*3/0.869

an 0^1

2/i « a£/0.932

Cfl

0.005

2.36

0.0118

0 0136

0 012

0 0283

0 030

0.744

1 0

0 744

0.856

0.835

0 835

0 896

0.251

0.45

0.113

0.130

0.152

0 0684

0 074

0.8688

0.9317

KT - 0.856/0.744

Tw ~ H6°C.

1.15 KT - 0.896/0.835 = 1.07
Ti - 113.4°C.

«110#2

2/2 » <*c/0.976

0-3

<*110#3

2/3 - ax /1. 02

CT
Cs

0.0254
0,868
0.106

0.06
0.868
0.0477

0.061
0.890
0.049

f.*.*

0.0508
0 864
0.086

0.13
0.864
0 039

0.117
0.845
0.038

0.9757

1.023

KT - 1/0.9757 - 1.025

KT - 0.978
r3 - 110,2°C.

«iio3U

y< « ax/1. 086

auo£«

2/6 «• aaJ/1.192

C6
C7
C*

0.096
0.826
0.076

0.226
0.826
0.034

0.208
0.760
0.031

0.171
0.757
0.071

0.403
0.757
0.032

0.338
0.636
0.027

1.086

1.192

K - 0.922
T4 « 108°C.

K *0.84
f • - 104.8'C.

RECTIFICATION OF MULTICOMPONENT MIXTURES 235

A comparison of the values calculated above with those previously
obtained shows a very close agreement, as must be the case, since both
calculations are fundamentally identical. In these calculations, the
temperature has been determined on each plate by taking the K for
toluene corresponding to the plate and determining the temperature
from Fig. 9-7. Thus, K's were determined for the still and first plate
by dividing the calculated y values by the value of #, KT « IJT/XT]
thus for the still, XT is 0.744, and yT was calculated as 0.856, so that
KT = 0.856/0.744 « 1.15. From Fig. 9-7, the temperature is 116°C.
at K *= 1.15. A little consideration will show that KT also is equal
to I/ Sax, since y = ax/1,ax} then y/x = «/2ax; but for the com-
ponent Relative to which the volatilities are taken, a is equal to 1, and
y/x = K = I /'Sax for this component. This latter method was used
for the second plate upward.

Continuing in this manner, the a's should probably be shifted when
the temperature becomes about 100°C. Taking the new values of a at
90°C. should be satisfactory for finishing the column. Thus, no trial
and error is needed, and only two sets of a values are employed. In
order to speed computations, further modifications can be made. If,
instead of calculating ym, one calculates Vym, where V is the mols of
vapor per mol of residue, then simply adding Xw to these values gives
OmXm+i, where Om is the mols of overflow per mol of residue. Then
Oaxm+i is calculated, and Vym+i = VOaxm+i/I,Oaxm+i, which mate-
rially shortens the time necessary per plate but has the disadvan-
tage that the actual x and y values do not appear. Continuing
by this method, V per mol of bottoms is 180.3/39.9 = 4.52, and
O = 220.2/39.9 = 5.52.

The values of XQ are essentially those obtained previously, indicating
that the trial-and-error calculation to determine plate temperature is,
in general, not necessary and that the Lewis and Matheson method
when carried out in such a manner does not possess any great obstacles.
The use of the relative-volatility method also offers other advantages
than the ease of determining the plate composition. Consider the
fractionation in a vacuum column in which the overhead pressure is
fixed and the pressure drop per plate is an appreciable percentage of the
total pressure causing the absolute pressure to vary widely. In gen-
eral, the K values are approximately inversely proportional to the
pressure and therefore would vary with the changes in pressure as well
as with the changes in temperature. On the other hand, the relative
volatility is often mainly a function of the temperature and only
slightly affected by the moderate changes in pressure. In such cases,

236

FRACTIONAL DISTILLATION

2/6

4.52t/j

5.52x0 • x^ 4~
4.52^/6

««

(5.52az8/7.411)(4.52)

Ce
C7
C8

0 005
0.744
0.251

0.338
0 636
0 027

1 55
2.88
0.12

1 535
3.624
0.371

3 62
3.624
0 167

2 21
2 21
0.098

7 411

K - 5.52/7,411 = 0.746 T6 - 100.7

5.52z7

«90

5.52az7

4.52^/7 ='(5.52<*z/8.56)4.52

5.52s8

c«

2 215

2 47

5.4fr

2 89

2 895

2.954

1.0

2.954

1 56

2.304

0 349

0 42

0 147

0 078

0 329

8 561

K = 5.52/8.56 - 0.646' T7 = 95.8

'5.52a$8

4,522/8

5,52z9

c«

Cs *

7 15
2 304

' 0 158

3 36
1 09
0.065

3 365
1 834
0 316

9 592

K - 0.577 T* - 92.2

C6
C7

0 609
0 334
0 057

it is therefore possible to proceed by the a method, as in the constant-
pressure calculations, without troubling with the pressure variation.

Ta^r Acid Fjractionati&n. As another example of such calculations,
consider the Iractionation of a 35 mol per cent phenol, 15 mol per cent
o-djresol, 30 mol pef cent m-cresol, 15 mol per cent xylenols, and 5 mol
per ce;ttt heavier. The overhead is to be 95 mol per cent phenol, and
the phenol recovery is to be 90 per cent. The still pressure will be 250
mm. Hg abs., and 4 mm. Hg pressure drop will be allowed per theoreti-
cal plate. A reflux ratio 0/D equal to 10 will be employed.

The equilibriuAx data obtained by Rhodes, Wells, and Murray
(Ref . 7) for this type of system indicate that RaoulVs law is followed,
and thus the relative volatilities are independent of the pressure and a
function of the temperature only. Thus, the relative-volatility

RECTIFICATION OF MULTICOMPONENT MIXTURES 237

363.5

330.5

430.5

mols %

. C6 31.50 95.30
OC7 1.50 4.55
mC7 0.05 OJ5

33.05

method will be most suitable for estimating the number of theoretical
plates.

The result of over-all material bal-
ances is given in Fig. 9-8. The ratio
of o-C? to ra-C? in the distillate was
assumed as 30 to 1.

Figure 9-9 gives the volatilities
relative to o-cresol as well as the
vapor pressure of o-cresol. In the
calculations, the temperature is
checked occasionally by determining
the vapor pressure of o-cresol, P0, on
the plate. Since y0ir = P0x0> then

where TT is corrected for pressure drop
in the column. Below the feed, per
mol of bottoms, the mols of vapor are

mols %

C6 3.50 524

oC7 13.50 20.20 •

mC7 29.95 44.70

C8 1500 22.40

R 5.00 7.50

6695
FIG. 9-8.

120 140 160

Te mperatu re , deg . C

FIG. 9-9.

238

FRACTIONAL DISTILLATION

5.43, and the mols of liquid are 6.43; above the feed, the corresponding
figures per mol of distillate are 11 and 10, respectively.

<*160

axw

5.43^ <- a*iK5.43)/0.685

6.43zi

0.0524

1 25

0 0656

0 521

0.573

o-C7

0.202

1.0

0.202

1 60

1 80

n-C7

0 447

0 7

0.312

2 48

2 93

0.224

0.44

0 099

0.79

1.01

0.075

0.087

0 006

0.048

0.123

0 6846

- 250/0,685 - 365 mm.

165°C.

6.43.x,

5.4%,

6.43*2

6.43ax2

5.432/2

0.716

0.775

0 828

1 035

1 045

o-Cr

1 80

1 95

2.152

2 152

2 18

Wl-C?

2.05

2 22

2.667

1 865

1 89

0 444

0 48

0 704

0 310

0 31

0 Oil

0.012

0.087

0 008

5.021

5 370

6.43z3

6.43ax3

5.43^/3

6.43x4

6.43«x4

c«

1.097

1.37

1 32

1 372

1.72

o-Cr

2 38

2.30

2 50

m-Gt

2 34

1 635

1 575

2 02

1 41

0.53

0.233

0 225

0 45

0 198

0.083

0 007

0 082

0 007

5 625

5 835

5.43^4

6.43z6

6.43^X6

5.432/s

6.43x6

Cfl

1.60

1.652

2 07

1 87

1.92

o-C7

2 33

2.53

2 53

2 28

2.48

w-C7

1.31

1 76

1 23

1 11

1.56

0.184

0.41

0.18

0.16

0.38

0.006

0.081

0.007

0.006

0.081

6.017

230(6.43)/6.017 - 246

* 153°C.

RECTIFICATION OF MULTICOMPONENT MIXTURES

239

6.43«x6

5.432/6

6.43x7

6.43ax7

5.431/7

2.40

2.12

2 17

2 72

2.36

o-Cr

2.48

2.20

2.40

2 40

2 08

W-Cz

1.09

0 96

1.41

0 99

0.86

0.17

0 15

0 37

0 163

0.14

0 007

0 006

0.081

0 007

0.006

6.147

6.280

6.43x8

«140

6.43az8

5.43^/8

6.43x9

6.43<*Xj)

2.41

1 26

3 04

2.61

2 66

3 35

0-C7

2.28

1 0

2 28

1 95

2 15

w-C?

1.31

0 675

0 88

0 76

1 21

0 82

0.36

0 392

0.14

0 12

0 34

0 13

0.081

0.087

0.007

0.006

0 081

0 007

6 347

6 457

5.43i/9

6.43xio

6.43<*xio

5.43?yio

6.43xu

6.4301X1!

2.82

2 87

3 62

3 01

3.06

3 86

o-C7

1 81

2.01

1 67

1 87

WI-C7

0.69

1 14

0 77

0.64

1 09

0 74

0 112

0 34

0 13

0.11

0.33

0 13

0 006

0.081

0 007

0.006

0.081

0 007

6.537

6 607

5.43yii

6.43xi2

6.43«Xi,

5.43t/i2

Ce
o-C7

C8
R

3 17
1 54
0.61
0 11
0.006

3.22
1 74
1.06
0 33
0.081

4.06
1.74
0.71
0.129
0.007

3.32
1 42
0.58
0 105
0 006

6.646

Po - 202(6.43)/6.65 - 195;

146.5°C.

6.43x18

6.43«xi3

y™

3.37

4.25

0 634

o-C7

1.62

1 62

0 242

m-C?

1.03

0.70

0.104

0.33

0.129

0.019

0.081

0.007

0.001

6.706

240

FRACTIONAL DISTILLATION

The ratio of phenol to o-cresol in the liquid on the thirteenth plate is
essentially that in the feed, and this plate was used as the feed plate.
The calculations are then completed using a basis of one mol of dis-
tillate. On such a basis, the operating line for each component
I0xn = ll^n-i — • %D and the remainder of the table is set up in this
manner.

2/13

112/13

10X14

10c*X14

Hi/14

C«

0 634

6 974

6 024

7 59

7 54

o-C7

0 242

2 662

2 617

2.617

2.60

77t-C/7

0.104

1 144

1 14

0.77

0 77

0.019

.209

0 082

0 081-

0.001

Oil

.011

0 001

11 060

10*»

lOaXis

102/15

10*16

lOorXie

H2/16

10*17

lOttXiT

H2/17

6.59

8 3

8.0

7 05

8 89

8.38

7.42

9 36

8 73

o-C7

2 55

2 46

2 41

2.41

2.27

2.22

2 22

2 07

m-C?

0 765

0 516

0.498

0.493

0 333

0.314

0.31

0 209

0 195

0.081

0 032

0 031

0.031

0 012

0.011

0 Oil

0 004

0.001

9 X 10-*

8 X 10~6

8 X 10-6

7 X10-*

7 XlO-«

7 X10-6

6 X10-7

6 X10"7

11.398

11.645

11 793

10^18

lOaXis

112/18

10*u

10aXi9

112/19

10X20

10o:X2o

112/20

7 78

9 81

9 02

8.07

10.2

9 3

8 35

10 55

9 53

o-C:

2.02

2 02

1 86

1.81

1 65

1 60

1 6

1 44

WrCj7

0.19

0 128

0 118

0.113

0 076

0 069

0 064

0 043

0 0399

0.004

0 002

2 X lO-3

2 X 10-*

8 X 10-*

7 X 10~4

7 X 10-'

3 X 10-4

3X10~4

11.960

12.086

12.193

P, - 178(10/11.96) = 149 Tit = 138°C.

10^21

10o:£2l

112/21

10^22

10a*22

lly»*

10x23

10aX23

112/23

Ce
o-C7

W-C7

8.58
1.39
0.035

10.8

1 39
0 024

9.74
1 25
0.022

8.79
1 20
0 018

11.08
1.20
0 012

9.91
1 07
0.011

8 96
1.02
0.007

11.30
1 02
4 7 X 10~3

10.1
0.91
4.2 X 10-s

12 214

12.292

12 32

RECTIFICATION OF MULTICOMPONENT MIXTURES 241

10z24

10aX24

112/26

10^26

10«£25

1 If/26

10X26

10tX#28

2/26

9 15

11 52

10 24

9 29

11 70

10.37

9 42

11 88

0 952

o-C7

0.86

0.76

0.71

0 63

0 58

0 047

12 38

12 41

12 46

154(10/12,38) * 124

133°C.

The concentration of m-cresol in the distillate is less than the
assumed value, but a recorrection of this value would not make enough
difference to be significant, and a material balance on this component
is in essence satisfied. The results of the calculations are plotted in
Fig. 9-10.

It is interesting to consider what would happen if the feed had not
been introduced on the thirteenth plate. This calculation has been
carried out and the results plotted in Fig. 9-1 1. Up to the thirteenth
plate, the results are obviously identical with those given in Fig. 9-10;
but above this plate, the change of concentration per plate is much less
in Fig. 9-11. By the twenty-sixth plate, all the components have
become almost asymptotic, and increasing the plates to an infinite
number would make little difference in the concentrations from those
for the twenty-sixth plate. Thus it is impossible to obtain the desired
separation without having plates above the feed plate, since the
asymptotic ratio of phenol to o-cresol is less than the desired ratio in
the distillate. The limit to this asymptotic ratio is obvious from Fig.
9-1 1 ; since the o-cresol, m-cresol, xylol, and residue must all flow down
the column, their concentrations cannot decrease below the value
necessitated by material balance for their removal from the still.
Although the concentration of the phenol is not limited by the same
factor as the heavier components, it is limited by the fact that its
value cannot exceed 1 minus the sum of the concentration of the
heavier fractions; and since a minimum limit for the heavier com-
ponents is fixed, a maximum for the phenol is likewise fixed. The
condition illustrated in Fig. 9-11 around twentieth to twenty-sixth
plate is termed " pinched in"; i.e., conditions are so pinched that
effective rectification is not obtained. As soon as the feed plate is
passed, this pinched-in condition would be relieved, since the heavier
components would decrease rapidly, as in Fig. 9-10, thereby allowing
the phenol to increase.

At a lower reflux ratio, the enrichment per plate would be reduced,
and more plates required for a given separation Figure 9-12 gives the

242

FRACTIONAL DISTILLATION

RECTIFICATION OF MULTICOMPONENT MIXTURES 243

results for the same design conditions as Fig. 9-10, except that 0/D was
7 instead of 10. The number of plates increases from 26 to 35. It is
to be noted that the asymptotic values of the concentrations of the
heavier components in the lower part of the column also increase;
this results from the fact that there is less overflow in this section of
the tower.

24
22
20
18
16
Sl4

£ 12

10
8
6
4
Z

-Creso/

ry/eA

Residue

)

7-C

resol

/Phenol

V A

' V

sswh-

"•"*»" .-».

bam

.c^

}

0.01 0;02 0.04 0.06 O.I 0.2

Mol Fraction in Liquid
FIG. 9-11.

0.4 0.6

Minimum Theoretical Plates at Total Reflux. The minimum num-
ber of theoretical plates for a given separation is obtained at total
reflux, the same as for a binary mixture. This minimum can be calcu-
lated by the stepwise method, using the operating line y = x for each
component for both sections ,of the column. Such a calculation will
give the concentration and conditions through the tower. The
assumptions made on page 175 for the calculation of the minimum
number of plates in the development of Eq. (7-52) apply to any two of
the components of a multicomponent mixture, and by its application
this limiting condition can be calculated. In general, when applying
the total reflux equation to a multicomponent mixture, it is desirable
to use the two components whose concentrations are most accurately
known in the distillate and residue, and most often these two com-
ponents are the kev components. However, the equation applies to

244

FRACTIONAL DISTILLATION

^ CM
fO fO

RECTIFICATION OF MULTICOMPONENT MIXTURES 245

any two components in the mixture. The same limitations as to the
constancy of relative volatility would apply as for the binary mixtures*
Feed-plate Location. The criterion for the optimum location
of the feed plate is that the relative enrichment of the key components
should be a maximum. As with binary mixtures, the feed plate corre-
sponds to the step that passes from one operating line to the other.
The change from one operating line to the other should be made just
as soon as it will give a greater enrichment than continuing on the same
operating line. In coming up from the still, the feed plate is the last
step on the lower operating line; calling this the nth plate, and the key
component vapors entering, yik(n~\} and yhk(n-i), for the light, or more
volatile, and heavy, or less volatile, components, respectively, the fore-
going criterion states that if the nth plate is the optimum position for
the feed, then the x ratio on this plate should be greater when calcu-
lated by the lower operating line from the yn~i values than by the
upper operating line, or

\Xhk/n yhk(n-l) -T (W/Vm)Xwhk yhk(n-l) —

where yn-i = mol fraction in vapor from plate n — 1 when feed is

added to plate n

y'n-i = raol fraction in vapor from plate n — 1 when feed is
added to plate n — 1

For a binary mixture, yn-i, would equal y'n_^ assuming that the
bottoms concentration is kept constant. However, the two values are
not equal for multicomponent mixtures because of the presence of
the components lighter and heavier than the key components. Owing
to the presence of these components, the total vapor is not available for
fractionating the key components. In order to reduce the interference
of the components lighter than the light key component, it is desirable
to utilize the lower operating line to a higher value of the ratio of the
key components than would be the case for yn~i = y'n-i- Owing to
the interference of the heavy components, it would be desirable to
change to the upper operating line at a lower ratio of the key com-
ponents. However, as a first approximation, it will be assumed that
i/n_i ss y^ and a correction for this assumption will be made later.

Combining Eq. (9-4) with

Fn - Vm + (P + 1)F
and

Wxw + DXD — FZP

246 FRACTIONAL DISTILLATION

gives

~* - 1 )
—

Also, by this criterion the x ratio on the (n + l)th plate should be
greater when calculated by the upper than by the lower opera ting line:

(Xlk
Xhk

— (D/Vn)XDhk y'nhk + (W/Vm)Xwhh

where yn = mol fraction in vapor from plate n when feed is added to

plate n
y'n = mol fraction in vapor from pla/te n when feed is added to

plate n + 1
Again using yn = y'n gives

ZFlk +

The right-hand sides of Eqs. (9-5) and (9-7) are equivalent and
equal to (xik/Xhk)t as given by the intersection of the operating line;
thus, since n is now the optimum feed plate, the subscript may be
changed, and the criterion for the feed-plate step becomes

^ >

~ W/ v ;

where (xik/xhk)i is the ratio of the key components as given by the
intersections of the operating lines. However, it should be emphasized
that the feed plate does not necessarily step across the intersection of
the operating lines, as it does for a binary, but simply that the ratio of
the keys for the optimum feed-plate step passes over the ratio of the
values given by the operating-line intersections. The absolute value
of both key components may be several times the values given at the
intersection, provided the ratio satisfies Eq. (9-8).

The derivations of Eqs. (9-5) and (9-7) neglected the effect of the
changing concentrations of the light and heavy components. The
light components have a relative constant concentration above the
feed plate, and this fact can be used to calculate their value. Thus,

and, assuming

RECTIFICATION OF MULTICOMPONENT MIXTURES 247
giving

n Vn - (On/Kl)

As shown on page 228, the concentration of a light component
changes approximately by a factor of Om/VmKi per plate below the
feed plate. Thus the change in the concentration of a light component
per plate at the feed plate is *

- (Q./F.JC,)

VJt

and for most cases DXDI =

Let Aj equal the sum of the changes per plate for all light compo-
nents, then

(9-10)

where ^ is the sum of the term for all components more volatile than

the light key components.

In a similar manner the sum of the changes in the vapor concentra-
tions for the heavy components per plate at the feed plate, AA, can be
calculated.

where ^ is the summation for all terms less volatile than the heavy

**+
key components,

The corrections due to these changes can be utilized (1) to calculate
terms to be added to the intersection ratio (#«,)/#**)* °r (2) to modify
the expression for the intersection ratio. The relations can be formu-
lated so that both methods give essentially the same optimum ratio for
the key components at the feed plate. The latter method is believed
to be the more convenient, and using <£ as the optimum ratio, an
approximate expression is

248

FRACTIONAL DISTILLATION

In using this expression, it is recommended that K i and KH be cal-
culated as (oti/aik) and («/»/«*&), respectively. This expression is simi-
lar to that for the intersection ratio, but if A* is large, <£ will be larger
than (xik)/(Xhk)%y if A* is large, <f> will be smaller. Using these correc-
tions, the optimum feed-plate location is such that

(**) ^ + <; (V)

\W/ \W/

Optimum Feed-plate Location. The use of Eq. (9-12) will be illustrated by the
examples already considered.

1. Benzene-Toluene-Xylene example

~ - 0.18;

- 120.2;

- 180.3

220.2

0593 ~~ 0 29>

29-7

In this case <#> is essentially the same as the intersection ratio of the operating
line which was 2.0.

2. Phenol-Cresol
D - 33.05;
mC7: #, =9^-0.54

330.5; On - 430.5;

3.6

1.26
0.392
1.26

_ 0.087
R. Kh - | 2Q

C8: #*

430.5

- 0.31

- 0.07

0.54(30)

1 -

0.54(363.6)
330.5

1 -

0.54(363.6)

430.5

4- 0.07(5)

0.31(363.6)

1 330.5

0.31(363.6)

430.5

0.07(363.6)
330.5

1 -

0.07(363.6)
430.5 .

0.039

^ »

(sm ~ 0 3-5

15 + (sm ~ 0 13-5

RECTIFICATION OF MULTICOMPONENT MIXTURES 249

As compared to the intersection ratio of 8Ms *" 2.33. In the stepwise calcula-
tion, the ratio of the key components in the feed plate was 2,08 and 2.31 on the
plate above. Thus the feed-plate composition utilized agrees with Eq. (9-12)
satisfactorily.

In the foregoing derivations the feed could be vapor, liquid, or a
mixture of the two, but it was assumed that the vapor and liquid leav-
ing the feed plate were in equilibrium. In the case of an all-vapor feed
that mixes with the vapor from the feed plate but does not react with
the liquid on the plate, a similar derivation gives (Ref . 1)

{Xhk/f+i \Xhk/% \XhkJ

This indicates the ratios of the concentrations of the key components
on the plate above the feed plate, and the plate above that should
straddle the intersection ratio for the key components. Or, if the
nomenclature is changed such that the feed plate is the first plate with
which this feed vapor reacts (i.e.j the plate above) and not the plate
into which it enters, then the criterion becomes the same as before.

Minimum Reflux Ratio. As in the case of binary mixtures, there
is a reflux ratio below which it is not possible to obtain the desired
separation of a multicomponent mixture even when an infinite number
of plates is used. The calculation of this minimum for a multicom-
ponent mixture is much more involved than for the corresponding
binary mixture. The condition of the minimum reflux ratio requires
that to perform the given separation an infinite number of plates must
be needed, which means that there must be a pinched-in region where
there are a large number of plates having the same composition, but
for multicomponent mixtures of normal volatility this region usually
does not occur at the feed plate as it does in the case of binary mix-
tures. Under this condition usually a relatively few plates above the
feed serve to reduce the concentrations of the components less volatile
than the heavy key component to negligible values, and then a true
pinched-in condition does occur with only the heavy key and more
volatile components present. Likewise, below the feed plate a rela-
tively few plates reduce the concentrations of the components more
volatile than the light key component to negligible values, and a true
pinched-in condition occurs with only the light key and less volatile
components present. Thus the tower operating at the minimum
reflux ratio might be considered as being composed of five sections
(Ref. 2): - v

(1) Starting at the still the only rtrmrmruients present in significant

250 FRACTIONAL DISTILLATION

amounts are the light key and the less volatile components, and in
proceeding up the tower in this bottom section, the concentration of
the light key component increases relative to the concentrations of the
heavy key and heavier components.

2. Above section 1 is a pinched-in region where the concentrations of
the light key and the less volatile components are all con^iant, and an
infinite number of plates is required to produce a finite change in the
plate composition.

3. Next there is an intermediate region where the concentrations of
the components less volatile than the heavy key component decrease
to negligible values and where the concentrations of the components
more volatile than the light key component increase to significant
values.

4. Above section 3 is another pinched-in region where the concen-
trations of the heavy key and the more volatile components are all con-
stant, and an infinite number of the plates is required to produce a finite
change in plate composition.

5. A section exists where the concentration of the heavy key com-
ponent decreases relative to the concentration of the more volatile
components until the overhead composition is obtained.

Actually there is no sharp line of demarcation between these five
sections, but this division serves as a useful picture for considering the
case of the minimum reflux ratio. The feed to the fractionating column
would be introduced on some plate in intermediate section 3, and the
true criterion for the minimum reflux ratio should be based on matching
the ratio of the concentrations of the key components above and below
the feed plate under conditions such that a pinched-in section occurs
both above and below the feed plate. For mixtures of normal vola-
tility, a pinched-in region in only one section does not necessarily mean
that an infinite number of plates would be required to perform the
desired separation at the reflux ratio under consideration, since by
relocating the feed plate, such as to shift the ratio of the concentra-
tions of the key components at this plate, the section that was not
limited could be made to do more separation and thereby relieve the
load on the pinched-in section. In other words, for mixtures of normal
volatilities the condition of the minimum reflux ratio is not determined
by either the fractionation above or below the feed plate alone, but is
determined such that the separation is limited both above and below
the feed.1 The conditions in the intermediate feed section lead to

1 For mixtures with abnormal volatilities the pinched-in condition may be due to

RECTIFICATION OF MULTICOMPONENT MIXTURES 251

calculational difficulties because the concentrations are not constant in
this region. Actually in this region the ratios of the concentrations of
the key components generally change in the opposite direction from
that desired for the separation; thus, proceeding up the column from
the feed plate at the minimum reflux ratio, the ratio of the concentra-
tions of the light key to the heavy key component decreases instead of
increases. No satisfactory method of estimating the extent of this
" retrograde" rectification has been developed. However, it is rela-
tively simple to calculate the plate composition for a region where the
concentrations are the same on successive plates, and such calculations
can be made the basis for estimating the minimum reflux ratio.

Case I. It can be assumed that the concentrations of all com-
ponents are constant for a number of plates above and below the feed
plate. This requires that the pinched-in condition between the key
components must occur with all the components present in significant
amounts; actually, as pointed out before, only certain of the compo-
nents are present at the pinched-in condition. The presence of these
extra components makes the separation more difficult, and for that
reason the minimum reflux ratio calculated on the basis of the assump-
tions of Case I will be equal to or greater than the true minimum
reflux ratio.

Case II. Alternately it can be assumed that the ratio of the concen-
trations of the key components for Jbhe pinched-in region below the feed
plate (sectio^JjIc^ the same ratio for tlie pinc£ed-in

region above the feed plate (section 4), which amounts to neglecting
tfieTSlerme^at^ region^section 3) . However, for the actual operation
sectiori"Sn^usTBepresent and in this section the ratio of the concentra-
tions of the key components changes in the opposite direction to that
desired for the separation; for that reason, the assumptions of Case II
correspond to an easier separation than the actual case, and the mini-
mum reflux ratio calculated by these assumptions will be equal to or
less than the true minimum reflux ratio.

Calculation of the minimum reflux ratio for these two cases will give
limits for the true minimum reflux ratio. To evaluate these cases, it
is necessary to calculate the concentrations of the various components
for the pineh<gd-m or constant composition regions. Such composi-
tions are"easlly cjH<^ tKe reTaHve^volatility, a, and

the fact that the ratio of the concentrations of any two components are

a tangent contact between the equilibrium curve and the operating line, and the
separation will not be limited both above and below the feed plate.

252 FRACTIONAL DISTILLATION

the same on successive plates; thus for the volatile components above
the feed plate,

J2L £L = !L = (°/D)Xi + XDI - 2i \ (v/D)yi

othk xhk ™ yhk (0/D)xhk + xDhk <*hk [ (V/D)yhk - >

giving

_ ahkxDi(D/0) _

^ - «» + a,
and

01 - <*A*
and, for less volatile components below the feed plate,

Xwtk

- xwh ah((Vm/W)yh + xwh]
gvng

' \0m) xlk

and

. - (9_ig)

In these equations the last term of the denominator is usually small
relative to the other factors and can be neglected for the first estima-
tions, making only the flow quantities, relative volatilities, and termi-
nal concentrations necessary for the calculation of the asymptotic
values. Corrections can then be made for the last term in the denomi-
nators, but usually this is not necessary.

These equations can be used to calculate the concentrations of the
key components for evaluating the minimum reflux ratios for Cases I
and II, which would involve matching the ratio of the concentrations
of the key components above the feed plate with the same ratio below
the feed plate. This leads to a quadratic solution for the minimum
reflux ratio. Thus for Case I the ratio of the key components above
and below the feed plate are equated, and allowance is made for all
components at their asymptotic values in both sections.

y XK - y

Xhk /n 1 - xn - xh -

RECTIFICATION Of1 MULTICOMPONENT MIXTURES 253
On substituting the values of Eqs. 9-13 and 9-15 and rearranging,

0\ -- (9-17)

In Case II, below the feed plate only the light key and less volatile
components are considered; above the feed plate, only the heavy key
and more volatile components are considered :

— Xhk — y

_ Xlk -

Y Xi

The solution of this equation is the same as Eq. (9-17), but instead of
/ and J, I' and «/', respectively, are used. Where

** i r JL (X»**\(D\\

C* = - aik — oihk + an [ — - J I 77 j
XDikl \XM/\U/I

D = mols of distillate per unit time

F — mols of feed per unit time

T pF . aik
I = +-

•2[-

» I C^ZA """""

h l_

/ = ^ + aWl^)l^r

(o*\ V r

i»/WZ/

* r*~

254 FRACTIONAL DISTILLATION

Xwh

(See nomenclature at the end of this chapter.)

These equations should not be applied to mixtures having minimum
reflux ratios determined by a tangent contact in one section of the
tower only. It is customary to use relative volatility values corre-
sponding to approximately the feed-plate temperature. Actually
some of the values correspond to the pinched-in region above the feed
and the others to the lower section, but usually this refinement is not
made although it can be done if necessary.

In addition to Eq. (9-17), it is possible to develop a large number of
alternate relations. The most helpful of those are obtained by equat-
ing the ratio of the key components at the pinched-in region to the
optimum feed-plate ratio.

Matching the optimum feed-plate ratio, 4, with the pinched-in
region,

yik\ _ [ (0/D)xik + XDIJC 1 aw
L (0/D)xhk + xDhk] alk

and solving for 0/D,

0 __ (cthkXplk/<t>) •""

Equation (9-18) is applicable to either binary or multicomponent
nixtures, and the problem is one of evaluating XM.
Tor Case I:

- xik - 2, xi -

1 - Ssi -

Xh = —

For Case II:

1 -

- 1 ~ *» - Xl ""

TTT
or, in general,

RECTIFICATION OF MULTICOMPONENT MIXTURES 255

where ft = 1 f or a binary

- 1 - 2xi for Case II

= 1 — 2xi — Zxh for Case I

The terms 2xt and 2xA can be evaluated by Eqs. (9-13) and (9-15) to
give

Oihk%Dlk

(
B

(9-20)

The first term is the minimum reflux ratio for a binary mixture with
the same ratio of key components in the distillate and <f>. Equation
(9-20) can be written

(g) -(

\/>/min \D

Equation (9-20) involves no trial and error if On = Ow, but it does
for other cases. By equating the liquid-phase ratios instead of the
vapor-phase ratios of Eq. (9-18), one obtains

(n
g
-'J

(9.22)

This equation requires no trial and ^error for F» = Fm. The first
bracket again corresponds to the binary mixture case for the same <f>
and ratio of key components in the distillate.

Equations (9-20) and (9-22) are given for Case I, and for Case II the
last term involving a;^ is eliminated.

Similar equations can be derived by equating the ratio of key com-
ponents below the feed plate to 0. Thus,

^\0ni <1 +

256 FRACTIONAL DISTILLATION

and
0\ VnW

xwlk\ /1

~ -T) V

These two equations correspond to Case I; Case II is obtained by
omitting the last term involving XDI in each equation. The bracketed
terms are again the minimum reflux ratios for the binary for the same
<t> and ratio of key components.

If there are no components heavier than the heavy key component,
Eqs. (9-20) and (9-22) should give essentially the true minimum reflux
ratio; if there are no components lighter than the light key, Eqs. (9-23)
and (9-24) should give the desired answer. In case both light and
heavy components are present, the choice should be based on the rela-
tive size of the terms involving the heavy and light components. If
the term involving the summation of the heavy terms is greater than
the summation of the light terms, then Eqs. (9-23) and (9-24) are pre-
ferred to Eqs (9-20) and (9-22), and vice versa. The choice between
Eqs. (9-20) and (9-22) and between Eqs. (9-23) and (9-24) is purely a
matter of convenience.

In the use of Eqs. (9-17) to (9-24) the values of the concentration
needed were calculated by difference. For example, in deriving Eq.
(9-17), the concentration of the heavy key component was calculated
as,

Instead of this relation, it is possible to use

and other equations would be obtained but they give essentially the
same results (Ref. 8).

As a result of the various approximations involved, a large number
of approximate equations can be derived. For the general case, none
of them gives the exact answer. The ones given here have proved to
be of real utility. Equation (9-17) is probably the most general and,
using the proper relative volatility, it should give answers for Cases I
and II that are above and below the true minimum reflux ratio. The
other equations presented involve an approximate expression for the

RECTIFICATION OF MULTICOMPONENT MIXTURES 257

ratio of the key components at the feed plate. It is therefore not so
certain that the upper and lower values will bracket the true minimum
reflux ratio but in most cases they do. Equations (9-20) to (9-24) are
in general considerably easier to employ than Eq. (9-17) and, for that
reason, are more commonly used. When it becomes possible to pre-
dict exactly the extent of the retrograde fractionation in the feed-plate
section, it will be possible to develop exact equations for the minimum
reflux ratio.

The foregoing derivations were made on the assumption that there
were no components heavier than the heavy key components at the
upper pinched-in region and no components lighter than the light key
components at the lower pinched-in region. This is not completely
true. If there are components only slightly heavier than the heavy
key components or only slightly more volatile than the light key, it is
possible for them to be present at the pinched-in region. This can be
shown as follows:

Consider the pinched-in section below the feed plate, then xm = xm+i
for all components present and

= Om+ixm+i - Wxw
= Kxm

and combining and solving for xm the value at the pinched-in region,

- (w/°^x^
Xm "" 1 - (VmK/Om)

For the light key and the heavier components the value of xm is posi-
tive and finite, which necessitates VmK/Om being less than 1.0. For
the light key components Wxw/0m is usually small in comparison to
Xm which results from VmK/Om being only slightly less than 1.0. If
a more volatile component has a K value significantly larger than K^,
then for that constituent VmK/Om would be greater than 1.0 and the
only solution for the above equation is xw ~ 0 (assuming that this
component is not being added below this region which could give nega-
tive values of xw). The zero value for xw was the condition assumed
for the light components in the derivations of Eqs. (9-17) to (9-24),
and in most cases it applies to all components having K values 20 per
cent or more greater than Km. However, it is possible for a compo-
nent to have a K value only slightly greater than Kik and such that
VmK/Om is still less than 1.0. This type of component could appear
at the lower pinched-in region in significant amounts, and its concen-

258 FRACTIONAL DISTILLA TION

tration should be included in calculations for Case II. Case I equa-
tions have already included a provision for such a component.

The derivation of the minimum reflux ratio equations does not take
into account the possibility of components intermediate in volatility
to the two key components. When such components are present, they
distribute themselves between the distillate and the bottoms, and dis-
tribution is a function of the reflux ratio; i.e., the percentage of such a
component that goes overhead will be different at total reflux than at
the minimum reflux ratio. Such components can be included in the
minimum reflux ratio equations either as light or heavy components
but to do so requires their concentration in the distillate or bottoms.
It is suggested that this distribution be made such that the values of
their mol fraction as calculated by Eqfc. (9-13) and (9-15) are equal.

These equations will be applied to some of the previous examples.

Minimum Reflux Ratio Examples. 1. Benzene-Toluene-Xylene (page 219). The
key components are benzene and toluene, and the design conditions are given
below. Since there are no components lighter than the light key and since
Vn » Fm, Eq. (9-24) will be employed.

F - 100 mols; W - 39.9 mols; D - 60.1 mols
otik « 2.5; ahk » 1.0; ah =- 0.45
4> = 1.98 (see page 248)

rio/071i 0>005\

/o\ 39.9 LOV°-744 Lfl8;

Wmin + ~" 60.1 L 2.5 - 1.0

[1 -f 2.5(1.98)]
39.9 ["0.45(0.251)1

1.0

This value should be nearly exact because, in addition to the key components
only heavy components were present and in small amounts. The value of <f> used
was for a reflux ratio (0/D) » 2.0 and should be rechecked.

_ 0-18(120.2)
0.18 „,., 60.1

160.1

30-f (K-K -1)29

0.992 V

so the assumed value was satisfactory.

As a comparison, use Eq. (9-17) and neglect the small correction terms. Assume
- 1-0.

RECTIFICATION OF MULTICOMPONENT MIXTURES
I

259

E -
G -

Tf ess

0.744
2.5 _
2.02 ~

1.0
1.505
-100

39.9

(2.5 - 1.0)
« 1.24
* 0.665

2.02

/-0665+25r39-9(6(U)'U °'251 ^-077

«/ — • \j, \j\jo —p tu,^j I rti-k'"' 1 / -t r»rk~"T~\ I \ o r i n f "~"

I7 = -0.84
J' - 0.665

Case I:

39.9
60.1

(-0.84) 4-0.77 - 0.213

-OH - -0.84(0.77) - 1.24(0.665) - -1.47
0.213

/ON
\D)

(-1.47)

1.09

The calculated value is close enough to the assumed value of 1.0, and the trial
will not be repeated.
Case II:

The calculation is the same except that /' and J' are employed.

1.01

In this example, the two cases give approximately the same result, but the
Case II value should be essentially correct because no light components were
present.

Values obtained with the other equations are summarized in the following table:

(O/
Case I

Case II

Eq. (9-24)

1.0

1 0

Eq. (9-17)

1.09

1.01

Eq. (9-23)

1.02

1 02

Eq. (9-20)

1.07

0 97

Eq. (9-22)

1 0

0.96

2. Phenol-Cresol (page 236).

F,« 100 W «

F« - Vm, aik - 1.26, aM

a» - 0.087; <A

66.84 D - 33.16

- 1.0, awC7 - 0.66,

- 2.26 (for ^ -

- 0.39

260

Using Eq. (9-24),

'FRACTIONAL DISTILLATION

Q\ 5^84 (l.O (0.202 -|g) [1+2.26(1.26)])

D/min * 33.16 I 1.26 - 1 J l

66.84 [-0.66(0.444) 0.39(0.224) 0.087(0.075) 1 _ _

33.16 Li. 26 - 0.66 "*" 1.26 - 0.39 "*" 1.26 - 0.087 J ~~ "*" ~

Rechecking 4 at (0/D) « 5.6,

A _ 0.54(219)
0.54(30)

JL_
286

186

0.54(219)

286

1 -

+ 0.31(15)

0.31(219)
186

1 -

0.31(219)

0.07(5) -

286

35 + (eg - 0 3'5

1 -

0.07(219)

186
0.07(219)

286

0.05

15 +

016 ~0

2.24

13-5

This is close enough to the value of 2.26 employed so that no recalculation is
necessary. In other cases, such as an all vapor feed, the value of <£ varies much
more with the reflux ratio.

3. If, in the preceding example, the feed condition had been such that 0,, « Om,
a higher reflux ratio would have been required. As a first assumption, (0/D)mm
will be taken as 7.0

Calculating <

232

232 » Om;
0.54(265)

265; Vm = 165
0.31(265)

Using Eq. (9-23),

66.84
i " 33.16 I

)* 232 1 03K

1 ^ 232

0.07(265)'
232

' 0.54(165) ' U'JU

D; 0.31(165)

232
35 1 f 265

232

1 -

4- 0 07(5")

0.07(165)

1 1 0 C

232 J
0.075 '

, d° ' L 165(0.94)

1 J o.O
-J 1 eo

15 i r 265

10 e

' L 165(0.94)
1"6(0"0") °-052l

+ 1.52)
, °'224 ,

*.-v,Vw.-w-/ 1.52 1 (1

1.26 - 1 J
/66.84\ / 0.447

006

^33.16/ V1.26 - 0.66 ^ 1.26 - 0.39

6.9

RECTIFICATION OF MULTICOMPONENT MIXTURES 261

This value is near enough to the original assumption and no additional retrials
will be made. »

Gasoline-stabilization Example. As a further illustration of the use
of these methods, a gasoline-stabilization operation will be con-
sidered. The feed composition is given in the table on this page and the
tower is to operate at 250 p.s.i.g. A reflux ratio of 2 will be used in the
upper portion of the tower, arid the feed will enter such that £0/F)m
below the feed will be 1.5. It is desired to recover 96 per cent of the
normal butane with the stabilized gasoline, but this bottoms product
is to contain not over 0.25 mol per cent propane.

In preparing this table, it was assumed that the concentrations of all
components lighter than propane were negligible in the residue and
that all components heavier than n-C* were negligible in the distillate.
The isobutane is intermediate to the propane and n-butane and there-
fore will appear in appreciable quantities in both the distillate and
residue. Since the i-C4 is more volatile than n-C4, the following table
was prepared on the assumption that 20 per cent of the i-Ci in the feed
would appear in the overhead. The volatilities relative to n-C4 are
given in Fig. 9-13. These relative volatilities are based on the fugacity
data of Lewis and coworkers (Ref. 6). The equilibrium constant K

Feed

Residue

Distillate

Mol,
per cent

Mols/100 feed

Mol,
per cent

Mols/100 feed

Mol,
per cent

CH4

2.0

—

2.0

6 33

C2H6

10.0

—

10 0

31 60

C8H6

6.0

—

6 0

19 00

C3H8

12.5

0 0025TF

0.25

12.5 -0 0025T7

39 00

i-C4Hio

3 5

2 8

4 10

0 7

2 2

n-CiELio

15.0

14.4

21 10

0.6

1.9

15.2

15 2

22.20

C«

11.3

11 3

16.50

9.0

13.20

8.5

8 5

12.40

360°F.

7.0

10.20

68.2 +0.002517

31.8 -0.0025TF

68,2 4- 0.0025TF - W
W - 68.4; D - 31.6,

for n-C4 is also plotted in this figure. Since the overhead is very
volatile, it will be removed as a vapor, only enough liquid being pro-

262

FRACTIONAL DISTILLATION

0.004

0.002

0,00?

150

200 250 300

Temperature ,deg.F.

Fia. 9-13.

350

400

RECTIFICATION OF MULTICOMPONENT MIXTURES 263

duced in the partial condenser to furnish reflux. It is assumed that the
reflux from the condenser leaves in equilibrium with the product vapor.

Starting at the composition of the overhead vapor, the calculations
are carried down the column by the use of the equations given on page
232 and the results are summarized in Table 9-2. The first coulmn of
this table gives the components, the second column lists the vapor con-
centrations for the plate in question, and the third column gives the a
values at the assumed temperature. The next column gives the values
of the vapor concentrations divided by the relative volatility, and by
using Eq. (9-3) on page 233, the liquid concentrations for this plate are
obtained by dividing the values of the fourth column by the sum of all of
the values in the fourth column. On the basis of 1 mol of overhead
vapor or product, there are 2 mols of reflux, and for this reason the fifth
column listst twice the concentrations obtained from column four and is
therefore the actual mols of overflow for the basis chosen. There will
be 3 mols of vapor to the plates, and the mols of each component'in the
vapor to any plate above the feed plate must equal the sum of the mols
of that component in the product and in the overflow from that plate;
i.e., the sum of the values in column five plus the values in y0 H.. These
vapor values for the plate below are listed in the last column of the
table. In Table 9-3, for the calculation beginning at the still, a similar
procedure was used employing Eq. (9-2) on page 233 and using a basis
of 1 mol of residue.

A temperature of 100°F. was assumed for the partial condenser, and
the calculated temperatures based on K n-c4 are given for each plate.
At the second plate below the top plate, the a values are shifted to
150°F. The liquid on the second plate below the top plate has a ratio
of CaHs/n-C^ a little higher than the feed ratio, and this plate will be
made the last plate above the feed; i.e., the feed plate will be the fourth
from the.toj) of the column. If an attempt is made to carry the calcu-
lations farther down the tower, a serious difficulty will be met in that
no components heavier tha$ ^Q± have been considered, but they are
much too large to be neglected below the feed plate. The most satis-
factory solution to this difficulty is to carry out calculations starting at
the still and continuing up to the feed plate. These calculations are
presented in Table 9-3. Such calculations are continued until the
ratio of C3H8/n-C4 in the vapor from some plate is approximately the
same as the ratio in the vapor calculated from the feed plate in Table
9-2. Thus, it is found that the vapor from plate 8 of Table 9-3 gives a
ratio approximately equal to the ratio on the T— 3 plate of Table 9-2.

264

FRACTIONAL DISTILLATION

TABLE 9-2
(Basis: 1 mol overhead vapor; 0/D = 2)

2/O.H.

<*100

2/0 H

2yo H

3yT

o _ a

"XR *2y/ct

0.0633

36.5

0 00173

0.012

0.075

0.316

7.4

0 0427

0.296

0.612

C8 —

0.190

3.0

0 0633

0.440

0.630

C8 +

0 390

2.7

0 144

1.000

1 390

i-C4

0.022

1.3

0 0169

0.117

0.139

rc-C4

0.019

1.0

0 019

0.132

0.151

0 2876

Kn-c< - 0.2876 T = 98°F.

3i/r

«ioo

3i/r/<*

2xr

32/r-i

0.075

36 5

0.0021

0.004

0 067

0.612

7 4

0 0826

0 155

0 471

C8~

0.630

3.0

0.210

0.394

0.584

C8+

1.390

2 7

0 515

0.965

1 355

*-C4

0.139

1 3

0 107

0.200

0 222

7l-C4

0.151

1.0

0 515

0 283

0 302

1.067

1.067/3 = 0.356 T = 120°F.

SVT-I

<*100

3yT-i/a

2XT-1

3?/r-2

0 067

36 5

0 0018

0 0029

0.066

0.471

7.4

0.0637

0.103

0.419

C3-

0.584

3 0

0 195

0.316

0.506

c,+

1.355

2.7

0.501

0.810

1.200

^-€4

0.222

1.3

0 171

0.277

0.299

w-C4

0.302

1.0

0.302

0.487

0.508

1.2345

n-c4 - 0.41 T - 130°F.

3l/r-2

«150

32/r~2/«

2zr-2

32/r.-8

0.066

0 0025

0.003

0 066

0.419

0.070

0.091

0 407

C3-

0.506

2.6

0.195

0.253

0.443

C84-

1 200

2.3

0.522

0.678

1 068

i-C4

0,299

1.23

0.243

0.315

p. 337

n-C*

0.508

1.0

0.508

0.660

0.679

1.5405

c< - 0.51

150°F.

RECTIFICATION OF MULTIGOMPONENT MIXTURES 265

Approximately eleven theoretical plates in addition to the still and
partial condenser are required.

Feed-plate Matching. In the gasoline stabilization problem just
considered, the vapor compositions obtained by calculating down from
the condenser do not appear to match very satisfactorily with those
obtained by calculations up from the still. This is due to the fact that,
with the exception of propane,^ isobutane, and n-butane, the compo-
nents involved in the two cases are different. In general, this condi-
tion is always encountered when there are components both more vola-
tile than the light component and less volatile than the heavy compo-
nent. In order to make the compositions match more exactly, it is
necessary to introduce the light components into the calculations
below the plate and the heavy components into the calculations above
the feed plate. It is not necessary to repeat the calculations all the
way from the still with all the light components. It is only necessary
to drop back a sufficient number of plates such that the concentration
of the component or components to be added will be small and can be
added without altering the accuracy of the material balance.

Thus, in Table 9-4 the results of Table 9-3 are dropped back to the
seventh plate and small concentrations of C2 and C3 are added; and
then on the eighth plate the Ci is introduced. It is obvious that the
concentration of these light components should be added such that the
vapor from the eighth plate will match the composition of these com-
ponents as determined by the calculations for the top section.

This matching is complicated by the fact that the values of light
components obtained for the top section of the tower in the first calcu-
lations will be reduced somewhat by the introduction of the heavy
components into this section, and it is therefore a matter of successive
approximations to obtain an exact match. The quantity of the
components to be added on a given plate to obtain a desired value
requires trial and error but can be simplified by the fact that for
a light component below the feed plate the operating line is essentially
VnJUm = (WiSm+i, since the value of Wxw is negligible. This can be
combined with the equilibrium constant to give own = Km(V/O)mXm.
Thus the increase in concentration per plate for a light component in
the lower portion of the tower is essentially equal to the equilibrium
constant times the 7/0 ratio. Instead of K the value of a/(2ax) can
be used. These relations make it relatively easy to estimate the
number of plates that should be recalculated to obtain the desired
values. Above the feed plate, a similar relationship can be developed
for the heavy components, and the decrease in concentration of such a

^ ^ „• ____ T

266

FRACTIONAL DISTILLATION

TABLE 9-3
Basis: 1 mol residue; (0/F)n

1.5

<*300

aXw

2yw *» 2<xxw/2<xx

3zi

C.+

0 0025

2.0

0 005

0 020

0.0225

z-C4

0 041

1 18

0 048

0 191

0.232

n-C4

0 211

1 0

0 211

0.840

1.051

0 222

0 58

0.129

0.513

0 735

0.165

0.38

0 0627

0.249

0.414

0 132

0 215

0 0284

0 115

0 247

0 125

0.12

0.0150

0 060

0.185

360°F.

0.102

0.038

0.0039

0.016

0 118

0.503

« 1/0.503 » 1.99

333°F.

3zi

«300

Saxi

2yi

3z2

C8 +

0 0225

2 0

0 045

0 044

0.0465

*-C4

0.232

1.18

0 273

0.269

0 310

w-C4

1 051

1.0

1 051

1.035

1 246

0 735

0.58

0 426

0.420

0 642

0 414

0 38

0 157

0 155

0.320

0.247

0 215

0 053

0 052

0 184

0.185

0 12

0 022

0.147

360°F.

0 118

0.038

0.005

0 005

0 107

2 032

3/2.032 = 1.48

285°F.

3*2

&300

3o#2

22/2

3#3

C8 +

0.0465

2.0

0.093

0 082

0 0845

t-C4

0.310

1 18

0 365

0 323

0 364

n-C4

1.246

1 0

1 246

1 103

1 314

0.642

0 58

0.372

0.329

0 551

0.320

0 38

0.121

0 107

0 272

0.184

0 215

0.040

0.035

0 167

0.147

0 12

0 018

0 016

0.141

360°F.

0.107

0.038

0.004

0.106

2.259

#n.C4 - 3/2.259 » 1.33 T - 273°F.

RECTIFICATION OF MVLTICOMPONENT MIXTURES 267
TABLE 9-3 (Continued)

3#8

a 2 50

3ax3

2t/3

3£4

c,+

0.0845

2.03

0 172

0 147

0 1495

*'-C4

0.364

1 18

0 430

0.367

0.408

n-C4

1 314

1.0

1.314

1.120

1.331

0.551

0 55 ,

0.302

0 258

0.480

cfl

0.272

0.30

0.082

0.070

0.235

0 167

0 172

0 029

0 025

0.157

0.141

0 090

0 013

0 Oil

0 136

360°F.

0.106

0.022

0 002

0 104

2 344

1.28 T - 265°F.

3z4

«260

3a£4

2?/4

3x6

C.+

0 1495

2 03

0 304

0 244

0 2465

i'-C4

0.408

1 18

0 481

0 386

0.427

7^C4

1.331

1.0

1 331

1.070

1.281

0 480

0 55

0 264

0.212

0 434

0 235

0.30

0 071

0 057

0 222

0 157

0 172

0.027

0 022

0 154

0 136

0 090

0 012

0 010

0 135

360°F.

0 104

0.022

0.002

0 002

0.104

2.492

1.2

T - 256°F.

3zB

<*260

3aZ8

2|/6

3z6

C3 +

0.2465

2 03

0 500

0 380

0 382

i-C4

0 427

1.18

0.504

0.383

0 424

n-C4

1.281

1.0

1.281

0.975

1 186

0.434

0.55

0.238

0 181

0 403

0.222

0.30

0 067

0 051

0 216

0.154

0 172

0.027

0 021

0.153

0.135

0.090

0 012

0 009

0 134

360°F.

0 104

0.022

0 002

0.104

2.631

1.14 T - 250°F.

268

FRACTIONAL DISTILLATION
TABLE 9-3 (Continued)

3*«

<*250

3«z6

2ye

3x7

c,+

0.382

2.03

0 776

0.556

0.558

i-d

0 424

1 18

0 501

0.359

0 400

n-Ct

1 186

1 0

1.186

0.850

1 061

0 403

0 55

0 222

0 159

0 381

0.216

0.30

0.065

0.047

0 212

0.153

0.172

0.026

0.019

0.151

0 134

0.090

0 012

0.009

0 134

360°F.

0.104

0 022

0 002

0.001

0 103

2.790

d - 1-075

241 °F.

3x7

«260

3a#7

2?/7

3x8

C3 +

0 558

2 03

1 135

0 762

0 764

f-C4

0.400

1.18

0 472

0 317

0 358

n-C4

1 061

1 0

1 061

0 713

0 924

0 381

0 55

0 210

0 141

0 363

0 212

0 30

0 064

0.043

0 208

0 151

0 172

0 026

0 018

0 150

0.134

0 090

0 012

0 008

0 133

360°F.

0 103

0 022

0.002

0 001

0 103

2 982

t-CU

1.01

232°F.

3x8

«260

3aZ8

2j/8

C3 +

0.764

2 03

1.550

0 970

i-C«

0 358

1 18

0 422

0 264

n-C4

0 924

1 0

0.924

0 578

0.363

0 55

0.200

0 125

0.208

0.30

0 062

0 039

0.150

0 172

0 026

0.016

0 133

0 090

0 012

0 008

360°F.

0.103

0.022

0 002

0 001

3 198

- 0.94 T = 220°F,

RECTIFICATION OF MULTICOMPONENT MIXTURES 269
TABLE 9-4. REMATCHING FEED PLATE FROM BELOW

3z7

«225

3cu7

2t/7

3#8

—

19 3

0 0045

0.036

5.1

0.181

0 100

C3-

0.183

2.37

0.432

0 240

C3 +

0 559

2 07 ,

1 155

0 642

0 644

i-C*

0 400

1 18

0 472

0 262

0 303

n-C4

1 061

1 0

1 061

0.590

0.801

0 381

0 53

0 202

0.112

0 334

0 212

0.27

0.057

0 032

0 197

0 151

0 15

0.023

0 013

0.145

0 134

0 072

0 010

0 006

0 131

360°F.

0 103

0 016

0 002

0 001

0 103

3 595

c4 - O.S33 T = 205°F.

3z8

«200

Sotfs

2/8

0 0045

0 094

0 024

0 100

5.3

0 530

0 134

C3-

0 240

2.4

0.576

0 146

0 644

2.1

1.350

0 341

i-C*

0 303

1.2

0 364

0 092

n-C4

0 801

1.0

0 801

0 203

0.334

0.5

0 167

0 042

0.197

0 24

0 047

0 012

0.145

0 125

0 018

0 004

0 131

0 057

0 007

0 002

360°F.

0.103

0 012

0 001

0 0003

3 955

0.76

188°F.

The rematching calculations for the gasoline-stabilizer problem for
above the feed plate are given in Table 9-5.

By such rematching procedure, it is possible to obtain exact agree-
ment at the feed plate for all the light components and all the heavy
components because they are arbitrarily chosen in one portion of the
tower. However, it may be impossible to obtain an exact match of
the key components because an even number of theoretical plates will
not be consistent with the design chosen. In this case, it is possible
to bracket the required number of plates within a difference of one,
plate. While this rematching operation gives a more consistent set of

270

FRACTIONAL DISTILLATION

compositions, in most cases it does not alter the conclusion as to the
number of theoretical plates required, and the first calculations of the
type illustrated by Tables 9-2 and 9-3 are usually sufficient for deter-
mining the number of theoretical plates.

After such adjustments, it is noted that the vapor ys of Table 9-4
and the vapor i/r-3 of Table 9-5 give a very satisfactory match. The
i-C4 from Table 9-5 is a little higher than in Table 9-4, indicating that
a little less than 20 per cent of the i-d would go overhead, but the dif-
ference is so small that it does not justify readjusting.

These concentrations given in the tables are plotted in Fig. 9-14.

TABLE 9-5. REMATCHING FEED PLATE FROM ABOVE

2x>r-i

3?/r_2

AUO

3?/r_2/a

2xr~2

2/T-3

0 0029

0.066

0.0025

0.003

0 022

0.103

0.419

0 070

0.084

0.133

CB-

0 316

0 506

2.6

0 195

0.233

0 141

C8 +

0.810

1.200

2.3

0 522

0 624

0 338

t-C4

0 277

0 299

1 23

0 243

0 290

0 104

n-C4

0 489

0 508

1 0

0 508

0.606

0.208

0.045

0 045

0 43

0 105

0.125

0 042

0.005

0 005

0.18

0.028

0.033

0.011

—

0 012

0 004

—

0.006

0 002

360°F.

—

0 0009

0 0003

1 674

Kn^< - 0.56 T - 163°F.

A heat balance around the feed plate indicates that the feed should
enter as a liquid at about 130°F. to give the vapor and liquid flows
assumed.

Optimum Feed-plate Location. The feed plate was chosen so that the
ratio of the key components was approximately the same as in the
feed. Equation (9-12) indicates the optimum feed-plate ratio for
Cs + /n-Ct should be 1.14 as compared to 0.83 for the ratio in the feed.
After rematching, the ratio of the key components for plate 8 in Table
9-3 is 0.803, and for plate 9 (plate T-2) in Table 9-4 the ratio is 1.03.
The ratio for these two plates should bracket the value of 1.14. These
values indicate that adding the feed to plate 9 would give more effec-
tive rectification than the plate that was employed.

Theoretical Plates at Total Reflux. The relative volatility of the key
components does not vary too widely from the still to the condenser,

RECTIFICATION OF MULTICOMPONENT MIXTURES 271

272

FRACTIONAL DISTILLATION

and a satisfactory answer for the number of theoretical plates at total
reflux can be obtained by the use of Eq. (7-54). Using an arithmetic
average of the relative volatility at the still and condenser,

2.0 + 2.7 OQP,

OJav = o = 2.35

N + 2 =

0.211

= 8.7

In 2.35
N = 6.7 (or 7) theoretical plates

Minimum Reflux Ratio. The minimum reflux ratio will be calcu-
lated for this separation by Eq. (9-22).
Basis: 100 mols of feed. For actual case,

v' -1-6

Om =d.5Fm = Vm + W
Vm = 136.8

- = 2
Fn = 3D = 94.8

Therefore, assume Vm — Vn = 42 for minimum reflux conditions.
Values of the relative volatility corresponding to the feed plate will
be used.

24 0

0 46

5 6

0 21

C8-

2.5

0.11

C8+

2.2

0.05

*-C4

1.21

360°F.

0.01

n^C4

1 0

In calculating <t> by Eq. (9-12), it is necessary to have values of
and Vm. As a first trial, assume (0/D)miB = 0.9.

Vn = 1.9(31.6) = 60
Vm = 102

On = 28.4
Om = 170.4

RECTIFICATION OF MULTICOMPONENT MIXTURES
in. fQ-10Y

By Eq. (9-10),

273

: 60

2.0

170.4

- 170-4
102(15.8)

_ 28.4
60(15.8)

+ 10

1 170.4

102(3.68)
_ 28.4
60(3.68)

6.0

- 170.4
102(1.66)

9ft /t

__ 28.4
60(1.66)

K values at a temperature of 175°F. were employed.
ByEq. (9-11),

r i - 0-30(60)

I30(15-2> , 0.308(1402)
170.4

= 0.148

I7O4

. 0-14(60)

OO A

* U.JL^^WJ

•*• 00 A x

+ 0-14(11.3) — 0-^ + 0.07(9.0)-
170.4 l

_ 0.07(60)
28.4

J- I

1 0-03(60)

- °-°3(8-5) , 0.023(102l
170.4

T?^. fC\ 10\

170.4
0.007(60)
28.4

a».4
0.007(102)

17O A

170.4

By Eq. (9-12)3

160(0.852)'^ I

A = 1102(0.976) 1J _ ^ j g

iKn^r60(0-8S2)2 .T...

15'0 + [ 102(0.976) -T4-4

Checking distribution of t-C4 for assumed reflux ratio.
By Eq. (9-13),

1 O^*
. _ L°28l

= 0.024

1.2
By Eq. (9-15),

OJ04Dxi>

2.2 - 1.21 + 1,

o2 ^IF

2-2170

91 / 68.4 \ /O.OQ25\

274 FRACTIONAL DISTILLATION

Equating these two values of art.c4 and using Wxw + DxD = 3.5,

DxD = 0.39
Wxw « 3.11

About 11 per cent of the isobutane would go overhead. The calcu-
lations of D and W were based on the original assumption of 20 per
cent i-C4 overhead, but the difference is not large and no correction
will be made. The values just calculated will be used in the minimum
reflux ratio calculations.

By Eq. (9-22)

- 0.019 j[l +2.2(1.8)]
+ 1

/0.39V

L21 V3L6;

24.0(0.063) 5.6(0.316) 2.5(0.19)

24.0 - 1 "^ 5.6-1 "*" 2.5 - 1 "*" 1.21 - 1

60(68.4) [0.46(0.222) 0.21(0.165) 0.11(0.132)

102(31.6) [2.2 - 0.46 "*" 2.2 - 0.21 + 2.2 - 0.11

, 0.05(0.124) , 0.01(0.102)

1 2.2 - 0.05 ' 2.2 - 0.01
~) = 0.75 for Case I

= 0.75 - 0.11 = 0.64 for Case II

The assumed value for 0/D of 0.9 is close to the calculated value for
Case I. A recalculation for Case II, assuming 0/D = 0.7, gave a
calculated value of 0.63.

Equation (9-22) was employed because the summation term for the
light components was greater than for heavy components. The term
for the isobutane was included with the light components, but it could
equally well have been included with the heavy components, and the
total would have been essentially the same.

It is interesting to note that a binary mixture of propane and butane
of the same ratio as in the feed would have required only one-half as
much vapor per mol of Cs + C* separated as is necessary for this
multicomponent mixture.

For a comparison, the minimum reflux ratio will also be calculated
by Eq. (947).

By Eq. (9-17), assume

RECTIFICATION OF MULTICOMPONENT MIXTURES 275

By Eq. (9-15),

2.2(14.4/170.4) A1KK

By Eq. (9-13),

Xlk = - 1.0(12.33/28.4) _ _

"-"+*&) Css)

For Eq. (9-17),

C = ~ (2'2 ~ 1'° + °-30) = 3'84

E = gj (2.2 - 1.0) = 5.68
O = M = 0.387

= 0-26

°'165

/ = + 0.387 + 2.2 +

68.4 ' v-""§ ' ' [2.2 - 0.46 ' 2.2 - 0.21

0.132 0.124 0.102 1

"*" 2.2 - 0.11 + 2.2 - 0.05 "*" 2.2 - 0.01 J

31.6/170.4\r 0.063

^ 68.4 V 28.4 / [24 - 1 + 0.12(24)
, 0.316 0.19

I K. a i i f\ -i ci t e /2\ i

5.6 - 1 + 0.12(5.6) ' 2.5 - 1 + 0.12(2.5)

+ 1

0.39/31.6 1

.21 - 1 +0.12(1.21)J

= -2.08 + 0.387 4- 0.825 + 0.617 = -0.251

J = 0.26 + 1^1 (^^ ) (0.825) + 0.205 = 0.762

If = -0.868
J' - 0.465

Case I:

W 68 4

IL / + j = 5^Z (-0.251) + 0.762

« 0.218

0.218 + ^/(0.218)2 - ^fP [(~0.251)(0.762) - (0.387) (0.26)1
= 0.91

276 FRACTIONAL DISTILLATION

Case II:

-1.415

1 4.1 K _L_ //'_

1 J.1 KN2 _

4(68.4)

r o sfiR/'n ifi^

/"n ^ft7^^^ 9 AM

1.4J.O -f 'V/V

31.6 '

[ U.oOOViU.4:OO^

(\}.oot ) \\},Zi\y)\

0.55

These values were not recalculated for a new assumed value of 0/D
since both are reasonably close to the first value assumed.

Constancy of Molal Flow Rates. The design calculations pre-
sented have been based on the use of constant molal overflow and
vapor rates. Enthalpy balances similar to Eqs. (7-13) and (7-27) can
be written for each of the components of a multicomponent mixture
and, if the data are available, they can be applied plate by plate in the
Stepwise calculations. This procedure usually requires trial and error
because, in calculating up or down the tower, the temperature on the
next plate is needed to complete the enthalpy balance, and it must be
assumed and checked in a later calculation. The most serious diffi-
culty is the lack of the necessary enthalpy data, but in most cases
these can be approximated by the methods on pages 139 to 142.

The same general considerations relative to the constancy of flow
rates apply to multicomponent and binary mixtures. Thus the latent
heat of vaporization at various positions in the column should serve
as the principal criterion, although in multicomponent systems there is
more possibility of large sensible heat effects changing the overflow and
vapor rates.

The modified latent heat method (M.L.H.V.) given on page 158 is
applicable to multicomponent mixtures, and it probably is the most
satisfactory procedure for handling such calculations. It should give
good results for the examples considered in this chapter, because the
heats of mixing for the mixtures involved would be small, even though
the latent heats of vaporization of individual components for the
gasoline stabilization problem differ several fold. In this latter exam-
ple, there is undoubtedly considerable variation in the molal flow
rates, and the M.L.H.V. method will give more satisfactory results
than the method employed. The calculations were repeated by the
M,L,H.V, method using arithmetic average values of latent heats

RECTIFICATION OF MULTICOMPONENT MIXTURES 277

of vaporization at the still and the condenser to calculate the val-
ues of ft for each component. Table 9-6 gives the latent heat values
and the ft terms based on the latent heat of n-butane. The values
of ft are used to calculate ftzF, ftyo.n., ftxw, and z'F = ftzF/2(ftzF),
2/o.H. ** 02/o.H./2(/ty0.H.), and x'w = ftxw/2(ftxw). The values of the
terminal flow quantities are given below:

W' = 68.4(1.467) = 100
D' = 31.6(0.650) = 20.5
F' = 100(1.205) = 120.5

Table 9-7 presents the plate-to-plate calculations; the values of
2/o H are taken from Table 9-6, and the values of t/o.n and 2xR are taken
from Table 9-2. The calculations will be made for 0/D = 2.0 and
for Vm — Vn = 42 which are the same as the values used in Tables
9-2 and 9-3. For F = 100, D = 31.6, 0K = 63.2, and the fourth col-
umn of Table 9-7 gives ftxR = 0.751 making 0'R = 0.751(63.2) = 47.5,
and O'JD' = 47.5/20.5 = 2.32. The remaining columns of Table
9-7 are based on D' » 1.0, 0' = 2.32, V = 3.32. The values of the

fifth column are 2.82^ = 2.32 ,*R • the values of 332y'T are equal

•^ (pxx)

to 2.32x'R + 2/o.H.- In order to go to yT, the values of 3.32y'T are
divided by the corresponding ft terms and yT = (y'T/ft)/2(y'T/ft). The
yT values are used to calculate XT, using the relative volatilities of
Table 9-2, and these are converted to x'T.

The general flow quantities are shown in Fig. 9-15 which gives both
the normal and modified values. For F = 100, the value of V'n is 68
throughout the top section, but the value of Vn will vary from plate to
3late. Thus VT = 94.8, but 7r-i = V^^yr-i/ft) = 91, and this
/alue decreases further for the next plate. This decrease in vapor
'ate is due to the fact that the average latent heat of vaporization is
ncreasing from plate to plate down the column, resulting in lower
fapor and liquid rates and making the separation more difficult.

The calculations are carried in the same manner down to yT~x- The
calculations were given in detail to show the values of y and x as
veil as those for y* and re', but this is not necessary because it is possi-
)le to go from y' to x' directly by xf » (yf / a) / I,(y' / a) in which the
lormal a values are employed.

The calculations for the lower section of the tower are given in
Table 9-8 on the basis W « 1.0, O'm = 2.186, and V'm = 1.186. The
'alue of Vw for F - 100 is calculated from VjV(fow) - 101. This

278

FRACTIONAL DISTILLATION

represents a large decrease in vapor from the top of the tower where
VT •» 94.8. With Vm - Vn « 42, the vapor from the still would have
been 136.8 on the usual basis. The decrease of 136.8 - 101 = 35.8 is
due to variation in the latent heats of vaporization. The calculations
up to x( are given in Table 9-8. The calculations up to x* were made
in detail, but from x% to x'6 the values of l.lSQy' were calculated from

f'100

(- /
Om

W'68.4

W'*tOO
FIG. 9-15. Comparison of flow quantities for gasoline-stabilization example.

1.186(2.186az')/:S(2.186az'). This method requires only three col-
umns per plate, and the calculations are as simple as the normal type
of calculations given in Table 9-3. The calculations were continued,
and yj gave a good match with y'T_$ of Table 9-7, indicating that 12
theoretical plates would be required as compared to 11 plates for con-
stant flow rate conditions. The difference would be greater for reflux
ratios nearer the minimum.

The minimum reflux ratio can be calculated on the modified latent
heat basis using any of the equations derived for constant molal flow
rates with the x* and y' values.

The calculations for (OfR/Df)min by Eq. 9-22 were made in an analo-
gous manner to calculations on page 274. The values are summarized

RECTIFICATION OF MULT1COMPONENT MIXTURES 279

below:

A{ = 0.069; Ai = 0.026; <#>' = 1.055
For isobutane,

D'x'D = 0.3
W'x'w = 2.96
By Eq. (9-22),

= 1.0, Case I
= 0.93, Case II

TABLE 9-6

Latent heat of vaporization,

B.t.u./lb. mol

ftZF

98°F.

333°F.

A.-V

1,100

200

650

0.113

0.02

0.002

0.0016

3,500

2,200

2,850

0.496

0.10

0.0494

0 0413

Ca —

4,700

3,500

4,100

0 713

0.06

0 0425

0 0354

Cs-f-

5,200

4,000

4,600

0 80

0.125

0.10

0 0826

i-C4

5,800

4,900

5,350

0.93

0 035

0.0321

0 027

n-C4

6,200

5,300

5,750

1.0

0.150

0 124

7,300

6,600

6,950

1.21

0.152

0.182

0 152

8,500

7,700

8,100

1.41

0.113

0.159

0.131

10,000

9,100

9,550

1.66

0.090

0.150

0.124

11,900

10,700

11,300

1.97

0 085

0.166

0.139

360°F.

14,900

13,500

14,200

2.47

0.070

0 172

0 143

1.205

fix*

* x*

fixw

0.0633

0.007

0.011

_*_

—

0 3160

0.157

0.242

—

c,-

0.1900

0.135

0 208

—

v/3 ~|

0.3900

0.312

0.480

0.0025

0.002

0 0014

$~O4

0 022

0.020

0.031

0.0410

0.038

0.026

n-C*

0.019

0,029

0.2110

0.211

0 144

—

0.2220

0.268

0.183

—

0.1650

0.233

0.152

—

0.1320

0.219

0 149

—

0.1240

0.244

0 166

360°F.

—

0.1020

0.252

0.172

0.650

1.467

280

FRACTIONAL DISTILLATION
TABLE 9-7

2/O.H.

J/O.H.

2xR

20afc

2.324

3.322/y

3.32^

2/r

0 Oil

0 0633

0 012

0.0014

0.0022

0 0132

0 117

0 025

0.242

0.316

0.296

0 147

0.227

0.469

0.945

0 204

C3-

0.208

0.190

0.440

0.314

0 484

0.692

0 970

0.210

tJ,+

0.480

0 390

1.0

0 80

1.232

1.712

2 14

0 463

i-C4

0.031

0 022

0.117

0.109

0.168

0.199

0 214

'0 046

n-C4

0.029

0.019

0.132

0 132

0 2035

0 2325

0 050

1 503

4.618

VT - 94.8

ftXT

2.324

3.322/;_1

3.32^

2/r-i

OflQQ

«150

0.00068

0 00191

0 00022

0 0006

0 0116

0 102

0 023

0 0008S

0.0276

0 0777

0 0385

0 112

0 354

0.714

0 161

0 027

C8-

0.070

0 197

0.141

0 409

0 617

0.866

0 195

0 075

C3 +

0 172

0 484

0 387

1 12

1.60

2 0

0 45

0 196

i-C4

0 035

0.0985

0.0915

0 266

0.297

0.319

0 072

0 059

n-C4

0 050

0.141

0 141

0 409

0 438

0.438

0.099

0 099

0 3553

0 799

4 439

0 4569

VT-I - 91

XT-I

ftXT-l

2.324-!

3.32^_2

J/r-a

2/r-2

«160

0 00195

0 00022

0 00062

0 0116

0 102

• 0 023

0 00088

0 059

0.0293

0.082

0.324

0 653

0 150

0 025

C3-

0 164

0.117

0 328

0 536

0 753

0 173

0 0665

C3 +

0 430

0.344

0 965

1 445

1 81

0 416

0 181

*-C4

0 129

0 120

0.336

0 367

0 395

0 091

0 074

w-C4

0.217

0 217

0 608

0 637

0 147

0 8275

4.350

0 . 4944

XT-Z

PXT~2

2.324-a

3.32^-8

0 0018

0.0002

0 00054

0 0115

0 051

0.025

0.068

0.31

C3-

0.135

0.096

0 260

0.468

C8 +

0 366

0.296

0 803

1.283

i~C4

0 150

0 140

0 380

0 411

w-C4

0 298

0 808

0 837

0 855

RECTIFICATION OF MULTICOMPONENT MIXTURES 281

TABLE 9-8

1.1862/17

l.lMpyw

1.186^

2.186^

C3-h

0.0014

0.0025

0 0118

0 0095

0 0068

0 0081

0.0095

;-c4

0.026

0 041

0.113

0 105

0.0755

0.090

0.116

rc-C4

0.144

0.211

0.497

0.357

0.424

0.568

0.183

0.222

0.304^

0 368

0.265

0.315

0 498

0 152

0 165

0 148

0 209

0 150

0.178

0 330

0.149

0.132

0 067

0.111

0 080

0 095

0.244

0 166

0.125

0.035

0 069

0 050

0.059

0 225

360°F.

0 172

0.102

0 009

0 022

0.016

0 019

0.191

1.390

Vw = 101

2.186*;

«30C#1

2/i

ftyi

1.186^

2.186a4

Ca-h

0.0119

0 0071

0 0142

0.0215

0 0172

0 0186

0.02

<-C4

0 1245

0.074

0.087

0.132

0 123

0 133

0 159

^-C4

0.568

0.338

0 338

0.512

0 512

0 552

0 696

0 411

0 244

0 141

0.214

0 259

0.280

0 463

n
^6

0 234

0 139

0 053

0.080

0 113

0 122

0 274

'"i

0.141

0.084

0 018

0 027

0.045

0.049

0 198

^
^8

0 114

0.068

0.008

0 012

0 024

0.026

0 192

560°F.

0 077

0.046

0 002

0.003

0.007

0.179

1.681

0.661

1.101

108

2.186^

<*300#2

2/2

fty*

1.186^;

2,186/3

?•+

0 025

0 014

0.028

0 0384

0.0307

0 034

0.035

-C4

0.171

0.097

0.114

0.156

0.145

0.161

0 187

^C4

0.696

0.396

0.543

0.602

0 746

-1
-"6

0.383

0.218

0.126

0.173

0,210

0 232

0 415

"1
^6

0.194

0.110

0.042

0 058

0.082

0 091

0 243

"i
•^7

0.119

0.068

0.015

0 021

0.035

0 039

0 188

"^
-'S

0 097

0.055

0.007

0.010

0 019

0 021

0 187

*60°F.

0.072

0.041

0.002

0 005

0.006

0 178

1.757

0.730

1.070

v - in

282

FRACTIONAL DISTILLATION
TABLE 9-8 (Continued)

2.186a80ozJ

1.186t/a

2.186x4

2.186^260^4

1.186/4

2.186^

c,+

0.070

0.058

0.059

0.120

0.098

0.099

i-C4

0.221

0.182

0.208

0.246

0.200

0.226

n-C*

0 746

0.616

0 760

0.760

0.619

0.763

0.241

0.199

0.381

0 210

0.171

0.354

0.092

0.076

0.228

0 068

0.056

0.208

0 040

0 033

0.182

0 031

0.025

0.174

0.022

0.018

0.084

0.017

0.014

0.180

360°F.

0.007

0.006

0.178

0.004

0.003

0.175

1.439

1.456

In calculating these values, the same relations used on pages 277 to
278 were employed with the same relative volatilities. Converting the
values of (OfK/Df}^ to (0«/Z>)»in by the factors given in Table 9-7 gives

( *?£ ) = 0.86, case I

\ ///rain

= 0.8, case II

These compare with 0.75 and 0.64, respectively, obtained on page
274, indicating that the variation of the flow rates results in a value of
(0«/Z>)min about 20 per cent greater than would be estimated on the
basis of constant molal flow rates. However, the mols of vapor that
would need to be generated in the still are approximately the same by
both methods.

In this example, the use of the modified values did not make a large
difference in the results, but if the design conditions had been nearer
the optimum, e.g., (0R/D) = 1.26(0^0)^ = 0.94, then calculations
based on the constant molal rates would be seriously in error. When
the average latent heats of vaporization at the bottom and top of the
column differ appreciably, calculations based on the modified basis are
to be preferred.

Another method of handling unequal molal overflow rates for multi-
component mixtures is to make the calculations on the constant molal
flow rates and then to calculate the heat load required to give these
rates for a few positions in the column. If the heat input to the still
is made equal to the largest requirements, and the condenser and
tower cross section correspondingly corrected, a conservative design
will be obtained. The check points most commonly employed are (1)
around the still, (2) the entire section below the feed plate, (3) the feed

RECTIFICATION OF MULTICOMPONENT MIXTURES 283

plate and section below, and (4) around the entire column. In general,
the modified latent heat of vaporization method is a more satisfactory
procedure.

Nomenclature

C « number of components
C* ** factor in Eq. (9-17)

D — distillate rate, mols per unit time

E ** factor in Eq. (9-17)

F « feed rate, mols per unit time

G - factor in Eq. (9-17)

H = factor in Eq. (9-17)
HL « molal enthalpy of liquid
Hv ** molal enthalpy of vapor
/ = factor in Eq. (9-17)

/' - factor in Eq. (9-17)

/ *• factor in Eq. (9-17)

/' « factor in Eq. (9-17)

K » equilibrium constant « y/x

m » number of plates below feed plate (including feed plate)

N « number of plates in column

n = number of plates above feed plate

0 — overflow rate, mols per unit time
P « vapor pressure

p - (0/44 - 0/)/F
Q « heat input
T = temperature

V « vapor rate, mols per unit time
W = bottom product rate, mols per unit time
x •» mol fraction in liquid
y «= mol fraction in vapor
z « mol fraction
a ** relative volatility

<t> « optimum ratio of key components on feed plates
TT « total pressure

Subscripts:

B refers to benzene
D refers to distillate

F refers to feed

/ refers to feed plate

h refers to heavy component
hk refers to heavy key component

1 refers to light component

Ik refers to light key component
m refers to conditions below feed plate
n refers to conditions above feed plate
R refers to reflux

284 FRACTIONAL DISTILLATION

T refers to toluene
W refers to bottom product
X refers to xylene

References

1. GILLILAND, Ind. Eng. Chem., 32, 918 (1940).

2. GILLILAND, Ind. Eng. Chem., 32, 1101 (1940).

3. GILLILAND and REED, Ind. Eng. Chem., 34, 551 (1942),

4. LEWIS and COPE, Ind. Eng. Chem., 24, 498 (1932).

5. LEWIS and MATHESON, Ind. Eng. Chem., 24, 494 (1932).

6 LEWIS et al, Ind. Eng. Chem., 26, 725 (1933); Oil Gas J., 32, No. 45, pp. 40,
114 (1934).

7. RHODES, WELLS, and MURRAY, Ind. Eng. Chem., 17, 1200 (1925).

8. ROBINSON and GILLILAND, "Elements of Fractional Distillation," 3d ed.,
McGraw-Hill Book Company, Inc., New York, 1939.

CHAPTER 10
EXTRACTIVE AND AZEOTROPIC DISTILLATION

The design engineer frequently must separate mixtures for which
normal distillation methods are not practical due either to the forma-
tion of azeotropes or to a very low relative volatility over a wide con-
centration region. In the first caserne separation is impossible unless
some special method of by-passing the azeotrope is employed (page
196), an3 in the second case excessive heat consumption and equipment
size are involved. For a large number of such mixtures, it has been
found possible to modify the relative volatility of the original com-
ponents by the addition of another component (or components). This
technique has been classified into two categories: extractive distillation
and azeotropic distillation.

In Essentially all separations carried out to date by these techniques
the effect of the added component is in the liquid phase, although it is
possible to modify the vapor-phase properties for systems operating at
high pressure.

The added component by being present in the liquid^ghase can alter
the activity coefficient of the various components, and unless the com-
ponents already present are identical in physical and chemical proper-
ties, {Kep^entsige^^gem^e activity coefficients will be different
for each' component, thereby altering their relative volatility. This
technique is effective only when the components in the original mix-
ture do not obey Raoult's law, and in general the greater the deviation
from Raoult's law, the easier it becomes to alter the relative volatility
significantly by the addition of another component. Thus a considera-
tion of the deviations from Raoult's law is essential for an understand-
ing^oTextractive and azeotropic distillation. While no exact relation-
ship has been developed for the predictions of such deviations, the
Van Laar equation does give a qualitative picture that is useful.

As the simplest case, consider a binary mixture of components 1 and
2 which is to be modified by the addition of component 3.

The relative volatility of component 1 to 2 is

(X = - — —

xi y*

285

286 FRACTIONAL DISTILLATION

By Eq. (3-58), page 73,

r £3.4.32'

Tin 72

+ X* +

Over moderate temperature ranges the ratio of the vapor pressures
of the pure components does not change appreciably. The main
possibility of modifying the relative volatility lies in altering the ratio
of the aci^itj_QQefl5cient. By subtracting the two activity coefficient
equations, it is possible to obtain the ratio,

Tin 2

Consider the case in which compounds 1 and 2 are similar, i.e.,
almost obey Raoult's law, e.g., ethanol and isopropanol. For this case,
J5i2 will be small and An will be approximately equal to unity. Equa-
tion (10-1) can be simplified for these conditions to the approximate
relationship,

(10-3)

where V is the effective volume fraction of the component in the
mixture.

The significance of this equation can be better shown by comparing
it with the activity coefficient ratio, (71/72)0, for the binary mixture
without the added agent. By Eqs. (3-44) and (3-45)

Tlnf—

where Fio, Fao are the volume fractions of components 1 and 2, respec-
tively, without agent present, and

T In T^V - Bit [2 ^|jj (F8) - Flo ~ ^uHo] (10-4)

EXTRACTIVE AND AZEOTROPIG DISTILLATION 287

For any effective addition agent the first term of the right-hand
side of the equation will be large in comparison to the last term and,
approximately,

T In T^V = 2 VB^ VW* V, (10-5)

(71/72)0

For a given binary mixture, the^ffectiveness of the added component
is indicated by how much the activity coefficient group of Eq. (10-5)
can be made to differ from unity, and the effectiveness is increased by
large absolute values of F3, and \/#i3. The derivation of Eq. (10-5)
was for A 12 approximately equal to unity and for small values of #12.
A more detailed analysis of Eq. (10-1) for the general case leads to
essentially the same conclusions; i.e., the ratio of the activity coeffi-
cients is changed the most when (1) \^B\2 is large, (2) \/#i3 is large,
and (3) Fs is large. However, in this case it will make a greater dif-
ference whether \/5i3 and \/5i2 are of the same or opposite sign.

A high value of Fs is obtained by using a large quantity of the added
agent, but it is usually not economically feasible to use a value greater
than 0.8 to 0.9. To obtain these values for F3 requires a 4 to 9 volume
of the addition agent per volume of the components to be separated,
and to obtain a value of Fs equal to 0.95 would require a ratio of 19.
For most cases, the small increase in the relative volatility obtained by
increasing Fs from 0.9 to 0.95 does not justify utilizing twice as much
of the added agent. In some cases, values of Fs as low as 0.4 to 0.5
may be sufficient, but higher values are usually required.

It should be noted that, for low values of \/Bi2, the effective change
in the ratio of the activity coefficient will be small. Thus this tech-
nique would not be effective for ideal solutions (B^ = 0, Ai2 = 1),
and the larger the value of \/rBn, the easier it would be to obtain large
effects.

The value of the \/2?i3 is probably the most important variable
under the control of the designer. To obtain a large modification of
the activity coefficient ratio, it is desirable that the absolute value of
the term be as large as possible. However, there is an upper limit
because a value of B from 1,200 to 1,800 corresponds to immiscibility.
While it is necessary for the agent to dissolve only 10 to 20 volume per
cent of the mixture, it is doubtful whether the value of BU can exceed
2,000 to 3,000. Qualitatively, the value of B is a function of the dif-
ference in polarity of the two compounds under consideration. Thus
a large value of B is associated with a large difference in polarity, and
for the case in question it would be desirable to add an agent which

288 FRACTIONAL DISTILLATION

differed as much as possible in polarity from component 1. This dif-
ference could be either a compound with greater polarity or one with a
lower polarity. In the first case, the A/Bis would be negative and in
the second case positive. Assuming that the \/rBn is positive, i.e.,
compound 1 more polar than 2, a positive value of V^is would increase
the relative volatility while a negative value would decrease it. It is
therefore possible either to decrease or to increase the relative vola-
tility by adding another component. The general rule is (1) if the
added material is more polar than the components of the original mix-
ture, it will increase the relative volatility of the less polar component
relative to the more polar, and (2) if the added material is less polar, the
reverse will be true. For example, consider a mixture of acetone and
methanol, which forms an azeotrope at atmospheric pressure. In this
mixture methanol is more polar than acetone, and by adding a more
polar component such as water, it is possible to increase the relative
volatility of acetone to methanol to such a degree that an azeotrope no
longer exists. However, if a material of low polarity, such as a hydro-
carbon, is added to the mixture, the volatility of methanol will be
increased relative to that of acetone. Water would probably be pre-
ferred to a hydrocarbon, not only because of cheapness and ease of
separation from the components, but also because the vapor pressure
of acetone at a given temperature is greater than that of methanol.
Thus water will act to aid the natural difference in vapor pressure while
a hydrocarbon will work against it.

For large values of Fs, components 1 and 2 behave as if they were
each in a binary mixture with component 3, and there is essentially no
interaction between 1 and 2. For these two "binary" systems, Eq.
(3-44) can be applied; for Fa approximately equal to unity, they reduce
to

T In ^ = J513 - B23 (10-6)

This equation can be modified to give

T In 1 - VBT* (VET* + V5^) (10-6a)

Equation (10-6) is equivalent to Eq. (10-5) for small values of Bu
and A 12 approaching unity, but it is not limited to these conditions.
The use of the binary principle for each component with the added
agent is a useful guide, both for selecting effective agents and for esti-
mating the quantitative vapor-liquid relationships. For example,

EXTRACTIVE AND AZEOTROPIC DISTILLATION 289

data are available on the vapor-liquid equilibria for the systems ace-
tone-water and methanol-water. The activity coefficients for acetone
and methanol can be calculated for a low concentration, and these
values will be close to the values for a dilute mixture of acetone and
methanol in water, thus allowing their relative volatility to be calcu-
lated for these conditions. Frequently, it is easier to determine the
vapor-liquid equilibria for the tw.o binaries with the added agent than
it is to investigate the three-component system.

While theoretically it is possible to add a material of either higher
or lower polarity, both cases are not usually equally attractive or
practical. For example, in separating a binary mixture of a paraffin
and an olefin of the same number of carbon atoms, it is difficult to find
practical agents of lower polarity, and the only real possibility is to use
a material of higher polarity. Such an agent will increase the vola-
tility of the paraffin relative to that of the olefin, and it is unfortunate
that the natural volatility of the olefin is often greater than that of the
paraffin. Thus, adding small quantities of the agent actually makes
the separation more difficult, and large enough quantities must be used
to reverse the normal volatility completely. Obviously, the reversal
must be sufficient to make the separation by distillation appreciably
easier than without the agent. For example, aqueous acetone has
been used for the separation of butadiene 1,3 from butylenes. In this
case, the solvent added is of high polarity and increases the volatility
of the olefines relative to the diolefine. Normally the relative vola-
tility of cis-butene-2 (one of the constituents of the mixture) to
butadiene is 0.78, and the value increases to 1.3 with 80 volume per
cent of aqueous acetone. A relative volatility of 1.3 gives an ease of
separation about equal to a value of 0.78, and the use of less than 80
volume per cent of aqueous acetone would make the separation of
these components more difficult than for the hydrocarbons alone.
/ Extractive and azeotropic distillations both employ this technique
of adding a component to modify the volatilities. They differ chiefly
in the fact that the agent added in the case of extractiy^distillatipn is
relatively nonvolatile as compared to the other components in the mix-
ture, ~lvTmejii^^ distillation the volatility is essentially the
same asjbhato^ therefore, forms an azeotrope
with one or moze^ifjbhem due to the differences in polarity. In extrac-
tive distillation the agent is usua3ly'tfcdded:iiesir^eTop of the column,
and most of it is removed with the liquid at the bottom. In azeo-
tropic distillation the agent is also added near the top of the column,

290 FRACTIONAL DISTILLATION

but in this case most of it is removed with the overhead vapor. The
distinction is not sharp. In the separation of pentane and amylenes
using acetone as the added agent, most of the acetone is removed from
the still, but it forms an azeotrope with the pentane and a small portion
is carried overhead. For the purposes of this text, the term extractive
distillation will be applied to those cases in which the agent is appre-
ciably less volatile than the components to be separated and in which
the concentration of the agent is relatively constant from plate to plate
except as affected by additions or withdrawals from the column.
Azeotropic distillation will be used to define those operations in which
the added material has a high concentration in the upper portion of the
column and then decreases to a relatively low value in the lower portion
of the unit.

In any given case there are usually a number of compounds that are
effective as azeotropic- or extractive-distillation agents, and the choice
depends on a number of factors:

L Effectiveness for modifying normal volatility.

2. Solubility relationships with system in question.

3. Cost of agent.

4. Stability of agent.

5. Volatility of agent.

6. Corrosiveness of agent.

7. Ease of separating agent from original components.
Extractive distillation will be considered first because the mathe-
matical analysis for it is simpler than for azeotropic distillation.

EXTRACTIVE DISTILLATION

This type of operation has been used for several important separa-
tions, and it is one of the most valuable techniques in fractional dis-
tillation. Its earliest use probably was in the distillation of nitric and
hydrochloric acids using sulfuric acid as the added agent to aid in the
separation. It is extensively used, and some of the separations that
have been carried out commercially are given in Table 10-1.

The general arrangement normally used is shown in Fig. 10-1. The
feed is introduced into the main extractive distillation tower, and the
extractive agent is introduced a few plates below the top of the column.
These top plates serve to remove the agent from the overhead product.
In certain cases, such as the isoprene-amylene separation with acetone,
the agent cannot be completely eliminated from the overhead product
by this method owing to azeotrope formation. In such cases, other

EXTRACTIVE AND AZEOTROPIC DISTILLATION

291

TABLE 10-1. COMMERCIAL EXTRACTIVE DISTILLATION OPERATIONS
System Extractive Agent

HC1~H20 .... H2S04

HN03-H2O H2S04

Ethanol-HaO , . . Glycerine

Butene-butane . . Acetone, furfural

Butadiene-butene Acetone, furfural

Isoprene-pentene , Acetone

Toltiene-paraffinic hydrocarbons Phenol

Aeetone-methanol Water

means, such as extraction, must be employed to separate the agent.
The bottoms from the tower are treated to separate the agent and the
bottoms product. Occasionally, a portion of the agent is added with
the feed in order to maintain its concentration essentially the same
above and below the feed.

Feed

Top product
Extractive agent

Bottom
product

FIG. 10-1. Schematic diagram of extractive distillation system.

The design calculations for such systems are straightforward, assum-
ing that the physical-chemical data on the systems are available. In
the limiting case of an essentially nonvolatile extractive agent, the
problem reduces to a standard binary or multicomponent problem
depending on the feed to the unit, except for the modification of the

292

FRACTIONAL DISTILLATION

volatilities. In case the agent is volatile, the problem is more com-
plicated, but it can be handled by methods analogous to those used fox
regular distillations.

Total Reflux. This limiting condition is not equivalent to that for
regular rectification because to maintain a given concentration of the
extractive solvent in the liquid phase would require the addition of an
infinite amount per unit of feed. Thus the concentration of the com-
ponents being separated would approach zero in the bottoms from the

Feed

•* — EJ *~ Top product

Bottom
product

Fio. 10-2. Extractive distillation diagram for total reflux.

extraction tower. It would require an infinite number of plates to
reduce the concentrations from the finite values in the upper part of
the column to zero at the bottom, except for the case where the solvent
was nonvolatile. This limit of an infinite number of plates and infinite
heat consumption does not aid in orienting the design calculations.
A useful limit for orientation purposes can be based on the desired
separation of the key components from the overhead to bottoms.
Thus the calculations should be carried down the column until the
desired bottoms ratio of the key components is obtained. Actually
the system can be considered as shown in Fig. 10-2. Column A is the
usual extractive distillation unit, which obtains the desired degree of
separation of the key components, and produces a bottoms containing

EXTRACTIVE AND AZEOTROPIC DISTILLATION

293

a finite concentration of the key components in the desired ratio. B

is some type of unit that produces the vapor for column A and reduces

the concentration of the key components without changing their ratio.

Column C is the tower that separates the solvent from the bottom

product. In case the solvent is nonvolatile, unit B is a still. The

number of theoretical plates for tower A is the desired answer. The

minimum number of plates at total reflux

will therefore be calculated on the basis of

the desired ratio of the key components in

the overhead and bottoms. If the relative

volatility is reasonably constant over the

concentration range involved, Eq. (7-52) f6ej

should be satisfactory. In any case,

the plate-to-plate calculation methods of

Chap. 9 can be applied.

Minimum Reflux Ratio. The calcula-
tion for this case is similar to that for the

T W

usual multicomponent mixture, but with I 1^.

the extractive agent included at both FIG. 10-3. Extractive distiila-
the pinched-in regions. The asymptotic lon iagram-
value is calculated in the same manner as Eqs. (9-13) and (9-15).
Thus for Fig. 10-3, the asymptotic concentration above the feed is

Oihk

XSn =

Oihk — OiS — OiS

and, below the feed,

XSm =

as + oiS(Wxwik/OmXik)

(10-7)

(10-8)

where x8n, xsm = asymptotic values of solvent above and below the

feed plate respectively

Sxs = solvent added to system at top of column
In many cases, the mixture to be separated is a binary, since it is
usually desirable to separate all but two components by regular dis-
tillation and then subject them to the extractive operation. Generally,
the amount of solvent added at the top of the tower is varied with the
reflux ratio in order to maintain a constant mol fraction of solvent in
the total liquid returned to the top region of the column. An alterna-
tive method of operation is to employ a given solvent rate independent

294 FRACTIONAL DISTILLATION

of the reflux ratio, which would give a solvent concentration in the
tower that varied widely with the reflux ratio. The first method of
operation appears to be more desirable for most cases, in order to
obtain a relative constant concentration of solvent in the tower that
approximates the desired value. The equations for the minimum
reflux ratio have therefore been derived to be most convenient for the
first method of operation. For the general case in which the feed con-
tains light and heavy components in addition to the key components,
it is recommended that the minimum reflux ratio be calculated by

0« =

where A =
H =

E =
G <=

JL J^DS \^K8 ^DSJ

Oihk — oiS

1 — XDS — ~

1 ~ XRS
aik (x

XDS)

Fs- D

XDS~\^ ,

I _J— Tit

\XRS

f I WXwM

1 XRS

, ^(Wx*

a|*Lai* ~" <*>

hk Lj \a'ik —

Of-ik '

*'* J + P

T —

( DXIM

• DXDI

DS \

Oihk I
\<Xlk ~ 0

•1"

ahk -

- as/

OR = reflux from condenser before solvent is added at top
XDS, XRS = mol fraction of solvent in distillate and total liquid added

to the top of tower, including reflux and solvent
Fs = solvent added with feed
a, af = relative volatilities for solvent concentrations above and

below feed, respectively

If the feed is binary mixture, the terms involving XDI and XWH
are dropped. For certain special cases, simpler equations can be
employed. For example, if solvent is added with the feed such that
its' concentration is the same above and below the feed plate, the mini-
mum vapor required can be calculated on a solvent-free basis by the
regular minimum reflux equations, and then this vapor requirement is

EXTRACTIVE AND AZEOTROPIC DISTILLATION 295

increased to allow for the solvent in the vapor. In case the solvent
is so high boiling that its concentration in the vapor is negligible, no
correction to the solvent-free calculation is necessary. In making the
calculation for the solvent-free conditions, the relative volatilities
employed are those for the pinched-in region with the solvent present.
By equating Eqs. (10-7) and (10-8) it is possible to calculate the
amount of solvent that must be added with the feed to obtain equal
concentrations above and below the feed plate. For simplification,
the last terms in the numerator can be neglected because they are usu-
ally small.

sr - — - — (1(MO)

where SF = mols of solvent added with feed

A = Om — On — SF = increase in overflow due to feed to tower

The value of xan can be obtained from Eq. (10-7), and On can be

obtained from a balance on the upper portion of the tower. For the

case in which the solvent rate is maintained proportional to the reflux

rate,

I — XRS

Feed-plate Location. In case solvent is added with the feed to
maintain the same concentration in the upper and lower sections, the
optimum key ratio can be calculated by the same method as for a
regular distillation. If the solvent concentration is not the same in the
two sections, there are usually two main factors tending to modify the
optimum ratio. If the solvent concentration is lower below the feed
plate than above, then (1) the relative volatility for the key compo-
nents may be less favorable in the lower section and (2) the mol fraction
of the solvent in the vapor is greater in the upper section. The first
factor would make it desirable to use a low key component ratio for the
feed plate to take advantage of the better relative volatility. Owing tc
the higher solvent vapor concentration above the feed plate, it would
be desirable to use a higher key ratio than normal. The safest method
is to calculate the feed-plate ratio for a regular distillation, and in the
plate-to-plate analysis test several plates in the region to determine the
optimum condition.

296

FRACTIONAL DISTILLATION

Concentration of Nitric Acid by Extractive Distillation. The con-
centration of nitric acid by the use of sulfuric acid will be used as an
example of extractive distillation employing a nonvolatile agent. A
typical flow sheet of a commercial unit is shown in Fig. 10-4. By the
oxidation of ammonia and the absorption of the nitrogen oxides a 62
weight per cent nitric acid is made. A portion of this feed acid is
mixed with 92 weight per cent sulfuric acid and added to the top of
the tower. The remainder of the feed is vaporized and introduced
into the middle region of the column. Direct steam is added at the

nnnn
Vaporizer

FIG. 10-4.

Si earn
Concentration of nitric acid

bottom. The overhead vapors are 99 weight per cent HN03, and an
over-all recovery of nitric of 99 per cent is obtained. The sulfuric acid
removed from the bottom is 65.0 weight per cent. The addition of
the feed to the top of the tower is unusual and, in general, is not good
distillation practice, but in this case the strong sulfuric acid would cause
decomposition of concentrated nitric acid and is therefore diluted by
the feed. The feed to the middle region is vapor so that there will be
less dilution of the sulfuric acid. 1.2 Ib of 92 per cent sulfuric is used
per pound of 62 per cent nitric concentrated. The mixed acid added
at the top is colder than the tower temperature, and the condensate
produced in the column to heat it serves as reflux.

The operation of this nitric acid system will be analyzed to deter-
mine the number of theoretical plates involved. An analysis of the
enthalpy values indicates that the usual simplifying assumptions will

EXTRACTIVE AND AZEOTROPIC DISTILLATION

297

not be satisfied, but the deviations are not large and the assumptions
will be used to simplify the calculations. The problem can be solved
more exactly on an enthalpy-composition diagram. In general such
a diagram is not suitable for a three-component mixture, but due to the
negligible volatility of the sulfuric acid, a modified form of the diagram
can be used. Equilibrium data (Refs. 2, 4) are given in Fig. 10-5.

FIG. 10-5.

02 03 04 05 0.6 07 0.8 0.9
Mot fraction nitric acid in licjuid,su!furic acid-free basis

Vapor-liquid equilibrium system, nitric acid-water-sulfuric acid.

1.0

These equilibrium data illustrate the effect of the added agent on the
relative volatility. With no sulfuric acid present, nitric acid and
water form a maximum boiling azeotrope containing 38 mol per cent
acid. The 62 weight per cent feed available is 31.8 mol per cent nitric
acid. It is therefore impossible to make 99 weight per cent nitric acid
from this feed unless some method of passing the azeotrope is available.
The addition of sulfuric increases the volatility of nitric acid relative to

298 FRACTIONAL DISTILLATION

water, and the equilibrium curves for liquid phases containing various
mol fractions of sulfuric acid are given in Fig. 10-5. The units of the
ordinates are expressed on a sulfuric acid-free basis. When the liquid
phase contains 10 mol per cent sulfuric acid, the volatility of nitric
acid is increased and the azeotrope composition becomes 12 mol per
cent nitric acid. This sulfuric acid strength could be employed in a
two-tower system to give the desired separation. The 31.8 mol per
cent acid could be treated to extractive distillation with 10 mol per
cent H2S04 in the liquid to give the desired concentrated product and
a bottoms containing about 15 mol per cent nitric acid. These bot-
toms would be fractionated without sulfuric acid being present to give
water overhead and 31.8 per cent HN03 as bottoms which would be
recycled. Instead of this two-tower arrangement, it is found more
practical to use more sulfuric acid and make the complete separation in
one step. In order to obtain a satisfactory relative volatility of nitric
acid to water at the low end of the curve requires 20 to 25 mol per cent
sulfuric acid in the liquid phase. The actual acid consumption for 25
mol per cent sulfuric acid is approximately the same as for 20 mol per
cent, because the lower relative volatility for the latter requires more
stripping steam which increases the acid requirement.

Solution. Basis: 100 mols of 62 weight per cent nitric acid.

xD - 0.965
n . 31.8(0.99)
D 0.965 -

Pounds of 62 per cent HN08 - 31.8(63) + 68.2(18) - 3,225
Pounds of 92 per cent H2SO4 - 1.2(3,225) - 3,870

Mols H20 in with H2S04 - 3>87°1(8°'08) . 17.2
Mols H20 in bottoms -

Mols H80 in feed - 68.2
Mols H2O in overhead - 32.6(0.035) - 1.14
Calculating the steam rate, 8, by over-all water balance,

8 * 106.5 + 1.14 - 17.2 - 68.2 - 22.2

It is assumed that sufficient nitric acid will be mixed with the sulfuric acid at the
top of the tower so that the combined stream contains 60 weight per cent sulfuric
acid.

EXTRACTIVE AND AZEOTROPIC DISTILLATION 299

Let xN » weight fraction of HNO8 in top mixture and 0.4 — XN •" weight frac-
tion of H2O in top mixture.
H2O balance:

XN - 0.215
0.4 - XN - 0.185

TTXT/A ^ i ^ 3,870(0.92) («y)

62 per cent IINO3 added at top - 0 62(0 60)

= 9,570zAr Ib. - 2,060
~ 297zjv Ib. mols

- 64(20.35 mols 100 per cent HN08)

Vapor and liquid rates on H^SO^free basis:

Vn * 22.2 + 36 - 58.2

On » 58.2 - 32.6 -f 64 + 17.2 - 106.8

Vm - 22.2

Om « 106.8

xD - 0.965

31.8(0.01) mq
** - 106.8 " °-003

- It is interesting to note that for £ = 22.2 an enthalpy balance gives Vn to the
top plate of 53.2 as compared to the 58.2 obtained on the basis of constant 0 and V
rates.

The mol fraction of sulf uric acid in the liquid phase is

36'4 =0.254

36.4 -f 106.8

For vapor-liquid equilibria the curve for a mol fraction of sulfuric acid equal to
0.25 given in Fig. 10-5 will be used. This curve is replotted in Fig. 10-6, and the
operating lines for the calculated flow rates are given.
The upper operating line for nitric acid is

58.2?/n - 106.8ztt+i 4- 31.5 - 20.35
- 106.8*n+i + 11.15

The lower operating line is

- 0.32

ym - 4.8zm+i - 0.0144

The plates are stepped* off starting at yT «" 0.99 and continuing down to
x « 0.003 at y =• 0. Approximately eight theoretical plates are required.

A more conventional extractive distillation would be to add all of the feed in the
middle region of the tower and to return a portion of the overhead product as reflux
with the sulfuric acid. Figure 10-7 shows such a system.

An exact comparison with the system of Fig. 10-4 can not be made, but several
cases will be evaluated using 25*4 mol per cent E^SO* in the liquid phase, and the
same (0/V) ratio below the feed plate,

300

FRACTIONAL DISTILLATION

Original system

— Case I

— CaseH
x— CoseM

02 03 0.4 0.5 06 07 0.8 0.9 1.0
Mol fraction nitric acid in liquid, sulfunc acid-free basis

FIG. 10-6. Design diagram for nitric acid concentration.

Case I:

Feed all vapor and using 92 per cent HaSCh at top. Basis : 100 mols of feed.

Vn - 122.2 On - 106.8
Vm = 22.2 Om - 106.8

Composition of acid mixture refluxed to tower, 3,870 Ib. 92 weight per cen
H2SO4 -f 89.6 Ib. mols 96.5 mol per cent HNO8

Weight

*Mol

per cent

H2SO*

3,560

38.0

25.4

H20

366

3.9

14.2

HN08

5,430

58.1

60.4

9,356

EXTRACTIVE AND AZEOTROPIC DISTILLATION

301

Lower operating line, same as before.
Upper operating line,

122.22/n - 106.8zn+i + 31.5
yn ** 0.873.Cn+i -f 0.2575

The upper operating line is shown on Fig. 10-6. it requires about one less
theoretical plate, but the heat requirements would be greater because of the
necessity of vaporizing all of the feed.

Feed

rl— 6

I Sulfur ic acid

Steam
FIG. 10-7.

Dilute suffuric acid
Concentration of nitric acid.

Case II:

Feed 36 per cent vapor and using 92 per cent H2SOi at top. Basis: 100 mols of
feed. In order to maintain the sulfuric acid concentration constant, 72 percent
of it will be added at the feed plate.

Vn - 58.2
Vm - 22.2

On
Om

30.4
106.8

Lower operating line, same as before but extends up the diagram farther because
of the part liquid feed.
Upper operating line,

i + 31.5
y. - 0.522ajn+i + 0.541

This case requires the same heat and acid consumption as the original example
but needs one additional theoretical plate. With the feed partly vaporized it
would have been better to separate the liquid and vapor and introduce each at its

302 FRACTIONAL DISTILLATION

optimum location. If this change is made, the system reduces to the original
system of Fig. 10-4, because the optimum feed-plate composition for the liquid
portion of the feed is approximately the same as the top plate composition.

Case III:

Feed 36 per cent vapor and using 85 weight per cent H2S04 at top. Basis : 100
mols of feed. The sulfuric acid will be split as for Case II.

For 85 per cent acid,

Mol fraction H2S04 - 0.51

For bottom acid strength,

H2SQ4

_
H2S04 + (0.49/0.51)H2S04 + (100 - 32.5) + 8

and, for same slope of lower operating line,

106.8 (0.49/0.51)H2S04 + (100 - 32.6) -f 8
22.2 "" 8

S «• 30.2 mols
H2S04 - 49.4 mols '17 mols at top, 32.4 mols at feed)

HaSOrfree basis,

Vn * 30.2 + 36 - 66.2 On - 49.95
Vm - 30.2 Om - 145.1

Lower operating line, same as before but the intersection of the two lines will
be at a different position.
Upper operating line,

66.22/n - 49.950ft+i 4- 31.5
yn - 0.755*n+i + 0.476

These operating lines are shown in Fig. 10-6. This case requires fewer plates
than Case II but requires more steam. It requires both more plates and steam
than the original example. The use of additional steam is objectionable both
because of the increased steam consumption and because it must be removed from
the sulfuric acid in the concentrator.

The original system is more desirable than any of the three cases. It is instruc-
tive to analyze the possibilities of improving the original distillation system.
Below the feed plate it would be desirable to reduce the steam consumption as
much as possible, but for the 25 mol per cent sulfuric acid, Fig. 10-6 indicates that
the ratio of 0/7 for this section cannot be increased significantly without increas-
ing the difficulty of fractionation excessively. Using this same slope (0/V «• 4.8),
the steam consumption is

a H20 (with H2S04) -f 67.4 (from feed)

8 -- 0 -

For the same strength nitric acid feed and the given acid recovery and overhead
concentration, the only way to reduce 8 is by reducing the water brought in by the
sulfuric acid. Higher strength sulfuric acid would reduce 8 but increase the
difficulty of reconcentrating the acid. A higher mol per cent sulfuric acid in the

EXTRACTIVE AND AZEOTROPIC DISTILLATION

303

liquid phase would make it possible to reduce S but would increase the acid recir-
culation. None of these alternatives for the lower section appears to offer any
great advantage over the original system. The concentration change per plate is
less below the feed plate than above, and it is desirable to shift to the upper operat-
ing line at a low concentration. The original system accomplishes this result by
using an all-vapor feed. If all the feed is added as a vapor (see Case I), the frac-
tionation in the enriching section is very easy but requires vaporizing all of the feed.
The original system reduces the heat consumption by vaporizing only a portion of
the feed and adding the remaining feed as liquid at the top which is approximately
the optimum feed-plate location for the liquid feed. This still gives the favorable
intersection of the operating lines and reduces the heat required for vaporizing the

FIG. 10-8. Relative volatility of isopropanol to ethanol in presence of water.

feed. It makes the separation a little more difficult than using all of the feed as
vapor, but it is obvious that the extra theoretical plate required is well justified
by the savings of heat. On the basis of Fig. 10-6, it would appear that it might be
advantageous to add more of the feed as liquid at the top and thereby reduce the
amount to be vaporized. This would necessitate preheating the mixed acid added
at the top by an amount equal to the reduced heat input with the vapor feed, but,
low-pressure waste steam might be used for this purpose. Such a change would
need to be carefully analyzed on an enthalpy basis in order to determine whether a
pinched-in condition was being encountered at the top of the column. In the case
of the original system, one portion of the feed was vaporized and the other added
as liquid at the top. Some improvement would be obtained by vaporizing under

304

FRACTIONAL DISTILLATION

equilibrium conditions such that the vapor feed would be more dilute in nitric'
acid and the liquid stronger. These two fractions could then be added as in Fig.
10-4, and the operating lines would be more favorable.

In most extractive distillation cases the added component is volatile.
The following example will illustrate the application of general equa-
tions developed in this chapter when using an extractive agent of
appreciable volatility.

Separation of Ethanol and Isopropanol by Extractive Distillation. A mixture
containing 20, 4, and 76 mol per cent of water, isopropyl alcohol, and ethyl alcohol,
respectively, is to be separated into an ethyl alcohol product containing not over

0.30

0.26

I 022

0.18

0.14

0.10

0.06

1.0

0.2

0.4

0.6
mols C2H5OH

0.8

1.0

E moIsC2H5OH + mo!siC3H7OH
Fio. 10-9. Relative volatility of water to ethanol.

3.2 mol per cent isopropyl alcohol on a water-free basis with an ethanol recovery of
)8 per cent. It has been decided that water will be used as an extractive distilla-
,ion agent, and enough water will be added to the reflux to make the liquid added
,o the tower 85 mol per cent water. The feed will be diluted to 85 mol per cent
vater before it is added to the tower, and it will be heated such that Vn «=* Vm.
Making the usual simplifying assumptions, calculate:

1. The minimum number of plates at total reflux.

2. The minimum reflux.ratio, 0/D.

EXTRACTIVE AND AZEOTROPIC DISTILLATION

305

Wafer

'Dilute isopropanol
Mo I fraction of water -0,85

20% wafer

4% isopropano/ /'

76% ethanol

Mo/ fraction of(
water '085

FIG. 10-10.

Dilute ethanof
Extractive distillation system for isopropanol-ethanol.

3. The number of theoretical plates required for 0/D equal to 1.5 minimum
0/D.

The equilibrium data (Refs. 5, 6) for this system are given in Figs. 10-8 and 10-9.
A schematic diagram is shown in Fig. 10-10.

Solution:

Let / » isopropyl alcohol
E «" ethyl alcohol
H = water

Basis: 100 mols of undiluted feed.

Since, in the presence of water, isopopanol is more volatile than ethanol, the
latter will be largely in the bottoms. By an ethanol balance

76(0.98) = WXE - 74.48

For design conditions,

Wxi

Wxi + WXE

= 0.002; Wxi~ 0.149

and, by difference,

DxE - 1.52; Dxi - 3.85

The water in the distillate cannot be determined until the composition of the
top plate is known, and this calculation is complicated by the fact that the water
concentration on the top plate is not known. The water content of the reflux to
the tower is 0.85, but there will be some change of this on the top plate; however,
as a first approximation, a mol fraction of water equal to 0.85 will be assumed for

306

FRACTIONAL DISTILLATION

this plate. Then using the relative volatilities from Figs, 10-8 and 10-9 a balance
is applied to the top plate.

Component

ytop

1.52

1 n

1.52

D
3.85

i >tQ

D
2.58

D
D -5.37

099

D
4.54D - 24 4

Y2/ 4. 547) -20. 3

A« D

XH « 0.85 =

4.54P - 24.4
4.54D - 20.3 '

D - 10.5

Water in distillate equals 10.5 — 5.37 ** 6.13, and for the distillate XE — 0.145,

xi - 0.367 xH - 0.488

The mols of water added to dilute the feed « (80/0.15) - 100 « 433. Mols of
water added at top of tower «* OR f ' , — l.Oj » 2.410#

Solution of Part 1. For the condition of total reflux the operating lines for the
alcohols are y »* 3.41$, and for water y = 3.41# — 2.41. The water concentra-
tion at the top of the tower will be approximately 0.85 and will increase slightly
down the tower because the relative volatility of water to the alcohols increases as
the ethanol concentration increases. The concentration at the bottom of the
tower is assumed as 0.86.

For these water concentrations the relative volatility of isopropanol jx> ethanol is
1.5 at the top and 1.64 at the bottom. Thus,

w , ! log (0.367/0.145) (0.998/0.002)

A\ -f- I =» .. , as ^Q-

log 1.57

Therefore, approximately 15 theoretical stages are required.

It has been pointed out in the discussion that this case is inconsistent in actual
bottoms concentration, but the answer just calculated gives a valuable limit for
orientation, and the actual theoretical plates for 0/D «• 1.5(0/D)min will be about
50 per cent greater.

Solution of Part 2. The minimum reflux ratio will be calculated by three
methods:

Method 1. Since the amount of water in both the liquid and vapor phases is
relatively constant from plate to plate, the system may be treated as a binary to

EXTRACTIVE AND AZEOTROPIC DISTILLATION 307

find (0/D)min. Using the mol fractions of the alcohols on a water-free basis, which
will be indicated by primes, with Fn * Vm, at the feed plate, the concentration of
isopropyl alcohol in the feed is Xj «• 4/80 » 0.05 and a/jj. —1.6 (mol fraction
water - 0.86).

,; . *•*«>•<»)
Vl 1+0.6(0.05)

on this basis,

0.0776 «

_ 0.716 - 0.0776
0.716 - 0.050
0.959 ,9
x " 0^41 " 23'2
D' - 5.37
and

(O^)mm =* 23.2(5.37) = 125 mols of alcohol in overflow to feed plate
125 + 5.37 = 130.4 mols of alcohol in vapor

___ = 833 (assume XH at feed plate =» 0.85)
0.15

By Eq. (10-7) the asymptotic concentration of the solvent is

»]

L0p.41(2«)-5,

0-875

*«» - i.o - 0.2

as compared to 0.85 assumed.

This value is essentially independent of the value of On, since the numerator of
the bracket is 2.410^ — 5.1, the denominator is 3.410#, and the 5.1 is only a small
correction.

Thus, with xsn - 0.875

KM • - 125 •
^»Mm ^0.125 '

- 293

10.5

The value of a;^ is a little lower than xSn, and Eqs. (10-7) and (10-8) indicate
that, for this reflux ratio, approximately 530 mols of water should be added with
the feed to make them equal instead of the 453 mols used to make the mol fraction
of water in the feed equal to 0.85.

Method 2. By Eq. (10-9) using same values for relative volatilities above and
below the feed plate,

XBS « 0.85; XDS « 0.49

(1 - o.49) - *'°n3 (0.85 - 0.49)

0.4

308 FRACTIONAL DISTILLATION

(1 - 0.49) - 1fiL6fto (0.85 - 0.49)

»- SnnS a667

>i - 1.0(3.86) _ lfi
A ~ (1.6 - 1.0X0.4)
u 1.6(74.48) ^

F " (1.6 - 1.0)(0.667)

0667 °-268

0.4

OR . 268 + V(268)24-4(16)(298) _ 2g5

OR __ 285 _ 07 t
T3T ~ 10.5

Using this value of the reflux ratio it is possible to calculate the ratio of the key
components at the feed plate. In this case, the value of the ratio is 0.055 or only
slightly higher than the ratio in the feed.

Method 3. A third method is equating the pinched-in ratio for the key
components to the intersection ratio, <j>. This is not a general method because the
intersection ratio may be considerably different from the optimum matching ratio.

A f 74.48 453 - 5.1\ _

U.6 - 1.0 + 1.6 - 0.2;

_ _-

ln/ 3.85 5.1 \

V1'6 " l'Q L0 "" °'2/ =

<f>

0.04

076 " I -xik- xsn

calculating xtk by Eq. (9-13) and x8n by Eq. (10-7).

1.0 (8.85/Qn)
0.0526

1 ~

1.6 - 1.0 1.0 - 0.2

6.41
°'0526 "

_
3.410u - 6.41 - S.OlOa + 6.41

The value for OJB by Method 3 is higher because the optimum key component
ratio is greater than that used.

EXTRACTIVE AND AZEOTROPIC DISTILLATION

309

A value of $ equal to 0.055 instead of 0.0526 would make the reflux ratio calcu-
lated by Method 3 equal to those of the other two methods.
Solution of Part 3. Using a value of 27.5 for (Ou/Z))mm gives

and

1.5(27.5) - 41.3

OR - 41.3(10.5) - 433 mols

Water added at top - 2.41(433) - 1,042 mols

Water out with bottom = 1,042 -f 453 - 5 = 1,490 mols

On - 433 + 1,042 = 1,475 mols

Vn « Vm - 433 4- 10.5 - 443 mols

Om - 1,475 + 533 - 2,008 mols

TABLE 10-2

Bottoms

Wxw

Mols

Mol fraction

2,008

74.5

0 0476

1 0

0 0476

0 067

0 037

0 149

0 0000955

1 96

0.000187

0 000263

0 00007

1,490.0

0.952

0.114

0.1085

0.1525

0 742

1,564 6

0 1563

. 00 , Wxw
Xl - 0.220, + 2^8

OiXi

0.22yi

0.222/2

0 104

1 0

0.104

0 0966

0 1335

1 0

0 101

0 000333

1 62

0 00054

0 0005

0 00057

1.65

0 000715

0 895

0.148

0.1325

0 123

0 865

0 18

0 118

0 237

0.227/3

0.138

1.0

0.101

0.138

0 137

0 000785

1.635

0.000935

0.0010

0.00127

0 0016

0 00196

0 86

0.188

0.1185

0.86

0.860

#10

Xlt

Xl*

0.137

0 1365

0.136

0 1355

0.135

0.134

0.0024

0.00293

0.00355

0.0042

0.00505

0.006

0 0071

0.860

0.859

0.869

0.859

310

FRACTIONAL DISTILLATION

Operating lines:
Above feed plate for alcohols,

4432/n •" l,475a;«+i + DXD
DxD

1,475' 1,475
Above feed plate for water,

0.00261 for isopropanol

- Xn+i

« 0.00103 for ethanol
5.13 - 1042

1,475
- xn+i - 0.703

Below the feed plate for alcohols,

2,008zm+i - Wxw

-
« - xm+i -

where W zir/2,008 equals 0.037 for ethanol, 0.00007 for isopropanol, and 0.742 for
water.

The calculations are carried up from the still in the usual manner, with relative
volatility values from Figs. 10-8 and 10-9. The results are summarized in Table
10-2.

The intersection ratio of the key components is 0.0526, and the actual ratio on
the thirteenth plate is 0.0444 and 0.053 on the fourteenth plate. In this case, the
optimum ratio is slightly higher than the intersection ratio, and the fourteenth plate
will be used as the feed plate. The calculations are then continued using the
enriching line equations. The results are presented in Table 10-3.

TABLE 10-3

Xl6

a?™

#18

X\9

£20

£21

0.130

0.126

0.123

0.118

0.1105

0.101

0.0886

0 0087

0 0113

0.0154

0 0215

0.0301

0.042

0 0563

0.861

0.862

0.860

0 859

0 857

0 855

323

2/28

#24

J/24

I
W

0.0744
0.0723
0.853

0.060
0 0889
0.851

0 1575
0.354

0.488

0.0463
0.104
0.849

0 119
0.401
0 480

0.145
0 367

0 488

The x and y values are given for plates 23 and 24. It will be noted that yn is
not quite up to #D, but that yu exceeds it. Thus between 23 and 24 theoretical
plates in addition to the still are required. The water concentration on the top
plate is close to the assumed value of 0.85. If this assumption had not checked
with the calculated value, it would be necessary to estimate whether the error
would materially affect the result. If the correction was large, the calculations
might need to be repeated to obtain a satisfactory result.

EXTRACTIVE AND AZEOTROPIC DISTILLATION

311

CM o oo to

J-d

312 FRACTIONAL DISTILLATION

The calculated values for the liquid phase are plotted in Fig. 10-11, There is a
rapid decrease in the water concentration from the still to the first few plates,
and then the value is relatively constant in the remainder of the tower. The
decrease through the tower is small because (1) the relative volatility of water
relative to the alcohols decreases as the ratio of isopropanol to ethanol increases
and (2) the water concentration in the still is high. The concentration of water in
the bottoms is higher because the vapor removed from the still contains a much
higher ratio of alcohol to water than the liquid to the still.

Because the water concentration is relatively constant, approximate calculations
could be made on a water-free basis. The total available vapor and liquid should
be decreased by an amount equal to that required for the water, and the remaining
vapor and liquid used to separate the alcohols as a binary mixture using relative
volatility from Fig. 10-8 at the assumed water concentration. For this case, the
result should be reasonably close to the more rigorous method employed in this
section, but it is doubtful whether the calculation is much simpler or less time-
consuming.

AZEOTROPIC DISTILLATION

The first commercial application of azeotropic distillation was the
use of benzene by Young (Ref. 7) for the azeotropic dehydration of
aqueous alcohol, which is still one of the most important applications
of this type of operation.

It has been pointed out that this system differs from extractive dis-
tillation chiefly in the behavior of the agent. For example, consider
the continuous dehydration of ethyl alcohol by the use of benzene as
the azeotropic agent, as shown in Fig. 10-12. Tower 1 serves to
remove the water from the alcohol, and tower 2 serves to recover the
alcohol and benzene. Essentially anhydrous alcohol is produced as
bottoms in tower 1, and sufficient plates are used above the feed plate
to produce an overhead vapor that will give two liquid layers on con-
densation. The benzene-alcohol layer is used as reflux for tower 1,
and the water layer containing small amounts of alcohol and benzene
is stripped to recover these constituents. In such an operation, the
agent, benzene, must vary from essentially zero in the still to a rela-
tively high concentration in the tower. Thus there is a wide variation
in the solvent concentration in the tower, and some of the approxima-
tions made for extractive distillation would lead to serious errors.

The benzene-alcohol-water system can produce an overhead vapor
that will give two liquid phases on condensation which makes it possi-
ble to by-pass the azeotrope in a manner analogous to that which was
shown for partly miscible binary distillations, and the same type of
two-tower system is applicable. It should be noted that the system
does not produce the ternary azeotrope as the overhead composition,

EXTRACTIVE AND AZEOTROPIC DISTILLATION

313

but it is essential that the condensate is two layers. With some azeo-
tropic systems the overhead will not give two liquid layers, and some
other type of operation must be used to split the overhead, such as
extraction or dilution.

In order to illustrate the phenomena involved in azeotropic distilla-
tion, an example will be considered first and then the various limiting

FIG. 10-12.

Anhydrous
alcohol
Azeotropic system for the production of absolute ethanol using benzene.

conditions will be reviewed. Consider the production of anhydrous
ethanol using benzene as the azeotroping agent. In such cases, it is
found most economical to concentrate the alcohol by normal distilla-
tion to almost the binary azeotrope concentration before it is intro-
duced into the azeotropic system.

Production of Absolute Alcohol by Azeotropic Distillation with Benzene.
For the purposes of this example it is assumed that the feed to the dehydration
system contains 89 mol per cent alcohol and 11 mol per cent water A two-tower
system will be employed similar to that illustrated in Fig. 10-12, and only a single
liquid layer will be refluxed to each tower. Both towers will be designed for an
overflow rate below the feed plate of 125 mols per 100 mols of vapor, and the usual
simplifying assumptions will be made. The feed to the alcohol tower will be such
that Vn = Vm, and it is assumed that any condensation due to the reflux liquids

314

FRACTIONAL DISTILLATION

Alcohol

, tower No.t '

Alcohol -water
azeotrope

-L iquid compositions
below feed plate

\ - - ~ Liquid compositions
~\ above feed plate

/T * * - -Benzene • alcohol
azeotrope

xw, tower
No <?

Water

'Benzene

Fia. 10-13. Diagram for system, ethanol-benzene-water.

XA+XH mol alcohol + mots water
Fio. 10-14. EquUibrium data for system, ethanol'ibenaene-water.

EXTRACTIVE AND AZEOTROPIC DISTILLATION

315

being at a lower temperature than their boiling point is negligible. The

from the water tower are to contain not over 0.01 mol per cent alcohol,

anhydrous alcohol is to contain not over 0.01 and 0.1 mol per cent

and water, respectively. Theoretically,

it is not possible to set the exact bot-

toms concentration because the compo-

sition of the refluxes is limited by the

solubility relationships; however, it is

found that rather wide latitude is pos-

sible in selecting the composition of the

bottoms product. For this example it

is assumed that the bottoms are 99.9,

0.01, and 0.09 mol per cent alcohol,

benzene, and water, respectively.

The physical-chemical data for this
system are taken from Cook (Ref. 3)
and Barbaudy (Ref. 1). The solubility
data for 25°C. are given in Table 10-4
and plotted in Fig. 10-13. The vapor-
liquid equilibrium data for the system
at atmospheric pressure are presented
in Figs. 10-14 and 10-15.

It should be noted that equilibrium
data available were not so complete or
so consistent as would be desired, and
that these two figures represent a
smoothing, extrapolation, and inter-
polation of the data.

Solution. Basis: 100 mols of feed.

Alcohol balance,

bottoms
and the
benzene

0.999Tfi + 0.0001 W
Over-all balance,

Wi + W2 - 100
Wi - 89.1
Wz - 10.9

- 89

FIG. 10-15. Equilibrium data for system,
ethanol-benzene-water.

TABLE 10-4. PAIKS OP TIB-LINE COORDINATES

Alcohol

0 3 and 0.315

0.225 and 0 23

0 18 and 0.13

0.08and0.04

Benzene
Water

0.068 and 0.465
0.632and0.22

0.025 and 0.655
0.75and0.115

0.015and0.82
0 805 and 0.05

0.007and0.94
0.914 and 0 02

Tower 1:

" IM

o!**Vm+ 89.1 - 1257*
Vm - 356 Om ~ 445
Vn - 356 On « 345

316

FRACTIONAL DISTILLATION

The calculations will be started from the still, using a basis of Om — 1.0,
V* « 0.8, and W * 0.2. The results are given in Table 10-5.

The feed ratio of the key components is l%$ «• 0.123, and the change from the
lower to the upper section should be made at about this ratio. The ratios for
plates 20, 21, and 22 are 0.075, 0.0935, and 0.114, respectively, from which it would

TABLE 10-5
(B - Benzene, A = Alcohol, H - Water)

0.2z

axw

0.82/TT

B
A
H

0.0001
0.999
0.0009

0.00002
0.1998
0.00018

3.6
0,89
1.0

0.00036
0.89
0 0009

0 00032
0.7989
0.00081

0.000344
0.9987
0.00099

0.89126

OiXl

0.82/1

aX2

0.82/2

B
A
H

3 6
0 89
1.0

0 00124
0.899
0.00099

0.00111
0.798 *
0.00089

0.00113
0.9978
0.00107

0.00406
0.888
0.00107

0.00364
0 7954
0.00096

0.00366
0.9952
0 00114

0.89123

0.89313

txXz

0.82/3

ax 4

0.82/4

0.0132

0.0117

0.01172

3.4

0.0398

0.0354

0.03542

3 2

0.886

0.7873

0.9871

0.87

0.859

0.764

0 9638

0 82

0.00114

0.00101

0.00119

1.0

0.00119

0.00106

0.00124

1.0

0.89934

0.89999

ax&

0.82/5

<xXo

0.82/5

B
A
H

0.113
0.79
0.00124

0.10
0.70
0.0011

0.10
0.8998
0.00129

2 5
0.73
1.0

0 25
0.657
0.00129

0.22
0.579
0.00114

0.22
0.7788
0.00132

0.90424

0.90829

oe.X^

0.8j/7

aXs

0.82/8

B
A
H

1.58
0.62
1.0

0.348
0.483
0.00132

0.335
0.464
0.00127

0.335
0.664
0.00145

0.98
0.54
1.0

0.328
0.358
0.00145

0.382
0.417
0.00169

0.83232

0.68745

EXTRACTIVE AND AZEOTROPIC DISTILLATION
TABLE 10-5 (Continued)

317

aX9

0.82/9

XlQ

aXiQ

0,8i/io

0.382

0.82

0.313

0 396

0.396

0.76

0 301

0.398

0.617

0.515

0.317

0.402

0.602 «

0.5

0.301

0.398

0.00187

1.0

0.00187

0.00237

0 00255

1.0

0.00255

0.00337

0 63187

0.60455

aXn

0.83/11

Zl2

aXiz

0.82/12

0.398

0 302

0 400

0 304

0.400

0 598

0 299

0 396

0.596

0 298

0 393

0.00355

0 00355

0 0047

0.0049

0 0049

0 00645

0 60455

0.6069

2?1S

«Si3

0.87/13

Si4

as H

0.8yi4

S16

0.400

0.304

0 400

0 304

0.399

0 593

0.297

0 392

0.592

0 296

0 389

0 589

0.0066

0 0087

0.0089

0 0089

0.0117

0.0119

0.6076

0 6089

aXu

0.82/15

OtXu

0.82/16

aXn

B 0 303

0.398

0 302

0.395

0 395

0.76

0.301

A 0 295

0 387

0.587

0 294

0.384

0.584

0 51

0.298

H 0.0119

0.0156

0.0158

0.0206

0.0208

1.0

0.0208

0.6099

0.6118

0 6198

0.82/

Sl8 <*Sl8 0.82/18 0?19

aXig

0.8yi»

B 0 389

0.389 0.296 0.382 0.382

0 82

0 314

0 385

A 0 385

0.585 0.298 0.384 0.584

0.52

0.304

0 373

H 0.0268

0.027 0.027 0.0348 0.035

1 0

0.035

0.0429

0.621

0.653

S20

aXzQ

0.82/20

aXzi

0.82/21

S22

B 0.385

0.316

0.385

0.82

0.316

0.38

A 0.573

0.298

0.363

0.563

0.53

0.298

0 358

0.558

H 0.0431

0 043

0.0524

0.0526

1.0

0.0526

0.063

0.0632

0 657

0 6666

315

appear that plate 22 would be the best feed plate. By trial it is found that plate
21 is more desirable. This is largely because, below the feed plate, the benzene
liquid-phase concentration is asymptotic at a value less than 0.4, but above the feed
plate, this composition rises rapidly to above 0.5. This increased concentration
increases the relative volatility of water to ethanol making the fractionation easier,
and it is advantageous to do more of the fractionation above the feed plate.

The plate-to-plate calculations (Table 10-6) are carried above plate 21 in a
similar manner. A basis of 1 mol of liquid is used, making D
0.032 and F * 1.032.

TABLE 10-6. FEED PLATE 21

1.032^/21

0.032xD

£22

a#22

1.0322/22

#23

axn

0.49

—

0.49

0 52

0.255

0.507

0.475

0.241

0.462

—

0,462

0.465

0 215

0.428

0 428

0.46

0.1965

0 0815

0.032

0.0495

1 0

0 0495

0 0985

0.066S

1.0

0 0665

0.5195

0 5040

1.0322/23

£24

a#24

1.032^/24

£25

«Z25

1.032s/26

#26

0.493

0 493

0 55

0.271

0.495

0,495

0 69

0 343

0.52

0.402

0.475

0.191

0.348

0.52

0 181

0.273

0.136

0 104

1 0

0 104

0 190

0 158

I 0

0 158

0 239

0 207

0 566

0 682

The top plate of the tower is determined by the fact that the reflux to it must
correspond to the benzene layer of the two-phase region. In order to illustrate this
condition, the liquid compositions are plotted on the triangular diagram in Fig.
10-13. The liquid composition #25 is in the single-phase region while aj2e is in the
two-phase region; thus neither of them satisfies the condition. This means that
the exact design conditions are not fulfilled by an even number of theoretical plates
and that between 24 and 25 theoretical plates are required since £25 would be the
reflux to plate 24 and a?26 the reflux to plate 25. Theoretically, the exact condi-
tions could be satisfied by using a different reflux ratio, feed-plate location, or
bottoms composition, but the trial-and-error procedure involved does not justify
the effort. It is sufficient to know that between 24 and 25 theoretical plates are
required. The x*t value could be satisfied exactly if a mixture of two liquid layers
was refluxed, but this is not advantageous.

While it is not necessary to obtain an even number of theoretical plates in the
calculations, it is essential to have a reasonably accurate estimate of the composi-
tion of the overhead vapor and the reflux in order to make the balances on the
condenser. A satisfactory method of evaluating these compositions is to plot the
compositions, as in Fig. 10-13, and use the composition where the curve cuts the
two-phase boundary as the composition of the reflux to the alcohol tower, x^
and the reflux (or feed) to the water tower, XR^ will be the liquid in equilibrium
with XRV From Fig. 10-13,

EXTRACTIVE AND AZEOTROPIC DISTILLATION

319

XRl

XR>

1.032yri

0 51

0.053

0.51

0 298

0.282

0.298

0.192

0.665

0.224

As would be expected, the values of xRl are intermediate between Xss and $«.

Using as basis an overflow rate of 1 mol of liquid per unit time for the water
tower, W* » 0.2 and F = 0.8. The mols of each component in the overhead
vapor from the water tower are equal to the difference between the mols in the
reflux and the bottoms. Thus,

3*2

0.2ZTF

O.Syr

0.053

—

0.053

0 282

0 00002

0.282

0.665

0.2

0.465

While both 1. 0322/1^ — xRl and #/?2 — 0.8g/r have been made equal to the
bottoms from the water tower, they are on different bases and it is interesting to
consider the relative quantity of the two refluxes.
For tower 1 :

VT - OB + W a

IF.

OBI

0.032

For tower 2:

Therefore,

- VT

0.2

0Rl - 6.

and VTl -

or, on the basis of 100 mols of feed,

0Rl - 345 and

- 55.2

The plate-to-plate calculations for tower 2 are given in Table 10-7.

The mol fraction of alcohol in #r~3 is 1.5 times the maximum value specified while
in a?r~4, the concentration is much lower, and between four and five theoretical
plates are required. Due to the high volatility of benzene in water, its concen-
tration in the bottoms of this tower would be quite low.

If two liquid layers had been refluxed to match #26 for tower 1, the composition
of the two layers would be the terminal points of the solubility tie line through the
composition of £*«. The reflux to the water tower would have the composition
of the water-layer end of the tie line. In this case this latter layer would hav0
been lower in benzene and alcohol, thereby making the f ractionation in the water

320

FRACTIONAL DISTILLATION
TABLE 10-7

Q.SyT

O.Syr

O.S^r-i

Q.8yT-.i

XT-l

0.8yr-2

A
H

0 053
0 282
0.465

150
8.0
1.0

0.00035
0.035

0.465

0.0007
0 070
0.93

0 0007
0.07
0 73

200
9.7
1.0

0.0000035
0.0072
0.73

5 X 10~«
0 0097
0.99

6 X 10-«
0.0097
0.79

200
10
1

0 50035

0.737

0.8yr-2
a

XT~Z

0.8s/r_3

200
10
1

Q.Syr-8

XT-3

0.8yr_«

0.8yr-4

XT~4

B
A
H

2 5 X lO-s
0 00097
0 79

3 X 10-8
0 0012
0 999

3 X 10-8
0.0012
0.799

1.5 X 10-JO
0 00012
0 799

2 X 10-"
0.00015
0 99985

2 X 10-io
0 00013
0 79984

1Q-12

0 000013
0 79985

10-12

0 000016
0.99998

0.79

0 799

0.79986

tower easier. The reflux to tower 1 would be two layers, but the liquid from the
top plate would be a single phase of composition #25. However, it is possible to
operate with two liquid phases on a plate, provided the mechanical design is satis-
factory. Thus in the present case the calculations can be carried past #28, but
allowance must be made for the two liquid phases. For example, the compositions
of the two phases corresponding to ic26 are as follows:1

fee)i

(£20) 2

Oil

<*(#26)l

1.032^26

#27

0.56

0 04

0.54

0.302

0.527

0.275

0.26

0.45

0.124

0.217

0.165

0.7

1.0

0.165

0.288

0.256

0 591

The composition of the vapor was calculated on the basis of (#26)1 because it is
believed that the equilibrium data are more reliable in the high benzene region
than in the high water region. If 26 theoretical plates were employed and two
liquid layers were refluxed to match #27, then plate 26 would have two liquid layers
present. Usually only one liquid layer is refluxed.

The tables of data and the liquid compositions plotted on Fig. 10-13 illustrate
the factors involved. Starting at the bottom of tower 1, the system behaves like a
mixture of benzene and alcohol, and the benzene concentration increases rapidly.
The relative volatility of water is low and does not increase significantly until the
benzene concentration has built up enough to increase the volatility of the water.

1 These calculations are not exact, because the solubility data given in Fig. 10-13
are for 25°C., while the temperature on plate 26 is about 66°C. The solubilities
are somewhat different at the two temperatures, but the 25°C. data are used to
illustrate the principle.

EXTRACTIVE AND AZEOTROPIC DISTILLATION 321

The water concentration then increases rapidly until the feed plate is reached.
Above the feed plate the heavy key component (alcohol) decreases rapidly, and the
benzene and water attain values that result in two-layer formation. It will be
noted that the liquid compositions are heading for ternary azeotrope composition
as reported by Young (Ref. 7). The composition of the mixed vapors to the
condenser is a point on the tie line through XR^ and XR^ and the relative distances
from this composition to XR and XR% are inversely as 0^ and 0#8.

The limiting conditions for azeotropic conditions are not easily
expressed in analytical equation, but they can be evaluated for each
specific case.

Minimum Number of Theoretical Plates at Total Reflux. Owing to
the wide variation of the relative volatility, equations of the type of
(7-53) are not applicable. The number of theoretical plates required is
calculated best by the plate-to-plate method using y = x as the operat-
ing line for each component. For the benzene-alcohol-water system
considered in the preceding section, this plate-to-plate method indi-
cates that between 12 and 13 theoretical plates are required at total
reflux.

Minimum Reflux Ratio. This limit corresponds to a pinched-in
position, or positions, in the tower. Because of the wide variation in
relative volatilities with composition, this limit frequently corre-
sponds to a tangent condition of the operating lines and equilibrium
values rather than an intersection. In such cases it is difficult to cal-
culate the exact tangent condition, and each system is essentially a
new problem. However, in a number of cases, the minimum reflux
ratio is determined by intersections of the operating lines and the
equilibrium values, and these often occur near the feed plate, because
the mixture to be treated is usually a binary and the azeotrope agent is
approximately constant above and below the feed plate. For these
cases, the general principles employed for multicomponent mixtures
can be applied. As an example, consider the benzene-alcohol-water
system already studied, which has this type of limiting condition. The
asymptotic concentrations below the feed plate are given by equations
of the type of (9-15). Solving for the values between water and
benzene,

r _

Xa ~

s - an + aff(W/Om)(xWB/xB)

The concentration of benzene in the bottoms, XWB, is much smaller
than the asymptotic value, xs, and the last term of the denominator

322

FRACTIONAL DISTILLATION

will be neglected. The value of x« is much larger than the numerator
of the right-hand side of the equation, and this necessitates <XB being
essentially equal to a#. Thus, for this case where XWB and XWB are
very small, the pinched-in condition corresponds to the relative vola-
tility of benzene to water being unity; i.e., a^n = 1.0. A study of Fig.
10-15 indicates that OLBH == 1 for only a limited concentration range for
benzene. Above the feed plate the net alcohol and benzene removals
are very small, and the same type of analysis leads to the conclusion that
<XB «* &*• The conditions as a = 1 below the feed plate and GAB = 1.0

Alcohol

Wafer

* Benzene- alcohol

FIG. 10-16.

above the feed plate can be used to evaluate the minimum reflux
ratio. One approximation for this limit can be obtained by equating
the concentration ratio of alcohol to water for otB — <XH to the feed
ratio. The composition for this condition can be obtained by drawing
a line through XA = 0.89, XH = 0.11, and the benzene corner of the
diagram. Where this line cuts the an = as line gives the desired
values. This construction has been carried out in Fig. 10-16, and the
intersection gives XA « 0.55, xa « 0.07, and XB - 0.38. From Figs.
10-14 and 10-15, aAa « 0.54 and etas « 1.0. By Eq. (9-15),

EXTRACTIVE AND AZEOTROPIC DISTILLATION 323

n ss _ WOm) (0.999)
0.55 -- ! _ 0.54

5 = 0.253

vflj

Om = 352 and Vm = 263

t = 26-3 = L34
or, taking the net distillate, D', as 11 mols,

On = 252 Vn = 263
Tfr ) = 22.8

The value of Om/Vm - 1.25 employed in the plate-to-plate calcula-
tions corresponds to

--
D' " TT ~ <

A similar calculation can be made above the feed plate using ou = afl,
and the value obtained is 0Rl/D' = 17.5. This difference is due to
the fact that the ratio of the key components was taken the same as in
the feed. It has already been pointed out that the optimum feed-
plate composition corresponds to a higher ratio of alcohol to water than
in the feed.

Another method of calculating the minimum reflux ratio is to equate
the ratio of the key components for the two pinched-in regions. This
involves a trial-and-error procedure to find a composition on the
OLE = oiB line that gives the same 0RJD' value as a composition on the
<XB = OLA line when the ratio of XA/X a is the same at both. This calcu-
lation gave 0Rl/Df = 21 for a key component ratio of 11 to 1 as com^-
pared to 8 to 1 in the feed. This last answer for the minimum reflux
ratio should be near the true value.

The minimum reflux ratios for other cases can be handled in a simi-
lar manner.

Nomenclature

A,B — constants in Van Laar equation
F ** feed rate
0 «* overflow rate
P » vapor pressure

324 FRACTIONAL DISTILLATION

p - (On - Om)/F

T «• temperature

V «• vapor rate

x «• mol fraction in liquid

y ** niol fraction in vapor

a. «" relative volatility

y = activity coefficient

Subscripts:

A refers to alcohol
B refers to benzene
D refers to distillate
E refers to ethanol
F refers to feed
H refers to water
h refers to heavy component
hk refers to heavy key component
/ refers to isopropanol
I refers to light component
Ik refers to light key component
m refers to below feed plate
n refers to above feed plate
R refers to reflux
S refers to extractive agent
T refers to top plate
1,2,3, refer to component or plate number

References

1. BARBAUDY, Sc.D. thesis in physical sciences, University of Paris, 1925.

2. CARPENTER and BABOR, Trans. Am. lust. Chem. Engrs., 16, Part 1, III (1924).

3. COOK, M.S. thesis in chemical engineering, M.I.T., 1940.

4. "International Critical Tables," Vol. Ill, 306, McGraw-Hill Book Company,
Inc., New York, 1928.

5. SHARPE and SIEGFRIED, M.S. thesis in chemical engineering, M.I.T., 1947.

6. TOBIN, M.S. thesis in chemical engineering, M.I.T., 1946.

7. YOUNG, "Distillation Principles and Process," Macmillan & Co., Ltd., London,
1922.

CHAPTER 11
RECTIFICATION OF COMPLEX MIXTURES

The design methods considered for multicomponent mixtures in
Chap. 9 were based on a limited number of definitely known compo-
nents. In some cases, the mixtures are so complex that the composi-
tion with reference to the pure component is not known. This is par-
ticularly true of the petroleum naphthas and oils which are mixtures
of many series of hydrocarbons, many of the substances present having
boiling points so close together that it is practically impossible to sepa-
rate them into the pure components by fractional distillation or any
other means. Even if it were possible to determine the composition
of the mixture exactly, there are so many components present that the
methods of Chap. 9 would be too laborious. It has become customary
to characterize such mixtures by methods other than the amount of the
individual components they contain, such as simple distillation or
true-boiling-point curves, density, aromaticity (or some other factor
related to types of compounds), refractive index, etc.

The simple distillation curve is the temperature as a function of the
per cent distilled in a simple or Rayleigh type of distillation. This
type of distillation is approximated by the laboratory A.S.T.M. dis-
tillation which is widely used to characterize petroleum fractions.
The A.S.T.M. procedure gives some reflux and rectification, and the
results are not exactly equal to the simple batch distillation, although
the difference is not large. The temperature normally measured is
the condensation temperature of the vapor flowing from the still to the
condenser. Curve A of Fig. 11-1 is typical for the simple distillation
of a complex mixture. The temperature at any point is the averaged
result of a large number of components and includes all the effects of
nonideality in the solutions. Thus in most cases it is impossible to
relate such a curve to the volatility of the individual components
involved. As a result, such simple distillation curves are not of much
direct value for the solution of rectification problems.

The true-boiling-point curve is an attempt to separate the complex
mixture into its individual components. Actually it is a batch dis-

325

326

FRACTIONAL DISTILLATION

tillation carried out under rectification conditions. Usually a labora-
tory distillation column equivalent to a large number of plates is
employed, and the separation is made at a high reflux ratio to obtain
efficient fractionation. Because it is a batch operation, low liquid
holdup in the column is important, and packed columns are generally
employed. Ideally a true-boiling-point curve for a mixture corre-
sponding to the simple distillation curve A of Fig. 11-1 might be repre-
sented by curve B of this figure, showing individual horizontal lines for
each component with sharp increases in temperature in going from one
constituent to the next. In most petroleum mixtures the curve
obtained is similar to B of Fig. 11-2, and no definite steps are obtained.

P«r C«n* DisKIIed
FIG. 11-1.

100

15940

Per Cen+ Drilled Over
FIG. 11-2.

100

This result is obtained because (1) the number of components is very
large and no single step would be very significant and (2) the degree of
fractionation usually, employed is not sufficient to give the sharp breaks
in the curve. The sharpness of the fractionation between the different
components is also lowered by the formation of azeotropes and by
other solution abnormalities. However, the true-boiling-point curve
probably represents a fairly high degree of separation in most cases.
It is interesting to compare curves A and B of Fig. 11-2. The true-
boiling-point curve begins at a lower and ends at a higher temperature
than the simple distillation curve because the latter gives an averaging
effect. Actually a simple distillation should give the same final tem-
perature as the true-boiling-point distillation, because the last material
to be vaporized should be the pure highest boiling component in both
cases, but in general the distillations cannot be carried to 100 per cent
distilled. A number of methods have been proposed for calculating

RECTIFICATION OF COMPLEX MIXTURES

327

the true-boiling-point and simple distillation curves from each other,
and they are useful in some cases, but if solution abnormalities are
involved, they can be in error.

It has been found possible to use true-boiling-point curves to define
the compositions for distillation calculation. A fraction distilling over
a narrow range is taken as an individual component. Thus the frac-
tion coming over as distillate between 39 and 40 per cent in curve B of
Fig. 11-2 might be considered as a component, the boiling point of
which, at the pressure at which the distillation was carried out, being

Per Cent Distilled Over
FIG. 11-3.

100

the average of the two temperatures corresponding to 39 and 40 per
cent. Ill this manner the curve can be divided into any desired num-
ber of "components" with estimated vapor-liquid characteristics cor-
responding to their distillation temperature. These components can
then be employed in the distillation calculations using the various
methods given in Chaps. 9 and 12. The components in the various
fractions can be recombined to give the true-boiling-point curve of the
products.

If the distillate during a true-boiling-point distillation were to be
divided into two fractions at some convenient point, A, corresponding
to the temperature t\, and simple distillation and true-boiling-point
curves obtained for the two fractious, the results would resemble the

328

FRACTIONAL DISTILLATION

curves shown in Fig. 11-3 where curvet is the original true-boiling-
point curve and Bf the A.S.T.M. distillation curve for the same mix-
ture. The true and the A.S.T.M. curve of the two fractions are
shown a$ (7 and t> dttfves for the more volatile and the less volatile
fractions, respectively.

It will be ndted that the initial temperature of curve D' is consider-
ably highei* than the final boiling point of curve C". This difference,

Bottoms io Succeeding Mill

FIG. 11-4. Flow sheet of still and tower used in test.

the so-called "gap," is frequently used as design specification for the
separation desired. The averaging effect of the simple distillation
technique tends to give large temperature gaps even though the lower
boiling fraction may have components that boil higher than some of
those in the less volatile fraction. The fractionation in actual cases
will not be so good as assumed for Fig. 11-3, and there will always be
some of each component in each fraction. Thus, theoretically, the
true-boiling-point curve of all fractions in a given system would begin
and end at the same temperature. However, with reasonably good
rectification it is possible to obtain fractions that will give considerable
temperature gaps by an A.S.T.M. distillation. In the case of a low
degree of separation, the A.S.T.M. curves for two successive or adjacent
fractions may give initial and final temperatures that overlap.

RECTIFICATION OF COMPLEX MIXTURES

329

Lewis and Wilde Method. These authors (Ref. 2) applied the
Sorel-Lewis method described in Chap. 9 to complex petroleum frac-
tions employing the true-boiling-point curves combined with Raoult's
law. Data were obtained in a test on a fractionating column used in a
petroleum refinery. The plates in the column Were 9 ft. in diameter
and fitted with the usual type of bubble caps. A schematic diagram
of the unit is shown in Fig. 11-4, and some of their data are summarized
in Table 11-1. In the table, the column called " Average boiling
point'7 is the temperature at which the fraction as a whole boils and
not the average that would be obtained during an A.S.T.M. distillation.

Weight Per Cent Over
2.4 6 . 6 10 l£ . M. 16 (8 20 22 Z4

.0 6

96100

24- 32 40 46 06 H 72 60 86
Weight Per Cent Over

FIG. 11-5. True-&oiling-point curves of feed residuum distillate.

The true-boiling-point curves for the feed, the distillate, and the resi-
due are given in Fig. 11*5. The curves for the liquids sampled from
the plates are given in Fig. 11-6.

Lewis and Wilde's method consists of breaking the true-boiling-point
curve of the feed up into fractions boiling within narrow tempera-
ture limits. Thus the feed is divided into 10 or 20°F. fractions and
expressed as a component boiling between definite temperature limits,
such as 420 to 430°F. fraction which is present to the extent of 1.5
weight per cent. Such cuts are then used as individual components
by the methods given in Chap. 9. The true-boiling-point curve on
any plate in the tower is constructed from the calculations for that
plate, by -simply recombining the cuts in the proportion that the calcu-
lations indicate.

The vapor above the plate of an actual column is not in equilibrium
with the liquid leaving the plate owing to inadequate contact. In

330

FRACTIONAL DISTILLATION

JS88

I 8 § I I I

RECTIFICATION OF COMPLEX MIXTURES 331

TABLE 11-1. SXTMMABY OF DATA. OBSBBVBD AT BATTBBY AND IN LABOBATOEY

Item

Column
temp., °F.

Gravity,
°A.P.I.

Average
boiling
point, °F.

Molecular
weight

Rate,
gal. per
hr.

Feed to battery

38.6

28,920

Total gasoline produced . .

58.4

10,200

Gasoline from still 4

330

50.6

320

112

1,585

Feed to still 4

446

30.8

460

230

feesiduum from still 4

490

29.2

515

250

Kerosene from still 5
Liquid on plate 1

447

45.3
32.7

405
463

2,000

Liquid on plate 2

442

32.6

457

Liquid on plate 3

438

34.0

444

212

Liquid on plate 4

391

44.8

402

141

Liquid on plate 5.

373

46.9

388

140

Liquid on plate 6

370

47 5

380

Liquid on plate 7

360

47 8

377

Liquid on plate 8

358 -

48.1

372

Liquid on plate 9

343

48.8

360

Liquid on plate 10
Reflux to top plate

340

49.0
49.4

357
340

Vapor from still to bottom of
tower . . „ . . ...

485

43.2

419

150

Gravity of cold oil through partial condenser 38.6° A.P.L

Average rate of cold oil through partial condenser 10,600 gal. per hr.

Average temperature of oil into partial condenser . . . 77°F.
Average temperature of cold oil out of partial condenser. . . 181 °F.

Total steam in vapor from tower 100 gal. per hr.

Steam used in heating feed to tower 26 gal. per hr.

Barometric pressure 758 mm. of Hg

Pressure at bottom of tower 16 mm. Hg above barometer

Pressure at top of tower 23 mm. Hg below barometer

analyzing the behatior of an actual column, this plate efficiency must
always be included. Using a plate efficiency of 65 per cent, Lewis and
Wilde estimated the proportion of the 420 to 430°F. component on the
several plates above the bottom, and compared it with the actual
amounts found in the test as a measure of the accuracy of their calcu-
lations. This comparison is given in Fig. 11-7 where the curve repre-
sents the calculated concentration and the points the actual ones as
found. It will be noted that very satisfactory agreement was obtained,
indicating the utility of this method.

Graphical Method. An alternate method (Eefs. 1, 3, 4) has been
proposed by which the complex mixture is treated as a binary mixture

332

FRACTIONAL DISTILLATION

of components, consisting of the fraction above and below the tem-
perature 8/t which the cut is being made. The vapor-liquid equilibria
are constructed from the characteristics of the true-boiling-point analy-
sis or A.S.T.M. distillation curves, and the calculation is carried out
as in the McCabe-Thiele method.

Laboratory Studies of Complex Mixtures. Where laboratory space
and facilities are available, it is very wise to design petroleum equip-
ment on the basis of laboratory experiments, using the data thus
obtained as a starting point in calculations of the sort just indicated.

0.10

0.09
0.08
0.07

.1 0.05

420-430 1)eg.f>
Stock

\ i ' *

360-370 !)<&.<

i i

f''

0,02

01 23456 789 10
Number of Plate above bottom
FIG. 11-7.

It is believed that a rational analysis of laboratory data collected with
a thorough understanding of the requirements for subsequent calcu-
lations offers the safest method for the study of commercial problems.

An illustration of such laboratory data, taken by Smoley (Ref. 5), is
given in the following pages.

A large-scale laboratory column with 10 plates was operated with
total reflux, so that all of the distillate was returned to the top of the
column. Under this condition, with total reflux and no distillate, the
column was operating with maximum separation per plate.

A mixture of benzene and toluene was distilled in this apparatus,
and the composition of the liquid on the several plates determined,
with the results shown by the solid line in Fig. 11*8. The effect of the

RECTIFICATION OF COMPLEX MIXTURES

333

efficiency of the actual plate as contrasted with the theoretical plate
is shown in the same figure. The dotted line was obtained by plate-
to-plate calculations at total reflux using theoretical plates. Thus, to
produce a 90 mol per cent distillate requires about 10 steps in the
actual column, whereas the same effect is obtained in 6 steps in the
perfec^ column, indicating a plate efficiency of around 60 per cent.

Th^s sai&0 column was then operated in the same way but using a
mixture of benzene, toluene, and xylenes so as to produce as high a
concentration of benzene as possible in the condenser and to segregate
the xylenes in as concentrated a form as possible at the bottom. The

Condenser
Top PI ate
Plate $

7
6

6
4
3
Z

Plate J
Still

1.0

O.Z 03 0.4- 0.5 0.6 0.7 0.& 0.9

Mol. Fraction of Benzene
FIG. 11-8. Operation of benzene-toluene column.

results are shown in Fig. 11-9. The amount of xylenes was small so
that a large proportion of toluene was present on the bottom plate.
The highest concentration of toluene occurred on the fifth plate, this
component thus tending to segregate in the column, Under ordinary
conditions, a column would be operated at a lower temperature level so
that the benzene at the top would have contained less toluene, thus
delivering the toluene and xylene together from the bottom for subse-
quent separation in a second column. This experimental column had
insufficient plates to do this.

The column was then operated with total reflux on a cracked petro-
leum distillate obtained from a Winkler-West Texas crude oil. The
liquid samples from the several plates in the column were then analyzed
in a true-boiling-point still, being separated into components of 5°C.

334

FRACTIONAL DISTILLATION

boiling-point range. Each of these components was indicated by its
mid-temperature. Thus a component boiling on the true boiling-
point apparatus between 75 and 80°C. was called the 77.5°C. com-
ponent. The results of this experiment are given in Fig. 11-10, where
each component is indicated by a concentration curve.

It will be noted that each component tends to segregate in the col-
umn, the segregation point depending on its boiling point. This segre-
gation of a component in a continuous column is the basis for the type
of still frequently found in petroleum refineries where streams or cuts

JopPlcrh
Phte 9
8

5
4
3
2
Mate /

ft*

FIG. 11-9.

0 0.1 02 03 OA 0.5 0£ 0.7 0.6 09 1.0

Mot. Fraction of Coroponenf
Operation of column on benzene-toluene-xylenes mixture.

are taken from the central portions of the column as well as from the
top and bottom. It is evident from Fig. 11-10 that such side cuts
cannot be all pure or free from other components; in the commercial
column, where total reflux is n6t employed, the segregation is much
less pronounced than is indicated in Fig. 11-10.

The maxima ift concentrations of the fractions shown in Figs. 11-9
and 11-10 are definitely related to the volatility of the component in
question and the temperature in the distillation column. These curves
were obtained at total reflux, and for this condition the composition
of the vapor entering a plate is equal to the liquid leaving the plate for
all components, i.e.,

yn =» #n-fl

and

y* « K «a»

giving

RECTIFICATION OF COMPLEX MIXTURES

For the position in the tower at which a component is going through
a maximum concentration, the value of the liquid composition on sue-

Curve No. Ave. B.P cfCo/nporterti

/ ?7.5*C.
2 67.S*C.
J 97.5'C.
4 f07.59C.

6 I27.$°C.
8 Mf>C

Condenser

TopP/at*
Ha+e 9

3
2

f>/crht
WJ/

\--

^-^

1 /

J*"*"^

aoa ao4 0.06 aoa aio aiz O.M- o,i& aie

Mol Fraction <tf Component
FIG. 11-10. Operation of column on petroleum distillate.

cessive plates is approximately the same, making Kn = 1.0. Thus the
maximum occurs at the position in the column where the temperature
is such that the equilibrium constant of the component is equal to 1.0.

References

1. BROWN, Chem. Eng. Congress, World Power Conf,, 2, 324 (1936).

2. LEWIS and WILDE, Trans. Am. Inst. Chem. Engrs., 21, 99 (1928).

3. PETERS and OBBYADCHIKOV, Nestyanoe, Klozyaistro, 24, 50 (1933),

4. SINGBB, WILSON, and BROWN, Ind. Eng. Chtm,, 28, 824 (1926),

5. SMOLEY, Sc.D. thesis in chemical engineering, M.I.T., 1930.

CHAPTER 12

ALTERNATE DESIGN METHODS
FOR MULTICOMPONENT MIXTURES

In Chap. 9 the Lewis and Matheson procedure for SorePs plate-to-
plate method was presented. Many other design methods have been
proposed based on alternate methods of analysis or approximations.
None of them illustrates the phenomena involved in multicomponent
rectification so well as the Lewis and Matheson ihethod. A number
of the methods require less effort to obtain certain design factors than
the stepwise procedure and are useful in cases where similar systems are
to be analyzed repeatedly. When a new type of problem is to be con-
sidered, the information obtained by the plate-to-plate method is well
worth the effort involved. Actually a detailed analysis by methods of
Chap. 9 does not usually require over a few hours, and the confidence
in the result and the insight obtained of the operation justify the
effort involved.

The space available in this text does not allow a detailed analysis of
these various design methods, but a brief review will be given of some
of them. A number of the methods involve assumptions that are not
justified in many cases, and the design engineer must appreciate these
limitations or misleading results will be obtained.

PLATE-TO-PLATE METHODS

In addition to the Lewis and Matheson, and Lewis and Cope meth-
ods given in Chap. 9, plate-to-plate procedures have been given by
Thiele and Geddes (Eef. 14) and Hummel (Ref. 9).

The Thiele and Geddes method is a stepwise procedure based on
using a ratio of the concentration of a component to its terminal con-
centration. Starting at the top of the column, for any component,

XT — %£- = «?• (for total condenser)
/Vr /ir

336

ALTERNATE DESIGN METHODS 337

and

XD
In general,

j/n __ Qn+1 /Wl i\ , 1

— • x t -j— j.

XD Vn \XD /

and

? = 7TT- (12-3)

XD Jtv nXo

Thus, if the values of the equilibrium constants and the ratios of
0/V are known for each plate, it is possible to calculate the ratio
XH/XD for any plate above the feed plate without knowing the value

ofXD.

Below the feed plate a similar analysis can be made.

Xw
and, from the operating line,

— = ^ ( — - 1 ) + 1 (12-4)

xw 0i \xw / v '

in general,

jfe « A

Xw /

+ 1

The calculations can be carried up from the still and down from the
condenser to the feed plate, giving values of ym/xw and yn/xo. Assum-
ing that the feed plate is such that the vapor and liquid leaving are in
equilibrium, then the values of the vapor composition in the two ratios
must be equal and the value of XD/XW can be calculated for each com-
ponent. This ratio can be used to calculate D and W.

338 FRACTIONAL DISTILLATION

Thus, for each component,

DXD + Wxw = FzF

Wxw * — ^

1 4- F~D txw

"*" D
Summing the Wxw and Da:i> terms for all components,

i —

(12-7)
(12-8)

(12-9)

where S indicates the sum of the terms for all components. With the
value of XD/XW for each component, W/F or D/F can be evaluated.
As compared to the Lewis and Matheson method, this method has
the advantage that it is easier to calculate the separation to be obtained
for a given number of theoretical plates at a specified reflux ratio. In
case the separation and reflux ratio are specified, the Lewis and
Matheson method is the easier to apply.

Thiele and Geddes Calculation for Benzene-Toluene-Xylene Separation. To
illustrate the application of this method, consider the separation of the benzene-
toluene-xylene mixture of page 219 in a tower having five theoretical plates with
the feed entering the middle plate. The reflux ratio 0/D will be 2.0, Vn * 7Wj
the vapor leaving the feed plate will be in equilibrium with the liquid leaving, and
the usual simplifying assumptions will be made.

Solution. Basis: 100 mols feed. 0/D « 2, (0/V)n - 0.667.

Mols
feed

Distil-
late
comp.

Assume
T -
85°C.
KT

Xr = *
XD KT

3K£=1 . 0.667 ( ^ -A+l
XD \XD J

Benzene. . . .

0,60

XDB

1.15

0.87

0.913

Toluene ....

0.30

XDT

0.452

2,21

1.807

Xylene

0.10

XDX

0.184

5.44

3.96

ALTERNATE DESIGN METHODS

339

Assume
T - 85°C.
Kr-i

XT-I ^ 1 (yT-i\

XD ** /Cr-i \ #£> /

yr-a
a?z>

Assume
T - 90°C.
•Kr-2

sr-a

«D

gr-t
JCD

1.15

0.793

0.862

1.38

0.649

0.166

0.452

3.99

3.0

0.533

5.63

4,09

0.184

21.5

15.35

0.221

69.4

46.6

From still up, assume D « 66; then Vn » 198 - Vm,0m ** 232, (FW/0TO) - 0.853.

C6
Cr

Assume
T - 115°C.

Xw ' \Xw )

Assume
T = 110°C.

Assume

T m 110°C.

2.62

1.12 r

2.38

1.102

2.29
0.975

5.45
1.075

4 8
1 064

2.29
0.925

0.513

0 587

0.435

0 255

0.365

0.435

2/2

a?8

Assume T * 100°C.

11.0

9.53

1.745

16.65

1 038

1.032

0.735

0.76

0.159

0.282

0 316

0.104

Equating y9 =* 2/r-s gives

By Eq. (12-9),

1 «

0,6

21.8
0.186
0.0022

0.3

-f;

0.1

~20.8(ff /F) -f 21.8 ^ 0.814(TT/F) + 0.186 ' 0.9978(TT/F) + 0.0022

- 0.334

The section below the feed was calculated on the basis W/F «» 0.34 compared
to the calculated value of 0.334, and it will not be rechecked. This is one of the
difficulties with the Thiele and Geddes method, i.e., specifying the reflux ratio and
feed condition still leaves trial and error for both the plate temperatures and the
ratio of 0/V below the feed plate.

340 FRACTIONAL DISTILLATION

The temperature assumptions will now be reviewed.

XD, by Eq. (12-8)

#r-i

ZT-I

sr-s

C6
C7

0.88
0.12
0.00065

0.765
0.265
0.0035

0.803
0.217
0.0025

0.698
0.479
0 014

0.571
0.676
0.045

1.0335

1.0225

1.191

1.292

The fact that the sum of XT is greater than 1.0 indicates that the actual tempera-
ture is slightly higher than 85°C. The assumed temperatures for both plates T — 1
and T— 2 are considerably low as indicated by the sums of XT-I and XT-Z being
greater than 1.0, and the results should be recalculated for a satisfactory design.

Below the feed plate,

y«

2/1

2/2

2/3

Cr
C8

0.042
0.658
0.300

0.11
0 737
0.153

0.229
0.707
0.076

0.462
0.693
0.048

0.699
0.50
0.031

1.000

1.012

1.203

1.230

The assumed temperatures for the still and first plate are satisfactory, but the
assumed temperature of plate 2 is too high. The sum of 3/3 is larger than 1.0, and
this would appear to indicate that the assumed temperature was much too high,
actually most of the excess is from plate 2. The fact that Sy2 > 1.0 made
Sa?a > 1.0 and, even if the assumed temperature for plate 3 were correct, Sg/s will
be greater than 1.0. Thus an error in the assumed temperature for one plate
carries through succeeding plates. It is obvious that the calculation requires con-
siderable trial and error.

A method similar to the Thiele-Geddes method has been proposed
by Hummel (Ref. 9). In this method plate-to-plate calculations are
made for a few plates at each end and around the feed plate to estab-
lish the temperatures. With the values and the known number of
theoretical plates the temperature gradient in the tower is drawn.
This then gives the temperatures to employ in a Thiele-Geddes type of
calculation. For a given number of plates and a given reflux ratio,
this method requires an estimation of the distillate, bottoms, and feed-
plate compositions. Basically, Hummel's method furnishes a sys-
tematic method of successive approximations for the plate tempera-
tures to be used in evaluating the equilibrium constants.

ALTERNATE DESIGN METHODS 341

REDUCED RELATIVE VOLATILITY METHODS

Underwood (Ref. 15) and Gilliland (Ref. 5) have proposed design
methods for applying a total reflux type equation with a reduced rela-
tive volatility. For the enriching section of the tower,

-a,

where T refers to top plate. For a total condenser,

XB/D XB/T

and, by the operating material balances,

=
i 1 + (D/0)(XBD/XST) \XB

= fr-i(fO (12-10)

\XB/T

and

where

«/+A ( «/+_i\ ( ^A (%A\

/W \ /»/W V /s// W/

Mn " 1 + (D/On)(xBD/XBn+l)

and below the feed plate,

xj\ ctf-i(af-2\ (ai\(ai\(<x w\ (x A

A"S^WV WWWWr

where

To simplify the calculation, average values of a and 0 are employed,

where ATn ^ number of theoretical plates above feed plate

JVTO « number of theoretical plates below feed plate including
feed plate

342 FRACTIONAL DISTILLATION

Arithmetic averages have been employed for a and /3

a, + (q/0),

2 (12-17)

(12-18)

The values of £/ and /9/_i involve the composition on the feed plate
and the plate above. The feed-plate composition is determined by
assuming that the components more volatile than the key components
are negligible in the bottoms, that those less volatile are negligible in
the distillate, and that the concentration of both the light and heavy
components are asymptotic at the feed plate.

Thus, for the more volatile components,

Vy/ = Oxf+i + DxD
Using y/ = Kfxf and x/+i = x/,

VKfXf = Ox/ + DXD

Xf -

(KfVn/On) -
Similarly, for the less volatile components,

(W/Om)xw

The values of x/+i needed for the calculation of ft/ are obtained by
stepwise calculation from #/. For approximate values, £/ can be cal-
culated with Xf instead of x/+i.

Reduced Relative Volatility Calculations for Benzene-Toluene-Xylene Separa-
tion. The benzene-toluene-xylene example of page 219 will be solved by this
method.

Solution. Basis: 100 mols of feed. (See page 220 for design quantities.)
Estimation of feed-plate composition. By Eq. (12-20) the mol fraction of xylene
in the feed plate is

„
Xfx

1 - (Kf-iVJ/OJ

10/220.2
* 1 - Kf-i (180.3/220.2)

Assuming Kf^ ~ K/ - 0.22 (t - 90°C.),

x/x •» 0.0555

ALTERNATE DESIGN METHODS

343

This compares with a value of 0.058 obtained by the stepwise calculations, page
223. A better check would be obtained using 2C/-i.
By mol fraction balances,

X/B + XfT - 1 - 0,0555 - 0.9445
y/js + y/T - 1 ~ 0.22(0.0555) - 0.9^8

Using KfB - 1.33, KfT * 0.533 (t - 90°C.),

1.332/s -h 0.533s/r - 0.988

S/B •" 0.609 and X/T =* 0.336

By plate-to-plate calculation,

3/+1

C6
CT
C8

0.609
0.336
0 0555

1.33
0.533
0 22

0 811
0.179
0 Oil

0.718
0.267
0.016

1 001

For benzene relative to toluene

0.995

2(0.718) _

-.
"" 2(0.267)

_ 1 - (39.9/180.3) (0.005/0.609)
/-I i _ (39.9/180.3) (0.744/0.336)

From page 223,

« 2.63

= 2.36
- 2.49

~^
1.69

2.49

fa\ __ 1.

Wnav "

.47 -f 2.63

- 2.05

1.81 + 2.36

1.84

By Eq. (12-15),
By Eq. (12-16)

r 4- l - lQg (0.995/0.005) (0.336/0.609) ^ Q 6
Ar log (0.609/0.336) (0.744/0.005) o 9

J\ m as r- — • » \J,6

344 FRACTIONAL DISTILLATION

N «• Nn + Nm =* 14.8 theoretical plates vs. 16 found on page 225 by stepwise
calculation. Because («/£)/ and (a/£)/-i are near to 1.0, the geometric mean
should be more conservative.
By Eq. (12-21),

- 1 + V0.47(1.63T - 1.87

\M/n av

and

- 1 + V0.31(1.36) = 1.65

Nn + I - 7.5

Nm • 11.2
Nn + Nm = 17.7 theoretical plates

The xylene in the distillate can be estimated by Eq. (12-15).
For toluene relative to xylene,

j , 0.005

2(0.267) _1Q

XDX
).0555)

O /iQ I O /4O

- 2.45

z z

i™ m nn?i x^^^/n nKKK /n 9
7.5

2 2

log (0.005 /gpy) (0.0555/0.336)

log 2.45
» 0-005(0.0555) ^ 10^6
^DX " 830(0.336)

An approximate check on the assumed feed-plate temperature can be obtained
by assuming that the temperature gradient is linear from the still to the condenser.
From pages 222 and 225, tw - 116°C. and to = 80°C.

7-5 (116 - 80) + 80

" 18.7
- 94.4°C.

as compared to the assumed temperature of 90°C.

The results given by these equations are only approximate, and
their accuracy increases as the reflux ratio, 0/D, increases. For reflux
ratios near the minimum value, /?/ becomes equal to a, and the equa-
tions should be applied with caution because they give too few the-
oretical plates under these conditions. In fact, owing to the method
of obtaining (a/£)av, they can indicate a finite number of plates at
values of the reflux ratio less than the true minimum. For values of
(a/ ft) near to 1.0 a better average is obtained by

ALTERNATE DESIGN METHODS 345

These averages force the equation to give an infinite number of
plates for (a/ ft) = 1.0.

ABSORPTION FACTOR METHOD

Brown and Souders (Ref . 3) suggested the use of the absorption fac-
tor method of Kremser (Ref. 11) as a design procedure for multicom-
ponent mixtures.

By a material balance on one component starting at the bottom of
the column,

where S - KV/0.

V (V \

xs = -Q 2/2 - r -Q yw - xil

™ yw - 2/TT -

and

— ( Q 2/TT - XlJ

Assuming S is a constant and letting Nm equal the number of theo-
retical plates below feed,

{ + s 4. $2 + . . . + ##,») (12-23)

__ z

+ (1 + S + S* + • • • + S»*-i)

346

FRACTIONAL DISTILLATION

(1 + 8 +

Xf """"" X\

I V*

8(1 + S +

1 + S + S* +

(12-24)

Multiplying the numerator and denominator of the right-hand side
by 1 — 8 gives

xf - xi _ SNm+1 - S

xf - yw » -

A similar analysis above the feed plate gives

y/-Vr _ AN^ - A
Vf -

(12-25)

(12-26)

where XR *= composition of reflux to top of tower
A = 0/KV

Equation (12-26) is applied to the heavy key component above the
feed plate, and Eq. (12-25) to the light key component below the feed
plate.

An average value of K is employed and, above the feed,

K av =

below feed plate,

Absorption Factor Calculation for Benzene-Toluene-Xylene Separation. This
method will be applied to the benzene-toluene-xylene example (data from pages
220 and 343).

»-«.

*,+,

3/TT

JTr

K«

— .

—

0.0131

0.0116

0.605

2.63

1.52

0.005

0.38

0.48

0.180

— —

— .

—

•— "

• —

ALTERNATE DESIGN METHODS 347

Above feed plate for toluene,

„ 0.38 + 0.48

/v.v = - g -

A ° 2 1 „

A * TV - 3(043) = IM*
By Eq. (12-26),

0.18 - 0.005 1.55***1 - 1.55

0.18 - 0.38(0.005) " 1.55"n+i - 1
Nn -f 1 - 7.95
Nn - 6.95

Below feed plate for benzene,

_ 2.63 + 1.52

Aav sss - S - -6.U/

5 - 2.07 ("%ao) - 1.7
By Eq. (12-25),

0.605 - 0.0116 i.7*m+i - 1.7

0.605 - (0.0131/2.63) 1.7^*»+l - 1
tf« + 1 - 9.3
#m = 8.3

Total plates « 7 + 8.3 + 1 - 16.3.

Edminster (Ref. 4) has presented a modified absorption factor
method that determines the molal quantities for each component as a
fraction of their values in the distillate and bottoms in a manner some-
what similar to the Thiele and Geddes equations. The geometric
mean of the absorption and stripping factors at the ends of the section
under consideration is employed, and empirical correction terms are
applied to these averages.

GRAPHICAL CORRELATIONS

When the number of theoretical plates is plotted as a function
of reflux ratio, the curve is hyperbolic in type with asymptotes at
Nnn and (0/Z))min. These two limiting conditions as asymptotes are
useful in drawing such a curve, but they would be more helpful as defi-
nite points on the diagram. By modifying the variables, they can be
made definite points; in fact they can be made the same points for all
cases. There are many ways in which the variables can be modified,
and one that has been useful (Ref. 6) is shown in Fig. 12-1. The ordi-
nate is (S — Sm)/(S +1), where S is the total theoretical steps includ-
ing any enrichment in the still and condenser, and Sm is the value of S

for total reflux, 0/D =00. The abscissa is | - +

348

FRACTIONAL DISTILLATION

At total reflux the ordinate is 0.0 and the abscissa is 1.0, while at the
minimum reflux ratio the ordinate is 1.0 and the abscissa is 0.0. As
the reflux ratio is increased from the minimum to total reflux, a given
design problem will give a curve that goes from 1,0 to 0,1. It was
expected that a series of curves between these two points would be
obtained, depending on (1) the degree of separation, (2) the relative

!0r

0.8

0.6

0.4

0.2

0.4

tO/

f •

FIG. 12-1. Graphical correlation for design calculations.

volatilities, and (3) the components lighter and heavier than the key
components. The results of plate-to-plate calculations were plotted
and gave a narrow band which could be reasonably represented by a
single line. It can be shown theoretically that a single line cannot
represent all cases exactly, and the correlation can be improved by
using more than one line. For example, the position of the line is a
function of the fraction of the feed that is vapor. The best line drawn
through the all-vapor feed cases on a plot such as Fig. 12-1 is lower
than the corresponding line for all-liquid feeds. It is also possible to

ALTERNATE DESIGN METHODS

349

improve the correlation by changing the variable groups, but it is
doubtful whether the increased accuracy justifies the added complica-
tions. The accuracy of such a correlation will always be limited by
the errors in 8m and (0/D)^ It is believed that it is of real value
when it is applied as (1) a rapid but approximate method for pre-
liminary design calculation or (2) a guide for interpolating and extra-

I.U

08
0.6

04
0.3

•»•

C/5

0.1
008

0.06

0.04
0.03

0.02
0

SSBMMM.

— «••

BBSS

—

-«—

•— 1

•«,

— «

— •

"^1

""^1^^

"Si,

01 0,02 0.03 0.04 0.06 0.08010 02 03 0.4 0.6 OB 1.

JB..IS
0

Fio. 12-2.

WofO

Graphical correlation for design calculation.

polating plate-to-plate calculations. In this latter case, if only one
plate-to-plate result is available at a reflux ratio from 1.1 to 2.0 times
(0/D)min, this point can be plotted on the diagram and a curve of simi-
lar shape to the correlation curve fitted to it. Such a method should
give good results for other reflux ratios, assuming the values of Sm and
(0/D)min are reasonably accurate.

Use of Graphical Correlation for Benzene-Toluene-Xylene Separation.

ing this correlation to the benzene-toluene-xylene example:
From page 259, (0/D)mi» - 1.0.

Apply-

350 FRACTIONAL DISTILLATION

Calculation of &»in. From Fig. 9-7, page 233, for benzene relative to toluene,

as « 2.6, aw * 2.3

«av - V2^(2T) - 2.45

0.00fi/\0.005 _ .
.4

(0/D) - (0/D)m 2.0 - 1.0

(0/D) + 1 " 2.0 4- 1.0 - °'333

From Fig. 12-1,

^~Y - 0.32

8 - 17.2

N « 16.2 theoretical plates

This compares with 16 plates as determined by the stepwise procedure.

An interesting fact is that the commonly used design reflux ratios of
1.2 to 1.5 times (0/D) mi* usually correspond to values of (S — Sm)/
(S + 1) from 0.4 to 0.6, and because S is usually large in comparison
to 1.0, a rough working rule is that the number of theoretical steps
required at the optimum reflux ratio will be twice the number needed
at total reflux. This is a rule that can be applied for orientation pur-
poses and requires only an estimation of the number of theoretical
plates at total reflux.

The correlation also illustrates why the economic reflux ratio is usu-
ally so close to (0/Z))min. A small initial increase in the reflux ratio
group above 0.0 makes a large decrease in the theoretical plate group,
but further increases become less effective.

A modification of the above correlation has been suggested by
Schiebel (Ref. 12), and a somewhat different graphical correlation has
been published by Brown and Martin (Ref. 2).

MODIFIED EQUILIBRIUM CURVES AND OPERATING LINES
Several methods have been given which treat a multicomponent
mixture as a modified binary mixture of the key components that can
be analyzed graphically on a y,x type diagram.

Jenny (Ref. 10) published a graphical method for multicomponent
design calculation. A few plate-to-plate calculations are made at the
top and bottom of the column and above and below the feed plate.
For the section below the feed plate, a yfx diagram is made for the light
key component using the calculated values to place the effective equi-
librium curve for this component on the diagram. The operating line
is drawn in the usual manner and the plates determined by the stepwise

ALTERNATE DESIGN METHODS 351

procedure. Above the feed plate a diagram is constructed for the
heavy key component in the same manner.

Hengstebeck (Ref. 8) developed a graphical method for multicom-
ponent mixtures .which employed the key components on a binary-
type diagram. The mol fractions for the distillate and bottoms were
calculated on a key component basis

r/ _ %wik

xwlk —

XWlk + Xwhk

and the equilibrium curve for a binary mixture of the key components
is employed. The operating lines are corrected for the quantity of the
light and heavy components present.

Schiebel and Montross (Ref. 13) developed a method for making
multicomponent design calculations on the basis of a pseudobinary
mixture. A modification of this method by Bailey and Coates (Ref. 1)
will be reviewed.

The analysis is made on a key component basis, and plate-to-plate
calculations are made above and below the feed plate to determine
xf+i> £/+i> 2//> x'f, if) and 2//_!. The feed-plate composition is obtained
by the method given on page 342. The operating line in the stripping

section is drawn as a straight line through y'wlk = x'wik = —^ and

Xwlk + Xwhk

i x/ik t y(f—i)ik j • M j

Xflk — 9 2/</_i)Zfc = i 9 an(* similar procedures

are used to calculate x'Dik, y'ftk, and x[f+l)ik, where the primed mol frac-
tions are on a key component basis. The relative volatility of the key
components is plotted vs. xr using the values at, x'D and a/+i, x'f+i to
give a curve for volatility above the feed plate and aw, x'w and a/, x'f
to give a curve below the feed plate. These relative volatility curves
are used to calculate the yf vs. x' equilibrium curves by

(a - IX + 1

In general, the equilibrium curves calculated from the two curves
do not match exactly, and each is used for its respective portion of the
column. Plates are stepped off between the operating lines and equi-
librium curves in the usual manner.

Bailey and Coates (Ref. 1) also modified the Schiebel and Montross
method of calculating the minimum reflux ratio for a multicomponent

352 FRACTIONAL DISTILLATION

mixture. They give

<*">

|W«r+ «u)J

(12-27)

where (0/D)'M is the pseudo-minimum reflux ratio.

This equation appears somewhat similar to Eq. (9-21), and the
nomenclature is the same. (0/D)fM is calculated by

(1228}

( ]

where x'D,

= mol fractions on key component basis for distillate and

at intersection of pseudo operating lines
= average relative volatility

m)[(0/D)'M

- x'D

(0/D)'U

(12-29)

where m ==

ML -

Mv -

= mol fraction on key component basis for feed

= mols of liquid in feed

= mols of vapor in feed

= mols of components heavier than heavy key component in

feed
= mols of components lighter than light key component in

feed

Values of x'D are calculated from the terminal conditions, and x[ and
(0/D)'M obtained by simultaneous solution of Eqs. (12-28) and (12-29).
The calculation of (0/D)'M is frequently laborious, and Eq. (9-21) is
easier to use.

Solution of Benzene-Toluene-Xylene Example by Modified Equilibrium Curve
Method. The use of this method will be illustrated by the benzene-toluene-xylene
example. Benzene and toluene are the key components and the necessary feed-
plate calculations were made fox this example on page 343.

The results are summarized for the key components:

s/+i

y/-i

C6
C7

0 609
0.336

0.811
0.179

0.718
0.267

0.739
0.246

- 90°C.

/-i - 95°C.

ALTERNATE DESIGN METHODS

353

f.o

0,8

8.
5

I 0.6
.8

c
o

I 04-

0.2

02 04 06 08

x'-pseuclo mol fraction of benzene in liquid

FIG. 12-3.

Z.O

Q>
£

2.0,

^*-

fr ii**

.^ -"•

^— —

.^ — — -

^ — —

) 02 0.4 0.6 0,8 I.
x', pseudo mol fraction

FIG. 12-4.

354 FRACTIONAL DISTILLATION

Coordinates for benzene for lower operating line,

0-°°67

, 0.739 _
'/-' ' 0985 = °'75

' °'609 n <u*
*' ~ 0945 - a645

Coordinate for enriching line,

/ . 0.995

0.995

0.718 7
0985 " °'73

Straight lines are drawn through these points on Fig. 12-3. The relative volatil-
ity values corresponding to at, «/+i, «/, and aw were plotted as a function of xf in
Fig. 12-4. These values were used to calculate the equilibrium curve of Fig. 12-3.
The steps are made on this diagram in the usual manner, and the procedure gives
between 16 and 17 theoretical plates. This method is particularly good in this
case because the components other than the key components are present only in
small quantities.

The minimum reflux ratio by this method is obtained as follows. By Eq. (12-28),

By Eq. (12-29),

m~— o °°

100 - 10
m =»

, 0.6

Xj, »

By Eq. (12-27),

, --0*7 and

,1Q1 , 3MF 0.4(01) 1
n ^ 60.6 [2.34 - 0.4 J

- 1.02

This compares closely with the values given on page 259.

ANALYTICAL EQUATIONS

Exact mathematical equations for the case of constant molal over-
flow rate and constant relative volatilities have been presented by
Harbert (Ref. 7) and Underwood (Ref. 16). Underwood's equation
for a three-component mixture can be arranged as follows:

ALTERNATE DESIGN METHODS 355

Above feed plate,

XDI _ [ 01 ] * XFI
Xpt \02/ Xpt

Xp9 _ I 03 \ " ^Ty8
Xpl \0i/ XFl

Similar ratio equations can be written for any two components of a
multicomponent mixture where

i B i cpC .

— —

OLA — 0i
XD = distillate composition
XD» XDa = XDl with 0i replaced by 02 and 03, respectively

XF = ^D with XDA, XDBK XDC replaced by XFA, XFB, and XFC,

respectively

XF = feed-plate composition
Nn = number of plates above feed plate
0i, 02, 03 = roots of the equation

D D D

- XDA Ois -r- XDB Oic -- XDC

OLA — 0 OJJ3 — 0 «C — 0

(12-31)

The roots will be bracketed by the relative volatilities; thus, for a
three-component mixture,

CiA > 01 > OtB, OiB > 02 > «C, <*C > 02 > 0

Below the feed plate similar equations are obtained.

~w~ ~~ i ~77 / V' (12-32)

AFa \02/ ^TF,

Similar relations can be written for the other components where

X'F — XF except 0' used instead of 0

X'w = Xp with XDA, XPB, XDC replaced by XWA, XWB, Xwc, respec-
tively, and using 0' instead of 0

Nm — number of theoretical plates, feed plate and below
0i> 02> 03 = roots of the equation

WWW
otA TV- XWA OLB fr- XWB etc TF~" Xwc

356 FRACTIONAL DISTILLATION

In many cases, the use of these equations is complicated by the fact
that a trial-and-error procedure is involved. If the terminal composi-
tions are known, the trial-and-error operation involves matching at
the feed plate. Usually the terminal conditions are not completely
known, and additional trial and error may be required. In most cases
a three-component problem can be solved just as rapidly by the usual
stepwise procedure, and variations in the relative volatility can be
included.

Underwood has used these equations to calculate the minimum
reflux ratio.

t + lss sfzrg (12"34>

where 0 is the root of the equation.

_j _| _|_ . . . a- 1 -J- p (12-35)

CtA — V &B — 0 OLc — 0

where p = (0» - Om)/F.

This equation has several roots. The one employed in Eq. (12-34)
is the value lying between aik and othk-

Solution of Benzene-Toluene-Xylene Example by Analytical Equations. These
equations will be applied to the benzene-toluene-xylene example. Relative vola-
tilities for the three components will be used as as - 2.45, <XT ~ 1.0, and ax =0.4.
By Eq. (12-31),

2.45(0.995/3) , 0.005/3 ^ .

2.45 - 0 + 1 - 0
2.45 > 0i > 1.0, 1.0 > 02 > 0

The term for xylene is neglected because its concentration in the distillate is
unknown, and it will make little difference in the values of 0i and 02. Solution
of this equation gives

0i - 1.64, 02 = 0.9963
By Eq. (12-33),

2.45 (0.005) (39.9/180.3) 1.0 (0.744) (39.9/180.3) 0.4(0.251) (39.9/180.3) _ .
2.45 - 0' 1-0' 0.4-0'

0J > 2.45, 2.45 > 02 > 1.0, 1.0 > 0J > 0,4
0'8 - 0.4173, 02 - 1.169, 0J « 2.45308

Using these values for the key components,

2.45 - 2.45308 " 1 - 2.45308 "" 0.4 - 2.45308
- 0.1952**

ALTERNATE DESIGN METHODS 357

It is obvious that the first term will be the important factor.

X'Wi « -795(0.005) - 0.688(0.744) - 0.195(0.251)

1 = -4.64

v' _ 2.45xfa , XFT , , 0.4^^
AF* ~ 2.45 - 1.169 "^ 1 - 1.169 "*" 0.4 - 1.169

« I.QI&XPB — 5.92a*r - QM9xFx
X'w% - -4.53

By Eq. (12-32),

2.45 \y- /-4.64\
l-169/ V-4.53/

- Q.G88XFT - 0.

By a similar procedure for XFl and XF8,

/ 1.169 V^ / -4.53 \
\0.4173/ V-4.514/

-f 1.

With XFB + XFT + ZFX = 1.0, there are three equations and four unknowns.
By using two similar equations for the section above the feed, only one additional
unknown, Nn, is introduced, and the equation can be solved. However, with
XDX unknown, only one equation is available. XDX is small, and to satisfy Eq.
(12-31) it can be shown that <fo = 0.4 — 0.334#z>;x-, and the correction term
— 0.334o?i>jr can be neglected in all cases except the (<£3 — 0.4) term. This intro-
duces an additional unknown, and an additional specification can be added such
as the ratio of the key components on the feed plate or that Nn + Nm is to be a
minimum. The trial-and-error solution for the latter case is time-consuming;
instead, the feed-plate compositions obtained by the stepwise procedure on page
223 will be employed; XPB =• 0.605, XPT = 0.336, and XPX * 0.058.

For ratio of 1 to 2, the relation reduces to

-4.53

/ 481 \
V-0.86/

1.169/ -4.64

Nm = 8.5 theoretical plates
For ratio of 2 to 3,

/ 1.169 YM _ 4.514 / 0.86 \
V0.4173/ 4.53 V0.0447

Nm - 2.9

The last value for Nm is very sensitive to the value employed for XFX, and a
value of 0.0562 for xylene at the feed plate instead of 0.058 would make the last
equation give Nm — 8.5. This value of Nm « 8.5 compares with nine plates by
the stepwise procedure. The difference is due to the fact that the relative vola-
tility in the lower section of the tower averaged less than 2.45.

A similar calculation for the upper section using the same feed composition gave
Nn + 1 - 8.6, Nn - 7.6 or total plates,

N - Nn -f Nm - 8.5 4- 7.6 * 16.1
as compared to 16 plates by the Lewis and Matheson method.

358 FRACTIONAL DISTILLATION

For the minimum reflux ratio, by Eq. (12-33),

2.45(0.6) 1(0.3) 0.4(0.1)
2.45 - 0 ~l" 1^1 "*" 0.4 - 0 * U

The desired $ is between 1 and 2.45.

e - 1.25

2.45

a ' 2.45 - 1.25

= 2,02

(£)-. - ^

This compares closely with the values given on pages 259 and 354.

SUMMARY

The Lewis and Matheson method appears to be the most satisfac-
tory method of handling the general multicomponent design problem.
In most cases, it requires relatively little trial and error and will handle
cases of normal and abnormal vapor-liquid equilibria. It is especially
well suited to the cases in which the reflux ratio and the separation of
the key components are specified and the problem is to determine the
number of theoretical plates and the component concentrations in the
column. In some specific cases, other methods may have advantages,
but unless a number of problems of the same type are to be handled, it
is more desirable to have one method that will apply to essentially all
cases.

The Thiele and Geddes method is advantageous when the number
of theoretical plates and the reflux ratio are specified and the calcula-
tion of the separation is desired. Even in this case, the trial and error
involved in obtaining the proper equilibrium constants for each plate is
formidable.

The reduced relative volatility and absorptipn factor methods are
rapid but can be appreciably in error because of the approximations
involved. If the design engineer understands their limitations, they
can be useful. In cases involving abnormal vapor-liquid equilibrium
conditions or at reflux ratios near to the minimum, these methods may
be so in error that the results are of little value.

The methods based on modified equilibrium curves and operating
lines are simple to use and can give satisfactory results for many cases.
They do not give information on the light and heavy components
without considerable additional effort. In general, they are not satis-
factory for mixtures with abnormal vapor-liquid equilibria.

ALTERNATE DESIGN METHODS 359

The analytical solutions for the case of constant molal overflow and
constant relative volatility are a mathematical accomplishment, but
they do not appear to be well suited for the average design problem.
The greatest contribution of these methods will probably be as an aid
in studying the process of rectification, particularly the minimum reflux
ratio condition.

The graphical correlations are useful in obtaining approximate
answers rapidly. They are not applicable to all cases and can be
seriously in error under some conditions.

The various approximate methods are helpful for orientation pur-
poses, but the greater confidence that the design engineer can place in
the rigorous plate-to-plate calculation justifies any greater effort
involved.

Nomenclature

A = absorption factor - On/KVn
D — molal distillate rate
F «• molal feed rate
K = equilibrium constant =» y/x
m — ratio, see page 352
ML = mols of liquid in feed
My = mols of vapor in feed

^MHF = mols of components heavier than heavy key component in feed
XMiF — mols of components lighter than light key component in feed
Nn = number of theoretical plates above feed plate
Nm = number of theoretical plates feed plate and below
0 = molal overflow rate
8 «* stripping factor « KVm/Om
t •» temperature
V = molal vapor rate
W = molal bottoms rate
X = mol fraction in liquid
X « composition factor in Eq. (12-30)
a/ = mol fraction in liquid based on key components only
y «= mol fraction in vapor

yf = mol fraction in vapor based on key components only
a « relative volatility
0 =» relative operability
<M' - roots of Eqs. (12-32) and (12-33)
0 « root of Eq. (12-35)

Subscripts:

A refers to component A
B refers to benzene or component B
Df refers to distillate
F refers to feed

360 FRACTIONAL DISTILLATION

f refers to feed plate

h refers to heavy component
hk refers to heavy key component
I refers to light component

Ik refers to light key component

n refers to section above feed plate

m refers to section below feed plate

T refers to toluene or to top plate
W refers to bottoms

X refers to xylene

References

1. BAILEY and COATES, Petroleum Refiner, 27, 30, 87 (1948).

2. BROWN and MARTIN, Trans. Am. Inst. Chem. Engrs., 35, 679 (1939).

3. BROWN and SOUDERS, Trans. Am. Inst. Chem. Engrs., 30, 438 (1933).

4. EDMINSTER, Chem. Eng. Progress, 44, 615 (1948).

5. GILLILAND, Ind. Eng. Chem., 27, 260 (1935).

6. GILLILAND, Ind. Eng. Chem., 32, 1220 (1940).

7. HARBERT, Ind. Eng. Chem., 37, 1162 (1945).

8. HENGSTEBECK, Trans. Am. Inst. Chem. Engrs., 42, 309 (1946).

9. HUMMEL, Trans. Am. Inst. Chem. Engrs., 40, 445 (1944).

10. JENNY, Trans. Am. Inst. Chem. Engrs., 36, 635 (1939),

11. KREMSER, Nat. Petroleum News, 22, No. 21, 48 (1930).

12. SCHIEBEL, Ind. Eng. Chem., 38, 397 (1946).

13. SCHIEBEL and MONTROSS, Ind. Eng. Chem., 38, 268 (1946).

14. THIELE and GEDDES, Ind. Eng. Chem., 25, 289 (1933).

15. UNDERWOOD, J. Soc. Chem. Ind., 52, 223 (1933).

16. UNDERWOOD, J. Inst. Petroleum, 32, 614 (1946).

CHAPTER 13
SIMULTANEOUS RECTIFICATION AND CHEMICAL REACTION

In most rectification systems, chemical reactions among the com-
ponents to form new stable compounds do not usually occur, but in a
few special cases such a condition is involved. In some cases, the
combination of rectification and reaction is beneficial as in the prepara-
tion of esters; while in other cases, it may be detrimental in that it
decreases the effectiveness of the separation or the yield of the desired
component.

In the preparation of esters such as ethyl acetate from acetic acid and
ethanol, the equilibrium is such that only a moderate conversion is
obtained. If the reaction mixture is brought to equilibrium and then
fractionally distilled to remove the ethyl acetate, leaving water, unre-
acted acetic acid, and ethanol, it would give only a low conversion on
further reaction. The reaction can be carried out in a fractionating
column with the esterification occurring in the liquid on the plates.
The acetic acid is added to the upper portion of the column, and the
alcohol is introduced in the lower portion. The plates below the
alcohol addition are used to strip the alcohol out of water. The middle
section between the plates where the alcohol and acetic acid are added
is the chief esterification section. The upper portion of this middle
section has a high ratio of acetic acid to alcohol and gives good cleanup
of the ethanol. The lower portion is high in alcohol and gives a rapid
reaction of the acetic acid. The top part of the column fractionates
the ethyl acetate out of the acetic acid. A small amount of acid
catalyst is added at the top of the column. The unit produces the
ethyl acetate and water and, by separating them, carries the reaction
almost to completion.

In the esterification system, if the liquid level on the plates is low
and a normal vapor rate is employed, the amount of reaction per unit
time will be small relative to the vapor rate and a high reflux ratio will
be necessary. This will result in a high heat consumption per gallon
of ethyl acetate produced. If a very deep liquid level is employed on
the plate to increase the amount of reaction relative to the vapor rate,
it may be that the rate at which the ethyl acetate is removed is so low
that the liquid on the plates is near equilibrium and the reaction rate

361

362 FRACTIONAL DISTILLATION

will be retarded. Thus the large volume of column required because
of the deep liquid level will be used ineffectively. The volume of the
liquid should be proportioned to the vapor rate such that there is an
adequate supply of ester for separation but not to the extent that it
seriously retards further reaction.

The large liquid volume required can be obtained by using a deep
liquid level on the plates or by having the overflow from a plate pass
through a holding tank for the chemical reaction before it is added to
the plate below. Deep liquid levels give undesirable action with nor-
mal-type bubble plates, but tall caps and risers can be used to obtain
satisfactory operation.

In other cases, the chemical reaction may be undesirable. For
example, in the distillation of an aqueous solution of alcohols and alde-
hydes, aldols can form in the upper portion of the column and, being
of lower volatility, go down the column and be hydrolyzed by the
water at the bottom. This cycling action is undesirable. Polymeri-
zation and thermal decomposition occur in some cases and are usually
objectionable. In most of these cases, the undesirable reactions take
place in the liquid phase and can be minimized by using low liquid
depth on the plates and by keeping the temperatures as low as practi-
cal. Frequently inhibitors can be added to the fractionating system
to reduce the amount of reaction. For example, in the f ractionation of
styrene from ethyl benzene or of butadiene from butylene, the poly-
merization of the styrene and butadiene can be reduced by adding
inhibitors such as sulfur or tertiary butyl catechol with the reflux.
The styrene rectification is also carried out under vacuum to reduce
the temperature. The separation of styrene from ethyl benzene
requires a large number of plates, and even with specially designed
bubble-cap plates the pressure drop would be so great that the still
temperature would be excessive. To avoid this difficulty the rectify-
ing column is made in two sections. The vapor from the section that
serves as the bottom portion of the tower is liquefied under vacuum in
a condenser and then pumped, vaporized, and added at the bottom of
the other section. The liquid from the bottom of the top section is
added to the top of the lower section. A vacuum condenser is also
employed for the top section. In this manner the average pressure
and temperature in the column can be made lower than for the single
tower.

The design calculations for these systems can be made by the usual
stepwise procedure making allowance for the chemical reaction on each
plate. The calculations frequently require trial-and-error methods,

SIMULTANEOUS RECTIFICATION AND CHEMICAL REACTION 363

because the terminal conditions and flow rates are a function of the
chemical reaction. The method of calculation will be illustrated by
the fractional distillation of cyclopentadiene.

Example on Preparation of Cyclopentadiene. Cyclopentadiene is obtained in
high-temperature vapor-phase petroleum cracking operations. It is mixed with
other hydrocarbons, and its separation is complicated by the fact that it will
dimerize and the dimer will depolymerize at about normal distillation conditions.
One method of operating is to dimerize the pentadiene and then remove all remain-
ing constituents below Cr. The residue is then given a thermal treatment which
will depolymerize the dimer, and the mixture is distilled to obtain the cyclo-
pentadiene. A unit is to be designed for this final fractional distillation, and an
estimate is to be made of the amount of polymerization that will be obtained.

As a basis for the estimate, it is assumed that the feed is a binary mixture con-
taining 20 mol per cent cyclopentadiene and 80 mol per cent C7. The overhead
product is to contain 98 mol per cent cyclopentadiene, and the cyclopentadiene
content of the bottoms is to be 0.5 mol per cent. The column will operate with a
still and total condenser at atmospheric pressure. A reflux ratio, 0/D, three times
the minimum reflux ratio for the separation of the binary mixture with no poly-
merization will be used.

Data and Notes

1. Holdup in the condenser is negligible.

2. Holdup per plate is equivalent to a liquid depth of 3 in. on the superficial
area.

3. Holdup in the still is equivalent to one plate.

4. Superficial vapor velocity is equivalent to 1.0 f.p.s. (S.T.P.)

5. The liquid on each plate is well mixed, and the plate efficiency is 100 per cent.

6. The feed enters such that Vn » Vm.

7. Neglect polymerization in vapor.

8. For simplification assume the relative volatilities of cyclopentadiene and
dicyclopentadiene to C7 are 6.0 and 0.1, respectively.

9. The rate of depolymerization at the temperature of distillation is negligible.

10. The rate of polymerization in the liquid phase is a function of temperature,
but for simplification an average value of the rate constant will be employed in the
equation — (dr/d6) = 2£V2, where r — g. mol of cyclopentadiene per liter, 0 « sec.,
K - 6 X 10-fil./sec.-g. mol.

11. Assume that liquid volumes are additive. Densities of CB, CT, and Cio
average 0.80, 0.68, and 0.92, respectively.

12. Assume that the same number of mols of vapor enters each plate per unit of
time.

Solution. Minimum reflux ratio for binary mixture: Because cyclopentadiene
is the more volatile component, the mol fractions for the binary mixture will be
used on the basis of this component. With Vn «• Vm, the intersection of the
operating lines will be at x ~ XF « 0.2.

With a constant relative volatility of 6.0, the pinched-in condition will occur at
the intersection of the operating lines. The y coordinate of the intersection is

F °* ___ 6(0.20)

1 + (« - D* 1 + (6 - 1)(0.20)

364 FRACTIONAL DISTILLATION

From these coordinates and because XD ** 0.98, the slope of the enriching line
and the reflux ratio are found.

0\ 0.98 - 0.60 0.38

F/min 0.98 - 0.20 " 0.78

°'38 0.95

<D/min. 0.78 - 0.38
Design case:

For this case, O/D » 3(O/D)mm.

~ - 2.85 and ~ = 3.85

Because the amount of polymerization is unknown and because the complete
composition of neither the distillate nor the bottoms is known, the calculations are
a series of trial-and-error calculations.

Basis: 1 sq. ft. of plate area and 1 sec. of time.

The volume of liquid on a plate * J£ cu. ft. = Y± (28.32) « 7.08 1.

The molal volumes of each of the three components may be found by dividing
their respective molecular weights by their densities.

C6:
C7:

Cio:

v - 66.1/0.80 - 0.0826 l./g. mol'
v - 100.2/0.68 - 0.1475 l./g. mol
v - 132.2/0.92 - 0.144 l./g. mol
1

0.0826z8 + 0.1475*7 + 0.144zi0 0.0826

or the number of mols of Cs that polymerize on any plate in 1 sec. is given by
p - 6(7.08)10~B r* - (4.25)10-4 r*

From the superficial vapor rate of 1 f .p.s. at standard temperature and pressure,
the conversion factor of 454 g./lb., and the fact that the pound molal volume of a
gas is 359 cu. ft. (S.T.P.), the vapor rate V may be found.

V = |g - 1.265 g. mob/sec.

V 1 9fi^

* D- 775 -iS- 0-828 g-moh/aeo.

0R _ 7 - D - 1.265 - 0.328 - 0.937 g. mols/sec.

Top plate:

Because the conditions at the top of the tower are much better known than are
those at the bottom of the tower, the calculations will be started at the top. The
volatility of Cio is so low that it is assumed first that the Cio in the distillate is
negligible.

To calculate the amount of polymerization on the top plate, the ratios XT/X& and
XM/XS must be known.

SIMULTANEOUS RECTIFICATION AND CHEMICAL REACTION 365

The value of Xi0/x* is not known until the amount of polymerization is known, so
it is assumed as 0.0252.

- 0.0826 -f 0.1475(0.1225) + 0.144(0.0252) « 0.104

With Vt~i = Vt (subscript tt t — 1 refer to top plate and plate below, respec-
tively), the polymerization will make Ot less than OR by the decrease in mols, p/2.

0< . 0R - % m 0.936 - 0.0195 » 0.916
&

Assuming that 0.00082 more mols of Cio enter the top plate in the vapor than
leave it in the distillate,

Ozio - + 0.00082 - 0.02027

+ Ox, = 0t- Ozio = 0.916 - 0.020 - 0.896

U.OoO

,* t—r\n

°'798

& LT225
n SQR

0x7 = (°-1225) - °-

0.02027/0.798 — 0.0254, which checks the assumption satisfactorily.

#10
2/io = —

0.020
: 0.0976

(0.02) (0.10) - 0.00041

which checks the assumption that it is negligible.

The following table may now be constructed for the top plate :

Comp.

Otxt

DxD

0 98

0.798

0 3215

0.02

0 0976

0.00656

0 00041

0.02027

0.000134

Plate* - 1:

Calculations may proceed downward to the next plate by making material bal-
ances on components Cg and €7 and by assuming, and later checking, vapor values
for Cio.

For C8,

DXD + Oxt + pt - 0,3215 + 0.798 + 0.0389 - 1.159

366
For C7,

FRACTIONAL DISTILLATION

Vyt-i - DxD 4- #*t - 0.00656 + 0.0976 - 0.1042
Knowing the ratio of the y's, the ratio of the a?'s may be found, using a

x7 0.1042

1,159

and assuming XIQ/X& » 0.0495,

- 0.0826 + 0.1475(0.540) 4- 0.144(0.0495) - 0.1694

Assuming that 0.0006 more mols of Cio enter plate (t — 1) in the vapor than leave
in the vapor,

Oxio - 0.02027 + 0.0074 -f 0.0006 - 0.0283

0M - Ot - ^ - 0.916 - 0.0074 - 0.909
2

Ox6 -f Ox1 - 0.909 - 0.028 - 0.881

0 ^4.
Oz7 - j=~ 0.881 - 0.309 and Ox, - 0.572

These calculated values may now be tabulated.
Tabulating for t — 1 :

Comp.

1.159

0.572

0.1042

0.309

0.00095

0 0283

The value of Vyw is calculated using

(0.1042)(0.10) - 0.00095

Thus, 0.00095 - 0.00013 - 0.00082, which checks the assumptions made for the
top plate. XIQ/XI — 0.0283/0.572 «• 0.0495, which checks the assumption used
to calculate r.
Plate t - 2:

For C6,
For Or,

- 0.3215 + 0.0389 + 0.0148 + 0.572 - 0.947
Vyt-* - 0.00656 -f 0.309 » 0.3155

5 -•TOT-1'"8

SIMULTANEOUS RECTIFICATION AND CHEMICAL REACTION 367

Assume that X

« 0.0986.

~ - 0.0826 + 0.1475(1.998) + 0.144(0.0986)

0.391

Assume that 0.0008 more mols of Cio are in. the vapor from plate (t — 2) than in
the vapor to this plate. Therefore,

Ozio - 0.0283 4- 0.0014 - 0.0008
On - 0,909 - 0.0014 - 0.908
Ox, + Ox, - 0.908 - 0.029 - 0.879

0.0289

Ox, -
Tabulating for t — 2:

0.293

and Ox, - 0.586

Comp.

0.947

0.293

0 3155

0 586

0.00155

0.0289

72/1

(°-10)(°-3155) - 0.00155

The Cio condensing on plate (< — 1) - 0.00155 - 0.00095 - 0.0006, which
checks the assumption used in finding Oxio for plate (t — l).

XIQ/XS » 0.0289/0.293 « 0.0986, which checks the value used to calculate n_2.
Plate i -3 - feed plate:

The ratio of x*/xi is approaching that in the feed, and trial calculations show that
plate (t — 3) should be the feed plate. To calculate 0 values for this plate, the
value of F must be known. To find F, assume that 0.0571 mol of the Cs fed to the
tower polymerizes. Over-all balances give

F - W + D

+ 0.357

B balance, assuming xw ** 0.004,

0.20F - 0.3215 4- 0.004TF + 0.057 - 0.004TT -f 0.3786
Solving,

F - 1.915 and W ** 1.56
For C5 + 2Cio,

TF&TT - 1.915(0.20) - 0.3215 - 0.0615
For CT,

Wxw - 1.915(0,80) - 0.00656 - 1.526

368

FRACTIONAL DISTILLATION

Calculations for plate (t— 3), the feed plate:
For C8,

Vyt-i - 0.3215 4 0.293 4 0.0389 4 0.0148 4 0.0028 » 0.671
For C7,

< 0.00656 4 0.586
n 0.5925

0.5925

0.671

5.30

Assume

— - 0.0661; - - 0.0826 4 0.1475(5.30) 4 0.144(0.0661) - 0.875

#6 T

(4.25)10-

0.000555; | = 0.000277

^~* (0.875)2
Assume that 0.002 mol of Cio condenses on t— 3, then

0z10 = 0.0289 4 0.00027 4 0.0002 - 0.0293
Ot-t - 0.908 - 0.0003 4 1.915 = 2.823
OzB 4 Ox-i ** 2.823 - 0.029 - 2.794
2.794

^ 6W
Tabulating for t-3:

•• 0.443 and Oz7 = 2.351

Comp.

0 671

0 443

0 5925

0.351

0 00074

0 0293

(0.5925) (0.10) - 0.000739

Mols Cio vaporizing from J-2 - 0.00155 - 0.00074 = 0.0008
Checking the assumption used in calculating Ox™ for plate (£—2),

xio 0.0293
xi 0.443

= 0.0661

which checks the assumption used to find n_3.

The calculations for the succeeding plates are carried out in the same manner and
the results are given in the following table:

Comp.

(VJ0I-4

(0aO,-4

(Vy)t~*

(Ox)t-t

(Vy)*-*

(OaOi-,

0.439

0.227

0.222

0.0956

0.0912

0.0357

0.825

2 567

1.041

2 697

1 171

2.757

0 00095

0 0295

0 00114

0.0296

0.0013

0.0306

0.00013

Neglected

SIMULTANEOUS RECTIFICATION AND CHEMICAL REACTION 369

Comp.

(7y)«

Wxw

0.0313

0.00646

0.004

1.231

0 526

,0.978

0 0023

0 0283

0.018

The value of 0.057 mol polymerized is approximately correct, and the per cent of
cyclopentadiene polymerized is

0.057(100)

1.915(0.20)

1.49 per cent

The number of plates required is seven.

The solution required a large amount of trial and error. An approximate value
could be obtained by treating the mixture as a binary of Ct> and C? on the usual y,x

01 02 03 0.4 0.5 06 0.7 0.8 0.9 10

x-mol fraction cyclopentadiene in liquid
FIG. 13-1. Fractionation of cyclopentadiene.

diagram and then calculating the amount of polymerization from the plate com-
position so obtained. Such a diagram is shown in Fig. 13-1, and six plates are
required. The plates are fewer because in the actual case the polymerization
decreases the effective reflux ratio for the C5 + C7. The polymerization was
calculated neglecting the Cio in the liquid and was found 0.057 mol of Cs. This
value agrees well with the value found by the plate-to-plate calculations.

CHAPTER 14
BATCH DISTILLATION

Continuous operation is normally employed when the material to be
distilled is large in quantity and is available at a reasonably uniform
rate. Under such conditions, it is usually cheaper than batch dis-
tillation, but there are a large number of cases which are not suited
to continuous operation and which are handled on the batch basis.

A batch distillation with rectification involves charging the still with
the material to be separated and carrying out the fractionation until
the desired amount has been distilled off. The overhead composition
will vary during the operation, and usually a number of cuts will be
made. Some of the cuts will be the desired products, while others will
be intermediate fractions that can be recycled to subsequent batches to
obtain further separation.

The equipment employed and the method of operation are similar
for batch and continuous distillation, but in the latter the mathematical
analysis is based on the ^ssujnoptimi that in all portions joj^e
the compositions and flowlra^^ tkpyfij

tion. TEeie^ohditiOM^o^ n^*"af3>p1y^t o aT batch 9Tstillation, and
TJecause of the continuous variations involved, it must be analyzed on
a differential basis. As a result, the calculations are much more diffi-
cult, and satisfactory design methods have been developed for only a
few simple cases.

BINARY MIXTURES

The batch distillation of binary mixtures will be considered fpr the
cases of (1) no rectification, (2) rectification without liquid holdup in
the column, and (3) rectification with holdup.

No Rectification. Batch distillation without rectification cor-
responds to the simple distillations of Chap. 6, and the calculations of
the concentrations as a function of the amount distilled can be made
by Eqs. (6-3) and (6-7).

Rectification without Liquid Holdup in Column. Finite Reflux Ratio,
In this case, it is assumed that the distillation is carried out with a
fractionating column, that the holjaqxQf liquid in the column is negligi-

- — ~** ' ~"~ • — — — *~-

BATCH DISTILLATION

371

ble in comparison to the liquid in thej^ysiem, and that the rate of
£ha5^^ plate is negligible in com-

parison to the rate of flow of that component through the plate. Thus
the change. in the quantity of each component in the column can be
neglected in the differential material balances.

Consider the system shown in Fig. 14-1. On the basis of the
assumptions made, an over-all differential
material balance gives

dD - -dL

where L represents the mols of liquid in
still, and a component balance gives

= -~d(LxL)

= — L dxL — XL dL

= — L dxL + XL dD
dD _ dxL
L

XL — XD

(14-1)

This equation is equivalent to the Ray-
leigh equation but differs in that the denom-
inator is XL — XD instead of XL — VL. The
integration of Eq. (14-1) involves determin-
ing a relation between XL and XD. Neglecting
the rate of change of holdup of a component
on the plates and in the condenser, a balance
between the n and n + 1 plates gives

FIG. 14-1. Schematic diagram
of batch distillation system.

At any time these equations are identical to those for the continuous
distillation and can be applied to determine the relation between XL
and XD for that instant. By applying the equations repeatedly, the
value of XL — XD can be obtained as a function of XL and the integration
of Eq. (14-1) performed. There are a number of ways such a dis-
tillation can be made, but the two most common cases involve (1)
operating at constant reflux ratio and taking cuts that average the
desired composition and (2) operating at variable reflux ratio to give
a constant product composition while making the desired product.
In the first case the value of (On/V^i) will remain constant during the
distillation, and a series of lines of this slope can be drawn on the usual
y,x diagram for various assumed values of XD, and the value of XL can
be determined by stepping down each line the number of theoretical

372

FRACTIONAL DISTILLATION

FIG. 14-2. Constant reflux ratio case.

FIG. 14-3. Constant distillate composition case.

plates equivalent to the column. This procedure is illustrated in
Fig. 14-2 for a column and still equivalent to four theoretical plates.
In the second case the value of XD is fixed, a series of operating lines of
different slopes is drawn through it, and the plates are stepped off on
each line to determine the value of XL. This procedure is illustrated in
Fig. 14-3. By these procedures, Eq. (14-1) can be integrated, giving
the relation between the amount distilled and the composition of the

BATCH DISTILLATION 373

liquid remaining in the still. In the second case with XD constant, the
integration of Eq. (14-1) amounts to only simple material balance, and
the data from Fig. 14-3 are not needed for this purpose. In addition
to the information obtained by integration of E(j. (14-1), it is frequently
necessary to have data on the vapor required or the average composi-
tion of a fraction produced.

For the constant reflux case, the calculation of the vapor require-
ment can be made as follows:

F = ^ + ljD (14-2)

and the average composition of any fraction is

vi/av ~~~" " '""" 7-

For the variable reflux case,

dV = dO + dD

and, by material balance,

LXL =

r Lo(&o -

XL — XD

where L0 = original mols in still

x0 = original composition of liquid in still
Substituting these values in Eq. (14-1) gives

— •b0(x0 — Xp) dXL

Form Fig. 14-3 the relation between XD, XL, and the slope of the
operating line, dO/dV, can be obtained and Eq. (14-4) integrated to
give the vapor requirement.

The use of these methods will be illustrated by the following example.

Batch Fractionation of a Binary Mixture. An equimolal mixture of A and B is
to be fractionated in a batch column equivalent to three theoretical plates plus a
still. The still will operate at atmospheric pressure, with a total condenser, and the

374

FRACTIONAL DISTILLATION

holdup in the column and condenser is negligible. The company desires to obtain
an overhead A product containing 95 moi per cent A, and two methods of operation
have been suggested: (1) Operate the column at a constant reflux ratio (0/D)
equal to 5.0 and continue the distillation until the average composition of the
distillate is 95 per cent A; (2) Operate the column at a variable reflux ratio to
give a distillate of constant composition.
Using data and notes given below, calculate:
1. For Method 1,

a. The mol per cent of the original charge to the still that can be obtained as

the 95 per cent distillate.

6. The mols of vapor per 100 mols of original charge to obtain the distillate
of Part a.

1.U

0.9
0.8
07
|0.6

<0.5
§

^04

J0.3
02

at
°(

•fit)

) at 0,£ 03 04 05 0.6 017 0.8 0,9 t.(
Mol fraction A in liquid

FIG. 14-4.

2. For Method 2, the mol per cent of the original charge to the still that can be
obtained as the 95 per cent distillate, using a total vapor-to-charge ratio equal to
that of Part 1, 6.

Assume «AB is constant at 2.5.

Solution. By trial and error the relations between XL and XD could be deter-
mined analytically by Eq. (7-62) because the relative volatility is constant and the
usual simplifying assumptions apply. However a y,x diagram is probably simpler
and will be employed. The mol fractions of component A will be used in the
calculations.

The equilibrium curve was calculated from the relative volatility and is given
in Fig. 14-4.

Method 1. The slope of the operating line - 0/V «• % « 0,833. The rela-
tion between any x& and the corresponding XD is found by taking four steps* from

BATCH DISTILLATION

375

XD on the operating line. The case for XD =* 0.94 is shown in Fig. 144 and gives
XL = 0.41. The values for other cases are given in Table 14-1.

TABLE 14-1

lnL<

XL — XD

ln L

0 98

0 658

~3 11

0.96

0.490

-2 13

0 022

0 978

0.962

0 94

0,410

-1 89

0.182

0 834

0 955

0 92

0.344

-1 73

0 299

0 742

0 948

0.90

0.293

-1.65

0 379

0.678

0 937

The values of I/ (XL — XD) are plotted as a function of XL, and the graphical
integration is performed from XL = 0.5 to XL- The resulting values of the integral
are equal to In (L0/L) and are tabulated in the fourth column of the table. The
fifth column gives the values of L/L0 and the average composition of all of the
distillate from the start of the distillation as calculated by Eq. (14-3). By plotting
the average value of XD vs. (L/L0), it is found that (£z>)av ** 0.95 for (L/L0) «• 0.82.
Therefore, mol per cent of original charge recovered as 95 per cent distillate is 18.0.

For
and

Lo - 100 mols, D - 18

V - total vapor = 0 + D - 6Z> - 6(18)
- 108 mols

Method 2. In this case it is necessary to calculate the per cent recovery of the
original charge as 95 per cent distillate for a total heat supply equal to 108 mols of
vapor when operating on the variable reflux basis. This result can be evaluated by
integrating Eq. (14-4). For various assumed values of the slope of the operating
line dO/dV, the lines were drawn through XD = 0.95 and the plates stepped down
to obtain XL- The values are summarized in Table 14-2 for L0 = 100.

TABLE 14-2

dO
dV

100(0.5 - 0.95)

(0.95 - xL)*(l - ^)

1.0

0.330

0.95

0 360

-2,590

176 2

0 90

0.390

-1,430

121.5

0 85

0.435

-1,130

65.7

0.80

0.480

- 968

19.1

0.75

0.510

-; 937

(•— «•

376 FRACTIONAL DISTILLATION

The values given in the third column of Table 14-2 were plotted vs. XLt and the
area under the curve from XL — 0.5 to XL is equal to the total vapor necessary ta
reduce the still concentration to XL. A plot of XL against V gives XL = 0.398 at
V - 108 and

0.398L + 0.95 (L0 - L) - 0.5L,
~ - 0.815

LIO

Recovery as 95 per cent distillate = 18.5 per cent of original charge. This
would indicate that Method 2 gave a slightly better recovery for the same heat
consumption, but the accuracy of the calculations is not sufficient to make the
difference significant. Method 1 would be more practical from an operating
viewpoint.

Total Reflux. Limiting conditions can be calculated for batch dis-
tillation that are useful for orientation purposes. The total reflux
limit applies to the constant reflux category and can be used to deter-
mine the minimum number of theoretical plates necessary to give the
desired product recovery or to determine the maximum possible recov-
ery for a given number of plates. Consider an equimolal mixture of
A and B having constant relative volatility of 2, and make the usual
simplifying assumptions. The desired distillate is to contain 95 mol
per cent A, and the limits to be estimated are (1) the number of theo-
retical plates for a 50 per cent recovery of A in the distillate and (2) the
maximum per cent recovery of A with a tower equivalent to five theo-
retical plates.

For total reflux and constant relative volatility, by Eq. (7-53),

XD " *

Substituting this value in Eq. (14-1),

^ » fa* (14-6)

L aN+lXL

\QL ~— " L)XL I J-

This equation is identical with Eq. (6-5) except that a**1 has
replaced a, and by integration from x0, L0 to x, I/,

BATCH DISTILLATION 377

for 50 per cent recovery of A,

(L. - L)0.95 = 0.5Le(0.5)
- 0.737

- °'339

Using these values with Eq. (14-7) gives TV = 3.8 theoretical plates.
For maximum per cent recovery with five theoretical plates,

, L 1 , 3(0.5) , . 0.5

- - '

*" T 96 — i n *n — ^ ' "x

JL/o A — 1 U.O^l — X)

and

LZ + 0.95(Lo - L) = 0.5L0
L 0.45

Lo 0.95 - 3
Solving these equations simultaneously gives

~ = 0.492, x = 0.0352
L/o

-P 0.5L0 - 0.0352L ^ lrkA
Recovery = 7r^? X 100

U.Olvo

= 96.3 per cent

It will be noted that the per cent recovery increases rapidly as the
theoretical plates are increased in number.

If the relative volatility is not constant, the total reflux condition
can be solved using the y — x line as the operating line and determining
XL as a function of XD graphically.

Minimum Vapor Requirements. Another limit that is instructive is
the minimum vapor requirement for a given separation and recovery.
This limit will correspond to the minimum reflux ratio condition for
continuous distillation and will require an infinite number of theoretical
plates. For this case, the operating line must contact the equilibrium
curve at some point. For the general case, the limit can be deter-
mined graphically, for either the constant or variable reflux conditions,
by drawing such operating lines and obtaining XL as a function of XD or
dO/dV and integrating Eqs. (14-1) and (14-4).

The general principles can be illustrated analytically for the case of
constant relative volatility. For the constant reflux method, the
operating line will intersect the equilibrium curve at either XD = 1 or

378 FRACTIONAL DISTILLATION

at x *• XL. The overhead product will be pure volatile component,
Xj> •• 1, until XL decreases to a value such that the operating line
through XD » 1 intersects the equilibrium curve at the composition of
the still; for smaller values of XL> the intersection will be at #L. This
limiting value of XL can be calculated by

0 _ i - yl
V ~ i - x*

where x* =* value of XL such that operating line intersects equilibrium

curve at XD = 1 and x*
y* = vapor in equilibrium with xj
For constant relative volatility,

L ~ o/v(« - i)

for values of XL equal to or greater than #*, XD will be equal to unity.
For values less than x%,

With constant relative volatility, substituting for yL gives

a 0~

(a -

[(a - l)afc + 1] 1 - rr

This relation ean be combined with Eq. (14-1) and integrated from
% to x and L* to L to give

which is the same as Eq. (6-5) except for the (1 — 0/V) term. This
equation can be rewritten

(14-12)

BATCH DISTILLATION 379

The average composition for the distillate from L0 to L is

and the total vapor generated from L0 to L,

By a similar procedure an equation can be developed for the variable
reflux case.

V XQ- XD [2(« - 1) + (1 + XD) , (I - X)(XQ - XD)
Lo 2(«-l)L XD-I m(l-x0)(x-xD)

H^l

/ J

Minimum Vapor Requirements for Batch Distillation. Consider the batch dis-
tillation of an equimolal mixture of A and B. The relative volatility, <XAB, is
constant at 2; and the average distillate is to be 95 mol per cent A. Calculate the
minimum mols of vapor for 50 mol per cent recovery 'of A in the distillate for both
the constant reflux ratio and the variable reflux ratio cases.
Solution

0.95(L0 - L) » 0.25L,

T- - 0.737

Final liquid composition, xL « 0.25/0.737 — 0.339. For the constant reflux
method, assume that the value of 0/D is such that x% will be less than 0.5; then
by Eq. (14-8)

* 1 - 0/7 D

XL . ___ _ ^

By over-all material balance

L* I - x0 0.5

Lo 1 - *J 1 - D/0

For the portion of the distillation when the liquid in the still is decreasing from
x*L to 0.339, Eq. (14-12) gives

\ - 1 474 (l - -^ - fl"i>/Q [0-880(1 -
V ^'^V O/ t 0.661 L D/0(0.661) /

Solution of this relation by trial and error gives 0/D ** 2.12. This value
gives #J « 0.472, indicating that the assumption made was correct. Then, by •

380 FRACTIONAL DISTILLATION

Eq. (14-14),

~ - (2.12 + 1)(0.263) - 0.82

For variable reflux operation,

, 0.661 (-0.45) . (0.339)*(- 0.45) (0.5) 1 _ ft 7fi7
0*5(-0.611) (0.5)2( -0.611) (0.661) J ~~ U*7b7

V_ -0.45 [2 + 1.95, 0.661 (-0.45)
Lo " 2 L .-0.05

As was found in the example page 376, the variable reflux method requires less
vapor for a given separation than the constant reflux method.

Rectification with Liquid Holdup in Column. In continuous distilla-
tion holdup does not harm the degree of separation obtainable; in some
cases it may be an advantage in that it gives "flywheel" action and
tends to smooth out the operation. The effect of column holdup in
batch distillation is not completely clear because a satisfactory method
of analyzing the operation has not been developed. Rose, Welshans,
and Long (Ref . 3) present approximate equations for total reflux, and
Colbura and Stern (Ref. 1) have discussed the case of finite reflux.
Rose et al. concluded that holdup was detrimental at total reflux, and
Colburn and Stern indicated that holdup in some cases improved the
sharpness of separation as compared to no holdup. In order to inves-
tigate this difference of opinion, a number of cases were studied analyti-
cally, and holdup was not advantageous in any of them. For the same
number of theoretical plates, reflux ratio, and total mols vaporized,
the no-holdup system was superior in all the cases tried. It is believed
that the improved results reported with column holdup were due to the
fact that, for the starting condition, a concentration gradient was
already established in the rectifying column which had required the
expenditure of considerable vapor which should be but was not included
in the evaluation. The amount of vapor required to establish this
concentration gradient can be estimated from Fig. 18-3. For a sys-
tem with a relative volatility of 2.0, the "prerun" vapor would be
about eight times the column holdup. Thus a column holdup equal
to 15 per cent of the charge to still would require a prerun vapor gen-
eration of approximately 1.2 times the charge to the still. If the
column holdup was obtained by charging it with the fresh still liquid
and including all vapor from the beginning of the distillation, the
holdup results would have been less favorable. Alternately, if
the no-holdup case was given the extra vapor corresponding to the
"tune-up" period for the holdup case, it would be more favorable.
It is concluded that in general holdup in the rectifying column is unde-
sirable in batch distillation.

BATCH DISTILLATION 381

There is an intermittent type of batch operation for which column
holdup has an apparent advantage. In this case the column is run at
total reflux, and the top plates are filled with liquid rich in the more
volatile component. Product is then withdrawn at a high rate for a
short time after which the column is put back on total reflux to reestab-
lish a concentration gradient. Most of the product withdrawn during
the short time interval comes from the accumulation of volatile com-
ponent on the top plates. For this type of operation, improved
results would be obtained without liquid holdup in the column, but
with a reservoir in the reflux line. The column would operate at total
reflux until the liquid in the reservoir was rich in the volatile compo-
nent. This liquid would then be withdrawn completely, and the col-
umn returned to total reflux operation to prepare the next fraction.
This type of operation is frequently convenient for laboratory distilla-
tions but is not often advantageous for large-scale operation.

A detailed differential stepwise integration for a batch distillation
with holdup can be made, but the time and effort involved are usually
not justified by the value of the result. The method is useful in
developing the principles of batch distillation with holdup. The fol-
lowing section will consider the basic differential equations which are
helpful in obtaining a qualitative picture of the process. By an analy-
sis similar to that for Eq. (14-1),

xDdD = -d(LxL) - d(Hxff) (14-15)

where H = holdup in column

XH = average composition of holdup

The integration of this equation requires a relationship between HXH
and L and XL. This relationship is very complex and has not been
expressed in a form suitable for direct integration. It has been cus-
tomary to simplify this equation by assuming that H is constant. In
some cases this may be a reasonable assumption, but on a molal basis
it could be greatly in error. For example in the distillation of an iso-
propanol-water mixture, a given volume of holdup could give a fourfold
variation in molal holdup. If Eq. (14-15) is applied on a weight and
weight fraction basis, the variation of holdup during the distillation
will be less, but it can still be large. Analysis based on the constancy
of H will be only approximate for most cases.

An operating line for this case can be written by taking a material
balance on one of the components,

yn dVn » xn+i dOn+i — d(LxL) - d(HmxHm) (14-16)

382 FRACTIONAL DISTILLATION

dVn - xn+i dO^i + XD dD - d(HnVan) (14-17)

where Hmxffm and HnXan equal holdup of component for plates from
still to plate n, and for plate n + I to top of column, respectively.

Because the last terms, d(Hnpcan?) and d(JHn##n), vary from plate to
plate, the operating line is curved and cannot be evaluated without a
knowledge of the variation of these terms. At total reflux, these terms
are negligible in comparison to the first two terms, and the operating
line is the y = x diagonal with holdup the same as it was for no holdup,
but for finite reflux ratio the operating line with holdup is curved and
difficult to establish on the y,x diagram.

When a batch distillation is carried out by the variable reflux ratio
method to give a constant value of XD, and the distillation is continued
until the reflux ratio is essentially total reflux, the amount of holdup in
the column at the end of the distillation can be easily calculated by
using the y = x line as the operating line. Such a procedure gives the
composition of the liquid on each plate, and a correction can be applied
for effect of, the holdup on the percentage yield of a given fraction.

Batch Rectification of Binary Mixture. Assume that the equimolal mixture
already considered, page 376, is to be distilled in this manner, and it is desired to
evaluate the effect of the column holdup.

Solution. Basis: 100 mols liquid charged to still, no liquid in column at start.

At the end of the distillation the results will correspond to total reflux and, by
Eq. (14-5),

2«(0.05) + 0.95
- 0.229

By material balance,

0.5(100) - 0.95D + 0.229L -f Y (hx)

l
5

100 - D -f L + Y h
T

5
w*

here } (hx) — total mols of A in the holdup on the plates at total reflux

y (h) •• total mols of holdup on the plates

BATCH DISTILLATION 383

For the case in which the total holdup per plate is constant, the value h y x

can be obtained approximately in a mathematical form (Ref. 2), but it is just as
easy in most cases to evaluate it plate by plate. In the present case, assuming the
holdup per plate is constant,

)** h(xi H- xz + x* + #4 + *e)

- M0.371 -f 0.541 -f 0.701 + 0.823 + 0.903)

- 3.35/1
and

50 - 0.95D + 0.229(100 - D - 5A) + 3.3
27.1 - 2.205fc - 0.721D

n 27.1 - 2.205fc

If the holdup per plate were 2 per cent of the original charge to the still, then

h - 2
and the mols of the distillate would be

n - 27.1 - 2.205(2)

0.721
- 31.5

as compared to D = 37.5 for no holdup. If the value of holdup per plate was as
high as 12.3 per cent of the charge, no 95 per cent distillate could be produced in the
5-plate column. If the percent of A in the original feed had been smaller, the
effect of holdup would have been even more serious.

A similar analysis has been applied by some writers to the case where the dis-
tillation has been discontinued at a finite reflux ratio. The usual straight operat-
ing line was employed to determine the plate composition at the end of the distilla-
tion, but in view of the fact that the operating line is curved for such a case, the
calculations are probably of little value. If the operating lines corresponding to
Eq. (14-16) or (14-17) could be evaluated for the end of the distillation, then an
analysis of the effect of holdup could be made.

MULTICOMPONENT MIXTURES

The state of the art for the batch distillation of multicomponent
mixtures is even less satisfactory than for binary mixtures, and except
for total reflux, no accurate and practical method is available even
without liquid holdup in the column. This difficulty arises from the
fact that, starting with a given liquid composition in the still, it is
possible to calculate the equilibrium vapor leaving the still, but the
composition of the overflow to the still from the first plate cannot be
calculated without knowing the composition of the distillate leaving
the system. In a binary system, it is possible to choose the composi-

384 FRACTIONAL DISTILLATION

tion of the distillate arbitrarily and calculate back to the liquid in the
still, and the liquid composition in the actual distillation must pass
through this condition. However, in a multicomponent mixture, the
still composition calculated in that manner for an assumed overhead
product will probably not occur in an actual distillation. For exam-
ple, consider the distillation of a mixture of components A, B, and C
and assume an overhead composition of XDA.J #Z>B, and XDC. For these
assumed values, the calculations can be carried down the column for
a given reflux ratio assuming no holdup, and values of #LA, #LB, and
XLC will be obtained. However, in the actual distillation when com-
ponent A has the value XL±, it is very improbable that the ratio of B to
C will be the same as calculated on the basis of the assumed overhead
composition. By a laborious trial-and-error procedure, a consistent
calculation could be made, but it is doubtful that it would justify the
effort. A laboratory distillation would probably give a better and
cheaper evaluation.

As a guide to the characteristics of multicomponent batch distilla-
tion, the case of (1) total reflux with no liquid holdup in the column
and (2) finite reflux ratio with no liquid holdup by an approximate
method, will be considered.

Rectification without Liquid Holdup in Column. Total Reflux. For
the case of no liquid holdup, Eq. (14-1) applies to each of the com-
ponents, but the difficulty involves the evaluation of XD as a function

of XL.

In this case the stepwise calculations can be carried out starting at
the still, because the composition of the distillate does not affect the
operating line. Thus for a given XL the value of XD can be calculated,
but the calculation is still difficult for the general case because the con-
centration pattern followed by the liquid in the still is unknown. The
pattern can be approximated by a stepwise integration, but the calcu-
lations are tedious. The general principles will be developed for the
case in which all the relative volatilities remain constant, because in
this case direct integration is possible. It is simpler to apply the rela-
tive volatility form of the simple distillation of Eq. (6-6) than to use
Eq. (14-1). Thus, for any two components of a multicomponent mix-
ture at total reflux,

BATCH DISTILLATION 385

and, for a column with a total condenser,

(£).-(£),-*•().

Let a differential amount, dV, be vaporized and set

ywdV = — dA
y^T dV = -dB

where A, B are mols of A and B in still, respectively.

(14-19)
integrating from A0to A, B0to B

A /B\a

T. ~ (I.)

Likewise,

and the same for the other components.

For a given fraction of A vaporized, it is possible to calculate (1) the
fraction of all components vaporized, (2) the composition of the liquid
remaining in the still, (3) the instantaneous composition of the distil-
late, and (4) the average composition of all of the distillate.

Batch Rectification of Multicomponent Mixture at Total Reflux. As an example,
consider the fraction ation of an equimolal mixture of A, B, C, and D at total reflux
with relative volatility «AB = 2.0, «AC — 4.0, and a AD = 8.0. The column is
equivalent to three theoretical plates and holdup of liquid will be neglected.

Solution. Basis: 100 mols originally charged to still.

a. First distillate composition:

^ . 2* - 16

2/B

^ - 4* - 256

^ - 84 « 4,096

2/D *

2/A 4- 2/B -f yc + 2/D =* 1.0

4.096 rt n0o

"A ' 4,096+256 + 16 + 1 = °'938

#B «• 0.0586
yG ** 0.00366
2/D » 0.00023

386 FRACTIONAL DISTILLATION

b. 50 per cent of A distilled:
B

Bo
C

* 0.5Ho » 0.958

53*»* - 0.9973
-£ « 0.51/4'09' - 0.99983

The compositions of (1) liquid remaining in still, (2) average distillate, and (3)
instantaneous distillate are given in Table 14-3.

TABLE 14-3

(2)

(3)

Mols

XD(*V)

0.125

0 145

0.922

0.900

0.24

0.278

0.074

0.0931

0.2494

0.288

0.004

0.0061

0.25

0,289

—

0.004

0.864

By repeating this type of calculation, the values of the instantaneous distillate
composition given in Fig. 14-5 were obtained.

0.4 0.6 0,8 1.0

Mo! fraction distilled
FIG. 14-5. Fractionation, curves for multicomponent mixture at total reflux.

3ATCH DISTILLATION

387

In the example on p&ge 385, about 60 per cent of the original charge
could be obtained as fractions containing one of the components in at
least 85 concentration. It will be noted that the least volatile material
can be obtained in the highest purity because \t is taken over when the
still contains the l<*ast amount of other components. By operating
an inverted batch 'Jolumn as shown in Fig. 14-6, it is possible to remove
the less volatile components and obtain the most volatile material in
high purity. In this case, the batch is charged to the reservoir, and

Reservoir

Vaporizer* ^Separator
FIG. 14-6. Inverted batch distillation system.

liquid is continuously added to the top of the column from this tank.
The liquid from the bottom of the column is partly vaporized, and the
unvaporized portion is removed as product. The vapor is passed back
up the column to strip out the more volatile components, and the
overhead vapor is condensed and returned to the reservoir. The most
volatile component collects in the reservoir and is obtained as the last
fraction after essentially all the other components have been elimi-
nated. An intermediate fraction can be increased in concentration

388

FRACTIONAL DISTILLATDN

by fractionating first in a normal batch operation to remove the lighter
components and then in an inverted batch unit &o remove the heavy
constituents; or both operations can be combinec - simultaneously tak-
ing the lighter components overhead and the he& Wer components out
the bottom with the reservoir between. This combined operation can
save heat but requires additional equipment.

Finite Reflux Ratio. It was pointed out on page 384 that this case
was difficult because, if the plate-to-plate calculations were carried
down from the top for an assumed overhead product, the still compo-
sition would probably not correspond to any liquid composition
encountered in the actual distillation. Starting with a given liquid
composition in the still, it is possible to make calculations for an
assumed overhead composition. If the calculated overhead composi-
tion checks the assumed value, this gives one set of corresponding
distillate and still compositions. Thus a laborious trial-and-error pro-
cedure is necessary to obtain this one set of values, but the design cal-
culation requires a number of such sets for liquid compositions that
follow a definite pattern. Thus the overhead vapor corresponding to
the original charge could be calculated by the above trial-and-error
procedure and a small increment of this composition removed, leaving
a new liquid composition. By a large number of such small steps, the
distillation curve can be established, but making the trial-and-error
plate-to-plate calculation for each step is almost prohibitive.

The total reflux analysis of the preceding section was relatj^ely
simple, and an approximate method for finite reflux ratios can be
developed in an analogous manner. For the operating lines for any
two components, the liquids on a plate are related to the vapors from
below by

1 +

»»)•(£)

CM.

n+1

(14-21)

n+l

Thus, for a batch distillation,

'B/T

(14-22)

(14-23)

BATCH DISTILLATION

389

for a total condenser,

These equations are the same as some of those presented in Chap.
12, page 341, for the approximate design of contiguous columns.
Using an average value of a/j3 gives

(14-25)

For the average value it is possible to use either

'a\ GLT + (CLL/PL)

or the ratio of (a^/^v) calculated as

OiT +

ftnr =

Equation (14-25) is analogous to Eq. (14-18) and the resulting
integrated form is equivalent to Eq. (14-20) except that a is replaced
by (a/ft).

The evaluation of £L requires a trial-and-error procedure that can
be best illustrated by the following example.

Batch Rectification of Multicomponent Mixture at Finite Reflux Ratio. The
example given on page 385 will be repeated for a constant reflux ratio, O/D, equal
to 5.0. The tower will operate with a total condenser, and the usual simplifying
assumptions are made. The calculation for the initial vapor overhead is given in
Table 14-4.

TABLE 14-4

0L (rela-

Comp.

Assumed

tive to
compo-

a (relative to
component D)

/3av

(a.

G)~

XD oalo

nent D)

0 85

1 68

1 34

5 96

315

0.851

0.14

1.112

1.06

3.78

50.8

0 138

0.01

1.008

1 00

2.0

4 0

0.011

0.0

1.0

1.00

1.0

0.25

0 0006

390

FRACTIONAL DISTILLATION

The second column Table 14-4 gives the assumed values of the overhead com-
position, and the third column is the PL relative to component D calculated from
the assumed XD values, for example, for component A,

, . D (XA.)D

PL -

This is not the correct value of PL which should be calculated using XL+I values
instead of XL values. For purposes of simplification this concentration in the
still was employed.

The fifth column gives the values of /3av, which divided into the relative volatili-
ties are presented in the sixth. These values of (a//3) can be used with the liquid
composition to calculate the distillate concentrations. The procedure used in
column seven is the same as Eq. (9-3). The individual values of (<*/|8)4 XL are
divided by 370.05 » S(a/0)4 XL to give XD. The calculated values are close enough
to the assumed value, and no retrial is necessary <

Equation (14-20) using (a//3)av instead of a cannot be applied over a wide range
due to the variation of the exponent as the distillation proceeds, but it can be used
over a limited region averaging the values of (a//3)av for the two ends of the range.
For example, assume that the first increment is to remove 40 per cent of the A, i.e.,
at the end of the step, FA =• A/A0 « 0.6. In order to carry out the calculations,
the values of (<x/£)av corresponding to this final condition are assumed. These
calculations are summarized in Table 14-5.

TABLE 14-5

Comp.

As-
sumed

(<*/0)av

Mols re-
maining
in still
- Q.25F

(D>

/3av

(*}

V/Vav

5.5

0.6

0.15

0.171

155.5

0 762

1 89

1.445

5.53

3.6

0.915

0.229

0.261

43.8

0.215

1 165

1.083

3.68

1.95

0.993

0.248

0.283

4.1

0.020

1.014

1.007

1.98

1.0

0 25

0.285

0.001

1.001

1.0

0.877

203.69

The assumed values of (<*/j8)av are given in the second column, and averaging
these with those of Table 14-4 gives values that are used to calculate the values of
F by Eq. (14-20). At the start of the distillation (a/0)av for component A was
5.96, and the assumed value is 5.5 giving an arithmetic value of 5.73 over the incre-
ment. The corresponding value for component B is 3.69 and

•f* » 0.915

BATCH DISTILLATION

391

A similar procedure is employed for the other components. The composition
of the liquid in the still is calculated fronn the values of the fractions unvaporized,
F, and is given in the fifth column of the table. The following column contains
the values of (a/j9)av of the second column raised to the fourth power and multi-
plied by the mol fraction in the liquid. The XD, PL, 0av, and («/0)av values are
calculated in the same manner as in Table 14-4. The calculated values of («/0)av
are in good agreement with the assumed values. The calculations for the next
increment are made in a similar manner using F' as the fraction unvaporized for
the increment. The results are given in Table 14-6.

TABLE 14-6

Comp.

As-
sumed

(«/W.T

Mols re-
maining
in still
- 0.25F

ex.-

(*)

W»v

5.0

0.5

0 3

0.075

0.098

60.6

0 58

2.19

1.595

5.01

3.5

0.86

0.786

0 197

0 257

38.9

0.372

1.29

1.145

3.49

1.97

0 986

0.98

0 245

0 320

4.77

0 0456

1.028

1.014

1.97

1.0

0.999

0 999

0 25

0 325

0.325

0 003

1 002

1.001

1.0

0.767

In this case the values of («/0)av were assumed, and F' for A was taken as 0.5.
This latter value indicates that the increment is to reduce the mols of A in the still
left after the step of Table 14-5 by one-half. The values of F' for the other com-
ponents were calculated from F^ == 0.5 and the average of the assumed value of
(a//J)av of Table 14-6 and the calculated values given in the last column of Table
14-5. The values of F were obtained by multiplying F' by the corresponding F
value of the preceding table. The rest of the calculations are made in the same
manner as t}iose of Table 14-5.

The values of yr calculated by this method are plotted in Fig. 14-7 as a function
of mol fraction of the original charge distilled. The total reflux results of Fig.
14-5 are also included in the dotted curves for comparison. As would be expected,
the separation with the finite reflux ratio is not so sharp as for total reflux. This
decrease in the degree of separation is particularly marked for the intermediate
fraction and is least serious in the case of the least volatile fraction. The inverted
and split-towers systems discussed for total reflux would be effective in increasing
the purity of any given fraction.

This approximate method is most accurate for high values of (0/D) and rela-
tively few total plates. It reduces to the total reflux method for (0/D) «• *>.
When the number of plates is large, they tend to pinch at the equilibrium curve
and the arithmetic average for ft is not satisfactory.

Finite Reflux Ratio with Column Holdup. The conditions in this
case are similar to those discussed on page 380 for binary mixtures.
The operating lines for each component are curved except for total

392

FRACTIONAL DISTILLATION

reflux, and it is very difficult to evaluate the position of these lines.
The total reflux relations given on page 382 for a binary mixture do not
apply exactly in this case because it was assumed that the unit was
operating with a variable reflux ratio, producing a constant overhead
composition. In the case of a multicomponent mixture, it is generally

t.O

02 0.4 0.6

Mol fraction distilled
FIG. 14-7. Fractionation curve for multicomponent mixture at finite reflux ratio.

impossible to keep the overhead composition constant for all com-
ponents as product is withdrawn. In certain cases, the distillate may
be reasonably constant, and a similar analysis can be made.

The design methods for batch distillation allowing for liquid holdup
in the column are very unsatisfactory, and it is a field that should be
actively studied in view of the importance of the operation. Improve-
ments in the design calculations for multicomponent mixtures with no
liquid holdup in the column are also needed.

References

1. COLBUKN and STERN, Trans. Am. Inst. Chem. Engrs., 37, 291 (1941).

2. EDGEWORTH-JOHNSTONE, Ind. Eng. Chem., 35, 407 (1943); 36, 482, 1068 (1944).

3. ROSE, WELSHANS, and LONG, Ind. Eng. Chem., 32, 673 (1940).

CHAPTER 15
VACUUM DISTILLATION

Reduced pressures are frequently used in distillation processes for
the purpose of lowering the temperature required. This is frequently
important in the distillation of organic substances that are thermally
unstable and would decompose if boiled at normal pressures. In addi-
tion to the reduced thermal degradation, the lower temperature often
modifies the relative volatility or the degree of separation involved.

In the design methods considered in the preceding chapters, vapor-
liquid equilibrium was used as a basic criterion. The equilibrium con-
dition between a vapor and a liquid is a dynamic condition in which
equal numbers of molecules of each species enter and leave the liquid
per unit time. Langmuir (Ref. 5) derived an expression for the rate
of this interchange based on kinetic theory considerations. The num-
ber of molecules striking a unit of surface per unit time for a perfect
gas is

Pu /i c 1 \

(15-1)

where P = pressure

u = average velocity of molecules
E = gas-law constant
T = temperature

n = number of molecules striking a unit area per unit time
The arithmetical average velocity of the molecules is

(15-2)

where M is the molecular weight, and the mass, m, striking a unit area
per unit time is

Langmuir has investigated the fraction of the molecules striking a sur-
face that rebound, and his results would indicate that Essentially all
the molecules striking the surface would enter and not be reflected.

394 FRACTIONAL DISTILLATION

Assuming that none are reflected, Eq. (15-2) can be used to calculate
the. rate of evaporation from a liquid at equilibrium with a vapor.
The same relation has been used to predict the rate of evaporation
from a liquid even when the vapor is not in equilibrium with a liquid.
This may be approximately true for the nonequilibrium case, but there is
undoubtedly some interference of the molecules in the vapor with those
evaporating, and the use of Eq. (15-3) to calculate the absolute rate of
evaporation probably gives results that are somewhat low. For the
purposes of this discussion, Eq. (15-3) will be used as the basis of esti-
mating the absolute evaporation rate from a liquid.

A consideration of Eq. (15-3) indicates that the vapor pressure is the
most important factor in determining the rate of evaporation. Molecu-
lar weight and temperature are of less importance. The same reason-
ing that was used to develop this equation can be applied to each com-
ponent in a mixture, giving for component A,

It is instructive to compare this absolute rate of evaporation with
the rate of mass transfer obtained in an atmospheric pressure distilla-
tion of benzene and toluene. Assume that the liquid phase is equi-
molal in benzene and toluene and that a vapor bubble % in. in diam-
eter changes 10 mol per cent in composition for a 0.1-sec. contact with
the liquid. This liquid composition corresponds to an equilibrium
temperature of 92.4°C. and an equilibrium vapor composition of 0.71
mol fraction benzene. By Eq. (15-3) the rate of evaporation of ben-
zene is

/ 2(3.1416) (8.316 X 107) (365.4)
= 14.5 g. per sq. cm. per sec.
= 53.5 tons per sq. ft. per hr.

A similar calculation for toluene gives 6.4 g. per sq. cm. per sec.
In the actual experiment it is assumed that the exchange is equimolar,
and the rate of exchange is

r/2.54V.ir 273 1/O.A
L\ 2 / 6 J [22,400(365.4) JVO.l/

•• 3.5 X 10~5 g. mols per sq. cm. per sec.

e 2.7 X 10~8 g. benzene per sq. cm. per sec.

o 3.2 X 10~8 g. toluene per sq. cm. per sec.

VACUUM DISTILLATION 395

The actual rates of mass transfer are about 0.03 per cent of the theo-
retical evaporation rates. Under the conditions of mass transfer, the
true vapor-liquid equilibria do not exist at the interface because mole-
cules are leaving the liquid phase at a different rate than they are
returning. In the case of benzene, more molecules are leaving than
return while for toluene the reverse is true. In the case just con-
sidered, the net removal from the interface is so small in comparison
to the interchange rate that equilibrium should be closely attained at
the interface. If the net removal is made a large percentage of the
interchange rate, normal vapor-liquid equilibrium will not be obtained.
In the extreme case all the molecules that evaporate could be removed,
and the relative rate for two components would be

( .

( }

If an equilibrium vapor were removed, the ratio of the components
would be PA/PB, and the relative evaporation rate differs by the molecu-
lar weight term. Thus, the composition of an equilibrium vapor and
the one obtained by removing all the molecules that evaporate will
generally be different unless the molecular weights are the same. It
should be possible in some cases to separate normal azeotropic mix-
tures by the evaporative technique, while other mixtures that would
give no separation by this method could be handled by equilibrium
vaporization. The azeotrope for the system ethanol-water should
give a vapor by the evaporative method of considerably different com-
position than the liquid due to the high ratio of the molecular weights.

It should be possible to obtain a vapor composition anywhere
between the true equilibrium value and that given by Eq. (15-5) by
adjusting the net removal rate relative to the evaporation rate. A
system with a net removal rate equal to the evaporation rate has been
termed "molecular distillation" by Fawcett (Ref. 2) and "unob-
structed-path distillation" by Hickman (Ref. 3). Hickman has used
the term "molecular distillation" for systems having mean free path
for the vapor molecules comparable to the distance between the evapo-
rating and condensing surfaces. In this text the term molecular dis-
tillation will be applied to those cases in which the net removal rate is
a high percentage of the absolute evaporation rate.

The benzene-toluene example gave low rates of transfer relative to
the absolute evaporation rate, because (1) the interchange process
introduces diffusional resistance and (2) the vapor approaches equilib-

396 FRACTIONAL DISTILLATION

rium with the liquid thereby decreasing the net rate of transfer. In
order to approach molecular distillation conditions, it is necessary to
increase the net rate at which molecules are removed relative to the
rate at which they evaporate. This can be accomplished by increasing
the removal rate and decreasing the evaporation rate.

For either type of vaporization, the general consideration of solution
laws apply and can be used to predict the results of modifying the
liquid phase. Thus it would be possible to modify the composition of
the vapor removed in molecular distillation just as in azeotropic or
extractive distillation.

VACUUM AND STEAM DISTILLATION

For pressures down to about 1 mm. Hg abs., the distillation opera-
tion can be carried out in a manner similar to those at higher pressures,
and the problems relate to reducing the pressure drop for vapor flow
through conventional equipment. The pressure drop of bubble-cap
plates can be reduced to the order of 2 mm. Hg per plate (see page
404), and special spray-type plates can give pressure drops as low as
0.5 mm. Hg per plate. With such contacting units it is obvious that
even a few plates will necessitate a still pressure of several millimeters
even with a high vacuum at the condenser. Packed towers can give
low pressure drops, and still pressures of the order of 5 to 10 mm. Hg are
obtainable with reasonable tower sizes and rates of distillation. How-
ever, owing to the poor vapor-liquid contact, they are not widely used
for such operations. The contact is particularly poor in this case
because of the low volumetric ratio of liquid to vapor.

Lower distillation temperatures can be obtained by the use of steam.
In this case the steam does not usually condense in the tower, and the
plates contain only the high boiling organic material. The steam acts
as an inert carrier that is easily condensed and does not have to pass
through the vacuum pump. In some cases the temperature gradient
in the distillation column may be so great that the steam will con-
dense in the upper portions, and it should be withdrawn. If too much
water collects on a plate, it may seriously interfere with the fractiona-
tion; if the water runs down the tower, it will vaporize on the lower
plates and this steam recycle in the tower can overload the unit and
seriously interfere with the fractionation. Theoretically it is possible
to fractionate a material with a very low vapor pressure by the use of
steam distillation, but the steam consumption increases as the vapor
pressure on the component decreases. Reduction of the total pressure
reduces the steam consumption, but if the vapor pressure of the com-

VACUUM DISTILLATION 397

ponent at the distillation temperature becomes less than 0.1 mm. Hg,
the steam consumption becomes excessive for most cases. The rectifi-
cation calculation for such distillations can be made in the usual plate-
to-plate manner.

The vapor-liquid equilibria at pressures doWn to 1 mm. Hg are not
greatly different from those at higher pressure. The relative volatility
of a binary system may either increase or decrease as the pressure is
reduced. For example, in a mixture of oleic and stearic acids, oleic is
the more volatile at temperatures above 100 to 110°C., while below
these temperatures it is the less volatile. For mixtures that obey
Raoult's law the relative volatility generally increases as the tempera-
ture is decreased because the less volatile constituent has the higher
latent heat resulting in a high temperature coefficient of vapor pressure.

The calculations of the separation to be expected as a function of
the reflux ratio and the number of theoretical plates for vacuum dis-
tillation are completely analogous to those for higher pressure opera-
tion. The difficult design problems are those related to obtaining
efficient contact between the liquid and vapor with the low pressure
drops available.

MOLECULAR DISTILLATION

This type of operation has been applied to the distillation of mate-
rials that have very low vapor pressures at the maximum operating
temperature. The available pressure drops in such cases would be too
low to obtain practical production rates in conventional equipment,
but by operating such that the rate of distillation is approximately
equal to the absolute evaporation rate of the liquid reasonable capaci-
ties can be obtained. The most common method of obtaining the
molecular distillation conditions is to carry out the operation at a high
vacuum (0.01 mm. Hg or less) and to place the condensing surface so
that it is parallel to the evaporating surface and in close proximity to
it. The condenser is operated at a low temperature to limit the
reevaporation. In order to obtain satisfactory absolute rates of evapo-
ration, it has been found that as an approximate rule the temperature
should not be lower than 100°C. below the temperature at which
the vapor pressure of the substance being evaporated is 1 to 5 mm.
Hg abs.

Even with molecular distillation, the rates of evaporation obtained
are low when the vapor pressure is less than 0.01 mm. Hg abs. Thus
for a material having a molecular weight of 400 and a vapor pressure of
0.01 mm. Hg at 100°C., the absolute evaporation rate by Eq. (15-3)

398 FRACTIONAL DISTILLATION

would be only 0.02 X 10~* g. per sec. per sq. cm. This is much lower
than the rate estimated on page 394 for normal vapor-liquid inter-
change. High-molecular-weight polymers would have low evapora-
tion rates regardless of the vacuum.

It should be possible to obtain results similar to molecular distillation
at higher total pressures by a high degree of turbulence in the space
between the condenser and the evaporating liquid in order to obtain
rapid mass transfer.

In the high-vacuum method of operating, it is usually suggested that
the distance between the condenser and the evaporating surface should
be of the order of the mean free path of the molecules in the vapor.
Jeans (Ref. 4) gives the following equation for the mean free path
(M.F.P.) of a molecule:

where M.F.P. = mean free path, cm.

p = molal density = molecules, cu. cm.
B 1.75 X 1019(P/!T) for a perfect gas
d » diameter of molecule, cm. (As an approximate rule
use cube root of 6/rr times the liquid volume per
molecule)

P = absolute pressure, mm. Hg
T ~ absolute temperature, °R.

Thus for a material that has a molecular weight of 500 and a liquid
density of 0.9, the mean free path at a pressure of 10~3 mm. Hg and
400°F. would be

1023)T
» 0.77 cm.

However, it does not appear to be necessary to make the mean free
path as large as the distance between the condenser and the evaporat-
ing surface to obtain molecular distillation conditions. Bronsted and
Hevesy (Ref. 1) obtained separations of mercury isotopes that cor-
responded closely to molecular distillation rates under conditions where
the condenser was separated from the evaporating mercury surface by
a distance approximately 100 times the mean free path. Taylor (Ref,
7) distilling petroleum fractions found the rate of evaporation to be
independent of the total pressure over a range corresponding to mean

VACUUM DISTILLATION 399

free paths of 0.01 to 10 times the clearance between the condenser and
the evaporating surface. He also found that noncondensable residual
gas at pressures up to the vapor pressure of the liquid being distilled did
not materially lower the distillation rate. Higher residual gas pressure
can cause appreciable lowering of the rate.

The most common type of high vacuum molecular distillation still is
the vertical-tube falling-film unit, a schematic diagram of which is shown
in Fig. 15-1. The liquid to be distilled is first degassed. This is essen-
tial if splashing in the distillation unit is to be avoided. This liquid
then flows down in a film on the outside of the inner tube which is
internally heated. The inner surface of the outer tube is the condenser
which can be air- or water-cooled. For a high rate of distillation, the
clearance between the two surfaces should be relatively small, but if
they are too close, any noncondensable gas released at the bottom of
the still will have difficulty flowing out of the unit. A clearance of 6.4
to 1.0 in. appears to be about optimum for a unit 2 to 4 ft. long.

In such a falling-film unit a molecule moving from the evaporating
liquid to the condenser encounters a number of resistances: (1) diffu-
sional resistance from the interior to the surface of the liquid, (2)
evaporational resistance, (3) resistance to transfer in the vapor, (4)
resistances in condensation.

The resistance to condensation is small, and the resistance to vapor
transfer is made small by the use of low pressure and by keeping the
condenser close to the evaporating surface. The evaporating rate of
the molecules in the surface is chiefly a function of temperature which
should be kept as high as possible without thermal degradation or
bubbling of the liquid which throws unvaporized material over to the
condenser. In most cases, the limiting rate is diffusion in the liquid
phase. Owing to the large size of the molecules and the viscosity of
the liquid, the mass-transfer rate is very low. The outer surface of the
liquid is depleted of the more rapidly evaporating molecules a short
distance from the top of the unit, and the surface then has a higher con-
centration of the less volatile molecules than the average composition
of the liquid. This reduces both the evaporation rate and the degree
of separation obtained. A small amount of large-size, essentially non-
volatile, material in the liquid can give a serious blocking of the surface.
Owing to this effect, increasing the length of the apparatus does not
give a proportional increase in the amount of evaporation. For this
reason, the falling-film units are seldom made over 2 to 4 ft. high.
Several methods can be used to reduce the effect of this surface block-
ing: (1) Various mechanical devices have been proposed to cause mix-

400

FRACTIONAL DISTILLATION
V

Vacuum
system

To switch
board

to aspirator and
pressure control

Pressure

measuring

equipment

Oistilhte
receiver
FIG. 15-1. Diagram of falling-film unit.

ing of the falling film. (2) The liquid circulation rate can be increased,
resulting in a lower percentage evaporation per pass through the unit.
A. high percentage evaporation can be obtained either by recirculating
the liquid or by using several units in series with mixing between each
unit. (3) A high liquid flow rate can be used to cause the outer por-
tion of the falling film to be in turbulent rather than laminar flow.
This type of operation may also necessitate recirculation or series

VACUUM DISTILLATION 401

operation to obtain a high percentage of the liquid distilled. (4) The
unit may be modified to give a continually increasing surface. For
example, the inner tube can be made conical with the small end at the
top. As the liquid flows down, it must increase in area which will
present fresh surface for evaporation. Hickman (Ref. 3) has devel-
oped a spinning-plate type of film unit in which the liquid is fed at the
center and flows across the spinning plate due to centrifugal force.
The plate is heated, and the condenser is placed parallel to it. Because
of the high centrifugal force, it is possible to obtain thin films which
mean large surface area per unit volume of liquid, and the increasing
diameter of the plate requires the formation of new surface as the
liquid flows outward. The spinning-plate unit is effective for the pur-
pose of increasing evaporation rate, but other methods would appear
to be simpler for large-scale units.

Besides keeping the temperature at a low level, molecular distilla-
tion holds only a small volume of the liquid at the evaporation tempera-
ture and thereby reduces the thermal degradation. The spinning-
plate type of still is particularly effective in this respect because of the
thin film obtained.

The thermal efficiency of a molecular distillation is low. Fawcett
(Ref . 2) has given a heat balance on a unit distilling triolein at 240°C.
with the condenser at 25°C. The data are summarized in the follow-
ing table:

Per Cent

Preheat of liquid . . .8

Radiation ... ... 59

Latent heat of evaporation 9

Conduction ... 24

Only 17 per cent of the total heat is usefully employed, the other 83
per cent is lost by heat transfer. The loss by radiation could be
reduced somewhat by increasing the temperature of the condenser, but
this might reduce the effectiveness of the distillation.

Another drawback to molecular distillation is the fact that an effec-
tive rectification system has not been developed. Greater separations
than are equivalent to a single distillation stage have been obtained by
repeated distillations in the manner described on page 102, but they
are tedious and difficult to perform.

Schaffner, Bowman, and Coull (Ref. 6) have described a vertical
falling-film multiple distillation column that can be employed for frac-
tionation under vacuum. The wall is made up of short sections with

402 FRACTIONAL

alternate sections being heated and cooled. Vapor condenses on the
cooled section and flows down to the heated section below where it is
partially vaporized, and the action is repeated on succeeding sections.
By this series of partial condensations and vaporizations, an enrich-
ment of the vapor in the more volatile component is obtained. Because
vapor must flow between sections, molecular-type distillation is not
possible. Such a unit is very sensitive to operating conditions, and
the heat added and removed in the successive sections must be well
balanced or the vapor will all condense or the liquid will all vaporize.
Owing to the temperature changes during distillation, the heat supplies
will need frequent readjustments to obtain optimum operation.

The very low pressures involved almost preclude an effective contact
similar to that obtained in normal rectification. It would appear that
the use of higher pressures with a high degree of turbulence to obtain
the necessary rate of distillation at the low temperatures involved
would offer better possibilities for rectification than the use of high
vacuum. By the use of a low molecular weight gas, such as hydrogen,
to maintain the pressure, it should be possible to obtain high transfer
rate with a moderate degree of gas turbulence.

Because of the high cost per unit of product distilled, the use of
molecular distillation has been limited in its application to the separa-
tion of relatively expensive materials that are sensitive to thermal
degradation.

References

1. BRdNSTED and HEVESY, Phil Mag. (VI), 43, 3 (1922).

2. FAWCETT, /. Soc. Chem. Ind., 58, 43 (1939).

3. HICKMAN, Chem. Rev., 34, 51 (1944).

4. JEANS, "An Introduction to the Kinetic Theory of Gases," pp. 131-155, The
Macmillan Company, New York, 1940.

5. LANGMTJIB, Phys. Rev., 8, 149 (1916).

6. SCHAFFNER, BOWMAN, and COULL, Trans. Am. Inst. Chem. Engrs.j 39, 77 (1943) .

7. TAYLOR, Sc.D. thesis in chemical engineering, M.I.T., 1938.

CHAPTER 16
FRACTIONATING COLUMN DESIGN

The design calculations given in the previous chapters involved the
mathematical problem of determining the number of theoretical plates
and did not consider whether or not such performance could be
obtained. This chapter will consider the mechanical design problems
encountered in attempting to attain the desired degree of contact
between the vapor and liquid and the necessary fluid and vapor flow.
The fractionating column must bring the liquid and vapor into counter-
current contact (or some approximation of this type of flow), and it
must be constructed to furnish the necessary pressure drops (or liquid
heads) to give the desired flow conditions.

The liquid gradients and the pressure drops encountered by the
liquid and the vapor are important for several reasons:

1. The conditions may be such that the liquid will not flow down the
column, countercurrent to the upflow of the vapor. Such action pre-
venting rectification is termed "flooding" or "priming." Any column
has a maximum operating capacity, above which this condition will be
encountered, but the capacity of a column of given diameter with fixed
plate spacing will be a function of the design.

2. The contact between the vapor and liquid is a function of the
liquid gradients and the gas pressure drops.

3. The pressure drop per plate is of vital importance in vacuum
rectification.

BUBBLE-CAP PLATES

The bubble-cap plate is the most common vapor-liquid contacting
device employed for fractional distillation. Its purpose is to bring the
vapor and liquid into intimate contact so that the necessary mass
transfer can be effected. This requires means for bringing the liquid
down the column from plate to plate, across the plates, and into contact
with the vapor. The design of such a unit involves obtaining the
desired flow conditions of the liquid and vapor and the contact between
the two. Figure 16-1 gives a schematic drawing of the cross section of
a bubble-cap plate. The liquid flows onto a plate from the down pipes,
flows across the plate contacting the vapor, flows over the outlet weir

403

404

FRACTIONAL DISTILLATION

and through the down pipe to the plate below. The vapor flows up
through the liquid on the plate and to the space above. .For these
flows to follow the desired pattern, the necessary pressure drops and

hydraulic heads must be available.
The liquid meets resistance in
the down pipes, in flowing across
the plate, and in flowing over the
weirs. The frictional resistance in
the down pipes is handled by mak-
ing them of adequate cross section
and height to take the liquid load.
For a given cross section, in general
the liquid-handling capacity of the
down pipes will increase with in-
creasing height but it is desirable
to keep the height low in order to
reduce the plate spacing. On the
other hand, down pipes of large
cross section reduce the available
plate area for vapor liquid contact. In flowing across the plate the
liquid decreases in depth owing to the frictional and kinetic effects
giving the so-called hydraulic gradient. An overflow weir is employed

FIG. 16-1. Schematic cross-sectional
diagram of bubble-cap plate column.

Tffff

Riser ->-

gPHrtj*^ Bottom of cap

/////////77/7///7/^ ~- Pi» t*

FIG. 16-2. Cross section of bubble cap.

to maintain the liquid level at approximately the desired level. These
various factors are considered in detail in later paragraphs.

Pressure Drop for Vapor Flow. The vapor meets its main resistance
in passing through the bubble cap and the liquid on the plate. Con-
sider the section of a cap shown in Fig. 16-2. The vapor from the plate

FRACTIONATING COLUMN DESIGN 405

below enters the riser at (1) and encounters a pressure drop due to the
reduction in cross section. There is a frictional drop in the riser from
(1) to (2), a reversal loss (2) to (3) between the top of the riser and the
cap, and then a frictional drop against the ca£ from (3) to (4). The
vapor then passes through the slot and up to the vapor space above
the liquid. It is convenient to group these into three pressure drops:
the pressure drop through the riser and cap, hc] the loss in pressure in
flowing through the slots, hs; and the pressure drop due to the liquid
head above the slots, hL.

Pressure Drop through Risers and Cap. This loss is chiefly a kinetic
velocity effect due to the changing cross-sectional areas. The pressure
drop in inches of the liquid equivalent to the kinetic head is

fc,-^* (16.1)

20C PL ^ '

where ha = kinetic head, inches of liquid of density pL

VR = maximum velocity in riser, between top of riser and cap,

or in annulus between riser and cap, f.p.s.
gc = conversion constant

= 32.2[(ft.)(lb force)]/[(sec.2)(lb. mass)]
pv = density of ga&, same units as pL

The actual loss in pressure, hc, from (1) to (4) should be a function of
hH. The data of Mayer (Ref. 24), Schneider (Ref. 28), and Dauphin6
(Ref . 5) on several 3-, 4-, and 6-in.-diam.eter caps with risers from 2 to 4
in. in diameter gave ratio of hc/hn from 4.7 to 6.3. Souders (Ref. 34)
gave results indicating a ratio of 2.9, Kirkbride (Ref. 20) recommended
3.2, and Edminster (Ref. 7) suggested 7.8 but included the pressure
drop through the slots. In view of the extensive data by the first three
investigators, a value of the ratio equal to 6.0 will be used giving

hc = 1.1F| (16-2)

Pressure Drop through Slots. The pressure drop through the slot,
h8j is evidenced by the liquid level in the cap being lower than the top
of the slots. The value of ha will be taken as equal to the difference in
the pressure of the vapor in the cap at position (4) and the liquid out-
side the cap at the top of the slot. The slot action varies with the rate
of flow. At low rates of flow an intermittent type of bubbling action is
obtained. Owing to the surface tension, the pressure within the cap
rises until the liquid under the cap is depressed an appreciable distance
below the top of the slots. When the pressure is sufficient to overcome

406 FRACTIONAL DISTILLATION

the surface tension, there is a rapid flow of vapor reducing the pressure,
and the cycle is then repeated. Thus a pressure drop across the plate
greater than the height of the liquid above the slots is necessary to
initiate vapor flow. At higher rates of vapor flow, bubbling becomes
continuous, and the pressure drop through the slots becomes greater
than that necessary to initiate flow. At still higher vapor rates, the
vapor blows open channels through the liquid. For all cases, the
pressure drop through the slots is greater than for flow through the
slots on a dry plate. The calculation of h, is also complicated because
the velocity of the vapor in the cap approaching the slots may be equal
to or greater than the slot velocity, and a simple orifice-type equation
is not applicable.

A definite value of h. is necessary to initiate flow. It then increases
slowly with the rate of vapor flow as the slot opening increases, and
when the slots are completely open, the pressure drop increases rapidly
with increase in vapor rate through the slots.

For values of he less than the height of the slot, the data of Carey
(Ref. 3), Griswold (Ref. 14), Mayer, Schneider, and Dauphin^ can be

correlated by

. 0>67

h, = 0.12 i + K (hfV. J— ^-) (16-3)

PL \ \PL — pv/

where h8 — slot opening or pressure drop through slots, in.
7 = surface tension, dynes/cm.
PL = liquid density, Ib. per cu. ft.
pv = vapor density, Ib. per cu. ft.
fe* = slot height, in.

7, « velocity based on total slot area, f.p.s.
K = see Table 16-1

^-^ = cu. ft. per sec. per total ft. of slot width
12

The variation in the values of K is believed to be due mainly to the
effect of the ratio of velocity in the cap to slot velocity. In the case of
the small caps, this ratio was high, and it is suggested that for other
cases the following relation be employed.

The pressure-drop values for a number of different caps are plotted
in Fig. 16-3 on the basis of the groups of Eq. (16-3)* The data for,the
large caps are higher than for the small caps, and the slope of the best

FRACTIONATING COLUMN DESIGN

407

0.8

TABIiB 16-1

Cap K

6-in.-diameter galvanized-iron cap:

Slots, H by (0.5, 1.0, 1.5, and 2.0) in.

Slots, Ke by, (0.5, 1.0, and 1.5) in.
6~in.-diameter cast-iron cap:

Slots, H by (0.5 and 1.0) in 1.0

4-in .-diameter copper cap:

Slots, ^2 by M in 0.6

4-in .-diameter cast-iron cap:

Slots, Ke by (0.5 and 1.0) in 0.7

Tunnel cap, slots KG by 0.875 in

4-in. caps, tangential slots }/% by 0.75 in.

0.55

10
08
06

rf
* 01

fr—^

-9

* j

°*+

•^ */

< Q

'& *

^008
006

0.04
002
0.0.

./£

Legend

;

(14
&
(26
(5,
' (5
(5
(St
(5
(5

fICG

.X"^

Tunnel 'cap-slots*/* 'w/da XQBTS'high

<. Styles /*%"*,&
\ 6* galvanized iron cap *sM$ fox Q5*
> 6'galvamzed iron cap-slots '/8"x/'f5'anc/20
7 6'galvanlzed iron cap -s/ofs Vie'xOS '
3 6'casfironccp'Stofs%"X05'andJO*
k 4" copper cap - slots %/ X 05 *
f 4' casf iron cap • slots fa XQ5 and /O *

^X^ c

)

/ * \

a)
) """

; ._
;
;

I
1

)
)

1 02 04 0.6 OB 1.0 2 468 10 ZQ 40 6C

FIG. 16-3. Pressure drop through bubble-cap slots.

line through the data for any one cap is approximately % for most
cases. The line drawn for all the caps is

h* = 0.12 3- + 0.058(/i*F,)0-7fi (16-5)

and, including the square root of the density terms, Eq. (16-5) becomes

(/ \0.75
h*V, J— e^-J (16-6)

PL — Pv"/

408 FRACTIONAL DISTILLATION

It is believed that Eq, (16-3) with a value of K by Eq. (16-4) gives
better results for specific cases, but that Eq. (16-6) is helpful for general
cases.

With the usual plate design the slots become completely open for
vapor flow at values of hs less than A* due to the fact that the average
density of the vapor-liquid mixture around the cap is only ^ to J^
that of the liquid. For studies of single caps this lowered-density effect
is of less importance. At vapor rates higher than those corresponding
to complete opening of the slots, the relation between hs and V8 should
be modified. In some cases the bottoms of the caps are raised above
the plate to allow excess vapor to escape. With this arrangement,
high capacities can be obtained without excessive pressure drops,
although the vapor-liquid contact for such operation is probably of low
effectiveness. In other cases, the bottoms of the caps are sealed to the
plates, and any excess vapor is forced through the slots. For this con-
dition the following equation is suggested for h8 greater than h* :

(16-7)

where 7? is vapor velocity in Eq. (16-3) for h8 equal to h* and hf is
the value of h8 for slots completely open. For conventional plate
arrangements it is suggested that hf be taken equal to ^h*.

Pressure Drop Due to Liquid Head above Slots. The pressure drop
due to the liquid head above the top of the slot is customarily taken as
equal to the actual liquid depth above the slots. This liquid depth is
frequently taken as weir height, hw, plus the weir crest, hcr, minus the
height of the top of the slot, hap, thus

AL = hw + her — hap (16-8)

In some cases a correction is added for the additional liquid head at
the cap in question owing to the liquid gradient. This method of
evaluating hL is simple, but the data of Seuren (Ref. 29), Ghormley
(Ref. 11), and Kesler (Ref. 19) show that the actual liquid head in the
aerated section of a plate can be considerably less than that at the weir.
This effect is the result of liquid flow from a nonaerated section to an
aerated section and back to a nonaerated section. This condition will
be discussed in the section on Hydraulic Gradient. The use of Eq.
(16-8) with a correction for the hydraulic gradient will give high values
for the liquid head.

The over-all pressure drop for a plate, hp, is calculated by

hp — hc + ht + hi< (16-9)

FRACTIONATING COLUMN DESIGN

409

Liquid Flow. Weirs and Down Pipes. The liquid depth on a
plate is controlled by exit overflow weirs. The action of the liquid
flowing over the weir is complicated by the action of the caps and by
the restrictions due to the wall. The last row of caps may blow a con-

A Cross flow
Single downpfpes

B Cross flow
Multiple downpipes

C Radial flow

0 Cross flow
Chord weirs

This space
blanked

Partition
Success 7i

^ ^ plates rotated
•* by this amount

Circumferential flow

FIG. 16-4.

F Split flow
Chord weirs
Downflow pipe arrangements on bubble-cap plates.

siderable quantity of liquid over the weir as spray and in surges. The
walls may be so close to the downstream side of the weir that they
interfere with the liquid flow.

A variety of different weir and down-pipe arrangements are employed
(see Fig. 16-4).

In small columns, the overflow from plate to plate is usually carried
in pipes, the upper end of the pipe projecting above the plate surface to
form an overflow weir and maintain a liquid seal on the plate. The
lower end extends into a well on the plate below, thereby sealing the

410 FRACTIONAL DISTILLATION

pipe so that the vapor may not pass upward through it. In larger
columns, straight overflow weirs placed on a chord across the tower
are often used.

Locke (Ref . 22) from a study of circular down pipes with the tops act-
ing as weirs concluded that at least three types of liquid flow were
possible in circular down pipes with liquid seals at the bottom. At
low rates of liquid flow, the top of the pipe acted as a weir, and the
liquid flowed down in a film. As the liquid head was increased, the
pipe became full and sucked vapor bubbles down with it; at still higher
liquid rates, the pipe ran full but did not entrap vapor. The first type
flow occurred for liquid head less than one-sixth to one-fifth of the pipe
diameter, and this type of flow could be represented by the familiar
Francis weir formula:

her - fc m (16-10)

where Q = cu. ft. of liquid per sec.

L = perimeter of inside surface of pipe, ft.
her = head of liquid above top of pipe, in.
k ~ constant, increasing from 3.9 to 4.3 as pipe size was

increased from 0.87 to 2.07 in.

Rowley (Ref. 27) recommends the following equation for the last two
types of flow:

where hf = head of liquid necessary to overcome down-pipe friction

and entrance and exit losses, in.
gc = proportionality constant, 32.2
D = inside diameter of pipe, ft.
V0 * linear velocity of liquid in pipe, f .p.s.
/ = proportionality constant of Fanning friction equation (See
Walker, Lewis, Me Adams, and Gilliland, "Principles of
Chemical Engineering," 3d ed., p. 78.)

For large straight weirs when the downspouts are not running full,
the Francis weir equation may be used. This weir formula is fre-
quently employed neglecting the approach velocity, and in some cases
this may not be a satisfactory approximation. The equation for zero
approach velocity is

Units same as for Eq. (16-10).

FRACTIONATING COLUMN DESIGN

411

To allow for the approach velocity, a correction factor is given in
Fig. 16-5, as a function of Q/L &nd hor + hw, where hw is the height of
the weir. This correction was calculated on the basis that the liquid
approaching the weir had a velocity corresponding to the unaerated
depth of hcr + hw. The actual velocity of approach is probably higher
owing to the caps and vapor. The correction factor is multiplied into
the right-hand side of Eq. (16-12) to calculate hcr.

When the head on the weir becomes high, the liquid carries past the
dam a considerable distance, and the wall may interfere with the flow.
In circular pipes, Locke reported that the streams from the various
sides interfered with each other when the head on the weir at the top
of the pipe was 0.2 of the pipe diameter. This would indicate an over-

FIG. 16-5. Correction factor for Eq. (16-12).

shoot of over twice the head. The effect for straight weirs is probably
less, and Edminster (Ref . 7) has suggested neglecting the portion of the
weir that is closer to the wall than the value of hcr- This would appear
to be a reasonable assumption.

For large downspouts running full, it is recommended that Eq.
(16-11) be used, employing for D four times the hydraulic radius, which
is equal to the cross-sectional area of the downspout divided by the
perimeter.

When the liquid head over the weir is low, the discharge from one
side to the other will vary if the top edge is not level. This variation
can be reduced by using a V-notched top which reduces the effective
weir length at low rates of flow. In general, it is desirable to keep the
head over the weir low because this reduces the variation of liquid
depth on the plate for different operating rates. Values of her of 1.0
in. are common, but they seldom exceed 3 in.

In some cases, under- and overweirs are used at the outlet to handle

412 FRACTIONAL DISTILLATION

two liquid layers on a plate. Such an arrangement can also be used to
hold back foam from the down pipe. The clearance between the
underweir and the plate must be adequate to handle the liquid load.

The bottoms of downflow pipes must have enough clearance to allow
the liquid to flow easily, but they should not allow vapor to by-pass up
through them. This vapor by-passing is usually prevented by a posi-
tive seal which holds liquid above the bottom of the down pipe (see
Fig. 16-1). In large columns, this seal is often made an inlet weir
which serves to distribute the liquid as well as seal the down pipe.

For the reversal loss at the bottom of the down pipe, it is recom-
mended that a loss equal to one kinetic head be used with a coefficient
of 0.6.

= 0.5F1,

where ho = loss in head at bottom of down pipe, in.

VD = maximum velocity at bottom of down pipe, f.p.s.

Liquid Gradient. One of the important factors that must be con-
sidered in the design of a bubble plate is the liquid gradient across the
plate. It is obvious that the liquid level will normally be higher at the
liquid inlet than at the outlet. In small towers, this difference in level
offers no serious difficulties, but in towers of moderate and large diam-
eters it can become so great that the vapor distribution is poor and the
overflow may by-pass the plate by dumping through the caps.

The gradient is due to the resistance to liquid flow across the plate
and results from (1) friction with the caps and the plate, (2) eddy losses
in the liquid due to repeated acceleration and deceleration, and (3)
resistance due to the effects of vapor flow.

Experimental data on this gradient have been published by several
investigators (Refs. 10, 11, 12, 13, 18, 19,29). Gonzales and Roberts
(Ref. 12) studied a plate with 4%~in.-diameter caps which were 6J^ in.
tall using air and water. Their column was rectangular in shape with
12 rows of caps in the direction of liquid flow. Based on liquid flow
around staggered pipes, they developed the following equation for the
gradient as a function of the number of rows of caps:

ft2.8 = A - En (16-14)

where h ~ liquid depth on plate

A,B SB constants
Their data were taken at liquid levels less than the top of the caps

FRACTIONATING COLUMN DESIGN 413

and agreed well with the equation for both aerated and unaerated con-
ditions. A and B were functions of both the water and air rates.

Bijawat (Ref. 2) reviewed the data obtained by Gonzales and
Roberts and, on the basis of orifice-type flow of the liquid between
the caps, suggested

h* = A' - B'n (16-15)

This equation correlated the data about as well as Eq, (16-14).
Seuren (Ref. 29), Ghormley (Ref. 11), and Kesler (Ref. 19) obtained
data on a rectangular plate with 10 rows of 4-in. caps in the direction
of liquid flow. Good, Hutchinson, and Rousseau (Ref. 13) investi-
gated the liquid gradient on a rectangular plate having 12 rows of 3-in.
caps for a number of operating conditions.

Klein (Ref. 20<z) has investigated the factors that cause the loss in
head of the liquid flowing across the plate. His data indicate that
with no vapor flow there is essentially no hydraulic gradient even at
liquid rates considerably greater than those normally employed. It
was concluded that the loss in head was due to high f rictional losses for
the flow of the vapor-liquid mixture. The drag per unit area was
determined by measuring the force on a plate suspended in the aerated
liquid, and values ten times as large as for unaerated liquid flowing at
the same linear velocity were found. Abnormally high f rictional
losses have also been reported for the flow of mixtures of vapor and
liquids in pipes.

Klein confirmed the observation of Ghormley and Kesler that the
hydrostatic head of the liquid in the aerated section was frequently
less than that at either the inlet or outlet to the plate. Klein, Kesler,
and Bloecher (Ref. 2a) measured the potential head of the liquid by
determining the hydrostatic head as a function of depth and then
integrating up from the bottom of the plate. The potential head is
equal to the number of inches above the tray at which all the water
present would give the same total potential energy as the aerated
liquid. For a nonaerated liquid the potential head, according to this
definition, is equal to one-half the liquid depth. It was shown that
the potential head decreases progressively across the plate. It is this
head, and not the hydrostatic head, which is the driving force for
liquid flow. The flow across a bubble plate can be considered in terms
of five zones. (1) There is the inlet unaerated section in which the
potential head is twice the hydrostatic head. (2) There is the transi-
tion section from the first zone to the aerated region. This transition
occurs a short distance upstream from the first row of caps to about

414 FRACTIONAL DISTILLATION

the second row of caps. In this region (a) the apparent depth of liquid
rises sharply owing to the aeration, (b) the hydrostatic pressure drops
abruptly, and (c) the potential head remains almost constant. For the
potential head to remain constant the hydraulic head must decrease
because some of the liquid is at a higher level. Near the plate the
pressure in the liquid in Zone 1 is higher than in the transition section
and liquid flows towards the caps, but at higher levels the reverse is
true and the liquid flows back (against the normal flow). (3) The
third zone is the main aerated section in which the hydrostatic head
remains essentially constant and there is a progressive decrease in the
potential head. (4) The fourth zone is the outlet transition zone which
extends from the last two rows of caps to the calming section before the
outlet weir. The phenomena are similar to those for the inlet transi-
tion zone, i.e., fa) the apparent depth of the liquid drops sharply, (b)
the hydrostatic head increases abruptly, and (c) the potential head
remains almost constant. In this case there is a flow of liquid back
toward the caps near the plate and toward the outlet section at the
higher levels. (5) The final region is the unaerated calming section
before the outlet weir. These effects are illustrated in Fig. 16-6. At
low air rates there is a decrease in hydrostatic head in the aerated
zone, but a clear liquid layer on the plate extends from the inlet to the
outlet. At higher vapor rates the liquid becomes "completely aer-
ated," and no apparent clear liquid layer remains. This condition is
shown in diagram A of Fig. 16-6. Data on the hydrostatic and poten-
tial head for a completely aerated condition are shown in diagram B.
The sharp drop and rise in the hydrostatic head at the two ends of the
plate are clearly shown, but there is essentially no change in the aerated
section. The value of twice the potential head is plotted so that it
will be numerically equal to the hydrostatic head in the two nonaerated
sections. The potential head decreases regularly across the plate.

Referring again to diagram 5, Fig. 16-6, the hydrostatic head given
is the value at the bottom of the plate. If values are determined in
planes parallel to, but above, the plate, it is found that the hydrostatic
head decreases more rapidly with height in the nonaerated sections
than in the bubbling zone owing to the lower density of the vapor-
liquid mixture. Eventually, at a height less than h<» the hydrostatic
head in the outlet transition zone (Zone 4) becomes equal to that in
Zone 5, and at higher heights above the plate the aerated section has
the higher head. Thus, the hydrostatic head is a complicated func-
tion of the position and the distance above the plate.

Klein found that for a given liquid rate the hydraulic gradient

FRACTIONATING COLUMN DESIGN

415

increased with increasing vapor rate until the liquid-vapor mixture in
the aerated section had an apparent density approximately one-third
that of the liquid but further increases in vapor rate did not appreciably
change the loss in head. He termed the condition for the mixture
density equal to one-third the liquid density "complete aeration."

•§.§
H

Liquid vapor mixture

rn n n n n

Zone 3

Diagram A

Inches of wafer

0 — N> 01 ^ 4s

-+.-.

*— — -.

-fJ:

>**.

oote*
~-+^

rf/at

hea

Hyd

**o$h

vtic t

heaa

-~<*p«,
f

-~~ -,

"•**•-*.

5 '

k .

, P

Iming sect

Exi
Wai-
fool
Supe

t~ we
er re

-Off

fffc/,

ir *L
ite *
>/a/<?
&/ ai

'2S$

widi
> vei

ipm
to
octfy

per
>*2.8

1 Inlet cot

ft./s

ec.

Outlet ca

23456789 10
Number of rows of caps in direcfion of liquid flow

Diagram B
FIG. 16-6.

Higher air rates or foaming agents increased the apparent liquid height,
but static pressure probes indicated that for these cases the density
was approximately one-third that of the liquid for the main region of
fluid flow just above the plate and on top of this layer there was a
light froth containing essentially no liquid. Practically all the liquid
flow was accounted for in the layer with the one-third normal density.
The light froth did not appear to have a significant effect on the
hydraulic gradient.

416 FRACTIONAL DISTILLATION

Klein correlated his data and that of other investigators on the basis
of a Fanning-type friction equation.

F = (16-16)

QcTh

n = ^ (16-17)

where F = loss in head from inlet to exit calming sections, ft.
/' = friction factor
Vf = velocity of liquid in foam, f .p.s.

- Qw/(pfL0b)

Qw = liquid rate, Ibs. per sec.
pf = density of foam, Ibs. per cu. ft.
Lo = foam height, ft.
6 = width of plate, ft.
B = length of bubbling section, ft.
gc = conversion constant = 32.2

Th = mean hydraulic radius, ft. = , , °j

The values of /' were calculated from the experimental data and cor-
related as a function of a modified Reynolds number, Ref = (r*7/p/)//*/,
where M/ = viscosity of foam. In making the correlation it was
assumed (1) that the average foam density was one-third the density
of the normal liquid, (2) that the viscosity of the foam, /*/, was one-
third the true viscosity of the liquid, and (3) that L0 was equal to two
times the hydrostatic head in the outlet calming section. Thus,

#e> =!^Z/^ (16-18)

ML
Vf = ^r (16-19)

PLLOO
and

Lo = 2h0 (16-20)

where PL = normal density of liquid, Ibs. per cu. ft.
ML = normal viscosity of liquid, Ibs. per f .p.s.
h0 — hydrostatic head in outlet calming section, ft.
Klein found that the value of/' was a function of the exit-weir height
relative to height of the slot and used an empirical relation to allow

/ h \«

for this effect. He correlated /' ( ? — *2_ ) as a function of Re1,

\h0 — hsp/

where h8p « height of top of slot above plate.

FRACTIONATING COLUMN DESIGN

417

The hydraulic gradient results of several investigators are plotted
in Fig. 16-7. In view of the number of factors and assumptions
involved, the correlation is reasonably good. Most of the results

4.0

1.0

a- 04

0.1

0.01

•P4!

-*•

+\
i<

£k

— JXg-

Klein - T'weir
Klefn-3"\Mefr
Klein -5" weir
Kemp and Py/e
Gooa, Hutch/risi
Ghorm/ey
Kemp and Py/e

0
D

(a/r- water)
>/7 Rousseau

(air - Perch/oroefhy/en

1,000 4,000 10,000 40,000

Re- Modified Reynolds number
FIG. 16-7.

correlated were for essentially standard caps and plate arrangements,
and the relation should be used with caution for plate layouts that
differ significantly.

It will be noted for Fig. 16-7 that /' is essentially inversely propor-
tional to the value of Re'. Combining this relationship with the fact

418 FRACTIONAL DISTILLATION

that for large plates 2L0 is small in comparison with 6 gives

This relation would indicate that the hydraulic gradient is directly
proportional to the liquid flow rate per foot of plate width and to the
length of the bubbling section. The outlet hydrostatic head is prob-
ably the most important factor determining the hydraulic gradient,
and high outlet weirs should be an effective method of reducing the
loss in head.

Equation (16-21) indicates that the gradient is approximately pro-
portional to the liquid viscosity. The liquid viscosities studied ranged
from that of water at room temperature to glycerine. Gardner (Ref .
10) studied the hydraulic gradient on a plate with tunnel caps using
water at different temperatures and concluded that the liquid viscosity
was not a factor. The plate design was unusual in that the liquid flow
was across rather than along the tunnel caps. However, until addi-
tional data are available it is recommended that PL be taken equal to
0.00067 pounds per f .p.s. for all liquids of lower viscosity and equal to
the actual viscosity for those having higher values.

Liquid Head. As shown in Fig. 16-6 the hydrostatic head in the
bubbling section can be less than at the outlet weir. However, the
liquid head above the top of the slots may be greater or less than the
difference in the height of the outlet liquid and the top of the slots
owing to the increased depth of the vapor-liquid mixture. The value
of the liquid head calculated by Eq. (16-8) should be satisfactory for
most cases, but the actual value may differ somewhat due to the aera-
tion effects.

Plate Stability. The term " plate stability" has been used to
describe the vapor and liquid distribution on a plate. , A stable plate
has been defined as one in which all the caps are handling vapor,
although the quantity of vapor per cap may vary widely from one side
of the plate to the other. The plate is stable in the sense that liquid is
flowing across the plate and not by-passing by "dumping" through
the cap risers, but the distribution of vapor may be quite poor.

The term "stable plate" does not differentiate between the various
types of plate action and in this text the following terms will be used.

Uniform vapor distribution — will indicate the condition when each
cap on the plate handles the same amount of vapor per unit time.
Active cap — will indicate that vapor is passing through the cap.

FRACTIONATING COLUMN DESIGN 419

Inactive cap — will indicate that vapor is not passing through the cap.
Completely active plate — will indicate that all caps are active.
Partly active plate — will indicate that only part of the caps are active.
Plate dumping — will indicate that liquid is flowing to the plate below
through some of the cap risers.

The distribution of the vapor among the various caps is a function of
the pressure drops involved. The lateral pressure difference in the
vapor space above a plate is usually small and, when this condition is
true,

hp = hc + h8 + hL = constant

Because hL varies across the plate, hc + h8 must vary. Substituting
the values from Eqs. (16-2) and (16-3) in the above equation gives

(16-22)

PL/ PL \ \PL-~PvJ

and, for a given plate, this can be condensed to

fciF2 + kzV* + hL = hp - 0.12 -£• (16-23)

The right-hand side of Eq. (16-23) would be constant for all parts of
the plate on the basis of the assumptions made, and if the value of hL
varies across the plate, then the vapor flow per cap must vary. This
relation indicates that a given cap will become inactive when ht at
that point is equal to hp — Q.l2(y/pL) ; i.e., when the liquid head over a
slot becomes equal to the total pressure drop minus the pressure drop
necessary to initiate bubbling against the surface tension.

Consider the case for the caps at the inlet side just inactive. By
Eq. (16-23),

where hi is the head of liquid above the slots at the inlet row of caps
where the pressure drop is just sufficient to cause bubbling.

Assuming that the plate is rectangular and that the two terms involv-
ing velocity can be combined,

fc,72 + ktV* - k*Vm t (16-25)

where V ~ vapor rate per row of caps perpendicular to direction of

liquid flow
m » constant between % and 2.0

420 FRACTIONAL DISTILLATION

Then, for any row of caps,

k*Vm « hp - hL - 0.12 -1 (16-26)

Equation (16-26) could be used to evaluate the vapor distribution if
the variation in hi, across the plate were known. It has already been
shown that the variation of the hydrostatic head across the plate where
the liquid is aerated is a function of the height above the plate at which
the head is determined, and the relation for the plane corresponding
to the top of the slots could vary significantly depending on the height
of the slots relative to the hydrostatic heads in the inlet and outlet
calming sections. However, as a first approximation it will be assumed
that

where h£, h°L = liquid head above top of slots at inlet and outlet of

plate

N = rows of caps on plate
n = number of rows of caps from outlet weir
When the inlet row of caps is just inactive, the condition for the out-
let row of caps can be obtained by combining Eqs. (16-24) and (16-26),
giving

k*V? = hi - h°L (16-28)

where V0 is the vapor rate at the outlet row of caps. Thus the pres-
sure drop due to vapor flow for the outlet row of caps is equal to the
hydraulic gradient when the inlet row of caps is just inactive. Equa-
tions (16-27) and (16-28) can be combined to evaluate the pressure
drop and the vapor distribution. In order to simplify the analysis, it
is assumed that the number of rows of caps is large enough that con-
ditions vary approximately continuously across the plate so that inte-
gration instead of stepwise summation can be employed. Thus, for
the total vapor flow,

dn
KS / JN\ &/

A* - h°\l/m (° t n\lf
V dn - - I L L 1 I I 1 — — 1

Yd \ k* ) JN\l N)

(hj - h

\ &3

where VT equals total vapor through plate.

FRACTIONATING COLUMN DESIGN 421

Combining this relation with Eq. (16-28) gives

If there were uniform vapor distribution, V0 would equal (Vr/N),
and the above relation indicates that the pressure drop is increased by
the liquid gradient. Where the inlet row of caps is just inactive, the
pressure drop due to velocity is greater by the factor [(m + l)/m]w,
and this term varies only from 1 .84 to 2.25 for m from % to 2. A value
of 2.0 is within the accuracy of the assumption made.

The plate pressure drop when the inlet caps just become inactive,
hp, is

hy = k,V? + hi + 0.12 -^ (16-31)

while, for a plate with uniform vapor distribution, it would be

+ &S, + 0.12-2- (16-32)

and using the factor of 2.0 obtained for Eq. (16-30) gives

hy = 2/c3 (%H +hl + 0.12 -2.
\M / PL

In cases where the value of the bracketed term is small, the pressure
drop for a plate with caps just becoming inactive is equal to approxi-
mately twice the pressure drop for the same plate with the same total
vapor load uniformly distributed. If the last terms are not small, the
pressure drop will be increased by a factor less than 2.0. This increased
pressure drop is one of the objections to hydraulic gradient.

The relations given in Eqs. (16-24) to (16-32) were based on the
condition that the inlet row of caps was just inactive. Similar analysis
can be made for other conditions. Thus, all caps on a plate will be
active if

Il~L "t~ V.L&
7

hp > (hi - hi) + hi + 0.12 -L (16-33)

For an active plate, with k*V™> hL — h°L, it is frequently desirable

422 FRACTIONAL DISTILLATION

to estimate the actual pressure drop in terms of that for a plate with
uniform vapor distribution. By the same type of integration employed
for Eq. (16-29), an approximation can be obtained for V0 and the value
of hp. Thus,

Va . EOV^L (16_34)

2 (*;-** -0.12 £)

and, by Eq. (16-23),

! + W + h'L + 0.12 £ (16-35)

The term (h'9 - h°L - 0.12 %\ is equal to ki(VT/N)* + kz

and will be termed the velocity head for uniform vapor distribution,
hy. It is equal to the pressure drop due to the vapor flow in the riser,
cap, and slots for a plate operating with a uniform velocity distribution.
For a plate with the inlet row of caps just inactive, 2hv is equal to
(Aj, — h°L) and Eq. (16-34) reduces to the previous criterion. When

(/4, — h°L) is small in comparison to 2hv, the equation gives V0 = -r£;

i.e., the vapor distribution is uniform. To use these equations, hfp is
calculated using V equal to (Vr/N) and TIL — /&!, and F0 is obtained
from Eq. (16-34). The pressure drop for the plate is then calculated
by Eq. (16-35).

The distribution of the vapor among the caps is important, and an
approximation can be obtained by the following relation:

Vn _ i (hj - kj)(n/N)

To ~ x Wv

where F« = vapor rate through inflow cap.

In order to obtain a velocity through the first row of caps equal to
one-half that through the last row would require that 2hv be twice the
hydraulic gradient, and this would appear to be about the minimum
ratio of velocities that should be considered for design purposes.

Plate Dumping. This condition can occur in extreme cases. When
the upstream row of caps becomes inactive owing to an increase in the
hydraulic gradient, the liquid level in the caps will be depressed below
the top of the slots by an amount equal to the surface tension effect.
If the gradient is increased further, the liquid level under the inactive
caps rises and the pressure drop for the same rate of vapor flow

FRACTIONATING COLUMN DESIGN 423

increases. Eventually a condition is reached where the liquid level
under some of the inactive caps reaches the top of the riser and liquid
spills to the plate below. In extreme cases, essentially all of the liquid
flowing to a plate will dump down through ,the risers of the first few
rows of caps, and very little will flow across the plate. Under these
conditions, there is frequently an abrupt drop in liquid level on the
plate between the section of inactive and active caps. The momentum
of the vapor issuing from the active caps acts as dam for the liquid
giving a "Red Sea " effect. Plate dumping is undesirable because with
the usual column, with liquid flowing in opposite directions on succes-
sive plates, the liquid that spills through the risers by-passes the vapor
on two plates.

Equations for the condition of plate dumping, similar to those for
inactive caps, can be derived but, because it is an undesirable condition,
it is better to control conditions such that all the caps are active. This
will ensure against plate dumping. Plate dumping conditions in a
large commercial tower have been described by Harrington et al.
(Ref. 15).

General Design Considerations. It is usually desirable to have a
bubble plate that gives (1) a low pressure drop and (2) a reasonably
uniform vapor distribution. These two conditions are partly incom-
patible because a high pressure drop usually gives more uniform vapor
distribution.

The vapor distribution relation of Eq. (16-36) indicates that the two
main factors involved in obtaining a low variation in (V/V0) across the
plate are (1) low value of the hydraulic gradient and (2) a high value of
hv. Equation (16-16) shows that a low liquid gradient will be obtained
for a given liquid flow rate by increasing h0, or decreasing N.

Increasing the liquid depth should be very effective in lowering the
gradient due to the increase in rh and the decrease in 7/. A deep liquid
level above the top of the caps is not desirable because it allows surges
and wave action to occur. Raising the caps to allow liquid flow below
the skirts is probably one of the most effective ways of reducing the
hydraulic gradient. It does introduce the possibility that part of the
liquid will cross the plate without intimate contact with the vapor.
This condition would be most serious when the column is operating at
reduced capacity. This method has the advantage that it reduces the
gradient without increasing the pressure drop.

The space above the caps should be as unobstructed as possible to
allow free passage of the liquid. Hold-down bars for the caps or other
mechanical devices should be arranged in the direction of liquid flow.

424 FRACTIONAL DISTILLATION

Decreasing the value of N is a common method of improving vapor
distribution, and this result is obtained by the use of split plates or
multiple downspouts. This shortened path may be accomplished by
having the inlet downspout on one plate in the center and the exit
downspouts placed uniformly around the circumference (Fig. 16-4(7).
Thus, the liquid flows out radially and crosses only half of the plate.
On the next plate below, the liquid would flow radially into the center.
Other designs bring the liquid in at opposite sides and flow across half
the tower to central weirs that extend across the tower perpendicular
to the direction of flow (Fig. 16-4F). On the plate below, the liquid
flow is outward. This latter method can be arranged to give any frac-
tional distance across to plate desired, such as ^, J^, ^, or ^.

These arrangements tend to decrease the hydraulic gradient, but
they may lower the plate efficiency by reducing the cross flow effect,
and they offer difficulties in properly proportioning the overflow

between the different sections.

Another arrangement that at-
tempts to obtain complete cross
flow, but reduced hydraulic gradi-
ent, is the staggered plate illus-
trated in Fig. 16-8. In this case;
the plate is broken up into narrow
segments, each with its own over-
flow weir. If the segments are nar-
row and the weirs are all adjusted

FIG, ie-8. Cascade plate. to tbe proper height, the level at

the caps can be kept uniform. The
plate also has the advantage of complete cross flow. The main diffi-
culty is the constructional complexity.

Increasing the value of hv by increasing the pressure drop will be
effective in improving vapor distribution. The use of excessive pres-
sure drops to obtain this result is not desirable, although it may be the
simplest method of correcting a column that has already been built.
In this latter case, some of the caps can be removed, or constrictions can
be placed in the risers to increase the pressure drop. This increase in
pressure drop increases the possibility of flooding the column by liquid
backing up the down pipes. Good original design should give low
pressure drop and good vapor distribution.

Entrainment. Entrainment is the carrying of the liquid from one
plate to the plate above by the flow of the vapor. It is usually defined

FRACTIONATING COLUMN DESIGN 425

as the weight of liquid entrained per weight of vapor. Entrainment is
undesirable, since it reduces plate efficiency by tending to destroy the
countercurrent action of the tower, and it also may affect the distillate
adversely from the standpoint of color or othjer nonvolatile impurities.

The entrainment of the liquid is due to two main causes: (1) the
carrying of liquid droplets due to the mass velocity of the gas and (2)
the splashing of the liquid on the plate. These depend on the slot-
vapor velocity, the superficial column velocity, and the plate spacing.

Several investigators have published quantitative data on the
amount of entrainment in bubble-plate columns. Most of their inves-
tigations have been on systems involving air and water.

The data of Volante (Ref. 36) are given in Fig. 16-9 where the
entrainment, expressed as pounds of liquid per pound of vapor, is
plotted as a function of the superficial velocity, Ve, in feet per second,
multiplied by the square root of the vapor density in pounds per cubic
foot. The entrainment increases rapidly with the vapor throughput
and with a decrease in plate spacing. An entrainment of 0.01 Ib of
liquid per pound of vapor does not seriously lower the plate efficiency
(see page 454), although it may give contamination. The superficial
vapor velocity for these data is lower than commercial practice, and
values of the abscissa of 0.3 and 0.7 would be more comparable. Other
data are given in Fig. 16-10. The curves labeled A are based on the
data of Peavy and Baker (Ref. 26) for the entrainment in an 18-in.-
diameter column with ten 3-in.-diameter caps per plate. They investi-
gated the entrainment when distilling an alcohol-water mixture for
plate spacings of 12 and 18 in.

Curves B arc based on the air-water results of Sherwood and Jenny
(Ref. 31). The tower contained two plates and was 18 in. in diameter.
Four-inch caps were employed having 33 notched-type slots. The
slots were ^ in. high and tapered from %Q in. at the bottom to % in.
at the top.

Holbrook and Baker (Ref. 16) studied entrainment in an 8-in.
bubble-plate column using steam and water. Curves C are based on
a portion of their data. They conclude that the plate spacing and
vapor velocity were the main factors in determining the amount of
entrainment and that the amount of liquid flow and slot-vapor velocity
were of less importance.

Curve E is based on the data of Ashraf, Cubbage, and Huntington
(Ref. 1) for the entrainment in a 7- by 30-ft. commercial absorber.
The tower contained 10 trays, 22 in. apart. The tower was operating

426

FRACTIONAL DISTILLATION

on a gas oil-natural gas system at 45 1 p.s.i.a. The investigators
obtained a maximum entrapment of 0.0017 at a mass velocity of 23.4
Ib. per min. per sq. ft.

0,1
fc 0.01

^late spacing ~

/-*'

/«

-f-

1
1

"c 0.001

0.0001

, ^

°/

~r —

— A

1—

0 01 0.2

vc/v

Fio. 16-9. Entrainment of water by air. %

Entrainment separators and baffles have been suggested, and tests
of various arrangements have shown that they are effective in reducing
the amount of liquid carried by the vapor, but they have not been used
to any extent in industrial rectifying towers.

Plate Spacing. Rectifying columns are built with the plates spaced
as close as 6 in. to as much as 4 to 6 ft* There are a numbersr of facto

FRACTIONATING COLUMN DESIGN

427

that influence this spacing, such as the proper flow of vapor and liquid

in the column and the necessity of a man working between the plates.

To obtain the proper flow of liquid down the column, it is essential

1.0

0.8

0.6
04

0.2

| 0.08
£ 0.06

I 004
I-

-g 0.02
I

c 0.01
** 0.008

0.006
0.004

0.002

0.00 1
0

>4-/%
B-M
C-Ht
0-Sf
E-As

ffVffffJ

erwoo
tibrook
vang-
hrafti
laiur

tfBakt
d ana
\anclt
air-*
Cubba
a /go*

r-a/co
'Jenrq
b*er-
wfer
yean
t-ya*

ho/'M
y -air-
sham

c/Mjn
-oil

rfar

/A t*

late
•xncing

wafer
-wafa

tingh.

}

-t

•/dm
plah
spacl

ch —

_A-

B/i

'/]

-4

-A

-f-

-y

•^^

/r^

/*^

<^22inchf.
spacing

i/ate

2 04 0.6 0.8 1.0 12 1.4

FIG. 16-10. Entrainment in bubble-cap plate columns.

that there be sufficient liquid head in the down pipes. Figure 16-11
shows schematically the liquid head in the down pipes and its causes.
The plate spacing must be great enough to allow for the sum of these
heads plus some extra height to handle short periods of excess flow.
Values of hp normally range from 2 to 4 in. of fluid; the various liquid
heads on the plate (hw + hcr + ho) amount to 3 to 4 in. A plate

428

FRACTIONAL DISTILLATION

spacing of 6 in. leaves little margin of safety and requires the use of
low liquid heads on the plate and low vapor velocities to give low
pressure drop and reasonable entrainment. A plate spacing of 12 to
24 in. would appear to be more feasible.

In addition to the items included in Fig. 16-11, there are the factors
of foaming and cross flow of the vapor. Foaming of the liquid may be

hw «• height of exit weir

her = height of crest on weir

ho *» liquid gradient

ho ** reversal loss in down pipe

hf = friction in down pipe

Ac =* pressure drop through riser and inside cap

ha «= pressure drop through slot

hi ** liquid head above slot

hp *» pressure drop through plate

FIG. 16-11. Schematic drawing of the liquid heads in a bubble-plate column.

so great that it extends from one plate to the next and results in high
pressure drop and entrainment, but it is usually not that serious.
However, a relatively small amount of foam may cause difficulties by
filling the upper part of the down pipe and hindering the liquid flow.
In many cases, foam blocking the down pipes is the limiting factor in
column capacity. The foam is produced largely by the vapor on the
plate, and its plugging effect in the down pipes can be reduced by
including a short calming section before the down pipes or by baffles

FRACTIONATING COLUMN DESIGN 429

that will hold the foam back on the plate and prevent it from flowing
into the down pipes. If relatively stable foams are produced, it may
be necessary to add some foam-breaking agent to the column. Experi-
mental data indicate that the average depsity of the liquid and
entrained vapor mixture is about one-half that of the liquid itself.
For design purposes, it is therefore desirable to have a downspout
height equal to approximately twice the value of the calculated liquid
head.

Vapor flow across the column in the vapor space results from the
nonuniform vapor distribution and from the usual reverse direction of
liquid flow on succeeding plates. On a given plate % to % of the total
vapor may flow up the downstream half of the plate. This necessi-
tates from J^ to }^ of the vapor flowing across the center of the plate to
enter the other half of the plate above. Using the factor of J^, for
purposes of illustration,

VcpHsD = ^~ (16-37)

where VCF = cross-flow velocity
Vc = superficial velocity
Hs — free clearance between plates
D — column diameter

VcF * 5 W~ V*
o /z s

In small-diameter columns, D/HS is usually so small that the effect
of cross flow is negligible, but in large columns it may become so great
that the kinetic head equivalent to VCF may be significant in terms of
liquid head. Under these conditions, the pressure in the vapor space
is not constant across the cross section, and its variation is such that it
forces an increase in the hydraulic gradient. In order to make Ha as
large as possible, any beams or projection on the bottom of a plate
should be positioned to aid the vapor cross flow. Good vapor distribu-
tion on the plate will eliminate the effect of vapor cross flow.

An effect equivalent to the action of vapor cross flow is frequently
obtained when the vapor is introduced through the side of the frac-
tionating column. For example, the vapor from the reboiler is nor-
mally introduced into the side of the column near the bottom. If this
vapor pipe terminates at the column wall, the kinetic energy of the
vapor may be so great that it will cause a high impact pressure on the
opposite side of the column. This type of action can result in very

430

FRACTIONAL DISTILLATION

poor vapor distribution for plates up the column, and some type of dis-
tributor should be employed. These frequently take the form of baf-
fles which serve to direct the vapor over the whole cross section.

Allowable Vapor Velocity. A factor closely related to plate spacing
is the allowable superficial vapor velocity that can be employed. The
limiting factor can be either the liquid-handling capacity of the down
pipes or the loss of rectification efficiency due to entrainment. The
limiting capacity in the first case is calculated on the basis of the pres-

0.3

§ 0.2

I-
1

—•*
^
X

£ O.f

**»

•»*•

*~*\

+**

S3"

t**

V**

+**

'^*

-*•

<^+
^

-y»
r*-

-— '

O^J

*>*

.0 ti

' /

tL^

' /

0 10 20 30 40 5C

Fm. 16-12.

Plate spacing, Inches
Allowable superficial velocity.

sure drops and the liquid heads as outlined in the preceding sections.
For the capacity limited by entrainment, correlations have beeti pre-
sented by Souders and Brown (Ref. 33) and Peavy and Baker (Ref . 26).
Their results plus other data have been used to construct Fig. 16-12.
The superficial vapor velocity, Fc, in feet per second multiplied by the
square root of the density ratio, is plotted as a function of the plate
spacing, taken as the distance from the liquid surface to the plate
above. Lines for different values of liquid seals above the top of the
slots are given. High liquid levels above the slots give increased
splashing and entrainment. The values given by this figure are reason-
able design factors for most cases.

FRACTIONATING COLUMN DESIGN 431

Plate Layout. A large number of types of bubble caps have been
employed, but in most cases circular caps 4 to 6 in. in diameter are
used. The slots are usually rectangular, although triangular and
trapezoidal shapes are also employed. In some cases, the slots have
been cut through the caps shell tangentially, but in the majority of
cases they are cut radially. Tunnel caps are employed in some cases,
but long caps of this type are sensitive to hydraulic gradient because
the corrective action of the pressure drop through the risers is less
effective. Hexagonal caps have been used but do not appear to have
any advantage over circular ones.

Caps are usually arranged on triangular centers, and they are most
commonly placed so that the liquid flow is at right angles to the rows
of caps. This requires the liquid to follow a staggered path across the
plate, and it has been assumed that this gave good contact of the liquid
with the vapor.

Small-diameter caps can give more slot area and more free space for
liquid flow than large caps. Caps larger than 6 in. in diameter are
seldom employed and, except in small laboratory columns, caps less
than 3 in. are not used because of the mechanical problem of handling
the large number of them needed even for plates of moderate diameter.
With the usual cap design and plate layout, the ratio of total slot area
to superficial area generally falls between 0.1 to 0.2.

The caps should not be placed too near the weirs or walls. A clear-
ance of 2 to 3 in. between the caps and the weirs and 1 to 2 in. at the
walls is usually adequate. Clearances of 1 to 2 in. between caps are
employed; i.e., 4-in.-diameter caps might be placed with centers at the
corners of equilateral triangles with sides 5% to 6 in.

OTHER TYPES OF PLATES

Bubble-cap plates are the most common plate type of contacting
device, but other arrangements are used. Perforated or sieve plates
are effective vapor-liquid contacting devices and are frequently used
for liquids containing suspended solids, such as the beer mashes in
alcohol production. At their rated capacity, their efficiency is equal
to that of bubble-cap plates, but they have two disadvantages that
have limited their utility. (1) Because the pressure drop through the
openings of the perforated plate is proportional to the square of the
vapor velocity, it varies .more rapidly with changes of vapor rate than
a bubble-cap plate, and this reduces its flexibility. (2) The liquid on
the plate can dump if the vapor rate is momentarily stopped. Various
arrangements have been proposed to make the opening of the perfora-

432 FRACTIONAL DISTILLATION

tions vary with the vapor in a manner similar to the slot action of a
bubble cap, but their use has been limited.

The design problems of the weirs and downflow pipes for perforated
plates are the same as those for bubble-cap plates. However, the
pressure drop due to vapor flow through the perforation will be con-
siderably different than for a bubble cap. The small holes behave like
a submerged orifice, but their cross section will not vary with the vapor
rate in a manner similar to the slot action of a cap. For the pressure
drop through these perforations, the following equation is suggested,

h, = 0.04 X + er 7,» (16-39)

dpL PL

where h* = pressure drop through perforation, in. of liquid
7 «= surface tension, dynes per pm.
pv = vapor density, Ib. per cu. ft.
pi, = liquid density, Ib. per cu. ft.
V« = velocity based on total perforation area, f.p.s.
d = diameter of perforation, in.

The total pressure drop across the plate will be equal to hv plus the
liquid head on the plate. The problem of calculating the liquid head
is similar to that for a bubble-cap plate, and the actual depth at the exit
weir is a good approximation in most cases. The hydraulic gradient
would increase the liquid head, but the aeration effect would reduce it.

The allowable vapor velocities for perforated plates should be com-
parable to those for bubble-plate trays, and Fig. 16-12 should give
suitable design values. Perforations from 0.1 to 0.5 in. are employed,
and they are usually placed on triangular centers spaced such that the
total area of the openings is 0.10 to 0.15 times the tower cross-sectional
area. The larger openings have more of a tendency to by-pass liquid
and give plate dumping but are less likely to become plugged. Except
for the reduction of liquid head due to aeration, dumping should occur
when hr for the downstream row of perforations is equal to the hydraulic
gradient. Actually the dumping will occur before the gradient is this
great, owing to aeration effects and surges.

Spray-type plates have been described for low-pressure operations.
In order to obtain high rates of mass transfer, the liquid is collected
periodically and resprayed. Kraft (Ref. 21) has described shower-type
trays in which the liquid is allowed to rain down from one plate to the
next through small perforations, and the vapor flows across the shower
but does not bubble through the liquid. These plates are reported to
give pressure drops of 0.75 mm. Hg per plate at practical operating

FRACTIONATING COLUMN DESIGN 433

rates. This is about one-half the pressure drop of the best bubble-cap
plates. Disk and doughnut-type plates have also been used, and
these also pass the vapor through a liquid shower.

PLATE DESIGN EXAMPLE

As an example of the use of the methods outlined in this chapter, consider the
design of a bubble plate for a column to handle a benzene-toluene mixture. For the
separation 20 actual plates will be used, and the plate design will be based on the
bottom plate. This plate will handle essentially pure toluene, and the liquid and
vapor rates will be 480 and 400 Ib. mols per hr., respectively.

Data and Notes.

The pressure at the bottom plate will be 2.5 p.s.i.g.

Use chord overflow weir.

Cap, 4 in. O.D. by 3><j I.D. by 4Ji6 in. high cast iron.

Caps placed on equilateral triangle centers of 6 in., 3-in. minimum clearance at

weirs and 1-in. minimum clearance at walls.
Slots, 33 per cap, % by 1^ in.
Minimum seal on slots at no flow, J4 in*
Minimum liquid depth at no flow, 2 in.
Minimum down-pipe seal, 1J4 in.

Plate will have cross flow with only one down pipe per plate.
Plate spacing, 24 in.

Solution. The seal on the caps at no flow is K in-> and the head on the weir at
operating conditions will be taken as 1.0 in., making a total liquid seal at the outlet
of \y± in. A hydraulic gradient of 0.1 in. is assumed, making the inlet seal 1.35 in.
An average seal of 1.3 in. will be used with Fig. 16-12, to determine the allowable
superficial velocity. The plates are spaced 24 in. apart, but this will be reduced to
20.5 in. to allow for liquid level and plate thickness.

From Fig. 16-12,

jz—Y - °-164

i - PV)

At 2.5 p.s.i.g. the boiling point of toluene is 694°R., and pL « 52.8 Ib. per cu. ft.
Assuming the perfect gas laws apply to toluene vapor,

pv 359(<^92)(14.7/17.2)
«* 0.212 Ib. per cu. ft.

Allowable vapor velocity,

= 2.6 f.p.s.

48'2

Net area of plate - ^ - 18.6 sq ft.

434 FRACTIONAL DISTILLATION

Allow 2 aq. ft. for the two down-pipe areas

~20lT

D -

Weir Length. By Eq. (16-12),

1.0 - 5.4

5.2 ft.

f 480(92) "IH

L53(3,600)LJ
and, using a factor of 0.96 from Fig. 16-5,

480(92)

L - [5.4(0.96)P-6
« 2.73 ft.

53(3,600)

In the text it was suggested that the weir within a weir head of the wall should
be neglected, and the actual weir length will be longer than just calculated. The
following sketch shows these conditions:

+ 26.8 - 912
A - 25.8 in.
C - 5.4 in.

A* + B*

~(¥x12)2

Ba - 974 - 644
B - 18.2 in.

974

Length of weir - 2 f ~~ J - 3.03 ft.
Let </> ** angle subtended by weir chord

sinl^^ »o.582

<f> sac 71 2°

A , j » /* OM /71.2\ 3.03(5.2/2) cos (0/2)
Area of downcomer «• | (5.2)a ( 355 ) " — ^ ^ZL- •

- 1.0 sq. ft.

Total area of downcomers on each side of plate equals 2 sq. ft,, which is the
assumed value.

FRACTIONATING COLUMN DESIGN

435

A schematic plate layout is shown in Fig, 16-13. The center line of the row of
caps nearest the weir is 25.8 — 5 in. from the center with a length of 46.6 in. The
number of caps in this row will be 46.6/6, or 7 caps. The distance between the
inlet and outlet rows of caps is 41.6 in., and this will allow 9 rowsof caps with 1,6 in.
over which will be used for the inlet weir. The total number of caps is 77.

Weep holes — - -

/Down flow pfpe to
X>SSs^ Pfaf* befow

O O O O O O ON

oooooooo
ooooooooo

OOOOOOOOOOl

ooooooooo

OOOOOOOOOOi

ooooooooo

oooooooo

ooooooo

FIG. 16-13.

*-//7/e/ weir

*^ ~ Down f low pipe
f romp/ate above
Layout of bubble-cap plate.

Hydraulic Gradient* The section with the bubble caps approximates a rec-
tangular section, and Eq. (16-16) will be employed for the calculation of the
hydraulic gradient.

The average foam velocity by Eq. (16-19) is

3(480) (92)
3600(53) (0.5) (4.6)
0.30 f .p.s.

The value of 6 *» 4.6 feet was taken as the average of the diameter of the column
md the width at the first row of caps.
By Eq, (16-17),

ind from Eq. (16-18), using the viscosity of the liquid equal to 0,00067 (see p, 418),

Re'

0.41 (0.8) (53)
0.00067

9730

436 FRACTIONAL DISTILLATION

Using this value with Fig. 16-7 gives

f ( *•> V* « 0.3

\h0 — hap)

The loss in head is calculated from Eq. (16-16)

(0.24) (0.30) 2(3.7)
* 32.2(0.41)

« 0.0061 ft. of toluene
= 0.073 in. of toluene
* 0.062 in. of water

The gradient for this plate is somewhat smaller than the assumed value of 0.1 in.
of toluene. However, the difference is not large and no corrections will be made.

Pressure Drop through Caps.

1. Assume uniform vapor distribution with no gradient,

48 2
,Cu. ft. vapor per second per cap = ~- ^ 0.626

a. Pressure drop through risers and inside cap

V = °-626(144)
R 2.352(0.7854)

By Eq. (16-2),

= 1.9 in. of toluene
6. Pressure drop through slots

At 694°R., T for toluene = 18 dynes /cm.

_ 0.626(144)
v. - 6 2 - 14.0

By Eq. (16-4),

This compares with a value of 0.7 for a slightly different 4-in. cap given on
page 407. To be conservative use 0.7.
From Eq. (16-3),

h. - 0.12(i%3) + 0.7 [ 1.5(14.5

- 0.04 + 0.86

» 0.90 in. of toluene

c. Liquid head above slots «• 1.25 in. of toluene.

FRACTIONATING COLUMN DESIGN 437

d. Pressure drop for ideal plate with no liquid gradient,

hp « 1.9 -f 0.90 + 1.25
» 4.05 in. of toluene

2. Pressure drop for actual plate, ,

hv - 4.05 - 0.04 - 1.25
» 2.76 in. of toluene

The value of the gradient is so small relative to hy that Eq. (16-34) indicates
essentially uniform vapor distribution, and the pressure drop will be the same as
calculated for an ideal plate.

Height of Liquid in Down Pipe.

1. Loss in head at bottom of downpipe. Assume the chord down pipe comes
within 1.0 in. of plate. For Eq. (16-13),

092f.p.s.

hD - 0.5(0 92) 2

= 0.42 in. of toluene

2. Friction loss in down pipe. Because the velocity in the down pipe is only
0.23 f.p.s., the friction will be negligible.

3. Head of liquid in down pipe above plate level (see Fig. 16-11).

JiDp « 2.0 + 1.0 + 0.07 -f 0.42 + 0 + 4.05
— 7. 54 In. of toluene

The plate spacing of 24 in. should be adequate unless foaming is excessive.

General Consideration. The general arrangement of the plate is shown in Fig
16-13. Weep holes have been added to allow the column to drain when shut down.
Three %-in. weep holes were placed in the plate before the exit weir. These should
allow the column to drain in about 5 hr. At rated load, the weep holes handle
about 2.5 per cent of the total liquid, but even this amount does not short-circuit
because it flows down on the proper side. These weep holes can be placed in the
inlet and outlet weirs instead of the plate.

PACKED TOWERS

Pressure Drop. Although the pressure drop through packed towers
is usually small at atmospheric pressure, it may become a limiting fac-
tor in vacuum distillations.

Chilton and Colburn (Ref . 4) have published a method for predicting
such pressure drop for solid packings, based on the Fanning equation
for friction in pipes. They modify the friction equation to

438

FRACTIONAL DISTILLATION

where Ap = pressure drop in height h

Aw = correction factor for wall effect

A i « correction factor for wetting of the packing, by the liquid
pv = density of the vapor
g = conversion factor for consistent units
u « superficial gas velocity, i.e., linear gas velocity based on

the total cross section of the tower
d* « size of packing, nominal
ji » viscosity of vapor

/ = function of (d*upfii) as given in Eqs. (16-41) and (16-42)
Chilton and Colburn gave a plot of / as a function of the Reynolds
number (d*up//i). The data of this plot may be approximated by the
following equations:

For (d*up/n) less than 40, use

For (d*up/fji) greater than 40,

850

d*Ufi/IJL

(16-41)

(16-42)

All these equations are dimensionally sound, and any consistent set
of units may be used. The following table contains values of the
factor Aw:

Packing diameter

Tower diameter

(*?)<«

(*?)>•

1.0

1 0

0.1

0.83

0 72

0.2

0.74

0.65

0.3

0 71

0.57

The correction for the wetting of the packing, AL, is 1.0 for dry pack-
ing and is greater than 1.0 for wet packing because the liquid decreases
the free volume. The factor is greater than 1,0 for wet packing even
without liquid flow, and it increases with the liquid rate. As an
approximation, the following relation is suggested for AL for wet

FRACTIONATING COLUMN DESIGN 439

packing,

AL = 1.4 + 0.0005F (16-43)

where F == liquid rate, Ib. per hr. per sq. ft.

Liquid rates above 5,000 to 10,000 Ib. per h». per sq. ft. give values of

AL so large that the tower may flood.

The pressure drop with hollow packings is less than given by the
equation for solid packings. The data on this effect are not very con-
clusive as to absolute magnitude but do indicate that, for hollow pack-
ing, the smaller the packing size the larger the pressure drop. For a
detailed discussion of this factor, the reader is referred to Sherwood's
summary (Ref, 30).

Allowable Gas and Liquor Velocities. The capacity of packed
towers is limited by the tower's becoming flooded with liquid. The
flooding can be caused by increasing either the liquid or the gas flow.
This flooding is a result of the pressure drop through the tower exceed-
ing the gravity head of the liquid flowing down.

These pressure drops per foot of height are given by the modified
Fanning equation:

<~ 2gm

where / = proportionality factor, a function of

Vo = actual linear gas velocity = uG/FAo, f.p.s.
VL = actual linear liquid velocity = UL/FAL, f.p.s.
p = density, Ib. per cu. ft.

g = conversion factor— 32.2 (Ib. force) (ft.)/(lb. mass)(sec.2)
n = viscosity, poises

F = fraction of tower that consists of voids1
S = surface of packing,1 sq. ft. per cu. ft. of packing
m = hydraulic radius = free volume /contact area = F/S, ft.
A0i AL = fraction of free cross section occupied by gas and liquid,

respectively

u = superficial velocity, f.p.s.

The flooding occurs when Ap* = PL — Apa is a small fraction of Apo,
since under such conditions a slight increase in the rate of flow of either
stream or an uneven surge in the tower will increase AL and decrease
Ao, because of increased liquid holdup. This decrease in Ao will
increase VG and thereby Ap<?. If Api, is large relative to Ap*, this

1 For data on the values of S and F for various packings, the reader should con-
sult Ref s. 9 and 30 at the end of the chapter.

440

FRACTIONAL DISTILLATION

X-fe-

*3"

FRACTIONATING COLUMN DESIGN

441

increase in Ape will affect the liquid flow only slightly; however, if
is small compared to Ap<?, a small increase in Ap<? will make a large per-
centage decrease in ApL, causing the holdup to increase and the tower
to flood.
Combining the pressure-drop equations,

Setting the ratio (PL — Apc?)/Ap(? equal to b and noting that at flooding
Ap<? becomes approximately equal to PL modifies the previous equation
to give

'Vu?-, (16-44)

At low values of (UG/UL^PV/PL, the denominator of the right-hand side
becomes 1, and the limiting gas velocity is a function of (UQ/UL)(PV/PL),
the tower dimensions, and/L, the latter term being chiefly a function of
PL. It is interesting to note that in this region the gas viscosity is not
a factor; but when the last term of the denominator is not negligible,
the viscosity of the gas becomes a factor in the limiting gas velocity.
At very high values of this group, the right-hand side reduces

TABLE 16-2

Packing

No. 19 aluminum jack chain
1-in. Raschig rings . . .
1-in. Berl saddles. .
8-mm. Raschig rings . . .
J^-in. Raschig rings.

J^j-in. Raschig rings

J^-in. Raschig rings . . . .

J«2-in. Berl saddles

3^-in. Raschig rings

J^-in. Rachig rings

J^-in. Raschig rings

}£-in. Raschig rings

H2-itt- carding teeth .
M-in. bent carding teeth
0.23- by 0.27-in. glass rings. .

0.18-in. glass rings

0.47-in. glass rings

System

Heptane-methyl cyclohexane
Air-water
Air-water
Air-water
H2-water
Air- water
COrwater
Air-water
Air-methanol

Air- (50% H2O -f 50% CH3OH)
Air-glycerin

Water -f- butyric acid-air
Benzene-carbon tetrachloride
Benzene-carbon tetrachloride
Benzene-carbon tetrachloride
Ethanol-water

Quinoline (distillation at 10 mm.
Hg abs.)

Reference

23
23
23
32
32
32
32
32
32
32
32
9
9
9
17

442 FRACTIONAL DISTILLATION

This would be the case when a very low liquid rate was employed and
the gas occupied essentially the whole free cross section of the tower.
For convenience in plotting, the right-hand side will be taken as a func-
tion of (UQ/UL) (PV/PL)H/I**L. The data of a number of investigators are
correlated in this way in Fig. 16-14 using a value of a = 0.21. The
systems involved are given in Table 16-2.

The data are seen to correlate well except at high values of the
abscissa. This deviation may be due to the fact that the f0 factor in
the denominator is neglected in the method of plotting of this figure.

Nomenclature

AL ** correction factor for liquid on packing in packed tower

A« — area under skirt of caps for liquid flow

Aw ** correction factor for wall effect in packed towers

B — length of bubbling section, ft.

6 *• width of plate, ft.

C « pressure recovery factor for orifice

D « inside diameter, ft. '

d «• diameter, in.
d* « size of packing, nominal

F » loss in head from inlet to exit calming sections, ft.

/ « friction factor for fluid flow

/' «• friction factor
g,gc «• conversion constant - 32.2 [(ft.)(lb.force)]/[(sec.2) (Ib. mass)]

h «• liquid depth above surface of plate
he «• pressure drop through riser and undercap, in. of liquid
her ** height of liquid above top of weir, in.

ho •• pressure drop for reversal of flow at bottom of down pipe, in. of liquid
hop » liquid level in down pipes above surface of plate, in.
ha «. kinetic head, in. of liquid
1 hi « liquid depth above surface of plate at inlet weir
hi «* liquid depth above top of slots, in.
h*L ** value of KL at inlet row of caps
hi ** value of hi at outlet row of caps
h0 «• liquid depth above surface of plate at outlet weir, in.
h* » slot height, in.

h* «• value of h, for slots completely open
hj » over-all pressure drop for plate, in. of liquid
h^ » over-all pressure drop for uniform vapor distribution, in. of liquid
hp — over-all pressure drop when inlet row of caps is just inactive, in, of liquid
Ht «* free clearance between plates^ ft.
h, ** pressure drop through slots, in. of liquid
htp — height of top of slot above surface of plate
hy •» velocity head for uniform vapor distribution, in. of liquid
hw «« height of weir above surface of plate
h* •» pressure drop through perforations* in. of liquid
K,k —constants

FRACTIONATING COLUMN DESIGN 443

L * perimeter of weir, ft.
Lo « foam height, ft.
m «* exponent

N « number of rows of caps from outlet to inlet weir
n «• number of rows of caps, counting from the overflow weir
Q » liquid flow rate, cu. ft. per sec.
Qw = liquid rate, Ibs. per sec.
q = liquid flow rate, g.p.s.

'•'" '

/*/

mean hydraulic radius, ft.

u =* superficial velocity in packed tower, f ,p.s.
UQ «• superficial velocity of gas
UL « superficial velocity of liquid

V — vapor flow rate
Ve =* superficial vapor velocity, f .p.s.
VCF = cross flow velocity, f.p.s.

VD •» maximum velocity at bottom of down pipe, f .p.s.
V/ = velocity of liquid in foam, f.p.s. «= Qw(p/L0b)
Vo — linear gas velocity in packed tower, f.p.s.
VL *• linear liquid velocity in packed tower, f ,p.s.
V0 *» liquid velocity in down pipe, f .p.s.
VR — maximum velocity under cap, f.p.s.
V9 «• slot velocity based on total slot area, f.p.s.
V% =• slot velocity for ha equal to h*
VT «* total vapor flow rate for plate
Vjf «• velocity based on total perforation area

7 ss surface tension, dynes per cm.
Pf ~ density of foam, Ibs. per cu. ft.
pv — vapor density
PL =* liquid density

0 » viscosity
/i/ = viscosity of foam
ML » normal viscosity of liquid, Ibs. per f.p.s.

References

1. ASHBAF, CUBBAGE, and HXJNTINGTON, Ind. Eng. Chem., 26, 1068 (1934).

2. BIJAWAT, S.M. thesis in chemical engineering, M.I.T., 1945.
2a. BLOECHER, S.B., thesis in chemical engineering, M.I.T., 1949.

3. CAREY, Sc.D. thesis in chemical engineering, M.I.T., 1929.

4. CHILTON and COLBURN, Trans. Am. Inst. Chem. Engrs., 26, 178 (1931).

5. DAUPHINE, Sc.D. thesis in chemical engineering, M.I.T., 1939.

6. DAVIES, Ind. Eng. Chem., 39, 774 (1947).

7. EDMINSTER, "Hydrocarbon Absorption and Fractionation Process Design
Methods," reprinted from The Petroleum Engineer.

8. FENSKE, LAWROSKI, and TONGBERG, Ind. Eng. Chem., 30, 297 (1938).

9. FENSKB, TONGBERG, and QTTIGGLB, Ind. Eng. Chem., 26, 1169 (1934).

444 FRACTIONAL DISTILLATION

10. GARDNER, Sc.D. thesis in chemical engineering, M.I.T., 1946.

11. GHORMLBY, S.M. thesis in chemical engineering, M.I.T., 1947.

12. GONZALES and ROBERTS, S.M. thesis in chemical engineering, 1943.

13. GOOD, HUTCHINSON, and ROUSSEAU, Ind. Eng. Chem., 34, 1445 (1942).

14. GRISWOLD, Sc.D. thesis in chemical engineering, M.I.T., 1931.

15. HARRINGTON, BRAGG, and RHYS, Petroleum Refiner, 24, 502 (1945).

16. HOLBROOK and BAKER, Ind. Eng. Chem., 26, 1063 (1934).

17. JANTZEN, Dechema Mon., 6, No. 48 (1932).

18. KEMP and PYLE, Chem. Eng. Progress, 45, 435 (1949).

19. KESLER, S.M., thesis in chemical engineering, M.I.T., 1949.

20. KIRKBRIDE, Petroleum Refiner, 23, 321 (1944).

20a. KLEIN, Sc.D. thesis in chemical engineering, M.I.T., 1950.

21. KRAFT, Ind. Eng. Chem., 40, 807 (1948).

22. LOCKE, S.M. thesis in chemical engineering, M.I.T., 1937.

23. MACH, Forschungsheft, 375, 9 (1935).

24. MAYER, S.M. thesis in chemical engineering, M.I.T., 1938.

25. 10.45 Notes M.I.T., 1945.

26. PEAVY and BAKER, Ind. Eng. Chem., 29, 1056 (1937).
26a. ROGERS and THIELE, Ind. Eng. Chem., 26, 524 (1934).

27. ROWLEY, S.B., thesis in chemical engineering, M.I.T., 1938.

28. SCHNEIDER, S.M. thesis in chemical engineering, M.I.T., 1938.

29. SEUREN, S.M. thesis in chemical engineering, M.I.T., 1947.

30. SHERWOOD, "Absorption and Extraction," p. 141, McGraw-Hill Book Com-
pany, Inc., New York, 1937.

31. SHERWOOD and JENNY, Ind. Eng. Chem., 27, 265 (1935).

32. SHIPLEY, S.M. thesis in chemical engineering, M.I.T., 1937.

33. SOUDE;RS and BROWN, Ind. Eng. Chem., 26, 98 (1934).

34. SOUDERS, HUNTINGTON, CoRNEiL, and EMERT, Ind. Eng. Chem., 30, 86 (1938).

35. STRANG, Trans. Inst. Chem. Engrs., 12, 169 (1934).

36. VOLANTE, S.B. thesis in chemical engineering, M.I.T., 1929.

CHAPTER 17
FRACTIONATING COLUMN PERFORMANCE

The design calculations considered in the preceding chapters were
based on theoretical plates. In order to complete the design, it is
necessary to have the relationship between these idealized values and
the actual performance of the contacting device. The vapor and
liquid brought into contact with each other in the tower are not at
equilibrium, and the rate of mass transfer determines the effectiveness
of the unit. This chapter will consider the methods of predicting the
effectiveness of the vapor-liquid contact for the various types of units.

PLATE-TYPE COLUMNS

Plate Efficiency Definitions and Relations. Over-all Column
Efficiency. The relation between the performance of actual and theo-
retical plates is expressed as plate efficiencies. A number of different
plate efficiencies have been proposed, but the two most commonly used
are the "over-all column efficiency" which was proposed by Lewis
(Ref. 20) and "plate" or "point" efficiencies suggested by Murphree
(Ref. 24). The over-all column efficiency, E°, is the number of theo-
retical plates necessary for a given separation divided by the number of
actual plates required to perform the same separation; i.e., it is the
factor by which the number of theoretical plates is divided to give the
actual number of plates. This efficiency has no fundamental mass-
transfer basis, but it serves as an easily applied and valuable design
factor and is therefore widely used. v ,

Murphree Efficiencies. The Mur-
phree efficiencies are based on more
fundamental concepts than E°y but
even in this case the basic relations
employed are more qualitative than
quantitative. Murphree developed
two cases: one employing vapor-phase relations and the other liquid-
phase conditions. For the vapor-phase derivation he assumed that
a bubble of vapor in rising through the liquid on a plate was in con-
tact with a liquid of constant composition, and that the composition

445

446 FRACTIONAL DISTILLATION

in the bubble changed continuously by mass transfer. Consider the
bubbles shown in the simplified schematic diagram of Fig. 17-1. These
bubbles enter with a composition y* pass up through a liquid of com-
position x, and leave with a composition yf0. Using a simplified expres-
sion for the instantaneous mass transfer to one of the bubbles,

-(? dy' - PK0a(yf - y.) de (17-1)

where G = mols of gas in bubble

yf = mol fraction of component in bubble
2/« = vapor in equilibrium with liquid
P = total pressure
KQ ~ over-all mass-transfer coefficient, (mols)/(unit time) (unit

pressure difference) (unit interfacial surface)
a = interfacial area of bubble
0 = contact time of bubble with liquid

Assuming (?, P, ye, and K&a are constant, this equation can be
integrated to give

PK0ae
G

Murphree applied the relation to the whole plate, assuming that con-
ditions were the same at all points. In this case (G/0) is replaced by
the vapor rate, and a becomes the average total interfacial area of all
the bubbles on a plate at any instant.
This equation can be rearranged to

where m

EMV - ** __ f ° _ 1 - e~m (17-3)

PKGaO

In many cases there are considerable differences in the compositions
of the liquids at various points on a plate, and the conditions assumed
in the derivation are not satisfied for the whole plate. However, for
convenience, the value of the Murphree plate efficiency is defined as

(17-4)

where y»-, y* * average composition of vapor entering • and leaving
plate, respectively ' '

y* =* composition of vapor in equilibrium witji liquid flowing
to plate below

FRACTIONATING COLUMN PERFORMANCE 447

Equations (17-3) and (17-4) appear similar, but the former uses the
values for a small region of the plate, while the latter uses average
values for the various streams entering and leaving a plate. U£F is
not equal to 1 — e~m.

The derivation of Eq. (17-3) should apply to a limited region of the
plate, and this has been termed the Murphree point efficiency, E%v.
In this case "

(17-5)

where y', y0 = vapor compositions entering and leaving the local

region
y< = vapor composition in equilibrium with the liquid in the

local region

"the Murphree plate efficiency is the integrated effect of all the
Murphree point efficiencies on the plate.

The derivation of the Murphree equation is based on a very qualita-
tive picture of the mass transfer. At low vapor rates individual bub-
bles are obtained, but a study of mass transfer for such systems indi-
cates that it cannot be expressed as a simple rate equation involving a
constant multiplied by a driving force in mol fraction units. The
experimental data indicate that mass transfer is very rapid while the
bubble is being formed and then is relatively slow while the bubble is
rising through the liquid. At higher vapor rates channels are blown
through the liquid, and a large quantity of .spray is thrown up into the
vapor space giving additional mass transfer. The simple rate Eq.
(17-1) cannot be any more than a crude expression of the phenomena
involved, and K <?a must be a complicated function of a large number of
variables including by and 0. y

The value of E°f E°MV, and E %v are the most commonly used design
factors for plate efficiencies. The over-all tower plate efficiency is
simpler to apply than the Murphree efficiencies because only terminal
conditions are required; whereas in the calculation of the Murphree
plate efficiency, plate-to-plate compositions are required, and for the
Murphree point efficiency, complete liquid- and vapor-composition
traverses are required on each plate. However, the Murphree effi-
ciencies are probably on a more fundamental basis than the over-all
efficiencies.

By definition, a theoretical plate is one on which the average compo-
sition of the vapor leaving the plate is the equilibrium value for the
liquid leaving the plate. If the vapor and liquid upon a plate were

448 FRACTIONAL DISTILLATION

completely mixed, it would be impossible to obtain better separation,
than that given by a theoretical plate. However, when there is a con-
centration gradient in the liquid across the plate, the average concen-
tration of the more volatile component in the liquid on the plate may
be appreciably greater than the concentration of the liquid leaving the
plate; as a result of this greater concentration, the vapor actually leav-
ing the plate may exceed the concentration of the vapor in equilibrium
with the liquid leaving. It is thus possible for the concentration-gradi-
ent effect to give over-all and Murphree plate efficiencies greater than
100 per cent; but since such gradients do not apply to the Murphree
point efficiency, this latter efficiency should never exceed 100 per cent.
The theoretical effect of the concentration gradient has been studied
by a number of investigators (Refs. 15, 19, 21). Three cases were con-
sidered by W. K. Lewis, Jr. : Case I, vapor completely mixed, liquid
unmixed; Case II, vapors do not mix, and the overflows are arranged
such that the liquid flows in the same direction on all plates; Case III,
the vapor rises from plate to plate without mixing, and the liquid flows
in the opposite direction on successive plates.

Lewis assumed that (1) E^v was constant over all of the plate, (2)
the equilibrium curve is a straight line over the concentration range
involved, ye — Kx + b, and (3) the liquid flows across the plate with-
out mixing.

The results of this analysis for the three cases are given in Fig. 17-2
in which the ratio (E°MV/E^V) is plotted as a function of E^v and the
ratio of the slope of equilibrium curve, K, to the slope of the operating
line, (0/F). The slope of the equilibrium curve, K, should be the
average slope, dye/dx, over the concentration region involved. These
calculated values indicate that it should be possible to obtain high
plate efficiencies by preventing the liquid from mixing. The usual
bubble-cap plates probably fall between Cases I and III as far as the
vapor is concerned, but they give considerable liquid mixing which
would lower the value of (E°UV/E^V) as compared to the values given
by the plot. Case II has the possibility of giving higher plate effi-
ciencies than the other two cases, but in practice it is difficult to arrange
the downflow pipes such that the liquid flows in the same direction on
all plates. The circumferential flow plate, Fig. 16-4$, gives essen-
tially this type of flow, but it is not a desirable construction in most
cases.

The relationship between the over-all column efficiency and E°MY can
be derived in a similar manner and assuming

1. Constant 0/7

FRACTIONATING COLUMN PERFORMANCE

449

FIG. 17-2. Relation between EMV°

450

FRACTIONAL DISTILLATION

2. Constant slope of the equilibrium curve, i.e., dy,/dx <= K

3. E°ur same for all plates considered.
Lewie (Ref. 21) obtained

pa_\nll+E°uvlK/(0/V)-l]}
ln(K/(0/V)]

(17-6)

This equation is plotted in Fig. 17-3. It will be noted that in gen-
eral E°/E0MV is close to 1.0. In cases where the rectifying system

0.3

FIG. 17-3. Relation between E° and

operates from low to high concentrations, the value of K/(0/V) will

average out to around 1.0, making E° approximately equal to E°uv.

For cases where K/(0/V) is widely different from 1.0 and E°uv is

low, the ratio of E° can be either much larger or smaller than Efo

FRACTIONATING COLUMN PERFORMANCE

451

Thus Fig. 17-4 illustrates a case where K/(0/V) is very small, and one
theoretical plate would go from yn to a composition of almost 1.0. If
the over-all column efficiency for this region were 0.5, two actual
plates would give the same increase, but the diagram indicates that
about five plates with E°MV ~ 0.5 would b6 required. Actually the
efficiencies employed were incompatible, and if E°MY = 0.5 then E°

FIG. 17-4.

would be small for the low value of K/(0/V). The first step with
E°MV = 0.5 does make a change in the vapor composition equal to
one-half of that for a theoretical plate but, because of the convergence
of the equilibrium curve and the operating line, the available potential
decreases and the succeeding plates do not make so large a change in
the vapor composition.

Murphree also gave a derivation for a plate efficiency based on
liquid-phase compositions. The basic differential equation was

-L' dx « KUQ,(X -

(17-7)

where x ** mol fractiovn i» liquid

x* s* liquid in equilibrium with Vapor leaving
Ki. = mass-transfer coefficient
V « liquid in slug Under consideration
The equation was integrated with Kt, a, L', and x* constant to give

(17*8)

452

FRACTIONAL DISTILLATION

where x09 Xi « liquid composition to and leaving plate, respectively
EML = liquid-phase plate efficiency
Z//0 = liquid rate

It is difficult to picture any mass-transfer process on a bubble-type
plate that corresponds to the derivation of Eq. (17-8). For example,
consider the application of these equations to a local section of the
plate. The liquid flowing across this section contacts the vapor rising
through it, but the vapor composition varies with the liquid depth
while the integration assumed that x* was constant. The mol fraction
ratio of Eq. (17-8) will be used as the definition of EML, but the mass-
transfer portion of the equation does not apply to bubble-cap plate
conditions.

Equations (17-3) and (17-8) can be related by the operating line and
the equilibrium curve. Assuming that the equilibrium curve is such
that Ke = y0/x* = y*/x0 (Ke is equal to K if the equilibrium curve is
a straight line through the origin) gives

Ke E°ML(l -

0/V E°MV(l ~

(17-9)

The assumptions made in the derivation of Eq. (17-3) appear to be
on a somewhat sounder mass-transfer basis than Eq. (17-8). If this
is true, E^v depends on the operating conditions only to the extent
that they affect the mass-transfer conditions while EML in addition is a
function of the ratio Ke/(0/V).

TABLE 17-1

Location in column

Bottom

Just below
feed plate

Just above
feed plate

Top

o/v

1.2

0.7

»K

2.5

1.0

0.4

K/(om

2.08

0 883

1.43

0 57

E°MV/E*aiV "

» 1.9

1.38

1.67

1.18 (Fig. 17-2)

EMV

1 14

0.83

1.0

0.71

E°/E°MV

0.96

0.98

1.0

0 91 (Fig. 17-3)

E°

1 09

0.81

1.0

0.65

2.5

1.5

1.0

EML

1.06

0.86

1.0

0.78

The vapor efficiency E°MV is much more commonly used than E°ML.
This preference is justified in view of the derivation of the latter.

FRACTIONATING COLUMN PERFORMANCE

453

As an illustration of these relationships, consider the separation of a
benzene-toluene mixture with (0/F)n = 0.7 and (0/F)m = 1.2 for
Eft? = 0.6. The flow conditions correspond to Case III.

The feed-plate region was arbitrarily chosen to make K = 1.0.
This corresponds to a liquid composition of about 40 per cent benzene.

The values given for E°MV would be higher than actually obtained
because of the liquid mixing on the plate. It is interesting to note that
E°/E°MV is near to 1.0 at all positions, and the use of E° = E°MV would
be a reasonable approximation. The values of Ke are larger than K
except at the bottom of the column. For usual cases, Ke/(0/V) is
greater than 1.0, and E°ML will be between E°MV and 1.0.

FIG. 17-5. Application of Murphree plate efficiencies.

From the calculational viewpoint, E° is the easiest to apply, but
E°MV and E°ML can be used without much additional effort either in
algebraic or graphical design calculations. Thus in algebraic calcula-
tions, the usual theoretical plate calculations will give y* for a known
2/n_i, and yn = t/n-i + E°MV(y% — 2/n_i). Thus the values of yn can be
calculated, if E°MV is known and the calculation is repeated for the next
plate. The efficiencies can also be applied to the graphical calculation
for binary or multicomponent mixtures. This is illustrated in Fig.
17-5. The use of E°MV is illustrated on the upper operating line : yn-i is
the actual vapor composition entering plate n and by material balance
the composition xn is fixed on the operating line; the vapor in equilib-

454

FRACTIONAL DISTILLATION

rium with xn is y* and the actual increase in vapor composition is
I/ft — y^i, which is obtained by a vertical step of a height equal to
E*Mv(y% -~ #n~i); i.e., a fraction of the theoretical plate increase equal
to El[V is taken. The use of E°M L for plate m is shown on the lower
operating line. In this case a fractional horizontal step equal to E°ML
times the theoretical plate change is used.

Effect of Entrainment on Efficiency. Entrainment can lower the
apparent plate efficiency because the vapor-liquid mixture carried to
the plate above will have a lower average concentration of the more
volatile components than the vapor alone. Colburn (Ref . 6) has given
an equation relating the measured efficiency with entrainment to that
for a plate giving the same change in vapor composition but without
entrainment.

Jpo

&UT —

1 +

oTv

(17-10)

where E°MT
E

apparent efficiency with entrainment
efficiency for same change in vapor composition
e = mols of entrained liquid per mol of vapor
For most cases the effect of entrainment does not become serious
until e is 0.1 or greater. Referring to Fig. 16-10, it will be noted that
e « 0.1 corresponds to VcpQ* of about 0.9 or for atmospheric pressure
a velocity of 2 to 4 f .p.s. with 12-in. plate spacing and to several fold
higher velocities for 24-in. spacing. Equation (17-10) is normally used
with E taken as the expected value of E°MVJ but this is in error due to

TABLE 17-2

Bottom

Just below
feed plate

Just above
feed plate

Top

*fcr

1.04

0.77

0.87

the mass transfer that takes place between the vapor and liquid drop-
lets above the main liquid body on the plate. The composition of the
liquid employed was that of overflow to the plate below while the
actual composition will be different because of mass transfefc^nd con-
centration gradients. This equation will give a too high value for E^ T
when Eltr is used f or E. For a value of e » 0.1 and using E°ur * E;
these values of E«MT corresponding to Table 17-1 are given in Table 17-2.
In this case the loss in efficiency for e « 0.1 is relatively small, and

FRACTIONATING COLUMN PERFORMANCE

455

larger values of e are not commonly found in commercial practice
because the operation of the column becomes unstable before the cor-
responding velocities are obtained.

Experimental Data on Plate Efficiencies. A considerable number
of investigations of plate efficiencies have been reported, but it is diffi-
cult to obtain a coherent picture from the data because of the number
of unknown factors usually involved. Thus some of the results are
reported on an over-all column basis while others are given on a plate
basis. They involve unknown amounts of entrainment, unknown
amounts of liquid mixing in the plate, unknown hydraulic gradients,
unknown vapor distribution, unknown interface temperatures, and in
some cases unknown degrees of liquid by-passing or dumping. As a
result, most of these data are not suitable for correlation purposes but
are useful for giving a general picture of the results obtained. The
following discussion reviews some of these data for orientation pur-
poses, but it is not intended to be a comprehensive survey.

TABLE 17-3

System

Average Murphree plate efficiency,

E°MV

Average superficial vapor velocity, f.p.s . . .

Methanol-water. . .

n-Propanol-water .

Isobutanol-water .

3V&8thanol-n-propanol .

Methanol-isobutanol

Benzene-carbon tetrachloride

Gadwa (Ref. 12) has studied the plate efficiency in the fractionation
of mixtures of (1) benzene-carbon tetrachloride, (2) methanol-iso-
butanol, (3) methanol-ra-propanol, (4) isobutanol-water, (5) n-pro-
panol-water, and (6) inethanol-water. A small four-plate column
containing one bubble cap per plate was employed. The bubble caps
were 3J^ in. in diameter and 2 in. high containing 38 slots % in. wide
by % in. high per cap. A vapor space of 5 by 5 in. was partitioned off
from the overflow pipes, giving a ratio of slot area to superficial area
of 0.12. The plates were spaced 11 in. apart, and overflow weirs were
employed. Plate samples were taken so that the Murphree plate effi-
ciencies could be Calculated. Some of these results are given in Table
17-3. The efficiencies i& this table were calculated for the vapor phase.

456 FRACTIONAL DISTILLATION

Gadwa concluded that, for the mixtures he studied, the Murphree
plate efficiency was substantially independent of the concentration and
of the vapor velocity so long as foaming and entrainment were not
appreciable but that, when foaming and entrainment did occur, the
efficiency decreased with increasing velocity.

Brown et al. (Ref . 4) and Gunness (Ref . 14) both report Murphree
plate efficiencies of 100 per cent or greater for large commercial gasoline
stabilizers. The tower studied by Gunness, operated at 250 p.s.i.g.,
was 4 ft. 8^i in. in diameter, and contained 28 plates each having 27
cast-iron bubble caps. The bubble caps were 6^[ in. in diameter and
contained 32 1- by -Hrin. rectangular slots per cap. The plate spacing
was 18 in. In these columns, there were a number of bubble caps per
plate, and the liquid flowed in opposite directions on successive plates.
Gunness analyzed his data by Lewis's cross-flow enrichment method
(page 448) and concluded that the Murphree point efficiency was
between 70 and 80 per cent.

Lewis and Smoley (Ref. 22) studied the plate efficiency in the rectifi-
cation of mixtures of (1) benzene-toluene, (2) benzene-toluene-xylene,
and (3) naphtha and mixtures of pinene and aniline in naphtha. An
experimental column 8 in. in diameter with 10 plates spaced 16 in.
apart was used. The bubble cap was rectangular, being 2 in. high and
2 in. wide, and extended across the column. There were 24 slots %
by ^{6 in. on each side of the cap, giving a ratio of slot area to super-
ficial area of about 0.16. The investigators found average plate effi-
ciencies of 60 per cent for the benzene-toluene mixture, 75 per cent for
the ternary mixture, and 80 to 95 per cent for the naphtha mixtures.

In the same tower, Carey, Griswold, Lewis, and McAdams (Ref. 5)
found an average Murphree efficiency of 70 per cent when fractionating
benzene-toluene. They found the efficiency substantially constant for
superficial velocities from 0.2 to 4.5 f.p.s. and independent of liquid
composition. The same investigators report efficiencies of 50 to
99.75 per cent for the fractionation of an ethanol-water mixture in a
6-in.-diameter tower containing one plate. The logarithm of 100
minus the plate efficiency was found to be a linear function of the
depth of submergence of the slots. A benzene-toluene mixture in the
same one-plate tower gave an average Murphree efficiency of 58 per
cent. A distillation of an aniline-water mixture in the 10-plate tower
gave an average plate efficiency of 58 per cent at a vapor velocity of
2.77 f.p.s.

Lewis and Wilde (Ref. 23) found an average plate efficiency of 65 per
cent at a vapor velocity of 2.8 f.p.s. for the rectification of naphtha in a

FRACTIONATING COLUMN PERFORMANCE 457

10-plate column 9 ft. in diameter. There were 115 bubble caps per
plate containing slots Y± by 1 in. The ratio of slot area to superficial
area was 0.10, and the plate spacing was 2 ft.

Brown (Ref. 3) reports efficiencies as high as 120 per cent for a com-
mercial beer column using perforated plates. The same efficiency was
reported for the rectification of an ethanol-water mixture in a special
laboratory column. The same investigator reports efficiencies of about
20 per cent for naphtha-absorption towers.

Atkins and Franklin (Ref. 1) found an over-all column efficiency of
18 per cent for a natural gasoline absorber using gas oil as the absorbing
liquid. Walter (Ref. 30) obtained Murphree vapor plate efficiencies
from 80 to 95 per cent in a 2-in. laboratory column for air humidifica-
tion. Data taken in the same unit on the absorption of propylene
and isobutylene in gas oil, heavy naphtha, and mixtures of gas and lube
oil, gave plate efficiencies on the vapor basis of 5 to 36 per cent.

Horton (Ref. 16) studied the absorption of carbon dioxide and
ammonia in water on a single 18-in. -diameter plate and reported values
of E°MV about 3 per cent for CC>2, and 70 per cent for ammonia. Fair-
brother (Ref. 9) studied the absorption of carbon dioxide in aqueous
solutions of glycerine and obtained values of the Murphree plate effi-
ciency of 0.65 to 4 per cent.

Peavy and Baker (Ref. 27) investigated the rectification of ethanol-
water mixtures in an 18-in.-diameter column, and obtained plate vapor
efficiency from 80 to 120 per cent for superficial vapor velocities
between 1 and 3 f.p.s. For other data on plate efficiency, see Refs. 7,
8, 13, 18, 25, 26, 28.

The values of the plate vapor efficiency vary from less than 1 to over
100 per cent. The absorption systems with gases of low solubility and
liquids of high viscosity have low efficiencies; while most of the distilla-
tion systems give values of 60 to 100 per cent. In most cases the
efficiency is relatively independent of vapor rate until appreciable
foaming and entrainment are encountered. For the distillation sys-
tems there does not appear to be any significant effect of liquid
composition.

Plate Efficiency Correlations. It has been pointed out that the
correlation of the plate efficiency data is difficult because of the large
number of unknown conditions involved in most cases. However,
some correlations have been developed that are helpful in estimating
the plate efficiency.

Gunness (Ref. 14) analyzed the data for several columns and con-
cluded that the liquid-film resistance was a major factor, and he sug-

458

FRACTIONAL DISTILLATION

gfcsted that the plate efficiency be correlated with the viscosity of the
liquid, because this characteristic is a major factor in liquid-phase mass
transfer. He plotted the efficiency as a function of the operating pres-
sure, on the basis of the fact that the viscosities of liquids are approxi-
mately the same at a given vapor pressure. This correlation on a vis-
cosity basis is given in Fig. 17-6.

6
6

4
2

1.0
08

1 { 1 II — le^u —

-4 — —

11—

A -Watte

r and Sherwood Q
r and Sherwood /Q(ft

S, I

ft . U/ftfJ-a

N \

/* . fir/nna

Ss >L

D ~ Or/earner and Bradford • —

E ' O'Connell (absorber) 10,000
F'Q'ConrnN (absorber) 0
G 'Q'Cortne/tffrQcffonofor) ft *2.Q —

0.6

1 0.2
0.!

i —

--.

•'Sb_

•«•

•*•-.

*>,

IN-

">s

s X.

s-^^

008
0.06

).04
101

\-\

_^_

Jisi^

V1 "'

x ,

0.01 Q02 Q04Q06 01 0.2 04 060810 2 4 6 8 10 20

Liquid viscosiiy-centipoises
FIG. 17-6. Comparison of plate efficiency correlations.

40 60

Walter and Sherwood (Ref. 31) gave a correlation for the plate effi-
ciency based on the derivation of Eq. (17-3). Using the two-film
absorption concept (Ref. 33), they separated the over-all mass-trans-
fer resistance into a liquid- and vapor-film resistance.

wiiere KQ& « same as for Eq. (17-1)

koa * gas-film transfer coefficient
kid » liquid-filin transfer coefficient
H « Henry's law constant

(17-11.

FRACTIONATING COLUMN PERFORMANCE 459

The basic assumptions of the two-film theory are not satisfied by the
action of a bubble-cap plate, but Eq. (17-11) is probably a reasonable
approximation for the division of the resistance between the two
phases, indicating that solubility is a major factor in the relative
resistances. *

Walter and Sherwood assumed (1) that KG was proportional to
G/0, (2) that the total interfacial area of all the bubbles, a, was pro-
portional to the liquid depth from the center of the slots to the top of
the overflow weir, (3) on the basis of the data of Carey, Griswold,
Lewis, and Me Adams (Ref. 5), that KQCL was proportional to the cube
root of the slot width, and (4) that both k0a and kLa were proportional
to the 0.68 power of the liquid viscosity. Their equation for the value
of m in Eq. (17-3) is

HIS)

„(). 68,^0. 33

where H = Henry's law constant, Ib. mols per cu. ft. per atm.
P = pressure, atm.
p = viscosity of liquid, centipoises
w = slot width, in.

h = height from center of slots to top of weir, in,
El v - 1 - er*

Equation (17-12) can be made more suitable for distillation calcula-
tions by replacing Henry's law constant by the equilibrium constant
Ko, (« y/x), giving

where Ke = equilibrium constant = yjx

M = molecular weight of the liquid
d * specific gravity of liquid relative to water
tie terms 2.5 and O.Q0591£«Af /d are the relative resistances of the
vajzfor tod liquid phases, respectively.

/Ke and M are properties of the system and, for a given mixture, it
is difficult to make any major changes. In distillation, the values
of Ke for the key components usually range from 1.0 to a maximum
of approximately 5, but for the absorption of relatively insoluble
gases, Kg may be as large as 1,000. The latter systems give small
Values of m and low EMV. By increasing the total pressure, the values

460 FRACTIONAL DISTILLATION

of K e for the absorption cases can be reduced, but the values for
distillation cases remain about the same.

The viscosity term plays a major part in determining the value of
w, and it is subject to some control. In the case of distillation systems,
the liquids are essentially at their boiling points under the pressure
involved. Under such conditions most common liquids have viscosi-
ties of the same order of magnitude. However, raising the operating
pressure increases the temperature and lowers the viscosity. Thus,
high-pressure towers tend to give high plate efficiencies, but the gain is
usually not great enough to justify such operation for this purpose
only. In the case of absorption towers, the operating temperature can
often be varied independently of the pressure, and at a given operating
pressure a high temperature gives a larger value of Ke and a lower
value of ju. These two counterbalancing effects usually work to pro-
duce an optimum operating temperature.

In most distillation systems, the value of the viscosity is from 0.15 to
0.5 centipoise, and with slots J^$ to J^ in. wide they give point effi-
ciencies of 60 to 95 per cent. However, in the case of absorption, the
liquid viscosities may be as high as 20 centipoises, and the point effi-
ciencies may be 10 per cent or lower.

For a binary mixture it can be shown that the value of EMV must be
the same for both components in order to make the mol fractions in
the liquid and the vapor add up to unity. In the case of multicom-
ponent mixtures, each of the components can have a different value of
EMV for a given plate. On the basis of Eq. (17-13) it might be assumed
that the heavier components with lower values of Ke would have the
highest efficiency. There are no data which prove conclusively the
relative efficiencies in a multicomponent mixture on a given plate.
The inaccuracies of the measurements are such that they leave the
trend in doubt. However, they do indicate that for the mixtures so
far tested the difference in the value of E°MV between components is not
large. On the basis of diffusion theory, it would be expected that the
h^atjlfef;ij^mponents would approach equilibrium more slowly and
therefore have lower values of EMV.

Tfe^&lues calculated from these equations are given in Fig. 17-6,
for h « l.o, w = 0.25, and various values of KM/d.

Drickamer and Bradford (Ref . 7) analyzed the test data for a num-
ber of rectifying towers separating hydrocarbons and gave a plot of
E* as a function of the molal average liquid viscosity. The data pre-
sented by these investigators correlated reasonably well over a con-
siderable variation in liquid viscosity. They did not include systems

FRACTIONATING COLUMN PERFORMANCE 461

that would have large values of Ke = y/x, and the correlation would
probably not be suitable for such cases. Their relation is given in Fig.
17-6.

A correlation similar to that of Drickamer and Bradford was devel-
oped by O'Connell (Ref. 26), but in this caste the plate efficiency was
plotted as a function of ap (a = relative volatility) for fractionating
towers and of p,/HP or K6M^/d for absorption towers. These correla-
tions include both of the main factors, solubility and liquid viscosity,
found to be important by Walter and Sherwood. The inclusion of the
solubility factor directly with the viscosity is probably not so sound as
the type of grouping used in Eq. (17-13). It would be expected that
O'ConnelFs correlation would break down for extreme or unusual
variations in the solubility factor.

Geddes (Ref. 13) has presented a semitheoretical method for the
estimation of plate efficiency. The method is complicated to apply,
and several of the assumptions made in the derivation are question-
able. He obtained good results by the method in cases involving
widely differing conditions. For the present state of the art, it is
believed that the Walter and Sherwood equation gives as satisfactory
results and is easier to apply.

A more detailed analysis of the mechanism of the mass transfer
between the two phases of a bubble-cap plate has been made by
Etherington (Ref. 8). He presented correlations for both kGa and kLa.
These values would then be combined with Eq. (17-11) to give KGa
which would be used to calculate E^v. The method has been tested
on only a few mixtures, and a more detailed evaluation is needed to
determine whether the added complications are justified.

The correlations proposed by Gunness, Walter and Sherwood,
Drickamer and Bradford, and O'Connell are compared in Fig. 17-6.
The ordinate is — In (1 — E), and the abscissa is the viscosity of the
liquid in centipoises. The correlations are not all comparable.
Walter and Sherwood's relation was based on small laboratory units
and probably corresponds to point conditions, while the other rela-
tions were based on plate or over-all column efficiencies. For Eq.
(17-13) h was taken = 1.0, w = 0.25, and two curves are given with
KeM/d = 0 being used to approximate fractionating conditions and
KM Id = 10,000 to correspond to absorber conditions. Three curves
are given for O'ConnelPs relations; curves E and F are for the absorber
correlation with KeM/d = 0 and 10,000, respectively; and curve
G is for the fractionator correlation with a = 2.0. The curves for
Gunness and Drickamer and Bradford agree with the K6M/d = 0

462 FRACTIONAL DISTILLATION

curve of Walter and Sherwood at low viscosities and approach the
KM I A ** 10,000 curve at high viscosities. This is because the data
used in these correlations for the high efficiencies were for fractionating
columns having low values of liquid viscosities and of KeM/d, while
the low efficiency points were for absorbers with high values of liquid
viscosity and KeM/d. While in many cases the solubility and vis-
cosity factors tend to parallel each other, it is possible to vary the
viscosity widely for the same value of the solubility. Etherington's
data indicate that the viscosity and solubility factors should be sepa-
rated. The curves based on O'ConnelPs correlation for absorbers
with KeM/d = 10,000 are in reasonable agreement with curve B, but
the values for high solubilities (KeM/d near 0) do not appear to repre-
sent the data. At higher values of KeM/dj the curves agree approxi-
mately with those of Walter and Sherwood.

From a review of the available data, it is recommended that Walter
and Sherwood's equation be used for values of E^v. In using this
equation, it is suggested that for fractionating towers the relative vola-
tility of the light key to heavy key components be used instead of Ke to
calculate the Murphree point vapor efficiency for the light key com-
ponent and that this value of the efficiency be used for all components
in the mixture. The experimental data for fractionation systems do
not vary with compositions to the extent that would be indicated by
the use of Ke values. The viscosity term can be taken as molal aver-
age viscosity of the liquid. In the case of absorbers, it is suggested
that K = dy/dx be used instead of Ke = y/x, since the former is more
consistent with the derivation of Eq. (17-11). The efficiency should
be calculated for the key component, and it is suggested that the same
value be employed for the other components. The value of h should
be a function of tfye liquid depth instead of the " fixed " value given for
the correlation. It is suggested that this term be replaced by (&')°'5>
where hf is the distance from the center of the slot opening to the top
of the liquid level over the weir. With these changes, the relations
become

where m' - -j ^ r (17-14)

' , 0.00594OA 068 083

2,5 .] 1 ^0.68^0.83

h' = distance from top of liquid level at weir to center of slot

opening
K «• use relative volatility of key components for fractionating

FRACTIONATING COLUMN PERFORMANCE 463

columns, and K = (dy/dx) for equilibrium curve for
absorbers

jj = viscosity, centipoises
w = slot width, in.

For values of the Murphree plate vapor* efficiency in commercial
size towers, the following equation is recommended:

where m" = 7 - *\ - (17-16)

7 \

/0 K , 0.005JOf \ 0 68 o 33

/ 35 _| -- - 1 ^0.88^0.33
\ d /

The units are the same as those of Eq. (17-14).

Plate Efficiency Example. As an example of the use of these equations, the
plate efficiency for the plate design example of Chap. 16, page 433 will be calcu-
lated. The values of the various terms from this previous example are summarized
below:

K « relative volatility « 2.4
M -92

d - 52.8/62.4 - 0.86
w «* M in.
0/V - 1.2

At 6WfR. viscosity of toluene « 0.24 centipoise.
The liquid depth at the outlet weir is hw -f- hcr — 2.0 + 1.0 — 3.0 in.
The top of the slots is 1.75 in. above the plate and the slot opening, h9 «* 1.13,

1 13
giving the center of the slot opening at 1.75 -- ^—' » 1.2 in. above the plate.

The value of V is 3,0 - 1.2 - 1.8 in.
By Eq. (17-14).

(1.8)0'5

- 1.75

E^y - 1 -

= 0.83
By Eq. (17-16),

,// d-8)"

1.47

«- 1.02

464 FRACTIONAL DISTILLATION

EFFICIENCY OF PACKED TOWERS

The efficiency of packed towers is generally expressed as the height
equivalent to a theoretical plate. Most of the reported values of
H.E.T.P.'s are for small laboratory columns, since this is one of the
largest uses of packed columns. H.E.T.P. is a function of the packing
dimension and construction, tower size, vapor velocity, and system
being rectified. The efficiency of packed towers may be seriously
impaired by the liquid's tending to pass down one side while the vapor
flows up the other. This channeling of vapor and liquid prevents effec-
tive interaction between the vapor and liquid.

Baker, Chilton, and Vernon (Ref . 2) report the results of tests on the
distribution of water over various packing materials with air flowing up
through the packing. The water rate was 500 Ib. per hr. per sq. ft. in
all tests. They found that a ratio of tower diameter to packing size
greater than 8 to 1 gave a fairly uniform liquid distribution. At values
of the ratio less than 8, the liquid tended to run down the tower walls
and leave the center of the column nearly dry. A multiple-point
liquid distributor at the top improved the liquid distribution at the top
portion of the tower. The results of these investigators indicate the
desirability of having the tower diameter over eight times the size of
the packing material and of using multiple-point liquid distributors;
this latter is most important in short towers.

Fenske, Tongberg, and Quiggle (Ref. 11) give the results of a large
number of tests on packed laboratory towers. A comparison of some
of their results for the distillation of a carbon tetrachloride-benzene
mixture is given in Table 17-4.

The investigators conclude that (1) the best packings are one-turn
and two-turn wire or glass helices, carding teeth, and No. 19 jack
chain, (2) the efficiency of the packing decreases when the tower
diameter is increased or when the height of the packed section is
increased, and (3) different hydrocarbon mixtures give approximately
the same value for H.E.T.P.

Weimann (Ref. 32) has published results on the fractionation of
ethanol-water mixtures in packed towers. Using a superficial velocity
of 1 f.p.S. at 0/D = 1.0 with 8- by 8-mm. porcelain Raschig rings in a
0.11-m. (43^-in.) diameter tower, H.E.T.P. values of 6 to 8.5 in. were
obtained for packing heights of 3.5 to 13 ft. A larger tower, approxi-
mately 1 ft. in diameter, using a 7-ft. depth of the same packing gave
an H.E.T.P. of 8.5 in. at a superficial vapor velocity of \Y± f.p.s.

Jantzen (Ref. 17) has presented the results of fractionating an

FRACTIONATING COLUMN PERFORMANCE

465

ethanol-water mixture in a 13.5-cm. (1.37-in.) tower packed to a depth
of 1 m. with either 1- or 0.46-cm. Raschig rings. The values of
H.E.T.P. calculated from his data range from about 3 to 6 in., for
superficial vapor velocities ranging from 0.15 to 2 f .p.s. The H.E.T.P.
values increased as the 0.2 power of the vapor velocity and were about
50 per cent larger for the large than for the small rings. These values
were found to be independent of the liquid concentration and of the
reflux ratio (experimental values of 0/D ranged from 4 to 10).

TABLE 17-4

Packing

Tower dimensions,
in.

H.E.T.P.,
in.

Straight %2-i11* carding teeth. .
Straight JHj2-in. carding teeth.
Bent H-in« carding teeth ...
Miscellaneous carding teeth . . .
Double-cross wire form ....
Hollow-square wire form
No. 20 single-link iron jack chain
No. 2 cut tacks .... ...

0 76 by 27
0 76 by 27
0.76 by 27
0 76 by 27
0.76 by 27
0.80 by 55
2.0 by 53
0 8 by 55

1.5
1.7
1.7
2.2
2.1
5.4
5.2
2.4

6-turn No. 24 Lucero wire helix
No. 18 single-link iron jack chain. .
Glass tubes
No. 16 single-link iron jack chain. ... ...

0.8 by 66
2 0 by 53
0 78 by 27
0.76 by 27

8.0
6.5
5 5
4.2

Fenske and coworkers (Ref . 10) have also made an extensive study
of the efficiency of packings when used for the separation of a n-hep-
tane-methyl cyclohexane mixture in a 2-in,-diameter glass tower at
total reflux. The tower was 114 in. high and was operated at atmos-
pheric pressure. A few of their results are summarized in Table 17-5.

These data indicate that H.E.T.P. values as low as 1.5 in. have been
obtained, making it possible to obtain the equivalent of a large number
of theoretical plates in a relatively short height. Because of their effi-
ciency and simplicity, packed towers are widely used for laboratory
columns. However, when larger packed towers are used, the efficiency
in general decreases, and H.E.T.P. values of a few feet are more com-
mon for columns of commercial size.

The cause of the poor results in large-diameter columns is apparently
poor liquid distribution. The use of liquid redistributor plates every
few feet can increase the effectiveness of the units, but they increase the
pressure drop. With a number of such liquid distributors, the values

466 FRACTIONAL DISTILLATION

of H.E.T.P. can be made reasonably low and reproducible, but the
tower has become almost equivalent to a perforated plate column.

The values of H.E.T.P. for large columns are so random that any
correlation gives only a qualitative idea of what might be expected.
The following equation is presented as a rough guide only, and it would
not be surprising if actual values in some cases were several fold
different.

H.E.T.P.

0 5

(17-17)

where H.E.T.P. = feet, for nominal packing size less than one-tenth

tower diameter
dt = tower diameter, ft.

MQ = average molecular weight of vapor phase
0 = superficial mass velocity of vapor, Ib. per hr. per

sq. ft.

P. =8 absolute pressure, atm.

D = diffusivity of solute gas in liquid, sq. cm. per sec.
L = liquid rate, Ib. per hr. per sq. ft. tower cross section
p' as liquid viscosity, poises
p an liquid density, g. per cu. cm.
H = Henry's law constant (Ib. mols/cu. ft.) per atm.
HP = 0.016 (KMi/p), where K - y/x, ML = molecular
weight of liquid

The individual film coefficients are based mainly on the results of
Sherwood and Holloway (Ref . 29), but they have been combined with
other factors and the equation should be considered purely empirical-
If the values obtained from Eq. (17-17) are greater than the distance
between distributor plates, it is suggested that this latter difference be
employed.

Packed Tower Example. Equation (17-17) will be used to estimate the
H.E.T.P. values for the atmospheric distillation of a benzene-toluene mixture in a
packed tower 5 ft. in diameter. The liquid and vapor rates are 480 and 400 Ib.
mols per hr., respectively. The calculation will be made for the section near the
bottom of the column where the liquid and vapor are essentially toluene.

Solution

Area of column - 5»(0.7854) « 19.6
0 - ffl)(^2) - 1,880 Ib. per hr. per sq. ft.

FRACTIONATING COLUMN PERFORMANCE 467

L - -^p - 2,250 Ib. per hr. per sq. ft.

MG -92
dt - 5.0

HP - 0.016 (MA - 0.016 [2Q(g52-1 (see page 463 for values)
- 4.15

The diffusivity for the system was estimated to be 1.0 X 10~"5 sq. cm./sec. at
t°C., and it was corrected to the higher temperature assuming that the diffusivity
paried inversely as the 1.5 power of the liquid viscosity.1 This calculation gave
> « 3.5 X 10~5 sq, cm. per sec.

ia»V_ 3o)2(X)

fjL\*'' f °-OQ24 1°'B
\PD) ** [o.85(3.5 X 10~6) J

By Eq. (17-1?),
H.E.T.P. - A [l2(i,880)o • + (3.5 x ^JS) (30,200) (9)] ~ 6'5 ft'

The pressure drop for this tower can be estimated from Eq. (j#-40). It will be
ssumed that the packing size is 3.0 in.

AP
h

The Reynolds number, »^ - 00024(242) " 81°
By Eq. (16-42),

/ « 38

* 810°-18

From the table on page 438, Aw = 0.86.

By Eq. (16-43), ^-14+ 0.0005(2,250) - 2.5

The superficial gas velocity, assuming that absolute pressure is 1.0 atm.,

400(359) (385)
U " 19.6(3,600)(273)

- 2.87 f .p.s.

AP 2(14) (0.86) (2.5) (0.21 ) (2.87)*
h * 32.2(H)

- 12.9 p.s.f . per ft.

» 2,9 in. of liquid toluene per ft. of length

This pressure drop is high indicating that the tower is operating near the flooding
ondition. Figure 16-4 would give a flooding velocity of about 4 f .p.s. for these
onditions.

1 See ARN6UD, Sc,D. thesis in chemical engineering, M.I.T., 1931.

468

FRACTIONAL DISTILLATION
TABLE 17-5

Packing

Vapor
velocity, f .p.s.

H.E.T.P.,
in.

Open tower

0 25 to 1 7

25 5 to 29 2

H- by J^-in. carbon Raschig rings ...
H- by H-in- stoneware Raschig rings .
%. by JHj-in. stoneware Raschig rings . . ...
}4r by Ji-in. carbon Raschig rings . .
H- by J£-in. glass Raschig rings . .
No. 19 aluminum jack chain
J£- in. clay Berl saddles
^-in. aluminum Berl saddles. .

0.25 to 1.45
0 8 to 1.55
03 to 1 . 1
0 15 to 0 5
04 to 0 9
0.1 tol 65
0.05 to 1.6
03 to 1 75

6 0 to 11 3
5 0 to 85
3.8to 7.3
4 7 to 60
4 3 to 6.8
4 2 to 89
5 8to 7.0
4 1 to 70

6-mesh carborundum

01 to 0.35

1 6 to 51

J4-in. aluminum single-turn helices
JKe-in. aluminum single-turn helices
JHj2-in- single-turn stainless-steel helices ,
y^i-\VL. single-turn nickel helices. . . ^v
J^-in. single-turn nickel helices ...
%2-fa- single-turn stainless-steel helices
%- by %2-in- carding teeth .

0.7 to2.1
0.1 to 1.8
0 3 to 1.25
0 55 to 1.65
01 to 1.0
0 15 to 0.95
0.4 to 1 35

5.0 to 10 7
3 7 to 62
4 Oto 5.4
2 9 to 55
2.9 to 5 5
1.5 to 2 3
2. Oto 4 2

Nomenclature

a «• interfacial area of bubble or total contact area per plate
D « diffusivity, cm.2/sec.
d «* specific gravity, relative to water
dp * diameter of packing, ft.
dt «• tower diameter, ft.
E «• plate efficiency
E° *• over-all plate efficiency
E°MV *• Murphree vapor-phase plate efficiency
E*MV ** Murphree point efficiency
E°ML s« Murphree liquid-phase plate efficiency
E°MT « apparent plate efficiency with entrainment
e *» mols of liquid entrained per mol of vapor

O «• mols of gas in bubble or superficial mass velocity of vapor, lb./(hr.) (sq. ft.)
H » Henry's law constant, (Ib. mols)/(cu. ft.) (atm.)
h * height from center of slots to top of weir, in.
hf » distance from top of liquid level to center of slot opening
K « slope of equilibrium curve « dy/dx
K. - y/x

KG •» over-all mass-transfer coefficient
koa «• gas-film mass-transfer coefficient
KL •• over-all mass-transfer coefficient
kua « liquid-film mass-transfer coefficient
27 ** liquid in slug

FRACTIONATING COLUMN PERFORMANCE 469

L «• liquid rate, lb./(hr.)(sq. ft. of tower cross section)
M »• molecular weight of liquid
I o » average molecular weight of vapor phase
P » total pressure, atm.
w — slot width, in.
a; =• mol fraction in liquid
x* ** equilibrium with vapor leaving
#0 =* liquid leaving plate
#* = liquid entering plate
y = mol fraction in vapor
yr = mol fraction in bubble
ye — equilibrium with liquid
2/» « entering plate
y0 «« leaving plate

2/J » equilibrium with liquid leaving plate
M = viscosity of liquid, centipoise
M' = viscosity of liquid, poise
p « liquid density, g./cu. cm.
0 = contact time of bubble

References

L. ATKINS and FRANKLIN, Petroleum Refiner, 16, No. 1, 30 (1936).
!. BAKER, CHILTON, and VERNON, Trans. Am. Inst. Chem. Engrs., 31, 296 (1935).
8. BROWN, World Power Conference, 1936, Trans. Chem. Eng. Congress, 2, 330.
L BROWN, SOUDERS, NYLAND, and HESLER, Ind. Eng. Chem., 27, 383 (1935).

1. CAREY, GRISWOLD, LEWIS, and McADAMs, Trans. Am. Inst. Chem. Engrs.,
30, 504 (1934).

>. COLBURN, Ind. Eng. Chem., 28, 526 (1936).

T. DRICKAMER and BRADFORD, Trans. Am. Inst. Chem. Engrs., 39, 319 (1943).

5. ETHERINGTON, Sc.D. thesis in chemical engineering, M.I.T., 1949.

). FAIRBROTHER, S.M. thesis in chemical engineering, M.I.T., 1939.

). FENSKE, LAWROSKI, and TONGBERG, Ind. Eng. Chem., 30, 297 (1938).

L. FENSKE, TONGBERG, and QUIGGLE, Ind. Eng. Chem., 26, 1169 (1934).

5. GADWA, Sc.D. thesis in chemical engineering, M.I.T., 1936.

J. GEDDES, Trans. Am. Inst. Chem. Engrs., 42, 79 (1946).

L GTTNNESS, Sc.D. thesis in chemical engineering, M.I.T., 1936,

>. HAUSEN, Forschungsheftt 7, 177 (1936).

5. HORTON, S.M. thesis in chemical engineering, M.I.T., 1937.

r. JANTZEN, Dechema Mon., 6, No. 48 (1932).

*. KEYES and BYMAN, Univ. Illinois Eng. Expt. Sta., Butt. 329 (1941).

J. KIRSCHBAUM, Forschungsheft, 8, 63 (1937).

). LEWIS, Ind. Eng. Chem., 14, 492 (1922).

L. LEWIS, Ind. Eng. Chem., 28, 399 (1936).

2. LEWIS and SMOLEY, Bull. Am. Petroleum InsLt 11, Sec. 3, No. 1, 73 (1930).
I LEWIS and WILDE, Trans. Am. Inst. Chem. Engrs., 21, 99 (1928).

L MURPHREE, Ind. Eng. Chem., 17, 747 (1925).

5. NORD, Trans. Am. Inst. Chem. Engrs., 42f 863 (1946).

5. O'CoNNELL, Trans. Am. Inst. Chem. Engrs., 42, 741 (1946).

470 FRACTIONAL DISTILLATION

27. PEAVY and BAKER, Ind. Eng. Chem., 29, 1056 (1937).

28. RAOATZ and RICHARDSON, Calif. Oil World, October, 1946,

29. SHERWOOD and HOLLO WAY, Trans. Am. Inst. Chem. Engrs., 36, 39 (1940).

30. WALTER, Sc.D. thesis in chemical engineering, M.I.T., 1940.

31. WALTER and SHERWOOD, Ind. Eng. Chem., 33, 493 (1941).

32. WEIMANN, Chem. Fabrik, 6, 411 (1933); Z. Ver. deut. Chem., No. 6 (1933).

33. WHITMAN, Chem. Met. Eng., 29, No. 4 (1923).

CHAPTER 18
FRACTIONATING COLUMN AUXILIARIES

In addition to the fractionating column, there are certain auxiliaries
necessary for its operation. These include the condenser, reboiler,
pumps, instruments, valves, and receivers.

Condensers. The condensers are tubular heat exchangers used to
liquefy the overhead vapor to produce the necessary reflux. For small
columns, the condensers are mounted above the column, and the con-
densate returns to the system by gravity. For larger installations, the
condensers are generally mounted below the top of the column, and the
reflux is pumped back to the system. Placing the condensers at the
lower level reduces the size of the supporting structure and makes them
more accessible for cleaning. The design of condensers is largely a
problem of heat transfer, and the reader is referred to "Heat Trans-
mission" by Me Adams (Ref. 1). In addition to the heat removal
problems, the various pressure drops should be evaluated to determine
whether the vapor and condensate will flow properly.

Reboilers. In small units the reboiler is frequently mounted
directly below the column and consists of a large kettle containing a
steam coil. In large installations it is customary to build the reboiler
as a separate unit. This allows easier servicing of the heat-transfer
surface. Where fouling of the heating tubes is rapid, two reboilers are
often included to allow a spare that can be used while the other is being
cleaned. The design of a reboiler is also mainly a heat-transfer prob-
lem, but pressure drops and adequate settling space above the liquid
should be considered. Where separate reboilers are employed, the
vapor is piped to the column, and the pressure drop may be such that it
hinders the liquid flow to the reboiler. Considerable splashing of the
liquid in the still is often encountered due to the deep liquid level, and
a settling space of at least 2 ft. should be employed with a cross-sec-
tional area two to three times that of the column. In order to obtain
a large cross-sectional area, horizontal drums are frequently used foi
reboilers, and these units can be arranged to give longitudinal flow of
the liquid. This can give additional separation by a Rayleigh-type
distillation. The reboiler should be designed to obtain the desired

471

472 FRACTIONAL DISTILLATION

vapor production, and any extra rectification in the still should be a
secondary consideration.

When live steam is employed, it can be added directly under the bot-
tom plate of the column, or a still can be used and the steam intro-
duced under the liquid through some type of distributor. Steam is
the most common medium for the stills, but in some cases it is not
desirable because (1) the still temperature is too high, (2) the leakage
of water into the system may be undesirable and hazardous, or (3)
the still temperature may be so low that freezing may be encountered.
In some cases, organic liquids or vapors, such as Dowtherm, are used
as the heating medium, or the liquid at the bottom of the column may
be pumped through a direct-fired heater and partly vaporized. In
this last case, the liquid-vapor mixture from the heater is passed
through a separator, with the vapor going to the column and the
liquid being removed as bottoms, although it may be recycled through
the heater to obtain additional vaporization.

Fractionating Column Control. In order to obtain satisfactory
performance from a fractional distillation unit, it is necessary to con-
trol the operating conditions. In the case of a continuous rectification
system, it is desirable, and almost essential, to introduce a feed of rela-
tively constant composition and temperature at a reasonably constant
rate. In general the system is to make a specified separation with the
given column and a given feed plate, and this leaves two main variables
to be controlled: (1) the reflux ratio and (2) the heat supply to the still.
For example, consider the separation of a binary mixture and assume
that the desired separation is not being obtained. The reflux ratio
can be increased which should increase the degree of separation, but
if the same heat supply to the still is maintained, the overhead product
rate will decrease. To obtain the desired percentage of the feed as
overhead product, it may be necessary either to decrease the feed rate
or to increase the heat supply if the column will handle the added load.
The problem of fractionating tower control becomes one of maintaining
the proper balance of the feed rate, reflux ratio, and heat supply to the
still. Manual control can be used, but automatic control instruments
generally do a better job because operators tend to overcorrect. The
problem of overcorrecting is especially serious in rectification because
the large liquid holdup introduces long time lags. In columns with a
large number of plates, the mols of liquid held up in the system may be
the same order of magnitude as the vapor or liquid throughput per
hour and may correspond to several times the feed rate per hour,
This large quantity of liquid acts like a flywheel and, smooths out the

FRACTIONATING COLUMN AUXILIARIES

473

operation, but it makes the response to corrective action very slow and
leads to overcorrection.

Instrumentation. One method of instrumentation of a fractionating
column is shown in Fig. 18-1. The feed rate to the column is held con-
stant by the feed-flow controller. The overhead vapor is condensed

Feed flow
controller?

Feed

Reboiler
vapor
controller i

Reflux
drum

Vent

Liquid level
controlhr^

„„

Reflux

IX,

product

Reflux flow
controller

Liquid level
controller-*

Steam

^ Steam trap Bottom product

FIG. 18-1. Diagram of instrumentation for continuous rectification system.

and runs to a reflux drum accumulator, and a portion of the condensate
is pumped back to the column for reflux. The excess condensate is the
overhead product and is removed from the drum by the liquid level
controller. The reflux rate is regulated by the reflux-flow controller
which attempts to hold the temperature near the top of the column at
a fixed value. The steam rate to the still is controlled by a reboiler
vapor controller operating on a temperature indicator, and the unva-
porized liquid is removed from the still by a liquid-level controller.
In the operation of the system illustrated in Fig. 18-1, the feed-flow,

474 FRACTIONAL DISTILLATION

reflux-flow, and reboiler vapor controllers would be set at the desired
values. If the temperature at the top control point became lower
than the desired value, the reflux-flow controller would reduce the
reflux rate which would increase the product withdrawal rate and raise
the temperature. If the temperature were too high, the controller
would take the opposite action. If the temperature at the reboiler
control point becomes lower than desired, the controller will increase
the steam supply to the still, resulting in a higher vapor rate. This will
give better stripping of the light components and will raise the tem-
perature at the control point. This action of the reboiler vapor con-
troller will lower the reflux ratio by increasing the overhead product
rate and will probably require the reflux-flow controller to increase the
reflux rate to hold the top temperature down. In this system the
reboiler vapor controller operates to give the desired temperature at
the bottom control point which presumably gives the specified bottom
product. The reflux-flow controller operates to give the desired tem-
perature difference between the two control points. The system is
operable because for a given number of plates the degree of separation
can be varied by changing the reflux ratio. The feed rate and the tem-
perature values that are fixed must obviously be within the fractionat-
ing capabilities of the system.

A number of alternate control systems can be employed. Thus a
fixed reflux ratio could be used with the top control point regulating
the feed rate to the column with the bottom control operating the same
as before. In case the overhead product is difficult to liquefy, a partial
condenser may be employed, and this requires a different method of
control. A typical case of this type is gasoline stabilization where a
small amount of €2, Ca, and C4 hydrocarbons are removed to obtain
the desired volatility of the bottoms product. For this case, the sys-
tem of Fig. 18-1 could be modified as shown in Fig. 18-2 by removing
the overhead product through the vent line. This vent line would be
equipped with a pressure controller which would adjust the vapor
removal rate to maintain the desired operating pressure. The liquid-
level controller would be used to adjust the cooling water rate to the
condenser instead of the product withdrawal rate. The reflux-flow
controller would operate to maintain the top temperature as before,
and the liquid-level controller would decrease the cooling water rate if
the level became too high, thereby decreasing the rate of condensate
production and increasing the overhead vapor rate.

Control Variables. One of the difficult problems in the automatic
control of fractionating towers is finding an easily measured charac-

FRACTIONATING COLUMN AUXILIARIES

475

teristic that will ensure the desired separation. Temperature is the
most commonly used factor, but it is not always a satisfactory criterion.
Thus, if the product is of high purity and contains only a small amount
of other constituents, these can vary several fold without significantly
changing the equilibrium temperature. This is particularly serious
in the separation of close boiling constituents. In the case of multi-
component mixtures, temperature is not a good criterion of composi-
tion, but it can be a satisfactory indication of volatility. The problem

Overhead
product

^Pressure
controller

Liquid level
controller

Reflux flow
controller

FIG. 18-2. Diagram for partial condenser operation.

is particularly difficult in extractive distillation systems where the
presence of the large quantity of solvent masks the effect of composi-
tion of the key components on the temperature. Temperature control
is also sensitive to the column pressure.

In some of these cases, the effectiveness of temperature can be
improved by proper location of the temperature control point. For
example, in the case of high product purity, the temperature control
point can be placed several plates f rom^the end of the column at a point
where the minor constituents have higher concentrations. This results
in a larger temperature variation which makes the instrumentation
easier, but it removes the direct control on the product.

Other control factors besides temperature have been used, such as
density of the liquid, refractive index of the liquid, infrared or other
spectrographic types of analysis, and freezing point. The factor should
be one which gives an indication of product composition and which can
be easily and rapidly determined by a relatively inexpensive, stable

476

FRACTIONAL DISTILLATION

instrument capable of operating electronic or pneumatic equipment.
The spectographic type of instruments should be very useful because
they can frequently be adjusted to indicate the amount of an impurity
present in extremely small amounts. Unfortunately they are some-
what delicate and need frequent adjustment, but these defects can
undoubtedly be eliminated.

0.1

20 30

FIG. 18-3.

Rate of Approach to Equilibrium. Corrective action or other
changes in the operating conditions introduce a transient condition into
the system. Consider a section of a column operating at steady-state
conditions. Assume that a change is made on one of the plates (such
as a change in the feed composition to the feed plate), and the problem
is to determine how rapidly other plates will approach their new
equilibrium condition. The exact mathematical solution of the gen-
eral case is very complex, but solutions based on certain approxima-
tions can be obtained. The curves given in Fig. 18-3, based on a
number of stepwise integrations, should be helpful in obtaining approxi-
mate results. In this figure the ordinate, F, is the fractional approach

FRACTIONATING COLUMN AUXILIARIES 477

to the new equilibrium conditions of a plate which is n plates from the
one on which the change in composition was made. The value of F is
defined as,

(yn)* - y°n *

where yn = vapor leaving plate at time 0

(2/n)oo = new equilibrium value of yn, i.e., value of yn at 0 = °o
y°n = equilibrium vapor composition before change in condi-

tions were made

The abscissa is the dimensionless ratio of liquid flow through the
section to liquid holdup in the section. This group is

00
nh

where 0 = overflow rate, mols per unit time
0 = time
n = number of plates in section between where conditions were

modified at 6 = 0 and plate n
h = holdup per plate, mols

The value of F decreases rapidly as (00/nh) increases, indicating that
the section approaches the new equilibrium after a change, more rap-
idly for large values of the overflow rate, small values of holdup per
plate, and a small number of plates. Lines are given on Fig. 18-3 for
different relative volatilities between the key components, and for mul-
ticomponent mixtures the value of F should be applied to the light key
component. For a change in condition that affects all plates almost
simultaneously, such as a change in reflux ratio, it is suggested that the
correlation of Fig. 18-3 be employed with n = 2.0.

To bring a section of a column to a reasonable approach to the new
equilibrium would require, according to Fig.' 18-3, a liquid throughput
of two to ten times the liquid holdup in the section. For usual design
conditions, a tower operating at atmospheric pressure will have a liquid
throughput per minute equal to about the holdup per plate. For high-
pressure towers the ratio of liquid holdup to liquid flow per plate may
be as low as 0.1 ; for vacuum towers it may be as high as 5.0. Thus an
atmospheric pressure column with 20 plates above the feed plate may
require from 0.5 to 3 hr. to steady down after a change in feed composi-
tion, while a high-pressure tower would adjust itself much more rapidly.

Figure 18-3 indicates that a change in the vapor composition entering
a section will make a difference in the plate just above much sooner

478 FRACTIONAL DISTILLATION

than on plates farther removed, while a change in the reflux ratio would
change (y)«> for all the plates almost simultaneously. Thus changes
in feed composition will be noticeable much sooner at plates just above
and below the feed plate, and instruments located in these positions
would be able to correct for such changes in composition much more
rapidly. However, in nmlticornponent mixtures it may be very diffi-
cult to relate conditions at these locations to the desired separation.
While the instruments can be set to control the temperature at the feed
plate region, this may not give control of the product compositions.
Control points near the ends of the columns are less affected by changes
in feed compositions but do not anticipate variations so rapidly and are
generally less sensitive. Changes due to reflux ratio or vapor rate are
most apparent where the change in composition of the key components
per plate is greatest. This usually occurs in the intermediate section
somewhat above and below the feed plate.

In most cases with temperature regulation, the control point for the
top should be down enough plates from the top to gain the amplified
temperature difference and the anticipatory effect of being closer to the
feed plate, but it should not be so close to the feed plate that it is
appreciably affected by the components heavier than the heavy key
component. Similar considerations apply to the section below the
feed plate.

Reference

1. McADAMS, "Heat Transmission," 2d ed., McGraw-Hill Book Company, Inc.,
New York, 1942.

APPENDIX

TABLE OF ENTHALPIES, on LATENT HEATS OF VAPORIZATION

Substance

Boiling point,
°C.

Molecular
Weight

Latent heat in
calories per gram
at boiling point

Acetal .

104.0

118.1

66 2

Acetaldehyde

20 8

44.0

134.6

Acetic acid

118 7

60.0

89.8

Acetic anhydride

136 4

102.1

66.1

Acetone . .

56.6

58.1

125.3

Acetyl chloride

55 6

78.5

78.9

Ammonia . .

- 34 7

17.0

341 0

Aniline ....

183.9

93,1

109 6

Benzaldehyde

178 3

106.1

86.6

Benzene . .

80.2

78.1

93.5

Benzyl alcohol

205 0

108.1

98 5

Brombenzene

155.5

157.0

57.9

Butyl alcohol (n)

117.6

74.1

143 3

Butyl alcohol (iso)

107 9

74.1

138 9

Butyric acid (iso)

162.2

88.1

114 0

Carbon tetrachloride

76.8

153.8

46 4

Chlorbenzene .

131.8

112.5

75.9

Chloroform .

61.2

119.4

58 9

Cresol (m) ,

200 5

108.1

100 5

Ethyl bromide .

38.2

109.0

60 4

Ethyl iodide —

72.3

155.9

47 6

Formic acid . .

100.8

46.0

120 4

Glycol

197.1

62.1

190 9

Heptane . .

98.4

100.2

74.0

Hexane

69.0

86.1

79 2

Iso amyl alcohol . .

130.1

88.1

125 1

Iso propyl alcohol .

82.9

60.1

161.1

Methyl alcohol . .

64.7

32.0

261.7

Methyl chloride

- 24.1

50.5

96. 9 (at 0°C.)

Methyl iodide .

42.4

141.9

46.0

Methyl ethyl ketone

81.0

72.1

103.5

Methyl aniline . . .

193.8

107.1

95 5

Nitrobenzene

208.3

123.1

79.2(atl51.5°C.)

Octane

125.8

114.2

71.1

Pentane ...

36.3

72.1

85.8

Propyl alcohol (w) .

97 4

60.1

162.6

Toluene

110.4

92.1

86.8

o-Toluidene .

203.3

107.1

95.1

Water. . .

100.0

18.0

536.6

o-Xylene . . ....

144 0,

106 1

82.5

479

APPENDIX

Vapor Pressure, Pounds per Square Inch

10 100 LOOQ IO.OOCL

K£Y

Numbers- No of carbon
atoms per molecule
Unpnmed Hos -Paraffins
Primed Nos«0lefms
6" s Benzene
44 W * Water

IpOO IQOOO7 ^^ ^t!OQ

Pressure, Millimeters """

Vqpor

FIG. 1. Cox chart for extrapolating vapor-pressure temperature curves. (Walker,
Lewis, Me Adams, and GUliland, "Principles of Chemical Engineering," 3d ed., McGraw-
Hill Book Company, Inc., New York, 1937.)

481

482 FRACTIONAL DISTILLATION

Specific heat* Btu/(lb)(DegF)« Pcu/(Lb)(DegC)

LIQUID

RANGE DE6.C

SPECIFIC
HEAT

ACETIC ACID I00°7o
ACETONE
AMMONIA

-7^1

AMYL ALCOHOL

— 50— 25

•»

AMYL ACETATE

0— \00

DEGF
400 —

30
23
27

ANILINE
BENZENE
BENZYL ALCOHOL

10- 80

^-02

BRINE^STo Na C?

-40- 20

iol2<

44
2

BUTYL ALCOHOL
CARBON' DISULPHIDE
CARBON TETRACHLORIDE

^LL0°RToERNMZENE

0-100

"?§: II
8:'°5°o

E-0.5

300 -i

4
1

DECANE
\ D1CHLOROETHANE
DICHLORO METHANE
DIPHENYL
DIPHENYLMETHANE
DIPHENYL OXIDE

-80- 25
-30- 60
-40- 50
80- 120
30- 100
0-200

60 06A

0 ?0 9 010
08 0

^-0.4

DOWTHERM A
ETHYL ACETATE
ETHYL ALCOHOL 100°To

0-200
-50- 25
30- 80

HO OQ O'3A

200 -E

46
50
25

ETHYL ALCOHOL 95 7o
ETHYL ALCOHOL 50%
ETHYL BENZENE
ETHYL BROMIDE
ETHYL CHLORIDE
ETHYL ETHER
ETHYL IODIDE

20- 80
20- 80

o-ioo

-50- 40
-100- 25
0-100

M ^Do O9I
2^001719 Z] 024

250 23
7A026

27 2^00 31 34

r-as

_;.r;—

ETHYLENE 6LYCOL

-40-200

t-oQ wfivoD
37 ^ ??

AO 4IQ 45 ° ^l

3-0.0

100-i

•j/ O /% AA O

^"S ~> T"4 Q *IV

-Z

46° 047

LIQUID

WN6EDEGC

049

rREON"IICCCf3F)

-20- 70

— 07

FREON-I2(CCI2F2)

-40- 15

—

•E

•REON-2l(CHCI2t)
rREON-22(CHCTF2)

-20- 70
-20- 60

0 -r

"REON-II3(CCI2F-CCIF2)
SLYCEROL

-20- 70
-40- 20

HEPTANE

0- 60

— .

35
48

HEXANE
-iYDROCHLORICACID.30%

-80- 20
20- 100

050 510

Z-Q8

41 !

SOAMYL ALCOHOL

0- 100

„,-

SOBUTYL ALCOHOL

0- 100

—

SOPROPYL ALCOHOL

-20- 50

—

SOPROPYL ETHER

-80- 20

—

Y| ETHYL ALCOHOL
METHYL CHLORIDE

:8§: io

-too -i

14
12

^T^/Nf^

T ?o8

Z~a9

34 (
33

^ONANE
XTANE

*: 8

5ERCHLORETHYLENE

•30- 140

PROPYL ALCOHOL
3YRIDINE

-20- 100
^50- 25

sui:ffii/^oio)aoi *8%

-20- Al

O 55O

E-u>

TOLUENE

0- 60

••

WATER

10- 200

!•>

0- 100

(YLtNt MElA

(YLENE PARA

0- 100

FIG. 2. True specific heats of liquids. (Ch&ton, Cdbwrn, and Vernon, based mainly
on data from "International Critical Tables," McGraw-Hill Book Company, Inc., New
York, 1926-1930.)

APPENDIX

483

0 10 20

Mol per cenf hydrocarbon
30 40 50 60 70 *

0 90

o.'

j"4

— 60

60 V o

— «to

50 —

"^W1^"

— 1ft

40 — -

* yfc

r"\

— 30

30 ~

'*-

\ 1

\^*gA

\[ *

>^ ,Q

— 70

20 -

Sjo

[Z^2v4~-

, «

N-i^Sv-l-

\ XS

IX I ^

>^ jrsf

<^(

— fl

H**

-*•

i^a*

rv \. ' »^

_ <

_ —

i-i^ik

I* &\u >VB

issS

-JO

11 " IV

1 1

X<( >i

v '\ ^J

-eo-'

• A t

rs -Atmospheric bor/ing

rH * At/nosphefc boiling
ternperafare, hydro cttf

Ljb^ ^X

N N

S. i

L -,„- - ou, rr

-9A

rTV

"i <> i? 'i *^\

**i!sJSl.

\ \.

-1 h

'ton *X

_j_ ^|r

90 ^^

• X *»

-40 —

l/fo

Hy^foCorbofi$ /w

>v^

jrecarbons

2^J

> »

V w

~^KffoT$ F

^< 0

Kehnes

0 *

-60

AMydes

Phenols &

(

0 0

"*" /v

~^T crgsoh

]

. M ( , ,f

""80

JO" 2f '30' 40 50 60 70 80 90 100
Mo) per cert hydrocarbon

i'jto. 3. Composition of binary azeotropes— hydrocarbons and various solvents.
[Courtesy of Meissner and Greenfield, Industrial and Engineering Chemistry, 40, 438
(194S).}

484

FRACTIONAL DISTILLATION

60
50

40
30

20
10

S
1

-20
-30
-40
-50
-60
-70
-80,

o Hydrocarbons ana

velds

• Hatogenated hydrocarbons and adds

>TJ

•

/°

•

%vj

•

'•

*•

>•

>o°y

(

0°

°/

t_^

;

°°/

T •

«•

Atmospheric boiling

Atmospheric boiling
kmperati/re, hydrocarbonf *K

-'0

•

,,^-K

I c

) 10 20 30 40 50 «, 60 70 80 90 101

Mol per cent '

* Abscissa is mol per cent halogen a ted hydrocarbons for halogenated hydrocarbons-acids sys-
tems and mol per cent acids in hydrocarbons-acids systems.

FIG. 4. Composition of binary azeotropes — hydrocarbons and carboxylic acids
[Courtesy of Mewmer and Greenfield, Ind. Eng. Chem., 40, 438 (1948).]

APPENDIX

485

Table of "K* values

Alcohols

ill

•

Aromattc

Symbol

•

Cyclic
nonaromafic
Monoolefins

-2
-5

-5
•8

-15

+/J

Diolefins
Aromatics

•10

-10
-/J

•20

-5

+/5

/:/

T - No data ® • Not plotted

**ffi-(W)*K

•

Atmospheric boiling points, *K

TI • Low boiling component
TT * High boiling component
Ts 'Solvent
T2 » Azeotrope

•^

LSs__^

nnS

rfjd

&«£

^»

0 0 10 20 30 40 50 60 70 00
X

FIG. 5. Atmospheric boiling points of binary azeotropes — hydrocarbons and various
solvents. [Courtesy of Meiaaner and Greenfield, Ind. Eng. Chem., 40, 438 (1948).]

INDEX

Absolute alcohol, azeotropic distillation
of, 313

two-tower system for, 206
Activity coefficients, 50, 54
Adsorption factor method, 345
Allowable gas and liquid velocities, 439
Allowable vapor velocity, 430
Amagat's law, 38

Analytical equations, binary mixtures,
174

finite reflux ratio, 176

minimum reflux ratio, 176

multicomponent mixtures, 354

straight equilibrium curve, 183

total reflux, 174

use of, 181

A.S.T.M. distillation, 325
Auxiliaries for fractionating column, 471
Azeotrope, 21

effect of pressure on, 204

example, 205

Margules equation for, 21

maximum boiling-point type, 21

minimum boiling-point type, 20, 95

pseudo-, 20

separation of binary, 196
Azeotropic distillation,* 285, 312

absolute alcohol, 313

agents for, 287

diagram for, 313

minimum reflux ratio, 321

total reflux, 321

Van Laar equation for, 286
Azeotropic mixtures, 21, 192

Batch distillation, 108, 370
constant distillate composition,
constant reflux ratio, 374

4*7

Batch distillation, equations for 108,

110

example, 108
inverted, 387
with liquid holdup, 380
without liquid holdup, 370
minimum vapor requirements 377
multicomponent, 383
split-tower system, 387
total reflux, 376, 384
Benzene, dehydration of, 89
Binary mixtures, analytical equations

for, 174

batch distillation of, 370
rectification of, 118
separation of azeotropes, 196
special, 192

work of separating, 162
Bubble caps, design of, 404

pressure drop through, 396
Bubble-cap plates, diagram of, 403
liquid flow on, 409
liquid gradient on, 412
pressure drop of, 396
due to liquid head, 408
through risers, 405
through slots, 405
for vapor flow, 404
Bubble-point curves, 7, 8, 79

Calculation of vapor-liquid equilibria,

Cascade plate, 424
Chemical reactions and rectification,

361

Clark equation, 60
Completely miscible mixtures, 26
Complex mixtures, laboratory studies

of, 332
rectification of, 325

488

FRACTIONAL DISTILLATION

Condensation, differential partial, 116

equilibrium partial, 116

fractional, 102

partial, 115
Condenser, 441

leak in, 192

partial, 116, 132, 475

total, 132

Constant-boiling mixtures, 20
Control of fractionating column, 472
Control variables, 474
Cost of fractionation, 130
Critical region, 8, 79
Cross flow in column, 448

Dalton's law, 28, 46

Dal ton's and Raoult's law, 28, 44

Definitions, of fractional distillation, 1

of plate efficiency, 445

of theoretical plate, 119, 122
Degrees of freedom in distillation sys-
tem, 215

for rectifying column, 216
Determination of vapor-liquid equilib-
ria, 3
Dew-point and bubble-point curves,

7, 8, 79
Diagrams, Henry's law, 28

Raoult's law, 27

temperature-composition, 16

vapor-liquid, 16
Differential distillation, 107
Distillation, A.S.T.M., 325

azeotropic, 285

batch, 108, 118, 370

continuous, 118

degrees of freedom in, 215

differential, 107, 108

efficiency of, 162

extractive, 285

multiple, 103

simple, 107

steam, 111

successive, 101

true-boiling-point curves, 325

use of instrumentation in, 362

vacuum, 113, 362
Down pipes, 409

Duhem equation, 47

application of, 49

constant pressure, 48

constant temperature, 47

example for, 50

generalization of, 48

use of, 50
Dumping of liquid in column, 419

Efficiency, of bubble-cap plates, 445
correlation plot, 457
effect on, of cross flow, 448

of entrainment, 454
equation for, 462
experimental data on, 455
Murphree, 445
over-all column, 445
of distillation column, 162

methods of increasing, 167
of packed towers, 464
Enthalpy balances, 120, 139
below feed plate, 145
general case, 145
Enthalpy calculations, 141, 142
of liquid, 139
vapor, 140
Entrainment, 424

Equilibrium, calculation of vapor-
liquid, 26

modification of, 286
presentation of vapor-liquid, 16
rate of approach to, 476
vapor-liquid, 3, 18

Equilibrium constants, 43, 215, 223
Equilibrium-curve design method, 350
Escaping tendency, 26
Evaporation rate, 394
Examples, adsorption-factor method,

345

analytical equations, 181, 354
azeotrope separation, 205
azeotropic distillation, 313
batch distillation, 108, 373, 385, 387,

389

benzene-toluene-xylene, 219
binary batch distillation, 373
bubble-cap plate design, 433
chemical reactions in rectification, 363

INDEX

489

Examples, condenser leak, 192
Duhem equation, 50
enthalpy-composition method, 141,

150, 152

extractive distillation, 304
feed-plate location, 241, 248
fugacity calculations, 44
gasoline stabilization, 261
graphical correlations for column

design, 347
immiscible liquids, 85
isopropyl alcohol stripping, 194
Margules equation, 65
minimum reflux ratio, 258
modified equilibrium-curve method,

350

net rate of evaporation, 394
nitric acid concentration, 298
optimum reflux ratio, 130
packed-tower design, 466
phenol-water separation, 198
plate efficiency, 463
Raoult's and Dalton's law, 30
reduced relative volatility method,

341

steam distillation, 113
tar-acid distillation, 236
Thiele-Geddes method, 326
Van Laar equation, 65
Extractive distillation, 285, 290
choice of agents for, 287
diagram for, 291
example, 304
feed-plate location, 295
minimum number of plates for, 293
minimum reflux ratio, 293
vapor-liquid equilibrium for, 286

6, f 1,

148, 241,

Falling-film unit, 399

diagram of, 400
Feed condition, 126
Feed-plate location, 126,

248, 270, 295
Feed-plate matching, 265
Fractional condensation, 102
Fractional distillation, definition of. 1
Fractionating column, 118

Fractionating column, auxiliaries for,
471

control of, 472

design of, 121, 401

diagram of, 404

performance of, 445

transients in, 476
Fractionation, cost of, 130

diagram for, 102

efficiency of, 165

general methods of, 101

Gas-law deviations, 36, 39
Gasoline stabilization, 261
Graphical correlation method, 347

Heat of mixing, 143

Van Laar equation for, 144
Heat economy, 162

Height equivalent to theoretical plate
(H.E.T.P.), 187, 464

equation for, 466

Height of transfer unit (H.T.U.), 188
Henry's law, 27, 93

diagram for, 28
Hydraulic gradient on plate, 412

Immiscible liquids, 85

example for, 85

plot for, 86

Inhibitors used in distillation, 362
Instrumentation, 473

K values, 41, 42
Key components, 217

Latent heat of vaporization, 142
Lewis method, 178
Lewis and Cope method, 229
use of, 231

490

FRACTIONAL DISTILLATION

Lewis and Matheson method, 219
Lewis and Randall rule, 37, 39, 56, 83
Lewis and Wilde method, 329
Liquid, enthalpy of, 140
Liquid-air fractionation, 150
Liquid head on bubble plate, 408, 418

gradient, 412

in column, 428
Liquid phase, 39
Logarithmic plotting, 127

McCabe-Thiele method, 123

diagram for, 124
Margules equation, 50, 54, 55
for azeotropic composition, 204
corrected for temperature, 56
% evaluation of, 62
example of use of, 65
plot for, 72
Mass transfer, 399
Maximum boiling azeotrope, 21, 95
Maximum boiling mixture, 21
Mean free path, 398
Methods, alternate design, for multi-
component mixtures, 336
fractionation, 101
graphical correlation, 347
Lewis and Cope, 229

use of, 231

Lewis and Matheson, 219
Lewis and Wilde, 329
McCabe-Thiele, 123

diagram for, 124
Ponchon-Savarit (see Ponchon

Savarit method)
reduced relative volatility, 341
Sorel-Lewis, 122
Sorel's, 118
Thiele-Geddes, 326
Minimum boiling azeotrope, 20
Minimum number of plates, 128, 174,

243, 270, 292, 321, 376, 384
Minimum reflux ratio, 128, 147, 170,
172, 176, 253, 258, 272, 293, 321,
379

equation for, 129, 253, 258, 294
Mixing, heat of, 143

Mixtures, azeotropic, 21, 192

completely miscible, 26

partially miscible, 14, 21
Modified latent heat of vaporization

method, 158, 276
Mol fraction, definition of, 16
Molecular distillation, 395, 397

thermal efficiency of, 401
Multicomponent mixtures, alternate
design methods for, 336

batch distillation, 383

equation for minimum reflux ratio,
249, 255, 258

equation for total reflux, 243

feed-plate location, 241, 248

rectification of, 214
Multicomponent systems, 72
Multieffect systems, 168
Multilayer systems, 14
Multitower systems, 168

Nitric acid, concentration of, 296
Number of transfer units (N.T.U.), 188

Open steam, 134

Operating lines, 119, 192, 214

intersection of, 125
Optimum reflux ratio, 129, 131

example, 130

Packed towers, 183

allowable gas and liquid velocities,
439

design of, 184, 466

efficiency of, 464

example, 466

flooding velocity, 440

pressure drop in, 437
Partial condensation, 115

equilibrium of, 116

differential of, 116
Partial condenser, 116
Partial pressure, 26

INDEX

491

jrtially miscible systems, 14, 21, 88,
197

rforated plate, 431

ase rule, 16, 81, 216

enol-water fractionation, 197

,te efficiency, 105, 445

jorrelations, 457

equation for, 462, 463

a^ample, 463

ixperimental data, 455

tie-design, example, 433

f}te dumping, 419

j|te layout, 431

ite spacing, 426

ate stability, 418

tnchon-Savarit method, 146

feed-plate location, 148

minimum reflux ratio, 147

side streams, 149

total reflux, 147

iynting's rule, 34, 48

esentation of vapor-liquid equilib-
rium data, 16

essure drop, for bubble-cap plate, 396

due to liquid head, 408

in packed towers, 433

through risers and caps, 405

through slots, 405

tor vapor flow, 404

essure, partial, 26

reduced, 43

sudo-azeotrope, 21

eudo mol fraction, 90

eudo critical pressure, 37

eudocritical temperature, 37

?T relations, 36

Cult's law, 26, 30, 46, 64, 93, 220

•Dalton's and, 28, 44

deviation from, 27

'diagram for, 27

ate of evaporation, 394

relative, 395

ayleigh's equation, 108

^boilers, 471

ectification, 104

of binary mixture, 118

Rectification, with chemical reaction,
361

of complex mixture, 325

degrees of freedom in, 216

of mulibicomponent mixture, 214
Reduced pressure, 43
Reduced relative volatility method, 341
Reduced temperature, 43
Reflux ratio, minimum, 128

optimum, 129

total, 129

Relative rate of evaporation, 395
Relative volatility, 30, 395

modification of, 203

use of, in multicomponent calcula-
tions, 232
Retrograde phenomena, 81

condensation, 81

Scatchard equation, 62
Simple distillation, 107
Solution deviations, 46, 65
Sorel-Lewis method, 122
Sorel's method, 118
Steam distillation, 111, 135, 396
Systems, acetic acid-water, 109

acetone-carbon disulfide, 19

acetone-chloroform, 19, 66

ammonia-water, 152

aniline-water, 202

benzene-n-propanol, 51

benzene-toluene, 19

benzene-water, 89, 202

butane-hexane, 80

carbon dioxide-sulfur dioxide, 82

carbon tetrachloride-carbon disulfide,

17
temperature-composition diagram,

16
yt x curve, 18

ethanol-benzene-water, 314

ethanol-isopropanol-water, 303

ether-water, 90

ethyl alcohol-water, 46, 52, 163, 165,
205-207

gasoline, 261

isobutahol-water, 19, 202

492

FRACTIONAL DISTILLATION

Systems, isobutylene-propane, 44
isopropyl alcohol-water, 196
methanol-water, 130
methyl ethyl ketone-water, 203
raulticomponent, 72
multieffect, 168
multilayer, 14
multitower, 168
nitric acid-water, 297
oleic-stearic acids, 397
phenol-o-cresol, 30
phenol-water, 94, 198
split-tower, 387
toluene-water, 86
two-tower, 202

Tar acid fractionation, 236
Temperature-composition diagram, 16

for acetone-carbon disulfide, 19

for acetone-chloroform, 19

for benzene-toluene, 19

for carbon tetrachloride-carbon disul-
fide, 17

for isobutanol-water, 19
Theoretical plate, 119, 122

height equivalent to, 187
Thermal efficiency, methods of increas-
ing, 167

in molecular distillation, 401
Thermodynamic relations, 32
Thiele and Geddes method, 336
Total reflux, 128, 147, 243, 270, 292,
321, 376, 384

analytical equation for, 174
Transfer unit, height of, 188

number of, 188

Transients in fractionating column, 476
True boiling-point distillation, 325
Two-tower system, 202

Unequal molal overflow, 138

Vacuum distillation, 113, 236, 393, 396
Van Laar equation, 50, 56-59, 63, 98

for azeotropic distillation, 286

evaluation of, 62

example of use, 65

for extractive distillation, 286

for multicomponent mixtures, 73
Van der Waals equation, 57
Vapor, distribution of, 418

enthalpy of, 140

velocity allowable, 430
Vapor-liquid composition diagram, 16
Vapor-liquid equilibria, 3, 18

calculation of, 26

data on, 22-24

determination of, 3

experimental determination of, 3

presentation of, 16
Vapor phase, 34
Vapor pressure, of ethanol, 51

of water, 51

Vapor recompression system, 170
Vapor reuse system, 170
Vaporization, heat of, 143
Volatility, 29

abnormal, 29

normal, 29

relative, 30

Weirs, 409

Wetted-wall tower, 116

Work for separation, of binary mix*

tures, 162
of ethanol-water, 163

y,x curve, 21
for CC14 and CSa, 18
ether-water, 90-92
normal, 20
propane-isobutylene, 44

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