
Citation 
 Permanent Link:
 http://ufdc.ufl.edu/AA00000383/00086
Material Information
 Title:
 Chemical engineering education
 Alternate Title:
 CEE
 Abbreviated Title:
 Chem. eng. educ.
 Creator:
 American Society for Engineering Education  Chemical Engineering Division
 Place of Publication:
 Storrs, Conn
 Publisher:
 Chemical Engineering Division, American Society for Engineering Education
 Publication Date:
 Spring 1985
 Frequency:
 Quarterly[1962]
Annual[ FORMER 19601961] quarterly regular
 Language:
 English
 Physical Description:
 v. : ill. ; 2228 cm.
Subjects
 Subjects / Keywords:
 Chemical engineering  Study and teaching  Periodicals ( lcsh )
 Genre:
 periodical ( marcgt )
serial ( sobekcm )
Notes
 Citation/Reference:
 Chemical abstracts
 Additional Physical Form:
 Also issued online.
 Dates or Sequential Designation:
 1960June 1964 ; v. 1, no. 1 (Oct. 1965)
 Numbering Peculiarities:
 Publication suspended briefly: issue designated v. 1, no. 4 (June 1966) published Nov. 1967.
 General Note:
 Title from cover.
 General Note:
 Place of publication varies: Rochester, N.Y., 19651967; Gainesville, Fla., 1968
Record Information
 Source Institution:
 University of Florida
 Rights Management:
 All applicable rights reserved by the source institution and holding location.
 Resource Identifier:
 01151209 ( OCLC )
70013732 ( LCCN ) 00092479 ( ISSN )
 Classification:
 TP165 .C18 ( lcc )
660/.2/071 ( ddc )

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3M FOUNDATION
CHEMICAL ENGINEERING EDUCATION
wito a do alio o/ jaunds.
EDITORIAL AND BUSINESS ADDRESS
Department of Chemical Engineering
University of Florida
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LIBRARY REPRESENTATIVE
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Chemical
VOLUME XIX
Engineering
NUMBER 2
Education
SPRING 1985
4UAa'Sd Iectwte.
72 Semiconductor Chemical Reactor Engineer
ing and Photovoltaic Unit Operations,
T. W. F. Russell
The Educator
54 Joe Hightower of Rice University,
Joyce Taber
Department of Chemical Engineering
58 Cornell University, Julian C. Smith and
Paul H. Steen
Classroom
68 The Nature of Adjoint Variables and Their
Role in Optimal Problems, O. K. Crosser
78 The Use of Computer Graphics to Teach
Thermodynamic Phase Diagrams,
C. D. Naik, Paulette Clancy, and
Keith Gubbins
Laboratory
84 An Improved Design of a Simple Tubular
Reactor Experiment, A. A. Asfour
Lecture
62 Classical Solution Thermodynamics: A
Retrospective View, H. C. Van Ness and
M. M. Abbott
88 The B. C. (Before Computers) and A. D. of
EquilibriumStage Operations,
J. D. Seader
67 Books Received
71 Positions Available
82,83 Book Reviews
CHEMICAL ENGINEERING EDUCATION is published quarterly by Chemical
Engineering Division, American Society for Engineering Education. The publication
is edited at the Chemical Engineering Department, University of Florida. Secondclass
postage is paid at Gainesville, Florida, and at DeLeon Springs, Florida. Correspondence
regarding editorial matter, circulation and changes of address should be addressed
to the Editor at Gainesville, Florida 32611. Advertising rates and information are
available from the advertising representatives. Plates and other advertising material
may be sent directly to the printer: E. O. Painter Printing Co., P. O. Box 877,
DeLeon Springs, Florida 32028. Subscription rate U.S., Canada, and Mexico is $20 per
year, $15 per year mailed to members of AIChE and of the ChE Division of ASEE.
Bulk subscription rates to ChE faculty on request. Write for prices on individual
back copies. Copyright 1985 Chemical Engineering Division of American Society
for Engineering Education. The statements and opinions expressed in this periodical
are those of the writers and not necessarily those of the ChE Division of the ASEE
which body assumes no responsibility for them. Defective copies replaced if notified
within 120 days.
The International Organization for Standardization has assigned the code US ISSN
00092479 for the identification of this periodical.
SPRING 1985
W educator
of Rice University
JOYCE TABER
Rice University
Houston, TX 77251
t 'VE BEEN DELIGHTED to be where I am," says
Dr. Joe Hightower in regard to his decision
17 years ago to become a chemical engineer and
an educator as well.
Joe Hightower, of the chemical engineering
department at Rice University in Houston, says
he started out like many other members of engi
neering faculties: "I started as a child by taking
things apartbicycles, motorcycles, clocks, every
thing!" As early as the fourth grade he proceed
ed to take his clarinet apart and to rebuild it
shortly after he began taking music lessons. Then
throughout high school, in addition to becoming
an accomplished musician as a member of the all
state band, he made a veritable career out of re
pairing the instruments of the other band mem
bers.
While he was in high school Joe also decided
to study chemistry. Later he obtained his masters
and PhD in chemistry from Johns Hopkins but
couldn't decide if he wanted to do industrial re
search or academic work. It was during a three
year stint at the Mellon Institute that he decided
to teach. "I found that I enjoyed the interaction
with the students, the stimulation of the faculty,
and the flexibility of the job."
He found he had to make another decision,
howeverwhether to go into a department of
chemistry or chemical engineering. "All my edu
cational background was in chemistry, but I had
a philosophical desire to work on things that have
very practical uses," he says.
However, chemistry departments
were moving in the direction of quantum
mechanics and other more esoteric areas while
chemical engineering was moving from unit operations
into engineering science. Thus, chemical
engineering embraced catalysis, and Joe
Hightower embraced engineering.
It happened at that time that heterogeneous
catalysis, the research area in which Joe was
interested, was a field that had been explored
primarily in chemistry departments. However,
chemistry departments were moving in the di
rection of quantum mechanics and other more
esoteric areas while chemical engineering was
moving from unit operations into engineering
science. Thus, chemical engineering embraced
catalysis, and Joe Hightower embraced engineer
ing.
Since then, Joe's research philosophy has been
directed toward providing new insights into how
existing catalysts work as opposed to discovering
new catalysts. "We try to ask the question 'Why?' "
he says. Using his chemical training, Joe has
worked at gaining information about the chemical
nature and concentration of active sites, the in
fluence of solid state parameters in determining
activity and selectivity, and the mechanisms of re
actions that occur over solids that are of interest
to the petroleum and petrochemical industries. He
and his students have extensively used isotopic
Copyright ChE Division, ASEE, 1985
CHEMICAL ENGINEERING EDUCATION
tracers (both stable and radioactive) to study the
kinetics, reaction networks, rate limiting steps,
and incorporation of surface species into product
molecules. (Some of his work has been sum
marized in an earlier issue of this journal, Vol.
XVI, No. 4, p. 148, Fall 1982). A few of the
catalyst systems that his group has studied in
clude cracking catalysts, auto emission control
catalysts, partial oxidation catalysts, and zeolites.
But Joe's research does not stop at the gradu
ate level. Several years ago he incorporated some
research techniques into a sophisticated under
graduate experiment. While taking his kinetics
and reactor design course, junior level students
now investigate all the kinetic parameters for cu
mene dealkylation over a silicaalumina cracking
catalyst, explore the reaction mechanism with
deuterium tracers and a mass spectrometer, and
determine the surface area of the material. From
their results the students are able to calculate the
true surface reaction rate constant, the concentra
tion of active sites, the turnover frequency, and
the role of intraparticle diffusion on the kinetics
(Chem. Eng. Educ., p.118, Summer, 1969). This
experiment allows the students to determine ex
perimentally many of the parameters that are
useful in scaling up such reactions from labora
tory to plant size.
Joe's research has led him into other situations
which he has especially enjoyed. In the early 70's,
for example, he was chairman of several National
Academy of Science panels which were assigned
the task of assessing the feasibility of using
catalytic converters to decrease pollutants from
automobiles. "No one had ever applied catalysts
in this way, and we were assigned the task of de
termining if these devices would in fact work,"
he says. "It was fascinating. There was a lot of
secrecy. No company would tell us directly what
they were doing, but they would tell us what they
thought the other companies were doing, and we
had to try to put together a clear picture. Once I
was asked to testify before the House of Repre
sentatives Committee on Science and Technology
which was chaired by Representative George
Brown of California. The congressman from
Detroit would say things like, 'I want you to know
that people from my district are being put out of
jobs because of government regulation and
control.' Brown would respond, 'People in my
district in California are dying because we don't
have enough controls, and pollutants are killing
people.' How can one give an objective testimony
in an atmosphere like that!"
Involvement in professional societies has been
another rewarding part of Joe's career. He is cur
rently chairman of the 24member Petroleum Re
search Fund Advisory Board, a group that is re
sponsible for a corpus of $150 million. This year
the foundation will donate $11 million for uni
versity research in petroleumrelated areas. In
1971 he received the National Award in Petroleum
Chemistry from the American Chemical Society.
Joe finds time to enjoy simple, relaxing activities, such
as blowing glass in his lab.
As a councilor for the American Chemical So
ciety, he represents the southeastern Texas local
section. He has been on the national research
committee and is on the national awards commit
tee of the American Institute of Chemical Engi
neers. He has also served as chairman of the
petroleum chemistry division of the American
Chemical Society.
Dr. Hightower has over 50 publications to his
credit and is also very much involved in presenting
short courses for industry. "Catalysis," he says,
"is a field that is not taught in many universities as
an area of specialization. Yet, 80 or 90% of all
commercially important chemical reactions are
catalytic reactions. People are trained as organic
chemists, physical chemists, or chemical engi
neers and then learn about catalysis on the job.
This creates a great demand for the types of short
courses that we instituted at Rice years ago and
that are being continued in cooperation with other
schools such as the University of Houston."
It would appear that Joe's day would have to
last more than 24 hours in order to accomplish
his many activities. Yet, there is still another part
of his life that is as important to him as his pro
SPRING 1985
. there is still another part of his life that is as important to him as his professional work.
In 1968 he was a leader in establishing the Human Resources Development Foundation (which) provides free
temporary housing for needy families who come for treatment to the Texas Medical Center hospitals.
fessional work. In 1968 he was a leader in es
tablishing the Human Resources Development
Foundation. The foundation provides free tempor
ary housing for needy families who come for
treatment to the Texas Medical Center hospitals.
The foundation started as a project at Dr. High
tower's church and has expanded to serve over 700
families from over 38 states and 26 foreign
countries in the last 16 years. Joe is president of
the foundation whose facilities have grown from
an old army barracks into 15 beautiful apart
ments. He heads a group of approximately 40 dedi
cated volunteers who minister daily to the needs
of families who are under enormous stress.
"All a person needs to move in are pajamas
and a toothbrush!" Joe laughs. But his statement
Human Resources Development Foundation apartments.
is very accurate. The apartments are furnished
down to the pots and pans. A local church even
provides meat once a week for the residents. Social
workers, ministers, and even former residents
refer potential patients. Selection is made on a
firstcome, firstserved basis without regard to
race, creed, sex, religion, age, or level of disability.
Need is the sole criterion used to determine eligi
bility. Residents are allowed to stay for up to
three months.
"The project is valued at over a halfmillion
dollars, and most of it has been given because of
something Joe has done," states Marge Norman,
VicePresident of the foundation. "Joe never hesi
tates to go speak to a group if there is some chance
that they might have an interest in the founda
tion. His work takes him to the far reaches of
the U.S. and overseas; on every airplane trip he
takes he makes sure his seat partner is very well
acquainted with his pet project, and it often leads
to very good things for this foundation."
Senator Orrin Hatch was one of the latest
people to hear about the foundation because of
one of Joe's "airplane contacts." Joe sat next to
a member of Senator Hatch's staff on one flight,
and later he received a letter from the senator,
who had been informed by the staff member about
the foundation. The letter commended Dr. High
tower for his charitable work.
Continues Mrs. Norman, "Even though we
have a foundation board which is functioning very
well, without Joe I don't know if the Board would
have been as effective or if this place would have
become what it is today. He loves it so much, and
he works so hard. A couple of times each year we
have a work day when all our volunteers come to
clean, repair, and paint. All kinds of people are
represented in the workers. Joe is always the first
here and the last to leave, working at anything
that needs to be done. Even before he comes over,
he gets up before dawn and bakes bread so that at
10 a.m. we can have hot bread and butter and
coffee. We've been acquiring land to build more
apartments next door, so you can be sure he's on
the campaign trail again! We have parties for
our residents, and again, Joe is always here with
bread he has baked. He even brings his mandolin
and plays and sings. There's not much Joe can't
do!"
On any given day the facility may house people
from Florida, or from various towns in Texas,
Columbia, or Indiana. There are no class dis
tinctions. Last winter a brain surgeon from Main
land China was allowed to leave his country with
only $200 when he came to care for his quadri
plegic daughter in the medical center. As a resi
dent of the foundation's apartments, he scrubbed
floors and took a lot of goodnatured ribbing when
he painted an outside door with interior paint.
Something all residents share, however, is grati
tude to Hightower, who personally greets each
newcomer with a loaf of bread, and gratitude to
the foundation he helped establish.
CHEMICAL ENGINEERING EDUCATION
Wrote one resident, "What a tremendous help
this facility has been to us. Each of us is faced
with an extremely serious medical problem, and
the expenses are staggering. To be sure, the
financial savings are important, but even more,
we have a place to call 'home' and people with
whom we can talk as friends. I'm one of the lucky
ones. Two weeks ago I had a kidney transplant,
and now I am hoping and praying that my body
will not reject it. I have been so impressed by the
consistent care and visits that I've had from the
jogger (Dr. Hightower) and his friends even
during my recovery from surgery. I hope in some
way I can repay the favors done for me. Right
now, though, I'm going to sit back and enjoy an
other slice of the hot bread Joe Hightower has
brought me before it has time to lose its flavor!"
In his professional life also, colleagues have
only good things to say about Joe Hightower. Dr.
T. W. Leland who was department chairman when
Joe came to Rice says, "I was impressed with him
right from the start, and I had a great interest in
getting him to come to Rice. I think it's been a
firstrate choice. He's done a remarkably good job
over the years. He's an excellent teacher and has
had an active research career. He is well thought of
by his students and has perfected a graduate
course in kinetics and catalysis to a high degree.
He has been effective in giving short courses and
he is outstanding in his volunteer public service.
Personally, he is outgoing, friendly, and extreme
ly wellorganized. He has excellent rapport with
people in all walks of life, from the top of the
technical ladder in terms of ability to students not
doing too well in their courses. He's a remarkable
individual who has been a great addition to our
department."
Joe is modest in describing his daily activi
ties. "I just enjoy it all," he says. He gets up at
4:45 every morning to jog and share breakfast and
a Bible reading with his wife Ann, a chemical
engineer who works for the Exxon Chemical
Company. By 6:30 a.m. he is at work, doing all
the things he loves to do. "From the very be
ginning I couldn't make up my mind about what
I wanted to do. I wanted something in both in
dustry and the academic world. Now I have both,
and I'm grateful for that. I probably border on
getting involved in more things than I should ..
but they're all so interesting! I guess I just like
being where I can interact with people and see
them grow, whether it's at the university or
whether it's with people who are hurting." E
SPRING 1985
TRANSPORT PROCESSES
MOMENTUM, HEAT AND MASS
Christie J. Geankoplis
University of Minnesota
1983 casebound 350pages
This text takes a unified approach to basic transport
processes: Geankoplis points out the similarities of
basic equations and calculation methods, and the
differences which occur in the actual physical
processes.
Each chapter of this classtested text is divided into
elementary sections, followed by more sophisticated
Selected Topics allowing you to expand or focus
your course according to the needs of your students
and the time limits of your course.
SI Units are used throughout, with important equa
tions and tables in dual units. Over 150 example
problems and more than 340 homework problems
emphasize applications as well as theory.
This text offers complete coverage of more essentials
than any other book you'll find. Look for these new
topics: diffusion in solids and porous solids, bioengi
neering transport, nonNewtonian fluids, numerical
methods for steady and unsteadystate diffusion and
conduction, design and scaleup of agitation systems,
mass transport, and an introduction to engineering
principles.
TRANSPORT
PROCESSES
AND UNIT
OPERATIONS
SECOND EDITION
Christie J. Geankoplis
University of Minnesota
1983 casebound 650 pages
This fullyrevised Second Edition includes TRANSPORT
PROCESSES: MOMENTUM, HEAT AND MASS as
Part One of the text, plus the unit operations (Part
Two) so essential to chemical engineers.
TRANSPORT PROCESSES AND UNIT OPERATIONS,
Second Edition offers an optimal balance of theory and
application. Geankoplis emphasizes the diversity of
practical applications in chemical, ceramic, mechanical,
civil, food process, and bioengineering. Over 220
example problems and 500 homework problems
illustrate both theory and applications.
The Second Edition features new sections on multi
component distillation, unit operations of membrane
processes, nonNewtonian fluids, diffusion in solids,
porous solids and biosystems, freezing, freeze drying,
and sterilization of biomaterials.
For examination copies or more information
on these two titles, write to Ray Short, Engineering
Editor, Allyn and Bacon, Inc., 7 Wells Avenue, Newton,
MA 02159.
Allyn and Bacon, Inc.
College Division
Olin Hall from the west.
no department 
CHEAT CORNELL UNIVERSITY
JULIAN C. SMITH AND PAUL H. STEEN
Cornell University
Ithaca, NY 14853
F OUNDED IN 1868 AND with a long tradition in
engineering, Cornell is almost unique in being
both private and statesupported; about half the
divisions, including engineering, are privately en
dowed while the other half are funded by the
State of New York. An awkward arrangement,
it would seem, but it works surprisingly well.
Cornell, a medium sized university with a total
enrollment of some 18,000 students, is set on a
hill overlooking the city of Ithaca, and the waters
of Cayuga Lake, the largest of the Finger Lakes.
Ithaca is small but strongly cosmopolitan. The
setting is semirural; the scenery is beautiful; the
air is clean. Almost the only drawbacks are a
modicum of cold gray weather on occasion, and
some possible difficulties in travelling in and out.
Ithaca has been called "the most centrally iso
lated city in the Northeast," but as a graduate
student from Greece recently remarked, "If it
wasn't for the weather, Ithaca would be Paradise!"
The School of Chemical Engineering has 18
faculty members, about 100 undergraduate
students (3rd and 4th years only), and over 65
graduate students. During the past twelve years
research activity and expenditures have greatly
increased, and strong research programs have
been established in fluid mechanics, polymers, sur
face science and catalysis, thermodynamics, and
biochemical engineering. The number and quality
of MS and PhD candidates (especially PhD's)
have risen rapidly. The growth in research, how
ever, has not reduced the traditional concern for
undergraduate and professional graduate teach
ing. All faculty members are expected to teach
undergraduate courses, and many participate in
advanced design projects in the professional
Master's program. The school occupies its own
building (shared with a few other tenants) with
CHEMICAL ENGINEERING EDUCATION
Copyright ChE Division, ASEE, 1985
a total area of some 90,000 square feet, 54,000 of
which is exclusively chemical engineering.
A BRIEF HISTORY
At Cornell, as at many institutions, chemical
engineering began in the Chemistry Department,
but its development was somewhat unusual. Very
early (before 1900) courses were offered in in
dustrial chemistry which had a considerable
practical flavor; as taught by Fred H. "Dusty"
Rhodes in the 1920's they dealt with the principles
and practice of chemical engineering. By 1930
Dusty had established both undergraduate and
graduate programs in chemical engineering, but
because of rivalries between Chemistry and the
Engineering College the undergraduate program
had to be a 5yearlong hybrid: four years in Arts
and Sciences (leading to the degree Bachelor of
Chemistry) followed by one year in engineering
(for the degree Chemical Engineer).
In 1938 the department with its three faculty
members became part of the Engineering College
and the 5year program led to the degree Bachelor
of Chemical Engineering. In 1942 chemical engi
neering moved to Olin Hall, the first building
on what was to become engineering's new
quadrangle. It was at a considerable distance
from chemistry and the old ties quickly weakened.
After World War II all the undergraduate engi
neering programs at Cornell were lengthened to
five years. This lasted until 1965, when the present
4year BS programs, including that in chemical
engineering, were established.
Dusty Rhodes was director of the school until
he retired in 1957 and Charles C. "Chuck" Wind
ing took over. Ken Bischoff, now at the University
of Delaware, was director from 1970 to 1975;
Julian Smith from 1975 to 1983; and Keith
Gubbins from 1983 to date.
For many years chemical engineering at
Cornell was known for its strong undergraduate
program. Rhodes felt that good teaching was the
most important thing required of a faculty mem
ber and while ability to do research should be
considered in reviews for promotion or tenure, it
should not be a major factor. This is not to say
that there was no research or graduate work. Be
tween 1932 and 1970 the school awarded 140 MS
and 104 PhD degrees, and many of the recipients
have had distinguished careers in industry and
academia, including John Prausnitz (Berkeley),
Ed Lightfoot (Wisconsin), and a string of past
or present heads of chemical engineering depart
ments: Bob Coughlin (UCONN) ; Howard Greene
(Akron) ; Deran Hanesian (NJIT) ; Will Kranich
(Worcester Poly) ; Larry McIntire (Rice) ; Steve
Rosen (Toledo); Julian Smith (Cornell); Tom
Weber (SUNY Buffalo) ; and Jacques Zakin (Ohio
State). Bob Finn's pioneering work in biochemi
cal engineering was begun in the 1950's.
Dusty's policies set a pattern for the school
which persisted until the early 1970's. By then it
was clear that the research effort had to be greatly
The Fred H. Rhodes Student Lounge, redecorated
through a gift from Joseph Coors, '40.
expanded. Beginning in 1970, new faculty mem
bers were added who developed, or brought with
them, strong research programs in several areas.
This attracted additional researchminded faculty
and increasingly stronger graduate students.
During an 8year period research expenditures in
creased by a factor of six. The number of gradu
ate students has risen to 67; more significantly,
a majority of them are now PhD candidates. And
collaboration with Chemistry and other depart
ments of the university is once again close and
extensive.
RESEARCH GROWTH
From 1976 through 1982, the annual research
expenditures, in dollars per faculty member,
climbed at a rate that was second highest and
reached a level that was fourth highest among all
chemical engineering departments in the country.*
Total sponsored research costs for 198384 were
over 1.5 million dollars, for an average of $94,000
*Journal of Engineering Education, March issues, 1976
1983.
SPRING 1985
per fulltime faculty member. This is especially
striking since only about a third of the faculty
was responsible for 75% of the total expenditures.
As the current younger faculty develop their pro
grams and additional researchminded faculty re
place retiring senior professors, the overall re
search program should continue its strong ex
pansion.
Biochemical Engineering
Biochemical engineering research has grown
from Bob Finn's early studies of microbes and
... many of the PhD recipients have
had distinguished careers in industry and
academia, including John Prausnitz (Berkeley), Ed
Lightfoot (Wisconsin), and a string of
past or present heads of chemical
engineering departments.
microbial populations. The goal was, and is, to
develop new and more efficient biochemical con
versions. One project seeks to find economical
ways of producing ethanol from pentose sugars,
a second to develop better treatment methods for
wastes containing pentachlorophenol (PCP), and
a third to exploit an unusual bacterium which can
rapidly ferment arabinose.
Mike Shuler shares several specific interests
with Bob Finn including the treatment of waste
water by specialized microorganisms. Mike's di
verse interests are tied together by a view of the
living cell as a "catalyst" waiting to be used in
chemical reactors. His research embraces studies
of plantcell tissue culture, reactors with solid
substrates (e.g. mold growth on solid surfaces),
photobioreactors, biofilm formation, and the con
tinuous protein production from bacteria with re
combinant DNA. Particularly noteworthy have
been his group's experimental demonstration of
the feasibility of hollowfiber membrane units for
entrapment of microbial populations (necessary
groundwork for the development of hollowfiber
reactors) and the construction of a mathematical
model of the organism Escherichic coli.
Doug Clark, who joined the faculty in 1984,
brings the point of view that enzymes rather than
the whole cell can be the building blocks for bio
chemical reactors. He is studying how the im
mobilization, or attachment to a foreign matrix
support, affects the structure and function of an
enzyme. A related interest is the transport of bio
logical macromolecules through porous media; this
transport is an essential step in enzyme immobili
zation, gel permeation and affinity chromatogra
phy, and ultrafiltration. In collaboration with Bill
Street, Doug has initiated a study of methane
producing bacteria which live at extreme tempera
tures and pressures in deepsea hydrothermal
vents.
Polymers and Materials Science
For a million circuit elements to fit on a tiny
silicon chip linewidths must be on the order of a
tenth of a micron. In one technique for achieving
such precisionelectronbeam lithographythe
silicon surface is covered with a polymer film
polymethyll methacrylate, for example), then ir
radiated by an electron beam creating a pattern
of soluble polymer. The soluble polymer is washed
away leaving a precision mask and the chip is
ready for the final step, silicon modification.
Ferdinand (Rod) Rodriguez is directing an inter
departmental program on polymers for advanced
lithography, to improve the performance of the
polymer "resist" used in the masking process. This
is a good example of Rod's research on polymeric
materials which has the broad goal of understand
ing the processes of polymerization and gelation
(crosslinking) and degradation (chain scission)
in order to produce better materials.
Claude Cohen uses macromolecular science to
interpret the physical properties of polymer
systems and to understand the structures that
develop during industrial processing. On the
fundamental level, predictions from models of
macromolecules are used to understand rheo
logical and lightscattering behavior, with ex
periments to complement the theoretical work and
to test the adequacy of the models. On the applied
level, the orientation of glass fibers in composite
thermoplastics during the molding process is
being investigated. This work is in conjunction
with an interdepartmental program on injection
molding.
Surface Chemistry, Catalysis and Reactor Engineering
The surface chemistry and physics of hetero
geneous systems which have direct technological
application is the central concern of Bob Merrill's
studies. Examples are CO oxidation on noble
metals (automobile exhaust converters), the de
composition of hydrazine (rocket monopropel
lant), the oxidation of aluminum (catalyst sup
CHEMICAL ENGINEERING EDUCATION
port technology, corrosion protection, and elec
tronic insulators in microcircuitry) and hydrode
sulfurization catalysis (sulfur removal from pe
troleum). On the one hand, Bob's group answers
practical questions; on the other they are develop
ing and sharpening several types of analytical
tools. These include the use of lasers in surface
chemistry, the use of synchrotron radiation
(EXAFS) to study the dynamics of gassolid re
actions, and the use of spectroscopy in real
catalyst systems (highsurfacearea configurations
and high pressures).
Peter Harriott studies the influence of mass
transfer, heat transfer and mixing on the per
formance of chemical reactors as well as the
kinetics of reactions in heterogeneous systems.
One project concerns the regeneration of catalysts
used in the pyrolysis and gasification of coal. An
other examines the heat and mass transfer and
the overall kinetics in limeslurry droplets used
in the "dry scrubbing" of SO, from flue gas; the
goal is to pin down the ratelimiting step and im
prove the design of commercial units.
Joe Cocchetto's recent work on catalytic re
action kinetics has concentrated on the fuel cell.
By controlling the structure of a porous electrode,
a better understanding of the interplay between
transport and reaction has been gained and tech
niques for improving efficiency have emerged. Joe
returned to industry in early 1985.
Bob Von Berg is interested in the use of gamma
radiation in various chemical processes: ammonia
synthesis and the reaction of hydrocarbons and
liquid nitrogen. Bob has also collaborated with
Herb Wiegandt on a longterm project involving
the desalination of water by freezing, as described
later.
Fluid Dynamics and Stability: Rheology
Bill Olbricht concentrates on problems in fluid
mechanics and rheology with applications in en
hanced oil recovery, biomedical fluid mechanics,
and the production of semiconductor materials.
He is studying the lowReynoldsnumber motion
and coalescence of immiscible drops in tubes of
various geometries (characteristic of porous
media) for critical evaluation of methods for en
hanced oil recovery. In the biomedical area, in
conjunction with the University of Rochester
Medical School, he is modelling the motion of red
blood cells in microcapillaries to predict the dis
tribution of these cells within tissue. A third area
Cornell's Chemical Engineering Faculty, 1984. Back
row: Shuler, Finn, Scheele, Steen, Smith, Winding.
Middle row: Zollweg (Research Associate), Cocchetto,
Harriott, Jolls (Visiting from Iowa State), Von Berg,
Rodriguez. Front row: Olbricht, Merrill, Clark, Wie
gandt, Thorpe, Clancy, Gubbins, Streett.
of research examines the momentum, heat, and
mass transfer involved in silicon film growth by
chemical vapor deposition with the aim of pre
dicting rates of film growth in lowpressure
deposition reactors.
Paul Steen, who joined the faculty in 1982,
studies fluid motions and their stabilities. Buoy
ancydriven convection patterns, generated in
fluidsaturated porous media, are examined as
prototypes of fluid motions susceptible to transi
tions in which strong nonlinear effects are domin
ant. This work involves the development of tools
in applied mathematics. In another area, motions
induced at fluid/fluid interfaces due to tempera
ture gradients (thermocapillary effects) are being
investigated by experiment, with relevance to the
floatzone crystalgrowth process and the break
up of thick films.
George Scheele's study of liquidliquid immis
cible systems focuses on the coalescence of drop
lets and the breakup of jetsboth at relatively
high Reynolds numbers. He also has interests in
the computer simulation of chemical processes,
particularly in computer graphics.
Molecular Thermodynamics and Computer Simulation
Keith Gubbins and Bill Streett have coordinat
ed their efforts towards understanding, predicting,
Continued on page 103.
SPRING 1985
P lecture
CLASSICAL SOLUTION THERMODYNAMICS
A Retrospective View
H. C. VAN NESS AND M. M. ABBOTT
Rensselaer Polytechnic Institute
Troy, NY 12181
T HE PRIMARY VARIABLES of classical thermo
dynamics for fluid systems are temperature T,
pressure P, and the molar properties volume V,
internal energy U, and entropy S. Temperature
is a primitive, having no definition in terms of
anything simpler. Pressure and molar volume are
defined directly by three other primitives: force,
mass, and length. These primitivestemperature,
force, mass, and lengthare subject to direct
sensory perception, and we have little difficulty
accepting them as meaningful. Internal energy and
entropy, however, are primitives not associated
with direct detection by the senses. Nor are they
directly measurable; we have no energy meters,
no entropy meters. Energy and entropy are
mental constructs which have meaning only as
mathematical functions. Accepting this, we then
need to know what they are functions of.
We find by experiment that the molar volume
of a homogeneous phase is a function of its
temperature, pressure, and composition. Generaliz
ing, we postulate that the molar internal energy
and entropy of a homogeneous phase are likewise
functions of temperature, pressure, and com
position. When this is true, the first and second
laws lead to a fundamental property relation
among the primary thermodynamic variables
d(nU) = Td(nS) Pd(nV) + Zidni (1)
The ni are mole numbers of the species present,
Nor are they directly measurable;
we have no energy meters, no entropy meters.
Energy and entropy are mental constructs which have
meaning only as mathematical functions.
Copyright ChE Division, ASEE, 1985
H. C. Van Ness is Distinguished Research Professor of Chemical
Engineering at Rensselaer Polytechnic Institute, where he has been
a faculty member since 1956. He is coauthor with J. M. Smith of
Introduction to Chemical Engineering Thermodynamics and has co
authored a number of research papers on thermodynamics with
M. M. Abbott, in addition to two books, Schaum's Outline of Theory
and Problems of Thermodynamics and (with M. W. Zemansky as a
third coauthor) Basic Engineering Thermodynamics. (L)
Michael M. Abbott is Associate Professor of Chemical Engineering
at Rensselaer Polytechnic Institute, where he has been a faculty
member since 1969. Prior to that he was employed by Esso Research
and Engineering. His teaching interests are mainly in the thermal
sciences and in chemical process design. He is coauthor with H. C.
Van Ness of the books Schaum's Outline of Theory and Problems of
Thermodynamics and Basic Engineering Thermodynamics (with M. W.
Zemansky). (R)
n = Ini is the total number of moles, and the
[i are chemical potentials. Written for n = 1, Eq.
(1) becomes
dU = TdS PdV + J.dx.
showing that
U = U(S,V,x)
Thus, in general, the natural independent (canoni
cal) variables for U are entropy, volume, and
composition.
New thermodynamic properties can be defined
that are functions of alternative sets of inde
pendent variables. In particular, the enthalpy H
and the Gibbs function G are defined as
CHEMICAL ENGINEERING EDUCATION
H E U + PV (2)
and
G H TS (3)
Then
nG = nU + P(nV) T(nS)
and
d(nG) = d(nU) + Pd(nV) + (nV)dP Td(nS) (nS)dT
Substitution for d (nU) by Eq. (1) gives
d(nG) = (nS)dT + (nV)dP + ZJ.dn. (4)
This equation is equivalent to Eq. (1), and repre
sents an alternative fundamental property rela
tion. Written for one mole of material, it becomes
dG = SdT + VdP + lZidxi (5)
whence
G = G(T,P,x)
Because temperature, pressure, and composition
are subject to direct measurement and control,
the Gibbs function is a defined thermodynamic
property of great potential utility.
An equation such as Eq. (4) is too general
for direct practical application. Its value is in
storing much information. Thus, we write by in
spection
s = pl (6)
V (7)
F T,x
and
i = n (8)
ST,P,n.
where the subscript nj indicates that all mole
numbers are held constant except ni. Application
of Eqs. (6) through (8) presumes knowledge of
G as a function of T, P, and x; given this, then
Eqs. (6) and (7) yield S and V as functions of
T, P, and x. Other properties come from defining
equations; for example, by Eq. (3)
H= G + TS
Thus, if we know how G is related to its canonical
variables, we can by simple mathematical opera
tions evaluate all the other thermodynamic
properties; given G = G (T,P,x), we can also find
S, V, ,it H, Cp, etc. as functions of temperature,
pressure, and composition.
All this is the legacy of J. W. Gibbs and in
principle nothing more is needed. An expression
giving G = G (T,P,x) is an example of a canonical
equation of state. Such an equation serves as a
generating function for the other thermodynamic
properties, and implicitly represents complete
property information.
For realfluid mixtures, canonical equations of
state are unknown. The problem is that such an
equation must be based on experimental data. Un
fortunately, there are no G meters and no con
venient experimental measurements that lead
easily to values of G. Without a canonical equation
of state, we can make no direct practical use of a
fundamental property relation. The slow evolu
tion of solution thermodynamics since Gibbs'
time has led to new formulations that relate much
more directly to experiment. Our purpose here is
The slow evolution of solution thermodynamics... has
led to new formulations that relate much more
directly to experiment. Our purpose is to
rationalize the structure of modern
solution thermodynamics.
to rationalize the structure of modern solution
thermodynamics.
In the early years of this century, G. N. Lewis
introduced several concepts basic to all subse
quent developments: the partial property, the
fugacity, and the ideal solution.
A partial property is defined by the equation
=. (nM) (10)
i n. (10)
1 T,P,nj
where M is the molar value of any extensive
property. The simplest interpretation of Eq. (10)
is that it apportions a mixture property among
the constituent chemical species. Thus, MR has the
characteristics of the property of species i in the
mixture. Indeed, a mathematical consequence of
Eq. (10) is the relation
M = iX
which shows that the partial properties combine
in the simplest rational way to yield the mixture
property. We see by comparison of Eq. (8) with
Eq. (10) that
Ui = G
SPRING 1985
Thus, the chemical potential is identified with
partial Gibbs function.
The fugacity is an auxiliary thermodyna
property related to the Gibbs function. Thus,
a mixture, the fugacity f is defined by the e
tions
dG = RT d in f
(const T,x)
the
i,:
dGa = RT d in (xiP)
(const T)
IIIlU The actual properties of a fluid may be com
for pared with the properties the fluid would have as
lua an ideal gas at the same temperature, pressure,
and composition. The comparison by subtraction
(13) gives rise to residual properties. Thus, by defini
tion
f
lim = 1 (14)
P.0
For the special case of pure species i, these become
and
dG = RT d in fi
(const T)
lim 1
P0
For species i as a constituent of a mixture, the
fugacity ?. is defined by the equations
1.
MR M M'
MR E M. Mi (26)
Applying this concept to the Gibbs function,
we subtract Eq. (19) from Eq. (13)
d(G 6') = RT d in
P
dG = RT d in p
(const T,x)
(const T,x)
(const T)
and
i
lim P 1 (18)
P0O i
For an idealgas mixture one replaces V in
Eq. (5) by RT/P; then
dG' = RT d in P
(const T,x)
where the prime (') denotes an idealgas proper
From Gibbs' theorem for such mixtures, we I
G' = xiGi + RT xi in xi
where 0 is the fugacity coefficient, defined as
f
Integration of Eq. (27) gives
Integration of Eq. (27) gives
G = RT in i
(19) The integration constant vanishes, because for
P = 0, GR = 0 by assumption and In 4) = 0 by
rty. Eq. (14). For the special case of pure species i,
have this becomes
(20)
R
G. = RT in n
1i
By Eqs. (6), (7), and (9), we get
For species i as a constituent of a mixture, we
subtract Eq. (24) from Eq. (17)
S' = xiS R Axi in xi
V, = zxiv!
f.
d(Gi GI) = RT d in
i .P
R
dGi = RT d n i
H' = xiH!
L 1
(const T)
(const T)
Each of these is implicit in Eq. (20). Moreover,
Eq. (8) yields
S= G = G + RT in x
whence
dG' = dG' + RT d in xi
x
(const T)
By Eq. (19) written for pure species i, this be
comes
where i. is the fugacity coefficient of species i
in the mixture, defined as
$ 
1 X.P
Integration of Eq. (31) gives
.R = RT n i
3. i
CHEMICAL ENGINEERING EDUCATION
dG. = RT d in f.
1 3.
Unlike a formulation based on a canonical equation of state, the
residualproperty formulation cannot provide complete property information. One needs in addition
the heat capacities necessary for evaluation of properties for the idealgas state.
where again the integration constant vanishes.
An alternative form of Eq. (4) derives from
the mathematical identity
) 1 nG
d nG= d(nG) dT
RJ RT RT 2
Substituting for d (nG) by Eq. (4) and for G by
Eq. (3) gives
d dT + dP dn (34)
R 2I R I T
RT2
For the idealgas state, Eq. (34) becomes
'd = dT + W dP + dn. (35)
RT RT2 RT RT
With ti replaced by Gi in Eq. (34) and p' replaced
by G: in Eq. (35), we subtract these two equa
tions:
SR
d nG dT + nV dP + dni (36)
where the definitions of Eqs. (25) and (26) have
been invoked. This is the fundamental property
relation for residual properties. In view of Eq.
(33), it may also be written
R R VR
d T = 2 dT + C dP + in i. dn (37)
Working relations for the residual properties can
now be written by inspection
H ( ( JRT)
RT aT p,x
 (a Zn 4
I aT )Px
VR f (GRT ,x a ki
RT P JTx P JT,x
in n= ( /RT)
n Oi @ani T,P,nj
Sa(n an ,,
I 3ni T,P,n.
an3
where the second form in each case follows from
Eq. (29).
Equation (39) may be written
vR
d In =, dP
RT
(const T,x)
where by definition
VR = V V' = V RT
P
Values of VR come directly from experimental
PVTx data, and Eq. (41) then allows calculation
of In 0; Eqs. (38) through (40) yield other
properties of interest. This close link to experi
ment is the major reason for a residualproperty
formulation of solution thermodynamics. Given
a PVT equation of state
V = V(T,P,x)
we can evaluate all residual properties. Because
of its direct relation to experiment, a PVT equa
tion of state is far more easily developed than is a
canonical equation of state. Furthermore, the
principle of corresponding states allows the
generalization of PVT data and the development
of generalized correlations for the residual proper
ties, thus greatly extending the usefulness of
available experimental data. Unlike a formulation
based on a canonical equation of state, the residual
property formulation cannot provide complete
property information. One needs in addition the
heat capacities necessary for evaluation of
properties for the idealgas state.
In principle, PVT equations of state apply
equally to gases and to liquids. In practice, how
ever, the accurate representation of liquid proper
ties proves much more difficult. Thus, an alterna
tive formulation of solution thermodynamics has
developed for liquids. The key idea is that of an
ideal solution. By definition
fid x.f (42)
1 l 1
where the superscript id denotes an idealsolution
property. Expressions for all of the properties
of an ideal solution follow from this equation.
Integration of Eq. (17) from the state of pure
i to the state of i in solution at the same T and P
gives
SPRING 1985
I
fo
G Gi = RT n (43)
For an ideal solution, this becomes
id
Gi = G + RT n x (44)
and by Eq. (11)
Gid = xii + RT
Equations (6), (7), and (9) in this case yield
Sid = xiSi Rxi n xi (46)
Vid= xIVI (47)
and
Hid = xiHi (48)
Just as we may compare the actual properties
of a fluid with its idealgasstate properties, so
may we compare the actual properties of a fluid
mixture with its idealsolution properties at the
same temperature, pressure, and composition.
Thus, we have definitions of excess properties
quite analogous to those for residual properties
ME =M Mid
E id
M. =M. M.
1 1 1
Equation (49) applied in turn to the properties
whose idealsolution expressions are given by
Eqs. (45) through (48) becomes
GE = G Zx.Gi RTx.i An x. (51)
SE = S x.iSi + Rjxi in xi (52)
VE = V xiVi (53)
HE = H xiHi (54)
The excess properties are closely related to
property changes of mixing
AM = M xi.M. (55)
These quantities measure the changes that occur
when one mole of mixture is formed from the
pure constituent species by a mixing process at
constant T and P. The definition of Eq. (55)
allows Eqs. (51) through (54) to be written
GE = AG RTyx. in x.
E
SE = AS + Rx.i n x.
VE V
HE = AH
Thus, the excess properties are readily calculated
from property changes of mixing and vice
versa. Interest in property changes of mixing is
focused on AV and AH, because these quantities
can be experimentally determined by direct
measurement.
Unfortunately, measurements of AV = VE and
of AH = HE for liquid mixtures do not allow
calculation of GE. For this, we need vapor/liquid
equilibrium data, which are related to GE as
follows. Subtraction of Eq. (44) from Eq. (43)
gives
X.
E id i
Gi = Gi G = RT nf
1 1 1 x.f.
i3
or
E
= (60)
in Yi RT
where the activity coefficient yi is defined by
f
Yi (61)
1ii
In accord with Eq. (11)
H E
GE GE
RT= x RT
and by Eq. (60) this becomes
GE
R = x.i n yi (62)
Values of yi are calculated from experimental
vapor/liquidequilibrium measurements by the
equation
yiPDi
Yi sat (63)
ii
Here, Ii is a secondary factor of order unity
that can be readily evaluated from volumetric
data for the equilibrium phases or from correla
tions of such data.
The fundamental property relation for the
excess properties follows from Eq. (34). For an
ideal solution, this equation is written
CHEMICAL ENGINEERING EDUCATION
TABLE 1
Summary of key equations
rnc nH nV i
d nj = dT d + T dP + dn
FRTJ RT2 RT RT
InC I II nVT
d = dT + nR dP
RT j RT2 T
+ I in i.dni
1 1
(34)
(37)
d nGrE nHE nVE
d dT + __T dP + L n idni (66)
RT2 RT R21
G Pi
R
GE G i E
RT ^i RT J in" y,
SnGid nlid nVid
d (R n I dT + dP
RT2 RI
id
+ I dni
RT i
With ii replaced by G. in Eq. (34) and ~d replaced
id
by Gi in Eq. (64), we subtract these two equa
EiE
E nHE E 6
d = n dT + + IdP + dn. (65)
RT2 T RT i
tions where the definitions of Eqs. (49) and (50)
have been invoked. In view of Eq. (60), this equa
tion may also be written
n _nHE nVE
d Ii nH dT + n dP + I n y dn (66)
RT2 RT 1 i
Equation (66) is analogous to both Eqs. (34) and
(37); analogous to Eqs. (38) through (40), we
have
HE E
S= T (/RT)1 (67)
RT ^ T JP,x
VE (G/RT) (68)
RT [ P J T,x
n (nGE/RT) T (69)
A i ni T,P,n
;
SPRING 1985
The formulation of solution thermodynamics
through excess properties derives its usefulness
from the fact that HE, VE, and yi can all be found
by experiment. This relative abundance of experi
mental entries provides alternative measurements
that yield property data. However, the excess
property formulation provides even lesscomplete
property information than the residualproperty
formulation, because it tells us nothing about the
properties of the pure chemical species.
In Table 1, we bring together for comparison
the parent fundamental property relation for the
Gibbs function and the two analogous property
relations which follow from it for the residual and
excess Gibbs functions. Included as well are the
equations which relate the three mixture Gibbs
functions to their respective partial properties.
These are particular applications of Eq. (11). [
10 books received I
Fundamentals of Chemistry, Second Edition, James E.
Brady, John R. Holum; John Wiley & Sons, Inc., New
York; $34.95 (1984)
Handbook of Powder Science and Technology, Edited by
M. E. Fayed and L. Otten; Van Nostrand Reinhold, 135
West 50th Street, New York, 10020; 850 pages, $79.50
(1984)
Analytical Pyrolysis: Techniques and Applications, Edited
by Kent J. Voorhees; Butterworths, 80 Montvale Ave.,
Stoneham, MA 02180; 486 pages, $69.95 (1984)
Heat and Mass Transfer in Rotating Machinery, Darryl
E. Metzger, Naim H. Afgan; Hemisphere Publishing Co.,
79 Madison Ave., New York 10016; 713 pages, $74.50
Cheaper, Safer Plants or Wealth and Safety at Work,
Trevor A. Kletz; Institution of Chemical Engineers, 165
171 Railway Terrace, Rugby, England; (1984)
Engineering Information Resources, Margaret T. Schenk
and James K. Webster; Marcel Dekker, Inc., New York
10016; 232 pages, $24.75 (1984)
From Technical Professional to Corporate Manager; A
Guide to Career Transition, David E. Dougherty; John
Wiley & Sons, Somerset, NJ 08873; 279 pages, $19.95
(1984)
Natural Product Chemistry: A Mechanistic and Biosyn
thetic Approach to Secondary Metabolism, Kurt B. G.
Torssell; John Wiley & Sons, Inc., Somerset, NJ 08873;
401 pages, $24.95 (1984)
The Wiley Engineer's Desk Reference, Sanford I. Heisler;
John Wiley & Sons, New York 10158; 567 pages, $34.95;
(1984)
Laboratory Manual of Experiments in Process Control,
Editor, Ch. Durgaprasada Rao; ChE Education Develop
ment Center, Indian Institute of Technology, Madras 600
036 India, $20 (1984)
classroom
THE NATURE OF ADJOINT VARIABLES
AND THEIR ROLE IN OPTIMAL PROBLEMS
0. K. CROSSER
University of MissouriRolla
Rolla, MO 65401
A DJOINT VARIABLES ARE frequently arbitrarily
introduced into the textbook discussion of op
timal or extremal theory. For example, Bryson
and Ho [1] "adjoin" them to the optimization prob
lem, Denn [2] introduces them as a "convenience,"
and Leitman [11] regards them as a transforma
tion to a "useful" vector space basis. Only Jackson
[10] has shown that they are desirable as a general
transformation from one set of variables which
appear naturally during the formulation of the
problem to the set of interest in the solution
search problems. Adjoint variables are the
sensitivity coefficients in optimal search problems.
Adjoint variables exist because the coefficient
matrix of every system (of describing equations)
has a transpose, and there are, therefore, two
independent solutions to the homogeneous form
of the system.
However, it was the late Professor F. M. J.
Horn who in 1958 most directly presented the
fundamental nature of the adjoint variables and
their role in optimal reactors in chemical engi
neering [5]. The original papers [6, 7, 8] and the
more complete elaboration in his thesis were in
German, with results published in English [9] by
1967. Publications about Pontryagin's Principle
[8, 10] became the standard literature reference,
and the directness of Horn's approach became less
available for the beginning student to appreciate.
Furthermore, this appreciation or understanding
of adjoint variables makes much of Horn's later
work in optimal chemical reactorseffect of by
passing, cyclical operation of nonlinear process
esmuch easier to follow.
This demonstration makes use of the example
presented in detail in appendix 1.11 of his Thesis
[5]. One asks for the optimal temperature profile
for a plug flow reactor with several independent
chemical reactions. The set of independent chemi
cal reactions is any set of the smallest number of
time dependent stoichiometric equations sufficient
to define all reaction compositions uniquely in
time. The proper interpretation of independent
is also clearly discussed in this thesis, although a
more formidable presentation is now available
[12]. We suppose a tubular plug flow reactor with
several chemical reactions and arbitrary kinetics
(Arrhenius)
Xe
0
dx./dz = Vi(xlx2 ..."m, T),
1 1..x )
(i = 1 to m)
O. K. Crosser received his PhD from Rice University in 1955 and
is currently professor of chemical engineering at the University of
MissouriRolla. His primary interests include optimization of pre
liminary plant design and fixed bed separations.
and presume that the objective function has the
form
M=M(xle,x ,...xme) = M(xe) (2)
M depends only upon the exit composition (ex
tents) x. and a straightforward solution to the
problem would be to assume a temperature pro
file, calculate the exit composition vector to give
Copyright ChE Division, ASEE, 1985
CHEMICAL ENGINEERING EDUCATION
M, then presume another temperature profile and
continue to adapt the profile in some beneficial
way until an extreme in M was obtained. Suppose
we had two such solutions (we use x for the vector
of extents of the independent reactions and V for
the vector of reaction rates, and the super dot to
imply differentiation with respect to z)
x + xl = V(xl,T1) and x2 x2 = V(x2,T2) (3)
so that for sufficiently small differences between
T, and T, a first difference (perturbation) is
sufficient. Then
x2 xI = [3Vi/3x.](x2 x1) + (3Vi/ T)(T2 T1)
or
y = [3Vi/ax.]y + (aVi/aT) T (4a)
where y stands for the perturbation in x caused
by the perturbation T in T. We will also have
the perturbed response
m aM
M = M(x2) M(Xl) = a Yk(Ze)
k=1 ke
= (M/axie)' y (Ze) (4b)
Since both x, and x, are zero at z equal zero, y(0)
is zero. Note that the matrix, [aVi /xj] and vector,
(aVi/aT) are functions of z only, so that the
system of Eqs. (4a,b) is a set of linear differential
equations in which the coefficients are functions
only of the independent variable z. y is the re
sponse of the system to T What we desire is to
solve Eqs. (4a,b) subject to the restriction that M
be an extreme, so that it is necessary that
dM = (aM/x. e)' dx = 0 (5)
(aM/axie)' is the transpose of the vector of
partial derivatives of M with respect to xie, that
is, x at the end of the reactor. A system like (4)
is usually solved by Variation of Parameters
(Boyce & DePrima [3] or Hochstadt [4]), finding
first the homogeneous (complementary) solutions.
The form of these solutions is more conveniently
manipulated if we use the solution matrix rather
than the solution vector (in contrast with usual
forms for systems with constant coefficients).
Therefore
Y = [B] yC
C c
where B.. = 3V./ax.
ij 1 J
has the homogeneous solution matrix [Y] such
that
[Y] = [B][Y] (7)
and we take [Y(ze)] to be [I] the identity matrix.
Any other boundary condition on [Y] may be ob
tained directly from this one. Using the Variation
of Parameters we suppose y = [Y]c and hope to
find the vector c to fit the inhomogeneous part,
which is the second term of Eq. (4)
y = ([Y]c) = [Y]c + [Y]c = [B][Y]c + (aVi/3T)T (8)
Substituting from Eq. (6) we have
1
[Y]c = (aVi/aT) T c = [Y] (V.i/aT) T (9)
and we see that the vector of the particular solu
tions c is directly related to the temperature pro
file, T Now, these functions depend only on z,
and we intend to keep the same inlet temperature
but to alter the shape of the profile. Therefore any
one of the particular solutions c must have the
property
z
e
Ck() = ck(z)dz
0
and since
y = [Y]c and [Y]
with ck(O) = 0
= [I] at z = z
e
y(ze) = c(z )
Then from Eq. 4b, using t as a dummy variable
and recalling that (DM/Dxie) does not depend on z
f=
m
M dt =
k=l
0
(aM/aXke)ck(e)
S(M/axke )ck(t) dt
k=l
M = (3M/aXke)k(Ze) = (aM/3aie)'c
k=1
(Note that M is the derivative, with respect to
the independent variable of the response M to the
perturbation T .)
We now have to solve simultaneously m + 1
linear equations involving c There are m inde
pendent chemical reactions, and Eq. (11) for
M Since these equations must be linearly de
SPRING 1985
Thus, the differential equations for the adjoint variables and their corresponding
conditions at the end of the reactor show that the influence of the exit extents upon the value of
the objective function can be obtained for any entering conditions to the reactor by integrating their
adjoint variable differential equations from the end of the reactor to the entrance.
pendent, their determinant must be zero
M
(av /aT)T
(WM/xie)'
=0
[Y]
i u'
or = 0
w [Y]
using vectors u and w for notational convenience,
then
S 0 u'
M JYJ + Y = 0
w [Y]
M/ax1 aM/ax2
(e 0 1
l(Ze) = o
3M/axi 3M/8x2
2(Z e)= 1 0
Hence
0 u'
and M=
IYl
A(ze) =
Ii
= u = (aM/xid )
As we expand the numerator determinant of Eq.
(12) first about w, the first column (deleting the
ith row in Y) and then about u' in the first row
(deleting the jth column in Y), we will obtain
the cofactors of the elements Yj, in Y, which we
label aij and Eq. (12) can be written
i uy w'[a ]u w[Adj(Y')]u
M Y i = w'A (13)
IYI IY1 IYI
because [aij] is the adjoint matrix of the transpose
of [Y], (a sign change occurs as the i + 1 index
in the determinant decreases to the i index for w).
The adjoint variables, A, are defined by Eq. (13)
and
1
S= [Y']lu [Y']X u
Since
S= (aM/xie) =
(for the extreme in M)
{[Y'IX} = 0 = [Y']x + [Y']J = [y'] 1 'l,]
But
[Y] = [B][Y] + [Y']X = {[B][Y]}' [Y'][B']
then
S= [Y']'[Y'][B']
and A = [B']X
These are the differential equations for the
adjoint variables. The boundary conditions of
[Y] = [I] at z = Ze imply (for m = 2 for clarity)
because Iij is the unit ij cofactor from the identity
matrix.
Thus, the differential equations for the adjoint
variables and their corresponding conditions at
the end of the reactor show that the influence of
the exit extents upon the value of the ob
jective function can be obtained for any enter
ing conditions to the reactor by integrating their
adjoint variable differential equations from the
end of the reactor to the entrance. These
functions, therefore, explain how the optimal re
sult is affected by changing the values of the ex
tents of reaction at any point along the reactor
such as the entrance. Since there is a direct cor
respondence between length in a plug flow re
actor and time, it is equally clear how the adjoint
variables apply to time optimization as well.
The adjoint variables are therefore nothing
more than the additional homogeneous solution
for the linear perturbation. Had the problem been
cast in the form of time optimal control, they
would have indicated the switching functions; in
troduced with an Hamiltonian or Lagrange multi
plier problem, they would have been the cor
responding multipliers [13].
The thing to see is that all of these structures
rely essentially only on a Cramer's rule for solving
a dependent set of linear equations and that the
adjoint variables appear naturally as the added
homogeneous solutions to the transpose of the
system coefficient matrix, and they show how
temperature changes along the reactor affect the
objective function, which depends on the con
version at the exit from the reactor.
CHEMICAL ENGINEERING EDUCATION
ACKNOWLEDGMENTS
One of the original reviewers of this article
observed that the subject of this paper is contained
in modern control theory texts. It is a pleasure to
suggest to students that Linear Systems by
Thomas Kailath (PrenticeHall 1980) is an ex
cellent reference with good examples and exercises.
The most directly relevant part is section 9.1 pp
598606 and example 9.13 p. 605, but there are
many other items of interest throughout the entire
text.
The University of MissouriRolla awarded the
Faculty Sabbatical during which this note was
written. D. W. T. Rippin and his Systems Engi
neering Group of ETH Zurich provided the
affectionate welcome and gentle scholarly support.
Don MacElroy offered a most helpful suggestion
toward the end of the work.
 POSITIONS AVAILABLE 
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Chemical Engineering .. Tenure system faculty positions.
Doctorate in Chemical Engineering or closely related field.
A strong commitment to teaching and the ability to de
velop a quality research program is expected. Preference
will be given to candidates with research interests in the
areas of Biochemical Engineering, Surface Science, Solid
State Phenomena, or Polymeric Materials. However, ap
plicants with outstanding credentials and research interests
in other fields related to Chemical Engineering are en
couraged to apply. Teaching and/or industrial experience
desirable but not essential. Michigan State University is
an affirmative actionequal opportunity employer and wel
comes applications from women and minority groups. Send
applications and names of references to Chairperson,
Faculty Search Committee, Department of Chemical Engi
neering, Michigan State University, East Lansing, Michi
gan 488241226.
LITERATURE CITED
1. Bryson, A. E., Y. Ho, Applied Optimal Control,
Halsted Press, New York (1975) [chapter 2, and pp
4750, pp 149150].
2. Denn, M., Optimization by Variational Methods,
McGrawHill, New York (1969), [pp 102109].
3. DiPrima, R. C., W. E. Boyce, Elementary Differential
Equations and Boundary Value Problems, John Wiley
& Sons, New York (1977).
4. Hochstadt, H., Differential Equations, Dover Press,
New York (1975).
5. Horn, R. "Optimalprobleme bei kontinuierlichen
chemischen Prozessen," Thesis, Tech. Hochsch. Wien,
Ostereich (1958).
6. Horn, F., Discussion of "Optimum Temperature Se
quences in Reactors," K. G. Denbigh, Chem. Eng.
Sci., Special Supplement, 8, 131 (1958).
7. Horn, F., U. Troltenier, "uber den Optimalen
Temperatur verlauf im Reaktionsrohr," Chem. Ing.
Tech, 3S, 382 (1960).
8. Horn, G., "Adjungtierte Variable und Maximum
prinzip in der Theorie Chemischer Reaktoren,"
Ostereichische ChemikerZeitung, 6, 186 (1967).
9. Horn, F., "Mathematical Models in the Design and
Development of Chemical Reactors," Ber. Buns. Ges.,
74, 8189, (1970) (in English).
10. Jackson, R., "Optimization of Chemical Reactors
With Respect to Flow Configuration," J. Theo. App.
Opt., 2, 240259 (1968).
11. Leitman, G., An Introduction to Optimal Control,
McGrawHill, New York (1966), [p 26, Chapter 2].
12. Smith, W. R., R. W. Missen, Chemical Reaction
Equilibrium Analysis, John Wiley & Sons, New York
(1982), [Chapter 2also see the reference by these
authors in Chem. Eng. Educ. which contains a re
view of the literature to 1976].
13. Report: SEG/R/128(83), Systems Engineering
Group, Tech. Chem. Labor, ETH, Zurich.
NOMENCLATURE
I I Determinant
[ ] Matrix (square)
( ) Vector (column)
( )' Transposed vector (row)
z Length of plug flow reactor
x Extent of reaction
V Vector of reaction rates
y Perturbation in x
T Temperature
M Objective function to be optimized
m Number of independent chemical reactions
B Coefficient matrix from partial derivatives
of rates V
Y Matrix of homogeneous solutions to Eq. 4a
I Identity matrix
c Vector of particular solutions for Eq. 4a
u' Row vector (aM/axie)'
w Column vector (aVi/aT)T
X Vector of adjoint variables
Subscripts
o
e
i,j,k
Entrance to reactor
Exit from reactor
Row column indexes
Superscripts
i Index to independent chemical reactions
(1 to m)
S Differentiation with respect to length
1 Inverse matrix
Transpose
 Indicates perturbation value
SPRING 1985
4waEd .2eceOND
SEMICONDUCTOR CHEMICAL REACTOR ENGINEERING
The Chemical Engineering
Division Award Lecturer for
1984 is T. W. Fraser Russell.
The 3M Company provides
financial support for this an
nual lectureship award. The
lecture has been presented at
the University of Florida,
N the University of Michigan,
and Colorado School of Mines.
A native of Moose Jaw,
Canada, Fraser Russell re
ceived his BSc and MSc in
chemical engineering from
the University of Alberta and
his PhD from the University
of Delaware. He joined the department at Delaware in
1964 and is presently the Allan P. Colburn Professor of
Chemical Engineering.
Prior to beginning his academic career, he spent two
years with the Research Council of Alberta, where he did
early development work with the Athabasca Tar Sands.
He later joined Union Carbide Canada as a design engineer,
where he completed the reactor and process design for all
of Union Carbide's ethylene oxide derived chemicals. His
innovative process design for these oxide derivative units
became the first multipurpose continuous processing unit
built in Canada.
In research, Russell's efforts have been directed into
two major areas: design of gasliquid systems; and semi
conductor chemical reaction engineering and photovoltaic
unit operations. His research in gasliquid system design
has resulted in over 25 publications which have been
widely used by industrial concerns and have led to improved
design of gasliquid contractors and reactors and biological
waste treatment systems.
Russell is recognized as a respected and inspiring teach
er and has received the University of Delaware's Excellence
in Teaching Award. His efforts in education have resulted
in the publication of two texts, Introduction to Chemical
Engineering Analysis with Morton M. Denn, and The
Structure of the Chemical Process Industries with J. Wei
and M. Swartzlander.
In the research and development of thinfilm photo
voltaic cells, his efforts have centered on the need to apply
chemical reaction engineering principles to improve the
design and operation of reactors used in making the semi
conductor material, and to ensure that solar cells developed
at the laboratory scale can be manufactured in commercial
quantities. He carries out this research as Director of the
Institute of Energy Conversion, a laboratory with a staff
of some fifty people devoted to the development of thin
film photovoltaic cells.
T. W. F. RUSSELL
Institute of Energy Conversion
University of Delaware
Newark, DE 19716
THE QUANTITATIVE ANALYSIS of a reactor pro
ducing semiconductor film can be termed semi
conductor chemical reactor engineering if the
analysis creates procedures which improve the de
sign, operation, and product quality of laboratory
or larger scale reactors. The creation of a thin
film semiconductor, or indeed any thin film, re
quires an understanding of both molecular and
transport phenomena. The process is analogous to
that encountered in a typical catalytic reacting
system (Fig. 1). Molecules must reach the surface
of a substrate, adsorb on the substrate, diffuse
and/or react on the substrate to produce a film
possessing specified material and electronic
properties. In a catalytic system, the product de
sorbs, leaving the substrate for the surrounding
fluid phase.
A variety of reactors have been employed to
move molecules or atoms to the substrate but much
can be learned by considering two general types:
Physical vapor deposition reactors
Chemical vapor deposition reactors
In a physical vapor deposition reactor the re
quired solid or liquid phase species are placed in a
source. Energy is supplied to vaporize these species
causing molecular beams to impinge on the sub
strate [1].
In a chemical vapor deposition reactor the
molecular species are continuously supplied in a
vapor phase which flows over the substrate. A
SURFACE
ADSORPTION
SURFACE
DIFFUSION
SUBSTRATE SURFACE
REACTION
FILM OR CRYSTAL
GROWTH
FIGURE 1. Surface molecular phenomena.
CHEMICAL ENGINEERING EDUCATION
0 Copyright ChE Division, ASEE, 1985
kND PHOTOVOLTAIC UNIT OPERATIONS
quantitative understanding of transfer from the
bulk vapor to the surface is required and it may
be necessary to contend with complex reactions in
the vapor phase [2].
The performance of a reactor which produces
a semiconductor film is judged by the quality of
the film produced. Much effort is being devoted to
ascertaining film quality by measuring optical and
Electronic & Optical
Properties
Material Properties
Design & Operation
of Reacting System
FIGURE 2. Simplified logic diagram.
electronic properties but film quality is ultimately
determined by the performance of the semicon
ductor in some type of electronic device. A success
ful semiconductor chemical reaction and reactor
analysis should provide experimentally verified
models linking the electronic properties of the film
to the design and operation of the reactor through
a detailed understanding of the material proper
ties of the film and the mechanism of film growth
(Fig. 2).
The logical sequence summarized in Fig. 2 has
been followed by the integrated silicon circuit com
munity of researchers and industrial practitioners
in dealing with the key step in integrated circuit
manufacture of dopant diffusion into a film. It has
not been a trivial task and well over two decades
of effort have gone into the development of models
relating device performance to doping concentra
tion profile and doping concentration profile to the
design and operation of the furnace [3]. Growing
a polycrystalline or amorphous film, predictably,
with the desired electronic properties is an even
more difficult task; one which remains an active
integrated circuit research area today.
If one is interested in applications which could
require millions of square meters per year of semi
conductor film, the task of effectively predicting
film growth becomes an order of magnitude more
complex. Semiconductor films covering an area on
the order of a square meter or more are needed for
photovoltaic panels for power generation
electrophotography
electronic devices for thinfilm displays
For these largearea applications it is necessary
to carry out research on a scale between that used
in a typical laboratory and that required for com
mercial operations. This unit operations scale re
search needs to have both a theoretical and experi
mental component which builds upon the labora
tory scale research. The position of chemical re
actor engineering and unit operations scale experi
mentation in the research logic is shown in Fig. 3.
I will illustrate the application of semicon
ductor chemical reactor engineering with research
we have underway in physical vapor deposition of
CdS at both the laboratory and unit operations
LABORATORYSCALE
EXPERIMENTATION
MATERIAL DEVICE DESIGN
DEVELOPMENT AND ANALYSIS
T T
CHEMICAL REACTOR
ENGINEERING ANALYSIS
MATHEMATICAL UNIT OPERATIONS
DESCRIPTION EXPERIMENTATION
PROCESSING EQUIPMENT
DESIGN
COMMERCIALSCALE
MANUFACTURE
FIGURE 3. Role of chemical reactor analysis.
SPRING 1985
We originally became interested in the semiconductor research because of a
need to design larger scale reacting systems. However, the last five years of research has
taught us that the chemical engineering analysis is very useful in the laboratory scale research effort,
and indeed essential, if such research is to be done efficiently and with minimum
expense (a key issue with today's research costs).
bell jar
heater box
radiant heater
i ffoil substrate
substrate holder
S top heat shield
Sheat shield
source bottle
I 1 "^ tantalum heater
current lead
and support
base plate
thermocouple
thickness monitor
FIGURE 4. Physical vapor deposition reactor.
scale and chemical vapor deposition of amorphous
silicon at the laboratory scale.
PHYSICAL VAPOR DEPOSITION
Laboratory Scale Research
A typical laboratory scale physical vapor de
position unit is shown in Fig. 4. The rate of evapo
ration of any material is determined by the surface
temperature of the source material. For thermal
evaporation this is a function of bottle geometry,
the material surface area, and the design of the
source heater. To make a semiconductor film, the
material of interest is placed in the source bottle,
heated to the point at which it evaporates or sub
limes, flows out of the source bottle to the sub
apv
=T,
pVCp, tpg AHR FVF (TTI4)A.
FIGURE 5. Model equations.
state, and then condenses on the substrate, the
temperature of which is carefully controlled.
The modeling and experimental verification of
a model describing the rate of effusion for CdS
which dissociates and sublimes has been thorough
ly discussed by Rocheleau et al [4]. The mass and
energy balance equations written for the material
to be evaporated are shown in Fig. 5 (nomencla
ture in [4]). These equations can be solved
numerically, given the initial dimensions of the
material in the source bottle and the appropriate
constitutive equations for the flow through the
orifice in the source bottle. Fig. 6 gives the required
equations in terms of the mass flux, r, related to
FLOW REGIME ORIFICE PIPE
Free gcR2p 2
Molecular r= (plgc/2 ")12 ( lP) r=  (P2 P2 )[4( ~) xm
(l\,/R>l) .16pL R "
gR2p1
Viscous r= CY[Zp gc(PlP2)]"2 r=  ( p2P22)
(Xm/R<00 ) 16pL
FIGURE 6. Constitutive flow equations.
pgq through the area available for flow. The solu
tion method is somewhat complex and complete de
tails are given by Rocheleau et al [4] and Roche
leau [6]. Solving the equations yields the rate of
effusion versus charge temperature, T1. A com
parison of model prediction (solid lines) and ex
perimental data (horizontal bars marked with the
wall temperature, T2) are shown for two different
orifices in Fig. 7. The heat transfer from the source
bottle walls to the subliming surface is the key
issue in predicting rate of effusion from the source
bottle.
Another type of experiment in which cadmium
and sulfur are used in separate source bottles can
be used to obtain information on the surface phe
nomena (Fig. 1). An extensive set of data has been
obtained by Jackson [5] who also was able to pre
dict the impingement rate of the molecular beam
at any point on the substrate. The impingement
rate of cadmium and sulfur on the substrate was
calculated and the corresponding rate of CdS film
CHEMICAL ENGINEERING EDUCATION
growth measured. About 1000 separate pieces of
data were obtained to verify the predicted model
behavior. The model equations for each species are
shown in Fig. 8. The rate of reaction of cadmium
to the CdS comprising the film is assumed to be
r(rxt, Cd) = k(CdS) [Cde][Ss]
This expression combined with the equations
shown in Fig. 8 yields
r(rxt,Cd) = K(CdS) [8(Cd) r(i,Cd)
r(rxt, Cd) ] [8(S) r (i, S) r (rxt, S)]
The parameter, 8, is a condensation coefficient;
r (i, Cd or S) is the rate of molecular impingement
of Cd or S; r (rxt, Cd or S) is the rate of reaction
of Cd or S; and K is a modified specific reaction
rate constant whose detailed form is given by
Jackson [5].
Comparison of the model behavior with some
of the data is shown as Fig. 9 where the rate of
deposition of CdS is plotted as a function of the
1.2 1
1334
numbers indicate wall temperature in oK
1.0 /
1309
.5 .8
E
"E orifice set 4 1271
.4
12 1 ~orifice set 3
.2 1179
1198
OL
1150
1200
charge temperature, K
FIGURE 7. Comparison of model behavior with ex
perimental data.
rate at which cadmium is delivered to the sub
strate. At low values of r (i, Cd) the rate of film
growth is proportional to the rate at which cad
mium is delivered; as the film growth becomes sur
facereaction dependent, the lines curve. The hori
zontal line indicates a region in which there is not
enough sulfur to react with all the cadmium being
delivered to the substrate.
Experimental evidence indicates that photo
voltaicgrade CdS can only be made when the rate
of film growth is controlled by the rate at which
Cadmium
r(r xt, Cd)=r(i, Cd)r(e, Cd)r(r, Cd)
Sulfur
r(rxt, S)=r(i, S)r(e, S)r(r, S)
Cadmium Sulfide
I dM(CdS)
I dM(CdS) == r(rxt, CdS)
M,(CdS)A, dt
FIGURE 8. Component mass balances.
cadmium and sulfur react on the surface to form
CdS. Furthermore, if sulfur is not present when a
cadmium molecule arrives at the surface, the
cadmium will reflect and not adhere. We are
just beginning to try to relate these observa
tions to film properties. This second laboratory
scale study of Jackson's is an attempt to learn more
about the semiconductor chemical reaction engi
neering necessary for the field to progress in an
orderly fashion. A much more complicated set of
chemical equations will be considered in the section
on chemical vapor deposition.
Unit Operations Scale Research
For large area applications uniform defect
free film with the required properties must be de
posited over areas on the order of meters in
dimension. It may be necessary to deposit on a
moving substrate to lower costs to the level re
quired to make a large area application like photo
voltaics economically feasible. In this section the
cooperative research efforts in photovoltaic unit
operations between the Department of Chemical
Engineering and the Institute of Energy Con
E2
4)
0
E
8
0
t."
3.3I I I I I
5
39
67 n 22
19
Model Predictions
8.3
Constant Sulfur Flux Indicated x 108 mi se
220 C Substrate Temperature
0 o I
0 8.3 16.7 25 33.3 41.7 50 58.3 66.7 75
r(i, Cd) x 108 Kgmoles/m2 sec
FIGURE 9. Deposition versus incident rate of cadmium
in cadmium sulfide.
SPRING 1985
I ( ( I i I I I
83.3 91.7 ICO
^
The theoretical and experimental
work of Rocheleau, Rocheleau et al, and Jackson
provide the verified models of the laboratory scale
batch experiments that can be used to design
apparatus and experiments at the
unit operations scale.
version at the University of Delaware will be de
scribed.
CdS is the wide bandgap window semicon
FIGURE 10. ThinFilm polycrystalline solar cell.
ductor for the following polycrystalline hetero
junction cells (Fig. 10)
CdS/CuInSe2
CdS/CdTe
CdS/Cu2S
All of the above devices have achieved con
version efficiencies (percentage of sun energy con
verted to electricity) of just over 10%, although
in the case of CuS cells some ZnS had to be al
loyed with the CdS. At this conversion efficiency,
inexpensive electrical power generation begins to
become feasible if modules containing the indi
vidual cells can be made cheaply. A first step in
meeting this goal is to find a way to continually
TABLE 1
Approximate Throughput and Size Specifications
Unit Ops.
Lab. Scale Scale
Exps. Expts.
Throughput
(m2/year)
Deposition
Chamber (m3)
Commercial
Scale
Production
1100 1,00020,000 100,0001,000,000
0.10.5 13
1030
FIGURE 11. Unit operations scale deposition system.
deposit the CdS on a moving substrate.
The theoretical and experimental work of
Rocheleau [6], Rocheleau et al [4] and Jackson
[5] provide the verified models of the laboratory
scale batch experiments that can be used to design
apparatus and experiments at the unit operations
scale. Throughputs and chamber sizes for typical
units are shown in Table 1 for the three scales of
operation of interest. The laboratory scale ex
periments are almost always batch experiments
on a static substrate. The unit operations and com
mercial scale equipment for photovoltaics need
to be designed for continuous deposition on a mov
ing substrate.
A sketch of the unit operations scale equip
ment used at the Institute of Energy Conversion
TABLE 2
Deposition Unit Specifications
Chamber1.28 m diameter X 1.34 m long
VacuumPumpdown to 5 X 106 torr in 2 hours
Web
Capacity500 mm maximum width
250 mm roll diameter (200 m Cu foil)
Web
Speed1.2 to 12 cm/min
Deposition
Zone45 cm X 10 cm
SourceA proprietary design (U.S. Patent
4,325,986) providing:
Constant rate2 micron/min
Uniformity over 20 cm wide zone
80% CdS utilization
Web
Temperature200 to 2250C
Throughput0.6 to 6 m2/shift
CHEMICAL ENGINEERING EDUCATION
Windup Roll
Web or /
Substrate.
Substrate
Temperature
Control
Payoff Roll
Vacuum Chamber
SGuide Roll
 Evaporation
Source
is shown in Fig. 11. This piece of equipment was
designed using model equations similar to those
presented as Fig. 5 and 6 and modified for a multi
orifice geometry and the different sourcesubstrate
geometry of the unit operations scale equipment.
It was also necessary to expand the energy balance
analysis to include radiative heat transfer between
the source and substrate. The model equations,
their behavior and their influence on the design
and operation of the unit operations scale reactor
are given by Rocheleau [6] and Griffin [7]. The
specifications determining the equipment are
shown in Table 2 and a photograph of the equip
ment is shown as Fig. 12.
FIGURE 12. Photograph of unit operations scale deposi
tion system.
The unit operations experimentation consisted
of controlled deposition of CdS on rolls of zinc
plated copper foil. Throughput of the foil ranged
from 180 to 600 cm2 per hour with film growth
rates ranging from 0.5 to 2 Mm/min. Substrate
temperatures were varied between 200C and
250C. At throughputs of 400 cm2/hour, up to
3000 cm2 of 25 im thick CdS was prepared in a
single run. Fig. 13 compares a crosssection of con
tinuously deposited CdS with that of photovoltaic
grade batch deposited CdS. Xray diffraction con
firms predominantly caxis orientation for the
continuously deposited CdS. Resistivities of the
continuously deposited CdS films ranged from
about 1 to 100 ohmcm. Resistivity of the best
laboratory CdS ranges from 1 to 10 ohmcm.
4j
FIGURE 13. CrossSection of CdS film. Crosssection of
continuouslydeposited CdS on left, crosssection of
batchdeposited CdS on right.
The principal means of evaluating the CdS was
to measure the photovoltaic response of cells fabri
cated using the material from the unit operations
experiments. Results are summarized in Table 3
which shows the efficiencies of CdS/Cu2S cells
made using both laboratory scale and unit opera
tions scale CdS. The Cu2S layer can be made using
a wet process by dipping CdS into a CuCl solution
or it can be made using a dry process in which
CuCI is evaporated onto the CdS and then allowed
to react with CuS. A quantitative description is
given by Brestovansky et al [8]. Cells made by the
dry process in the unit operations scale equipment
had both layers, CdS and CuS, continuously de
posited on the moving substrate. Cells made by
the wet process had to have the Cu2S layer made
in a batch operation. All cells had an evaporated
gold front contact.
The efficiency figures show that the unit opera
tions scale continuously deposited CdS is virtually
of the same photovoltaic quality as the laboratory
scale batch deposited CdS. It took some ten years
of research to achieve the efficiencies shown for
the batch deposited CdS. The continuously deposit
ed CdS reached the efficiency shown in well under
two years of unit operations experimentation. This
could only have been achieved by drawing heavily
Continued on page 106.
TABLE 3
Cell Efficiencies (CdS/Cu2S)
Laboratory
Scale
(Batch)
Wet Process
(CdS Only)
Dry Process
(CdS/CuS)
Unit Operations
Scale
(Continuous)
8%
7%
SPRING 1985
Sn classroom
THE USE OF COMPUTER GRAPHICS
TO TEACH THERMODYNAMIC PHASE DIAGRAMS
CHANDRASHEKHAR D. NAIK
Singer Corporation
Silver Springs, MD 20904
PAULETTE CLANCY AND
KEITH E. GUBBINS
Cornell University
Ithaca, NY 14853
THE TEACHING of thermodynamic phase dia
grams poses problems which affect both the in
structor and the students. The usual approach in
which the threedimensional pressuretempera
turecomposition diagrams for binary fluid mix
tures are represented on a twodimensional page is
difficult for students to visualize. Traditionally, in
order to simplify this complex situation, 'cuts' at
constant pressure, temperature, or composition are
made to show a truly twodimensional diagram de
scribing the relationship between two of the three
independent variables. However, the interre
lationship of all the variables involved is lost with
this approach, and the problem of comprehension
intensifies as the complexity of the phase behavior
increases. Construction of threedimensional
Chandrashekhar D. Naik ob
tained his B. Tech. degree in
chemical engineering from the
Indian Institute, India, in 1981.
He obtained an MS degree in
chemical engineering from
Cornell University in 1983 and
is currently employed at Singer
Link Simulations Systems Di
vision in Silver Spring, Mary.
land. (L)
Paulette Clancy is current
ly an assistant professor in
chemical engineering and as
sociate director of the Manu
facturing Engineering Program at Cornell University. She received
her BS degree at the University of London and a D.Phil. degree at the
University of Oxford. She held fellowships at Cornell University and
at London University before joining the faculty at Cornell in 1984.
(C)
Keith E. Gubbins is currently the Thomas R. Briggs Professor of
Engineering and director of chemical engineering at Cornell Uni
models offers an alternative solution, but they are
difficult and timeconsuming to produce and offer
no possibility for student interaction.
At Cornell an alternative to traditional ap
proaches was sought to improve the quality of
teaching and the level of comprehension of the
students. Computer graphics offers an innovative
solution to these difficulties: presentday graphics
hardware can perform rotational transformations
of threedimensional images almost instantaneous
ly and allows extensive manipulation of the viewed
image by the user, making this an extremely
powerful tool eminently suited to the task at hand.
During the past two years a highly interactive
"user friendly" graphics package has been de
veloped depicting the threedimensional phase be
havior of binary fluid mixtures, and it has been
used in both undergraduate and graduate courses
with great success.
THE GRAPHICS WORKSTATION
The Computer Aided Design Instructional
Copyright ChE Division, ASEE, 1985
versity. He received his BS and PhD degrees at the University of
London, and was on the staff at the University of Florida from 1962
76, when he moved to Cornell. He has held visiting appointments at
Imperial College, London, at Oxford University, and at the University
of California at Berkeley. He has coauthored two books, Applied
Statistical Mechanics (Reed and Gubbins) and Theory of Molecular
Liquids (Gray and Gubbins). (R)
CHEMICAL ENGINEERING EDUCATION
FIGURE 1. An example of a typical workstation show
ing the Evans and Sutherland vector refresh graphics
monitor with VT100 terminal, electronic tablet and
stylus.
Facility (CADIF) at Cornell houses "stateofthe
art" computer graphics equipment used solely for
teaching (and developmental work towards edu
cation). The central computers for the facility are
Digital Equipment Corporation (DEC) VAX ma
chines, an 11/780 and an 11/750, running the
VMS operating system, with DEC PDP 11/44 ma
chines as postprocessors. Attached to these ma
chines are two different types of graphical display
equipment for student use: vector refresh stations
with dynamic threedimensional capabilities and
color raster stations for applications requiring
color. In this application, the vector refresh work
stations were used exclusively, these being the
highly sophisticated Evans and Sutherland Multi
picture Systems. Each workstation has a digitiz
ing tablet and electronic stylus as the primary in
put peripheral for cursor control, with a DEC
VT100 terminal for alphanumeric input. A typi
cal workstation is shown in Fig. 1. An electrostatic
plotter is also available for hardcopy output, a use
ful and necessary addition allowing students to
submit a record of their progress to the instructor.
The software, which is the heart (or perhaps
more appropriately, the brains) of this application,
was written in FORTRAN making use of system
graphics software routines developed at CADIF.
The consideration of ergonomic factors to produce
a well designed application in terms of its "user
friendliness" was considered essential to promote
ease and clarity of use of the graphics package as
well as increased flexibility. Some of the ways this
was achieved include the following points: ex
tensive 'help' messages and prompts for required
input were made available, clear consistent
"menus" for optional choices of interactive re
sponse by the program were produced, and the
ability to recover from mistakes or unintentional
"miskeying" was provided. It was an original
tenet of this study that students should not have
to read a computer manual before using the pro
grams. The emphasis is thus on learning engineer
ing principles without requiring prior expertise
in computing.
REPRESENTATION OF THE PHASE DIAGRAMS
The phase equilibrium data for binary mix
tures needed for the representation of the phase
diagram (i.e. pressures, temperatures and com
positions) were generated using a theoretical
model. The original RedlichKwong equation of
state was employed for this purpose, chiefly be
cause of the simplicity of its representation (since
only two adjustable parameters are involved) and
the reasonably realistic description of binary phase
behavior it provides. This approach was also used
It has proven to be extremely
popular with the students, and has raised the
level of comprehension of this potentially difficult
subject above that achieved previously by
using conventional means.
by Willers and Jolls [1] who produced three
dimensional phase diagrams on a Cal Comp plotter
using the same equation of state.
The wellknown RedlichKwong expressions
describing the conditions for vaporliquid or liquid
liquid equilibrium in terms of the pressures and
chemical potentials of both phases were used to
generate data points P, T, VL, VG, X1, and y1 cover
ing a region from the higher of the pure com
ponent triple points to a temperature above both
critical temperatures. The nonlinear equations in
volved were solved using a multidimensional
NewtonRaphson [2] technique. Close to the criti
cal region, however, convergence problems were
encountered which were due, we believe, to
singularities in the Jacobian matrix. These difficul
ties were overcome by using the Marquardt [3]
method which combines the advantages of New
tonRaphson and Steepest Descent algorithms.
Here Argonne National Laboratory's 'MINPAK'
SPRING 1985
Computer graphics offers an innovative solution present day graphics hardware can
perform rotational transformations of threedimensional images almost instantaneously and allows
extensive manipulation of the viewed image by the user an extremely powerful tool .
software package provided the subroutine for a
Marquardt method of solution. Solving for vapor
liquid critical lines also provided a challenge.
Neither of the previous techniques mentioned was
able to reproduce these highly nonlinear equa
tions, and a specialized algorithm due to Deiters
[4] was employed for their solution.
Scott and Van Koynenburg [5, 6] classified the
experimentally observed types of fluid phase dia
grams into six classes, based on the presence or
absence of threephase lines and their connection
with the critical lines. So far we have been able
to cover the two simplest classes, I and II, although
extension of the programs to cover the other
classes is well underway. These more complex
systems will provide an even more striking visual
illustration of the advantage of using computer
graphics. In classes I and II both components have
similar critical temperatures with the vaporliquid
critical line passing continuously between them.
In class II, however, the mixture is more nonideal
and exhibits liquidliquid immiscibility at low
temperatures. For this class, in addition to the
vaporliquid region encountered for class I, two
other regions exist in the phase diagram, those of
liquidliquid equilibrium and a threephase liquid
TABLE 1
Examples of Classes I and II Type Behavior Available
for Display By the User
CLASS I
Binary
Mixture
PentaneNonane
Cyclopentane
Nonane
Pentane
Ethylbenzene
Acetone
Trichloromethane
MethaneTetra
fluoromethane
Perfluoropen
tanePentane
Azeo Temp Pressure
tropet Range K+ Range Barst
N 425590(594) 2.3824.30(33.7)
N 425590(596) 2.3921.92(45.2)
N 425615(617) 4.1836.70(37.4)
Y 420530(535) 11.8651.18(55.6)
CLASS II
N 80224.5(228)
Y,Het 240505(506)
,+ Horn
1.25x104
38.76(46.0)
4.18x102
39.28(39.7)
tN,Y = no, yes; + = positive or negative deviation from
Raoult's law; Het, Hom = heterogeneous, homogeneous
azeotrope.
+The figures in parentheses are the highest values of T,
and Pc occurring along the critical line.
FIGURE 2. Threedimensional phase diagram for a typi
cal class I system, pentanenonane. The solid and
dashed lines show the vapor and liquid boundaries,
respectively.
FIGURE 3. PTx diagram for the class II system, me
thanetetrafluoromethane, showing the coexisting
vaporliquid equilibria (solid lines for the vapor, dashed
for the liquid) and the region of liquidliquid immisc
ibility (shown as solid vertical lines). Superimposed on
the diagram (shown in bold) is a Tx cut at a pressure
of 0.03 bars.
CHEMICAL ENGINEERING EDUCATION
liquidgas line. Examples of the binary systems
chosen to illustrate the phase behavior of classes I
and II are shown in Table 1. Some of the available
systems exhibit azeotropic phenomena with either
positive or negative deviations from Raoult's Law,
and of either a heterogeneous or homogeneous
nature. Photographs depicting some of the com
FIGURE 4. A 3D view of another class II system, CF,, 
CH12. The original display has been rotated by 180
degrees and tilted downward so that the view is from
the hightemperature end and somewhat above the
phase diagram. The regions of vaporliquid equilibria
(showing an azeotrope) and liquidliquid equilibria
(solid vertical lines) are clearly visible.
SYSTEM: C5F12(1)Pentane(2)
I0o mTrArP 9479, F VArni
o.o
0.0
X(W)
HELP
(CUT)
ULTAIN EAS[
MUL. CUT
FIGURE 5. TwoDimensional xy diagram for the system
C5F12C5sH, derived from the threedimensional phase
diagram at 247.26K. This diagram shows the character
istic behavior of an azeotropic system with liquidliquid
immiscibility, as shown by the horizontal portion of the
curve.
putergenerated phase diagrams are reproduced
in Figures 26 illustrating the kind of image dis
played for the user to manipulate.
USER INTERACTION WITH THE GRAPHICS
SOFTWARE PROGRAMS
The image of the phase diagram (e.g. as in
Figs. 26) can be manipulated by the user by
means of an electronic tablet and stylus (pen). As
the pen is moved over (and slightly above) the
surface of the tablet, a cursor in the form of cross
hairs moves over the display. When the pen is
pressed down onto the tablet the graphics program
is activated and performs an operation appropri
ate to the area of screen chosen, given that such an
area is one of the several specially designated parts
of the screen called "windows" on a socalled
"menu" of options.
In this application of computer graphics the
menu contained the following list of 'entrees' for
the user to select
a) READ: Allows the user to choose different binary
systems to examine by supplying one of a given
set of data file names via the terminal.
b) ORBIT: This allows the phase diagram to be ro
tated about its pressure and composition axes in a
continuous fashion as required.
c) PAN: Allows horizontal or vertical translation of
the phase diagram.
d) ZOOM: Provides closer examination of a chosen
area of the image by scaling the diagram up or
down.
SYSTEM: CSF12(1)Pentane(2)
L I :i!N.
HELP
ULN CUT
MUL CUT
FIGURE 6. A view of the CsF12C5Hi2 phase diagram as
it appears in the initial orientation on the screen. Solid
and dashed lines have the same meaning as in Figure
2. A Pxy cut is shown superimposed in bold on the
diagram at a temperature of 259.9 K.
SPRING 1985
_1
1.0
e) STRETCH: Scales either of both of the P, T axes
relative to the composition axis for ease of viewing.
f) HELP: Summons the HELP text.
g) RESET: Voids all previous manipulations and resets
the system to the beginning of the program.
h) SNAP: Produces a hardcopy image of the screen
on a nearby plotter.
i) EGRESS: Allows the user to terminate his or her
session.
j) TX, PX, PT: Each of these windows allows a
particular highlighted "cut" of the phase diagram
to be chosen by the user as shown in Figure 3 for a
Tx cut at 0.03 bars for the system methanecarbon
tetrafluoride, and in Figure 6 for a Px cut at
259.9K for CF12pentane.
k) CUT: Produces a Px, PT or Tx "cut" displayed
alone (i.e. not superimposed on the whole phase
diagram) depending on which of these three
windows (PX, PT or TX) was last active.
Multiple cuts (of Px at different temperatures for
example) may be displayed simultaneously.
1) Produces an xy plot at constant temperature, as
shown in Fig. 5 for the system CF,,pentane at
247 K.
A 16mm movie lasting approximately thirteen
minutes has been prepared to illustrate the cap
abilities of this graphics package; this was pre
sented at the 1983 AIChE annual meeting in
Washington, D.C.
SUMMARY
The interactive graphics package illustrating
the phase behavior of binary mixtures which has
been described in this paper has been used within
the chemical engineering curriculum at Cornell
since the fall semester of 1982. It has proven to be
extremely popular with the students, and has
raised the level of comprehension of this potential
ly difficult subject above that achieved previously
using conventional means. The major advant
age lies in the suitability of computer graphics as
a means of visualizing threedimensional objects
(here the PTx phase space) ; the capability of the
hardware to perform rapid and continuous rota
tions of the image; and, perhaps most importantly,
the opportunity to interact, manipulate and con
trol the image observed on the screen, brought
about by flexible "userfriendly" software. All
these features combine to contribute to the success
of this technique in undergraduate instruction. O
ACKNOWLEDGMENTS
It is a pleasure to thank the Gas Research
Institute for partial support of this work.
REFERENCES
1. K. R. Jolls and G. P. Willers, Cryogenics, 329, June
1978.
2. J. Stoer and R. Bulirsch, Introduction to Numerical
Analysis, Springer Verlag (1980).
3. D. W. Marquardt, J. Soc. Ind. & App. Math, 11, 431
(1963).
4. U. K. Deiters, Diplomarbeit, Univ. of Bochum, West
Germany (1976).
5. R. L. Scott, Ber. Buns. Phys. Chem., 76, 296 (1972).
6. R. L. Scott and P. H. Van Koynenburg, Disc. Fara.
Soc., 49, 87 (1970).
Qs book reviews
FOUNDATIONS OF BOUNDARY LAYER
THEORY FOR MOMENTUM, HEAT AND
MASS TRANSFER
by Joseph A. Schetz
Prentice Hall, Inc., NY (1983)
Reviewed by
O. T. Hanna
University of California, Santa Barbara
This book on Boundary Layer Theory is indi
cated by the author to be applicable for students
in mechanical, aerospace, chemical, civil, and
ocean engineering. Some people would doubt that
anyone could succeed in such a broad task. The
author's stated goals for this book include (i) pro
viding an understandable coverage of advances in
turbulence modeling, (ii) presenting application
of large digital computers to boundary layer prob
lems, and (iii) treating mass transfer in an inte
grated manner with momentum and heat transfer.
It would appear that the first goal has been met
reasonably well; achievement of the second goal
is questionable, and the third goal has definitely
not been met to the satisfaction of chemical engi
neers.
The book is generally well written and well
organized. The coverage of laminar flows includes
chapters on integral and differential equations of
flow together with approximate integral solutions
and exact similarity solutions. Unfortunately al
most all of this material is available in a number
of other sources and hardly any of it is more
recent than 1960. The meager coverage of mass
CHEMICAL ENGINEERING EDUCATION
transfer is likely to be of little interest to chemical
engineers. Chapters 4 and 5 do contain some use
ful discussions of numerical solutions of bound
ary layer problems. However, there are no example
problems or computer programs.
The major contribution of this book would ap
pear to lie in Chapters 6 through 9, on turbulence
modeling, which constitute more than half the
length of the book. This material includes a useful
historical perspective and spans the complete
range of engineering approaches in this area up
to the present time. The chronological discussion
of work in turbulence modeling beginning with
early mixinglength theory and progressing up
to algebraic and various partial differential equa
tion models should be of interest and value to
chemical engineering. This discussion also inte
grates well the contributions to modeling from
both experimental and theory.
In summary, the present book seems somewhat
disappointing in its treatment of laminar bound
ary layers, but in contrast it contains material on
turbulent momentum transfer which should be of
interest to chemical engineers. In this context the
book can be recommended as a useful reference. E
INDUSTRIAL HYGIENE ASPECTS OF PLANT
OPERATIONS
by L. J. Cralley, L. V. Cralley, and J. E. Mutchler
Macmillan Publishing Company, New York,
1984: $60.00
Reviewed by
Klaus D. Timmerhaus
University of Colorado
This is the second volume of a new three
volume series that is being prepared to
provide recognition, measurement, and control
of potential hazards normally present in various
industrial plant operations. The first volume
covered process flows while the third volume will
treat equipment selection, layout and building de
sign.
This volume, divided into two major sections
of unit operations and product fabrication, en
compasses a broad range of industries with au
thoritative information contributed by specialists
from these industries. In the first section twenty
three contributors discuss unit operations as dis
tinct entities along an industrywide concept.
Some of the unit operations considered include
filtration, clarification, mixing, blending, grind
ing, and spray, vacuum, freeze and fluidized bed
drying. The second section includes thirteen con
tributions which cover the operations and pro
cedures for assembling parts and materials into
final products. The industries considered in this
latter survey range from such basic industries as
storage battery and tire manufacturing to the
high technology industries of semiconductor and
liquid scintillation counter manufacturing. One
may argue with the manufacturing processes that
were selected by the editors; however, the breadth
of the selected processes and the hazards associ
ated with these processes should provide a good
introduction to the hazards associated with those
manufacturing processes that were not included.
Even though most contributors to this second
volume have adequately described each step in the
unit operations and product fabrication flow of a
specific manufacturing process and have included
a discussion of the various health hazards that
may be encountered with suggestions for their
monitoring and control, many engineering read
ers will be disappointed by the qualitative ap
proach taken by the contributors to this important
subject. Only a few of the chapters in the volume
have included quantitative information that
would be necessary in the design and construction
of process equipment that minimizes or eliminates
identified industrial hygiene hazards. Where such
quantitative information is included, it is general
ly quite sketchy and incomplete forcing the design
engineer to consult other literature sources. Un
fortunately, no guidance to such quantitative data
is included by any of the contributors. Chemical
engineers will also be somewhat disappointed in
this volume because the "unit operations" portion
of the title implies that some of the contributions
will examine the conventional unit operations as
sociated with heat, mass and momentum transfer.
However, many of the key unit operations such
as distillation, absorption, extraction, evaporation,
heat transmission, etc. found in most typical
petroleum and chemical processing plants have not
been included.
Nevertheless, this volume does manage to bring
together a wealth of experience in a broad range
of industries and will aid engineers, managers,
and industrial hygienists to more fully recognize
potential hazards of industrial processes. This, in
turn, will permit these professionals to evaluate
such hazards and take the necessary steps to
effectively control the problem. C
SPRING 1985
laboratory
AN IMPROVED DESIGN
OF A SIMPLE TUBULAR REACTOR EXPERIMENT
ABDULFATTAH A. ASFOUR
University of Windsor
Windsor, Ontario, Canada N9B 3P4
DESIGNING A TUBULAR FLOW reactor experi
ment for an undergraduate laboratory is not
a simple task. This is because the experiment will
have to meet certain criteria, viz
It is safe
It is simple and cost effective
It is instructive
Its analytical needs must be simple and easy, to meet
the time constraints of an undergraduate laboratory
Anderson [1] developed a tubular flow reactor
experiment for an undergraduate laboratory at
Princeton that utilized the system acetic an
hydridewater. This particular experiment re
quires relatively elaborate safety precautions.
Moreover, since the reaction is exothermic, rather
expensive temperature control equipment is re
quired. Samples taken at the reactor inlet and
outlet are analyzed by the anilinewater method
which is relatively lengthy and subject to errors.
Hudgins and Cayrol [2] utilized the basic de
sign of Anderson in developing a simple and
interesting experiment. They utilized the classical
reaction system of crystal violet dye neutraliza
tion with sodium hydroxide. This system was
studied earlier by other investigators, mainly in a
batch reactor (Carsaro [3]). The two novel aspects
of the HudginsCayrol experiment compared with
that of Anderson are
A colour change can be seen between the inlet and
outlet of the reactor
The temperature constraint is removed. This makes
the experiment operable at room temperature
Also, from the safety standpoint, a relatively dilute
sodium hydroxide solution (0.04 N according to
Hudgins and Cayrol) is used.
However, the experimental setup design
given by Hudgins and Cayrol can be significantly
Copyright ChE Division, ASEE, 1985
A. A. Asfour received his B.Sc. (Hon) and M.Sc., both in chemical
engineering, from Alexandria University, Egypt, and a Ph.D. from
the University of Waterloo, Waterloo, Ontario, Canada. He joined
the research department of Imperial Oil, Sarnia, Ontario for the
period 19791981. In 1981 he joined the Chemical Engineering De
partment at the University of Windsor, Windsor, Ontario, Canada. His
research interests are in the area of mass transfer in threephase
fluidized beds and in membrane processes.
improved. The design improvements suggested in
this article should make the experiment easier to
run and control, significantly improve the repro
ducibility of results, and expedite the process of
data collection within the time constraints of an
undergraduate laboratory.
The main objectives of this experiment are
To study the effect of residence time on conversion
in a tubular flow reactor
To compare the experimental conversions with those
obtained from plugflow and laminarflow reactor
models
THEORY
As it was established by Corsaro [3], the re
action between crystal violet dye and sodium hy
droxide is of the first order in the concentration
of each of the reactants, i.e., the reaction is of
the second order. However, the reaction can be
made pseudofirst order if sodium hydroxide is
used in great excess with respect to crystal violet
dye. In other words
rdye = k' [dye] (1)
CHEMICAL ENGINEERING EDUCATION
The value of the rate constant, k', is needed for this experiment. Students are requested to
run a batch experiment to determine the value of k' at the same temperature of the flow experiment.. (and) to
prepare their own calibration curve of the dye concentration versus absorbance.
For the purpose of this experiment, 0.02 N sodium
hydroxide solution is used with 6.86 x 104 M dye
solution, i.e., the sodium hydroxide concentration
would be about 282 times that of the dye, if equal
volumes of reactants are used.
Experimental conversions are calculated, as
will be described later, and compared with theo
retical conversions predicted from the plugflow
model and the laminarflow model.
For a first order reaction in a plugflow re
actor, the following equation applies assuming
constant density of reaction mixture:
V 1
7= = In(1x) (2)
Vo k
= In CA (3)
k' CAo
If CA is taken as [dye]e, i.e., the dye concentration
at reactor exit and CAo as [dye]i, i.e., the dye con
centration at reactor inlet, then one can rewrite
Eq. (3) as follows:
V 1 [dye],
T  n (4)
vo k [dye]e
For a first order reaction in a laminarflow
reactor, the following equation applies assuming
no change in volume by reaction as well as no
mixing in both radial and axial directions (4)
x= 1i 2 E ( ) + ( 1)exp(Nn/2)
(5)
where
NR =k'r
V Lrr2
V, Vo
The function E (y) is defined by:
exp (4)
E(y) = d
y
The function E (y) is tabulated in standard tables
as Ei(X).
EXPERIMENTAL
A schematic diagram of the proposed experi
mental setup is given in Fig. 1. The experimental
apparatus is comprised of the following compo
nents.
Reservoirs
Constant Head
Tanks
Pumps
Mixer
Rotameters
FIGURE 1. Schematic diagram for the experimental
setup. M: mixer, P: pump, R: rotameter, RES: reser
voir, S: sampling point, T: constant head tank, TFR:
tubular flow reactor.
:(RES 1) 200L polyethylene tank for
the sodium hydroxide solution
(RES 2) 20L polyethylene tank for
the dye solution
: (T1) 20L polyethylene tank for the
sodium hydroxide solution
(T2) 4L polyethylene tank for the
dye solution
:(P1) Magnet drive gear pump; Model
P/N 81152 manufactured by
Micropump Corp., Conford, Cali
fornia. Purchased from Cole
Parmer Co.
(P2) Centrifugal pump. Cole Parmer
catalogue No. K700430
: (M) Little Giant Pump. Model 2E
38NT. Purchased from Can Lab
: (R1) Size R615A rotameter. Max
flow 450 ml/min with SSfloat.
Purchased from Brooks Instru
ment Co.
(R2) Size R615B rotameter. Max
flow 1300 ml/min with SSfloat.
SPRING 1985
Purchased from Brooks Instru
ment Co.
: 40 meters of 3/8 in. I.D. Tygon tubing,
wound on spool (made of lexan),
28 cm in diameter and 55 cm in
length*
: needle valves to adjust flow
: Spectronic 20 (Bausch & Lomb) modi
fied, as will be described later, to
provide continuous measure
ments.
Two polyethylene tanks (RES 1 and RES 2)
of capacity 20 liters and 200 liters serve as reser
voirs for the crystal violet dye and sodium hy
droxide solutions, respectively. Two pumps (P1
and P2) are employed to pump the reactants to
two constant head tanks (T1 and T2). The over
flows from the constant head tanks are returned
to their respective reservoirs. The underflows from
the constant head tanks go via rotameters (R1
and R2) to a small pump that acts as a mixer
(M). The reactant streams are mixed in the mixer,
M, and are pumped through the reactor. The
tubular reactor is in the form of a helical coil
wound on support. Connections are provided at
the inlet and outlet of the reactor to the flow
through curvettes of the spectrometers.
The main advantages of the proposed experi
mental setup over that suggested by Hudgins
and Cayrol are
Reservoirs and constant head tanks are used. This
arrangement provides more stable rotameter opera
tion, especially at low flow rates.
A flowthrough accessory which is simpler in design
and operation than that suggested by Hudgins and
Cayrol has been used. The flowthrough accessory
shown in Fig. 2 allows one to use Spectronic 20 for
continuous measurements.
PROCEDURE
Due to the limitation of the headroom in most
undergraduate laboratories, the constant head
tanks (T1 and T2) are placed about 3 meters above
the rotameters' level. This limitation makes it
only possible to attain maximum flow rate of 1300
ml/min of NaOH. The maximum flow rate of the
dye is set at about 135 ml/min.
*One of the reviewers suggested the use of polyethylene
instead of Tygon tubing, which discolors to a deep purple
making it difficult to observe gradual colourchange
along the reactor. It is believed that polyethylene is
more resistant to the dye than Tygon tubing.
Cap made 
of blue glass
= FLOW IN
F FLOW OUT
Parafilm used as
a seal
Spectronic 20
Cuvette
FIGURE 2. Flowthrough cuvette for the Spectronic 20.
The flow rates of sodium hydroxide and the
dye solution are set such that the ratio is 9:1. One
should start at the highest possible flow rate to
expel all air bubbles from the reactor.
One should wait for slightly longer than the
residence time, for a particular flow rate, for
steadystate to be reached. The reaction mixture
is then allowed to flow through the Spectronic 20
flowthrough cuvettes and the readings are re
corded. Usually, one waits for two minutes and
takes another reading as a duplicate. Experience
has shown that the Spectronic 20 readings are
highly reproducible. Other flow rates of NaOH and
dye solution are chosen, keeping the flow rates
ratio 9:1 as before, and the Spectronic 20 read
ings are recorded. The experiment usually lasts
for one hour provided that the solutions are pre
pared prior to the laboratory period.
TABLE 1
Holding Time and Conversion Data
Holding Exp.
Time Reynolds PFRM LFRM Conversion
min Number conv. conv. x
6.42 791 71.7 63.3 71.5
4.3 1181 57.1 50.3 53.3
3.23 1651 47.1 41.6 44.5
2.54 2000 39.4 34.96 36.9
2.13 2385 34.3 30.7 33.85
CHEMICAL ENGINEERING EDUCATION
Reactor
Valves
Spectrometers
The value of the rate constant, k', is needed
for this experiment. Students are requested to run
a batch experiment to determine the value of k'
at the same temperature of the flow experiment.
This has proven worthwhile, since temperature
fluctuations in most undergraduate laboratories
do not allow conducting a batch experiment at the
beginning of the semester and giving the value
of k to the students to perform the required calcu
lations. Also, students are required to prepare
their own calibration curve of the dye concentra
tion versus absorbance. This leads to better results.
RESULTS AND DISCUSSION
Table 1 reports the residence time and the con
versions from the plugflow reactor model
(PFRM), laminarflow reactor model (LFRM)
and the experimental conversions. Fig. 3, also,
depicts the conversions against the residence
time. The data reported in Table 1 and Fig. 3 were
obtained from an experiment conducted on the
setup available in Windsor.
It is clear from Fig. 3 that, as expected, the
experimental conversions fall between the con
0.7
0.6 +
0.5
S0.4
0.3 II I
2 3 4 5 6 7
HOLDING TIME, 7 (min.)
FIGURE 3. Comparison between conversion obtained
from experiment and those obtained from LFRM and
PFRM.
REQUEST FOR FALL ISSUE PAPERS
Each year CHEMICAL ENGINEERING EDUCATION publishes
a special fall issue devoted to graduate education. This issue
consists 1) of articles on graduate courses and research written
by professors at various universities, and 2) of announcements
placed by ChE departments describing their graduate pro
grams. Anyone interested in contributing to the editorial
content of the fall 1985 issue should write the editor, indicating
the subject of the contribution and the tentative date it can be
submitted. Deadline is June 15th.
versions obtained from the two theoretical models,
viz., the PFRM and LFRM.
It is worthwhile to note here that the data re
ported by Hudgins and Cayrol indicate that the
experimental conversion curve crosses the LFRM
curve at short holding times, i.e., experimental
conversions are lower than those predicted by
LFRM, which is not possible. Such results may be
attributed to the obvious design flaws in the set
up reported by those authors.
The change of colour of the reaction mixture
between the inlet and outlet of the reactor is due
to the conversion along the reactor. Such a visual
effect helps the students to integrate the labora
tory experiment with what they learned in the
lecture part of the course about conversion in
tubular flow reactors. O
REFERENCES
1. Anderson, J. B., "A Chemical Reactor Laboratory for
Undergraduate Instructions," Princeton University,
1968.
2. Hudgins, R. R., and B. Cayrol, "A Simple Tubular
Reactor Experiment," CEE, XV, 1, 26, 1981.
3. Corsaro, G., Chem. Educ., 41, 48, 1964.
4. Holland, C. D. and R. G. Anthony, Fundamentals of
Chemical Reaction Engineering, PrenticeHall, Engle
wood Cliffs, N.J., 1979.
NOTATION
o0
i,e
CA
k'
L
NR
r
(r)
Vo
V
x
r
= subscript symbol for initial
= subscript symbols for reactor inlet and
exit, respectively
= concentration of component A, (mole/L)
= pseudofirst order rate constant, (min1)
= length of reactor tube, (m)
= k' = reaction number for a first order
reaction
= inside radius of reactor tube, (m)
= reaction rate, (mol/L.min)
= volumetric flow rate, (L/min)
= reactor volume, (m3)
= conversion
= V/vo = space time (min)
SPRING 1985
OnI lecture
The B. C. (Before Computers) and A. D. of
EQUILIBRIUMSTAGE OPERATIONS*
J. D. SEADER
University of Utah
Salt Lake City, UT 84112 1
T HE ART OF DISTILLATION and other multi
component, multistage separation operations
has been practiced since antiquity. Although de
scribing equations for distillation were formu
lated before 1900, flexible, efficient, and robust
procedures for solving the equations did not ap
pear in the literature until after the availability
of digital computers beginning in 1951. This
paper is keyed to that date with B.C. referring to
"before computers."
In 58 B.C., Sorel developed the first theoretical
equations for simple, continuous, steadystate
distillation, but they did not find wide application
until 30 B.C., when they were adapted to a rapid
graphical construction technique for binary
systems by Ponchon and then Savarit. This was
quickly followed in 26 B.C. by a much simpler, but
restricted, graphical technique by McCabe and
Thiele. Computer methods have largely replaced
the rigorous PonchonSavarit Method, but the
graphical McCabeThiele Method is so simple and
so illustrative, it continues to be popular.
A modern version of Sorel's equations (shown
in Fig. 1) includes, in the case of a partial con
denser, total and component material balances and
an energy balance around the top section of the
column. Phase equilibrium on each tray is ex
The development of a separation
process can be an exciting experience when
computers and computer programs are available to
perform the tedious calculations and allow
time for more consideration of synthesis
and optimization aspects.
*Tutorial Lecture presented at 92nd ASEE Annual Con
ference, Salt Lake City, Utah, June 2228, 1984.
J. D. Seader has been a faculty member at the University of Utah
since 1966. Prior to that, he was employed by Chevron Research and
Rocketdyne. His principal technical interests are equilibriumstage
operations, process synthesis, process simulation, and synthetic fuels.
He is a Director of the AIChE and was the Annual Institute Lecturer in
1983 for AIChE. He prepared the section on distillation for the sixth
edition of the Chemical Engineers' Handbook and is a Trustee of
CACHE.
pressed in terms of component Kvalues and one
mole fraction sum per stage for either vapor or
liquid is needed. Similar equations are written
for the bottom section of the column and for the
feed stage.
DEGREES OF FREEDOM ANALYSIS
A degrees of freedom analysis for the equations
was first developed by Gilliland and Reed in 9
B.C. A more thorough treatment for all types of
separations and other operations was reported
by Kwauk in 5 A.D.
If the equations and variables are counted,
for a column with N stages (including the con
denser and reboiler) to fractionate a feed with
C components, it is found that the number of
equations is N (2C + 3), while the number of vari
ables is N(2C + 4) + C + 7. Variables include
stage temperatures, pressures, vapor and liquid
flow rates and component mole fractions; feed
Copyright ChE Division, ASEE, 1985
CHEMICAL ENGINEERING EDUCATION
The nature of the equations of Sorel and the difficulty of their solution for
multicomponent systems has long been recognized. The set of equations can be large in
number. For example, with 10 components and 30 equilibrium stages, the equations number 690. Sixty
percent of the equations are nonlinear, making it impossible to solve them directly.
III II
flow rate, composition, temperature and pressure;
reboiler and condenser duties; and number of theo
retical stages above and below the feed. The
thermodynamic properties, K and H, are not
counted as variables because they can be written
explicitly in terms of the other variables just
mentioned. The degrees of freedom or number of
variables that must be specified equals the differ
ence between the number of variables and the
number of equations or N + C + 7.
A simple set of specifications would include
feed flow rate, composition, temperature, and
V, TOPDOWN
I TVN+ t = LN + VI
IN+1 VN+1 Xl,N LN + Y1,1 V]
HVN,1 VN+ L HLN Vi V1 0c
SKI'N YI,/X ,
X1, M 1 = YIM V + X+ N LN
1HLM L0 + O; = HVH V, V HLN LN
KIM= YI,/X,1M
X,N
FEED STAGE
SF + LF1 + VF+1 = L + V
XIF1 Y ZI, F + X,F LF1 YIF+ VF+
F XI,F F ,, YI,, VF
HFF + HLF1 LF1 + F+ VF+1 HLF LF + HV VF
+ I VF KFK = F YI F XIVF
XIF YI ,F+ XFF = 1 F
FIGURE 1. Modern version of Sorel's equations.
pressure; number of trays above the feed and
below the feed; the pressure of each stage; and
the reflux flow rate, L1. This totals N + C + 6,
which is one short of the number of degrees of
freedom. From my own practical experience in
5 A.D., failure to supply the one additional
specification can result in a calculational pro
cedure that will never converge. The additional
specification might be the distillate flow rate V1.
NATURE OF SOREL'S EQUATIONS AND
SPARSITY PATTERNS
The nature of the equations of Sorel and the
difficulty of their solution for multicomponent
systems has long been recognized. The set of
equations can be large in number. For example
with 10 components and 30 equilibrium stages,
the equations number 690. Sixty percent of the
equations are nonlinear, making it impossible to
solve them directly. The magnitude of the values
of the variables can cover an enormous range.
For example, the mole fraction of a very volatile
component at the bottom of the column might be
very small, perhaps 1050. The value of a total
flow rate might be 104.
Commonly used procedures for solving such
sets of equations, as discussed by Henley and
Seader, are iterative in nature, requiring start
ing guesses for some or all of the variables.
Early procedures were complete equationtearing
methods, suitable for manual calculations, wherein
the equations were solved oneatatime in a se
quential manner. With the advent of the digital
computer, partial tearing methods appeared,
wherein small groups of equations as well as
single equations were solved at a time. Most re
cently, with the availability of larger and faster
digital computers, very flexible simultaneous
correction methods appeared wherein all the equa
tions were solved simultaneously by a modified
Newton's method.
An additional characteristic of Sorel's set of
equations is sparsity. That is, no one equation
contains more than a small percentage of the
variables. For example, for the case of 10 com
ponents and 30 stages, no equation contains even
SPRING 1985
7 percent of the variables. This sparsity is due to
the fact that each stage is only directly connected
to, at most, two adjacent stages.
The nature of this sparsity has been exploited
in the development of the abovecited methods by
seeking certain sparsity patterns which are best
observed by the use of incidence matrices. The
rows of the incidence matrix represent the differ
ent functions or equations being solved. The
columns of the matrix represent the different vari
ables contained in the equations. Thus, for N =
30 and C = 10, the matrix is of size 690 by 690.
If a certain equation contains a certain variable,
some nonzero entry, such as an X, is placed in
the matrix at the corresponding location. Other
REARRANGEMENT OF A SPARSE MATRIX
TO OBTAIN A MORE DESIRABLE PATTERN
12345678
C X X XX
X X X XX
X X X X X X
xx
x x x
x x
UNDESIRABLE PATTERN
5 7 1 3 8 4 2 6
57138426
6 X
8 X X
4 X X
7 X X X
5 X X X X
2 X X X X
I X X X X X
3 X X X X X X
DESIRABLE PATTERN
(LOWER TRIANGULAR)
FIGURE 2. Incidence matrices.
wise a zero or no entry is made.
The sparsity pattern depends upon the order
in which the columns and rows of the matrix are
arranged. The arrangement shown at the left in
Fig. 2 appears to be random without pattern. How
ever, by interchanging certain columns and rows,
the lower triangular pattern shown at the right
is obtained. Such an organized pattern, if it can be
achieved, is highly desirable because it indicates
that the equations can be solved oneatatime
starting with the equation for the first or top row,
solving for the only unknown, and then proceed
ing down the rows, equationbyequation, solving
for one unknown atatime, but where necessary,
using values of previously computed variables.
Since at least 10 B.C., a number of other de
sirable sparse matrix patterns have been recog
nized. Shown at the left in Fig. 3 is a block
diagonal pattern. The nonzero entries are all
contained within the interior boldlined region.
Shown in the middle is a banded matrix, where
BLOCK DIAGONAL
BANDED
BLOCKED AND BORDERED
FIGURE 3. Desirable sparse matrix patterns.
all nonzero entries are contained on the main
diagonal and a few adjacent diagonals. Shown at
the right is a blockedandbordered matrix. Such
organized sparse matrix patterns, when they exist,
can be found readily by computer algorithms
such as the MA28 subroutine of the Harwell li
brary.
EQUATIONTEARING STRATEGIES
For the four organized patterns just dis
cussed, specialized sparsepattern computer al
gorithms have been developed to solve, in an
efficient manner, linearized forms of the nonlinear
equations that describe the system. These sparse
matrix methods strive to: (1) eliminate storage
of zero coefficients and certain repetitious nonzero
elements, (2) reduce arithmetic operations, in par
ticular those involving zeros, and (3) maintain
sparsity during computations.
Lessorganized sparsematrix structures can
sometimes utilize organized sparsematrix methods
in an iterative manner by employing equation
tearing strategies. The structure shown in Fig. 4
is lower triangular, except for an additional non
1
2
3
4
5
EQUATION 6
7
8
9
10
11
VARIABLE
1 2 3 4 5 6 7 8 9 10 11
EQUATIONSOLVING
ORDER:
1
10
11
Qy TEAR VARIABLE
 BORDER OF INNER CYCLE
FIGURE 4. Application of tearing to a sparse matrix.
CHEMICAL ENGINEERING EDUCATION
I N\ \ 
zero entry at column 8 in row 3. The linearized
equations cannot be solved directly, oneatatime,
starting with Equation 1, because when Equation
3 is reached, the value of variable 8 is not known;
thus, Equation 3 cannot be solved for variable 3.
A tearing strategy can be employed to overcome
this difficulty, but an iterative calculational loop
or cycle, shown by the dashed border, involving
Equations 3 through 8, is necessary. Variable 8
shown as a circled X, is the single tear variable
which, when given an estimated value, results in
the tearing apart of that subset of equations so
they can be solved individually in order. At Equa
tion 8 in the cycle, variable 3 is calculated and
the value obtained is compared to the value used
in Equation 3. If the two values are sufficiently
close, the cycle is converged and variable 9 in
Equation 9 is computed, followed by solution of
Fr = 2x{l + X2 85 = 0
F1 2x12 35 0
F2o201 2 ,* 035=0
TEARING STRATEGY #1:
1 2
S
2 X X
TEARING STRATEGY #2t
2 1
2 X
GUESSED X1 FROM X2 FROM
ITERATION X2 F1 F2
1 8 110 5,9 x 108
2 5.9 x 108 3.0 x 1034
ITERATION
1
2
3
GUESSED
500
4.029936
4.000010
X2 FROM
6.346546
8.999170
9.000000
X1 FROM
4.029936
4.000010
4.000000
BETTER RESULTS WITH THIS STRATEGY BECAUSE F1 IS RELATIVELY
SENSITIVE TO X2 BUT NOT TO X0v WHILE THE OPPOSITE IS TRUE FOR F2
FIGURE 5. Sensitivity of the tearing method.
Equations 10 and 11 to complete the system.
Otherwise, a new approximation for the tear
variable must be determined and another iteration
of the cycle completed.
For the tearing strategy to be successful and
efficient, it is necessary that Equation 3 not be
sensitive to the assumed value of Variable 8. If
too sensitive, it is best, if possible, to reorder the
equations and variables to obtain a less sensitive
situation. As a simple example of this sensitivity,
consider the two equations shown in Fig. 5. If x,
is the tear variable and Equation 1 is solved first,
SUCCESSIVE SUBSTITUTION
BOUNDED WEGSTEIN (16 A.D.)
1 2 3 4 5 6
x x
x x[
BEFORE PATITIONI
BEFORE PARTITIONING
DELAYED WEGSTEIN (28 A.D.)
DOMINANT EIGENVALUE (20 A.D.)
1 3 5 2 4 6
AFTER PARTITIONING
FIGURE 6. Partitioning when convergence acceleration
methods ignore interactions among the variables.
convergence is impossible to achieve from any
starting guess of the tear variable. For example,
if the initial guess for x2 is 8, the sequence ob
tained quickly diverges, as shown. After only one
iteration, x, has increased in value to 5.9108,
which is far from the solution.
Alternatively, in tearing strategy #2, where
the two columns of the matrix are interchanged
to make x, the tear variable, convergence is readily
achieved, as shown, from any initial guess, even
x, = 500. The solution x, = 4, x2 = 9 is obtained
in just three iterations. These two drastically
different results are obtained because Equation 1
is very sensitive to the value of x, and almost in
sensitive to the value of x,. Thus, in this example,
x, should not be the tear variable when Equation
1 is solved before Equation 2.
In the example just considered, the guess for
x,, at the beginning of each iteration is set equal
to the value computed from Equation 2 in the
previous iteration. This procedure, called succes
sive substitution, can be slow to converge, and,
therefore, a number of some simple and some
complex procedures have been developed to ac
celerate convergence. These procedures are par
ticularly useful when more than one tear variable
must be used as in Fig. 6. The example at the left
has two tear variables. Some methods, including
successive substitution, bounded Wegstein, de
layed Wegstein, and dominant eigenvalue, ignore
interactions among the variables. When inter
actions do not exist, it should be possible to inter
change columns and rows of the matrix to obtain
SPRING 1985
Chemical engineering educators
need to closely examine courses on separation
processes to make sure that students are
being instructed in modern and
efficient computational tools.
NEWTON
BROYDEN (18 A.D.)
1 2 3 4 5 6
1 TI (X)
2 xl
2 X 0
I
5 X XX
6 X X X X
FIGURE 7. Convergence acceleration methods that ac
count for interactions among the variables.
a block diagonal structure, shown at the right
of Fig. 6, which allows separate computations of
the two individual partitions, each containing just
a single tear variable.
The more common case is when interactions
among the tear variables exist, as shown in Fig.
7, where the two tear variables are 5 and 6. The
iterative cycle includes all equations. Then, par
titioning is not possible and, if the interactions
are strong enough, convergence acceleration by
Newton's method or a quasiNewton method, such
as that of Broyden, may be desirable.
SIMPLE AND COMPLEX SEPARATION OPERATIONS
The nature of the sparsematrix pattern ob
tained from Sorel's equations and the correspond
ing calculational procedure depends on a number
of factors, including: (1) selection of the work
ing equations, (2) selection of the variables, (3)
degree of flexibility in the specifications, (4)
order of arrangement of the equations, (5) order
of arrangement of the variables, (6) functionality
of the physical properties, and (7) method by
which any equations are linearized.
An additional factor that influences the nature
of the sparse matrix pattern of Sorel's equations
is the type of separation operation. In simple
distillation, a single feed is separated into two
products, a distillate and a bottoms; energy re
quired to separate the species is added in the form
of heat by a reboiler at the bottom of the column
where the temperature is highest. Also, heat is
removed by a condenser at the top of the column
where the temperature is lowest. This frequently
results in a large energy input requirement and
low overall thermodynamic efficiency, which was
of little concern (except for cryogenic and high
temperature processes) before 22 A.D. when
energy costs were low. With recent dramatic in
creases in energy costs, complex distillation opera
tions (described by Seader in the 6th edition of
Perry's Chemical Engineers' Handbook) and sys
tems are being explored that offer higher thermo
dynamic efficiency and lower energy input re
quirements. Complex columns and systems may
involve multiple feeds, sidestreams, intermediate
heat transfer, multiple columns that may be inter
linked, and in some cases, all or a portion of the
energy input as shaft work.
COMPLETE TEARING METHOD
Simple and complex distillation operations
have two things in common: (1) both rectifying
and stripping sections are provided such that a
separation can be achieved between two compon
ents that are adjacent in volatility, and (2) the
separation is effected only by the addition and
removal of energy and not by the addition of any
mass separating agent (MSA) such as in liquid
DISTILLATE (YB = 0.75)
V1 = 50 (YT 0.25)
L "10
FEED
BUBBLEPOINT LIQUID
F = 100
X( = 0.5
X, = 0.5
(L2 = 10)
(T3 = 200'F)
ALL PRESSURES =
1 ATM
L BOTTOMS
FIGURE 8. Example of specifications and tear variables
for top down, bottomup, stagebystage method.
liquid extraction. Sometimes, other related
multiplestage vaporliquid separation operations,
such as refluxed rectification, reboiled stripping,
absorption, stripping, reboiled absorption, re
fluxed stripping, extractive distillation, and azeo
tropic distillation, as described by Seader, may be
more suitable than distillation for the specified
task. All these separation operations can be re
ferred to as distillationtype separations because
they have much in common with respect to calcula
tions of thermodynamics properties, vaporliquid
equilibrium stages, and column sizing. For calcu
CHEMICAL ENGINEERING EDUCATION
lations involving such operations, prior to digital
computers, the factors influencing the nature of
the sparse matrix pattern from Sorel's equations
were chosen so that a complete tearing method
could be employed so the equations could be solved
sequentially oneatatime. Many techniques were
proposed, with the stagebystage methods of
Lewis and Matheson in 19 B.C. and Thiele and
Geddes in 18 B.C. being the most useful. In 6
A.D., features of these two methods were com
bined into a single method, applicable to computa
tions with a digital computer. Specifications are
the simple set discussed previously and illustrated
in Fig. 8 by an example involving two compon
ents, benzene and toluene, and five theoretical
stages. The tear variables (2C + Nl) in number,
and typical initial guesses for them are shown in
parentheses. These tear variables include the com
ponent mole fractions in the distillate.
The initial guesses for the distillate composi
tion are conveniently obtained by using a rear
rangement of the Fenske equation with the mini
mum number of equilibrium stages set equal to
onehalf of the total number of specified equilib
rium stages. The sum of the component flow rates
in the distillate must equal the specified total dis
tillate flowrate and, for each component, the sum
of the flow rates in the distillate and bottoms must
equal the feed rate.
The equations used are modifications of Sorel's
equations, and include total material balances,
component material balances, bubblepoints, dew
points, energy balances, an adiabatic flash for the
feed stage, and equations for reestimating distil
late mole fractions.
The incidence matrix, shown in Fig. 9, is lower
triangular except for the six tear variables, which
are represented as circled X's. They appear in
vertical columns in the order L2, Va, V4, Ta, Y1,1,
and Y2,1. The latter are the assumed distillate mole
fractions. The variables across the top and the
equations corresponding to the rows are ordered
by stage number, as shown, where the stages are
numbered from the top down, with 1 as the partial
condenser and 5 as the partial reboiler. The calcu
lations begin in the upper left corner and proceed
down the diagonal. At the upper left corner, the
first two equations, which each contain only a
single unknown, are solved directly for the bottoms
rate L,, and the toptray vapor rate V2. All remain
ing equations are enclosed within the dashedline
border, which contains all six tear variables. This
large, squared region is the outer loop that con
STAGE CALCULATION:
j 1 1 2 5
i 4 1 3 1
x x 
xxx 42
I X X X _
x xxx x
I X X X 2 I
xx XI
1I x I1
I xxx x x x 1
Sxx
x x x x xx x xL132_
XI X X X X XT
/ X X XX V5
x X x 1, 1
Xx xix Xx 11
x x
x x xx x xxLxXx 3
x l X X I X
I xx x x xx x x x x I
I xx xx xxxx
o Tear Variable
 Border of Inner Cycle
! Nonlinear Equatlon
FIGURE 9. Incidence matrix for topdown, bottomup,
stagebystage method (6 A.D.).
tains 40 equations to converge.
The matrix is 42 x 42, rather than 35 x 35
(calculated from N[2C + 3]) because the feed
stage temperature is computed with three differ
ent equations, and V, and the vapor and liquid
mole fractions for the feed stage are computed
with two different equations.
Calculations for the outer loopinner cycle
begin with stage 1, the partial condenser. All cal
culations involve just linear equations in the case
of compositionindependent properties, except for
T2, which is computed iteratively from a non
linear dewpoint equation. Variables computed
from nonlinear equations are boxed. Calculations
for stage 2 are completed next. Seven equations
are involved, with the last five contained in a
dashed inner loop, containing a single tear vari
able, L,. At this step in the calculations, the stage
above the feed stage has been completed and the
calculation steps are now transferred to stage 5,
the partial reboiler. Here, two nonlinear bubble
point equations are encountered for T, and T4, and
another tear variable, V,, is contained in a 5 x 5
matrix. Similar calculations are made next for
stage 4. Finally, the feed stage (3) is computed
by making an adiabatic flash calculation. The last
two equations at the lowerright corner are used
to compute a new estimate of distillate composition
by comparing the feed flash conditions with those
computed from the topdown and bottomup stage
SPRING 1985
The method works best for feeds of narrowboilingrang e components. Otherwise, estimates of
distillate composition may be too uncertain and cause difficulty in convergence. If feeds are wideboiling,
the bubble and dewpoint calculations become sensitive and convergence is jeopardized.
calculations, with an adjustment made to main
tain the specified distillate rate.
In all, four inner loops are contained within
one major outer loop. Thirteen equations are in
fluenced directly by the guesses for the distillate
composition and ten others are influenced in
directly by the corresponding bottoms mole
fractions. Thus, although the complete tearing
method is suitable for manual calculations, the
method is relatively inefficient and limited to in
sensitive cases of simple distillation of nearly ideal
solutions with low reflux ratios.
The method works best for feeds of narrow
boilingrange components. Otherwise, estimates of
distillate composition may be too uncertain and
cause difficulty in convergence. If feeds are wide
boiling, the bubble and dewpoint calculations be
come sensitive and convergence is jeopardized. In
any event, convergence may be slow, unless special
acceleration techniques are used. However, the
calculation by hand of just a few stages for a
multicomponent mixture by this method is a very
worthwhile learning experience; one not to be
missed.
EQUILIBRIUM FLASH METHOD
Some of the limitations of the previous method
were eliminated by McNeil and Motard (23 A.D.)
in their development of a stagebystage algorithm
that utilizes adiabatic or percent vaporization
flash calculations. Their technique requires
[(N1) (2C + 2) 2] tear variables, but, as shown
in Fig. 10, initial guesses for all but (N2) of
these variables can be set to zero. The (N2) vari
ables are interior stage temperatures, which are
relatively easy to estimate.
If the feed is predominantly liquid, the pro
cedure starts with an adiabatic flash at the feed
stage followed by stagebystage adiabatic flashes
in a downward direction until the partial reboiler
is reached, where a percent vaporization flash is
made. Subsequent adiabatic flashes are made
moving up the column until the partial condenser
is reached and another percent vaporization flash
is made. Additional sequences of flash calcula
tions are made moving down, and then up, the
column until convergence is achieved.
The method is not particularly suitable for
manual calculations because adiabaticflash compu
tations are tedious. However, flash computer sub
routines are readily available, and it is relatively
easy to construct an executive routine to apply
the method. It is, therefore, another very worth
while learning exercise, particularly because
column startup is simulated. The method is ap
plicable to complex distillation, and initial guesses
for the tear variables are easily obtained from
vapor pressure data. The flash calculations are
usually not sensitive, but convergence, although
DOWN AND UP STARTING FROM FEED STAGE
FEED
BUBBLEPOINT LIQUID
F = 100
XB = 0.5
XT = 0.5
DISTILLATE
V = 50
*C
L = 10
(T2 = 200"F)
(L2 = 0)
(T3 = 200"F)
(T4 = 200*F)
(T5 = 200'F)
i BOTTOMS
FIGURE 10. Example of specifications and tear vari
ables for equilibriumflash method of McNeil and
Motard (23 A.D.)..
almost certain, can be very slow, particularly for
large ratios of internal traffictofeed flow rate.
MATRIX METHODS
Rather than use an equationbyequation com
plete tearing technique for Sorel's equations,
Amundson and Pontinen in 7 A.D., in a significant
development, showed how the equations could be
combined in a manner such that they could be
solved in the order of type of variable, rather than
by stage. However, only partial tearing was
achieved and the method involved solving C sets
of N x N simultaneous linear algebraic equations.
CHEMICAL ENGINEERING EDUCATION
To do this, they used full matrix inversion, which
often led to computational difficulties. These diffi
culties were overcome by taking advantage of the
sparse tridiagonal form of the matrices and apply
ing Gaussian elimination or LU decomposition in
EQUATIONS AND VARIABLES ORDERED BY TYPE
FEED
BUBBLEPOINT LIQUID
F = 100
XB = 0,5
X, = 0.5
DISTILLATE
S V1 = 50
(Ti = 170F)
= 10
(T2 = 185F)
(T3 = 200F)
(T4 = 215'F)
(T5 = 230F)
I BOTTOMS
FIGURE 11. Example of specifications and tear variables
for bubblepoint method of Wang and Henke (15 A.D.).
the manner of Thomas. In 15 A.D. Wang and
Henke applied the Thomas algorithm to narrow
boiling feeds, while Burningham and Otto re
formulated some of the equations in 16 A.D., fol
lowing the work of Sujata in 10 A.D., to apply
them to wideboiling feeds typical of absorbers and
strippers. The need for two such partialtearing
methods was shown clearly in 13 A.D. by Friday
and Smith, who referred to the two procedures
as the bubblepoint and sumrates methods.
An N x N tridiagonal equation for each com
ponent is formed by combining the component
material balance, phase equilibrium, and a total
material balance to form an equation in liquid
phase mole fractions, stage temperatures, and
vapor flow rates. By choosing the temperatures
and vapor flow rates as tear variables, the equa
tions become linear in the mole fractions, with no
more than three mole fractions contained in any
one equation, because one stage is connected to
no more than two adjacent stages. For each com
ponent, the linear equations are ordered by stage.
The result is a tridiagonal matrix equation, where
the nonzero coefficients are contained only on the
three principal diagonals. The solution of the
matrix equation is easily achieved by Gaussian
elimination, as shown e.g. by Carnahan, Luther,
and Wilkes, in no more than 20 lines of FORTRAN
code.
BUBBLEPOINT METHOD
The bubblepoint method of Wang and Henke
in 15 A.D. utilized the tridiagonal matrix al
gorithm to obtain a computer method for solving
distillation problems for relatively narrowboiling
feeds. The specifications for the same 5stage, 2
component example used above are shown in Fig.
11. The tear variables are the stage temperatures
and vapor flow rates. The distillate rate V1 and
the reflux rate L1 are specified. Initial guesses for
these tear variables are obtained with a minimum
of effort by assuming constant molar overflow, in
the manner of the McCabeThiele method. Esti
mates of the stage temperatures are obtained by
linear interpolation of the distillate and bottoms
temperatures, which may be computed by dew
point and bubblepoint calculations assuming the
most perfect split of the feed components, con
sistent with the specified distillate rate.
Wang and Henke utilized a form of Sorel's
equations that permits the solution by variable
type rather than by stage as in the first two
methods described. The equations include a total
material balance to compute liquid traffic, a com
ponent material balance combined with phase
TYPE VARIABLE:
Border of inner   Set of Linear Equations
cycle i
FIGURE 12. Incidence matrix for bubblepoint method.
SPRING 1985
Tear Variable
O Nonlinear Equation
BI x xI I T I VT I I I L TI
S i
x x x
X XX x 
x x xx
x x x
S x xx I
X X I
X XX X
x x xx x
x x x x I
x x
XX X X
X X XX
I X X X
X X
X X X *X X
I X X XX X X
X X X I
X X I
x x
XX X XX X XX XX
equilibrium and total material balance to compute
liquidphase mole fractions, bubblepoint equa
tions to obtain stage temperatures and vaporphase
mole fractions, and energy balances to compute
vapor traffic.
Although the same variables are computed,
starting from Sorel's equations, the incidence
matrix, shown in Fig. 12, is quite different from
the stagebystage method. The matrix is lower
triangular except for the circled tear variables and
N x N (5 x 5 in this case) block sub matrices for
each component (two in this case). The overall
size of the matrix for the example is only 35 x 35
because no variable is computed from more than
one equation. At the upper left corner, the first
two variables are the same as before and are com
puted directly as before. All but the last two of
the remaining equations are contained in one large
loop. Then the two tridiagonal submatrices are
solved separately to obtain the liquidphase mole
fractions. All remaining variables are computed
oneatatime. Stage temperatures are computed
from the nonlinear bubblepoint equation. This is
followed by computation of vaporphase mole
fractions. Energy balances give the vapor traffic
and total material balances give the liquid traffic.
The cycle is repeated until the tear variables are
converged. Then the duties of the reboiler and con
denser are computed.
For narrowboiling feeds, the initial estimate
of stage temperatures and vapor traffic will be
EQUATIONS AND VARIABLES ORDERED BY STAGE
FEED
BUBBLEPOINT LIQUID
FB = 50
FT = 50
DISTILLATE
T = 12.5
(T1 = 170'F)
(T2 = 185'F)
(T3 = 200'F)
(T4 = 215"F)
(T5 = 230'F)
. BOTTOMS
BB = 12.5
FIGURE 13. Example of specifications and starting
guesses for simultaneouscorrection method of Naphtali
and Sandholm (20 A.D.).
quite close to the final result and convergence is
rapid using successive substitution for the tear
variables. For widerboiling feeds, convergence is
quite sensitive to the initial estimates of T and V
and may not be rapid. In that event, use of a de
layed Wegstein [Orbach and Crowe (20 A.D.)] or
dominant eigenvalue technique [Rosen (29 A.D.)],
rather than successive substitution, can reduce the
number of iterations required.
The bubblepoint method is not suitable for
manual calculations because of the need to solve
matrix equations. However, it is easily pro
grammed if algorithms are available for solving
single nonlinear equations and the tridiagonal
matrix equation. The method is applicable to simple
and complex distillation columns. Convergence
may not be possible if the liquid phase is highly
nonideal. The method provides no flexibility in
specifications. The user must specify the reflux
rate or ratio and the total distillate rate; however,
these specifications almost always permit a real,
positive solution. An exception can occur where
the reflux rate is too small, such that it diminishes
to zero at some stage down from the top. The
bubblepoint method can be successfully applied
to complex distillation e.g. two feeds, two side
streams, and one intercooler. Such problems are
difficult for stagebystage tearing algorithms, but
are relatively easy for partial tearing algorithms
like the bubblepoint method, where variables are
computed by type.
NEWTON'S METHOD
More advanced computer methods that can
handle a complete range of boilingpoint of feed
components and nonideal liquid solutions, as well
as offer more flexibility in problem specifications,
involve handling the nonlinear equations simul
taneously without the use of tear variables.
Newton's method, and variants thereof, has long
been the popular technique. The use of a simple
twoequation manual exercise is sufficient to il
lustrate to students the basic concept of Newton's
*method, which may then be applied with com
puter programs to hundreds of equations when
solving a multicomponent, multistage separation
problem.
Computer methods that solve all of Sorel's dis
tillation equations simultaneously may be referred
to as simultaneouscorrection methods. Many such
methods have been proposed and the Newtonbased
NaphtaliSandholm technique of 20 A.D. is repre
CHEMICAL ENGINEERING EDUCATION
sentative of one of the better ones. The equations
and variables are ordered by stage. To reduce the
size of the matrix to be handled, component flow
rates replace mole fractions and total flow rates.
Thus, for the fivestage, twocomponent example,
shown in Fig. 13, the number of equations to be
solved is 25, rather than the 35 previously. The
specifications are different from previous ones
in that distillate and bottoms purities replace re
flux and distillate rates. Such specifications should
be used with caution and the Fenske (19 A.D.)
minimumstage equation should be checked to
make sure that the minimum number of required
stages for the specified purities is less than the
specified number of five. Theoretically, initial
guesses must be provided for all 25 unknowns, but
these guesses can be generated by the program
based on guesses of just a few temperatures and
vapor rates as shown. These guesses are called
tear variables here, but are really not tear vari
ables in the strict sense. The initial guesses are
generated easily from the T and V guesses by
solving the tridiagonal matrix equations of the
WangHanke method for the liquidphase mole
fractions, from which the initial guesses for the
component flow rates are obtained readily from
their definition and the component material
balances.
The NaphtaliSandholm method only involves
three types of equations, namely stage component
material balances, phase equilibrium in terms of
Kvalues, and stage energy balances. The form of
the equations is almost identical to the original
equations of Sorel. Because bubblepoint, dew
point and flash calculations are absent, sensitivity
problems are largely avoided.
With equations and variables ordered by stage,
the incidence matrix, shown in Fig. 14, is block
tridiagonal in shape. The blocks are 5 x 5 in this
example. The matrix is for the linearized form of
the equations, which permits the application of
Newton's method. Thus, the matrix is the
Jacobian of partial derivatives and an X entry
signifies a nonzeroderivative. The entire matrix
is iterated to convergence.
The solution of the block tridiagonal matrix
is obtained readily by modifying the previously
mentioned Thomas algorithm for a tridiagonal
matrix. The only significant changes are the re
placement of matrix multiplication for scalar
multiplication and matrix inversion and multipli
cation for division. If large numbers of components
are present, the (2C + 1) x (2C + 1) submatrices
may be large and timeconsuming to invert. The
convergence criterion is based on the sum of the
squares or socalled square of the Euclidean norm
of the three different types of functions. Early
iterations are often damped to avoid corrections
that are too large.
Because of the block nature of the matrices in
the NaphtaliSandholm method, it is not at all
suitable for manual calculations. Furthermore, the
computer program is rather complex. Consequent
ly, it is best to obtain the code from one of several
STAGE CALCULATION:
1 2 3 5
X XXX
X X X
X X X
x x xx
x x x
x x x x
xxxxxxx x
xxx x x xx X X
XXXXXX
X X X X
X X X X X
XXXX_ ___
xxxxxxxxXXX
X X X X
X X X X
xxx xx
xx xx
XXX X
x x x x x x xx x
X X X X
XXXXX
x x x x x
x x x x x
FIGURE 14. Incidence matrix for simultaneouscorrec
tion method.
sources, e.g. Fredenslund et al (26 A.D.). Versions
ranging from PC to Mainframe codes are avail
able. The method is applicable to all singlecolumn,
complex multistage operations, including those
with highly nonideal liquid solutions. Flexibility
in specifications is provided at the top and bottom
of the column by substituting specification equa
tions for the condenser or reboiler energy balances.
For example, specification options at the top in
clude condenser duty, reflux rate, reflux ratio, dis
tillate rate, component molefraction purity and
component distillate rate. Convergence is rapid
from good initial guesses, but may otherwise be
slow and require damped corrections. The method
can fail, particularly if initial guesses are very
poor.
CONTINUATION METHOD AND MULTIPLE
STEADYSTATE SOLUTIONS
Ideally, especially in practice, one would prefer
SPRING 1985
a multistage, multicomponent separation com
puter method that would offer complete flexibility
in specifications and would always converge to
a correct solution. Newton's method and most of
its variants are known to be only locally con
vergent. That is, the initial guesses must be within
a certain region of the variable space or con
vergence will not be achieved. This region can be
expanded by adjusting the Jacobian in Newton's
method or employing a hybrid method such as that
of Powell (19 A.D.) or Marquardt (12 A.D.),
Equations:
20 x + x2 = 17
I(8X) 3 + x2=
1 H omotopy
x2 Path
FIGURE 15. Example of regions of convergence for
some methods of solving nonlinear equations.
which combine the best features of Newton's
method with steepest descent. To achieve complete
robustness, however, it is necessary to employ a
globally convergent technique, such as differential
homotopy continuation, for which four algorithms,
in FORTRAN, have become available starting in
25 A.D. The one by Kubicek is particularly easy to
understand and apply, but is not written for sparse
matrices.
The regions of convergence for a simple two
nonlinearequation example are shown in Fig. 15.
Newton's method will converge to X1 = 1 and
X, = 4 from an initial guess of X1 = 2, X2 = 5.
As seen, another root exists at X, = 4.07 and X2 =
0.65, which can be reached by Newton's method
from a nearby initial guess. With Newton's
method, the initial guesses must lie within the
rather narrow and confined crosshatched regions.
With Powell's hybrid method (which is available
in the Harwell library, the IMSL library, and
MINPACK) the region of convergence is expanded
outward to the dashed lines. Both methods will
fail badly from a starting guess of 15 and 15. The
use of differential homotopy continuation gives
convergence from any starting guess, with a typi
cal homotopy path to one of the two roots, shown
as a dashdot line starting from (15, 15).
Many types of homotopy paths have been pro
posed, with the linear homotopy being common.
The linear homotopy, h, is set equal to the function
to be solved, f(X), multiplied by a homotopy pa
rameter, t, and added to a function g(X), whose
solution is known, multiplied by the function
(1 t). The calculations start from the known
solution at t = 0, where h = g and X = Xo and
move along the path of h vs. t as t is gradually in
creased to a value of one, at which point h = f,
whose solution X* is to be determined.
Choices for the function g(X), with a known
solution, Xo, are almost unlimited. For consistency
with Newton's method, the Newton homotopy is
useful, where g(X) is set equal to f(X) f(Xo),
where Xo can be selected arbitrarily. With this
homotopy, h equals f(X) minus (1 t) times
f(Xo). Alternatively, (1 t) can be replaced by a
new homotopy parameter, X, to obtain a slightly
more compact form for the homotopy expression.
The path will then be from X = 1 to X = 0.
If the homotopy path is simple, without turns
or rapid changes in X with t, classical continuation
can be employed by selecting a sequence of values
of t at 0, t,, t2, t, etc., and 1, with X being solved
from h at each step by Newton's method using an
initial guess equal to the solution from the pre
vious step. Thus, Newton's method is embedded
into classical continuation. This technique of using
continuation is not common though because it is
not globally convergent and can not, in general,
solve problems that fail with Newton's method
alone, which amounts to moving in one step from
t = 0 to t = 1. To be robust, one must closely
follow the homotopy path and not just continually
take steps in t with Newton corrections in Xspace.
For example, classical continuation as well as
Newton's method will fail on the cubic equation,
x3 30x2 + 280x 860 = 0, because of two
singular points at about x = 7.418 and 12.582,
where the derivative of the function becomes zero.
CHEMICAL ENGINEERING EDUCATION
C1 40
C2 = 30
C 3= 30
R 3
HP 30
L_. ) Stage 19
FIGURE 16. Example of specifications for an interlinked
system of Petlyuk towers.
For initial guesses of x less than about 12.6,
Newton's method fails to find the single real root
at about 15.55. The homotopy path for this cubic
function depends on xo the choice of g(x). For
g(x) = x x0 or g(x) = f(x) f(xo), the two
singularity points become turning points in the
path, and it is important that the continuation
method follow the path closely around these turn
ing points to avoid cycling in the manner of Moses
in the Sinai.
Consider the application of the homotopycon
tinuation method to the Petlyuk system of two
interlinked towers shown in Fig. 16. The feed is a
ternary mixture, which is to be separated into
three products. A sloppy split is made in the pre
fractionator, with the final three products being
produced in the second tower. Reflux and boilup
for the first tower are provided by the second
tower. The two towers in the system can be solved
by alternating back and forth between separate
iterations on the individual towers. But numerous
studies have shown that it is more efficient to con
verge the two towers simultaneously. The stages
for the two towers are ordered as shown starting
at the top of the second tower, switching to the
first tower after stage 11, and then switching back
to the second tower after stage 15. Specifications
include interlink flows from stage 4 to stage 12
and from stage 16 to stage 15; the reflux ratio;
the middle product flow rate; and the bottoms flow
rate.
The types of equations solved are as in the
NaphtaliSandholm method except that balances
include interlink flows and provision, if desired,
for entrainment of liquid droplets, occlusion of
vapor bubbles, and chemical reaction. Phase
equilibrium equations can include a Murphree
plate efficiency that can be specified by component
and tray location.
A FORTRAN computer code for applying
differential homotopycontinuation to such a prob
lem was reported by Wayburn and Seader in 32
A.D. Considerable flexibility in specifications is
provided, including at any stage, total flow rates
or ratios, component flow rates or purities, and
stage temperatures or heat transfer rates. The
equations are linearized in the Newton manner
and ordered by stage to a bordered, blockdiagonal
structure, which is processed by an efficient and
stable blockrowreduction algorithm. An attempt
is first made to solve the equations by Newton's
method, using a line search on the Euclidean norm
of the function residuals to determine the best
damping factor. If Newton's method fails, then
differential homotopy continuation with a linear
f(x) xf(x) = 0
IVP: df(x) dx d )
1 a x fdx dp ) 
I dx, + x (x
IC: =0 ) = 1, x =
1C: p= 0, Az 1, x = x0
FIGURE 17.
tinuation.
Equations for differential homotopy con
Newton homotopy is employed.
The differential form of homotopy continuation
was first proposed by Davidenko in 2 A.D. As
shown in Fig. 17, the homotopy function, f(X) 
Xf(Xo), is differentiated with respect to arc
length, p (i.e., distance along the path), to con
vert a system of M nonlinear equations to a system
SPRING 1985
of M + 1 ordinary differential equations that
constitute an initial value problem. Because of the
extra variable, p, an additional equation is need
ed. This is provided by the Pythagorean theorem
applied in (M + 1)dimensional space. Fortunate
ly, the set of differential equations is not stiff.
Rather than simply integrating the differential
equations by, say, a RungeKutta method, it is pre
NEARLY BLOCK TRIDIAGONAL AND BORDERED FORM
S2 3 4 5 6 7 8 910111213141516171819
1 BC
2 ABC
3 ABC
4 ABC C
5 ABC
6 ABC
7 ABC
8 ABC
9 ABC
10 ABC
II A C
12 A BC
13 ABC
14 ABC
15 ABC
16 A ABC
17 ABC
18 ABC
19 AB
s4
FIGURE 18a. Example of incidence matrix for homotopy
continuation method.
ferable and more efficient to follow the homotopy
path by alternating between an Euler predictor
for the differential equations and two Newtonstep
correctors for the nonlinear homotopy equations.
The Euler step moves the variables somewhat off
the path, but the Newton step corrects the vari
ables back sufficiently close to the path. An im
portant aspect of such a technique is the size of the
Euler step, for which a number of stepsize al
gorithms have been proposed, as discussed by
Seader and Wayburn.
The incidence matrix for the Petlyuk column
example is in Fig. 18a, where the organization is
by stage. Each letter, A, B, or C actually repre
sents, in this example, a nonzero 7 x 7 submatrix,
which applies to both the coefficients of the differ
ential homotopy equations and the Jacobian
partial differentials of the nonlinear homotopy
equations. The matrix is almost block tridiagonal
with bottom and rightside borders, which contain
the nonstandard specifications. The four disperse
submatrices of A and C, located above and below
the three principal diagonals represent the inter
links.
By moving the number 4 and 16 rows and
columns, which contain the disperse submatrices,
to the borders, the block diagonal and bordered
matrix form, shown in Fig. 18b, is obtained. Solu
tion of the corresponding matrix is achieved block
by block, starting at the upper lefthand corner,
by a blockrow reduction algorithm, which treats
the righthand border as part of the righthand
side vector.
The differential homotopycontinuation method
has been applied to the interlinked system shown
in Fig. 19 for a ternary aromatic system, over a
range of reflux ratios from 4.55 to 5.75. Purity
specifications of between 90 and 95 mole% are
made for each product, and a bottoms rate of 380
is specified. The program must compute the re
quired interlink flow rates, including L, the liquid
interlink recycle from the second column back to
the first column. In some cases, Newton's method
converged, for this system, while in other cases,
the differentialhomotopycontinuation method had
BLOCKED AND BORDERED FORM
I 2 3 5 6 7 8 91011121314151718194 16
I BC
2 ABC
3 AB C
5 BC A
6 ABC
7 ABC
8 ABC
9 ABC
10 ABC
II AB C
12 BC A
13 ABC
14 ABC
15 AB C
17 BC A
18 ABC
19 AB
4 AIC C B
16 A AC
FIGURE 18b. Permuted incidence matrix for homotopy
continuation method.
CHEMICAL ENGINEERING EDUCATION
to be applied to obtain a solution.
In Fig. 20, a plot of L, the liquid interlink rate
versus the reflux ratio, shows unexpected multiple
solutions, three in number. For example, at a re
flux ratio of 5, the specifications were achieved
with three different liquid interlink rates of about
110, 330, and 420 lbmoles/hr. Such multiple solu
tions have long been known to exist for certain
cases of an adiabatic reaction in a CSTR reactor,
but have not been observed previously for distilla
tion. When such solutions are close together, as for
solutions 1 and 2 at low reflux ratios, possible
control problems could arise.
BubblePoint
Lqulid Fed
nzee (B) zo00O
Toluen (T) 200
o*Xyler. {X 400
XB = 0.95
5 o 5.75
X = 0.95
B = 380
FIGURE 19. Example that gave multiple solutions.
The continuation method is not at all suitable
for manual calculations. The computer code is
lengthy, but is applicable to all kinds of complex
multistage operations, including interlinked
columns. Except for tray numbers, complete flexi
bility in specifications is permitted. When the
homotopy is constructed properly, convergence is
always achieved. The method is best suited for
cases where the NaphtaliSandholm fails or can't
be applied. The method can find multiple solutions
if they exist.
TRANSPORT MODEL
Sorel's equilibriumstagemodel of almost 100
years ago has served us well in the calculation of
multicomponent, multistage separation opera
tions. However, that model has always been sus
pect for applications to systems of known moder
S07
.
4. 5 4. 9 5. 1 5. 3 5. 5 7
REFLUX RATIO
FIGURE 20. Multiple solutions to Petlyuk towers.
atetolow stage efficiency. For that reason, some
programs, such as the SC method of Naphtali
and Sandholm and the differentialhomotopycon
tinuation method of Wayburn and Seader in
corporate a Murphree tray efficiency, which ac
counts to some degree for masstransfer effects.
However, the plate efficiencies must be specified,
and heat transfer effects are ignored. A better
approach is to apply a transport model to handle
nonequilibrium directly. Such a model has just
been developed by Krishnamurthy and Taylor,
who account for multicomponent masstransfer
interactions and heat transfer. Their modeling
equations are written separately for the vapor and
liquid phases with coupling by liquid and gas mass
transfer rates, and energy transfer rates. These
transport rates are estimated from carefully
formulated mass and energy transfer coefficients,
applicable to multicomponent systems. For non
interlinked columns, the resulting equations lead
to an incidence matrix that is similar to that of
the NaphtaliSandholm method, for which a solu
tion technique is well established. Krishnamurthy
and Taylor have applied their method, with good
success, to several sets of experimental data from
the operation of small laboratory columns. Data
from commercialsize columns are now being
sought to make further comparisons of predicted
and measured compositions so as to evaluate the
usefulness and applicability of this transport
model.
CONCLUSIONS AND RECOMMENDATIONS
The digital computer has been responsible for
sweeping changes in the manner in which multi
stage separation operations are synthesized and
SPRING 1985
A
c
P
I
I
f
P
~ ~t'
TABLE 1
Recommended Additions to Content of Undergraduate
Courses
1. Numerical methods for
A. Linear algebraic equations
B. Sparse matrices
C. Systems of nonlinear equations
2. Application of numerical methods to
A. Complete tearing, partial tearing, and simul
taneous correction methods for multicomponent
separation processes
3. Use of computeraided simulation programs to
A. Analyze, correlate and predict multicomponent
thermodynamic properties
B. Solve openended separation process problems in
volving energy integration
4. Secondlaw analysis
analyzed. Chemical engineering educators need to
closely examine courses on separation processes
to make sure that students are being instructed
in modern and efficient computational tools. Some
recommended additions to the content of under
graduate courses are listed in Table 1, where many
of the items should prove useful in other chemical
engineering subjects as well.
Numerical methods should be stressed for
linear algebraic equations, including efficient
handling of sparse matrices, and systems of non
linear equations. These methods should then be
applied using computers to utilize partial tearing
and simultaneous correction methods for multi
component separation processes. However, some
manual calculations on simple examples should be
performed using complete tearing methods to help
develop a basic understanding.
More complex and openended separation
problems should be assigned that stress energy
integration. A secondlaw analysis [see Denbigh
(5 A.D.), and de Nevers and Seader (28 A.D.)] of
a process should be required, and attempts should
be made to improve the process by finding eco
nomical means to reduce the lost work.
The development of a separation process can
be an exciting experience when computers and
computer programs are available to perform the
tedious calculations and allow time for more con
sideration of synthesis and optimization as
pects. Ol
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Naphtali, L. M. and D. P. Sandholm, AIChE J., 17, 14
(1971).
Orbach, 0., and C. M. Crowe, Can. J. Chem. Eng., 49, 509
513 (1971).
Ponchon, M., Tech. Moderne, 13, 20, 55 (1921).
Powell, M. J. D., "A Hybrid Method for Nonlinear Equa
tions," in "Numerical Methods," Ed. P. Rabinowitz,
Gordon and Breach, New York (1970).
Rosen, E. M., "SteadyState Chemical Process Simulation:
A Stateofthe art Review," Computer Applications to
Chemical Engineering, R. G. Squires and G. V. Reklaitis,
editors, ACS Symp. Ser. No. 124 (1980).
Savarit, R., Arts et Metiers, pp. 65, 142, 178, 241, 266, 307
(1922).
Seader, J. D., Section 13 of Perry's Chemical Engineers
Handbook, 6th ed., McGrawHill, New York (1984).
Sujata, A. D., Hydrocarbon Processing, 40, No. 12, 137
(1961).
Thiele, E. W., and R. L. Geddes, Ind. Eng. Chem., 25, 289
(1933).
Wang, J. C., and G. E. Henke, Hydrocarbon Processing,
45 No. 8, 155 (1966); Hydrocarbon Processing, 45
No. 9, 169 (1966).
Wayburn, T. L., and J. D. Seader, "Solutions of Systems
of Interlinked Distillation Columns by Differential
HomotopyContinuation Methods," Proceedings of the
Second International Conference on Foundations of
ComputerAided Process Design, June 1924, 1983,
Snowmass, Colorado (available from CACHE Corp.,
P. O. Box 7939, Austin, Texas 787137939).
NOMENCLATURE
A, B, C, Coefficients in a tridiagonal matrix equa
CHEMICAL ENGINEERING EDUCATION
B
f
F
HF
h
H
H,
H,
K
L
MP
p
Q
R
s
tion; submatrices of partial deriva
tives in a block tridiagonal matrix
Bottoms product molar flow rate
An arbitrary function
Molar feed rate to a stage; mathematical
function
Molar enthalpy of feed to a stage
The homotopy function whose arguments
are x and t
The homotopy function whose arguments
are x and X
Molar enthalpy of vapor leaving a stage
Molar enthalpy of liquid leaving a stage
Vaporliquid equilibrium ratio
Molar liquid flow rate leaving a stage
Middle product molar flow rate
Path length
Heat duty (R for reboiler; C for con
denser)
Reflux ratio
Ratio of liquid drawoff to primary liquid
(liquid not withdrawn or entrained)
DEPARTMENT: Cornell
Continued from page 61.
and measuring the properties of liquids and
liquid mixtures using theory, computer simula
tion, and experiment. Cornell is one of very few
institutions with strength in all three areas.
Keith guides the theory and the computer
simulation (with help from Senior Research As
sociate Steve Thompson,) making use of recently
developed accurate theories for dense fluids of
complex molecules as well as improved computer
simulation methods and computer hardware.
Typically, highly nonideal substances (in the
thermodynamic sense) are chosen for study; sub
stances for which traditional methods of pre
diction fail. Examples include mixtures occurring
in coal gasification and liquefaction, hydrogen
energy technology, synthetic fuel processing and
supercritical fluid extraction. Other research
underway or planned includes studies of ad
sorption at gasliquid, liquidliquid and solidfluid
interfaces, nucleation and droplet phenomena,
polarization in polar fluids, and surfactant effects.
Bill Streett and Senior Research Associate
John Zollweg carry out experimental studies of
dense fluids. In progress are (i) experiments in
vaporliquid, liquidliquid, and gasgas equilibria
at temperatures from 70 to 500 K and pressures
to 10,000 atmospheres; (ii) equationofstate
(PVT) measurements of pure liquids and mix
tures at temperatures from 70 to 500 K and pres
sures to 4,000 atmospheres; and (iii) measure
ments of enthalpy of mixing in samples of lique
fied gases at temperatures from 70 to 300 K and
pressures to 20 atmospheres. Bill is currently de
veloping new experiments to measure the surface
and interfacial tensions and the velocity of sound
in fluids under pressure.
The researches of Paulette Clancy, who became
a member of the faculty in 1984, range from a
statistical mechanical study (using perturbation
theory) of multicomponent highly polar fluid
mixtures to a development of phase diagrams
(based on molecular thermodynamics) of semi
conductor materials. In addition, she is involved
in the application of computers to chemical engi
neering.
Herb Wiegandt's interest in desalting sea
water, using a freezing process based on direct
contact with butane, goes back to 1958. Recent
efforts, with Bob Von Berg as a partner, have
aimed at overcoming the problems associated
with washing and separating the ice crystals
which are typically very small.
Julian Smith, past Director of the School in a
period of unprecedented growth, seasoned edu
cator and coauthor of Unit Operations of Chemi
cal Engineering (now in its fourth edition, with
Pete Harriott as coauthor), has expertise in
mixing, centrifugal separation, and handling of
granular solids. He is teaching fulltime and is
active in the guidance of the school.
Ray Thorpe, who has advised graduate
students in the areas of phase equilibria and
separations processes, splits his time between
SPRING 1985
S Ratio of vapor drawoff to primary vapor
(vapor not withdrawn)
t Homotopy parameter.
T Temperature; when used as a superscript
denotes matrix transpose
V Molar Vapor flow rate leaving stage
x The vector of independent variables (un
knowns) for the distillation equations;
liquidphase mole fraction
x The starting vector for the nonlinear
equation solver
x* The solution to the set of nonlinear equa
tions
X Mole fraction in liquid of a component;
variable
y Vaporphase mole fraction
Y Mole fraction in vapor of a component
Z Mole fraction in feed of a component
Greek Letters
X Homotopy parameter
teaching and university administration: he is di
rector of the Division of Unclassified Students.
Research Interactions
Many research projects involve active col
laboration with other researchers at Cornell or
elsewhere. Some are directly with other depart
ments; some are through Cornell's numerous in
disciplinary programs, centers, and institutes
many industrially supportedthat facilitate inter
action among departments and with industry.
Examples are the Biotechnology Institute; the In
jection Molding Project; COMEPP (Cornell
Manufacturing Engineering and Productivity
Program); Applied Mathematics Center; Theory
and Simulation Center (established by Ken
Wilson, Cornell's 1982 Nobel laureate in physics) ;
Materials Science Center; National Facility for
Submicron Studies. Strong ties have been es
tablished with other departments and colleges of
Cornell, and with workers at other universities
around the world.
Paulette Clancy, Associate Director of COM
EPP, is joined by Professor Scheele in a study
of ways to improve the interface between the
user and ASPEN software chemical process syn
thesis and design.
UNDERGRADUATE PROGRAM
Undergraduate chemical engineering enroll
ments at Cornell were almost constant during the
twenty years before 1975, with about 40 bache
lor's degrees awarded annually. Then, although
freshman admissions to the engineering college
were held constant, the number of students opt
ing for chemical engineering roughly doubled, and
for nine years the number of BS degrees awarded
was between 65 and 75. After 1985, however, the
number will return to 40 or so and is expected to
stay at that level for the next several years.
The subject matter of the undergraduate pro
gram is much the same as at other institutions.
For the first two years the students are not in
chemical engineering but are enrolled in the "com
mon curriculum" of the engineering college. Never
theless, their curriculum has much that is differ
ent from that of other engineering students. In
the freshman year chemical engineers take two
semesters of chemistry, not one. Sophomores take
two semesters of physical chemistry, with labora
tory each terma special course taught by Chemis
try almost exclusively for chemical engineers
and the required introductory course in mass and
energy balances. Organic chemistry (two semes
ters, one with laboratory) is given in the third
year, as are chemical engineering thermody
namics, rate processes and separation processes.
The fourth year includes required courses in re
action kinetics, process evaluation, process con
trol, and unit operations laboratory, and a spring
term course in process design. Overall, 132 credit
hours are required for the BS degree, including
two courses in computer programming and ap
plications, four engineering distribution courses,
and six courses in humanities and social sciences.
Ten of the required courses (32 credit hours) are
in chemical engineering subjects.
The senior laboratory course is considered the
most demanding by students and faculty alike.
Each student reports on only five experiments
during the term, but each report is thoroughly
edited for both form and content by the faculty
member in charge of that experiment and nearly
always must be extensively revised by the student
before it is accepted. The emphasis is on technical
accuracy, completeness, and clarity of expression.
Oral presentations are stressed in the senior
design course, in which each team of students
makes weekly oral presentations before two faculty
members or industrial visitors. In recent years
experienced engineers from industry have been
hired for fulltime assistance in this course and
in the senior laboratory. Their contributions have
been supplemented, during shortterm visits, by
those of people from Exxon, Union Carbide, and
other firms. Despite this, the laboratory and de
sign courses demand large contributions of time
by senior faculty members, and pose the most
difficult problems for future staffing.
A Special Cooperative Program
For the past ten years the better students in
the sophomore year have been encouraged to enroll
in an unusual industrial cooperative program
which gives them practical experience without
lengthening their time at the university. Typical
ly 15 to 20 students are accepted into the program
after company interviews exactly like those for
permanent employment. Coop students take the
fallterm thirdyear courses during the summer
following their sophomore year; they work in in
dustry during the fall and return to Cornell in
the spring; work again for the same sponsor the
following summer; and complete their senior year
CHEMICAL ENGINEERING EDUCATION
in the regular sequence. Industrial assignments
are carefully monitored to insure appropriateness,
and each student is visited at the worksite by a
Cornell person at least once during the course of
the program.
THE PROFESSIONAL MASTER'S PROGRAM
This is a twosemester nonthesis master's
program leading to the degree Master of Engineer
ing (Chemical). It requires 30 credit hours of
advanced technical work, including a substantial
design project, with emphasis on practical ap
plications. Most of the matriculants are not from
Cornell or other U.S. schools; instead the program
is attractive to foreign students, especially from
developing countries such as the Dominican Re
public, Guatemala, India, Kuwait, Taiwan and
Venezuela. Over the years a chemical company
in India has sent, one after another, three of its
top technical employees to this program.
Required courses for the MEng (Chemical)
degree include equipment design and selection,
numerical methods, reactor design, the design pro
ject, and a chemical engineering elective. The
remaining credit hours can be filled by elective
courses in science or engineering or in the Gradu
ate School of Management. The choice of subjects
for MEng design projects is much wider than in
the typical undergraduate design course, and
more initiative and originality are expected of
the students. Some of the projects are done in
close collaboration with industrial firms.
RELATIONS WITH INDUSTRY
The school has always had close relations with
industry and an unusually supportive group of
alumni. Industry helps us in many ways: in the
design courses; in a "Nonresident Lecture Series"
(zero credit, but compulsory), given to seniors on
the various kinds of professional careers; in un
restricted grants; in scholarships, fellowships, and
sponsored research. Continuing fellowship sup
port has been provided by Amoco, Chevron, Dow,
DuPont, Exxon, Shell, Stauffer and Union Car
bide, and recent large research projects came
from IBM, Kodak and Mobil. In 1981 the Sun
Company gave $250,000 over three years to sup
port research initiation on ideas too new and ill
defined to merit submission of a proposal to NSF
or other granting agencies. This unusual grant
led to a number of publications and several con
tinuing sponsored research programs.
ADVISORY COUNCIL
An advisory council, largely from industry,
was formed a few years ago. It meets in Ithaca
twice a year to review progress and help the di
rector steer a course for the school. About half
the members are alumni. Recently expanded to
15 members, the council now includes four aca
demic people: Andy Acrivos (Stanford), Gus Aris
(Minnesota), Gary Leal (CalTech), and Bill
Schowalter (Princeton). We don't always agree
with the council's suggestions, of course, but as
a group it has been marvelously effective in pro
viding an "outside" viewpoint and keeping us from
being too provincial or selfsatisfied.
WHAT OF THE FUTURE?
Cornell is facing many of the same problems
that face other chemical engineering departments
around the countryfaculty retirements and fu
ture faculty development, staffing of design and
laboratory courses, the optimum use of computers
for teaching, expansion of research and the gradu
ate program, and renovation of aging facilities.
The five professors hired right after World War II
are nearing retirement, so for the next several
years an average of one new faculty member per
year will have to be hired to keep the number
constant. Because of the loss of professors with
industrial experience one or more people with an
industrial background will probably be hired on
a nontenuretrack basis to teach design and to
supervise the laboratory courses.
A related problem is in the use of computer
software. How much emphasis should be placed
on teaching the use of ASPEN, for example?
More generally, as personal computers become
ubiquitous, what will happen to teaching methods?
Will the course in mass and energy balances, for
example, become a course in the use of available
canned programs?
The total number of graduate students in the
school, and the fraction going for a PhD rather
than an MS, should rise somewhat over the next
ten years, depending on the availability of
financial support. This will increase the need for
equipment and laboratory and office space. Rela
tively speaking, the Chemical Engineering School
has a lot of space, but much of it is virtually un
usable for modern research. A comprehensive
building renovation plan, made by a firm of archi
tects, proposes a complete reallocation and rear
rangement of available space and the conversion
SPRING 1985
of the enormous unit operations laboratory into
offices and small research labs. New electrical and
other services will be provided, along with central
air conditioning. The average faculty office will
shrink from over 400 to a more modest 200 square
feet and the offices will be grouped more closely,
to stimulate greater interaction among the oc
cupants. The total estimated cost is some fifteen
times the original cost of the building. A fund
drive for the first stage is being launched. D
AWARD LECTURE
Continued from page 77.
on the batch experience and using verified mathe
matical models to both design the equipment and
direct the experimentation.
CHEMICAL VAPOR DEPOSITION
Laboratory Scale Research
A low pressure chemical vapor deposition
(LPCVD) system for amorphous silicon is shown
in Fig. 14 and the simplified process flow diagram
as Fig. 15. Reactants, Si2H,, and material for
doping the film, PHs and B2He, in a stream of argon
are controlled by valves at the inlet to the reactor.
The tubular quartz reactor is temperature con
trolled inside an electric furnace. System pressure
is controlled manually with a valve at the exit.
Effluent gas can be analyzed by gas chromatogra
phy and unreacted material is decomposed in a
furnace before venting. The detailed operation of
this system is described by Bogaert [9].
This effort in amorphous silicon research, spon
FIGURE 14. Photograph of Low Pressure Chemical
Vapor Deposition unit (LPCVD).
scored by the Department of Energy through the
Solar Energy Research Institute, is ongoing at
the present time and is far from being complete. I
am discussing it here to allow the reader to con
trast and compare with the physical vapor depo
sition reacting systems just described.
The chemistry is much more complex for
amorphous silicon than for CdS and not well
FIGURE 15. Simplified process flow diagram of LPCVD.
understood. The present stateoftheart is shown
below:
Gas Phase
Si2H, ;SiH4 + SiH2
SisHs Si2He + SiH,
SiHloSiiH, + SiH2
SisHijsSiHo + SiH2
SieH,,4SisH,1 + SiH2
SiH1,,SiH,, + SiH,
SisHs8SiH,, + SiH2
Film Formation
SisH2*3SiHo.os + 2SiH, + 1.88H2
SiH2 *SiHo.os + 0.96H,
This is a preliminary set of chemical equations.
The gas phase equations are based on the results
of Ring [10], John and Purnell [11], and Bowery
and Purnell [12]. The film formation equations are
based upon our own preliminary research.
The component mass balance equations for this
tubular reactor system are given below:
Gas Phase
D4q j dCi= Ir(rxt,i) + kga (CiCs)
\v;rD'j dZ
CHEMICAL ENGINEERING EDUCATION
Surface
0 = kga(Ci Ci,) ka(yiCi)
Film
1 dprV' = ka(yiCi)
MW, dt
Both the gas phase composition and the film
growth rate are functions of axial position. Film
growth rate (i.e., amount of amorphous silicon
deposited) at any axial position can be determined
but it has been possible to measure gas composition
only at the reactor exit. The gas and solid phase
mass balance equations are coupled through the
chemistry of film formation and the transfer from
the bulk gas to the surface.
Solution of the model equations produces the
gas phase exit composition versus reactor holding
time plots shown as Figs. 16 and 17. The solid
lines were obtained using our present "best"
estimates of the specific reaction constants. This
"best" estimate is now obtained by using the ex
perimentally determined growth rate in the solu
tion of the set of mass balance equations. The
agreement between data and the predicted values
is only fair but we expect to improve the model
loo
100 I I
NT.400*C NSSl2H6
80 P= 24 Torr SIH4
SSi1H,8
S60 S12H *
S40
SSIH4
20 /S13H
0 o0 20 30 40 50 60 70
Holding Time. (sec.)
FIGURE 16. Normalized molar percentages versus hold
ing time: Major silanes.
predictions as we learn more about the system.
This research on the chemical reactor and re
action engineering for amorphous silicon in the
LPCVD reactor is closely coupled with studies of
the material and electronic properties of the film
and much effort has been devoted to finding the
best conditions for good photovoltaic amorphous
silicon. To date, we have been able to make a 4%
solar cell using material from the LPCVD reactor.
These efforts are described in the work of Hegedus
et al [13].
Holding Time, (sec.)
FIGURE 17. Normalized molar percentages versus hold
ing time: Minor silanes.
CONCLUDING REMARKS
Incorporating chemical reactor and reaction
engineering analysis into a semiconductor research
effort requires the researchers to achieve a quanti
tative understanding of both the molecular phe
nomena and the transport phenomena associated
with the creation of the semiconductor materials.
A test of this understanding is the ability to write
useful mathematical descriptions of the laboratory
scale reacting system. Mathematical descriptions
are an essential part of the analysis because they
provide the language which allows the profes
sionals doing the research to effectively and un
ambiguously communicate with each other. Com
munication is easier if the models are simple and,
of course, the model predictions must be verifiable
by experiment. In fact, the model behavior should
be used to plan the experimental program because
an enlightened use of a chemical reaction and re
actor engineering analysis will identify critical
molecular and transport phenomena problems and
direct experimental attention to them with the
proper priorities.
We originally became interested in the semi
conductor research because of a need to design
larger scale reacting systems. However, the last
five years of research has taught us that the chemi
cal engineering analysis is very useful in the
laboratory scale research effort, and indeed es
sential, if such research is to be done efficiently and
with minimum expense (a key issue with today's
research costs).
It is not possible, in our view, to effectively de
sign and operate larger scale systems without re
action and reactor engineering analysis. In photo
SPRING 1985
voltaic applications it is also necessary to carry out
analysis and experimentation at the unit opera
tions scale. Those who have attempted to scale up
without following these procedures have wasted
time and money building equipment which is in
adequate for the commercial scale processing of
solar cells.
A useful start has been made in applying
chemical engineering analysis to the deposition of
thinfilm semiconductors but much effort now must
be devoted to the task of relating electronic and
optical properties to the design and operation of a
reacting system. When we have learned to do this
properly, we can begin to "tailormake" material
with any desired property.
ACKNOWLEDGMENTS
Semiconductor chemical reaction and reactor
research requires a team effort involving a number
of professionals. I am particularly indebted to
B. N. Baron, R. E. Rocheleau, S. C. Jackson and
R. J. Bogaert, my chemical professional colleagues
at the Institute of Energy Conversion. Their
analysis, their effective experimentation and their
discussions with me have been essential to the de
velopment of this field. None of the research could
have been carried out without the excellent semi
conductor material development and analysis and
device design and analysis that my other col
leagues at the Institute of Energy Conversion do
so well. I am also in their debt for their willing
ness to educate a chemical engineer in the art and
science of applied solid state physics.
Science and engineering research today re
quires some considerable management talent. The
Department of Energy's photovoltaic office and the
Solar Energy Research Institute have worked very
hard to develop a rational plan for photovoltaic
research that both produces results and handles
the political pressures that arise in a budget con
scious government. The management group within
the Institute of Energy Conversion is unique in
its capability to protect the director from ad
ministrative detail and to allow me to put most of
my effort into technical work. I would like to thank
S. Barwick and M. Stallings for this gift. Ol
REFERENCES
1. Thornton, J. A., Annual Rev. of Material Science, 7,
p. 239 (1977).
2. Kern, W. and V. S. Ban, Thin Film Processes, (J.
Vossen and W. Kern, editors) Academic Press, New
York (1978).
3. Dutton, R. W., "Modeling of the Silicon Integrated
Circuit Design and Manufacturing Process," IEEE
Trans. Electron Dev., 30, 9, p. 968 (1983).
4. Rocheleau, R. E., B. N. Baron and T. W. F. Russell,
"Analysis of Evaporation of Cadmium Sulfide for the
Manufacture of Solar Cells," AIChE Journal, 28, 4,
p. 656 (1982).
5. Jackson, S. C., PhD Thesis, "Engineering Analysis of
the Deposition of CadmiumZinc Sulfide Semicon
ductor Film," University of Delaware (1984).
6. Rocheleau, R. E., PhD Thesis, "Design Procedures
for a Commercial Scale Thermal Evaporation System
for Depositing CdS For Solar Cell Manufacture,"
University of Delaware (1981).
7. Griffin, A. W., MChE Thesis, "Modeling and Control
of a Unit Operations Scale System to Deposit Cad
mium Sulfide for Solar Cell Manufacture," Uni
versity of Delaware (1982).
8. Brestovansky, D. F., B. N. Baron, R. E. Rocheleau
and T. W. F. Russell, "Analysis of the Rate of Vapor
ization of CuCl for Solar Cell Fabrication," J. Vac. Sci.
Technol. A, 1, 1, p. 28 (1983).
9. Bogaert, R. J., PhD Thesis, "Chemical Vapor Deposi
tion of Amorphous Silicon Films," University of
Delaware (1985).
10. Ring, M. A., "Homoatomic Rings, Chains and Macro
molecules of Main Group Elements," Elsevier, N.Y.,
1977, Ch. 10.
11. John, P. and J. H. Purnell, Faraday Trans. I, 69, p.
1455 (1973).
12. Bowery, M. and J. H. Purnell, Proc. Roy. Soc. Lond.,
A821, p. 341 (1971).
13. Hegedus, S. S., R. E. Rocheleau and B. N. Baron,
"CVD Amorphous Silicon Solar Cells," Proceedings
of the 17th IEEE Photovoltaic Specialists Confer
enceOrlando, p. 239 (1984).
NOMENCLATURE
a
Ci
D
K
k
kg
MW
q
r(e)
r(i)
r(r)
r (rxt,i)
V
Z
Greek
area
concentration of species i
diameter of reactor
effective reaction rate constant
reaction rate constant
mass transfer coefficient
molecular weight
volumetric flow rate
rate of evaporation
impingement rate, species i
rate of reflection
net rate of reaction, species i
volume
axial position in tubular reactor
8 condensation coefficient
y stoichiometric coefficient
p density
Subscripts
f film property
g gas phase
i molecular species
s denotes on the surface
CHEMICAL ENGINEERING EDUCATION
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z 0 < u ::, C w 0 z 0:: w w z (5 z w 0:: 0 LL. w u 0 V) z < u c2 w < LL. 0 z 0 CJ') > c 0 0:: w w z <3 z w _, < u w J: u VOLUME XIX NUMBER 2 JOE HIGHTOWER of Rice University SPRING 1985 Semiconductor Chemical Reactor Engineering and Photovoltaic Unit Operations T. W. F. RUSSELL ruui ... Classical Solution Thermodynamics: A Retrospective View VAN NESS, ABBOTT An Improved Design of a Simple Tubular Reactor Experiment ASFOUR Use of Computer Graphics to Teach Thermodynamic Phase Diagrams NAIK, CLANCY, GUBBINS The Nature of Adjoint Variables and Their Role in Optimal Problems CROSSER The B.C. (Before Computers) and A.O. of EquilibriumStage Operations SEADER CHEAT CORNELL UNIVERSITY
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3M FOUNDATION CHEMICAL ENGINEERING EDUCATION wdli a o/ /u111cil.
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EDITORIAL AND BUSINESS ADDRESS Department of Chemical Engineering University of Florida Gainesville, Florida 32611 Editor: Ray Fahien (904) 3920867 Consulting Editor: Mack Tyner Managing Editor: Carole C. Yocum (904) 3920861 Publications Board and Regional Advertising Representatives: Chairman: Lee C. Eagleton Pennsylvania State University Past Chairman: Klaus D. Timmerhaus University of Colorado SOUTH: Homer F. Johnson University of Tennessee Jack R. Hopper Lamar University James Fair University of Texas Gary Poehlein Georgia Tech CENTRAL: Robert F. Anderson UOP Process Division Lowell B. Koppel Purdue University WEST: William B. Krantz University of Colorado C. Judson King University of California Berkeley Frederick H. Shair California Institute of Technology NORTHEAST: Angelo J. Perna New Jersey Institute of Technology Stuart W. Churchill University of Pennsylvania Raymond Baddour M.I.T. A. W. Westerberg CarnegieMellon University NORTHWEST: Charles Sleicher University of Washington CANADA: Leslie W. Shemilt McMaster University LIBRARY REPRESENTATIVE Thomas W. Weber State University of New York SPRING 1985 Chemical VOLUME XIX Engineering NUMBER 2 Education SPRING 1985 72 Semiconductor Chemical Reactor Engineer ing and Photovoltaic Unit Operations, T. W. F. Russell The Educator 54 Joe Hightower of Rice University, Joyce Taber Department of Chemical Engineering 58 Cornell University, Julian C. Smith and Paul H. Steen Classroom 68 The Nature of Adjoint Variables and Their Role in Optimal Problems, 0. K. Crosser 78 The Use of Computer Graphics to Teach Thermodynamic Phase Diagram s, C. D. Naik, Paulette Clancy, and Keith Gubbins Laboratory 84 An Improved Design of a Simple Tubular Reactor Experiment, A. A. Asfour Lecture 62 Classical Solution Thermodynamics : A Retrospective View, H. C. Van Ness and M. M. Abbott 88 The B. C. (Before Computers) and A. D. of EquilibriumStage Operations, J. D. Seader 67 Books Received 71 Positions Available 82, 83 Book Reviews CHEMICAL ENGINEERING EDUCATION i s published quarterly by Chemical Engineering Division American Society for Engineering Education. The publication i s edited at the Chemical Engineering Department, University of Florida. Secondclass postage i s paid at Gainesville, Florida, and at DeLeon Spring s, Florida. Correspondence regarding editorial matter, circulation and changes of address should be addressed to the Editor at Gainesville, Florida 32611. Advertising rates and information are available from the advertising representatives. Plates and other advertising material may be se nt directly to the printer: E. 0. Painter Printing Co., P 0. Box 877, DeLeon Springs, Florida 32028. Subscription rate U S ., Canada, and Mexico is $20 per year, $15 per l[ear mailed to member s of AIChE and of the ChE J?ivision '?f ~~EE. Bulk s ubscription rates to ChE faculty on request. Write for price s on individual back copies Copyright 1985 Chemical En g ineerin g Division of American Society for Engineering Education. The state ments and opinions expressed in this periodical are those of the writers and not necessarily those of the ChE Division of the ASEE which body assumes no responsibility for them. Defectiv e copie s replaced if notified within 120 days. The International Organization for Standardization ha s assigned the code US ISSN 00092479 for the identification of thi s periodical. 53
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[iJ ;j :I educator /oet:11~ of Rice University JOYCE TABER Rice University Houston, TX 77251 {'{'I 'VE BEEN ~ELIGHTED to be where I am," says Dr. Joe Hightower in regard to his decision 17 years ago to become a chemical engineer and an educator as well. Joe Hightower, of the chemical engineering department at Rice University in Houston, says he started out like many other members of engi neering faculties : "I started as a child by taking things apartbicycles, motorcycles, clocks, every thing!" As early as the fourth grade he proceed ed to take his clarinet apart and to rebuild it shortly after he began taking music lessons. Then throughout high school, in addition to becoming an accomplished musician as a member of the all state band, he made a veritable career out of re pairing the instruments of the other band mem bers. While he was in high school Joe also decided to study chemistry. Later he obtained his masters and PhD in chemistry from Johns Hopkins but couldn't decide if he wanted to do industrial re search or academic work. It was during a three year stint at the Mellon Institute that he decided to teach. "I found that I enjoyed the interaction with the students, the stimulation of the faculty, and the flexibility of the job." He found he had to make another decision, howeverwhether to go into a department of chemistry or chemical engineering. "All my edu cational background was in chemistry, but I had a philosophical desire to work on things that have very practical uses," he says. However, chemistry departments were moving in the direction of quantum mechanics and other more esoteric areas while chemical engineering was moving from unit operations into engineering science. Thus, chemical engineering embraced catalysis, and Joe Hightower embraced engineering. 54 It happened at that time that heterogeneous catalysis, the research area in which Joe was interested, was a field that had been explored primarily in chemistry departments. However, chemistry departments were moving in the di rection of quantum mechanics and other more esoteric areas while chemical engineering was moving from unit operations into engineering science. Thus, chemical engineering embraced catalysis, and Joe Hightower embraced engineer ing. Since then, Joe's research philosophy has been directed toward providing new insights into how existing catalysts work as opposed to discovering new catalysts. "We try to ask the question 'Why?' he says. Using his chemical training, Joe has worked at gaining information about the chemical nature and concentration of active sites, the in fluence of solid state parameters in determining activity and selectivity, and the mechanisms of re actions that occur over solids that are of interest to the petroleum and petrochemical industries. He and his students have extensively used isotopic Copyright ChE Divis i on, ASEE 1985 CHEMICAL ENGINEERIN(; EDUCATION
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tracers (both stable and radioactive) to study the kinetics, reaction networks, rate limiting steps, and incorporation of surface species into product molecules. (Some of his work has been sum marized in an earlier issue of this journal, Vol. XVI, No. 4, p. 148, Fall 1982). A few of the catalyst systems that his group has studied in clude cracking catalysts, auto emission control catalysts, partial oxidation catalysts, and zeolites. But Joe's research does not stop at the gradu ate level. Several years ago he incorporated some research techniques into a sophisticated under graduate experiment. While taking his kinetics and reactor design course, junior level students now investigate all the kinetic parameters for cu mene dealkylation over a silicaalumina cracking catalyst, explore the reaction mechanism with deuterium tracers and a mass spectrometer, and determine the surface area of the material. From their results the students are able to calculate the true surface reaction rate constant, the concentra tion of active sites, the turnover frequency, and the role of intraparticle diffusion on the kinetics (Chem. Eng. Educ., p.118, Summer, 1969). This experiment allows the students to determine ex perimentally many of the parameters that are useful in scaling up such reactions from labora tory to plant size. Joe's research has led him into other situations which he has especially enjoyed. In the early 70's, for example, he was chairman of several National Academy of Science panels which were assigned the task of assessing the feasibility of using catalytic converters to decrease pollutants from automobiles. "No one had ever applied catalysts in this way, and we were assigned the task of de termining if these devices would in fact work," he says. "It was fascinating. There was a lot of secrecy. No company would tell us directly what they were doing, but they would tell us what they thought the other companies were doing, and we had to try to put together a clear picture. Once I was asked to testify before the House of Repre sentatives Committee on Science and Technology which was chaired by Representative George Brown of California. The congressman from Detroit would say things like, 'I want you to know that people from my district are being put out of jobs because of government regulation and control : Brown would respond, 'People in my district in California are dying because we don't have enough controls, and pollutants are killing people.' How can one give an objective testimony SPRING 1985 in an atmosphere like that!" Involvement in professional societies has been another rewarding part of Joe's career. He is cur rently chairman of the 24member Petroleum Re search Fund Advisory Board, a group that is re sponsible for a corpus of $150 million. This year the foundation will donate $11 million for uni versity research in petroleumrelated areas. In 1971 he received the National Award in Petroleum Chemistry from the American Chemical Society. Joe finds time to enjoy simple, relaxing activities, such as blowing glass in his lab. As a councilor for the American Chemical So ciety, he represents the southeastern Texas local section. He has been on the national research committee and is on the national awards commit tee of the American Institute of Chemical Engi neers. He has also served as chairman of the petroleum chemistry division of the American Chemical Society. Dr. Hightower has over 50 publications to his credit and is also very much involved in presenting short courses for industry. "Catalysis," he says, "is a field that is not taught in many universities as an area of specialization. Yet, 80 or 90 % of all commercially important chemical reactions are catalytic reactions. People are trained as organic chemists, physical chemists, or chemical engi neers and then learn about catalysis on the job. This creates a great demand for the types of short courses that we instituted at Rice years ago and that are being continued in cooperation with other schools such as the University of Houston.'' It would appear that Joe's day would have to last more than 24 hours in order to accomplish his many activities. Yet, there is still another part of his life that is as impo:rta,nt to him as his pro55
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... there is still another part of his life that is as important to him as his professional work. In 1968 he was a leader in establishing the Human Resources Development Foundation ... (which) provides free temporary housing for needy families who come for treatment to the Texas Medical Center hospitals. fessional work. In 1968 he was a leader in es tablishing the Human Resources Development Foundation. The foundation provides free tempor ary housing for needy families who come for treatment to the Texas Medical Center hospitals. The foundation started as a project at Dr. High tower's church and has expanded to serve over 700 families from over 38 states and 26 foreign countries in the last 16 years. Joe is president of the foundation whose facilities have grown from an old army barracks into 15 beautiful apart ments. He heads a group of approximately 40 dedi cated volunteers who minister daily to the needs of families who are under enormous stress. "All a person needs to move in are pajamas and a toothbrush!" Joe laughs. But his statement Human Resources Development Foundation apartments. is very accurate. The apartments are furnished down to the pots and pans. A local church even provides meat once a week for the residents. Social workers, ministers, and even former residents refer potential patients. Selection is made on a firstcome, firstserved basis without regard to race, creed, sex, religion, age, or level of disability. Need is the sole criterion used to determine eligi bility. Residents are allowed to stay for up to three months. "The project is valued at over a halfmillion dollars, and most of it has been given because of something Joe has done," states Marge Norman, VicePresident of the foundation. "Joe never hesi tates to go speak to a group if there is some chance that they might have an interest in the founda56 tion. His work takes him to the far reaches of the U.S. and overseas; on every airplane trip he takes he makes sure his seat partner is very well acquainted with his pet project, and it often leads to very good things for this foundation." Senator Orrin Hatch was one of the latest people to hear about the foundation because of one of Joe s "airplane contacts." Joe sat next to a member of Senator Hatch's staff on one flight, and later he received a letter from the senator, who had been informed by the staff member about the foundation. The letter commended Dr. High tower for his charitable work. Continues Mrs. Norman, "Even though we have a foundation board which is functioning very well, without Joe I don't know if the Board would have been as effective or if this place would have become what it is today. He loves it so much and he works so hard. A couple of times each year we have a work day when all our volunteers come to clean, repair, and paint. All kinds of people are represented in the workers Joe is always the first here and the last to leave, working at anything that needs to be done. Even before he comes over, he gets up before dawn and bakes bread so that at 10 a.m. we can have hot bread and butter and coffee. We've been acquiring land to build more apartments next door, so you can be sure he's on the campaign trail again! We have parties for our residents, and again, Joe is always here with bread he has baked. He even brings his mandolin and plays and sings. There's not much Joe can't do!" On any given day the facility may house people from Florida, or from various towns in Texas, Columbia, or Indiana. There are no class dis tinctions. Last winter a brain surgeon from Main land China was allowed to leave his country with only $200 when he came to care for his quadri plegic daughter in the medical center. As a resi dent of the foundation's apartments, he scrubbed floors and took a lot of goodnatured ribbing when he painted an outside door with interior paint. Something all residents share, however, is grati tude to Hightower, who personally greets each newcomer with a loaf of bread, and gratitude to the foundation he helped establish CHEMICAL ENGINEERING EDUCATION
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Wrote one resident, "What a tremendous help this facility has been to us. Each of us is faced with an extremely serious medical problem, and the expenses are staggering. To be sure, the financial savings are important, but even more, we have a place to call 'home' and people with whom we can talk as friends. I'm one of the lucky ones. Two weeks ago I had a kidney transplant, and now I am hoping and praying that my body will not reject it. I have been so impressed by the consistent care and visits that I've had from the jogger (Dr. Hightower) and his friends even during my recovery from surgery. I hope in some way I can repay the favors done for me. Right now, though, I'm going to sit back and enjoy an other slice of the hot bread Joe Hightower has brought me before it has time to lose its flavor!" In his professional life also, colleagues have only good things to say about Joe Hightower. Dr. T. W. Leland who was department chairman when Joe came to Rice says, "I was impressed with him right from the start, and I had a great interest in getting him to come to Rice. I think it's been a firstrate choice. He s done a remarkably good job over the years. He's an excellent teacher and has had an active research career. He is well thought of by his students and has perfected a graduate course in kinetics and catalysis to a high degree. He has been effective in giving short courses and he is outstanding in his volunteer public service. Personally, he is outgoing, friendly, and extreme ly wellorganized. He has excellent rapport with people in all walks of life, from the top of the technical ladder in terms of ability to students not doing too well in their courses. He's a remarkable individual who has been a great addition to our department." Joe is modest in describing his daily activi ties. "I just enjoy it all," he says. He gets up at 4 :45 every morning to jog and share breakfast and a Bible reading with his wife Ann, a chemical engineer who works for the Exxon Chemical Company. By 6 :30 a m. he is at work, doing all the things he loves to do. "From the very be ginning I couldn't make up my mind about what I wanted to do. I wanted something in both in dustry and the academic world. Now I have both, and I'm grateful for that. I probably border on getting involved in more things than I should ... but they're all so interesting! I guess I just like being where I can interact with people and see them grow, whether it's at the university or whether it's with people who are hurting." SPRING 1985 TRANSPORT PROCESSES MOMENTUM, HEAT AND MASS Christie J. Geankoplis University of Minnesota 1 983 casebound 350 pages This text takes a unified approach to basic transport processes : Geankoplis points out the similarities of basic equations and calculation methods, and the differences which occur in the actual physical processes Each chapter of this classtested text is divided into elementary sections, followed by more sophisticated Selected Topics allowing you to expand or focus your course according to the needs of your students and the time limits of your course SI Units are used throughout, with important equa tions and tables in dual units Over 150 example problems and more than 340 homework problems emphasize applications as well as theory. This text offers complete coverage of more essentials than any other book you'll find. Look for these new topics : diffusion in solids and porous solids, bioengi neering transport, nonNewtonian fluids, numerical methods for steady and unsteadystate diffusion and conduction, design and scaleup of agitation systems, mass transport, and an introduction to engineering principles TRANSPORT PROCESSES AND UNIT OPERATIONS SECOND EDITION ChristieJ. Geankoplis University of Minnesota 1983 casebound 650 pages This fullyrevised Second Edition includes TRANSPORT PROCESSES: MOMENTUM, HEAT AND MASS as Part One of the text, plus the unit operations (Part Twol so essential to chemical engineers TRANSPORT PROCESSES AND UNIT OPERATIONS, Second Edition offers an optimal balance of theory and application. Geankoplis emphasizes the diversity of practical applications in chemical, ceramic, mechanical, civil food process, and bioengineering. Over 220 example problems and 500 homework problems illustrate both theory and applications The Second Edition features new sections on multi component distillation, unit operations of membrane processes, nonNewtonian fluids, diffusion in solids, porous solids and biosystems, freezing, freeze drying, and sterilization of biomaterials. For examination copies or more information on these two titles, write to Ray Short, Engineering Editor, Allyn and Bacon, Inc., 7 Wells Avenue, Newton, MA02159 Allyn and Bacon, Inc. College Division .. 57
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Olin Hall from the west. [9;j Hi department I CHE AT CORNELL UNIVERSIT'I JULIAN C. SMITH AND PAUL H. STEEN Cornell University Ithaca NY 14853 F OUNDED IN 1868 AND with a long tradition in engineering Cornell is almost unique in being both private and statesupported; about half the divisions, including engineering, are privately en dowed while the other half are funded by the State of New York. An awkward arrangement it would seem, but it works surprisingly well. Cornell, a medium sized university with a total enrollment of some 18,000 students, is set on a hill overlooking the city of Ithaca, and the waters of Cayuga Lake, the largest of the Finger Lakes Ithaca is small but strongly cosmopolitan. The setting is semirural; the scenery is beautiful; the air is clean. Almost the only drawbacks are a modicum of cold gray weather on occasion and some possible difficulties in travelling in and out. C o p yr i ght ChE D iv i si on, A SEE, 19 85 58 Ithaca has been called "the most centrally iso lated city in the Northeast," but as a graduate student from Greece recently remarked, "If it wasn't for the weather, Ithaca would be Paradise!" The School of Chemical Engineering has 18 faculty members, about 100 undergraduate students (3rd and 4th years only), and over 65 graduate students During the past twelve years research activity and expenditures have greatly increased, and strong research programs have been established in fluid mechl'.i.nics, polymers, sur face science and catalysis, thermodynamics, and biochemical engineering. The number and quality of MS and PhD candidates (especially PhD's) have risen rapidly. The growth in research, how ever, has not reduced the traditional concern for undergraduate and professional graduate teach ing. All faculty members are expected to teach undergraduate courses, and many participate in advanced design projects in the professional Master's program. The school occupies its own building (shared with a few other tenants) with CHEMICAL ENGINEERING EDUCATION
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a total area of some 90,000 square feet, 54,000 of which is exclusively chemical engineering. A BRIEF HISTORY At Cornell, as at many institutions, chemical engineering began in the Chemistry Department, but its development was somewhat unusual. Very early (before 1900) courses were offered in in dustrial chemistry which had a considerable practical flavor; as taught by Fred H. Dusty" Rhodes in the 1920's they dealt with the principles and practice of chemical engineering. By 1930 Dusty had established both undergraduate and graduate programs in chemical engineering, but because of rivalries between Chemistry and the Engineering College the undergraduate program had to be a 5yearlong hybrid: four years in Arts and Sciences ( leading to the degree Bachelor of Chemistry) followed by one year in engineering ( for the degree Chemical Engineer) In 1938 the department with its three faculty members became part of the Engineering College and the 5year program led to the degree Bachelor of Chemical Engineering. In 1942 chemical engi neering moved to Olin Hall the first building on what was to become engineering's new quadrangle. It was at a considerable distance from chemistry and the old ties quickly weaken ed. After World War II all the undergraduate engi neering programs at Cornell were lengthened to five years. This lasted until 1965, when the present 4year BS programs, including that in chemical engineering, were established. Dusty Rhodes was director of the school until he retired in 1957 and Charles C. "Chuck" Wind ing took over. Ken Bischoff, now at the University of Delaware, was director from 1970 to 1975; Julian Smith from 1975 to 1983; and Keith Gubbins from 1983 to date. For many years chemical engineering at Cornell was known for its strong undergraduate program. Rhodes felt that good teaching was the most important thing required of a faculty mem ber and while ability to do research should be considered in reviews for promotion or tenure, it should not be a major factor. This is not to say that there was no research or graduate work. Be tween 1932 and 1970 the school awarded 140 MS and 104 PhD degrees, and many of the recipients have had distinguished careers in industry and academia, including John Prausnitz (Berkeley), Ed Lightfoot (Wisconsin), and a string of past SPRING 1985 or present heads of chemical engineering depart ments: Bob Coughlin (UCONN); Howard Greene (Akron); Deran Hanesian (NJIT); Will Kranich (Worcester Poly); Larry McIntire (Rice); Steve Rosen (Toledo); Julian Smith (Cornell); Tom Weber (SUNY Buffalo); and Jacques Zakin (Ohio State). Bob Finn's pioneering work in biochemi cal engineering was begun in the 1950's. Dusty's policies set a pattern for the school which persisted until the early 1970's. By then it was clear that the research effort had to be greatly The Fred H. Rhodes Student Lounge, redecorated through a gift from Joseph Coors, '40. expanded. Beginning in 1970, new faculty mem bers were added who developed, or brought with them, strong research programs in several areas. This attracted additional researchminded faculty and increasingly stronger graduate students. During an 8year period research expenditures in creased by a factor of six. The number of gradu ate students has risen to 67; more significantly, a majority of them are now PhD candidates. And collaboration with Chemistry and other depart ments of the university is once again close and extensive. RESEARCH GROWTH From 1976 through 1982, the annual research expenditures, in dollars per faculty member, climbed at a rate that was second highest and reached a level that was fourth highest among all chemical engineering departments in the country.* Total sponsored research costs for 198384 were over 1.5 million dollars, for an average of $94,000 *Journal of Engin ee ring Education, March issues, 19761983. 59
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per fulltime faculty member. This is especially striking since only about a third of the faculty was responsible for 75 % of the total expenditures. As the current younger faculty develop their pro grams and additional researchminded faculty re place retiring senior professors, the overall re search program should continue its strong ex pansion. Biochemical Engineering Biochemical engineering research has grown from Bob Finn's early studies of microbes and ... many of the PhD recipients have had distinguished careers in industry and academia, including John Prausnitz (Berkeley), Ed Lightfoot (Wisconsin), and a string of past or present heads of chemical engineering departments. microbial populations. The goal was, and is, to develop new and more efficient biochemical con versions. One project seeks to find economical ways of producing ethanol from pentose sugars, a second to develop better treatment methods for wastes containing pentachlorophenol (PCP), and a third to exploit an unusual bacterium which can rapidly ferment arabinose. Mike Shuler shares several specific interests with Bob Finn including the treatment of waste water by specialized microorganisms. Mike's di verse interests are tied together by a view of the living cell as a "catalyst" waiting to be used in chemical reactors. His research embraces studies of plantcell tissue culture, reactors with solid substrates (e.g. mold growth on solid surfaces), photobioreactors, biofilm formation, and the con tinuous protein production from bacteria with re combinant DNA. Particularly noteworthy have been his group's experimental demonstration of the feasibility of hollowfiber membrane units for entrapment of microbial populations (necessary groundwork for the development of hollowfiber reactors) and the construction of a mathematical model of the organism Escherichia coli. Doug Clark, who joined the faculty in 1984, brings the point of view that enzymes rather than the whole cell can be the building blocks for bio chemical reactors. He is studying how the im mobilization, or attachment to a foreign matrix support, affects the structure and function of an enzyme. A related interest is the transport of bio60 logical macromolecules through porous media; this transport is an essential step in enzyme immobili zation, gel permeation and affinity chromatogra phy, and ultrafiltration. In collaboration with Bill Streett, Doug has initiated a study of methane producing bacteria which live at extreme tempera tures and pressures in deepsea hydrothermal vents. Polymers and Materials Science For a million circuit elements to fit on a tiny silicon chip linewidths must be on the order of a tenth of a micron. In one technique for achieving such precisionelectronbeam lithographylthe silicon surface is covered with a polymer film (polymethyl methacrylate, for example), then ir radiated by an electron beam creating a pattern of soluble polymer. The soluble polymer is washed away leaving a precision mask and the chip is ready for the final step, silicon modification. Fe rdinand (Rod) Rodriguez is directing an inter departmental program on polymers for advanced lithography, to improve the performance of the polymer "resist" used in the masking process. This is a good example of Rod's research on polymeric materials which has the broad goal of understand ing the processes of polymerization and gelation ( crosslinking) and degradation ( chain scission) in order to produce better materials. Claude Cohen uses macromolecular science to interpret the physical properties of polymer systems and to understand the structures that develop during industrial processing. On the fundamental level, predictions from models of macromolecules are used to understand rheo logical and lightscattering behavior, with ex periments to complement the theoretical work and to test the adequacy of the models. On the applied level, the orientation of glass fibers in composite thermoplastics during the molding process is being investigated. This work is in conjunction with an interdepartmental program on injection molding. Surface Chemistry, Catalysis and Reactor Engineering The surface chemistry and physics of hetero geneous systems which have direct technological application is the central concern of Bob Merrill's studies. Examples are CO oxidation on noble metals (automobile exhaust converters), the de composition of hydrazine (rocket monopropel lant), the oxidation of aluminum (catalyst supCHEMICAL ENGINEERING EDUCATION
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port technology, corrosion protection, and elec tronic insulators in microcircuitry) and hydrode sulfurization catalysis (sulfur removal from pe troleum). On the one hand, Bob's group answers practical questions; on the other they are develop ing and sharpening several types of analytical tools. These include the use of lasers in surface chemistry, the use of synchrotron radiation (EXAFS) to study the dynamics of gassolid re actions, and the use of spectroscopy in real catalyst systems (highsurfacearea configurations and high pressures) Peter Harriott studies the influence of mass transfer, heat transfer and mixing on the per formance of chemical reactors as well as the kinetics of reactions in heterogeneous systems. One project concerns the regeneration of catalysts used in the pyrolysis and gasification of coal. An other examines the heat and mass transfer and the overall kinetics in limeslurry droplets used in the "dry scrubbing" of S0 2 from flue gas; the goal is to pin down the ratelimiting step and im prove the design of commercial units. Joe Cocchetto s recent work on catalytic re action kinetics has concentrated on the fuel cell. By controlling the structure of a porous electrode, a better understanding of the interplay between transport and reaction has been gained and tech niques for improving efficiency have emerged. Joe returned to industry in early 1985. Bob Von Berg is interested in the use of gamma radiation in various chemical processes: ammonia synthesis and the reaction of hydrocarbons and liquid nitrogen. Bob has also collaborated with Herb Wiegandt on a longterm project involving the desalination of water by freezing, as described later. Fluid Dynamics and Stability: Rheology Bill Olbricht concentrates on problems in fluid mechanics and rheology with applications in en hanced oil recovery, biomedical fluid mechanics, and the production of semiconductor materials. He is studying the lowReynoldsnumber motion and coalescence of immiscible drops in tubes of various geometries ( characteristic of porous media) for critical evaluation of methods for en hanced oil recovery. In the biomedical area, in conjunction with the University of Rochester Medical School, he is modelling the motion of red blood cells in microcapillaries to predict the dis tribution of these cells within tissue. A third area SPRING 1985 Cornell's Chemical Engineering Faculty, 1984. Back row: Shuler, Finn, Scheele, Steen, Smith, Winding. Middle row: Zollweg (Research Associate), Cocchetto, Harriott, Jolls (Visiting from Iowa State), Von Berg, Rodriguez. Front row: Olbricht, Merrill, Clark, Wie gandt, Thorpe, Clancy, Gubbins, Streett. of research examines the momentum, heat, and mass transfer involved in silicon film growth by chemical vapor deposition with the aim 0f pre dicting rates of film growth in lowpressure deposition reactors. Paul Steen, who joined the faculty in 1982, studies fluid motions and their stabilities. Buoy ancydriven convection patterns, generated in fluidsaturated porous media, are examined as prototypes of fluid motions susceptible to transi tions in which strong nonlinear effects are domin ant. This work involves the development of tools in applied mathematics. In another area, motions induced at fluid / fluid interfaces due to tempera ture gradients (thermocapillary effects) are being investigated by experiment, with relevance to the floatzone crystalgrowth process and the break up of thick films. George Scheele's study of liquidliquid immis cible systems focuses on the coalescence of drop lets and the breakup of jetsboth at relatively high Reynolds numbers He also has interests in the computer simulation of chemical processes, particularly in computer graphics. Molecular Thermodynamics and Computer Simulation Keith Gubbins and Bill Streett have coordinat ed their efforts towards understanding, predicting, Continued on page 103. 61
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[eJ;jplll Iecture CLASSICAL SOLUTION THERMODYNAMICS A Retrospective View H. C. VAN NESS AND M. M. ABBOTT Rensselaer Polytechnic Institute Troy, NY 12181 THE PRIMARY VARIABLES of classical thermodynamics for fluid systems are temperature T pressure P, and the molar properties volume V, internal energy U, and entropy S. Temperature is a primitive, having no definition in terms of anything simpler. Pressure and molar volume are defined directly by three other primitives: force, mass, and length. These primitivestemperature, force, mass, and lengthare subject to direct sensory perception, and we have little difficulty accepting them as meaningful. Internal energy and entropy, however, are primitives not associated with direct detection by the senses. Nor are they directly measurable; we have no energy meters, no entropy meters. Energy and entropy are mental constructs which have meaning only as mathematical functions. Accepting this, we then need to know what they are functions of. We find by experiment that the molar volume of a homogeneous phase is a function of its temperature, pressure, and composition. Generaliz ing, we postulate that the molar internal energy and entropy of a homogeneous phase are likewise functions of temperature, pressure, and com position. When this is true, the first and second laws lead to a fundamental property relation among the primary thermodynamic variables d(nU) = Td(nS) Pd(nV) + L idni (1) The n 1 are mole numbers of the species present, Nor are they directly measurable; we have no energy meters, no entropy meters Energy and entropy are mental constructs which have meaning only as mathematical functions Copyright ChE Di vi sion, ASEE, 198 5 62 H C. Van Ness is Distinguished Research Professor of Chemical Engineering at Rensselaer Polytechnic Institute, where he has been a faculty member since 1956. H e is coauthor with J. M. Smith of Introduction to Chemical Engineering Thermodynamics and has co authored a number of research papers on thermodynamics with M M. Abbott, in addition to two books, Schaum's Outline of Theory and Problems of Thermodynamics and (with M W. Zemansky as a third coauthor) Basic Engineering Thermodynamics. (L) Michael M. Abbott is Associate Professor of Chemical Engineering at Rensselaer Polytechnic Institute where he has been a faculty member since 1969 Prior to that he was emp lo yed by Esso Research and Engineering. His teaching interests are mainly in the thermal sciences and in chemical process design. He is coauthor with H C. Van Ness of the books Schaum's Outline of Theory and Problems of Thermodynamics and Basic Engineering Thermodynamics (with M. W. Zemansky). (R) n = Ini is the total number of moles, and the are chemical potentials. Written for n = 1, Eq. (1) becomes dU = TdS PdV + L .d x ]. ]. showing that U = U(S,V,x) Thus, in general, the natural independent ( canoni cal) variables for U are entropy, volume, and composition. New thermodynamic properties can be defined that are functions of alternative sets of inde pendent variables. In particular, the enthalpy H and the Gibbs function G are defined as CHEMICAL ENGINEERING EDUCATION
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H :: U + P V a nd G = H TS Then nG =nu+ P(nV) T ( n S) and ( 2) (3) d ( n G ) = d ( nU) + Pd ( nV) + (nV)dP Td(nS) (nS)dT Substitution for d(nU) by Eq. (1) gives d ( n G ) = ( n S ) d T + (nV)dP + d n. l. l. (4) This equation is equivalent to Eq. (1), and repre sents an alternative fundamental property rela tion. Written for one mole of material, it becomes (5) whe n c e G = G ( T P x) Because temperature, pressure, and composition are subject to direct measurement and control the Gibbs function is a defined thermodynamic property of great potential utility An equation such as Eq. (4) is too general for direct practical application. Its value is in storing much information. Thus, we write by in spection (6) V = [a G ) aP T, x (7) and = [a(nG)) i a n 1 T,P,n. J (8) where the subscript ni indicates that all mole numbers are held constant except n 1 Application of Eqs. (6) through (8) presumes knowledge of G as a function of T, P, and x; given this, then Eqs. (6) and (7) yield S and V as functions of T, P, and x. Other properties come from defining equations; for example, by Eq. (3) H = G + TS (9) Thus, if we know how G is related to its canonical variables, we can by simple mathematical opera tions evaluate all the other thermodynamic properties; given G = G (T,P,x), we can also find S, V, .i, H, Cp, etc. as functions of temperature, pressure, and composition. SPRING 1985 All this is the legacy of J. W. Gibbs and in principle nothing more is needed. An expression giving G = G(T,P,x) is an example of a canonical equation of state. Such an equation serves as a generating function for the other thermodynamic properties, and implicitly represents complete property information. For realfluid mixtures, canonical equations of state are unknown. The problem is that such an equation must be based on experimental data. Un fortunately, there are no G meters and no con venient experimental measurements that lead easily to values of G. Without a canonical equation of state, we can make no direct practical use of a fundamental property relation. The slow evolu tion of solution thermodynamics since Gibbs' time has led to new formulations that relate much more directly to experiment Our purpose here is The slow evolution of solution thermodynamics ... has led to new formulations that relate much more directly to experiment. Our purpose is to rationalize the structure of modern solution thermodynamics. to rationalize the structure of modern solution thermodynamics. In the early years of this century, G. N. Lewis introduced several concepts basic to all subse quent developments: the partial property, the fugacity, and the ideal solution. A partial property is defined by the equation M = [a CnM)) (10) l. ani T,P ,n j where M is the molar value of any extensive property. The simplest interpretation of Eq. (10) is that it apportions a mixture property among the constituent chemical species. Thus, i\ has the characteristics of the property of species i in the mixture. Indeed, a mathematical consequence of Eq. (10) is the relation M = rx.M. l. l. {11) which shows that the partial properties combine in the simplest rational way to yield the mixture property. We see by comparison of Eq. (8) with Eq. (10) that ( 12 ) 63
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Thus, the chemical potential is identified with the partial Gibbs function. The fugacity is an auxiliary thermodynamic prop e rty related to the Gibbs function. Thus, for a mixture, the fugacity f is defined by the equa tions dG = RT d fo f (const T x) (13) and lim i = 1 (14) P+O p For the sp e cial case of pure species i, these become and dG. = RT d R. n f 1 i lim = 1 P+0 p (const T) (15; (16) For species i as a constituent of a mixture, the fugacity t. is defined by the equations l. and dG = RT d Rn f. 1 l. A lim .:.L = 1 P..,0 xiP (const T) (17) (18) For an idealgas mixture one replaces V in Eq. (5) by RT/P; then dG' = RT d R,n P (const T,x) (19) wh e re th e p r i me (') d e notes an idealgas property. Fr o m Gibbs' theore m for such mixtures, we have G' = Ix .G 1 + RTix. Rn x. 1 1 1 By Eqs (6), (7), and (9), we get and ( 20) (21) (22) (23) Each of these is implicit in Eq. (20). Moreover, Eq (8) yields whence dG~ = dG~ + RT d Rn x 1 (const T) By Eq (19) written for pure species i, this be comes 64 (cons~ T) (24) The actual properties of a fluid may be com pared with the properties the fluid would have as an ideal gas at the same temperature, pressure, and composition The comparison by subtraction gives rise to residual properties. Thus, by defini tion and (26) Applying this concept to the Gibbs function, we subtract Eq. (19) from Eq. (13) d(G s') = RT d R.n (const T,x) or (const T,x) where cf, is the fugacity coefficient, defined as (27) (28) ( 29) The integration constant vanishes, because for p = o, GR = 0 by assumption and ln cf, = 0 by Eq. (14). For the special case of pure species i, this becomes (30) For species i as a constituent of a mixture, we subtract Eq. (24) from Eq. ( 17) or f. d(G G'.) = RT d R.n ~p 1. 1. x 1 (const T) (cons t T) (Jl) where ~. is the fugacity coefficient of species i l. in the mixture, defined as (32) Integration of Eq. (31) gives (33) CHEMICAL ENGINEERING EDUCATION
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Unlike a formulation based on a canonical equation of state, the residualproperty formulation cannot provide complete property information. One needs in addition the heat capacities necessary for evaluation of properties for the idealgas state. where again the integration constant vanishes An alternative form of Eq. (4) derives from the mathematical identity Substituting for d (nG) by Eq. ( 4) and for G by Eq. (3) gives d (nGJ = nH dT + nV dP + l i dn (34) RT RT2 RT RT i For the idealgas state, Eq. (34) becomes d (nG ') = nH' dT + nV' dP + L i dn. RT RT2 RT RT i (35) With f1i replaced by G. in Eq. (34) and'. replaced i i by c'. in Eq. (35), we subtract these two equai tions: R R G.R d (nRGTR] = nH dT + !!Y..._ dP + l _i_ dn. (36) RT2 RT RT i where the definitions of Eqs. (25) and (26) have been invoked. This is the fundamental property relation for residual properties. In view of Eq. (33), it may also be written (37) Working relations for the residual properties can now be written by inspection HR= T[a(GR/RT)) = T(~J (38) RT 3T P,x a T P,x VR = (a (GR /RT) J RT 3P T,x [ ~] ap T, x (39) and .l'.n ~= (a(nGR/RT)) = (a(n .Q. n ) (40) i ani T,P,n. ani T,P,n J J where the second form in each case follows from Eq. (29). SPRING 1985 Equation (39) may be written (const T,x) where by definition VR = V V' = V RT p (41 ) Values of VR come directly from experimental PVTx data, and Eq. (41) then allows calculation of ln cf>; Eqs. ( 38) through ( 40) yield other properties of interest. This close link to experi ment is the major reason for a residualproperty formulation of solution thermodynamics. Given a PVT equation of state V = V(T,P,x) we can evaluate all residual properties. Because of its direct relation to experiment, a PVT equa tion of state is far more easily developed than is a canonical equation of state. Furthermore, the principle of corresponding states allows the generalization of PVT data and the development of generalized correlations for the residual proper ties, thus greatly extending the usefulness of available experimental data. Unlike a formulation based on a canonical equation of state, the residual property formulation cannot provide complete property information. One needs in addition the heat capacities necessary for evaluation of properties for the idealgas state. In principle, PVT equations of state apply equally to gases and to liquids. In practice, how ever, the accurate representation of liquid proper ties proves much more difficult. Thus, an alterna tive formulation of solution thermodynamics has developed for liquids. The key idea is that of an ideal solution. By definition (42) where the superscript id dertotes an idealsolution property. Expressions for all of the properties of an ideal solution follow from this equation. Integration of Eq. ( 17) from the state of pure i to the state of i in solution at the same T and P gives
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For an ideal solution, this becomes id Gl.. = G. + RT tn x l. i and by Eq. (11) Equations (6), (7), and (9) in this case yield id I.' S = lxisi Rixi in xi and (43) (44) (45) (46) (47) (4 8 ) Just as we may compare the actual properties of a fluid with its idealgasstate properties, so may we compare the actual properties of a fluid mixture with its idealsolution properties at the same temperature, pressure, and composition. Thus, we have definitions of excess properties quite analogous to those for residual properties (49) and E id M. = M. M. l. l. l. (SO) Equation (49) applied in turn to the properties whose idealsolution expressions are given by Eqs. (45) through (48) becomes GE= G IxiGi RTixi i n xi SE= S IxiSi + RLxi tn xi (51) (52) (53) (54) The excess properties are closely related to property changes of mixing LIM = M Ix .M l. ]. (55) These quantities measure the changes that occur when one mole of mixture is formed from the pure constituent species by a mixing process at constant T and P. The definition of Eq. (55) allows Eqs. (51) through (54) to be written 66 (56) (57) (58) (5 9) Thus, the excess properties are readily calculated from property changes of mixing and vice versa. Interest in property changes of mixing is focused on /l V and /lH, because these quantities can be experimentally determined by direct measurement. Unfortunately, measurements of /lV = V E and of /lH = H E for liquid mixtures do not allow calculation of GE. For this, we need vapor/liquid equiltbrium data, which are related to GE as follows. Subtraction of Eq. (44) from Eq. (43) gives E id f i G. = G. G. = RT t n x f l. l. l. i i or where the activity coefficient 'Yi is defined by fi y i xifi In accord with Eq. (11) and by Eq. (60) this becomes GE RT= Ixi tn Yi (60) (61) (6 2) Values of y 1 are calculated from experimental vapor / liquidequilibrium measurements by the equation (63) Here, i is a secondary factor of order unity that can be readily evaluated from volumetric data for the equilibrium phases or from correla tions of such data. The fundamental property relation for the excess properties follows from Eq. (34). For an ideal solution, this equation is written CHEMICAL ENGINEERING EDUCATION
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TABLE 1 Summary of key equations [ nG) nH nV i d RT = RT2 dT + RT dP + l RT dni (34) d [nRTGR) = nHR dT + nVR dP + l R, d (37) RT2 RT n i 0 1 d [nRGTEJ = nHE dT + nVE dP + R, (66) RT2 RT l n yidni ( nGid) d RT nHid nVid ~d = 2 dT + RT dP + l RT dn 1 RT (64) With .1 replaced by G. in Eq. (34) and d replaced i i id by G 1 in Eq. (64), we subtract these two equaE E E G E d ( nRTG ] = nH dT + !!Y__ dP + _i_ dn RT2 RT l RT i ( 6 S) tions where the definitions of Eqs. (49) and (50) have been invoked. In view of Eq. (60), this equa tion may also be written (66) Equation (66) is analogous to both Eqs. (34) and (37); analogous to Eqs. (38) through (40), we have HE = T [cl (GE /RT)) RT clT P,x (67) VE (cl (GE /RT)] RT= clP T ,x (68) (69) SPRING 1985 The formulation of solution thermodynamics through excess properties derives its usefulness from the fact that H E, V E, and y; can all be found by experiment. This relative abundance of experi mental entries provides alternative measurements that yield property data. However, the excess property formulation provides even lesscomplete property information than the residualproperty formulation, because it tells us nothing about the properties of the pure chemical species. In Table 1, we bring together for comparison the parent fundamental property relation for the Gibbs function and the two analogous property relations which follow from it for the residual and excess Gibbs functions. Included as well are the equations which relate the three mixture Gibbs functions to their respective partial properties. These are particular applications of Eq. (11). [e) n #I books received Fundamentals of Chemistry, Second Edition, James E. Brady, John R. Holum; John Wiley & Sons, Inc., New York; $34.95 (1984) Handbook of Powd er Science and Technology, Edited by M. E. Fayed and L. Otten; Van Nostrand Reinhold, 135 West 50th Street, New York, 10020; 850 pages, $79.50 (1984) Analytical Pyrolysis: Techniqu es and Applications, Edited by Kent J. Voorhees; Butterworths, 80 Montvale Ave., Stoneham, MA 02180; 486 pages, $69.95 (1984) Heat and Mass Transfer in Rotating Machinery, Darryl E. Metzger, Nairn H. Afgan; Hemisphere Publishing Co., 79 Madison Ave New York 10016; 713 pages, $74.50 Cheaper, Safer Plan ts or Wealth and Safety at Work, Trevor A. Kletz; Institution of Chemical Engineers, 165171 Railway Terrace, Rugby, England; (1984) Engineering Information Re sources Margaret T. Schenk and James K. Webster; Marcel Dekker, Inc., New York 10016; 232 pages, $24.75 (1984) From Technical Prof essional to Corporate Manager; A Guide to Career Transition, David E. Dougherty; John Wiley & Sons, Somerset, NJ 08873; 279 pages, $19.95 (1984) Natural Product Chemistry: A Mechanistic and Biosyn thetic Approach to Secondary Metabolism, Kurt B. G. Torssell; John Wiley & Sons, Inc., Somerset, NJ 08873; 401 pages, $24.95 (1984) The Wiley Engineer's Desk Referenc e Sanford I. Heisler; John Wiley & Sons, New York 10158; 567 pages, $34.95; (1984) Laboratory Manual of Experiments in Process Control, Editor, Ch. Durgaprasada Rao; ChE Education Develop ment Center, Indian Institute of Technology, Madras 600 036 India, $20 (1984) 67
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tiJn:>11 classroom THE NATURE OF ADJOINT VARIABLES AND THEIR ROLE IN OPTIMAL PROBLEMS 0. K. CROSSER Uni v ersity of Missou ri Rolla Rolla MO 65401 ADJOINT VARIABLES ARE frequently arbitrarily introduced into the textbook discussion of op timal or extremal theory. For example, Bryson and Ho [1] "adjoin" them to the optimization prob lem, Denn [2] introduces them as a "convenience," and Leitman [11] regards them as a transforma tion to a "useful" vector space basis. Only Jackson [10] has shown that the y are desirable as a general transformation from one set of variables which appear naturally during the formulation of the problem to the set of interest in the solution search problems. Adjoint variables are the sensitivity coefficients in optimal search problems. Adjoint variables exist because the coefficient matrix of every system (of describing equations) has a transpose, and there are, therefore, two independent solutions to the homogeneous form of the system. However, it was the late Professor F. M. J. Horn who in 1958 most directly presented the fundamental nature of the adjoint variables and their role in optimal reactors in chemical engi.. C>. K Crosser received his PhD from Rice University in 1955 and is currently professor of chemical engineering at the University of Missouri Rolla. His primary interests i nclude optimization of pre liminary plant design and fixed bed separations 68 neering [5]. The original papers [6, 7, 8] and the more complete elaboration in his thesis were in German, with results published in English [9] by 1967. Publications about Pontryagin's Principle [8, 10] became the standard literature reference, and the directness of Horn's approach became less available for the beginning student to appreciate. Furthermore, this appreciation or understanding of adjoint variables makes much of Horn's later work in optimal chemical reactorseffect of by passing, cyclical operation of nonlinear process esmuch easier to follow. This demonstration makes use of the example presented in detail in appendix I.11 of his Thesis [5]. One asks for the optimal temperature profile for a plug flow reactor with several independent chemical reactions. The set of independent chemi cal reactions is any set of the smallest number of time dependent stoichiometric equations sufficient to define all reaction compositions uniquely in time. The proper interpretation of independent is also clearly discussed in this thesis, although a more formidable presentation is now available [12]. We suppose a tubular plug flow reactor with several chemical reactions and arbitrary kinetics (Arrhenius) x1 _______ lx:0 o l. le 1 2 m dxi / dz = Vi(x ,x ... x T) (i = 1 to m) (1) and presume that the objective function has the form ( 2) M depends only upon the exit composition ( ex tents) x and a straightforward solution to the problem would be to assume a temperature pro file, calculate the exit composition vector to give C o p y ri ght ChE D ivi s i crn, A SEE 1 9 85 CHEMICAL ENGINEERING EDUCATION
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M, then presume another temperature profile and continue to adapt the profile in some beneficial way until an extreme in M was obtained. Suppose we had two such solutions (we use x for the vector of extents of the independent reactions and V for the vector of reaction rates, and the super dot to imply differentiation with respect to z) and ( 3) so that for sufficiently small differences between T 1 and T 2 a first difference (perturbation) is sufficient. Then o r y = r av ./ ax ]y + ( a v. /a T ) T 1. J l. ( 4a) where y stands for the perturbation in x caused by the perturbation r in T. We will also have the perturbed response m a M M = M(x2e) M ( x le) = l a;zy k (ze) k=l ke = ( a M/axie)' y (ze) (4b) Since both X 2 and x 1 are zero at z equal zero, y(0) is zero. Note that the matrix, [oV i/ axi] and vector, (oV i/ aT) are functions of z only, so that the system of Eqs. ( 4a,b) is a set of linear differential equations in which the coefficients are functions only of the independent variable z. y is the response of the system to r What we desire is to solve Eqs. ( 4a,b) subject to the restriction that M be an extreme, so that it is necessary that (5) (oM / axi )' is the transpose of the vector of partial derivatives of M with respect to x 1 0 that is, x at the end of the reactor A system like ( 4) is usually solved by Variation of Parameters (Boyce & DePrima [3] or Hochstadt [4]), finding first the homogeneous (complementary) solutions. The form of these solutions is more conveniently manipulated if we use the solution matrix rather than the solution vector (in contrast with usual forms for systems with constant coefficients). Therefore y = (B] ye wher e Bi j = a v /a x j (6) C has the homogeneous solution matrix (Y] such that SPRING 1986 [ Y ) = [BJ [ Y ] (7) and we take [Y (z )] to be [I] the identity matrix. Any other boundary condition on [Y] may be ob;. tained directly from this one. Using the Variation of Parameters we suppose y = [Y]c and hope to find the vector c to fit the inhomogeneous part, which is the second term of Eq. ( 4) y = { [ Y ic } = [Y] c + [ Y ]~ = [B ] [ Y ]c + ca v./ a T)T (8 ) l. Substituting from Eq. (6) we have (9) and we see that the vector of the particular solu tions c is directly related to the temperature pro file, r Now, these functions depend only on z, and we intend to keep the same inlet temperature but to alter the shape of the profile. Therefore any one of the particular solutions c must have the property z c k(ze) = f e ~k(z)dz 0 a nd since with y = [Y) c and [Y] = [I] then at z = z e (10) Then from Eq. 4b, using t as a dummy variable and recalling that ( O M / O x ; ) does not depend on z J ze m M = M dt = l k=l 0 then M = (11) (Note that is the derivative, with respect to the independent variable of the response M to the perturbation r .) We now have to solve simultaneously m + 1 linear equations involving There are m inde pendent chemical reactions, and Eq. (11) for M Since these equations must OE;! linearly de69
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Thus, the differential equations for the adjoint variables and their corresponding conditions at the end of the reactor show that the influence of the exit extents upon the value of the objective function can be obtained for any entering conditions to the reactor by integrating their adjoint variable differential equations from the end of the reactor to the entrance. pendent, their determinant must be zero M M u' = 0 or = 0 [Y) w [Y) using vectors u and w for notational convenience, then IO u I M I y I + w [Y l I = 0 and M = l o u' I w [Y) jYJ (12) As we expand the numerator determinant of Eq. (12) first about w, the first column (deleting the ith row in Y) and then about u' in the first row (deleting the jth column in Y), we will obtain the cofactors of the elements Y1 i in Y, which we label a 1i and Eq. (12) can be written !.=_,: ~,_w'[aijJu_w[Adj(Y')Ju_ 1 M IYI IYI IYI WA (13) because [a1i] is the adjoint matrix of the transpose of [Y], (a sign change occurs as the i + 1 index in the determinant decreases to the i index for w). The adjoint variables, A, are defined by Eq. (13) and Sin c e (for the extreme in M) {[Y'h} = o = [Y' ]A+ [Y' ]~+= [Y' 1\y ]A But [Y] = [Bl[Y]+[Y 1 ]A = {[Bl[Y]}' a [Y 1 l[B 1 ] then = [Y' (.l.[Y' ][B' l ~=LB']>. These are the differential equations for the adjoint variables. The boundary conditions of [Y] = [I] at z = z imply (form = 2 for clarity) because Iii is the unit ij cofactor from the identity 70 Al (ze) laM/:x 1 aM/:x 2 1 I I I A2(ze) laM/:xi aM/:x 2 1 I I I Hence A(ze) 1u I = u = (aM/ax. ) I ie ij matrix. Thus, the differential equations for the adjoint variables and their corresponding conditions at the end of the reactor show that the influence of the exit extents upon the value of the ob jective function can be obtained for any enter ing conditions to the reactor by integrating their adjoint variable differential equations from the end of the reactor to the entrance. These functions, therefore, explain how the optimal re sult is affected by changing the values of the ex tents of reaction at any point along the reactor such as the entrance. Since there is a direct cor respondence betwe en length in a plug flow re actor and time, it is equally clear how the adjoint variables apply to time optimization as well. The adjoint variables are therefore nothing more than the additional homogeneous solution for the linear perturbation. Had the problem been cast in the form of time optimal control, they would have indicated the switching functions; in troduced with an Hamiltonian or Lagrange multi plier problem, they would have been the cor responding multipliers [13]. The thing to see is that all of these structures rely essentiall y only on a Cramer's rule for solving a dependent set of linear equations and that the adjoint variables appear naturally as the added homogeneous solutions to the transpose of the system coefficient matrix, and they show how temperature changes along the reactor affect the objective function, which depends on the con version at the exit from the reactor. CHEMICAL ENGINEERING EDUCATION
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ACKNOWLEDGMENTS One of the original reviewers of this article observed that the subject of this paper is contained in modern control theory texts. It is a pleasure to suggest to students that Linear Sjjstems by Thomas Kailath (PrenticeHall 1980) is an ex cellent reference with good examples and exercises. The most directly relevant part is section 9 1 pp 598606 and example 9.13 p. 605, but there are many other items of interest throughout the entire text. The University of MissouriRolla awarded the Faculty Sabbatical during which this note was written. D. W. T. Rippin and his Systems Engi neering Group of ETH Zurich provided the affectionate welcome and gentle scholarly support. Don MacElroy offered a most helpful suggestion toward the end of the work. LITERATURE CITED 1. Bryson, A. E., Y. Ho, Applied Optimal Control, Halsted Press, New York (1975) [chapter 2, and pp 4750, pp 149150 ). 2. Denn, M., Optimization by Variational Methods, McGrawHill, New York (1969), (pp 102109). 3. DiPrima, R. C., W. E. Boyce, Elementary Differential Equations and Boundary Value P roblems, John Wiley & Sons, New York (1977). 4. Hochstadt, H., Differential Equations, Dover Press, New York (1975). 5. Horn, R. "Optimalprobleme bei kontinuierlichen chemischen Prozessen," Thesis, Tech. Hochsch. Wien, Ostereich ( 1958). 6. Horn, F., Discussion of "Optimum Temperature Se quences in Reactors," K. G. Denbigh, Chem. Eng. Sci., Special Supplement, 8, 131 (1958). 7. Horn, F., U. Troltenier, "uber den Optimalen Temperatur verlauf im Reaktionsrohr," Chem Ing Tech, 32, 382 (1960). 8. Horn, G., "Adjungtierte Variable und Maximum prinzip in der Theorie Chemischer Reaktoren," Ostereichische ChemikerZeitung, 6, 186 (1967). 9. Horn, F., "Mathematical Models in the Design and Development of Chemical Reactors," B er. B uns Ges., 74, 8189, (1970) (in English). 10. Jackson, R., "Optimization of Chemical Reactors With Respect to Flow Configuration," J. Th eo App. Opt., 2, 240259 (1968). 11. Leitman, G., An In troduction to Optimal Control, McGrawHill, New York (1966), (p 26, Chapter 2). 12. Smith, W. R., R. W. Missen, Chemical Reaction Equilibrium Analysis, John Wiley & Sons, New York (1982), [Chapter 2also see the reference by these authors in Chem Eng. Educ. which contains a re view of the literature to 1976). 13. Report: SEG / R / 128 (83), Systems Engineering Group, Tech. Chem. Labor, ETH, Zurich. SP~ING 1985 POSITIONS AVAILABLE Use CEE's reasonable rates to advertise. Minimum rate page $60; each additional column inch $25 MICHIGAN STATE UNIVERSITY Chemical Engineering ... Tenure system faculty positions. Doctorate in Chemical Engineering or closely related field. A strong commitment to teaching and the ability to de velop a quality research program is expected. Preference will be given to candidates with research interests in the areas of Biochemical Engineering, Surface Science, Solid State Phenomena, or Polymeric Materials. However, ap plicants with outstanding credentials and research interests in other fields related to Chemical Engineering are en couraged to apply. Teaching and / or industrial experience desirable but not essential. Michigan State University is an affirmative actionequal opportunity employer and wel comes applications from women and minority groups. Send applicatio ns and names of references to Chairperson, Faculty Search Committee, Department of Chemical Engi neering, Michigan State University, East Lansing, Michi gan 488241226. NOMENCLATURE I ( z X V y T M I ] ) ) Determinant Matrix (square) Vector (column) Transposed vector (row) Length of plug flow reactor Extent of reaction Vector of reaction rates Perturbation in x Temperature Objective function to be optimized m B Number of independent chemical reactions Coefficient matrix from partial derivatives of rates V y I Matrix of homogeneous solutions to Eq. 4a Identity matrix C u w A Vector of particular solutions for Eq. 4a Row vector (0M / ax1 ) Column vector cav. /clT)T ]_ Vector of adjoint variables Subscripts o Entrance to reactor e Exit from reactor i,j,k Row column indexes Superscripts i Index to independent chemical reactions 1 (ltom) Differentiation with respect to length Inverse matrix Transpose Indicates perturbation value 71
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SEMICONDUCTOR CHEMICAL REACTOR ENGINEERINC The Chemical Engineering Division Award Lecturer for 1984 is T. W Fraser Russell. The 3M Company provides financial support for this an nual lectureship award. The lecture has been presented at the University of Florida, the University of Michigan, and Colorado School of Mines. A native of Moose Jaw, Canada, Fraser Russell re ceived his BSc and MSc in chemical engineering from the University of Alberta and 1::,._ his PhD from the University of Delaware. He joined the department at Delaware in 1964 and is presently the Allan P. Colburn Professor of Chemical Engineering. Prior to beginning his academic career, lie spent two years with the Research Council of Alberta, where he did early development work with the Athabasca Tar Sands. He later joined Union Carbide Canada as a design engineer, where he completed the reactor and process design for all of Union Carbide's ethylene oxide derived chemicals His innovative process design for these oxide derivative units became the first multipurpose continuous processing unit built in Canada. In research, Russell's efforts have been directed into two major areas: design of gasliquid systems; and semi conductor chemical reaction engineering and photovoltaic unit operations. His research in gasliquid system design has resulted in over 25 publications which have been widely used by industrial concerns and have led to improved design of gasliquid contactors and reactors and biological waste treatment systems. Russell is recognized as a respected and inspiring teach er and has received the University of Delaware's Excellence in Teaching Award His efforts in education have resulted in the publication of two texts, Introduction to Chemical Engineering Analysis with Morton M. Denn, and The Structure of the Chemical Process Industries with J. Wei and M. Swartzlander. In the research and development of thinfilm photo voltaic cells, his efforts have centered on the need to apply chemical reaction engineering principles to improve the design and operation of 1eactors used in making the semi conductor material, and to ensure that solar cells developed at the laboratory scale can be manufactured in commercial quantities. He carries out this research as Director of the Institute of Energy Conversion, a laboratory with a staff of some fifty people devoted to the development of thin film photovoltaic cells. Copyright ChE Div i sion, ASEE, 1985 72 T. W. F. RUSSELL Institute of Energy Conversion University of Delaware Newark, DE 19716 THE QUANTITATIVE ANALYSIS of a reactor producing semiconductor film can be termed semi conductor chemical reactor engineering if the analysis creates procedures which improve the de sign, operation, and product quality of laboratory or larger scale reactors. The creation of a thin film semiconductor, or indeed any thin film, re quires an understanding of both molecular and transport phenomena. The process is analogous to that encountered in a typical catalytic reacting system (Fig. 1). Molecules must reach the surface of a substrate, adsorb on the substrate, diffuse and / or react on the substrate to produce a film possessing specified material and electronic properties. In a catalytic system, the product de sorbs, leaving the substrate for the surrounding fluid phase. A vari ety of reactors have been employed to move molecules or atoms to the substrate but much can be learned by considering two general types: Physical vapor deposition reactors C hemical vapor deposition reactors In a physical vapor deposition reactor the re quired solid or liquid phase species are placed in a source. Energy is supplied to vaporize these species causing molecular beams to impinge on the sub strate [1]. In a chemical vapor deposition reactor the molecular species are continuously supplied in a vapor phase which flows over the substrate. A u SURFACE ADSORPTION SURFACE DIFFUSION SURFACE REACTION FILM OR CRYSTAL GROWTH FIGURE 1. Surface molecular phenomena CHEMICAL ENGINEERING EDUCATION
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~ND PHOTOVOLTAIC UNIT OPERATIONS quantitative understanding of transfer from the bulk vapor to the surface is required and it may be necessary to contend with complex reactions in the vapor phase [2]. The performance of a reactor which produces a semiconductor film is judged by the quality of the film produced. Much effort is being devoted to ascertaining film quality by measuring optical and Electronic & Optical Properties j ,r Material Properties ,r Design & Operation of Reacting System FIGURE 2. Simplified logic diagram. electronic properties but film quality is ultimately determined by the performance of the semicon ductor in some type of electronic device. A success ful semiconductor chemical reaction and reactor analysis should provide experimentally verified models linking the electronic properties of the film to the design and operation of the reactor through a detailed understanding of the material proper ties of the film and the mechanism of film growth (Fig. 2). The logical sequence summarized in Fig. 2 has been followed by the integrated silicon circuit com munity of researchers and industrial practitioners in dealing with the key step in integrated circuit manufacture of dopant diffusion into a film. It has not been a trivial task and well over two decades of effort have gone into the development of models relating device performance to doping concentra tion profile and doping concentration profile to the design and operation of the furnace [3]. Growing a polycrystalline or amorphous film, predictably, SPRING 1985 with the desired electronic properties is an even more difficult task; one which remains an active integrated circuit research area today. If one is interested in applications which could require millions of square meters per year of semi conductor film, the task of effectively predicting film growth becomes an order of magnitude more complex. Semiconductor films covering an area on the order of a square meter or more are needed for photovoltaic panels for power generation electrophotography electronic device s for thinfilm displays For these largearea applications it is necessary to carry out research on a scale between that used in a typical laboratory and that required for com mercial operations. This unit operations scale re search needs to have both a theoretical and experi mental component which builds upon the labora tory scale research. The position of chemical re actor engineering and unit operations scale experi mentation in the research logic is shown in Fig. 3. I will illustrate the application of semicon ductor chemical reactor engineering with research we have underway in physical vapor deposition of CdS at both the laboratory and unit operations LABORATORYSCALE EXPERIMENTATION MATERIAL DEVICE DESIGN DEVELOPMENT AND ANALYSIS ~ ~r CHEMICAL REACTOR ENGINEERING ANALYSIS MATHEMATICAL UNIT OPERATIONS DESCRIPTION EXPERIMENTATION ~, PROCESSING EQUIPMENT DESIGN COMMERCIALSCALE MANUFACTURE FIGURE 3. Role of chemical reactor analysis. 73
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We originally became interested in the semiconductor research because of a need to design larger scale reacting systems. However, the last five years of research has taught us that the chemical engineering analysis is very useful in the laboratory scale research effort, and indeed essential, if such research is to be done efficiently and with minimum expense (a key issue with today's research costs). radiant heater foil substrate substrate holder top heat shield heal shield s ource bottle tontalum heater current lead and support base plate FIGURE 4. Physical vapor deposition reactor. scale and chemical vapor deposition of amorphous silicon at the laboratory scale. PHYSICAL VAPOR DEPOSITION Laboratory Scale Research A typical laboratory scale physical vapor de position unit is shown in Fig. 4. The rate of evapo ration of any material is determined by the surface temperature of the source material. For thermal evaporation this is a function of bottle geometry, the material surface area, and the design of the source heater. To make a semiconductor film, the material of interest is placed in the source bottle, heated to the point at which it evaporates or sub limes, flows out of the source bottle to the subFIGURE 5. Model equations. 74 strate, and then condenses on the substrate, the temperature of which is carefully controlled. The modeling and experimental verification of a model describing the rate of effusion for CdS which dissociates and sublimes has been thorough ly discussed by Rocheleau et al [4 ] The mass and energy balance equations written for the material to be evaporated are shown in Fig. 5 (nomencla ture in [4]). These equations can be solved numerically, given the initial dimensions of the material in the source bottle and the appropriate constitutive equations for the flow through the orifice in the source bottle. Fig. 6 gives the required equations in terms of the mass flux, r, related to F L OW RE GIME O R I F IC E P I PE F ree Mol e cul ar r = { P l gc/2 ) 112 ( P 1 p 2 l { X m / R> l) V i sco us r = C 0 Y[2p 9c
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growth measured. About 1000 separate pieces of data were obtained to verify the predicted model behavior. The model equations for each species are shown in Fig. 8. The rate of reaction of cadmium to the CdS comprising the film is assumed to be r(rxt, Cd) = k(CdS) [Cd][S ] This expression combined with the equations shown in Fig. 8 yields r(rxt,Cd) = K(CdS)[8(Cd)r(i,Cd) r(rxt, Cd)] [8(S)r(i, S) r(rxt, S)] The parameter, 8, is a condensation coefficient; r (i, Cd or S) is the rate of molecular impingement of Cd or S; r ( rxt, Cd or S) is the rate of reaction of Cd or S ; and K is a modified specific reaction rate constant whose detailed form is given by Jackson [5]. Comparison of the model behavior with some of the data is shown as Fig. 9 where the rate of deposition of CdS is plotted as a function of the 1.2 ,....r,,,~~~ 1.0 .S .8 E ....... E .6 Q) 4 numbers indicate wall temperature in K .2 1179 1198 133 4 0 1'"=1=s:o'l.L.l..12 ...1.0_0_1_ _,__'_j charge temperature, K FIGURE 7. Comparison of model behavior with ex perimental data. rate at which cadmium is delivered to the sub strate. At low values of r (i, Cd) the rate of film growth is proportional to the rate at which cad mium is delivered; as the film growth becomes sur facereaction dependent, the lines curve. The hori zontal line indicates a region in which there is not enough sulfur to react with all the cadmium being delivered to the substrate. Experimental evidence indicates that photo voltaicgrade CdS can only be made when the rate of film growth is controlled by the rate at which SPRING 1985 Cadmium r{rxt, Cd)=r{i, Cd)r(e, Cd)r{r, Cd) Sulfur r(rxt, S)=r{i, S)r(e, S)r(r, S) Cadmium Sulfide I dM{CdS) M {CdS)A dt = r(rxt, CdS) FIGURE 8. Component mass balances. cadmium and sulfur react on the surface to form CdS. Furthermore, if sulfur is not present when a cadmium molecule arrives at the surface, the cadmium will reflect and not adhere. We are just beginning to try to relate these observa tions to film properties. This second laboratory scale study of Jackson's is an attempt to learn more about the semiconductor chemical reaction engi neering necessary for the field to progress in an orderly fashion. A much more complicated set of chemical equations will be considered in the section on chemical vapor deposition. Unit Operations Scale Research For large area applications uniform defect free film with the required properties must be de posited over areas on the order of meters in dimension. It may be necessary to deposit on a moving substrate to lower costs to the level re quired to make a large area application like photo voltaics economically feasible. In this section the cooperative research efforts in photovoltaic unit operations between the Department of Chemical Engineering and the Institute of Energy Con0 ., en N 33 3 r.,.,.,,,,,,, E 25 ..... en ., 0 E I 39 16 7 "'o >< ij 8 3 Constant Sulfur FluK Indicated x 10 8 K!z:~!es 220 C Substrate Temperature u :i 0 ~'''l.''..l.....'.J._'.J.__J 0 8 3 16 7 25 33 3 41.7 50 58 3 66 7 75 83 3 91.7 100 r(i, Cd) x 10 8 Kgmoles /m 2 sec FIGURE 9. Deposition versus incident rate of cadmium in cadmium sulfide. 75
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The theoretical and experimental work of Rocheleau, Rocheleau et al, and Jackson provide the verified models of the laboratory scale batch experiments that can be used to design apparatus and experiments at the unit operations scale. version at the University of Delaware will be de scribed. CdS is the wide bandgap window semiconEncapsulant A/ R Transparent Contact Window Semiconductor (CdZn)S FIGURE 10. ThinFilm polycrystalline solar cell. ductor for the following polycrystalline hetero j unction cells (Fig. 10) CdS / CulnSe 2 CdS / CdTe CdS / Cu 2 S All of the above devices have achieved con version efficiencies (percentage of sun energy con verted to electricity) of just over 10 % although in the case of Cu 2 S cells some ZnS had to be al loyed with the CdS. At this conversion efficiency, inexpensive electrical power generation begins to become feasible if modules containing the indi vidual cells can be made cheaply. A first step in meeting this goal is to find a way to continually TAB,LE 1 Approximate Throughput and Size Specifications Throughput (m 2 / year) Deposition Chamber (m 3 ) 76 Unit Ops. Lab. Scale Scale Exps. Expts. Commercial Scale Production 1100 1,00020,000 100,0001,000,000 0.10 .5 13 1030 P ayo ff R o ll W e b o r Substr a t e ...._ Substrat e T e mp e r a tur e Control Guid e R o ll FIGURE 11. Unit operations scale deposition system. deposit the CdS on a moving substrate. The theoretical and experimental work of Rocheleau [6], Rocheleau et al [4] and Jackson [5} provide the verified models of the laboratory scale batch experiments that can be used to design apparatus and experiments at the unit operations scale. Throughputs and chamber sizes for typical units are shown in Table 1 for the three scales of operation of interest. The laboratory scale ex periments are almost always batch experiments on a static substrate. The unit operations and com mercial scale squipment for photovoltaics need to be designed for continuous deposition on a mov ing substrate. A sketch of the unit operations scale equip ment used at the Institute of Energy Conversion TABLE 2 Deposition Unit Specifications Chamber1.28 m diameter X 1.34 m long VacuumPumpdown to 5 X 10 1> torr in 2 hours Web Capacity500 mm maximum width 250 mm roll diameter (200 m Cu foil) Web Speed1.2 to 12 cm / min Deposition Zone45 cm X 10 cm SourceA proprietary design (U.S. Patent 4,325,986) providing: Constant rate2 micron / min Uniformity over 20 cm wide zone 80 % CdS utilization Web Temperature200 to 225 C Throughput0.6 to 6 m 2 / shift CHEMICAL ENGINEERING EDUCATION
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is shown in Fig. 11. This piece of equipment was designed using model equations s imilar to those presented as Fig. 5 and 6 and modified for a multi orifice geometry and the different sourcesubstrate geometry of the unit operations scale equipment. It was also necessary to expand the energy balance analysis to include radiative heat transfer between the source and substrate. The model equations their behavior and their influence on the design and operation of the unit operations scale reactor are given by Rocheleau [6] and Griffin [7 ] The specifications determining the equipment are shown in Table 2 and a photograph of the equip ment is shown as Fig. 12 FIGURE 12. Photograph of unit operations scale deposi tion system. The unit operations experimentation consisted of controlled deposition of CdS on rolls of zinc plated copper foil. Throughput of the foil ranged from 180 to 600 cm 2 per hour with film growth rates ranging from 0 5 to 2 m / min. Substrate temperatures were varied between 200 C and 250 C. At throughputs of 400 cm 2 / hour, up to 3000 cm 2 of 25 .m thick CdS was prepared in a single run. Fig. 13 compares a crosssection of con tinuously deposited CdS with that of photovoltaic grade batch deposited CdS. Xray diffraction con firms predominantly caxis orientation for the continuously deposited CdS. Resistivities of the continuously deposited CdS films ranged from about 1 to 100 ohmcm. Resistivity of the best laboratory CdS ranges from 1 to 10 ohmcm. SPRING 1985 Ul 'O u FIGURE 13. CrossSection of CdS film. Crosssection of continuouslydeposited CdS on left, crosssection of batchdeposited CdS on right. The principal means of evaluating the CdS was to measure the photovoltaic response of cells fabri cated using the material from the unit operations experiments. Results are summarized in Table 3 which shows the efficiencies of CdS / Cu 2 S cells made using both laboratory scale and unit opera tions scale CdS. The Cu 2 S layer can be made using a wet process by dipping CdS into a CuCI solution or it can be made using a dry process in which CuCl is evaporated onto the CdS and then allowed to react with Cu 2 S. A quantitative description is given by Brestovansky et al [8]. Cells made by the dry process in the unit operations scale equipment had both layers, CdS and Cu 2 S, continuously de posited on the moving substrate. Cells made by the wet process had to have the Cu 2 S layer made in a batch operation. All cells had an evaporated gold front contact. The efficiency figures show that the unit opera tions scale continuously deposited CdS is virtually of the same photovoltaic quality as the laboratory scale batch deposited CdS. It took some ten years of research to achieve the efficiencies shown for the batch deposited CdS. The continuously deposit ed CdS reached the efficiency shown in well under two years of unit operations experimentation. This could only have been achieved by drawing heavily Continued on page 106. TABLE 3 Cell Efficiencies (CdS / Cu 2 S) Wet Process (CdS Only) Dry Proces s (CdS / Cu 2 S) Laboratory Scale (Batch) 9 % 7 % Unit Operations Scale (Continuous) 8 % 7 % 77
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[iJ ;j a classroom THE USE OF COMPUTER GRAPHICS TO TEACH THERMODYNAMIC PHASE DIAGRAMS CHANDRASHEKHAR D. NAIK Singe r Co r po r ation Sil v er Sp r ings, MD 20904 PAULETTE CLANCY AND KEITH E. GUBBINS Cornell Uni v ersity Ithaca, NY 14853 T HE TEACHING of thermodynamic phase dia grams poses problems which affect both the in structor and the students. The usual approach in which the threedimensional pressuretempera turecomposition diagrams for binary fluid mix tures are represented on a twodimensional page is difficult for students to visualize. Traditionally, in order to simplify this complex situation, 'cuts' at constant pressure, temperature, or composition are made to show a truly twodimensional diagram de scribing the relationship between two of the three independent variables. However, the interre lationship of all the variables involved is lost with this approach, and the problem of comprehension intensifies as the complexity of the phase behavio r increases. Construction of threedimensional Chandrashekhar D. Naik ob tained his B. Tech degree in chemical engineering from the Indian Institute, India, in 1981. He obtained an MS degree in chemical engineering from Cornell Univers i ty in 1983 and is currently employed at Singer Link Simulations Systems Di vision in Silver Spring, Mary. land. (L) Paulette Clancy is current ly an assistant professor in chemical engineering and as sociate director of the Manu facturing Engineering Program at Cornell University She received her BS degree at the University of London and a D.Phil. degree at the University of Oxford. She held fellowships at Cornell University and at London University before joining the faculty at Cornell in 1984. (Cl Keith E. Gubbins is currently the Thomas R. Briggs Professor of Engineering and director of chemical engineering at Cornell Uni 78 models offers an alternative solution, but they are difficult and timeconsuming to produce and offer no possibility for student interaction At Cornell an alternative to traditional ap proaches was sought to improve the quality of teaching and the level of comprehension of the students. Computer graphics offers an innovative solution to these difficulties : presentday graphics hardware can perform rotational transformations of threedimensional images almost instantaneous ly and allows extensive manipulation of the viewed image by the user, making this an extremely powerful tool eminently suited to the task at hand. During the past two years a highly interactive "user friendly" graphics package has been de veloped depicting the threedimensional phase be havior of binary fluid mixtures, and it has been used in both undergraduate and graduate courses with great success. THE GRAPHICS WORKSTATION The Computer Aided Design Instructional C o py r ig ht C hE D ivision ASE E, 198 5 versity. He received his BS and PhD degrees at the Univers i ty of London and was on the staff at the University of Florida from 1962 76, when he moved to Cornell. He has held visiting appointments at Imper i al College, London at O x ford University and at the University of California at Berkeley. He has coauthored two books Applied Statistical Mechanics (Reed and Gubbins ) and Theory of Molecular Liquids (Gray and Gubbins ) (R) CHEMICAL ENGINEERING EDUCATION
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FIGURE 1. An example of a typical workstation show ing the Evans and Sutherland vector refresh graphics monitor with VTl 00 terminal, electronic tablet and stylus. Facility (CADIF) at Cornell houses "stateofthe art" computer graphics equipment used solely for teaching (and developmental work towards edu cation). The central computers for the facility are Digital Equipment Corporation (DEC) VAX ma chines, an 11 / 780 and an 11 / 750, running the VMS operating system, with DEC PDP 11 / 44 ma chines as postprocessors. Attached to these ma chines are two different types of graphical display equipment for student use: vector refresh stations with dynamic threedimensional capabilities and color raster stations for applications requiring color. In this application, the vector refresh work stations were used exclusively, these being the highly sophisticated Evans and Sutherland Multi picture Systems. Each workstation has a digitiz ing tablet and electronic stylus as the primary in put peripheral for cursor control, with a DEC VTl00 terminal for alphanumeric input. A typi cal workstation is shown in Fig. 1. An electrostatic plotter is also available for hardcopy output, a use ful and necessary addition allowing students to submit a record of their progress to the instructor. The software, which is the heart ( or perhaps more appropriately, the brains) of this application, was written in FORTRAN making use of system graphics software routines developed at CADIF. The consideration of ergonomic factors to produce a well designed application in terms of its "user friendliness" was considered essential to promote ease and clarity of use of the graphics package as SPRING 1985 well as increased flexibility. Some of the ways this was achieved include the following points: ex tensive 'help' messages and prompts for required input were made available, clear consistent "menus" for optional choices of interactive re sponse by the program were produced, and the ability to recover from mistakes or unintentional "miskeying" was provided. It was an original tenet of this study that students should not have to read a computer manual before using the pro grams. The emphasis is thus on learning engineer ing principles without requiring prior expertise in computing. REPRESENTATION OF THE PHASE DIAGRAMS The phase equilibrium data for binary mix tures needed for the representation of the phase diagram (i.e. pressures, temperatures and com positions) were generated using a theoretical model. The original RedlichKwong equation of state was employed for this purpose, chiefly be cause of the simplicity of its representation (since only two adjustable parameters are involved) and the reasonably realistic description of binary phase behavior it provides. This approach was also used It has proven to be extremely popular with the students, and has raised the level of comprehension of this potentially difficult subject above that achieved previously by using conventional means. by Willers and Jolls [1] who produced three dimensional phase diagrams on a Cal Comp plotter using the same equation of state. The wellknown RedlichKwong expressions describing the conditions for vaporliquid or liquid liquid equilibrium in terms of the pressures and chemical potentials of both phases were used to generate data points P, T, V L V G Xi, and Y 1 cover ing a region from the higher of the pure com ponent triple points to a temperature above both critical temperatures. The nonlinear equations in volved were solved using a multidimensional NewtonRaphson [2] technique. Close to the criti cal region, however convergence problems were encountered which were due, we believe, to singularities in the Jacobian matrix. These difficul ties were overcome by using the Marquardt [3] method which combines the advantages of New tonRaphson and Steepest Descent algorithms. Here Argonne National Laboratory's 'MINP AK' 79
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Computer graphics offers an innovative solution ... present day graphics hardware can perform rotational transformat i ons of threedimensional images almost instantaneously and allows extensive manipulation of the viewed image by the user ... an extremely powerful tool software package provided the subroutine for a Marquardt method of solution. Solving for vapor liquid critical lines also provided a challenge. Neither of the previous techniques mentioned was able to reproduce these highly nonlinear equa tions, and a specialized algorithm due to Deiters [4] was employed for their solution. Scott and Van Koynenburg [5, 6] classified the experimentally observed types of fluid phase dia grams into six classes, based on the presence or absence of threephase lines and their connection with the critical lines. So far we have been able to cover the two simplest classes, I and II, although extension of the programs to cover the other classes is well underway. These more complex systems will provide an even more striking visual illustration of the advantage of using computer graphics. In classes I and II both components have similar critical temperatures with the vaporliquid critical line passing continuously between thm. In class II, however, the mixture is more nonideal and exhibits liquidliquid immiscibility at low temperatures. For this class, in addition to the vaporliquid region encountered for class I, two other regions exist in the phase diagram, those of liquidliquid equilibrium and a threephase liquidSYSTEM : Pentane( 1 ) Nonane ( 2 ) IE GRESS I HELP (c u T) ~RE S X{YV (IEHP X(YV zyR[S llHP ) ( x y ) ~mm rn~)[ MUL CUT FIGURE 2. Threedimensional phase diagram for a typi cal class I system, pentanenonane. The so '. id and dashed lines show the vapor and liquid boundar i es, respectively. 80 TABLE 1 Examples of Classes 1and 11Type Behavior Available for Display By the User CLASS I Binary A zeo, Temp Pressure Mixture trope t Range K t Range Bars + PentaneN onane N 4 25590 (594) 2.3824.30(33.7) CyclopentaneNonane N 425590(596) 2.3921.92( 4 5 2) PentaneEthyl benzene N 425615(617) 4.1836 70(37.4) Acetone Trichloromethane Y 420530(535) 11 8651.18(55.6) CLASS II MethaneTetra N 8022 4 .5 (228) 1.25xl0 4 fluoromethane 38 76(46.0) PerfluoropenY,Het 240505 (506) 4.18xl0 2 tanePentane + Hom 39.28(39 7) t N,Y = no, yes; + = po s itive or negativ e deviation from Raoult's law; Het, Hom = heterogeneou s homo g eneou s azeotrop e + Th e figur e s in par e nthese s a r e th e high es t valu es of T c and P c occurring along th e critical lin e SY STEM : METHANE ( 1 )ff 4 (2) T I E GRE SSJ H ELP (cuT) ( P RES. xm) ~1 0[ :;;;;:::M P :::;;:;: rn ;;;.;i v l zy R 1 S. l[HP ) ( x y ) ~mm rn~ ) [ MUL .CU T FIGURE 3. PTx diagram for the class II system, me thanetetrafluoromethane, showing the coexisting vaporliquid equilibria (solid lines for the vapor, dashed for the liquid) and the region of liquidliquid immisc ibility (shown as solid vertical lines). Superimposed on the dia g ram (shown in bold) is a Tx cut at a pressure of 0.03 bars. CHEMICAL ENGINEERING EDUCATION
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liquidgas line. Examples of the binary systems chosen to illustrate the pha s e behavio r of classes I and II are shown in Table 1. Some of the available systems exhibit azeotropic phenomena with either positive or negative deviations from Raoult's Law, and of either a heterogeneous or homogeneous nature. Photographs depicting some of the com1 E GRE 55 I HELP (cuT) ~RES. mv (TEHP. rnv zyRES. IEHP.) ( x y ) i, : ~rn rn rn~~[ MUL CUT FIGURE 4. A 3D view of another class II system, C 5 F 1 2 C 5 H 12 The original display has been rotated by 180 degrees and tilted downward so that the view is from the hightemperature end and somewhat above the phase diagram. The regions of vaporliquid equilibria (showing an azeotrope) and liquidliquid equilibria (solid vertical lines) are clearly visible. SYST E M : C5F12(1)Pentane(2) '!'(\) 0 0 1 0 X(I) I E @ E 55 I HELP (cuT) \PRES X ( Y l) (T EMP X(YV (PRES ll H P) k X y ) / ~rnrn rn~)r MUL CUT FIGURE 5. TwoDimensional xy diagram for the system C 5 f 1 2 5 H 1 2 derived from the threedimensional phase diagram at 247.26K. This diagram shows the character istic behavior of an azeotropic system with liquidliquid immiscibility, as shown by the horizontal portion of the curve SPRING 1985 putergenerated phase diagrams are reproduced in Figures 26 illustrating the kind of image dis played for the user to manipulate. USER INTERACTION WITH THE GRAPHICS SOFTWARE PROGRAMS The image of the phase diagram (e.g. as in Figs. 26) can be manipulated by the user by means of an electronic tablet and stylus (pen). As the pen is moved over (and slightly above) the surface of the tablet, a cursor in the form of cross hairs moves over the display. When the pen is pressed down onto the tablet the graphics program is activated and performs an operation appropri ate to the area of screen chosen, given that such an area is one of the several specially designated parts of the screen called "windows" on a socalled "menu" of options. In this application of computer graphics the menu contained the following list of 'entrees' for the user to select a) READ: Allows the user to choose different binary s ystems to examine by supplying one of a given set of data file names via the terminal. b) ORBIT: This allows the phase diagram to be ro tated about its pressure and composition axes in a continuous fashion as required. c) PAN: Allows horizontal or vertical translation of the phase diagram. d) ZOOM: Provide s closer examination of a chosen area of the image by scaling the diagram up or down. SYSTEM : C S F 1 2 ( 1 1 ) ~p e::::n::t::::a =n~e ;, ( 2~)~ Tr[:::;:E=;::t;;;:;R:;:::E ;;;55=J ; T H EL P ( c uT) \~RE S rnv\ (IEMP X\YV zyRES IEHP ) ( x y ) ~rnrn rn~ J/: MUL C UT FIGURE 6. A view of the C 5 F 12 C 5 H 12 phase diagram as it appears in the initial orientation on the screen. Solid and dashed lines have the same meaning as in Figure 2. A Pxy cut is shown superimposed in bold on the diagram at a temperature of 259.9 K. 81
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e) STRETCH: Scales either of both of the P, T axes relative to the composition axis for ease of viewing. f) HELP: Summons the HELP text. g) RESET: Voids all previous manipulations and resets the system to the beginning of the program. h) SNAP: Produces a hardcopy image of the screen on a nearby plotter. i) EGRESS: Allows the user to terminate his or her session. j) TX, PX, PT: Each of these windows allows a particular highlighted "cut" of the phase diagram to be chosen by the user as shown in Figure 3 for a Tx cut at 0.03 bars for the system methanecarbon tetrafluoride, and in Figure 6 for a Px cut at 259.9K for C 5 F 12 pentane. k) CUT: Produces a Px, PT or Tx "cut" displayed alone (i.e. not superimposed on the whole phase diagram) depending on which of these three windows (PX, PT or TX) was last active. Multiple cuts (of Px at different temperatures for example) may be displayed simultaneously. 1) Produces an xy plot at constant temperature, as shown in Fig. 5 for the system C 5 F 1 2 pentane at 247 K. A 16mm movie lasting approximately thirteen minutes has been prepared to illustrate the cap abilities of this graphics package; this was pre sented at the 1983 AIChE annual meeting in Washington, D .C. SUMMARY The interactive graphics package illustrating the phase behavior of binary mixtures which has been described in this paper has been used within the chemical engineering curriculum at Cornell since the fall semester of 1982. It has proven to be extremely popular with the students, and has raised the level of comprehension of this potential ly difficult subject above that achieved previously using conventional means. The major advant age lies in the suitability of computer graphics as a means of visualizing threedimensional objects (here the PTx phase space) ; the capability of the hardware to perform rapid and continuous rota tions of the image; and, perhaps most importantly, the opportunity to interact, manipulate and con trol the image observed on the screen, brought about by flexible "userfriendly" software. All these features combine to contribute to the success of this technique in undergraduate instruction. 82 ACKNOWLEDGMENTS It is a pleasure to thank the Gas Research Institute for partial support of this work. REFERENCES 1. K. R. Jolls and G. P. Willers, Cryogenics, 329, June 1978. 2. J. Stoer and R. Bulirsch, Introduction to Numerical Analysis, Springer Verlag (1980). 3. D. W. Marquardt, J. Soc Ind. & App. Math, 11, 431 (1963). 4. U. K. Deiters, Diplomarbeit, Univ of Bochum, West Germany (1976) 5. R. L. Scott, Ber. Buns. Phys. Chem ., 76, 296 (1972). 6. R. L. Scott and P. H. Van Koynenburg, D isc Fara. Soc., 49, 87 (1970). t.; ft j book reviews FOUNDATIONS OF BOUNDARY LAYER THEORY FOR MOMENTUM, HEAT AND MASS TRANSFER by Joseph A. Schetz Prentice Hall, Inc., NY (1983) Reviewed by 0. T.Hanna University of California, Santa Barbara This book on Boundary Layer Theory is indi cated by the author to be applicable for students in mechanical, aerospace, chemical, civil, and ocean engineering. Some people would doubt that anyone could succeed in such a broad task. The author's stated goals for this book include (i) pro viding an understandable coverage of advances in turbulence modeling, (ii) presenting application of large digital computers to boundary layer prob lems, and (iii) treating mass transfer in an inte grated manner with momentum and heat transfer. It would appear that the first goal has been met reasonably well; achievement of the second goal is questionable, and the third goal has definitely not been met to the satisfaction of chemical engi neers. The book is generally well written and well organized. The coverage of laminar flows includes chapters on integral and differential equations of flow together with approximate integral solutions and exact similarity solutions. Unfortunately al most all of this material is available in a number of other sources and hardly any of it is more recent than 1960. The meager coverage of mass CHEMICAL ENGINEERING EDUCATION
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transfer is likely to be of little interest to chemical engineers. Chapters 4 and 5 do contain some use ful discussions of numerical solutions of bound ary layer problems. However, there are no example problems or computer programs. The major contribution of this book would ap pear to lie in Chapters 6 through 9, on turbulence modeling, which constitute more than half the length of the book. This material includes a useful historical perspective and spans the complete range of engineering approaches in this area up to the present time. The chronological discussion of work in turbulence modeling beginning with early mixinglength theory and progressing up to algebraic and various partial differential equa tion models should be of interest and value to chemical engineering. This discussion also inte grates well the contributions to modeling from both experimental and theory. In summary, the present book seems somewhat disappointing in its treatment of laminar bound ary layers, but in contrast it contains material on turbulent momentum transfer which should be of interest to chemical engineers. In this context the 1 )0ok can be recommended as a useful reference INDUSTRIAL HYGIENE ASPECTS OF PLANT OPERATIONS by L. J. Cralley, L. V. Cralley, and J. E. Mutchler Macmillan Publishing Company, New York, 1984: $ 60.00 Reviewe d by Klaus D. Timmerhaus University of Colorado This is the second volume of a new three volume series that is being prepared to provide recognition, measurement, and control of potential hazards normally present in various industrial plant operations. The first volume covered process flows while the third volume will treat equipment selection, layout and building de sign. This volume, divided into two major sections of unit operations and product fabrication, en compasses a broad range of industries with au thoritative information contributed by specialists from these industries. In the first section twenty three contributors discuss unit operations as dis tinct entities along an industrywide concept. Some of the unit operations considered include filtration, clarification, mixing, blending, grindSPRING 1985 ing, and spray, vacuum, freeze and fluidized bed drying. The second section includes thirteen con tributions which cover the operations and pro cedures for assembling parts and materials into final products. The industries considered in this latter survey range from such basic industries as storage battery and tire manufacturing to the high technology industries of semiconductor and liquid scintillation counter manufacturing. One may argue with the manufacturing processes that were selected by the editors; however, the breadth of the selected processes and the hazards associ ated with these processes should provide a good introduction to the hazards associated with those manufacturing processes that were not included. Even though most contributors to this second volume have adequately described each step in the unit operations and product fabrication flow of a specific manufacturing process and have included a discussion of the various health hazards that may be encountered with suggestions for their monitoring and control, many engineering read ers will be disappointed by the qualitative ap proach taken by the contributors to this important subject. Only a few of the chapters in the volume have included quantitative information that would be necessary in the design and construction of process equipment that minimizes or eliminates identified industrial hygiene hazards. Where such quantitative information is included, it is general ly quite sketchy and incomplete forcing the design engineer to consult other literature sources. Un fortunately, no guidance to such quantitative data is included by any of the contributors. Chemical engineers will also be somewhat disappointed in this volume because the "unit operations" portion of the title implies that some of the contributions will examine the conventional unit operations as sociated with heat, mass and momentum transfer. However, many of the key unit operations such as distillation, absorption, extraction, evaporation, heat transmission, etc. found in most typical petroleum and chemical processing plants have not been included. Nevertheless, this volume does manage to bring together a wealth of experience in a broad range of industries and will aid engineers, managers, and industrial hygienists to more fully recognize potential hazards of industrial processes. This, in turn, will permit these professionals to evaluate such hazards and take the necessary steps to effectively control the problem. 83
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[eJ ;j #I laboratory AN IMPROVED DESIGN OF A SIMPLE TUBULAR REACTOR EXPERIMENT ABDULFATTAH A. ASFOUR Uni v e r sity of Windso r W i ndso r Ontario Canada N9B 3P4 DESIGNING A TUBULAR FLOW reactor experi ment for an undergraduate laboratory is not a simple task. This is because the experiment will have to meet certain criteria, viz It is safe It is simple and cost effective It i s instructive It s analytical needs must be simple and easy, to meet the time constraints of an undergraduate laborator y Anderson [l] developed a tubular flow reactor experiment for an undergraduate laboratory at Princeton that utilized the system acetic an hydridewater. This particular experiment re quires r elatively elaborate safety precautions. Moreover, since the reaction is exothermic, rather expensive temperature control equipment is re quired. Samples taken at the reactor inlet and outlet are analyzed by the anilinewater method which is relatively lengthy and subject to errors. Hudgins and Cayrol [2] utilized the basic de sign of Anderson in developing a simple and interesting experiment. They utilized the classical reaction system of crystal violet dye neutraliza tion with sodium hydroxide. This system was studied earlier by other investigators, mainly in a batch reactor ( Carsaro [3]) The two novel aspects of the HudginsCayrol experiment compared with that of Anderson are A colour change can be seen between the inlet and outlet of the reactor The temperature constraint is removed. This makes the experiment operable at room temperature Also, from the safety standpoint, a relatively dilute sodium hydroxide solution (0.04 N according to Hudgins and Cayrol) is used. However, the experimental setup design given by Hudgins and Cayrol can be significantly C o pyrig ht C h E D ivision, AS EE 1 985 8 4 ~ :, < ... t t A. A. Asfour received his B Sc (Hon) and M.Sc ., both in chemical engineering, from Alexandria University, Egypt and a Ph D from the University of Waterloo Waterloo, Ontario, Canada He joined the research department of Imperial Oil Sarn i a Ontario for the period 19791981. In 1981 he joined the Chemical Engineering D partment at the Univers it y of Windsor Windsor, Ontar i o Canada His research interests are in the area of mass transfer in threephas e fluidized beds and i n membrane processes improved. The design improvements suggested in this article should make the experiment easier to run and control, significantly improve the repro ducibility of results, and expedite the process of data collection within the time constraints of an undergraduate laboratory. The main objectives of this experiment are To study the effect of residence time on conversion in a tubular flow reactor To compare the experimental conversions with those obtained from plug flow and laminarflow reactor model s THEORY As it w as established by Corsaro [3], the re action between crystal violet dye and sodium hy droxide is of the first order in the concentration of each of the reactants, i.e., the reaction is of the second order. However the reaction can be made pseudofirst order if sodium hydroxide is used in great excess with respect to crystal v iolet dye. In other w ords r dye = k [ dye ] (1) CHEMICAL ENGINEERING EDUCATION
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The value of the rate constant, k', is needed for this experiment. Students are requested to run a batch experiment to determine the value of k at the same temperature of the flow experiment ... (and) to prepare their own calibration curve of the dye concentration versus absorbance For the purpose of this experiment, 0.02 N sodium hydroxide solution is used with 6.86 x 10 4 M dye solution, i.e., the sodium hydroxide concentration would be about 282 times that of the dye, if equal volumes of reactants are used. Experimental conversions are calculated, as will be described later, and compared with theo retical conversions predicted from the plugflow model and the laminarflow model. Fo r a first order reaction in a plugflow re actor, the following equation applies assuming constant density of reaction mixture: T =~ = 1 ln(l x) (2) V 0 k = l ln (3) k C:Ao If C A is taken as [dye ]. i.e., the dye concentration at reactor exit and C Ao as [dye] 1 i.e., the dye con centration at reactor inlet, then one can rewrite Eq. (3) as follows: T =l = 1 ln [dye }1 (4) V 0 k [dye] For a first order reaction in a laminarflow reactor, the following equation applies assuming T1 T2 R1 R2 RES1 RES2 P2 S1 S2 TFR FIGURE 1. Schematic diagram for the experimental setup. M: mixer, P: pump, R: rotameter, RES: reser voir, S: sampling point, T: constant head tank, TFR: tubular flow reactor. SPRING 1985 no change in volume by reaction as well as no mixing in both radial and axial directions ( 4) x = 1 ( ~R) 2 E ( ~ R ) + ( ~ R 1 )exp( NR / 2) (5) where N R =k r V L1rr 2 7"= V o Vo The function E (y) is defined by: (X) E(y) =Jexp(cf>)dif> q, y The function E (y) is tabulated in standard tables as E 1 ( X) EXPERIMENT AL A schematic diagram of the proposed experi mental setup is given in Fig. 1. The experimental apparatus is comprised of the following compo nents. Reservoirs Constant Head Tanks Pumps Mixer Rotameters (RES 1) 200L polyethylene tank for t he sodium hydroxide solution (RES 2) 20L polyethylene tank for the dye solution (Tl) 20L polyethylene tank for the sodium hydroxide so lution (T2) 4L polyethylene tank for the dye solution (Pl) Magnet drive gear pump; Model P / N 81152 manufactured by Micropump Corp., Conford, Ca li fornia. Purchased from Cole Parmer Co (P2) Centrifugal pump. Cole Parmer catalogue No. K700430 (M) Little Giant Pump. Model 2E 38NT. Purchased from Can Lab (Rl) Size R615A rotameter. Max flow 450 ml / min with SSfloat. Purcha s ed from Brooks Instru ment Co. (R2) Size R615B rotameter. Ma x flow 1300 ml / min with SSfloat. 85
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Purchased from Brooks Instru ment Co. Reactor 4 0 meters of 3 / 8 in. I.D. Tygon tubing, wound on spool (made of lexan), 28 cm in diameter and 55 cm in length Valves Spectrometers needle valves to adjust flow Spectronic 20 (Bausch & Lomb) modi fied, as will be described later, to provide continuous measure ments. Two polyethylene tanks (RES 1 and RES 2) of capacity 20 liters and 200 liters serve as reser voirs for the crystal violet dye and sodium hy droxide solutions, respectively. Two pumps (Pl and P2) are employed to pump the reactants to two constant head tanks (Tl and T2). The over flows from the constant head tanks are returned to their respective reservoirs. The underflows from the constant head tanks go via rotameters (Rl and R2) to a small pump that acts as a mixer (M). The reactant streams are mixed in the mixer, M, and are pumped through the reactor. The tubular reactor is in the form of a helical coil wound on support. Connections are provided at the inlet and outlet of the reactor to the flow through curvettes of the spectrometers. The main advantages of the proposed experi mental setup over that suggested by Hudgins and Cayrol are Reservoirs and constant head tanks are used. This arrangement pro vi des more stable rotameter opera tion, especially at low flow rates. A flowthrough accessory which is simpler in design and operation than that suggested by Hudgins and Cayrol has been used. The flowthrough accessory shown in Fig. 2 allows one to use Spectronic 20 for continuous measurements. PROCEDURE Due to the limitation of the headroom in most undergraduate laboratories, the constant head tanks (Tl and T2) are placed about 3 meters above the rotameters' level. This limitation makes it only possible to attain maximum flow rate of 1300 ml / min of NaOH. The maximum flow rate of the dye is set at about 135 ml / min. One of the reviewers suggested the use of polyethylene instead of Tygon tubing, which discolors to a deep purple making it difficult to observe gradual colourchange along the reactor. It is believed that polyethylene is more resistant to the dye than Tygon tubing. 86 Cap made of blue glass I r.::====>FLOW IN ===FLOW OUT Parafilm used as a seal 1.Spectronic 20 Cuvette FIGURE 2. Flowthrough cuvette for the Spectronic 20. The flow rates of sodium hydroxide and the dye solution are set such that the ratio is 9 :1. One should start at the highest possible flow rate to expel all air bubbles from the reactor. One should wait for slightly longer than the residence time, for a particular flow rate, for steadystate to be reached. The reaction mixture is then allowed to flow through the Spectronic 20 flowthrough cuvettes and the readings are re corded. Usually, one waits for two minutes and takes another reading as a duplicate. Experience has shown that the Spectronic 20 readings are highly reproducible. Other flow rates of NaOH and dye solution are chosen, keeping the flow rates ratio 9 :1 as before, and the Spectronic 20 read ings are recorded. The experiment usually lasts for one hour provided that the solutions are pre pared prior to the laboratory period. Holding Time min 6.42 4.3 3.23 2.54 2.13 TABLE 1 Holding Time and Conversion Data Reynolds Number 791 1181 1651 2000 2385 PFRM conv. 71.7 57.1 47.1 39.4 34.3 Exp. LFRM Conversion conv. x 63.3 71.5 50.3 53.3 41.6 44.5 34.96 36.9 30.7 33.85 CHEMICAL ENGINEERING EDUCATION
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The vaiue of the rate constant, k', is needed for this experiment. Students are requested to run a batch experiment to determine the value of k' at the same temperature of the flow experiment. This has proven worthwhile, since temperature fluctuations in most undergraduate laboratories do not allow conducting a batch experiment at the beginning of the semester and giving the value of k to the students to perform the required calcu lations. Also, students are required to prepare their own calibration curve of the dye concentra tion versus absorbance. This leads to better results. RESULTS AND DISCUSSION Table 1 reports the residence time and the con versions from the plugflow reactor model (PFRM), laminarflow reactor model (LFRM) and the experimental conversions. Fig. 3 also, depicts the conversions against the residence time. The data reported in Table 1 and Fig. 3 were obtained from an experiment conducted on the setup available in Windsor. It is clear from Fig. 3 that, as expected, the experimental conversions fall between the con0 7 0 6 )< i 0 vi a:: w > z 0 5 0 u < z 0 ;:: u <( a:: IL 0 4 0 3 '2''3 ___ 4 .1....__15 __ .1..6 __ ....J7L..........J HOLDING TIME r I m i n.) FIGURE 3. Comparison between conversion obtained from experiment and those obtained from LFRM and PFRM. SPRING 1985 REQUEST FOR FALL ISSUE PAPERS Each year CHEMICAL ENGINEERING EDUCATION publishes a special fall issue devoted to graduate education. This issue consists 1) of articles on graduate courses and research written by professors at various universities, and 2) of announcements placed by ChE departments describing their graduate pro grams. Anyone interested in contributing to the editorial content of the fall 1985 issue should write the editor, indicating the subject of the contribution and the tentative date it can be submitted. Deadline is June 15th versions obtained from the two theoretical models, viz., the PFRM and LFRM. It is worthwhile to note here that the data re ported by Hudgins and Cayrol indicate that the experimental conversion curve crosses the LFRM curve at short holding times, i.e., experimental conversions are lower than those predicted by LFRM, which is not possible. Such results may be attributed to the obvious design flaws in the set up reported by those authors. The change of colour of the reaction mixture between the inlet and outlet of the reactor is due to the conversion along the reactor. Such a visual effect helps the students to integrate the labora tory experiment with what they learned in the lecture part of the course about conversion in tubular flow reactors. D REFERENCES 1. Anderson, J. B. "A Chemical Reactor Laboratory for Undergraduate Instructions," Princeton University, 1968. 2. Hudgins, R. R., and B. C a yrol, "A Simple Tubular Reactor Experiment," GEE XV, 1, 26, 1981. 3 Cor s aro G., C hem. Ed uc ., 41, 48, 1964. 4. Holland, C. D. and R. G Anthony, F u ndam e ntal s of Ch e mi c al R e a ct ion E n gin ee ring PrenticeHall, Engle wood Cliffs, N.J., 1979. NOTATION 0 i,e r (r) V o V X T subscript symbol for initial subscript symbols for reactor inlet and exit, respectively concentration of component A, (mole/L) pseudofirst order rate constant, (min 1 ) = length of reactor tube, (m) = k' = reaction number for a first order reaction = inside radius of reactor tube, (m) = reaction rate, (mol / Lmin) volumetric flow rate, (L / min) reactor volume, (m 3 ) = conversion V / vo = space time (min) 87
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[e)ftl lecture T h e B C. (B e for e C om p u te r s) and A. D of EQUILIBRIUM ST AGE OPERATIONS J. D. SEADER Uni v e r sity of Utah Salt Lake City, UT 84112 T HE ART OF DISTILLATION and other multicomponent, multistage separation operations has been practiced since antiquity. Although de scribing equations for distillation were formu lated before 1900, flexible, efficient, and robust procedures for solving the equations did not ap pear in the literature until after the availability of digital computers beginning in 1951. This paper is keyed to that date with B C. referring to "before computers." In 58 B C Sorel developed the first theoretical equations for simple, continuous, steadystate distillation, but they did not find wide application until 30 B.C when they were adapted to a rapid graphical construction technique for binary systems by Ponchon and then Savarit. This was quickly followed in 26 B.C by a much simpler, but restr i cted, graphical technique by McCabe and Thiele. Computer methods have largely replaced the rigorous Ponchon Savarit Method, but the graphical McCabeThiele Method is so simple and so illustrative, it continues to be popular. A modern version of Sorel's equations (shown in Fig 1) includes, in the case of a partial con dens e r, total and component material ba l ances and an energy balance around the top section of the column. Phase equilibrium on each tray i s exThe development of a separation process can be an exciting experience when computers and computer programs are available to perform the tedious calculations and allow time for more consideration of synthesis and optimization a s pects. Tutorial Lecture pr e sented at 92nd ASEE Annu a l Con ference, Salt Lake City Utah, Jun e 2228, 1984. 88 J D S e ade r has been a faculty member at the Un i versity of Utah since 1966 P r io r to that he was employed by Chevron Research and Rocketdyne H is principal technical interests are equilibriumstage operations process synthesis, process s i mulation and s ynthetic fuels. He is a Director of the AIChE and was the Annual Institute Lecturer in 19 8 3 for AIChE He prepared the section on distillation for the sixth edition of the Ch e mical Engineer s' H a ndbook and is a Trustee of CAC H E. pressed in terms of component Kvalues and one mo l e fraction sum per stage for either vapor or liquid is needed Similar equations are written for the bottom section of the column and for the feed stage. DEGREES OF FREEDOM ANALYSIS A degrees of freedom analysis for the equations was first developed by Gilliland and Reed in 9 B.C. A more thorough treatment for all types of separations and other operations was reported by Kwauk in 5 A.D If the equations and variables are counted, for a column with N stages (including the con denser and reboiler) to fractionate a feed with C components, it is found that the numb e r of equations is N (2C + 3), while the number of vari ab l es is N (2C + 4) + C + 7. Variables include stage temperatures, pressures, vapor and liquid flow rates and component mole fractions; feed Copy r ig h t C h E Division ASEE, 1 985 CHEMICAL ENGINEER I NG EDUCATION
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The nature of the equations of Sorel and the difficulty of their solution for multicomponent systems has long been recognized. The set of equations can be large in number. For example, with 10 components and 30 equilibrium stages, the equations number 690. Sixty percent of the equations are nonlinear, making it impossible to solve them directly. flow rate, composition, temperature and pressure; reboiler and condenser duties; and number of theo retical stages above and below the feed. The thermodynamic properties, K and H, are not counted as variables because they can be written explicitly in terms of the other variables just mentioned. The degrees of freedom or number of variables that must be specified equals the differ ence between the number of variables and the number of equations or N + C + 7. A simple set of specifications would include feed flow rate, composition, temperature, and TOPDO W N v.,l t,. Y l N+ l X l ,N v. BOTTO/'iUP Yl,M FEE D S TAGE l,,1 v, F l l,Fl v, zl,F L,, v,,1 l: X 1 ,F I FIGURE 1. Modern version of Sorel's equations. SPRING 1985 pressure; number of trays above the feed and below the feed; the pressure of each stage; and the reflux flow rate, L 1 This totals N + C + 6, which is one short of the number of degrees of freedom. From my own practical experience in 5 A.D., failure to supply the one additional specification can result in a calculational pro cedure that will never converge. The additional specification might be the distillate flow rate V 1 NATURE OF SOREL'S EQUATIONS AND SPARSITY PATTERNS The nature of the equations of Sorel and the difficulty of their solution for multicomponent systems has long been recognized. The set of equations can be large in number. For example with 10 components and 30 equilibrium stages, the equations number 690. Sixty percent of the equations are nonlinear, making it impossible to solve them directly. The magnitude of the values of the variables can cover an enormous range. For example, the mole fraction of a very volatile component at the bottom of the column might be very small, perhaps 10 5 0 The value of a total flow rate might be 10 4 Commonly used procedures for solving such set~ of equations, as discussed by Henley and Seader, are iterative in nature, requiring start ing guesses for some or all of the variables. Early procedures were complete equationtearing methods, suitable for manual calculations, wherein the equations were solved oneatatime in a se quential manner. With the advent of the digital computer, partial tearing methods appeared, wherein small groups of equations as well as single equations were solved at a time. Most re cently, with the availability of larger and faster digital computers, very flexible simultaneous correction methods appeared wherein all the equa tions were solved simultaneously by a modified Newton's method. An additional characteristic of Sorel's set of equations is sparsity. That is, no one equation contains more than a small percentage of the variables. For example, for the case of 10 com ponents and 30 stages, no equation contains even 89
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7 percent of the variables. This sparsity is due to the fact that each stage is only directly connected to, at most, two adjacent stages. The nature of this sparsity has been exploited in the development of the abovecited methods by seeking certain sparsity patterns which are best observed by the use of incidence matrices. The rows of the incidence matrix represent the differ ent functions or equations being solved. The columns of the matrix represent the different vari ables contained in the equations. Thus, for N = 30 and C = 10, the matrix is of size 690 by 690. If a certain equation contains a certain variable, some nonzero entry, such as an X, is placed in the matrix at the corresponding location. OtherREARRANGEMENT OF A SPARSE MATRIX TO OBTAIN A MORE DESIRABLE PATTERN 1234 5 678 571384 26 1 xxxxx 6 X X X X X 8 X X X X X X X 4 X 4 X X X X 5 X X X 5 X X X X X X 7 X 1 X X X X X 3 X X X X X X UNDESIRABLE PATTERN DES I RAB LE PATTERN ( LOWER TR I ANGULAR) FIGURE 2. Incidence matrices. wise a zero or no entry is made. The sparsity pattern depends upon the order in which the columns and rows of the matrix are arranged. The arrangement shown at the left in Fig. 2 appears to be random without pattern. How ever, by interchanging certain columns and rows, the lower triangular pattern shown at the right is obtained. Such an organized pattern, if it can be achieved, is highly desirable because it indicates that the equations can be solved oneatatime starting with the equation for the first or top row, solving for the only unknown, and then proceed ing down the rows, equationbyequation, solving for one unknown atatime, but where necessary using values of previously computed variables. Since at least 10 B.C., a number of other de sirable sparse matrix patterns have been recog nized. Shown at the left in Fig. 3 is a block diagonal pattern. The nonzero entries are all contained within the interior boldlined region. Shown in the middle is a banded matrix, where 90 BLOCK DIAGONA L BANDED BLO CK ED AND BORDERED FIGURE 3. Desirable sparse matrix patterns. all nonzero entries are contained on the main diagonal and a few adjacent diagonals Shown at the right is a blockedandbordered matrix. Such organized sparse matrix patterns, when they exist, can be found readily by computer algorithms such as the MA28 subroutine of the Harwell li brary. EQUATIONTEARING STRATEGIES For the four organized patterns just dis cussed, specialized sparsepattern computer al gorithms have been developed to solve, in an efficient manner, linearized forms of the nonlinear equations that describe the system. These sparse matrix methods strive to: (1) eliminate storage of zero coefficients and certain repetitious nonzero elements, (2) reduce arithmetic operations, in par ticular those involving zeros, and (3) maintain sparsity during computations. Lessorganized sparsematrix structures can sometimes utilize organized sparsematrix methods in an iterative manner by employing equation tearing strategies. The structure shown in Fig. 4 is lower triangular, except for an additional non[OUATION 10 11 VARI ABL E EOU ATION SoLVI NG 1 2 3 4 5 6 7 8 9 10 11 O RDE R: @, I I I I I TEAR VARIABLE BORDER OF INNER CYCLE [j 10 11 FIGURE 4 Application of tearing to a sparse matrix. CHE MICAL ENGINEERING EDUCATION
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zero entry at column 8 in row 3. The linearized equations cannot be solved directly, oneatatime, starting with Equation 1, because when Equation 3 is reached, the value of variable 8 is not known; thus, Equation 3 cannot be solved for variable 3. A tearing strategy can be employed to overcome this difficulty, but an iterative calculational loop or cycle, shown by the dashed border, involving Equations 3 through 8, is necessary. Variable 8 shown as a circled X, is the single tear variable which, when given an estimated value, results in the tearing apart of that subset of equations so they can be solved individually in order. At Equa tion 8 in the cycle, variable 3 is calculated and the value obtained is compared to the value used in Equation 3. If the two values are sufficiently close, the cycle is converged and variable 9 in Equation 9 is computed, followed by solution of Fl 2Xl~ X~ 85 0 2 2x/ + x 2 %_ 35 o G UE SSED TEARIN G STR ATEGY #1: ITERATION __2_ Xl FROM X2 FROM __2 8 110 5,9 X 10 8 5,9 X 10 8 3,0 X 10 3 GUESSED X2 FROH TEARING STRATEGY #2 : 1TERATION 11 500 6, 3"6506 o,029936 8 999 170 0 000010 9.000000 BETTER RESULTS WITH THI S S TRATEGY BECAUSE Fl IS RELATIVELY SENSITIVE TO )(2 BUT NOT TO Xl' WHILE THE OPPOSITE IS TRUE FOR Fz )(1 FR OM 2 o.02993 6 0 000010 0 ,00 000 0 FIGURE 5 Sensitivity of the tearing method. Equations 10 and 11 to complete the system. Otherwise, a new approximation for the tear variable must be determined and another iteration of the cycle completed. For the tearing strategy to be successful and efficient, it is necessary that Equation 3 not be sensitive to the assumed value of Variable 8. If too sensitive, it is best, if possible, to reorder the equations and variables to obtain a less sensitive situation. As a simple example of this sensitivity, consider the two equations shown in Fig. 5. If X 2 is the tear variable and Equation 1 is solved first, SPRING 1985 Chemical engineering educators need to closely examine courses on separation processes to make sure that students are being instructed in modern and efficient computational tools. SUCCESS I V E S UB S TITUTI ON BO UNDED WE GS TEJN (16 A,D,) I 2 3 4 5 6 I X C!'.f~ 3 X 5 X @ 1 I I I I BEFORE PARTITIONING DELAYED WEGSTEJN (28 A,D,) DoMJNANT E I G ENVALUE (20 A,O,) 1 3 5 2 4 6 .. X 0 1 I AFTER PARTITIONING FIGURE 6. Partitioning when convergence acceleration methods ignore interactions among the variables convergence is impossible to achieve from any starting guess of the tear variable. For example if the initial guess for X 2 is 8, the sequence ob tained quickly diverges, as shown. After only one iteration, X 2 has increased in value to 5.9,10 8 which is far from the solution. Alternatively, in tearing strategy #2, where the two columns of the matrix are interchanged to make X1 the tear variable, convergence is readily achieved, as shown, from any initial guess, even X1 = 500. The solution X 1 = 4, x 2 = 9 is obtained in just three iterations. These two drastically different results are obtained because Equation 1 is very sensitive to the value of x 2 and almost in sensitive to the value of X1. Thus, in this example, X 2 should not be the tear variable when Equation 1 is solved before Equation 2. In the example just considered, the guess for X1, at the beginning of each iteration is set equal to the value computed from Equation 2 in the previous iteration. This procedure, called succes sive substitution, can be slow to converge, and, therefore, a number of some simple and some complex procedures have been developed to ac celerate convergence. These procedures are par ticularly useful when more than one tear variable must be used as in Fig. 6. The example at the left has two tear variables. Some methods, including successive substitution, bounded Wegstein, de layed Wegstein, and dominant eigenvalue, ignore interactions among the variables. When inter actions do not exist, it should be possible to inter change columns and rows of the matrix to obtain 91
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3 4 5 6 NEWTON BROYDEN (18 A,D ,) FIGURE 7 Convergence acceleration methods that ac count for interactions among the variables. a block diagonal structure, shown at the right of Fig. 6, which allows separate computations of the two individual partitions, each containing just a single tear variable. The more common case is when interactions among the tear variables exist, as shown in Fig. 7, where the two tear variables are 5 and 6. The iterative cycle includes all equations. Then, par titioning is not possible and, if the interactions are strong enough, convergence acceleration by Newton's method or a quasiNewton method, such as that of Broyden, may be desirable. SIMPLE AND COMPLEX SEPARATION OPERATIONS The nature of the sparsematrix pattern ob tained from Sorel's equations and the correspond ing calculational procedure depends on a number of factors, including: (1) selection of the work ing equations, (2) selection of the variables, (3) degree of flexibility in the specifications, ( 4) order of arrangement of the equations, (5) order of arrangement of the variables, (6) functionality of the physical properties, and (7) method by which any equations are linearized. An additional factor that influences the nature of the sparse matri x pattern of Sorel's equations is the type of separation operation. In simple distillation, a single feed is separated into two products, a distillate and a bottoms; energy re quired to separate the species is added in the form of heat by a reboiler at the bottom of the column where the temperature is highest. Also, heat is removed by a condenser at the top of the column where the temperature is lowest. This frequently results in a large energy input requirement and 92 low overall thermodynamic efficiency, which was of little concern ( except for cryogenic and high temperature processes) before 22 A.D. when energy costs were low. With recent dramatic in creases in energy costs, complex distillation opera tions ( described by Seader in the 6th edition of Perry's Chemical Engineers' Handbook) and sys tems are being explored that offer higher thermo dynamic efficiency and lower energy input re quirements. Complex columns and systems may involve multiple feeds, sidestreams, intermediate heat transfer, multiple columns that may be inter linked, and in some cases, all or a portion of the energy input as shaft work. COMPLETE TEARING METHOD Simple and complex distillation operations have two things in common: (1) both rectifying and stripping sections are provided such that a separation can be achieved between two compon ents that are adjacent in volatility, and (2) the separation is effected only by the addition and removal of energy and not by the addition of any mass separating agent (MSA) such as in liquidF EED BUBBLEPOINT L I QUID F = 100 \ = 0 .5 XT = 0,5 DIST I LLAT E V 1 = 5 0 BOTTOM S (YB 0,75) (YT 0,25) ALL PRESSURE S l ATM FIGURE 8. Example of specifications and tear variables for top down, bottomup, stagebystage method. liquid extraction. Sometimes, other related multiplestage vaporliquid separation operations, such as refluxed rectification, reboiled stripping, absorption, stripping, reboiled absorption, re fluxed stripping, extractive distillation, and azeo tropic distillation, as described by Seader, may be more suitable than distillation for the specified task. All these separation operations can be re ferred to as distillationtype separations because they have much in common with respect to calcula tions of thermodynamics properties, vaporliquid equilibrium stages, and column sizing. For calcuCHEMICAL ENGINEERING EDUCATION
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lations involving such operations, prior to digital computers, the factors influencing the nature of the sparse matrix pattern from Sorel's equations were chosen so that a complete tearing method could be employed so the equations could be solved sequentially oneatatime. Many techniques were proposed, with the stagebystage methods of Lewis and Matheson in 19 B.C. and Thiele and Geddes in 18 B.C. being the most useful. In 6 A.D., features of these two methods were com bined into a single method, applicable to computa tions with a digital computer. Specifications are the simple set discussed previously and illustrated in Fig. 8 by an example involving two compon ents, benzene and toluene, and five theoretical stages. The tear variables (2C + N1) in number, and typical initial guesses for them are shown in parentheses. These tear variables include the com ponent mole fractions in the distillate. The initial guesses for the distillate composi tion are conveniently obtained by using a rear rangement of the Fenske equation with the mini mum number of equilibrium stages set equal to onehalf of the total number of specified equilib rium stages. The sum of the component flow rates in the distillate must equal the specified total dis tillate flowrate and, for each component, the sum of the flow rates in the distillate and bottoms must equal the feed rate. The equations used are modifications of Sorel's equations, and include total material balances, component material balances, bubblepoints, dew points, energy balances, an adiabatic flash for the feed stage, and equations for reestimating distil late mole fractions. The incidence matrix, shown in Fig. 9, is lower triangular except for the six tear variables, which are represented as circled X's. They appear in vertical columns in the order L 2 V 5 V 4 T 3 Y 1,11 and Y 2,1 The latter are the assumed distillate mole fractions. The variables across the top and the equations corresponding to the rows are ordered by stage number, as shown, where the stages are numbered from the top down, with 1 as the partial condenser and 5 as the partial reboiler. The calcu lations begin in the upper left corner and proceed down the diagonal. At the upper left corner, the first two equations, which each contain only a single unknown, are solved directly for the bottoms rate L s and the toptray vapor rate V 2 All remain ing equations are enclosed within the dashedline border, which contains all six tear variables. This large, squared region is the outer loop that conSPRING 1985 S TAGE CALCULATION: ~ :, xxx xx ,, @ Te a rVariable X ,. xx Border of Inner Cycle xxxx QNonlln earEquati o n f)I I I I 00 v I 1 1 Yz ,1 1 FIGURE 9. Incidence matrix for topdown, bottomup, stagebystage method (6 A.D.). tains 40 equations to converge. The matrix is 42 x 42, rather than 35 x 35 (calculated from N[2C + 3]) because the feed stage temperature is computed with three differ ent equations, and V 3 and the vapor and liquid mole fractions for the feed stage are computed with two different equations. Calculations for the outer loopinner cycle begin with stage 1, the partial condenser. All cal culations involve just linear equations in the case of compositionindependent properties, except for T 2 which is computed iteratively from a non linear dewpoint equation. Variables computed from nonlinear equations are boxed. Calculations for stage 2 are completed next. Seven equations are involved, with the last five contained in a dashed inner loop, containing a single tear vari able, L 2 At this step in the calculations, the stage above the feed stage has been completed and the calculation steps are now transferred to stage 5, the partial reboiler. Here, two nonlinear bubble point equations are encountered for T s and T and another tear variable, V s is contained in a 5 x 5 matrix. Similar calculations are made next for stage 4. Finally, the feed stage (3) is computed by making an adiabatic flash calculation. The last two equations at the lowerright corner are used to compute a new estimate of distillate composition by comparing the feed flash conditions with those computed from the topdown and bottomup stage 93
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The method works best for feeds of narrowboilingrange components. Otherwise, estimates of distillate composition may be too uncertain and cause difficulty in convergence. If feeds are wideboiling, the bubble and dewpoint calculations become sensitive and convergence is jeopardized calculations, with an adjustment made to main tain the specified distillate rate. In all, four inner loops are contained within one major outer loop. Thirteen equations are in fluenced directly by the guesses for the distillate composition and ten others are influenced in directly by the corresponding bottoms mole fractions. Thus, although the complete tearing method is suitable for manual calculations, the method is relatively inefficient and limited to in sensitive cases of simple distillation of nearly ideal solutions with low reflux ratios. The method works best for feeds of narrow boilingrange components. Otherwise, estimates of distillate composition may be too uncertain and cause difficulty in convergence. If feeds are wide boiling, the bubble and dewpoint calculations be come sensitive and convergence is jeopardized. In any event, convergence may be slow, unless special acceleration techniques are used. However the calculation by hand of just a few stages for a multicomponent mixture by this method is a very worthwhile learning experience; one not to be missed. EQUILIBRIUM FLASH METHOD Some of the limitations of the previous method were eliminated by McNeil and Motard (23 A.D.) in their development of a stagebystage algorithm that utilizes adiabatic or percent vaporization flash calculations. Their technique requires [(N1) (2C + 2) 2] tear variables, but, as shown in Fig. 10, initial guesses for all but (N2) of these variables can be set to zero. The (N2) vari ables are interior stage temperatures, which are relatively easy to estimate. If the feed is predominantly liquid, the pro cedure starts with an adiabatic flash at the feed stage followed by stagebystage adiabatic flashes in a downward direction until the partial reboiler is reached, where a percent vaporization flash is made. Subsequent adiabatic flashes are made moving up the column until the partial condenser is reached and another percent vaporization flash is made. Additional sequences of flash calcula tions are made moving down, and then up, the column until convergence is achieved. 94 The method is not particularly suitable for manual calculations because adiabaticflash compu tations are tedious. However flash computer sub routines are readily available, and i t is relatively easy to construct an executive routine to apply the method. It is, therefore, another very worth while learning exercise, particularly because column startup is simulated. The method is ap plicable to complex distillation, and initial guesses for the tear variables are easily obtained from vapor pressure data. The flash calculations are usually not sensitive, but convergence, although DOWN AND UP STAR T I N G F ROM FEED STAGE F E ED BUBBLE POINT LIQUI D F = 100 XB = 0.5 XT = 0,5 DJ S TI LLA T E v 1 = 50 L 1 = 10 n 2 = 2 00 Fl ( L 2 = O) n 3 = 200Fl n 4 = 2 00 F l n 5 = 200Fl FIGURE 10. Example of specifications and tear vari ables for equilibriumflash method of McNeil and Motard (23 A.O.) .. almost certain, can be very slow, particularly for large ratios of internal traffictofeed flow rate. MATRIX METHODS Rather than use an equationbyequation com plete tearing technique for Sore l 's equations, Amundson and Pontinen in 7 A.D., in a significant development, showed how the equations could be combined in a manner such that they could be solved in the order of type of variable, rather than by stage. However, only partial tearing was achieved and the method involved solving C sets of N x N simultaneous linear algebraic equations CHEMICAL ENGINEERING EDUCATION
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To do this, they used full matrix inversion, which often led to computational difficulties. These diffi culties were overcome by taking advantage of the sparse tridiagonal form of the matrices and apply ing Gaussian elimination or LU decomposition in EQUATIONS AND VARIABLES ORDERED BY TYPE FEED B UBBL E POINT LIQUID F = 100 X 8 = 0,5 XT = 0,5 DISTILLATE v 1 = 50 (Tl = 170 F) ,._.__........, L 1 = 10 n 2 = 185 'F) CT 3 = 200 Fl CT 4 = 215'F) CT 5 = 230 F) FIGURE 11. Example of specifications and tear variables for bubblepoint method of Wang and Henke (15 A.D.). the manner of Thomas. In 15 A.D. Wang and Henke applied the Thomas algorithm to narrow boiling feeds, while Burningham and Otto re formulated some of the equations in 16 A.D., fol lowing the work of Sujata in 10 A.D., to apply them to wideboiling feeds typical of absorbers and strippers. The need for two such partialtearing methods was shown clearly in 13 A.D. by Friday and Smith, who referred to the two procedures as the bubblepoint and sumrates methods. An N X N tridiagonal equation for each com ponent is formed by combining the component material balance, phase equilibrium, and a total material balance to form an equation in liquid phase mole fractions, stage temperatures, and vapor flow rates. By choosing the temperatures and vapor flow rates as tear variables, the equa tions become linear in the mole fractions, with no more than three mole fractions contained in any one equation, because one stage is connected to no more than two adjacent stages. For each com ponent, the linear equations are ordered by stage. The result is a tridiagonal matrix equation, where the nonzero coefficients are contained only on the three principal diagonals. The solution of the matrix equation is easily achieved by Gaussian elimination, as shown e.g. by Carnahan, Luther, SPRING 1985 and Wilkes in no more than 20 lines of FORTRAN code BUBBLEPOINT METHOD The bubblepoint method of Wang and Henke in 15 A.D. utilized the tridiagonal matrix al gorithm to obtain a computer method for solving distillation problems for relatively narrowboiling feeds. The specifications for the same 5stage, 2component example used above are shown in Fig. 11. The tear variables are the stage temperatures and vapor flow rates. The distillate rate V 1 and the reflux rate L 1 are specified. Initial guesses for these tear variables are obtained with a minimum of effort by assuming constant molar overflow, in the manner of the McCabeThiele method. Esti mates of the stage temperatures are obtained by linear interpolation of the distillate and bottoms temperatures, which may be computed by dew point and bubblepoint calculations assuming the most perfect split of the feed components, con sistent with the specified distillate rate. Wang and Henke utilized a form of Sorel s equations that permits the solution by variable type rather than by stage as in the first two methods described. The equations include a total material balance to compute liquid traffic, a com ponent material balance combined with phase 1X X I X I X X Ix x I X X I X x 1 x x I X X TYPE VARIABLE: X X X X X X X X X X X X X XX X X X X X X X X X X X X X X X X X X X X X X X X X X XX L a X X X X X X X X X X X X X X X I XX X X X X I L _ _ ____ x _x; X X X XX X XX X X X XX XX X XX Tear Variable D Nonlinear Equation Border of inne r Set of Linear Equations cyc le FIGURE 12. Incidence matrix for bubblepoint method 95
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equilibrium and total material balance to compute liquidphase mole fractions, bubblepoint equa tions to obtain stage temperatures and vaporphase mole fractions, and energy balances to compute vapor traffic. Although the same variables are computed starting from Sorel's equations, the incidence matrix, shown in Fig. 12, is quite different from the stagebystage method. The matrix is lower triangular except for the circled tear variables and N x N ( 5 x 5 in this case) block sub matrices for each component (two in this case). The overall size of the matrix for the example is only 35 x 35 because no variable is computed from more than one equation. At the upper left corner the first two variables are the same as before and are com puted directly as before. All but the last two of the remaining equations are contained in one large loop. Then the two tridiagonal submatrices are solved separately to obtain the liquidphase mole fractions. All remaining variables are computed oneatatime. Stage temperatures are computed from the nonlinear bubblepoint equation. This is followed by computation of vaporphase mole fractions Energy balances give the vapor traffic and total material balances give the liquid traffic. The cycle is repeated until the tear variables are converged. Then the duties of the reboiler and con denser are computed. For narrowboiling feeds, the initial estimate of stage temperatures and vapor traffic will be EQUATIONS AND V A RIA B LES ORDERED BY STAGE FEED B UBBLEPO I NT LIQUI D D IS TILLATE C T = 1 2,5 (Tl = 170 F) n 2 = 1 85'F) (T 3 = 200'Fl (T 4 = 215'Fl
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sentative of one of the better ones. The equations and variables are ordered by stage. To reduce the size of the matrix to be handled, component flow rates replace mole fractions and total flow rates. Thus, for the five stage, twocomponent example shown in Fig. 13, the number of equations to be solved is 25, rather than the 35 previously. The specifications are different from previous ones in that distillate and bottoms purities replace re flux and distillate rates. Such specifications should be used with caution and the Fenske (19 A.D ) minimumstage equation should be checked to make sure that the minimum number of required stages for the specified purities is less than the specified number of five. Theoretically, initial guesses must be provided for all 25 unknowns, but these guesses can be generated by the program based on guesses of just a few temperatures and vapor rates as shown. These guesses are called tear variables here, but are really not tear vari ables in the strict sense. The initial guesses are generated easily from the T and V guesses by solving the tridiagonal matrix equations of the WangHanke method for the liquidphase mole fractions, from which the initial guesses for the component flow rates are obtained readily from their definition and the component material balances. The NaphtaliSandholm method only involves three types of equations, namely stage component material balances, phase equilibrium in terms of Kvalues, and stage energy balances. The form of the equations is almost identical to the original equations of Sorel. Because bubblepoint, dew point and flash calculations are absent, sensitivity problems are largely avoided. With equations and variables ordered by stage, the incidence matrix, shown in Fig. 14, is block tridiagonal in shape. The blocks are 5 x 5 in this example. The matrix is for the linearized form of the equations, which permits the application of Newton's method. Thus, the matrix is the Jacobian of partial derivatives and an X entry signifies a nonzero derivative. The entire matrix is iterated to convergence. The solution of the block tridiagonal matrix is obtained readily by modifying the previously mentioned Thomas algorithm for a tridiagonal matrix. The only significant changes are the re placement of matrix multiplication for scalar multiplication and matrix inversion and multipli cation for division. If large numbers of components are present, the (2C + 1) x (2C + 1) submatrices SPRING 1985 may be large and timeconsuming to invert. The convergence criterion is based on the sum of the squares or so called square of the Euclidean norm of the three different types of functions. Early iterations are often damped to avoid corrections that are too large. Because of the block nature of the matrices in the Naphtali Sandholm method, it is not at all suitable for manual calculations. Furthermore, the computer program is rather complex. Consequent ly, it is best to obtain the code from one of several STAGE CALCULATION: 1 2 3 I 4 I 5 X X X X X X X X X X X X X X X X X X X X X X X X X X X X X X X X X X X X X X X X X X X X X X X X X X X X X X X X X X X X X X X X X X X X X X X X X X X X X X X X X X X X X X X X X X X X X X X X X X X X X X X X X X X X X X X X X X X X X X X X X X X X X X X FIGURE 1 4 In cidenc e m a t r ix f o r s i multaneousc orrec ti on met h od sources, e.g. Fredenslund et al (26 A.D.). Versions ranging from PC to Mainframe codes are avail able. The method is applicable to all singlecolumn, complex multistage operations, including those with highly nonideal liquid solutions Flexibility in specifications is provided at the top and bottom of the column by substituting specification equa tions for the condenser or reboiler energy balances. For example, specification options at the top in clude condenser duty, reflux rate, reflux ratio, dis tillate rate, component molefraction purity and component distillate rate. Convergence is rapid from good initial guesses, but may otherwise be slow and require damped corrections The method can fail, particularly if initial guesses are very poor. C ONTINUATIO N METHOD AND MULTIP L E S T E AD YSTA TE S O LUTI ON S Ideally, especially in practice, one would prefer 97
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a multistage multicomponent separation com puter method that would offer complete flexibility in specifications and would always converge to a correct solution. Newton's method and most of its variants are known to be only locally con vergent. That is, the initial guesses must be within a certain region of the variable space or con vergence will not be achieved. This region can be expanded by adjusting the Jacobian in Newton's method or employing a hybrid method such as that of Powell (19 A.D.) or Marquardt (12 A.D.), ""0 St artin g Newton guess Powell 10 0 Root s 10 20 FIGURE 15. Example of regions of convergence for some methods of solving nonlinear equations which combine the best features of Newton's method with steepest descent. To achieve complete robustness, however, it is necessary to employ a globally convergent technique, such as differential homotopy continuation, for which four algorithms, in FORTRAN, have become available starting in 25 A.D. The one by Kubicek is particularly easy to understand and apply, but is not written for sparse matrices. The regions of convergence for a simple two nonlinearequation example are shown in Fig. 15. Newton's method will converge to X1 = 1 and X 2 = 4 from an initial guess of X 1 = 2, X 2 = 5. As seen, another root exists at X1 = 4.07 and X 2 = 0.65, which can be reached by Newton's method from a nearby initial guess. With Newton's method, the initial guesses must lie within the 98 rather narrow and confined crosshatched regions. With Powell's hybrid method (which is available in the Harwell library, the IMSL library, and MINP ACK) the region of convergence is expanded outward to the dashed lines. Both methods will fail badly from a starting guess of 15 and 15. The use of differential homotopy continuation gives convergence from any starting guess, with a typi cal homotopy path to one of the two roots, shown as a dashdot line starting from (15, 15). Many types of homotopy paths have been pro posed, with the linear homotopy being common. The linear homotopy, h, is set equal to the function to be solved, f (X), multiplied by a homotopy pa rameter, t, and added to a function g(X), whose solution is known, multiplied by the function (1 t). The calculations start from the known solution at t = 0, where h = g and X = X 0 and move along the path of h vs. t as t is gradually in creased to a value of one, at which point h = f, whose solution X* is to be determined. Choices for the function g (X), with a known solution, X 0 are almost unlimited. For consistency with Newton's method, the Newton homotopy is useful, where g(X) is set equal to f (X) f (X 0 ), where X 0 can be selected arbitrarily. With this homotopy, h equals f(X) minus (1 t) times f (X 0 ). Alternatively, (1 t) can be replaced by a new homotopy parameter, 'A., to obtain a slightly more compact form for the homotopy expression. The path will then be from 'A. = 1 to 'A. = 0. If the homotopy path is simple, without turn s or rapid changes in X with t, classical continuation can be employed by selecting a sequence of values of t at 0, t 11 t 2 t s etc., and 1, with X being solved from h at each step by Newton's method using an initial guess equal to the solution from the pre vious step. Thus, Newton's method is embedded into classical continuation. This technique of using continuation is not common though because it is not globally convergent and can not, in general, solve problems that fail with Newton's method alone, which amounts to moving in one step from t = 0 to t = 1. To be robust, one must closely follow the homotopy path and not just continually take steps int with Newton corrections in Xspace. For example, classical continuation as well as Newton's method will fail on the cubic equation, x 3 30x 2 + 280x 860 = 0, because of two singular points at about x = 7.418 and 12.582, where the derivative of the function becomes zero. CHEMICAL ENGINEERING ED UC ATION
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Stag e 1 14 M P 3 0 II 18 FIGURE 16. Example of specifications for an interlinked system of Petlyuk towers. For initial guesses of x less than about 12.6, Newton's method fails to find the single real root at about 15.55. The homotopy path for this cubic function depends on x 0 the choice of g (x). For g(x) = x x 0 or g(x) = f(x) f(x 0 ), the two singularity points become turning points in the path, and it is important that the continuation method follow the path closely around these turn ing points to avoid cycling in the manner of Moses in the Sinai. Consider the application of the homotopycon tinuation method to the Petlyuk system of two interlinked towers shown in Fig. 16. The feed is a ternary mixture, which is to be separated into three products. A sloppy split is made in the pre fractionator, with the final three products being produced in the second tower. Reflux and boilup for the first tower are provided by the second tower. The two towers in the system can be solved by alternating back and forth between separate iterations on the individual towers. But numerous studies have shown that it is more efficient to con verge the two towers simultaneously. The stages for the two towers are ordered as shown starting at the top of the second tower, switching to the first tower after stage 11, and then switching back to the second tower after stage 15. Specifications SPRING 1985 include interlink flows from stage 4 to stage 12 and from stage 16 to stage 15; the reflux ratio; the middle product flow rate; and the bottoms flow rate. The types of equations solved are as in the NaphtaliSandholm method except that balances include interlink flows and provision, if desired, for entrainment of liquid droplets, occlusion of vapor bubbles, and chemical reaction. Phase equilibrium equations can include a Murphree plate efficiency that can be specified by component and tray location. A FORTRAN computer code for applying differential homotopycontinuation to such a prob lem was reported by W ayburn and Seader in 32 A.D. Considerable flexibility in specifications is provided, including at any stage, total flow rates or ratios, component flow rates or purities, and stage temperatures or heat transfer rates. The equations are linearized in the Newton manner and ordered by stage to a bordered, blockdiagonal structure, which is processed by an efficient and stable blockrowreduction algorithm. An attempt is first made to solve the equations by Newton's method, using a line search on the Euclidean norm of the function residuals to determine the best damping factor. If Newton's method fails, then differential homotopy continuation with a linear f(x) Af(x 0 ) = O IVP: df (x) g__Px crxuu T ($) (i) + (9) 1c: P = o~ A = L x = x 0 0 2 I FIGURE 17 Equations for differential homotopy con tinuation. Newton homotopy is employed. The differential form of homotopy continuation was first proposed by Davidenko in 2 A.D. As shown in Fig. 17, the homotopy function, f(X) H(X 0 ), is differentiated with respect to arc length, p (i.e., distance along the path), to con vert a system of M nonlinear equations to a system 99
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of M + 1 ordinary differential equations that constitute an initial value problem. Because of the extra variable, p, an additional equation is need ed. This is provided by the Pythagorean theorem applied in (M + 1)dimensional space. Fortunate ly, the set of differential equations is not stiff. Rather than simply integrating the differential equations by, say, a RungeKutta method, it is preNEARLY BLOC K TRIDIAGONAL AND BORDERED FORM ~Sis I 2 3 4 5 6 7 8 910ll 1213141516171819 :r IBC 2 ABC 3 ABC 4 A BC C 5 ABC 6 ABC 7 ABC 8 ABC 9 ABC 10 A BC II AJ3.]. 12 A 7Efc C 13 ABC 14 ABC ffi ABC A ABC 17 ABC 18 ABC AB FIGURE 18a. Example of incidence matrix for homotopy continuation method. ferable and more efficient to follow the homotopy path by alternating between an Euler predictor for the differential equations and two Newtonstep correctors for the nonlinear homotopy equations. The Euler step moves the variables somewhat off the path, but the Newton step corrects the vari ables back sufficiently close to the path. An im portant aspect of such a technique is the size of the Euler step, for which a number of stepsize al gorithms have been proposed, as discussed by Seader and Wayburn. The incidence matrix for the Petlyuk column example is in Fig. 18a, where the organization is by stage. Each letter, A, B, or C actually repre sents, in this example, a nonzero 7 X 7 submatrix, which applies to both the coefficients of the differ ential homotopy equations and the Jacobian partial differentials of the nonlinear homotopy equations. The matrix is almost block tridiagonal with bottom and rightside borders, which contain 100 the nonstandard specifications. The four disperse submatrices of A and C, located above and below the three principal diagonals represent the inter links. By moving the number 4 and 16 rows and columns, which contain the disperse submatrices, to the borders, the block diagonal and bordered matrix form, shown in Fig. 18b, is obtained. Solu tion of the corresponding matrix is achieved block by block, starting at the upper lefthand corner, by a blockrow reduction algorithm, which treats the righthand border as part of the righthand side vector. The differential homotopycontinuation method has been applied to the interlinked system shown in Fig. 19 for a ternary aromatic system, over a range of reflux ratios from 4 55 to 5. 75. Purity specifications of between 90 and 95 mole % are made for each product, and a bottoms rate of 380 is specified. The program must compute the re quired interlink flow rates, including L, the liquid interlink recycle from the second column back to the first column. In some cases, Newton's method converged, for this system, while in other cases, the differentialhomotopycontinuation method had I 2 3 5 6 7 8 9 10 II 12 13 14 15 17 18 19 4 16 BLOCKED AND BORDERED FORM I 2 3 5 6 7 8 9 10 II 1213 14 15171819 4 16 BC ABC AB C BC A ABC ABC ABC ABC ABC AB C l:i C A ABC ABC AB C BC A ABC AB A = C M A AC B FIGURE 18b. Permuted incidence matrix for homotopy continuation method. CHEMICAL ENGINEERING EDUCATION
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to be applied to obtain a solution. In Fig. 20, a plot of L, the liquid interlink rate versus the reflux ratio, shows unexpected multiple solutions, three in number. For example, at a re flux ratio of 5, the specifications were achieved with three different liquid interlink rates of about 110, 330, and 420 lbmoles / hr. Such multiple solu tions have long been known to exist for certain cases of an adiabatic reaction in a CSTR reactor but have not been obse r ved previously for distilla tion When such solutions are close together, as for solutions 1 and 2 at low reflux ratios, possible control problems could arise l.0 1 U1. r ttu rt T hr o 1,1 gllout lubt11 ., o1nt llqu l df tt d L~ l tt/h r h n zt lll! ( Bl 200 Tolu t ne ( T l 2 00 Olyl e ne ( I ) 4 00 15 X T = 0 90 X 8 = 0 95 B = 380 FIGURE 19. Example that gave multiple solutions. The continuation method is not at all suitable for manual calculations. The computer code is lengthy, but is applicable to all kinds of complex multistage operations, including interlinked columns. Except for tray numbers, complete flexi bility in specifications is permitted. When the homotopy is constructed p r operly, convergence is always achieved. The method is best suited for cases where the NaphtaliSandholm fails or can't be applied. The method can find multiple solutions if they exist. TRANSPORT MODEL Sorel's equilibriumstagemodel of almost 100 years ago has served us well in the calculation of multicomponent, multistage separation opera tions. However, that model has al w ays been sus pect for applications to systems of known moderSPRING 1985 a s~~, 07 < 06 z .05 U D 4 0 E,, 03 A s a 02 A C f .~s 4 5 4 7 4. 9 5 1 5 3 5 5 5 7 REFL UX R A TIO FIGURE 20. Multiple solutions to Petlyuk towers. atetolow stage efficiency For that reason, some programs, such as the SC method of Naphtali and Sandholm and the differentialhomotopycon tinuation method of Wayburn and Seader in corporate a Murphree tr a y efficiency, w hich ac counts to some degree for masstransfer effects. Howe v er the plate efficiencies must be specified, and heat transfer effects are ignored. A better approach is to apply a transport model to handle nonequilibrium directly. Such a model has just been developed by Krishnamurthy and Taylor, who account for multicomponent masstransfer interactions and heat transfer Their modeling e q uations are w ritten separately for the vapor and liquid phases wi th coupling by liquid and gas mass transfer rates, and energy transfer rates. These transport rates are estimated from carefully formulated mass and energy transfer coefficients, applicable to multicomponent systems. For non interlinked columns, the resulting equations lead to an incidence matrix that is similar to that of the NaphtaliSandholm method for which a solu tion technique is well established Krishnamurthy and Ta y lor have applied thei r method, with good success to se v eral sets of experimental data from the operation of small laboratory columns. Data from commercialsize columns are now being sought to make further comparisons of predicted and measured compositions so as to evaluate the usefulness and applicability of this transport model. CONCLUSIONS AND RECOMMENDATIONS The digital computer has been responsible for s w eeping changes in the manner in which multi stage separation operations are synthesized and 101
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TABLE 1 Recommended Additions to Content of Undergraduate Courses 1. Numerical methods for A. Linear algebraic equations B. Sparse matrices C. Systems of nonlinear equations 2. Application of numerical methods to A. Complete tearing, partial tearing, and simul taneous correction methods for multicomponent separation processes 3. Use of computeraided simulation programs to A. Analyze, correlate and predict multicomponent thermodynamic properties B. Solve openended separation process problems in volving energy integration 4. Secondlaw analysis analyzed. Chemical engineering educators need to closely examine courses on separation processes to make sure that students are being instructed in modern and efficient computational tools. Some recommended additions to the content of under graduate courses are listed in Table 1, where many of the items should prove useful in other chemical engineering subjects as well. Numerical methods should be stressed for linear algebraic equations, including efficient handling of sparse matrices, and systems of non linear equations. These methods should then be applied using computers to utilize partial tearing and simultaneous correction methods for multi component separation processes. However, some manual calculations on simple examples should be performed using complete tearing methods to help develop a basic understanding. More complex and openended separation problems should be assigned that stress energy integration. A secondlaw analysis [see Denbigh (5 A.D.), and de Nevers and Seader (28 A.D.)] of a process should be required, and attempts should be made to improve the process by finding eco nomical means to reduce the lost work. The development of a separation process can be an exciting experience when computers and computer programs are available to perform the tedious calculations and allow time for more con sideration of synthesis and optimization as pects. REFERENCES Amundson, N. R., and A. J. Pontinen, Ind. Eng. Che m ., 50, 730 (1985). Burningham, D. W and F. D. Otto, Hydrocarbon Pro102 cess ing 46 (10), 163170 (1967). Carnahan, B., H. A. Luther and J. 0. Wilkes, Appli e d Num eri cal M e thod s John Wiley, New York (1969). Davidenko, D Dokl. Akad. Nauk USSR, 88, 601 (1953) Denbigh, K. G., Ch e rn. Eng. Sci. 6, 19 (1956). de Nevers, N., and J. D. Seader, "Mechanical Lost Work, Thermodynamic Lost Work and Thermodynamic Efficiencies of Processes," paper presented at the AIChE 86th National Meeting, Houston, Texas April 15, 1979. Fenske, M. R., Ind. Eng. Ch e m., 2 4, 482485 (19 3 2). Fredenslund, A J. Gmehling, and P. Rasmussen, "Vapor Liquid Equilibria Using UNIF AC, A Group Contribu tion Method." Elsevier, Amsterdam, (1977). Friday, J. R., and B. D. Smith, AIChE J., 10, 698 (1964). Henley, E. J., and J. D. Seader, EquilibriumStage S e para tion Operation s in Ch e mical Engin ee ring, John Wiley and Sons, New York (1981). Krishnamurthy, R., and A. Taylor, AIChE J., 31, 449465 (1985). Kubicek, M., "Algorithm 502," ACM Trans. on Math. Soft ware, 2, No 1, 98 (1976). Lewis, W. K., and G. L. Matheson Ind. Eng. C h em 24 496 498 (1932). Marquardt, D. W., SIAM J 11, 43141 (1963). McCabe, W. L., and E. W. Thiele, Ind. Eng. Ch em ., 1 7, 605611 (1925). McNeil, L. J and R. L. Motard, "Multistage Equilibrium Systems," Proc e edings of GVC / AIChE Meeting at Munich, Vol. II, C5, 3 (1974) Naphtali, L. M. and D. P. Sandholm, AIChE J., 17, 14 (1971). Orbach, 0., and C. M. Crowe, Can. J. Ch e m. Eng., 49, 509513 (1971). Ponchon, M., T ec h. Modern e 13, 20, 55 (1921). Powell, M. J. D., "A Hybrid Method for Nonlinear Equa tions," in "Numerical Methods," Ed. P. Rabinowitz, Gordon and Breach, New York (1970). Rosen, E. M "SteadyState Chemical Process Simulation: A Stateofthe art Review," Comput e r Applications to Ch e mical Engineering, R. G. Squires and G. V. Reklaitis, editors, ACS Symp. Ser. No. 124 (1980). Savarit, R., Arts et Metiers, pp. 65, 142, 178, 241, 266, 307 (1922). Seader, J. D., Section 13 of P e rry' s Chemical Engin ee r s Handbook, 6th ed., McGrawHill, New York (1984). Sujata, A. D., Hydrocarbon Proces s ing, 40, No. 12 137 (1961). Thiele, E. W., and R L. Geddes, Ind. Eng. Ch e m., 2 5, 289 (1933). Wang, J. C., and G. E. Henke, Hydrocarbon Proces s ing 45 No. 8, 155 (1966); Hydrocarbon Proc essi ng, 45 No. 9, 169 (1966). Wayburn, T. L., and J. D. Seader, "Solutions of Systems of Interlinked Distillation Columns by Differential HomotopyContinuation Methods," Proc e edings of the Second International Conference on Foundations of ComputerAided Process Design, June 1924, 1983, Snowmass, Colorado (available from CACHE Corp., P. 0. Box 7939, Austin, Texas 787137939). NOMENCLA JURE A, B, C, Coefficients in a tridiagonal matrix equaCHEMICAL ENGINEERING EDUCATION
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tion; submatrices of partial derivatives in a block tridiagonal matrix B Bottoms product molar flow rate f An arbitrary function F Molar feed rate to a stage; mathematical function H F Molar enthalpy of feed to a stage h The homotopy function whose arguments are x and t H The homotopy function whose arguments are x and;\. H v Molar enthalpy of vapor leaving a stage H L Molar enthalpy of liquid leaving a stage K Vaporliquid equilibrium ratio L Molar liquid flow rate leaving a stage MP Middle product molar flow rate p Path length Q Heat duty (R for reboiler; C for condenser) R Reflux ratio s Ratio of liquid drawoff to primary liquid (liquid not withdrawn or entrained) D E PAR T M E NT : C o rn e ll' Con t inu e d fr om p ag e 6 1. and measuring the properties of liquids and liquid mixtures using theory, computer simula tion, and experiment. Cornell is one of very few institutions with strength in all three areas. Keith guides the theory and the computer simulation (with help from Senior Research As sociate Steve Thompson,) making use of recently developed accurate theories for dense fluids of complex molecules as well as improved computer simulation methods and computer hardware. Typically, highly nonideal substances (in the thermodynamic sense) are chosen for study; sub stances for which traditional methods of pre diction fail. Examples include mixtures occurring in coal gasification and liquefaction, hydrogen energy technology synthetic fuel processing and supercritical fluid extraction. Other research underway or planned includes studies of ad sorption at gasliquid, liquidliquid and solidfluid interfaces, nucleation and droplet phenomena, polarization in polar fluids, and surfactant effects. Bill Streett and Senior Research Associate John Zollweg carry out experimental studies of dense fluids. In progress are ( i) experiments in vaporliquid, liquidliquid, and gasgas equilibria at temperatures from 70 to 500 K and pressures to 10,000 atiospheres; ( ii) equationofstate (PVT) measurements of pure liquids and mix tures at temperatures from 70 to 500 K and pres sures to 4,000 atmospheres; and ( iii) measureSPRING 1985 s Ratio of vapor drawoff to primary vapor (vapor not withdrawn) t Homotopy parameter. T Temperature; when used as a superscript denotes matrix transpose V Molar Vapor flow rate leaving stage X The vector of independent variables (unknowns) for the distillation equations ; liquidphase mole fraction x o The starting vector for the nonlinear equation solver x The solution to the set of nonlinear equations X Mole fraction in liquid of a component; variable y Vaporphase mole fraction y Mole fraction in vapor of a component z Mole fraction in feed of a component Gre ek L ette rs >.. Homotopy parameter ments of enthalpy of mixing in samples of lique fied gases at temperatures from 70 to 300 K and pressures to 20 atmospheres. Bill is currently de veloping new experiments to measure the surface and interfacial tensions and the velocity of sound in fluids under pressure. The researches of Paulette Clancy, who became a member of the faculty in 1984, range from a statistical mechanical study (using perturbation theory) of multicomponent highly polar fluid mixtures to a development of phase diagrams (based on molecular thermodynamics) of semi conductor materials. In addition, she is involved in the application of computers to chemical engi neering. Herb Wiegandt's interest in desalting sea water, using a freezing process based on direct contact with butane, goes back to 1958. Recent efforts, with Bob Von Berg as a partner, have aimed at overcoming the problems associated with washing and separating the ice crystals which are typically very small. Julian Smith, past Director of the School in a period of unprecedented growth, seasoned edu cator and coauthor of Unit Operations of Chem i cal Engineering (now in its fourth edition, with Pete Harriott as coauthor), has expertise in mixing, centrifugal separation, and handling of granular solids. He is teaching fulltime and is active in the guidance of the school. Ray Thorpe, who has advised graduate students in the areas of phase equilibria and separations processes, splits his time between 103
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teaching and university administration: he is di rector of the Division of Unclassified Students. Research Interactions Many research projects involve active col laboration with other researchers at Cornell or elsewhere. Some are directly with other depart ments; some are through Cornell's numerous in disciplinary programs, centers, and institutes many industrially supportedthat facilitate inter action among departments and with industry. Examples are the Biotechnology Institute; the In jection Molding Project; COMEPP (Cornell Manufacturing Engineering and Productivity Program) ; Applied Mathematics Center; Theory and Simulation Center ( established by Ken Wilson, Cornell's 1982 Nobel laureate in physics) ; Materials Science Center; National Facility for Submicron Studies. Strong ties have been es tablished with other departments and colleges of Cornell, and with workers at other universities around the world. Paulette Clancy, Associate Director of COM EPP, is joined by Professor Scheele in a study of ways to improve the interface between the user and ASPEN software chemical process syn thesis and design. UNDERGRADUATE PROGRAM Undergraduate chemical engineering enroll ments at Cornell were almost constant during the twenty years before 1975, with about 40 bache lor's degrees awarded annually. Then, although freshman admissions to the engineering college were held constant, the number of students opt ing for chemical engineering roughly doubled, and for nine years the number of BS degrees awarded was between 65 and 75. After 1985, however, the number will return to 40 or so and is expected to stay at that level for the next several years. The subject matter of the undergraduate pro gram is much the same as at other institutions. For the first two years the students are not in chemical engineering but are enrolled in the "com mon curriculum" of the engineering college. Never theless, their curriculum has much that is differ ent from that of other engineering students. In the freshman year chemical engineers take two semesters of chemistry, not one. Sophomores take two semesters of physical chemistry, with labora tory each terma special course taught by Chemis try almost exclusively for chemical engineers104 and the required introductory course in mas s and energy balances. Organic chemistry (two semes ters, one with laboratory) is given in the third year, as are chemical engineering thermody namics, rate processes and separation processes. The fourth year includes required courses in re action kinetics, process evaluation, process con trol, and unit operations laboratory, and a spring term course in process design. Overall, 132 credit hours are required for the BS degree, including two courses in computer programming and ap plications, four engineering distribution courses, and six courses in humanities and social sciences. Ten of the required courses (32 credit hours) are in chemical engineering subjects. The senior laboratory course is considered the most demanding by students and faculty alike. Each student reports on only five experiments during the term, but each repo r t is thoroughly edited for both form and content by the faculty member in charge of that experiment and nearly always must be extensively revised by the student before it is accepted. The emphasis is on technical accuracy, completeness, and clarity of expression. Oral presentations are stressed in the senior design course, in which each team of students makes weekly oral presentations before two faculty members or industrial visitors. In recent years experienced engineers from industry have been hired for fulltime assistance in this course and in the senior laboratory. Their contributions have been supplemented, during shortterm visits, by those of people from Exxon, Union Carbide, and other firms. Despite this, the laboratory and de sign courses demand large contributions of time by senior faculty members and pose the most difficult problems for future staffing. A Special Cooperative Program For the past ten years the better students in the sophomore year have been encouraged to enroll in an unusual industrial cooperative program which gives them practical experience without lengthening their time at the university. Typical ly 15 to 20 students are accepted into the program after company interviews exactly like those for permanent employment. Coop students take the fallterm thirdyear courses during the summer following their sophomore year; they work in in dustry during the fall and return to Cornell in the spring; work again for the same sponsor the following summer; and complete their senior year CHEMICAL ENGINEERING EDUCATION
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in the regular sequence. Industrial assignments are carefully monitored to insure appropriateness, and each student is visited at the worksite by a Cornell person at least once during the course of the program. THE PROFESSIONAL MASTER'S PROGRAM This is a twosemester nonthesis master's program leading to the degree Master of Engineer ing (Chemical). It requires 30 credit hours of advanced technical work, including a substantial design project, with emphasis on practical ap plications. Most of the matriculants are not from Cornell or other U.S. schools; instead the program is attractive to foreign students, especially from developing countries such as the Dominican Re public, Guatemala, India, Kuwait, Taiwan and Venezuela. Over the years a chemical company in India has sent, one after another, three of its top technical employees to this program. Required courses for the MEng (Chemical) degree include equipment design and selection, numerical methods, reactor design, the design pro ject, and a chemical engineering elective. The remaining credit hours can be filled by elective courses in science or engineering or in the Gradu ate School of Management. The choice of subjects for MEng design projects is much wider than in the typical undergraduate design course, and more initiative and originality are expected of the students. Some of the projects are done in close collaboration with industrial firms. RELATIONS WITH INDUSTRY The school has always had close relations with industry and an unusually supportive group of alumni. Industry helps us in many ways: in the design courses; in a "Nonresident Lecture Series" (zero credit, but compulsory), given to seniors on the various kinds of professional careers; in un restricted grants; in scholarships, fellowships, and sponsored research. Continuing fellowship sup port has been provided by Amoco, Chevron, Dow, DuPont, Exxon, Shell, Stauffer and Union Car bide, and recent large research projects came from IBM, Kodak and Mobil. In 1981 the Sun Company gave $250,000 over three years to sup port research initiation on ideas too new and ill defined to merit submission of a proposal to NSF or other granting agencies. This unusual grant led to a number of publications and several con tinuing sponsored research programs. SPRING 1985 ADVISORY COUNCIL An advisory council, largely from industry, was formed a few years ago. It meets in Ithaca twice a year to review progress and help the di rector steer a course for the school. About half the members are alumni. Recently expanded to 15 members, the council now includes four aca demic people: Andy Acrivos (Stanford), Gus Aris (Minnesota), Gary Leal (CalTech), and Bill Schowalter (Princeton). We don't always agree with the council's suggestions, of course, but as a group it has been marvelously effective in pro viding ari "outside" viewpoint and keeping us from being too provincial or selfsatisfied. WHAT OF THE FUTURE? Cornell is facing many of the same problems that face other chemical engineering departments around the countryfaculty retirements and fu ture faculty development, staffing of design and laboratory courses, the optimum use of computers for teaching, expansion of research and the gradu ate program, and renovation of aging facilities. The five professors hired right after World War II are nearing retirement, so for the next several years an average of one new faculty member per year will have to be hired to keep the number constant. Because of the loss of professors with industrial experience one or more people with an industrial background will probably be hired on a nontenuretrack basis to teach design and to supervise the laboratory courses. A related problem is in the use of computer software. How much emphasis should be placed on teaching the use of ASPEN, for example? More generally, as personal computers become ubiquitous, what will happen to teaching methods? Will the course in mass and energy balances, for example, become a course in the use of available canned programs? The total number of graduate students in the school, and the fraction going for a PhD rather than an MS, should rise somewhat over the next ten years, depending on the availability of financial support. This will increase the need for equipment and laboratory and office space. Rela tively speaking, the Chemical Engineering School has a lot of space, but much of it is virtually un usable for modern research. A comprehensive building renovation plan, made by a firm of archi tects, proposes a complete reallocation and rear rangement of available space and the conversion 105
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of the enormous unit operations laboratory into offices and small research labs. New electrical and other services will be provided, along with central air conditioning. The average faculty office will shrink from over 400 to a more modest 200 square feet and the offices will be grouped more closely, to stimulate greater interaction among the oc cupants. The total estimated cost is some fifteen times the original cost of the building. A fund drive for the first stage is being launched. AWARD LECTURE Continued from page 77. on the batch experience and using verified mathe matical models to both design the equipment and direct the experimentation. CHEMICAL VAPOR DEPOSITION Laboratory Scale Research A low pressure chemical vapor deposition (LPCVD) system for amorphous silicon is shown in Fig. 14 and the simplified process flow diagram as Fig. 15. Reactants, Si 2 H 6 and material for doping the film, PH 3 and B 2 H a in a stream of argon are controlled by valves at the inlet to the reactor. The tubular quartz reactor is temperature con trolled inside an electric furnace System pressure is controlled manually with a valve at the exit. Effluent gas can be analyzed by gas chromatogra phy and unreacted material is decomposed in a furnace before venting. The detailed operation of this system is described by Bogaert [9]. This effort in amorphous silicon research, sponFIGURE 14. Photograph of Low Pressure Chemical Vapor Deposition unit (LPCVD). lOo sored by the Department of Energy through the Solar Energy Research Institute, is ongoing at the present time and is far from being complete. I am discussing it here to allow the reader to con trast and compare with the physical vapor depo sition reacting systems just described. The chemistry is much more complex for amorphous silicon than for CdS and not well @ 0 REACTOR Ar I I I I I Vacuum VENT FIGURE 15. Simplified process flow diagram of LPCVD. understood. The present stateoftheart is shown below: Gas Phase Si 2 H a ~SiH 4 + SiH 2 Si 3 H s ~Si 2 H a + SiH 2 Si 4 H 10 ~Si 3 H s + SiH 2 Si s H ui ~Si 4 H 10 + SiH 2 Si a H 14 ~Si s H 12 + SiH 2 Si 1 H 16 ~Si s H 14 + SiH 2 Si s H 1s ~Si 1 H 16 + SiH 2 Film Formation Si s H1 2 3SiH o.os + 2SiH 4 + l.88H 2 SiH 2 SiH o.os + 0.96H 2 This is a preliminary set of chemical equations. The gas phase equations are based on the results of Ring [10], John and Purnell [11], and Bowery and Purnell [12]. The film formation equations are based upon our own preliminary research. The component mass balance equations for this tubular reactor system are given below: Gas Phase { 1T4J 2 ) = Ir(rxt,i) + kga(CiC1 s ) CHEMICAL ENGINEERING EDUCATION
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Surface O = kga ( C1 C 1. ) k.a ( 'Y1C 1 ) Film 1 dpNt k ( C) MWt ~. a '}'1 i Both the gas phase composition and the film growth rate are functions of axial position. Film growth rate (i.e., amount of amorphous silicon deposited) at any axial position can be determined but it has been possible to measure gas composition only at the reactor exit. The gas and solid phase mass balance equations are coupled through the chemistry of film formation and the transfer from the bulk gas to the surface. Solution of the model equations produces the gas phase exit composition versus reactor holding time plots shown as Figs. 16 and 17. The solid lines were obtained using our present "best" estimates of the specific reaction constants. This "best" estimate is now obtained by using the ex perimentally determined growth rate in the solu tion of the set of mass balance equations. The agreement between data and the predicted values is only fair but we expect to improve the model 100 ~~.r,.., 80 6 j 60 .., _; "i 40 ls z 20 N T 400 C Sl2He P 24 Torr SIH4 S1 1 H 8 HoldlnQ Time. ( sec ) 70 FiGURE 16. Normalized molar percentages versus hold ing time: Major silanes. predictions as we learn more about the system. This research on the chemical reactor and re action engineering for amorphous silicon in the LPCVD reactor is closely coupled with studies of the material and electronic properties of the film and much effort has been devoted to finding the best conditions for good photovoltaic amorphous silicon. To date, we have been able to make a 4 % solar cell using material from the LPCVD reactor. These efforts are described in the work of Hegedus et al [13]. SPRING 1985 4 0 3 0 ... 0 0 ::i; ..,2 0 ~ 0 0 z 1 0 0 10 20 30 40 N T =400 C P 24 Torr Si 4 H 10 e : Si5H12 50 HoldinQ Time. (sec ) 70 FIGURE 17. Normalized molar percentages versus hold ing time: Minor silanes. CONCLUDING REMARKS Incorporating chemical reactor and reaction engineering analysis into a semiconductor research effort requires the researchers to achieve a quanti tative understanding of both the molecular phe nomena and the transport phenomena associated with the creation of the semiconductor materials. A test of this understanding is the ability to write useful mathematical descriptions of the laboratory scale reacting system. Mathematical descriptions are an essential part of the analysis because they provide the language which allows the profes sionals doing the research to effectively and un ambiguously communicate with each other. Com munication is easier if the models are simple and, of course, the model predictions must be verifiable by experiment. In fact, the model behavior should be used to plan the experimental program because an enlightened use of a chemical reaction and re actor engineering analysis will identify critical molecular and transport phenomena problems and direct experimental attention to them with the proper priorities. We originally became interested in the semi conductor research because of a need to design larger scale reacting systems. However, the last five years of research has taught us that the chemi cal engineering analysis is very useful in the laboratory scale research effort, and indeed es sential, if such research is to be done efficiently and with minimum expense (a key issue with today's research costs) It is not possible, in our view, to effectively de sign and operate larger scale systems without re action and reactor engineering analysis. In photo107
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voltai c applications it is also necessary to carry out analysis and experimentation at the unit opera tions scale. Those who have attempted to scale up without following these procedures have wasted time and money building equipment which is in adequate for the commercial scale processing of solar cells. A useful start has been made in applying chemical engineering analysis to the deposition of thinfilm semiconductors but much effort now must be devoted to the task of relating electronic and optical properties to the design and operation of a reacting system. When we have learned to do this properly, we can begin to "tailormake" material with any desired property. ACKNOWLEDGMENTS Semiconductor chemical reaction and reactor research requires a team effort involving a number of professionals. I am particularly indebted to B. N. Baron, R. E. Rocheleau, S. C. Jackson and R. J. Bogaert, my chemical professional colleagues at the Institute of Energy Conversion. Their analysis, their effective experimentation and their discussions with me have been essential to the de velopment of this field. None of the research could have been carried out without the excellent semi conductor material development and analysis and device design and analysis that my other col leagues at the Institute of Energy Conversion do so well. I am also in their debt for their willing ness to educate a chemical engineer in the art and science of applied solid state physics. Science and engineering research today re quires some considerable management talent. The Department of Energy's photovoltaic office and the Solar Energy Research Institute have worked very hard to develop a rational plan for photovoltaic research that both produces results and handles the political pressures that arise in a budget con scious government. The management group within the Institute of Energy Conversion is unique in its capability to protect the director from ad ministrative detail and to allow me to put most of my effort into technical work. I would like to thank S. Barwick and M. Stallings for this gift. REFERENCES 1. Thornton, J. A., Annual Rev. of Material Science, 7, p. 239 (1977). 2. Kern, W. and V. S. Ban, Thin Film Processes, (J. 108 Vossen and W. Kern, editors) Academic Press, New York (1978). 3. Dutton, R. W., "Modeling of the Silicon Integrated Circuit Design and Manufacturing Process," IEEE Trans. Electron Dev., 30, 9, p. 968 (1983). 4. Rocheleau, R. E., B. N. Baron and T. W. F. Russell, "Analysis of Evaporation of Cadmium Sulfide for the Manufacture of Solar Cells," AIChE Journal, 28, 4, p. 656 (1982). 5. Jackson, S. C., PhD Thesis, "Engineering Analysis of the Deposition of CadmiumZinc Sulfide Semicon ductor Film," University of Delaware (1984). 6. Rocheleau, R. E., PhD Thesis, "Design Procedures for a Commercial Scale Thermal Evaporation System for Depositing CdS For Solar Cell Manufacture," University of Delaware (1981). 7. Griffin, A. W., MChE Thesis, "Modeling and Control of a Unit Operations Scale System to Deposit Cad mium Sulfide for Solar Cell Manufacture," Uni versity of Delaware (1982). 8. Brestovansky, D. F., B. N. Baron, R. E. Rocheleau and T. W. F. Russell, "Analysis of the Rate of Vapor ization of CuCl for Solar Cell Fabrication," J. Vac. Sci. Technol. A, 1, 1, p. 28 (1983). 9. Bogaert, R. J., PhD Thesis, "Chemical Vapor Deposi tion of Amorphous Silicon Films," University of Delaware (1985). 10. Ring, M.A., "Homoatomic Rings, Chains and Macro molecules of Main Group Elements," Elsevier, N.Y., 1977, Ch. 10. 11. John, P. and J. H. Purnell, Faraday Trans. I, 69, p. 1455 (1973). 12. Bowery, M. and J. H. Purnell, Proc. Roy. Soc. Land., A321, p. 341 (1971). 13. Hegedus, S. S., R. E. Rocheleau and B. N. Baron, "CVD Amorphous Silicon Solar Cells," Proceedings of the 17th IEEE Photovoltaic Specialists Confer enceOrlando, p. 239 (1984). NOMENCLATURE a Ci D K k kg MW q r(e) r (i) r(r) r(rxt,i) V z Greek area concentration of species i diameter of reactor effective reaction rate constant reaction rate constant mass transfer coefficient molecular weight volumetric flow rate rate of evaporation impingement rate, species i rate of reflection net rate of reaction, species i volume axial position in tubular reactor 8 condensation coefficient 'Y stoichiometric coefficient p density Subscripts f film property g gas phase i molecular species s denotes on the surface CHEMICAL ENGINEERING EDUCATION
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ACl(NOWLEDGMENTS Departmental Sponsors: The following 150 departments contributed to the support of CHEMICAL ENGINEERING EDUCATION in 1985 with bulk subscriptions. University of Akron University of Alabama University of Alberta Arizona State University University of Arizona University of Arkansas University of Aston in Birmingham Auburn University University of British Columbia Brown University Bucknell University University of Calgary California State Polytechnic California State University, Long Beach California Institute of Technology University of California (Berkeley) University of California (Davis) University of California (Los Angeles) University of California at San Diego CarnegieMellon University CaseWestern Reserve University University of Cincinnati Clarkson University Clemson University Cleveland State University University of Colorado Colorado School of Mines Colorado State University Columbia University University of Connecticut Cornell University Dartmouth College University of Dayton University of Delaware Drexel University University of Florida Florida State University Florida Institute of Technology Georgia Technical Institute University of Houston Howard University University of Idaho University of Illinois (Urbana) Illinois Institute of Technology Institute of Paper Chemistry University of Iowa Iowa State University Johns Hopkins University University of Kansas Kansas State University University of Kentucky Lafayette College Lamar University Laval University Lehigh University Loughborough University of Technology Louisiana State University Louisiana Tech. University University of Louisville University of Lowell University of Maine Manhattan College University of Maryland University of Massachusetts Massachusetts Institute of Technology McMaster University McNeese State University University of Michigan Michigan State University Michigan Tech. University University of Minnesota University of Missouri (Columbia) University of Missouri (Rolla) Monash University Montana State University University of Nebraska University of Newcastle University of New Hampshire University of New Haven New Jersey Inst. of Tech. New Mexico State University City University of New York Polytechnic Institute of New York State University of N.Y. at Buffalo North Carolina State University University of North Dakota Northeastern University Northwestern University University of Notre Dame Nova Scotia Tech. College Ohio State University Ohio University University of Oklahoma Oklahoma State University Oregon State University University of Ottawa University of Pennsylvania Pennsylvania State University University of Pittsburgh Princeton University Purdue University Queen's University Rensselaer Polytechnic Institute University of Rhode Island Rice University University of Rochester RoseBulman Institute Rutgers University University of South Alabama University of South Carolina University of Saskatchewan South Dakota School of Mines University of Southern California Stanford University Stevens Institute of Technology University of Surrey University of Sydney Syracuse University Tennessee Technological University University of Tennessee Texas A&I University Texas A&M University University of Texas at Austin Texas Technological University University of Toledo TriState University Tufts University Tulane University University of Tulsa University of Utah Vanderbilt University Villanova University University of Virginia Virginia Polytechnic Institute Washington State University University of Washington Washington University University of Waterloo Wayne State University West Virginia Col. of Grad Studies West Virginia Inst. Technology West Virginia University University of Western Ontario Widener University University of Windsor University of Wisconsin (Madison) Worcester Polytechnic Institute University of Wyoming Yale University Youngstown State University TO OUR READERS: If your department is not a contributor, please ask your department chairman to write CHEMI CAL ENGINEERING EDUCATION, c / o Chemical Engineering Department, University of Florida, Gainesville, Florida 32611.
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Our name has been synonymous with engineering education for over 150 years. Here are eight more reasons why. ECONOMIC ANALYSIS AND INVESTMENT DECISIONS Chi U. Ikoku, Pennsylvania State University (0471814555) April 1985 323 pp. PRINCIPLES AND PRACTICE OF AUTOMATIC PROCESS CONTROL Carlos A. Smith, University of South Florida Armando B. Corripio, Louisiana State University (0471883468) January 1985 600 pp. NATURAL GAS PRODUCTION ENGINEERING Chi U. Ikoku, P en n sy l va ni a State Unive r sity Solution s Manual available (04 71894834) 1984 517 pp. FUNDAMENTALS OF MOMENTUM, HEAT, AND MASS TRANSFER, 3rd Edition Jam es R. Welty, Charles E. Wicks, and Rob er t E. Wilson, a ll of Oregon State University Solutions Manual available (0471874973) 1984 832 pp. NUMERICAL METHODS AND MODELING FOR CHEMICAL ENGINEERS Mark E. Davis, Virgi ni a P o l ytec hni c Institute a nd State U ni versity Solutions Manual availab l e (0471887617) 1984 320 pp. INTRODUCTION TO MATERIAL AND ENERGY BALANCES Gintaras V. R ek l aitis, Purdu e Univers it y Solutions Manual availab l e (0 471 04131 9 ) 1984 683 pp. A GUIDE TO CHEMICAL ENGINEERING PROCESS DESIGN AND ECONOMICS Gae l D. Ulrich, University of New H a mp s hir e (0471052767) 1984 480 pp NATURAL GAS RESERVOIR ENGINEERING Chi U. Ikoku, P en n sylva nia State Un i vers it y Solutions Manual available (04718:4826) 1984 498 pp. Wlley THE ENGINEERING PUBLISHER To be cons id ered for complimentary cop ies, pl ease write to L e Ro y Da vis, D e pt. JW4154. Pl ease includ e course n a m e, enro llm e nt, and titl e of present text. JOHN WILEY & SONS, Inc. 605 Third Avenue, NewYork, N.Y. 10158

