THE UNIVERSITY OF MICHIGAN
INDUSTRY PROGRAM OF THE COLLEGE OF ENGINEERING
NON-IDEAL STAGE MULTICOMPONENT ABSORBER CALCULATIONS
BY AUTOMATIC DIGITAL COMPUTER
Arthur E. Ravicz
A dissertation submitted in partial fulfillment
of the requirements for the degree of
Doctor of Philosophy in the
University of Michigan
1958
February 1959
IP-354

Doctoral Committee:
Professor Julius T. Banchero, Chairman
Associate Professor John W. Carr III
Assistant Professor Kenneth F. Gordon
Professor Joseph J. Martin
Assistant Professor M. Rasin Tek
Professor G. Brymer Williams
ii

ACKNOWLEDGMENTS
The author wishes to express his appreciation to the following
individuals and organizations for their contributions to the research
which was the basis of this dissertation:
Professor J. T, Banchero, chairman of the doctoral committee,
for his wise counsel, his wholehearted cooperation, and his willingness
to give freely of his time and effort on every occasion.
Associate Professor John W. Carr III, for his advice on the
mathematical aspects of the problem and his help in securing the use of
the IBM 704 computero
The other members of his doctoral committee for their advice
and encouragement.
The General Motors Corporation and the International Business
Machines Corporation for their generous contributions of time on the
IBM 704 at the General Motors Technical Center.
The Phillips Petroleum Company and the Natural Gasoline
Association of America for their donations of absorber test data.
The California Research Corporation for the use of their
multicomponent distillation reports.
Mr. Wayne C. Edmister (California Research Corporation) and
Mr. Robert Lo McIntire (then with Phillips Petroleum Company) for their
personal interest and advice on the engineering aspects of the problem.
Mr. Go Ho Lolmaugh and Mrs. Shirley Callahan (General Motors
Research Staff, Data Processing Group) and Dr. B. A. Galler (University
of Michigan) for their help in coding, assembly, and checkout of the
machine program.
His wife, Patricia Ravicz, for her help in coding the program,
assembling data, and preparing the dissertation.
The Standard Oil Company of California for their Fellowship
in Chemical Engineering during the academic years 1955-56 and 1956-57.
iii

TABLE OF CONTENTS
Page
ACKNOWLEDGMENTS..,... i, o o o o,ii
LIST OF TABLES. o.o..o.....o....Oeoo o......o........... ii
LIST OF FIGURESo...o....o. o.o.o..o o.......o... o...... o ix
I. INTRODUCTION.......o.....o...,o o o.... o. o... 1
A. -Purpose of Investigation...................... 1
B. Description of a Petroleum Gas Absorber.....o.. o... 2
C. Statement of the Problem.... o, -.oo oooooo.o.. oooooo. o 4
D. Summary...O.. o. O o o. o. o o o e 4
II. BACKGROUNID. o.. o o. o o.o o.o. o o o.. o. o o. o. o 6 o
A. General Remarks..o................... oo.. 6
Bo Absorption Factor Methods.....o o.......o.o 6
C, Plate-to=Plate Calculation Methods o......o.....0. 12
D. Machine Computation Methods o.o o.. o.... o........ o.... 16
IIIo ENGINEERING CONCEPTSo....oo...O,...,.....o.o ooo.. oo 28
A. The Non-Ideal Stage o.....oo. o.. oo.o.o........... 28
Bo Mass Transfer Relationships o...o. o..... o........ o 29
The Problem.. O..........o........ o o,................ 29
The Murphree Plate Efficiency.o. o. o.................. 30
An Expression Based on Mass Transfer Coefficients...... 34
C. The Mass Transfer-Heat Transfer Analogyo...... o.....o 38
A Consequence of the Colburn Analogy.,..o.o.....,..,.0 38
The Heat Transfer Efficiency.....o.. t.......... 40
D. Prediction of the Murphree Plate EfficiencyO..... O *..0 41
EMV Based on Separate Phase Transfer Units.........42
The Effect of System Properties..O o..o...,..o..., 43
The Effect of Operating Variables..........,....... 44
The Plate Efficiency Correlation..o...o,.o.o.....o...., 46
Comparison of Predicted Values with Test Data......0o 47
E. Enthalpy Procedures.... 6,o oo........ o.,..50
Vapor Enthalpy...........,............. 52
Liquid Enthalpy, Method Ioo..ooo...oo... ooo......oo 52
Liquid Enthalpy, Method IIo.o oo...oao.. o........o 55
iv

TABLE OF CONTENTS (CONT D)
Page
F. K Values and Convergence Pressureo..... o oo, o o o. 57
Functional Dependence o..o.o.o...................... 57
Convergence Pressure............o. o....o.oo0...o.o 58
K as a Function of Convergence Pressure o.O.. O....o. 58
IV. PHYSICAL PROPERTIES.........o o o o o o o......o..o o.............o o o 61
A. Vapor Density.... o...*....... o o o o. 61
B. Liquid Densityo o............... o..62
C. Vapor Diffusivity o o.........o o o o.........o....o o64
D, Liquid Diffusivity.......o oo o o.......o..o...*....*o 69
E Liquid Viscosityo....... o.................... 69
F.o Vapor Thermal Conductivityo.........o......... o,.. 71
Go Ideal Gas Enthalpy. o o o o. o. o............ ~... o 71
H. Pressure Correction to Ideal Gas Enthalpy..........o.. 73
I. Isobaric Heat of Vaporization..... o....... o o o o... o 81
J. Isothermal Heat of Vaporization...o.o............... 81
K. Enthalpy Change of Subcooled Liquid.....oo........... oo 82
L O. K Value e o o o o o o o o o a.. o o o o.. o o..... o.. a a............ o 84
M. Convergence Pressure o o... o o o............ o.. o......... 85
V o PROBLEM-SOLVING PROCEDURE.o..... o. o o o... o........ o...... 92
A. The Overall Procedureo... o.,,................o o. 92
B o Data Conversion.o............o.. 6.....o........., oooo.o.oo.o 92
C. Initial Approximation....... o.... o........o. 101
Do Plate-to-Plate Calculation..oo.....o ooo......o..o...o.o.ooo...o 106
E. Output e.00 0 6 0 0 0 0 0 0o0 0 0 00 0 0 0 0 0 0 0 0 0 0 0 o 0 o 118
Fo The Liquid Temperature Subroutine o..o,.....o......... 120
G. The Dew Point Subroutine..., O O O,.....o....o. o.. o 122
VI. MATHEMATICAL ANALYSIS o o o o o o o o o o o o o o.............................. 124
A. The System of Simultaneous Equations........ o 124
Bo The Iterative Solution. o.oo.oo o..o oo. o.o...O.... o 128
C. Experience with the Iterative Solution.............o... 131
D. The Aitken "Delta-Squared" Method......o..oooo..o.o..o o 132
E. Wegstein's Method. o o o o..o...o.o..o.o... oo.........o. 134
VIIo RESULTSoo..ooo.0 ooooooooooo............ o...... o o oo06 0o0.o o0 o o, 139
A. An Ideal Stage Column.o.....o.ooo,.......o.... 139
Bo. Column A.... o.o.. o o o o o o o * o e o o.. o. o. o..... o.... 140
C. Column B o.. o..o...0....... o.. a. o.. o o o o o o 141
Do Column C e oo.ooO...o o oo..e..o o.......o.....oo........,43
E. Column Do.o o.. o...............o........... o.....o..... 143
v

TABLE OF CONTENTS (CONT'D)
Page
VIII. ANALYSIS OF RESULTSo.000o o o....o.o o.o....... o o. o... 180
A. The Ideal Stage Calculation... o.... O O. O.... O.. o o 180
B Recovery of Components... o.......o... o,o..o o. o.. 180
C. The Average Vapor Flow Rate o..o....oo oo o..oooo 81
D. The Effect of Plate Efficiency............ o. o.. 181
Eo The Effect of Lean Oil Composition,.. o......*. o.. o.. 185
Fo The Effect of Intercooling..o....oo....,......o o *o 186
GO Convergence Pressure...,....1......o............ 186
H. Dew Points and Bubble Points.................... 187
Io Heat Transfer Efficiency.. o..o..o....oo.o...o.oo 188
J. Prediction of the Heat Transfer Efficiency Factor,.... 191
Ko Estimation of Plate Efficiencieso. o...o..oo. o...o.oo 194
IX. CONCLUSIONS...... o...o.1,..... 0........ 197
X. APPENDIX - MACHINE UTILIZATIONo..,. o........2......o200
A. Description of Machine and System O.............o 200
B. Machine Run Procedure o.......o O...o. o i,.. *..... 202
C. Programming and Coding Policy o.o...o6*..o.... 206
D. User's Instructions. oo o o o...,.. 0 o........ 209
E. Subroutine Specifications............................. 8
F. Availability of Program...o...o..o.......... o 230
XI. REFERENCESN e.... o... o. o. o o.. o o,....... 231
XII. NOMENCLATURE....o....o. o..............o..... 236
vi

LIST OF TABLES
Table Page
1. Sample Calculation of Murphree Plate Efficiencyo....o.o 48
2o Sample Enthalpy Calculations. o.ooo.o0o..oo, o., o. 53
3. Beattie-Bridgeman Constants..., o....,..... 63
4. Vapor Diffusivity Constants....,. o.............0. 66
5. Ideal Gas Enthalpy Constants o......o... 74
6o Cross-Correlation Constants for Pressure Correction
Curves.....,oo.....,.o^O...a...,,....O, 7o 77
7. Sample Calculation of Pressure Correction to Ideal
Gas Enthalpy..o.,o...... o,.o.... *.......o,, 80
8. Convergence Pressure Correlation Constants........... 88
9. Consolidated Physical Properties.......o......, 90
10o Results for an Ideal Stage Column.,..,............ 145
11. Component Results for Column Ao.......,... o...a... 146
12. Plate Results for Column A,.......,....o o........... 147
13* Component Results for Column B,..o o,.oo o.oo 155
14. Plate Results for Column B,.oo..o....... o.........o 156
l5o Component Results for Column B with Well-Stripped
Lean Oil.... ooo io...Oo a o.... a O 0 O a, o.o.., aO..... O 165
16o Plate Results for Column B with Well-Stripped
Lean Oil o o o o o. ao o, o o o o * o.... O oO 166
17. Component Results for Column B with Well-Stripped
Lean Oil and Intercooler.... o.. o............ 168
18, Plate Results for Column B with Well-Stripped Lean
Oil and Intercoolero..oo............ o.o...... 169
19. Component Results for Column C,............,...., o 172
20 Plate Results for Column C........o...o,..,,... o o 173
21o Component Results for Column D..,...............,. 176
vii

LIST OF TABLES (CONT D)
Table Page
22. Plate Results for Column D..o * o.. o.. a.. o o.. o. 177
23. The Effects of Average Vapor Rate and Plate Efficiency.. 182
24~ Prediction of the Factor Cqoo,.o.ooo....,,.o...,.o,..oo 192
25. Composition of Program Decks. o o...... o........o 204
260 A Typical Set of Column Data.........o.................. 216
viii

LIST OF FIGURES
Page
1. Schematic Diagram of an Absorber......,.........~.a. 3
2, Enthalpy Evaluation Steps......................... 51
3. Liquid Enthalpy - Method II............................. 56
4. Critical Region of a Binary Mixture..................... 59
5. Problem-Solving Procedure.........a.................. 93
6. Flow Diagram - Subprogram A.......................... 94
7.~ Flow Diagram - Subprogram Bl..p.............*....^.. 102
8. Flow Diagram - Subprogram C...............~... 107
9, Flow Diagram - Subprogram D,,...,........,.....,. 119
10, Flow Diagram - Liquid Temperature Subroutine............ 121
11. Flow Diagram - Dew Point Subroutine....*,,,,*4......., * 123
12. The Wegstein Method..,.....,,.... a...............a 135
13, Results for Column A.............,.,............ 149-153
14. Plate Efficiencies in Column A.,,,.,.............,... 154
15, Results for Column B...,...,.,.............,,..., 158-163
16. Plate Efficiencies in Column B..................,,..,a 164
17. Results for Column B with Well-Stripped Lean Oil*....**. 171
18. Results for Column C.1.......,....................... 175
19 Results for Column D... a a 179
20, Operating and Equilibrium Lines...........,........, 183
21, Variation in Temperature Profiles,..................... 90
22. Prediction of the Factor C......................... 193
23. Machine Run Procedure,.......,.,.....,...........,.. 203
ix

I. INTRODUCTION
A. Purpose of Investigation
The advent of high-speed, electronic computing machinery has
caused a great deal of interest in the automatic computer solution of
many chemical engineering problems. One of the first problems to be
investigated was vapor-liquid mass transfer such as occurs on the plates
of a distillation or gas absorption column. Most of the effort spent
thus far has been directed to the machine adaptation of methods already
developed for hand calculationz methods involving the simplifying assumption of the so-called "ideal", "theoretical",. or "equilibrium" stage.
The purpose of the present investigation was fourfold: (1) to
formulate a mathematical model of an actual or non-ideal vapor-liquid
contact device, such as a bubble-cap platea, and to make this model suitable for use with an automatic computer; (2) to develop a computer procedure for predicting the mass transfer relations involved in the nonideal stage; (3) to investigate the use of enthalpy, equilibriums and
physical property correlations more rigorous and accurate than those
currently in use, methods too tedious for hand calculation; and (4) to
demonstrate the soundness and applicability of the non-ideal stage approach by writing a computer program for its use and comparing predicted
performance of actual equipment with test data.
The equipment selected for investigation was the multicomponent
petroleum gas absorber. Computation was done on the IBM 650 magnetic
drum data processing machine at the University of Michigan and the IBM
704 electronic data processing machine installed at the General Motors
Technical Center.
-1

-2B. Description of a Petroleum Gas Absorber
A petroleum gas absorber is a device for recovering certain
hydrocarbons from a natural gas or refinery gas streamo The main components of the stream, methane and ethane, carry with them smaller percentages of materials in the LPG (liquefied petroleum gas) and natural
gasoline ranges, propane through hexane or heptane, which we desire to
recover. Small quantities of hydrogen, nitrogen, carbon dioxide, hydrogen sulfide, etc,, may be present alsoO
Recovery of the desired hydrocarbons is effected by absorbing
them in a heavy petroleum fraction called absorber oily a material with
average molecular weight equal to that of kerosene or heavy diesel oilo
Transfer of material between gas and absorber oil streams occurs in a
counter-current multi-stage contacting device called an absorption column.
Figure 1 gives a schematic diagram. The rich gas stream, V0, enters the
bottom of the column the lean gas, Vn, leaves at the top. Lean absorber
oil, Ln+l enters the top of the columns rich absorber oil, L1, leaves at
the bottom. At each stage or plate of the column, m, liquid and vapor
streams are br ought into intimate contact, the vapor bubbling upward
through the liquid stream as it passes across the plateo Plates are
numbered from the bottom upo
Absorption columns may have as many as forty or fifty plates,
plate height or spacing of one to three feet, and column diameter of two
to ten feeto Operating pressures range from 50 to 3000 psia, temperatures from 50 to 2000~Fo The absorptison of mteerisal liberates heattherefore, cooling may be provided by withdrawing a liouid stream from
the column, cooling it, and returning ite or by putting cooling coils.on
the plateso Heat loss by radiation and convection from the column also
occurs o

-3LEAN GAS LEAN OIL
Vn
vn I -I-ILn+i
n __
Vnl Ln
n-I
> Vn2 L
INTERCOOLER
m+1
CONVECTION
Hc l Vm Lm+l
fm. Lm+i Lm+l
o I.=., -m - mlm-'
VI Lme
RICH GAS. V'RICH OIL, L
Figure 1. Schematic Diagram of an Absorber

-4C. Statement of the Problem
In order to demonstrate the non-ideal stage calculation, a
computer program was written to solve the following problem:
Given the temperatures, compositions, and flow rates of the
rich gas and the lean oil, the column pressure (assumed constant)^ the
number, diameter, and other pertinent dimensions of the bubble-cap plates,
and data on intercooling and other heat losses- compute the fractional
recovery of all components and the complete properties of the lean gas,
rich oil, and all internal streams of the column
D. Summary
Following are brief statements on the purpose, content, and
importance of the various sections of this dissertation,
Section II, Background, contains discussion of past and present
methods for the design and performance evaluation of multicomponent distillation columns and gas absorberso Particular emphasis is given to
aspects of these methods which are of interest or utility in the development of the non-ideal stage calculation and its employment with automatic
computers
Section III, Engineering Concepts is probably the most important section of the dissertationo It contains the theoretical engineering bases for the computational procedure employed.
Section IV, Physical Properties., contains detailed information
on the ways in which the various physical and thermodynamic properties
are correlated for machine useo
Section V, Problem-Solving Procedure, consists of a stephbystep account of the procedure followed in the automatic computation.

-5
The procedure is illustrated by flow sheete which present a somewhat
more machine-oriented view than the text.
Section VI, Mathematical Analysis, treats certain subjects of
interest to the more mathematically inclined reader. Consideration of
the whole procedure as the solution of a set of simultaneous equations
is discussed here.
Section VII presents, in tabular and graphical form, the results
of the investigation of an ideal stage column and four operating absorbers
for which test data are available,
Section VIII, Analysis of Results, contains discussion of information and experience gained in the investigation of the various
absorbers. With the exception of the treatment of the ideal stage calculation, the discussion has been outlined according to phenomenon,
effect, or trend. This arrangement is considered preferable to discussion
of each of the calculations in turn.
Section IX, Conclusions, consists of statements based not only
on the results of computation, but on the entirety of experience gained
in the research effort.
The Appendix is of importance only to the person with interest
in and knowledge of automatic computation,
For the reader with limited time to devote to this dissertation
and whose interest is in the general engineering (as opposed to mathematical and computational) aspects of the problem, the following sections
and subsections are recommended: Introduction; Engineering Concepts;
subsection A of Problem-Solving Procedure; Results; subsections A through
Et I, and K of Analysis of Results; and Conclusions.

II. BACKGROUND
A. General Remarks
The inclusion of a section entitled "Background" is of only
limited utility in this dissertationo The computational procedure developed here differs from previous work in one fundamental way - it is
based on the non-ideal stage. To the author's knowledge, there has been
no previously published material dealing with non-ideal stage, multicomponent, plate-to-plate distillation or absorption calculations 0 In
other particulars, ego., the use of material and energy balances, the
present method is based on previous worko'1Te general subject of absorption and distillation will be
discussed in terms of the well-known hand computation procedures. Distillation calculation methods will be included because of their similarity
or application to absorption methods. Machine procedures, which now
consist mainly of adaptations of hand procedures, will then be mentioned
briefly. It should be borne in mind that all the methods to be discussed
are based upon the ideal stageo
B. Absorption Factor Methods
The absorption factor of component i on plate m is defined
Lm LIM xmi-,
-Aml Vm Km,i Vm Ym,i
where
Am is the absorption factor
mi
Lm is the li uid flow rate from plate m
Vm is the vapor flow rate from plate m
-6

-7Kmi is the vapor-liquid equilibrium ratio of component i on
plate m
Xm i is the mole fraction of component i in the liquid Lm
Ym.i is the mole fraction of component i in the vapor Vm.
Considering just one component and dropping the subscript i from now on,
we write- the absorption factor on the next higher plate, mI+l, as
AB^mi Vm+l
ANL+I Vin+l K1 (1')
A component material balance around plate m gives
Ln+ (Xm - Xm+l) = VO (Yn -. Ym) (2)
where
Ln+l is the lean oil flow rate
Xm and Xm+1 are the quantities of component i in Lm and L+1
per mole of Ln+1
V0 is the rich gas flow rate'm-l and Ym are the quantities of component i in V.1- and Vm
per mole of V0.
Recalling the equilibrium relation
m = K Xm 3)
we may rewrite it in the form
Ym V0 Xm Ln+l
Ym VO- = K X n(4)
and solve for Xm to get
Ym VOm L ( ) Ym
VXm n= ~v- Ln Km

-8Solving Equation (2) for Ym, we obtain
Ym= Y,1 Ln+l (Xm -Xm+l) (6)
V0
and substituting Equation (5)
Ym Y - Am Ym + Am+l Ym+l (7)
Finally-,
Y-m l'+ A m+l + (8)
Ym 1 +A (
If'mis the bottom plate of an absorbers, we may again incorporate
Equation (5) and write
_Y + (X2 Ln+l/Vo)
1 1 + A,
or, for a one-plate absorber
Yo + (x2 LJ/Vo)
Y1 ^ + O 1 ) (1o)
Equation (9) may be written for the upper plate of a two-plate
absorber as
Y+ (X3 L3/Yo)
-1+A2 (11)
and Equations (10) and (11) may be combined to give
Yo + A2 Y2 A+ [Y1 + (X3 L3/V0)]
1 + Al I + Al + A1) (1 + A2)
whence
YA2 (Y0)(1 + A2) + (A2)(X3 L3/V) (13)
(1 + A1)(1 + A (1 + A1)(1 + A2)
and
y (A2 + )(Y0)+ (A2)(X3 L3/V0) ()
A2 A1 + A1 + 1

-9~
The equation for an absorber of three plates may be developed
(A3 A2 + A2 + 1)(YQ) + (A3 A2)(X4 L4/V) ()
A3 A2 A1 + A2- A1 + Al + 1
and for one of any number of plates, n
(An An, 1 * A2 + An1l. + A+2 + o. + A2 + 1)(YO) +
(An An,.1. A2)(Xn+i Ln+I/V0)
1 Aa An_ r. A1 + An^l.. AI + *, 4 Al + 1
A material balance around the whole absorber may be written:
(Ln+) (X1 ~ Xn+) = (VO)(YO - Yn) (17)
Replacing X1 by its equivalent from Equation (5) and solving for Yj,
Y + (Ln+tl X+l/VO) (I8)
y, =-.. - - _ ",~- (18)
Al
Combining Equations (16) and (18),
YO = n An 1. An -iAn1 *- - A1 + *.A + Al
nl n An-.. An + An-1 + An + + A + l
- r n^i^X nl r~n-lA *S. A^1. An.'.~,o A > + *-',",t'-A~,-'
VO yA A- A + An-... l + + Ai +
(19)
and the term on the left may be designated the fractional recovery of
component i from the rich gas stream.
Based upon the theoretical analysis of Kremser(63) Brown and
Souders (18) developed a calculation method involving the assumption
that the absorption factor of each component was constant throughout

-10the column. A development yielded the equation
O Yn An+l A
(20)
0O Yn+l An+l 1
where
A was the average absorption factor
Yn+l was the number of monles of a component in equilibrium
with the lean oil per mole of rich gas
n was the number of theoretical plates
YO and Yn were defined as beforeo
Horton and Franklin(55) assumed that an effective absorption
factor might be used for each component, a factor equal to an actual
absorption factor occurring a certain relative distance from the top of
the column. A table was furnished, indicating that the effective factor
of the light components should be evaluated quite close to the bottom of
the column, that of heavy components toward the top, etCo In the shortcut procedure, the further assumptions were made that (I) a constant
percentage "knock-out" occurred on each plate, iaeo
V 1/n V
( /n) = (21)
0 Vm-1
and (2) the temperature difference from plate to plate was proportional
to the amount of absorption occurring
tl -_ _0- Vo Vml /(22)
tl - tn+l " VO Vn

-11=
Edmister(28) devised an effective absorption factor method in
which Ea^ the absorption efficiency of each component, was evaluated
by the expressions
Ae =(A)(An + 1) + 0.25 - 0.5 (23)
A (A1)(An + 1)
A'-' = -. (24)
Yo - Yn Ln+l Xnl. A,. (25)
Harbert(46), in a manner quite similar to the rigorous absorption factor derivation given- above, developed expressions for the
quantity of'a component leaving plate m,. based on the terminal streams
of the column.
(Lx)m = (Am + Am Am+i +.+. + Am Am+l.. An An)- (VY)O +
(Lx),....l..:A,....: A;..... ) ^) +
t. X Wl. Ai A-ln) (l + A1 + A1 Ag + f - + A1 A2... Am.)
1 + A 1 + A A +., + A 2... An
(26)
(Vy)_, = (LX)m (27)
An energy balance around plate m gave the following equation:
Vm-1 HVm. + m+l+l = Vm.Vm + Lm 8)
where B, and HI were molal enthalpies in the liquid and vapor phases.
The following rigorous procedure was suggested: Given the properties
of the inlet streams to the column, (1) assume L/V ratios and temperatures on each plate, (2) calculate K values and absorption factors for
each component on each plate, (5) compute the quantity of each component

-12in each liquid and vapor stream in the column using Equations (26) and (27),
(4) sum the component quantities to obtain new values of Lm and V on each
plate, (5) recalculate tm on each plate from an energy balance around that
plate, and (6) repeat steps (2) through (5) until the desired accuracy is
obtained
C. Plateto-PLate Calculation Methods
The papers of Lewis and Matheson(68) and Thiele and Geddes(89)
are usually credited %with providing the ground-work for continuous, multicomponent,, plate-to-plate distillation and absorption calculationso
Lewis and Matheson considered a distillation column in which feed, overhead, and bottoms streams were specified, the primary unknowns being
reflux ratio and the number of theoretical plates above and below the
feed plateo Thiele and Geddes dealt -with the case where the feed stream
and the number of plates in the enriching and stripping sections were
specified,: the compositions and quantities of the distillate and bottoms
and the reflux ratio being the primary unknowns~ In both methods, the
temperatures throughout the column and the pressure were secondary
unknowns o
The Lewis-Matheson and Thiele-Geddes methods, being concerned
with distillation columns contain some material which has no application
to absorbers. The following features, however, are of interest in both
absorbers and distillation col umns (1) the use of material balances
around a section of the column for relating qcuantities of components
in adjacent or "passing" streams, (2) the use of energy or enthalpy
balances for calculating the qantities or flow rates of streams and L/V
ratios, (5) the use of the dew point calculation for determining the

-13temperature of a vapor stream and the composition of a liquid stream in
equilibrium with it, and (4) the use of the bubble point calculation for
determining the temperature of a liquid stream and the composition of a
vapor stream in equilibrium with it.
The utility of the material balance in both distillation and
absorption calculations is obvious. A typical material balance applied
around the reboiler and several stripping section plates in the bottom-up
calculation of a distillation column (plates numbered from bottom up)
might be,+li = vm,i + bi (29)
where
2m+,i is the flow rate of component i in the liquid
from plate m+l
Vm i is the flow rate of component i in the vapor
from plate m
b, is the flow rate of component i in the bottoms stream.
In the bottom-up calculation of an absorber, the corresponding equation
would be,m+l,i + Voi = ai + i (3~)
where v,i and Il i refer to the rich gas and rich oil streams, respectively.
The liquid and vapor rates and the Vt ratios may not change
appreciably in the enriching and stripping sections of distillation
columns, and it is often permissible to consider them constant. Lewis
and Matheson did allow for the use of energy balances,^ howevere and

-14suggested that a constant change in the quantity of reflux per degree of
temperature rise be used. The flow rates of vapor and liquid in an
absorber do change markedly, even radically, and the use of a constant
L/V ratio is usually a very gross assumptiono
It is in the use or attempted use of dew point and bubble point
calculations for setting temperatures that a fundamental difference between absorbers and distillation columns ariseso In a distillation
column, for a "sharp separation", we usually assume that no component
heavier than the heavy key appears in the overheado Even if there are
one or more distributed components as in a "sloppy separation", the
heavy key is usually not a relatively heavy component; that is, its
volatility relative to that of the lightest component in the column is
not nearly as small as the volatility of absorber oil relative to methane
or nitrogen, for exampleo In the enriching section, the dew point calculation, involving the equality
Z = 1 (31)
is not especially sensitive to the presence of the heavy key, and it is
not until we approach the feed plate that the heavier components of the
feed appear. Even then, they are present in small concentrations and on
only the bottom few plates of the enriching section, and so their effect
on the stability of the temperature calculations is not serious0 A
similar argument holds with respect to the stripping section, the
bottom-up calculations the bubble point calculation,
Kx = 1 (32)
and the presence of components lighter than the light key0

-15b
When one attempts to use bubble point calculations for setting
stream temperatures of absorbers, light components in liquid streams
throughout the column and the sensitivity of the bubble point to light
components make the temperature profile quite unstable, A similar instability in dew point calculations is caused by the presence of heavy
components in the vapor. Even if one assumes that the heaviest component
in the column, the absorber oil, is non-volatile, the heaviest component
of the rich gas may still cause trouble.
Holcomb(54) used a platei-to-plate absorber method which exploited the advantages, but not the disadvantages, of the dew point
calculation. Given the properties of input and output streams and the
column.'pressure, he desired to calculate the number of ideal stages*
The absorber oil was assumed to be non-volatile. Employing the equilibrium relation,
XM n (33)
Xm,i.t-"
which is the basis of the dew point calculation, he computed the mole
fraction of each component except the absorber oil in the liquid from
the top plate, no The mole fraction of the absorber oil was obtained by
difference. Since Xn of the absorber oil, which we may designate as
xnjj represented the total quantity of that component entering the
column, the quantity of stream L was obtained by the equation
Ln =n^, (54)
n xn,j Xn,j
and the quantities of the other components from the equation
2n,i n X ni (35)

-16where i, j. The quantities of all components except absorber oil in
the vapor to the top plate, Vn-l were obtained by material balance,
Vn-l, Vn,i + In,i -,n+l (6)
and the total quantity of the stream as the summation of Vn l,i~ Since
the temperatures and thus the molal enthalpies of streams Ln, Ln, and
Vn were known, the molal enthalpy and then the temperature of Vn1l could
be obtained by an energy balances
Vn H + Ln Ln LnLn+
n-, =.-.L (37)
Vn-l Vnl1
Thus, the complete set of properties of Vn was obtained The above
sequence of steps was repeated down the column, adding the heavier
components at intervals, until a point was reached where the mole fracation of the key component in the vapor stream was about equal to that in
the rich gas. This point was designated the bottom plate. The places
where heavier components had been added to the vapor stream were changed,
based on a comparison of the computed rich gas stream with the known
rich gas, and the procedure was repeated until reasonable values for all
components were obtained. Thus the procedure was not a complete method
for the design of absorbers, but merely a way of checking material and
energy relationships and determining the number of ideal stages furnished
by a given absorber under certain conditions.
D. Machine Computation Methods
A rather large number of machine methods for continuous,
multicomponent distillation and absorption calculations have appeared

-17'
in recent years. A few will be discussed here, the purpose being either
to illustrate machine adaptation of hand methods mentioned earlier or to
show current trends in the art of plate-to-plate computation.
McIntire and Shelton(69) described two fractionator design
programs for use with the Datatron. One, the "product mesh" method,
used to determine the number of ideal stages necessary to make a given
separation, was roughly the LewisoMatheson procedure programmed for
machine use. The other, the "feed mesh" method, used to calculate the
separation achieved by a given column, corresponded to the Thiele-Geddes
procedure. Both methods used Antoine-type equations for K values and
third-order polynomial approximations for enthalpies.
The "feed mesh" method consisted of the following steps:
Given the properties of the feed and the number of ideal stages in each
section of the column, the enthalpy and separation of the feed were
obtained by a flash calculation* A distribution of components in the
product streams was assumed, and some of each component, even if an
infinitesimal amount, was included in each product stream* A reflux
ratio was assumed, and the heat flux in each section of the column was
determined. Starting with the assumed overhead stream, a plate-to-plate
calculation downward through the enriching section was made. Heat and
material balances and equilibrium calculations were made on each step.
The procedure was terminated when the liquid from the feed plate was
reached. A bottom-up calculation through the stripping section was
made, again terminating at the feed plate liquid, A feed plate mesh
ratio was used to provide a revised estimate of the quantity of each

-18component in the product streams,
xrf., (down)
Ri - (UP) (38)
where
Ri was the feed plate mesh ratio of component i
xf i was the mole fraction of component i in the liquid
calculated for the feed plate.
Ri would be less than 1 if too little of component i had been assumed in
the overhead, greater than 1 if too much had been assumed. The mass
fluxes of each light component, 56 were then revised to give bt according
to the equations
6' = Ri 6s i (39)
and
D1 = i, f0 (40)
e,i s,i i
and the mass fluxes of each heavy component according to the equations
6,i (l/Ri) (e,i) (41)
and
so e + +f (42)
s,i e,i i
where the mass flux was defined as the quantity of component i in any
liquid stream minus the quantity in the adjacent or passing vapor, and
6s i was the mass flux of component i in the stripping section
6e, was the mass flux of component i in the enriching section
fi was the quantity of component i in the feed.

-19Using the established heat flux and the revised product estimates, the
plate-to-plate calculations were repeated. The procedure was continued
until convergence was obtained, usually within one or two hours of
machine computation.
Two interesting features of the feed plate mesh method are
(1) the use of quantity ratios for correcting, in effect, the assumed
quantities of components in the product streams and (2) the necessity of
having at least a token quantity of each component in every stream. The
avoidance of discontinuities such as the sudden introduction of a component, however small the quantity of it, in a stream partway through
the column is typical of machine computation methods.
Bonner(l6) presented an IBM 650 program similar to the feed
plate mesh method described above. Relative volatilities, rather than
K values, were used to describe the equilibrium relationships, and
"activity coefficient corrections" were employed to account for the
different volatility characteristics found with aromatic and paraffinic
absorber oils.
Greenstadt, Bard, and Morse(42) wrote a distillation program
for the IBM 704. Interesting facets of the procedure include (1) the
use of the Benedict-Webb-Rubin equation of state(^) for computing K
values, enthalpies, and their derivatives; (2) the formal consideration
of the distillation problem as a series of simultaneous equations; and
(3) the use of the Newton method for forcing convergence of an iterative
procedure. Discussion of points (2) and (3) follows.
The equations describing a distillation or absorption column
can be considered a set of n simultaneous equations in n unknowns:
fi(xl, x xn) = 0, (i 1, 2,..., n) (43)

-20We wish to solve them for the set of roots, the vector xi, satisfying
all the equalities. (The vector xi without the superscript bar denotes
a set of values which occur during the iterative solution and still
contain errors.) According to the Newton approximation,
c3f
fi(x) = fi(x) + ZE ( ) (j - Xj) (44)
J
Setting fi(x) equal to zero and denoting (xj - xj) by bxj, we obtain
ZJ (E)(8X:) - fi(x) (45)
Equation (45) is only an approximation, but it may be iterated in order
to obtain as close an approximation as is desired.
We may consider certain of the variables as final variables
(superscript F)
fF (xl) = 0 (46)
Others may be considered initial variables, xok In a distillation
column, initial variables might be the quantities of each component
assumed in the product streams; the final variables, the quantities
calculated for the feed plate by the enriching and stripping section
calculations; and the final functions, the requirements that the two
final quantities of each component be equalo Similar functions may be
found in absorption columns.
I
Given a set of initial variables, xk, we may, by solving a
F
series of equations, arrive at values of the final variables, xi, so
that the final variables are functions (ultimately, but not explicitly)
of the initial variables, denoted functionally
F x 4 (xi) (47)

-21and further
fF [xF (x) ] = 0 (48)
To employ the Newton approximation, we must solve
6F
Z (|^) (6Xk) = -f (x) (49)
k 6fjIk)
The partial derivatives may be obtained by the so-qalled "chain rule"
for partial derivatives,
IfF afF aXF
(50)
ak ci aF a I
and the problem reduces to that of finding the derivatives of the final
variables with respect to the initial variables.
In a plate-to-plate calculation, we might assume the properties
of the bottoms stream and, by the usual mass and energy balances and
equilibrium relations, obtain the properties of the streams above the
first or bottom plate of the column. We write the relationships
fj (Xi xk) = 0 (51)
and solve them iteratively by Newton's method,
m ( ) f( ) - - (X1> xI) (52)
m
Equation (51) may be differentiated,
ar l ax1 afl
and having obtained the values of xm, we may evaluate the partial
derivatives of fr with respect to x and x. at the correct values of xm

-22and then solve for the set -F Proceeding to plate 2, we write the
k
equations
f 2 (x2, xm, x) 0 (54)
solve them iteratively as before for the values x2, and then differentiate Equation (54) to obtain
af2 6x2 af2 1x af2
c~rax2 ()(I),+ c ( ~~ )( ~XzIk + 0 (55)
r (x2) (X) + ax) (ax7 ~ (5x)
r k m k k
and again we may solve for the set of partial derivatives ~,
A F
dxi
Proceeding tray-to-tray, we ultimately arrive at values of axI and f(xk)
and are able to solve Equation (49) for the 6Xk, obtain new approximaI
tions to the set of initial variables Xk, and finally obtain the values
-I -l -2
Xk, xm' xr etc., satisfying the complete set of equations.
Using the above method, Greenstadt and associates have obtained
the solution of a four-component, twenty-plate column in about three
minutes (41)
Amundson and Pontinen(7) recently described a distillation
method developed on the Univac Scientific Computer Model 1103. If
properties of the feed and the number of ideal stages in both sections
of the column are known, the flow rates and compositions of bottoms and
overhead may be estimated. Constant molal overflow may be assumed and
later corrected by an energy balance, With plates in the stripping
section numbered from the bottom up, a material balance around the
reboiler (plate O) and the bottom m plates gives
Lm+l = Vm + B (56)

-23where B is the bottoms product flow rate. A component balance around
plate m is
Vm-1 Ym-Li - Lm Xmi Vm Ymi + Lm+l Xm+l,i = 0 (57)
Employing the equilibrium relation
Ym,i = Km,i Xmi (58)
and Equation (56), we obtain
(Vm-l)(Kl,i m,i) (Vm.l + B)(xmi) - (Vm)(Kmi Xmi) +
i(m + B)(xm+,i) = (59)
or, collecting terms,
(Vm_1 Kml,i)(xm_l,i) -(Vml + B- Vm Km,i)(Xmi)+
(Vm + B)(Xm+,1i = 0 (60)
We now have a component material balance around plate m in terms of the
liquid mole fractions on plates m-l, m, and m+l. If there are n plates
plus reboiler and j components, a system of (n+l)(j) equations may be
written, the equations for the top plate, general enriching section
plate, feed plate, and bottom plate differing only slightly from the
one given. If the total stream flow rates and temperatures are known
(Kmi assumed to be a function only of temperature and the constant
pressure), the system of equations may be solved for the mole fractions,
Xm i. By a somewhat similar procedure, a system of m heat balance
equations in terms of the variables Vm may be derived,
The author8t method of determining the temperature profile and
flow rate profiles necessary to produce the equality
fm = 4 (Xm,i) = i (Km,i Xm,i) = 1 (61)

-24on each plate is of interest, They began by assuming a temperature on
each plate m of tm. Computing the K values by power series approximations
Kmi = +i + Pi tm + 7i t bi t (62)
and solving the set of simultaneous equations produced a set of values
of x and a value of the function fm on each plate. The values of xm,i
in general did not sum to 1 on each plate. They were "normalized" by
dividing each by the sum. Then the partial derivative of fm with
respect to tm was computed
(; m Z)t Xmi[i + 2Yi t + 35i(tm1)2] (63)
and a new value of tm was computed by the Newton approximation
tm t+ 1 - fm(t )] [m (64)
In a typical 15-plate column, the authors alternated re-estimation
of temperatures with recalculationof vapor rates and found 15 iterations
to be sufficient for convergence.
Rose, Sweeny, and Schrodt(83) discussed a relaxation method
for continuous distillation suitable for intermediate-size computers.
Given feed conditions, flow rates of product streams, number of plates,
and reflux ratio, the product compositions were calculated using certain
simplifying assumptions and a material balance equation for the holdup
of each component on each plate. The assumptions (which may be replaced
by rigorous calculation) were (1) constant molal overflow, (2) ideal
stages, (5) negligible vapor holdup, and (4) the composition of liquid
holdup on plate m being identical to the composition of stream Lme The

-25"
holdup equation for any component on plate m was
(A (m)k+l = (A) (x)k + (Vml Ym-l)k + (Lm+l Xm+l)k
(Vm Ym)k- (L Xm)k (65)
where
A was the total holdup on plate m
k and k+l were successive time intervals
V and L were in the units of moles/time interval,
Each equation was solved for (xm)k+l,
(Xm)k+ = (Xm)k +_ [(Vm-l Ym-l)k + (Lm+l Xm+l)k
(Vm Ym)k (LmXm)k] (66)
and the process was repeated for successive time intervals. It was
found necessary to make the time interval short enough that the quantity
of flow during the interval was one-fifth to one-tenth of the total
holdup.
Continuing now to the category of absorber calculation by
digital computer, Mieth and Moore(70) reported using the Edmister
effective absorption factor method(28) on a medium size computer. This
program was part of a larger program, the object being the design of an
entire absorption plant.
Norman(76) programmed a modification of the Harbert method(46)
for the IBM 650, The program used second-order polynomial approximations
for K values and enthalpies. Program specifications included the number
of equilibrium stages, the complete properties of inlet streams, heat

-26loss, and allowable errors in L/V ratios and enthalpy balances. The
supplying of initial approximations of L/V ratios and the temperature
profile was optional; the program generated its own initial values if
none were furnished. Beginning with these approximations, the machine
computed K values and absorption factors for each component on every
plate. Using Equations (26) and (27), the quantity of each component
in every liquid and vapor stream was computed, the quantities were summed
to give the total liquid and vapor streams from each plate, and the L/V
ratios were re-computed. These new L/V ratios were compared with the
previous values, and if the maximum difference exceeded the allowable
error, the procedure was repeated until all L/V ratios were within the
bound. Having converged on a mass balance for a given temperature distribution, the machine then computed the enthalpy balance error on each
plate, picked the largest one, and compared it to the maximum allowable
error. If the error was too large, the temperature on that one plate
was revised by the formula
(tm)k+l (tm)k - Em/F (67)
where
(tm)k+l was the temperature on plate m for iteration k+l
(tm)k was the temperature used in iteration k
Em was the enthalpy balance error on plate m
F was an empirical correlation parameter.
The mass balance convergence and enthalpy balance were then repeated
until final convergence was obtained. Machine computation time varied
from 15 minutes to two hours depending upon the complexity of the
problem.

-27An absorber program using a method similar to that of Holcomb(54)
was discussed by Cobb and McIntir.e(22) The machine calculated the
recovery obtained with specified input streams and number of plates.

III. ENGINEERING CONCEPTS
A. The Non-Ideal Stage
The character of the non-ideal or non-equilibrium vapor-liquid
stage contact may be explained by comparison with the ideal or equilibrium
stage. The term "equilibrium stage" implies that equilibrium is reached
with respect to both mass and heat transfer. In the non-ideal stage,
neither of these conditions need be the case.
In the ideal stage (1) Vm is a dew point vapor, (2) Lm is a
bubble point liquid, (3) the composition of Vm is related to that of Lm
by the expression
Ym,i = im, xm,,i (3)
(4) the temperatures of Vm and Lm are the same and e'qual to the temperature at which Km,i is evaluated, (5) the compositions of the liquid on
plate m and the liquid leaving plate m are the same, and a similar statement holds for the vapor, and (6) time is not a factor in any of the mass
or heat transfer relationships. In the non-ideal stage, (1) Vm is at
its dew point temperature or higher, (2) Lm is at its bubble point
temperature or lower, (3) Vm and Lm are not necessarily in equilibrium
with respect to mass transfer, but their compositions may be related by
expressions based on mass transfer rates, (4) the temperatures of Vm
and Lm are not necessarily the same, but they may be related by expressions
based on heat transfer rates, and (5) differences in both vapor and liquid
compositions at various points in the zone of vapor-liquid contact may
need to be considered.
It will be shown in the following sections that the non-ideal
stage is more complicated than might be supposed, and that attempts to
-28

-29
use existing efficiency concepts according to their original definitions
have impeded, rather than aided, the present investigation.
B. Mass Transfer Relationships
The Problem
For purposes of discussion, let us consider the following
problem: We are performing a plate-to-plate calculation upward through
a multicomponent, non-ideal stage gas absorber. Pressure is constant
and known. There is no heat gain or loss from the column. Having arrived
at plate m, we have complete information on streams Vm-1 and Lm; i.e.,
composition, temperature, and flow rate; and desire to calculate the
properties of streams Vm and Lm+l. We wish to avoid simplifying
assumptions as much as possible.
Material and energy balances around plate m may be made:
~m+li + Vm-l i = m,i + Vm,i (68)
HI l + HVml HL + HVm (69)
(H without underlining designates the total enthalpy of a stream) and
by rearrangement,
2m+l,i = m,i + (mi- Vm-l,i) mt + i (70)
HLm HVm + (HLm - HVml) =HV + q (71)
Thus we may relate the adjacent or "passing" streams above plate m by
the same "5" quantities that apply to the streams below the plate. The
problem of relating streams Vm and Lm remains.

-30O
The Murphree Plate Efficiency
The classic equation for specifying the degree to which
conditions on an actual plate approach those of an equilibrium plate
was proposed by Murphree(71) and is called, when based on the vapor
phase, the Murphree vapor plate efficiency. For component i on plate m,
E MV, i Ym-l,i (72
or in words, the change in mole fraction of component i which actually
takes place in the vapor stream as it rises through the froth on plate
m, divided by the change which would take place if plate m were an
equilibrium contact. (The term "froth" is defined as the volume of
continuous liquid phase containing entrained vapor as distinguished from
the continuous vapor phase with entrained liquid which exists just above
the froth.) Equation (72) is written for a complete plate and applies
to concentrations in the average or completely mixed streams. If the
expression is applied to a particular location on a plate, it is called
the Murphree point efficiency, Ep. Various authors(34'67) have proposed
methods of estimating the effect of liquid mixing on a plate and thus
connecting EMV and Ep
Returning to our problem and recalling that we wish to compute
the properties of stream Vm, knowing those of Vm-l and Lm, we may solve
Equation (72) for ym,i
Ym,i = Yml,i EMV (Y* mi Ym-l,i)=
Sm-l,i + EM,i (9mi Xmi " Ym-1,i) (73)

-31where Km,i is the equilibrium ratio (Ym i/xm,i). Assuming that the
EMV i are either constant and known or may be estimated for each plate,
Ym i for each component may be calculated. Now only fortuitously will
the Ym,i sum to 1, and several methods of adjusting the Ymi to make
them sum to 1 have been tried:
(1) An additional variable, tm, the temperature at which K
values were evaluated, was introduced. Changing tm, and thus the K,i,
caused the sum of the ym,i to equal 1, but the difference between tm
and either tLm or tVm was often many times the temperature variation
which might be. expected to occur on any plate. Furthermore, the use of
this adjustment introduced computational instability, and so it was
abandoned.
(2) The adjustment
m,i
Y m, = (74)
where the yi were obtained from Equation (75) was employed. Again,
computational instability prevented the obtaining of a solution.
(3) A method was tried which involved adjusting the Km,iy
in effect, in such a way that constant relative volatility among the
components was preserved. Equation (73)may be written and then summed,
term by term, for all components to give
IYmi = IYm-li + 4i(EMv,i Kmi Xm,i)- Z (EMV,i m-li) (75)
The consequence of both Ziy i and IYmli being equal to 1 is that
Zi (EMVi Km,i Xm,i) = Z(EMVi ym-l,i) (76)

-32s
The products (EMV i Km,i Xm,i)t (before adjustment) were then "normalized"
to produce the equality (76) by the equation
(EMV,i Km,i Xmi)L E(EMV,i Ym-l,i)
E K x -
MV!,i Mi M'i ) 4(EMV,i Km,i Xmi) ()
and then Equation (73) was used to calculate the Ym,io Method (3), a
direct calculation, was much faster than method (1), an iterative procedure. Furthermore, it produced a convergent sequence in many cases.
(4) The necessity of adjusting the ym,i produced by Equation
(73) led to the conclusion that only ill of the set of EMVi may be set
independently. Consider the binary mixture of components a and b. Any
mole fraction of component b will always be equal to 1 minus the mole
fraction of component a in that mixture. Then
(1 Yma) (1 Ym.l,a) Ym,a - Ym-l,a
MV,.b = (1 Ya) Ymla CI Yma ) Ym-l a - MVa (78)
The interdependence, but not necessarily equality, of the EMVi for
the various components of a multicomponent mixture may be similarly
demonstrated. A fourth method of producing a set of m i summing to 1
involved computing ym,i of the major component, methane, by difference
and in effect adjusting EV of methane to produce the desired result,
This method also resulted in convergence in some cases.
Having computed the composition of stream Vm, we may estimate
its temperature with an equation similar to the plate efficiency expressions. For the present, let us state it in general terms
tVm = f(tVml, tL, Eq) (79)

-33where Eq is some measure of the efficiency of heat transfer. The problem
of determining the quantity of Vm then remains. Several methods have
been tried, each of them involving the postulation of a step-wise series
of conditions through which Vm_1 and Lm+^ might pass, ultimately emerging
as Vm and Lm. In all cases where convergence has been obtained using
the methods of determining compositions and quantities just discussed,
the result has been the calculation of negative quantities of the lighter
components in the liquid streams. It is for this reason that the use of
the Murphree plate efficiency has proved unsatisfactory for absorbers,
Examination of the efficiency expression and the conditions occurring in
gas absorbers will furnish more conclusive evidence.
According to the definition of the Murphree vapor plate
efficiency,
E Ymi - Ym-l,i (72)
MV.i y, Y
if net absorption of component i is to take place on plate m, the denominator will be negative. If EM i is to be positive, the numerator must
be negative and the mole fraction of component i must decrease as the
vapor stream passes through the plate. Yet it has been found, both from
machine computation and experimental results, (e.g., the Phillips absorber
data - see Results) that the lightest components of a vapor stream may
undergo absorption as the vapor passes through the upper stages of the
column, yet increase in mole fraction because of the more rapid absorption
of heavier components and consequent decrease of the quantity of the
vapor stream. This would produce a negative plate efficiency, which
is contrary to the general use of the word "efficiency", the existing

-34r
methods of estimating plate efficiency, and the supposed intention of
Murphree, who derived his efficiency expression for an "isothermal"
column with negligible change in the quantity of the vapor streamo
The conclusion reached by the foregoing experience is that the
Murphree plate efficiency (1) may be useful in the analysis of binary
distillation or the case of a carrier gas and one absorbed component,
and (2) may be a convenient index of the behavior of a key component in
a multicomponent mixture, but it is an oversimplification for multicomponent systems generally,
An Expression Based on Mass Transfer Coefficients
Continuing with the problem of relating the properties of Vm,
the vapor leaving plate m, with those of Lom the liquid leaving the plate,
we may write an expression for the rate of mass transfer of a component
from the body of the vapor phase to a vapor-liquid interface in the
froth layer and thence to the body of the liquid phase. Considering one
component and dropping the subscript i,
dN - d(Vy) dv kGa(fG f)Acsdz kLa(fI fL)Acsdz (80)
where
N is the number of moles/sec of the component transferred
kG is the vapor phase mass transfer coefficient in
moles/sec ft2 atm
kL is the liquid phase mass transfer coefficient in the
same units
fG, fI, and fL are the fugacities of the transferred component
in the body of the vapor, at the vapor-liquid interface,
and in the body of the liquid. The unit is atm.
a is the surface area per unit volume of the froth, ft2/ft3
Acs is the horizontal crossesectional area of the froth, ft2
z is vertical distance in the froth, ftn

Implicit in the above expression is the assumption that there is no resistance to mass transfer at the interface, If we make the usual assumptions
that fugacities are proportional to concentrations and (to justify: use of
the overall coefficient, KOG) that the operating lines (concentration
gradients) are straight lines, we have
- dv kGa(y - YI)pACdz i kLa(fI - fL)Asdz
KOGa(y - y*)pAcdz (81)
where
KOG is the overall mass transfer coefficient based on
the vapor phase
y* is the mole fraction in the vapor which would be in
equilibrium with the mole fraction in the liquid,
Assuming that the liquid phase is completely mixed, that the vapor
phase is uniform in any horizontal section, and that y* Kx, we have
dvy KoGapAc dz (82)
- Kx
Dividing both sides by'V and integrating,
f dv f KO csd KoGaPPAc dz (8)
V(y Kx) V c
Since Equation (83) is really i such equations, one for each
component, and V is the sum of the v of each component, Equation (83)
must- be integrated numerically, taking increments, z sufficiently small
and re.coaputing V for each increment. Such a process would be quite
tedious and would not be justified by the accuracy with which KOQ is
estimated, Instead, let us use a constant vapor rate V defined as the
value of V which, when considered constant,s will result in the same value
of the integral as would be obtained if the integration were done in

-36small increments with V a variable. Note that this is not equivalent
to considering the vapor rate to be a constant. With this approximation,
d -v a dv KOGapAcsz
-' (Vy - "VKx)' (v..- VKx) - NOG (84)
NOG is the number of overall gas transfer units on the plate as proposed
OG
by Chilton and Colburn(20). One further approximation should be noted
in passing, that of letting Vy equal v, when actually Vy = v. Integrating
the left side of Equation (84) through the froth layer,
m dv Vm - VKxm
-fl_ vm —1r -.V — ] - NOG (85)
- v VKxV vml- VKx NOG
and
vm VKx mOG
Vm_1 - VKxm
- _ e (86)
Solving for vm,
-NOG
vm = VKxm + eNOG(vm Km)
v vml + V-Kxm + e vOGm eoG VKxm v_
= Vm_1 + (1 e-NOG)(VKxm vm) (87)
The expression (1 e O) will be recognized as the formula
for Ep derived for the case of a carrier gas and one absorbed component,
with suitable simplifying assumptions. Equation (87) should not be
considered the result of multiplying the equation
Ym = Ym-1 + Ep (Kxm - ym) (88)
by V since V occurs only in the term VKxm and is not a common factor
of vm and vml.

-37In using Equation (87), one might first let V be equal to Vm1,
compute Vm as the sum of the Vm, replace V by some sort of average of
Vm.1 and the first approximation of Vm, recompute the values of the vm
and of Vm, etc. This procedure was found to be undesirable since (1) it
resulted in a much longer calculation than the direct computation of vm,
and (2) the sequence of values of vm sometimes became non-convergent.
Instead, V was set equal to Vmnl, and it was found that the value of Vm
produced by the direct calculation did not differ appreciably from that
produced by the iterative procedure mentioned above. The reason for this
small difference is that for light components, where the K values and
therefore the terms VKxm are large, the factor (1 - e ) is small;
for the heavy components, where (1 - e OG) is appreciable, K and VKXm
are small.
One final alteration may be made. The various methods of
accounting for the effect of liquid phase concentration gradients and
relating Ep and EMV are compatible with the derivation above. We may,
therefore, relax the restriction of constant liquid composition and
include the slight increase of mass transfer this affords. The final
equation is
vm = Vm.1 + (EMV)(Vm.lKXm- vml)
Vm-l [Ymi_ + (EMV)(KXm - m-l)] (89)
and the mass transfer factor is given the name EV for convenience and in
recognition of the way in which its values are predicted, but not because of being used as Murphree defined it.

-38C. The Mass Transfer-Heat Transfer Analogy
The subject of heat transfer in vapor-liquid stage contact
has received very little attention by comparison with mass transfer.
Nord(75) restricting his discussion to the case of equimolar counterdiffusion in a binary system -with vapor phase resistance controlling,
showed how the Colburn "j" factor analogy(9) led to a temperature
efficiency expression. Kirschbaum(60) discussed temperature differences
and heat transfer in a distillation column; his model, however, was
restricted to a binary system, ideal stages, and constant liquid-tovapor ratio. The fact remains, as stated by Curtiss and Hirschfelder(25)
that the effect of mass transfer on heat transfer is not well-known.
One may postulate the release of heat at a vapor-liquid interface caused
by absorption of material and even estimate the quantity of heat involved.
The disposition of this heat and its effect on the convective transfer
of heat are not known.
A Consequence of the Colburn Analogy
Colburn(l9) defined "j factors" for heat and mass transfer
as follows -
f h )( ( h )(p)2/3 (90)
5 - )27 -)(Pr (90)
2 cpu k CPU -
^i-^ )273'^ 27K s)/
f a) () )2/3 (p)(Sc)2/3 (91)
JD f'x D) p 1
where
jH is the j factor for heat transfer
JD is the j factor for mass transfer (diffusion)
f is the Fanning friction factor

-39h is the heat transfer coefficient
c is specific heat
p is density
u is linear velocity
is viscosity
k is thermal conductivity
K is the individual phase mass transfer coefficient
p is pressure
D is the diffusion coefficient or diffusivity
Pr is the dimensionless Prandtl number
Sc is the dimensionless Schmidt number
The equality of jH and jD has been found to hold even when they are not
mutually equal to f/2. Equating jH and jD on the basis of the vapor
phase and making the very important assumption that the Colburn analogy
may be applied to overall coefficients in a two-phase system,
h _ KOG Sc 2/3 (KOG )(Le)2/3
= Xo ( )25 ( ~'( -v Leu/ (92)
Cp PV u PV Pr PV u
where h and KOG are now the overall heat and mass transfer coefficients
based upon the vapor phase. The ratio of the Schmidt to the Prandtl
numbers is sometimes called the Lewis number. Since
V
PV u A (93)
Acs
then
h Acs KOG P Acs(Le2/3 (94)
c v=...- (9)
p ~ ~ - (')/

40o
and multiplying both sides by az,
h Acs a z KOG a p Acs z )2 (5
--— V.. (Le)2/ (NOG)(Le)2/' (95)
p
h As a z
Equation (95) suggests that the quantity (- V- ) might be given the
symbol NGH, the number of heat transfer units based on the vapor phase,
and that it might be related to NOG by the Lewis number to the arbitrary
2/3 power. We shall now see that the symbol NGH is appropriate.
The Heat Transfer Efficiency
If the vapor passes through a liquid phase of constant temperature and if we consider heat transfer without reference to mass transfer taking place, then in a form similar to the differential equation
for mass transfer,
dQ h a Acs (tL - tV) dz
C V dtV (96)
where Q is the heat flow rateo Rearranging,
dtV h a Acs dz
tV tL c V(97)
Integrating,
tV tm^ h a Acs z
n[ tV - tLm c ~ NGH (98)
and
tVm tLm + (eNGH)(tVm, - tLm)
- tVl + (L - eNGH)(tL tvm ) (99)
and the factor (1 eNGH) might be called the heat transfer efficiency,
Egq

-41In view of Equation (95), a reasonable way of evaluating Eq for
a multicomponent system might be by way of NOG:
NGH = (Yi NOG,)(Le)i (100)
However, since (1) EMV is readily available, (2) the process of computing logarithms and exponentials is relatively slow, and (3) EMV of
the lighter components (which form the bulk of the vapor phase) does not
differ greatly from NOG because both are in the range 0.001 to 0.100,
nor are the Lewis numbers in the vapor phase much different from 1, the
following formula has been selected:
Eq = C R Z(yi EMV,i)(Le)2i/ (101)
where Cq is an empirical factor necessitated by the fact that this particular use of the Colburn analogy is not exact.
D. Prediction of the Murphree Plate Efficiency
It should be emphasized that there is presently no proven
method of predicting Murphree plate efficiencies for hydrocarbon mixtures.
Methods have been proposed by Bakowski(9, Chu(2), Geddes5), and
Walter and Sherwood(90), but none has found general acceptance.
In the present investigation, the Murphree vapor plate efficiency, defined
Ym- Ym-1
E ym - ym (72)
MV Y m- Ym-1
was computed as a function of the point efficiency, Ep, itself a function
of NOG, the number of transfer units based on the vapor phase. NOG was
in turn computed from the separate resistances of the vapor and liquid
phases, and these were estimated from data obtained with other systems

-42in laboratory-size apparatus. The charge may be made that data extrapolated this far are of little value and no defense can be made except
to'point out the following facts (l) the correlation to be discussed
was based on physical properties believed to be the basic ones in mass
transfer operations, and (2) the basic purpose of the investigation was,
after all, to make the fullest possible use of present data and to show
where the need for further knowledge exists.
EMV Based on Separate Phase Transfer Units
The Murphree efficiency, as originally intended, applied only
to a "point", that is, to a particular location on a plate where a vapor
stream, whose composition does not vary in any horizontal direction,
flows up through a liquid of unvarying compositiono The plate efficiency, EMV, which is applied to the completely mixed vapor and liquid
streams before and after their contact on the plate, is thus seen to
refer to an "integration" of changes which occur in the various parts
of the vapor stream as they contact liquid whose composition varies
both horizontally and vertically on the plate. The integration may be
expressed by computing a "point efficiency", Ep, based on the average
conditions existing on the plate and correlating the plate efficiency
with the point efficiency and other pertinent variables.
Gautreaux and OConnell have proposed such a correlation(34):
L EEpmVn
EMV mV[ n ( L + n ) 1] (102)
where m is the slope of the equilibrium curve, m e dys/dx, and n is
the number of stages in series

-43In this case, the stages are imaginary "mixing pools" across the width
of the plate. A graphical correlation of n versus x, the length of the
liquid path in feet, has been fitted by the equation
n = 1 + 0.7 x (103)
Further, in multicomponent systems the slope of the equilibrium curve
becomes a rather nebulous concept and, for small mole fractions, the
slope of a chord from the origin to a point on the equilibrium curve,
which is of course the K value, y*/x, may be used as an approximation
to the slope of a tangent to the curve, dy*/dx.
Gerster, Colburn, et al(38) have related Ep to NOG, the number
of overall mass transfer units based upon the vapor phase, by the
equation
Ep 1 - NOG (104)
and NOG is in turn related to NG and NL, the number of mass transfer
units in the vapor and liquid phases, respectively.
1N +^ (+ )(1 + (K( 1) (105)
NOG - NG 1 NL NG L N
The substitution of K for m is used as in Equation (102).
The Effect of System Properties
The AoI.Ch.E. Committee on Tray Efficiencies in Distillation
Columns(l) indicates that N, the number of transfer units in the liquid
or vapor phase of system 2 may be related to N in the corresponding
phase of system 1 by the equation
[ (S )l 12 (106)

i44under similar conditions of vapor and liquid rates and in columns of
similar construction~ Sc, equal to (I/pD), is the dimensionless Schmidt
number, and,iy p, and D are the absolute viscosity, density, and diffusion
coefficient, respectively, all in consistent units, The Committee also
states, however, that the effect of density might be represented better
by an exponent of 0.13 rather than 05 and that the effect of viscosity
is supposed to be small, though appreciable(5)= Equation (106) may also
be used to correlate values of kG and kL, the mass transfer coefficients,
and Gerster(37) has suggested that for mass transfer coefficients, the
following equation serves equally wello
-kG (kG)2 [ l/2 (107)
The following relationships were finally chosen for the
present correlation
(NL) -I (CNL)l [e ]/ (108)
where system 1 was the air-water-carbon dioxide system with (Sc), of 550
under the conditions at which the correlation of (NG), was made, and
(NG)2 (NG)1 [()2]1/2 (109)
(DG)l
where subscript 1 referred to the air-water-ammonia system with (DG)1
of 096 ft2/hr or 0.000267 ft2/sec,
The Effect of Operating Variables
In the air-water-ammonia system, NG has been correlated graphically against 9GG the vapor residence time in seconds, with u, the
linear vapor velocity in ft/secas a parameter(2)0 The graphs are

-45straight lines through the origin and the slope of each line is a simple
function of u. The same applies to NL as a function of'LI the liquid
residence time in seconds, and u in the air-water-carbon dioxide system.
Algebraic correlations were developed:
(NG)l = (+.8)G (110)
and
= 0.8 u
(NL)l ( + 8 )L (111)
The vapor residence time is the quotient of the quantity of
vapor in the froth layer and the vapor rate. The liquid residence time
involves the quantity of liquid in the froth layer and the liquid rate.
Z Acs P PL
9L " --- ^ --- I(112)
z Acs ( - p) PV
%G = (113)
where
p is the "froth density", actually the volume fraction of
liquid in the froth
~L and 0G are the liquid and vapor phase residence times.
Correlations of froth height and froth density in the actual
columns operating with hydrocarbon streams were required. They were not
available, and so data from other systems were used. In the air-waterammonia system, a curve fit of graphical data(6) gave
z = w + 0.14 + 0.0071 F (114)
where
w is the weir height in ft
F is the quantity (u)(pV)1/2 and has the dimensions
(ft/sec)(lb/ft)1l/2/

-46In the air-water system(4),
0,20 F + 0.20 (5
('115)
F + 0.20
Correlations of the form of Equations (110), (111), and (115)
were chosen in preference to functions involving exponentials or fractional powers for two reasons: (1) their values are bounded for large
and small values of the independent variable and are thus not able to
give grossly unreasonable values, and (2) the expressions may be evaluated on a digital computer much more quickly than those containing
logarithms or exponentialso
The Plate Efficiency Correlation
The equations given in the previous paragraphs made possible
a correlation of EMV based upon characteristics of the column and
properties of the vapor and liquid streams being contacted on a particular plate. The correlation was developed in the following way4
From Equations (110), (113), (114), and (115),
(w + 0.14 + 0.0071 F)(Acs)(0.80 F)(PV)
Q = (116)
G (V)(F + 0.20)
(20 u)(w + 0.14 + 0.0071 F)(Acs)(F)(PV)
(N G) (u + O,8)(V)(F + 0.20)
From Equations (111), (112), (114), and (115),
(w + 0.14 + 0.0071 F)(Acs)(0.20)(F + 1)(PL)(
@ - - (L) + (F+(1120) )
NL) (L)(F + 203)(L)(F + - 20) (
(ONl u)( + o1+007 )(c)F+I(L

-47From Equations (105), (108), (109), (117), and (119),
1 1 rirJ-V I 1 [ [(DG)1]1/2 1KV 1 (Sc)2 1/2
NOG (NG)2' Er- I(NL ) 2 [(NG)I [(D G)2 + I[TL ][(NL)lI(Sc)l
(u + 0.8)(V)(F + 0.20)(0.01635)
= (20 u)(w + 0.14 + 0.0071 F)(Acs)(F)(pv)(DG)/2
(K) (V) (u + 3) (L) (F + 0.20) (Sc)//2
(L)(0.16 u)(w + 0.14 + 0.0071 F)(Acs)(F + 1)(pL)(23.5)
(u + 0.80)(v)(F + 0.20)
=12 +
(1223 u)(w + 0.14 + 0.0071 F)(Acs)(F)(pv)(DG)-/2
(K)(V)(u + 3)(F + 0.20) (Sc)2/
(3*76 u)(w + 0.14 + 0.0071 F)(ACs)(F + 1)(p)
= FN/(DG)/2 + (FNL)(K)(Sc)l/2 (120)
En and EMV were then calculated from Equations (104) and (102).
The factors FNV and FNL were common to all components on a
particular plate. The quantities (DG)2 and (Sc)2 varied for each
component. The quantities FNV/(DG)1/2 and (FNL)(K)(Sc)1/2 are sometimes called the vapor and liquid phase resistances.
Comparison of Predicted Values with Test Data
The sample calculation found in Table 1 occurred in a computation based on a set of test data designated as column B. Data with
which to compare the estimated efficiencies are quite scarce. Walter
and Sherwood(90) submit the following data for hydrocarbons in bubble
cap absorbers:

-48TABLE 1
SAMPLE CALCULATION OF MURPHREE PLATE EFFICIENCY
Vapor
ResistComponent xm Ym-1 K DG ance DL
Nitrogen 0o0010 0o0410 18,95 35502 x 10-6 1.754 4.797 x 10-8
Methane 0 1164 o O8026 4.900 2.922 1 920 4.272
Ethane 0,0531 0 0723 1,167 2.914 1.922 35413
Propane 0.o800o 00597 0.4489 2 466 2.090 2.813
Isobutane 0.0158 0.0059 0.2345 2.092 2.269 2.418
n-Butane 0 0440 000135 o.1804 2o115 2.257 2.435
Isopentane 0O0095 0.0016 0o09266 1.889 2.388 2.164
n-Pentane 0o0112 0.0014 0.07408 1.886 2.390 2.155
Hexane 0 0074 ooo0004 003203 1.o726 2.498 1.942
Heptane 00007 0o0016 0.01292 1.6o6 2.590 1.775
Octane 0.0011 0. O00511 1.500 2.679 1.640
Nonane o.o018 - 0.00215 lo414 2o760 1.524
Decane 0.0200 - ooo00096 1.339 2,836 1,425
Abs.Oil 1 oo1084 - ooo0008 1.251 2.934 1,312
Abs.Oil 2 05196 - ooo00006 1.108 3o117 1.131
t = 114~F u - 0.4470 ft/sec
p = 740 psia F = 0.7352
L = 0.3790 lb moles/sec pL = 45.66 lb/ft3
V = 0.9324 lb moles/sec w = 0.25 ft
Acs = 15.708 ft2 z 0. 395 ft
pT = 2o705 lb/ft35 p - 0.3711

-49TABLE 1
SAMPLE CALCULATION OF MURPEREE PLATE EFFICIENCY (CONT'D)
Liquid
ResistComponent (Sc)L ance NOG Ep EMV
Nitrogen 247.0 156.20. 00633 oo. 0063 0.0070
Methane 277.4 42.79 0.02237 0.0221 0.0244
Ethane 347.2 11.4o 0.07505 0.0723 0.0780
Propane 421.2 4.831 0,.1445 0,1345 0.1421
Isobutane 490.1 2.722 0.2004 o.1816 0.1887
n-Butane 486.7 2.087 0.2302 0.2056 0.2127
Isopentane 547.7 1,137 0.2837 0.2470 0.2522
n-Pentane 549.9, 0.911 0.3030 0.2614 0.2660
Hlexane 610.3 0o.41 00.3433 0.2906 0.2930
Heptane 667.6, 0.175 0.3617 0.3035 0. 3046
Octane 722.7 0.072 0.3634 0.3047 0.3052
Nonane 777.4 0.031 0.3583 0.3011 0.3013
Decane 831.3 0,014 0.3508 0.2959 0.2960
Abs.Oil 1 902.9 0.006 0. 3402 0.2884 0.2884
Abs.Oil 2 1047.3 0.001 0.3207 0.2743 0.2742
= 0.557 sec
0L- == 1.926 sec
FNV = 0.00282
FNL = 0.5243
= 0.8051 cp
x = 4.00 ft (liquid path)
n 3.-8 mixing stages

-50EMV of isobutane in heavy naphtha 36' %
EMV of isobutane in gas oil 17
EMV of isobutane in gas oil and lube oil 9 - 10
EMV of propane in heavy naphtha 22 - 24
EMV of propane in gas oil 11 - 13
EMV of propane in gas oil and lube oil 5 - 11
E. Enthal Procedures
Estimation of the enthalpy of vapor and liquid hydrocarbon
mixtures has been the subject of many publications in recent
years(10,3177,82) Each method involves certain assumptions and
approximations, and no one method can be considered best for all
purposeso In the present investigation, methods were evolved which
would optimize a function of rigor, accuracy, machine computation time,
and machine storage
The datum point for the enthalpy procedures was the ideal gas
state at 0~Ro Enthalpy was evaluated by (1) postulating a series of
"steps" or changes of the variables temperature and pressure, each of
which could be characterized thermodynamically, and (2) computing the
enthalpy change associated with each step. Reference to Figures 2 and 3
will clarify description of the various steps, and the detailed evaluation of the enthalpy changes is covered in the section on physical
properties. Sample calculations are given in Table 2.

-51I i
I I
2- PHASE
>a. I REGION
Hf| 1 3
z I
77~F
TEMPERATURE
Figure 2. Enthalpy Evaluation Steps

-52
Vapor Enthalpy
The enthalpy of a vapor mixture at temperature tV and
pressure p was computed as the sum of the enthalpy changes involved
in (1) heating the ideal gas mixture to the temperature tv and then
(2) raising the pressure isothermally to p.
H + A (121)
where
HtT is the molal enthalpy of the vapor
_H is the ideal gas enthalpy at temperature tV
AHp is the pressure correction to the ideal gas enthalpy
at constant temperature tv.
Liquid Enthalpy, Method I
The liquid enthalpy procedure which was found to be successful consisted of the following steps (1) heating of the ideal gas at
zero pressure to the arbitrary standard temperature 77 ~F (25 ~C), followed
by isothermal compression of the gas mixture (2) to the dew point
pressure, then (3) to the bubble point pressure, and finally (4) to the
condition of subcooled liquid at the existing pressure of the column,
and (5) isobaric heating or cooling of the liquid phase to the temperature tL. Enthalpy changes were computed only for steps (1), (3),
and (5), the other two changes being neglected.
L =- H - HV + SsHC (122)
where
_- Iis the molal enthalpy of the liquid

-53TABLE 2
SAMPLE ENTHALPY CALCULATIONS
Vapor Enthalpy
A typical rich gas at 88~F, 740 psia:
Ideal gas enthalpy at 880~F' 4680 Btu/lb mole
Pressure correction - 519
4161 Btu/lb mole
A typical lean gas at 112~F, 740 psia:
Ideal gas enthalpy at 112 F 4643 Btu/lb mole
Pressure correction - 339
4304 Btu/lb mole
Liquid Enthalpy
A typical lean oil at 103~F, 740 psia:
Ideal gas enthalpy at 77~F 25,069 Btu/lb mole
Heat of vaporization - 26,265
Subcooled liquid enthalpy change 2,378
1,182 Btu/lb mole
A typical rich oil at 114~F, 740 psia
Ideal gas enthalpy at 77~F 19.078 Btu/lb mole
Heat of vaporization - 19,656
Subcooled liquid enthalpy change 2,603
2,025 Btu/lb mole
* For detail, see Table 7.

554AHV is the heat of vaporization at constant temperature
AHSC is the enthalpy change accompanying heating or
cooling of the subcooled liquid.
Since the liquid phase usually contained large quantities of heavy components, the dew point pressure at 77~F was a rather high vacuum, and
the enthalpy change over this very small pressure change was considered
negligible. The enthalpy change caused by compression of the liquid
phase, although not negligible, was ignored because (1) it was small by
comparison with the probable errors in the other enthalpy changes, (2)
the temperature was considerably below the critical point of the mixture,
(3) the enthalpy change would not change much for the various streams
found in a given column and its effects would thus be minimized, and (4)
no convenient methods were available for evaluating the enthalpy change.
A variation of the above method was found to be unsuccessful.
It consisted of using the molal average boiling point as the reference
temperature (instead of 770F) and evaluating the heat of vaporization
with an equation proposed by Kistyakowsky.(61) for non-polar liquids:
aHv - (Tb)(7y58 + 1.985 gn Tb) (123)
where
MHv is in Btu/lb mole
Tb is the normal boiling temperature in ~R,
Enthalpies for wide-boiling-range liquids, e.g., rich oils, were inaccurate and incompatible with enthalpies of narrow-boiling-range liquids.
The conclusion was that for purposes of using the Kistyakowsky equation,
the molal average boiling point was not a sufficiently sensitive indicator of the composition of wide-boiling-range liquids~

-55Liquid Enthalpy, Method II
An alternate method of computing liquid enthalpy was investigated and found to have some merit, but was finally abandoned. It consisted of calculating the dew point and bubble point temperatures of
the liquid at the column pressure and then evaluating the changes of
enthalpy corresponding to (1) heating the ideal gas to the dew point
temperature, (2) increasing the pressure isothermally to the dew point,
(3) lowering the temperature to the bubble point and thus condensing the
vapor at constant pressure, and (4) subcooling the liquid to the
temperature tL.
= H~ + Aipc Efv + AsC (124)
_ -p --
where AHv is the isobaric heat of vaporization.
-p
Enthalpies computed for lean, well-stripped oils were found to
be inconsistent with those for rich oils, and reference to Figure 3
will help to explain the difficulty. In a typical case, both streams
might have dew points in the vicinity of 9000F. The bubble point of the
lean oil might be 8500~F that of the rich oil, 100~F. Thus, the rich
oil enthalpy includes a large heat of vaporization, the lean oil enthalpy
includes a large enthalpy of subcooling, and both include a large ideal
gas enthalpy change and are computed as the difference of large numbers.
Accuracy of the individual step enthalpy correlations is not sufficient
to cover these wide temperature ranges. Furthermore, the pressure correction to the ideal gas enthalpy must span the range of zero pressure
to the column pressure, and the upper limit of the reduced pressure in
the correlation is 2.0, or about 400 psia for the heaviest components,
which are the major constituents of the liquid streams. Finally, the

-56RICH OIL =
////
_d
-_ LEAN OIL 1
HS I HiEPFATR
tL
TEMPERATURE
Figure 3. Liquid Enthalpy - Method II

"57bubble point is quite sensitive to the presence of light components,
and the use of Method II was found to be unstable computationally.
Method II is unsuitable for gas absorbers, particularly those
operating at high pressures, but it is thermodynamically sound and may
find application in distillation columns at low or moderate pressure.
F. K Values and Convergence Pressure
The K value or equilibrium vaporization ratio, defined
K = (125)
x*
is a most useful property, having been used in the present investigation
for determining vapor compositions, calculating bubble points and dew
points, computing the isobaric and partial molal heats of vaporization,
and estimating individual component plate efficiencies.
Functional Dependence
K values, strictly speaking, are functions of temperature,
pressure, and the compositions of both the vapor and liquid in equilibrium. Methods for predicting K values subject to the above conditions
are available(26'32'58) For most applications, dependence of K on both
phases produces lengthy trial-and-error calculations, and usually little
error will be introduced by neglecting the effect of the vapor composition.
Most well-known K value correlations(4',72,95) involve only the liquid
phase composition through the artifice of the "convergence pressure"
(This is one point in favor of plate-to-plate calculations from the
bottom of the column upward -- liquid phase compositions are used to
calculate vapor compositions.) For rough work, particularly at low
pressures, the liquid composition need not be considered either, provided

-58that the set of K values used be determined using liquid phase compositions of approximately the same composition as those in the problem at
hand. Most intermediate size computer programs use K values or relative
volatilities on this basis.
Convergence Pressure
For binary systems, the locus of critical points of mixtures
is a continuous line between the critical points of the two pure compounds.
With three or more components, a multiplicity of binary loci is produced,
and any prediction of the critical point of a mixture becomes unreliable.
The convergence pressure of a binary has been described as the
pressure coordinate of the intersection of a constant (system) temperature
line with the locus of critical points. See Figure 4o For multicomponent mixtures, convergence pressure becomes just a simple correlation
parameter defined as the pressure at which K values of all components in
the system appear to converge to unity.
Iadden(45) proposed a method for estimating convergence pressure, which consists of treating a multicomponent mixture as a binary
mixture composed of the lightest component of the mixture and a heavy
"pseudo-cmponent" which is a function of all other components of the
mixture. Lenoir and White(65) devised a method utilizing two pseudocomponents in a manner easily adaptable to machine computation, and this
latter method was used in the present investigation.
K as a Function of Convergence Pressure
Winn(95) has presented K values as a function of convergence
pressure through the use of a parameter "grid pressure". The functional

-59LOCUS OF BINARY CRITICAL POINTS
k _ __ — -_ -- -
/ I
PC // C.P.
/ I \
/ I \
/ I \
/CRIT. PT.
r4r 4. Ii R
q II
4, ~ I
"'4
Figure 4, Critical Region of a Binary Mixture
Ftguxe 4, Critical Reglon of a, Blnaxy Hlxture

-60relationship is given by nomograph, and Winn has stated(96) that it is
(44)
probably too complex for algebraic representation. Haddenn has discussed the dependence of K on convergence pressure in the vicinity of
the critical pressure for binary systems. The NoGoA.A. Equilibrium
Ratio Committee(73), having furnished K values at a set of nine different
convergence pressures, merely implies that one should use the set at
convergence pressure closest to that of the system being investigated,
Hinshaw(52) using a "correlating pressure" equivalent for most purposes
to convergence pressure, suggests plotting three values of Rn(K) versus
the corresponding values of en(Pk) and interpolating graphically.
The K value subroutine correlated the natural logarithm of K
as a function of temperature for a fixed value of a system pressure at
two convergence pressures, the expected extremes of those found in the
gas absorber. When K was required at an intermediate convergence
pressure, interpolation was performed according to the formula
pk nPk2) i n(pkl)
Xn(K~p =n(K) ~ + n(pk) _[npk)_n(K) n(K)p.] (126)
)Pk Prkl + Yn (Pk2) - -nk ] [n(K)pk2 kl

IV, PHYSICAL PROPERTIES
A. Vapor Density
The volume-explicit form('") of the Beattie-Bridgeman equation
of state(12) was used for calculating vapor density.
V= [ RT+B+ B(l - - RT - ] ( )
The constants for a mixture were computed from those for pure components
with the following formulas proposed by Beattie, Stockmayer, and
Ingersoll(14)
A = ( Yi yi J )2 (128)
Bo = (1/4)(i yi Bo,i) + (3/4) ( yi Boil )(I yi Bo,i2/) (129)
a F=> yi ai (130)
b = 2_ Yi bi (131)
c (4 yi fi)2 (132)
where
V is molal volume in liters/g mole
R is the universal gas constant in liter atm/g mole ~K
T is temperature in ~K
p is pressure in atm
Ao, Bo, a, b, and c are parameters in units consistent
with the above.
Values of Aol/2, Bo/, a, b, and cl/2 as tabulaated by Beattie
and Stockmayer(l3) from the work of many investigators, were available
6l1

-62for hydrogen, nitrogen, methane, ethylene, ethane, propane, isobutane,
normal butane, normal pentane, heptane, oxygen, and carbon dioxide.
Values for propylene, isopentane, hexane, and octane were estimated by
plotting values of neighboring compounds versus number of carbon atoms
in the molecule and interpolating on a smooth curve. The implication
here is not that quantities so obtained are of any value in predicting
the behavior of the pure components; merely that a smaller error is
introduced by using these values in calculating constants for a mixture
than by leaving them out entirely and in effect assuming that the unknown values are equal to those of the mixture,
Table 3 contains Beattie-Bridgeman constants used in computing
vapor density.
Be Liquid Density
Liquid density was assumed additive volume-wise and the
following expression was derived:
P _ 7; 6 (133)
L (xi M(/p1)
where
6o
PL was the density of the liquid at 60~F
L
M was molecular weight.
The mean coefficient of expansion was assumed equal to 0,0005 as suggested by Nelson(74) for 50.9 to 3550 ~API materials, and the liquid
density at temperature t was then given by the expression' 60
PL 1 i 000)- (. 5)(t - 60) (14

-63TABLE 3
BEATTIE-BRIDGEMAN CONSTANTS
Component (Ao)/2 (B)1/ a b c
Hydrogen. 44441 0.275717 -0.00506 -0.04359 22.45
Nitrogen 1.15953 0.369529 0.02617 -0.00691 204.94
Methane 1.50894 0.382290 0.01855 -0.01587 358.19
Ethylene 2.48032 0.495371 0.04964 0.03597 476.24
Ethane 2.42487 0.454684 0.05861 0.01915 948.68
Propylene 3.5 0.58 0.07 o0.6 1000.
Propane 3.45254 0.565665 007321 0.04293 1095.45
Isobutane 4.07476 0.617451 0.11171 0.07697 1732.05
n-Butane 4.21829 0.626752 0.12161 0.09423 1870.83
Isopentane 5.2 0.72 0.142 0.131 1950.
n-Pentane 5.31601 0.733104 0.15099 0.15960 2000.
Hexane 6.35 0.81 0.178 0.168 2000.
Heptane 7.38377 0.891341 0.20066 0.19179 2000.
Octane 8.5 o.98 0.22 0.213 2000.
Oxygen 1.22111 0.358927 0.02562 0.004208 219.09
Carbon Dioxide 2.23752 0.471410 0.07132 0.07235 812.40

-64The effect of pressure was neglected since temperatures were considerably below the pseudocritical,
The average apparent densities suggested by Brown, et al(7
were used for methane and ethane The value suggested for ethane was
used for ethylene also. Specific gravities for the other hydrocarbons
(84)
were obtained from Rossini Table 9 contains the values used.
C, Vapor Diffusivity
Gilliland(39) has proposed the following equation for the gas
diffusion constant of component a in a mixture of a and b:
T3/j2 (Ma + Mb/M, Mb)1/2
Dab o 0.o043 (/ 1)2 (155)
p (Va +Vb +
where
Dab is the diffusion coefficient in cm2/sec
T is the temperature in ~K
Ma and Mb are the molecular weights of a and b
p is the pressure in atm
va and vb are molecular volumes at the normal boiling
point in cm3/g mole.
If the units of Dab are changed to ft2/sec, those of T to ~R, of
p to psia, and those of v left unchanged, the constant,becomes
2.79 x 10O5, The exponent of T is not constant, and the equation is
recommended only for the range 0 - 1000~C
Wilke(93) has given an expression for the coefficient of
diffusion of component i through a stagnant multicomponent layer in
terms of the various possible binaries.
DGi 1 -i (136)

-65and when Equation (135) is substituted,
(2.79 x lo-5)(T3/2)(1/p) l yi)
D -(279 x. 10 i (1 _ - i
G =i (vl/3 + Vl/3)2 (137)
Zjs^i yj Mj.1/2
Mi Mj
Eighteen components were included in the computation: hydrogen,
nitrogen, and the hydrocarbons methane through decane plus two absorber
oil components. Other non-hydrocarbon components were neglected on the
grounds that they occurred in small concentrations and that Equation
(136) was intended for components of similar molecular configuration.
If values for these components were needed, values of others were substituted in the following manner: the diffusion constant of nitrogen
for that of carbon monoxide, oxygen, and water vapor; ethylene for
hydrogen sulfide; and propylene for carbon dioxide. The criteria for
selection of components whose coefficients were to be substituted were
these: (1) Use a compound of molecular weight close to the one for which
the coefficient is substituted. (2) Use a component likely to occur in
low concentration and thus not be affected much by the factor (1 - Yi).
M. M 1/2
The values of the expression (vil/3+vjl/3) (Mi Mj) were
-L < Mi + M
constant, and it was expedient to use them in tabular form. The two
absorber oil components were not included since their values changed
with whatever absorber oil properties were used. They were recomputed
and inserted for each absorber oil. The values are given in matrix
form as Table 4. The matrix is, of course, symmetric and has a zero main
diagonal. The molecular weights and molecular volumes, given by Rossini(86)
are shown in the consolidated table of physical properties, Table 9.

-66TABLE 4
VAPOR DIFFUSIVITY CONSTANTS
Component Hydrogen Nitrogen Methane Ethylene Ethane Propylene
Hydrogen -- 42.61 44.75 50.91 53.35 59.14
Nitrogen 42.61 -- 135 15 173.89 184.o6 2135.69
Methane 44.75 135.15 -- 157.61 165.86 189.71
Ethylene 50.91 173.89 157.61 -- 195.90 247.74
Ethane 53535 184.06 165.86 195.90 -- 261.88
Propylene 59.14 215.69 189-71 247.74 261.88
Propane 61.61 225.62 197.69 258.56 273.25 322.67
Isobutane 69.03 261.43 224,83 297.70 314.84 374.80
n-Butane 68.67 260.22 223582 296.41 313.49 373.25
Isopentane 75.24 291o26 247.29 330o16 349.29 418531
n-Pentane 75o36 291o70 247 59 330,62 349.81 418.87
Hexane 81.71 320 76 269 54 362 02 383 10 460.67
Heptane 87062 347.46 289.67 390o74 413.54 498.86
Octane 93533 372.78 308.80 417.89 442,26 534.85
Nonane 98.88 397 10 327.22 443.90 469.74 569.21
Decane 104o37 420.81 345,21 469.16 496.43 602.46
Abs. Oil 1* 118,26 480.76 390.50 532.90 563.69 686.41
Abs. Oil 2* 142,86 584.42 469o10 642.42 679.04 829.50
Absorber Oil 1 is a typical light absorber oil with molecular weight
of 184 and molecular volume of 304 cm3/g mole.
* Absorber Oil 2 is a typical heavy absorber oil with molecular weight
of 269 and molecular volume of 444 cm3/g mole,

-67TABLE 4
VAPOR DIFFUSIVITY CONSTANTS (CONT'D)
Component Propane Isobutane n-Butane Isopentane n-Pentane Hexane
Hydrogen 61.61 69.03 68.67 75.24'75.36 81.71
Nitrogen 225.62 261.43 260.22 291.26 291.70 320.76
Methane 197.69 224.83 223.82 247.29 247.59 269.54
Ethylene 258.56 297.70 296.41 330.16 330.62 362.02
Ethane 273.25 314.84 313549 349.29 349.81 383.10
Propylene 322.67 374.80 373.25 418.51 418.87 460.67
Propane -- 391.38 389.79 436.93 437.51 481,19
Isobutane 391.38 -- 455.68 513.14 513.79 566.98
n-Butane 389.79 455.68 -- 511.20 511.84 564.91
Isopentane 436.93 513.14 511.20 -- 578.48 640.27
n-Pentane 437.51 513.79 511.84 578.48 -- 641.03
Hexane 481.19 566.98 564.91 640.27 641.03
Heptane 521.08 615.63 613543 696.87 697.67 775.36
Octane 558.62 661.38 659.08 750.12 750.97 835.88
Nonane 594,44 704.95 702.56 800.81 801.69 893.51
Decane 629.11 747.03 744.54 849.73 850.65 949.07
Abs. Oil 1* 716.58 853.52 850.81 973.85 974.85 1090.51
Abs. Oil 2* 865.41 1033.77 1030.73 1183.45 1184.58 1328.98
* Absorber Oil 1 is a typical light absorber oil with molecular weight
of 184 and molecular volume of 304 cm3/g mole.
*A Absorber Oil 2 is a typical heavy absorber oil with molecular weight
of 269 and molecular volume of 444 cm3/g mole.

-68TABLE 4
VAPOR DIFFUSIVITY CONSTANTS (CONTID)
Component Heptane Octane Nonane Decane Abs.Oil 1* Abs.Oil 2*
Hydrogen 87.62 93.33 98.88 104.37 118.26 142.86
Nitrogen 347.46 372.78 397.10 420.81 480.76 584.42
Methane 289.67 308.80 327.22 345.21 390.50 469.10
Ethylene 390.74 417.89 443.90 469.16 532.90 642.42
Ethane 413.54 442.26 469,74 496.43 563.69 679.04
Propylene 498.86 534.85 569.21 602.46 686.41 829.50
Propane 521.08 558.62 594.44 629.11 716.58 865.41
Isobutane 615.63 661.38 704.95 747.03 853.52 1033.77
n-Butane 613.43 659O08 702.56 744.54 850 81 1030.73
Isopentane 696.87 750.12 800.81 849.73 973.85 1183.45
n-Pentane 697.67 750.97 801,69 850.65 974.85 1184.58
Hexane 775.36 835.88 893051 949.07 1090.51 1328.98
Heptane - 914.18 978.29 1040.10 1198.08 1464.23
Octane 914.18 -- 1058.40 1126.20 1300.07 1593.10
Nonane 978.29 1058.40 - 1208.40 1397.68 1716.92
Decane 1040o10 1126.20 1208o40 -- 1491.98 1836.91
Abs. Oil 1* 1198.08 1300.07 1397.68 1491.98 -- 2152.49
Abs. Oil 2* 1464.23 1593.10 1716.92 1836.91 2152.49
* Absorber Oil 1 is a typical light absorber oil with molecular weight
of 184 and molecular volume of 304 cm3/g mole.
* Absorber Oil 2 is a typical heavy absorber oil with molecular weight
of 269 and molecular volume of 444 cm3/g mole.

-69"
D. Liquid Diffusivity
The liquid diffusivity was estimated by a formula of Wilke and
Chang(94)
DLi = 74 x 10-8 0 (158)
kLVi
where
DL is the liquid diffusivity of component i in the mixture
x is an "association factor"
M is the molecular weight of the solvent
T is the temperature in ~K
i4 is the viscosity of the solvent in centipoises
vi is the molal volume of the solute in cm3/g mole.
When the dimensions of DL,i' T, and 4 are changed to those involving
pounds, feet, seconds, and degrees Rankine, the constant becomes
-14
2.97 x 1014 The factor x may be taken as 1 in dilute solutions of
unassociated liquids. Furthermore, little error is introduced by computing M as the molecular weight of the entire liquid phase.
E, Liquid Viscosity
According to Partington(79), one of the more satisfactory of
the convenient additive formulas for viscosity was proposed by Kendall(59).
1/53 i xi 1/3 (139)
where rq is the kinematic viscosity in centistokes. By definition,
IQ = 4-/P (140)

-70where 1 is the absolute viscosity in qentipoises and p is in g/cm3. A
relation for the temperature dependence of the kinematic viscosity was
proposed by Cornelissen and Waterman(24)
an(i) = - + B (141)
where A and B are empirical constants. The exponent x, for moderate
changes of temperature, is approximately equal to 2.
Values of the kinematic viscosity were obtained from Rossini(85)
at 60 and 200~Fo The kinematic viscosities of a mixture at each of
these temperatures were then computed by Equation (139), the constants
A and B were evaluated from the equations
2n%1q/r2) T12 T22
A n(/2 T1 T2 7.11847 x 10-5 (in T60 - In r200) (142)
2 - 2
and
B = n2= In 1 = n 60 - (3.7026 x 10 6)(A) (143)
and the kinematic viscosity at temperature t was computed from Equation
(141). Multiplying r in cs by the density in lb/ft3 and dividing by the
constant 92,900 produced the absolute viscosity in lb/ft sec.
One further correction for the effect of pressure was then
made.
4 = 40 exp(o.o00014 p) (144)
where lo is the viscosity "at zero pressure" and does not differ appreciably from that at 1 atm, and p is the pressure in psia. The form of
Equation (144) is given in Perryts Handbook(80) and the value of the

-71~
constant was based on the statement, "At 1000 atm the viscosities of
many common organic liquids are approximately doubled in the temperature
range 30~ to 750~C.o"
F. Vapor Thermal Conductivity
The temperature dependence of the thermal conductivity of gases
was computed by the equation of Sutherland(88)
491.7 + Cs )( T )3/2
kT =(k32 F)(49 + Cs)( )9 (145)
where Cs is the Sutherland constant for a particular compound. Values
of the thermal conductivities of the various compounds were taken from
the collection in Perryts Handbook (81), and Sutherland constants for
the compounds hydrogen, nitrogen, methane, ethylene, propane, n-butane,
carbon monoxide, oxygen, carbon dioxide, and water vapor were given by
Partington(78). The others were calculated from the approximate formula
of Arnold(8),
Cs = 1.47 Tb (146)
G. Ideal Gas Enthalpy
In the ideal gas state, the pure component enthalpy is equal
to the partial molal enthalpy, the molal enthalpy of a mixture then
being equal to the summation, by mole fraction, of the individual component enthalpies.
HO = 0 i Y Ii Hi (147)
where
Ci is the partial molal enthalpy of component i in the
ideal gas state
i is the molal enthalpy of pure component i in the
ideal gas state.

-72The ideal gas enthalpy of a component may be correlated with
temperature by a fifth-order power-series approximation
HI -= Z Ai (t/oo)j (148)
4 j~=0 j i
the expression (t/100), where t is in ~F, being used for convenience.
The ideal gas enthalpy of a mixture then becomes
_f=~ Z. (i 9)
H0 = iZi Zpj=o Aj,i (t/100)j (149)
Values of Aj i for hydrogen and methane through n-octane,
based on the data of A.PoI. Project 44 7) were furnished by Edmister
Values for the non-hydrocarbon gases were derived from heat capacity
equations by Kobe and Long(62), and the constants for nonane, decane,
and the absorber oil components were obtained by using an equation for
hydrocarbon specific heats given by Fallon and Watson(3)
c = (0.0450 K' - 0.233) + (0.440 + 0.0177 K')(10-3 t) -
(0.1530 x 10-6)(t2) (150)
where Kt is the characterization factor. In cases where heat capacity
equations were used, integration produced constants Al i through A5,i
for nonane, decane, and the absorber oils, A4 i and A5 i being zero;
and A1,i through A4 i for the other components, A5 i being zero. For
components other than absorber oil, A0 i the constant of integration
(equal to the molal enthalpy at 0~F) was obtained from the literature(62'87). A, i for the absorber oils was arbitrarily set at
(103)(Mi), where i was the molecular weight, any error thus introduced
being cancelled by the fact that enthalpies were used for energy balances,

-73where enthalpy changes, rather than absolute values, are important. The
value of 103 is quite close to the enthalpies, in Btu/lb, of the heavier
paraffin hydrocarbons. Values of Aji are given as Table 5.
H. Pressure Correction to Ideal Gas Enthalpy
The pressure correction to the ideal gas enthalpy is given
thermodynamically as
pc f |(c)t dp [V T() plIp (151)
The integration may be carried out using a suitable equation of state,
but this method is cumbersome and not recommended for machine use,
Edmister and Canjar(31) have published a graphical correlation
of the partial molal enthalpy correction for pressure (in the form
-S/Tc) as a function of the reduced temperature, pressure, and normal
boiling point.
(Tr)i -T (152)
(p-r)i (15)
r j XJ (Tb) J molal average normal boiling point (15
(B )ino ~ (T ) = - g (Trm ~-~ p (154)
(Tb) i (Tb)i
where
Tr is the reduced temperature
Tc is the critical temperature in ~R
Pr is reduced pressure
Pc is critical pressure in psia
Br is reduced boiling point
Tb is normal boiling point in ~R

-74TABLE 5
IDEAL GAS ENTHALPY CONSTANTS
Component A0 A1 A2 A3 A4 A5
Hydrogen 3,109. 675.02 14.127 -4.955 0.879 -0.057
Nitrogen 3,192. 692.2 1.323 0.1445 -0.0029
Methane 3,667. 825.5 9.33 8.375 -1.333 0.092
Ethylene 3,781. 920.8 93508 -15.417 3.917 -0.350
Ethane 4,214. 1146.3 65.62 7.125 -1.125 0.058
Propylene 4,708. 1372 7 89.75 7.750 -1.250 0.083
Propane 5,043 1568.2 114.17 9.250 -1.667 0.083
Isobutane 6,014. 2044 0 164.83 8.875 -1.833 0.092
n-Butane 6,639. 2125.8 120.92 19.500 -3.417 0.200
Isopentane 7,476. 2479.1 231.33 0.375 -0.833 0.058
n-Pentane 7,951. 2743.6 50.12 62.750 -11.125 0.700
Hexane 9,439. 3012.8 267.29 -1.792 -0.292 0.008
Heptane 10,823. 3508.6 300.25 -0.583 -0.250 -0.017
Octane 12,270. 3889.6 372.83 3.208 -2.833 0.225
Nonane 13,671. 4326.3 426.00 -6.541
Decane 15,066 4805.3 472.72 -7.256
Abs. Oil 1 18,988 6334.7 614.63 -9.402
Abs. Oil 2 27,656 9726.6 905.05 -13.694

-75TABLE 5
IDEAL GAS ENTHALPY CONSTANTS (CONT'D)
Component AO Al A2 A3 A4 A5
Carbon 3,194 690.3 2.644 0.0902 -0.0023
Monoxide
Oxygen 3,188 690.6 7.83 -0.1511 0.0013
Carbon 3,358. 844.8 28.78 -0.720 0.0076
Dioxide
Hydrogen 3,449. 794.6 10.20 0.0783 -0.0034
Sulfide
Water 3,642. 796.8 1.588 0.1917 -0.0037

-76Twenty charts, one for each value of Br from 0.3 to 2.2 at increments of
0,1, contain graphs of the function (H/Tc) versus Tr with Pr as a
parameter. Values of Pr ranged from 0.1 to 1.1 (in some cases 0.8) at
increments of 0.1; values of Tr, from 0.3 to 2,4, although no one graph
covered the whole range. The charts of Br of 1.3 and greater were actually
the writerts cross-plots of data given as (H/Tc) versus Tr with Br as the
parameter.
The curves, 178 in all, were fit with equations of the form
(AT/Tc) = cl Tr-2 + c2 Tr-1 + 3 + c4 Tr (155)
using the method of least squares with from six to fifteen points per
curve spaced at approximately equal distances along the curve, slightly
closer in regions of maximum curvature. Each of the constants was then
further correlated against Pr with the equation
cj = klj Pr2 + k2j Pr + k3 (156)
giving, for each value of Br, a set of twelve constants required to reproduce the set of eight or eleven curveso Table 6 contains values of k;
Table 7, a sample calculation; Table 9, the critical data used in the
calculation
The pressure correction is given by the expression
Z6pC = E (Yi) (T()i (T?/Tc)i (157)

-77TABLE 6
CROSS-CORRELATION CONSTANTS FOR PRESSURE CORRECTION CURVES
Br kl,l k2,l k3l kl,2
0.3 0.76515201 -1.0896985 0.048774930 -2.5260888
0.4 0.65641733 -1.8188528 0.0076648226 -0.79829508
0.5 0.053625970 -2.0889816 0.10234174 0.012211400
o.6 -2.7204796 -1.2274600 -0.029616518 6.0421819
0.7 6.8463816 -10.905157 1.4236963 -16.755093
0.8 -7.2901857 -1.0717085 -0.56703191 10.957736
0.9 -21.957863 5.2381433 -1.3307538 37.844393
1.0 -4.9646741 -14.042670 0.19602834 5.3078301
1.1 -13.001823 4.0844344 -1.1619453 24.366139
1.2 -24.619161 16.703230 -2.6743881 43.492558
1.3 -15.707064 10.345823 -1.1954760 28.052020
1.4 -9.1706785 2.3486746 -0.15448517 19.607486
1.5 -5.0820381 -3.2345288 0.0025136118 9.6589914
1.6 -26.281748 8.7038198 -2.6813022 48.158575
1.7 -12.221984 -40.224737 4.5784289 14.952675
1.8 -47.576168 -81.148598 13.192791 55.946835
1.9 707.86697 -558.01626 71.988096 -1055.1484
2.0 1017.0506 -750.86950 96.835324 -1354.2093
2.1 136.63509 -91.321110 18.977083 -128.21982
2.2 -104.80550 96.437031 14.509830 170.78796

-78TABLE 6
CROSS-CORRELATION CONSTANTS FOR PRESSURE CORRECTION CURVES (CONT'D)
Br k2 k3,2 kl,3 k2,3
0.3 1.6424300 -0.36092828 2.7965679 -1.7766868
0o4 1.6141758 -0.052154370 -0.78930334 0.16169519
0.5 1.8278866 -0.43011532 -0.38639856 -0.87145890
0.6 -1.0986707 -0.081923870 -4.3500451 0.68778619
0.7 20.553221 -3.5128029 11.801998 -14.253674
0.8 -0.85629380 0.95173645 -5.5902836 0.25957340
0.9 -12.116760 2.0630605 -21.535105 6.6800258
1.0 24.352870 -0.55029913 -1.8256805 -15.202776
1.1 -12.710789 2.1576869 -14.819421 8.3165842
1.2 -32.864472 4.6783490 -24.953644 18.711362
1.3 -18.654170 1.8166865 -143.45968 8.4351253
1.4 -359948665 0.42651519 -11.718348 1.6332848
1.5 8.9914864 0.25360696 -4.9646483 -5.8217500
1.6 -10.848954 5.6598528 3-0.341967 8.4778520
1.7 83.637982 -6,7041671 -8.0142457 -47,282593
1.8 164,54885 -19.217993 -235512913 -94.294868
1.9 852.77109 -101o73664 498.65291 -420.67737
2.0 1056.0718 -127.91437 592.46645 -481.80108
2.1 132.71598 -19.060167 30 318558 -53.165091
2.2 -94.595823 -17 957405 -92 /486516 40.103620

-79TABLE 6
CROSS-CORRELATION CONSTANTS FOR PRESSURE CORRECTION CURVES (CONT"D)
Br k3,3 kl4 k2,4 k3,4
0.3 0.64216386 -1.0992639 0.48136410 -0.31815297
0.4 -0.0035042948 0.76794306 -0.84538628 0.050396749
0.5 0.42106418 0.10041700 -0.00055905000 -0.10956390
o.6 0.18301313 0.88428910 -0.073278290 -0.050436775
0.7 2.5661826 -2.6288551 3.1468000 -0.56924327
0.8 -0.53552344 0.91443227 -0.025887490 0.10964841
0.9 -0.98585531 4.0084568 -1.2217063 0.14369424
1.0 0.59329410 0.24077358 2.9094673 -0.070760689.1 -1.2553654 2.9583135 -1.7954403 0.24046920
1.2 -2.5654896 4.7181227 -3-5407984 0.45245450
1.3 -0.71359150 2.2243009 -1.1973324 0.064486269
1.4 -0.31806367 2.2550089 -0.45210724 0.082852748
1.5 -0.31050288 0.68967455 0.93653583 0.090312239
1.6 -3.8718402 6.2494725 -2.5319863 0.84497597
1.7 3.2100072 1.6771271 7.9035915 -0.49863798
1.8 9.2360967 3.4846976 16.501281 -1.4641740
1.9 47.507386 -79.379934 67.576609 -7.3347124
2.0 55.755917 -85.244240 71.647827 -8.0187174
2.1 5.5848900 -0,32240363 5.7549133 -0.38698325
2.2 7.2650072 16.415901 -7.2277083 -0.95281385

-80TABLE 7
SAMPLE CALCULATION OF PRESSURE CORRECTION TO IDEAL GAS ENTHALPY
Component y Tr Pr Br /Tc -yH
Nitrogen O.0410 2.4118 1.5031 1.6374 0.4486 4.177
Methane 0.8026 1.5954 1.0994 1.1343 0.8766 241.539
Ethane 0.0723 0.9962 1.0448 o.6864 2.3711 94.249
Propane 0.0597 Oo8224 1.1986 0.5481 2.9205 116.111
Isobutane 0.0059 0,7455 1o3986 0.4845 3.3258 14.416
n-Butane Oo0135 0.7156 1.3437 o.4646 3.2259 33.329
Isopentane o.0016 o.6600 1.5321 0.4208 3.6019 4.782
n-Pentane 0.0014 0.6477 1.5117 0.4096 3.4785 4.118
Hexane* ooo 0004
Heptane o.oo006 0.5633 1.8644 0.3409 4,0546 6.308
519.029
519-029 519.237
0.9996
tv = 88~F
p = 740 psia
Z1 (yi)(Tb,i) 228.006
* The machine program, to save time, neglects
mole fractions less than 0.001 and "normalizes"
the result.

-81
I. Isobaric Heat of Vaporization
The isobaric heat of vaporization was computed with an equation proposed by Edmister(29):
v BT1T2 K2,i +0
- [T2 Tl2n Kl,i158
where
pAH is the isobaric heat of vaporization in Btu/lb mole
R is the universal gas constant in Btu/lb mole OR
T1 is the bubble point temperature in OR
T2 is the dew point temperature in OR
K2,i is the K value of component i at the dew point
K1,i is the K value of component i at the bubble point
zi is the mole fraction of component i
AHi is the ideal gas enthalpy change of component i over the
range T1 to T2.
J. Isothermal Heat of Vaporization
According to the definition of the partial molal heat of
vaporization,
Hvt = z atti (159)
where SH is the partial molal isothermal heat of vaporization of
component i. These partial quantities may be calculated from the
vantt Hoff equation
Z7 = RT 2( ) (160)
and the values for nitrogen and the various hydrocarbons at 77~F (25~C)
are shown in Table 9, These values were calculated by (1) reading K

-82
values at 10 psia from the N.GoAoA. K value charts, (2) plotting their
natural logarithms versus temperature, (3) drawing smooth curves, (4)
computing the slopes of the tangent lines at 77~F, and (5) applying
Equation (159).
K values for absorber oils were not sufficiently accurate for
the calculation of the partial molal heat of vaporization, so an approximation of the pure component heat of vaporization was used. Kistyakowsky's
equation was employed to estimate the heat of vaporization at the normal
boiling point,
AHv = (Tb)(7.58 + 1.985 In Tb) (123)
and an equation suggested by Watson(91) was used to compute the heat of
vaporization at 770F as a function of the normal boiling point, the critical temperature, and the heat of vaporization at the normal boiling point.
v Tc T 0.38 (
TjT b (161)
A typical absorber oil with molecular weight of 223, boiling point of
5355 F, and critical temperature of 845~F had a heat of vaporization of
21,200 Btu/lb mole at 535~F and 29,900 Btu/lb mole at 77~F.
K. Enthalpy Change of Subcooled Liquid
The change of enthalpy accompanying the temperature change of a
liquid hydrocarbon mixture may be calculated by Fallon and Watson's
(33)
specific heat equation
c = [(03.55 + 0.128 x 10-2 ~API) + (0.503 + 0.117 x 10-2 ~API).
(10-5 t)][0.05 K' + 0.41] (162)

-83where
Cp is the specific heat in Btu/lb ~F
~API is the API gravity
K' is the characterization factor.
Like many other equations for hydrocarbon fluids, Equation (162) is
probably more accurate for narrow-boiling fractions than for wideboiling ones. Since the equation is seldom applied over more than a
40-degree temperature change, the inaccuracy is probably no greater
than that associated with many other correlations.
When the specific heat is evaluated at the arithmetic average
of temperatures tl and t2 to give cp, the formula for the enthalpy
change becomes
LHsc = (p)(ML)(t2 - tl) (163)
It can easily be demonstrated that this is equivalent to an integration
over the temperature range (t2 - t1). Generally,
I t2 + tl
M = (p)(t2 - tl) = [a + b( —2 )][t2 - t] =
(a)(t2 - t1) + (b/2)(t2 - tl2)
Integrating, we have
H t2 t2 (b)(t2
— = c dt = f (a + bt)(dt) = [(a)(t) + 2 ]t2 -
ML t tP tl
(a)(t2 - t1) + (b/2)(t22 - tl2)

-84L. K Value
The Gram orthogonal polynomial approximation as discussed by
Hildebrand(48) was used to correlate the natural logarithm of K as a
function of temperature. K values at nine equally-spaced values of
temperature were used, and for each component a set of five correlation
coefficients was produced, corresponding to a fourth order polynomial
approximation. For component i (subscripts on the K values indicate the
nine temperature points and the subscript i has been dropped),
1
a0 = -[n K1 + In K2 + nn K3 + Jn K4 + nn K5 + nn K6 + In K7 +
in K8 + Jn Kg] (164)
al = -- [ 4 In K1 3 In K2 - 2 In K3 - In K4 + 0 + 2n K6 +
15
2 In K7 + 3 In K8 + 4 in Kg] (165)
1
a2 = - [28 In K1 + 7 In K2 - 8 In K3 - 17 In K4 - 20 an K5
17 Jn K6 - 8 n K7 + 7 In K8 + 28 In Kg] (166)
a3 = F [- 14 Jn K1 + 7 In K2 + 13 in K3 + 9 n K4 + 0 -
9 nn K6 -13 n K7 - 7 nn K8 + 14 in Kg] (167)
a4 = -7 [14 In K1 - 21 In K2 - 11 n K3 + 9 In K4 + 18 In K5 +
a4 = 9659 In K6 - 11 nn K7 - 21 in K8 + 14 n Kg] (168)
Then at any temperature t the function i was evaluated,
t - t5
A- At (169)
where
t5 was the fifth or center temperature point
At was the temperature interval between points,

-85and five functions of the parameter 4,
P = 1 (170)
i = (171)
P2 - 20 (172)
28
5P 3 - 590]
P3 = 58384 598; (173)
P4 = - 840o - (174)
P4 35?4 - 5 5 - 1080 (14)
Then for any component i
K = exp [ 0 aj i Pj] (175)
The K value data of Hinsha(53) were used for absorber oils,
Winn's nomograph(95) for water vapor, and the N.G.A.A. data(72) for all
other components,
M. Convergence Pressure
An article by Lenoir and White(65) is the basis of the convergence pressure correlation. The authors define an effective light
component, which is a function of all components in the mixture except
the heaviest, and an effective heavy component, which is a function of
all except the lightest.
Letting tRe and the be the light component and heavy component
effective boiling points in OF, and Tb2 and Tbh be the absolute (OR)
boiling points of the lightest and heaviest components of the mixture,
then
= i i (e /T i) 459.69 (176)
t< ~ xi (Tb2/Tb,i)5"57

-86where i does not include the heaviest component, and
i Xi Tb,i(Tbi/Tbh)2"0 (
the 1(Tb/Tb)'0 - 459.69 (177)
E i-Xi (Tb, i/Tbh
where i does not include the lightest component. In evaluating te and
the on the machine, the exponent of the ratio (Tbj/Tb,i) in Equation (176)
was changed to 6 and that of the ratio (Tb,i/Tbh) in Equation (177) was
changed to 2. These changes resulted in a considerable saving of
machine time, and an examination'of the authors' figures indicated that
the accuracy was not seriously affected.
Lenoir and White gave two graphical correlations of an(pk)
versus the temperature of the system, t, with the as a parameter, one of
the charts with t.e corresponding to methane, -259~F, the other with te
corresponding to ethane, -128~Fo On each chart, en(Pk) was correlated
with t by an equation of the type
In(Pk) = (a)(t/100)3 + (b)(t/100)2 + (c)(t/100) + d (178)
using the method of least squares. Each of the four constants was then
cross-correlated with the by an equation of the type
a = (al)(the00)2 + (a2)(the/100) + a3 (179)
and thus twelve constants served to reproduce each of the charts.
Values of the constants are given in Table 8.
To find In(Pk) with a value of te other than -259~F or 128~F,
an(pk) was evaluated with t2e equal to each of these values, and then
linear interpolation was performed. The authors have recently published

-87a more complete article(66) furnishing graphs for twelve values of the
effective light component, and it seems reasonable to expect that
nn(pk) may be made a smooth function of t;e also.

-88TABLE 8
CONVERGENCE PRESSURE CORRELATION CONSTANTS
tge the/100 a b c d
-259 0 0.0224 -0.1971 0,1651 7.4077
1 o -0.0126 -o.o666 0.1597 7.7413
2.0 -0.0088 -0.0457 0.1533 8.0347
3.0 -0.oo4i -0.0536 0.1821 8.3041
4.0 -0.0039 -0.0402 0.1620 8.5793
5.0 -0.0037 -0.0271 0.1316 8.8070
6.o -0.0029 -0.0301 0.1670 8.9733
7.0 -0.0193 -0.2798 1.5663 12.2065
8.0 -0.0210 -0.3873 2.6868 15.7365
-128 0 -0.0215 -0.0177 0.2279 6.4141
1.0 -0.0076 -o01045 0.5497 6.2478
2.0 -0.0038 -0.1488 0.8883 5.9164
3.0 -0.0132 -0.0378 0o6730 6.0768
4.0 0.0028 -0.1567 1.0186 5.8432
5.0 0.0009 -0.1272 0.9838 5.9152
6.0 0.00o36 -0.1568 1.1600 5.7595
7.0 -0.0012 -0.0744 0.8209 6.2386
8.0 0.0015 -0.0968 0.8791 6.3042

-89TABLE 8
CONVERGENCE PRESSURE CORRELATION CONSTANTS (CONT'D)
tge j jl2 j3
-259 a -0.00056 0.00587 -0.018
b 0 0.00718 -0.0696
c 0.000611 -o00035 0.162
d -0.015 0.348 7 41
-128 a -0.00090 0.00887 -0.0193
b 0.0070 -0.0593 -0.023
c -o 00262 0.289 0.250
d 0.0205 -0.205 6.37

-90TABLE 9
CONSOLIDATED PHYSICAL PROPERTIES
Molecu- Normal Critical Kinematic Kinematic
Component lar Boiling Tempera- Critical Viscosity'Viscosity
Weight Point ture Pressure at 60~F at 200~F
OR pR psia cs cs
Hydrogen 2.016 36.828 59.87 188.11
Nitrogen 28.020 139.25 227.09 492.532
Methane 16. 042 201.01 343530 673.1
Ethylene 28.052 305.01 509.51 742 o1
Ethane 30oo68 332o16 549.77 708.3
Propylene 42.078 405.83 657.20 667.0
Propane 44.094 415.96 665.95 617.4 0.230 0.145
Isobutane 58,120 470.58 734.65 529.1
n-Butane 58.120 490.79 765.31 550.7 0.314 0.185
Isopentane 72.146 541.82 829.80 483.0
n-Pentane 72.146 556.62 845.60 489.5 0.387 0.220
Hexane 86.172 615.42 914.10 439.7 0.4913 0.270
Heptane 100.20 668.86 972.31 396.9 0.6379 0.3551
Octane 114.22 717.89 1024.9 362.1 0.8177 0.4201
Nonane 128.25 763.13 1071.0 331.0 1.057 0.4975
Decane 142 28 805.11 1114.0 306.0 1.558 0.5857
Carbon 28.010 146.09 239.69 507.01
Monoxide
Oxygen 32.000 162.29 278.57 736.27
Carbon 44o010 350.39 547.49 1071.3
Dioxide
Hydrogen 34.080 384 41 672o41 1306.5
Sulfide
Water 18.016 491.69 1165,2 3208.1 1.172 0.318

-91TABLE 9
CONSOLIDATED PHYSICAL PROPERTIES (CONT D)
Heat of Thermal
Component Density Vaporization Molecular Conductivity Sutherland
at 60 ~F.at;t77 OF Volume at 2 ~F Constant
g/cm3 Btu/lb mole cm3/g mole Btu/sec ft ~F ~R
Hydrogen --- 14.3 2.78 x 10-4 139
Nitrogen -- 1,339 31.2 3.89 x 10-5 212
Methane 0.25 1,477 37.85 4.86 322
Ethylene 0.4o 4,453 49.32 2.81 457
Ethane 0.40 5,478 55.03 2.94 473
Propylene 0.5215 7,292 69.13 2.30 603
Propane 0.5068 6,308 75.94 2.41 599
Isobutane 0.5626 8,786 97.75 2.22 660
n-Butane O.5840 9,576 96.60 2.17 670
Isopentane 0.6241 10,490 117.94 2.00 733
n-Pentane 0.6303 11,210 118.40 2.05 774
Hexane 0.6633 12,020 140.90 2.00 876
Heptane 0.6875 12,930 163.58 1.94 980
Octane 0.7061 13,720 186.87
Nonane 0.7210 16,080 210.87
Decane 0.7335 19,270 235.79
Carbon -- -- 30.7 3.75 212
Monoxide
Oxygen -- - 25.6 3.95 248
Carbon -- 2,220 34.0 2.36 493
Dioxide
Hydrogen -- 5,080 2.9 2.11 596
Sulfide
Water 0.9991 18,920 18.90 3.67 995

V. PROBLEM-SOLVING PROCEDURE
A. The Overall Procedure
The procedure followed in solving a problem, as distinguished
from the machine run procedure discussed in the Appendix, is shown in
flow sheet form as Figure 5. Recalling the statement of the problem,
given the complete properties of the rich gas and lean oil streams, the
column pressure, the number and dimensions of the bubble plates, and
data on heat losses from the column, we wish to calculate the properties
of the output streams and the internal streams of the column. We do this
by (1) making a first approximation to the properties of the rich oil
stream, (2) computing the properties of the lean gas stream by material
and energy balance, (3) computing these same properties by a plate-toplate calculation upward through the column, (4) comparing the values
obtained in steps (2) and (3) and, on the basis of this comparison, adjusting the properties of the rich oil, and (5) repeating steps (2)
through (4) until convergence is obtained.
The complete sequence may be broken down into four phases:
(1) data conversion, (2) initial approximation, (3) plate-to-plate calculation, and (4) output. Each of these phases corresponds to a subprogram and is covered by a detailed flow sheet. The two complex subroutines are also discussed.
B. Data Conversion
The data conversion phase (contained in Subprogram A) begins
with the reading of a set of K values and absorber oil data. If the
machine detects an error in reading the set, a remark is written to that
-92

-93Prepare correlation of K values8 Store absorber
oil data and functions of them,
If necessary, convert column data into units used
by remainder of program.
Compute column cross-section for vapor flow and convection heat loss per plate.
Estimate overall column efficiency. Given approximate outlet
stream compositions,
compute recovery.
Compute absorption factors..
Initial estimate of reEstimate recovery of components. covery has been furnished,
Estimate outlet stream quantities, compositions, and
temperatures based on recovery of rich gas components
and overall heat balance.
Plate-to-plate calculation from bottom to top of
column, Comparison with results of overall mass
and energy balances.
Has convergence been obtained?
Yes
jPrint results.
Figure 5. Problem-Solving Procedure.

-94_Start
Read. set of K values and absorber oil data
Damp set of K values (if desired)
Write absorber oil data d
Compute At and tbase in K value correlation
Compute natural logarithms of all K valueso
Compute a0 through a4 for K values of'each component at both
convergence pressures and store.
|[ Dump constants of K value correlation (if desired),
-- i
Store molecular weight, normal boiling point and. critical temperature (converted to degrees Rankine), and critical pressure of
absorber oils in appropriate places.
Compute cube roots of kinematic viscosities of absorber
oils at 60~F and 200~F and store.
Store specific gravities of absorber oils in proper places,
If negative, indicating that API gravities are given, convert to specific gravities,
Compute ideal gas enthalpy correlation constants
for both absorber oils and store,
(Continued)
Figure 60 Flow Diagram-Subprogram A.

-95 —
Compute molecular volume to powers of 1.0, 0o,6, and 1/3 for each
absorber oilr Store, Compute vapor diffusivity constants and store.
\ Y
Dump com onstants (if desired).io
Intia estimad set of rcolumnvery is givendata
Write column data.
Is there data for -e side stream cooling
cooling? _sdtata t
Should machine compute first approximation to recovery?
No
No
Is first approximation to outlet compositions given?
Yes
Dump first approximation
lInitial estimate of recovery is given
Diimp initial estimate |
L.~ ""t ~Is VO given in terms of seconds? YeI
O^ No
| Put in terms of seconds ~
Figure 6 (Continued)

-96Is VO given in terms of poundsT s
Yes
No
|Is composition in weight fraction?
Yes
Compute number of moles per seconde Vo in moles per secondu
and yo n in mole fracrctiOn
y is given in mole fraction.
Yes,.0 m iue.u onied
Compute number of pounds of each component per secmole molecularnd
eight of V0 in moles p er second.'
Composition given in mle fractiondard ubi feet
Ye
VQ in moles per second.
Compute moles per pound of each component, total number
of moles per pound of VO., and yo in mole fraction,
Figure 6. (Continued)

-97No
Is energy content of Vo given as enthalpy in Btu
per pound?
Yes
Cbmpute molecular weight of V0 and enthalpy
in Btu per pound mole,
Is energy content of VO given as Btu per pound m.ole.?
Yes No
Find tVo as a function of. 1 tVo given.
Have the properties of Ln+l been analyzed in the same way
as those of VO?
No Yes
Modify machine program to analyze properties
of Ln+lIDump transformed input (if desired).
Yes
-.- Are all column dimensions given in fee-t?j
No
[Convert those that may optionally be given in inches
If heat loss rate data are given in terms of hours or
days, convert to terms of seconds
Figsre 6. (Cont~inued)
Figure 6 ~ (Continued)

-98Convert n (number of plates in column) from fixed point
to floating point notation.
Is there a value given for percentage of column
cross section for vapor flow'?
I
No Yes
-use 80%
Compute column cross sectional area for vapor flow.
Is convection heat loss per square foot given? No
Yes
Compute column surface area per plate and heat
loss in Btu per second per plate.
Is heat loss for total column surface area given? -
Yes No
Compute heat loss per plate No heat loss,
Dump converted input (if desired),
Transfer to Subprogram B
Figure 6. Flow Diagram-Subprogram A.

-99G
effect, all sets of column data following the set are ignored, and the
machine attempts to read another set of K value and absorber oil data.
If none is available, the entire program is transferred off the machine.
The set of data, if correctly read, may be dumped (printed in a very
simple and compact form) if desired for record purposes. The machine
then proceeds to the correlation of the K values.
The fifth or middle one of the nine temperature values is
designated tbase the basic or datum temperature, and the difference
between adjacent values is called At. The K values are correlated as
the natural logarithm, so the machine then employs the GMLOG1 subroutine
to obtain the natural logarithms. GMLOG1 gives an immediate error return if the argument is zero, and this feature is used to store zeros
in all locations which originally contained zeros, i.e., for components
not present in the column. All non-zero numbers are replaced by their
natural logarithms. Note that K of 1.0 is also replaced by a zero. The
machine then computes the orthogonal polynomial coefficients for each
set of nine values of en(Ki). If en(Kli) is zero, in(K2,i) is examined.
If both are zero, the machine concludes that the component is missing,
stores zeros for coefficients, and precedes to the next component. The
machine may dump the coefficients if desired,
The molecular weight and critical pressure of each of the
absorber oils are stored in the table of physical properties. Critical
temperature and normal boiling point are converted to ~R and stored.
The cube roots of the kinematic viscosities at 60~ and 200~F are computed
and stored. If the API gravity of the absorber oils is given (designated
by a negative sign), it is changed to specific gravity. Specific
gravities are stored.

=100The machine then computes the characterization factor of
each of the absorber oils and the constants for the ideal gas enthalpy
correlation. The fifth and sixth constants are zero. The molecular
volume is estimated as equal to 1.65 times the molecular weight (an
average value for the heavy paraffin hydrocarbons), and the one-third
and six-tenth powers are stored. The vapor diffusivity constants are
computed and stored in the 18 x 18 matrixo The computed functions may
be dumped if desired. This completes processing of the K values and
absorber oil data.
The first set of column data is read. If an error is detected, a remark is printed and the machine reads the next set. If no
set of column data is available, the machine returns to the reading of
another set of K value and absorber oil data.The column data are
printed for reference. If side stream cooling (intercooling) data is
included, it is dumped. A first approximation of either outlet stream
compositions or fractional recovery of components is dumped if supplied;
otherwise the machine writes a remark saying that it will compute the
first approximation.
The machine proceeds to an examination of the dimension and
the column code flags and converts the input data to the standard units
used by the program. These are seconds, feet, mole fractions, moles per
second, and degrees Fahrenheit. The flow sheet (Figure 6) adequately
explains this procedure.
The number of plates in the column, originally in fixed
point form, is converted to floating point also. The mean column crosssectional area and the area for vapor flow are computed. If no value

-101l
for the fraction of the column area for vapor flow has been given, 0.8
is used. If data for convection and radiation heat loss have been given,
they are converted to the units Btu/sec per plate, Hc/n.
This completes the data conversion phase, and control is
transferred to Subprogram B.
C. Initial Approximation
The computing of a first approximation to the recovery of the
various components or the use of such information if provided, together
with preparation for the iterative, plate-to-plate procedure, constitutes
the work of Subprogram B.
The machine begins by filling out the complete list of properties of the rich gas and lean oil streams. This consists of computing
the flow rate of each component (the product of the mole fraction and
the total stream flow rate), the molal enthalpy, and the total enthalpy.
The properties of stream V0 (the rich gas) are transferred to the corresponding locations for stream Vm_l in preparation for the plate-to-plate
calculation.
The average temperature in the column, t, is computed as the
arithmetic average of tVO and tL+l plus 15~F, This has been found to
be a fair approximation for most columns.
The computation code flag is examined. If the machine is to
compute the first approximation, the overall column efficiency is required, and the Drickamer and Bradford formula(27) is used to approximate
it.
Eo = - 1.78 - 0.268 an(tf) (180)
oc~~~~

-102Control transferred from Subprogram A
YO -Ym-l (Y)(V O) v0 - Vm-l3 (nl)(In+ l) "- Pn+1'
Compute -om' pu -t HVm (Vo) (VO) - HVo HVm-l
Compute %1~o (e+l) (1L+,) a H+.l
(tn+ + tVo + 0)(1/2) t ): Is approximate recovery given? es
No
Are approximate outlet compositions given?!_Yes
No.(Compute liquid density and liquid viscosity
Figure 7 of streams Ln+l and Vmo B
(Vo) (LVO) + (n+1) ('+Ln) -,' Vo + L%+
_| -1o78 - 0.268 xn('T) - Eoc, n/Eoc - nx'
[Obtain p^ and. K- values at t ]
(K)(V0) — A' Y0 [An'+l V n
(Continued)
Figure 7o Flow Diagram~Subprogram Bo

-103"
lYO R 3
(1 ~ R)(vO) - vn, Zvn - Vnt, nn yn
Vn
V*'n+l - Vn L1, vo + n+l - Vn - * 1 ~-*x h
Find component i for which [ + + 1 is a minimum..'- - ___-__.(yn)i..iQ.
[(n+l)i + )i (VO ( + Ln+l)(xl)i
(Y-.. —— 1 -' —...-.. — _Vn, Vn, V + Ln+l - Vn - L1.
(yn)i. xi
(Vn)(Yn) n...(Li'
YO
Insure that all components which appear in either
Vn or Ll appear in the other stream also, If a
component is missing, insert token concentration
(mole fraction) of 10-3.
Figure 7. (Continued)
Figure 7. (Continued)

-104Compute EHnM HVn,' L1 HL1
1 -'vo - 6s, HL, - Vo -H lc
Dump first approximation (if desired)
Transfer to Subprogram C
Figure 7. Flow Diagram-Subprogram B.

1055
where 4f is the viscosity of the feed in lb/ft sec. The viscosity of the
"feed" is taken as the average of the liquid viscosities of the rich gas
and lean oil, weighted by the flow rate of each stream. The number of
theoretical plates, n', is obtained as the quotient of n divided by Eoc,
and a set of K values is computed at t.
The machine then makes use of an absorption factor technique
quite similar to the Kremser-Brown method(l8'63) to calculate Yn of
each component, the quantity of each component leaving the column in
stream Vn per mole of rich gas entering. The absorption factor is
computed as
A = Ln (181)
A =?(K)(Vo)
and
A- 1
Ynl = YoC] i(182)
Yn'= Y0[An+l 1 (182)
The sum of the Ynt' multiplied by V0, is equal to Vn. The recovery of
any component is defined
R y - Yn. (183)
Y0
The remaining quantities and compositions of streams Vn and L1 are obtained by an overall column material balance. All material entering in
the lean oil is assumed to be completely recovered. The average temperature, t, is used as the first approximation of tVn and tL1.
If the approximate recovery of each component has been given
with the column data, the machine proceeds with the overall column
material balance as in the paragraph above and again uses t for approximating tVn and tLl.

-lo6If a first approximation of the compositions of the outlet
streams has been furnished, the machine picks out the component most
likely to afford an accurate material balance. This component is taken
as the one for which the sum [(l/Yn) + (l/x1)] is a minimum, the criterion
being that it occur in appreciable concentration in both streams. An
overall column material balance yields the quantities of the streams Vn
and L1. Temperatures of Vn and L1 are furnished with the column data
under the option of providing outlet compositions.
At this point, the three routes to making the first approximation of the product streams all join again. The machine insures that all
components entering the column occur in both outlet streams and inserts a
token mole fraction of 10-3 for any that are missing. The molal and
total enthalpies of streams Vn and L1 are computed, the delta quantities
for heat and for each component are computed, and the machine then
proceeds to the plate-to-plate calculation, Subprogram C.
D. Plate-to-Plate Calculation
Subprogram C begins with the transfer of the properties of
stream L1 to those of the liquid from the general plate, Lm. At this
point, the machine enters the column calculation loop and the calculation
loop for plate 1o
Using the temperature tL1 (now tLm), and the properties of
streams L1 and V0 (now Lm and Vml), the machine computes values of convergence pressure, K values, and plate efficiencies.'The quantity of
each component in the stream V1 (Vm) is computed from the formula
Vm,i = Vml[Ym-l + (EMV,i)(K,i Xm,i - (184)
~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~(mi Xm m-L~

-107Control transferred from Subprogram B
Set column iteration counter to zero
Move properties of stream Ll to those of Lp,
&q -,6qmi
Set plate counter to zero (k
Compute convergence pressure, pk
Compute K values at tLM
Compute plate efficiencies, EMV
Compute heat transfer efficiency, Eg
Hc
5m1 + b qm
vm-l + EMV (Vm-l Kxm - vml) vm (each component)
Vm
SVM — V> Vw 7-M'^ M
(Continued)
Figure 8. Flow Diagram-Subprogram C.

-1087,.1 m+l
Vm + 5s - em+l Z m+l - yl+ a1 l 4 Xm+1
Lm+1
(H)(Vm) ~ Vm' 1Vm I + + ~ —> ~ —
|tVm-I+ E (t Lm- tm)< 1 t0 m
(mV ) (V,) ~ HVmY tV m + _ Hm - HHL+ m ~ Lm +
Is detailed p late printoutert required
m, Rn(pk)p Vnme tV^m -T1 la LzH^ t49 ^'W ^n ^table|
Yes Is detailed plate prlntout requaired%?
Compute tbp of Lm and tdp of Vm
n,0 L
Print detailed results for plate m t
Figure 8. (Continued)

-109Is there intercooling data? I.No
Yes
nm+l + a HLm+l - HL +l + tLm tLm m+l
Recompute _+1, tL%+l Recompute -Lr+l HIi+l|
E —- HIm 3l - 1 rf _' All properties of Lm+1 moved to those of Lm
All properties of Vm moved to those of Vml
i Plate counter advanced by 1 (m+l - m)
|Has top plate been re ached (Does m = n+l?)
TNo JYes
Figure 8 (Continued)

-110I-Yes I Is isi > O? INo
Is m+l,*i <? Is v <
Yes No Yes
100 -rij Is en+lji > O?-1 - 100'r
Yes
n- > ri Is vni< 07
I No Yes
Vn,i _
Vmi - ri 01 -ri
Is ri >.01?.
Yes No
Is ri >.999? -- No _ l01 - ri
1 Yes
Is ri > 1.001? N-I
Yes
I f
Is r > 100? NI o
Yes
100 -+.rit —i Is o999 < ri < 1o001 for each component?
Yes No
Figure 8, (Continued)

-111Machine notes that mass convergence attained
s(^Y" (B1 i)(ri ) (m -- >m,i (each component)
i
[E ~msi -e L1
Is this the end of the 1st iteration? No
Yes
}^- ^j L> K^i) X1 el, i (li )k |
(r\~ ~ Is.999 < ri < 1.001?
No
-- --- - - m-i " - 1 k ei -1 i ^ ik-l
N o, i ( (, 1 ) k N o
| mi __ (w&i- m,i _li |
Yes Yes
(O~9)(mji) -,i (ll)(m,i) -, i
Figure 8. (Continued)

-112 -
\li (ii )k-l (mi i- (li)k-* l,i
Lis) —~ l, 1, -- i 18-o
~il i i
-n-o) ~ xl~i-~xli - x, i L i
Iais mass convergence been attained?
Yes No
J ItLn+l - tLm+ < 0.2? 1 Yes l
No
tL1 + (2)( - ) )(tLn+1- tLm+l) -tL1
tL1 - tLm...
Compute HL L1 HL -I HLm' (HL )(L1) -eHL1 e HL
Figure 8. (Continued)

-113Vo - Vm- l Yo - Ym-1 1 - Vvo "~6s' ~n+l - ns Vnj
Z Vn -n, V Yn
Vn
Vo -,Vmt-1 t tVml, Vo -* Vml o HV0 Vml
IDump ri; t.l+l;j ll
Have both mass and temperature convergence
been attained?
Yes No
Has machine been printing NPrint co'umn summary
only column'summaries? if ordered
Yes
Print column summary Maximum number of
iterations
Change instruction to print Yes No
detailed plate results
---. REMiiARK (
_ - O -n ^- R Subprogram D
Figure 8. Flow iagramSubprogra C.
Figure 8. Flow Diagram-Subprogram C.

-114and the sum of them is Vm. Mole fractions, m,i are computed, and then
the quantity of each component in stream Lm+l is computed from the stream
5si quantities.
)m+l,i =Vm,i+ i + (185)
The sum of the lm+l,i is equal to Lm+ and the mole fractions, Xm+l,i,
follow.
Using the properties of stream Vm, the machine obtains the
heat transfer efficiency and computes the temperature of the vapor, tVm*
tVm = tm + (E)(tLm - tVml) (99)
The molal and total enthalpy of Vm are obtained, and the corresponding
properties of L+1 by energy balance.
HL + = HVm + qm (186)
where
bqm = 6qml1 + Hc/n (187)
The temperature of the liquid stream, tLm+l is obtained as a function
of HLm+1 and Xm+l,i
Having completed a plate calculation, the machine checks the
size of Lm+1. If the condition
0.25 < Lm+l < 2.00 (188)
Ll
is not met, the machine advances the plate counter to n. The next time
the plate counter is checked, plate m will be treated as the top plate,
the rich oil properties will be revised, and the plate-to-plate calculation will be started again. Eight key properties of plate m are moved
to the column summary table, and the tag of the computation code flag is

-115examined. If it calls for a detailed plate printout, the bubble point of
Lm and the dew point of Vm are computed and placed in the proper locations, and the plate printout is made. The machine examines the column
code flag. If it indicates that there is any intercooling, the table
of data is checked to see if there is any cooling just above or on
plate m. If data have been given as a change in temperature of stream
Lm+l, tLm+l is increased by AtLm+l and ELm+l and HLm+l are recomputed
at the higher temperature. If the cooling has been expressed as a change
of enthalpy, Hl+l is increased and H+ and tLm+l are revised. The
heat flux, 5qm, is recalculated.
The plate calculation is now complete, and the machine prepares
to continue with the next plate. The heat flux, 6qm, is moved to bqm-,,
the complete properties of stream Lm+l are moved to those of Lm, the
properties of Vm are moved to those of Vml, and the plate counter is
advanced by 1.
The plate counter is examined to see if the top plate has been
reached. If not, the plate calculation is repeated; if so, the machine
inquires about convergence of the mass balance, taking each component in
order. The quantity 5si is examined. If it is identically zero, the
component is not present in the column. If it is positive, indicating a
heavy component whose net flow is downward, the ratio (Xn+li/2m+l,i)
will be examined; if it is negative, indicating a light component with
net flow upward, the ratio (vn,i/vm,i) will be used. For illustration,
we assume the latter case. The quantity vmi is first examined. If it
is negative, the machine concludes that too little of component i has
been assumed in the rich oil and sets the correction ratio ri equal to

-116m
the upper limit of all correction ratios, 102. Otherwise, the ratio
r. Vni (189)
1 v
Vmi
is computed. If it is less than the lower limit (including being negative), the lower limit, 102, is used instead. A ratio for each component is obtained, and if one or more do not meet the condition
0.999 < ri < 1.001 (190)
the mass balance has not converged.
The assumed quantity of each component in the rich oil is
revised by the following procedure: The convergence ratios are used
in the equation gi
R 2
(2l,i)k+l = (7l,i)k (ri)k (191)
where
(1,i)k+l is the quantity of component i proposed to be
used in the rich oil for the next iteration (k+l)
(li.)k is the quantity of component i in the rich oil
used in the iteration just concluded (k)
(ri)k is the convergence ratio obtained in the
iteration just concluded
gi, the numerator in the exponent on (ri)k, is an empirical
parameter for each component
m is the highest plate reached during iteration k.
The significance of the bar over (B, i)k is that it denotes a quantity
actually used as the input of a plate-to-plate column calculation. Now
if k equals 1, the quantity (1, i)k+l is used for the next iteration.
(,i)2 = (1,i )2 (192)

-117On all succeeding iterations, the Wegstein extrapolation procedure is
employed (see Mathematical Analysis),
(7l,i)k+l - (l,i)k (193)
(l,i)k+l = (l,i) + i) (,i)k+l] (l94)
and, lest any vagary in the extrapolation produce an unreasonable value,
a further condition is imposed:..,i )k+l
0.9< (71-k-l< 1.1 (195)
Revision of the quantities in the rich oil stream occurs whether mass
convergence has been reached or not, but the extrapolation is not used
on those components for which convergence has been obtained.
If mass convergence has been reached, the criterion of
temperature convergence is applied,
ItLnl (t+l)kl < 02 F (196)
If this requirement is not met, tL1 is adjusted according to the equation
Ln+1 Ln+l
(tL)k+l = (tL1)k + (2) V ](L k[ tL - (tL+l)k] (197)
If mass convergence has not been reached, the assumed temperature tL
is not changed.
The machine now computes the total quantity of stream L1, the
mole fractions ofa the various components enthalies, the mass flux or
&si quantities, and the properties of Vn obtainable by overall column

-LlSmaterial balance. The properties of L1 are transferred to those of Lm
and the properties of the rich gas to those of Vml1 in preparation. fbr
another column iteration. If the machine has been printing only column
summaries on each previous iteration and mass and temperature convergence
have been obtained, one more column iteration will definitely be made,
printing detailed plate results. If one of the other two printout
options has been employed and convergence has been obtained, the plateto-plate phase of the computation is finished. If convergence of both
mass and temperature has not been realized, another iteration will be
made conditionally. The machine dumps the current set of values of ri,
tLm+, and (-1 i)k+l and then gives a column summary unless the option
of printing only final results is in effect. If another conditional
iteration is to be made, it is made unless the maximum specified number
of column iterations has already been made, in which case the machine
prints a remark to this effect and proceeds to terminate the plate-toplate phase. The machine computes the fractional recovery of each component present and then transfers control to Subprogram D.
E. Output
Writing of the output in a convenient, tabular form is the
function of Subprogram D. The machine transfers properties of the exit
streams to their proper locations, transfers the compositions and temperatures to the locations used for the initial estimate of outlet compositions and temperatures, and changes the computation code flag to indicate
that these data are furnished, This transferring of data is done to
facilitate the running of a group of similar columns.

-119~
Control transferred from Subprogram C
Eject sheet
Move properties of exit streams to output locations
Move computed outlet compositions into storage
locations of initial estimate of compositions
Replace prefix of computation code flag with a 2
(Initial estimate of outlet compositions given).Write column number, input stream data, product stream data, composition of' lean gas and rich oil, and recovery of components
Write individual plate data on last,f:olumn iteration (if requested)
Figure 9,.Flow Diagram-Subprogram D.

-120The machine then writes the results of the column calculation
and returns control to Subprogram A for the reading of the next set of
column data or K value and absorber oil data.
F. The Liquid Temperature Subroutine
The graph of L versus tL is a smooth curve without inflection
points. If we imagine that tL is now a function of HL, we can write an
extrapolation formula based upon the Newton method. Given a starting
value tk and desiring to find t corresponding to H, we can write
t tk + dt (H -k) (198)
Approximating the derivative by the slope of a chord cutting the curve
at tk - At and tk + At, the points k-l and k+l, instead of the tangent
at tk,
(2)(At)(H - k)199)
b ^ttk+ ^ + -~- ^(199)
k Hk+l -1
We may substitute the point (Hk l, tk - At) for the point (Hk, tk)
and obtain
(2)(At)(H - Hk-1)
t X (tk - t) + (2 t)(H - (200)
Hk+l - Hk -
or, in the notation used in the rest of the program,
t' = t + (2)(At)[ t- t - 1 (201)
It+At -.t-At 2
The liquid temperature subroutine operates in the following
way: Given HL and a first approximation to the corresponding temperature,
tL, and required to find the exact temperature tL, the machine sets At

-121"
| LTEMP
Is first approximation of t given? 0o t
Yes
S.'
|H IH -'.- t -— 1 t Is this first iteration?'
t +2 A t( - 2-) - tt "' "
-LIQEW|QS —---- |IQSC
Yes_ Is t - At < t, < t + Att? Io
tf -ot V it -\t
Does At = Ol? | YeB | Maximrum of 10 iterations? |
No Yes
-I % -- 0 t
Yes!, No, -!- _Is -o00 <,t < 8oo?
Normal Return Replace t with exceeded bound
Off Range Return|
Figure 10. Flow Diagram-Liquid Temperature Subroutine.

-122"
equal to 5~F, uses formula (201) to compute a closer approximation, t',
and then checks to see if t. lies between t - At and t + At. If t'
does not, t' replaces t and the process is repeated. As soon as a t' does
lie within t + At, tt replaces t, t is set equal to 0.1~F and the above
sequence is repeated until t' is again obtained as an interpolation or
until a maximum of ten iteration of the process have been made. This
tT is returned as the stream temperature, tL. If tL lies outside the
range -100~F to 800~F, the off-range return is made.
G. The Dew Point Subroutine
The dew point temperature of a multicomponent mixture is that
temperature at which
Y.. = 1i (202)
1 Ki
We may therefore define the function
Yi
f(Yi, t) = 1= 0 (203)
(since K is a function of temperature and composition) and find the value
of t which satisfies the equality. (The subroutine actually uses y'Yi
instead of 1 and ignores negative mole fractions.)
The logic of the dew point subroutine is given in Figure 11.
Briefly, the machine uses the method of false position to find the root,
tdp. This consists of computing values of the function f(Yi, t) for a
series of temperatures (differing by Zt) until two values on either side
of zero are found. Linear interpolation is performed to approximate tdp.
the interval At is shortened from 50~F to 2~F, and the procedure is
repeated once more. The result of the second interpolation is taken as
the dew point temperature.

-123IDEWP7T
Is first approximation to temperature given? — No - 300 - t
Yes
0 -, iteration counters 50 -.At
Modify program to compute convergence pressure and K values as a
function of vapor composition.
Compute pk
Compute set of K values at t
Compute Sy, Z- Store values of - as liquid mole fractions,
K K
Eyt A'AZ. Is sAz =O? Yes
Is Az >= 7O
YeY
I Is this the first iteration? Is this the first iteration?
|_NTo TNo Yes
Was.A z > 0 on last No
iteration? No Was Az < 0 on last
iteration?
Yes
Yes
- t + Qt -~ t, Az Ailr | i —-- t - At > t, Az A 4z'
N
Maximum of 25 iterations l ( l f > —----
FFoa Poit-S SubrotL t + t
Yes JAz _ Az' |
_'Mqdify program to use liquid _ Yes Is program using vapor compoaicompositions. 0 ->counter. tions for computing K, p.?
No~
I2 At ^ —---— t Is A.t = 50?
Yes No
Figure 11, Flow Diagram-Dew Point Subroutines

VI. MATHEMATICAL ANALYSIS
A. The System of Simultaneous Equations
In company with a great many other chemical engineering calculations, the solution of a multicomponent absorption or distillation
problem may be considered as the solution of a set of simultaneous equations. The requirement for the existence of a unique solution is that
the number of equations or relationships be equal to the number of variables, and that the equations be linearly independent; in other words,
that no one of the equations be a linear combination of any others, The
plate-to-plate procedure described here is an example of an iterative
solution, exemplified (for a system of linear equations) by the GaussSiedel method, as opposed to the direct solution illustrated by the
Crout matrix inversion.
The problem of fixing the values of just the right number of
variables in order to leave the number of remaining variables exactly
equal to the number of equations has been the subject of articles by
Gilliland and Reed(40) and lately by Kwauk(64). Using the nomenclature
and technique of Kwauk, it will now be shown that the non-ideal stage
absorber problem is a completely specified one.
Kwauk has defined an element as "a single stage equipment,
such as a flash drum, a total condenser, a theoretical plate, etc." For
any element, the number of variables available for process specification,
Nea, is equal to the total number of variables, NV, minus the number of
conditions inherent to the system, Ne, minus the number of normally fixed
x m
Ne = N - Ne N| (204)
-12o

-125%
The phase rule states that for one phase or for two or more in equilibrium,
N C + 2 - (205)
that is, the number of degrees of freedom, N, is equal to the number of
independent components, C, plus 2, minus the number of phases, 0. The
degrees of freedom are "intensive" ones, and to them may be added one
more, an "extensive" one, the rate of flow. Therefore, either for one
phase or for two phases in equilibrium, where the flow rate of either
phase may be varied independently, the number of variables is C + 2. In
consideration of any element, one may introduce yet one more variable,
the heat gain or loss to the surroundings.
Kwauk, in discussing the theoretical plate, states that there
are C + 2 variables associated with each of the streams entering the
plate, C + 2 with the two equilibrium phases leaving, and one more with
the heat transfer aspect, for a total of 3C + 7.
Ne- = (2)(C + 2) + (C + 2) + 1 = 3C + 7 (206)
The inherent conditions are C material balances and one heat balance
for all streams, so
Ne = C + 1 (207)
The normally fixed variables are the compositions and flow rates of the
two entering streams, the pressure, and the heat exchange with the
surroundings, so
Ne = (2)(C + 2) + 14 + 1 = 2C + 6 (208)
x\

-126Therefore,
N = (5C + 7) - (C + i) - (2C + 6) = 0 (209)
a
and the theoretical plate just described is invariant.
Let us now consider the non-ideal stage. In place of the two
equilibrium phases leaving the plate, we now have two which are not in
equilibrium and.NJ = (4)(C + 2) + 1 = 4C + 9 (210)
As before, inherent conditions include C material balances and one heat
balance, but there are others also. The C + 2 variables associated with
a single stream consist of C - I compositions, pressure, temperature, and
total flow rate, or (equivalently) C component flow rates, pressure, and
temperature. The reason that C + 2 variables define a two-phase equilibrium system, even though either stream alone has C + 2 variables
associated with it, is that the condition of equilibrium implies C + 2
additional restrictions: C mass transfer equilibrium relations, equal
temperatures in both phases, and equal pressures. Returning to the nonideal stage, we introduce an additional C + 2 inherent conditions: C mass
transfer rate or non-equilibrium relations, a heat transfer equation relating temperatures in the two phases, and again the requirement that
pressures in the two phases be equal. Therefore,
N_ = (C + 1) + (C + 2) = 2C + 5 (211)
As before,
Ne = 2C + 6 (212)
x
and so the single non-ideal stage is also an invariant system.
Ne = (4C + 9) - (2C + 5) - (2C + 6) = 0 (215)

-127Proceeding now with Kwauk's analysis of a multiplate contactor,
we note that the total number of variables associated with a system of n
elements is 1 (for the option of deciding the value of n) plus n times
the number of independent variables associated with each element. Now
Ne = N - Ne = (4C + 9) - (2C + 3) = 2C + 6 (214)
and so
E = 1 + (n)(2C + 6) = 2nC + 6n + 1 (215)
V
and the capital E used as a superscript signifies a complex element or
system of elements. The inherent conditions are summarized by the notion
of interstreams. If plate m is neither the top nor the bottom plate of
the column, then the two feed streams to plate m are themselves the
product streams of other plates. To avoid double counting of the variables associated with these interstreams, we recognize these restrictions
as inherent conditions.
N = (2)(n - 1)(C + 2) = 2nC - 2C + 4n - 4 (216)
The normally fixed variables are the properties of the feed streams, the
pressure on each plate (assumed constant throughout the column in our
case), the heat transfer with the surroundings on each plate, and the
total number of plates.
NE = (2)(C + 2) + n + n +1 = 2C + 2n + 5 (217)
Therefore,
N = (2nC + 6n +) - (2nC - 2C + 4n- 4) (2C + 2n + 5) = 0
(218)
and the non-ideal stage absorber is proved to be invariant or completely
specified,

-128B. The Iterative Solution
The set of equations characterizing conditions in an absorber
may be thought of as a system of n non-linear equations in the variables
x1 through xn with xj the general variable. The equations 1 through n
may be solved for variables 1 through n respectively to give another
set of equations
xj = fj( x2, x,..., xn) ( = 1, 2,219)
Two general methods exist for the iterative solution of this set of
equations —the total step and the single step. In the total step procedure, when k is any iteration,
(xj)k+l = fj[(Xl)k, (x2)k,..., (xn)k] (220)
In the single step procedure,
(xj)k+l = fj[(Xl)k+l..., (xj-l)k+l, (xj)k,..., (xn)k] (221)
If the set (xj)k is considered a vector, then for the total-step procedure,
k+1 T xk (222)
and for the single-step procedure,
Xk+1 = T (xk+l, Xk) (223)
where "T" means "a transformation of". In the case of linear equations,
the above procedures will be recognized as the Jacobi and the Gauss-Siedel
methods, respectively.
Collatz(23) has investigated iterative solution of non-linear
equations and has developed sufficient conditions for convergence of

-129=
total- and single-step methods. Let the set of equations (219) have
continuous partial derivatives in the region F, and let ar be the maximum absolute value of the partial derivative of fj with respect to each
of the variables r.
ar.
air IlaFITl (224)
Ajr maxF (224)r
These ajr may be considered to form a matrix. Now if
n
k = max. Z ajr (225)
r=l
and k is less than 1, the "row sum" criterion is met and the sequence
(220) or (221) will be a convergent one. If
n
k - maxr ar (226)
r j=l j
and k is less than 1, the "column sum" criterion is met and the sequence
likewise converges.
One further criterion, and the most important for our purposes,
applies only to single-step procedures. Let
n
I = C oals (227)
s=l
and
j-l n
Pj = ajsps + Z ajs (228)
s=l s=j
Now if k is the maximum value of Pj and is less than 1, the single-step
procedure is convergent.
Unfortunately, the proofs given by Collatz cannot be applied
directly to the absorber problem because the condition of continuous

-130partial derivatives cannot be met. The region F must include all possible
values taken on by the variables during the iterative solution, not just
the ones which may be expected in operation of the equipment. Two of the
most frequent sources of discontinuity in the partial derivatives are
negative quantities of certain components (which may cause a total stream
to become negative and thus abruptly reverse the signs of all mole fractions) and the exceeding of bounds in the correlations of various physical properties.
examination of Equations (224) through (228) will produce some
valuable principles even though the equations themselves may not be
applicable. First, the set of equations should be arranged so that all
the partial derivatives are as small as possible. (The equivalent
statement in the case of a set of linear equations is that one should
order the equations to produce a large main diagonal in the matrix of
coefficients.) Secondly, if one has equations with large jr, he should
put them toward the end of the list, so that as many of the ajr as possible may be multiplied by Pj less than 1. Further, if one has equations
which involve only a few of the variables, they should be placed in the
list in such a way that as many as possible of the ajr fall below the
main diagonal of the matrix of ajr.
The "natural" sequence of equations used in a plate-to-plate
calculation will usually satisfy the second and third principles. One
usually does not have machine storage enough to store two sets of values
for each of the variables involved, and so one begins at one end of the
column and proceeds from one plate to the next, using values calculated
on plate m as a basis for the evaluation of the variables on plate m+l,
etc.

-131C. Experience with the Iterative Solution
As explained previously, the iterative solution of the absorption column is carried out by (1) assuming the properties of the rich oil
stream, L1, (2) performing a plate-to-plate calculation upward through
the column until the stream Vm corresponding to the lean gas, Vn, is
reached, (3) calculating the properties of Vn by overall material and
energy balance with L1, (4) comparing corresponding properties of V
and Vn, which should ultimately be equal, of course, and (5) on the
basis of this comparison, revising the estimate of the properties of L1.
The most mathematically straightforward method of revising the
properties of L1 would be by material and energy balance with Vm For
example,
(2l,i)k+l = VOi + n+al,i (vmi)k (229)
This method has been found to be grossly unstable for the lighter components. For nitrogen in a typical column, Il might be on the order of
0.0008 moles/sec and vm about 0.21 moles/sec. A small percentage change
of 21 might result in only a small percentage change in v, but the absolute change of vm relative to the size of jL and hence the partial
derivative of vm with respect to 21, would be quite large. This "direct
substitution" method was found to be unsuitable.
The percentage change of v caused by perturbation of 21 was
m
found to be the key to forcing convergence of the iterative sequence.
The revision
(r(Vni)k) (230)
(l,i)k+l = [(,l, i)k ] [ (20)k
\m~ i/ i

-132made simultaneously to all components, was successful with a simplified
three-plate column. For longer columns, the formula was
91
(Vn.,i)k
(1l,i)k+l [(= i vkn, )k( (231)
(VM'i)k
for those components whose net rate of flow was upward (the lighter
components) and
(X,=i)k+l [ I( i)k]M[( gi (252)
C(Rl~ik (y (m+l,i)k
for those whose net flow was downward (the heavier components). This
choice resulted in the more stable of the two possible ratios being used,
i.e., the one involving larger numbers. The divisor in the exponent, mn,
was chosen in order to produce a greater correction in the first few
iterations, when the calculation proceeded only part-way up the column.
The parameters gi in the exponents were determined experimentally, and the following set was found to be satisfactory for all
problems run: 0.2 for the non-hydrocarbon gases and methane, 0.5 for
hexane through decane, and 1.0 for ethylene through n-pentane and the
absorber oils.
At this point, a procedure had been developed which was stable
and convergent but quite slow. A thousand iterations might have been
required for the solution of a difficult problem, and the need for some
method of speeding convergence was evident.
D. The Aitken "Delta-Squared" Method
Aitken's 2 method has been adequately discussed elsewhere (4 9,56)
and so no detailed description of it will be given here. If one has a

-133convergent function in one variable
xk+l = f(xk) (233)
and the sequence of approximations xk-l, Xk, xk+l to the root x, then
rather than use the value xk+l as the argument for the next iteration,
one may speed up the convergence by using an extrapolated value xk+l
defined
- (xk+l) (k-l) (xk)2 (xk+l - xk)
xk+ xk+l - (2)(xk) + xkl x k+l xk+l (2)(xk) + xkl
(sk )2
- Xk+l y - (x) (254)
The extrapolation may be applied every other iteration to give the
sequence xk3, Xk-2 Xk-l, xk, Xk+l,.., the bar indicating an argument
obtained by extrapolation.
It was found that the A2 process could be applied to each of
the sequences of values of 0l i in spite of the fact that a system of
simultaneous equations was involved, provided that certain precautions
were taken. The use of the extrapolation on all 1 at once would
cause a disruption of the slow, smooth convergence, and certain of the
l, i would begin an accelerating approach to the final values. It was
found necessary to delay the next application of the A method to any of
the variables until the acceleration had ceased for all, and then apply
the extrapolation once more to all the variables at once'. Eight or ten
iterations usually occurred between applications of the A2 method, and
so it was not particularly useful even though it did speed up convergence
somewhat.

-134The difficulty encountered with an increasing absolute value of
the first difference of the sequence of I1 i can be readily understood
from an examination of the "symmetrical" form of Equation (234). The
quantities xk-l and xk+l are interchangeable, and the extrapolation is
made in the direction of decreasing absolute value of the first difference.
E. Wegstein's Method
Recently, Wegstein(92) described a technique which, when applied to an iterative procedure, accelerates the rate of convergence if
the iteration converges and very frequently results in convergence of
otherwise divergent iterations.
Suppose that we have the iterative sequence
xk+l = (xk) (233)
and that
-l < f'(x) < O (235)
in the vicinity of the root, causing convergence in an oscillating manner.
In order to change the rate of convergence, the successive values of xk
must be changed. We denote this by
Xk+l = (q)(xk) + (1 - q)(xk+l) (26)
where
Xk is the argument used in iteration k+l
Xk+1 is the value of f(xk) produced by that iteration
xk+l is the value to replace xk+l as the argument of
iteration k+2
q is determined anew for each iteration.
Let the convergence be as shown in Figure 12.

-135W<^ /
f(XK+l) I
Xf
igure 12. he Weten Mtod
Figure 12. The Wegstein Method

-136The ideal value of q is such that xk+1 would equal the abscissa
of point C; i.e., we would like to have
q BC
=- CA (237)
1 - q'CA
To get an approximation to q, we note that PC = BC and PC/CA = - w,
where, by the Mean Value theorem, w is the value of f' (x) at some point
between P and A. Thus
1 = - w (238)
i q
and
w-w 1 (239)
Since a more convenient method is lacking, w can be approximated by a
suitable difference quotient
f(xk) - f(xk-l) Xk+l - Xk
W = - (240)
k - Xk-1 Xk - xk-1
At this point, the method is equivalent to the Aitken A2 method, which
may easily be proved. Beginning with Equations (236) and (239),
Xk+l (w )(xk) + (w-. )(k+l) (241)
(Xk+l - x)(k - Xkl)(xk) (k - xk-l)(xk+l)
xk+l k -l x )(k+x l k - k + Xkl) (k+l - - xk + xk-1)
(xk+l - xk)(xk) (Xk - xk-) (xk+l) (xk+l)(xk-l) - (Xk)2
k2-1 Xk-l k^-1
-A2xkl A2xk1 A2Xkl
(242)

-137A variation of Equations (236) and (240) results in the Wegstein method.
Xk+l = ()(x) + ( - q)(xk+l) (243)
Xk+! - xk
w + k (244)
xk - Xk1
Whereas the A2 procedure may be used (at most) every other iteration, we
now have a procedure which may be used every iteration after the first.
(On the second iteration, it is equivalent to the A2 method.) Its increased utility may be rationalized from this fact and also from the
fact that the Wegstein method produces an extrapolation based on four
previous values, rather than three.
The Wegstein procedure was applied independently to each
sequence..., ( i) k-l' (1,i)k' (, i)k+l'..' for each component i
and was found to accelerate convergence by a factor of 15 or 20 over
the basic, slowly convergent sequence. The greatest improvement occurred
during the initial phases, when the values of 1,i were relatively distant from their converged values. In order to avoid difficulties caused
by round-off error in the calculation of very small second differences,
(1) the successive values of lii were required to satisfy the condition
(~l,i)
0.9< < 1.1 (195)
and (2) use of the extrapolation method was discontinued for each variable when the error in that variable had been reduced within allowable
limits. The success of this very naive use of the extrapolation method
on a system of perhaps twelve or fifteen variables can be credited

-138to the fact that the dependence of the variables on one another is quite
small, that is,
fj(xl, x2^,... ( xn) j(X...,) (25)
c) xr o xj

VII. RESULTS
Following are results obtained in the investigation of five
different columns, the material being arranged in the chronological
order of the work. Where pairs of values occur in the tables, the top
line is the computed value; the bottom line, the value with which comparison is made, usually the test value. The solid dots on the figures
are computed values; the circled points, test results. Various other
symbols —short arrows, triangles, dashed lines, and asterisks —occur on
the figures. Their significance is explained under Analysis of Results.
On the average, fifty iterations of the column calculation
loop were required for the solution of each of the problems. Machine
time varied between 45 minutes and an hour.
A. An Ideal Stage Column
During the early stages of the investigation, a test was made
to determine if setting Eq and EMV of all components equal to 1.00 would
produce a reasonably close approximation of an ideal stage calculation,
that is, if the non-ideal stage calculation would reduce to the logical
limiting case. Examination of Equation (89) shows, of course, that the
calculation will be identically an ideal stage one only if the vapor
stream rate does not change.
= (Ym)(Vm) =Vm-l [ym-1 + EMV (Km - Ym_1)] (89)
A test problem involving a three-plate ideal stage absorber
at 100 psia was devised, and the problem was solved using the ideal
stage, IBM 650 program developed by Norman(76). The IBM 704 non-ideal
-139

-14o0
stage program was then modified to use efficiencies of 1.00 and simplified correlations of enthalpies and K values in order to approximate the
correlations used in the Norman program insofar as was convenient. A
comparison of the results is given in Table 10.
B. Column A
The Phillips Petroleum Company donated a set of test data for a
20-plate, bubble cap column operating at 232 psia. Plate detail drawings
indicated that the column diameter was 6.o ft, plate spacing 27 in.,
length of the liquid path 3.59 ft, and fractional cross-section for
vapor flow 0.768. The column was assumed to operate adiabatically. The
"'mineral seal oil" fraction had a molecular weight of 223.
A comparison of machine computation results with the test data
is given in Tables 11 and 12, and in Figure 13. Computed plate efficiencies appear in Figure 14. The values of y upon which the machine
calculation was based appear in Table 11. Reference to Figure 13 will
show that in some cases they are not the same as any of the test analysis
points. Apparently these values of yO were chosen as representing average values or most reliable estimates. The mole fractions of nitrogen,
methane, and ethane inr the rich oil appearing as test data were actually
values calculated by the Phillips pe6ple. They assumed that (1) the
rich oil was a bubble point liquid and (2) the ratios (y, CH4/Y N2) and
(Y, CH4/Y1,cH6) were the same as the ratios (YOCH4/Y0,N2) and
(YoCH4/YOC2H6), respectively, i.e., the mole fraction ratios in a
vapor stream in equilibrium with the rich oil were equal to those same
ratios in the rich gas. If the K values used had been those which the
N.G.A.A. data would predict, the mole fractions of nitrogen, methane,

-141and ethane in the rich oil would have been approximately 0.0030, 0.0607,
and 0.0298 instead of 0.0014, 0.0458, and 0.0276.
C, Column B
A data set contributed by "X" Petroleum Company (name withheld
by request) contained results obtained with a 27-plate, Koch Kaskade-type
column operating at 740 psia. Inside diameter was 4.5 ft, plate spacing
26 in., and the Kaskade plates were estimated to give an equivalent
liquid path of 4.5 in., weir height of 2.25 in., and a fractional
column cross-section for vapor flow of 0.390. The plate efficiency
subroutine is intended for bubble cap plates only, but the data set was
used because it was one of the few which included compositions of in.
ternal streams.
The unit was actually a combined accumulator and absorber,
and the properties of the vapor stream entering the absorber were pbtained by mass balance around the accumulator. The volumetric flow rate,
API gravity, A.S.T.M, distillation, and temperature. of the lean oil were
furnished, and the composition and temperature of the rich oil, Two
components listed as "156 M.W." and "170 M.W." were combined as absorber
oil 1. The "absorption oil" fraction of the lean oil, designated absorber oil 2, was estimated to have a molecular weight of 210; the entire
lean oil, a molecular weight of 190. (The lean oil had a 10% - 90%
slope of 1.2~F per percent, volumetric average boiling point of 483 ~F
and gravity of 41.3~API. The 80% - 98% fraction distilled between 513~F
and 60o9F, and its gravity was 39.5~API,) With these estimates, the remaining properties of the lean oiland the rich oil were calculated by
overall column material balance.

-142Results of the initial Column B run are given in Tables 13
and 14 and Figures 15 and 16. Significance of the short arrows in
Figure 15 and the small triangles in Figure 16 will be explained under
Analysis of Results.
When it became apparent that the estimate of lean oil composition had contained excessive amounts of certain of the rich gas components (n-butane through hexane), another run was made using a wellstripped lean oil. This lean oil contained only 0.001 mole fraction of
hexane and heptaneo Results are given in Tables 15 and 16 and Figure 17.
Since Column B operated at the highest temperatures of any of
the columns investigated, a third run was made to demonstrate the effect
of an intercoolero The intercooler was placed above the 10th plate
(just above the hottest part of the column) and produced a 10~F temperature change. Results are presented in Tables 17 and 18 and Figure 17.
In Table 17, the comparison is of the machine-computed results with and
without the intercooler. Because of machine time limitations, computation was discontinued at a point where mass balance convergence had been
obtained for an assumed rich oil temperature which was still a bit too
high. The computed lean oil temperature was 1.57 F above the given
value, and it was known from previous experience with this column that
lowering the assumed rich oil temperature by about half this amount would
lower the lean oil temperature by the required 157~F and the temperatures
of intermediate streams proportionately. The original results are found
in Table 18, but the revised temperatures are plotted in Figure 17 in
the interest of presenting an orderly picture.

-143D. Column C
A set of test data was obtained through the courtesy of the
N.G.A.A. Absorption Committee. The 30-plate, bubble cap column operated
at 1475 psia, the highest pressure investigated. Column diameter was
5.5 ft. plate spacing 2.0 ft, length of liquid path 3.0 ft, and weir
height 3.0 in. Heat loss by convection and radiation was estimated to
be 60 Btu/ft2 hr. Molecular weight of the absorber oil component was
estimated to be 207 (molecular weight of the total lean oil stream was
given as 205).
A rather severe degree of instability was experienced in the
machine computation, showing up most strongly in the various quantities
of carbon dioxide. It is probable that this component did not cause
the instability, but merely reflected it in the greatest degree because
of its high volatility and low concentrations. The program was modified
to omit application of the Wegstein extrapolation to carbon dioxide,
and the calculation then converged slowly and with no further difficulty.
Results are given in Tables 19 and 20 and Figure 18.
E. Column D
An absorber data set designated as Unit 3 of the N.G.A.A. Low
Pressure Data was investigated in an attempt to predict the value of Cq,
the empirical constant in the heat transfer efficiency equation. (This
aspect will be discussed under Analysis of Results.) The column operated
at 400 psia, was 5.0 ft in diameter, had plate spacing of 2.0 ft, liquid
path of 2.917 ft, weir height of 1-7/8 in., and a fractional crosssection for vapor flow of 0*687. Molecular weight of the absorber oil
was 207, and this was the only column investigated whose lean oil

-144consisted entirely of the absorber oil component. The original data set
specified 0.012 mole fraction of air in the rich gas and 0.013 in the
lean gas. This small amount was eliminated in the interest of expediting
convergence of the iterative calculation, and the remaining mole fractions were normalized. Results are given in Tables 21 and 22 and
Figure 19.

-145TABLE 10
RESULTS FOR AN IDEAL STAGE COLUMN
Property_ _ Rich Gas Plate 1 Plate 2 Plate 3 Lean Oil
Vm (moles/sec).6000oo 0.5911 0.5855 0.5683
0. 6000 0.5902 0.5846 0.5670
L (moles/sec) - - 0.1517 0.1428 0.1372 0.1200
m — 0.1550 0.1433 0.1376 0,1200
tm (~F) 80.00 101.19 103.36 100.49 90.00
80,00 101.20 103.39 100,88 90.00
Ym (methane) 0.9000 0.9131 0.9216 0.9413
0.9000.9143 0.9230 0 9435 -
(n-butane) 0.0900 0.0851 0.0774 0.0577
0.0900 0.0842 0.0761 0.0557
(heptane) 0.0100 0.0018 0.0010 0.0009 -
0.0100 0- 0013 0.0007 0.0007
(abs. oil) -0 10-5 10-5 10-5
0- -0 -0- 0- --
xm (methane) -- 0.0531 0.0334 0.0340 -0--.0329 0,0331 0,0339 -0(n-butane) -- 0.1476 0.108 00998 08.0100
- o 0.1541 0.1348 0.1025 0,0100
(heptane) -- 0.0677 0.0573 0.0354 0.0400
-- 0.0679 0.0363 0.0349 0.0400
(abs. oil) -- 07516 0.7986 0.8308 0O9500
0 7451 0.7958 o,8289 0.9500
Lm/Vm ratio - 0.2567 0.2438 0.2414 -
0.2598 0.2458 0.2431
Bub. Pt. of Lm - 105.33 106.07 106.76
Dew Pt. of Vm 101l90 1035.80 101.44
Fractional recovery of methane 0,0093 n-butane 0.3924 heptane 0.9120
0.0093. 4145 0.9525
In each pair of values: top line-value by non-ideal stage calculation,
bottom line-value by ideal stage calculation.

-146TABLE 11
COMPONENT RESULTS FOR COLUMN A
Component YOe. Ew
Component Vy Xn+l Yn xL Rec. Avg, EMV
Nitrogen 0,1540 0 o.l604 0.0030 0.0037 0.0012
0,1540 0 0.1615 0 0014*
Methane 0.7211 0 0.7423 0.058 0.0157 o,0046
0.7211 0 0.7434 0.0458* 0,0130
Ethane o,0650 0 0.0622 0.0283 0.0842 0,0214
0,0650 0 0.0621 0.0276* 0.0853
Propane 0.0375 0 0.0289 0.0509 0.2626 0.0562
0,0375 0 0.0289 0.0510 0.2621
Isobutane 0.0045 0 0.0022 0.0123 0.5296 0.0987
o.o0045 0 0.0021 0.0129 0.5532
n-Butane o00104 0.00011 0.0036 0.0363 0.6729 0.1192
0.0104 0.00011 0.0 016 0.0462 0.8528
Pentane 0.0047 0,00251 0.0003 0.0246 0.9339 0.2083. 0047 0.00251 0.o0004 0.0244 0, 9186
Hexane + 0.0028 0.00087 0.00001 0.0151 0.9957 0.2830
(Heptane) 0.0028 0,00087 - 0.0152 1,0
Abs, Oil 0 0.99651 0.00001 0.7709 - 0.2554
0 0.99651 -- 0.7755 -
* Estimated values. In each pair of values: top line-computed value,
bottom line-test value,

-147TABLE 12
PLATE RESULTS FOR COLUMN A
Plate Vm tvm Lm tLm Bub.Pt. Dew Pt. E Conv.
Lm Vm Pres.
LOil.1 -- 0.2044 67. 0 0. O 0. 2044 67.00
20 1.3068 74.19 0.2095 68,87 995.85 77.42 0.1338 7125.
1.3094 72.00
19 1.3119 7'501 0.2148 70.44 791.71 77.24 0.1453 7576.
18 1o3172 75.79 002196 71L85 597.37 77.66 0.1571 7564.
7600 --
17 1.3220 76.52 0.2240 73.18 404.85 78.06 0.1684 7561.
16 1.3264 77.20 0.2280 74.40 251.41 78.41 0.1793 7552.
15 1.3304 77.81 0.2315 75 51 189.59 78.70 0.1897 7545.
14 1.3339 78.35 0.2347 76.50 156 40 78.91 0.1996 7537....... ~0~ O0........
13 1.3371 78.81 0.2375 77.39 135.71 79.03 0.2089 7530.
12 1.3399 79.18 0.2401 78.18 121.85 79.05 0.2177 7521.
11 1,3425 79.46 0.2424 79.14 112.20 78.93 0.2260 7516.
10 1.3448 79.56 0.2445 79.70 105.33 78.53 0.2337 7507..... 80o.00o o
9 1.3469 79 52 0.2464 80o14 100.37 77.89 0.2411 7498.
8 1.3488 79 32 0.2483 80.47 96 77 77.96 0.2484 7488.
7 1.3507 78.93 0.2501 80.67 94.14 76,76 0.2557 7476.
6 1.3525 78.34 0.2520 80.74 92.24 75.06 0.2632 7464.
5 1.3544 77.48 0.2539 80,64 90.88 72.68 0.2712 7448.
4 1.3563 76 30 0.2560 80.77 90.00 69 37 0.2798 7433.
3 1.3584 74.56 0.2583 80.28 89.43 64.70 0.2891 7411.

-148'
TABLE 12
PLATE RESULTS FOR COLUMN A (CONT'D)
Plate Vm tVm Lm tLm Bub.Pt. Dew Pt. EQ Conv.
Lm Vm P-res.
2 1.3607 72.24 0.2610 79.48 89.10 57.81 0.2998 7383.
8300oo
1 1.3634 69.14 0.2642 78.26 88.97 48.64 0.3124 7347.
- -- 0.2627 80.00 -- - n
R.Gas 1.3666 65.00 —.- --..
1.3666 65.oo0 -
In each pair of values: top line-computed value,
bottom line-test value,

-14920 _ o 0 0- 0
19 0 0 O
18 L 1 NITROGEN
17 - METHANE
ETHANE
16
15 - L I B I o +
14
13 O i
12
11 -0 0
w 10
a)0
D 9
w8
I3 7_ j, O o
5 3 00 0
3 0
2
I -I 0
0- 010 0
Io - i odo o I
0.15 0.16 0.165 0.72 0.73 0.74 0.75 0.06 0.07
MOLE FRACTION IN VAPOR.
Figure 13-a. Results for Column A

-15020 Q
19 o
18 L PROPANE
17 0 b(
16 ISOBUTANE
15 00\ 0
14
13 0 0
12
10
I 8.t 7 0 0.5 6
5 (- O E o
G:~b~~ 0 I
4 _
3- 0G0
2 _
0. I I _..... I,.,, I_ _,, I I.I1-.
0.03 0.04 0.002 0,003 0,004 0005
MOLE FRACTION IN VAPOR
Figure 13-b. Results for Column A

-15120 o Q
19 Q
18
17 0
N-BUTANE
16
15 o
14
13 F
12
II - \
a 10
w
9, \
co
8
7 - \
16
3 -O \
2.
I a~ \
0 0b
I | |. I I!.., I I _ I I - I
0 0.002 0.004 0.006 0.008 0.010 0.012
MOLE FRACTION IN VAPOR
Figure 13ec. Results for Column A

-15220 0
19 0O,18 PENTANE
16 THEPTANE
15 0o
14
13
12
10
z 8
3- e \
3_
0 I I
0 0.002 0.004 0.006 0 0.002 0.004 0.006
MOLE FRACTION IN VAPOR
Figure 13-d. Results for Column A

-153L.O.
20 _
\ LIQUID RATE
19
16.20 02 0206LIQUID
\t I
15 M TEMP.E
iure 13. sults for Colmn A
12
I
5 1
4_ I
i2 _
Figure 33~. Reeults for Column A

-15420 ~
16
4
t
0
0.001 0.002 Q000 0010 0020 0.060 0.10 0.20 060 100
MV, E
Figure 14. Plate Efficiencies in Column A

-155r
TABLE 13
COMPONENT RESULTS FOR COLUMN B
Component YO - Xn+l xL Rec, Avg. EMV
Nitrogen 0.0410 0 o.0466 0,00204 0,0194 0.0030,
0.0410 0 0.0536
Methane 0.8026 0 o,8610 0.15201 0.0739 0.0105
0.8026 0 0.8915 0.11638 0,0325
Ethane 0.0723 0 0.0595 0.05360 0.2894 0.0360
0.0723 0 o 0442 0.05314 o,4674
Propane 0,0597 0 0.0269 0.09352 0o 615 0.0707
0.0597 O0 0.002 o,08000 0.8654
Isobutane 0.0059 0 " 0.0012 0.01255 0. 8302 0,1020
0.0059 0 0.0007 0.01581 0.9091
n-Butane 0.0135 0.0123 0.0033 0.03534 0.7909 0.1187
0.0135. 0123 0.0005. 04400 0.9682
Isopentane -0.0016 0.0080. 0006 0.00784 0.6458 0.1518
0.0016 o.0080 0.0001 0,00952 0.9333
n-Perntane 0.0014 0.0110 0.0007 0.00919 0. 5689 0.1637
0.0014 0.0110 0.0001 0, 01124 0 9231
Hexane 0.0004 0.0093 0.0002 0.00651 0.4552 0.1921
o.0004 0.0093 0,0001 0.00743 0.7500
Heptane 0.0016 0.0010 0.00002.00471 0.9918 0.2070
00016 0.0010 -- 0.00057 1
Octane 0 0.0017 0.00001 0.00109 O- 0.2111
0 0,0017 -- 0. 00114 --
Nonane 0 0.0169 0.00002 0.01093 0.2096
0 0.0169 --. 01181
Decane 0 0.0287 0.00002 o 0 860 -- O.2063
0 0.0287 - 0.02000
Abs. Oil 1 0 0.1573 0.00007 0.10204 -- 0.2009
0 0.1573 - 0.10838 -
Abs. Oil 2 0 0.7538 -- 0.49003 -- 0.1906
0 0.7538 0.52058
In each pair of values: top line - computed value,
bottom line - test value,

-156TABLE 14
PLATE RESULTS FOR COLUMN B
Plate v tvm Lm tLm Bub.Pt. Dew Pt. E Conv.
Lm Vm L VPres.
L.Oil - - 0.2365 103.00
0.2365 103.00- --
27 0.8049 109.12 0.2423 104.33 -- 127.81 0.0772 4809.
0.8121 112.00 - -- -- -- -- -
26 0.8107 109.52 0.2486 105.17 969.67 119.18 0.0812 6391.
25 0.8170 109.91 0.2547 106.00 909.69 117.50 0.0858 6820.
24 0.8231 110.28 0.2608 106.83 822.22 117.09 0.0907 6910.
-- -.- o108.oo -00 - -- -
23 0.8291 110.62 0.2666 107.62 708.17 116.94 0.0955 6899.
22 0.8350 110.94 0.2723 108.39 646.17 116.79 0.1003 6888.
21 0.8407 111.22 0.2778 109.11 598.48 116.62 0.1050 6877.
20 0.8462 111.47 0.2832 109.82 555.87 116.44 0.1097 6866.
19 0.8516 111.67 0.2883 110.50 514.75 116.22 0.1143 6855.
18 0.8567 111.82 0.2933 111.14 471.29 115.96 o.1188 6842... _ — -- 117.00 - --
17 0.8617 111.92 0.2981 111.75 422.18 115.63 0.1232 6831.
16 0.8665 111.94 0.3028 112.31 369.22 115.23 0.1274 6818.
15 0.8712 111.89 0.3073 112.83 319.40 114.73 0.1316 6805.
14 0.8757 111.74 0.3116 113.29 279.89 114.11 0.1356 6792.
13 0.8800 111.50 0.3158 113.71 250.71 113.33 0.1396 6778.
12 0.8842 111.14 0.3199 114.07 229.09 112.36 0.1434 6764.
-- 118.0oo -- -
11 0.8883 110.65 0.3239 114.36 212.64 111.16 0.1471 6750.
10 0.8923 110.01 0.3278 114.58 199.86 109.66 0.1508 6735.
9 0.8962 109.20 0.3317 114.71 189.70 107.77 0.1544 6719.

-157TABLE 14
PLATE RESULTS FOR COLUMN B (CONTID)
Bub:.Pt. Dew Pt. Conv
Plate Vm tVm Lm t i m Vm E Pres
8 0.9001 108.20 0.3355 114.76 181.50 105.40 0,1579 6702,
7 0.9039 106.97 0.3393 114,81 174.84 l02.41 0.1611 6685.
6 0.9077 105.46 0.3432 114.62 169.37 98.57 0.1645 6666.
e, 116. 00
5 0.9116 103.66 0.3470 114.42 164.86 93.69 0.1676 6647.'4 0.9154 101.49 0.3510 113.93 161.13 87.38 0.1710 6625.
3 0.9194 98.92 0.3551 113.47 158.06 79.63 0,1740 6603.
2 0.9235 95.86 0,3594 112.62 155.55 75,45 0.1772 6577,
1 0.9278 92.25 0.3640 111.53 153.50 69.97 0.1805 6549.
_ — 0,3568 114.00
R.Gas 0.9324 88.oo - o, -
0,9324 88.oo 00
In each pair of values: top line computed value,
bottom line - test value,

-1586 0 0
26
24
22
NITROGEN
w 20
ZB
Z 18 _, 00,
" 16
METHANE
14
12 0 0
0
I6 I1 I I L
0.04 0.06 0.08 0.80 0.85 0.90
MOLE FRACTION IN VAPOR
Figure 15-a. Results for Column B

-1590 t 0
26
24 0 0
22
ETHANE PROPANE
20
18 _ o \ 0
16
W l4
In.
6 10
8
6 a
- I0
0.04 0.05 0.06 0.04 0.08 0.01 0.02 0.03 0.04 0.05 0.06
MOLE FRACTION IN VAPOR
Figure 15-b. Results for Column B

-16oO 0 *
26
24
22
20 _ \ \ N- BUTANE
ISOBUTANE
18
16
14
I12 -(g 0
12:-0. \ ~ \
8
6 e 0
4
0
I v I I I
0 0.003 0.006.0 0.005 0.010 0014
MOLE FRACTION IN VAPOR
Figure 15-c. Result8s for Column B

0 0
26
24
22
ISOPENTANE 1 N- PENTANE
ISOPENTANE
20
18
16
14
w
ca
12 0' \
- 10
8
6 0 0
0 I. l I 0 0
o 0.0010 0.0020 0 0.0010 0.0020
MOLE FRACTION IN VAPOR
Figure 15w.d. Results for Column B

-16226
24
22
20 HEXANE HEPTANE
18
6 0
J14
12
- I I I I0
0 0.003 0 0.002 Q005
MOLE FRACTION IN VAPOR
Figure 15-e. Results for Column B

-163L.O.
26
LIQUID RATE
24
22
LIQUID \
20
2~~0 -T~\ |'TEMR
18
W 16
Z 14
w
\ 12
00.25 0.30 0.35 105 10 115 20
8
4
2
0
0.25 Q30 0.35 105 110 115 126
MOLES/SEC TEMPERATURE OF
Figure 15-f. Results for Column B

-16427
26
24
22
". i!L I a A i t i!
14
52
eny
Figure o6. Plate Efficiencies in Column B
0.091 p000 0 0o020 00.00 oo0 0.20 00 5t00

-165TABLE 15
COMPONENT RESULTS FOR COLUMN B WITH WELL-STRIPPED LEAN OIL
Component YO +1Yn Rec.
Nitrogen o.o04o o. 0467 0.00 oo204 0.0196
o.o41o o 0.0536
Methane O. 8026 0 0.8630 0.15079 0.0737
0.8026 0 0.8915 0.11638 0.0325
Ethane 0.0723 o0 0.0598 0.05289 0.2869
0.0723 o0 0.0442 0.05314 0.4674
Propane 0.0597 0 0.0275 0.09191 0.6038
0o.0597 0 0.0092 o. 08000 0.8654
Isobutane O. 0059 0 0.0012 0.01238 0.8229
0.0059 0 0.0007 0. 01i581 0.9091
n-Butane 0.0135 0.0017 0.03066 0.8905
0.0135 0 0.0005 o.o440oo.9682
Isopentane 0.0016 0 o0.ooo6 0.00395 0.9688
0.0016 0 o.ooo1 0.00952 0.9333'
n-Pentane 0.0014 0 0.00003 0.00350 0.9804
00014 0 0.0001 0.01124 0.9231
Hexane 0.oo004 0.0010 0.00003 0.00161 0.9408
0.oo4 0.0010 0.0001 0.00743 0.7500
Heptane 0.006 o.oo000o 0.00470 0.9992
0.0016 0.0010 0 —0.00057 1
Octane 0 0.0017 0.00001 0.00108
0 0.0017 -- 0.00114
Nonane 0 0.0169 0.00003 0.01087
o 0.0169 -- 0.011.81
Decane 0 0.0287 0.00002 O. 01851
0 0.0287 -- 0.02000
Abs. Oil 1 0 0.1573 0.00004 0.10166
0 0.1573 - 0.10838
Abs. Oil 2 0 0.7934 -- 0.51343
0 0.7934 -- 0.52058
In each pair of values: top line - computed value,
bottom line - test value.

-166TABLE 16
PLATE RESULTS FOR COLUMN B WITH WELL-STRIPPED LEAN OIL
Plate Vm tVm L m tL, Conv. Pres.
L.Oil - 0.2365 103.00
0.2365 103.00
27 0.8078 108.76 0.2469 104.09 7031.
0.8121 112.00
26 0.8136 109.17 0.2526 104.89 7021.
25 0.8194 109.56 0.2583 105.67 7011.
24 0.8250 109.95 0.2638 106.46 7002...- -.. -lO- 108.00
23 0.8305 110.31 0.2692 107.22 6991.
22 0.8359 110.65 0.2745 107.96 6981.
21 0.8412 110.96 0.2796 108.68 6970.
20 0.8464 111.24 0.2846 109.37 6959.
19 0.8514 111.48 0.2895 110.04 6947.
18 0.8563 111.67 0.2943 110.68 6935.......o- -~117.00
17 0.8610 111.81 0.2989 111.29 6924.
16 0.8656 111.89 0.3034 111.86 6911.
15 0.8701 111.89 0.3077 112.40 6898.
14 0.8745 111.81 0.3120 112.88 6885.
13 0.8787 111.64 0.3161 113.33 6871.1
12 0.8829 111.35 0.3202 113.72 6858....-.. -. 118.00oo
11 0o8870 110.94 0.3242 114.06 6842.
10 0.8910 110.37 0.3282 114.32 6827.,
9 0.8949 109.63 0.3321 114.53 6811.'

-167TABLE 16
PLATE RESULTS FOR COLUMN B WITH WELL-STRIPPED LEAN OIL (CONT'D)
Plate Vm tVm Lm tm Conv.Pres.
8 0.8988 108.68 0.3360 114.62 6794e
7 0.9027 107.50 0.3399 114.61 6776.
6 0.9066 106.04 0.3438 114.47 6757....116.o00
5 0.9106 104.27 0.3479 114.18 6736.
4 s0.946 102.12 0.3520 113.83 6714.
3 0.9188 99.51 0.3563 113.50 6690.
2 0.9231 96.34 0,3608 112.64 6664.
1 0.9276 92,54 0.3656 111*50 6634...0. O3568 114.00
R.Gas 0.9324 88.0 -0 -
0.9324 88,00 -.
In each pair of values: top line - computed value,
botcom line - test value.

-168TABLE 17
COMPONENT RESULTS FOR COLUMN B WITH WELL-STRIPPED
LEAN OIL AND INTERCOOLER*
Component Y xn+l n Rec.
Nitrogen o.o4i 0 00.0467 0.00204 0.0196
o.o410 0 0.0467 o.00204 0.0196
Methane 0.8026 0 0.8642 0.15068 0.0739
0.8026 0 0.8630 0.15079 0.0737
Ethane 0.0723 0 0.0594 0.05389 0.2934
0.0723 0 0.0598 0.05289 0.2869
Propane 0.0597 0 0.0267 0.09331 0.6152
0.0597 0 0.0275 0.09191 0.6038
Isobutane 0.0059 0 0.0012 0.01244 0.8300
0.0059 0 0.0012 0.01238 0.8229
n-Butane 0.0135 0 o.oo60 0.03071 0.8954
0.0135 0 0.0017 0.03066 0.8905
Isopentane. 0016 0 o.oooo6 0.00394 0.9702
0.0016 0 o.oooo6 0.00395 0.9688
n-Pentane 0.0014 0 0.00003 0.00349 0.9812
0.0014 0 0.00003 0.00350 0.9804
Hexane.o0004 0.0010 0.00003 0.00159 0.9339.ooo4 o0010 0.00003 o.oo00161 0.9408
Heptane 0.0016 0.0010 0.00001 0.00468 0.9916
0.0016 0.0010 -- 0.00470 0.9992
Octane O 0.0017 0.00001 0.00108
0 0.0017 0.00001 0.00108
Nonane 0 o.0169 0.00003 0.01083
0 0.0169 0.00003 0.01087
Decane 0 0.0287 0.00002 o.01845
0 0.0287 0.00002 0.01851
Abs. Oil 1 0 0.1573 0.0000oooo4 0.10129
0 0.1573 o.00oooo4 0.10166
Abs. Oil 2 0 0.7934 -- 0.51156
0 0.7934 -- 0.51343
* Figures on the second lines are computed values without intercooler.

-169TABLE 18
PLATE RESULTS FOR COLUMN B WITH
WELL-STRIPPED LEAN OIL AND INTERCOOLER
Plate Vm tVm Lm tLm Conv.Pres.
L.Oil - - 0.2365 103.00 -
27 0.8019 107.96 0.2406 105.43
26 o.8060 108.17 0.2466 106.12 5853.
25 0.8120 108.35 0.2524 106.81 6651.
24 0.8178 108.50 0.2582 107.49 6961.
23 0.8236 108.60 0.2638 108.12 7002.
22 0.8292 108.65 0.2693 108.73 6990.
21 0.8347 108.64 0.2746 109.29 6980.
20 0.8400 108.56 0.2798 109.80 6968.
19 0.8452 io8.41 0.2848 110.29 6956.
18 o.8503 108.17 0.2897 110.72 6944.
17 0.8552 107.82 0.2945 111.10 6932.
16 0.8599 107.36 0.2991 111.42 6919.
15 0.8645 106.76 0.3036 111.67 6905.
14 o.86o90 106.02.3080 111.85 6891.
13 0.8734 105.09 0.3123 111.94 6876.>
12 0.8777 103.98 0.3165 111.94 6861.
11 0.8819 102.63 0.3206 111.86 6845.
10 o.8860 101.02 0.3250 102.40 6780.
9 o.8904 100.79 0.3294 102.92 6766.
8 0.8948 100.42 0.3338 103.35 6749.
7 0.8992 99.88 0.3382 103.72 6731.

-170TABLE 18
PLATE RESULTS FOR COLUMN B WITH WELL-STRIPPED
LEAN OIL AND INTERCOOLER (CONT'D)
Plate Vm tm Lm Conv.Pres.
6 0.9036 99*15 0.3427 104.02 6712..
6 O.9036 99.15 0.3427 104.02 6712.
5 0.9081 98.20 0.3472 104.21 6692.
4 0.9127 96.98 0.3519 104.38 6670.
3 0.9173 95.44 0.3567 104.63 6648.
2 0.9221 93.45 0.3617 104.43 6622.
1 0.9271 91.00 0.3670 104.o4 6594.
R.Gas 0.9324 88.00

-171LO.
26 _ LIQUID RATE
LIQUID
24 _ TEMP.
22
20
18
16
12'A
10
0.25 0.30 0.35 105 10 115
MOLES/ SEC. TEMPERATURE'F
Figure 17. Results for Column B with Well-Stripped Lean Oil

-172TABLE 19
COMPONENT RESULTS FOR COLUM C
Component Yo Xn+1 Yn xi Rec. Avg. EMV
Methane 0.8471 0 0.9039 0.2941 0.,0925 0.0280
0.8471 0 0.9083 0.2383 0.0826
Ethane 0.0737 0.040 0.0619 0.0807 0.2854 0.0700
0.0737 0.o004o oo616 0.0833 0.2850
Propane 0.0383 0.0010 0.0198 0.0812 0.5614 0.1078
0.0383 0.0010 0.0172 0.0901 o.6158
Isobutane.00o6o 0.0020 0,0017 0.0180 0.7601 0,1359
o.oo6o 0.0020 0.0005 0.0221 0.9287
n-Butane o,01l6 0.0020 0.0019 0.0382 0.8572 0,1519
0,0116 0.0020 0.0001 0.0441 0.9926
Pentane + 0.0123 0,0040 0.00015 0.0474 0.9897 0,221
(Hexane) 0.0123 0,0040 00003 o.o8o6 09791 -
Abs. Oil 0 0.9870 0*00002 0.4328 - 0,2284
0 0.9870 0 o.4415 --
Carbon,00110 0 o.0106 0,0075 0.1828 0,0526
Dioxide 0.011 0 0 0.0120 - -
In each pair of values; top line - computed value,
bottom line - test value,

-173TABLE 20
PLATE RESULTS FOR COLUMN C
Plate Vm tV L tL Bub.Pt. Dew Pt. E Conv..... m "Vm ^ Lm V^m Pres.
L Oil - - 0.1592 81.o00
-.. 0.1592 81.oo 00 --
30 1.1764 94.86 0o1863 82.79 996.32 100.24 0.0992 7258.
1.1662 94.oo00 - -
29 1.1863 96.19 0.1964 85.33 975.88 100.67 0.1148 7213....... 94.oo0 -- --
28 1.1963 97.60 0.2064 87.90 954.32 101.08 0.1316 7180.
27 1.2063 99.07 0.2163 90.46 787.89 101.51 0.1494 7153.
102.00- - --
26 1.2162 100.59 0.2260 92.98 649.49 101.96 0.1679 7130.
25 142259 102.12 0.2353 95.44 560.77 102.43 0.1867 7107...... 106.00 --..
24 1.2353 103.66 0.2442 97.80 473.40 102.90 0.2054 7087.
23 1.2441 105.17 0.2525 100.06 353.06 103.37 0.2239 7067.
-. -- 108.00oo -- — 8.
22 1.2524 106o65 0.2603 102.19 26o.40 103.81 0.2418 7047.
21 1.2602 108.07 0.2674 1o4.19 216.89 104.21 0.2588 7028..-.- -- 109.00 -- -- --
20 1.2674 109.43 0.2740 106.o6 191.87 104.56 0.2750 7009.
19 1.2739 110.71 0.2800 107.79 175.47 104.81 0.2901 6991.
110.00 -
18 1.2799 111.90 0.2855 109.38 164.02 104.96 0.3042 6974.
17 1.2854 113.00 0.2905 110.84 155.73 104.98 0.3173 6956.
-- -- -- 111.00.
16 1.2904 114.01 0.2951 112.17 149.54 104.83 0.3294 6938.
15 1.2950 114.91 0.2993 113.38 144.92 104.47 0.3404 6921.. o.... 112.0O0..1200 -
14 1.2992 115.70 0.3032 114,45 141.46 103.85 0.3505 6904.

-174TABLE 20
PLATE RESULTS FOR COLUMN C (CONT'D).~Plate VM tmtLm BBub.Pt Dew Pt E Conv,
Plate Vm, t^V 1^ ti^ V V ~M Pres.
13 1.3032 116.37 0.3069 115.71 138.94 110.16 0.3595 6888.
- 5- -112.50 --
12 1.3068 116.74 0.3103 116.42 137.09 109.30 0.3676 6869.
3.1 1.3102 116.93 0.3137 116.96 135.76 108.00 0.3749 6849.
-- - -- 113*50o -.
10 1.3136 116.91 0.3170 117.33 134,86 106.15 0,3815 6828.
9 1.3169 116.66 0.3204 117.50 134.31 103.58 0.3874 6805.
-- -- -- 113.0 - r0- - -
8 1.3203 116.13 0.3239 117.44 134.06 100.09 0.3928 6779.
7 1.3238 115.28 0.3276 117.14 134.08 95.42 0.3975 6750,
3i13.00
6 1.3275 114.05 0.3317 116.54 134,34 89,29 0,4015 6717,
5 1.3316 112.37 0.3362 115.61 134.85 81.58 0.4049 6678,.-... - -. 112.50 - -
1.3361 110.18 0.3414 114.28 135.62 72.84 0.4075 6633,
1.3413 107.35 0.3474 112.48 136.70 65.11 0.4093 6579.
- 111, 00 - _
2 1.3474 103.80 0,3546 10o.14 138,12 64.41 0.4100 6516,
1 1.3545 99.39 0.3631 107.17 139.99 66.41 0.4094 6439....~ o0.3561 1o04.oo
R.Gas 1.3630 94.00 -
1.3630 94.00
In each pair of values: top line - computed value,
bottom line - test value,

-175L.O.
30
28
26
LIQUID RATE
24
22
LIQUID TEMP.
20
18
16
w
_14
z
12
00 0
8
6
4
2
0.20. 0.30 0.40 80 100 120
MOLES/SEC TEMPERATURE ~F
Figure 18. Results for Column C

-176TABLE 21
COMPONENT RESULTS FOR COLUMN D
Component Y n+ln Rec.
Methane 0.7529 0 0.81254 0.0946 0.0382
0.7529 0 0.81413 0,0850 0.0344
Ethane 0.0564 0 0.05192 0.0334 0.1796
0.0564 0 0.05448 0.0253 0.1371
Propane 0.0505 0 0.02883 o.0817 o.4912
0.0505 0 0.02740 0.0854 0.5159
Isobutane 0.0092 o 0.00200 0.0244 0.8064
0.0092 0.000o46 0.0288 0.9555
n-Butane 0.018l 0.00214 0.0533 0.8945
0.0181 0 0.00009 0.0591 0.9947
Pentanes + 0.0127 0 0.00003 0.0417 0.9981
(Hexane) 0.0127 0 0.00010 0.0412 0.9924
Abs. Oil 0 1.0000 -- 0.6426
0 1.0000 0 0.6471
Carbon 0.1002 0 0.10280 0.0282 0,0856
Dioxide 0.1002 0 0.10334 0.0260 0.0781

-177TABLE 22
PLATE RESULTS FOR COLUMN D
Plate Vm t Lm tLm E Conv.Pres.
L.Oil. - 0.1895 71. 00
0.1895 71.00
24 0.8664 81.47 0.1947 73.70 0.1174 6346.
0.8690 78.00 - 76.00 -- -
23 0.8716 82.50 0.2014 75.44 0.1289 6626.
22 0.8782 83.55 0.2078 77.14 O.1406 6666.
21 0.8846 84.6o 0.2138 78.79 0.1524 6672.
20 0.8906 85.64 0.2194 80.38 0.1642 6668........ 89.oo00
19 0.8963 86.67 0.2248 81.92 0.1759 6658.
18 0.9016 87.68 0.2297 83.40 0.1874 6645.
17 0.9066 88.67 0.2343 84.82 0.1987 6630.
16 0.,9112 89.63 0.2386 86.17 0.2098 6614.
94.00oo
15 0.9155 90.55 0.2426 87.47 0.2208 6597.
14 0.9195 91.42 0,.2464 88.71 0.2315 6579.
13 0.9232 92.24 0.2499 89.89 0.2421 6560.
12 0.9268 92.99 0.2532 91.01 0.2525 6541.... — -- 93.00 --
11 0,9301 93.66 0.2564 92.07 0.2630 6520.
10 0.9332 94.23 0.2594 93.34 0.2730 6502.
9 0.9363 94.56 0.2624 94.26 0.2834 6480,o.
8 0.9393 94.68 0.2654 95.07 0.2941 6457.
95.00
7 0.9423 94.52 0,2685 95.75 0.3052 6431.

-178TABLE 22
PLATE RESULTS FOR COLUMN D (CONT'D)
Plate V L E ConvPres
nLm
6 0.9453 93.99 0.2717 96.25 0.3169 6402.
5 0.9486 92.94 0.2752 96.53 0.3295 6369.
4 0.9520 91.17 0.2790 96.46 0.3432 6330.. — -- - 98.00 - --
3 0.9559 88.4o 0.2835 95.88 0.3584 6283.
2 0.9603 84.23 0.2888 94.52 0.3751 6222.
1 0.9656 78.06 0.2953 92.00 0.3937 6143.. —. 0.2926 92.00 -- -
R.Gas 0.9721 69.00
0.9721 69.00. -

-179LO.
24
LIQUID RATE
22
20
18
16
LIQUID: TEMP.
w 14 \
Z 12
a8
6
4
2
0 - 1' I. I I I I I I I....I I1, 1
0.20. 0.25 0.30 70 80 90 100
MOLE /SEC. TEMPERATURE OF
Figure 19. Results for Colmn D

-180VIII. ANALYSIS OF RESULTS
A. The Ideal Stage Calculation
The results, although not in particularly close agreement with
those of the "legitimate" ideal stage calculation, do seem to indicate
that no serious discrepancies exist. Some measure of the programs
inability to produce equilibrium streams may be gathered from examination
of the computed bubble and dew points.
Sources of the disagreement, in probable order of importance,
are (1) the mass transfer relationship used in the writer's program;
(see the discussion under Results —An Ideal Stage Column), (2) the different ways in which enthalpies and K values were correlated in the two
programs, and (3) the fact that neither set of results represents a true
or completely converged solution of the equations involved. Both contain
truncation and round-off error within "reasonable" limits specified by
the user.
B. Recovery of Components
Probably the most striking thing common to the results for all
four non-ideal stage columns is the discrepancy between computed and
test recoveries of the medium-volatility components of the rich gas.
The most likely causes of this error are (1) the approximation of using
V_-1 in place of V in the mass transfer expressions,
vm = Vm-1 + (EMV)(Vm-_lKm - vm-l) (89)
(2) the use of plate efficiencies which are too low, and (3) the estimation of lean oil compositions (in some cases) which contain excessive

-181amounts of some components of the rich gas. These three possibilities
will be discussed in order.
C. The Average Vapor Flow Rate
The bottom plate of Column B (where the vapor flow rate
changes most rapidly) was selected as the basis for a short hand calculation to demonstrate the justifiability of using V 1 in place of
(some average of Vm-l and Vm) in the mass transfer equations. In Table
23, V0 and yO are the input flow rate and composition to plate 1, and
V1 and y, are the corresponding quantities leaving the plate as calculated by the machine. The arithmetic average of V0 and V1 is V, and
when V is used in the mass transfer expressions, V' and y' result. The
1 1
difference in all cases is quite small, amounting to about 0.0044% for
V1 and V{.
D. The Effect of Plate Efficiency
In absorbers, plate efficiency and number of plates are critical only for the "key" component and those immediately adjacent to it in
volatility. The key component is defined as the one whose average absorption factor is closest to 1.00, or equivalently, the one whose operating line is most nearly parallel to its equilibrium line. In Column A,
n-butane, the key, has a K value of 0.21 at 80~F, and the L/V ratio on
plate 10 (the middle of the column) is 0.182. Figure 20 illustrates the
case for propane, n-butane, and pentane. Note the rapid change of propane concentrations near the top of the column where the driving force
is greatest, the opposite effect with pentane, the almost uniform change
for n-butane, and the fact that the curve in the operating lines reflects

-182TABLE 23
THE EFFECTS OF AVERAGE VAPOR RATE AND PLATE EFFICIENCY
Component EV Y1 * Y'
Nitrogen 0.0410 0.003862 0.041191 0.041193 0.041387
Methane 0.8026 0.013544 0.805711 0.805722 0.808874
Ethane 0.0723 0.045267 0.072169 07216215 0.072029
Propane 0.0597 o.o86171 0.058380 0.058374 0.057034
Isobutane 0.0059 0.118339 0.005568 0.005567 0.005231
n-Butane 0.0135 0.135166 0,012573 0.012572 0.011595
Isopentane.o0016 0.164772 0.001459 0.001459 0.001316
n-Pentane 0.0014 0.175153 0.001276 0.001276 0.001150
Hexane o.0004 0.196579 0.000363 0.000363 0.000325
Heptane 0.0016 0.206374 0.001288 0.001288 0.000973
Octane 0 0.207433 0.000001 0.000001 0.000002
Nonane 0 0.204857 0o000005 0.000005 0.000009
Decane 0. 0.201063 0.000003 0.000003 0.000007
Abs. Oil 1 0 0.195577 0.00 7.000007 0.000007 o 14
Abs. Oil 2 0 0.185291 0.000005 0.000005 0.000011
V0 = 0.932400
V1 = 0.927834
V = 0.930117
vl- = 0.927793
* V tt 0.923186
* yl and V1 were computed using values of EMV twice as large as those
used in computing Yj, y1, V1, and V1.

-1830.04
_/ (y, xI)
(YoX,)
0.03
QO>F ~~ I~ /I/
//
/
IYn, Inm/./
0.002 1 ^ o
/ 00
/
/
/
0 0.01 0.02 0.03 0.04 005 0.06
Figure 20, Operating and Equilibrium Linea

-184the decrease of L/V ratios as m increases. The equilibrium lines represent K values at 80oF, 232 psia, and 5000 psia convergence pressure.
Components much lighter than the key component usually reach a concentration in the rich oil which is nearly in equilibrium with the rich gas,
i.e., the "pinch" or zone of diminished driving force occurs in the lower
part of the column; components much heavier than the key reach a concentration in the lean gas nearly in equilibrium with the lean oil, i.e.,
the pinch is in the upper part of the column. This latter case is
illustrated by pentane in Column A, Figure 13-D, whose computed absorption is much slower than the test results indicate but almost as complete.
In both Columns A and B, vapor composition test data for
lighter components in the internal streams scatter too badly to indicate
anything except trends. Data on the medium-volatility components show
that they are absorbed out of the vapor stream much more quickly than
the machine computation predicts. A liquid rate curve prepared by the
Phillips people from material balance calculations (shown dotted because
it is derived rather than primary data) has a much more pronounced "S"
shape than the one computed by the machine and shown in Figure 13-E.
Returning to Table 23, the values V and y were computed
by hand using values of EMV twice as large as the machine computed ones.
The difference in this case is significant, and the short arrows on
Figure 15 show that the use of larger plate efficiencies would produce
computed concentration profiles much more nearly in accord with the
test data.
The importance of plate efficiency to the key component and
neighboring ones is illustrated in Column A by the fact that the

-185discrepancy between computed and test recoveries is greatest for n-butane
and less serious for isobutane and pentane. In Column B with wellstripped lean oil, the keys are propane and isobutane., and here again
the discrepancies are greatest for the keys, less serious for those
close to the keys. Similar situations hold in Columns C and D.
E. The Effect of Lean Oil Composition
When the results for Column B showed large discrepancies
between computed and test recoveries of components heavier than the keys,
a well-stripped lean oil was substituted in order to determine the effect
of the lean oil composition. As expected, recovery was markedly increased and exceeded the test recoveries in most cases.
The fact that lean gas concentrations of heavier components
approach equilibrium with the liquid from the top plate is illustrated
by the asterisks in Figures 13 and 15, which show the equilibrium compositions. Approach to equilibrium is only a contributing factor to
the lpw recovery of n-butane in Columns A and B, however. In the highpressure Column C, the key component was isobutane; the concentrations
of ethane, propane, isobutane, n-butane, and hexane in the lean oil were
0.06115, 0.01958, 0.001649, 0.001936, and 0.000147, respectively; the
vapor mole fractions which would have been in equilibrium with the lean
oil were 0.01021, 0.00254, 0.000424, 0.000416, and 0.000120; and the
actual lean gas concentrations, found in Table 19, were 0o06l9, 0.0198,
0*0017, 0.0019, and 0.00015. Thus, it appears that only the recovery of
hexane (which was satisfactory) was influenced significantly by the lean
oil composition. The lean oil to Column D was completely stripped,

-186F. The Effect of Intercooling
Examination of Table 17, which compares machine computation
results of Column B (well-stripped lean oil) with and without an intercooler, indicates that the effect of the intercooler was about what might
have been expected. The lowered temperatures caused lower K values and
greater recoveries of all rich gas components except nitrogen (unchanged),
hexane, and heptane. Recoveries of the key and adjacent components were
increased the most..
G. Convergence Pressure
Convergence pressure is computed from liquid compositions and
is a function of temperature and the boiling points of the effective
light and heavy components, The light component effective boiling point
is influenced most strongly by the mole fractions of the lightest components in the liquid, as Equation (175) indicates; a similar statement
holds for the effective heavy component. In the range of conditions
found in gas absorbers, an increase in the heavy component boiling point
or a decrease in the light component boiling point will raise the convergence pressure. The effect of system temperature is variable and
depends on the values of the other two variables, but it is small and
should not be significant over the small temperature ranges found in
absorbers.
The convergence pressure profiles of all columns seem to follow
approximately the same pattern, They increase slowly from the bottom
of the column to a plate either at the top or near the top and then (in
the latter case) decrease rather abruptly. Flow rates of the heavy

-187components of the lean oil are almost constant in the liquid streams
throughout a column. Since Lm decreases as m increases, mole fractions
of the heavy components increase, the heavy component effective boiling
point increases, and this would seem to explain the slow rise of pi as
m increases. Mole fractions of the lightest components decrease slowly
as m increases until the top few plates are reached, at which point they
drop off to zero in the lean oil stream. (The degree of approach to
convergence acceptable to the machine may even permit very small negative
mole fractions of nitrogen or methane in Ln or Ln_1.) This sudden disappearance of light components raises the light component effective
boiling point and probably accounts for those decreases of convergence
pressure which do occur. Excluding the top and bottom plates, the
percentage changes of pk do not exceed 10% through any of the columns,
H, Dew Points and Bubble Points
The dew points and bubble points computed for Columns A, B,
and. C show the trends that might be expected of them. The lean oil
enters the column well-stripped of light and medium-volatility components,
The liquid stream has extremely high bubble point temperatures at the top
of the column, the temperatures drop as the liquid flows downward absorbing material from the vapor stream, and the bubble point temperatures
near the bottom approach the liquid stream temperatures. The dew point
of the vapor stream rises slightly as the stream proceeds up the column.
The vapor loses its heavier components, but very small amounts of the
heaviest components of the absorber oil are vaporized. These small
quantities, because of their extremely low K values, are sufficient to
increase the dew point temperature,

-188A curious anomaly arises in thecomputation of dew point temperatures in that they become higher than the vapor stream temperatures in
the upper parts of the columns. This condition, if it were found in test
data, might be considered an indication that the vapor contained entrained
liquid, but the machine program is not equipped to detect or predict
entrainment. In view of the high sensitivity of the dew point calculation to heavy components in the vapor stream, this inversion of dew
point temperature and vapor stream temperature must be considered a
computational phenomenon with no physical significance.
I. Heat Transfer Efficiency
In all four columns investigated, the heat transfer efficiencies decrease markedly as m, the plate number, increases. The lower
efficiencies on the upper plates result in (1) less curvature in the
graphs of computed liquid temperatures than in those of the measured
temperatures, and (2) higher computed lean gas temperatures than the
test data contain. The importance of the one exception to the second
stated general result, Column B, may be discounted somewhat by the facts
that the computed temperatures are generally low in that column and the
lean oil rate is only an estimated value. The two effects mentioned
above would seem to be consequences of the lowered heat exchange rates
computed for the upper parts of the columns.
The variations of the Eq values produce one worthwhile result,
however. The liquid temperature profiles computed for the lower parts
of thecolns have um shapes in fairly good agreement with those plotted
from test data; the maximum liquid temperatures occur reasonably close
to the test data locations, for instance, and this agreement would seem

-189to be desirable since absorption of the most valuable components takes
place in these regions. The values of C for Columns A, B, and C were
I q
adjusted to make the computed differences between tL (maximum) and tL
approximately equal to the differences which existed in the actual
columns. The value of C for Column D was predicted with the same
objective in mindo
The effect of using a constant value of E is illustrated by
Figure 21, which shows the test data liquid temperature profile, the
final computed one, and the results obtained with various constant values
of E, all for Column Ac The arithmetic average of the twenty computed
values of E is 0.2260. The temperature profiles obtained with E of 002
and 0.3 agree well with test values on the upper plates of the column,
but agreement is poor on the middle plates, and the maximum temperatures
occur too high in the column. The poor results obtained with E of 0.8
show that values approaching 1.00 (usually tacitly assumed) are not
suitable, at least not with the machine program used.
The shapes of the test liquid temperature curves seem to indicate that Eq, if it is a valid measure of heat transfer at all, must have
large values at the top and bottom of a column and somewhat smaller values
in between. This observation leads one to two statements made earlier~
(1) The "S" shapes of the liquid flow rate curves indicate that more
absorption takes place in the upper and lower sections of the columns
than in the middle. (2) The heat transfer efficiency expression was derived without reference to the effect of mass transfer on heat transfer.
The cause of the decrease in Eq with increase of m is easily
understood from examination of Equation (101).
Eg = Cqj(Yi EMV,i)(Le)2/3 (101)

-190L.O.
20
19
18
17 - *'.0.3
16
Eq'0.2
15
14
FINAL COMPUTATION
12
L 10
65 70 75 80 65 90
6
TEMPERATURE OF
Figure 21. Variation in Temperature Proftle8

-191~
The plate efficiencies of nitrogen and methane are much lover than those
of the heavier components. As the vapor stream proceeds up the column,
the heavier constituents are absorbed and the mole fractions of nitrogen
and methane increase. Since methane is the principal constituent of all
the vapor streams, Eq follows the trend of EMV of methane quite closely,
as shown by the plot of Eq on Figure 14. Eq is plotted as Eq/100 in
order to avoid cluttering the right side of the figure and to juxtapose
it to the methane curve. This transformation (on a logarithmic scale)
changes position but neither shape nor orientation.
J. Prediction of the Heat Transfer Efficiency Factor
After three columns had been investigated with values of Cq
adjusted to give the desired liquid temperature profiles, a study was
made to determine if a value of Cq might be accurately predicted for a
fourth absorber, Column D. Table 24 presents the various quantities
involved.
The arithmetic average values of Eq were computed and compared
to the ratios of Ln+l/V in Columns A, B, and C. The product of these
two quantities was found to be very nearly a constant, the average value
being 0.348. Using 0.35 as the constant and the known Ln+l/VO ratio of
Column D, 0.195, Eq was estimated to be 0.179. The implication here is
not that such a simple empirical relationship adequately describes the
functional dependence of Eq, merely that lack of knowledge and scarcity
of data points make a simple function necessary.
In an attempt to correlate C and E, the quotients Cq/E were
computed and plotted as the large dots in Figure 22. The quantity
Cq/E is inversely proportional to the quantity which would be produced

-192TABLE 24
PREDICTION OF THE FACTOR C
Column A D B C
Pressure 232 psia 400 psia 740 psia 1475 psia
Ln+i/VO o.156 0.195 0.254 0.117
Avg. temp. t 81.O F 85.0 oF 110.5~F 102.5~F
C Ci24 17 9 9
Avg. E 0.2260 0.179 0.1328 0.3030
(L/v)(E) 0.0352 0.035 0.0337 0.0354
C /E 106.2 96 67.8 29.7
At 19.0 15.0 -10.5 -2.5
A(Cq/E) -7.6 -6.o 4.6 1.0
% change -7.16 -6.25 6.78 3.37
New C /-E 98.6 90.0 72.4 30.7
Ci Ci

4193.140
120 Z
0
-r ~
o z
- ^\ I
100 0rC).1
80 S'UF
\ C\
^-o \ ^
^*60-
N 1A ~
F: z~~~~~~~~~~~~~~2.o 6
c 40 — j
0~~~~~~~~~
w
CY 40~~~~~~~~~~~~~C
I -
20
0~~~~~~~~~~
0 I I ll l l
0 400 800 1200 1600 2000
COLUMN PRESSURE, PSIA.
Figure 22. Prediction of the Factor Cg

-194by the mass transfer-heat transfer analogy without use of a factor C,
In lesser degree, it is inversely proportional to EMV of methane and,
in turn, proportional to the K value of methane. The "average temperatures", t, of the columns were computed as the arithmetic averages of
tv and tL plus 15~F and were compared to an arbitrary reference, 1000F.
ad tL1
If t of Column A, for instance, were 100"F instead of 810F, it is reasonable to assume that the K value and EMV of methane would be larger and
Cq and Cq/ q smaller. The points representing the three columns were
moved by distances proportional to (T - 100~F) to produce three points
lying in a straight line, and the short arrows indicate these displace.
ments, There is, of course, no particular reason why the points should
lie on a straight line; the percentage changes of the quotients necessary
to produce this convenience are not large, however.
Reading a corrected quotient of 90.0 for Column D and carrying
out the above procedure in reverse produced a value of 17 for C. The
machine computation gave a liquid temperature profile which conformed to
the test results (in the lower half of the absorber) more closely than
that of any other column, On the other hand, the arithmetic average of
computed heat transfer efficiencies was 0.248 instead of 0.179 ae had
been predicted,
K. Estimation of Plate Efficiencies
As mentioned in the section The Effect of Plate Efficiency,
it appears that the computed plate efficiencies in all columns were too
low, Those in Column B were particularly low because of the equivalent
bubble cap plate properties estimated for the Kaskade plates. When the
properties of a typical bubble cap plate were used instead, as in the

-195sample plate efficiency calculation, Table 1, the efficiencies for the
bottom plate shown as triangles in Figure 16 were computed. Components
are in the same order as they occur on the curves.
By comparison with the large amount of work which has been
done on the subject of predicting plate efficiencies, the few brief
remarks made here are little more than conjecture. It would seem that
the most likely reasons for the high computed values of vapor and liquid
phase resistances and low plate efficiencies are the following:
(1) A factor involving m, the "slope of the equilibrium
curve", is used to convert NL, the number of liquid phase mass transfer
units, to the vapor phase basis, This is inherently an inaccurate procedure for light components (where the liquid phase resistance is the
larger of the two), but the same basis must be used for all components.
Furthermore, m is an ambiguous quantity in a system of more than two
components, and its approximation by the K value is still a further source
of error. The large K values of nitrogen and methane produce extremely
small plate efficiencies. Unfortunately test analyses of vapor streams
show the most scatter in the concentrations of light components (see
Figures 13-A and 15-A), and no conclusions may be drawn from them.
(2) Liquid phase resistances in the hydrocarbon systems were
compared to those of the reference system (air-water-carbon dioxide)
by using the square roots of the ratios of Schmidt numbers. In the
sample calculation, Table 1, these comparison factors always lie between
0.67 and 1.38. In comparing vapor phase resistances with the reference
system (air-water-ammonia), the square roots of the ratios of vapor
diffusivities were used, and these factors were in the range 8,7 to 15.5.

-196This represents quite a large increase of vapor phase resistance due to
change of system, probably too large an increase to be an accurate estimate. Had the ratios of Schmidt numbers been used, the range would have
been 1.27 to 2.26, and hindsight indicates that this would have been
the more cautious comparison, even though, as discussed under Engineering
Concepts, there is no conclusive evidence to dictate the use of one comparison in preference to the other,
(3) A third source of error lies in the fact that the plate
efficiency procedure takes no account of plate size or configuration,
other than to specify bubble cap plates. Properties of the reference
systems were determined in laboratory-size apparatus, and it is doubtful
that these data can be extrapolated to large columns without adjustment.

IX. CONCLUSIONS
(1) The non-ideal stage calculation appears to be a very useful tool for those seeking better ways of understanding, analyzing, and
predicting the operating conditions of vapor-liquid stage contacting
devices. The calculation, as presented here, contains many assumptions,
none of which is less reasonable than those embodied in the ideal stage
calculation. Some of these assumptions —driving forces or potentials
proportional to mole fractions, constant liquid composition, the use of
an "average" vapor rate in one term of the mass transfer expression —
might be eliminated if one had more complete knowledge of the systems
involved and were willing to undertake the additional computation
necessary.
(2) The utility of the computer procedure is demonstrated by
its ability to duplicate, at least in an approximate way, the conditions
on each actual plate and the general trends and effects found in operating absorption columns. Greatest discrepancies between computed and
test results occur in the fractional recoveries of key components and
the liquid temperature profiles. The cause of these deficiencies appears
to be inaccurate values of EMV and Eq.
(3) The Murphree plate efficiency, as employed in the mass
transfer relations developed here, is a very useful characterization of
the behavior of the various components in stage-wise separation processes.
It is not suitable for plate-to-plate calculations in absorbers when used
as originally defined,
(4) Prediction of Murphree plate efficiencies from separate
phase resistances apparently has found more favor than the more patently
i197

-198"
empirical procedures. In the present work, the two most difficult
aspects of the procedure are relating resistances or numbers of transfer
units in different systems, and accounting for the effect of changing
column size or configuration.
(5) The heat transfer or "vapor temperature" efficiency expression used in this investigation is a suitable concept per se. Its
theoretical basis, an equation for heat transfer in the absence of mass
transfer, appears to be as strong as the basis for the Murphree plate
efficiency. The fact remains (and this applies equally to the use of Eq
and EMV) that heat transfer and mass transfer do not occur independently,
and their interaction must ultimately be learned.
(6) Better methods of predicting the heat transfer efficiency,
Eq, must be found. Perhaps more astute use of the Colburn analogy, such
as applying it to both phases separately and then combining resistances,
will provide the needed improvement. Alternatively, empirical correlation based on test data may be satisfactory.
(7) The inherent computational instability of either the
bottom-up or the top-down plate-to-plate calculation of absorbers has
been mitigated by an iterative procedure based on the reduction of relative, rather than absolute errors. The Wegstein extrapolation procedure
has been successful in speeding up the otherwise slow convergence of the
iterative solution.
(8) The vapor and liquid enthalpy correlations employed in
this investigation are somewhat more complicated and at least potentially
more complete and accurate than those generally used in hand calculation
or on intermediate-size computers. The computation done so far does not

-199constitute a complete test of the correlations, but an estimate of their
worth may be obtained from the usually favorable comparison of test and
computed values of flow rates and temperatures of the exit streams from
the absorbers.
(9) In view of the small percentage change of convergence
pressure found throughout each of the absorbers investigated, the use
of a constant value of pk appears to be in order.
(10) For industrial use, machine computation time might be
reduced by a factor of five through the use of constant values of plate
efficiency, heat transfer efficiency, and convergence pressure. Present
accuracy in the prediction of EMV and E does not justify the added
expense of computing these efficiencies for each plate (on each iteration
of the column calculation) except in research work.

X. APPENDIX - MACHINE UTILIZATION
A. Description of Machine and System
The IBM 704 Electronic Data Processing Machine
The IBM 704 is a large-scale, high-speed electronic calculator
(57)
controlled by an internally stored program of the single address type(57).
The machine executes instructions at an average rate of about 10,000 per
second on engineering-type problems, and operates parallel in the binary
number system. The machine installed at the General Motors Technical
Center contains 8,192 magnetic core storage registers, 8,192 units of
storage on four logical high-speed magnetic drums, and eight magnetic
tape units, each with 2400 feet of tape.
The Input-Output System
The General Motors Research I/O system is a method of integrating the 704 main frame, attached equipment, peripheral equipment, coordinators, and programmers into an efficient, smoothly-functioning
group. Objectives of the system are 36):
"(1) Minimizing input-output time by utilizing magnetic
tape as the input-output medium.
(2) Tying together of many jobs to be run as a multi-job
entity in order to reduce machine time costs.
(3) Reducing programmer effort and time required to
process a job through the check-out stage into the production
stage.
(4) Simplifying operating procedures so that machine operators need not be trained specifically for each job and programmers need not be present.
(5) Standardizing procedures relative to layout and
programming for greater staff efficiency.
(6) Providing the programmer with input, output, and
debugging routines without the usual storage liability."
"200"

-201~
In order to use the I/O system, the programmer must relinquish
use of locations (O - 299)10 in core storage, tapes 1, 2, 3, and 7,
logical drum number 1, and all sense switches, For all except the very
longest problems, the cost of using the I/O system is more than repaid
by the advantages.
CRUTCH
An adjunct to the I/O system is a group of subroutines called
CRUTCH, occupying locations (7000 - 8191)10 of core storage. CRUTCH
contains the common functional subroutines of exponential, logarithm,
square root, etc., plus several subroutines used in transfer of instructions or data between the various storage media of the machine.
Operation of the System
The sequence of events involved in making a machine run under
control of the I/O system is as follows: (1) The input deck, containing
program, data, etc., in proper order is delivered to the 704 coordinator.
(2) The coordinator prepares the BCD input tape, utilizing the peripheral
card reader and tape unit. (3) The BCD input tape, comprising one or
more machine runs together with suitable control flags, is delivered to
the machine operator. (4) The 704, in input-translation phase, translates instructions and data into the binary mode, prepares the binary
input tape, and automatically enters the compute phase, (5) The I/O
system, in compute phase, brings the programs, one at a time, into core,
brings in CRUTCH if indicated, and transfers control to the programs.
The programs write output on the binary output tape and return control
to the system when finished. When all programs have been run, the 704

-2022
automatically enters the next phase. (6) During output-translation
phase, the binary output tape is translated and written on the BCD output
tape, which is then dismounted and taken to the peripheral room..(7) The
BCD output tape is printed on the peripheral printer, and the results are
returned to the originators of the machine runs.
B. Machine Run Procedure
Figure 23 illustrates the control sequence for a typical production run. The contents of Decks 1 and 2 are given in Table 25. The
composition of the input deck for a machine run is the following:
(1) "A" Record - A single blue card containing information
required by the I/O system. Column 63 contains an 11 or minus punch,
the signal that CRUTCH is to be brought into core just before transferring
control to the program.
(2) Deck 1 Manilla-colored column binary program cards with
identification numbers UOOZ5001 through UOOZ5146, the last card of which
is an "ENDI' or transition card and is the signal for transfer of control
to the Executive subprogram.
(3) Permanent Operating Data -Green class 2 (specified field,
floating point decimal) data cards, containing the following data:
molecular weights, normal boiling points, critical temperatures, critical
pressures, cube roots of kinematic viscosities at 60~F, cube roots of
kinematic viscosities at 200 ~F specific gravities, molecular volumes to
the 0.6 power, molecular volumes to the 1/3 power, vapor diffusivity
constants, ideal gas enthalpy coefficients, Beattie-Bridgeman constants,
and pressure correction constants. Identification numbers are B67E4001

-203
General Motors Input-Output System reads Deck 1
and transfers control to Executive Subprogram.
Executive Subprogram stores zeros where necessary,
stores portions of Deck 1 on drums, reads Permanent
Operating Data and Deck 2 and stores them on drums,
and transfers control to operating program.
Subprogram A reads a set of K values and absorber
oil data, prepares correlation of K values, stores
absorber oil properties and functions of them in
appropriate locations.
Machine reads a set of data for a particular column
computes properties of output streamsa recovery of
components, and conditions on each plate of column,
and-: print s- result sYes
Is there another set of column data?|
Yes- Is there another set of K values and absorber oil data?
No
Control transferred back to Input-Output System|
Figure 23. Machine Run Procedure.

-204"
TABLE 25
COMPOSITION OF PROGRAM DECKS
Block Title/Purpose Symhol Locations
1-E Executive Subprogram E 5030-6999
1-C Plate-to-Plate Calculation Subprogram C 0360-1877
1-D Output Subprogram D 2150-2404
1-R Plate Efficiency Subroutine PLEFF 2410-2936
1-S Vapor Density Subroutine VDENS 2940-3278
1-U Vapor Diffusivity Subroutine VDIFF 3280-3456
1-V Liquid Diffusivity Subroutine LDIFF 3460-3574
1-X Heat Transfer Efficiency Subroutine HTEFF 3595-3793
1-Y Thermal Conductivity Subroutine TCOND 3795-3923
1-N Dew Point, Bubble Point Subroutine DEWPT 3925-4156
BUBPT
1-P Convergence Pressure Subroutine CONVP 4160-4499
l-Q K Value Subroutine KVALU 4505-4774
1-T Liquid Density Subroutine LDENS 4780-4874
1-W Liquid Viscosity Subroutine LVISC 4880-5025
2-A Data Conversion Subprogram A 0360-1424
2-B Initial Approximation Subprogram B 1425-2409
2-F Stream Temperature (Vapor or Liquid) VTEMP 2410-2578
Subroutine LTEMP
2-G Vapor Enthalpy Subroutine VAPEN 2580-2658
2-H Liquid Enthalpy Subroutine LIQEN 2660-2765
2-J Ideal Gas Enthalpy Subroutine IDGAS 2895-2993

-205=
TABLE 25
COMPOSITION OF PROGRAM DECKS (CONT'D)
Block Title/Phrpose Symbol Locations
2-K Pressure Correction PCORR 2995-3596
2-L Heat of Vaporization Subroutine HVAPN 3600-3686
2-M Enthalpy Change of Subcooled Liquid SUBCL 3740-3906
Subroutine
CRUTCH: Transfer instructions or data from GOREAD 7050-7078
magnetic tape into core memory
READOV 0252
READTR 0253
CRUTCH Transfer instructions or data between DRMTAP 7282-7321
core and magnetic tape or drums
CRUTCH Print output GODUMP 7187-7255
NODUMP 7187-7255
WRITER 7112-7186
REMARK 7094-7111
CRUTCH Exponential GMEXP1 7418-7490
CRUTCH Natural Logarithm GMLOG1 7365-7417
CRUTCH Square Root GMSQT2 7342-7364
CRUTCH Turn off accumulator overflow, multiplier- TRGOFF 7088-7093
quotient overflow, divide check, and
tape check triggers
CRUTCH Space printer to top of next page EJECT1 7322-7338
CRUTCH Space printer to top of next 1/6 page EJECT6 7322-7338

-2o06
through B67E4159o Cards B67E4118 and B67E4159 have a minus punch in
dolumn 1 to indicate returns from the READOV subroutine.
(4) Deck 2 - Manilla column binary program cards with identification numbers UOOZ6001 through UOOZ6112 The last card is a transition
card returning control from the READTR subroutine to the Executive subprograme
(5) K Value and Absorber Oil Data - Green class 2 data cards
with identification numbers in the series UOOZ5 —-. K values must be
given for every component appearing in following column data sets. One
or two absorber oil components must be specified. The last card contains
a minus punch in column 1. One or more sets, each followed by one or
more sets of column data, may appear.
(6) Column Data - Green class 6 (variable field; octal, fixed
point decimal, floating point decimal) data cards with identification
numbers in the series UOOZ6 —-. One or more sets may be used with each
set of K value and absorber oil data, The last card of the last set
contains a plus punch (12) in column 1; last cards of all other sets
contain a minus punch in column 1.
C, Programming and Coding Policy
Assembly
The SHARE Assembly Program (SAP), originally written by the
United Aircraft Corporation and subsequently modified by the Data
Processing Group, General Motors Research Staff, was used in coding and
assembly of the problem. The program consisted of two decks, each containing several "blocks" of coding~ Each block was a separate subprogram

-207or subroutine and was distinguished by its own alphabetic character in
column 76 of each of the decimal symbolic program cards.
Transfer of Control
Transfer between subprograms was made by the direct, unconditional transfer instruction (TRA). Transfer between subprograms and
subroutines or between two subroutines was accomplished by the Transfer
and Set Index (TSX) instruction, which is the standard method of transfer
to subroutines.
Subroutine Conventions'The program was arranged in blocks, called subprograms or
subroutines, in an attempt to facilitate the checking out, modification,
and replacement of portions of the program. Because of the size of the
program, a rigid set of conventions had to be devised and followed in
order to avoid confusion in the use of subroutines. These rules were as
follows:
(1) Index register 4 was used in the calling sequence of all
subroutines. (This is a SHARE convention.)
(2) The machine had to store the contents of index register 4
before executing a TSX instruction, but the subroutine to which the
machine transferred was required to store the contents of index registers
1 and 2 and to replace them before the return transfer.
(3) Index register 1 was usually used for handling the 25
components; index register 2, for counting iterations around a loop.
(4) All iterative procedures had a counting arrangement, even
if there were other exits from the loop, in order to insure that an exit
would eventually be made.

-208(5) If data were to be used or treated by more than one subprogram or subroutine, they were stored in block E (Executive Subprogram),
Otherwise, they were stored in the block using them.
(6) The calling sequence of a subroutine contained all information that the subroutine (and any other subroutines which it employed)
required for its operation. No exchange of information between blocks,
other than by means of a calling sequence, was allowed.
(7) Blocks of data, such as a group of 25 mole fractions,
were designated by the first address, e.g., xm. If a subroutine required incremented addresses for an indexed instruction, e.g., xm + 25,
the subroutine incremented it.
(8) If a subroutine returned one or two items of information,
they were placed in the accumulator and the multiplier-quotient registers
before the return. Otherwise, the calling sequence specified the first
location of the block in which the data were to be stored.
(9) If the indication that a certain function or piece of
information occurring in a group of locations had not been calculated
was to be zero, the subroutine first stored zeros in all locations of
the group, then filled the appropriate ones.
(10) Subroutines checked input data for reasonableness and
gave an error or off-range return if indicated.
(11) In the event an error occurred in a subroutine, the subroutine gave an indication of it with a remark or dump, but the parent
program took necessary corrective action if possible.
(12) All subroutines (except library subroutines not written
by the author) performed formal initiating and terminating procedures.

-209O
The initiating procedure included (1) storing contents of accumulator,
multiplier-quotient register, and index registers, (2) transferring information to calling sequences of other subroutines, (3) incrementing
and storing addresses furnished by the calling sequence, and (4) executing TRGOFF. The terminating procedure included (1) checking error triggers and giving an error return if indicated, (2) restoring contents of
index registers, (3) loading answers into the accumulator and multiplierquotient registers (if applicable), and (4) returning to the parent
subroutine or subprogram.
D. Userts Instructions
The term "user" is used in place of "operator", which occurs
in most discussions of this type, in recognition of the roles which
various persons play in the operation of the I/O system. The program
operates entirely within the system, so neither the coordinator nor the
machine operator need have any knowledge of the problem. These instructions are for the person who desires to use the program to solve a
problem. Knowledge of the IBM 704 and the I/O system, which may be
obtained from References (36) and (57), is assumed.
Machine Storage
In the following discussion, all locations will be given in
the decimal number system. In the analysis of a word or storage unit,
the octal representation will be used.
Components
The order of components is (1) hydrogen, (2) nitrogen, (3) methane, (4) ethylene, (5) ethane, (6) propylene, (7) propane, (8) isobutane,

-210l
(9) n-butane, (10) isopentane, (11) n-pentane, (12) hexane, (13) heptane,
(14) octane, (15) nonane, (16) decane, (17) 1st absorber oil component,
(18) 2nd absorber oil component, (19) carbon monoxide, (20) oxygen, (21)
carbon dioxide, (22) hydrogen sulfide, (23) water vapor, (24) and (25)
not used. If there is one absorber oil component, it should be the first;
if there are two, the less volatile should be the second. The order or
identity of components should be changed only with extreme care and a
complete knowledge of the program.
K Values
Class 2, five-digit data cards have been found most convenient
for loading K data, but actually any applicable type card may be used to
load the indicated data into the proper locations. All numbers are in
floating point form.
Location Datum
5979 Pressure (psia) of K value correlation
5980 t1 in K correlation (0~F)
5981 t2 (100~F)
5988 t9 (800~F)
5989 1st (lower) convergence pressure, Pkl
5990 2nd (higher convergence pressure, Pk2
5991 K1,1 (component 1 at tl) at Pkl
5992 K2,1 (component 1 at t2)
5999 Kg9, (component 1 at t9) at Pkl
6000 K, 2 (component 2 at t1)
eo o a

-211"
Location Datum
6008 K9,2
6215 K9,25 (component 25 at t9) at Pkl
6216 K1,1 (component 1 at tl) at pk2
6440 K9,25 at Pk2
The temperature range 0~F to 800~F was found necessary for
computing bubble points of lean oils. The program is flexible, however,
and any temperature range above O0F may be used, provided only that the
temperatures tI through t9 be equally spaced. K values need be given
only for the components that appear in the column data sets which follow.
The program writes over the original data, so a complete set must be
given each time.
Absorber Oil Data
A complete set of data must be given for each absorber oil
component used; only one need be used in any problem, however. All
numbers are in floating point form.
Location Datum
6441 Molecular weight of 1st absorber oil
6442 Normal boiling point in ~F
6443 Critical temperature in ~F
6444 Critical pressure in psia
6445 Kinematic viscosity at 60~F
6446 Kinematic viscosity at 200~F

-212
Location Datum
6447 Specific gravity (or API gravity with negative sign)
6448 Heat of vaporization at 77~F in Btu/lb mole
6449 Molecular weight of 2nd absorber oil
6456 Heat of vaporization of 2nd absorber oil
The last card of the absorber oil data must have a minus punch
in column 1.
Column Data
Class 6 data cards have been found most convenient for loading
column data, but again any appropriate data card type may be used.
Numbers are in floating point decimal form unless specified otherwise.
Location Datum
6459 Column identification number (octal)
6460 Rich gas dimension code flag (octal)
6461 Lean oil dimension code flag (octal)
6462 Column dimension code flag (octal)
6463 Computation code flag (octal)
6464 Number of actual plates (fixed decimal)
6465 Quantity of rich gas
6466 Thermal energy indication of rich gas
6467 Quantity of lean oil
6468 Thermal energy indication of lean oil
6469 Composition of component 1 in rich gas
6470 Composition of component 2 in rich gas
* * a a

-213Location Datum
6493 Composition of component 25 in rich gas
6494 Composition of component 1 in lean oil
6518 Composition of component 25 in lean oil
6519 Column pressure in psia
6520 Column diameter in ft
6521 Plate spacing
6522 Length of liquid path across plate in ft
6523 Weir height
6524 Heat loss by convection and radiation
6525 Fraction of column cross section for vapor flow
6526 1st side stream cooling (intercooling) data (octal)
6535 10th side stream cooling (intercooling) data (octal)
6536 Initial estimate mole fraction of component 1 in lean
gas, or initial estimate fractional recovery of
component 1
6537 Initial estimate.., component 2...
6560 Initial estimate.., component 25...
6561 Initial estimate mole fraction of component 1 in rich oil
6585 Initial estimate.., component 25...
6695 Initial estimate temperature of lean gas in ~F
6749 Initial estimate temperature of rich oil in ~F
Data need not be loaded into locations 6526 - 6585, 6695, and 6749 unless
specified by code flags.

-214Code Flags
A code flag is a twelve-digit octal number. The left-most
digit, together with the sign, which may be omitted if positive, is
called the prefix (P); the next five digits are the decrement (D); the
next one, the tag (T); and the right-most five, the address (A). The
significance of the various code flags is given below.
Flag Code Meaning
Gas P - 1 Thermal energy indication given by temperature in ~F
P = 2 Energy indication is enthalpy in Btu/lb
P = 4 Energy indication is enthalpy in Btu/lb mole
D = 1 Composition in mole fraction
D = 2 Composition in weight fraction
T = 1 Unit time is one second
T = 2 Unit time is one hour
T = 4 Unit time is one day
A = 1 Quantity of stream in moles/unit time
A = 2 Quantity of stream in SCF/unit time
A = 4 Quantity of stream in lb/unit time
Oil Same as Rich Gas Dimension Code Flag
Column P = 1 All column dimensions in ft
P = 2 Weir height and plate spacing in in., others in ft
D = 1 No intercooling
D = 2 Intercooling in Btu/unit time
D = 4 Intercooling given by temperature change in ~F
T = 1 Unit time is one second
T = 2 Unit time is one hour
T = 4 Unit time is one day

-2150
Flag Code Meaning
Column A = 1 No convection and radiation heat loss
A 2 Heat loss in Btu/unit time for whole column
A 4 Heat loss in Bti/ft2 (unit time)
Compute P = 1 Machine is to compute initial estimate of outlet
compositions and temperatures
P = 2 Initial estimate of outlet compositions and
temperatures is given
P = 4 Initial estimate of fractional recoveries of
components is given
D Maximum number of complete column iterations
T = 1 Print final results and column summary on last
iteration only
T = 2 Print column summary after each iteration and
complete plate results on last iteration
T = 4 Print complete plate results and column summary
on each iteration
At the conclusion of a column run, the machine will have changed input
data into the standard units of the program —moles, mole fractions,
feet, seconds, and OF, The output stream compositions will be loaded
into locations 6536 through 6585, the temperatures will be in locations
6695 and 6749, and the prefix of the Computation Code Flag will be +2.
Table 26 contains a typical set of column data.
Dumps
Dumps in the 306 — series, originally diagnostic, have been
left in operation to give a running account of progress at the end of
each column iteration (every 30 to 60 seconds, usually). Numbers in
locations 1126 - 1150 are the values of ri of the various components;
locations 6692 - 6716 contain values of e qi.for the next iteration;

2216TABLE 26
A TYPICAL SET OF COLUMN DATA
Addr.* Opn. Data Identification
6459 OCT 001005001001 UOOE6701
6460 OCT 100001400001 UOOE6702
6461 OCT 100001400001 UOOE6703
6462 OCT 100001100001 UOOE6704
6463 OCT 200024200000 UOOE6705
6464 DEC 24 UOOE6706
6465 DEC 83992.,69.,16375.,71 UOOE6707
6469 DEC 0,0,.7529,0,.0564,0,.0505,.0092,.0181,o,o, UOOE6708.0127,0,0,0,0
DEC 0: 0,0,,,.1002,o00,0,0 U00E6709
6494 DEC 0,0,0,00,0,0,0,00,0,0,0,000,0,1.,0 UOOE6710
DEC 00,0,0,0,0,00 UOOE6711
6519 DEC 400,,5.,2.,2.917,.1562,0, 687 UOOE6712
6536 DEC 0,,.81413,0,.05448,0,.0274,.00046,.00009, U00E6713
0,,. 0001
DEC 0,0,00,,.00001,00,0O,.10334,o00,0,,0 UOOE6714
6561 DEC 0,0,.085,0,.0253,0,.0854,.0288,.0591,0,0, UOOE6715.0412
DEC 00,0,0,,6471,00,O0,.026,00, 0,0 UOOE6716
6695 DEC 78. UOOE6717
6749 DEC 92. UOOE6718
* A series of numbers loads sequentially;

-217location 675 contains the value of tLm+l on the last iteration where m
was the highest plate reached. Thus, if a machine error or other difficulty should interrupt computation, the series of dumps will give the
minimum information necessary for restarting the problem.
Dump 105 gives the initial estimate of the outlet compositions;
dump 106, the initial estimate of recoveries. Any other dumps indicate
trouble and should contain information sufficient for diagnosis.
Timing
A column of 20 plates containing 12 components will require
about 30 seconds per iteration. Iteration time is roughly proportional
to the number of plates and the number of components. About 50 to 100
iterations will be required for convergence, depending upon how good
the initial approximation is. It is advisable to make one or two test
runs of short duration to insure that the program is working properly.
Convergence Ratio Exponents
The parameters gi used in the equations
gi
(el,i)k+l = (1,i )k (ri) (190)
were determined experimentally and found to be suitable for all problems
investigated. It is quite possible, however, that they will not produce
convergence for some future problems. The parameters are found in locations 479 - 503 of deck 1 and may be changed by override cards. It is
recommended that the parameters be reduced by factors of 2 until a successful set is found. Examination of the 306-. series of dumps will
show which parameters need to be changed.

_218Heat Transfer Efficiency Constant
A value of C to be used in a given column calculation may be
estimated by the method discussed under Analysis of Results, Prediction
of the Heat Transfer Efficiency Factor.'E. Subroutine Specifications
All subroutines are restricted to floating point arguments.
Contents of index registers 1 and 2 will be restored. SHARE assembly
program (SAP) symbolic notation is used in calling sequences, i.e.,
(Location, Prefix, Address, Tag, Decrement). TRGOFF is executed at the
beginning of each subroutine, and if divide check, accumulator overflow,
or multiplier-quotient overflow is detected, an error dump and error
return will be given. Access to the table of physical properties is
assumed.
Plate Efficiency (PLEFF)
Purpose: Given temperature, pressure, compositions of vapor and liquid
streams, flow rates, and dimensions of the bubble cap plates;
compute the Murphree plate efficiency, EMV, for each component.
Restrictions: EMV are restricted to the range 0.001 to 1.2.
Method. NG and NL, based on systems air-water-ammonia and air-watercarbon dioxide, are combined to give NOG. Ep and EMV are
computed from the separate phase resistances.
Calling sequence: (tLm in AC, p in MQ)
d TSX PLEFF, 4
d + 1 PZE Addr of xm 1,0, Addr of Yml 1

-219d + 2 PZE Addr of Km 1. 0, Addr for EMV 1
d + 3 Value of Lm
d + 4 Value of Vml
d + 5 Error return
d + 6 Normal return (EMv stored where specified)
Storage required: Instructions — 250, data — 252, erasable — 25.
Other subroutines used: VDENS, LDENS, VDIFF, LDIFF, LVISC, GMSQT2,
GMLOG1, GMEXP1, TRGOFF, NODUMP.
Vapor Density (VDENS)
Purpose: Given temperature, pressure, and composition of a vapor
mixture, compute its density in lb moles/ft3
Restrictions: Mole fractions smaller than 0.001 are ignored.
Method. The volume-explicit form of the Beattie-Bridgeman equation
is used(2). Constants for pure components are combined
according to methods of Beattie, Stockmayer, and Ingersoll(14)
Calling sequence: (tv in AC, p in MQ)
d TSX VDENS,4
d + 1 PZE Addr of Ym-l,1
d + 2 Error return
d + 3 Normal return (pV in AC)
Storage required: Instructions — 156, data — 163, erasable — 20.
Other subroutines used: TRGOFF, NODUMP.

-220Liquid Density (LDENS)
Purpose: Given temperature and composition of a liquid mixture, compute
its density.
Restrictions: Mole fractions smaller than 0.001 are ignored.
Method: Density at 60~F is computed as a summation based on pure
component densities. Temperature correction is made.
Calling sequence: (tL in AC)
d TSX LDENS,4
d + 1 PZE Addr of Xm,1
d + 2 Error return
d. + 3 Normal return (PL in lb/ft3 in AC, PL in lb moles/ft3 in MQ)
Storage required: Instructions — 73, data — 12, erasable — 10.
Other subroutines used: TRGOFF, NODUMP.
Vapor Diffusivity (VDIFF)
Purpose: Given temperature, pressure, composition of a vapor stream,
and the composition of an adjacent liquid stream, compute the
vapor phase diffusivity of each component that appears in
either the vapor or the liquid. Units are ft2/sec.
Restrictions: The correlation is based on the components hydrogen,
nitrogen, and the hydrocarbons methane through absorber oil.
Method: Diffusivities of binaries are computed by Gilliland's formula(39)
Diffusivities of components in the multicomponent mixture are
computed from those of binaries by Wilke's formula 93)
Calling sequence: (tV in AC, p in MQ)
d TSX VDIFF,4

-221
d + 1 PZE Addr of Ym,-,1, O, Addr for DG,
d + 2 PZE Addr of xmi
d + 3 Error return
d + 4 Normal return (DG stored where specified)
Storage required: Instructions — 120, data — 30, erasable — 25.
Other subroutines used: GMSQT2, TRGOFF, NODUMP.
Liquid Diffusivity (LDIFF)
Purpose: Given temperature, viscosity, and composition of a liquid
mixture, compute the diffusivity of each component in ft2/sec.
Restrictions: None.
Method: Diffusivities are computed by the method of Wilke and Chang(94)
Calling sequence: (tL in AC, ~L in MQ)
d TSX LDIFF, 4
d + 1 PZE Addr of Xm., O, Addr for DLl
d + 2 Error return
d + 3 Normal return (DL where specified)
Storage Required: Instructions — 72, data — 8, erasable — 35.
Other subroutines used: GMSQT2, TRGOFF, NODUMP.
Liquid Viscosity (LVISC)
Purpose: Given temperature, pressure, density, and composition of a
liquid mixture, compute its viscosity in lb/ft sec.
Restrictions: None.
Method: Kinematic viscosities of pure components at 60~F and 200 F are
combined according to the formula of Kendall(59). Temperature

-222 -
dependence is expressed according to Cornelissen and
Waterman(24). Further correction is made for pressure.
Calling sequence: (tL in AC, p in MQ)
d TSX LVISC,4
d + 1 Value of liquid density
d + 2 PZE Addr of x
m,l
d + 3 Error return
d + 4 Normal return (pL in AC)
Storage required: Instructions — 108, data — 23, erasable — 15.
Other subroutines used: GMLOG1, GMEXP1, TRGOFF, NODUMP.
Heat Transfer Efficiency (HTEFF)
Purpose: Given temperature, pressure, composition and density of a
vapor phase, and individual component Murphree plate efficiencies, compute the heat transfer efficiency.
Restrictions4 Eq is restricted to the range 0.05 to 1.0.
Method: Specific heat of the vapor phase and vapor thermal conductivities of all components are computed. Efficiency is estimated
by the summation
Eq Cq ymi cp k,)2/3 (246)
Eq Cq z Ymi (cp PV DG,i
Calling sequence. (t in AC, p in MQ)
d TSX HTEFF 4
d + 1 PZE Addr of Ym,^, 0, Addr of EMV,
d + 2 PZE Addr of DG,
d 3 Value of vapor density
d 4+ 3 Value of vapor density

-223d + 4 Error return
d + 5 Normal return (E in AC)
Storage required: Instructions — 117, data — 66, erasable — 15.
Other subroutines used: VAPEN TCOND, GMLOG1, GMEXP1, TRGOFF, NODUMP,
Thermal Conductivity (TCOND)
Purpose: Given temperature and composition of a vapor mixture, compute
thermal conductivity of all components appearing. Units are
Btu/sec ft ~F.
Restrictions: None.
Method: Values at 32~F are stored. Temperature dependence is computed
according to the Sutherland equation.
Calling sequence: (tV in AC)
d TSX TCOND,4
d + 1 PZE Addr of Ym,l, O0 Addr for k,
d + 2 Error return
d + 3 Normal return (ki where specified)
Storage required: Instructions — 63, data — 56, erasable — 10.
Other subroutines used: TRGOFF, NODUMP.
Stream Temperature (VTEMP or LTEMP)
Purpose: Given the molal enthalpy and composition of a vapor or liquid
stream and the pressure, find the temperature.
Restrictions: Temperature must be in the range -100~F to 800~F. If
outside the range, the bound exceeded is given and the offrange return is made.

-224Method: The Newton method is used to find a temperature corresponding
to the molal enthalpy given.
Calling sequence, (H or H in AC, p in MQ)
d TSX VTEMP,4
d + 1 PZE Addr of Ym,1' O Addr of 1st approx. to tV (or zero)
d + 2 Error return
d + 3 Off-range return (bound in AC)
d + 4 Normal return (tV in AC)
The calling sequence for LTEMP is similar to the above.
Storage required Instructions — 140, data — 14, erasable — 15.
Other subroutines used: VAPEN (with VTEMP) or LIQEN and LIQSC (with
LTEMP), TRGOFF, NODUMP.
Vapor Enthalpy (VAPEN)
Purpose: Given temperature, pressure, and composition of a vapor
mixture, compute its enthalpy in Btu/lb mole.
Restrictions: If tV is outside the range -100~F to 800~F, the exceeded
bound is used instead of tV and the off-range return is made.
Method: HV is computed as the sum of HV and AHpc the ideal gas
enthalpy and the pressure correction to the ideal gas enthalpy
at tvy respectively.
Calling sequence: (tV in AC, p in MQ)
d TSX VAPEN,4
d + 1 PZE Addr of Ym,
d + 2 Error return
d + 3 Off-range return (_V in AC)
d + 4 Normal return (HV in AC)

-225Storage required: Instructions — 60, data — 9, erasable — 10.
Other subroutines used- IDGAS, PCORR, TRGOFF, NODUMP.
Liquid Enthalpy (LIQEN LIQSC)
Purpose: Given temperature, pressure, and composition of a liquid
mixture, compute its enthalpy in Btu/lb mole.
Restrictions: If tL is outside the range -100~F to 800~F, the exceeded
bound is used instead of tL and the off-range return is made.
Method: HL is computed as the sum of 47, A-v7, and Hsc, the ideal gas
enthalpy at 77~F, the isothermal heat of vaporization at 77~F
and the enthalpy change of the subcooled liquid between 77~F
and tL. If LIQSC (liquid enthalpy, short cut) is used, the
sc
machine computes a new value only of AHsC using the values of
0. V
H77 and 77 determined in an earlier use of LIQEN.
Calling sequence: (tL in AC, p in MQ)
d TSX LIQEN,4 (or LIQSC,4)
d + 1 PZE Addr of xm.
d + 2 Error return
d + 3 Off-range return (_L in AC)
d + 4 Normal return (H in AC)
Storage required: Instructions — 82, data — 14, erasable — 10.
Other subroutines used: IDGAS, HVAPN, SUBCL, TRGOFF, NODUMP.
Ideal Gas Enthalpy (IDGAS)
Purpose: Given temperature and composition of a vapor mixture, compute
the ideal gas enthalpy in Btu/lb mole.

-226Restrictions: Mole fractions less than 0.001 are ignored.
Method: Enthalpies of pure components are computed by fifth-order
polynomials and are then summed by mole fraction.
Calling sequence: (tv in AC)
d TSX IDGAS,4
d + 1 PZE Addr of Yml or Xm,l
d + 2 Error return
d + 3 Normal return (H~ in AC)
Storage required: Instructions — 76, data — 8, erasable — 15.
Other subroutines used: TRGOFF, NODUMP.
Pressure Correction (PCORR)
Purpose: Given temperature, pressure, and composition of a vapor
mixture, compute the pressure correction to the ideal gas
enthalpy in Btu/lb mole.
Restrictions: Parameters in the correlation are restricted to the
following ranges: 0 Pr % 2.0, 0.3 Tr < 3.0,
0.3 < Br ~ 2.2. A parameter exceeding a bound is replaced
by that bound. Mole fractions less than 0.001 are ignored.
Method: The partial molal pressure correction of each component is
computed as a function of reduced temperature, reduced pressure,
and reduced boiling point. Corrections are combined by mole
fraction.
Calling sequence: (tV in AC, p in MQ)
d TSX PCORR,4
d + 1 PZE Addr of Yml

-227d + 2 Error return
d + 3 Normal return (nHPC in AC)
Storage required: Instructions- 187, data — 395, erasable — 20.
Other subroutines used: TRGOFF, NODUMP.
Heat of Vaporization (HVAPN)
Purposes Given the composition of a liquid mixture, compute its isothermal heat of vaporization at 770F in Btu/lb mole.
Restrictions: Mole- fractions less than 0.001 are ignored.
Method: Partial molal heats of vaporization at 77~F are combined by
mole fraction.
Calling sequence:
d TSX HVAPN,4
d + 1 PZE Addr of xm,
d + 2 Error return.
d + 3 Normal return (AT7 in AC)
Storage required: Instructions — 52, data — 30, erasable — 5.
Other subroutines used: TRGOFF, NODUMP.
Enthalpy Change of Sub cooled Liquid (SUBCL)
Purpose: Given tl and t2 of a liquid mixture and its composition, compute
the enthalpy change in heating or cooling from t1 to t2. Units
are Btu/lb mole.
Restrictions: Mole fractions smaller than 0.001 are ignored.
Method: Specific heat at the average of tL and t2 is computed by the
formula of Fallon and Watson (33) Enthalpy change is computed

-228as the product of specific heat, molecular weight, and
temperature change.
Calling sequence: (t1 in AC, t2 in MQ)
d TSX SUBCL,4
d + 1 PZE Addr of xm,
d + 2 Error return
d + 3 Normal return (AHESC in AC)
Storage required. Instructions — 122, data — 25, erasable — 20.
Other subroutines used: TRGOFF, NODUMP.
K Value (KVALU)
Purposes Given temperature,-pressure, composition, and the natural
logarithm of convergence pressure, compute the K value of
every component appearing in the mixture.
Restrictions: Temperature it restricted to 1.5 times the range of the
correlation. Convergence pressure must lie between the two on
which K correlations are based. A variable exceeding a bound
is replaced by that bound.
Method. The natural logarithm of K is computed as a function of temperature and pressure at each of the two convergence pressures
used in the correlation. Interpolation of Xn(K) versus in(Pk)
is performed. The subroutine contains its own exponential
routine based on a formula of Hastings ).
Calling sequence: (t in AC, p in MQ)
d TSX KVALU,4
d + 1 Value of an(pk)

-229d + 2 PZE Addr for K1, 0, Addr of xml or Yml
d + 3 Error return
d + 4 Normal return (K values where specified)
Storage required: Instructions — 167, data-. 88, erasable — 20.
Other subroutines used: TRGOFF, NODUMP.
Convergence Pressure (CONVP)
Purpose: Given temperature and composition of a liquid mixture, compute
the natural logarithm of its convergence pressure.
Restrictions: If hydrogen is present, the convergence pressure is
immediately set equal to 20,000 psia. If there is no component
lighter than ethylene, pk of 1000 psia is used. Only hydrogen,
nitrogen, and the hydrocarbons are used in the correlation.
Method: The natural logarithm of convergence pressure is computed
according to the method of Lenoir and White(65)
Calling sequence: (tL in AC)
d TSX CONVP,4
d + 1 PZE Addr of xm
d + 2 Error return
d + 3 Normal return [an(pk) in AC]
Storage required: Instructions — 185, data — 133, erasable — 20.
Other subroutines used: TRGOFF, NODUMP.
Dew Point - Bubble Point (DEWPT - BUBPT)
Purpose: Given pressure, composition, and a first approximation to the
temperature sought (optional), compute the dew point or
bubble point temperature in ~F.

-230Restrictions? Negative mole fractions are ignored.
Method: In DEWPT, a temperature is found which produces the equality
Yj i = 1 (201)
Ki
In BUBPT, a temperature is found which produces the equality
2i (xi)(Ki) 1 (246)
The method of false position is used,
Calling sequence: (1st approx. of tr in AC, p in MQ)
d TSX DEWPT,4
d + 1 PZE Addr of Ym ij O'Addr for K1
d + 2 Error return
d + 3 Normal return (tp in AC)
The calling sequence for BUBPT is similar to the above*
Storage required: Instructions-^ 174, data — 47, erasable — 10.
Other subroutines used: CONVP, KVALU, TRGOFF. NODUMP.
F. Availability of Program
A listing of the machine program requires approximately one
hundred sheets of IBM 407 tabulator paper. A microfilm copy is in the
possession of Professor J. T. Banchero, Department of Chemical and
Metallurgical Engineering, University of Michigan. The machine program,
as punched cards, is filed with the Data Processing Group, General Motors
Research Staff, Detroit, Michigan

XI. REFERENCES
1. Am. Inst. of Chem. Engrs. Research Committee on Tray Efficiqncies
in Distillation Columns, 3rd Annual Progress Report, p. 7 (1955).
2. Am. Inst. of CAem. Engrs. Research Committee on Tray Efficiencies
in Distillation Columns, 3rd Annual Progress Report, p. 8 (1955).
3. Am. Inst. of Chem. Engrs. Research Committee on Tray Efficiencies
in Distillation Columns, 3rd Annual Progress Report, p. 21 (1955).
4. Am. Inst. of Chem. Engrs. Research Committee on Tray Efficiencies
in Distillation Columns, 3rd Annual Progress Report, p. 24 (1955).
5. Am. Inst. of Chem. Engrs. Research Committee on Tray Efficiencies
in Distillation Columns, 3rd Annual Progress Report, p. 34 (1955).
6. Am. Inst. of Chem. Engrs. Research Committee on Tray Efficiencies
in:.lDistillation Columns, 4th Annual Progress Report, p. 7 (1956).
7. Amundson, N. R., and A. J. Pontinen, Ind. Eng. Chem. 50, 730 (1958).
8. Arnold, J. H, J. Chem. Phys. 1, 170 (1933).
9. Bakowski, S., Chem. Eng. Sci. 1, 266 (1952).
10. Bauer, C..R., and J. F. Middleton, Petrol. Refiner 32,. 111
(January 1953).
11. Beattie, J. A., Proc. Natl. Acad. Sci. U.S. 16, 14 (1930).
12. Beattie, J. A., and 0. C. Bridgeman~ Proc. Am. Acad. Arts Sci.
63, 229 (1928).
13. Beattie, J. A., and W. H. Stockmayer in "Treatise on Physical
Chemistry", Taylor and Glasstone, vol. 2, p. 206, D. Van NQstrand
Company, Inc., New York (1951).
14. Beattie J. A., W. H. Stockmayer, and H, G. Ingersoll, J. Chem. Phys., 871 (1941).
15. Benedict, M., G. B. Webb, and L. C. Rubin, Chem. Engr. Progr. 47,
443 (1951).
16. Bonner, J. S., "An Integrated System for the Automatic Solution of
Distillation Problems", Am, Inst. of Chem, Engrs. meeting, Pittsburgh,
Pa. (September 12, 1956).
17. Brown, G. G., et alT "Natural Gasoline and the Volatile Hydrocarbons",
p. 54, Natural Gasoline Association of America, Tulsa, Oklahoma (1948).
-231~

-23218. Brown, G. G., and M. Souders, Jr., Ind..En. Chem. 24, 519 (1932).
19. Chilton, T. H., and A. P. Colburn, Ind. En. Chem. 26, 1183 (1934).
20. Chilton, T. H., and A. P. Colburn, Ind. Eng. Chem. 27, 255 (1935).
21. bhu, J. C., et al J. Appl. Chem. (London) 1, 529 (1951).
22. Cobb, J. R., and R. L. McIntire, Oil Gas J. 54, 164 (April 16, 1956)..23. Collatz, L., Z. angew. Math. U, Phys. 4, 327 (1953).
24. Cornelissen, J., and H, I. Waterman, Chem. Eng. Sci. 4, 238 (1955).
25. Curtiss, C. F., andC. 0. Hirschfelder, J. Chem. Phys. 17, 550 (1949).
26. DePriester, C. L., Chem. Eng. Progr. Symposium Ser. Vol. 49, no. 7,
p. 1 (1953).
27. Drickamer, H. G., and J. R. Bradford, Trans. Am. Inst. Chem, Engrs.
^3, 319 (1943).
28. Edmister, W. C., Ind. Eng. Chem. 35, 837 (1943).
29. Edmiater, W. C., A.I. Ch. E. Journal, 1, 38 (1955).
30. Edmister, W, C., private communication (1956).
31. Edmister, W. C., and L. N. Canjar, Chem Eng. Progr. Symposium Ser.
vol. 49, no. 7, p. 85 (1953).
32. Edmister, W. C., and C. L. Ruby, Chem. Eng. Progr. 51, 95F (1955).
33. Fallon, J. F., and K. M. Watson, Nat. Petrol. News, 3, R372 (1944).
34. Gautreaux, M. F., and H. E. O'Connell, Chem. Eng. Progr. 51, 232
(1955).
35. Geddes, R, L., Trans. Am. Inst, Chem. Engrs. 42, 79 (1946).
36. General Motors Research Staff, Special Problems Dept., Data
Processing Group, "704 Operations Manual I", p. 3-1-1 (1957).
37. Gerster, J. A., "Mass Transfer on Bubble Trays", Mass Transfer
Conference, Okla. A, and M. College, Stillwater, p. 9 (1956).
38. Gerster, J. A., A. P. Colburn, et al, Chem. Eng. Progr. 45, 716
(1949).

-23339. Gilliland, E. R,, Ind. Eng. Chem. 26, 681 (1934).
40. Gilliland, E. R., and C. E. Reed, Ind.Eng. Chem. 34, 551 (1942).
41. Greenstadt, J., private communication (1958).
42. Greenstadt, J., Y. Bard, and B. Morse, "Multicomponent Distillation
on the IBM 704", Am. Chem, Soc. Meeting, San Francisco, California
(April 15, 1958).
43. Hadden, S. T., Chem. Eng. Progr. 44, 37 (1948).
44. Hadden, S. T., Chem. Eng. Progr. 44, 135 (1948).
45. Hadden, S. T., Chem. Eng. Progr. Symposium Ser-.i vol. -49,.no. 7
P. 53 (1953).
46. Harbert, W. D., unpublished report, University of Michigan, Ann Arbor
(1942). An example appears in "Handbook of Natural Gas Engineering",
D.L. Katz et al, McGraw-Hill Book Company, Inc., New York (1958).
47. Hastings, C., Jr,, "Approximations for Digital Computers", p. 182,
Princeton University Press, Princeton, N. J. (1955).
48. Hildebrand, F B., "IrAtroduction to Numerical Analysis", p. 288,
McGraw-Hill Book Company, Inc., New York (1956).
49. Hildebrand, F. B., "Introduction to Numerical Analysis", p. 445,
McGraw-Hill Book Company, Inc., New York (1956).
50. Hildebrand, F. B., "Introduction to Numerical Analysis", p. 446,
McGraw-Hill Book Company, Inc., New York (1956).
51. Hildebrand, F. B., "Introduction to Numerical Analysis", p. 447,
McGraw-Hill Book Company, Inc., New York (1956).
52. Hinshaw, D. F., Ph.D. dissertation, vol 1, p. 52, University of
Michigan, Ann Arbor (1955).
53. Hinshaw, D. F., Ph.D. dissertation, vol 2, University of
Michigan, Ann Arbor (1955).
54. Holcomb, D. E., Ph.Dl, dissertation, p. 356, University of
Michigan:, Ann:.Arbor (1941).
55. Horton, G., and W. B. Franklin, Ind. Eng. Chem. 32, 1384 (1940).
56. Householder, A. S., "Principles of Numerical Analysis", pp. 117-8,
126-8, McGraw-Hill Book Company, Inc., New York (1953).
57. I.B.M. Corp., "Manual of Operation, 704 Electronic Data Processing
Machine", New York (1955).

"23458. M. W. Kellogg Co., "Equilibrium Constants, Polyco Data" (1950).
59. Kendall, J. V., and K. P. Monroe, J. Am. Chem. Soc. 39, 1787 (1917).
60. Kirschbaum, E., Angew, Chem. B-20, 335 (1948).
61. Kistyakowsky, W., Z, Physik. Chem. 107, 65 (1923).
62. Kobe, K. A., and E, G. Long, Petrol. Refiner 28, 127 (Nov., 1949).
29, 126 (January 1950).
63. Kremser, A., Nat. Petrol. News 22, 48 (May 21, 1930).
64. Kwauk, M., A.I. Ch. E. Journal, 2, 240 (1956).
65. Lenoir, J. M., and G. A. White, Petrol. Refiner 32, 115 (December 1953).
66. Lenoir, J. M., and G. A. White, Petrol. Refiner 37, 173 (March, 1958).
67. Lewis, W. K., Jr., Ind. Eng. Chem. 28, 399 (1936).
68. Lewis, W. K., and G. L, Matheson, Ind. Eng. Chem. 24, 494 (1932).
69. McIntire, R. L., and R. 0. Shelton, "Fractionator Design with
Automatic Computing Equipment", Am. Inst. Chem. Engrs. meeting,
Pittsburgh, Pa., (September 12, 1956).
70. Mieth, H. C., and A. C. Moore, Petrol. Refiner 37, 140 (April, 1958).
71. Murphree, E. V., Ind. Eng. Chem. 17, 747 (1925).
72. Nat. Gaso. Assn. of America, "Equilibrium Ratio Data Book", Thlsa,
Oklahoma.
73. Nat. Gaso. Assn. of America, "jquilibrium Ratio Data Book", p. X,
Tulsa, Oklahoma.
74. Nelson, W. L,, "Petroleum Refinery Engineering", p. 157, McGrawHill Book Company, Inc., New York (1949).
75. Nord, M., Ind. Eng. Chem. 38, 657 (1946).
76. Norman, R. L., program prepared for Nat. Gaso. Assn. of Am. Committee
on Absorber Efficiencies, University of Michigan, Ann Arbor (Jan. 1958).
77. Papadopaulos, A., R. L. Pigford, and L. Friend, Chem. Eng. Progr.
Symposium Ser., vol. 49, no. 7, p. 119 (1953).
78. Partington, J, R,, "An Advanced Treatise on Physical Chemistry", vol.
1, p 869, 873, Longmans, Green, and Co., New York (1949).

-235"
79. Partington, J. R,, "An Advanced Treatise on Physical Chemistry",
vol. 2, p. 117, Longmans, Green, and Co., New York (1951).
80. Perry, J. H. (Editor), "Chemical Engineers' Handbook", 3rd edition,
p. 374, McGraw-Hill Book Company, Inc., New York (1950).
81. Perry, J. H. (Editor), "Chemical Engineers' Handbook"' 3rd edition,
p. 461, McGraw-Hill Book Company, Inc., New York (1950).
82. Peters, H. J., Petrol. Refiner 28, 109 (May, 1949).
83. Rose, A., R. E, Sweeny, and V. N. Schrodt, Ind. Eng. Chem. 50, 737
(1958).
84. Rossini, F. D,, et al "Selected Values of Physical and Thermodynamic
Properties of Hydrocarbons and Related Compounds", p. 98, Carnegie
Press, Pittsburgh, Pa., (1953).
85. Rossini, F. D., et al, "Selected Values of Physical and Thermodynamic
Properties of Hydrocarbons and Related Compounds", p. 247, Carnegie
Press, Pittsburgh, Pa., (1953).
86. Rossini, F. D., et al, "Selected Values of Physical and Thermodynamic
Properties of Hydrocarbons and Related Compounds", p. 287, Carnegie
Press, Pittsburgh, Pa., (1953).
87.. Rossini, F D,, et al, "Selected Values of Physical and Thermodynamic
Properties of Hydrocarbons and Related Compounds", p. 579, Carnegie
Press, Pittsburgh, Pa., (1953).
88. Sutherland, W., Phil. Mag. 36, 507 (1893).
89. Thiele, E. W., and R. L. Geddes, Ind. Eng. Chem. 25, 289 (1933).
90. Walter, J. F., and T. K. Sherwood, Ind. Eng. Chem. 33, 493 (1941).
91. Watson, K. M., Ind. Eng, Chem, 35, 398 (1943).
92. Wegstein, J. H., Communications of the Assoc. for Computing Machinery
1, 9 (J-e 1958)o
93. Wilke, C, R., Chem. Eg. Progr. 46, 95 (1950).
94. Wilke, C. R., and Pin Chang, A.I. Ch. E, Journal, 1, 264 (1955).
95. Winn, F. W., Chem. Eng. Progr. Symposium Ser,, vol. 48, no. 2,
p. 121 (1952).
96. Winn, F, W., private communication (1956).

XII, NOMENCLATURE
Capitals
A A parameter used in evaluating the temperature dependence of i-.
Ae, A' Functions used in the Edmister effective absorption factor
method.
Acs Cross-sectional area of the froth layer, ft2.
Ax i Ideal gas enthalpy correlation coefficients.
Ami Absorption factor of component i on plate m.
Ao i First constant for component i in the Beattie-Bridgeman
equation of state.
~API API gravity of a liquid.
B Bottoms stream flow rate from a fractionator, lb moles/sec.
B A parameter used in evaluating the temperature dependence
of r.
Bo i Second Beattie-Bridgeman constant of component i.
Br Reduced boiling point.
C Number of components in a system.
C Heat transfer efficiency factor.
Cs Sutherland constant, 9R.
DG Diffusion coefficient in the gas or vapor phase, ft2/sec.
DL Diffusion coefficient in the liquid phase, ft2/sec.
Ea Absorption efficiency.
Em Enthalpy balance error on plate m.
EMV Murphree vapor plate efficiency.
E Murphree vapor point efficiency,
Eq Heat transfer efficiency.
-236

-237F "F" factor equal to (u)(pv)2, (ft/sec)(lb/ft3)2.
F Empirical factor used in Norman's absorber program,
~F A multi-dimensional region.
FN Liquid phase resistance factor.
NL
FNV Vapor phase resistance factor,
H Total enthalpy of a stream, Btu/sec.
H Molal enthalpy, Btu/lb mole.
H Partial molal enthalpy, Btu/lb mole.
K Vapor-liquid equilibrium ratio.'Characterization:. factr..
GK Overall mass transfer coefficient based on the vapor phase.
-OG
W ~Liquid flow rate, lb moles/sec.
M Molecular weight.
N Mass transfer rate, lb moles/sec.
N Number of variables in a system (used with subscripts and
superscripts)
NG Number of vapor phase transfer units.
N Number of overall heat transfer units based on the vapor phase.
NL Number of liquid phase transfer units.
NO Number of overall mass transfer units based on the vapor phase.
Q Heat transfer rate, Btu/sec.
R Universal gas constant (various units).
Ri Fractional recovery of component i in an absorber.
Ri Feed plate mesh ratio of component i in a fractionator,
T Temperature on the absolute scale, OR.

-238V Vapor flow rate, lb moles/sec.
V Average vapor flow rate through the froth layer.
V Molal vapor volume, ft3/lb mole.
Xm i Number of moles of component i in liquid stream Lm per mole
of lean oil, L+1.
Ym i Number of moles of component i in vapor stream Vm per mole
of rich gas, V c.
Lower Case
a Surface area per unit volume of the froth layer, ft2/ft3.
ai Third Beattie-Bridgeman constant for component i.
aj, Coefficients of the K value correlation for component i.
a, b The two components of a binary mixture.
b. Flow rate of component i in the bottoms stream of a
fractionator.
bi Fourth Beattie-Bridgeman constant for component i.
ci Fifth Beattie-Bridgeman constant for component i.
cp Specific heat at constant pressure, Btu/lb mole ~F.
d First instruction of a calling sequence.
f.Fugacity, atm.
f Fanning friction factor.
f A function
gi Emperical convergence promotion factor.
h Heat transfer coefficient, Btu/sec ft2 ~F.
i Any component of a mixture.
jD Colburn "j" factor for mass transfer (diffusion).
jH Colburn "j" factor for heat transfer,

-239,
k Thermal conductivity, Btu/sec ft ~F.
kG Mass transfer coefficient in the vapor phase, lb moles/sec ft2 atm.
kL Mass transfer coefficient in the liquid phase..i Flow rate of component i in a liquid stream, lb moles/sec.
m The general plate of a column.
m Slope of the equilibrium curve.
n Number of variables in a system of equations.
n Number of stages or "pools" on a plate.
n Number of actual stages in a column.
n' Number of theoretical or ideal stages in a column.
p Pressure, psia.
pj.Temperature function in the K value correlation.
pk Convergence pressure, psia.
q A ratio occurring in the Wegstein extrapolation procedure,
ri Convergence ratio of component i.
t Temperature, OF.
u Vapor linear velocity, ft/sec.
v Molecular volume, cm3/g mole.
Vi Flow rate of component i in a vapor stream, lb moles/sec.
w Weir height, ft.
Wi A ratio for component i occurring in the Wegstein procedure.
x A variable.
x Length of the liquid path on a bubble plate, ft.
x Association factor.
xi Mole fraction of component i in a liquid mixture.

-240yi Mole fraction of component i in a vapor mixture.
z Froth height on a plate, ft.
zi Mole fraction of component i in either a vapor or a liquid
mixture.
Greek Letters
a, f Functions of partial derivatives in a system of simultaneous,
non-linear equations.
aP7,y,& Coefficients in a polynomial series correlation for K values.
5 Mass flux in a fractionator, lb moles/sec.
5s Mass flux in an absorber, lb moles/sec.
5q Heat flux in an absorber, Btu/sec.
A An increment or difference.
I ~Kinematic viscosity, centistokes.
~g Residence time, sec.
Am Total holdup on plate m, lb moles
Absolute viscosity, lb/ft sec.
p Density, lb/ft3 and lb moles/ft3
~ "The summation of."
0 NNumber of phases in a system,
cp Froth density.
Temperature function in the K value correlation.
Subscripts and Superscripts
a Available for specification.
b Normal boiling point.
bh Normal boiling point of the heaviest component of a mixture.
bR Normal boiling point of the lightest component of a mixture.

-241c Critical property.
c Conditions of a system normally fixed.
e An element.
e Enriching section of a fractionator.
2E A complex element or system.
F:Final variable or function.
f Feed or feed plate.
G Gas or vapor phase.
he Effective heavy component.
I Initial variable or function.
~I Interface condition.
i The general component of a mixture.
i Independent variable.
j A component.
k An iteration or time interval.
L Liquid phase.
le -Effective light component.
m The general plate in a column.
n The top plate or total number of plates..o Ideal gas state.
p Constant pressure.
pc Pressure correction.
r Reduced property.
s Stripping section of a fractionator.
sc Subcooled liquid.

-242t Constant temperature.
V Vapor phase.
v Vaporization
v Total number of variables.
x NqrmaLly fixed variables.
(prime) A revised quantity.
* Equilibrium value.
(superscript bar) An average value; a value to be used as the
argument of a function.
(subscript bar) A molal quantity.
-, A vector.
I l Absolute magnitude or value.

UNIVERSITY OF MICHIGAN
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