Front Cover
 Table of Contents
 Larry Duda, of Penn State
 Howard University
 Division activities
 Fundamentals of chemical engin...
 Book reviews
 Knowledge structure of the stoichiometry...
 Letter to the editor
 Thermodynamics: A structure for...
 The basic concepts in transport...
 Book reviews
 An appetizing structure of chemical...
 Book reviews
 On letting the inmates run the...
 What works: A quick guide to learning...
 Czochralski crystal growth modeling:...
 Speaking of education
 Introducing statistical concepts...
 Purdue-industry computer simulation...
 An inexpensive and quick fluid...
 Helping students communicate technical...
 An interesting and inexpensive...
 Back Cover


Chemical engineering education
http://cee.che.ufl.edu/ ( Journal Site )
Full Citation
Permanent Link: http://ufdc.ufl.edu/AA00000383/00118
 Material Information
Title: Chemical engineering education
Alternate Title: CEE
Abbreviated Title: Chem. eng. educ.
Physical Description: v. : ill. ; 22-28 cm.
Language: English
Creator: American Society for Engineering Education -- Chemical Engineering Division
Publisher: Chemical Engineering Division, American Society for Engineering Education
Place of Publication: Storrs, Conn
Publication Date: Spring 1993
Frequency: quarterly[1962-]
annual[ former 1960-1961]
Subjects / Keywords: Chemical engineering -- Study and teaching -- Periodicals   ( lcsh )
Genre: periodical   ( marcgt )
serial   ( sobekcm )
Citation/Reference: Chemical abstracts
Additional Physical Form: Also issued online.
Dates or Sequential Designation: 1960-June 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., 1965-1967; Gainesville, Fla., 1968-
 Record Information
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Rights Management: All rights reserved by the source institution and holding location.
Resource Identifier: oclc - 01151209
lccn - 70013732
issn - 0009-2479
Classification: lcc - TP165 .C18
ddc - 660/.2/071
System ID: AA00000383:00118


This item has the following downloads:

Larry Duda, of Penn State, Written by His Colleagues ( PDF )

Howard University, Joseph H. Cannon, Ramesh C. Chawla, Dorian Etienne ( PDF )

Fundamentals of Chemical Engineering, Donald R. Woods, Rebecca J. Sawchuk ( PDF )

Book Reviews ( PDF )

Mathematics, Stuart W. Churchill ( PDF )

Knowledge Structure of the Stoichiometry Course, Richard M. Felder ( PDF )

Thermodynamics: A Structure for Teaching and Learning About Much of Reality, John P. O'Connel ( PDF )

The Basic Concepts in Transport Phenomena, R. Byron Bird ( PDF )

An Appetizing Structure of Chemical Reaction Engineering for Undergraduates, H. Scott Fogler ( PDF )

On Letting the Inmates Run the Asylum, Alva D. Baer ( PDF )

What Works: A Quick Guide to Learning Principles, Phillip C. Wankar ( PDF )

Czochralski Crystal Growth Modeling: A Demonstrative Energy Transport Problem, David C. Venerus ( PDF )

Speaking of Education, Richard M. Felder ( PDF )

Introducing Statistical Concepts in the Undergraduate Laboratory: Linking Theory and Practice, Annette L. Burke, Aloke Phatak, Park M. Reilly, Robert R. Hudgins ( PDF )

Purdue-Industry Computer Simulation Modules: 2. The Eastman Chemical Reactive Distillation Process, S. Jayakumar, R.G. Squires, G.V. Reklaitis, P.K. Andersen, L.R. Partin ( PDF )

An Inexpensive and Quick Fluid Mechanics Experiment, W.D. Holland, John C. McGee ( PDF )

Helping Students Communicate Technical Material, William R. Ernst, Gregory G. Colomb ( PDF )

An Interesting and Inexpensive Modeling Experiment, W.D. Holland, John C. McGee ( PDF )

Table of Contents
    Front Cover
        Front Cover 1
        Front Cover 2
    Table of Contents
        Page 65
    Larry Duda, of Penn State
        Page 66
        Page 67
        Page 68
        Page 69
        Page 70
        Page 71
    Howard University
        Page 72
        Page 73
        Page 74
        Page 75
        Page 76
    Division activities
        Page 77
        Page 78
        Page 79
    Fundamentals of chemical engineering
        Page 80
        Page 81
        Page 82
        Page 83
        Page 84
    Book reviews
        Page 85
        Page 86
        Page 87
        Page 88
        Page 89
        Page 90
        Page 91
    Knowledge structure of the stoichiometry course
        Page 92
        Page 93
        Page 94
    Letter to the editor
        Page 95
    Thermodynamics: A structure for teaching and learning about much of reality
        Page 96
        Page 97
        Page 98
        Page 99
        Page 100
        Page 101
    The basic concepts in transport phenomena
        Page 102
        Page 103
        Page 104
        Page 105
        Page 106
        Page 107
        Page 108
    Book reviews
        Page 109
    An appetizing structure of chemical reaction engineering for undergraduates
        Page 110
        Page 111
        Page 112
        Page 113
        Page 114
        Page 115
        Page 116
    Book reviews
        Page 117
    On letting the inmates run the asylum
        Page 118
        Page 119
    What works: A quick guide to learning principles
        Page 120
        Page 121
    Czochralski crystal growth modeling: A demonstrative energy transport problem
        Page 122
        Page 123
        Page 124
        Page 125
        Page 126
        Page 127
    Speaking of education
        Page 128
        Page 129
    Introducing statistical concepts in the undergraduate laboratory: Linking theory and practice
        Page 130
        Page 131
        Page 132
        Page 133
        Page 134
        Page 135
    Purdue-industry computer simulation modules: 2. The Eastman chemical reactive distillation process
        Page 136
        Page 137
        Page 138
        Page 139
    An inexpensive and quick fluid mechanics experiment
        Page 140
        Page 141
        Page 142
        Page 143
    Helping students communicate technical material
        Page 144
        Page 145
        Page 146
        Page 147
        Page 148
        Page 149
    An interesting and inexpensive modeling experiment
        Page 150
        Page 151
        Page 152
    Back Cover
        Back Cover 1
        Back Cover 2
Full Text

Knowledge Structure^^^^^^^^^^^^^^^

CLISO ...^f ^^>/7~?T~7jy7^^^^^^^^^^^^^^^^





Each year, CHEMICAL ENGINEERING EDUCATION publishes a spe-
cial fall issue devoted to graduate education. It consists of
articles on graduate courses and research, written by profes-
sors and
advertisements describing the graduate programs at various

Extra copies of this informative issue are then distributed to the reading
rooms and/or AIChE offices of bulk-subscribing universities, to be used as a
source of information by those students who are interested in going on to
graduate school.

Anyone interested in contributing to the editorial content of the 1993 fall
issue should immediately write to CEE (c/o Chemical Engineering Depart-
ment, University of Florida, Gainesville, FL 32611-2022). Please indicate
the subject of the contribution and the tentative date it will be submitted.

Deadline for submissions is June 15, 1993.


Chemical Engineering Education
Department of Chemical Engineering
University of Florida
Gainesville, FL 32611
FAX 904-392-0861

Ray W. Fahien (904) 392-0857
T. J. Anderson (904) 392-2591
Mack Tyner
Carole Yocum (904) 392-0861
James 0. Wilkes and Mark A. Burns
University of Michigan


E. Dendy Sloan, Jr.
Colorado School of Mines

Gary Poehlein
Georgia Institute of Technology
Klaus Timmerhaus
University of Colorado

George Burnet
Iowa State University
Anthony T. DiBenedetto
University of Connecticut
Thomas F. Edgar
University of Texas at Austin
Richard M. Felder
North Carolina State University
Bruce A. Finlayson
University of Washington
H. Scott Fogler
University of Michigan
J. David Hellums
Rice University
Angelo I. Perna
New Jersey Institute of Technology
Stanley I Sandler
University of Delaware
Richard C. Seagrave
Iowa State University
M. Sami Selim
Colorado School of Mines
James E. Stice
University of Texas at Austin
Phillip C. Wankat
Purdue University
Donald R. Woods
McMaster University

Chemical Engineering Education

Volume 27 Number 2 Spring 1993

66 Larry Duda, of Penn State, Written by His Colleagues

72 Howard University,
Joseph H. Cannon, Ramesh C. Chawla, Dorian Etienne
78 Introduction, Donald R. Woods
80 Fundamentals of Chemical Engineering,
Donald R. Woods, Rebecca J. Sawchuk
86 Mathematics, Stuart W. Churchill
92 Knowledge Structure of the Stoichiometry Course, Richard M. Felder
96 Thermodynamics: A Structure for Teaching and Learning About Much
of Reality, John P. O'Connell
102 The Basic Concepts in Transport Phenomena, R. Byron Bird
110 An Appetizing Structure of Chemical Reaction Engineering for
Undergraduates, H. Scott Fogler
118 On Letting the Inmates Run the Asylum, Alva D. Baer
120 What Works: A Quick Guide to Learning Principles, Phillip C. Wankat
144 Helping Students Communicate Technical Material,
William R. Ernst, Gregory G. Colomb
122 Czochralski Crystal Growth Modeling: A Demonstrative Energy
Transport Problem, David C. Venerus
128 Speaking of Education, Richard M. Felder
130 Introducing Statistical Concepts in the Undergraduate Laboratory:
Linking Theory and Practice, Annette L. Burke, Aloke Phatak,
Park M. Reilly, Robert R. Hudgins
136 Purdue-Industry Computer Simulation Modules: 2. The Eastman
Chemical Reactive Distillation Process, S. Jayakumar, R.G. Squires,
G. V. Reklaitis, P.K. Andersen, L.R. Partin
140 An Inexpensive and Quick Fluid Mechanics Experiment,
J.T. Ryan, R.K. Wood, P.J. Crickmore
150 An Interesting and Inexpensive Modeling Experiment,
W.D. Holland, John C. McGee
77 Division Activities
95 Letter to the Editor
85,109,117 Book Reviews

CHEMICAL ENGINEERING EDUCATION (ISSN 0009-2479) is published quarterly by the
Chemical Engineering Division, American Societyfor Engineering Education, and is edited at the
University of Florida. Correspondence should be sent to CEE, Chemical Engineering Department,
University of Florida, Gainesville, FL 32611-2022. Copyright 1993 by the Chemical Engineering
Division, 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, ASEE.
Defective copies replaced if notified within 120 days of publication. Write for information on
subscription costs and for back copy costs and availability. POSTMASTER: Send address changes
to CEE, Chemical Engineering Department., University of Florida, Gainesville, FL 32611.

Spring 1993

l =F educator

Pennsylvania State University
University Park, PA 16802-4400

.Larry Duda was born and raised in Donora,
Pennsylvania, a small steel-mill town twenty
miles down the Monongahela River from Pitts-
burgh. It was considered a good omen when he was
delivered by the high school football team physician
since his was a family in which most of the sons
went to college on football scholarships. In spite of
his lack of weight, skill, and interest, when he fi-
nally got to high school he too fulfilled the family
obligation of trying out for the "Donora Dragons,"
which had given the country not only such great
players as "Deacon Dan" Tyler, "Pope" Galiffa,
"Bimbo" Ceconi, but also Stan Musial. Although
Larry did not make the team, he did hear the first
pep talk wherein the coach indicated that there were
two paths down from the football field, which tow-
ered on the hill above the town-"One can graduate
from high school and go down into the mill, or one
can play good football and go to college." Fortu-
nately, Larry found a third path: a scholarship at
Case Institute of Technology. He decided to study
chemical engineering because he liked math and
chemistry and also because he was fascinated by
the old lead chambers which produced sulfuric acid
in the zinc works section of U.S. Steel.
He started out as a mediocre student at Case,



of Penn State

hampered by a poor high school background and a
dyslexia problem which he did not recognize at that
time. He was struggling along with Cs and some Bs
until he took his first chemical engineering course
in stoichiometry and found that his forte was in
solving problems. Even in high school, when he had
difficulty with formal algebra, he found he could
always solve the statement problems through his
own devious techniques. As a consequence, he ex-
celled in stoichiometry and became the top student
in the class. Upon graduating, he decided to go to
graduate school since he felt he did not yet fully
understand chemical engineering and was a little
fearful of going out and practicing the subject with
his limited knowledge.
Larry blossomed as a graduate student at the
University of Delaware and was particularly stimu-
lated by research and interactions with such chemi-
cal engineering greats as Bob Pigford, Art Metzner,
and Kurt Wohl. An outstanding group of graduate
students who were at Delaware at the same time
also contributed to the exciting intellectual climate.
In addition to learning how to do research under
the tutelage of Art Metzner, he honed his tennis
game, helped integrate restaurants in Delaware,
and met his future wife, Margaret Barbalich. He
worked in the area of catalysis with ion exchange
resins and likes to joke that he did so poorly
that neither he nor Art Metzner ever worked in
that area again. Larry reminisces that his years at
Delaware were the best, blending an intensity of
research studies, sports, and personal life. The spe-
cific subjects studied at Delaware were quite sec-

Copyright ChE Division ofASEE 1993

Chemical Engineering Education

Larry joined the Process Fundamentals Group of the Dow Chemical Company..., and...
his long and successful collaboration with Jim Vrentas began... it was [there] that they forged their
friendship and created one of the most productive teams in the profession. They
represented a contrast in styles and abilities, yet had an abiding
respect for each other's points of view and contributions.

ondary compared to the enthusiasm he
gained for learning and the creation of new
knowledge through research.

In 1963, Larry joined the Process Funda-
mentals Group of the Dow Chemical Com-
pany in Midland, Michigan, and it was there
that his long and successful collaboration
with Jim Vrentas began. Although the two
knew each other in graduate school, it was
in the Process Fundamentals Laboratory
that they forged their friendship and cre-
ated one of the most productive teams in
the profession. They represented a contrast
in styles and abilities, yet had an abiding
respect for each other's points of view and
Their differences were demonstrated by
an incident one Friday when they had a
very difficult problem which they could not
solve. Late that afternoon, Larry concluded
that they had been pounding on the prob-
lem too long and had actually begun recy-
cling potential solutions that they had al-
ready considered. He felt they were burned
out, and he was going to take the evening
off, see a play with Marge, have dinner, and
hopefully wake up the next morning with
fresh insight. In contrast, Jim decided to
stay on through the wee hours of the night,
continuing to work on the problem. When
Larry and Marge returned home later that
evening, they found Jim's solution nailed to
their front door! He wanted to make it crys-
tal clear that he had come up with the solu-
tion first, just in case Larry woke up in the
morning with a bright idea of his own.
Larry and Marge quickly established a
family in Midland, and within less than
three years had four children (twins John
and David, Paul, and Laura). Larry likes to
kid the Dow people that there was nothing
else to do in Midland in those days.
During the Dow days, Larry and Jim's
basic work in the area of diffusion in poly-
Spring 1993



Cleverly disguised as a young
student, circa 1962, at Brown
Laboratory, University of Delaware
... and...
as jolly old Saint Nick himself
(below), with co-disguised
John Phillips in an interesting
impersonation of an elf...

as the
Shiek of... ah ... University Park

as... whatever...
with sons Paul, John, wife Marge, son David, and daughter
Laura all getting in on the act.

mer systems was initiated. To their dismay, how-
ever, they were not free to continue along the paths
of scientific interest-instead, they had to respond
to the more direct economic needs of the company.
Nevertheless, they were at Dow during the golden
days when great advancements were being made in
polymer science, led by such individuals as Turner
Alfrey and Ray Boyer. It was natural that in this
environment they would be drawn into considering
problems associated with polymer production and
processing. In addition to Jim Vrentas, Turner Alfrey
and Art Metzner (as a Dow consultant) also exerted
great influence on Larry's professional development.
Despite Larry's successful career development at
Dow in the late 1960s, he and Jim decided that they
should consider academia if they wanted to con-

... it became clear to him
that he would rather stay at Penn State
without research and just teach than to take a
position where he could concentrate on
his desired research with no
opportunityfor teaching.

tinue along their main avenues of interest. In his
last years with the company, Larry made his most
successful contribution through his work on design-
ing insulation systems for the trans-Alaskan pipe-
line that would keep it from melting the permafrost
during the short Alaskan summer.
Although Larry left Dow for Penn State in 1971,
he has maintained strong contacts with Dow. In
many ways, Larry has tribal instincts and develops
a strong devotion to groups he lives and works with.
In addition to his family, he still has vital attach-
ments to his home town of Donora and to the Dow
Chemical Company. He stays in touch with several
friends at Dow, including Doug Leng and George
Shier. Larry was most recently named a charter
member of Dow's Academic Advisory Council.

Over the past twenty years, Larry has devoted
most of his energies to the development of the De-
partment of Chemical Engineering at Penn State.
The dominant characteristics of Larry's work are in
its diversity and its strong emphasis on collabora-
tion. He has conducted collaborative research work
with almost every member of the chemical engi-
neering faculty as well as with several researchers
outside the department. Besides his work with Jim
Vrentas in the general area of diffusion in polymer

systems, Larry has made contributions in many other
fields, most significant of which has been his joint
research with Elmer Klaus in the area of tribology.
Although Larry was attracted to academia because
he sought to define his own research work, he quickly
became enthralled by teaching. In fact, within
a few years it became clear to him that he would
rather stay at Penn State without research and
just teach than to take a position where he could
concentrate on his desired research with no oppor-
tunity for teaching.
The meetings of his research groups with their
inevitable interplay of ideas are the most enjoyable
parts of Larry's working schedule, and the most
attractive feature of these interactions comes from
the general thrill of exploring the natural world. In
these group meetings, Larry often makes bets with
Elmer Klaus that the results of the new experi-
ments will turn out a certain way. But Elmer is an
expert at oracle statements, and no matter which
way the results come out he can be counted on to
argue that he had already predicted the results.
Second only to chemical engineering is Larry's
continued interest in tennis. In fact, it is rumored
that Lee Eagleton originally hired him only be-
cause they were great tennis partners who together
could take on opponents from the chemistry depart-
ment. At present, Jack McWhirter and Larry offer a
standing challenge to take on any two students in
the department.
On the home front, Larry is proud of his belief
that the way to educate people is to help them be-
come themselves. This freedom of spirit is strongly
exhibited in his children. None of them has become
an engineer, or has even gotten close to engineer-
ing-they have been students of art, English litera-
ture, and medicine. Now that the children are grown,
Larry and Marge are able to nurture their interest
in international travel, and when at all possible
they try to couple it with Larry's technical interests
and Marge's photographic interests.

Through the years at Dow and at Penn State,
Larry's work has exhibited a common thread of re-
search on polymers and transport phenomena. His
well-known collaboration with Jim Vrentas on mo-
lecular diffusion in polymer systems has yielded
many results which have been presented in over
seventy journal publications.
At the time Duda and Vrentas initiated their work,
the area of diffusion in concentrated polymer solu-
Chemical Engineering Education

tions and melts was in a state of disarray, and no
techniques were available to the design engineer for
the prediction or even the correlation of diffusivity
data. In fact, available experimental data revealed
many apparent contradictions. Some experiments
showed that the binary mutual diffusion coefficient
in polymer-solvent systems were strong functions of
concentration, while in other studies these coeffi-
cients were found to be independent of concentra-
tion. Some investigators found that diffusion in poly-
mer systems depended strongly on temperature and
did not follow an Arrhenius-type behavior, while
other studies indicated that the data could be corre-
lated with the Arrhenius equation with relatively
low activation energies for diffusion. Superimposed
on this perplexing situation were the experimental
observations that, in some cases, diffusion in con-
centrated polymer systems did not even follow Fick's
law. Numerous investigations showed that anoma-
lous effects were present which were not consistent
with the classical diffusion theory.
In response to this situation, the studies of Duda
and Vrentas led to the concept that molecular diffu-
sion processes involved the coupling of migration
and relaxation of molecules. Up to this time it had
been implicitly assumed that the molecules partici-
pating in a diffusion process could relax very quickly
to new equilibrium states and that local thermody-
namic equilibrium was maintained. Duda and
Vrentas quantified their theory with the introduc-
tion of the diffusion Deborah number, which is the
ratio of the characteristic relaxation time of the mol-
ecule to the characteristic time of the diffusion pro-
cess. This dimensionless group revealed under what
conditions classical diffusion theory is appropriate
for the description of diffusion in polymeric systems.
Probably the most important outcome of the col-
laborative work of Duda and Vrentas is the develop-
ment of the free volume theory in which the viscous
behavior of polymer melts is coupled to the diffu-
sional behavior in binary solutions. Their theory
allows the prediction of diffusion coefficients as a
function of temperature and concentration from vis-
cosity and thermodynamic data obtained essentially
for pure component systems. The free volume theory
as developed by Duda and Vrentas has been shown
to be applicable up to at least 80 C above the glass
transition temperature and for concentrations as
high as 70 weight percent solvent. Interestingly, the
theory is also capable of predicting anomalous abrupt
changes in the diffusivity observed in the vicinity of
glass transition temperature.
Concurrently, Duda and Vrentas developed experi-
Spring 1993

mental techniques and associated analyses for the
determination of accurate diffusivity data over the
wide ranges of temperature and concentration
needed for various polymer processes. Their work
led to the development of a widely used high-
temperature sorption apparatus as well as a novel
oscillatory sorption experiment. The latter tech-

Duda and Mary Eagleton presenting the Lee and
Mary Eagleton Design Award to Heather Bergman.

nique is the only method available to study unam-
biguously the coupling of diffusional transport and
molecular relaxation.
Not well known to the academic chemical engi-
neering community are Larry's contributions to the
area of tribology and lubrication. Larry recognized
how fundamental principles of chemical engineer-
ing can be successfully applied to bring order into
a traditionally empirical field of research that
has remained largely proprietary over the years.
This led to his collaborative research with Elmer
Klaus, the results of which are summarized in over
forty publications.
Probably the most important outcome of this re-
search has been the development of a micro-reactor
technique to study the thermal and oxidative degra-
dation of lubricants under conditions that simulate
automotive engine tests, heavy-duty diesel engine
performance, electrical power generating equipment,
and gas turbine engines. The test has been adapted
by over fifteen industrial research groups as a way
to minimize costly engine tests and has been suc-
cessfully used to study the performance of lubricant
additives as well as the catalytic effects of metal
surfaces on lubricant degradation.
Another important result from this research is the
development of a novel lubricant delivery system,

for applications at elevated temperatures. In this
system, a lubricant film is formed on a hot surface
from a homogeneous vapor phase. The lubricant-
forming vapor is adsorbed on the solid surface and
reacts to form the lubricant film. This new lubricant
system is being evaluated for applicability in di-
verse areas, including the lubrication of an adia-
batic ceramic engine and metal-forming operations.
Duda's work has also led to development of methods
for the theological characterization of lubricants
under extreme conditions of temperature, pressure,
and shear rate.
By his work, Duda has taught his colleagues and
students how to carry out fundamental research that
leads directly and tangibly to industrially signifi-
cant results. A unique indicator of this success is
the fact that virtually all of his research support
comes from industry.

Larry has been a teacher with great impact. Over
the years he has developed a unique teaching style
and educational philosophy. For example, he starts
out each lecture in his graduate course with a quote
from a famous engineer, scientist, philosopher, or
religious leader. He feels that each lecture should
not only present some specific segment of technol-
ogy, but also should incorporate some thought or
philosophy concerning the general aspects of life it-
self. To illustrate, one of his classroom techniques is
role playing. He will introduce himself as an inven-
tor, while the students play the role of engineers in
a company that is considering buying the inventor's
latest creation. His "inventor" is usually a super-
salesman who is very close to playing a con game.
The students' roles are to analyze the proposed in-
vention for its scientific merits and to find its fatal
flaws, if any. A quote from Harold McMillan sets
the stage for this particular lecture: "Nothing that
you will learn in your studies will be of the slight-
est possible use to you in the afterlife. Save only
this: that if you work hard and intelligently, you
should be able to detect when a man is talking rot,
and that, in my view, is the main, if not the sole
purpose of education."
Larry's classroom emphasis is on creativity and
the ability to solve unique problems, as opposed to
the mere accumulation of specific knowledge or so-
lution of conventional problems. He stays awake at
night thinking up problems for homework or exams
which at first glance appear to be unrelated to the
topic at hand but that can be solved by using the
course fundamentals. For example, to illustrate the

use of the Flory-Reiner model for crosslinked poly-
mers, he considers the case of how the ancient Egyp-
tians cracked stones in their quarries. They drove a
wooden wedge into a crack, poured water on the
wedge, and let the swelling wedge crack the rock.
By giving properties of polymeric wood, students
can develop equations to predict the pressures that
such swelling wedges will develop.
Larry's success as a teacher stems not only from
his classroom lectures, but also from his close work
with undergraduate and graduate students in their
research. He has advised sixty-one masters' students
and thirty-two doctoral students at Penn State, in-
cluding the forty-six students who have worked with
him and Elmer Klaus in the area of tribology. It is
fair to say that almost every industrial tribology
researcher with a chemical engineering degree has
been trained at Penn State. Duda is a much sought-
after member of graduate student thesis commit-
tees. Over the last twenty years, in addition to the
students whose work he has supervised he has also
served on 100 doctoral committees and 120 masters'
committees. The students have been from chemical
engineering, chemistry, polymer science, petroleum
engineering, mineral processing, agricultural engi-
neering, fuel science, bioengineering, etc. An impor-
tant reason for the diverse backgrounds of students
seeking Duda's guidance is the active collaboration
Duda has developed and maintained with over
twenty faculty members in other disciplines. He has
also been unusually active in guiding over seventy
undergraduate students on their honors research
projects. Many of them have gone on to graduate
schools, inspired by their research experience.

Larry has been recognized for his teaching and
research through a number of awards. In 1980 he
received the Outstanding Research Award from the
Pennsylvania State Engineering Society, and in 1981
Larry and Jim Vrentas were co-recipients of AIChE's
William H. Walker Award in recognition of their
work on molecular diffusion in polymers and the
analysis of complex transport phenomena. Larry was
chosen by Penn State's senior ChE class as the Out-
standing Professor in 1983, and in 1989, along with
Jim Vrentas, he received the Charles M.A. Stine
Award in Materials Engineering and Sciences from
the AIChE Materials Division in recognition of their
development of the free volume theory and the oscil-
latory sorption technique. Also in 1989, Larry was
selected to receive the ASEE Chemical Engineering
Lectureship Award. The Pennsylvania State Engi-

Chemical Engineering Education

One oJ Duda's happier duties as Department Head-
receiving a check for the department!

neering Society honored Larry and Jim Vrentas in
1991 with its Premier Research Award, and also in
that year Larry was chosen as the Alumni Delegate
representing the Class of 1963 at the 1991 Com-
mencement Ceremony of the University of Delaware.

Larry has been an unusual department head for
the past ten years. He has demonstrated that taxing
administrative duties need not diminish one's in-
tense involvement in teaching, research, and guid-
ing students. Under his stewardship, the depart-
ment has increased its visibility, with many of the
faculty receiving national awards from AIChE, ACS,
ASEE, ASME, etc. The department has recruited
several outstanding young faculty in John Frangos,
Kristen Fichthorn, Lance Collins, Ali Borhan, Wayne
Curtis, and Themis Matsoukas. Two of these new
faculty, Frangos and Fichthorn, were honored with
Presidential Young Investigator Awards. The de-
partment also added two nationally prominent se-
nior scientists to its ranks in Art Humphrey and
Paul Weisz. Duda's leadership has been responsible
for the creation of a strong research program in
biotechnology at Penn State, capped by the recruit-
ment of Art Humphrey as the Director of the Bio-
technology Institute.

Larry has been a spokesperson for academic inter-
ests to industry. His own experience of having al-
most all of his research sponsored by industry gives
him unusual insights into the mechanisms for and
the benefits of university-industry collaboration. He
has been a member of the Council for Chemical
Research (CCR) for the past ten years and served on
its Governing Board and on the latter's Executive
Spring 1993

Committee. He has served on the CCR Committee
on Industrial College Relations and as CCR Liaison
with the NSF, and has also been active on the Aca-
demic Advisory Council of Dow Chemical. His ad-
vice as an educator has been sought after by other
departments of chemical engineering: he has served
as the external reviewer for Rutger's University;
he has served on the Promotion and Tenure Re-
view Committees of the Illinois Institute of Tech-
nology and the University of Rochester; he serves
on the advisory committees of the chemical engi-
neering departments at West Virginia University,
University of Delaware, and Carnegie-Mellon
University. An example of the esteem in which
Duda is held by other department heads is their
election of him as the Chair of the Board of Judges
for the McGraw-Hill Kirkpatrick Award in
Chemical Engineering.
Larry has been active in the AIChE in a number
of ways. He has served as a member of the National
Program Committee, Public Relations Committee,
Polymer Engineering Subcommittee of the Research
Committee, Walker Award Committee, Charles M.A.
Stine Award Committee, and the National Awards
Committee. He is currently a Director of the Materi-
als Engineering and Sciences Division. He also serves
on the Publications Committee of the ASEE.

Larry has made unique contributions to his field
through his role as a philosopher of graduate educa-
tion. His three articles in Chemical Engineering Edu-
cation, "Common Misconceptions Concerning Gradu-
ate School," "Graduate Studies: The Middle Way,"
and "Graduation: The Beginning of Your Education"
are necessary reading material for all graduate stu-
dents. They outline a philosophy that has guided
Duda's work and offer much-needed perspective to
beginning graduate students. To quote from the con-
clusion of the second article, Duda says
From my presentation, you might conclude that
there is a middle way in every aspect of graduate
work that is the most appropriate approach. Al-
though I have attempted to illustrate that this is
certainly true in many instances, there is one very
important exception. Some students say to them-
selves, "This is not the best that I can do but it's
good enough." Well, it's not good enough. Push
yourself-take time and make the effort to perform
at the very highest level of which you are capable.
There is no middle way when it comes to the pur-
suit of excellence.
Duda's contributions to chemical engineering epito-
mize the above philosophy in action. I

epartmen t


Frederick Douglass Hall-
houses Liberal Arts Departments and classrooms

Howard University
Washington, DC 20059
Howard University is a private, co-educational institution
located in Washington, DC. Named for General Oliver
Otis Howard, a Civil War hero who helped found the Uni-
versity, it was incorporated in 1867 by an Act of Congress, and its
founding mission was to help educate the four million freed slaves
and others to whom education had previously been denied.
The University offers degree programs in about two hundred
*Graduate Student in chemical engineering.

There is strong interaction among
faculty members across research areas
[that] provide three research focal
areas... transport phenomena in
environmental engineering,
separation processes, and
kinetics and reactor modeling.

specialties and its four campuses encom-
pass 241 acres. Most of its schools and
facilities, including the radio and televi-
sion stations, a full-service hotel, a hos-
pital, and a number of research centers,
are located on its eighty-nine-acre main
campus three miles north of the Capitol
in the heart of Washington, DC.
Howard's more than 1,200 full-time fac-
ulty members are a microcosm of the
world population of scholars, and its ap-
proximately 12,000 students come from
all over the United States and more than
one hundred countries.
Howard University introduced its en-
gineering programs in 1911 and they
were among the first accredited programs
in the United States. Historically,
Howard has been the nation's major
source of minority engineers, particularly
African American engineers. Bachelor's
and master's degrees are offered in
chemical, civil, electrical, and mechani-
cal engineering, and in systems and com-
puter science. The departments of elec-
trical and mechanical engineering also
have PhD programs. Each year about
850 undergraduate and 200 graduate
Copyright ChE Division ofASEE 1993
Chemical Engineering Education


Downing Hall of Engineering:
Chemical Engineering Wing on the left.

students enroll in the various programs offered.
Modern instructional and research laboratories,
together with computing facilities, support both
student and faculty research pursuits. The
Computer Learning and Design Center (CLDC),
the school's centralized computing facility, and
the Computer Laboratory for Instruction and
Design in Engineering (CLIDE), provide a full
spectrum of computer resources for faculty
and students. These include PCs, HP and DEC
VAX minicomputers, Sun Workstations, and
access to an Alliant mini-supercomputer. These
resources are linked via networks to each other,
to the university's IBM 3090 mainframe, and
The Chemical Engineering Department was es-
tablished in 1969 with the appointment of Dr.
Herbert Katz as the Chair and with five stu-
dents at the sophomore level. Between 1970 and
1972 three more faculty joined the department:
Pradeep Deshpande (now at the University of
Louisville), Joseph Cannon (current Chair of the
Spring 1993

ivegnoors of nowara umverslry: ne Lincol
Memorial, the Washington Monument, and the
U.S. Capitol building.

^lv t~lA-i

Graduate students reviewing laws of motion.

department), and Franklin King (now Chair at NCA&T
State University). In June 1972, the five original stu-
dents successfully completed the curriculum and were
awarded the first BS degrees granted by the depart-
ment. Presently, there are six full-time faculty posi-
tions (one vacant), one part-time faculty, approximately

one hundred undergraduate students, and twelve
MS students.
In 1975 a modern chemical engineering wing
was added to the L.K. Downing Hall of Engineer-
ing. This facility contains a number of research
laboratories, each equipped with state-of-the-art
equipment to meet the experimental research
needs of the faculty.
The Fluid and Thermal Engineering Laboratory
houses equipment and instrumentation for the
measurement of flow and heat transfer. A
Laser Anemometry System, a rotational vis-
cometer, along with other instruments enable
researchers to measure velocities, map shear-
stress patterns, and conduct routine measure-
ments of shear viscosity.

The undergraduate program is
structured to provide a broad background in
the fundamental areas of chemical engineering,
with special attention given to the development
of analysis and problem-solving skills.

The Biochemical Engineering Laboratory is
equipped to conduct research in microbial fermen-
tation, protein purification, bioremediation, and
protein adsorption. Specialized equipment in-
cludes an inverted phase contrast photo-capable
microscope, automated high-pressure and low-
pressure liquid chromatography systems with vari-
able wavelength detection capabilities, full-spec-
trum scanning spectrophotometry instrumentation,
and a microtome.
The Microelectronics Materials Processing Labo-
ratory features two chemical vapor deposition re-
actors (horizontal and vertical) and a sublimation
reactor, all for the growth of silicon carbide and
related materials. This laboratory is a part of the
NSF-funded Materials Science Research Center of
Excellence (MSRCE) located in Downing Hall of
Engineering where faculty and students from elec-
trical and chemical engineering, physics, and chem-
istry carry out interdisciplinary research.
The environmental engineering laboratories have
facilities dedicated to analytical instrumentation,
microbiology, incineration, water pollution, and air
pollution. Jointly shared by the environmental en-
gineering faculty in civil engineering, these labo-
ratories are the focus of several interdisciplinary
research projects. All the necessary equipment for
the growth, isolation, and analysis of microorgan-

Undergraduate unit operations laboratory
Undergraduate unit operations laboratory.

isms is located in the microbiology laboratory. Equip-
ment for studying the kinetics, chemistry, and heat
and mass transfer characteristics of high-temperature
reactions are available in the Incineration Laboratory.
This includes a Shirco infrared incinerator, a liquid/
gas combustion unit, and a fluidized bed high-tem-
perature reactor. Howard is one of a handful of aca-
demic institutions possessing the incineration facili-
ties capable of studying thermal degradation of waste
in all phases-solid, liquid, and gas.
The EPA-funded Great Lakes and Mid-Atlantic
Hazardous Substances Research Center (a con-
sortium of the University of Michigan, Michigan State
University, and Howard University) supports coop-
erative research efforts among chemical engineering
and other university faculty. These environmental
research programs are in the areas of bioremedia-
tion and composting.

The undergraduate program is structured to provide
a broad background in the fundamental areas of chemi-
cal engineering, with special attention given to the
development of analysis and problem-solving skills. The
breadth of the undergraduate program is intended to
prepare students to either enter the chemical engi-
neering profession upon graduation or to successfully
continue their education at the graduate level.
The curriculum is particularly strong in providing
comprehensive design experience, basic engineering
technology in separation processes, and the fundamen-
tals of transport processes. Computer use is integrated
throughout the curriculum, with special emphasis on
digital simulation and software for analysis and de-
sign. Laboratories support undergraduate instruction
in momentum, heat, and mass transfer, reaction kinet-
ics, process control, and process design. Electives of-
Chemical Engineering Education

The EPA-funded Great Lakes and Mid-Atlantic Hazardous Substances Research Center [is] a consortium
of the University of Michigan, Michigan State University, and Howard University [which] supports
cooperative research efforts among chemical engineering and other university faculty.

feared by the department include: polymer engineering, bio-
medical engineering fundamentals, bioprocess engineering,
processing of electronic materials, transport phenomena,
energy systems, and environmental engineering.
Most of the BS graduates have found employment in in-
dustry, while about one-third of them have gone on to pur-
sue advanced degrees in chemical engineering, environmen-
tal engineering, business, or other professional areas such
as medicine, law, and dentistry.

The goal of our Master's program is to provide the neces-
sary academic experiences to prepare students for challeng-
ing and responsible careers as practitioners and adminis-
trators in the chemical engineering profession and for the
numerous other opportunities associated with this level of
achievement. The program is intended to extend the student's
training in the mainstream areas of chemical engineering
at an advanced level, with sufficient in-depth study of a
n, selected area and involving both formal course work and a
osep thesis research project.
The instructional program is based on core courses in
TABLE 1 thermodynamics, transport phenomena, reaction kinetics,
Faculty and Research Areas advanced engineering mathematics, and elective courses re-
Joseph N. Cannon, P.E., Professor and Chair; lated to the student's area of specialization. Graduate the-
PhD, University of Colorado ses are generally based on faculty research.
Transport phenomena in environmental systems,
computational fluid mechanics, heat transfer FACULTY AND RESEARCH FOCUS
Ramesh C. Chawla, Professor;
PhD, Wayne State University Since ours is a small department, there is a great oppor-
Chemical kinetics, separation processes, bioremedia- tunity for interaction among students and faculty. Stu-
tion, incineration, environmental engineering dents feel comfortable visiting faculty at any time to seek
M. Gopala Rao, Professor; advice or assistance on matters related to their courses,
PhD, University of Washington, Seattle their research, or personal well-being. There is also strong
Separation processes, energy systems, radioactive
waste management interaction among faculty members across research areas.
Mobolaji E. Aluko, Associate Professor; These interactions provide three research focal areas for
PhD., University of California, Santa Barbara the department: transport phenomena in environmental
Process control, mathematical methods, reactor engineering, separation processes, and kinetics and
modeling, crystallization, microelectronic materials reactor modeling. Table 1 lists the research interests of
processing *
processing each faculty member.
John P. Tharakan, Assistant Professor;
PhD., University of California, San Diego Mobolaji Aluko's research is in three specific areas:
Reactor design and bioprocess engineering, protein experimental analyses and numerical modeling of gas-
separations, protein adsorption, biological hazardous phase deposition reactors for semiconductor materials; solu-
waste treatment
Robert J. Lutz, Visiting Professor; tion crystallization of ceramic materials; and control of non-
PhD., University of Pennsylvania linear chemical systems. Recent MS theses projects have
Hemodynamics, intra-arterial drug delivery focused on the analysis of mixed-suspension mixed-product
Herbert M. Katz, Professor Emeritus; removal (MSMPR) crystallizers and on the design of hetero-
PhD., University of Cincinnati generous catalytic reactors.
Environmental engineering
In addition, he directs the Engineering Coalition of Schools
Spring 1993 75

for Excellence in Education and Leadership (ECSEL)
program at Howard. This coalition consists of seven
universities funded by NSF to seek fundamental
changes in engineering education through active stu-
dent involvement in learning and by incorporating
interactive, open-ended teaching approaches. Dur-
ing the 1992-93 academic year he is spending a
sabbatical leave at two coalition schools-the Uni-
versity of Washington, Seattle, and the University
of Maryland, College Park.
In his spare time, he plays tennis, ping-pong, and
chess, and he is one-half of the 1991-92 Howard
University Tennis Doubles' Championship team. He
is always ready to argue politics and religion.
Joseph Cannon's research focuses on transport
phenomena with applications in environmental en-
gineering. He is currently studying the movement of
hazardous organic in soil and has both experimen-
tal and numerical work underway. He is also inter-
ested in the cooling of electronic equipment contain-
ing printed circuit boards. One of his students has
just completed a thesis on the numerical analysis of
conjugate heat transfer in electronic packages.
For over twenty years, Joe has been known to
frequently challenge a student to a one-on-one bas-
ketball game. It appears as though "Father Time"
has caught up with him, however, and he has re-
cently started taking tennis lessons. He also enjoys
pocket billiards and chess.
Ramesh C. Chawla's research combines the ap-
plication of the principles of mass transfer and ki-
netics to environmental systems. He has been in-
volved in an on-going EPA-funded research program
in hazardous waste treatment using physical and
chemical techniques such as soil washing, adsorp-
tion, acid protonation, and biodegradation of haz-
ardous wastes using indigenous microorganisms cul-
tured from the contaminated sites. He has also been
studying the combined technique of surfactant-as-
sisted biodegradation of hazardous wastes. His
projects on thermal treatment of hazardous wastes
deal with the assessment of organic emission and
heat and mass transfer limitations in incineration.
Ramesh loves to discuss politics and sports with
his colleagues and students. He is the faculty advi-
sor for the AIChE Student Chapter and frequently
conducts some of the chapter meetings with stu-
dents while bowling at the University Center.
Gopala Rao has been very active in the areas of
adsorption and ion exchange separation processes,
radioactive waste management, and alternative pro-
cess energy systems. His current research, funded

by the Office of Civilian Radioactive Management
(DOE), concentrates on sorption equilibrium mea-
surements of binary and ternary ionic systems of
radionuclides (such as cobalt, nickel, strontium, ce-
sium, and lead) in aqueous phases and on single
and mixtures of minerals such as clinoptilolite, mont-
morillonite, and goethite. These efforts are in sup-
port of the Yucca Mountain Site Characterization
Project being conducted at the Los Alamos National
Laboratory and the Sandia National Laboratory.
Gopala is an avid swimmer and jogger. When-
ever he is out of town for a meeting or a conference,
he can be found after hours on the jogging trails
around his hotel.
John Tharakan's protein separation research fo-
cuses on the effects of such parameters as ligand
distribution, resin structure, and flow configuration
on process efficiency. He has carried out cell culture
research to investigate the fundamental physico-
chemical parameters at the microenvironmental level
that affect cell viability and productivity in novel
bioreactor configurations for large-scale cell culture.
His bioremediation research focuses on the syner-
gistic effects of pathways and cofactors utilized by
individual and consortia of microbes in the biodeg-
radation of toxic wastes.
John enjoys discussing politics and is especially
interested in the interactions of science, technology,
and culture. He is a member of the local Chapter
of Science for the People. When not involved with
class or lab work, he can usually be found cooking
in the kitchen.
Robert Lutz has been studying methods of intra-
arterial drug delivery to achieve high concentrations
of an anticancer drug at the tumor site while main-
taining subtoxic levels at other sensitive sites in the
remainder of the body. He is investigating catheter
design and infusion methods that minimize
nonuniform drug distribution.
Bob has been teaching and participating in re-
search at Howard for the past twelve years. When
not working at NIH or Howard, he. can be found
playing basketball or golfing or participating in any
other sport for which the weather is suitable.
Howard University has come a long way since its
inception in 1867 with the mission of educating freed
slaves. Today it successfully conducts the daily busi-
ness of educating young people anxious to and ca-
pable of making significant contributions in many
ways and in many areas. We are proud of our gradu-
ates and look forward to the educational and soci-
etal challenges of the future. O
Chemical Engineering Education

lhE Division Activities

ASEE Annual Conference

When-June 20-24, 1993
Where-University of Illinois campus in Urbana-Champaign, Illinois

Since this is ASEE's centennial year, the overall
topic of this year's meeting will be the history of
engineering. With that theme in mind, the ChE di-
vision has organized three sessions pertaining to
our history: a lecture on the history of chemical
engineering education; a display of textbooks used
throughout our history; and a poster session on his-
tories of chemical engineering departments at the

various universities. Other sessions will deal with
new developments in education and research, such
as new environmental courses and curricula, high-
performance computing, new courses not tradition-
ally taught in chemical engineering, new chemical
engineering research areas, and new partnerships
with industry in chemical engineering The program
promises to be interesting and entertaining.






1613 Monday
2213 Tuesday
2413 Tuesday
2513 Tuesday



2713 Tuesday
3213 Wednesday
3413 Wednesday
3513 Wednesday

3613 Wednesday
4213 Thursday














The History of Chemical Engineering Education
Donald Dahlstrom, University of Utah
Moderator: James E. Stice, University of Texas-Austin
Chemical Engineering Chairpersons' Luncheon
Moderator: Richard Alkire, University of Illinois
Environmental Engineering Courses and Curricula
Moderator: Gary K. Patterson, University of Missouri-Rolla
High Performance Computing in Chemical Engineering
Moderator: Mark A. Stadtherr, University of Illinois
Chemical Engineering Executive Committee Meeting
Moderator: John Friedly, University of Rochester
Histories of Chemical Engineering Departments
Moderators: Ron Larsen, Montana State University
Susan Montgomery, University of Michigan
New Courses not Traditionally Taught in Chemical Engineering
Moderators: J. L. Zakin, The Ohio State University
L. S. Fan, The Ohio State University
Chemical Engineering Division Banquet and Division Lecture
Moderator: John Friedly, University of Rochester
New Chemical Engineering Research Areas
Moderator: Thomas Marrero, University of Missouri-Columbia
Chemical Engineering Division Luncheon, General Business Meeting
Moderator: John Friedly, University of Rochester
Textbooks in the History of Chemical Engineering
Moderators: Melanie McNeil, San Jose State University
Polly R. Piergiovanni, Lafayette College
New Partnerships with Industry in Chemical Engineering Research
Moderator: Neil Book, University of Missouri-Rolla
Tour of Research Facilities
Leader: To be announced

Spring 1993 7;



mnowo courtesy of ictoria nau, unwersaiy oT Uairornia, aong neacn
Participants in the AIChE session on Knowledge Structure in Chemical Engineering,
held in Los Angeles, November, 1991. Pictured, left to right: Scott Fogler, Stuart
Churchill, R. Byron Bird, Richard M. Felder, and John P. O'Connell
(not pictured, Donald R. Woods).

Chemical Engineering Education


Knowledge has structure;

structure to facilitate learning


structure to facilitate problem solving.

Research by cognitive psychologists has revealed the
characteristics of those different structures.

The Undergraduate Education Committee of the AIChE spon-
sored a session on knowledge structure at the Los Angeles
meeting in 1991. At that session (which I cochaired with Bill
Kroesser), Stu Churchill, Rich Felder, John O'Connell, Bob Bird,
and Scott Fogler shared their views of the fundamentals and
the structure of knowledge in the areas of mathematics, mass
and energy balances, thermodynamics, transport phenomena,
and reaction kinetics/reactor design.
The results were exciting and diverse. Some of the presenta-
tions focused on the importance of the subject and the process
of using the knowledge effectively, some on structure related to
learning, some on problem solving, and some on a combination
of these factors.
We are pleased to present those papers on
the following pages.

Don Woods, McMaster University
Guest Editor

Spring 1993




McMaster University
Hamilton, Ontario, Canada L8S 4L7

he subject "Chemical Engineering" has struc-
ture. It is not an unrelated collection of about
three thousand equations that we somehow
put together to solve problems. The subject is built
upon fundamental laws, concepts that allow us to
use those laws, models, theories, semi-empirical cor-
relations, and data. English and mathematics are
the languages we use to work within the subject.
Unfortunately, some surveys of our graduating
seniors reveal that many see the discipline as a
"collection of isolated equations to be memorized
and 'cooked' to solve problems." They see no rela-
tionship between such courses as thermodynamics
and heat transfer-the topics are seen simply as
different courses taught in different semesters by
different instructors. Students fail to recognize links
between the courses and the concepts in chemical
engineering, and consequently they see little struc-
ture to the subject.
There are two vital types of structure: we use
a structure of the knowledge to facilitate learn-
ing, and we use a structure of the knowledge to
solve problems.

Structures to Facilitate Learning
To facilitate learning, Ausubelm emphasized the
importance of providing students with "advanced
organizers." Such advanced organizers help students
see the structure of the subject and provide a "big
picture" of the route ahead. The structure, selected
to facilitate learning, provides a framework that we
can hang new knowledge on as we learn it. One
considers which concepts are easier to learn first
and notes a certain sequence of topics. Most texts
attempt to provide such structure, and most of us in
the field of teaching attempt to provide such struc-
ture to facilitate learning.
The structures and relationships are created to
facilitate learning. The structures may pertain only

Don Woods is a professor of chemical engi-
neering at McMaster University. He is a gradu-
ate of Queen's University and the University of
Wisconsin. His teaching and research inter-
ests are in surface phenomena, plant design,
cost estimation, and developing problem-solv-
ing skills.

Rebecca Sawchuk is a senior in McMaster
University's chemical engineering undergraduate
program. The goal of her senior thesis project is
to link the fundamentals of chemical engineering
to form an organized "structure" of the knowl-
edge. She plans to work at Dow Chemical Canada
Inc. after graduation.

to the course and the subject we are teaching.
Rarely does the structure interlink with other
courses. Novak and Gowin[21 suggest "concept map-
ping" as a useful way of displaying the structure.
Our work with seniors shows that they can create
reasonable concept maps that reflect the structure
used to help them learn. However, they provide
separate and unconnected maps for each course.
Furthermore, the maps are very detailed and tend
to classify the information on the basis of the se-
quence in which it was taught. As they develop the
maps they say, "First we had this, and then this,..."
Thus, what we and the textbooks are providing
seems to help their recall. On the other hand, they
rarely have thought previously about connecting the
maps to see the bigger picture of all the under-
graduate subject matter.

Structures to Facilitate Problem Solving
A crucial finding about problem solving is that the
problem a person solves is their own internal, men-
tal image, or representation of the problem. We do
not simply solve "problem 6.3 at the end of Chapter
6." Although one reads the problem statement, the
mental task is one of reformulating the words and
images into some mental image of what "we think
the problem is all about." The creation of that inter-
nal representation is dictated by the problem solver's
Copyright ChE Division ofASEE 1993
Chemical Engineering Education


internal structure of the subject knowledge.
For example, a student's internal representation
of chemical engineering for the purposes of problem
solving may be a "collection of unrelated equations."
Unsuccessful problem solvers tend to use a trial-
and-error tactic of using equations that will "use
up" the information they are given. For example, a
problem statement in Chapter 3 of a fluid mechan-
ics textbook included extraneous viscosity data. One
of the A+ students searched through the text until

he found, in Chapter 5, an equation that included
viscosity and all of the other information in the
problem statement!
This behaviour might be interpreted as being re-
lated to people whose grasp of the subject discipline
is only an unstructured collection of unrelated equa-
tions. Clement'31 and Larkin"'4 provide evidence in
the context of physics. Clement suggests that we
use four interconnected and hierarchical modes of
thinking with our internal knowledge: observations

Design of a solution
Write relations involving
known and desired Combine and solve

Figure 1. Unsuccessful problem-solver's script (From Larkin;151 reproduced with permission)

POINTER Rotating object.
Motion of various
points on that object.


Immediate Analytic s
Diagram showing object's rotation, Visualize separately motion of
path of center of mass center of mass and motion relative
to center of mass

| Motion of center of
v R mass is just like
motion of a particle.

to another
area of physics

Particle Lotion

mx=F F

Relate by integration


Figure 2. Successful problem-solver's script. (From Larkin;ts5 reproduced with permission)

Spring 1993


in series

R=R1+R2 C C1C
CC=1 C2

ap: i -tors
in neres


and practical knowledge-leading to qualitative
physical models-leading to concrete mathematical
models-leading to written symbol manipulation.
Successful problem solvers tend to start solving prob-
lems by checking the observations, qualitatively un-
derstanding what is going on, invoking mathemati-
cal models, and then manipulating symbols to ob-
tain a quantitative result. Thus, they start with

observations and a qualitative understanding of what
is going on. Unsuccessful problem solvers depend
solely on symbol manipulation. Larkin's research
uncovered key differences between unsuccessful and
successful problem solvers: the unsuccessful prob-
lem solver, as illustrated in Figure 1, selects "point-
ers" in a problem in DC circuits that lead to a broad
set of relationships that then had to be played around

Comparison Between Unsuccessful and Successful Problem Solvers' Use of Knowledge

Problem Solvers' Use of Subject Knowledge Problem Solvers' Use of Subject Knowledge
cannot quickly and accurately identify the pertinent subject knowledge; tend rapid and correct identification of the pertinent subject (usually within
to play around with many equations;[4,5] tend to manipulate symbols and seconds of completing the reading of the problem statement)[51
combine what they select as being a relevant relationshipI3'4]
misinterpret and misuse "pointers"l51 identify and use "pointers" to zero in rapidly on key principles and

redescription and creation of mental image is limited, formal, and often not redescription is rich, accurate, and uses assumptions and approximations
helpfull51 rapidly to identify key features;[5] use qualitative analysis to point to
crucial concepts[61

a particular relationship is recalled independently of any general relationship strong structure connecting concepts, principles and laws[51 apply
upon which it is based;15] no restructuring and chunkingg" of knowledge; related "chunks" of subject knowledge[6]
work with independently applied individual principles(61
do very little qualitative analysis[61 do extensive qualitative analysis of the situation16]
unwilling to guess, to make approximations, and have no memorized, order- have memorized "tacit" or order-of-magnitude experience factors that
of-magnitude values to assist them in doing a qualitative analysis110'11,121 allows them to do rapid and extensive qualitative analysisl0',l1,12]
have incomplete and imprecise knowledge about knowledge[8] have a complete set of knowledge181
lack an organized, hierarchical and abstract knowledge structure that is based possess an organized, hierarchical, and abstract knowledge structure that
on fundamentals and tied to the real world by pointers[9'10] is based on fundamentals and tied to the real world by pointers19'101
do not know when to apply general theory and when to apply specific
subsets of the general theory that seem to apply[6,81
confuse specific and special cases with generally applicable relationshipsl81
have difficulty recalling/identifying conditions under which special case
equations apply and hence try to apply these when they are inapplicable[7,8]
have difficulty identifying and formulating the specific information to which
the general principles applyl81
have difficulty reasoning from basic principles; instead rely on "beginning"
and "end" events without reflecting on the chain of events between the two;
depend on redescriptive activities which merely rephrase the problem
situation without advancing one's understanding of it; depend on inappropri-
ate arguments by analogy181
cannot distinguish between additive and non-additive quantities;[81 have
difficulty working with "intensive" properties[81
place more emphasis on collecting sample solutions and working examples
than on understanding the fundamentals when "learning" a subjectll'0
replace precise technical definitions with imprecise, everyday usage, e.g.,
fail to realize that once certain physical parameters are set, other measurable
quantities cannot be varied independently181
have conceptual difficulty applying calculus in physics[81

82 Chemical Engineering Education


with and "cooked" to see which one might apply.
The successful problem solver, illustrated in Figure
2, selects "pointers" in a problem on a falling disk
that show a direct and rapid connection with funda-
mental principles and methods. A summary of the
research on unsuccessful and successful problem
solver's use of subject knowledge is summarized in
Table 1. '3-13
More specifically, research has shown that suc-
cessful problem solvers have a structure to their
subject knowledge that-instead of being a collec-
tion of unrelated concepts and equations-is charac-
terized as follows:
1. The knowledge is structured hierarchically (with
fundamental laws and principles at the higher
levels and surface structure and pointers at the
lower levels. [6,9,14.15,16]
2. The highest levels in the hierarchy-or the under-
pinnings-are the fundamental laws, the abstrac-
3. Related to the fundamentals are concepts and
"chunks" of information that allow us to apply the
fundamentals effectively. The knowledge is encoded
to include conditions and constraints when the
knowledge is applicable.t4'7'1,6
4. The lower levels are the surface structure (key words
in a problem statement that trigger one to use
certain approximations or concepts or descriptions
of the everyday events that work because of the
fundamentals) and "pointers" that link the surface
structure to the fundamentals.5'"'7''16171
5. Encoded with the subject knowledge is "tacit" or
memorized, order-of-magnitude numerical values
that allow qualitative application of the knowl-
edge. 10-121
6. Subject knowledge is organized in block or "chunks"
convenient for mental processing.6'7'121

Concerning the types of knowledge, there are
* the fundamentals
* concepts or defined terms to allow us to use the
the procedural knowledge about how to work with
the information
the pointers or links
a rich set of episodic knowledge that gives us a
qualitative understanding of what is going on, as
opposed to a series of symbolic equations that one
manipulates. This includes memorized, numerical,
and order-of-magnitude knowledge.

Glaser114' suggests that the knowledge structure is
not static; rather, as new knowledge comes in it
should be embedded in the hierarchy, attached to
the fundamentals, and related to the episodic knowl-
Spring 1993

edge so that it relates to our past experience. This
embedding modifies the original structure.

Identifying the fundamentals is not easy. Some-
times the things we call "laws" are "wishes," not
laws; sometimes "principles" are really laws, etc.
Some terminology might be:
Law A universally applicable explanation of how things
behave; e.g., the conservation of mass.
Constrained Law An explanation that is applicable over a
defined set of circumstances; e.g., the ideal gas law.
Balance An equation applied to a conserved entity-thus
one would have a "mass balance," but not a "mole balance" or
a "volume balance."
Model A representation of a situation for the purpose of
explaining how it behaves.
Theory A mathematical relationship between the dependent
and independent variables that is almost completely based on
fundamental laws and constrained laws. There may be a few
constants that have to be used to tune the theory to the
specific situation. There may be many different theories for
one particular behaviour.
Empirical Correlation A mathematical relationship
between the dependent and independent variables. No theory
or fundamentals were used in creating the relationship. It
considers the system to be a "black box."
Semi-Empirical Correlation A mathematical relation-
ship between the dependent and independent variables that is
based on some fundamental laws and constrained laws.
Concept A general term for an entity or idea that is useful
in applying a law; e.g., the concept of "force."
Convention An agreed-upon set of rules; e.g., Gibbs
convention for the dividing surface in surface phenomena.
Postulate A simplifying set of agreed-upon conditions.
Examples of "laws" and "postulates" pertinent to
chemical engineers include1'18191

1. Law: Mass is neither created nor destroyed; it is
conserved; the total mass is conserved; the mass of
an element is conserved (unless nuclear reactions
occur or E=mc2 occurs, in which case, mass and
energy will exchange).
2. Law: Electrical charge is neither created nor de-
stroyed; it is conserved.
3. Law: Energy is neither created nor destroyed; it is
conserved (unless nuclear reactions occur or E=mc2
occurs, in which case mass and energy will ex-
4. Law: Momentum is conserved.
5. Law: The law of definite proportions is related to
compounds and their formation.
6. Law: The second law of thermodynamics-systems of
processes occur so as to minimize the total free
energy in the system. Concept: free energy.


7. Law: If a process proceeds spontaneously, the reverse
process can never proceed spontaneously.
8. Law: If a system is left alone, it will go to a state of
dynamic equilibrium that has equal forward and
reverse rates and no available free energy.
Extensive details are needed for each law or corre-
lation.'4'12' The details include
a statement of the fundamental principle law of
an identification of the meanings of all the concepts
used in the law
identification of the dependent and independent
numerical units of measurement
listing of the region of application, identification of
the limitations and assumptions
hints to prevent errors in the application
utility hints (tacit information) about when a
particular principle is most useful
In addition, we must have a qualitative under-
standing bf what is going on as predicted by the law.

To simplify our ways of thinking about nature and
how it behaves, we often define simplifying postu-
lates. Rase"1' provides the following examples of pos-
1. Postulate: Isothermal (constant temperature)
2. Postulate: Isobaric (constant pressure)
3. Postulate: Isochoric (constant volume)
4. Postulate: Isentropic (constant entropy); simplifica-
tion for a compressor or turbine
5. Postulate: Isenthalpic (constant enthalpy); simplifica-
tion for flow through a valve
6. Postulate: Adiabatic (no exchange of energy between
the inside and the outside of the system); simplifica-
tion for perfect insulation
7. Postulate: Equilibrium exists (assume an infinitely
fast rate)
8. Postulate: Reversibility (neglect friction)
9. Postulate: Ideality (this has many subcomponents);
ideal gas when the ideal gas law applies; ideal liquid
(could be zero viscosity or Newtonian depending on
how ideal is defined); ideal Hookean solid, ideal
isotropic solid, ideal solution, ideal mixture, ideal
crystal, ideal catalyst
10. Postulate: Models for mixing; plug flow or complete
11. Postulate:Incompressible flow (Vv) = 0
12. Postulate: Unidirectional flow
13. Postulate: Black body radiation and grey body

14. Postulate for shape and configurations: infinite
shape, semi-infinite shape, perfectly smooth surface,
zero thickness surface region, point source, constant
total cross-sectional area, and perfect geometrical
shapes (flat, cylindrical, spherical)
15. Postulate for time: steady state, pseudo steady state,
zero time, infinite time
16. Postulates about limiting cases
As we move from laws to models, through con-
cepts and through to postulates and conventions, we
move down the structure. Indeed, the pointers that
connect the real world to the structure are usually
connected to "postulates."

Knowledge has structure. Having the appropriate
structure facilitates learning and problem solving.
Key characteristics of the knowledge structure to
aid in problem solving are that knowledge is hierar-
chically organized with the fundamentals at the
higher levels and pointers at the lower levels. Knowl-
edge is "chunked" to include the bases, assumptions,
conditions of application, and tacit or experience
knowledge. Some example "laws" and "postulates"
have been given in this paper.

1. Ausubel, D.P., Educational Psychology: A Cognitive View,"
Holt, Rinehart, and Winston, New York (1968)
2. Novak, J.D., and D. Bob Gowin, Learning How to Learn,"
Cambridge University Press, Cambridge (1984)
3. Clement, J., "Some Types of Knowledge Used in Under-
standing Physics," unpublished manuscript, Dept. of Phys-
ics and Astronomy, University of Massachusetts (1977)
4. Larkin, J.H., "Developing Useful Instruction in General
Thinking Skills," Paper JL010276, Group in Science and
Mathematics Education, University of California, Berkeley
5. Larkin, J.H., "Cognitive Structures and Problem Solving
Ability," Paper JL060176, Group in Science and Mathemat-
ics Education, University of California, Berkeley (1976)
6. Larkin, J.H., "Processing Information for Effective Problem
Solving," unpublished paper, Group in Science and Math-
ematics Education, University of California; presented at
the Amer. Asso. of Physics Teachers, Chicago (1977)
7. Larkin, J.H., "Understanding Problem Representations and
Skill in Physics," Internal Report, Carnegie Mellon Univer-
sity (1980): Larkin, J.H., et al., "Expert versus Novice Per-
formance in Solving Physics Problems," Science, 208, 1335-
1342 (1980): Larkin, J.H., "Cognition in Learning Physics,"
Am. J. of Physics, 49(6), 534-541 (1980)
8. Lin, H.S., "Problem Solving in Introductory Physics: De-
mons and Difficulties," PhD Thesis, Department of Physics,
MIT, Cambridge, MA (1979)
9. Voss, J., "Problem Solving and the Educational Process," in
Handbook of Psychology and Education, R.Glaser and A.
Lesgold, eds., Lawrence Erlbaum Publishers, Hillsdale, NJ
10. Woods, D.R., et al., "56 Challenges to Teaching Problem
Solving," CHEM 13 News, 155, (1985); "Major Challenges
Chemical Engineering Education


to Teaching Problem Solving," Annals of Engr. Ed., 70(3),
11. Mettes, C.T.C.W., A. Pilot, H.J. Roossink, and H. Kramers-
Pals, "Teaching and Learning Problem Solving in Science,"
J. Chem. Ed., 57(12), 882-885 (1980) and 58(1), 51, 55
(1981); B. van Hout Wolters, P. Jongepier, and A. Pilot,
"Studiemethoden," AULA, Uitgeverij Het Spectrum, Utrecht
(in Dutch), and K. Mettes and J. Gerritsma, "Probleem
Oplossen," AULA, Uitgeverij Het Spectrum, Utrecht (in
Dutch) (1985)
12. Reif, F., and J.I. Heller, "Making Scientific Concepts and
Principles Effectively Usable: Requisite Knowledge and
Teaching Implications," Paper ES-13; "Cognitive Mecha-
nisms for Facilitating Human Problem Solving in Physics:
Empirical Validation of Prescriptive Model," Paper ES-14b;
and "Knowledge Structure and Problem Solving in Phys-
ics," Paper ES-18; Physics Department, University of Cali-

Sbook review

(Second Edition)
by D. Kunii, O. Levenspiel
Butterworth /Heinemann, Stoneham, MA 02180; 491
pages, $145 (1991)
Reviewed by
Roy Jackson
Princeton University
The first edition of this book, which appeared
over twenty years ago, enjoyed considerable success
in drawing together the research results available
at that time and synthesizing from them a con-
nected account of direct value to engineers involved
in the design of fluidized beds. It is, therefore, a
hard act to follow-but this second edition succeeds
in preserving (and even enhancing) the virtues of its
predecessor, while at the same time weaving many
newer ideas into the fabric of the text.
Though some passages from the earlier work are
retained, the present book is essentially a completely
rewritten text. Even where the material is similar
to the earlier presentation, it has been reorganized,
expanded, and supplemented with more worked ex-
amples. There is much more attention paid to mat-
ters such as the influence of the properties of the
particulate material on fluidization behavior, rest-
ing on concepts (such as the Geldart classification)
which have appeared since publication of the first
edition. Variants on the classical dense fluidized
bed are also treated; for example, a whole chapter
(entitled "High Velocity Fluidization") is devoted to
turbulent beds and fast fluidized beds, configura-
tions that have become increasingly important. On
the other hand, the many students and practitio-
Spring 1993

fornia, Berkeley (1982);
13. Woods, D.R., "Summary of Novice versus Experts Research
Results," PS News, 55, 55-2 to 55-21 (1988)
14. Glaser, R., "Education and Thinking: The Role of Knowl-
edge," Amer. Psychologist, 39(2), 93-104 (1984)
15. Boreham, N., "A Model of Efficiency in Diagnostic Problem
Solving: Implications for the Education of Diagnosticians,"
Instructional Sci., 15, 119-121 (1986)
16. Bransford, J., et al., "Teaching Thinking and Problem Solv-
ing," Amer. Psychologist, 41, 1078-1089 (1986)
17. Bhasker, R., and H. Simon, "Problem Solving in Semanti-
cally Rich Domains: An Example from Engineering Ther-
modynamics," Cognitive Sci., 1, 195-215 (1977)
18. Porter, S.K., "Ordinary Atoms Made in Stars," J. of Col.
Sci. Teach., Dec 1985/Jan 1986, p. 168 (1986)
19. Rase, H.F., Philosophy and Logic of Chemical Engineering,
Gulf Publishing Company, Houston, TX (1961) J

ners who have benefited from the information in
Chapter 3 of the first edition (which provided ex-
plicit instruction on how to estimate such elemen-
tary, but vital, properties as the terminal velocity of
fall and the minimum fluidization velocity) will be
happy to know that the same chapter of the second
edition provides the same help, but in an updated
and improved form.
My only criticism of the first edition was that the
very success of the authors in presenting the mate-
rial in such simple, clear exposition tended to give a
false impression that the material was well estab-
lished, reliable, and beyond controversy. In fact, this
was far from the truth. Many of the correlations
presented were extrapolations from limited data,
while the models, though reasonable and the best
available at the time, were gross simplifications
which had been subjected to only the most superfi-
cial testing. In short, the story was told so well that
it made the state-of-the-art seem much more firmly
based than it really was.
I have some of the same feeling about the second
edition. The unwary designer might easily be se-
duced into following the path so clearly marked out,
only to receive a rude awakening further down the
road. The subject remains today a very messy one,
in a state of continuing flux, with both the physical
principles and the tools available to apply them
changing very quickly.
But this is only a minor reservation about a book
which is likely to be as well received as was its
predecessor. We might even hope that the rapid
changes in the field will encourage the authors to
venture a third edition at some time in the future.




The University of Pennsylvania
Philadelphia, PA 19104-6393

he assigned objective for the presentation that
led to this paper was a discussion of the struc-
ture of knowledge in applied mathematics
which is appropriate to the undergraduate chemical
engineering curriculum. That presentation was, and
this paper is, actually focused on a limited aspect of
the assigned topic-namely, the form of exposition
of applied mathematics in the curriculum and its
reception and retention by students. The state and
consequences of current undergraduate preparation
in applied mathematics will be examined first, and
then proposals for improvement will be presented.

Doctoral students in chemical engineering have
sufficient time and a sufficiently narrow focus so
that they can master and utilize those aspects of
applied mathematics that are directly useful in their
research. On the other hand, undergraduate stu-
dents are currently exposed to a great amount of
material, including applied mathematics, in a form
and at a rate that precludes its mastery. The need
for such mastery as contrasted with exposure is first
considered, and then the degree to which it is cur-
rently accomplished. The superior preparation in
mathematics of students from Europe and Japan is
evident to all who encounter them in graduate
courses. Their preparation provides a measure of
what we might aspire to achieve.

Stuart W. Churchill is the Carl V.S. Patterson
Professor Emeritus at the University of Penn-
sylvania, where he has been since 1967. His
BSE degrees (in ChE and Math), MSE, and
PhD were all obtained at the University of Michi-
gan where he also taught from 1950-1967. Since
his formal retirement in 1990 he has continued
to teach and carry out research on heat transfer
and combustion. He is also currently complet-
ing a book on turbulent flow.
Copyright ChE Division ofASEE 1993


"It's staggering to consider how much one's
social acceptance depends upon being quadrati-
cally integrable."

Most of those in the academic community do not
need to be convinced that innovative applications of
mathematics have made great contributions to the
advancement of the practice of chemical engineer-
ing in recent decades. Unfortunately this contribu-
tion is largely ignored and grossly underestimated
by its principal beneficiary-the chemical and
process industries (CPI). The leaders of the CPI
take for granted the improvements in process de-
sign, process control, process analysis, and process
safety that are a direct consequence of better
modeling and better understanding. They do not see
the link between those improvements and the
articles in mathematical language which have
appeared in the literature of engineering science,
such as the AIChE Journal.
Faculty members who read this article (and in
particular, the authors of the associated papers of
this symposium) owe their academic and professional
achievements at least in part to a superior grasp of,
and facility with, applied mathematics. A good un-
derstanding of applied mathematics and a willing-
ness to extend that understanding, as stimulated by
new topics in research or teaching, is a necessary
condition for successful academic practice today.
We owe our students an appropriate preparation
in mathematics, not only for the problems that are
current or foreseeable, but also as a foundation for
the acquisition of those mathematical skills that
will become important during their lifetime. If we
are successful, our students will have sufficient con-
fidence in their mathematical background and a suf-
ficient vision of its value to exercise, maintain, and
extend this competence throughout their career, or
at least as long as that career is focused on technol-
Chemical Engineering Education


One of [my graduate students] devised a general method for deriving similarity transformations, and he
and others were among the first in engineering to use digital computers. They produced the first
numerical solutions of the partial differential equations governing laminar, transient, and
multidimensional natural convection, and subsequently they were among the
first to model the turbulent regime multidimensionally.

ogy as contrasted with management.
I assert that we now fail almost totally in this
respect-at least with those who do not pursue
graduate work in chemical engineering. The math-
ematical knowledge of those who enter industrial
practice with a bachelor's degree fades rapidly after
graduation owing to its disuse. An ancient but nev-
ertheless instructive study is re-examined below as
evidence thereof.

The 1968 Goals of Engineering Study,'" sponsored
by the American Society for Engineering Education,
is still instructive despite its date. Although it floun-
dered for several reasons, the findings and recom-
mendations with respect to mathematics will be ex-
humed here. As a first step of the study, a survey
was conducted to determine the retroactive self-as-
sessment of their education by engineers in indus-
try who had been practicing five, ten, and twenty
years since receiving their baccalaureate degree. The
participants were asked to identify those aspects of
their undergraduate studies which had proved the
most and the least useful. An increasing majority at
each level of experience asserted that they had never
used mathematics (and in particular, calculus) in
their professional career. The only cited deficiency
in their mathematical preparation was in statistics.
This response was totally misleading; most, if not
all, of the participants had used mathematics in the
first few years of practice, but at the current stage
of their career the ambitions of many had become
more focused on skills and credentials required by
management. The technical and mathematical skills
they had used to reach the point of consideration for
management had been forgotten.
The authors of the Goals report used the results
of the survey as justification for recommending
more courses in managerial-related topics at the
expense of courses in engineering science and math-
ematics. This survey is an example of obtaining the
wrong answer by virtue of asking the wrong ques-
tion. Fortunately, a number of recommendations of
the Goals report, including those related to math-
ematics, were eventually rejected by the AIChE and
Spring 1993

other professional societies as unacceptable criteria
for accreditation.
My informal conversations with undergraduate stu-
dents at the University of Pennsylvania over the
past twenty-five years suggest that when they gradu-
ate they still lack confidence in their working knowl-
edge of engineering science and, to an even greater
extent, advanced mathematics. This is undoubtedly
a factor in the choice by many of positions in mar-
keting, sales, etc., that will not expose this pre-
sumed deficiency. Similar conversations with our
graduate students from other schools (consisting pre-
dominantly of students who performed exception-
ally well in engineering science and mathematics)
indicates that this insecurity is not unique to any
one school or group of students. If the Goals survey
were reconducted with industrial practitioners to-
day, a quarter of a century later, I suspect the same
general response would be obtained.

The standard textbooks in chemical engineering
provide additional insight into the role of applied
mathematics in the undergraduate curriculum. In
Elements of Chemical Engineering, by Badger and
McCabe,[2' (out of which I studied as an under-
graduate) the models were almost wholly algebraic.
Even Unit Operations, by Brown and Associates, 31 a
generation later in 1950, utilized very few differ-
ential models and then only elementary ones. Two
outstanding books on applied mathematics in chemi-
cal engineering appeared in the interim-Applied
Mathematics in Chemical Engineering, by Sherwood
and Reed in 1939,[4] and Application of Differential
Equations to Chemical Engineering Problems, by
Marshall and Pigford in 194751--but the material
therein was not required to solve any of the prob-
lems considered in the undergraduate curriculum
and had no direct impact thereon. Indeed, to this
day material of the level of mathematical sophisti-
cation of these latter two books is hardly recognized
in industrial practice, at least outside research de-
partments or their equivalent. As an aside, it may
be significant to note that no mention was made of
the former pioneering book in the citation for the
recent award of the National Medal of Science to
Charles E. Reed.


Subsequent undergraduate texts, beginning in
1960 with Transport Phenomena by Bird, Stewart,
and Lightfoot,"6' have used more advanced notation,
models, and solutions, but otherwise have not greatly
extended the mathematical demands on students.

Several years ago, on the occasion of the 75th
Anniversary Meeting of the AIChE, I was asked to
review the role of mathematics in the history of our
profession and particularly in its publications.[7'
One of my conclusions was that our graduates were
no longer sufficiently prepared in mathematics to
read much of the AIChE Journal. That situation
has not changed significantly in the intervening
nine years. The mathematical sophistication of
our published research is certainly advancing more
rapidly than the preparation of our undergraduates
in this respect.
This inaccessibility of the articles in our archival
journals to our bachelor's graduates in terms of com-
prehension has at least two serious consequences:
1) the advances in knowledge, as described in the
journals, are a well-kept secret insofar as most of
our profession is concerned, and 2) the cultural gap
and the difficulty of technical communication be-
tween those with graduate education and those with-
out becomes ever greater.
How can we expect the results of analysis to
be implemented in industry if they are effectively
hidden from the practitioners and managers by
their expression in an unknown language? How can
we maintain professional continuity across the
division of degrees without a common language? This
is a relatively new phenomenon, and hence it is not
widely recognized.

A Personal Digression *
At this point I will take the liberty of citing some
of my own experiences as evidence of the impact on
chemical engineering practice of even modest skills
in applied mathematics. Such a personal digression
is perhaps tolerable since these experiences form
the basis for my commitment to, and proposals for,
improved instruction in this field.
As an undergraduate I majored in applied math-
ematics as well as in chemical engineering, and I
received a bachelor of science degree in each. The
decision to do this was one of the more felicitous
ones in my career since it set me somewhat apart
from my classmates in chemical engineering and
greatly enhanced my self-confidence. The added

Faculty members who read this article
(and in particular, the authors of the associated
papers of this symposium) owe their academic
and professional achievements at least in
part to a superior grasp of, and facility
with, applied mathematics.

mathematics was not very sophisticated by current
standards, and the topics themselves did not differ
greatly from those we now offer our undergraduates
in an elective or required course in "advanced math-
ematics for engineers." My courses were, however,
taught by mathematicians, and the classes included
students majoring in mathematics as well as in en-
gineering. Accordingly, the instruction included a
stronger focus on structure and rigor.
That focus has proven to be a long-term benefit-
one which is often denied our current students when
such courses are taught by engineers. For the short
term, including my first years of practice after gradu-
ation, the primary benefit of the added mathematics
was a capability to solve problems that my peers in
chemical engineering could not. Upon returning to
graduate school five years later, I was able to main-
tain this competitive edge by taking as many elec-
tives as possible in that lowly branch of mathemat-
ics known as analysis, and this had the unexpected
but happy collateral consequence of allowing me to
take advanced courses in physics.
In each phase of my subsequent academic career
I have been able to take advantage of this marginal
preparation in mathematics by my choice of and
approach to problems of research. My graduate
students have also been encouraged to undertake
problems involving advanced applications of ap-
plied mathematics. One of them devised a general
method for deriving similarity transformations, and
he and others were among the first in engineering
to use digital computers. They produced the first
numerical solutions of the partial differential equa-
tions governing laminar, transient, and multidimen-
sional natural convection, and subsequently they
were among the first to model the turbulent regime
multi-dimensionally. In the course of solving nu-
merically the integro-differential equations govern-
ing radiative transfer through dispersed media, we
were privileged to interact personally with Peter J.
W. Debye, John von Neumann, George Uhlenbeck,
and Subrahmanyan Chandrasekhar. My later stu-
dents solved models for radiatively stabilized com-
bustion that involved integro-differential equations
and complete free-radical kinetic models. My most
Chemical Engineering Education


recent students devised the first numerical solu-
tions for flow and heat transfer with secondary mo-
tion in double-spiral heat exchangers and for
thermoacoustic convection.
These exciting and productive experiences have
all been a consequence of undertaking problems for
which no method of solution was yet known. We did
not necessarily even know in advance what forms of
mathematics would prove helpful; we simply were
confident (or foolhardy) enough to believe in our
ability to identify, master, and apply the necessary
techniques, whatever they might be.
This is not a unique story; it could be told with
slight differences by a number of others in our pro-
fession.[8' The only excuse for reviewing my own ex-
periences is that they illustrate the advantages that
accrue from a working knowledge of advanced math-
ematics and the confidence to use that knowledge.
This confidence, particularly when it extends be-
yond classical solutions, leads to boldness in tack-
ling problems of unpredictable difficulty and in de-
vising new techniques for their solution. Our objec-
tive as teachers should be to prepare and encourage
students to undertake such challenges rather than
to avoid them.
Although the above experiences were generally in
the framework of doctoral research, in most instances
one or more undergraduates also participated and
they often made significant intellectual contribu-
tions. Insofar as it is possible, such an experience
should be provided for our undergraduates as a
supplement to their regular course work. (As an
aside-within-a-digression, there appears to be a
strong correlation between undergraduate partici-
pation in exciting research and a positive decision to
attend graduate school.)

The principal technological event of our genera-
tion has been the rapid development of computers
and their software. This development has shifted
the emphasis on mathematics rather than replacing
it. Students are more receptive to computer use than
to mathematics, but some analogous problems have
arisen. Symbolic manipulators and canned numeri-
cal algorithms are often a convenience, but they
isolate the user from the mathematics and technol-
ogy of the process itself. Most computer scientists
show the same disdain as do pure mathematicians
for engineering-hence, we should anticipate some
of the same difficulties in instruction in numerical
methods that have been endemic in mathematics.
Spring 1993

Said the Mock Turtle with a sigh, "I only took the
regular course."
"What was that?" inquired Alice.
"Reeling and Writhing, of course, to begin with,"
the Mock Turtle replied; "and then different
branches ofArithmetic-Ambition, Distraction,
Uglification, and Derision."
Alice's Adventures in Wonderland
The choice of appropriate topics in applied math-
ematics is difficult, but perhaps less important than
the methodology used in the classroom. I have re-
cently attained the position in academic life desig-
nated as "emeritus." This title apparently means
that you are welcome to assist, but not to advise or
to expect compensation. From that perspective, it
would be unseemly for me to make recommenda-
tions concerning specific topics in applied mathemat-
ics to be included in the curriculum of the future. In
any event, I do not have a crystal ball with which to
predict those topics of engineering science that will
either fade, continue, grow in importance, or newly
appear in the decades ahead. Recently, in this same
journal, Ramkrishna91 mentioned some aspects of
mathematics which may become important to chemi-
cal engineers, but his focus was presumably on re-
search rather than on undergraduate education. I
will limit myself to a single guideline.
The topics of chemical engineering science that
were included in the knowledge structure sympo-
sium (physical chemistry and stoichiometry, ther-
modynamics, transport phenomena, and reaction
engineering) constitute one obvious criterion for the
choice of topics of mathematics to be included in the
curriculum-that is, the essential elements are those
needed to describe the laws and relationships of
these and other engineering sciences in the form of
algebraic, differential, integral, or stochastic mod-
els, as well as to derive solutions for these models
for general or specific cases. The elements and pro-
cedures of mathematics that are common to more
than one of the engineering sciences are of particu-
lar importance. It is, after all, the latter commonal-
ity that gives mathematics an all-encompassing role
and importance in the curriculum.

One technique which I have found to be successful
in the classroom and in homework for providing an
overview of applied mathematics is to emphasize
the analysis of models rather than the process of
solution. (This is not to say that instruction in meth-
ods and solutions should be neglected.) The models



which serve as a starting point for such analyses
should ordinarily be the general partial differential
equations of conservation, together with equations
of state, kinetic mechanisms for chemical reactions,
etc. If these equations are not familiar to the stu-
dents from their previous courses in engineering
science, they must be derived at this point as a
preliminary step. A list of questions which can then
be posed is presented in Table 1.
In some complex processes, for example in turbu-
lent flow1131 and in the motion of a rising bubble,[141
considerable information can be attained, even if a
simple list of variables rather than a set of partial
differential equations is used as the model. Conven-
tional dimensional analysis is then applied to deter-
mine a minimal set of linearly independent dimen-
sionless groups incorporating all of these variables.

One or more procedures for identifying these group-
ings are described in most standard textbooks. They
generally fail, however, to indicate the possibility of
error owing to omissions or improper inclusions in
the list of variables. They also often neglect to indi-
cate the consequences of alternative choices of vari-
ables-for example, the shear stress on the wall
rather than the pressure gradient, and the mass
rate of flow rather than the mean velocity. They
frequently fail to stress the significance of alterna-
tive groupings and of the speculative deletion of
variables.[11"12141 These procedures often produce
asymptotic solutions whose only deficiency relative
to a complete analytical solution is a numerical value
for the leading coefficient. As a consequence, this
methodology may prove to be more useful to a stu-
dent and a practicing engineer than some particular

Illustrative Questions to Stimulate the Critical Analyses of Models
1. Which terms can be dropped in general? Why?
2. Which terms can be dropped for particular cases on mathematical grounds? Which ones can be dropped on physical grounds?
3. Which terms must be dropped and which must be retained to obtain the appropriate model for classical limiting cases? For example, for fluid
motion the following limiting cases might be considered:
inviscid flow incompressible flow developing flow open-channel flow
purely viscous flow one-dimensional flow boundary-layer flow buoyant motion
purely inertial flow rectilinear flow steady flow free-streamline flow
slightly inertial flow fully developed flow unconfined flow turbulent flow
4. Which of the reduced models are linear?
5. What is the form of the reduced models from a mathematical point of view (parabolic, elliptic, hyperbolic)? What is the significance of this division
by form?
6. What are the relative advantages of the Eulerian and Lagrangian forms? (This question provides an opportunity to contrast the approaches in
engineering and in pure science.)
7. How can the model be dimensionalized? (This provides an opportunity to introduce the method of Hellums and Churchill'"" or the equivalent.)
8. What are the advantages and disadvantages, if any, of dedimensionalization?
9. Is a similarity transformation possible? (Exposition of the method of Hellums and Churchill"0' is again suggested.)
10. Can the model be simplified by a transformation of variables other than a similarity transformation? (Examples are the stream function, the vector
potential, and the vorticity.)
11. What is the physical significance of each of the boundary conditions? Are they physically realistic?
12. What is the physical significance of each term in the model? What are the criteria for dropping a term?
13. Will different methods of solution, such as 1) separation of variables and expansion in Fourier series, 2) conformal mapping, and 3) the Laplace
transform, lead to different solutions?
14. How can the Laplace transform be used to derive asymptotic solutions?
15. Try to conceive as many types of asymptotic behavior as possible. What reductions in the model are appropriate for these limiting cases? (See
Churchill" .121)
16. To what extent can the behavior of interest be circumscribed by asymptotic solutions?
17. What are the consequences of time-averaging an equation for conservation?
18. Is dimensional analysis of the equations of conservation applicable for turbulent flow in both time-dependent and time-averaged forms?
19. Does a numerical method of solution yield exact results as the subdivisions are increased?
20. What are the relative advantages of finite-difference, finite-element, and Monte Carlo methods?
21. What are the advantages of generalized computer programs as compared to special purpose ones?
22. What is the significance of a pseudo-steady state? Suggest possible applications.
23. What is stiffness with respect to numerical solution of a differential equation?
24. What are the advantages and consequences of developing a solution for the "phase-plane"?
25. What is the consequence of simplifications in a model which reduce the required number of boundary conditions?
26. What is the consequence of a transformation such as the introduction of the stream function which raises the order of the derivatives?
27. How can the required number of boundary conditions be identified?
28. What is the significance and consequence of an integral approximation such as for a boundary layer?

) Chemical Engineering Education


analytical method such as conformal mapping.
The development of correlating equations that in-
corporate asymptotic solutions as components also
helps to provide an overview. The derivation of as-
ymptotic solutions may then be recognized as an
essential part of this process. The derivation of ei-
ther complete analytical or numerical solutions may
similarly be recognized as a means of generating
precise data with which the coefficients in the corre-
lating equation can be evaluated. In this context,
Shinnar~'15 has recently mentioned the need to base
correlations on the advances of engineering science,
and Churchill"161 has asserted that one of the princi-
pal roles of analysis is to support the construction of
correlating equations.
What are the advantages of the above approach?
It focuses on the model rather than on a
particular solution.
It focuses on the process and significance of
reducing the model rather than on the solu-
tion for some reduced model of unknown
It emphasizes the possibility of more than one
method of solution and suggests an objective
basis for comparison of competitive solutions.
It integrates the physics and chemistry with
the mathematics.
It stresses a method of pure reasoning which
will be applicable to new problems.
It reveals the asymptotic character and
significance ofparticular solutions. (Most
solutions in closed form actually fall in this
It avoids excesses by focusing on possible
simplifications and their significance rather
than on general solutions.
It helps to alleviate one of the principal
sources of frustration for students, namely the
evocation of simplified models by the teacher
without explanation or justification in ad-
How and where can we incorporate the above ap-
proach in the curriculum? The simplest procedure is
to incorporate it in each of our present courses in
engineering science as applicable, rather than add-
ing or replacing a course.
Arthur E. Humphrey, my previous departmental
chairman and dean, has asserted that I teach the
same material and methodology in whichever course
I am assigned. In regard to this approach to analy-
sis, I am willing to confess to the crime, if it be one.
Spring 1993

We are failing to prepare our undergraduates to
use mathematics in their professional work. This
failure has serious consequences relative to our
profession and to the recognition of new devel-
opments when they are expressed in the language
of advanced mathematics. The superior preparation
in mathematics of students from Europe and Japan
is a benchmark in this respect. Focusing on the
structure of models rather than only on detailed
analytical solutions is proposed as a partial correc-
tive. The increasing role of computers in the prac-
tice of chemical engineering can be expected to in-
fluence the choice of topics in applied mathematics
but not to eliminate the importance of proficiency
therein. The well-known problems of instruction in
mathematics can be expected to reappear in courses
in computer science.

1. "Goals of Engineering Education," Amer. Soc. Eng. Ed.,
Washington, DC (1968)
2. Badger, W.L., and W.L. McCabe, Elements of Chemical En-
gineering, McGraw-Hill, New York (1931)
3. Brown, G.G., and Associates, Unit Operations, John Wiley
& Sons, New York (1950)
4. Sherwood, T.K., and C.E. Reed, Applied Mathematics in
Chemical Engineering, McGraw-Hill, New York (1939)
5. Marshall, Jr., W.R., and R.L. Pigford, The Application of
Differential Equations to Chemical Engineering Problems,
University of Delaware, Newark, DE (1947)
6. Bird, R.B., W.E. Stewart, and E.N. Lightfoot, Transport
Phenomena, John Wiley & Sons, New York (1960)
7. Churchill, S.W., "The Changing Role of Applied Mathemat-
ics in Chemical Engineering," AIChE Symp. Ser. No 235,
79,142 (1948)
8. Aris, R., and A. Varma, Eds., The Mathematical Under-
standing of Chemical Engineering Systems: Selected Papers
ofNeal R. Amundson, Pergamon Press, Oxford (1980)
9. Ramkrishna, D., "Applied Mathematics: Opportunities for
Chemical Engineers," Chem. Eng. Ed., 24, 198 (1990)
10. Hellums, J.D., and S.W. Churchill, "Simplification of the
Mathematical Description of Boundary and Initial Value
Problems," AIChE J., 10, 110 (1964)
11. Churchill, S.W., "The Use of Speculation and Analysis in
the Development of Correlations," Chem. Eng. Commun., 9,
19 (1981)
12. Churchill, S.W., "Derivation, Selection, Evaluation and Use
of Asymptotics," Chem. Eng. Technol., 11, 63 (1988)
13. Churchill, S.W., "New and Overlooked Relationships for
Turbulent Flow in Channels," Chem. Eng. Technol., 13, 264
(1990); 14, 73 (1991)
14. Churchill, S.W., "A Theoretical Structure and Correlating
Equation for the Motion of Single Bubbles," Chem. Eng.
Proc., 26, 269 (1989); 27, 66 (1990); also see Viscous Flows:
The Practical Use of Theory, Chap 17, Butterworths, Bos-
ton, MA (1988)
15. Shinnar, R., "The Future of Chemical Engineering," Chem.
Eng. Prog., 87, No. 9, 80 (1991)
16. Churchill, S.W., "The Role of Analysis in the Rate Pro-
cesses," Ind. Eng. Chem. Res., 31, 643 (1992) 0




North Carolina State University
Raleigh, NC 27695-7905

Most chemical engineering curricula in North

America begin with the stoichiometry
course. The content of this course is fairly
standard: definitions, measurement, and estimation
of various process variables and physical properties
of process materials; descriptions and flow charts of
unit operations and integrated processes; gas laws
and phase equilibrium relations; laws of conserva-
tion of mass and energy; and the incorporation of all
of the above into material and energy balance calcu-
lations on individual chemical process units and
multiunit processes. The material is not intrinsi-
cally difficult, especially compared to the content of
later courses in transport processes and thermody-
namics, but the approach required to set up and
solve course problems takes most students consider-
able time to grasp.[1
The course may conveniently be divided into two
parts for the purposes of defining a knowledge struc-
ture-material balances and energy balances. Pos-
sible structures for each part are shown in Figures 1
and 2. The following paragraphs comment on those
Knowledge Structure
Material Balances
(Figure 1)
E The concept of conservation is essential to the
course and is generally accepted implicitly. The prin-

Richard M. Felder is Hoechst Celanese Pro-
fessor of Chemical Engineering at North Caro-
lina State University. He received his BChE
from City College of CUNY and his PhD from
Princeton. He has presented courses on chemi-
cal engineering principles, reactor design, pro-
cess optimization, and effective teaching to vari-
ous American and foreign industries and insti-
tutions. He is coauthor of the text Elementary
Principles of Chemical Processes (Wiley, 1986).
Copyright ChE Division ofASEE 1993

cipal concepts that all students must bring with
them from their prior education come from math-
ematics (basic algebraic and graphical manipula-
tions, solving sets of linear equations and word prob-
lems), physics (phases of matter, conservation of
mass), and chemistry (atoms and molecules, stoi-
chiometric equations, molecular weight, and molar
O The building blocks of the course are properties
of process systems and materials including mass,
volume, pressure, temperature, and (later) en-
ergy. These properties are conceptually taken for
granted in the course, although both professors and
students might be hard pressed to define most of
them. Fortunately, they are familiar enough for us
to be comfortable with them, which is all we need to
be able to build on them successfully.
E Other broad concepts that may be presented in
first-year courses but are more likely introduced in
this one include (a) multistep chemical processing
and graphical representation of chemical processes
flowchartss); (b) the idea that a system has a state,
defined as the collection of all its physical properties
and determined by the values of a subset of those
properties; (c) various physical laws and relations
among system variables that allow some variable
values to be determined from specified values of
others; (d) the notions of equilibrium, dynamic equi-
librium, phase equilibrium, reaction equilibrium, and
steady-state and transient operation of a system; (e)
the idea that variables must sometimes be estimated
approximately rather than calculated to six signifi-
cant figures; (f) the idea that problems must some-
times be solved by trial-and-error; (g) the idea that
there's nothing illegal or shameful about (e) and (f).
E Most of the content of this part of the course
consists of definitions of and relations among the
variables that characterize chemical process sys-
tems-temperature, pressure, volume, density,
flow rate, mass fractions and concentrations, frac-
tional conversion, compressibility factor, vapor
Chemical Engineering Education


pressure, relative humidity, etc.-and procedural
and computational algorithms for calculating values
of some of these variables from known values of
others. The sequence of the information flow is sug-
gested in Figure 1.

Knowledge Structure
Energy Balances
(Figure 2, next page)
All of the material listed in the previous section is
prerequisite to that which follows.
O Energy now takes its place as a basic course
concept that few involved with the course either

understand or question. Again, the fact that we think
we know what it is and feel comfortable about it
keeps us from hopelessly bogging down at this point.
D The integral of a function now joins the list of
prerequisite mathematical concepts and the prin-
ciple of conservation of energy joins the required
physical concepts.
ED Once energy and temperature are admitted as
working concepts, the three forms of energy (kinetic,
potential, and internal) and modes of energy trans-
fer (heat and work) can be introduced, setting the
stage for the first law of thermodynamics.
D A concept that arises in the context of the first

Figure 1. Knowledge structure-material balances

Spring 1993


law is enthalpy, defined for convenience as a fre-
quently occurring combination of other system vari-
ables (H = U + PV). Perhaps because of its strange-
sounding name, students never get fully comfortable
with enthalpy-they eventually learn to work with
it (as in Q = AH), but they always have the feeling
that there is something fundamental and mysteri-
ous about it that puts it beyond their intellectual
reach. Trying to convince them that enthalpy is re-
ally a simple concept and things like mass and en-
ergy are concepts much more worth worrying about

is generally futile. (Later they will get into thermo-
dynamics, which will finish the job of overwhelming
them with variables defined for convenience and
given strange names.)

0 The previously introduced notions of the state
of a system and state properties now reappear, lead-
ing to the concepts of reference states and process
paths for calculating AU and AH. All the necessary
ingredients for energy balance calculations are now
in place.

Figure 2. Knowledge structure-energy balances.
Chemical Engineering Education



E Most of the content of this part of the course
consists of (a) definitions of and procedures for
measuring and estimating the energy-related vari-
ables that characterize chemical process systems-
heat capacities, latent heats of phase change,
solution, and mixing, and heats of reaction, forma-
tion, and combustion; (b) procedural and computa-
tional algorithms for calculating internal energy
and enthalpy changes associated with transitions
from one system state to another; and (c) procedures
for solving the first law equation for unknown en-
ergy flows or changes in state in various unit pro-
cesses. The sequence of the information flow is sug-
gested in Figure 2.

Most stoichiometry textbooks contain a chapter on
transient system balances. A key concept here is
that of a derivative. While students in the course
can differentiate functions on demand, they have no
physical or intuitive understanding of derivatives,
in part because most mathematics professors seem
to fear that they would harm their reputations by
putting applications in the elementary calculus se-
quence. Covering transient balances in the stoichi-
ometry course could help the students make signifi-
cant progress toward this understanding; unfortu-
nately, the course almost never gets to transient
balances and most introductory transport courses

take the underlying knowledge structure of this topic
for granted.

Once a knowledge structure has been defined, the
next logical step is to consider how it can best be
transferred into students' brains. I believe that for
stoichiometry there are two keys:

1. Provide explicit instruction and lots of drill in
basic problem-solving procedures,
especially the systematic use of the flow chart
coupled with informal degree-of-freedom analy-
sis to organize the solutions of material and en-
ergy balance problems.

2. Establish an active, cooperative learning environ-
Have students work in teams on problems in
class and on the homework, identifying concep-
tual and procedural sticking points and finding
out how to get past them, teaching and learning
from one another.

For specific ideas about how to accomplish these
tasks, see Reference 1.

1. Felder, R.M., "Stoichiometry Without Tears," Chem Engr.
Ed., 24(4), 188 (1990) 0

RJR% letter to the editor

Dear Editor:
Due to an unfortunate oversight, the article on the Mark-
ovian approach to chemical kinetics (CEE, 27, 42-43) does
not discuss the importance of choosing properly the
duration of a stage for the sake of acceptable numerical
accuracy. In any discrete approximation to a contin-
uous phenomenon, the time increment in the former
must be sufficiently small, and Markov chains are no
In the numerical example of the article, the stage width
of 0.001 minutes is one appropriate choice, when 75% of
species A at a given time decompose to B and 5% of
species B at a given time decompose to species A in one
minute. With this choice, the Markov formulation
A(n+ 1) (0.99925 5.0e-5)(A(n))
B(n+ 1)) (7.5e-4 0.99995)(B(n))
and the integral rate equations
A(t) =0.075 +0.925 exp(-8.0 e 4t)
B(t)= 1.125-0.925 exp(-8.0 e 4 t)
Spring 1993

agree to at least a four-decimal accuracy when t=n=stage
number, as shown in the tabulation. Steady state condi-
tions are reached essentially at n = 10000.
The author regrets the omission of this material, and
wishes to thank Dr. Alan M. Lane at the University of
Alabama, Tuscaloosa, for drawing it to his attention.

n A(n) B(n)
Markov Rate Eq. Markov Rate Eq.
1 0.9993 0.9993 0.2007 0.2007
10 0.9926 0.9926 0.2074 0.2074
1000 0.4905 0.4906 0.7095 0.7094
5000 0.0919 0.0919 1.1081 1.1081
10000 0.0753 0.0753 1.1247 1.1247
inf. 0.075 0.075 1.125 1.125

Thomas Z. Fahidy
University of Waterloo



A Structure for Teaching and Learning About Much of Reality

University of Virginia
Charlottesville, VA 22903

Thermodynamics is an amazing discipline. Its
two brief laws contain the complete basis for
establishing the states of pure and mixed sys-
tems and their tendencies for change. The founda-
tion for scientific investigations into all forms of
matter lie in its rigor. Constraints on engineers to
interconvert heat and work, separate components
from solutions, and obtain ultimate yields from
chemical reaction arise with its symbolic manipula-
tion. Reliable screening for feasibility and optimiza-
tion of nearly every type of process can be guided by
its procedures. Thermodynamics is fundamental and
applicable to all technical endeavors.
Though thermodynamics focuses on natural phe-
nomena, it is really just a deductive structure devel-
oped by creative and systematic human minds. Na-
ture has carried out her diverse processes for eons
without being explicit about energy, entropy, and
fugacity. We find these mental constructs useful be-
cause they give us a chance to assimilate extensive
amounts of real behavior, rather than being over-
whelmed by its totality or misguided by less general
alternatives. Further, we can use "always true" ther-
modynamics with appropriate information and ap-
proximation to effectively describe and predict mul-
titudes of reality.
Modern thermodynamic ideas originated over 150
years ago, but the subject still evolves. Although
some scholars claim that "there is nothing new in
thermodynamics," a few still find challenges in its

John O'Connell is H.D. Forsyth Professor of
Chemical Engineering at the University of Vir-
ginia. He received his BA from Pomona College,
his BS and MS from MIT, and his PhD from the
University of California (Berkeley). He taught
thermodynamics and statistical mechanics as
well as materials science for chemical engineers
at the University of Florida from 1966-88. His
research on varieties of fluids involves theory,
molecular simulation, and experiment.
Copyright ChE Division ofASEE 1993

Though thermodynamics focuses on natural
phenomena, it is really just a deductive structure
developed by creative and systematic human
minds. Nature has carried out her diverse
processes for eons without being explicit
about energy, entropy, andfugacity.

abstractness, rigor, and universality as well as in
debating the "best" way to phrase the principles and
their limits of application. But most current engi-
neering work deals with the practical uses of
thermodynamics-predominantly with models of
reality. Modern computers enable testing of
quantitative expressions for phenomena at every
level of complexity. As a result, we find thermody-
namics being used to an unprecedented extent to
mimic and predict Nature's behavior more easily
(and often more reliably) than experiment-espe-
cially for mixtures.
Unfortunately, the word "thermodynamics" pro-
vokes uneasiness or frustration in many well-edu-
cated people, especially in alumni of engineering
thermodynamics courses. This often arises from an
incomplete or insecure classroom experience com-
bined with insufficient background in assimilating
all the basics. Becoming really comfortable with ther-
modynamic concepts and proficient in their use re-
quires a comprehensive appreciation of the subject
in addition to care, maturity, and intelligence. Thus,
major objectives of thermodynamics education should
include overcoming confusion and antipathy while
fully integrating the concepts, knowledge, and pro-
cedures. The process, though quite demanding, must
guide students to appreciate the structure and rel-
evance of thermodynamics and to become effective
in its use. It should also enrich their vision of
Nature's unity and diversity.
What follows is one teacher's view of fundamental
thermodynamic structure and how it leads to ap-
plications that could foster a useful and satisfying
learning experience for chemical engineering stu-
dents. While integration and connection require
Chemical Engineering Education


terms of study and often years of practice, the ideas
presented here may be valuable for others in en-
hancing learning.*

Because thermodynamics is a logical construct,
there are many ways that the subject can be devel-

* Jim Haile (Clemson) contributed much to these beginning

A Pedagogical Structure of Thermodynam
Category Elements
Observations PrimitjveConcepts ral Phenomer
S ns Mathema
Generalizations Measu les & Variables
& Constraints as kles & Vanables
onstraMaterial Conservation
( Initially For Energy Conservatio~ Nonequivlnce
\Fixed Composition I Degradation_ st& 2Tfaw Equatio
# ii eeident arables of Sti
Articulation & St te Prdperties & Boundaries' Conce pa
Quantification / fConnections Among Partial Deri
Pro rties........ ...Principles .of
Multicomponent Fluids :: Compoiion Variables Partial M

Equilibrium & Change Extrema of Conceptuals Pro
Phase Stability Diffe

Phases and Reactions Phas ilibum Phase Reaction
SRelations ia ams Relatio
ods Organization of Information
Models Data Tabulations Simulations
............................................................ ... ............ ...... ....... ........ .. ".
'-Mixing & Separation Chemic
Applications -, Transport Tendencie Heat &
"* Volumes and Heats
Direct Connection ------- Indirect Connec

Figure 1. A structure of chemical engineering thermod

Things Properties & State Changes Thermal and Material
System Force Input mechanical transfer of
Surroundings Length Output effects on substances
Boundaries Pressure Accumulation fluids via various
Volume Generation and solids mechanisms
Table 1. Observations

Definitions Mathemat

Identity Chemical species Algebraic equations a
System-surrounding interactions Functions of several
Quality Extensive and intensive Independent and depe
Reversible and irreversible Calculus derivatives
Quantities Mass Differential equations
Temperature Exact and inexact
Work Balance equations
Kinetic energy Path dependence of ii
Potential energy
Table 2a. Generalizations I
Spring 1993

oped. That is why there are so many textbooks. One
extreme approach is to begin with axiomatics and
mathematics, which then lead through formalisms
to applications (the most elegant and difficult trea-
tise on this is by Truesdellt11). The other extreme is
to note common observations, followed by generali-
zations and applications (Fenn has cleverly shown
this technique with a cartoon character called
"Charlie the Caveman"[21).
Present-day students tend to be
ics inexperienced in "the way things
work," so I find that discussing ob-
servations first can be motiva-
.--. ................. tional, informative, and organiza-
tional. The flow diagram I gener-
f Work & Heat ally follow is shown in Figure 1.
nns & Inequality
te ............. The way basics are initiated is
a Properties j through primitive concepts and
ves as Properties natural phenomena, as listed in
....... Table 1. The key to the concepts is
lar Properties
a high level of precision that will
ead easily to later mathematical
erty Differences .
rental Finite descriptions of the phenomena.
................These initial elements should
ns / be phrased and illustrated in what-
.... ever terms that will make the
Equation Solving
properties Flow group of learners relate to them.
alReactions Years ago, references to cars and
Work Machine farming worked. Nowadays it
needs to be connected to tele-
tion vision, sports, music, environ-
ment, and the materials and goods
dynamics. of affluence.

IOMENA The next step is generalizations
s and constraints, beginning with
Reactive definitions related to physics and
conversion chemistry as well as to mathemat-
with regular ics. Table 2a shows the kinds of
changes in things I define in familiar terms,
form and amount
in addition to the mathematical
tools that must be used with facil-
ity in the study and implementa-
ics tion of thermodynamics. One point
that I usually make is that we do
nd variables not know what temperature (hot-
1 variables
indent variables ness) really is-we only know it
ad integrals can make a difference to a system,
although not always. This uncer-
differentials tainty of what a property "really

integration is" often makes the later concep-
tual quantities of energy, entropy,
and fugacity less threatening since

something as familiar as temperature
is really unknown yet easily utilized
with experience. Another device I have
used to deal with the unfamiliar prop-
erties is to read the story of 37 sugar
cubes, a small boy, and his mother.[31
(Some students, however, find that
these challenges make the whole pro-
cess even less appealing!)
Table 2b lists a set of "convenient"
observations, definitions and equations
which encompasses the conservation
of mass, the number of variables
needed to completely describe a sys-
tem, the definition and conservation
of energy, and the definition and gen-
eration of entropy. These particular
choices are made because they lead di-
rectly to the most widely used state-
ments of the laws and open system
results, even though at this stage only
systems of fixed composition are dis-
cussed. The level and amount of time
on this part depends on the students'
prior exposure-obviously, less is done
with graduate students.
Because the fundamentals given to
this point are often alien and abstract,
they are articulated further, as shown
in Table 3. The objectives are under-
standings, procedures, and recognition
that users must assimilate. It is es-
sential that learners understand at this
point that people invented state prop-
erties and conceptuals for their gener-
ality and directness, even if they were
not measurable like the things we pre-
fer to (and ultimately must) deal with.
The next step separates chemical en-
gineers from the rest because the de-
velopment is extended to multicompo-
nent systems (see Tables 4a,b). This
leads to a morass of definitions and
complexities as the dimensionality of
the system grows, and I know of no
way to simplify this. By this time I
have begun to insist that students have
a notebook of definitions and I give
the first part of each exam as a closed-
book set of definitions, asking both
word and equation answers for the
quantities previously defined. I also al-

ways use the symbol for the definitions to distinguish them from
mere qualities.
Particularly troublesome is fugacity. Students must recognize it is
not a "corrected partial pressure"; it is a practical substitute for the
chemical potential. Also, fugacity must be connected to temperature,
pressure, and composition. The "Four Famous Fugacity Formulae"
(FFFF) of Mike Abbott'41 (I use Five FFF) assures students that there
are only a few options for every problem of phase and reaction equi-
libria. Also, they should be aware that it is now routine, though
complicated, to connect complex composition behavior of activity co-

Convenient Observations Definitions

Mass conserved
(except in nuclear changes)

# independent variables
is # interactions

Work depends on path
except if no thermal
interactions (Joule)

Work and heat are not
equivalent even in
reversible changes

Work can be degraded
to heat, but not
vice versa


Proper # variables
for describing system

Replace boundary-crossing
quantities with changes
of system state properties

Conceptual properties and
their "natural" variables

Connections among
properties via
partial derivatives

Partial derivatives
as properties
Property evaluation from
integration of
partial derivatives

Atomic and
molecular "weights"

Heat, work, and
material interactions

Closed system
AE WAdiabatic
Q = AE- W
U = E EK -EP

Closed system
dS- 8QRev / T
T is integrating factor
(Born and Caratheodory)

dSGen a dS- 0


Mass balance
MIn MOut = MAccumulated

(#Ind) = (#Work) + (# Species) + 1
P = P(T,V,x)
V' = V(T,P,N') = N'V(T,P,x)

Energy balance
EOut(Work,Heat,Material) =
Closed system
dE= TdS + WRev

Entropy balance
SOut (Heat,Material)=

Example Mechanisms and Equations

Determine total # variables; Count # equations
Difference is # independent variables =
# dependent (solved-for) variables is # equations
dUSys = 8Surroundings + WSurroundings
=dQRev +dWRev
= TSysdSSys PSysdVSys

dH=TdS+VdP dG= -SdT+VdP

Maxwell relations
(S / ap), = -(av / aT)p

Gibbs-Helmholtz relations
(G /T)/a(l/T)|p= H

T =_(aU / aS)V P = -(au / aV)S
Cp(T,P)=( QRev / dT)p = T(aS / aT)p = (aH / aT)p

AH = f(aH/aT)pdT + f(aH/aP)TdP

Path independent Path dependent

Path dependent

Table 3. Articulation
Chemical Engineering Education

Table 2b. Generalizations II


Composition Variations
Amounts or fractions of
species affect system

Nt =[N]t
xi [x]i

n variables
n -1 variables

Independent variables
Extensive = n + 2
Intensive = n + I
Mass fractions, volume fractions,
molarity, molality, etc.
Chemical potential

gi(S,V,X) =aut /aNt
St ,V t,Nj

Fluid Mixtures
Compressibility factor
Residual properties
Fugacity and fugacity coefficient
dGi = RTd In fi (Fixed T)
Condensed Phase Solutions
Excess properties

Reference fugacities and
activity coefficients
Lewis-Randall (LR) reference

Henry's Law (HL) reference
Famous Fugacity Formulae (FFF)
Have P in various terms
Use Poynting correction

Connections of residual to
excess properties

Partial Molar Properties

Fi F / N P
F Ft/ T,P,Nt

Fi= YxiFi
Gibbs-Duhem equation shows how n+2 intensive
variables (Fi,T, P) are related by one equation
(aF /aT)p,x dT + (aF / aP)T, dP = xidFi
to give n+l independent variables

Partial Molar Gibbs Energy

GiaGtc'/aNI =gi(T,P,x)

Nonidealities and Idealities
PV = zNtRT lim z=

fi xiiP

lim G i = 1

Real z, i from EOS

F(T,P,x) xiFio(T,P)+FE (T,P,x)

Ideal Solution FE = 0
fi (T,P,x) = xiifo Ideal Solution Yi =1

fio pure

lim yi-l

fi=Hi lim i=l1
#2 fi(T,P,x)=xiYi(T,P,x)fi(T,P)
#3fi(T,P,x)=xiYi(T,P,x)fio(T)exp[J( Vo/ RT)dP]

#4i(T,P,x)= i(Txxii(Tx)fi(T)exp[(Vi /RT)dP]

#5 fi(T,P,x)=xiYi(T,P,x)f(T)
FFF#2 f(T,P)= ?(T,P)P
Yi (T,P,x) = i (T,P,x) / (T, P)

Table 4b. Multicomponent systems II

efficient to parameter mixing rules
for PVTx equations of state (e.g.,
Heidemann and Kokals51).
By this point in the course, there
has been a tremendous amount of
abstractness and it is time to get to
applications which can be addressed
via change and equilibrium, as
shown in Tables 5a and 5b. This is
where the generalizations can be
made real with characterizations of
driving forces and entropy genera-
tion in interesting systems under-
going change as well as in equilib-
rium (and metastable!) cases. A key
is to make sure students overcome
the myth about entropy always be-
ing maximized.
The next section (Table 6) shows
the sets of relations that apply to
phase and reaction equilibria as well
as some of the physics of what Na-
ture can do when it settles down. I
use many plots of different variables
for different kinds of substances and
mixtures. The message is that "Na-
ture does all things easily; ain't Na-
ture grand!" I also insist that stu-
dents recognize that "for every equa-
tion there is a graph, and for every
graph there is an equation." If they
are not sure of what an equation re-
ally means, they should draw its
graph and vice versa.
At this point, the fundamentals are
done. It's time to use thermodynam-
ics. Table 7 shows the elemental
methodology. The ultimate goal is a
quantitative result that is reliable

Driving forces for change
Heat flows from high to low temperature
Work flows from high to low "force"
(e.g., PV work from high to low P)
Material diffuses from high to low chemical potential fugacityy)
Reversible changes Equilibrium changes (S,. =_ 0)
Only differential differences in T, P, Gi
Spontaneous changes Real changes (Se. > 0)
Finite differences in T, P, Gi
SG. increases with property difference
Greatest for heat flow with T difference
Table 5a. Change
Spring 1993

Criteria for Stability-What is Observed
Systems Only in isolated systems is entropy maximized
All others-minimization of an energy function
e.g., Minimum G if T,P fixed while varying x, phase, reaction
(Entropy not minimized for ambient 2-liquid oil/water!)
Phases Differential criteria
Pure fluids-Proposed phase unstable if (aP / aV)T > 0
Mixtures-Proposed phase unstable if stability matrix of
aGi /IN T,,N t not positive definite
J *
SMetastability satisfies differential but not global criteria
Can occur in solids, microstructured fluids (polymer, bio,
surfactant, colloid)
Table 5b. Equilibrium

Table 4a. Multicomponent systems I



and appropriately accurate for the case at
hand. This requires organization of
thermodynamic knowledge and introduc-
tion of models at the proper stage. Since
with models "where there's a way, there's
a will," the latter requires decisions.
While students feel unprepared and/or
unmotivated to choose among the myriad
of options, "life's like that," so the practice
is good. Modeling consists of a math-
ematical relation used to connect certain
properties to measurables that will allow
calculation of values for unknown
measureables. Generalized thermo-
dynamic models contain parameters that
depend on the substances) of interest. The
kinds of choices are suggested in Table 7.
Teachers must make sure that students
are given the tools to make these deci-
sions, including the common choices and
the usual rules of thumb.
The final requirement is to solve a set of
nonlinear algebraic equations. With cur-
rent computers and model software, stu-
dents can now solve numerically more "re-
alistic" problems, though the "black box
syndrome" can arise if the only student
input consists of numbers and "run."

This process can be formalized in differ-
ent ways; I prefer the concept of problem
solving.[6' Table 8 shows a method devel-
oped with Jim Haile (Clemson University)
called "PSALMS". It is a step-by-step tech-
nique that works well for typical chemical
engineering thermodynamic problems. The
two initial steps of "Problem" and "Sys-
tem" are the same as in essentially all
"PS" methodologies. The next uses the
power of thermodynamics that declares cer-
tain relations to be "ALways true" so us-
ers can initiate something valid and rel-
evant rather than stare at a blank piece of
paper or make an assumption too early.
The fourth step of "Model" is an essential
part of contemporary thermodynamics (the
table suggests some of the choices to be
made). The final step is "Solve and check,"
which is again a part of all PS methods.
By now this article has either lulled you
to sleep because all of this seems so
straightforward, or made you extremely

Phase Equilibrium Relations
Chemical Potentials
gi(T,P,x)=ii (T,P,x")=...

fi (T, P,x')= fi(T,P,x")...

Reaction Equilibrium Relations
Minimum Gibbs Energy
SvikGi =0 (Independent Reactions k)

Equilibrium Constant and Standard State (o)
Kk(T)= exp -vikG? / RT

= exp -vikAGO /RT

= [xiii(T,P,x)P] k (FFF#1)
Ref.()is pure IG at P=1

= I[fi(T,P,x)/fo(T) ik (FFF#3-5)
Ref.(o) is pure real (LR) or
hypothetical (HL) substance
Phase Diagrams
Pure P T
2-phase boundaries (S/V, L/V, S/L, S/S)
Fixed points (V/L, L/L critical; triple points)
Mixed multiphase systems
PTxy (multidimensional and projections)
Azeotropes, critical, 3- and 4-phases
Enthalpies ("Heats") of Vaporizing, Melting, Subliming
Related to P T 2-phase lines by Clapeyron equation
d i(PS) (H"-H')
d(/T) R -
d T) 2-phase -R z '
Clausius approximation for gases (") and liquids/solids (')
z =l>>z

Table 6. Phases and reactions

Organization of Information

Models and Data
Mathematical relations for
PROPERTIES (z, C GE, ) in terms of
MEASURABLES (T, P, x, V) for manipulation to get
ALL properties (e.g., conceptuals) (H, S, fi, etc.) needed for
SOLVING EQUATIONS for dependents (x, P, T, V, W, Q, etc) containing
PARAMETERS depending on molecular constitution
COMMON EXPRESSIONS requiring decisions
PVTx equations of state
GE = activity coefficients
Reference fugacities, f9
Mixing rules for pseudopure parameters of mixtures
Combining rules for unlike parameters from likes (often via kij)

Solve Nonlinear Algebraic Equations for
Intensive variables (T, P, x, etc.) in phases
Material flows (flash, etc.)
Phase existence and amounts

Table 7. Methods
Chemical Engineering Education



agitated because of how unrealistic it might seem to be. Cer-
tainly, it's not easy-but much of it is possible to achieve.

Problem Desired quantity, other variables
System Physical and chemical situation
Contents and constraints
Boundaries, work modes, species
Total variables, specified variables
ALways True Relevant generalized relations for specified system
Balances on mass, energy, entropy
Property differences
Fugacity, reaction equilibrium constant equations
Model Approximations to generalized relations
Choice of Famous Fugacity Formula
If FFF #1, EOS Type
Ideal, Virial, CSP, "Full"; P or V Independent
If FFF #2-5, Reference State, Pressure Effect
Specific choices, e.g.,
CSP parameterization; Cubic EOS;
fo values; GE correlation, group contribution method
Data (new or literature)
Solve and check Analytic, graphical, tabular, numerical
Table 8. A thermodynamics problem-solving strategy.

The Dilemmas of Beginners
There's a lot of material!
I've never done deduction before!
It's mostly abstract.
It's incredibly detailed!
It's a long way before real applications appear!
If I get started wrong, it takes a lot of work to get to the answer!
Table 9. How students respond to thermodynamics.

Suggestions to Keep Students Focussed
Stress procedure more than results, particularly in the beginning.
"Any fool, even a computer, can do a calculation."
Articulate that the goal is to quantitatively describe the richness of nature.
"Nature does all things and without any difficulty."
Emphasize exponential pattern of learning the subject.
"Hang in there. When things begin to click, you take off The question is
whether the end of the term happens first."
Connect equations to observable phenomena and pictures (graphs).
Student inexperience with natural behavior is pervasive and growing.
Insist upon precision of expression and thought, especially definitions.
"You gotta know what you're talking about!"
Minimize "understanding" and maximize "doing."
"I don't know what entropy andfugacity are, but I can tell you when to use them
and how they go."
Give practice problems involving only the setup steps (PSAL) of problem solving.
Assist students to develop their own PS style.
Have students read and report on the literature of physical properties.
"Hey guys! They actually use fugacity out there!"
Have students code a VLE program for real substances:
Forces decision-making and precision.
"Computers are unforgiving; they do all and only what they're told."
Undergraduates hard to teach fundamentals to; are not used to deduction, precision.
They want to "do" something immediately.
Hard to get graduates to unlearn old errors; they are reluctant to change old ways.
"They got me a B+!"

Table 10. Suggestions to keep students focused.
Spring 1993

Table 9 shows how I think the beginner in
this subject reacts. Of course, self-recogni-
tion is vital to making progress, so I confront
students with this soon after I begin to get
the "glazed-eyes syndrome" somewhere in the
multicomponent section. I also tell them that
I only became fully comfortable with thermo-
dynamics after the third time I taught it.
But that does not mean they can cop out-it
merely declares that progress is tough.
Finally, I have given in Table 10 a list of
some suggestions that seem to help keep us
on track. They are phrased as admonitions,
followed by salient quotes from instructors
and students.
My success as a teacher has fluctuated with
the class, my own distractions, the ions in
the air, and who knows what else. But it has
been tremendously satisfying when things
have clicked with students (the number in
class ranging from one to nearly all). Inter-
estingly, I think there is less a correlation
with intelligence than with commitment, at-
tentiveness, and willingness to move ahead
without being completely satisfied.

In addition to those already mentioned,
many other stimulating individuals have in-
fluenced my pedagogy in thermodynamics.
They principally include Martin Fuller
(Pomona College), Bob Reid (MIT), John
Prausnitz (Cal-Berkeley), Tim Reed (Florida),
Grant Wilson (Wiltec), Aage Fredenslund and
Peter Rasmussen (DTH, Lyngby, Denmark),
Anneke Sengers (NIST), Ed Glandt (Penn),
and Herb Cabezas (Arizona). Also, Warren
K. Lewis (MIT), John Biery (Florida), and
Verna O'Connell (wherever I am) have been
tremendous inspirations.

1. Truesdell, C., Rational Thermodynamics: A Course
of Lectures on Selected Topics, McGraw-Hill, New
York (1969)
2. Fenn, J.B., Engines, Energy and Entropy, W. H.
Freeman, New York (1982)
3. Van Ness, H.C., Understanding Thermodynamics,
Chap. 1, McGraw-Hill, New York (1969)
4. Abbott, M.M., personal communication, Rensselaer
Polytechnic Institute, Troy, New York (1978)
5. Heidemann, R.A., and S.L. Kokal, "Combined Ex-
cess Free Energy Models and Equations of State,"
Fluid Phase Equil., 56, 17 (1990)
6. Woods, D.R., ed., P.S. News, McMaster Univer-
sity, Hamilton, Ontario, Canada I




University of Wisconsin-Madison
Madison, WI53706-1691
he transport phenomena can be described at
three scales: the molecular, the microscopic
(continuum), and the macroscopic. At each
scale the conservation laws for mass, momentum,
angular momentum, and energy play a key role.
Also, at each scale empiricisms have to be intro-
duced to complete the description of the systems: an
intermolecular potential expression at the molecu-
lar scale, the flux expressions (constitutive equa-
tions) at the microscopic scale, and the transfer co-
efficient correlations at the macroscopic scale. The
three scales are intimately connected, with the re-
sults for each scale contributing to the understand-

The Equations of Change Based on Conservation Laws
(A) -pa = -(V pav)-(V ja)+ra a = 1,2,3...

(B) pv = -[V pvv]- [V ]+ paga
(C) p p[rx v] + )= -V pv([rx v]+ [V ]

V. rx w'} + [rx pga] +XPata
T a a
(D) tp (1v2 + =-(V.v[v2 + ) (V q)
(V [. v])- ((pav +ja) ga)
p. = mass concentration of species a
p = density of fluid mixture
v = mass-average velocity
j = mass flux of a with respect to v
r = mass rate of production of a by chemical reaction
zT = (total) stress tensor
g. = external force per unit mass acting on a
r = position vector
L = internal angular momentum per unit mass
X = couple stress tensor
t = external torque per unit mass acting on a
U = internal energy per unit mass
q = heat flux vector

ing of the next larger scale.
At the microscopic scale, some information about
the constitutive equations can be obtained from the
thermodynamics of irreversible processes. This ap-
proach is particularly important in understanding
multicomponent diffusion and the "cross-effects" in
energy and mass transport.
For the most part, the notation and sign conven-
tions here will be those used in references 1, 2, 3, 4,
and 5, hereinafter referred to as TrPh, DPL1, DPL2,
STTP, and MTGL, respectively.

The Equations of Change
The basic equations of transport phenomena are
the equations of change for the conserved quantities
as shown in Table 1:
(A) Conservation of mass for each species
TrPh Eq. 18.3-4; MTGL 11.1-1
(B) Conservation of momentum
TrPh Eq. 18.3-2; MTGL 11.1-3; DPL1, 1.1-8
(C) Conservation of angular momentum
MTGL, p. 831, Problem 6
(D) Conservation of energy
TrPh Eq. 18.3-6; MTGL 11.1-4;DPL1, 1.1-12
These equations can be obtained by writing conser-
vation statements over
(a) a region fixed in space through which the fluid is
moving (DPL1, Chapter 1)
(b) a material element of fluid (i.e., a "dyed" blob of
fluid) moving through space.[61

"Bob" Bird retired in 1992 after forty years of
teaching-one year at Cornell and thirty-nine
years at Wisconsin. The book Transport Phe-
nomena, which he wrote with colleagues Warren
Stewart and Ed Lightfoot, was the first textbook
on the subject specifically prepared for under-
graduate chemical engineering students. He also
coauthored Dynamics of Polymeric Liquids, with
Bob Armstrong (MIT), Ole Hassager (DTH), and
Chuck Curtiss (UW).

Chemical Engineering Education


The equations of change have been written in Table
1 in terms of the total stress tensor t which is
conventionally split into two parts: i = p5 + T
(where 8 is the unit tensor); p is the "thermody-
namic pressure"; and T is the "(extra) stress tensor"
which vanishes in the absence of velocity gradients.
No assumption has been made here that the stress
tensor be symmetric. Equations (A) through (D)
have to be supplemented with the thermal equa-
tion of state p = p(p,T,o,) and the caloric equation
of state U = U(p,T,o,)), where o. stands for the
mass fractions of all but one of the chemical
species a in the mixture.
By adding the equations in Eq. (A) over all spe-
cies, one gets the equation of continuity for the fluid
mixture (TrPh, Eq. 3.1-4]. By forming the cross prod-
uct of the position vector r with the equation of
motion, Eq. (B), one obtains Eq. (E); when the latter
is subtracted from Eq. (C) (the equation of conserva-
tion of total angular momentum), the equation for
internal angular momentum, Eq. (F), is obtained. 71
Similarly, by forming the dot product of the fluid
velocity v with the equation of motion, Eq. (B), one
obtains Eq. (G), the equation of change for the ki-
netic energy; when the latter is subtracted from Eq.
(D), the equation of change for the internal energy,
Eq. (H), is obtained. These various derived equa-
tions are tabulated in Table 2.
As pointed out in TrPh (page 314), the term
(nT:Vv) appears in Eq. (G) with a plus sign and in
Eq. (H) with a minus sign; it describes the
interconversion of mechanical and thermal energy.
Similarly, the term [e:'T] appears with a plus sign
in Eq. (E) and with a minus sign in Eq. (F), thus
Equations of Change for Nonconserved Quantities

(E) -p[rxv]=-[V.pv[rxv]]- V. rxnrTT]

+ [rxpag ]+ [e:rrT]
(F) pL= -VpvLp-[Vk]]- [ pt- _e:,]

(G) 2 v(2)) -)

(H) apt=-(v pvt)-(V-q) ( T:vv)+^(v.pag a)

NOTE: On page 831 of MTGL, Eq. (E) is given for a
symmetric stress tensor, but [r x p] should be re-
placed by [r x p]T.

Spring 1993

describing the interconversion of external and inter-
nal angular momentum. In fluid dynamics textbooks,
it is usually assumed that the stress tensor is sym-
metric (i = eT), so that the external and internal
angular momentum are conserved separately, since
[E:'T] is then identically equal to zero. The so-called
"proofs" that the stress tensor is symmetric (such as
in Problem 3.L in TrPh, pages 114-115) tacitly as-
sume that there is no interconversion of external
and internal angular momentum, and that the ex-
ternal angular momentum is conserved in the fluid.
The kinetic theory of dilute monatomic gases yields
a symmetric stress tensor, as does the kinetic theory
for dilute solutions of flexible and rodlike polymers.
So far there is no experimental evidence that a
nonsymmetric stress tensor is needed.
When all species are subjected to the same exter-
nal forces (so that all gn equal g), and when it is
assumed that the stress tensor is symmetric, Eqs.
(B) and (D) simplify to Eqs. 3.2-8 and 10.1-9 in
TrPh. If all species are subjected to the same exter-
nal torques so that all t, are equal to t, a similar
simplification occurs in Eq. (C).
We emphasize that the equations in Table 1 are to
be considered the fundamental equations at the mi-
croscopic scale, whereas those in Table 2 are de-
rived from those in Table 1. As explained in TrPh,
the equations of change may be put into many alter-
native forms; for example, they may equally well be
written in terms of the "substantial" (or "material")
derivative operator D/Dt (TrPh Eq. 3.0-2). The en-
ergy equation has always been a special problem
because it can be written in so many different ways
(see TrPh, pages 322-323, 582, for useful tabular
The Flux Expressions
(Also Called "Constitutive Equations")
In order to get solutions to the equations of change,
we need to have expressions for the fluxes j, T, and
q. The standard expressions for these are the "lin-
ear laws," in which the fluxes are proportional to
gradients, as shown in Table 3.
Equation (I) for the mass-flux vector is shown only
for the binary system A-B, and the thermal diffu-
sion, pressure diffusion, and forced diffusion terms
have been omitted (see Eq. (Z) for the complete ex-
pression). Equation (K) shows the conductive and
diffusive contributions to the heat-flux vector, but
the diffusion-thermo (Dufour) effect has been omit-
ted. In Eq. (J) we have included the two viscosity
coefficients I and K, although the latter is omitted
in most fluid dynamics texts since it is zero for mona-



tomic gases (we know this from kinetic theory-see
MTGL, Chapter 7) and since for liquids incompress-
ibility is often assumed, so that div v = 0 and the
term containing K is zero anyway.
The flux expressions do not have the exalted sta-
tus accorded to the conservation laws in Table 1.
They are empirical statements, proposed as the sim-
plest possible linear forms; they also emerge from
the kinetic theory of gases when one works to the
lowest orders in the gradients of concentration, ve-
locity, and temperature (see MTGL, Chapter 7). It is
well known that Eq. (J) does not describe the me-
chanical responses of polymeric liquids (see DPL1,
Chapter 2); various nonlinear expressions, and in-
deed time-dependent expressions, arise from the ki-
netic theory of polymeric liquids (see DPL2, Chap-
ters 13-16, 19-20). Furthermore, for some complex
materials it is found that the thermal conductivity
and diffusivity are tensors rather than scalars, so
that the fluxes and forces are not collinear. In order
to use Eqs. (I,J,K), one needs numerical values for
the diffusivity, viscosity, and thermal conductivity;
these are preferably obtained from experiments, but
in the absence of experimental values kinetic theory
results can be used.
Once the flux expressions have been substituted
into the equations of change, we then have a set of
equations which, when solved, will give the concen-
tration, velocity, and temperature distributions as
functions of time. There are many ways in which
these important equations can be used:
Analytical solutions can be found (for simple, ideal-
ized problems, in which transport properties are
assumed to be constant)[''10
Approximate solutions can be found with perturba-
tion theories1'l
Numerical solutions can be foundl12"15
Boundary-layer solutions can be found116
Time smoothing can yield the turbulent transport
Volume smoothing leads to the equations for po-
rous media'20'
Flows with chemical reactions can be analyzedt211
Approximate solutions can be used for lubrication
Mixing and chaos can be studied1221
Particulate motion, suspensions, and emulsions can
be described121
Interfacial transport equations can be established123'
Polymer fluid dynamics and transport phenomena
can be studied[DPL"1
This partial list of topics gives some idea as to the

breadth of the field of transport phenomena and the
extremely great importance of the equations of
change. All these topics are active research fields
in which chemical engineers are obligated to play
an important role.

The Macroscopic Balances
The statements of the laws of conservation of mass,
momentum, angular momentum, and energy can be
written down for a typical macroscopic engineering
system, with one entry port ("1") and one exit port
("2"); heat can be added to the system at the rate Q,
and the system can do work on the surroundings at
a rate Wm by means of moving parts (such as pistons
or rotatory devices). These conservation statements
are given in Eqs. (L-O) in Table 4. It is assumed
there that the fluid velocities at the inlet and outlet
planes are parallel to the directions of flow n, and
n2. It is also assumed that the extra stress tensor
does not contribute to the work done on the system
at the entry and exit planes. It is further assumed
that there are no mass-transfer surfaces in the mac-
roscopic system; such surfaces are considered in
TrPh, Chapter 22, and in STTP, Chapter 1.
Of course, Eqs. (L-O) can also be obtained by inte-
grating the equations of change in Eqs. (A,B,E,D)
over the entire volume of the flow system; in doing
this we must take into account the fact that the
shape of the volume is changing with time because
of the moving parts.[24,25] The macroscopic mechani-
cal energy balance (also called the engineering Ber-
noulli equation) cannot be written down directly since
there is no conservation law for mechanical energy.
It can be derived by integrating Eq. (G) over the
macroscopic system as outlined in Table 5. For the
sake of simplicity, we take the external forces go to
be all the same (g = -6, where D is the potential

The Flux Equations (or "Constitutive Equations")

(I) JA =-PDABVoA (binary system of A and B)
(J) T = -VV+(VV)T) + V)a
(K) q=-kVT +(H, /Ma)ja

DA = binary diffusivity
oA = pA/p = mass fraction of a
p = viscosity
K = dilatational viscosity
8 = unit tensor (with components 5.)
k = thermal conductivity
Ha = partial molar enthalpy of a
Chemical Engineering Education



Macroscopic Balances
no mass-transfer surfaces;
all species subject to same external forces and external torques;
7r contributions neglected at "1" and "2"

(L) mtot =wal-Wa2 +ra,tot

(M) Ptot 2 wi+PiSij wni ( 2 2+P2S2 -2 F+mtotg
(M) -tPtot= I f2

(N) -Mtot= )1 w+plS1 [rlxni]- w2+P2S2 [r2xn2]-T+[rcxmtotg]
dt (vl M2

()3 1 ____3) P
(0) (Ktot+ tot+Uto 2t) (V +il+1 W- 2 +2 W2 2+Q-Wm

(Q v3) 1 1 3 2 M2
(P) d(Ktot + _tot)= +1+ wl 21 v 2 + 2 -Ec-Ev -Wm
()2 (v)d 1w 2 M P2

(Q) dUtot=Uilwl-U2W2+Q+Ec+Ev

mt = total mass in flow system
Pt = total momentum in flow system
Mt = total angular momentum in flow
Ktot= total kinetic energy in flow system
tot = total potential energy in flow system
Utt = total internal energy in flow system
() = averages over tube cross section at
entry and exit
n,n2 = unit vector in flow direction at entry
and exit
w l,w = mass rate of flow of a at entry and

w,, w2 = mass rate of flow at entry
and exit
pl, p2 = thermodynamic pressure at
entry and exit
S,, S2 = cross-sectional areas of entry
and exit conduits
Q = heat added to system
through container walls
Wm = work done on surroundings
E,, Ev = quantities defined in Eq. (S)
rl,r2,r = location of the centers of the
entry and exit planes and
the center of mass of the
fluid in the system

energy per unit mass,
which is considered to be
independent of time). In
doing the integration, we
need to use the Gauss di-
vergence theorem (TrPh,
A.5-1) and the 3-dimen-
sional Leibniz formula
(TrPh, A.5-5).
This leads to Eq. (R), in
which S, and S2 are the
cross-sectional areas at
"1" and "2", Sf stands for
the fixed surfaces of the
system, and Sm stands for
the moving surfaces, by
means of which work can
be done on the surround-
ings. Since the velocity v
of the fluid equals the sur-
face velocity vs on the
fixed and moving sur-
faces, these surface inte-
grals contribute nothing
to the first term on the
right side. Also, since the
fluid velocity v is zero on
all fixed surfaces, the
fixed-surface contribution
to the second term on the
right side is zero; the in-
tegral over the moving
surfaces gives the work
transmitted via these sur-
faces, Wm, (sometimes
called the "shaft work").
The integrals of the extra

Intermediate Steps in Deriving the Macroscopic Mechanical Energy Balance

(R) A J (lpv2+p )dV=_-J(n.(pv2+p()(v vs))dS-J(n.[(p6 +T)v])dS-Ec-Ev
V(t) S=S,+S,+s+S,(t) S=S,+S,+S,+S,(t)

(S) inwhich Ec=- fp(V.v)dV and E =- J(Tr:Vv)dV
V(t) V(t)
(T) -t (Ktot+(Dtot)= P 3l(v13S1-lp2 v3 2S2+Pl(vl(1SI -p2(v)2 2S2+Pl(V)lS1-p2(v)2S2-Ec-Ev-Wm

V(t) = volume of engineering flow system
S, = fixed surfaces of flow system
Sm = moving surfaces of flow system
v. = velocity of surface (equals zero on S,,S,,S,)

6 = potential energy per unit mass
n = outwardly directed unit normal on surface S
7 = (extra) stress tensor

Spring 1993 10O


stress tensor T over S, and S2 are presumed
small and have been omitted here; they are
identically zero for laminar, Newtonian flow
when the fluid velocity vectors are parallel to
the walls of the entry and exit tubes. The inte-
grals labeled E, and E, are not evaluated; the
latter gives the rate at which mechanical en-
ergy is degraded into thermal energy. From
Eqs. (R) and (S) we get Eq. (T), which is easily
rearranged to give the mechanical energy bal-
ance in Eq. (P); the latter includes the two
special cases given in TrPh Eqs. 15.2-1 and 2.
Equation (P) is particularly convenient for in-
compressible fluids for which E, is exactly zero.
Equations (L)-(P) are easily generalized to sys-
tems with multiple inlet and outlet ports.
NOTE: In some textbooks it is stated that
the mechanical energy balance (Eq. P) is an
"alternative form" of the total energy balance
(Eq. O). Such a comment seems inappropriate
since Eq. (P) comes from the equation of con-
servation of momentum, whereas Eq. (0) has
its origins in the equation of conservation of
energy. In other textbooks some thermody-
namic "incantations" are offered to get from
Eq. (0) to Eq. (P). The arguments must essen-
tially involve Eq. (Q), obtained by subtracting
Eq. (P) from Eq. (0); of course, Eq. (Q) can also
be obtained from integrating Eq. (H)-which
is a consequence of the equations of energy
and motion-over the macroscopic flow sys-
tem. Certainly Eq. (Q) cannot be written down
directly, since there is no conservation law for
the internal energy in an open system with
dissipative processes. Furthermore, the ther-
modynamic arguments cannot yield the expres-
sions in Eq. (S), showing how E, and E, are
related to the velocities and stresses in the
system. Comments from textbook authors (and
others) on this point would be welcome; before
commenting, however, it would be advisable
to read Whitaker's historical essay.[6, pp 90-931
The Transfer Coefficients
Although the macroscopic balances can be
used, as shown in Table 4, it is often useful to
estimate some of the terms in them by using
dimensionless correlations:
F can be estimated by using friction factor
E, can be estimated by using friction loss
factor correlations
Q can be estimated by using heat-transfer

coefficient correlations
wa m can be estimated by using mass-transfer coefficient
These quantities are given in the form of dimensionless
correlations based on large amounts of experimental data;
they contain the transport properties and the density and
the heat capacity, as well as quantities describing the char-
acteristic length, velocity, temperature, etc.
For steady-state systems, the macroscopic balances form a
set of algebraic relations; for unsteady-systems they become
a set of differential equations, with time as the independent
variable. The macroscopic balances are the starting point for
calculations involving heat exchangers, separations equip-
ment, chemical reactors, and fluids-handling systems.

The Three Levels of Transport Phenomena
For many engineering applications, one starts with the
macroscopic balances in order to understand the overall
behavior of the system. One can often estimate some of
the quantities in the balances by using transfer coefficient
correlations, photographic or other visualization methods,
direct pressure, temperature, and density measurements
on the system, etc. Other quantities may be assigned by
crude methods.
In other problems, one needs to know more about the de-
tails of the pressure, velocity, temperature, and concentra-
tion distributions within the system. This calls for "moving
down" one level (see Table 6) and solving the equations of
change. Many analytical solutions are available, but there
are also modern computing techniques if numerical solu-
tions are needed (usually the case if pressure, temperature,

The Three "Levels" of Transport Phenomena
Basic Empirical
Equations Expressions Results

Macroscopic Dimensionless Solve to get
Balances over correlations relations among
b engineering f, h, k,, inlet, outlet, and
system and e, transfer quantities
S J T dimensional analysis
S Equations of Flux Solve to get
8' change for + expressions concentration,
o 0 conserved for pressure, velocity,
S quantities j,, T, q and temperature
Equation for Intermolecular Solve to get
dg time evolution + force D^, Ki, and
S of phase-space expression kT in the flux
distribution function expressions

Chemical Engineering Education


and concentration dependence of the physical prop-
erties have to be taken into account). Dimensional
analysis of the equations of change suggests the
form that the transport-coefficient correlations
should take, these being needed for the macroscopic
It should be recognized that there is still one scale
smaller than the macroscopic and microscopic scales,
namely the molecular scale. Although this part of
the subject normally lies in the domain of the theo-
retical physicist or the theoretical chemist, engineers
occasionally need some familiarity with the molecu-
lar aspects of transport phenomena. The basic equa-
tion at the molecular level is an equation for the
time evolution of a phase-space distribution func-
tion. One example of this is the Boltzmann equation
for dilute gases (MTGL, Chapter 7), and additional
examples may be found for dilute polymer solutions
and polymer melts in DPL2, Chapters 17-19.
From the differential equation for the phase-space
distribution function, one can obtain a "general equa-
tion of change," special cases of which are the usual
equations of change in Table 1; in developing these
equations, one makes use of the fundamental con-

Spring 1993

servation laws as applied to molecular collisions. As
a by-product of this derivation, formal expressions
are obtained for the fluxes in terms of the distribu-
tion function. In this way expressions are obtained
for the transport properties in terms of molecular
Recently there has been an interesting develop-
ment in connection with the kinetic theory of dilute
gases and the Boltzmann equation. This famous
equation, although over one hundred years old, has
been found to be in error in that it cannot be ob-
tained by starting with the quantum Boltzmann
equation and letting Planck's constant vanish.26" The
new "Boltzmann-Curtiss equation" does not suffer
from this defect since it accounts properly for the
contributions associated with bound pairs of mol-
ecules; the added terms in the equation are appar-
ently important at low temperatures; as a result the
table in TrPh, page 746, will have to be modified.
It is seen in Table 5 that at each of the three
scales, use is made of the basic conservation laws.
Also, at each scale some kind of empiricism is intro-
duced. Each scale can be better understood by going
to the next smaller scale in order better to appreci-
ate the origins of the equations

Thermodynamics of Irrevers-
ible Processes
If into Eq. (H), the equation
of change for the internal en-
ergy, we insert the thermody-
namic relation Eq. (U) (see
Table 7) for a binary mixture,
we get (after using the equa-
tion of continuity) the result
in Eq. (V)-an equation of
change for the entropy. In this
equation we can identify an
entropy flux s as the sum of
two contributions, one associ-
ated with heat conduction and
one with diffusion; we can also
identify a rate of entropy pro-
duction o, which is given as a
sum of terms, each of which is
the product of a flux and a
Then, according to the ther-
modynamics of irreversible
processes, every flux will de-
pend linearly on each of the

Thermodynamics of Irreversible Processes (Binary Systems)

(U) dJU=TdS-pd(1/p)+ AdOA
(V) pS=-(v.pvS)-(V.s)+o
in which s=l(q-IjA)

c =-((q- jA)- 1VT)'-(jA -(V + (gB -gA)))- rAr- (T: Vv)

( ) JA =-11 vA+(B-gA))- a12- V

(X) q-RJA=-a21(Va +(gB-gA))-a22-VT
2 T2
(Y) q= a12 A 22 1 VT=HA HB -kVT

(Z) jA =-PAB(VOA +kTVinT+kpVenp)+kF(gB-gA)

S = entropy per unit mass
A = (GA/MA)-(GB/MB)
s = entropy flux
a = rate of entropy production
a. = phenomenological coefficients
q = Dufour effect contribution to heat flux
kT = thermal diffusion coefficient
k = pressure diffusion coefficient
kl = forced diffusion coefficient

and their limitat s.


forces, with the restriction that fluxes must depend
on forces of the same tensorial order, or with order
differing by 2 (Curie's law). There is also the restric-
tion that the matrix of coefficients in the flux-force
relations be symmetric (Onsager-Casimir reciprocal
relations). This leads us to Eqs. (W) and (X), in
which a12 = a21. Then combination of Eqs. (W) and
(X) gives Eq. (Y) for the heat flux vector. In this
equation, the coefficient of -VT can be identified as
the thermal conductivity for the mixture. The other
term in Eq. (Y) leads to the second term on the right
side of Eq. (K), the diffusion term, plus one addi-
tional very small term associated with the Dufour
The term involving Vj in Eq. (W) can be expanded
by using the chain rule of partial differentiation
V= = (D / 3COA)Vw(A + ( 1/ T)VT + (/ / ap)Vp
The term in VT combines with the other VT term in
Eq. (W), and the final result is Eq. (Z); in this equa-
tion the coefficients kp and kF are completely deter-
mined from the thermodynamic properties of the
mixture, whereas the diffusivity DA and kw are two
phenomenological coefficients that have to be deter-
mined experimentally for each gas pair or estimated
by kinetic theory. As a result of the Onsager rela-
tions, the four phenomenological coefficients in Eqs.
(W) and (X) have been reduced to the three trans-
port properties: diffusivity, thermal diffusion ratio,
and thermal conductivity.
Equation (Z) shows clearly that a mass flux of
species "A" can result from a concentration gradient
(ordinary diffusion), a temperature gradient (ther-
mal diffusion), a pressure gradient (pressure diffu-
sion), and a difference of external forces (forced dif-
fusion) (TrPh, Chapter 18 and MTGL, Chapter 11).
A concise introduction to the thermodynamics of ir-
reversible processes has been given by Landau and
Lifshitz;'27' a more thorough discussion can be found
in the classic text by de Groot and Mazur.1281 The
thermodynamics of irreversible processes has been
found to be particularly useful in the systematiza-
tion of the flux expressions for multicomponent dif-
fusion as well as in linear viscoelasticity.[291 Although
this topic is not essential for undergraduate stu-
dents, perhaps graduate students can benefit from
the extra insight provided by the thermodynamic

It is essential that students of transport phenom-
ena recognize the central position occupied by the
conservation statements. The conservation laws for

mass, momentum, and energy applied to a large
engineering system through which a fluid is flowing
lead to the macroscopic balances; the two additional
balances for angular momentum and mechanical en-
ergy can be obtained from the integration of mo-
ments of the equation of motion. The utility of the
balances is enhanced by the use of empirical corre-
lations for the transfer coefficients. The five macro-
scopic balances are the starting point for many analy-
ses of unit operations and chemical reactors. They
are invaluable for making order-of-magnitude esti-
mates for engineering systems.
The conservation laws, when applied to a small
region of space through which a fluid is flowing,
lead to the equations of continuity, motion, and en-
ergy; the assumption of the symmetry of the stress
tensor is usually made, and this assumption makes
it unnecessary to deal with the interconversion of
external and internal angular momentum. The flux
expressions usually used in the equations of change
are the simplest possible relations that are linear in
the gradients. The vast literature dealing with solu-
tions of the equations of change should be familiar
to engineers, even though these solutions are for
idealized systems; they are, however, very useful for
making order-of-magnitude estimates and for check-
ing the computer programs used for obtaining nu-
merical solutions.
The conservation laws applied at the molecular
scale are used in kinetic theory developments. Ki-
netic theory provides expressions for the transport
properties in terms of intermolecular forces; these
expressions are highly developed for dilute mona-
tomic gases. In the last several decades the kinetic
theory of polymers has developed rapidly, so that
much more is now known about the transport prop-
erties of polymeric liquids.'301
The subject of transport phenomena can be useful
in many fields, including micrometeorology, zool-
ogy, analytical chemistry, nuclear engineering, tri-
bology, metallurgy, biomedical engineering, phar-
macology, and space science. Chemical engineering
departments are in a good position to provide gen-
eral service courses in transport phenomena for other
department on campus.

The author wishes to thank Professors W.E.
Stewart and T. W. Root, and Mr. Peyman Pakdel of
the Department of Chemical Engineering at the Uni-
versity of Wisconsin, and Professor J.D. Schieber at
the University of Houston for valuable suggestions.
Chemical Engineering Education


1. Bird, R.B., W.E. Stewart, and E.N. Lightfoot, Transport
Phenomena, Wiley, New York (1960)
2. Bird, R.B., R.C. Armstrong, and 0. Hassager, Dynamics of
Polymeric Liquids. Vol. 1: Fluid Mechanics, 2nd ed., Wiley,
New York (1987)
3. Bird, R.B., C.F. Curtiss, R.C. Armstrong, and 0. Hassager,
Dynamics of Polymeric Liquids. Vol. 2: Kinetic Theory, 2nd
ed., Wiley, New York (1987)
4. Bird, R.B., W.E. Stewart, and E.N. Lightfoot, Selected Top-
ics in Transport Phenomena, Chem. Eng. Prog. Symp. Se-
ries No. 58, Vol. 61, AIChE (1965)
5. Hirschfelder, J.O., C.F. Curtiss, and R.B. Bird, Molecular
Theory of Gases and Liquids, Corrected Printing with Added
Notes, Wiley, New York (1964)
6. Whitaker, S., in One Hundred Years of Chemical Engineer-
ing, ed. N.A. Peppas, Kluwer Academic Publ., Dordrecht,
pp. 47-109 (1989)
7. Dahler, J.S., and L.E. Scriven, Nature, 192,36-37 (1961)
8. Berker, R., "Integration des equations du movement d'un
fluide visqueux incompressible," in Enyclopedia of Physics,
ed. S. Fliigge, Springer, Berlin, pp. 1-384 (1968)
9. Carslaw, H.S., and J.C. Jaeger, Heat Conduction in Solids,
2nd ed., Oxford University Press (1959)
10. Crank, J., The Mathematics of Diffusion, Oxford University
Press (1956)
11. Leal, L.G., Laminar Flow and Convective Transport Pro-
cesses: Scaling Principles and Asymptotic Analysis,
Butterworth-Heinemann, Boston (1992)
12. Kim. S., and S.J. Karrila, Microhydrodynamics: Principles
and Selected Applications, Butterworth-Heinemann, Bos-
ton (1991)

13. Finlayson, B.A., The Method of Weighted Residuals and
Variational Principles, Academic Press, New York (1972)
14. Finlayson, B.A., Nonlinear Analysis in Chemical Engineer-
ing, McGraw-Hill, New York (1980)
15. Finlayson, B.A., Numerical Methods for Problems with Mov-
ing Fronts, Ravenna Park, Seattle, WA (1992)
16. Schlichting, H., Boundary Layer Theory, 4th ed., McGraw-
Hill, New York (1960)
17. Hinze, J.O., Turbulence, 2nd ed., McGraw-Hill, New York
18. Tennekes, H., and J.L. Lumley, A First Course in Turbu-
lence, MIT Press, Cambridge, MA (1972)
19. Speziale, C.G., in Ann. Rev. Fluid Mech., 23, 107-157 (1991)
20. Adler, P.M., Porous Media, Butterworth-Heinemann, Bos-
ton (1992)
21. Rosner, D.E., Transport Processes in Chemically Reacting
Flow Systems, Butterworth-Heinemann, Boston (1986)
22. Ottino, J.M., The Kinematics of Mixing: Stretching, Chaos,
and Transport, Cambridge University Press (1989)
23. Slattery, J.C., Interfacial Transport Phenomena, Springer,
Berlin (1990)
24. Bird, R.B., Chem. Engr. Sci., 6, 123 (1957)
25. Slattery, J.C., and R.A. Gaggioli, Chem. Engr. Sci., 17,893
26. Curtiss, C.F., J. Chem. Phys., 97, 1416, 1420, 7679 (1992)
27. Landau, L., and E.M. Lifshitz, Fluid Mechanics, Chap. VI,
Addison-Wesley, Reading, PA (1959)
28. de Groot, S.R., and P. Mazur, Non-Equilibrium Thermody-
namics, North-Holland, Amsterdam (1962)
29. Kuiken, G.D.C., Thermodynamica van de Irreversibele
Processen, T.U. Delft, Netherlands (1992)
30. Bird, R.B., and H.C. Ottinger, in Ann. Rev. Phys. Chem.,
43, 371 (1992) 0

r M book review

by Donald L. Katz, Robert L. Lee
McGraw Hill, New York, NY 10020; 760 pages, $54.95
Reviewed by
R. A. Greenkorn
Purdue University
This book covers most aspects of natural gas engi-
neering. It is a survey suitable for a short course to
introduce practicing engineers to the topic. The book
is descriptive and as such is much too broad to be
used as a textbook. The later half of the book is
essentially a monograph recording the senior author's
extensive experience in this area. Chapters 1-7 de-
scribe the material properties of the system, chap-
ters 8-13 contain the core of the material concerned
with the production and storage of natural gas, chap-
ters 14-15 mainly discuss operations, and chapters
16-17 contain miscellaneous topics.
Chapter 1 Natural Gas Technology and Earth
Spring 1993

Sciences. This chapter is a concise review of natural
gas engineering production and underground stor-
age of natural gas. Several subjects are covered,
e.g., the branches of petroleum industry, sources of
information for natural gas engineering, a brief dis-
cussion of geology and earth sciences, and earth
temperatures and pressures.
Chapter 2 Properties of Rocks. This chapter con-
tains some descriptions of the properties of rocks or
porous media, including a description of how these
properties are measured. The discussion is under-
standable and relatively clear-but very terse.
Chapter 3 Thermodynamics: Flow Equation,
Fluid Properties, Combustion. This chapter is basi-
cally descriptive. It is terse, explaining how the equa-
tions are derived and giving some limited informa-
tion on how to calculate combustion of natural gas.
Chapter 4 Physical Behavior of Natural Gas
Systems: Physical and Thermal Properties, Phase
Behavior, Analyses. The initial part of this chapter
is a review of pressure, volume, and temperature
relationships of pure fluids. The phase rule and the
behavior of complex mixtures are briefly discussed.
Continued on page 116.





The University of Michigan
Ann Arbor, MI 48109

Chemical reaction engineering (CRE) is fun to
teach, not only because it has extremely in-
teresting subject matter and is one of the
few courses that sets chemical engineering apart
from other engineering disciplines, but also because
it has a very logical structure. The six basic pillars
that hold up what could be called the "Temple of
Chemical Reaction Engineering" are shown in Fig-
ure 1.11] The four on the left are usually covered in
the majority of undergraduate reaction engineering
courses.[2' But diffusion effects, which include mass
transfer limited reactions, effectiveness factors, and
the shrinking core model, are covered in only a small
number of courses. Contacting, which includes reac-
tor characterization (e.g., residence time distribu-
tion) and modeling non-ideal reactors, is normally
left to graduate-level courses.
The pillar structure shown in the figure allows
one to develop a few basic concepts and then to
arrange the parameters (equations) associated with
each concept in a variety of ways. Without such a

structure, one is
faced with the possi-
bility of choosing, or
perhaps memorizing,
the correct equation
from a multitude of
equations that can
arise for a variety of
different reactions,
reactors, and sets of
conditions. We draw
a loose analogy with
dining at a Swedish
smorgasbord where it
is difficult to choose

H. Scott Fogler is the Ame and Catherine
Vennema distinguished professor of chemical
engineering at the University of Michigan. His
teaching interests are in the areas of reaction
engineering and problem solving. His research
interests are in the areas of colloid stability and
flow and reaction in porous media, in which he
has over one hundred research publications.

from a multitude of dishes in order to end up with a
satisfying, well-balanced meal that fits together. In
CRE, consider the number of equations that arise in
calculating the conversion in CSTRs, batch, plug
flow, and semibatch reactor-for zero, first, second,
and third order reactions-for both liquid and gas
phase systems-with and without pressure drop. The
number of equations (dishes) from all the above pos-
sible combinations which we must choose (memo-
rize) is then 4 x 4 x 3 = 48. If we also consider
catalyst decay with either first, second, or third or-
der decay laws, the number of dishes increases to
192. IfLangmuir-Hinshelwood kinetics are included,
the number of equations, or dishes (i.e., equations in
isothermal reactor design) increases to well over
1,000. Finally, if we add non-isothermal effects, the




T e I
O B 0


number of dishes in-
creases to such an ex-
tent that choosing the
right dish, or dishes,
becomes a task of un-
believable gastronomi-
cal proportions. The
challenge is to put ev-
erything in an orderly
and logical fashion so
that we can proceed to
arrive at the correct
equation (dishes that fit
together) for a given

Copyright ChE Division ofASEE 1993
Chemical Engineering Education

Figure 1. Pillars of the Temple of Chemical Reaction Engineering.



Fortunately, by structuring CRE using an
algorithm analogous to a fixed-price menu in
a fine French restaurant, we can eliminate
virtually all memorization (see Figures 2 and
4) and proceed in a logical manner to de-
velop the reaction engineering equation nec-
essary to describe the given situation. The
lower price (220 FF) menu corresponds to
isothermal reactor design, while the higher
price menu (280 FF) corresponds to non-iso-
thermal design.
Here we start by choosing one dish from
the appetizers listed. The analog is to choose
the mole balance from one of four reactor
types shown. Next, we choose our main course
from one of four entries: the main course
analog in CRE is to choose the appropriate
rate law. Continuing with our meal, we
choose cheese or dessert; the analogy in CRE
is stating whether the reaction is liquid or
gas phase in order to use the appropriate
equation for concentration.
The main difference between CRE and
ordering and eating a fine French meal as
we have just done is that in CRE we com-
bine everything together at the end; such
a mixing of the courses on a single plate
before eating a wonderful French meal would
be a disaster! The application of this struc-
ture to a first order gas phase reaction car-

Le Cataliste Flambi
344 Champs Elysees
AMenu 4 220TT

fatd de Canard (supplement 15F7)
Potage Crime de Cresson
Escargots & La Bouguine (supplment

Ragnons de Veau
Coq au 1in
'Boeufd a provenpale
(Tous nos plats sontgarnis)

Brie on Crime Anglais

1/2 bouteilfe
e vin b6anc ou vin rouge

Mole Balance
sBatch Pactor

Semibatcht actor

Rate Law
Power Law (e.g.)
Ist Order
2nd Order
N9n-Integer Order

Gas orLiquid

Mix together and digest with
1/2 bouteiife of PoLyMr'1

Service Compris
Figure 2. French Menu I: Isothermal reactor design
Spring 1993

ried out in a PFR (with a change in the total number of
moles) is shown in Figure 3. As an example, we will follow
the dark lines as we proceed through our algorithm. The
dashed lines represent possible pathways for other situa-
tions. Here we choose

1. the mole balance on species A for PFR,
2. the rate law for an irreversible first order reaction,
3. the equation for the concentration of A in the gas phase, and
4. combine to evaluate the volume necessary to achieve a given
conversion or the conversion that can be achieved in a
specified reaction volume.

For the case of isothermal operation with no pressure drop,
we were able to obtain an analytical solution. In the majority
of situations, however, analytical solutions of the resulting
ordinary differential equations appearing in the combine
step are not possible. Consequently, we include POLY-MATH131
in our menu. POLYMATH is analogous to vin blanc ou rouge
in that it makes obtaining solutions to the differential equa-
tions much more palatable.

dX V FAnX X =-rAV
dV FA -A dt NAO

-rA= kCA -rA -C-A


FA = FAO(1-X) NA =NAO(-X)
constant flowrate variable flow rate variable volume constant volume

S= o = ol( +sX)V-'- = Vo(l +X)- V=Vo

CA=CAO(I-X) CA= C(l-XPL C C"n(l-X)P], CCAo(1-X)
S(+X) P T (1+X) T

4. COMBINE (1st order Gas Phase Reaction in a PFR)
From mole balance Ifrom rate law from stoichiometry
dX -rA kC k C
dV FFA FA = AO (+eX)P. T

=- where y = (A)

Integrating for the case of constant temperature and pressure

Vi 3 li +EX)nISrea rsX

Figure 3. Algorithm for ISOTHERMAL reactors


Studying non-isothermal reactor design is analo-
gous to ordering from a more expensive (280 FF)
French menu (see Figure 4) in which we have an
extra category from which to make a selection. In
CRE this corresponds to choosing which form of the
energy balance to use (e.g., PFR, CSTR) and which
terms to eliminate (e.g., Q=0 for adiabatic opera-
tion). The structure introduced to study these reac-
tors builds on the isothermal algorithm by introduc-
ing the Arrhenius Equation, k = A e-E/RT in the rate
law step, which results in one equation with two
unknowns, X and T, when we finish with the com-
bine step. The students realize the necessity of per-
forming an energy balance on the reactor to obtain a
second equation relating X and T. For example, us-
ing again the PFR mole balance and conditions in
Figure 3 (Eq. A), we have, for constant pressure
dX Ae-E/RT (-X)(T) A)
dV vo (1+ X) ()
An energy balance on a PFR with heat exchange
yields the second equation we need relating our in-
dependent variables X and T
dT [UA (T T) + (rA)(AHR)] (

Le Cataliste Flamb6
344 Champs Elyses
Menu tt28OrP

Pati de Canard
Pouge Crime de Crsson
'Ecargots A La Bourguine



hroutruue qanmi
oufsBourauat ona
Tous nospfatssontgarnis)

Assiette de Fromge

Emvaroi a range
souffia u Cuoto at
fdtuaua eL Crps a [a "inarmi
Petit Tour _

revenz so me,

Mole Balance


RatLa w kt-sAe
Power Law

qas or Liquid

LEnerqg Bafancx-fm

5u(# Steadyj tats

Service Compris

Figure 4. French Menu II: Non-isothermal reactor design

With the emergence of extremely user-friendly
software packages, we can now allow students to
explore the problem much more effectively, to
develop an intuitive feeling for the
reactor/reaction behavior,...

1111111111Ei-: l

Figure 5. Paradigm shifts in chemical engineering education.
These simultaneous differential equations can be
readily solved with an ODE solver, as discussed

With the emergence of extremely user-friendly soft-
ware packages (see Figure 5), we can now allow
students to explore the problem much more effec-
tively, to develop an intuitive feeling for the reac-
tor/reaction behavior, and to obtain more practice in
creative problem solving.
To illustrate this point, consider an exothermic
reaction carried out in a plug flow reactor with heat
exchange. Obtaining the temperature and concen-
tration profiles requires the solution of two coupled
non-linear differential equations such as those given
by Eqs. (A) and (B). In the past, it would have been
necessary to spend a significant amount of time
choosing an integration scheme and then writing
and developing a computer program before any re-
sults could be obtained. Now, with the available
software programs (especially POLYMATH), it rarely
takes more than ten minutes to type in the equa-
tions and obtain a solution."31 As a result, the major-
ity of the time on the exercise can be spent explor-
ing the problem through parameter variation and
analysis of the corresponding observations. For ex-
ample, in the above exothermic reaction in a PFR
with heat exchange, the students can vary such pa-
rameters as the ambient and entering temperatures,
the flow rates, and the heat transfer coefficient, and
look for conditions where the reaction will "ignite"
and conditions for which it will "run away." By try-
ing their own different combinations and schemes,
the students are able to carry out open-ended exer-
cises which allow them to practice their creativity
and better understand the physical characteristics
of the system.
Chemical Engineering Education


As a result of the paradigm shift in the ease of
computation using ODE solvers, the study of a
wide variety of chemical reaction engineering sys-
tems just becomes a variation on our main theme
(menu?). Once the concepts of the four left-most pil-
lars are mastered, many important extensions can
be viewed as minor variations on the four basic
steps in our isothermal reaction design algorithm.
Table 1 shows the basic algorithm for solving
CRE problems (including the energy balance), along
with the steps that need to be examined to handle
the individual variation.
To reinforce how many different CRE problems
are minor extensions of the basic algorithm, we shall
discuss three in greater detail below.

Pressure Drop
If pressure drop is not accounted for in gas phase
reactions, significant under-design of the reactor size
can result. This variation is handled in the stoichi-
ometry step, where concentration is expressed as a
function of conversion, temperature, and total pres-
sure. The change in total pressure is given by the
Ergun equation'"

I Variations


Reactive Distillation
Membrane Reactors





Catalyst Decay

Pressure Drop
Multiple Reactions
Membrane Reactors

Multiple Steady States



What if...??

I '-------
Table 1. Variations on the basic algorithm
Spring 1993

dP_ G(1-0) 150(1-0)p .75G (C)
dL pgcDp3 Dp

This equation can be rearranged by lumping the
constant parameters to arrive at the following equa-
tion, giving the variation of the pressure ratio (y=P/
Po) with either reactor volume
dy al(l+eX) (D)
dV 2y
or catalyst weight
dy_ 2 (1+eX) (E)
dW 2y
Either of these equations can be coupled with the
combined mole balance, rate law, and stoichiometry
and solved numerically (e.g., with an ODE Solver).
For example, for isothermal conditions, Eq. (E) would
be coupled with Eq. (A) adopted to an isothermal
dX k(1-X)
dW vo(l+eX)y (F)

Catalyst Decay
For the case of separable kinetics, we simply in-
clude a catalyst decay law in the rate law step of our
algorithm. For example, for a straight through trans-
port reactor (STTR), the rate law might be given by
-rA =a(t) ACA (G)
where the catalyst activity, a(t), is

a(t)= 1 (H)

The algorithm for studying a catalytic reaction in a
straight through transport reactor is shown in Fig-
ure 6.



DECAY LAW a(t)= 1+


COMBINE dX k (1-X) 1
dV u(1+KACAo(1-X)) (1+3t1/2
Figure 6. Variations on a theme: Catalyst decay in a STTR


Membrane Reactors
The only trick in studying membrane reactors is
to make sure to write the equations in terms of
molar flow rates rather than conversion, and to ac-
count for the products that are leaving the reactor
through the sides of the membrane reactor in our
mole balance step. Consider the reaction
taking place in a membrane reactor
A- ) A, B, C

For the product that exits through the side of the
membrane reactor, C, the mole balance and stoichi-
ometry steps are

c _Fi

V=v~o-=V FAO+FB
Fo ( FTO

After writing a mole balance on A and B, tl
ing set of non-linear ODEs is solved numer
sample POLYMATH solution is
shown in Figure 7.

The steps that are varied when
multiple reactions occur are rate
law and stoichiometry. As with
membrane reactors, we work in
terms of the number of moles or
molar flow rates of each species
rather than concentration or con-
version. Figure 8 shows the appli-
cation of the algorithm to a sample
reaction problem.

One of the primary pedagogical
advantages of developing mecha-
nisms and rate limiting steps in
heterogeneous catalysis is that it
provides insight into how to ana-
lyze and plot the data in order to
evaluate the rate law parameters.
Most schools spend one-and-a-half
to two weeks on heterogeneous ca-
talysis. Once the basic concepts of

ie result-
rically. A

adsorption, surface reaction, and desorption are in-
troduced in conjunction with the idea of a rate limit-
ing step, one can derive many possible rate laws by
varying the mechanism and rate limiting step. For
example, consider the following dual site isomeriza-
tion mechanism
A+ S= A*S
A.S+So B-S+S
B-So B+S
One can write the rate laws for each step in this
mechanism and then show that if the catalytic reac-
tion (A B) in this example is surface reaction

19 *99

B *


4 *9

0.00 2.00 4.90 6.00 8.90 I19B.
r (Vi/o)
Figure 7. Composition profiles in a membrane reaction


A-4-B -rAl= kAICA

A--C+D -rA2=kA2CA Ci=
B+D-4E -rB3 = kB3CBCD
rA =A1 +rA2 =-kA1CA kA2C
rB = rB1 +rB3 = kACA -kB3CBCD
rc = rc2 = kA2C
rD = rD2 +D3 = kA2C -kB3CBCD
E = rE3 = kB3CBCD
dNAA =-kA NA-kA2 V2
=Adt V=-2 V

dNE = rE = kB3 V


Figure 8. Variation on a theme: Multiple reactions
Chemical Engineering Education


limited, a plot of the initial rate data in the form of

versus PAo should yield a straight line, as shown in
Figure 9. By formulating different mechanisms
and rate limiting steps, a variety of rate laws can
be developed which provide a number of options
on how to interpret the data and evaluate the rate
law parameters.
Currently, there is an added incentive to study
mechanisms in heterogeneous catalysis because of
the emergence of chemical vapor deposition (CVD).
CVD is widely used in the microelectronics indus-
try, and the mechanisms for CVD are very similar
to the mechanisms analyzed in heterogeneous ca-
talysis. By developing fundamental laws and
principles such as those in heterogeneous catalysis,
the students will be prepared to analyze chemical
reaction engineering problems in engineering tech-
nologies, e.g., CVD.
Other mechanisms and rate laws that can be eas-
ily incorporated into the original algorithm include:
* Enzyme kinetics (e.g., the Michaelis-Menten equations)
rs = 1+KCs


Figure 9. Gaining insight into how to
analyze the data.

Bioreactors (e.g., the Monod equation for bacteria
growth; see Figure 10)
r ^-C
^ T+KCs.CB
g = B

Pseudo Steady-State-Hypothesis

A+A<>A* +A
A* B
-r =0
-rA -j
-A 1+k2CA
Polymerization (long chain approximation)
In studying these topics, rules are put forth to
guide the student in the development of the under-
lying mechanism and of the rate law. Once the rate
law(s) is formulated by analyzing the particular re-
action mechanism, one can then use it (them) in
step 2 of the algorithm (menu) to study the particu-
lar system of interest.

Discussion of future directions in CRE with col-
leagues at Michigan and elsewhere is the same as
discussion on other academic issues. Where n fac-
ulty members are gathered to express an opinion on
an issue, there will be 1.5 n opinions. But my feeling
is that in the immediate future we will continue to
focus on developing problems that exploit software
packages such as POLYMATH, Maple, and
Mathematica. We will see materials processing, en-
vironmental reaction modeling, reaction pathways
and more applications on safety, batch processing,
mixing, ecology (see Figure 11), and novel reactors
(membrane batch reactors?) along with stochastic
approaches for analyzing reacting systems.
Finally, in the not-too-distant future I see a greater
emphasis on predicting the reactivity of different
species, first perhaps by using empirical means but

u Time

Figure 10. Phases of bacteria cell growth
Spring 1993

0 z
Figure 11. Using wetlands to degrade toxic wastesil4



later on from first principles. This direction will
lead us into what I would call molecular chemical
reaction engineering. These and other topics not men-
tioned here may first be covered (and in some cases
are currently covered) at the graduate level, but
they will filter down to the undergraduate level.
This filtering will occur much more rapidly than
have analogous topics in the past.

By arranging the teaching of chemical reaction
engineering in a structure analogous to a French
menu, we can study a multitude of reaction systems
with very little effort. This structure is extremely
compatible with a number of user-friendly ordinary
differential equation (ODE) solvers. Using ODE solv-
ers such as POLYMATH, the student is able to fo-
cus on exploring reaction engineering problems
rather than on crunching numbers. Thus, one is
able to assign problems that are more open-ended
and to give students practice at developing their
own creativity. Practicing creativity is extremely
important, not only in CRE but also in every course
in the curriculum, if our students are to compete in
the world arena and succeed in solving the relevant
problems that they will be faced with in the future.
1. Fogler, H.S., The Elements of Chemical Reaction Engineer-
ing, 2nd ed., Prentice Hall, Englewood Cliffs, NJ (1992)
2. Eisen, E.O., "The Teaching of Undergraduate Kinetics/Re-
actor Design," paper presented at the AIChE Annual Meet-
ing, Los Angeles, CA, November 14 (1991)
3. Shacham, M., and M.B. Cutlip, "Applications of a Micro-
computer Computation Package in Chemical Engineering,"
Chem. Eng. Ed., 121), 18 (1988)
4. Kadlec, R.H., "Hydrologic Factors in Wetland Treatment,"
Proc. Int. Conf. on Constructed Wetlands, Chattanooga, TN,
Lewis Pub., Chelsea, MI (1989)
a catalyst activity

A frequency factor, appropriate units
Ac cross sectional area, m2
Ci concentration of species i (i = A,B,C,D), mol/dm3
C heat capacity of species i, J/g/K
D, particle diameter, m
E activation energy, J/mol
F. entering molar flow rate of species i, mol/s
G superficial gas velocity g/m2/s
ge conversion factor
k specific reaction rate (constant), appropriate units
Ke equilibrium constant, appropriate units
L length down the reactor, m
N. number of moles of species i, mol
P pressure, kPa
ri rate of formation of species i per unit volume,
r' rate of formation of species i per unit mass of
catalyst, mol/s/g
t time, s
T temperature, K
U overall heat transfer coefficient, J/dm3.sK
V volume, dm3
W catalyst weight, g
X conversion
y pressure drop parameter, (P/Po)
yA mole fraction of A
A refers to species A
0 entering or initial condition





pressure drop parameter, g'1
catalyst decay parameter, s12
heat of reaction, J/mole A
change in the total number of moles per mole of A
volume change parameter = yA08
viscosity, cp
density, g/dm3
volumetric flow rate, dm3/s

REVIEW: Natural Gas Engineering
Continued from page 109
There is a discussion of the compressibility of natu-
ral gases with an explanation of the various correla-
tions that have been used, including the effects of
nitrogen, carbon dioxide, and hydrogen sulfide. There
are a few examples that show how to use these
particular charts.
Chapter 5 Gas Hydrates and Their Prevention.
The formation of hydrates is an important issue
associated with the production of natural gas, espe-
cially in colder circumstances. The discussion is quite

Chapter 6 Applications of Flow Equations: Pres-
sure Drop, Compression, Metering. The material in
this chapter is relatively standard on fluid flow.
However, the emphasis is on problems of natural
gas flow and two-phase flow. The problems associ-
ated with calculating vertical and horizontal flow
are useful-especially the hints on how to calculate
flow in such systems.
Chapter 7 Drilling and Completion of Wells.
Chapter 7 is an overview containing relatively con-
cise descriptions of gas fracturing and acidizing op-
erations. The discussion of well logging is a review
of most of the various kinds of logs that are used.
Chemical Engineering Education

Chapter 8 Flow in Reservoir and Adjacent Aqui-
fer. This is a strong chapter. It presents a discussion
of the flow of gas in reservoirs. The pressure, (pres-
sure)2, and pseudopressure methods of general flow
are discussed. The unsteady state solution for the
constant terminal rate case and for the steady-state
drainage radius case are discussed. In addition, there
is a good discussion of the skin effect, the effect of
high-velocity flow, and the well storage effects.
Chapter 9 Gas Well-Testing. Another strong chap-
ter, it begins with a good discussion of deliverability
tests. The second part discusses tests for determin-
ing reservoir parameters. Examples are given of how
to calculate the actual pressures. Tests for deter-
mining reservoir parameters include drawdown tests,
multi-rate tests, two-rate tests, and build up tests.
The discussion, though concise, is complete, and there
are several examples that show how to use these
particular tests to determine reservoir properties.
Chapter 10 Reservoir Engineering Applied to
Gas, Gas / Condensate, and Gas / Oil Fields. This sur-
vey chapter discusses determining initial estimates
of oil and gas reserves using either volumetric cal-
culations or early production history. The mecha-
nisms of oil recovery are discussed in very general
terms. The solution of the material balance equa-
tion for a reservoir is shown.
Chapter 11 Simulation: Field and Reservoir Per-
formance. This chapter and its first three sections
discuss the implicit, explicit, and Crank-Nicolson
numerical methods for solving the partial differen-
tial equations that approximate the flow in a reser-
voir. The discussion is concise and includes an ex-
ample of the one-dimensional situation. There is
also a brief discussion of the inverse problem.
Chapter 12 Conversion of Depleted Gas, Gas/
Condensate Fields to Gas Storage Reservoirs. The
gas storage problem is discussed at a survey level in
this chapter. There is a good description of why
storage is needed and how gas is stored. A detailed
case study is presented.
Chapter 13 Gas Storage in Aquifers. This is an
excellent chapter-the strongest in the book. It be-
gins with procedural steps in locating and develop-
ing an aquifer storage reservoir. The discussion of
locating such a reservoir is detailed, and there is a
series of discussions of the measurements that are
required. Predictions on the rate of bubble develop-
ment or water pushback are discussed. There is a
general discussion of the various studies of aquifers.
Chapter 14 Monitoring, Inventory Verification,
Deliverability Assurance, and Safety in Storage Op-
erations. This chapter surveys what must be done to
Spring 1993

run a gas storage operation and represents the tre-
mendous experience of the senior author. This chap-
ter and the previous one are condensed from Under-
ground Storage of Fluids, by Katz and Coats (1968).
Chapter 15 Natural Gas Liquid Recovery and
Gas for the Fuel Market. Natural gas liquids (ethane,
propane, butanes, and pentanes) are recovered by
refrigeration adsorption stripping or cryogenic ex-
Chapter 16 Storage in Salt Cavities and Mined
Caverns. The description in this chapter is narra-
tive, explaining the attributes of such caverns and
how one develops caverns. Also, there is a discus-
sion of creating cavities by dissolving materials or
leaching out the materials.
Chapter 17 Miscellaneous Topics. This chapter
opens with a narrative of compressed air energy
storage for electric power peaking cycles. Much of
the chapter discusses the design of a storage facil-
ity. It also contains calculations associated with
transcontinental pipelines, geochemical identifica-
tion of natural gas, superheat limit vapor explosion,
and the phase behavior associated with it.
Six appendices contain data on computer programs
for calculating flow, derivations of gas flow equa-
tions in reservoirs, a detailed discussion of the Peng-
Robinson equation of state, equilibrium constants,
and nomenclature. n

book review

by J. A. Wesselingh, R. Krishna
Ellis Horwood Ltd., Market Cross House, Cooper St.,
Chichester, West Sussex, P019 1EB England; 243
pages, $69.95 (1991)
Reviewed by
Phillip C. Wankat
Purdue University
This is an extremely interesting (but in many ways
frustrating) short book on the use of the Maxwell-
Stefan (M-S) approach for solving complicated mass
transfer problems. Since the authors assume con-
siderable familiarity with Fickian diffusion and with
various separation methods, this book is appropri-
ate for graduate students. A finite difference ap-
proximation to the differential equations is used
throughout the book, and the calculations required
are well within the capabilities of graduate students.
After an introductory chapter, Chapter 2 explores
Continued on page 126.

re, curriculum
-- .--------------



University of Utah
Salt Lake City, UT 84112

Please accept my apology for the deceptiveness
of the title of this paper. It should read "What
About Telling Students How and Why We
Run Engineering Colleges the Way We Do," or some-
thing like that-but that would really be too
long to be useful. The deception was designed to
attract your attention. (Besides, we professors do
let the inmates run the asylum by permitting
anonymous students' comments to affect RPT deci-
sions in the universities!)
Many engineering students consider an engineer-
ing college to be simply a super high school which
has as its main purpose subjecting them to a cur-
riculum of difficult and unrelated courses, or worse.
Such a belief is, of course, incorrect-but it is sel-
dom that anyone attempts to change the perception.
James Wei wrote a paper entitled "The Reju-
venation of Chemical Engineering" (first presented
in 1979 as a Phillips Petroleum Lecture at Okla-
homa State University and later given wider distri-
bution in CHEMTECIH1) that included an excellent
discussion of the rationale for most chemical engi-
neering curricula. This discussion was, and is, an
effective means of explaining to students the nature
of their progress toward an engineering degree. I
have given a one-page synopsis of the lecture to
undergraduate students for some time now, and the
perceptive students greatly appreciate Professor
Wei's observations.
One thing that is not obvious to students (or even

Copyright ChE Division ofASEE 1993

to some young faculty!) is the manner in which prob-
lem solving pervades the whole of engineering study
and practice. Although the list of successive courses
in the curricula bears titles of chemical engineering
topics, the nature of the problems treated in each
course is more complex than those that were en-
countered in the prerequisite classes. Table 1 at-
tempts to outline how problems change throughout
the education and career of an engineer.
This table is, for the most part, self-explanatory.
The word "paths" was used as a shortened term
to indicate strategies or approaches, and "practitio-
ner" is used in a broad sense to include students at
all levels. Also, the term "technician" implies the
semi-professional or industrial technician-operator.
The number of levels corresponds to about the nor-
mal yearly educational breaks in engineering edu-
cation and was selected to avoid any religious or
political implications.
The fact that such a progression is deliberate and
proceeds in an ordered fashion is something that
students need to know and to appreciate. When this
table is given to serious students who then take
the time to ponder its information, their acceptance
and understanding of the engineering approach to
problem solving becomes easier. Perhaps the
faculty is no longer seen as just being "difficult"
when they require more demanding and complex
assignments. Note also that it is here implied that
the introduction of truly "open-ended" problems is
later in the course of study than is normally sug-
gested by ABET requirements. However, it appears
that the undergraduate's later encounter with such
difficult problems corresponds to current engineer-
ing education practice.
Table 1 is not an outline for a problem-solving
algorithm. The manner in which complexity is in-
troduced into the problem solving process of very
quantitative disciplines is illustrated. The method
for introducing complexity into more qualitative
fields would be much different than the process
shown in the table. This is just one of the more
obvious differences between problem solving in en-
Chemical Engineering Education

Alva Baer has taught chemical engineering at
the University of Utah since earning his PhD
from that school in 1969. His academic and
industrial research efforts have been in support
of the areas of propulsion and combustion. He
has now reached the age where it is easier to
write short papers than it is to read long ones.

Possible Levels of Engineering-Type Problem-Solution Methods
NOTE: Each level is based on the lower-level method

Level Number of
Steps and Concepts


Characteristics of Solution Method

1 1 or 2 Given only the
data needed

2 2 or 3 May change units
of data

3 4 or 5

May be given
extraneous data

4 several Must find some
needed data

5 many Must decide on and
find needed data
in published

6 many

Must evaluate
conflicting data, the
data are incomplete,
and needed accuracy
depends on how the
data enter the problem.

7 many Measurement of data
paths by standard methods
may be required

8 most paths Measurement of
fail needed data may
require new techniques
or be impossible.

Numbers are inserted into a given equation

Numbers are inserted into given, combined, or
rearranged equations

Final relationships must be developed; the problem
has an exact answer; and one may work toward
a given solution

The problem has one unique solution and one may
work toward a known answer. At this point, trial-
and-error and iterative methods may be introduced.
Computer solution methods become practical.

The problem has a unique solution, but the answer
is not given. A method is required to verify the
solution. Progress is now based on personal efforts
mainly by learning from one's own mistakes.

The problem may have several possible solutions,
but there may be an optimum answer. Usually
approximations must be made. Time-critical
answers are required. Economic factors are

The type of problem may never have been solved
before. Economics must be considered at all
stages. Safety and environmental considerations
maybe important. The starting and ending points
are not obvious.

The problem selection is critical, but the selection is
usually by the practitioner. Often, the final
answer may not be recognized. Likely, an
acceptable solution is not possible.

High school students
and technicians

First-year science and
engineering students;

Second-year science and
engineering students;
advanced technicians

Second- and third-year
engineering students

Third-year engineering

Fourth-year engineering
students; graduate
students; design engineers

Graduate students; design
engineers; research

Researchers; creative

gineering and activities in less mathematical fields.
A result of this difference is that it is not likely
that a general problem-solving algorithm exists for
the range of human endeavors, and this fact is too
often missed by engineers. People with quantita-
tive training too often get into difficulties by
attempting to attach numbers to truly quali-
tative activities such as in the arts and humani-
ties, or for supervision and evaluation of other
people, etc. Engineering students should be made
aware of the limitations of their quantitative edu-
cation. Anyone who disagrees with this suggestion
should consider attaching a number to indicate the
affection of their dog.
Spring 1993

Finally, it should be noted that the details in Table
1 represent my observations and biases. Others
might organize the information in a different order
or with different emphasis. The important point,
however, is that there is a continuous progression in
complexity leading to the goal of efficiently solving
real-world problems. The table represents an origi-
nal effort, but if this notion or the information ap-
pears elsewhere the omission of a reference is the
result of ignorance, not perfidy.

1. Wei, James, "The Rejuvenation of Chemical Engineering,"
CHEMTECH, 15, 655, November (1985) 0


O classroom


A Quick Guide to Learning Principles

Purdue University
West Lafayette, IN 47907

Great teachers may be born that way, but the
vast majority of professors have to work to
improve their teaching. Fortunately, a re-
search base now exists that shows which teaching
methods work in a variety of situations. In this pa-
per, ten of the procedures which are known to work
will be briefly presented and applied to chemical
engineering education. More details and a variety of
earlier references are given by Wankat and
1. Develop a structured hierarchy of content
and guide the learner.
Content is king (or queen). The professor should
be sure that the content of his subject is both impor-
tant and up-to-date. Some structure should be evi-
dent to the students, and they should be guided in
their learning. Tell the students where they are go-
ing and why it is worth their effort to get there. For
example, thermodynamics is both beautiful and ex-
tremely useful, but many professors act as if the
beauty of thermodynamics alone should be suffi-
cient to hold their students' attention, and they ne-
glect to tell the students what they will be able to do
once they have mastered the subject. Be sure that
the students know what the objectives of the course
are. Actually developing some of the structure of
knowledge themselves helps students learn the ma-
terial; thus, an overly rigid structure should be
avoided. A good homework assignment is requiring
the students to prepare a "key relations chart" which
lists everything the student wants to know to solve
problems or for a test. Obviously, first-year students

...thermodynamics is both beautiful
and extremely useful, but many professors act
as if the beauty of thermodynamics alone should
be sufficient to hold their students' attention ...

Copyright ChE Division ofASEE 1993

need considerably more structure than graduate stu-
dents, and courses should be designed accordingly.
Since problem solving is a major part of chemical
engineering, both the structure and the method of
problem solving should be part of the course.1'"21 Much
of the structuring of content and guiding the stu-
dent can be done in lectures, although other teach-
ing methods work just as well if not better.
2. Develop images and use visual modes of
Most people prefer visual learning and remember
visual images much longer than words, but most
college instruction is auditory (e.g., see Felder and
Silverman[31). The McCabe-Thiele diagram has been
a successful teaching method for decades simply be-
cause it provides the student with a visual image.
Modern computer graphics and plotting calculators
can also be used to provide visual images-they are
particularly useful for three-dimensional plots and
for showing motion. Most students better under-
stand equations when they are plotted for a variety
of circumstances rather than simply looking at the
symbolic form. The professor should require that
the students develop their own visual images.
3. Make the students actively learn.
People learn best by actively grappling with infor-
mation; thus, some sort of classroom activity is re-
quired.[4] This activity can be external (such as dis-
cussing a question, solving a problem, developing a
structured flowsheet of knowledge, brainstorming,
or working in a group) or internal (such as reading,
questioning by oneself, pondering, etc.). Lecturing
without student interaction is active only for the
Chemical Engineering Education

Phil Wankat received his BSChE from Purdue
and his PhD from Princeton. He is currently a
professor of chemical engineering at Purdue Uni-
versity. He is interested in teaching and counsel-
ing, has won several teaching awards at Purdue,
and is Head of Freshman Engineering. His re-
search interests are in the area of separation
processes with particular emphasis on cyclic sepa-
rations, adsorption, preparative chromatography,
simultaneous fermentation and separation.

professor-which is one reason why professors often
feel they have learned more than anyone else in the
class. Most students will initially resist active in-
volvement in the classroom since it is not safe and
they have been trained to passively take notes. But
once the students become familiar with classroom
activity, they usually grow to like it-and most stu-
dents certainly benefit from it.
4. Practice and feedback.
The professor should provide the students with an
opportunity to practice what they learn while they
are still in a supportive environment. A variety of
different problems and questions should be tackled,
and it is important to have some (but not excessive)
repetition to increase both speed and accuracy. A
series of regular assignments with frequent feed-
back will elicit more work and higher levels of com-
mitment to the class than will one long assignment.151
Following this principle, a long design project can
be broken into several smaller parts with various
assigned due dates. Students should have feedback
during, or shortly after, their first practice so that
they do not keep practicing incorrect methods.12' Feed-
back a month later is not useful. The students should
have the opportunity to practice again-after they
have received the feedback. For instance, after
a laboratory or design report has been returned to
the student with the usual excessive amount of red
ink, it is most effective to require the student to
produce a final, corrected, clean copy. With the aid
of word processors, preparing a clean copy is much
less work than it used to be. Computer-aided in-
struction can provide very useful practice, particu-
larly if it is interactive.
5. Positive expectations and student success.
Studies have shown that when a teacher expects
students to do well, they usually respond by doing
well."41 When someone important believes in the
student, his or her expectations can be a major in-
fluence in the student's success. A very interesting
and accessible report on this topic, pertaining to
families, is given by Caplan, et al.16' Success is a
strong motivation in itself, and it leads to additional
success. When a student does not have the proper
educational background, he or she will be probably
be unsuccessful; so one useful activity is to pro-
vide background material for those students, and
then to make sure that they use it. For instance, a
lack of skill in algebraic manipulation will cer-
tainly sink a student in a mass and energy balance
course-extra help in algebra can be much more
effective in advancing that student than tutoring in
mass and energy balances. Many capable students
Spring 1993

The professor should provide the students
with an opportunity to practice what they learn
while they are still in a supportive environment.
A variety of different problems and
questions should be tackled...

leave engineering due to a lack of encouragement or
a lack of success."71
6. Develop a cooperative class with students
teaching each other.
Most students learn better in a cooperative envi-
ronment where a significant amount of the work is
done in groups.l41 Since modern engineering practice
usually involves groups of engineers, group classwork
can be good training for the students' professional
careers. Many students who leave engineering cite
the overly competitive atmosphere as a major rea-
son for leaving."' A number of successful programs
involving group work with engineering students have
been reported.1'2] A recent study at Harvard Univer-
sity found that the students who grow most aca-
demically and who are happiest structure their time
to include intense interpersonal interactions with
faculty or other students."5' In large classes the pro-
fessor may not have time to meet individually with
every student, but he or she can and should encour-
age group work both in and out of class. Study groups
should be set up with the understanding that each
group member must do the reading or problem as-
signments before coming to the study group. Opti-
mum group size appears to be from four to six stu-
dents. Competitive grading procedures using "the
curve" do not encourage cooperation; other proce-
dures, such as grading against an absolute standard
or mastery testing, will encourage more cooperation
with the professor and between the students them-
selves."11 One advantage of working in groups is that
students have an opportunity to informally teach
other students (which helps both of the students
learn better). Formal approaches to encouraging stu-
dents to teach other students (such as tutoring or
serving as the expert on a laboratory experiment)
also increase student learning. It is important to
note that teaching others should not be reserved for
only the best students.

7. Be enthusiastic-care about teaching.
Students respond to enthusiasm. It is important
that the professor cares about what he or she is
teaching. Those professors who put teaching on "au-
tomatic" cannot possibly do a good job. There is no
excuse for reading a book to the students in lec-
Continued on page 127.

jg class and home problems

The object of this column is to enhance our readers' collection of interesting and novel problems in
chemical engineering. Problems of the type that can be used to motivate the student by presenting a
particular principle in class, or in a new light, or that can be assigned as a novel home problem, are
requested, as well as those that are more traditional in nature and which elucidate difficult concepts. Please
submit them to Professors James O. Wilkes and Mark A. Burns, Chemical Engineering Department, Univer-
sity of Michigan, Ann Arbor, MI 48109-2136.



A Demonstrative Energy Transport Problem

Illinois Institute of Technology
Chicago, IL 60616

he development of new and interesting trans-
port phenomena examples and problems that
can be solved using relatively simple math-
ematical tools can be a challenge. This is especially
true when teaching an undergraduate course in
transport phenomena where the students have little
or no experience solving partial differential equa-
tions. There are, unfortunately, a finite number of
physically meaningful problems one can formulate
that lead to linear, ordinary differential equations
(even with three coordinate systems and several
types of boundary conditions to choose from).
This paper presents an energy transport problem
that is both instructive and interesting; it can be
used to demonstrate
the use of dimensional analysis
the quasi-steady state approximation
the fin approximation
As the example progresses, students see an "intimi-

David C. Venerus is an assistant professor of
chemical engineering at Illinois Institute of Tech-
nology. He received his BS degree from the Uni-
versity of Rhode Island and his MS and PhD
degrees from Penn State University He has been
at IIT since 1989 and conducts research in the
areas of polymer theology and processing. He
has taught courses in material and energy bal-
ances, unit operation, transport phenomena, and
polymer processing, and is faculty advisor to the
AIChE student chapter.
Copyright ChE Division ofASEE 1993

dating" PDE transformed into an innocent ODE
through the introduction of several physically rea-
sonable assumptions. The novelty of the example
arises from the fact that it applies to a technology
not traditionally associated with chemical engineers,
although it is one that virtually all engineers use.

The example arises from an energy transport
analysis of a process widely used in the semiconduc-
tor device industry. The process, known as
Czochralski Crystal Growth (CZCG), is used to
produce single-crystal, defect-free ingots of Si (and
similar materials) which are subsequently sliced
into thin disks (or "wafers"), polished, and used
as substrates in the fabrication of microelectronic
devices, or "computer chips." The example or prob-
lem might be introduced by giving a short descrip-
tion of the process.
CZCG is a batch process initiated when a seed
crystal is dipped into a melt of the same material so
that the liquid wets the seed crystal. As solidifica-
tion occurs, the seed is slowly withdrawn from the
melt so that a neck and shoulder are grown. Once
the desired radius is achieved, a nearly cylindrical
crystal is grown by manipulation of the pull rate
and/or melt temperature.
It should be noted that even for the simplified
description of CZCG given above, a high level of
complexity is required to develop detailed transport
models. The presence of a number of free bound-
Chemical Engineering Education

aries (at the crystal-melt, crystal-ambient, and melt-
ambient interfaces), radiative heat transfer, and tem-
perature-dependent physical properties all make the
problem highly non-linear. Detailed transport mod-
els which account for these phenomena require nu-
merical solution on large computers. An excellent
review of the CZCG and other crystal growth pro-
cesses along with discussions on the importance of
various transport processes can be found elsewhere.11

Before they are presented with this example, stu-
dents should have been exposed to the appropriate
energy transport phenomena fundamentals: conser-
vation laws (either by the shell balance[2] or Reynolds
Transport Theorem"" approach), constitutive equa-
tions, and boundary conditions.
For the example, we will consider a relatively
simple model of the CZCG process that describes
energy transport within the cylindrical crystal. A
schematic diagram of the process in shown in Fig-
ure 1, which indicates the position of the coordinate
system. Results from this analysis could be used to
estimate thermoelastic stresses (due to temperature
gradients), which can lead to crystal defects, or used
to find relationships between crystal length, growth
rate, and melt temperature. The assumptions to be
used in the development of the model are
1. Axial symmetry in the crystal.
2. All physical properties are isotropic and indepen-
dent of temperature.
3. Heat transfer between the crystal and the ambient



T-a L T




Figure 1. Schematic diagram of Czochralski Crystal
Growth (CZCG) process.
Spring 1993

can be described by a convective heat transfer law
to an ambient temperature that is independent of
time and position.
4. Heat transfer between the crystal and the melt can
be described by a convective heat transfer law to a
melt temperature that is independent of position.
5. The velocity of the crystal (pull rate) is constant.
6. The crystal-melt interface is planar and fixed at
the origin of the coordinate system.
Assumptions 1 to 6 lead to the following for the
thermal energy equation and boundary conditions:

_T 1T -1 aT a2 T]
at +Vz =ar ar =i (1)

T(r,z,0)= Tf 0 T(r,0,t)= Tf 0 r 0 (3)

-kzT(r,L(t),t) =ha[T(r,L(t),t)-Ta] 0 r0 (4)
-k (R,z,t)=ha [T(R,z,t)-Ta ] 00 (5)

where a = k / pCp. The (jump) energy balance at the
crystal-melt interface gives
-k ( (r,0,t))= hm[Tm (t)- Tf ]+ pVAHf t>0 (6)
where ( ) indicates a radially averaged quantity.
A list of the dimensional variables that appear
in this set of equations can be found at the end of
this paper.
Equations (1) to (5) define the linear boundary
value problem (BVP) for T(r,z,t). The boundary con-
ditions given by Eqs. (4) and (5) simply relate the
conductive and convective energy fluxes at the crys-
tal-ambient interfaces according to assumption 3.
Equation (6) is an energy balance for the crystal-
melt interface that must be satisfied so that a con-
stant radius crystal is grown by manipulation of the
melt temperature, Tm(t). If Tm is made constant,
then the crystal velocity V, rather than the melt
temperature, is manipulated to maintain a con-
stant radius crystal. In this case, Eq. (6) would be
solved for the crystal velocity which would be a func-
tion of the temperature gradient at z = 0 rather
than a constant. Hence, Eqs. (1) to (6) would consti-
tute a non-linear BVP since Eq. (1) would be non-
linear (due to the convective term) and because the
moving boundary L(t) would be a function of the
dependent variable rather than some external influ-
ence. In this case, assumptions 4 and 5 would, of
course, be modified.
Although the problem defined by Eqs. (1) through
(6) can be solved analytically, we will try to find

ways to simplify it using several additional assump-
tions. To begin, we first put the equations in dimen-
sionless form using
Sr z t T-Ta
R =R R2 Tf-Ta
Substitution of the above definitions into Eq. (1)

ao ao i a + 8 ao2
T+ ePe 1 (7)

where Pe is a dimensionless group
Pe = VR Peclet No.

The problem at hand would be much simpler if it
were a steady-state problem, but unfortunately no
steady-state exists because the length of the crystal
is a function of time: V = dL(t)/dt. Suppose, how-
ever, that we could neglect the unsteady term in Eq.
(7) but still allow the length of the crystal to change.
Under what conditions would this be a good as-
sumption? To answer, let us consider the time scale
for energy transfer (tE) and the time scale associated
with a change in the length of the crystal (tL). If
t << tL, i.e., conduction along the length of the crys-
tal is instantaneous compared to the time required
for the length to change, then neglecting the un-
steady term in Eq. (7) would seem reasonable. Of
course, what we are saying is that the quasi-steady
state approximation (QSSA) would be valid. If we
let tE = L2/a and tL = L/V, then we have

tE = VL
tL Oa

L L < < V

For CZCG growth of Si, a ~ 101 cm2/sec and
V ~ 103 cm/sec, so that the QSSA will be valid if
L < -100 cm. We will later see the QSSA will also
be valid for much greater crystal lengths. Thus we
will add the following assumption to our list:
7. The quasi-steady state approximation is valid.
The complete problem in dimensionless form is
now given by

P e +_ 1 a ( 2 (8)

(4, 0) =1 0 <1 (9)

-o (,A)+Bia (I,A)=0 050 <1 (10)

(1,)+Bia(1,)=0 0
(, ))+ Bi[-l1]= PeSt (12)

which includes the following dimensionless groups:
A = L Aspect Ratio

Bii = Biot No. for ambient (i=a) or melt (i=m)

St = CpAHf Stefan No.
Tf Ta
The solution to Eqs. (8) through (11) can now be
found by the separation of variables method for
#((,0), and Eq. (12) can be evaluated for the dimen-
sionless melt temperature, m,. For the case when
Pe -- 0, one might have the students find the solu-
tion in the literature.[41
Let us see if there are other physical arguments
that will further simplify our CZCG model. For stages
of the process when the crystal is long (A > > 1), it
would seem reasonable to expect the temperature
variation in the z-direction to be much greater than
in the r-direction. Does this mean that neglecting
the radial conduction term in Eq. (8) would be a
good assumption? While this seems like a good idea
at first glance, we remind the students that in doing
so we are in effect saying that the cylindrical sur-
face of the crystal is insulated and no energy is
transferred across it. For large A, this surface is
much larger (2A times) than the surface of the top of
the crystal and neglecting the heat transfer from it
would be a poor assumption.
Our original argument, however, still seems valid,
and it would be nice if we could find some way of
simplifying the radial conduction term. Since the
variation of T in the r-direction is probably small,
suppose we use an average radial temperature to
represent it? This approach is, of course, the "fin
approximation" which is frequently used to describe
finned heated transfer surfaces. In terms of dimen-
sional variables, the definition for average tempera-
ture we use is
2n R
fJ T(r,z)rdrd6 R
(T(z))= 0 0 =2 J T(r,z)rdr
21c R R2 (-0
J Jrdrd6
0o o0
or, in dimensionless form
<((())= 2\ $(,0))d (13)
Since A can be large (-102) in a typical CZCG
process, we will pursue this approach and add the
following assumption:
8. The fin approximation is valid.
Of course, the fin approximation can also be incor-
porated into the governing equations by the shell
Chemical Engineering Education

balance approach.2'
Integration of Eq. (8) according to Eq. (13) gives

Pe =2 (1, ) + 2 (14)
ao a4 ao
and substitution of the boundary condition given by
Eq. (11) in Eq. (14) leads to
a(4) a2()
Pe =-2 Bia(,o)+ a2

which, since (1,ao) =< 0(o) >, can be written as
d2(o) d(e)
dG2 Pe d -2 Bia() = 0 (15)
Equations (9), (10), and (12), in terms of the radially






0.00 0.20 0.40 0.60 0.80 1.00
Figure 2. Dimensionless axial temperature distribution
from Eq. (19) for the indicated values of Bio.
Solid lines: Pe = 0; dashed lines: Pe = 0.1.


/ ^-----_--___

0.1 06.02

0 1 2 3 4 5 6 7 8 9 10
Figure 3. Dimensionless temperature gradient at the crys-
tal-melt interface from Eq. (20) versus crystal length for
the indicated values of Bi..
Solid lines: Pe = 0; dashed lines: Pe = 0.1.
Spring 1993

averaged temperature, < (oc) >, can be written as

()(0) = 1 (16)

do (A)+ Bia()(A) = 0 (17)
d (O) + Bi [m 1] = Pe St (18)

Hence, utilization of the quasi-steady state and
fin approximations has transformed the original
problem (Eqs. 1-6) to the problem given by Eqs. (15)
through (18), which the typical junior or senior
chemical engineering student can solve.

For the sake of space, we present only the solution
to the last model, Eqs. (15) through (17), which is
given by

cosh[X(A-)]+ + 2 sinh (A-a )
(0(Y)) = exp[ (Y] P2. 2
((I)) exp[] cosh [A]+ +2 Bia sinh A]


where X= -Pe2 + Bia. The gradient at the crystal-
melt interface is given by

J0- (0)-

-Bia cosh[ A]+ Bia(Pe-4) sinh[ A]

cosh[A]+ Pe+2 Bi sinh[A]
2 X 2

which can be used in Eq. (18) to find 0.
This solution can be used to demonstrate various
aspects of the heat transfer processes in CZCG. Stu-
dents can see how the temperature distribution and
pull rate depend on the dimensionless groups that
arise in the model, which always provides physical
insight into their meaning. The temperature distri-
bution predicted by Eq. (19) is shown in Figure 2 for
typical values of Bia and Pe. One might also point
out that a crystal length A, can be found beyond
which the crystal has an effectively infinite length.
For A > A&, we can infer that tL -- so that the
QSSA will be valid for long crystals as was men-
tioned earlier. A, can be estimated by plotting the
interfacial temperature gradient 3 from Eq. (20) as
a function of crystal length A, as shown in Figure 3.
Another interesting exercise is to have students
find the range of Bia for which the fin approximation
is valid (this turns out to be Bia -0.2). This can be
done by comparing P from Eq. (20) to the radial
average of the crystal-melt temperature gradient
determined from the two-dimensional model, which

can be given in class or derived from results found
in the literature.[41
The simple model and its solution that have been
presented in this paper are most appropriate for an
undergraduate-level transport phenomena course.
When used in a lecture, it is a compact example that
demonstrates the use of two important engineering
approximations. At the graduate level, the two-di-
mensional models (both unsteady and steady) could
provide the basis for a good homework or exam prob-
lem. The validity of the QSSA can be determined by
comparing results from the transient and quasi-
steady models. A more realistic transient model could
be developed by allowing the crystal-melt interface
to move according to a mass balance on a melt of
finite volume. In this case, the crystal pull rate and
crystal velocity will not be the same.
This type of problem can also be useful to demon-
strate techniques for boundary immobilization. There
are, of course, many other ways to look at or use
this example; they are left for the reader to ponder.

The author is grateful to Daniel White, Jr., for
bringing the CZCG modeling problem to his atten-
tion during an excellent course Dr. White taught at
Penn State University in the fall of 1986.

C specific heat capacity of crystal
h convective heat transfer coefficient to ambient
h convective heat transfer coefficient to melt
AHf specific enthalpy of fusion
k thermal conductivity of crystal
L crystal length
r radial position
R crystal radius
t time
T crystal temperature distribution
Ta ambient temperature
T, melting temperature of crystal
Tm melt temperature
V crystal velocity or pull rate
z axial position
a thermal diffusivity of crystal
p density of crystal
1. Brown, R.A., "Theory of Transport Processes in Single Crys-
tal Growth from the Melt," AIChE J., 34, 881 (1988)
2. Bird, R.B., W.E. Stewart, and E.N. Lightfoot, Transport
Phenomena, John Wiley and Sons, Chap. 9 (1960)
3. Slattery, J.C., Momentum, Energy and Mass Transfer in
Continue, McGraw Hill, New York, Chap 5 (1972)
4. Carslaw, H.S., and J.C. Jaeger, Conduction of Heat in Sol-
ids, 2nd ed., Oxford University Press, Chap. 8 (1959) O

REVIEW: Mass Transfer
Continued from page 117
some problems in using Fickian theories of diffusion
when more than two species are present. Chapters 3
and 4 start formulating the M-S theory, but with
little mention of it by name. Chapter 3 discusses
driving forces for mass transfer, while Chapter 4
considers friction effects. The idea of a "bootstrap"
relationship to provide an absolute level of velocity
in the M-S equations is introduced in Chapter 4 and
is explained with examples in Chapters 5 and 6.
Chapter 5 shows several binary ideal solution ex-
amples and notes that the M-S theory gives the
same results as Fickian diffusion and is no more
difficult to use. Ternary examples are the subject of
Chapter 6. The example for distillation is a particu-
larly clear explanation of how the Murphree effi-
ciency can be infinite or negative.
Chapter 7, on combined mass and heat balances,
suffers from the need for a more complete descrip-
tion of the energy balance. Chapter 8, on
nonidealities, has an interesting example on etha-
nol water distillation, but the reasons for lack of
agreement with the exact solution are not spelled
out. This chapter also contains the first detailed
comparison between the M-S and Fickian ap-
proaches. Moving parts of this comparison to a place
much earlier in the book would help many readers.
Chapter 9 briefly presents theories for determin-
ing the M-S diffusivities. Driving forces other than
activity gradients are the topic of Chapter 10. These
include pressure gradients, centrifugal force, sup-
port forces (from solid matrices), and electrical forces.
With these additional forces, Chapter 11 can look at
the diffusion of ions in electrolytes. The end of Chap-
ter 11 is a natural break in the book since the fun-
damental ideas have all been presented.
The second part of the book briefly covers a vari-
ety of separation processes. Chapter 12 is an over-
view of various membrane separation processes,
while individual processes are covered in detail in
the following chapters: gas permeation (13), dialysis
and pervaporation (14), electrodialysis (15), and re-
verse osmosis and ultrafiltration (16). There is very
little detail on the nature of the membrane itself,
and statements such as "We shall treat the mem-
brane as homogeneous," (p. 125, referring to reverse
osmosis) could easily be misinterpreted. Supplemen-
tation of these sections with a book such as R.E.
Kesting, Synthetic Polymeric Membranes: A Struc-
tural Perspective (2nd ed., Wiley, New York, 1985) is
Continued on page 149
Chemical Engineering Education

Guide to Learning Principles
Continued from page 121
ture-it is boring and shows a tremendous lack of
respect for both the students and the material. En-
thusiasm and caring (both for the students and for
the material) are so important that they are suffi-
cient to help cover a variety of other teaching sins.
One advantage of professors teaching courses in their
research areas is that most of them are naturally
enthusiastic and caring about their subject. Of
course, someone still has to teach the beginning
courses, and it is vitally important to also show a
love of learning in them. Small classes can be a
big advantage since it is easier for professors to
show enthusiasm and caring when there are fewer
students. Small classes almost force personal inter-
action between students and faculty. It has been
shown that students who took small classes early in
their careers were much more likely to become en-
gaged in academics."5

8. Challenge the students-
ask thought-provoking questions.
Many students leave engineering because they see
it as boring!"7 To offset this, the professor should
find ways to provide some measure of challenge to
each of the students. One method is to ask ques-
tions which "stretch" the students, requiring that
they use their fundamental knowledge in new
ways to answer questions about real phenomena.
For example, ask what the temperature will be in a
car sitting on the street if the wind chill is -10C but
the ambient temperature is 200C-then ask the stu-
dent to explain what "wind chill" is. It is often a
good idea to leave a question unanswered during
class and to challenge the students to obtain an
answer within their study groups. The challenges
should be arranged so that each student can shine
once in a while.

9. Individualize the learning environment.
Since students have very different learning
styles,"['3 it is useful to employ a variety of teaching
styles throughout the course. In that way each stu-
dent will be able to use his or her favorite style at
some time during the course. The professor should
use both inductive and deductive approaches to teach
the material, although an inductive approach is usu-
ally more effective the first time through the mate-
rial. Use a variety of different exercises-when brain-
storming is one exercise and analysis is a second
exercise, you will often be able to observe that dif-
ferent students shine in the different exercises. Rich
Spring 1993

Felder's column, "Random Thoughts" in CEE, has
contained many examples of individual learning
styles and methods to individualize instruction in
chemical engineering.

10. Ifpossible,
separate teaching and evaluation.
Evaluation gets in the way of teaching since the
evaluator tends to be seen as the "enemy," particu-
larly if grading is done on the curve. The professor
who separates teaching and evaluation can then be-
come a coach who is there with the sole purpose of
helping the students learn. Someone else should do
the evaluation, or a mastery style course where ev-
ery student can succeed should be used. For ex-
ample, in a large multisection course with several
professors, one of the professors could write and be
in charge of scoring all tests and quizzes while the
other professors do the teaching. In a design course,
an industrial evaluation of the designs may well be
appropriate-it is certainly realistic.

These learning principles are certainly not inclu-
sive, but they do present a good start for improving
teaching and avoiding disastrous classes. Note that
most of the focus of these principles is on the stu-
dents. It is the students, not the professor, who have
to learn in order for the course to be a success.

This article was written while the author was en-
joying the hospitality of the Department of Chemi-
cal Engineering at the University of Florida while
on sabbatical there. The work was partially sup-
ported by NSF grant USE-8953587.

1. Wankat, P.C., and F.S. Oreovicz, Teaching Engineering,
McGraw-Hill, New York; (1993)
2. Hewitt, G.G., "Chemical Engineering in the British Isles:
The Academic Sector," Chem. Eng. Res. Des., 69 (Al), 79,
Jan. (1991)
3. Felder, R.M., and L.K. Silverman, "Learning and Teaching
Styles in Engineering Education," Eng. Ed., 78 (7), 674
4. Chickering, A.W., and Z.F. Gamson, "Seven Principles for
Good Practice in Undergraduate Education," American As-
sociation for Higher Education Bulletin, 3, March (1987)
5. Light, R.J., "The Harvard Assessment Seminars", Second
Report, Harvard University, Cambridge, MA (1992) [Free
copies of this report can be obtained by writing to: School of
Education Office, Harvard Graduate School of Education,
Larsen Hall, Cambridge, MA 02138]
6. Caplan, N., M.H. Choy, and J.K. Whitmore, "Indochinese
Refugee Families and Academic Achievement," Sci. Amer.,
266 (2), 36, Feb. (1992)
7. Hewitt, N.M., and E. Seymour, "A Long Discouraging
Climb," ASEE Prism, 1(6), 24, Feb. (1992) O

Random Thoughts...


North Carolina State University
Raleigh, NC 27695

What all great teachers appear to have in common is love of their subject, an obvious
satisfaction in arousing this love in their students, and an ability to convince them that
what they are being taught is deadly serious. (Joseph Epstein)

The only rational way of educating is to be an example-if one can't help it, a warning
example. (Albert Einstein)

Teaching is not a lost art, but the regard for it is a lost tradition. (Jacques Barzun)

A good education is not so much one which prepares a man to succeed in the world, as
one which enables him to sustain failure. (Bernard Iddings Bell)

If we desire to form individuals capable of inventive thought and of helping the society of
tomorrow to achieve progress, then it is clear that an education which is an active dis-
covery of reality is superior to one that consists merely in providing the young with
ready-made truths. (Jean Piaget)

The Romans taught their children nothing that was to be learned sitting. (Seneca)

There is nothing on earth intended for innocent people so horrible as a school. To begin
with, it is a prison. But it is in some respects more cruel than a prison. In a prison, for
instance, you are not forced to read books written by the wardens and the governor.
(George Bernard Shaw)

+ "We must remember," said a Harvard Classics professor at a meeting, "that professors
are the ones nobody wanted to dance with in high school." (Patricia Nelson Limerick)

I try not to let my schooling interfere with my education. (Mark Twain)

I learned three important things in college-to use a library, to memorize quickly and
visually, and to drop asleep at any time given a horizontal surface and fifteen minutes.
(Agnes de Mille)

It can be said unequivocally that good teaching is far more complex, difficult, and de-
manding than mediocre research, which may explain why professors try so hard to avoid
it. (Page Smith)
128 Chemical Engineering Education

+ Do not try to make the brilliant pupil a replica of yourself (Gilbert Highet)

At present the universities are as uncongenial to teaching as the Mojave Desert to a clutch
of Druid priests. If you want to restore a Druid priesthood you cannot do it by offering
prizes for Druid-of-the-year. If you want Druids, you must grow forests. (William

Examinations are formidable even to the best-prepared; for the greatest fool may ask more
than the wisest man can answer. (Charles Colton)

If you are given an open-book exam you will forget your book. If you are given a take-home
exam you will forget where you live. (Variant of Murphy's Law)

One of the great marvels of creation is the infinite capacity of the human brain to with-
stand the introduction of knowledge. (Theodore Roosevelt)

+ Universities are full of knowledge; the freshmen bring a little in and the seniors take none
away, and knowledge accumulates. (Abbott Lowell)

Ifyou want a track team to win the high jump, you find one person who can jump seven
feet, not seven people who can jump one foot. (Anonymous)

The best way to get a good idea is to get a lot of ideas. (Linus Pauling)

A first principle not formally recognized by scientific methodologists-when you run onto
something interesting, drop everything else and study it. (B. F. Skinner)

If you hear the word "Impossible!" spoken as an expletive, followed by laughter, you will
know that someone's orderly research plan is coming along nicely. (Lewis Thomas)

Four to six weeks in the lab can save you an hour in the library. (G. C. Quarderer)

Experience is not what happens to you; it is what you do with what happens to you.
(Aldous Huxley)

Believe those who are seeking the truth; doubt those who find it. (Andr6 Gide)

I arise in the morning torn between a desire to improve the world and a desire to enjoy the
world. This makes it hard to plan the day. (E. B. White)

We know that the most advanced computer in the world does not have a brain as sophisti-
cated as that of an ant. True, we could say that of many of our relatives but we only have to
put up with them at weddings or special occasions. (Woody Allen)

+ The only thing God didn't do to Job was give him a computer. (I. F. Stone)

Why, a four-year-old child could understand this. Someone get me a four-year-old child.
(Groucho Marx) C
Spring 1993





Linking Theory and Practice

University of Waterloo
Waterloo, Ontario, Canada N2L 3G1
Laboratory experiments are an integral part of
the chemical engineering curriculum because
they serve several purposes. Their primary
purpose is to reinforce key chemical engineering
concepts, but they are also supposed to teach stu-
dents about model development and how to obtain
reliable data in the presence of experimental error.
With the present-day emphasis on quality control
in chemical industries and manufacturing, these
skills are needed by every chemical engineer who
will collect and analyze data, and they are espe-
cially important for engineers involved in process
modeling and development.
Unfortunately, with the exception of one course
in statistics, we do very little to teach under-
graduates about data collection and analysis. At Wa-
terloo, an introductory course in statistics is given
in the second year, and topics include an introduc-
tion to probability distributions, properties of means
and variances, estimation, confidence intervals, sig-
nificance tests, and linear regression. These tools
provide a background in collecting and analyzing
data-but students forget most of the material be-
cause they never get a chance to apply it. As a
result, they complete their undergraduate training
without really grasping the connection between sta-
tistics and experimentation.
In the students' defense, most laboratory experi-
ments are not designed using the same principles
that we teach in class. For example, in a statistics
course we might emphasize the importance of re-
porting confidence intervals for a parameter that we
have estimated, but laboratory experiments are
rarely designed to allow students to do just that.
Copyright ChE Division ofASEE 1993

Annette Burke obtained her BASc in chemical engineering from the
University of Waterloo in 1990. She is working on the development of
improved methods for designing sequential model discrimination experi-
ments. Her research interests include a variety of issues related to process
modeling and experimental design.
Aloke Phatak obtained his BASc and MASc in chemical engineering from
the University of Waterloo. After working as a research scientist in the field
of rocket propellants for three years, he returned to UW for his doctorate.
He is currently working on applications of multivariate statistics in ChE.
Bob Hudgins holds degrees from the University of Toronto and Princeton
University. He teaches reaction engineering, staged operations and labo-
ratories that go with them. His research interests lie in periodic operation of
catalytic reactors and in the improvement of gravity clarifiers.
Park Reilly graduated from the University of Toronto in 1943 and worked
in industry until 1967 when he joined the faculty at the University of
Waterloo. He studied statistics at the University of London and received a
PhD in Statistics in 1962. His research and publications are in the area of
applied statistics.

Consequently, they are left wondering about the
practical "real-world" value of statistical techniques.
To bring statistics down from the blackboard and
onto the lab bench, therefore, we must include sta-
tistical concepts in undergraduate laboratory experi-
ments. If we do not reinforce this link between the
theory and the practice of statistics, we will be do-
ing a disservice to our students who, as practicing
engineers, will have to deal with measurement
error on a daily basis.

The purpose of this paper is to show that it is
possible to incorporate statistical ideas within exist-
ing experiments, while still respecting the need to
illustrate chemical engineering concepts. We have
made changes in two second-year physical chemis-
try experiments. These experiments are particularly
appropriate, not only because they are a part of
physical chemistry courses in many departments,
but also because here at Waterloo they are carried
out in the term following the introductory statistics
course. Thus, students begin applying statistical tools
very early in the curriculum. We hope that through
this early exposure they will come to view sound
Chemical Engineering Education

statistical analysis as a necessary part of all experi-
mentation. Our long-term objective is to incorporate
more advanced concepts, such as design of experi-
ments and response surface methodology, into all
laboratory courses-especially into the unit opera-
tions laboratories in the third and fourth years.
In the following paragraphs we outline the old
procedures, the changes we have made, and the sta-
tistical concepts that have been introduced. We be-
lieve that the new procedures are better, but we
also suggest additional modifications which could be
made to further improve the didactic value of the
experiments. Complete laboratory procedures, which
include laboratory questions and supplementary ma-
terial, can be obtained from the authors.

Determination of the Molecular Weight of
Polystyrene by Viscometry
In this experiment, students determine the viscos-
ity-average molecular weight (My) of a sample of
polystyrene by dilute solution viscometry. Some of
the concepts introduced in this experiment (the rheo-
logy of suspensions, for example) are also discussed
in the physical chemistry course and in a fluid me-
chanics course. In addition, the students learn a
little bit about polymers and polymer-solvent inter-
actions. The standard experimental procedure for
determining My is described by Smith and Stires."'
It is quite commonly used in both industrial and
research laboratories.
In dilute solution viscometry, the idea is to relate
My to the viscosity of a very dilute solution of poly-
mer and solvent. The viscosity of a polymer solution
increases with both the concentration and the mo-
lecular weight of the polymer. By measuring the
viscosity of a polymer solution at several concentra-
tions, however, and then extrapolating to zero con-
centration, the effect of molecular weight can be
isolated, thereby allowing us to estimate My.
The viscosity-average molecular weight of a poly-
mer is related to the intrinsic viscosity of a polymer
by the Mark-Houwink-Sakurada (MHS) equation[21
[ ]= KMa (1)
Here, [il] is the intrinsic viscosity, and K and a are
constants which depend upon the polymer, solvent,
and solution temperature. The intrinsic viscosity,
sometimes known as the limiting viscosity number,
is defined in terms of the Newtonian viscosity of a
polymer-solvent solution of concentration, c, as the
concentration approaches zero, e.g.,
[TI] = lim (T( / 11o 1) (2)
[J= im (2)
Spring 1993
Spring 1993

where Tr is the viscosity of the polymer-solvent
solution of concentration c, and io is the viscosity
of the solvent alone. Once we know the intrinsic
viscosity of a polymer in a given solvent and the
MHS constants K and a, we can calculate its viscos-
ity average molecular weight by solving Eq. (1) for
My. But how can we determine the intrinsic vis-
cosity in the first place?
The Newtonian viscosity of a polymer-solvent so-
lution depends on the concentration of the polymer.
For very dilute solutions this concentration depen-
dence can be described by the Huggins equation,"
which is written as
( /i -1) [l] + kH [l]2c (3)

where the constant kH is known as the Huggins
constant. Thus, to determine the intrinsic viscosity,
we first measure the viscosity of the solvent as well
as the viscosities of at least two polymer solutions of
known concentration. Then, assuming that the
Huggins equation is correct, we can use linear re-
gression to estimate [il] in Eq. (3).
In the experiment, however, 1" and rn are never
actually measured. In the viscometer used, the time
required for the polymer solution to flow through a
marked length of glass tubing is measured. It turns
out that in such a viscometer, the flow time is pro-
portional to the viscosity of the solution and in-
versely proportional to its density. However, because
the different polymer solutions used are very dilute,
their density is roughly the same, and flow time
depends on the viscosity of the solution only. Thus,
we can write i/To = t/to, where t is the flow time for a
polymer solution and to is the flow time for the pure
solvent. As a result, Eq. (3) can be written in terms
of flow times instead of viscosities, e.g.,
(t/t- 1) = []+kH[n]2c (4)

and we can carry out a linear regression as outlined
above to estimate [i].
Old Procedure: A 50-ml solution consisting of
0.5 g of polymer in solvent (toluene, for example) is
prepared and left for a day to allow the polymer to
dissolve. A 10-ml aliquot of pure solvent is then
placed in a Cannon-Ubbelohde viscometer, and the
flow time is measured three times. These measure-
ments are then averaged. All flow times are deter-
mined in this way since uncertainty in these mea-
surements is the major source of error in this ex-
periment. The solvent is removed and the viscom-
eter cleaned. Next, 10 ml of polymer solution is placed
in the viscometer and the flow time is measured.
The solution is diluted by the successive addition of

2, 5, 5, 10, and finally 20 ml of solvent. After each
addition the solution is mixed and the flow time
measured. Then, the data are plotted using the
Huggins equation, and by using linear regression,
[1i] is estimated. Figure 1 is a plot of typical data.
Once [il] has been determined, Mv can be calcu-
lated using the Mark-Houwink-Sakurada equation.
Students are supplied with appropriate values of
the constants K and a from the Polymer Handbook. 4
0.__ Why is such a
design inherently
0.48 flawed? Reilly, et
6 al.,[51 pointed out
that as more and
-0.44 more solvent is
o added to the ini-
0.42 tial polymer so-
0.40 lution, the error
in measuring flow
0.38 times increases,
o.36 as does the un-
0.0 0.2 0.4 0.6 0.8 1.0 certainty in the
c (g/mL) concentration. As
Figure 1. Huggins plot of viscosity a result, the error
data generated using the old ex- in the quantity
perimental procedure. The polymer (q/Tio 1)/c in-
system is polystyrene in toluene at creases as con-
300C. centration de-
creases, and making a large number of measure-
ments at low concentrations decreases the precision
with which we can determine [Tn]. Our objective in
modifying this experiment, therefore, was to imple-
ment an improved procedure suggested by Reilly, et
al.,-one which yields more precise estimates of [qr]
and My and which, more importantly, allows
students to construct confidence intervals for these
two quantities.
New Procedure: How many polymer-solvent so-
lutions should we run through the viscometer to
estimate the intrinsic viscosity with the greatest
precision? The answer, according to Reilly, et al.,51
is only two! The first solution has a concentration
given by c1, and the second a concentration of c1/2,
which we denote by c,2. Once the flow times of these
solutions and of the pure solvent have been mea-
sured, the problem of estimating [r] by regressing
(t/to 1)/c on c reduces to fitting a straight line
between two points. After a little bit of algebra, it is
easy to show that the intercept of this line, the
intrinsic viscosity, is given by
-t +4 tl/2 -3 t (5)
Here, to denotes the average flow time of pure sol-

vent, while t, and t/2 are the average flow times of
the solutions of concentration cl and c,2, respectively.
In the modified procedure, c, corresponds to an ini-
tial solution of 0.35 g polymer in 50-ml solvent. Two
solvents are used: toluene and 80/20 by volume mix-
ture of toluene and methanol. Thus, in all, students
measure the average flow times of six solutions,
where each average has been calculated from three
measurements. This allows them to estimate the
variance of the flow measurements as

S2- = 1 n[ k-i)] (6)-
n(k -1) i=1 (6)

where ti is the jth replicate measurement of the ith
solution, ii is the average of k(= 3) replicate mea-
surements of solution i, and n(= 6) is the total num-
ber of solutions. Our practice is to combine the
data from two groups of students to get a more
reliable estimate of the variance based on n = 12
Having determined s2, it is relatively straightfor-
ward to estimate the variance of [T1] by applying
standard formulas for the variance of the quotient
of random variables to Eq. (5). If we do so, it turns
out that
var []- tll 2 (7)

where V is a factor that depends on the number of
replicates of flow time measurements, el, and the
estimate of [rl] obtained by using Eq. (5). Then, con-
fidence intervals for [in] and Mv can be constructed
in the usual way.

Discussion; The new procedure is better in two
important respects: 1) the estimate of [i], and hence
of Mv, is more reliable, and 2) the students can now
construct confidence intervals based on an estimate
of the variance that is independent of the regression
that is carried out to estimate [rn]. In addition, we
also briefly discuss the old procedure so that the
students can understand why its design is flawed
and why the new procedure yields a more precise
estimate of intrinsic viscosity.
How could we further improve the didactic value
of the experiment? One way would be to explain to
the students why a design in which only two concen-
trations are used is optimal. In addition, we could
also make them derive Eq. (7), including the exact
value of the factor V. However, we have to strike the
right balance between illustrating statistical con-
cepts and illustrating physical principles. Although
we are convinced of the value of introducing statisti-
cal ideas into these experiments, we do not want to
Chemical Engineering Education

do so at the expense of the chemical engineering
concepts being illustrated. Thus, we leave it up to
the instructor to decide whether or not to incorpo-
rate the additions mentioned above.

Adsorption of Acetic Acid on Charcoal
This experiment illustrates the discussion of
adsorption from solution that is presented in the
second-year physical chemistry course. Like dis-
tillation, adsorption can also be thought of as a
chemical engineering unit operation.'61 Two expres-
sions-the Freundlich isotherm and the Langmuir
isotherm-are used to describe the adsorption of
acetic acid onto activated charcoal. Each isotherm
is based on different assumptions about the nature
of adsorption, and they apply under different condi-
tions. The Freundlich isotherm"7' is a purely em-
pirical model which works well at low solute con-
centration. It relates the mass of solute adsorbed (x)
on the adsorbent to the equilibrium concentration of
solute (C), e.g.,
_x = kC" (8)
where m is the mass of adsorbent, and k and n are
empirical constants. The Langmuir isotherm,'71 how-
ever, was derived assuming an explicit adsorption
mechanism. It may be written as
x (x / m)o KC
m 1+KC (9)
where K is the equilibrium constant, and (x/m)O is
the mass ratio required for monolayer coverage of
the surface of the adsorbent. Equations (8) and (9)
are nonlinear, and they are usually used in their
linearized form, e.g.,
n- = in k + n nC (10)
for the Freundlich isotherm, and
1 1 1 1 (11)
x/m K(x/m)0 C (x/m)0
for the Langmuir isotherm.
After measuring x/m for several different concen-
trations of acetic acid, the students are asked to
comment on the fit of Eqs. (10) and (11) to the data.
The procedure used in the past is based on the ex-
periment described by Ellis and Mills;1"' it is not
well designed to allow the students to quantitatively
assess which of the two isotherms better describes
adsorption of acetic acid onto charcoal. Furthermore,
note that the linearized forms of the equations are
used. Also, as we will see from the procedure below,
x and C are not statistically independent.
Old Procedure: Two grams of activated charcoal
Spring 1993

are placed in each of six flasks. Starting with 0.5 M
acetic acid, six 100-ml lots of acetic acid with con-
centrations ranging from 0.5 to 0.025 M are pre-
pared. The acetic acid solutions are added to the
charcoal, mixed, and left to stand overnight to
reach equilibrium. The solutions are then suction
filtered. Filtrate samples are titrated with 0.2 M
NaOH to determine the equilibrium concentrations.
Finally, the amount of acetic acid adsorbed onto the
charcoal is calculated.
Students then plot the data using Eqs. (10) and
(11). The correlation coefficient for the Freundlich
isotherm is typically 0.99, and for the Langmuir
isotherm it is typically 0.70. Figures 2(a) and 2(b)
show plots of representative data. Students notice
the curvature in the plot of the Langmuir isotherm
and then conclude that it is not due to chance alone,
but to systematic departure from the fitted model.
3.0 they rarely real-
2.8 ize that the ob-
2.6 served curvature
2.4 provides informa-
22 tion which is dif-
\2. ferent from that
12. provided by a low
1.8 correlation coeffi-
1.6 cient. As a result,
1.4 they often pro-
1.2 ceed in later
1.0 .5 years to rely
0.0 0.5 1.0 1.5 2.0 2.5 C30 3.5 4.0 4.5 5.0 heavily on the
1n (C)
Figure 2(a). Acetic acid adsorption correlation coeffi-
data generated using the old pro- cient as a mea-
cedure and plotted according to sure of model-fit
Freundlich isotherm. and sometimes
even neglect to
20 plot data. Our
s1 purpose in modi-
16 fying this experi-
14 i ment, therefore,
is to emphasize
12 the limitations of
10 the correlation co-
8 o efficient and to
6D give the students
4 experience in us-
2 ing other mea-
0 sures of model fit.
0 20 40 60 80 100 120 140 160 New Proce-
1/cdure: The proce-
Figure 2(b). Acetic acid adsorption dure: The proce-
data generated using the old proce- dure is un-
dure and plotted according to changed except
Langmuir isotherm, for the number of

solutions used. Instead of preparing six solutions of
different concentration, three independent replicates
of four different concentrations are prepared. The
concentrations used are between 0.5 and 0.025 M.
For each replicate, charcoal is weighed out and ace-
tic acid solution is prepared separately to ensure
independence. The twelve samples are left overnight
to reach equilibrium and are then suction filtered;
the filtrate from each is again titrated using 0.2 M
NaOH. It is tempting here to titrate a set of three
replicates sequentially, but this would invalidate
the estimate of the error variance. Filtrate samples
must be titrated in random order so that the corre-
lation between any two measurements is constant,
and the data may be treated as independent. Fi-
nally, students perform least-squares regression to
fit Eqs. (10) and (11) to the data, calculate the corre-
lation coefficients, plot the residuals, and perform
the lack-of-fit test described below. Figures 3(a) and
3(b) show typical results using Eqs. (10) and (11).
The lack-of-fit test is an extension of analysis of
variance in linear regression, which students learn
in their introductory statistics course. It is described
in standard texts such as Draper and Smith."19 If a
model is a good representation of the data, the re-
siduals, or prediction error, should reflect only ran-
dom error. If a model is a poor representation of the
data there is additional variation caused by lack-of-
fit, which manifests itself as a systematic departure
from the fitted line. This is evident when the data
from this experiment are plotted using the Langmuir
isotherm, but in the original experiment there is no
way to estimate random error independently of the
model or to confirm lack-of-fit quantitatively.
The introduction of replication allows us to esti-
mate the random error, or pure error, independently


0.0 0.5 1.0 1.5 2.0 2.5 3.0 3.5 4.0 4.5 5.0
Figure 3(a). Acetic acid adsorption
data generated using the new proce-
dure and plotted according to
Freundlich isotherm.


m 10



of the model that is postulated. The model predic-
tion errors can then be divided into random error
and lack-of-fit error. Comparison of the lack-of-fit
sum of squares to the pure error sum of squares
using an F-test serves as a quantitative measure of
the fit of the model.
The calculations for the lack-of-fit test are straight-
forward. Let yij, i = 1,2,...,k, j = 1,2,...,n, be the jth
measurement on the 'dependent' variable at the ith
concentration. In the modified procedure outlined
above, three replicates (ni = 3, Vi) are performed at
each of four (k = 4) different concentrations. Recall
that because we are using the linearized forms of
the Freundlich and Langmuir isotherms, y = ln(x/m)
for Eq. (10) and y = 1/(x/m) for Eq. (11).
The random, or pure, error can be estimated by

k ni 2
I I (Yij-Y)
2 i=l j=l

n= Yni

where yi is the average of the ni measurements at
the ith concentration. In much the same way, the
lack-of-fit sum of squares (LFSS) is estimated as

LFSS= ni(yi-i)2 (13)
where yi is the value predicted by the model (Eqs.
10 or 11) at the ith concentration. Once we have
fitted the Freundlich and Langmuir isotherms and
calculated the corresponding values of s2 and LFSS,
we can construct an F-statistic and compare it to
the critical F-value at the desired confidence level,
SLFSS/(k- 2)
2 Fk-_2,n-k (14)

If the calculated value is larger than the critical F
value, then lack-of-fit error is signifi-
cant at the chosen confidence level, and
o the model does not describe the data
adequately. Using such an F-test, stu-
dents find significant lack-of-fit for the
Langmuir isotherm, which provides
quantitative reinforcement of the con-
clusions they draw by simply observ-
ing the curvature in Figure 3(b).


The changes we have made to the
experimental procedure are minor; the
benefits reaped by the students, how-
ever, will be substantial. First, the stu-
dents will be introduced to replication,
which is essential in estimating ran-
Chemical Engineering Education


0 20 40 60 80 100 120 140 160
Figure 3(b). Acetic acid adsorption
data generated using the new proce-
dure and plotted according to
Langm uir isotherm.

dom error and in identifying problems such as non-
constant variance. More important, we hope that
students will realize just what experimental error
really is when they carry out repeat measurements
which do not yield the same results. Second, the
introduction of the lack-of-fit test and the analysis
of residual plots encourages students to use tools
other than the correlation coefficient in discriminat-
ing between competing models. Finally, students are
forced to review least-squares regression and analy-
sis of variance in order to understand the lack-of-fit
test and to interpret the residual plots.
We recognize, however, that some flaws which are
present in the old procedure remain in the modified
one. First, linearized forms of the original expres-
sions are still fitted, which may change the error
structure. Second, the amount of solute adsorbed is
determined from the change in solution concentra-
tion, which causes the variables x and C to be statis-
tically dependent. Finally, we can see in Figure
3(a) that the variance of the data, plotted according
to the Freundlich isotherm, increases as the con-
centration, C, increases. This violates one of the
assumptions of least-squares regression-that of
constant variance.
How can we remedy these deficiencies? One way
of doing so is to express the models in terms of the
actual quantities measured: the initial acetic acid
concentration, the equilibrium concentration, and
the mass of charcoal. Since the resulting model
will be nonlinear with error in all the variables, an
analysis using the error-in-variables method""l would
be most appropriate. It would be unrealistic, how-
ever, to expect second-year students to carry out
such an analysis. Here, we face a question that we
will no doubt encounter when trying to incorporate
statistical concepts into other experiments: how can
we adopt the best, "statistically correct" analysis of
a poorly designed experiment without burdening
our second-year students with statistical methodol-
ogy that would tax even a competent researcher?
Our solution here is a compromise-we have incor-
porated changes that we think are better, but we
also recognize the remaining deficiencies and en-
courage the students to think about and discuss
other ways of analyzing the data and why they
might be more appropriate. In this way we hope
that they will be able to recognize how the design of
an experiment can affect the statistical analysis of
data derived from it.

Our objective in modifying these two experiments
was to introduce statistical concepts into the under-
Spring 1993

graduate laboratory. The changes to the procedures
themselves are minor, but by modifying the analy-
sis of the data it is possible to include a wealth of
ideas which reinforce the connection between statis-
tics and experimentation. By introducing replica-
tion, we force the students to confront experimental
error-they see that measurement uncertainty is an
unavoidable fact of life. By showing them the means
to quantify this error, we show them a rational ba-
sis for dealing with it.
In the long term, our objective is not only to make
notions like replication and interval estimates an
essential element of all undergraduate laboratories
but also to include advanced concepts such as facto-
rial designs, especially in upper-year unit opera-
tions laboratories. As we saw with the analysis of
Experiment 2, however, it is sometimes difficult to
incorporate statistical concepts into existing proce-
dures that are poorly designed to begin with. Short
of redesigning all undergraduate experiments or in-
troducing students to advanced statistical techniques
which they may not be able to appreciate, our solu-
tion has been to incorporate statistical techniques,
but at the same time point out deficiencies where
they exist and encourage the students to discuss
alternative methods of data analysis. However we
choose to do it, it is clear that we must incorporate
statistical concepts into the undergraduate labora-
tory. By doing so as early as possible in the chemical
engineering curriculum, we hope to remove the mis-
taken notion of statistics as something complex and
mysterious when it is really fundamental to the
engineer's craft.
1. Smith, J.L., and A. Stires (Eds.), Experimental Physical
Chemistry, 7th Ed., McGraw-Hill, New York (1970)
2. Rudin, A., The Elements of Polymer Science and Engineer-
ing, Academic Press, New York (1982)
3. Huggins, M.L., "The Viscosity of Dilute Solutions of Long-
Chain Molecules. IV. Dependence on Concentration," J. Am.
Chem. Soc., 64, 2716 (1942)
4. Brandrup, J., and E.H. Immergut (Eds.), Polymer Hand-
book, Wiley-Interscience, New York (1975)
5. Reilly, P.M., B.M.E. Van der Hoff, and M. Ziogas, "Statisti-
cal Study of the Application of the Huggins Equation to
Measure Intrinsic Viscosity," J. Appl. Polym. Sci., 24, 2087
6. Mantell, C.L., Adsorption, 2nd Ed., McGraw-Hill, New York
7. Castellan, G.W., Physical Chemistry, 3rd Ed., Addison-
Wesley, Reading, MA (1983)
8. Ellis, R.B., and A.P. Mills, Laboratory Manual in Physical
Chemistry, McGraw-Hill, New York (1953)
9. Draper, N.R., and H. Smith, Applied Regression Analysis,
2nd Ed., John Wiley & Sons, New York (1981)
10. Reilly, P.M., and H. Patino-Leal, "A Bayesian Study of the
Errors-in-Variables Model," Technometrics, 23, 221 (1981)

1 laboratory



2. The Eastman Chemical Reactive Distillation Process*

Purdue University
West Lafayette, IN 47907

As described in previous papers,"1'21 a series of
computer modules for use in the chemical en-
gineering senior laboratory is being devel-
oped at Purdue University. The modules are meant
to supplement, not to replace, traditional laboratory
experiments. In our laboratory, for example, only
one of three month-long experiments may be the
use of a computer module. Computer simulated ex-
periments have a number of advantages over tradi-
tional experiments:
Processes that are too large, complex, or hazardous for the
university laboratory can be simulated with ease on the
Realistic time and budget constraints can be built into the
simulation, giving the students a taste of "real world" engi-
neering problems.
The emphasis of the laboratory can be shifted from the
details of operating a particular piece of laboratory equip-
ment to more general considerations of proper experimental
design and data analysis.
Computer simulation is relatively inexpensive compared to
the cost of building and maintaining complex experimental
Simulated experiments take up no laboratory space and are
able to serve large classes because the same computer can
run many different simulations.

The Eastman Chemical Reactive Distillation Pro-
cess is part of a series of steps for obtaining acetic
anhydride from coal. Acetic anhydride is an impor-
tant chemical intermediate used in the production
of cellulose acetate, which itself is used in the manu-

* The first paper in this series, "Purdue-Industry Computer
Simulation Modules: The Amoco Resid Hydrotreater Process,"

appeared in CEE, 25, 98 (1991).

S. Jayakumar is a post-doctoral research fellow in the School of Chemi-
cal Engineering at Purdue University. He received a B.Tech from Indian
Institute of Technology (1985), and his MS and PhD from Purdue Univer-
sity (1988, 1992). His research interests include process design, simula-
tion, optimization, and plant layout
R. G. Squires is a professor of chemical engineering at Purdue Univer-
sity. He received his BS from Rensselaer Polytechnic Institute (1957),
and his MS and PhD from the University of Michigan (1958, 1963). His
current research interests center on the educational applications of com-
puter simulation.
G. V. Reklaitis is Head of the School of Chemical Engineering at Purdue
University. He earned his BS from Illinois Institute of Technology (1965),
and his MS and PhD from Stanford University (1969). His research
interests include process systems engineering, process scheduling meth-
odology, and the design and analysis of batch processes.
P. K. Andersen is an assistant professor in the Department of Freshman
Engineering at Purdue University. He earned his BS from Brigham Young
University (1981) and his PhD from UC Berkeley (1987), both in chemical
engineering. His research has dealt with transport in multiphase flows and
the educational applications of computer simulation.
L. R. Partin is a Research Associate with Eastman Chemical Company in
Kingsport, Tennessee. He received his BS in chemical engineering from
the University of Kentucky in 1976 and his MS in chemical engineering
from Purdue University in 1977.

facture of photographic film base, fibers, plastics,
and other products.
Each day the plant converts 900 tons of coal to
acetic anhydride. Using conventional methodology,
the chemicals produced would require the annual
equivalent of one million barrels of oil. A brief de-
scription of the process is: synthesis gas produced
from coal is used for methanol production; methanol
(MeOH) is reacted with recycled acetic acid (HOAc)
to produce methyl acetate (MeOAc) and then acetic
anhydride; and finally, acetic anhydride is reacted
with cellulose to form cellulose acetate. The process
for formation of MeOAc was developed at the
Eastman Chemical Company3' and patented. The
reader is also referred to a paper by Agreda, Pond,
and Zoeller[41 for more details.
The formation of MeOAc is the focus of this project.
In this process, methanol reacts with recycled acetic

Copyright ChE Division ofASEE 1993

Chemical Engineering Education

This part of the project involves analyzing trays of the column. Students do not run a
tray to "obtain data," unlike the batch reactor or the equilibrium cell. Instead, they input the kinetic and
phase equilibrium parameters to observe tray performance and use the results of the simulation to
calculate the tray efficiency. In addition, for the same feed (vapor and liquid), they
are asked to predict the conversion that will be obtained in a CSTR.

acid in the presence of sulfuric acid catalyst to form
methyl acetate and water:
CH3COOH + CH3OH => CH3COOCH3 + H20 (1)
The reaction rate is given by

rMeOAc = koA exp(-E / RT){ CHOAcCMeOH

ko = preexponential factor

CMeOAcCH20 ()
K1 (2)

A = catalyst acidity function (0 < A < 1)
T = temperature
E = activation energy
and the equilibrium constant is

K = MeOAcCH20 (3)

Clearly, the maximum conversion of the reactants is
equilibrium limited.
This Reactive Distillation Process relies on the
fact that MeOAc is more volatile than water or the
reactants. Thus, if the reaction mixture is distilled
simultaneously, a significant amount of MeOAc prod-
uct will vaporize, forcing the reaction to shift to the
right and thereby allowing a much higher conver-
sion than would be possible in the absence of dis-
tillation (conventional process). An additional con-
sequence and advantage is that high purity MeOAc
(> 99.5 wt%) is produced as the overhead product.

The program is written entirely in the C language
and uses some IMSL routines for solving systems of
linear, non-linear, and ordinary differential equa-
tions. It can simulate
A batch reactor
An equilibrium cell
A tray in the distillation column
The program can run on any machine that sup-
ports the X Window System. At Purdue, it runs on
Sun Sparc workstations with 12 MB of memory.
Each module uses less than 10 MB of disk space.
An important feature of the program is its menu-
Spring 1993

BATCH REACTOR Cleanup $700
Sample analysis $50 / sample
All expenses are to be multiplied by 1.5 and 2 for Saturday and Sunday runs

driven graphical user interface. This enables any-
one to use it, regardless of his or her knowledge of
computers. An on-line help facility is provided to
further assist the user in navigating through the
program. The user can exit the program at any time.

Budget and Experiments
In the first part of the Eastman project, the stu-
dents are required to determine the following:
1. Activation energy
2. Preexponential factor
3. Catalyst activity function
4. Wilson parameters for liquid activity coefficients
To determine these quantities, the students simu-
late a batch reactor and an equilibrium pressure
cell, both described below. Also, a requirement that
the students work within a budget of $30,000 con-
tributes a sense of realism to the module. Table 1
shows costs associated with operating the reactor
and cell.
Laboratory Batch Reactor A batch reactor is
available for the students to study reaction kinetics
for the determination of items 1-3 above: activation
energy, preexponential factor, and the catalyst ac-
tivity function. As seen in Eq. (2), the reaction is
first order in the concentrations of each of the reac-
tants and products. Preliminary data obtained us-
ing laboratory batch apparatus suggest that in the
range of conditions used in the Eastman column,
the equilibrium constant is independent of tempera-
ture and catalyst concentration. The catalyst acidity

function, A, depends on the sulfuric acid concentra-
tion, as shown in Figure 1.
The students are to design experiments to deter-
mine the activation energy E, the preexponential
factor ko, and establish the functional dependence of
A on the catalyst concentration. The equilibrium
constant, Ke is known.
Equilibrium Cell The purpose of this appara-
tus is to determine liquid phase activity coefficients
in two-component systems. The cell is simply a closed
vessel with a pressure gauge and a temperature
sensor. Temperature control is provided, so isother-
mal runs can be made. The user can charge it with
two components and record the equilibrium pres-
sure. For all practical purposes, equilibrium can be
assumed to have been established in four hours.
There is no facility to measure or monitor the vapor
or liquid phase mole fractions. However, since the
vapor volume is very small compared to the volume
of the liquid, it can be assumed that the concentra-
tions of the components in the liquid phase at equi-
librium are equal to the corresponding concentra-
tions of the liquid charged into the cell. Note that
there are five components of interest: the reactants,
the products, and the catalyst, H2SO4.
When the critical region is not approached (as in
this case), we can assume that the liquid phase
activity coefficients and standard fugacities are in-
dependent of pressure. If the standard fugacity is
taken in the sense of the Lewis-Randall rule, we
have for each component i

y where for component i,
y = liquid phase activity coefficient
PO = vapor pressure
P = equilibrium pressure of the system
x = liquid phase mole fraction
yi = vapor phase mole fraction
0i, 0o = fugacity coefficients ofi, the latter calculated
at the saturation vapor pressure.
Quite often, however, 4, and 4o are nearly equal,
and Eq. (4) simplifies to
Yi P = xi y, P (5)
Since the total pressure equals the sum of the
partial pressures of the components, adding Eq. (5)
for each component results in
P = x,1P +X22 P (6)

The vapor pressures, Pi, of the pure components

Catalyst Acidity Function

Figure 1. The catalyst activity function
Figure 1. The catalyst activity function

may be calculated from the Antoine equation:

nP=A iB (7)
fn po = Ai (7)
i T+C(
where the Antoine constants, A,, Bi, C, are known
from the literature, and T is the temperature.
At low to moderate pressures, the Wilson equa-
tion[5'6' may be used to predict the activity coeffi-
cients as a function of temperature and composition
for a variety of liquid solutions comprising diverse
chemical species. The Wilson equation for compo-
nent i is

nT = 1-'n (xGij)- G(8)

The summations are over all components present
in the mixture. For an ij pair, there are two tem-
perature-dependent parameters, Gij and Gji. For
i = j, Gj = 1. Over the narrow temperature range of
interest, the temperature dependence of the param-
eters is weak enough to be ignored. The Wilson pa-
rameters for all of the component pairs except
MeOH MeOAc have been previously estimated and
are known. The assignment given to students is to
design a series of runs in the equilibrium cell to
determine the two Wilson parameters for the
MeOH MeOAc pair. It should be noted that acetic
acid was not chosen as a component in the binary
pair since it exhibits vapor phase association and
the simplification for Eq. (5) does not apply.

Application Problem-Tray Simulation
This part of the project involves analyzing trays of
Chemical Engineering Education

[~7~17~17 ~~7-rT1-TT~-~1Tr7

i i I i ~


the column. Students do not run a tray to "obtain
data," unlike the batch reactor or the equilibrium
cell. Instead, they input the kinetic and phase equi-
librium parameters to observe tray performance and
use the results of the simulation to calculate the
tray efficiency. In addition, for the same feed (vapor
and liquid), they are asked to predict the conversion
that will be obtained in a CSTR. It is to be observed
that the tray is actually a CSTR with an additional
operation, e.g., distillation. It is instructive to see
that the conversion obtained in a tray is much higher
(for the current process) than in a simple CSTR.
This is an important observation since the basic
idea in carrying out the methyl acetate process in a
distillation column was to increase the conversion.
Two representative trays, one in the upper portion
of the column and the other in the lower portion, are
simulated. The students must run both.
For the purpose of the current simulation, the
tray is assumed to operate isothermally at a known
temperature. Thus, material balance and equilib-
rium relationships are used to model the tray. A
tray efficiency is incorporated in the model. Given
the input flow rates of the vapor (from the bottom
tray), the liquid (from the upper tray), the composi-
tion of the input streams, the pressure, tempera-
ture, tray efficiency, and the kinetic and phase equi-
librium parameters, the model predicts the flow rates
and compositions of the output liquid and vapor
streams of the tray.
The tray model does not explicitly account for the
dependence of the acidity function on the acid con-
centration. For the steady state, it suffices to pro-
vide the simulator with the acidity value corre-
sponding to the steady-state acid concentration in
the liquid phase (output). This value, however, is
not known since the output acid concentration in
the liquid phase is not known. The student must
therefore resort to an iterative procedure. He must
guess a value of the acid composition in the output
liquid stream, calculate the value of A from his
catalyst acidity function, and input this value to
the simulator. The model solves the problem using
this value of A and predicts the output liquid-phase
acid concentration. This procedure must be repeated
until the entered acidity function value agrees
with the predicted acid concentration according to
the student's function.
In the event that students are unable to evaluate
the kinetic and phase equilibrium parameters within
the assigned time, or that they come up with im-
practical values or cannot establish the acidity func-
tion curve, they have the option of using the
Spring 1993

instructor's data (with the permission of the instruc-
tor, of course). In this case, the students do not have
to input the kinetic and phase equilibrium param-
eters; furthermore, the acidity function dependence
is implicitly taken into account in the model, so that
iteration is not required.

Our experience with the Eastman module has been
very positive. The module presents a challenging
problem that helps prepare students for the kinds of
problems they are likely to encounter in industry.
The simulated budget is especially effective in mak-
ing the project more true-to-life than conventional
lab experiments. Although the Eastman problem is
challenging, students report that the software itself
is very user-friendly.
One advantage of computer simulations is their
flexibility. The Eastman module was originally de-
veloped for the chemical engineering laboratory; how-
ever, it would also be useful in courses in thermody-
namics, chemical kinetics, and separations.

The Purdue-Industry ChE Simulation Modules are
being made available for educational use by the
CACHE Corporation. Anyone interested in obtain-
ing more information should contact Professor

This work has been supported by the National
Science Foundation (Grant No. USE-888554614), the
Eastman Chemical Company, and the CACHE Cor-

1. Squires, R.G., G.V. Reklaitis, N.C. Yeh, J.F. Mosby, I.A.
Karimi, and P.K. Andersen, "Purdue-Industry Computer
Simulation Modules: The Amoco Resid Hydrotreater Pro-
cess," Chem. Eng. Ed., 25(2), 98 (1991)
2. Squires, R.G., P.K. Andersen, G.V. Reklaitis, S.
Jayakumar, and D.S. Carmichael, "Multi-Media Based
Educational Applications of Computer Simulations of
Chemical Engineering Processes," Comp. Appns. Engr.
Ed., 1(1), 25 (1992)
3. Agreda, V.H., and L.R. Partin, U.S. Patent 4,435,595,
March 1984 (Assigned to Eastman Kodak Co.)
4. Agreda, V.H., D.M. Pond, and J.R. Zoeller, "From Coal to
Acetic Anhydride," Chemtech, 172 (1992)
5. Sander, S.I., Chemical and Engineering Thermodynam-
ics, John Wiley & Sons, New York, NY (1989)
6. Wilson, G.M., "Vapor Liquid Equilibrium. XI: A New
Expression for the Excess Free Energy of Mixing," J.
Am. Chem. Soc., 86, 127 (1964) 0




University ofAlberta
Edmonton, Alberta, Canada T6G 2G6

he first laboratory course in chemical engi-

neering at the University of Alberta is taught
to about seventy students in the first term of
their junior year, and its primary objective is to
improve and develop the students' writing skills.
They are required to write three reports. The first
two reports are each about ten pages in length and
deal with technical material which is familiar to the
student. Each of the first two experiments is per-
formed and the reports written within a period of
two weeks, and the corrected reports are returned
to the students within another two weeks. The En-
glish construction and the presentation of the re-
ports are brutally criticized by the academic staff
responsible for the course. The students then do a
standard engineering experiment and write their
third (hopefully readable) report.
The key to this approach is to present simple and
short experiments. Typically, the first two experi-
ments should each take less than ten minutes to
complete. Quick experiments have the advantage of
conveying to the students that even though the tech-
nical aspects are easy, describing them in clear, un-
derstandable English is often very difficult.
For the last six years the first experiment we have
used has been a computer simulation of a simple
concept that the students should be able to under-
stand. Usually, the simulated experiment is based
on a fundamental principle that was taught to the
students in the previous semester, i.e., the vapor
pressure of water as a function of temperature. The
students run the program, specifying the tempera-
ture, with the simulator returning a slightly inaccu-
rate value of the vapor pressure. The students are
asked to compare the simulated vapor pressures
with those in the steam tables and those calculated
Copyright ChE Division ofASEE 1993

from a published correlation of vapor pressure and
temperature. They then write a report about the
simulated experiment.
Experience has taught us that both of the short
experiments should not be simulations since, if they
are, the students will write a simulated second re-
port. Their attitude seems to be, "If you don't take
the experiment seriously, why should we take the
report seriously?" Clearly, this is not the objective of
the course. So, we make the second experiment a
real experiment-but still quick.
One experiment that is simple and fast is the
filling and blowdown of a tank of air. In spite of
its simplicity, this experiment is surprisingly rich in
its technical content, involving ideal gas ther-
modynamics, unsteady state material balances, and
simple fluid mechanics. All of these subjects have
been covered in the preceding term or are being
taken concurrently with this course. A further
advantage is that the complexity of the data analy-
sis can be adjusted to accommodate the technical
skills of the students.

The experiment consists of two stages: 1) filling
the tank with air at about 90 psig and 70'F, and 2)
emptying the tank by venting the compressed air to
the atmosphere. The equipment (shown in Figure 1)

J.T. Ryan is a professor and registered engineer. He has taught thermo-
dynamics, fluid mechanics, and process design for more than twenty-five
R. K. Wood is professor of chemical engineering at the University of
Alberta. His recent teaching responsibilities, in addition to the introduc-
tory laboratory course, have involved process analysis, optimization,
dynamic modeling, and simulation. His research is concerned with digital
simulation of the dynamic and control behavior of process systems and
the computer control of distillation columns.
P. J. Crickmore is an associate professor of chemical engineering at the
University of Alberta. He received his BSc (Chemical Engineering), MSc
(Mathematics), and PhD (Chemical Engineering) degrees from Queen's
University, Kingston. Research areas include oil sands, coal and envi-
ronmental sampling, and remediation.

Chemical Engineering Education


consists of a modified 30-pound propane bottle and
a manifold mounted at the top of the bottle. The
manifold has four nozzles, each of which is isolated
with a quick-acting ball valve. Standard 1/2-inch
copper tube and fittings are used for the manifold.
The nozzles are brass plugs which have holes drilled
to diameters ranging from 1/16 to 7/64 inch. These
plugs are soldered into the outlet of 1/2-inch unions
which are located above each of the ball valves.
While it is not required because of the limited air
supply pressure, a relief valve is installed on the
tank for educational purposes. A cheaper alterna-
tive would be to incorporate the relief valve in the
manifold. High-pressure air is supplied from the
building service air through a flexible hose. Another
quick-acting ball valve is installed on the tank at
the hose connection.
The instrumentation consists of a fast response
thermocouple, a pressure transducer, and associ-
ated signal conditioning equipment. The thermo-
couple is installed through the tank wall. Since one
of the objectives of the experiment is to finish the
experiment quickly, the data-logging was done with
a microcomputer using an OPTO 22 interface.
The experimental procedure is straightforward:
With the pressure in the tank at atmospheric, close
valves (VI V4). Start logging the pressure and tem-
perature using a one-second sampling time. Open V5.

Figure 1.
Spring 1993

The tank will reach the supply pressure in about ten
seconds. Close V5 and open one or more valves in the
manifold. The pressure in the tank will reduce to
near atmospheric in about one minute, depending on
which valves) is opened. At this time the data-log-
ging can be stopped or the experiment repeated.
The cost of the equipment is small (less than $100,
not counting the instrumentation and microcom-
puter), and the total shop time required for machin-
ing and welding is less than two hours. Our advice
is to purchase a new propane bottle and have the
propane relief valve removed upon purchase since
this valve is extremely difficult to remove. An entire
class can perform the experiment in about two days
by using scheduled 15-minute time slots, so the in-
strumentation and data-logging computer can usu-
ally be borrowed.

This part of the experiment focuses primarily on
thermodynamics. As it is usually presented in ther-
modynamics texts, the theory for filling a tank with
an ideal gas is correct-but the major assumption is
wrong. The standard assumption is that the process
is adiabatic. When the experiment is actually per-
formed, however, the dominant influence is the heat
transfer, not the thermodynamics. But, the students
believe the textbooks and their professors-who also
believe the textbooks!
Van Wylen and Sonntag"1 give the following equa-
tion as the appropriate form of the first law of ther-
modynamics over a control volume for a uniform
state, uniform flow process:

Qcv + mi{hi +( )v2+gZi

=-me{he +( )v2 +gZe+Wcv +m2{h2 +() +gZ2

S -m 2 h 1+( +)gZ1 (1)
When we apply this equation to the filling of a tank,
neglecting the heat transfer, work, potential energy
changes, and all kinetic energy terms except for the
input, we get
mi hi+ ()v2Jm2u2-miul (2)

When it is combined with the definitions ofh and u
for an ideal gas, this equation can be used to solve
all the cases appropriate to this experiment. The
simplest case occurs if the incoming kinetic energy
and the initial mass of the system is neglected. Then
hi = u2 (3)
If we assume constant heat capacities, then
T2 kTi (4)

where k Cp / Cv.
A better approximation of the final temperature
can be made, a priori, by correcting for the initial
mass of air in the tank. A further refinement can be
made, after the experiment, by including an esti-
mate of the incoming kinetic energy. Supposedly,
the theory can be expanded or refined-however,
our objective is to introduce the students to a simple
laboratory experiment and to the difficulty of writ-
ing a technical report on such a simple experiment.
Table 1 shows a comparison of the measured final
temperature and those calculated from Eq. (2),
corrected for both the initial mass and kinetic en-
ergy of the incoming air but still assuming no heat
transfer. The point to be made from this table is
that all of the calculations are simply wrong and
differ from the measurement by a minimum of ap-
proximately 200F.
As soon as the students see the difference between
the theory taught in thermodynamics lectures and
the results of the experiment, they question the ex-
periment. There is sufficient time to do multiple
runs, but they find that the repeated experiments
produce virtually the same results. The students
are faced with an experiment which has precision
but, in their minds, is of dubious accuracy. To ex-
plain the difference between the experiment and
the temperature calculated from Eq. (4), many stu-
dents do all the corrections indicated in Table 1.
They work hard to prove that the theory is right and
the experiment is wrong. Ultimately, they realize
Eqs. (2) and (4) are simply not true and are based on
a bad assumption, and finally they conclude that
the process is not adiabatic and that the heat trans-
fer is the dominant effect. A typical value of the
heat transfer is 14 18 Btu/cycle.

At this point the students are convinced that ex-
periments are worthwhile, but they are somewhat
skeptical of theory. Venting of the tank is designed
to resolve this conflict for them. The venting of air
illustrates the polytropic decompression of an ideal
gas and an unsteady state material balance.
The relationship between the temperature and the
pressure of air in the tank, as it empties, must be
established before the material balance is attempted.
Analysis of the data is relatively simple and is cov-
ered in most introductory courses in engineering
thermodynamics. The decompression of the air in
the tank is taken to be a polytropic process. The
appropriate equation in terms of the measured vari-
ables is

Comparison of Experimental and Calculated
Adiabatic, Final Air Temperatures
Method of Approximate Final
Determination Air Temperature
Experiment 100F
Equation 4 315F
Equation 2 (corrected for initial mass) 305F
Equation 2 355F

T (n-1)/n (5)
T0- P0
If the decompression is isothermal, n is equal to
one. The process is isentropic when n = k. The value
of n is found from the slope of the line through the
measured pressure and temperature data when plot-
ted on log-log paper. Many students use a regres-
sion program to estimate n from their data; how-
ever, we require the T versus P plot for educational
reasons. Using the equipment and procedure de-
scribed earlier, the typical experimental value of n
was 1.04 0.005. Even though the temperature of
the air in the tank drops by approximately 850F, the
students conclude that the decompression process is
better approximated by an isothermal process than
by an isentropic one. This conclusion simplifies the
mathematics of the material balance. A more accu-
rate analysis is possible but is not worthwhile given
the intent of the course.
After the students have established that the de-
compression process is approximately isothermal,
the unsteady state material balance equation fol-
lows easily
dm -= (6)
dt mN
Since the volume of the system is constant and the
temperature is nearly constant, the material equa-
tion for the air in the tank is
dm Vv dP 7
dt RT dt
The mass flow rate of air through the nozzle(s) is
rN = PN AN VN (8)
Provided that the air pressure in the tank is above
the critical pressure required for sonic velocity, the
velocity of the air through the nozzle is equal to
VN =(kRTN)12 (9)
The differential equation for the unsteady state ma-
terial balance reduces to
V dP (kRT)112
T dt -PN A (RTN)I (10)
The trick is to convert, in a simple way, the ther-
modynamic variables evaluated at nozzle conditions
Chemical Engineering Education

to those measured in the tank. The theory required
for this transformation is fully developed in most
mechanical engineering thermodynamics texts,
though not in many chemical engineering texts. Our
students do not deal with compressible flow in lec-
ture courses until after the experiment. This prob-
lem is solved by simply stating that the thermody-
namic variables at the nozzle can be evaluated at
tank conditions by applying a correction factor. Stu-
dents seem to like correction factors. The two rela-
tionships, shown below, are derived by Holman.[21

T=T 2 (11)
I k I
PN=P( 2k l- (12)\
k+ (12)
When the change in mass inventory is equated to
the mass flow rate out of the nozzle(s), the differen-
tial equation for the pressure in the tank is
A k k+1 l
dP AN (kRT) /2( 2 2(k-i)p (13)
dt V\v Ik+)I k ))
Since the students previously established that the
absolute temperature in the tank is approximately
a constant, they can now write this differential equa-
tion in a short form as
d--KlP (14)
where K, is the constant term in Eq. (13). The solu-
0.8 Theory K1 = 0.027
0.7 - Empirical K2 = 0.028
0.6 *\


0 10 20 30 40 50 60 70
Time (sec)
Figure 2.
Spring 1993

tion, shown below, is simple; however, it is valid
only until the critical pressure ratio is reached.

in a -Kt (15)

Another tack would be to regard this equation
merely as the basis of a correlating equation. The
equation would be the same as Eq. (15) but have a
leading coefficient of K2, as
In(- =--K2t (16)
The students are required to plot the ratio of the
measured pressure to the maximum pressure ver-
sus time on semi-logarithmic paper. They then com-
pare the slope of the line determined by the data
and Eq. (14) and that predicted by Eq. (13).
A typical set of data taken by the staff, a regres-
sion line based on Eq. (16), and the theoretical pre-
diction from the solution of Eq. (15) are shown in
Figure 2. A nozzle with an internal diameter of 3/
32-inch was used. The empirical value of K2 was
found to be 0.028 based on ten runs with a total of
462 data points. The value of K, predicted by theory,
at the average temperature, was 0.027.
About 90% of the students find that the difference
in the slope of the lines between theory and data is
5% or less. This difference is not statistically signifi-
cant given the inaccuracies in measurement of the
nozzle diameters and the volume of the tank and
manifold. Surprisingly, the other 10% of students,
who predict differences of up to 200%, make the
mistake of using the wrong nozzle(s) or recording
the nozzle diameter(s) incorrectly. None of the stu-
dents have difficulty with the mathematics, though
some think that Eqs. (14) and (16) apply even when
the velocity in the nozzle is subsonic.

The experiment described in this paper is ideal
when the experiment requirements are a quick turn-
around time, inexpensive equipment, and flexible
technical content.

A = Area; ft2
h = enthalpy; Btu/lb
Ki = constant; defined by Eq. (14)
KI = constant; defined by Eq. (16)
ke = kinetic energy; Btu/lb
m = mass flow rate; lb/s
m = mass of system; lb
n = polytropic exponent
P = pressure; psia
Q = rate of heat transfer; BTU/cycle
Continued on page 149.




Georgia Institute of Technology
Atlanta, GA 30332-0100
University of Illinois at Urbana-Champaign
Urbana, IL 61801

Communication skills are important to engi-

neers and to their employers, but the commu-
nication skills of graduates in engineering are
seldom as good as their technical skills.l1"3 In most
engineering curricula, laboratory and design reports
provide an opportunity to help students learn
how to communicate technical material.[4-8 We
miss that chance, however, if we evaluate the tech-
nical merit of students' reports but ignore how
well they are written.
The least we can do is to identify those places
where the reports communicate poorly, require that
the students rewrite them, and hope for improve-
ment. The best we can do is to show students why
their reports communicate poorly and how to make
the required improvements. This article will outline
a method of showing students how to write clearly
and will explain the principles behind that method.

Some writing problems are easy to spot and easy
to fix: errors in spelling, grammer, and punctuation;
problems in literature references; tables and figures
that lack legends or are not discussed in the text;
etc. Though important, these problems alone may
not determine how well a report communicates. By
focusing only on them, we do not help students to
master a skill crucial to employers and working en-
gineers-the ability to communicate technical infor-
mation in words as well as in numbers.
One key to effective communication is style-the
sentence forms in which students express technical
information. But when it comes to problems of style,

William R. Ernst is Professor of Chemical Engi-
neering at Georgia Institute of Technology and
has taught technical economics and the capstone
design course to seniors for the past twenty years.
His principal technical interests are kinetics and
reaction engineering. He is also interested in pipe-
line issues and has developed science and engi-
neering modules for pre-college education.

Gregory G. Colomb is Associate Professor of
English and Director of Business and Technical
Writing at the University of Illinois. With others,
he created the "Writing Across the Curriculum"
(WAC) program at the University of Chicago
and has conducted WAC faculty workshops at
more than fifty institutions. He has published on
writing theory, WAC pedagogy, and the relation-
ship between writing and critical thinking.

some of us have little to say. We might tell students
that their writing is unclear, indirect, abstract,
convoluted, flowery, awkward, etc., and advise
them to be "clear and direct" or to "write as you
speak." While such comments may be accurate,
they are far too general to be of much use. In order
to improve as writers, students need to know
both what causes their writing to be unclear or
convoluted and what they must change in order to
make it clear and direct.
Writing researchers have recently developed bet-
ter and more useful methods of responding to stu-
dents' writing-methods based on research on how
people process and understand what they read. Much
of that work can be found in the book Style, [9 which
presents a simple, but powerful, method of teaching
style. The research base of the book can be found in
Colomb and Williams,[o10 and the methodology de-
veloped by those authors is summarized and ap-
plied to scientific writing by Gopen and Swan."] In
this paper we will describe its most useful tools and
show how they can be used by teachers to help stu-
dents improve their communication skills.

Copyright ChE Division ofASEE 1993

Chemical Engineering Education

The First Principle
of Readable Writing

Express important actions as verbs, and the characters
associated with those actions as the subjects
of those verbs.
Consider the following "Conclusions and Recom-
mendations" section of a senior design report that is
technically sound but poorly written in typical ways.
From the study done regarding this process a fairly firm
conclusion may be stated affirming the feasibility of Case 1
in which only the n-butane rail imports are replaced. Be-
cause no modifications are made to the gas concentration
unit itself the specifications predicted may be obtained with
very little error....
For Case 2, in which all of the butane rail imports are to
be replaced, a feasible plan, which involves the purification
of the excess n-butane entering with the required amount of
isobutane in the NGL, has been developed. The introduction
of the NGL stream was made into the feed to the butane
splitter. The desired quantity of isobutane from the top of
the column was achieved by this method. The bottoms from
the butane splitter would then be sent to a packed column
which has been designed to separate pure n-butane which
meets industry specs. The bottoms from the new column
would then be returned to the blending butane product stream
which would then be producing an extra 10,000 Bbl/yr.
The style of this passage is typified by the sen-
la. The introduction of the NGL stream was made
into the feed to the butane splitter.
In order to see the distinctive features of the sen-
tence la, compare the following three variations on
a theme:
2a. The heating of the reaction mixture occurred
after the introduction of the catalyst.
2b. The reaction mixture was heated after the cata-
lyst was introduced.
2c. She heated the reaction mixture after she intro-
duced the catalyst.
Sentences 2a-c tell roughly the same story, but
most readers find 2a less clear and readable than
either 2b or 2c. Between 2b and 2c, most readers
find 2c slightly more readable, but readers with tech-
nical backgrounds are perfectly comfortable with 2b.
Note that 2a is most similar in feel to la.
These reactions are uniform among readers be-
cause these examples demonstrate key features of
the way we understand sentences. Sentences, even
the most technical ones, tell stories. With rare ex-
ceptions, sentences have two necessary elements:
subjects and verbs. Similarly, stories have two nec-
essary elements: characters and actions. Readers
Spring 1993

Writing researchers have recently
developed better and more useful methods of
responding to students' writing-methods based
on research on how people process and
understand what they read.

find that sentences are clearer, more direct, less
abstract, less complex-in short, more readable-
when the story elements line up with the sentence
elements: characters as subjects and actions as verbs.
In 2b and 2c, key actions are expressed as verbs:
was heated . was introduced and heated . .
introduced. Subjects are characters: reaction mix-
ture ... catalyst and She ... she. In 2c the character
is a person, while 2b treats the experimental mate-
rials reaction mixture and catalyst as characters.
Although sentences are usually clearer when the
subject/character is a person (preferably the agent
or "doer" of the action) readers with technical back-
grounds are accustomed to stories about such things
as reaction mixtures and catalysts, and they gener-
ally prefer not to have their stories focus on the
persons who do the heating and introducing.
In both la and 2a, however, the actions are ex-
pressed not as verbs, but as nouns. As a result,
readers must struggle through the grammar in or-
der to unpack the story. The la and 2a sentences
are built around nouns made from verbs (often by
adding a suffix: -tion, -ness, -ence, -ity, or -ing). These
nouns, called nominalizations, are usually a prob-
lem because they steal important action from the
verb, forcing writers to use a weak or empty verb.
Students tend to overuse nominalizations, and they
need our feedback in order to distinguish between
those that are necessary technical terms and those
that steal action from the verb.
So now, for sentences la and 2a we can
locate the problem
explain to the writer what caused the sen-
tences to seem to us unclear, indirect, and
tell the writer how to make them more read-
"Sentences la and 2a are unclear because the ac-
tions in the sentences are expressed as nouns rather
than verbs. As a result, the key sentence elements-
subject and verb--do not correspond to the key story
elements-character and action. You can make the
sentences more readable if you change the nouns
expressing actions into verbs (e.g., introduction into
introduce) so that the subjects express characters
and the verbs express actions."

The Second Principle
of Readable Writing
Keep subjects as short as possible
so that sentences move quickly from a short,
specific subject to an action verb.
Once again, we begin with sentences that tell
roughly the same story:
3a. The mixture, because it was vigorously stirred
and the temperature was maintained above 200
Deg. C, reacted rapidly.
3b. The mixture reacted rapidly because it was vig-
orously stirred and the temperature was main-
tained above 200 Deg. C.
While neither sentence is unreadable, most read-
ers find 3b more readable than 3a, and all readers
begin to struggle in passages with lots of sentences
like 3a. In this case, the story elements (character
and action) do line up with the sentence elements
(subject and verb). But in 3a the story in the main
clause, The mixture . reacted rapidly, is inter-
rupted by all the rest of the sentence. Readers must
process all of the intervening information before they
achieve the subject-verb closure that holds the story
together. In 3b, the subject-verb/character-action
pairs are all joined, so that we are able to process
the story in three discrete chunks connected by logi-
cal markers (because and and).
Technical writers are particularly prone to write
sentences with long, complex subjects or with infor-
mation intervening between subject and verb. Since
they so often use passive verbs in order not to focus
on the persons who perform the actions, technical
writers often push the verb toward the end of the
sentence. Here is an instance from our long example:
4a. For case 2, in which all of the butane rail imports
are to be replaced, a feasible plan which involves
the purification of the excess n-butane entering
with the required amount of isobutane in the NGL,
has been developed.
In this sentence, readers are forced to process quite
a lot of information before they attain subject-verb
Here too, the method allows us to locate the prob-
lem, explain to the writer what caused the sentences
to seem to us unclear, indirect, and difficult, and tell
the writer how to make them more readable:
"Sentence 4a is unclear because its subject is long
and complex. As a result, readers have to process
too much information before they can connect the
key sentence elements, subject and verb. You can
make the sentence more readable if you move quickly
from a short, specific subject to an action verb."

The Third Principle
of Readable Writing
Sentences should begin with old information
and end with new information.
It is not enough that our students write sen-
tences that are individually clear. The sentences
also have to "flow" together into a story that is
coherent as a whole. Sentences flow together be-
cause readers use the information they have al-
ready read and remembered to look forward to the
next sentence. If the next sentence surprises them
by beginning with something they did not expect,
they feel a little jolt of disorientation. If the sen-
tences in a passage consistently surprise readers,
their feeling of disorientation builds until it be-
comes hard to follow the story.
In order not to surprise readers, sentences should
begin with something that places them in the
context of the discussion: they should begin
with information that readers will already have
read and remembered. This familiar infor-
mation can be something in the immediately
previous sentence or any information that is as-
sumable or expectable, given what has come be-
fore. In short, sentences should begin with old
Once again, we find an instance in our long ex-
5a. The introduction of the NGL stream was made
into the feed to the butane splitter. The desired
quantity of isobutane from the top of the column
was achieved by this method.
These sentences do not "flow." They feel dis-
jointed, even after we eliminate the nominalization
from the first sentence and connect the subject
and verb in the second sentence:
5a'. The NGL stream was introduced into the feed
to the butane splitter. The desired quantity of
isobutane was achieved from the top of the col-
umn by this method.
Few readers of the first sentence in 5a or 5a'
would expect the second sentence to begin with
"The desired quantity ofisobutane." Isobutane had
been mentioned earlier, but not in a way that
readers would expect it to return. In the second
sentence, the phrase that most strongly refers
backward is "this method." Although "method" has
not occurred before, it is nevertheless old informa-
tion, because the whole passage has been describ-
ing the method.
The old-before-new principle has even greater
Chemical Engineering Education

effect in longer passages, as in these examples:

6a. We should consider employing multiple reactors in
parallel before we finalize our design. If one reactor
shuts down, the other reactors can operate. For this
reason, the parallel arrangement is flexible. Identical
controllers can be used on all the reactors, thus mak-
ing the parallel arrangement easier to control than a
series arrangement. Plant operators have more diffi-
culty in understanding series operation than parallel
6b. Before we finalize our design, we should consider
employing multiple reactors in parallel. The parallel
arrangement is highly flexible; if one reactor shuts
down, the others can continue to operate. Parallel re-
actors are easier to control than reactors in series be-
cause all of the parallel reactors utilize identical con-
trollers. Parallel operation can be understood by the
plant operators more easily than series operation.

7a. We should consider employing multiple reactors in
series before we finalize our design. More control
equipment but less volume at a given conversion are
required for a series of reactors as compared to a
single reactor. Three reactors in series would save us
about $1,000,000 in fixed capital, at our required con-
version of 90 percent. Higher quality separators could
be purchased with the saved capital.
7b. Before we finalize our design, we should consider
employing multiple reactors in series. A series re-
quires more control equipment than a single reactor
but requires less volume for the same conversion. At
our required conversion of 90 percent, three reactors
in series would save us about $1,000,000 in fixed
capital. These savings could be invested in higher qual-
ity separators.

In both pairs, the second passage feels "tighter"
and more organized because each new sentence be-
gins in a way we expect. In 6b, each sentence re-
turns to the same idea ("parallel"). In 7b, the sen-
tences begin differently, but with an idea from the
immediately previous sentence. Both arrangements
create an organized flow through the passage.

Thus far, we have described the methodology in
terms of an interrelated set of principles. Once teach-
ers understand the principles, the methodology can
be implemented through a series of simple decision
procedures. These procedures focus on the first five
or six words of the sentence, because the three prin-
ciples work together to put the key elements there.
If the first several words of a sentence include a
subject that names a character, a verb that expresses
a key action, and some old information, then that
sentence is likely to be in a readable style.
Spring 1993

What follows is a method of responding to the
style of student writing before you require a revi-
sion. (It is equally useful as a way to edit our own
work.) At first the method might feel counter-
intuitive-especially if you usually start reading,
red pen in hand, marking as you go. In the long
run, however, the method allows us to mark those
problems that matter most and to give students
useful feedback.
* Read once, very quickly, for an overview. If the
report is long, skim just the major sections. The
goal of this step is to determine the overall story
line and to run a first check on the technical
merit of the report. Do this quickly, without mak-
ing any marks on the page.
* Check that the report has the right sections and
the right results in the right places. If data tables
or figures are especially important to the results,
check them now. Comment on any problems.
* Read through the report. Let the "feel" of the
prose, more than your understanding, be your
guide. (Because you know the material so well,
you can often understand even poorly written pas-
sages, supplying from your knowledge the infor-
mation and connections that students leave out
or misstate.) Whenever you feel that you are be-
ginning to work too hard to read a passage, slow
down and give it the six-word test.
* The six-word test: Check the first four to six words
of each sentence (ignore short introductory
phrases). The first several words should include
a short, specific subject naming a character
a verb expressing a key action
old information that sets a context for the rest
of the sentence
If a sentence fails the test, especially if it begins
with a nominalization that is not used as a term
of art, the sentence is very likely to violate the
* Comment on passages or sentences that violate
the principles. Don't mark up the page too much;
if there are many problems, comment only on the
most important ones. Your comments can take
any form that makes you comfortable, but it is
generally best to give the student something to
do: 1) analyze a portion of a problem passage on
the page, and then direct the student to use the
six-word test to check the rest for him- or herself;
or 2) pick out the problem element in a sentence
or passage and suggest a specific kind of change
(e.g., "Make this word a verb" or "Make sure your
sentences begin with old information"). If you don't

trust the student to be able to make the change
and you have the time, you can edit the sentence
or passage and comment on the change you made
(e.g., "This is clearer with X as a verb").
If you have the time and energy to spare and you
have not already made many comments, you can
check grammar, punctuation, spelling, etc. Un-
less students have very serious problems, these
comments will be less important in helping them
to communicate effectively. Yet correctness does
count, and some teachers believe that students
ought to be held to industry standards. It is gen-
erally better to pick out a problem and require
the student to fix it rather than to fix it yourself.

One of the authors (Ernst) introduces the above
principles and report writing in general in a techni-
cal economics course, a prerequisite to the senior
capstone design course. The students are required
to submit a one- to three-page report every other
week, usually in the form of a letter discussing in
detail an assigned homework problem-its solution
and the implications of the solution. Each report is
graded on how well the student communicates the
information. If the report is poorly written, the writ-
ing style is checked and appropriate comments are
written in the margin as described above. Students
are given a chance to revise unsuccessful reports.
We have been pleased with the results of this
process for two reasons: 1) when it is applied to
reports assigned early in a quarter, most students
who initially submit poor reports produce well-writ-
ten reports after only one revision, and 2) toward
the end of the quarter, most students routinely sub-
mit reports that do not need revising.
For one assignment, students were asked to revise
a report previously written by another student-in
this case the "Conclusions and Recommendations"
section which we discussed earlier. Here is one of
the best revisions:
In Case 1, only the n-butane rail imports are to be
replaced by NGL. We have developed a feasible plan,
under which NGL would be transferred directly to
the blending butane product stream, yielding a com-
bined product which meets specification ....
In Case 2, all of the butane rail imports are to be
replaced by NGL. We have developed a feasible plan,
under which the NGL would be fed to the butane
splitter, where iso-butane would be removed as over-
head at the desired rate. The splitter bottoms would
be fed to a new packed column, designed to produce

a pure n-butane overhead stream which meets in-
dustry specs. The column bottoms would be returned
to the blending butane product stream at a rate that
would increase production by 10,000 Bbl lyr.
Often, students find it necessary to add informa-
tion as they revise the original work, which illus-
trates an additional feature of these principles: they
serve as a mental discipline that improves the qual-
ity of students' thinking. While any well-designed
writing assignment can help students consolidate
and improve their knowledge, students get an addi-
tional boost by writing and revising in accord with
these principles. Because the principles focus stu-
dents on the key elements of the story they have to
tell, they help students to think through those sto-
ries and discover missing information or gaps in
their logic. When students adhere to the principles,
they are encouraged to be complete, precise, and
logical. When teachers adhere to the principles and
follow three easy steps (locate the problem, explain
what caused it, explain how students can fix it), the
students' gain is threefold: they understand their
own research and its results more fully, they com-
municate their results to us more effectively, and
most of all, they learn how to do better next time.

1. Cranch, E.T., and G.M. Nordby, "Engineering Education:
At the Crossroads Without a Compass," Eng. Ed., 76(8),
742 (1986)
2. Bennett, A.,W., and D. McAuliff, "Integrating Communica-
tions Skills into the Engineering Curriculum," ASEE-IEEE
Frontiers in Ed. Conf. Proc., Vol 2, 693, November (1987)
3. Friday, C., "An Evaluation of Graduating Engineers' Writ-
ing Proficiency," Eng. Ed., 77(2), 114 (1986)
4. Frank, C.W., G.M. Homsy, and C.R. Robertson, "The Devel-
opment of Communications Skills Through a Laboratory
Course, Chem. Eng. Ed., 16(3), 122 (1982)
5. Bakos, Jr., J.D., "A Departmental Policy for Developing
Communication Skills of Undergraduate Engineers," Eng.
Ed., 77(2), 101 (1986)
6. Sullivan, R.M., "Teaching Technical Communication to Un-
dergraduates: A Matter of Chemical Engineering," Chem.
Eng. Ed., 20(1), 32 (1986)
7. Hudgins, R.R., "Tips on Teaching Report Writing," Chem.
Eng. Ed., 21(3), 130 (1987)
8. Hanzevack, E.L., and R.A. McKean, "Teaching Effective
Oral Presentations as Part of the Senior Design Course,"
Chem. Eng. Ed., 25(1), 28 (1991)
9. Williams, J.M., and G.G. Colomb, Style: Toward Clarity
and Grace, University of Chicago, Chicago (1990). Also pre-
vious editions of this book: Williams, J.M., Style: Ten Les-
sons in Clarity and Grace, 1st and 3rd eds., Scott Foresman,
Glenview, IL (1981)
10. Colomb, G.G., and J.M. Williams, "Perceiving Structure in
Professional Prose: A Multiply Determined Experience," in
Writing in Academic Settings, edited by L. Odell and D.
Goswami, Guilford, NY (1986)
11. Gopen, G.D., and J.A. Swan, "The Science of Scientific Writ-
ing," Amer. Sci., 78, 550, Nov-Dec (1990) C
Chemical Engineering Education

Fluid Mechanics Experiment
Continued from page 143.
R = gas constant; Btu/lb(R)
t = time; s
u = internal energy; Btu/lb
v = velocity; ft/s
V = volume; ft3
W = power input; BTU/cycle
p = density; lb/ft3
cv = control volume
e = exiting air
i = incoming air
N = nozzle
0 = time zero valve opening
1 = initial state
2 = final state

1. Van Wylen and Sonntag, Fundamentals of Classical Ther-
modynamics, 3rd ed., SI Version, John Wiley, New York
2. Holman, J.P., Thermodynamics, 4th ed., McGraw-Hill, New
York (1988) 0

REVIEW: Mass Transfer
Continued from page 126
strongly recommended.
Chapter 17, on sorption processes, discusses fixed
bed adsorption and ion exchange. The presentation
on why loading and elution in ion exchange are not
symmetrical is particularly clear and easy to under-
stand. In general, the authors assume that the reader
is familiar with these separation processes. Readers
who are not (particularly electrodialysis) will find
these chapters difficult, but readers familiar with
the processes will gain deeper insight.
A third part of the book starts with Chapter 18,
which compares the M-S, Fickian, and irreversible
thermodynamics approaches to mass transfer. This
is a very enlightening chapter, and sophisticated
readers should read it following Chapter 2 or 3.
Chapter 19 cites references. A rather complete list
of symbols starts on page 160. I found myself refer-
ring to this list often and wish it were in a more
prominent location.
The fourth part of the book consists of thirty-six
worked exercises (pages 163 to 238) which consider
some very interesting and challenging problems. Al-
though the solutions are not polished, they are cer-
tainly sufficient to show how to attack the problems.
A major problem with this book is highlighted in
the Guidelines to the Reader on page 11: "This text
was written to accompany overhead transparencies
Spring 1993

The 1993 (maroon) revised printing of the
is now out,
and it still costs $24.
Order at your bookstore,
or FAX your order and card number to
OSU Bookstores, Corvallis, OR at 503-737-3398

in a course on multicomponent mass transfer. So
the Figures are quite important." Unfortunately,
many readers will not pay enough attention to this
section and will find reading the book difficult until
they have learned the proper way to read it. Also,
since the figures are hand drawn, the reader needs
to learn how to decipher the authors' script. The
inclusion of equation numbers would be useful. Some
of the examples are confusing since the problem
statements are not clear (e.g., Figure 6.2) and data
or formulas are slipped into the solutions with little
explanation (e.g., Figures 3.7 and 5.5). Statements
such as "Qualitatively the reasoning should be clear,"
(page 91) will unintentionally demotivate readers
who are struggling, and they should be removed.
The basic ideas of the M-S approach are not sum-
marized until pages 64 and 65. A much earlier expo-
sition of this would help many readers. Also, since
the authors assume considerable familiarity with
mass transfer, Chapter 18 could appear earlier in
the text. If a second edition is planned, the authors
could aid readers by correcting these problems. One
hopes that the authors will make this effort since
the book presents a very important topic in a way
which will be accessible to most readers.
Where can this book be used in the curriculum?
The book is a curious mix of sophistication (M-S
theory and challenging problems) and of approxima-
tions (difference solutions and overly simplified ther-
modynamics). Because of the subject matter and the
assumed high degree of knowledge in mass transfer
and separations, this text is appropriate at the gradu-
ate level. However, the approximations and some
lack of rigor may cause difficulties. It book would be
a very useful supplement in a graduate-level course,
particularly if journal articles are used in most of
the course. It is also a very good source of problems
and examples for a graduate-level course.
Finally, for practicing professionals who missed
the M-S theory in their formal education, this book
would be very useful for self study. Wesselingh and
Krishna will stimulate and frustrate, but the reader
will never be bored. )

re. laboratory



Tennessee Technological University
Cookeville, TN 38505

In the search for new laboratory experiments, a
simple experiment that works well is always
welcome. In this paper we describe an inexpen-
sive apparatus, using simple and widely available
components, that will help student understanding
of process modeling. The equipment can be arranged
in a variety of configurations to allow study of dif-
ferent models. Many chemical engineering depart-
ments carry out mixed-tank experiments, some with
computer interfaces for data collection, that can be
rearranged and modified to include the models sug-
gested in this article.

In a text by Levenspiel1' several models are pre-
sented for long time scale behavior of real stirred
tanks. The models examined here are Levenspiel's
model L (which is described in more detail by Bischoff
and Dedrick121) and a modification of that model.
In these models, shown in Figure 1, flow enters a
perfectly mixed tank of volume aV, is interchanged
at a rate by with a second perfectly mixed tank of

William D. Holland is a professor of chemical
engineering at Tennessee Technological Univer-
sity. He has taught chemical engineering for
twenty-five years and has served as a consultant
at Oak Ridge National Laboratory in nuclear fuel

John C. McGee is a professor of chemical engi-
neering at Tennessee Technological University,
where he served as the first chairman of the
department for twenty years. He holds BS and
MS degrees from West Virginia University and a
PhD from North Carolina State University. He has
had industrial experience with Dupont.
@ Copyright ChE Division ofASEE 1993

Syphon tubes
PumpLL- L Notched weir
Rotameter Tank 2
Tank 1

Figure 1. Stirred tank model (Levenspiel's Model L)

volume (1-a)V, and is discharged from the first tank.
Nomenclature used here is consistent, where pos-
sible, with that used by Levenspiel. The total vol-
ume of the system is V.
In Levenspiel's model L, a unit impulse is imposed
in the feed to tank 1. If the concentration in tank 1
is C1 and the concentration in tank 2 is C2, the
material balances on the two tanks, assuming per-
fect mixing, yield for tank 1

v6(t)+ bvC2 bvC1 ClV1 = (1)

and for tank 2

d[(1- a)VC2]
Cbv -C2bv = dt 2)
Initial conditions for each tank reflect no tracer in
either tank with the initial condition in tank 1 a
formal property assigned to the Dirac delta function
as indicated by Churchill13"
C(0)= 0 and C2(0)= 0 (3)
These equations yield to rather simple Laplace trans-
form solution. The transformed equations are
1+bC2(s) (4)
1C( as +(1+b)

C2 (s) =E e Ed) (5)
(1- a)lts +b
Chemical Engineering Education

where t=V/v. Inverse transformation of these equations yields
the solutions in dimensionless time, 0, given by Levenspiel for
the two tanks

Ee=C1 1 [mi-am +b]em-(m-am2 +b)em20 (6)
a(1-a)(m1 -m2)

O2 b [emle (7m2)

= 1-a+b 1- 4ab(1-a)
ml,m2 =- a -1+ 1
2a(1-a) ( (1-a+b)2

The discussion in Levenspiel is necessarily brief, and stud
need to be sure they understand the equations describing
model, Eqs. (6) and (7) above, and the procedures to reduce
data to a similar form (or to change the equation for the m
to the data form). Fogler[41 also shows the development ol
equations for Levenspiel's model L.
Most of the long time scale models presented by Leveni
could be examined in the experimental apparatus with a ]
equipment rearrangement. A variation which has been trie
students in our laboratories is a modification of Levensy
model L in which the tracer or unit impulse is imposed in
"stagnant" compartment. The solution to this model as wo
out with Jones15' gives the following expressions:

Ee =Co a(1-a)(mlm2)1em-em2 ]

C2 am2)[(am +b+ 1)eml -(am2 +b+ 1)em2]

The stirred tanks for this experiment were one 5-gE
aquarium and one 10-gallon aquarium placed end-to-en,
schematic diagram of the apparatus is shown in Figure 2 a
photograph of the apparatus is shown in Figure 3. Water
fed to the larger aquarium through a small rotameter,
water was discharged from the larger tank via four syp
tubes into a small notched-weir overflow tank and then tc
drain. Flow from the larger tank to the smaller tank was
enabled by using four syphon tubes. Return flow from the sm;
tank to the larger tank was accomplished by using a sii
aquarium pump/filter device without the filter.* The fil
intake was positioned in the smaller tank and the dischE
which was adjustable with an integral valve, was made to
larger tank. The filter used in this case included a one-
hold-up tank which was filled with inert materials to elimi:
a possible third mixed tank in the apparatus. Tracer selec
could be dye, salt, or any tracer with detection capabil
available. In this work, the tracer selected for the quantity
work was sodium chloride because a YSI Scientific Mode
* In this case a Model 2 Secondnature Whisper Power Filter: catalog No. 6(
Willinger Bros., Inc., Wright Way, Oakland, NJ (201-337-0001).
Spring 1993

C1, aV

2, (1-a)V

Figure 2. Schematic diagram of apparatus

rked -
Figure 3. The experimental apparatus

(9) conductance meter was available. This
meter allowed for either a continuous record
(10) of conductance when used with a millivolt
potentiometric recorder or an instantaneous
reading. Total equipment cost excluding the
conductance meter, rotameter, and stirrers
illon was $85.00.
d. A
was Both tanks were initially filled with wa-
and ter. Then water flow at a rate of 3.1 liters/
)hon minute was initiated through the rotame-
the ter into the larger tank, and flow from the
also larger tank through the two syphon sys-
aller teams was started. Flow rates are typical
mple and, of course, may be set at any reason-
ter's able level. Stirrers, placed in both tanks,
Large, were activated. Flow through the pump/
the filter was started and measured using the
liter bucket-and-watch method on the outflow
nate of the power filter after a period of time to
.tion allow steady-state flow. A return water flow
cities rate to the larger tank of 4.8 liters/minute
Live was measured. At steady-state the volume
S35 of liquid in tank 1 was 36.2 liters, and the
)02, volume in tank 2 was 16.5 liters. The above
tank volumes and flow rates gave model

parameters of 0.678 for a and 1.55 for b. A one-
molar solution of sodium chloride was prepared for
use as tracer, and a calibration curve for the con-
ductivity meter was prepared. Before an experimen-
tal run was made using the salt tracer, a run was
made using dye as the tracer-this demonstrated
the flow patterns in the system and gave some in-
sight into the perfectly mixed tank assumptions.
Dye, instead of salt, has also been used in separate
experiments to monitor the tracer concentration.
When flows were properly established and steady-
state conditions were obtained, one liter of the 1 M
salt tracer was rapidly poured into the center of
the larger tank over a short period of time to
approximately replace the regular water flow in a
pulse-shaped input. The concentration of the mate-
rial in each tank was monitored alternately with
the single conductivity probe, with the probe rein-
serted into the tanks in approximately the same
location each time. Sampling was halted after ap-
proximately 37 minutes.

The experimental results are shown in Figure 4.
The response of Model L to a unit impulse input was
also determined by solving the equations numeri-
cally; these results are compared to the experimen-
tal results in Figure 4. The expected characteristic
shapes were obtained and agreement between
the experimental results and'the model were
within seven percent for tank 1 and within eleven
percent for tank 2. The maximum concentration in
tank 2 was eleven percent below the model and
about two minutes late. No attempt was made to
adjust model parameters.
Many variations of the experiment demonstrated
here could be studied including other models, effect
of tracer injection method, effect of adjusting the
model parameters, and effect of mixing. Because of
the flexibility derived from the ease of rearranging
the system, individual laboratory participants or
groups could study a number of different models or
a few models in depth.

An interesting and inexpensive process-modeling
experiment was demonstrated with qualitative and
quantitative results. Other models could be exam-
ined by simple modifications to the experimental
apparatus. The work integrates studies in chemical
reaction engineering courses with process modeling
and control courses and provides the students with
some insight into problems in modeling systems.

0 5 10 15 20 25 30 35 40
Time, minutes
Figure 4. Comparison of model with experiment
The authors gratefully acknowledge helpful criti-
cism by the referees of this paper.

a = model parameter, fraction of total volume in feed
b = model parameter, fraction of feed volume flowing
to "stagnant" region
C, = tracer concentration in tank 1
C2 = tracer concentration in tank 2
Ce1 = C-curve for tank 1 based on 6, C1 = Eg = tC1
C02 = C-curve for tank 2 based on 9, C02 = tC2
E, = exit age distribution for tank 1 in dimensionless
t = mean residence time, V/v
V = total system volume
v = volumetric feed rate
8(t) = Dirac delta function for unit impulse
8 = dimensionless time t /
1. Levenspiel, Octave, Chemical Reaction Engineering, 2nd
ed., Wiley, New York (1972)
2. Bischoff, KB., and R.L. Dedrick, J. Theor. Biol., 29, 63
3. Churchill, Ruel V., Operations Mathematics, 2nd ed.,
McGraw-Hill, New York (1958)
4. Fogler, H. Scott, Elements of Chemical Reaction Engineer-
ing, 2nd ed., Prentice Hall, Englewood Cliffs, NJ (1992)
5. Jones, S., private communication (1990) 01
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