THE EFFECT UPON THE MENIMUM
REFLEX EN MULTICQMPQNENT DISTILLATHON

OF VARIABLE LIQUID-VAPOR RATIOS
AND RELATIVE VOLATILITY

Thesis for "19 Degree of M. S.
MECHIGEN mm EM" l‘ITV
Manohar Shankar Raine

1961

 

IHESlS

This is to certifg that the

thesis entitled

The Effect upon the Minimum Reflux in Multicomponent
Distillation of Variable Liquid-Vapor Ratios and

Relative VOlatility
presented by

Manohar Shankar Rane

has been accepted towards fulfillment
of the requirements for

Mdegree in __Chemical Engineering

/ Major professo’rz7 6,116

Date June 23, 1961

 

LIBRAR Y.

Michigan State
University

 

ABSTRACT

THE EFFECT UPON THE MINIMUM
REFLUX IN MULTICOMPONENT DISTILLATION
OF VARIABLE LIQUID-VAPOR RATIOS
AND RELATIVE VOLATILITY

by Manohar Shankar Rane

Continuous fractionation by distillation requires a
certain quantity of the overhead product to be fed back to
the distillation column, in order that the separation may be
possible. The mole ratio of the quantity fed back to the
column to the quantity of the distillate taken off as a
product is the reflux ratio. For every fractionation, there
exists a reflux ratio below which the separation is not
possible even with an infinite number of plates. Though
no column is Operated at minimum reflux ratio, the value of
the minimum reflux ratio is very important. When the value
of the minimum reflux ratio is accurately known, the preper
operating reflux ratio can be selected and the size of the
column determined.

There are many methods for Calculation of the minimum
reflux ratio. Each is based on some or many simplifying
assumptions-~which may or may not be valid in many cases.
The most common, which can lead to errors, are the assump-
tions of constant relative volatility and constant liquid

and vapor rates in the column.

Manohar Shankar Rane

This work was undertaken to study the errors involved
in the values of the minimum reflux ratio calculated with
the above noted assumptions.

Calculation of the minimum reflux ratio, with the assump-
tions of constant relative volatility and constant liquid and
vapor rates, was made by the Underwood method. Calculations
of the minimum reflux ratio without these assumptions were
made by an iterative method utilizing the rigorous plate-to-
plate calculations from each end of the column to the pinch
regions. The relationship suggested by Colburn was used to
relate the compositions of the two pinch regions. An elec-
tronic computer calculation method was develOped to perform
the laborious calculations involved.

Values of minimum reflux ratios were calculated for
different feeds and different quantities and compositions of
products. It was observed that there are deviations of O
to 18.9 percent in the values obtained by the Underwood
method as compared to the plate-to-plate calculation with
the Colburn correlation. In 6 of the 7 cases, the values
obtained by the plate-to-plate Colburn method were higher
than the values obtained by the Underwood method. These
deviations in value, though small, can play an important
role in the design and operation of a fractionating column.

Further work, were more memory locations available in
the computer, could be carried out to relate the two pinch

regions by actual plate-to-plate calculations from one pinch

Manohar Shankar Rane

region to another, as Opposed to Colburn's correlation method.
By this procedure the possibility of error arising from the

Colburn correlation would be eliminated.

THE EFFECT UPON THE MINIMUM
REFLUX IN MULTICOMPONENT DISTILLATION
OF VARIABLE LIQUID-VAPOR RATIOS
AND RELATIVE VOLATILITY

BY

Manohar Shankar Rana

A THESIS

Submitted to
Michigan State University
in partial fulfillment of the requirements
for the degree of

MASTER OF SCIENCE
Department of Chemical Engineering

1961

ACKNOWLEDGMENT

The author wishes to express his deep gratitude to
Professor J. W. Donnell for his excellent guidance and
encouragement during the course of this work.

Thanks are due to Dr. C. M. COOper for his valuable
suggestions and guidance.

Thanks are extended to Mr. J. W. Hoffman, Director
of the Division of Engineering Research, and Dr. C. Fred
Gurnham, Head of the Department of Chemical Engineering,
for the partial financial support they gave during the
investigation.

ii

II.
III.

IV.
V.
VI.
VII.

TABLE OF CONTENTS

ACKNOWI‘EDGMENT 0 O O O O O O O O O O O O O O O O O 1 1
INTRODUCTION . . . . . . . . . . . . . . . . . . . 1
STATEMNT OF PROBLEM O O O O O O O O O O O O O O O 17

PROCEDURE USED TO CALCULATE THE MINIMUM REFLUX WITH
VARYING RELATIVE VOLATILITIES AND VAPOR RATES. . 18

MATHEMATICAL PROCEDURE FOR CALCULATION . . . . . 24
COMPUTER USED . . . . . . . . . . . . . . . . . . 37
CODING PROCEDURE . . . . . . . . . . . . . . . . . 38
DISCUSSION OF RESULTS AND CONCLUSIONS . . . . . . 41
APPENDIX . . . . . . . . . . . . . . . . . . . . . 45
NOMENCLATURE . . . . . . . . . . . . . . . . . . . 82
GLOSSARY . . . . . . . . . . . . . . . . . . . . . 83
BIBLIOGRAPHY . . . . . . . . . . . . . . . . . . . 85

iii

LIST OF TABLES

Table Page
1 0 [Data and Results 0 I O O O O O O O O O O O O O O 43

2. Coefficients of Equilibrium Constant
Polynomials . . . . . . . . . . . . . . . . . 46

3. Coefficients of Liquid Enthalpy Polynomials . . 46
4. Coefficients of Vapor Enthalpy Polynomials . . . 46
5. Coefficients of Cm and On Polynomials . . . . . 47

iv

LIST OF FIGURES

Figure Page
1. A Section Above the Feed Plate . . . . . . . . . 20
2. A Section Below the Feed Plate . . . . . . . . . 21
3. Fractionating Column . . . . . . . . . . . . . . 35

4. Dew Point Temperature Convergence Flow
Diagram . . . . . . . . . . . . . . . . . . . 48

5. Bubble Point Temperature Convergence
Flow Diagram . . . . . . . . . . . . . . . . . 49

6. Chart for Calculating CH1 and Cn . . . . . . . . 5C

I. INTRODUCTION

Sorel (15) was the first to apply a mathematical theory
to the rectifying column for binary mixtures. He calculated
the enrichment from plate to plate by making material and
energy balances around each theoretical plate.

The theoretical plate concept assumes that the vapor
leaving each plate is in complete equilibrium with the liquid
through which it escapes. Even to this day, all distillation
calculations are based on this theoretical plate concept.

For actual design calculations the efficiency factor of the
plate must be taken into account.

Sorel's method requires starting at either end of the
fractionating column with the desired and composition, thereby
fixing the composition at the other end as well, with a given
reflux ratio, and the calculation is made from plate to plate
down or up (depending on whether the calculations were started
at the condenser or at the reboiler) until the other end
product composition is obtained. In the Sorel method, the
theoretical plate concept, material balance, and energy bal-
ance are all taken into account. The compositions of the
liquid and vapor from a plate are calculated by vapor-liquid
equilibrium relationships. Assumptions are made of the com-
position (binary) of the liquid (or the vapor) entering from
the plate above (or below). These assumptions are corrected

1

2
with the help of material and energy balances around each
plate, by iterative calculations.

Lewis (10) modified.Sorel"s rigorous method of calcu-
lations for binary mixtures by making the following assump-
tions. He assumed that the latent heats of vaporization of
the components in the column were the same, and that molal
vapor rate from plate to plate were the same throughout the
height of the column. Lewis further assumed that the liquid
rates in the sections above and below the feed plate were
different because of addition of the feed at that point.
With these assumptions and modifications, Lewis calculated
the results by a graphical method. With a given reflux
ratio, a material balance around each plate was taken to be
the lepe of the liquid composition curve versus plate
number. From this curve the number of plates for a given
separation was found by graphical integration.

The McCabe and Thiele (l4) graphical method, in con-
Junction with the equilibrium curve, is one of the most
pOpular methods of tray calculation. McCabe and Thiele
assumed: I. The number of moles of vapor ascending the
column (and hence the molal overflow except for the change
at the feed plate) is constant for all the plates. 2. There
are no heat losses from the column to the surroundings.

3. Feed enters the column at a temperature equal to the
boiling point of the liquid on the feed plate. 4. Compo-
sition of the reflux is the same as that of the overhead

product. 5. The molal latent heats of vaporization of all

3
the components are the same and hence do not change with changes
in temperature.

The McCabe and Thiele method requires plotting of the
equilibrium curve X versus Y for the more volatile component
of the binary mixture. On this curve are also plOtted the
Operating lines. There are two Of these Operating lines; one
for the section above the feed plate and one for the section
below the feed plate. The slope Of each of these Operating
lines is equal to the ratio Of the liquid overflow to the vapor
rate in the respective sections.

The plates are found by a stepwise procedure consisting
Of drawing a horizontal line from the point on the diagonal
(XzY) at the distillate composition to the equilibrium curve,
thence vertically downward to the Operating line and thus
continuing the stepwise procedure until the bottom product
composition is reached. This procedure can also be started
from the bottom product composition point as well. One hor-
izontal and vertical step represents a plate.

At total reflux, the slopes Of the Operating lines are
one and the same. Both Of the Operating lines lie along the
diagonal XzY. Under such conditions, the number of plates
in the column is at a minimum.

At minimum reflux, the Operating lines meet on the
equilibrium curve on the line szF. From the slopes Of
these lines one calculates the minimum reflux.

The McCabe and Thiele method gives quick and relatively

accurate results within its limitations of the assumptions made.

4

The knowledge of the minimum number Of plates at total
reflux and of the minimum reflux at an infinite number of
plates for a given separation, places practical limits on the
design problem. Fenske (6) proposed relationships for calcu-
lating both the minimum number of plates and the minimum
reflux. His initial calculations were based on a binary mix-
ture Of components A and B, but it was further extended to
include multicomponent mixtures. Fenske assumed that the
moles of overflow and moles of vapor ascending the column are
the same, and that the relative volatility of the lighter
component with respect to the heavier component is the same
throughout the height of the column.

Fenske algebraically shows the minimum number Of plates
at total reflux for a mixture Of components A and B; A being

lighter than B, thus:

< XA ) _'< XA.) n+1
XB Distillate* XB Bottoms ‘*<A

where n represents the minimum number of plates above the

reboiler.
Calculation of the minimum reflux ratio can be made by
a material balance around the plates above the feed for the

components A and B, thus:

VYnA = Lx(n+1)A + DXDA
VYhB = Lx(n+1)B + DXDB
where X and Y : Mole fractions in liquid and vapor respectively
V, L, and D : Total moles Of vapor, liquid overflow, and

distillation product respectively

5
n and n+1 : Any plate and the plate just above it
respectively
XD 2 Mole fraction in the distillation product
At minimum reflux, the composition on the nth plate
equals the composition on the (n+l)th plate. Dividing one

of these equations by another, for the nth plate,

£115 ”Sm + DXDA

YnB _ LXnB + DXDB

 

 

But,
YnA ': «981A
YnB XnB
Substituting this in the above equation,
L = 1 ( XDA . XDB
D minimum G<-l) XnA XnA )
If the nth plate is considered to be the feed plate,
E = 1 (XDA ._ X1313 \
D minimum R-l) xFA xFB /

where XF : Concentration in mole fraction on the feed plate.

In arriving at the above relation, Fenske assumes that
the feed is introduced on a plate, having the same composition
as the feed.

Distillation problems involve both binary and multi-
component mixtures, and both are basically solved by material
and energy balances. As indicated earlier, the problems
involving binary distillation mixtures are not complicated;
however, the multicomponent distillation problems cannot be
solved so easily. This is due to the fact that a slight

change in the composition Of any one Of the components changes

6
the composition of all the others. The boiling point of
liquid on any plate cannot be determined by knowing the com-
position Of only one of the components, because for the same
composition of a particular component there can be several
boiling points corresponding to various compositions Of the
rest of the components. The boiling points and the vapor-
liquid equilibrium constants can vary over wide ranges. All
such contributing factors make the multicomponent distilla-
tion problems complicated.

In multicomponent mixture fractionation, two Of the fre-
quently used terms are the "Light key" and the ”Heavy key"
components. In a multicomponent mixture, if two components
A and B (B having a boiling point higher than that of A) are
the key components, then it would be desired that all com-
ponenushaving boiling points lower than that Of A, with
practically all of A and a small quantity Of B can be taken
as overhead distillation products, and that all components
with boiling points higher than that of B, with practically
all of B and a small quantity of A can be taken as bottom
products.

A separation in which the distillation product contains
a very small, almost negligible quantity Of the heavy key and
none at all Of any heavier component, and the bottom product
contains a very small, almost negligible quantity of the
light key and none at all of any lighter compound, is referred
to as "sharp separation." A separation in which both the

distillation product and the bottom product contain components

7
heavier than the heavy key and components lighter than the
light key respectively, is generally referred to as "leppy
separation."

Fenske (6) extended the previously mentioned method of
calculations for a binary mixture to multicomponent mixtures
in which A and B are the light key and the heavy key respect-
ively. Thus the relations in terms Of the light key and the

heavy key become:

< XLIN / XLK ) X «n+1
XHK/Distillate: \ XHK Bottoms LK
and

L». X LK_D XHKw

D minimum: RLK 1-1)(XLKF- °<LK XHK F)

The Lewis and Matheson (ll) distillation calculations
are based on plate-tO-plate calculations. They assume that
temperature changes but that liquid overflow and vapor rate
remain constant from one end to the other of the column
(except that, below the feed plate, the overflow changes by
the amount of feed introduced).

The Lewis and Matheson method of calculation starts
with the desired end products and involves calculation Of
the number Of plates. The reflux ratio used is determined
by economic factors within the minimum reflux limitations.
The temperature and composition Of the liquid on the top
plate are calculated. For this, some reflux ratio is assumed
and the composition Of the vapor arising from the tOp plate
is calculated. From this, the temperature and composition

Of the liquid on it is calculated. The temperature can be

8
said to be cOrrect when:£X:£E:l. Temperature Of the still and
composition of vapors arising from the reboiler are similarly
calculated. ‘By use Of material balances and equations Of the
type YzKX and X:%, the temperature and composition on each
plate are determined by working down from the top and up from
the bottom.

If the assumption Of the desired tOp and bottom products
are compatible with the possible separation, the results Of
temperature and composition on each plate obtained should
approach each other towards the middle of the column. If they
do not, the assumed compositions Of the end products are
changed and the calculations are repeated.

The effect Of the reflux ratio on the size Of the column
is studied by assuming various values Of the reflux ratio.

COpe and Lewis (4) applied the same method graphically.
For each Of the components, XPY diagrams, diagonals X:Y, and
operating lines are drawn similar to those in the binary mix-
ture problems. In general, it may not be possible to draw
such equilibrium curves for all components. The method then
is to draw a series of equilibrium lines, as Ysz, which are
straight lines at constant temperatures. For each component,
such lines are drawn at different temperatures. Similar to
the case in binary mixture problems, Operating lines and
vertical and horizontal steps are drawn for each component.
Different temperature values are tried until.E;Y'became unity
at each step. If the calculations are started at the bottom,

the same Operating line is used until the feed plate is

9
reached, and thereafter the Operating line corresponding to
the section above the feed plate is used. If the calcu-
lations are started at the tOp, the situation is reversed.

These are some Of the different methods Of calculations
utilized to solve the general distillation problems for
binary and multicomponent mixtures. One of the most impor—
tant problems Of multicomponent mixtures is the minimum
reflux ratio.

The relationship put forth by Fenske (6) has been dis-
cussed. The same relationship was also proposed by Underwood
(1?).

According to Gilliland (8), a fractionating column
Operating at minimum reflux can be divided into five imagin-
ary regions:‘ 1. Reboiler to stripping pinch. In this
region, the only components present in significant amounts
are the key and heavier components. The relative amount
Of the light key increases in the vapor going up the column
from the reboiler. 2. Stripping section pinch region.

Here the concentrations Of the keys and heavier components
are constant and an infinite number of plates is required to
make a finite change in concentration. 3. Intermediate
region between the pinch points. This region includes the
feed plate. In this region, the components more volatile
than the light key increase from negligible to significant
amounts in going from the stripping pinch to the enriching
pinch, and the components less volatile than the heavy key

decrease from significant to negligible amounts in going

10
from the stripping pinch to the enriching pinch. 4. Enrich-
ing section pinch region. In this region the concentration
Of the keys and more volatile components are all constant and
an infinite number of plates is required for a finite change
in composition. 5. Enriching section pinch to the condenser.
The only components present in this region in significant
amounts are the keys and lighter components. In this region,
the relative amount Of the heavy key increases in the liquid
going down the column from the condenser.

Gilliland proposed to find the minimum reflux ratio,
rather the high and low limits of the minimum reflux, for a
given separation. He assumed that the calculations Of mini-
mum reflux ratio for each Of the following cases would set
the limits mentioned earlier. Case 1. Gilliland assumed
that the concentrations of all the components are constant
for a number Of plates. This amounts to all the components
being present in significant quantities in the pinch regions.
The minimum reflux calculated for such a case will give a
value equal to or greater than the true minimum reflux ratio.
Case 2. He assumed that the ratio of the concentration Of
the key components for the pinch region below the feed plate
can be matched with the same ratio for the pinch region above
the feed plate. This amounts to having the pinch region Just
around the feed plate. The value Of the minimum reflux ratio
in this case will give a value equal to or less than the

true minimum reflux ratio.

ll

Thus the value of the true minimum reflux ratio is
bounded by these two limits. The composition Of the pinch
region is not calculated by plate-tO-plate calculations,
but by overall material balances with the help of the fact
that compositions on two consecutive plates do not change
in the pinch region. Relative volatilities and liquid and
vapor rates are assumed constant.

The two cases which give the limits of the true mini-
mum reflux ratio are not in many cases exact. The two
limits obtained have a wide range. The actual true value
of the minimum reflux ratio is again approximated or inter-
polated. This method, with all the assumptions and a wide
range between the limiting values, is likely to give con-
siderable error.

Thiele and Geddas (l6) develOped a method for multi-
component fractionation calculations. This method is use-
ful in calculating the tOp and bottom product compositions.
It is necessary to fix initially the number Of theoretical
plates and the heat input (or the reflux ratio) tO the
column. The temperature on each of the plates is first
assumed. Other factors assumed are the ratios of mole
fraction in the vapor phase (or in the liquid phase) to the
mole fraction of the same component in the distillate (or
the bottom product) for each component. Ratios are taken
with respect to the distillate composition in the part
above the feed plate and with respect to the bottom product

composition in the part below the feed plate. Calculations

12
are carried out from plate to plate from the reflux condenser
and from the reboiler to the feed plate. The results are
matched and the compositions of the distillate and the bottom
product are calculated. With this calculated product compos-
ition, calculations are repeated and the assumed temperatures
and ratios are checked. Temperatures are finally adjusted
and the end products and plate-tO-plate compositions are
determined.

Brown and Martin (2) took into account the material
balance Of a component around the portion above the feed
plate and calculated the composition Of a light component
in the enriching pinch as:

X5 _ DXD

_ (K-l) Lmin +4EB ’

 

and for a heavy component in the stripping pinch as:

= wa
f‘min (l-K) 4- KB

Xm

 

It should be noted that in this expression, K, the equili-
brium constant value, is used rather than OC, the relative
volatility. This can account for the variable«q5 not being
assumed constant.

As can be seen from the expressions for the composi-
tions in the pinch regions, the minimum reflux ratio can be
easily calculated if the pinch region compositions and tem-
peratures are accurately known. But there is no way to

check if the assumed values are correct.

13

Jenny's (9) method Of distillation calculations involved
in a multicomponent fractionation is a graphical one. Separ-
ate equilibrium curves for the enriching and stripping sec-
tions are drawn. For the enriching section, an equilibrium
curve for the light key is used and, for the stripping
section, an equilibrium curve for the heavy key. The equili-
brium curves are drawn by making calculations for a few
plates from both ends of the respective sections. This
method thus requires a knowledge of the feed plate compos-
ition and an assumption of the reflux ratio.

Jenny's method of calculation of the minimum reflux
ratio is based on the temperatures of the feed plate and the
pinch regions in both the enriching and the stripping sections.
This method involves calculation Of the minimum quantities of
the components in the vapor from the feed plate, with the
assumed value of the reflux ratio, and then checking these
with the estimated temperatures. This procedure, in fact,
joins the region from one pinch to the other, thus becoming
a rigorous method. The method, however, is not quite prac-
ticable from the vieWpOint of the various estimations and
the checking involved.

As indicated earlier, if the correct values of the com-
positions and temperatures in the enriching and stripping
pinch regions are known, the value Of the minimum reflux
ratio can easily be calculated. This relation is given by
Fenske and by Underwood. 8

In terms of the equilibrium constant values, for a

light component in the enriching section pinch:

l4

XD

X“: K+ (K-l)%’n_'

 

and for a heavy component in the stripping section pinch:

Kw

Y :
Lm K + (l-K)E§ ’

W
where Xn : Concentration Of a component in the enriching
section pinch.
Xm : Concentration of a component in the stripping
section pinch.
Lm = Liquid to bottoms ratio in the stripping section.
W
Ln = Liquid to distillate ratio in the enriching
D

section.

With an assumed reflux, the calculations of the respect-
ive pinch region temperatures are done by trial and error
until all the X values in the respective sections add up to
unity. The main problem, therefore, is how to know if the
assumed reflux ratio and the calculated temperatures are
correct.

On the basis of a large number of correct stepwise
solutions of minimum reflux, Colburn (3) found an empirical
rule to tell when the correct values Of the pinch region
compositions, and thereby of the temperatures, are Obtained.
According to Colburn,

12
Y.

2 4’ , where

l5

 

l
(1- z cmoc xm) (1.1011an
Ym : Ratio of the liquid concentration of the light key

to that of the heavy key in the stripping section
pinch.

{h : Ratio of the liquid concentration of the light key
to that of the heavy key in the enriching section
pinch. ‘

Cm, C are the correction factors given by the curve
(Figure 6).
EECme<Xm : Summation of Cm<><Xm for all the components heavier
than the heavy key in the stripping section pinch.
fiiCan : Summation of Can for all the components lighter
than the light key in the enriching section pinch.

The results obtained by Colburn were comparable with
those Obtained by stepwise calculations by Jenny's method,
hence there are good reasons to believe the relations sug-
gested by Colburn are most nearly correct.

Underwood (18) proposed a method to calculate the mini-
mum reflux for multicomponent fractionation. He assumed
that the relative volatilities of all the components remain
constant throughout the column, and also that the heavy key
and the heavier components are not present in the distillate
nor the light key and the lighter components in the bottom

product. With these assumptions he found,

o< : L + 1
Z 0-22.? (D ) minimum

16

where 9 satisfies
2 °<XF = 1 - q
o<-O
and where q depends on the condition of the feed, and is
numerically equal to the heat needed to convert one mole of
feed into saturated vapor, divided by the molal latent heat.

As seen from the relation which 63 must satisfy, there
are as many values of’ 6*as of components in the feed. The
value Of 63 to be used for calculation of the minimum reflux

ratio lies between the relative volatilities of the key

components.

II. STATEMENT OF THE PROBLEM

The calculation Of minimum reflux ratio in multicom-
ponent fractionation is important for the design and Operation
Of a fractionation‘column. The various methods of calcula-
tion usually used assume constant relative volatilities and
liquid and vapor rates throughout the column. This study
was undertaken to find the effect of variations in relative
volatilities and liquid and vapor rates on values Of the mini-

mum reflux ratio.

17

III. PROCEDURE USED TO CALCULATE
THE MINIMUM REFLUX WITH VARYING
RELATIVE VOLATILITIES AND
VAPOR RATES

In brief, the method used in this study consists of
taking a distillate composition obtained by calculating
rigorously plate to plate, while using a finite number Of
plates for the separation specifications at a given distill-
ate production. With this distillate composition and an
estimated minimum reflux, rigorous plateptO-plate calcu-
lations are carried out until no appreciable change in the
composition exists between two consecutive plates. The
same procedure is carried out from the reboiler to the strip-
ping pinch region. The reboiler composition is Obtained by
an overall material balance from the feed and distillate
compositions. If the ratio of the key components in the en-
riching section pinch region is in agreement with the ratio
of the key components in the stripping section pinch region,
as outlined by Colburn's (3) relation of the stripping pinch
key components ratio to the enriching pinch key components
ratio, the proper minimum reflux ratio has been found.
Otherwise, a new trial must be made and the above calculations
repeated. This iterative procedure is repeated until the

correct minimum reflux is Obtained.

18

19

The rigorous plate—to-plate calculations alluded to above
refers to the procedure where latent heats of vaporization,
heat contents of vapors, heat content Of liquids, and the
plate-tO-plate temperatures at equilibrium are taken into
account.

One word of warning should be given at this point regard-
ing the distillate composition Of the components heavier than
the heavy key. These heavier components that appear in the
distillate for a finite number of plates may not be accurate
enough when calculations on a large number of plates (as in
minimum reflux calculations) are involved. This may be true
even if the perfect matching of components at the feed plate
appeared satisfactory for a finite number of plates.

In case of Underwood's (18) method Of calculation of the
ntinimum reflux, the appearance Of the heavy key and heavier
Ccomponents in the distillate invalidates the calculation

amethod. The Underwood method is used for the comparison Of
tile effect of variable relative volatilities and the ratio
01? liquid to vapor rates in this work.

For plate-tO-plate calculations, an iterative method
is used. The calculations are started at a plate for which
Silffhnent though not complete information is available, in
Olflder to get complete information about that plate and some
1rliformation about the plate next to it. If the calculations
are started at the top Of the column, the tOp plate is such

a-Iplate; and for starting from the bottom, the reboiler is

311011 a plate.

20

Consider a plate just below the top plate. Let this

1th

plate be considered as nth plate, the plate above as n-

and the plate below as n+lth.

 

 

 

Vn-i
i Vn ~LLn-1
I v... in
lLn+1

 

 

Figure l.--A Section Above the Feed Plate

The composition and temperature of Ln-l are known, whereas
the quantity of Ln_1 is unknown. Similarly unknown are the

values Of Ln, Vn, Ln+1, Vh+1, Yhi, Xni' Y(n+1fi! X(n+i)i’
T

n’ Tn+i° However to know about the plates n and n-l, all
these values are not necessary. To complete the information
about the (n-l)th plate and to get for the nth plate all but
the quantity Ln, the tools which are used are the overall
material balance around plate n and the section above the
nth plate, the heat balance around this section, a material
balance for each component around the plate, equilibrium
constant equations for all these components, and the dew-
pOint equations. TO get the value of Ln-l’ a value is

assumed. The nearest value for assumption is Ln_2. This

assumed value is then corrected. Similarly, to find the

21
temperature Tn, the best first assumption is Tn-l’ which is
then corrected.
In a similar manner the plate-to-plate calculations for
the section“below the feed plate are started at the reboiler
as follows. Consider a plate m, the plate below being (m-l)th

and the plate above being (m+l)th.

 

 

 

 

 

Vm+l

in l Lm+l

firm i Lm
iLM _

Figure 2.-—A Section Below the Feed Plate

In this case Vi_l, Tm_1 and Y(m-l)i are known. Values Of

Lm+1, Tm+1. Tm, Xmi' Vm, le, Vm+1 and Y(m+l)i are unknown.

th and

But as before, to get the information about the m
(m-l)th plate, it is not necessary that all these values be
known. TO complete the information about the (m-l)th plate
and all but the quantity vm, for the plate m, the tools
which are used are the overall material balance about the
mth plate and the section below it, the heat balance around
this section, the material balance around each plate, equil-
ibrium constant equations, and the bubble point equations
for all the components. To get the values Of Vm, Tm+l these

values are first assumed. The first best approximate values

22
are Vm_1 and Tm° These approximations are later corrected.

In the calculations it was assumed that there are neg-
ligible heat losses from the column to the surroundings. It
is also assumed that the feed enters the column at its boil-
ing point, the condenser is a total condenser, and the Oper-
ation pressure of the tower is 100 psia.

The test of convergence is to find how accurate the
estimated value is. There should be no difference in the
assumed value and the correct value. Since for any case a
value of absolute zero can be achieved only after a prO-
longed time, a value of $0.001 as the approaching limit was
set to cut the time considerably and yet to get the answers
accurately enough.

All the calculations were based on one mole of feed.
The values of the distillation product composition and
quantity for each feed were taken from Foy's (7) work. All
these values are Obtained for a finite reflux and a finite
number of plates as already noted.

In order that the vapor-liquid equilibrium constant
(K:%) and the enthalpy of liquids and vapors for each com-
ponent be calculated easily by computer method, third degree
polynomials Of the temperature, of the form K:CO+ClT+02T2+

3 C were Obtained

2’ 3
from the literature and by computer calculations. Data used

CST were used. Coefficients Co, 01’ C
to get these coefficients was taken from Donnell and Cooper
(5) and Maxwell (13).

The coefficients of the polynomials were Obtained by
use of the MISTIC library routines K3 (Least Squares) and

23
L78 (Automatic Linear Equation Solver). The purpose of the
K3 routine is to fit a set of N experimental points by the
best polynomial of the form

n-1 s

F (X) : é- Z ASX
8:0

by the method of least squares. In order to Obtain this poly-
nomial, the routine K3 produces n simultaneous linear equa-
tions from data and then these equations are solved by L7S

to get the coefficients AS.

Data to be supplied for the use of routines K3 and L75
are, in this case, the temperature and the respective values
together with the number of coefficients required in a parti-
cular format. All the data except the number Of coefficients
required should be so scaled that no number is equal to or
greater than 00.5. Detailed method Of use of these routines
is available in the MISTIC library.

The values of Colburn's correction factors Cm and Cn
were eXpressed as seventh degree polynomials, using data
taken from Brown (1). I

All the coefficients are listed in Tables 2 to 4 in

the appendix.

IV° MATHEMATICAL PROCEDURE
FOR CALCULATION

Calculate the amount of the bottom product.

Wzl-D
Calculate the composition Of the bottom product.
X - DX
XWI = F1 W D1

(Next to be determined is the temperature Of the bottom
product. Since a total condenser is Considered, the
composition of the vapor from the top plate is the

same as that Of the distillation product.)

Assume temperature TTo- Calculate the equilibrium con-

 

stants.
2 3
KT01 = COi + CllTTO + C2iTTo + CBiTTo

Calculate X

25 D1

KToi
Calculate l - 2: XDi
KToi
Calculate IS. XD
I - ’1 - ZE‘L‘ . If this quantity

T
is greater than or equal to zegd, the assumed temper-
ature is correct. TTO = TT'

If TTO # TT as shown by step 5, the new value of the

temperature to be assumed is l +ZXDi , T . Steps

KToi . ‘59

3 onwards are repeated till TTO=TT'

24

10.

ll.

12.

13.

14.

15.

25
Next to be determined is the reboiler temperature.

Assume temperature TBO° Calculate the equilibrium con-

stants
K=C+CT+CT2+CT3
Boi oi ii BO 21 BO 31 Bo
Calculate K301XW1
Calculate l - §:_KB01XW1
Calculate [S’- | 1 - g KBoini . If this quantity

is greater than or equal to zero, the assumed tempera-
ture is correct. TBO : TB.
If TBO # TB as shown by step 9, the new value Of the

temperature to be assumed is 2 , TBO‘
1 + Z KBOiXW1

Steps 7 onwards are repeated till TBO : TB.
Calculate \JKTiKBi for each component and be denoted
by K1-°-°°K5 for the respective values for iC4°°°"C6.

Next to be determined is the root 6} Of the equation

 

Z K:lXFi : 1-q : 1-1 : O. The value of e to be deter-
Ki-e
mined should lie between the limits K2 and K3.
As a first approximation, assume K2+K3 = 90
2
Calculate K X ,
Ei'Ki.gi
1-
Calculate (5/ (2K K:XF1 T! If this value is

greater than or equal to zero, assumed e is correct,
eo=e.
If GU # 9’ as shown by step 14, value Of Q has to be

reassumed. If the value Of :2; KiXFi is positive,

Ki" 90

l6.

l7.

l8.
19.

20.

21.

22.

26
K +9
the new value of e to be assumed is .152 . If the

value of 2% is negative, the new value Of G to be
i-

assumed is Egggg

. Steps 13 onwards are repeated

till correct value of 9 is determined.

Calculate iii-f1? =< %)minimum + 1. This gives the

value Of the minimum reflux according to Underwood's

 

method.
Next to be determined is the temperature of the distill-
ate.
Assume temperature TDO' Calculate the equilibrim con-
stants
2 3

Knoi = Coi + CiiTDO + C2iTDO + C3iTDo °
Calculate 2i KD01XD1
Calculate 1 -ZK901XD1
Calculate lg]- (1 -zKDoixml. If this quantity is
greater than or equal to zero, the assumed temperature
is correct. TDO : TD.
If TDO # TD as shown by step 19, calculate 1 +2 Kpagipi.

The new value of the temperature to be assumed is

2 o TDD
l + "Z KDOiXDi

Steps 17 onwards are repeated till TDO = TD.
Determine the amount Of liquid fed back to the column
as reflux, L = R - D
Calculate the amount of vapor leaving the top plate,
VT : D (R+1)
The next few steps are for calculation of Qa, the

heat flux in the enriching section.

23.

24.

250

26.

27a

28.

29.

30.

31.

32.

27
Calculate the heat content per mole Of each component
in vapor phase at the temperature Of the top plate.

fii = 061 + CiiTT + C2iTT2 + 051%}
Calculate 2: fiiYTi Where YTi is the concentration of
the component i in the vapor leaving the tOp tray.
YIi = XDi
Determine the total heat leaving the column with the
vapor leaving the column. VT: EiYTi
Calculate the heat content per mole Of each component
in the liquid phase at the temperature Of the reflux
(the same as that of the distillate).

Hi = 081 + CiiTD + OgiTDQ + °§1T93
Calculate zhixm, since the composition of the reflux
is the same as that of the distillate.

Determine the heat entering the column with the reflux.
L 2‘. "151x91

Calculate Qa, the heat flux in the enriching section.
Qa = VT 2: fiiYTi - L2: 171x131

By heat balance around the column, heat flux in the

stripping section : Qa - Heat contents Of the feed.
Qb = Qa - Qf

Thus to calculate Qb, knowledge of Qf is necessary.

Assume temperature TF0. Calculate the equilibrium

constants

2
KFoi = COi + CllTFO + C2iTFO + C31TFO3

(38.101.113.136 2: KFoiXFi.
Calculate l - ZKFOiXFi"

33.

34.

35.

36.
37.
38.

39.

40.

41.
42.
43.

44.

28
Calculate \S‘ - \1 ~51 KFoiXFil- If this quantity is
greater than or equal to zero, the assumed value Of
temperature is correct. TF0 : TF

If TF0 i TF as shown by step 33, the new value of the

2 . T .
1 + Z KFOiXFi F0
Steps 31 onwards are repeated till TF0 : TF.

temperature to be assumed is

Calculate the heat content per mole in liquid phase at
the temperature Of the feed.
HFi = 081 + CiiTF + ‘3)2CiTF2 + 031T;
Calculate QF : Z HFiXFi
Calculate Qb = Qa - QF
Calculate the composition of the liquid on the tOp plate
XTi : §%% where YTi : XDi and KTi is calculated.
by step 3.
Calculate DXDi for each component.

Plate-tO-plate calculations are now started for the
enriching section. Temperature and composition Of the
liquid on the top plate are fixed by all the conditions
above. Calculations are started from the top plate
being considered as the nt'h plate.

Assume the temperature Tn Of the plate n to be also the
temperature T(n+1)o of the plate below.

Assume Lno' Best assumption is Ln : L(n-1)'

Calculate v(n+1)o : Lno + D.

Calculate y(n+1)i = Lnoxni + DxDi

(n+1)o
Calculate the equilibrium constants at the temperature
T(n+1)o° 2 3
K(n+1)oi z COi + C1T(n+1)o + C2T(n+1)o + C3T(n+1)o

45.

46.
47.
48.

49.

50.

51.

52.

53.

54.

55.

56.
57.

29

Calculate X(n+1)oi = an+1)oi
(n+1)oi

Calculate Z X(n+1)oi'

Calculate l - Z X(n+1)01.

Calculate IS] - I1 - Z x(n+l)01(. If this quantity is
greater than or equal to zero, T(n+1)o = T(n+1)'

If the assumed value of temperature T(n+1) is not correct
as shown by step 48, the new value of T(n+1) to be

1 + X
assumed is 'Z:2(n+1)01 ° T(n+1)0o

Steps 40 onwards are repeated till T(n+1)o : T(n+1)°
Determine the heat content per mole Of each component
in the liquid phase at temperature Tn.

— _ x x 2 x 3

hni _ C01 + CIiTn + CZiTn + C3iTn
Calculate the heat content per mole Of the liquid

leaving the nth plate. ‘2: finixni
Find the heat content of the liquid leaving the nth
plate. hn = Ih6:;hh1xn1
Calculate the heat content per mole Of each component
in the vapor phase at temperature T(n+1)'
— I I l I
H(n+1)i = Coi + C11T(n+1) + C21T(n§1) + C31T(n§1)
Calculate heat content per mole of vapor leaving plate
(n+1). 22H(n+1)iY(n+1)i
Calculate heat content of vapor leaving(n+1)t’.h plate.
H(n+1) : V(n+1)oE-TrH(n+1)iY(n-I-1)i
I
Heat flux Qa : H(n+1)' hn
Calculate Qa - DZH(n+1)1Y(n+1)1 where Qa is calculated

in step 29.

58.
59.

60.
61.

62.

63.

64.

3O

Calculate ZH(n+1)1Y(n+1)_1 " Z hnixni

Calculate L; : 3§.:_DZZH(H+1)1Y(H+1)1
:Z:H(n+1)iy(n+1)i 72:Hnixni

Find Lno - Ln.

Calculate lg‘]- ’Lno'Lni° If this quantity is greater
than or equal to zero, the value of Lno assumed is correct
Lno : Ln

If the assumed value of Ln is not correct as shown by

I
step 61, new value of Ln to be assumed is Lno + Ln .

2
Steps 41 onwards are repeated till the correct value

of Ln is Obtained.

These calculations complete the calculations on plate
n, at least for the first cycle. After the calcula-
tions are complete, then plate (n+1) is regarded as

the nth

plate and successive calculations are carried
out for the number of plates desired. After calculat-
ing for a large number of plates, a pinch region is
assumed to have been reached; this is later confirmed.
Calculate Yg, the ratio Of the liquid concentration Of
the light key to that of the heavy key in the pinch
region. I

Calculate the relative volatilitieSOCn1 of all the
components with respect to the heavy key at the pinch
region temperature.

Plate-tO-plate calculations are now started fOr the

stripping section. Temperature and composition on the

bottom plate (reboiler) are known. Calculations are

65.
66.

67.

68.

69.
7o.

71.

72.
73.
74.
75.

76.

31

started at the bottom plate, the bottom plate being

considered as the mth

)t

plate and the plate above as

h plate.

Calculate Ymi : Kmixmi°

Calculate WXWi for each component.

the (m+1

Assume temperature Tm on plate m, also to be the tem-

perature T( on plate (m+1).

m+1)o
Assume Vfio, the quantity of vapor going up from the
th _
m plate. First best assumption is Vmo _ V<m+1).
Calculate L(m+1)o : Vmo + W

Determine X = Vmoymi + WXWi
(m+1)oi L< ) .1.
m+1 0

Calculate the equilibrium constants at the temperature
T(m+1)o°
_ 2 3

K(m+1)oi 7 Col + C1iT(m+1)o + C2iT(m+1)O + C3E(m+1)o
Calculate Y(m+1)oi = K(m+1)oix(m+1)ci
Find 2 Y(m+1)01.
Calculate 1 -Z Y(m+1)01.
Calculate lgl- ll "ZY(m+l)oi)° If this quantity is
greater than or equal to zero, the assumed value of the
temperature is correct, T

(m+1)o : T(m+1)

If T(m+1)o # T(m+1)as shown by step 75, the new value
2
of T(m+1) to be assumed is 1 + §LY(m+1)Oi . T(m+l)o.

Steps 67 onwards are repeated till the correct value

of the temperature is found.

77.

78.

79.

80.

81.

82.

83.
84.
85.

86.
87.

88.
89.

90.

from the (m+1

32
Calculate the heat content per mole Of each component in
liquid phase at temperature T(m+1)'
H(m+1)i = 0:1 + CiiT(m+1) + CgiT(m§1) + 031T(m21)
Calculate heat content per mole of the liquid coming to

th

plate m from the (m+1) plate,2f 1--1m(m+1);tx(m+1)oi

Determine the heat content Of the liquid coming down
)th plate to the mth plate.
h(m+1)o : L(m+1)oz:h(m+1)ix(m+1)oi

Calculate heat content per mole for each component in
vapor phase at temperature Tm'

-— t t I t
Hmi = Coi + C11Tm + C21Tm2 + 031Tm3

Calculate heat content per mole of the vapor going up

from the mth plate,2: HQiYmi
Calculate heat content of the vapor going up from the

mth plate, Hm : VbéEZHhiYmi
l

Heat flux Qb : Hm - h(m+1)o

Calculate W 2:-h-(m+1)1x(m+1)Oi

Calculate Qb + WZH(m+1)oix(m+1)oi
Calculate 2: HmiYmi -Zh(m+1)oix(m+1)oi
: Q. + w'Z:--H(m+ X m+1 1
Calculate v = .21. .1 1%121_i___1_
m ZHmYmi - Z (m+1)oix(m+1)i

Calculate v — v5.

 

Calculate [S] 'leO - $1. If this quantity is greater
than or equal to zero, the assumed value of vm is correct,

V : Vm.

mo

If the assumed value of Vm is not correct as shown by

3
step 89, the new value of Vm to be assumed is ch_%_1m,

91.

92.

93.

94.

95.

33
Steps 68 onwards are repeated till the correct value Of
vm is found.

These calculations complete the calculations on plate
m, for the first cycle. After the calculations are
complete, the (m+1)th plate is considered as the mth
plate and successive calculations are carried out for
a large number of plates. After calculating for a
large number of plates, the pinch region is assumed to
have been reached; this is later confirmed .
Calculate‘fm, the ratio Of the liquid concentration of
the light key to that Of the heavy key in the pinch
region.

Calculate the relative volatilities<26m1 for all the

components at the pinch region temperature.

Calculate X(n+1)HK - XnHK in the enriching section

pinch region. If ‘S|- lx(n+1)HK - X'nHKl is greater
than or equal to zero, there is the pinch region in
the enriching section. If there is no pinch region,

the problem is rejected.
Calculate X(m+1)LK - XmLK in the stripping section

pinch region. If [El - lx(m+l)LK - XmLK‘ is greater
than or equal to zero, there is the pinch region in the

stripping section. If there is no pinch region, the
problem is rejected.
Ca1culate(31 = (“CmLK - l)o(fii for all the components

heavier than the heavy key in the stripping pinch region.

96.

97.

100.

101.

102.

103.

104.

105.

106.

34

ed -1
Calculate 91 : OCnLK( nLK ) for all the components

OOni
lighter than the light key in the enriching section

 

pinch region.
Determine Cmi factors for all the cOmponents heavier
than the heavy key, thus:

Cmi = C: + C1 1 + C281 + .... + C7Pi'
Calculate Cni factors for all the components lighter
than the light "key, thus:

Cm: CO +0179 + 0'2 1012+ + ($17.
Calculate Zcmixmixmi for all the components heavier
than.the heavy key in the stripping section pinch
region.

Calculate.Z:CniXh1 for all the components lighter than

the light key in the enriching section pinch region.
Find (1 'ZCmiQCmiXmi) (1 -ZCn1Xn1)

_ 1
Calculate \}2 - (1 -ZCm1°6miXmiT(1 -zCn1Xn15

Using the values onIn and Yh as calculated by steps

 

63 and 91 respectively, calculate 1E

Yn
Calculate IQ , _1_
Yn \V
Find. )8[- ‘ _%E..,g7, If this quantity is greater than
n

or equal to zero, the value Of R, the minimum reflux
with which the calculations are started, is correct.
If the step 105 shows that the assumed value of R is

not correct, the new value of R tO be assumed is

2 , R .(previously assumed).
1+ym.1

my

35

Step 21 onwards are repeated till the correct value Of
the minimum reflux is found.
107. Ratios of R, V, L as corrected up to step 106 to those

for Underwood’s method of calculation are taken.

36

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

— l Condenser
, i I v D
VT L 3 3?Di
1 1
fir Vn-11 an * (n-l)t’h plate
-1
VB: I I nth plate
V
n+1f 1:: (n+1)th plate
+1
-1 T fir-
9 1
F T
Km 1 feed plate
9 l
__ -L.
—— fir-
Vfi+1T 1 (m+1)th plate
Lm+1
V
m t I mth plate
Lm
VIII-11 ILm (m-1)th plate
-1
T
__ l 1.
_. T _,..
if Reboiler
W
1’-
Figure 3.--Fractionating Tower XWi

V. COMPUTER USED

The computer used during this work was the Michigan
State Integral Computer, better known as MISTIC.

The input section of the MISTIC is an electromechan-
ical device for translating information presented to it into
the language Of electrical pulses accepted by the computer.
The input media can be either perforated tape or punched
cards. The photoelectric reader pulls the tape past a beam
of light. Light passes through the holes to a light-sensi-
tive surface which converts the holes to electrical pulses
accepted by the computer.

The memory Of the MISTIC is made up of 40 Cathode ray
tubes. On the face of each tube is a grid Of 32 x 32
spots for a total Of 1024 spots. Thus, 1024 words 40 bits
long (in the binary number system) can be stored in the
memory. Each memory location can store either one order
pair or one number.

After execution Of the program Of mathematical steps,
it may be desired to get from the computer some information.
The output section of the computer translates the machine
language Of electrical pulses to some mechanical medium.
Three types Of output are available from the MISTIC: perfor-
ated tapes similar to the input tape, punched cards, and
printer output.

37

VI. CODING PROCEDURE

The MISTIC is a binary, fractional, single address
computer. An order for MISTIC contains two symbols chosen
from the decimal digits 0, l, 2 °°°°' 9 and the letters
K, S, N, J, F, L and one decimal number n in the range
0 g N £1023. Since MISTIC is a binary machine, the decimal
address must be converted to binary before entering the
machine. This can be done by converting all decimal addresses
to sexadecimal, each Of which is read as four bits by MISTIC.
It is not always possible to do this, and there is a program,
Decimal Order Input (DOI) which can be used. The Decimal
Order Input is a program which, when read into and stored in
MISTIC, reads in any program written in a prescribed format,
changes the orders to ordinary MISTIC format, and stores
this program in specified locations in the MISTIC.

The MISTIC is a fractional computer; and all the numbers
put in as data and all the results of calculations must be
between minus one and plus one. Hence, the programmer must
foresee every difficulty and program accordingly. In order
to avoid such exact foreknowledge Of the results and scaling
of the numbers supplied as data, MISTIC library routine A1
(Floating decimal arithmetic routine) was used. This routine
by itself takes care of the scaling and the input, output

numbers are decimal numbers and a power of ten exponent.

38

39
The accuracy of the routine is up to 9 decimal places.
Except for a few orders, the entire program is written in
A1. The details Of this routine and the orders used, are in
the description of the routine in the appendix.

In order to Obtain the geometric mean Of the equili-
brium constants at the top and bottom plate temperatures,
the MISTIC library routine RA1 (Floating decimal square root
auxiliary) was used.

More informamon about the MISTIC coding technique and
various routines available can be had from the MISTIC pro-
graming manual (12) or the MISTIC library.

Allocation of the computer memory locations for the
program is as follows:

3-4 Specifications of the floating accumulator location.
13-15 Specifications of data and answers storage locations.
16-17 Floating accumulator.

18-184 Floating decimal arithmetic routine A1.
187-202 Floating decimal square root auxiliary routine RA1.
205-673 Program
700-800 Data
810-974 Temporary storage used over and over again.

Data supplied is stored in memory as:
700-702 Constants 0-2.
703 Constant (0.001)
704 Amount of distillate
705-709 Distillate composition
710-714 Feed composition

715-734
735-736

737-756
757-776
777-784
785-792
793-800

40

Equilibrium constant polynomial coefficients
Initial assumptions of tOp and bottom plate tem-
peratures.

Liquid enthalpy polynomial coefficients.

Vapor enthalpy polynomial coefficients.
Constants 3-10.

C factor polynomial coefficients.

III

Cn factor polynomial coefficients.

VII. DISCUSSION OF RESULTS
AND CONCLUSIONS

The results of Table 1 show that values obtained by
Underwood's method differ from those obtained by plate-to-
plate calculation of the pinch region compositions (corre-
lation Of pinch regions by Colburn's method).

The factors which may contribute, individually or com-
bined, to the difference in the values can be considered to
be:

1. Change in relative volatilities, taken into
account in the modified Colburn method of calculation but
not in the Underwood method.

2. Change in liquid-vapor rate, also taken into account
in the modified Colburn method Of calculation but not in the
Underwood method.

3. Possibility of any particular problem being out of
the range of Colburn's correlation.

4. To calculate minimum reflux by Underwood's method,
the geometric mean value Of the equilibrium constants at
the top and bottom temperatures,of the column is used for
each component. This may or may not be the true average in
the column. However, it is realized that a mean could

always be selected to make Underwood‘s calculation correct.

41

42

The relative volatilities of the components depend on
the temperature. If the temperatures of the two pinch
regions are close, the relative volattuties for any component
at the respective temperatures also appear to be close or
constant. Results 1 to 3 indicate that the smaller the
difference in the pinch region temperatures, the closer are
the values of the minimum reflux ratios calculated. Under-
wood's method is based on constant relative volatilities.
Thus from these few results it appears that Colburn's method
gives values of minimum reflux ratio close to Underwood's
when there is not much change in relative volatilities or
liquid-vapor rates, and values are quite different from Under-
wood's when there are large changes in these factors. This
would point to the conclusion that differences are due to
failure of the Underwood method to take into account these
changes. 0n the other hand, a definite conclusion in this
respect should not be drawn until a completely rigorous

minimum reflux calculation is made.

43

 

 

COSGEGOU v30? mflfi on. mampuooom 0.3mm cannon gewnwz n GMEXM
@0503 PUOOBHCpGD op mnflpnooomnoflmh 025mm.“ 555322 .1. 555m
oooo.o ooo_.o so
88 .o ooom .o .0:
meg ash mien.o meow.m wacm.m Naom.o Leoo.o oooe.o son m
mmcc.o ooo~.o mos
ommm.o oooH.o «on
oooo.o oooa.o ao
oooo.o ooo~.o mos
men Hal Nico.o Nmoo.~ Leom.n oooe.o eooo.o ooom.o mom e
coee.o ooom.o .Os
cemm.o oooH.o .oa
oooo.o oooH.o .o
oooo.o oooH.o mos
sea mes oooo.L ssN.L es~.~ Laoc.o Hmoo.o ooo~.o son m
mlmw.o ooom.o .Os
ecca.o oooL.o .oa
oooo.o oooL.o .o
oooo.o ooml.o roe
men men Nmsa.o ooLL.L memo.H swam.o mooo.o oomL.o mom N
cwae.o ooom.o .Os
oHom.o ooom.o .oa
oooo.o ooom.o .0
0000.0 ooo~.o .Os
ems mam aama.o imam.n ngm.a some.o eooo.o oomL.o mom L
msmm.o oom~.o .Os
omee.o ooo~.o .oa
.m0 .m0
.QEOH £23m .QECB Aocmnm Gfimmm GL8 xm ”38.x Q ON bx unocomEoU
msEotsm mechanism BER 82:an ccsznoa coca.

 

 

mfidmom paw mqunu .H 03.95

44

EH03 m3» on wnwpnooom Ogden gfiwnez n ceased

poem—cg m.pooguopcb on. mnfipuooom 03mm HUGO.“ ggwfiz u Enrm

 

 

88 .o 82 .o .o
, 88 .o ooom .o mmos
E z: $.ch stem Mme: Solo 385 Some son a
2.? .o 88 .o .0:
mas... .o 82 .o .OL
88 .o ooom .o so
88 .o ooom .o mOs
u: .2: 535 £34 caomg 335 8.85 82.0 son c
$.35 . oommo .Os
cmee.o ooo~.o sun
so no
.dECH £03m. .QECH £65m EEXM Egonm 58.x Q QN hum acodOmEOU
meEoEsm menaaascm Esra ccoznmao sonata coca.

 

 

Aeossnsoovi .L case

APPENDIX

46

 

 

 

 

 

 

 

 

12 x 32.213 n.2 x 22.2.6- ...2 w 2.2.2 6+ .2 x 222 6+ .0
m72 x momma... 6- 72 x 2.66m 6+ N2 x $226+ .2 x 21.2 6+ .2.
.12 w 5236+ s-2 x 3336- N2 x 232 6+ .2 x 522 6+ .2
+-2 w No.22 6+ 72 x 21.22 6+ N.2 x :82 6+ m2 x 22.2 6+ .2.
.33 x owcwoe6+ 13 x ewewmm .o+ mg x 35: .o+ a: x 32.3 .o+ «02
m N H o uaoGOQEOU
mam-«Eocrom Lama-manna udomm> mo muaofloflmooUuu .v 3nt
#2 x 222 6+ 12 x $22 6+ N.2 x 22.? 6+ .2 w $236+ .0
...-2 x 2264.? .2 x 2.32 6- N2 x owomme6+ .2 x $226+ s.2.
...2 x 2826+ N.2 x 4262 6- .2 x $226+ .2 w Reef-6+ m2
T2 x $326+ 72 x 32: 6+ ...2 x 2.2.2 6+ +2 on $286+ +0:
12 x 2.62 6+ 72 x 32.3.6- N.2 x 6362 6+ .2 x $336+ .2
m N H o unecomEoU
.2289.an someone 22.3 no senescencoo: .m czoe
72 x Sol-36+ 12 x 325.6- m.2 x $336+ .12 x $236+ .0
F2 x mime... 6+ ...2 x 23: 6+ N.2 x 262 6- .2 x 82.: 6+ mOs
.-2 x Sam-2 6+ .72 x comes... 6+ .12 x 628... 6- .2 x 22.2 6+ .2
$2 x $22.6- 12.38213 N.2 x 232 6- .2 x 2.1.2: 6+ .2.
72 x 232.6- 12.. $826+ .72 x 222.6- .2 x $226+ .2
m N H o unccomEoU

 

1]

ll

harmonica osecasoo 633235 no 3:22:80-- .N 282.

Table 5. --Coefficients of Cm and Cn Polynomials.

47

 

 

Coefficient For Cm For Cn
o +0.277350 x10"l -0.857200 x 10'1
1 +0.494254 x 101 +0. 506012 x 101
2 -o.769972 x10l -0.905295 x101
3 +0.628226 x101 +0.869189 x101
4 -O.280358 x10l -0.461897 x101
5 +0.690212 x 10° +0. 135899 x 101
6 -0.879928 x10"1 -0. 206866 x10°
7 +0.453658 x10"2 +0. 126876 x10“l

 

48

Flow diagram

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

1L

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

Assume To
' ‘1
K1
V
Y1 Y1 Y1
E3:- Xi ‘ E; 32?;-
V
Y
1 -Z..£
K1
V
|S| i1 Y1| A
Zr
ve 4 fl ve
+ “* , _
’ {
emperature. + Y1
'€—— Correct 1 ER?
To = T W
A?
1+2E 1
r 7
1 .Tol
f"1§"

 

Figure 4.--Dew Point Temperature Convergence

49

Flow diagram

 

 

Assume To

 

 

 

A

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

K1
V

f K1X1 : Y1 |~_ K1X1,ZK1X1
v

1 - z Kix1
‘7

ISI- \1-2K1Xil

.+ve < '

 

 

 

 

Temperature
correct
T0=T

 

 

 

 

 

 

 

 

 

 

 

1+zK1x1

O

 

Figure 5.--Bubble Point Temperature Convergence

 

 

 

 

 

50

o.m

 

o.m

.60 2...... EU .6 $3.5 2.28.3.2. 662.320 .6 £st

Ilnohu-
xdbodTvvzvow xo Qoanq-vov
66 6.. w. 6.-. ... m. .N. 2.66.

00 .H «.0 .2. men

v..Gn.Q-.x.30g Tivnw h> EU

51
COMPUTER LABORATORY
LIBRARY ROUTINE A1-63__

TITLE: Floating Decimal Arithmetic Routine

TYPE: Interpretive routine with 18 interpretive
orders, entered as a closed routine, left
by an 8J interpretive order.

NUMBER OF WORDS: 168

PURPOSE: This routine manipulates numbers in the

degimal form, that is, numbers which are represented as A x
10 It is of the interpretive type. This means that it
selects parameters called interpretive orders which are
written by the user one at a time and performs a calculation
corresponding to each interpretive order. Interpretive
orders carry out normal arithmetic operations such as addi-
tion and multiplication and some red tape operations such as
counting and address changing.

In general, one will use this routine to
do computations which do not require the full Speed of the
computer but which are too time consuming to be done by
hand. It is especially effective for problems with scaling
difficulties. In a sense one may think of the floating deci-
mal routine as converting the MISTIC to a medium speed
floating decimal computer having a very convenient order
code.

ACCURACY: About 9 decimals
TEMPORARX STORAGE: O, l, 2.
PRESET PARAMETERS: 83 is used to specify two locations of

non-temporary storage, S} and 133, which are used for the
floating decimal accumulator.

METHOD OF USE: The floating decimal routine is entered

as a standard subroutine. Following the entry,ie., after

the transfer of control to the subroutine, one begins writing
interpretive orders. These orders each occupy one half word
and consist of a pair of function digits followed by a single
address. They therefore have the same form as standard
machine orders and may be read by the Decimal Order Input
with full use of the conventional terminating symbols.

52
The first of the two function digits of
an interpretive order describes the group characteristics
of the order and may take values 0, l, ..., 8. Normal arith-
metic interpretive orders have this digit equal to 8. The
second of the two function digits describes the type of
interpretive order.

INTERPRETIVE ORDER LIST WITH FIRST FUNCTION DIGIT b = 8
Let F be the floating decimal number in the floating

accumulator and let F(n) be the floating decimal number in

location n.

80 N Replace F by F - F(n).

81 n Replace F by -F(n).

82 n Transfer control to the right hand interpretive order
in n if F O. '

83 n Transfer control to the left hand interpretive order
in n if 0.

84 n Replace by F + F(n).
by F/F(n).
by F x F(n).

F
F

85 n Replace F by F(n).
86 n Replace F
F

87 n Replace

88 0 Replace F by one number read from the inlut tape
punched as sign, any number of decimal digits, sign,
and two decimal digits I8 represent the exponent.
For example, .8971 x 10 would be punched as +8971
+10.

89 n Punch or print F as a sign, n decimal digits, sign,
two decimal digits to represent the exponent and two
spaces. This print out may be reread by this routine.
After F has been punched or printed, it may not re-
main in the floating accumulator unmodified. n can
take values 2 to 9.

8K n Replace F by n if Os n < 200.

88 n Replace F(n) by F.

8N n Replace F by IF I - [F(n)].

8J n Transfer control to the ordinary MISTIC order on the

left hand side of n. This is used to escape from the
floating decimal subroutine.

53

8F n Give a carriage return and line feed and start a
new block of printing having n columns. This order
is only obeyed once for a particular block of print-
ing. At this time a counter is set up which will
cause a carriage return and line feed to occur auto-
matically from then on after every set of n numbers
that is printed.

INTERPRETIVE ORDERS WITH b 6 8. If the first function digit
of an interpretive order is O, 1, ..., 7 it will refer to
one of a set of control registers or b- registers in the
floating decimal routine which are similarly numbered.

These registers are used for counting the number of passages
through loops or cycles and for advancing addresses on suc-
cessive passages. For this purpose a particular b- register
which may be used in a particular cycle contains two count-
ing indices g and °b° These are both integers in the range
0 to 1023. T e index Cb is used for counting purposes to
determine the number of passages through a loop. The index
gb is used for advancing the addresses of interpretive
arithmetic orders. Although the interpretive order with
first function digit b is not actually altered in the memory,
it is obeyed as if gb were added to its address. The index
5b is increased by one upon each passage through the cycle.
The multiplicity of b-registers allows one to program many
lOOps within loops.

ORDER LIST WITH n ¢ 8

b0 n Replace F by F - F(n+gb).

b1 n Replace F by ~F(n+gb).

b2 n Replaceg by g? +1, cb+1. Then transfer control

b3 n to the right0 and (I f b2 n) or left hand interpretive
(if b3 n) order in n if 0 +1 is negative. This
transfer is used at the end of a loop.

b4 n Replace F by F + F(n + gb).

b5 n Replace F by F(n+gb).

b6 n Replace F by F/F (n+gb).

b7 n Replace F by F x F (n+gb).

bK n Replace gb , by O, -n. This interpretive order is
used for preparing to cycle around a loop n times.

bS n Replace F (n+gb) by F.

bN n Replace F by, F l - lF(n+gb)I .

bL n

8L n

54

Replace gb, Cb by gb+n, c +n, c . This interpretive
order is used when one Wighes t8 step addresses by
some increment other than +1 in a loop. If one
places bL 1022 in a 100p, the effect will be to
decrease addresses by two on each passage. bL 1

will increase them by 1, etc. If this order is

used in conjunction with a h2 or b} order, the chang-
ing address will be increased by a total of 1+n on
each passage.

Replace EECb by n, Cb, where b is the last b-register
0 b

referred y some previous interpretive order.

DURATION OF INDIVIDUAL INTERPRETIVE ORDERS

8N
8O
84

81
85

82
83

5 milliseconds + m x (3 2). Where m is the number
of shifts required to convert A, p back to standard
form.

2 milliseconds

3 milliseconds

milliseconds
milliseconds
milliseconds
milliseconds
milliseconds

milliseconds

\JJIDKJIKNKNONUT

milliseconds

When an interpretive order is preceded by b i 8,

add one millisecond t0 the above times.

When one wishes to repeat a cycle of interpretive orders

n times, the interpretive order bK n may be written before
entering the loop to set the counter Cb to -n. The interpre-
tive orders in the loop will be obeyed n times if the loop is
terminated by b2 0r b3 interpretive order to transfer con-

trol to the beginning of the 100p. This transfer of control
interpretive order will be obeyed n-l times and disobeyed the
nth time.

55

Use of Auxiliary_Routines. It is often convenient to
be able to leave the floating decimal routine so as to modify
interpretive orders or to perform calculations which may be
done more effectively outside of floating point. To leave
the floating decimal routine one uses an 8J n order. (All
standard floating decimal auxiliaries are entered in this
way.) To return to floating decimal one should transfer
control to the left hand side of word 29 of the floating
decimal routine. The interpretive order following the 8J n
order which was last obeyed will then be obeyed and so on.

In this way, it is not necessary to plant a link in auxiliary
subroutines. One may, in fact, think of the 8J n order as

a subroutine order. In case any changes are made in the
floating decimal accumulator while outside the floating
decimal routine, control should be returned to the left hand
side of word 19 rather than 29 so that this number may be
standardized before reentry.

Handling of Numbers. Each number is represented in
the form A x 101’ where 1>IA13 1/10, and 64 > pg, -64. In
a single register of the memory the number A is placed in
the 33 most significant binary digits (a0, a ,..., a ) in
the same way as an ordinary fraction is placed in th entire
register. An accuracy of between 8 and 9 decimal digits is
therefore achieved. The exponent p is stored as the integer
p + 64 in the 7 least significant digits of the same regis-
ter. For convenience, the floating decimal accumulato uses
two registers S3 and 183 for holding the number A x 10 . The
fraction A/2 is in $3 andthe integer p + 64 is in 183.

The only exception to the above rules is in thePnumber
zero which cannot, of course, be represented as A x 10 with
A 10. For this reason, zero is handled in a special way.
It is represented as a number with A.= O and p = -64. This
representation happens to correspond exactly with the
ordinary machine representation of zero.

After each arithmetic interpretive order is obeyed,
the number in the floating decimal accumulator is standard-
ized, i.e., the number in S3 representing A/2 is adjusted
so 1 >IAI>, 1/10 and p is changed accordingly. To accomplish
this, control is transferred to word 19 in the floating
decimal routine after each arithmetic order.

If an interpretive store order is attempted when F
has an exponent greater than 63, the machine will stop on
the order 34 p at location p, where p is word 72 of the
routine.

Important WOrds in the Routine. Word 2 in the float-
ing decimal routine determines the location of the current
interpretive order. When obeying the left hand interpretive
order in location n, this word is 50 nF SS 20F and when

56

obeying the right hand interpretive order in location n, it
is L5 nF OO 20F. Other words of interest are the b-re isters
which s.art at word 158 (for g0 and co) and go to 165 %g7

c These registers hold gband Cb in the form 80 ng

08%. 2048 + Cb)Fo

Warning: When the same number is continually added
to a sum, such as when an argument is being increased, the
error can be quite large, because it is additive over a
decade. For example, if we increaigb 10 to 100 by units, we
can get a maximum error of 90 x 2' cause he errors all
have the sane sign. If we increase 10 to 10 , we can have
a maximum error of 9000 x 2' '33. This can easily be prevented
by writing an auxiliary subroutine to stabilize the fractional
part of F, i.e., to replace it by the nearest multiple of
say 10'

PROGRAM

OOF OOIOF
OOF 0018F

O

OOF 00975F
OOF OO7OOF
OOF OOBIOF

O

22
5O

26
OK

88
OS

03
8J

0

003K

013K

0205K

L
L

S4
lOlF

F
SF

2L
210E

0210K

58

Directive

Specifies location of floating accumulator
Specifies location of A1 routine

Directive

Specifies location of temporary answer storage
Specifies location of Data storage

Specifies location of answer storage

Directive

Standard entry to A1.

Set to read 101 Words of data

L00p to left side of order 2.
Leave A1.

Directive

In this directive are all the title print out orders.

0

8K
80

88
IX

81
l7

14
86

IS
13

85
85

2K
OK

0230K

1F
48F

SL
5F

48F
58F

lOSF
SL

ISL
2L

358E
68L

F
5F

Directive
1
1-D:W

Set to loop 5 times

Calculate Xwi

Bring in TT assumed

Set to loop 5 times

I...)
\31
V

1...!
CO
v

’30)
L._L..

23)

22+)

0‘) 0

SJ

Cw H
L“ *4

r ’1“
J \

Li -
1:“ to
£1

UI ”I ’4 DJ
P11

.4 H
mm
W L1

'3]

Calculate K

Loop to left hand of order 7

Set to loop 5 times

a . "u _ 1.. “U- ""
' '1 0.1 "- ‘ .'3
VublVL\.L'\.JLJ(: Z /‘.L'--1_.

”Di ii

Loop to right hand of order 13

Temperature convergence test

TT reassumed
Put TT reassumed in 6 SL

Loop to left hand of order 6

assumed

Set to loop 5 times

Bring in C

31

to)

41)

2+2)

1+3)

’
81‘:
o
\J

87

95

8K
85
87

8J
88

OK
05

+3]

CO 0') K]
on m t“ m

FJH «HA
[4'11

N
.,-

2S
BSJ
1F

38?

BSJ
41

"I”?
‘—

QSJ

6O

L00p to left hand of order 24

Calculate Z KBiXWI

Loop to right hand of order 30

Temperature convergence test

TB reassumed

LOOp to left hand of order 23

Set to 100p 5 times

61

44) O7 l4SL ___
8d 187F Leave A1 to enter RA1. Calculate K K
Ti Bi
45) os l9SL
8K 1F

46) OO 43L

L,

47) 8K 1F
8K 1F

48) SK 1F

8K 1F

49) 85 218L
88 68J

w5 QOSL
oS SSJ

51) 81 1F

2K 5F Set to 100p 5 times
52) 25 iOSL
27 lOSF
53) 28 SJ
2} 52L LOOp to left hand of order 52
54) 85 58d
84 6SJ
55) 86 28F Assumee
88 24SL
56) 2K 5F Set to loop 5 times
25 198L

5?) 80 24SL
2S 7SJ

58) 22 56L
8K F

59) 88 128J
2K 5F

6o) 25 SJ
26 7SJ

61) 84 128J
88 l2SJ

62)

63)

64)

65)

66)

67)

68)

69)

7o)

71)

72)

73)

74)

l)

2)

3)

23

8N
83

85
83
85
88
8K
83

85
88

O
L;

8K
85

2K
25

’3

26

84
88

22
80

88
8K

60L
38F

128J
69L

128J
67L

24SL
6SJ

1F
54L

248L
SSJ

1F
54K

2SSL

5F
198L

58F

7SJ

258L
25SL

70L
18F

26SL
1F

OOBOOK

85
88

87
88

84
88

BK

26SL
37SL

48F
39SL

48F
388L

1F
1F

62

Convergence for e , test

New lower limit of 6
Loop to left hand of order 54
New upper limit of a

Loop to left hand of order 54

Set to loop 5 times

Calculate Rminimum + 1

L00p to right hand of order 70
Calculate Rminimum

Directive
R

R'D _ L

L+D=V

O)

l)

6)

7)

8)

10)

00310K

8K
8K

8K
86

87
88

85
87

88
84

88
BK

1F
1F

10F
848F

268L
268L

268L
48F

288L
48F

27SL

1F

00316K

8K
85

88
2K

OK
25

87
24

87
24

87
24

OS
2L

O2
8K

88
3K

35
37

84
83

1F
355?

298L
F

5F
158F

29SL
l68F

298L
l7SF

298L
188F

OSJ
4F

2L
F

SSJ
5F

OSJ
55F

SSJ
SSJ

63
Directive

L+D=V

Directive

TD assumed

Set to loop 5 times

Calculate KD1

Loop to right hand of order 2

Set to 100p 5 times

Calculate :ZIKD1XD1

11)

12)

13)

14)

15)

16)

24)

25)

26)

27)

28)

33
8K

80

88
85
8N
8K
84

88‘

8K
86

87

88

8K

82
85

2K
OK

25
87

24
87
24
87

24
OS

2L
03

8K
88

3K
35

37
84

9L
1F

SSJ
68J

38F
68J

19L
1F

SSJ
78J

2F
TSJ

298L
298L

1F
1L

68L
308L
5F

37SF
298L

38SF
29SL

39SF
29SL

4OSF
OSJ

4F
21L
SSJ

5F
OSJ

58F
SSJ

64

Loop to left hand of order 9

Temperature convergence test

TD reassumed

LOOp to right hand of order 1

Set to loop 5 times

Calculate Hi

L00p to left hand of order 21

Set to 100p 5 times

Calculate Ef'hixpi

as
32

85
87

Q r!
\J .La.

2K
OK

(p’o
01 P4

L»! \,\I
U‘I P31

(BK/N1
ewe

KN 60
TI) 0')

U} '11

\YlUl 0U]
0.4

avg

U) U)

WU}
\OUJ
WG-4

m
\1
C
If

U1
U)
L.

CH
U1

Loop to right hand of order 27

Calculate L -ZE.hiXDi

Set to loop 5 times

Calculate 81

Loop to left hand of order 53.

Set to loop 5 times

Calculate 2: HixDi

Calculate VTiEHiXDi

Calculate Qa

Set to loop 5 times

KI]
4>

U7
U)
V

('1‘?

1V Ii

83

8K
8 [1-

88
86
87

88
8K

83

2K
OK

66

Calculate K“
Di

LOOp to left hand of order 47

Set to 100p 5 times

Calculate ZE'KFiXFi

LOOp to right hand of order 53

Convergence of temperature test

TF reassumed

Loop to left hand of order 46

Set to 100p 5 times

65)

66)

67)

68)

69)

70)

71)

72)

73)

74)

75)

5)

6)

25
87

24
87

24
87

24
OS

2L
O3

8K
88

3K
35

37
84

88
32

85
8O

88
8K

37SF
348L

388?
348L

395F
548L

4OSF
OSJ

4F
65L

F
35SL

5F
OSJ

lOSF
355F

35SL
71L

33SL
35SL

36SL
1F

OO392K

OK
05

87
08

02
OK

05
OS

06
OS

03
85

88
85

5F
551?

48F
OSJ

OL
5F

58F
7OSL

98L
55SL

3L
288L

458L
308L

67

C lcul te h
a a Fi

LOOp to left hand of order 65

Set to 100p 5 times

Calculate

Calculate Qr

Directive

Set to 100p 5 times

Calculate DXDi

LOOp to right hand of order 0
Set to 100p 5 times

Calculate XT1

LOOp to left hand of order 3

2)

3)

4)

\O

10)

ll)

12)

13)

14)

15)

88
BK

5lSL
1F

00405K

8K
6K

8K
8K

85
88
85
84

88

OK

05

87

O4
86

OS
03

8K
8K

8K

(“Tr
LIL

8K
8K

2K
OK

25
87

24
87
24
87

24
08

1F
50F

1F
1F

518L
528L

458L
48F

498L
5F

55SL
458L

OSJ
498L

758L
5L

1F
1F

1F
1F

1F
1F
5F

l5SF
52SL

16SF
52SL

l7SF
52SL

188F
85SL

68

Directive

Set to loop 50 times

Tn+1 assumed

Ln assumed

Set to 100p 5 times

Calculate Y(n+1)i

L00p to left hand of order 5

Set to loop 5 times

Calculate K(n+1)i

16)

17)

18)

19)

2o)

21)

22)

2 )

24)

25)

8K

4F
12L
SSJ

5F
75SL

858L
6OSL

SSJ
SSJ

18L
1F

SSJ
lSF

6SJ
38F

6SJ
28L

1F
SSJ

28F
52SL

52SL
1F
lOL
1F

1F
1F

00435K

85
88

2K
OK

25
87

SF
lOSJ
5F

37SF
5lSL

69

Loop to left hand of order 12

Set to loop 5 times

Calculate X(n+1)i

Calculate 2::X(n+1)i

Temperature convergence test

T(n+1) reassumed

Loop back to left hand of order 10

Directive

Set to loop 5 times

ll)

13)

14)

15)

l6)

17)

18)

19)

2o)

21)

24
87

24
OS

07
84

8S
2L

O3
85

87

8K
8K

85
88

2K
OK

25
87

24
87

24
87
24
OS

07
84

88
2L

O3
85

87
88

80
88

388F
SlSL

4OSF
SSJ

555L
lOSJ

lOSJ
4F

2L
IOSJ

458L

1F
1F

SF
l28J
5F

57SF
52SL

588F
52SL

59SF
52SL

6OSF
58J

7SSL
l28J

l2SJ
4F

13L
12SJ

498L
138J

llSJ
l4SJ

7O

Calculate fihi

Calculate 2: Khixni

Calculate hn

Set to loop 5 times

Calculate fikn+1)i

Calculate ZZ 3(n+1)1Y(n+1)i

Calculate H(n+1)

Calculate Qa

22)

23)

24)

25)

26)

27)

28)

8K
8K

81
87

84
88

81
84

88
85

86
88

8K
8K

1F
1F

48F
l2SJ

33SL
SSJ
lOSJ
l2SJ
6SJ
58J

6SJ
YSJ

1F
1F

OO464K

81
84

8K
83

8K
8K

TSJ
458L

88J
38F

88J
6L

7SJ
458L

28F
458L

1F
408F

1F
1F

00471K

85
86

88
85

86
88

56SL
57SL

9OSL
7lSL

72SL
918L

71

Calculate LA

Directive

Test for equality of Ln and L;

Ln reassumed

Loop to left hand of order in location 408

Directive

Calculate XnLK/XnHK

Calculate YnLK/YnHK

3)

4)

O)

1)

OK
05

86
OS

02
8K

5F
858L

87SL
92SL

3L
1F

00477K
(All the orders under this directive are to regard

the (n+1)th

OK
05

OS
05

OS
05

OS
05

OS
02

85
88

85
88

85
88

85
8S

8K
63

5F
55SL

658L
6OSL

558L
7OSL

808L
758L

7OSL
OL

458L
46SL

498L

50 SL

5lSL
53SL

52SL
518L

1F
406F

OO488K

8K
OK

05
OS

07

08.

1F
5F

18L

105SL

148L

l2OSL

72
Set to 100p 5 times

Calculate o<ni
Loop to right hand of order 3

Directive

plate as the nthplate.)

LOOp to left hand of order in location 406

Directive

Set to l00p 5 times

Calculate YBi

8)

9)

4)

5)

6)

03
OK

05
87

OS
03

85
88

85
88

85
88

8K
8K

1L
5F"

lOSSL
OSL

OSJ
4L

78L
lO2SL

27SL
99SL

OSL
97SL

1F
1F

OO498K

8K
6K

8K
8K

85
88

85
84

88
OK

05
87

O4
86

OS
03

8K
8K

2K
OK

1F
50F

1F
1F

102 SL
lO3SL

99SL
OSL

988L
5F

l2OSL
99SL

OSJ
988L

llOSL
5L

1F
1F

5F

73
Set to loop 5 times
Calculate WX

W1

Loop to left hand of order 4

Directive

Set to 100p 50 times

Tm+1 assumed

Vm'assumed

Set to loop 5 times

Calculate X(m+l)i

Loop to left hand of order 5

Set to loop 5 times

lo)

11)

12)

13)

14)

15)

16)

17)

19)

20)

21)

22)

25)

26)

27)

25
87

24
87

24
87

24
OS

2L
03

88

OK
05

07
OS

84
88

02
8K

85
88
85

8N
82

85
84

88
85

86
87

88
8K

82
8K

ISSF
1038L

l6SF
1038L

l7SF
lO3SL

l88F
1358L

4F
lOL

SSJ

5F
llOSL

1358L
125SL

5SJ
SSJ

16L
1F

SSJ
18F

68J
38F

6SJ
27L

18F
SSJ

7SJ
28F

7SJ
lO3SL

103SL
1F

8L
1F

74

Calculate K(n+1)i

Calculate Y(n+1)i

Calculate Z Y(n+1 )1

Temperature convergence test

T(m+1) reassumed

10)

ll)

l2)

l3)

14)

15)

16)

17)

OO526K

85
8S

2K
OK

25
87

24
87

24
87

24
O7

84
88

2L

.03

85
87

88
8K

88
2K

OK
25

87
24

87
24

87
24

O7
84

88
2L

O2
85

SF
5SJ

5F

37SF
1038L

588F
lO3SL

39SF
1038L

4OSF
llOSL

5SJ
5SJ

4F
2L

58J
98SL

68J

78J

5F
57SF

lO2SL
588F

102SL
59SF

102SL
608F

l2OSL
7SJ

7SJ
4F

11L
7SJ

75
Directive

Set to 100p 5 times

Calculate K7n+1)1

Calculate h
(n+1)

Calculate Hmi

76

18) 87 998L Calculate Hm
CS ESJ
19) 85 88J '
80 68J Calculate QT
2 ) 8S CSJ
8K 1F
21) C5 OSL
87 SSJ
22) C4 368L
88 103J
23) 85 7SJ
8O SSJ
24) 88 llSJ
35 lOSJ
25) 86 llSJ Calculate v;
88 128
26) 8K IF
8; 'F
00554K Directive
o) 81 2SJ
84 ,98L
1) ”s 138J
83 38?
2) 8: 138J Test for equality of Vm and v;
8} 6L ‘
3) 35 l2SJ
84 QQSL
4) 86 28F Vm reassumed
8S 998L
5) 81: 1F
85 501F Loop to left hand of order in location 501
6) 8K 1F
8L 1F

6)

7)

8)

9)

00561K

85
86

88
85

86
88

OK
05

86
08

O2
8K

1068L
107SL

l4OSL
l2lSL

1228L
1418L

5F

135 SF

137SL
l42SL

3L
1F

00567K
(Ordergh 0-

(n+1)h

85
88

85
88

85
88

OK
05

08'

0.5

OS
05

OS
05

08
02

SK
63

85
86

993L
lOOSL

102SL
lO4SL

lO3SL
102SL

5F
105SL

1158L
12OSL

105SL
12OSL

13OSL
1258L

12OSL
3L

1F
499F

l4OSL
9OSL

77
Directive

Calculate XmLK/XmHK

Calculate YmLK/YmHK

Set to 100p 5 times

Loop to left hand of order 3.

Directive

8 under this directive are to regard the
plate as the nth plates.)

Calculate Yfi/Yh

10)

3)

4)

88 1478L
8K 1F

00578K

85 57SL
80 67SL

8S OSJ
85 38F

8N OSJ
83 5L

8F 1F
8K 6F

89 5F
83 667F

85 106SL
80 1168L

8S OSJ
85 38F

8N OSJ
83 10L

8F 1F
8K 9F

89 5F
83 667F

8K 1F
8K 1F

00589K

85 1438L
8O 18F

88 OSJ
OK 2F

05 l458L
87 OSJ

OS 1558L
O3 2L

85 933L
80 18F

78

Directive

Calculate (XnfiKX(n+1)HK)enriching pinch

Pinch condition test

Calculate (XmLK'X(m+1)LK) stripping pinch

Pinch condition test.

Directive

Calculate (°<LK‘1)°<i at stripping pinch for
calculation of Cmi’

10)

ll)

12)

13)

14)

15)

16)

17)

18)

19)

20)

21)

22)

87
86

8S
8K

7K
OK

8K
77

O4
O2

78
72

8K
OK

8K
87

O4
O2

88
8K

OK
05

O7
O7

08
O2

85
84

88
85

87
88
80

8S
8K

93SL
928L

157SL
1F

2F
8F
1558L

858F
8L

1588L
7L

1F
8F
1578L

93SL
12L

16OSL
1F

2F
1588L

lO88L
1458L

1618L
15L

1618L

162 SL

1638L
16OSL

558L
164SL

1F
1638L

28J
1F

79

Calculate (OC -1)qc% at enriching
pinch for ca culat%1of Cni‘

Set to loop 2 times
Set to loop 8 times

Loop to right hand of order 8

Calculate Cmi
Loop ti right hand of order 7

Set to loop 8 times

Loop to right hand of order 12

Calculate Cni

Calculate Cmixmieé

Calculate 210mlxm1851

Calculate Cn1Xh1

l)

80
87

8S
8K

86
88

8K
8K

1648L
2SJ

lSJ

fl

1:

18J
1488L

1F
1F

OO616K

85
86

1478L
1488L

1498L
1F

1498L
48J

38F
48J

lOL
1F

1498L
SSJ

2F
6SJ

68J
58J

26SL
26SL

1F
313F

1F
1F

OO627K

OK
05

06
08

3F
37SL

26SL
1508L

80

Calculate (8710119311) (l-Zcmixmixé)

Calculate #1

Directive

Ym/an

1-vh/ynwv

Test for equality of Ym/Yh and %/

l+‘%/&+’

Reflux ratio reassumed

LOOp to left hand of order in location 313

Directive

Calculate ratio of R' to RX

2)

81

BK 1F
02 CL

0063OK
(All the orders under this directive are the print
out orders)

OO667K Directive

8K 1F

8K 1F

8F 1F

8J 2L Leave A1 routine at left hand of order 2
2O 2L

50 2L Standard entry to A1

26 84

OK 6F Set to 100p 6 times

88 F Read

08 48F

05 4L Loop to left hand of order 4
8K 1F

82 208F Loop to right hand of order in location 208
82 208F

24205N Termination

GLOSSARY

Bubble point: The temperature at which a multicomponent
liquid is beginning to boil; also used to denote the
trial-and-error procedure by whichEKX is made equal
to unity.

Dew point: The temperature at which a multicomponent vapor
is beginning to condense; also used to denote the
trial-and-error procedure by whichZ-E. is made equal
to unity.

Enriching section: The section of a fractionating column
above the feed tray.

Heat flux: The net upward heat flow in a column as calcu-
lated by difference in heat content of passing streams.
This flow is constant in respective sections.

Pinch region: The region containing a large number of
plates on any two successive plates of which there is
no effective separation. Compositions on all the plates
in this region are constant.

Stripping section: The section of a fractionating column
below the feed plate.

Total condenser: A condenser which condenses all of the

entering vapor.

82

"db

, h
H’

9

12131:]:

”col—‘13

NOMENCLATURE

Equilibrium constant polynomial coefficient.

Liquid phase enthalpy polynomial coefficient.

Vapor phase enthalpy polynomial coefficient.

Cm factor polynomial coefficient.'

Cn factor polynomial coefficient.

Colburn's correction factor in the stripping section.
Colburn's correction factor in the enriching section.
Moles of distillation product.

Moles of feed.

Enthalpy in vapor and liquid streams respectively.
Enthalpy of a pure component as vapor and liquid respect-
ively.

Equilibrium constant (Y/X).

Moles of liquid.

Heat flux.

Reflux ratio (L/D).

Ratio of liquid concentrations of the key components
in the stripping section pinch region.

Ratio of liquid concentrations of the key components in
the enriching section pinch region.

Temperature (OF).

Moles of vapor.

83

'1] U 0 U1 0" SD +m§g~® R 1'4 >8 2

H

m+1

n+1

84

ioles of bottom product.

Mole fraction in liquid.

Mole fraction in vapor.

Relative volatility with respect to the heavy key.
Term used to calculate Cm values.

Term used to calculate Cn values.

Arbitrary value near zero.

Factor to be compared with ratio of Ym to 7&.

Subscripts
Enriching section.
Stripping section.
Bottom plate of the fractionating column.
Condenser.
Distfllate.
Feed.
Particular Component.
Any plate in the stripping section.
The plate Just above the mth plate in the stripping sec-
tion. ‘
Any plate in the enriching section.
Any plate just below the nth plate in the enriching _
section.
Assumed value.
T0p plate of a fractionating column.

Bottom product.

Superscripts

Indicates value obtained by preliminary assumptions.

10°
ll.

l2.

l3.

14.

15.

BIBLIOGRAPHY

Brown, C. G., Unit Operations. New York: John Wiley
and Sons, Inc., 1951.

Brown, G. G., and Martin H. 2., Trans. Am. Inst. Chem.
3118-, 31. 679 (1939). ..__.._ "“ “"'

Colburn, A. P., Trans. fig. Inst. Chem. Eng., 31, 805

COpe, J. Q. and Lewis, W. K., Igd. Eng. Chem.,.gfi, 498
(1932).

 

Donnell, J. W. and CCOper, C. M., Principles'gg Petro-
leum Refinery Engineering.

Fenske, M. R., Ind. Eng. Chem., gfl, 482 (1932).

Foy, R. H., "Optimum Feed Plate Location in Multicom-
ponent Distillation." Unpublished Master's
thesis, Department of Chemical Engineering,
Michigan State University, 1959.

Gilliland, E. R., Ind. Eng. Chem., 22, 1101 (1940).

Jenny, F. J., Trans. Ag. Inst. Chem. Egg., :5, 635
(1939).

Lewis, w. K., Ind. Eng. Chem., 14, 694 (1922).

Lewis, w. K. and Matheson, G. L., Ind. Eng. Chem., 23,
494 (1932) . "" "*7

MISTIC Pro ramin Manual, Michigan State University
1959).

Maxwell, J. B., Data Book 9g Hydrocarbons. New York:
D. Van Nostrand Co., Inc.

McCabe, W. L. and Thiele, E. W., Ind. Eng. Chem., 11,
605 (1925). -

Sorel,8E., "La rectification de l'alcool." (Paris)
1 93.

85

l6.

17.

18.

19.

Thiele, E. W.

289, (1933).

Underwood, A.
(London,

Underwood A.
1946).

Underwood, A.

86

and Geddes, R. L., In . Eng. Chem., 25,

 

V. J., Trans. Inst. Chem. Eng., 19, 112
1932).

V. J-,.£- Inst. Petroleum, fig, 614 (London,

v. J., Chem. Eng. Prcg., 35+, 603 (1948).

“All a 1!
I m .l e;
. afflka
tiara}
p a...

”1%..“
n1

 

MICHIGAN STATE UNIVERSITY LIBRARIES

111 IIIIIIIIII

3175 7879

l I)

3 1293